LMFDB - Embedded newform 90.7.g.f.37.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,7,Mod(37,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.37"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	90.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,32,0,0,-60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.7048675258$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1602x^{6} + 816401x^{4} + 140305200x^{2} + 7638760000 $ x^8 + 1602x^6 + 816401x^4 + 140305200*x^2 + 7638760000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.2
Root			$-25.3784i$ of defining polynomial
Character	$\chi$	$=$	90.37
Dual form			90.7.g.f.73.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.00000 + 4.00000i) q^{2} +32.0000i q^{4} +(-75.4773 - 99.6402i) q^{5} +(145.451 + 145.451i) q^{7} +(-128.000 + 128.000i) q^{8} +(96.6517 - 700.470i) q^{10} -1037.11 q^{11} +(2027.16 - 2027.16i) q^{13} +1163.61i q^{14} -1024.00 q^{16} +(-4670.49 - 4670.49i) q^{17} -9189.32i q^{19} +(3188.49 - 2415.27i) q^{20} +(-4148.42 - 4148.42i) q^{22} +(13577.9 - 13577.9i) q^{23} +(-4231.35 + 15041.2i) q^{25} +16217.3 q^{26} +(-4654.44 + 4654.44i) q^{28} -8478.24i q^{29} +17088.4 q^{31} +(-4096.00 - 4096.00i) q^{32} -37364.0i q^{34} +(3514.53 - 25471.1i) q^{35} +(-14956.3 - 14956.3i) q^{37} +(36757.3 - 36757.3i) q^{38} +(22415.0 + 3092.86i) q^{40} -50688.1 q^{41} +(-12486.6 + 12486.6i) q^{43} -33187.4i q^{44} +108624. q^{46} +(25787.6 + 25787.6i) q^{47} -75336.8i q^{49} +(-77090.0 + 43239.2i) q^{50} +(64869.0 + 64869.0i) q^{52} +(166649. - 166649. i) q^{53} +(78277.9 + 103337. i) q^{55} -37235.6 q^{56} +(33912.9 - 33912.9i) q^{58} +247188. i q^{59} -281194. q^{61} +(68353.5 + 68353.5i) q^{62} -32768.0i q^{64} +(-354991. - 48982.0i) q^{65} +(-304343. - 304343. i) q^{67} +(149456. - 149456. i) q^{68} +(115942. - 87826.2i) q^{70} +288398. q^{71} +(-487894. + 487894. i) q^{73} -119650. i q^{74} +294058. q^{76} +(-150848. - 150848. i) q^{77} -222009. i q^{79} +(77288.8 + 102032. i) q^{80} +(-202752. - 202752. i) q^{82} +(-749160. + 749160. i) q^{83} +(-112853. + 817885. i) q^{85} -99892.6 q^{86} +(132749. - 132749. i) q^{88} +689800. i q^{89} +589705. q^{91} +(434494. + 434494. i) q^{92} +206301. i q^{94} +(-915626. + 693585. i) q^{95} +(820944. + 820944. i) q^{97} +(301347. - 301347. i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{2} - 60 q^{5} + 388 q^{7} - 1024 q^{8} - 976 q^{10} + 4192 q^{11} + 7884 q^{13} - 8192 q^{16} + 1372 q^{17} - 5888 q^{20} + 16768 q^{22} - 14888 q^{23} + 17932 q^{25} + 63072 q^{26} - 12416 q^{28}+ \cdots + 301888 q^{98}+O(q^{100}) $ 8 * q + 32 * q^2 - 60 * q^5 + 388 * q^7 - 1024 * q^8 - 976 * q^10 + 4192 * q^11 + 7884 * q^13 - 8192 * q^16 + 1372 * q^17 - 5888 * q^20 + 16768 * q^22 - 14888 * q^23 + 17932 * q^25 + 63072 * q^26 - 12416 * q^28 + 2240 * q^31 - 32768 * q^32 + 78232 * q^35 - 69996 * q^37 + 71008 * q^38 - 15872 * q^40 - 169064 * q^41 + 164136 * q^43 - 119104 * q^46 + 109248 * q^47 - 280832 * q^50 + 252288 * q^52 + 480740 * q^53 - 409068 * q^55 - 99328 * q^56 - 2272 * q^58 + 266088 * q^61 + 8960 * q^62 - 903396 * q^65 + 92488 * q^67 - 43904 * q^68 - 228352 * q^70 + 2526032 * q^71 + 324464 * q^73 + 568064 * q^76 - 2822104 * q^77 + 61440 * q^80 - 676256 * q^82 - 213264 * q^83 - 1537292 * q^85 + 1313088 * q^86 - 536576 * q^88 + 3340176 * q^91 - 476416 * q^92 - 3084784 * q^95 - 854736 * q^97 + 301888 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	4.00000	+	4.00000i	0.500000	+	0.500000i
$2$	4.00000	+	4.00000i	0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$			32.0000i			0.500000i
$5$	−75.4773	−	99.6402i	−0.603818	−	0.797122i
$5$	−75.4773	−	99.6402i	−0.603818	−	0.797122i
$6$		0			0
$7$	145.451	+	145.451i	0.424057	+	0.424057i	0.886598	−	0.462541i	$-0.153062\pi$
$7$	145.451	+	145.451i	0.424057	+	0.424057i	−0.462541	+	0.886598i	$0.653062\pi$
$8$	−128.000	+	128.000i	−0.250000	+	0.250000i
$9$		0			0
$10$	96.6517	−	700.470i	0.0966517	−	0.700470i
$11$	−1037.11			−0.779193			−0.389596	−	0.920986i	$-0.627385\pi$
$11$	−1037.11			−0.779193			−0.389596	+	0.920986i	$0.627385\pi$
$12$		0			0
$13$	2027.16	−	2027.16i	0.922693	−	0.922693i	−0.0745260	−	0.997219i	$-0.523744\pi$
$13$	2027.16	−	2027.16i	0.922693	−	0.922693i	0.997219	+	0.0745260i	$0.0237444\pi$
$14$			1163.61i			0.424057i
$15$		0			0
$16$	−1024.00			−0.250000
$17$	−4670.49	−	4670.49i	−0.950640	−	0.950640i	0.0481978	−	0.998838i	$-0.484652\pi$
$17$	−4670.49	−	4670.49i	−0.950640	−	0.950640i	−0.998838	+	0.0481978i	$0.984652\pi$
$18$		0			0
$19$		−	9189.32i		−	1.33975i	−0.742476	−	0.669873i	$-0.766349\pi$
$19$		−	9189.32i		−	1.33975i	0.742476	−	0.669873i	$-0.233651\pi$
$20$	3188.49	−	2415.27i	0.398561	−	0.301909i
$21$		0			0
$22$	−4148.42	−	4148.42i	−0.389596	−	0.389596i
$23$	13577.9	−	13577.9i	1.11597	−	1.11597i	0.123638	−	0.992327i	$-0.460544\pi$
$23$	13577.9	−	13577.9i	1.11597	−	1.11597i	0.992327	−	0.123638i	$-0.0394561\pi$
$24$		0			0
$25$	−4231.35	+	15041.2i	−0.270807	+	0.962634i
$26$	16217.3			0.922693
$27$		0			0
$28$	−4654.44	+	4654.44i	−0.212028	+	0.212028i
$29$		−	8478.24i		−	0.347625i	−0.984779	−	0.173813i	$-0.944391\pi$
$29$		−	8478.24i		−	0.347625i	0.984779	−	0.173813i	$-0.0556087\pi$
$30$		0			0
$31$	17088.4			0.573609			0.286805	−	0.957989i	$-0.407407\pi$
$31$	17088.4			0.573609			0.286805	+	0.957989i	$0.407407\pi$
$32$	−4096.00	−	4096.00i	−0.125000	−	0.125000i
$33$		0			0
$34$		−	37364.0i		−	0.950640i
$35$	3514.53	−	25471.1i	0.0819716	−	0.594078i
$36$		0			0
$37$	−14956.3	−	14956.3i	−0.295269	−	0.295269i	0.543888	−	0.839158i	$-0.316952\pi$
$37$	−14956.3	−	14956.3i	−0.295269	−	0.295269i	−0.839158	+	0.543888i	$0.816952\pi$
$38$	36757.3	−	36757.3i	0.669873	−	0.669873i
$39$		0			0
$40$	22415.0	+	3092.86i	0.350235	+	0.0483259i
$41$	−50688.1			−0.735452			−0.367726	−	0.929934i	$-0.619864\pi$
$41$	−50688.1			−0.735452			−0.367726	+	0.929934i	$0.619864\pi$
$42$		0			0
$43$	−12486.6	+	12486.6i	−0.157050	+	0.157050i	−0.781258	−	0.624208i	$-0.785422\pi$
$43$	−12486.6	+	12486.6i	−0.157050	+	0.157050i	0.624208	+	0.781258i	$0.285422\pi$
$44$		−	33187.4i		−	0.389596i
$45$		0			0
$46$	108624.			1.11597
$47$	25787.6	+	25787.6i	0.248380	+	0.248380i	0.820306	−	0.571925i	$-0.193803\pi$
$47$	25787.6	+	25787.6i	0.248380	+	0.248380i	−0.571925	+	0.820306i	$0.693803\pi$
$48$		0			0
$49$		−	75336.8i		−	0.640352i
$50$	−77090.0	+	43239.2i	−0.616720	+	0.345914i
$51$		0			0
$52$	64869.0	+	64869.0i	0.461347	+	0.461347i
$53$	166649.	−	166649.i	1.11937	−	1.11937i	0.127542	−	0.991833i	$-0.459291\pi$
$53$	166649.	−	166649.i	1.11937	−	1.11937i	0.991833	−	0.127542i	$-0.0407087\pi$
$54$		0			0
$55$	78277.9	+	103337.i	0.470491	+	0.621111i
$56$	−37235.6			−0.212028
$57$		0			0
$58$	33912.9	−	33912.9i	0.173813	−	0.173813i
$59$			247188.i			1.20357i	0.798658	+	0.601785i	$0.205544\pi$
$59$			247188.i			1.20357i	−0.798658	+	0.601785i	$0.794456\pi$
$60$		0			0
$61$	−281194.			−1.23884			−0.619422	−	0.785058i	$-0.712633\pi$
$61$	−281194.			−1.23884			−0.619422	+	0.785058i	$0.712633\pi$
$62$	68353.5	+	68353.5i	0.286805	+	0.286805i
$63$		0			0
$64$		−	32768.0i		−	0.125000i
$65$	−354991.	−	48982.0i	−1.29264	−	0.178360i
$66$		0			0
$67$	−304343.	−	304343.i	−1.01190	−	1.01190i	−0.999928	−	0.0119749i	$-0.996188\pi$
$67$	−304343.	−	304343.i	−1.01190	−	1.01190i	−0.0119749	−	0.999928i	$-0.503812\pi$
$68$	149456.	−	149456.i	0.475320	−	0.475320i
$69$		0			0
$70$	115942.	−	87826.2i	0.338025	−	0.256053i
$71$	288398.			0.805780			0.402890	−	0.915248i	$-0.368006\pi$
$71$	288398.			0.805780			0.402890	+	0.915248i	$0.368006\pi$
$72$		0			0
$73$	−487894.	+	487894.i	−1.25417	+	1.25417i	−0.300338	+	0.953833i	$0.597100\pi$
$73$	−487894.	+	487894.i	−1.25417	+	1.25417i	−0.953833	+	0.300338i	$0.902900\pi$
$74$		−	119650.i		−	0.295269i
$75$		0			0
$76$	294058.			0.669873
$77$	−150848.	−	150848.i	−0.330422	−	0.330422i
$78$		0			0
$79$		−	222009.i		−	0.450286i	−0.974326	−	0.225143i	$-0.927715\pi$
$79$		−	222009.i		−	0.450286i	0.974326	−	0.225143i	$-0.0722850\pi$
$80$	77288.8	+	102032.i	0.150955	+	0.199280i
$81$		0			0
$82$	−202752.	−	202752.i	−0.367726	−	0.367726i
$83$	−749160.	+	749160.i	−1.31021	+	1.31021i	−0.388948	+	0.921260i	$0.627161\pi$
$83$	−749160.	+	749160.i	−1.31021	+	1.31021i	−0.921260	+	0.388948i	$0.872839\pi$
$84$		0			0
$85$	−112853.	+	817885.i	−0.183762	+	1.33179i
$86$	−99892.6			−0.157050
$87$		0			0
$88$	132749.	−	132749.i	0.194798	−	0.194798i
$89$			689800.i			0.978482i	0.872149	+	0.489241i	$0.162726\pi$
$89$			689800.i			0.978482i	−0.872149	+	0.489241i	$0.837274\pi$
$90$		0			0
$91$	589705.			0.782548
$92$	434494.	+	434494.i	0.557983	+	0.557983i
$93$		0			0
$94$			206301.i			0.248380i
$95$	−915626.	+	693585.i	−1.06794	+	0.808963i
$96$		0			0
$97$	820944.	+	820944.i	0.899494	+	0.899494i	0.995391	−	0.0958974i	$-0.0305721\pi$
$97$	820944.	+	820944.i	0.899494	+	0.899494i	−0.0958974	+	0.995391i	$0.530572\pi$
$98$	301347.	−	301347.i	0.320176	−	0.320176i
$99$		0			0
$100$	−481317.	−	135403.i	−0.481317	−	0.135403i
$101$	290048.			0.281518			0.140759	−	0.990044i	$-0.455046\pi$
$101$	290048.			0.281518			0.140759	+	0.990044i	$0.455046\pi$
$102$		0			0
$103$	291498.	−	291498.i	0.266762	−	0.266762i	−0.561032	−	0.827794i	$-0.689596\pi$
$103$	291498.	−	291498.i	0.266762	−	0.266762i	0.827794	+	0.561032i	$0.189596\pi$
$104$			518952.i			0.461347i
$105$		0			0
$106$	1.33319e6			1.11937
$107$	−792277.	−	792277.i	−0.646734	−	0.646734i	0.305468	−	0.952202i	$-0.401187\pi$
$107$	−792277.	−	792277.i	−0.646734	−	0.646734i	−0.952202	+	0.305468i	$0.901187\pi$
$108$		0			0
$109$		−	165909.i		−	0.128112i	−0.997946	−	0.0640559i	$-0.979596\pi$
$109$		−	165909.i		−	0.128112i	0.997946	−	0.0640559i	$-0.0204036\pi$
$110$	−100238.	+	726461.i	−0.0753103	+	0.545801i
$111$		0			0
$112$	−148942.	−	148942.i	−0.106014	−	0.106014i
$113$	750389.	−	750389.i	0.520057	−	0.520057i	−0.397532	−	0.917589i	$-0.630133\pi$
$113$	750389.	−	750389.i	0.520057	−	0.520057i	0.917589	+	0.397532i	$0.130133\pi$
$114$		0			0
$115$	−2.37774e6	−	328083.i	−1.56340	−	0.215720i
$116$	271304.			0.173813
$117$		0			0
$118$	−988752.	+	988752.i	−0.601785	+	0.601785i
$119$		−	1.35866e6i		−	0.806250i
$120$		0			0
$121$	−695974.			−0.392859
$122$	−1.12478e6	−	1.12478e6i	−0.619422	−	0.619422i
$123$		0			0
$124$			546828.i			0.286805i
$125$	1.81808e6	−	713653.i	0.930854	−	0.365390i
$126$		0			0
$127$	583066.	+	583066.i	0.284647	+	0.284647i	0.834959	−	0.550312i	$-0.185491\pi$
$127$	583066.	+	583066.i	0.284647	+	0.284647i	−0.550312	+	0.834959i	$0.685491\pi$
$128$	131072.	−	131072.i	0.0625000	−	0.0625000i
$129$		0			0
$130$	−1.22403e6	−	1.61589e6i	−0.557139	−	0.735499i
$131$	4.18868e6			1.86321			0.931607	−	0.363466i	$-0.118407\pi$
$131$	4.18868e6			1.86321			0.931607	+	0.363466i	$0.118407\pi$
$132$		0			0
$133$	1.33660e6	−	1.33660e6i	0.568128	−	0.568128i
$134$		−	2.43474e6i		−	1.01190i
$135$		0			0
$136$	1.19565e6			0.475320
$137$	−250638.	−	250638.i	−0.0974731	−	0.0974731i	0.656689	−	0.754162i	$-0.271957\pi$
$137$	−250638.	−	250638.i	−0.0974731	−	0.0974731i	−0.754162	+	0.656689i	$0.771957\pi$
$138$		0			0
$139$		−	4.91265e6i		−	1.82924i	−0.404309	−	0.914622i	$-0.632488\pi$
$139$		−	4.91265e6i		−	1.82924i	0.404309	−	0.914622i	$-0.367512\pi$
$140$	815075.	+	112465.i	0.297039	+	0.0409858i
$141$		0			0
$142$	1.15359e6	+	1.15359e6i	0.402890	+	0.402890i
$143$	−2.10237e6	+	2.10237e6i	−0.718956	+	0.718956i
$144$		0			0
$145$	−844774.	+	639914.i	−0.277100	+	0.209903i
$146$	−3.90315e6			−1.25417
$147$		0			0
$148$	478601.	−	478601.i	0.147635	−	0.147635i
$149$		−	3.77991e6i		−	1.14268i	−0.820715	−	0.571338i	$-0.806425\pi$
$149$		−	3.77991e6i		−	1.14268i	0.820715	−	0.571338i	$-0.193575\pi$
$150$		0			0
$151$	1.19813e6			0.347996			0.173998	−	0.984746i	$-0.444331\pi$
$151$	1.19813e6			0.347996			0.173998	+	0.984746i	$0.444331\pi$
$152$	1.17623e6	+	1.17623e6i	0.334937	+	0.334937i
$153$		0			0
$154$		−	1.20679e6i		−	0.330422i
$155$	−1.28979e6	−	1.70269e6i	−0.346356	−	0.457236i
$156$		0			0
$157$	2.93950e6	+	2.93950e6i	0.759582	+	0.759582i	0.976246	−	0.216664i	$-0.0695176\pi$
$157$	2.93950e6	+	2.93950e6i	0.759582	+	0.759582i	−0.216664	+	0.976246i	$0.569518\pi$
$158$	888035.	−	888035.i	0.225143	−	0.225143i
$159$		0			0
$160$	−98971.4	+	717281.i	−0.0241629	+	0.175118i
$161$	3.94986e6			0.946465
$162$		0			0
$163$	3.63502e6	−	3.63502e6i	0.839351	−	0.839351i	−0.149422	−	0.988774i	$-0.547741\pi$
$163$	3.63502e6	−	3.63502e6i	0.839351	−	0.839351i	0.988774	+	0.149422i	$0.0477413\pi$
$164$		−	1.62202e6i		−	0.367726i
$165$		0			0
$166$	−5.99328e6			−1.31021
$167$	738364.	+	738364.i	0.158533	+	0.158533i	0.781917	−	0.623383i	$-0.214242\pi$
$167$	738364.	+	738364.i	0.158533	+	0.158533i	−0.623383	+	0.781917i	$0.714242\pi$
$168$		0			0
$169$		−	3.39192e6i		−	0.702725i
$170$	−3.72295e6	+	2.82013e6i	−0.757776	+	0.574014i
$171$		0			0
$172$	−399570.	−	399570.i	−0.0785250	−	0.0785250i
$173$	−5.01441e6	+	5.01441e6i	−0.968459	+	0.968459i	−0.999518	−	0.0310584i	$-0.990112\pi$
$173$	−5.01441e6	+	5.01441e6i	−0.968459	+	0.968459i	0.0310584	+	0.999518i	$0.490112\pi$
$174$		0			0
$175$	−2.80321e6	+	1.57230e6i	−0.523048	+	0.293374i
$176$	1.06200e6			0.194798
$177$		0			0
$178$	−2.75920e6	+	2.75920e6i	−0.489241	+	0.489241i
$179$		−	7.77058e6i		−	1.35486i	−0.735587	−	0.677430i	$-0.763094\pi$
$179$		−	7.77058e6i		−	1.35486i	0.735587	−	0.677430i	$-0.236906\pi$
$180$		0			0
$181$	6.98198e6			1.17745			0.588726	−	0.808333i	$-0.299630\pi$
$181$	6.98198e6			1.17745			0.588726	+	0.808333i	$0.299630\pi$
$182$	2.35882e6	+	2.35882e6i	0.391274	+	0.391274i
$183$		0			0
$184$			3.47596e6i			0.557983i
$185$	−361387.	+	2.61911e6i	−0.0570766	+	0.413655i
$186$		0			0
$187$	4.84379e6	+	4.84379e6i	0.740732	+	0.740732i
$188$	−825202.	+	825202.i	−0.124190	+	0.124190i
$189$		0			0
$190$	−6.43684e6	−	888163.i	−0.938452	−	0.129489i
$191$	−1.75621e6			−0.252044			−0.126022	−	0.992027i	$-0.540221\pi$
$191$	−1.75621e6			−0.252044			−0.126022	+	0.992027i	$0.540221\pi$
$192$		0			0
$193$	−8.48929e6	+	8.48929e6i	−1.18086	+	1.18086i	−0.201343	+	0.979521i	$0.564530\pi$
$193$	−8.48929e6	+	8.48929e6i	−1.18086	+	1.18086i	−0.979521	+	0.201343i	$0.935470\pi$
$194$			6.56755e6i			0.899494i
$195$		0			0
$196$	2.41078e6			0.320176
$197$	7.08383e6	+	7.08383e6i	0.926552	+	0.926552i	0.997481	−	0.0709298i	$-0.0225966\pi$
$197$	7.08383e6	+	7.08383e6i	0.926552	+	0.926552i	−0.0709298	+	0.997481i	$0.522597\pi$
$198$		0			0
$199$		−	2.34311e6i		−	0.297326i	−0.988888	−	0.148663i	$-0.952503\pi$
$199$		−	2.34311e6i		−	0.297326i	0.988888	−	0.148663i	$-0.0474970\pi$
$200$	−1.38365e6	−	2.46688e6i	−0.172957	−	0.308360i
$201$		0			0
$202$	1.16019e6	+	1.16019e6i	0.140759	+	0.140759i
$203$	1.23317e6	−	1.23317e6i	0.147413	−	0.147413i
$204$		0			0
$205$	3.82580e6	+	5.05058e6i	0.444080	+	0.586245i
$206$	2.33199e6			0.266762
$207$		0			0
$208$	−2.07581e6	+	2.07581e6i	−0.230673	+	0.230673i
$209$			9.53029e6i			1.04392i
$210$		0			0
$211$	−4.10418e6			−0.436897			−0.218448	−	0.975848i	$-0.570100\pi$
$211$	−4.10418e6			−0.436897			−0.218448	+	0.975848i	$0.570100\pi$
$212$	5.33277e6	+	5.33277e6i	0.559687	+	0.559687i
$213$		0			0
$214$		−	6.33822e6i		−	0.646734i
$215$	2.18662e6	+	301712.i	0.220018	+	0.0303583i
$216$		0			0
$217$	2.48553e6	+	2.48553e6i	0.243243	+	0.243243i
$218$	663634.	−	663634.i	0.0640559	−	0.0640559i
$219$		0			0
$220$	−3.30680e6	+	2.50489e6i	−0.310556	+	0.235245i
$221$	−1.89356e7			−1.75430
$222$		0			0
$223$	−9.84643e6	+	9.84643e6i	−0.887901	+	0.887901i	−0.994321	−	0.106421i	$-0.966061\pi$
$223$	−9.84643e6	+	9.84643e6i	−0.887901	+	0.887901i	0.106421	+	0.994321i	$0.466061\pi$
$224$		−	1.19154e6i		−	0.106014i
$225$		0			0
$226$	6.00311e6			0.520057
$227$	−447932.	−	447932.i	−0.0382943	−	0.0382943i	0.687700	−	0.725995i	$-0.258620\pi$
$227$	−447932.	−	447932.i	−0.0382943	−	0.0382943i	−0.725995	+	0.687700i	$0.758620\pi$
$228$		0			0
$229$		−	2.13385e7i		−	1.77688i	−0.458991	−	0.888441i	$-0.651789\pi$
$229$		−	2.13385e7i		−	1.77688i	0.458991	−	0.888441i	$-0.348211\pi$
$230$	−8.19862e6	−	1.08233e7i	−0.673840	−	0.889560i
$231$		0			0
$232$	1.08521e6	+	1.08521e6i	0.0869064	+	0.0869064i
$233$	−1.19508e6	+	1.19508e6i	−0.0944775	+	0.0944775i	−0.752766	−	0.658288i	$-0.771281\pi$
$233$	−1.19508e6	+	1.19508e6i	−0.0944775	+	0.0944775i	0.658288	+	0.752766i	$0.271281\pi$
$234$		0			0
$235$	623103.	−	4.51586e6i	0.0480127	−	0.347966i
$236$	−7.91001e6			−0.601785
$237$		0			0
$238$	5.43464e6	−	5.43464e6i	0.403125	−	0.403125i
$239$		−	2.07007e6i		−	0.151632i	−0.997122	−	0.0758160i	$-0.975844\pi$
$239$		−	2.07007e6i		−	0.151632i	0.997122	−	0.0758160i	$-0.0241562\pi$
$240$		0			0
$241$	2.35841e7			1.68487			0.842437	−	0.538795i	$-0.181120\pi$
$241$	2.35841e7			1.68487			0.842437	+	0.538795i	$0.181120\pi$
$242$	−2.78389e6	−	2.78389e6i	−0.196429	−	0.196429i
$243$		0			0
$244$		−	8.99822e6i		−	0.619422i
$245$	−7.50658e6	+	5.68622e6i	−0.510439	+	0.386656i
$246$		0			0
$247$	−1.86282e7	−	1.86282e7i	−1.23617	−	1.23617i
$248$	−2.18731e6	+	2.18731e6i	−0.143402	+	0.143402i
$249$		0			0
$250$	1.01269e7	+	4.41769e6i	0.648122	+	0.282732i
$251$	8.83541e6			0.558734			0.279367	−	0.960184i	$-0.409875\pi$
$251$	8.83541e6			0.558734			0.279367	+	0.960184i	$0.409875\pi$
$252$		0			0
$253$	−1.40818e7	+	1.40818e7i	−0.869552	+	0.869552i
$254$			4.66453e6i			0.284647i
$255$		0			0
$256$	1.04858e6			0.0625000
$257$	6.09969e6	+	6.09969e6i	0.359342	+	0.359342i	0.863570	−	0.504228i	$-0.168223\pi$
$257$	6.09969e6	+	6.09969e6i	0.359342	+	0.359342i	−0.504228	+	0.863570i	$0.668223\pi$
$258$		0			0
$259$		−	4.35082e6i		−	0.250422i
$260$	1.56743e6	−	1.13597e7i	0.0891799	−	0.646319i
$261$		0			0
$262$	1.67547e7	+	1.67547e7i	0.931607	+	0.931607i
$263$	4.35686e6	−	4.35686e6i	0.239500	−	0.239500i	−0.577143	−	0.816643i	$-0.695832\pi$
$263$	4.35686e6	−	4.35686e6i	0.239500	−	0.239500i	0.816643	+	0.577143i	$0.195832\pi$
$264$		0			0
$265$	−2.91832e7	−	4.02673e6i	−1.56818	−	0.216379i
$266$	1.06928e7			0.568128
$267$		0			0
$268$	9.73898e6	−	9.73898e6i	0.505952	−	0.505952i
$269$		−	1.36102e7i		−	0.699208i	−0.936898	−	0.349604i	$-0.886316\pi$
$269$		−	1.36102e7i		−	0.699208i	0.936898	−	0.349604i	$-0.113684\pi$
$270$		0			0
$271$	−8.20889e6			−0.412455			−0.206228	−	0.978504i	$-0.566119\pi$
$271$	−8.20889e6			−0.412455			−0.206228	+	0.978504i	$0.566119\pi$
$272$	4.78259e6	+	4.78259e6i	0.237660	+	0.237660i
$273$		0			0
$274$		−	2.00510e6i		−	0.0974731i
$275$	4.38836e6	−	1.55993e7i	0.211010	−	0.750077i
$276$		0			0
$277$	2.56480e7	+	2.56480e7i	1.20674	+	1.20674i	0.972076	+	0.234665i	$0.0753995\pi$
$277$	2.56480e7	+	2.56480e7i	1.20674	+	1.20674i	0.234665	+	0.972076i	$0.424601\pi$
$278$	1.96506e7	−	1.96506e7i	0.914622	−	0.914622i
$279$		0			0
$280$	2.81044e6	+	3.71016e6i	0.128027	+	0.169012i
$281$	2.47592e7			1.11588			0.557940	−	0.829881i	$-0.311592\pi$
$281$	2.47592e7			1.11588			0.557940	+	0.829881i	$0.311592\pi$
$282$		0			0
$283$	−1.00351e7	+	1.00351e7i	−0.442753	+	0.442753i	−0.892936	−	0.450183i	$-0.851359\pi$
$283$	−1.00351e7	+	1.00351e7i	−0.442753	+	0.442753i	0.450183	+	0.892936i	$0.351359\pi$
$284$			9.22873e6i			0.402890i
$285$		0			0
$286$	−1.68190e7			−0.718956
$287$	−7.37266e6	−	7.37266e6i	−0.311873	−	0.311873i
$288$		0			0
$289$			1.94895e7i			0.807433i
$290$	−5.93875e6	−	819436.i	−0.243501	−	0.0335986i
$291$		0			0
$292$	−1.56126e7	−	1.56126e7i	−0.627085	−	0.627085i
$293$	−1.88230e6	+	1.88230e6i	−0.0748319	+	0.0748319i	−0.743532	−	0.668700i	$-0.766851\pi$
$293$	−1.88230e6	+	1.88230e6i	−0.0748319	+	0.0748319i	0.668700	+	0.743532i	$0.266851\pi$
$294$		0			0
$295$	2.46299e7	−	1.86571e7i	0.959392	−	0.726737i
$296$	3.82881e6			0.147635
$297$		0			0
$298$	1.51196e7	−	1.51196e7i	0.571338	−	0.571338i
$299$		−	5.50493e7i		−	2.05939i
$300$		0			0
$301$	−3.63238e6			−0.133196
$302$	4.79253e6	+	4.79253e6i	0.173998	+	0.173998i
$303$		0			0
$304$			9.40986e6i			0.334937i
$305$	2.12238e7	+	2.80183e7i	0.748037	+	0.987510i
$306$		0			0
$307$	−3.23684e7	−	3.23684e7i	−1.11868	−	1.11868i	−0.991935	−	0.126747i	$-0.959547\pi$
$307$	−3.23684e7	−	3.23684e7i	−1.11868	−	1.11868i	−0.126747	−	0.991935i	$-0.540453\pi$
$308$	4.82715e6	−	4.82715e6i	0.165211	−	0.165211i
$309$		0			0
$310$	1.65162e6	−	1.19699e7i	0.0554403	−	0.401796i
$311$	−1.96913e7			−0.654625			−0.327313	−	0.944916i	$-0.606143\pi$
$311$	−1.96913e7			−0.654625			−0.327313	+	0.944916i	$0.606143\pi$
$312$		0			0
$313$	1.21275e7	−	1.21275e7i	0.395493	−	0.395493i	−0.481147	−	0.876640i	$-0.659780\pi$
$313$	1.21275e7	−	1.21275e7i	0.395493	−	0.395493i	0.876640	+	0.481147i	$0.159780\pi$
$314$			2.35160e7i			0.759582i
$315$		0			0
$316$	7.10428e6			0.225143
$317$	−2.36698e7	−	2.36698e7i	−0.743048	−	0.743048i	0.230116	−	0.973163i	$-0.426090\pi$
$317$	−2.36698e7	−	2.36698e7i	−0.743048	−	0.743048i	−0.973163	+	0.230116i	$0.926090\pi$
$318$		0			0
$319$			8.79282e6i			0.270867i
$320$	−3.26501e6	+	2.47324e6i	−0.0996402	+	0.0754773i
$321$		0			0
$322$	1.57995e7	+	1.57995e7i	0.473232	+	0.473232i
$323$	−4.29187e7	+	4.29187e7i	−1.27362	+	1.27362i
$324$		0			0
$325$	2.19132e7	+	3.90684e7i	0.638344	+	1.13809i
$326$	2.90802e7			0.839351
$327$		0			0
$328$	6.48808e6	−	6.48808e6i	0.183863	−	0.183863i
$329$			7.50167e6i			0.210654i
$330$		0			0
$331$	3.78781e7			1.04449			0.522245	−	0.852796i	$-0.325095\pi$
$331$	3.78781e7			1.04449			0.522245	+	0.852796i	$0.325095\pi$
$332$	−2.39731e7	−	2.39731e7i	−0.655104	−	0.655104i
$333$		0			0
$334$			5.90691e6i			0.158533i
$335$	−7.35382e6	+	5.32958e7i	−0.195604	+	1.41762i
$336$		0			0
$337$	−2.24435e7	−	2.24435e7i	−0.586411	−	0.586411i	0.350247	−	0.936657i	$-0.386098\pi$
$337$	−2.24435e7	−	2.24435e7i	−0.586411	−	0.586411i	−0.936657	+	0.350247i	$0.886098\pi$
$338$	1.35677e7	−	1.35677e7i	0.351362	−	0.351362i
$339$		0			0
$340$	−2.61723e7	−	3.61129e6i	−0.665895	−	0.0918810i
$341$	−1.77225e7			−0.446952
$342$		0			0
$343$	2.80701e7	−	2.80701e7i	0.695602	−	0.695602i
$344$		−	3.19656e6i		−	0.0785250i
$345$		0			0
$346$	−4.01153e7			−0.968459
$347$	−3.01955e7	−	3.01955e7i	−0.722692	−	0.722692i	0.246461	−	0.969153i	$-0.420732\pi$
$347$	−3.01955e7	−	3.01955e7i	−0.722692	−	0.722692i	−0.969153	+	0.246461i	$0.920732\pi$
$348$		0			0
$349$			7.82272e7i			1.84027i	0.391600	+	0.920135i	$0.371922\pi$
$349$			7.82272e7i			1.84027i	−0.391600	+	0.920135i	$0.628078\pi$
$350$	−1.75021e7	−	4.92365e6i	−0.408211	−	0.114837i
$351$		0			0
$352$	4.24798e6	+	4.24798e6i	0.0973991	+	0.0973991i
$353$	9.45650e6	−	9.45650e6i	0.214984	−	0.214984i	−0.591397	−	0.806381i	$-0.701423\pi$
$353$	9.45650e6	−	9.45650e6i	0.214984	−	0.214984i	0.806381	+	0.591397i	$0.201423\pi$
$354$		0			0
$355$	−2.17675e7	−	2.87360e7i	−0.486545	−	0.642305i
$356$	−2.20736e7			−0.489241
$357$		0			0
$358$	3.10823e7	−	3.10823e7i	0.677430	−	0.677430i
$359$			2.13996e7i			0.462511i	0.972893	+	0.231256i	$0.0742834\pi$
$359$			2.13996e7i			0.462511i	−0.972893	+	0.231256i	$0.925717\pi$
$360$		0			0
$361$	−3.73977e7			−0.794919
$362$	2.79279e7	+	2.79279e7i	0.588726	+	0.588726i
$363$		0			0
$364$			1.88706e7i			0.391274i
$365$	8.54388e7	+	1.17889e7i	1.75702	+	0.242436i
$366$		0			0
$367$	−1.29658e7	−	1.29658e7i	−0.262302	−	0.262302i	0.563687	−	0.825989i	$-0.309383\pi$
$367$	−1.29658e7	−	1.29658e7i	−0.262302	−	0.262302i	−0.825989	+	0.563687i	$0.809383\pi$
$368$	−1.39038e7	+	1.39038e7i	−0.278991	+	0.278991i
$369$		0			0
$370$	−1.19220e7	+	9.03088e6i	−0.235366	+	0.178289i
$371$	4.84787e7			0.949356
$372$		0			0
$373$	3.92352e6	−	3.92352e6i	0.0756048	−	0.0756048i	−0.668293	−	0.743898i	$-0.732975\pi$
$373$	3.92352e6	−	3.92352e6i	0.0756048	−	0.0756048i	0.743898	+	0.668293i	$0.232975\pi$
$374$			3.87504e7i			0.740732i
$375$		0			0
$376$	−6.60162e6			−0.124190
$377$	−1.71867e7	−	1.71867e7i	−0.320752	−	0.320752i
$378$		0			0
$379$		−	3.61170e7i		−	0.663429i	−0.943380	−	0.331714i	$-0.892373\pi$
$379$		−	3.61170e7i		−	0.663429i	0.943380	−	0.331714i	$-0.107627\pi$
$380$	−2.21947e7	−	2.93000e7i	−0.404482	−	0.533970i
$381$		0			0
$382$	−7.02485e6	−	7.02485e6i	−0.126022	−	0.126022i
$383$	774461.	−	774461.i	0.0137849	−	0.0137849i	−0.700181	−	0.713966i	$-0.746897\pi$
$383$	774461.	−	774461.i	0.0137849	−	0.0137849i	0.713966	+	0.700181i	$0.246897\pi$
$384$		0			0
$385$	−3.64494e6	+	2.64162e7i	−0.0638716	+	0.462901i
$386$	−6.79144e7			−1.18086
$387$		0			0
$388$	−2.62702e7	+	2.62702e7i	−0.449747	+	0.449747i
$389$			4.56599e7i			0.775686i	0.921726	+	0.387843i	$0.126780\pi$
$389$			4.56599e7i			0.775686i	−0.921726	+	0.387843i	$0.873220\pi$
$390$		0			0
$391$	−1.26831e8			−2.12176
$392$	9.64311e6	+	9.64311e6i	0.160088	+	0.160088i
$393$		0			0
$394$			5.66707e7i			0.926552i
$395$	−2.21210e7	+	1.67566e7i	−0.358933	+	0.271891i
$396$		0			0
$397$	5.08606e7	+	5.08606e7i	0.812849	+	0.812849i	0.985060	−	0.172211i	$-0.0550912\pi$
$397$	5.08606e7	+	5.08606e7i	0.812849	+	0.812849i	−0.172211	+	0.985060i	$0.555091\pi$
$398$	9.37244e6	−	9.37244e6i	0.148663	−	0.148663i
$399$		0			0
$400$	4.33291e6	−	1.54021e7i	0.0677016	−	0.240658i
$401$	3.46994e7			0.538133			0.269066	−	0.963122i	$-0.413285\pi$
$401$	3.46994e7			0.538133			0.269066	+	0.963122i	$0.413285\pi$
$402$		0			0
$403$	3.46408e7	−	3.46408e7i	0.529265	−	0.529265i
$404$			9.28155e6i			0.140759i
$405$		0			0
$406$	9.86537e6			0.147413
$407$	1.55112e7	+	1.55112e7i	0.230072	+	0.230072i
$408$		0			0
$409$		−	6.63216e6i		−	0.0969360i	−0.998825	−	0.0484680i	$-0.984566\pi$
$409$		−	6.63216e6i		−	0.0969360i	0.998825	−	0.0484680i	$-0.0154339\pi$
$410$	−4.89909e6	+	3.55055e7i	−0.0710827	+	0.515162i
$411$		0			0
$412$	9.32795e6	+	9.32795e6i	0.133381	+	0.133381i
$413$	−3.59538e7	+	3.59538e7i	−0.510382	+	0.510382i
$414$		0			0
$415$	1.31191e8	+	1.81019e7i	1.83552	+	0.253268i
$416$	−1.66065e7			−0.230673
$417$		0			0
$418$	−3.81212e7	+	3.81212e7i	−0.521960	+	0.521960i
$419$			1.87567e7i			0.254985i	0.991840	+	0.127493i	$0.0406929\pi$
$419$			1.87567e7i			0.254985i	−0.991840	+	0.127493i	$0.959307\pi$
$420$		0			0
$421$	1.01392e8			1.35881			0.679406	−	0.733762i	$-0.262237\pi$
$421$	1.01392e8			1.35881			0.679406	+	0.733762i	$0.262237\pi$
$422$	−1.64167e7	−	1.64167e7i	−0.218448	−	0.218448i
$423$		0			0
$424$			4.26622e7i			0.559687i
$425$	9.00121e7	−	5.04871e7i	1.17256	−	0.657679i
$426$		0			0
$427$	−4.09001e7	−	4.09001e7i	−0.525340	−	0.525340i
$428$	2.53529e7	−	2.53529e7i	0.323367	−	0.323367i
$429$		0			0
$430$	7.53963e6	+	9.95332e6i	0.0948297	+	0.125188i
$431$	2.54338e7			0.317672			0.158836	−	0.987305i	$-0.449226\pi$
$431$	2.54338e7			0.317672			0.158836	+	0.987305i	$0.449226\pi$
$432$		0			0
$433$	−4.84883e7	+	4.84883e7i	−0.597273	+	0.597273i	−0.939586	−	0.342313i	$-0.888790\pi$
$433$	−4.84883e7	+	4.84883e7i	−0.597273	+	0.597273i	0.342313	+	0.939586i	$0.388790\pi$
$434$			1.98842e7i			0.243243i
$435$		0			0
$436$	5.30907e6			0.0640559
$437$	−1.24772e8	−	1.24772e8i	−1.49511	−	1.49511i
$438$		0			0
$439$			3.77933e7i			0.446706i	0.974738	+	0.223353i	$0.0717002\pi$
$439$			3.77933e7i			0.446706i	−0.974738	+	0.223353i	$0.928300\pi$
$440$	−2.32468e7	−	3.20762e6i	−0.272901	−	0.0376551i
$441$		0			0
$442$	−7.57426e7	−	7.57426e7i	−0.877149	−	0.877149i
$443$	2.16224e7	−	2.16224e7i	0.248710	−	0.248710i	−0.571731	−	0.820441i	$-0.693728\pi$
$443$	2.16224e7	−	2.16224e7i	0.248710	−	0.248710i	0.820441	+	0.571731i	$0.193728\pi$
$444$		0			0
$445$	6.87318e7	−	5.20642e7i	0.779970	−	0.590826i
$446$	−7.87715e7			−0.887901
$447$		0			0
$448$	4.76615e6	−	4.76615e6i	0.0530071	−	0.0530071i
$449$			9.43557e7i			1.04239i	0.853439	+	0.521193i	$0.174513\pi$
$449$			9.43557e7i			1.04239i	−0.853439	+	0.521193i	$0.825487\pi$
$450$		0			0
$451$	5.25689e7			0.573059
$452$	2.40124e7	+	2.40124e7i	0.260028	+	0.260028i
$453$		0			0
$454$		−	3.58345e6i		−	0.0382943i
$455$	−4.45094e7	−	5.87584e7i	−0.472517	−	0.623786i
$456$		0			0
$457$	−3.23415e7	−	3.23415e7i	−0.338853	−	0.338853i	0.517083	−	0.855936i	$-0.327018\pi$
$457$	−3.23415e7	−	3.23415e7i	−0.338853	−	0.338853i	−0.855936	+	0.517083i	$0.827018\pi$
$458$	8.53542e7	−	8.53542e7i	0.888441	−	0.888441i
$459$		0			0
$460$	1.04987e7	−	7.60876e7i	0.107860	−	0.781700i
$461$	−8.77131e7			−0.895286			−0.447643	−	0.894212i	$-0.647736\pi$
$461$	−8.77131e7			−0.895286			−0.447643	+	0.894212i	$0.647736\pi$
$462$		0			0
$463$	4.91700e7	−	4.91700e7i	0.495402	−	0.495402i	−0.414601	−	0.910003i	$-0.636079\pi$
$463$	4.91700e7	−	4.91700e7i	0.495402	−	0.495402i	0.910003	+	0.414601i	$0.136079\pi$
$464$			8.68171e6i			0.0869064i
$465$		0			0
$466$	−9.56063e6			−0.0944775
$467$	9.52295e7	+	9.52295e7i	0.935020	+	0.935020i	0.998014	−	0.0629938i	$-0.0200648\pi$
$467$	9.52295e7	+	9.52295e7i	0.935020	+	0.935020i	−0.0629938	+	0.998014i	$0.520065\pi$
$468$		0			0
$469$		−	8.85342e7i		−	0.858208i
$470$	2.05558e7	−	1.55710e7i	0.197989	−	0.149976i
$471$		0			0
$472$	−3.16401e7	−	3.16401e7i	−0.300892	−	0.300892i
$473$	1.29499e7	−	1.29499e7i	0.122372	−	0.122372i
$474$		0			0
$475$	1.38218e8	+	3.88832e7i	1.28968	+	0.362812i
$476$	4.34771e7			0.403125
$477$		0			0
$478$	8.28027e6	−	8.28027e6i	0.0758160	−	0.0758160i
$479$		−	2.04024e8i		−	1.85642i	−0.372062	−	0.928208i	$-0.621349\pi$
$479$		−	2.04024e8i		−	1.85642i	0.372062	−	0.928208i	$-0.378651\pi$
$480$		0			0
$481$	−6.06374e7			−0.544886
$482$	9.43362e7	+	9.43362e7i	0.842437	+	0.842437i
$483$		0			0
$484$		−	2.22712e7i		−	0.196429i
$485$	1.98364e7	−	1.43762e8i	0.173875	−	1.26014i
$486$		0			0
$487$	−1.00420e8	−	1.00420e8i	−0.869424	−	0.869424i	0.122984	−	0.992409i	$-0.460753\pi$
$487$	−1.00420e8	−	1.00420e8i	−0.869424	−	0.869424i	−0.992409	+	0.122984i	$0.960753\pi$
$488$	3.59929e7	−	3.59929e7i	0.309711	−	0.309711i
$489$		0			0
$490$	−5.27712e7	−	7.28143e6i	−0.448548	−	0.0618911i
$491$	7.28010e7			0.615025			0.307512	−	0.951544i	$-0.400503\pi$
$491$	7.28010e7			0.615025			0.307512	+	0.951544i	$0.400503\pi$
$492$		0			0
$493$	−3.95976e7	+	3.95976e7i	−0.330467	+	0.330467i
$494$		−	1.49025e8i		−	1.23617i
$495$		0			0
$496$	−1.74985e7			−0.143402
$497$	4.19478e7	+	4.19478e7i	0.341696	+	0.341696i
$498$		0			0
$499$			1.79824e8i			1.44726i	0.690188	+	0.723630i	$0.257528\pi$
$499$			1.79824e8i			1.44726i	−0.690188	+	0.723630i	$0.742472\pi$
$500$	2.28369e7	+	5.81784e7i	0.182695	+	0.465427i
$501$		0			0
$502$	3.53416e7	+	3.53416e7i	0.279367	+	0.279367i
$503$	5.58642e7	−	5.58642e7i	0.438965	−	0.438965i	−0.452699	−	0.891664i	$-0.649539\pi$
$503$	5.58642e7	−	5.58642e7i	0.438965	−	0.438965i	0.891664	+	0.452699i	$0.149539\pi$
$504$		0			0
$505$	−2.18921e7	−	2.89005e7i	−0.169986	−	0.224404i
$506$	−1.12654e8			−0.869552
$507$		0			0
$508$	−1.86581e7	+	1.86581e7i	−0.142324	+	0.142324i
$509$			2.14366e6i			0.0162556i	0.999967	+	0.00812779i	$0.00258718\pi$
$509$			2.14366e6i			0.0162556i	−0.999967	+	0.00812779i	$0.997413\pi$
$510$		0			0
$511$	−1.41930e8			−1.06368
$512$	4.19430e6	+	4.19430e6i	0.0312500	+	0.0312500i
$513$		0			0
$514$			4.87975e7i			0.359342i
$515$	−5.10465e7	−	7.04345e6i	−0.373718	−	0.0515661i
$516$		0			0
$517$	−2.67444e7	−	2.67444e7i	−0.193536	−	0.193536i
$518$	1.74033e7	−	1.74033e7i	0.125211	−	0.125211i
$519$		0			0
$520$	5.17085e7	−	3.91691e7i	0.367749	−	0.278570i
$521$	5.06571e7			0.358201			0.179101	−	0.983831i	$-0.442681\pi$
$521$	5.06571e7			0.358201			0.179101	+	0.983831i	$0.442681\pi$
$522$		0			0
$523$	−1.58594e7	+	1.58594e7i	−0.110862	+	0.110862i	−0.760362	−	0.649500i	$-0.774978\pi$
$523$	−1.58594e7	+	1.58594e7i	−0.110862	+	0.110862i	0.649500	+	0.760362i	$0.274978\pi$
$524$			1.34038e8i			0.931607i
$525$		0			0
$526$	3.48549e7			0.239500
$527$	−7.98112e7	−	7.98112e7i	−0.545296	−	0.545296i
$528$		0			0
$529$		−	2.20686e8i		−	1.49076i
$530$	−1.00626e8	−	1.32840e8i	−0.675899	−	0.892278i
$531$		0			0
$532$	4.27712e7	+	4.27712e7i	0.284064	+	0.284064i
$533$	−1.02753e8	+	1.02753e8i	−0.678597	+	0.678597i
$534$		0			0
$535$	−1.91437e7	+	1.38742e8i	−0.125016	+	0.906036i
$536$	7.79118e7			0.505952
$537$		0			0
$538$	5.44406e7	−	5.44406e7i	0.349604	−	0.349604i
$539$			7.81322e7i			0.498958i
$540$		0			0
$541$	4.43541e7			0.280119			0.140059	−	0.990143i	$-0.455271\pi$
$541$	4.43541e7			0.280119			0.140059	+	0.990143i	$0.455271\pi$
$542$	−3.28356e7	−	3.28356e7i	−0.206228	−	0.206228i
$543$		0			0
$544$			3.82607e7i			0.237660i
$545$	−1.65312e7	+	1.25223e7i	−0.102121	+	0.0773563i
$546$		0			0
$547$	1.89599e7	+	1.89599e7i	0.115844	+	0.115844i	0.762653	−	0.646808i	$-0.223897\pi$
$547$	1.89599e7	+	1.89599e7i	0.115844	+	0.115844i	−0.646808	+	0.762653i	$0.723897\pi$
$548$	8.02041e6	−	8.02041e6i	0.0487366	−	0.0487366i
$549$		0			0
$550$	7.99505e7	−	4.48436e7i	0.480544	−	0.269533i
$551$	−7.79092e7			−0.465730
$552$		0			0
$553$	3.22915e7	−	3.22915e7i	0.190947	−	0.190947i
$554$			2.05184e8i			1.20674i
$555$		0			0
$556$	1.57205e8			0.914622
$557$	1.03290e8	+	1.03290e8i	0.597712	+	0.597712i	0.939703	−	0.341991i	$-0.111101\pi$
$557$	1.03290e8	+	1.03290e8i	0.597712	+	0.597712i	−0.341991	+	0.939703i	$0.611101\pi$
$558$		0			0
$559$			5.06245e7i			0.289818i
$560$	−3.59888e6	+	2.60824e7i	−0.0204929	+	0.148519i
$561$		0			0
$562$	9.90369e7	+	9.90369e7i	0.557940	+	0.557940i
$563$	−1.84599e8	+	1.84599e8i	−1.03444	+	1.03444i	−0.0350521	+	0.999385i	$0.511160\pi$
$563$	−1.84599e8	+	1.84599e8i	−1.03444	+	1.03444i	−0.999385	+	0.0350521i	$0.988840\pi$
$564$		0			0
$565$	−1.31406e8	−	1.81316e7i	−0.728569	−	0.100529i
$566$	−8.02806e7			−0.442753
$567$		0			0
$568$	−3.69149e7	+	3.69149e7i	−0.201445	+	0.201445i
$569$			9.69662e7i			0.526361i	0.964747	+	0.263180i	$0.0847715\pi$
$569$			9.69662e7i			0.526361i	−0.964747	+	0.263180i	$0.915229\pi$
$570$		0			0
$571$	−8.24057e7			−0.442638			−0.221319	−	0.975201i	$-0.571036\pi$
$571$	−8.24057e7			−0.442638			−0.221319	+	0.975201i	$0.571036\pi$
$572$	−6.72760e7	−	6.72760e7i	−0.359478	−	0.359478i
$573$		0			0
$574$		−	5.89812e7i		−	0.311873i
$575$	1.46775e8	+	2.61681e8i	0.772055	+	1.37648i
$576$		0			0
$577$	1.75260e8	+	1.75260e8i	0.912336	+	0.912336i	0.996456	−	0.0841201i	$-0.0268079\pi$
$577$	1.75260e8	+	1.75260e8i	0.912336	+	0.912336i	−0.0841201	+	0.996456i	$0.526808\pi$
$578$	−7.79579e7	+	7.79579e7i	−0.403716	+	0.403716i
$579$		0			0
$580$	−2.04773e7	−	2.70328e7i	−0.104951	−	0.138550i
$581$	−2.17933e8			−1.11120
$582$		0			0
$583$	−1.72833e8	+	1.72833e8i	−0.872209	+	0.872209i
$584$		−	1.24901e8i		−	0.627085i
$585$		0			0
$586$	−1.50584e7			−0.0748319
$587$	5.98432e7	+	5.98432e7i	0.295870	+	0.295870i	0.839394	−	0.543524i	$-0.182910\pi$
$587$	5.98432e7	+	5.98432e7i	0.295870	+	0.295870i	−0.543524	+	0.839394i	$0.682910\pi$
$588$		0			0
$589$		−	1.57031e8i		−	0.768490i
$590$	1.73148e8	+	2.38911e7i	0.843065	+	0.116327i
$591$		0			0
$592$	1.53152e7	+	1.53152e7i	0.0738173	+	0.0738173i
$593$	−7.10812e7	+	7.10812e7i	−0.340872	+	0.340872i	−0.856695	−	0.515823i	$-0.827486\pi$
$593$	−7.10812e7	+	7.10812e7i	−0.340872	+	0.340872i	0.515823	+	0.856695i	$0.327486\pi$
$594$		0			0
$595$	−1.35377e8	+	1.02548e8i	−0.642680	+	0.486829i
$596$	1.20957e8			0.571338
$597$		0			0
$598$	2.20197e8	−	2.20197e8i	1.02969	−	1.02969i
$599$		−	1.93410e8i		−	0.899911i	−0.893051	−	0.449955i	$-0.851440\pi$
$599$		−	1.93410e8i		−	0.899911i	0.893051	−	0.449955i	$-0.148560\pi$
$600$		0			0
$601$	−2.04018e8			−0.939822			−0.469911	−	0.882714i	$-0.655714\pi$
$601$	−2.04018e8			−0.939822			−0.469911	+	0.882714i	$0.655714\pi$
$602$	−1.45295e7	−	1.45295e7i	−0.0665981	−	0.0665981i
$603$		0			0
$604$			3.83403e7i			0.173998i
$605$	5.25302e7	+	6.93470e7i	0.237215	+	0.313156i
$606$		0			0
$607$	2.70178e8	+	2.70178e8i	1.20805	+	1.20805i	0.971659	+	0.236388i	$0.0759636\pi$
$607$	2.70178e8	+	2.70178e8i	1.20805	+	1.20805i	0.236388	+	0.971659i	$0.424036\pi$
$608$	−3.76394e7	+	3.76394e7i	−0.167468	+	0.167468i
$609$		0			0
$610$	−2.71779e7	+	1.96968e8i	−0.119736	+	0.867774i
$611$	1.04551e8			0.458357
$612$		0			0
$613$	−1.48040e8	+	1.48040e8i	−0.642685	+	0.642685i	−0.951215	−	0.308529i	$-0.900163\pi$
$613$	−1.48040e8	+	1.48040e8i	−0.642685	+	0.642685i	0.308529	+	0.951215i	$0.400163\pi$
$614$		−	2.58947e8i		−	1.11868i
$615$		0			0
$616$	3.86172e7			0.165211
$617$	3.46749e7	+	3.46749e7i	0.147625	+	0.147625i	0.777056	−	0.629431i	$-0.216712\pi$
$617$	3.46749e7	+	3.46749e7i	0.147625	+	0.147625i	−0.629431	+	0.777056i	$0.716712\pi$
$618$		0			0
$619$			3.42044e8i			1.44215i	0.692858	+	0.721074i	$0.256351\pi$
$619$			3.42044e8i			1.44215i	−0.692858	+	0.721074i	$0.743649\pi$
$620$	5.44861e7	−	4.12731e7i	0.228618	−	0.173178i
$621$		0			0
$622$	−7.87651e7	−	7.87651e7i	−0.327313	−	0.327313i
$623$	−1.00332e8	+	1.00332e8i	−0.414932	+	0.414932i
$624$		0			0
$625$	−2.08332e8	−	1.27289e8i	−0.853328	−	0.521375i
$626$	9.70201e7			0.395493
$627$		0			0
$628$	−9.40641e7	+	9.40641e7i	−0.379791	+	0.379791i
$629$			1.39706e8i			0.561390i
$630$		0			0
$631$	8.36979e7			0.333140			0.166570	−	0.986030i	$-0.446731\pi$
$631$	8.36979e7			0.333140			0.166570	+	0.986030i	$0.446731\pi$
$632$	2.84171e7	+	2.84171e7i	0.112572	+	0.112572i
$633$		0			0
$634$		−	1.89358e8i		−	0.743048i
$635$	1.40886e7	−	1.02105e8i	0.0550233	−	0.398774i
$636$		0			0
$637$	−1.52719e8	−	1.52719e8i	−0.590848	−	0.590848i
$638$	−3.51713e7	+	3.51713e7i	−0.135434	+	0.135434i
$639$		0			0
$640$	−2.29530e7	−	3.16708e6i	−0.0875588	−	0.0120815i
$641$	−1.80537e8			−0.685475			−0.342737	−	0.939431i	$-0.611354\pi$
$641$	−1.80537e8			−0.685475			−0.342737	+	0.939431i	$0.611354\pi$
$642$		0			0
$643$	2.11026e8	−	2.11026e8i	0.793784	−	0.793784i	−0.188324	−	0.982107i	$-0.560305\pi$
$643$	2.11026e8	−	2.11026e8i	0.793784	−	0.793784i	0.982107	+	0.188324i	$0.0603054\pi$
$644$			1.26396e8i			0.473232i
$645$		0			0
$646$	−3.43349e8			−1.27362
$647$	−1.30845e8	−	1.30845e8i	−0.483109	−	0.483109i	0.423014	−	0.906123i	$-0.360972\pi$
$647$	−1.30845e8	−	1.30845e8i	−0.483109	−	0.483109i	−0.906123	+	0.423014i	$0.860972\pi$
$648$		0			0
$649$		−	2.56360e8i		−	0.937812i
$650$	−6.86209e7	+	2.43926e8i	−0.249871	+	0.888215i
$651$		0			0
$652$	1.16321e8	+	1.16321e8i	0.419676	+	0.419676i
$653$	3.15489e8	−	3.15489e8i	1.13304	−	1.13304i	0.143371	−	0.989669i	$-0.454206\pi$
$653$	3.15489e8	−	3.15489e8i	1.13304	−	1.13304i	0.989669	−	0.143371i	$-0.0457943\pi$
$654$		0			0
$655$	−3.16150e8	−	4.17361e8i	−1.12504	−	1.48521i
$656$	5.19046e7			0.183863
$657$		0			0
$658$	−3.00067e7	+	3.00067e7i	−0.105327	+	0.105327i
$659$		−	1.90964e8i		−	0.667260i	−0.942704	−	0.333630i	$-0.891726\pi$
$659$		−	1.90964e8i		−	0.667260i	0.942704	−	0.333630i	$-0.108274\pi$
$660$		0			0
$661$	4.57014e8			1.58243			0.791216	−	0.611536i	$-0.209448\pi$
$661$	4.57014e8			1.58243			0.791216	+	0.611536i	$0.209448\pi$
$662$	1.51512e8	+	1.51512e8i	0.522245	+	0.522245i
$663$		0			0
$664$		−	1.91785e8i		−	0.655104i
$665$	−2.34062e8	−	3.22961e7i	−0.795913	−	0.109821i
$666$		0			0
$667$	−1.15117e8	−	1.15117e8i	−0.387938	−	0.387938i
$668$	−2.36276e7	+	2.36276e7i	−0.0792667	+	0.0792667i
$669$		0			0
$670$	−2.42599e8	+	1.83768e8i	−0.806610	+	0.611006i
$671$	2.91628e8			0.965299
$672$		0			0
$673$	2.86067e8	−	2.86067e8i	0.938474	−	0.938474i	−0.0597400	−	0.998214i	$-0.519027\pi$
$673$	2.86067e8	−	2.86067e8i	0.938474	−	0.938474i	0.998214	+	0.0597400i	$0.0190272\pi$
$674$		−	1.79548e8i		−	0.586411i
$675$		0			0
$676$	1.08541e8			0.351362
$677$	−2.60104e8	−	2.60104e8i	−0.838266	−	0.838266i	0.150365	−	0.988631i	$-0.451955\pi$
$677$	−2.60104e8	−	2.60104e8i	−0.838266	−	0.838266i	−0.988631	+	0.150365i	$0.951955\pi$
$678$		0			0
$679$			2.38815e8i			0.762872i
$680$	−9.02442e7	−	1.19135e8i	−0.287007	−	0.378888i
$681$		0			0
$682$	−7.08898e7	−	7.08898e7i	−0.223476	−	0.223476i
$683$	9.24222e7	−	9.24222e7i	0.290078	−	0.290078i	−0.547033	−	0.837111i	$-0.684243\pi$
$683$	9.24222e7	−	9.24222e7i	0.290078	−	0.290078i	0.837111	+	0.547033i	$0.184243\pi$
$684$		0			0
$685$	−6.05614e6	+	4.38911e7i	−0.0188419	+	0.136554i
$686$	2.24560e8			0.695602
$687$		0			0
$688$	1.27863e7	−	1.27863e7i	0.0392625	−	0.0392625i
$689$		−	6.75648e8i		−	2.06568i
$690$		0			0
$691$	4.54828e8			1.37852			0.689261	−	0.724513i	$-0.257935\pi$
$691$	4.54828e8			1.37852			0.689261	+	0.724513i	$0.257935\pi$
$692$	−1.60461e8	−	1.60461e8i	−0.484230	−	0.484230i
$693$		0			0
$694$		−	2.41564e8i		−	0.722692i
$695$	−4.89498e8	+	3.70794e8i	−1.45813	+	1.10453i
$696$		0			0
$697$	2.36739e8	+	2.36739e8i	0.699150	+	0.699150i
$698$	−3.12909e8	+	3.12909e8i	−0.920135	+	0.920135i
$699$		0			0
$700$	−5.03136e7	−	8.97028e7i	−0.146687	−	0.261524i
$701$	3.98272e8			1.15618			0.578090	−	0.815973i	$-0.303798\pi$
$701$	3.98272e8			1.15618			0.578090	+	0.815973i	$0.303798\pi$
$702$		0			0
$703$	−1.37438e8	+	1.37438e8i	−0.395586	+	0.395586i
$704$			3.39839e7i			0.0973991i
$705$		0			0
$706$	7.56520e7			0.214984
$707$	4.21879e7	+	4.21879e7i	0.119380	+	0.119380i
$708$		0			0
$709$		−	1.94492e8i		−	0.545711i	−0.962055	−	0.272855i	$-0.912032\pi$
$709$		−	1.94492e8i		−	0.545711i	0.962055	−	0.272855i	$-0.0879680\pi$
$710$	2.78741e7	−	2.02014e8i	0.0778801	−	0.564425i
$711$		0			0
$712$	−8.82943e7	−	8.82943e7i	−0.244621	−	0.244621i
$713$	2.32025e8	−	2.32025e8i	0.640128	−	0.640128i
$714$		0			0
$715$	3.68163e8	+	5.07995e7i	1.00721	+	0.138977i
$716$	2.48659e8			0.677430
$717$		0			0
$718$	−8.55984e7	+	8.55984e7i	−0.231256	+	0.231256i
$719$			5.36699e8i			1.44392i	0.691934	+	0.721961i	$0.256759\pi$
$719$			5.36699e8i			1.44392i	−0.691934	+	0.721961i	$0.743241\pi$
$720$		0			0
$721$	8.47977e7			0.226245
$722$	−1.49591e8	−	1.49591e8i	−0.397460	−	0.397460i
$723$		0			0
$724$			2.23423e8i			0.588726i
$725$	1.27522e8	+	3.58744e7i	0.334636	+	0.0941393i
$726$		0			0
$727$	6.70897e7	+	6.70897e7i	0.174603	+	0.174603i	0.788998	−	0.614395i	$-0.210600\pi$
$727$	6.70897e7	+	6.70897e7i	0.174603	+	0.174603i	−0.614395	+	0.788998i	$0.710600\pi$
$728$	−7.54823e7	+	7.54823e7i	−0.195637	+	0.195637i
$729$		0			0
$730$	2.94599e8	+	3.88911e8i	0.757292	+	0.999727i
$731$	1.16637e8			0.298596
$732$		0			0
$733$	−2.35707e7	+	2.35707e7i	−0.0598495	+	0.0598495i	−0.736398	−	0.676549i	$-0.763475\pi$
$733$	−2.35707e7	+	2.35707e7i	−0.0598495	+	0.0598495i	0.676549	+	0.736398i	$0.263475\pi$
$734$		−	1.03727e8i		−	0.262302i
$735$		0			0
$736$	−1.11231e8			−0.278991
$737$	3.15636e8	+	3.15636e8i	0.788467	+	0.788467i
$738$		0			0
$739$		−	5.18563e8i		−	1.28490i	−0.766329	−	0.642448i	$-0.777919\pi$
$739$		−	5.18563e8i		−	1.28490i	0.766329	−	0.642448i	$-0.222081\pi$
$740$	−8.38114e7	−	1.15644e7i	−0.206827	−	0.0285383i
$741$		0			0
$742$	1.93915e8	+	1.93915e8i	0.474678	+	0.474678i
$743$	2.89312e8	−	2.89312e8i	0.705343	−	0.705343i	−0.260209	−	0.965552i	$-0.583792\pi$
$743$	2.89312e8	−	2.89312e8i	0.705343	−	0.705343i	0.965552	+	0.260209i	$0.0837916\pi$
$744$		0			0
$745$	−3.76631e8	+	2.85298e8i	−0.910851	+	0.689968i
$746$	3.13882e7			0.0756048
$747$		0			0
$748$	−1.55001e8	+	1.55001e8i	−0.370366	+	0.370366i
$749$		−	2.30476e8i		−	0.548504i
$750$		0			0
$751$	3.00670e8			0.709857			0.354928	−	0.934893i	$-0.384505\pi$
$751$	3.00670e8			0.709857			0.354928	+	0.934893i	$0.384505\pi$
$752$	−2.64065e7	−	2.64065e7i	−0.0620950	−	0.0620950i
$753$		0			0
$754$		−	1.37494e8i		−	0.320752i
$755$	−9.04319e7	−	1.19382e8i	−0.210126	−	0.277395i
$756$		0			0
$757$	−4.36582e8	−	4.36582e8i	−1.00642	−	1.00642i	−0.999979	−	0.00643784i	$-0.997951\pi$
$757$	−4.36582e8	−	4.36582e8i	−1.00642	−	1.00642i	−0.00643784	−	0.999979i	$-0.502049\pi$
$758$	1.44468e8	−	1.44468e8i	0.331714	−	0.331714i
$759$		0			0
$760$	2.84212e7	−	2.05979e8i	0.0647444	−	0.469226i
$761$	−5.85378e8			−1.32826			−0.664128	−	0.747619i	$-0.731197\pi$
$761$	−5.85378e8			−1.32826			−0.664128	+	0.747619i	$0.731197\pi$
$762$		0			0
$763$	2.41316e7	−	2.41316e7i	0.0543267	−	0.0543267i
$764$		−	5.61988e7i		−	0.126022i
$765$		0			0
$766$	6.19568e6			0.0137849
$767$	5.01089e8	+	5.01089e8i	1.11053	+	1.11053i
$768$		0			0
$769$		−	10186.9i		−	2.24008e-5i	−1.00000		1.12004e-5i	$-0.999996\pi$
$769$		−	10186.9i		−	2.24008e-5i	1.00000		1.12004e-5i	$-3.56520e-6\pi$
$770$	−1.20245e8	+	9.10850e7i	−0.263386	+	0.199515i
$771$		0			0
$772$	−2.71657e8	−	2.71657e8i	−0.590432	−	0.590432i
$773$	1.49946e8	−	1.49946e8i	0.324636	−	0.324636i	−0.525906	−	0.850542i	$-0.676274\pi$
$773$	1.49946e8	−	1.49946e8i	0.324636	−	0.324636i	0.850542	+	0.525906i	$0.176274\pi$
$774$		0			0
$775$	−7.23070e7	+	2.57029e8i	−0.155337	+	0.552175i
$776$	−2.10162e8			−0.449747
$777$		0			0
$778$	−1.82639e8	+	1.82639e8i	−0.387843	+	0.387843i
$779$			4.65789e8i			0.985319i
$780$		0			0
$781$	−2.99099e8			−0.627858
$782$	−5.07326e8	−	5.07326e8i	−1.06088	−	1.06088i
$783$		0			0
$784$			7.71449e7i			0.160088i
$785$	7.10270e7	−	5.14758e8i	0.146830	−	1.06413i
$786$		0			0
$787$	3.29898e8	+	3.29898e8i	0.676793	+	0.676793i	0.959273	−	0.282481i	$-0.0911573\pi$
$787$	3.29898e8	+	3.29898e8i	0.676793	+	0.676793i	−0.282481	+	0.959273i	$0.591157\pi$
$788$	−2.26683e8	+	2.26683e8i	−0.463276	+	0.463276i
$789$		0			0
$790$	−1.55511e8	−	2.14575e7i	−0.315412	−	0.0435210i
$791$	2.18290e8			0.441067
$792$		0			0
$793$	−5.70025e8	+	5.70025e8i	−1.14307	+	1.14307i
$794$			4.06885e8i			0.812849i
$795$		0			0
$796$	7.49795e7			0.148663
$797$	2.30664e8	+	2.30664e8i	0.455622	+	0.455622i	0.897215	−	0.441593i	$-0.145587\pi$
$797$	2.30664e8	+	2.30664e8i	0.455622	+	0.455622i	−0.441593	+	0.897215i	$0.645587\pi$
$798$		0			0
$799$		−	2.40881e8i		−	0.472240i
$800$	7.89402e7	−	4.42769e7i	0.154180	−	0.0864784i
$801$		0			0
$802$	1.38798e8	+	1.38798e8i	0.269066	+	0.269066i
$803$	5.05997e8	−	5.05997e8i	0.977241	−	0.977241i
$804$		0			0
$805$	−2.98125e8	−	3.93565e8i	−0.571493	−	0.754448i
$806$	2.77127e8			0.529265
$807$		0			0
$808$	−3.71262e7	+	3.71262e7i	−0.0703795	+	0.0703795i
$809$			1.15009e8i			0.217213i	0.994085	+	0.108607i	$0.0346389\pi$
$809$			1.15009e8i			0.217213i	−0.994085	+	0.108607i	$0.965361\pi$
$810$		0			0
$811$	7.36769e8			1.38124			0.690619	−	0.723218i	$-0.257338\pi$
$811$	7.36769e8			1.38124			0.690619	+	0.723218i	$0.257338\pi$
$812$	3.94615e7	+	3.94615e7i	0.0737064	+	0.0737064i
$813$		0			0
$814$			1.24090e8i			0.230072i
$815$	−6.36556e8	−	8.78327e7i	−1.17588	−	0.162250i
$816$		0			0
$817$	1.14743e8	+	1.14743e8i	0.210407	+	0.210407i
$818$	2.65287e7	−	2.65287e7i	0.0484680	−	0.0484680i
$819$		0			0
$820$	−1.61618e8	+	1.22426e8i	−0.293123	+	0.222040i
$821$	−5.00735e8			−0.904854			−0.452427	−	0.891802i	$-0.649442\pi$
$821$	−5.00735e8			−0.904854			−0.452427	+	0.891802i	$0.649442\pi$
$822$		0			0
$823$	6.80752e6	−	6.80752e6i	0.0122121	−	0.0122121i	−0.700974	−	0.713186i	$-0.747251\pi$
$823$	6.80752e6	−	6.80752e6i	0.0122121	−	0.0122121i	0.713186	+	0.700974i	$0.247251\pi$
$824$			7.46236e7i			0.133381i
$825$		0			0
$826$	−2.87631e8			−0.510382
$827$	2.59523e7	+	2.59523e7i	0.0458838	+	0.0458838i	0.729676	−	0.683793i	$-0.239671\pi$
$827$	2.59523e7	+	2.59523e7i	0.0458838	+	0.0458838i	−0.683793	+	0.729676i	$0.739671\pi$
$828$		0			0
$829$		−	3.24193e8i		−	0.569037i	−0.958671	−	0.284518i	$-0.908166\pi$
$829$		−	3.24193e8i		−	0.569037i	0.958671	−	0.284518i	$-0.0918337\pi$
$830$	4.52356e8	+	5.97172e8i	0.791127	+	1.04440i
$831$		0			0
$832$	−6.64259e7	−	6.64259e7i	−0.115337	−	0.115337i
$833$	−3.51860e8	+	3.51860e8i	−0.608744	+	0.608744i
$834$		0			0
$835$	1.78410e7	−	1.29300e8i	0.0306451	−	0.222096i
$836$	−3.04969e8			−0.521960
$837$		0			0
$838$	−7.50269e7	+	7.50269e7i	−0.127493	+	0.127493i
$839$			5.69147e8i			0.963692i	0.876256	+	0.481846i	$0.160034\pi$
$839$			5.69147e8i			0.963692i	−0.876256	+	0.481846i	$0.839966\pi$
$840$		0			0
$841$	5.22943e8			0.879157
$842$	4.05570e8	+	4.05570e8i	0.679406	+	0.679406i
$843$		0			0
$844$		−	1.31334e8i		−	0.218448i
$845$	−3.37972e8	+	2.56013e8i	−0.560157	+	0.424318i
$846$		0			0
$847$	−1.01230e8	−	1.01230e8i	−0.166594	−	0.166594i
$848$	−1.70649e8	+	1.70649e8i	−0.279844	+	0.279844i
$849$		0			0
$850$	5.61997e8	+	1.58100e8i	0.915118	+	0.257440i
$851$	−4.06151e8			−0.659021
$852$		0			0
$853$	2.70461e7	−	2.70461e7i	0.0435770	−	0.0435770i	−0.684982	−	0.728560i	$-0.740190\pi$
$853$	2.70461e7	−	2.70461e7i	0.0435770	−	0.0435770i	0.728560	+	0.684982i	$0.240190\pi$
$854$		−	3.27201e8i		−	0.525340i
$855$		0			0
$856$	2.02823e8			0.323367
$857$	−1.75013e8	−	1.75013e8i	−0.278052	−	0.278052i	0.554279	−	0.832331i	$-0.312994\pi$
$857$	−1.75013e8	−	1.75013e8i	−0.278052	−	0.278052i	−0.832331	+	0.554279i	$0.812994\pi$
$858$		0			0
$859$			3.42840e8i			0.540894i	0.962735	+	0.270447i	$0.0871716\pi$
$859$			3.42840e8i			0.540894i	−0.962735	+	0.270447i	$0.912828\pi$
$860$	−9.65479e6	+	6.99718e7i	−0.0151792	+	0.110009i
$861$		0			0
$862$	1.01735e8	+	1.01735e8i	0.158836	+	0.158836i
$863$	−7.99150e8	+	7.99150e8i	−1.24336	+	1.24336i	−0.284757	+	0.958600i	$0.591913\pi$
$863$	−7.99150e8	+	7.99150e8i	−1.24336	+	1.24336i	−0.958600	+	0.284757i	$0.908087\pi$
$864$		0			0
$865$	8.78111e8	+	1.21163e8i	1.35675	+	0.187206i
$866$	−3.87906e8			−0.597273
$867$		0			0
$868$	−7.95369e7	+	7.95369e7i	−0.121621	+	0.121621i
$869$			2.30247e8i			0.350860i
$870$		0			0
$871$	−1.23390e9			−1.86735
$872$	2.12363e7	+	2.12363e7i	0.0320280	+	0.0320280i
$873$		0			0
$874$		−	9.98177e8i		−	1.49511i
$875$	3.68243e8	+	1.60640e8i	0.549681	+	0.239789i
$876$		0			0
$877$	5.77037e8	+	5.77037e8i	0.855470	+	0.855470i	0.990800	−	0.135331i	$-0.0432097\pi$
$877$	5.77037e8	+	5.77037e8i	0.855470	+	0.855470i	−0.135331	+	0.990800i	$0.543210\pi$
$878$	−1.51173e8	+	1.51173e8i	−0.223353	+	0.223353i
$879$		0			0
$880$	−8.01566e7	−	1.05818e8i	−0.117623	−	0.155278i
$881$	2.68153e8			0.392153			0.196077	−	0.980589i	$-0.437180\pi$
$881$	2.68153e8			0.392153			0.196077	+	0.980589i	$0.437180\pi$
$882$		0			0
$883$	4.48535e8	−	4.48535e8i	0.651499	−	0.651499i	−0.301855	−	0.953354i	$-0.597606\pi$
$883$	4.48535e8	−	4.48535e8i	0.651499	−	0.651499i	0.953354	+	0.301855i	$0.0976057\pi$
$884$		−	6.05941e8i		−	0.877149i
$885$		0			0
$886$	1.72979e8			0.248710
$887$	−4.07398e8	−	4.07398e8i	−0.583779	−	0.583779i	0.352161	−	0.935940i	$-0.385447\pi$
$887$	−4.07398e8	−	4.07398e8i	−0.583779	−	0.583779i	−0.935940	+	0.352161i	$0.885447\pi$
$888$		0			0
$889$			1.69616e8i			0.241413i
$890$	4.83184e8	+	6.66703e7i	0.685398	+	0.0945720i
$891$		0			0
$892$	−3.15086e8	−	3.15086e8i	−0.443950	−	0.443950i
$893$	2.36970e8	−	2.36970e8i	0.332766	−	0.332766i
$894$		0			0
$895$	−7.74263e8	+	5.86503e8i	−1.07999	+	0.818089i
$896$	3.81292e7			0.0530071
$897$		0			0
$898$	−3.77423e8	+	3.77423e8i	−0.521193	+	0.521193i
$899$		−	1.44879e8i		−	0.199401i
$900$		0			0
$901$	−1.55667e9			−2.12825
$902$	2.10276e8	+	2.10276e8i	0.286529	+	0.286529i
$903$		0			0
$904$			1.92100e8i			0.260028i
$905$	−5.26981e8	−	6.95686e8i	−0.710967	−	0.938572i
$906$		0			0
$907$	−1.38639e8	−	1.38639e8i	−0.185808	−	0.185808i	0.608073	−	0.793881i	$-0.291943\pi$
$907$	−1.38639e8	−	1.38639e8i	−0.185808	−	0.185808i	−0.793881	+	0.608073i	$0.791943\pi$
$908$	1.43338e7	−	1.43338e7i	0.0191471	−	0.0191471i
$909$		0			0
$910$	5.69960e7	−	4.13071e8i	0.0756346	−	0.548152i
$911$	−7.12128e8			−0.941896			−0.470948	−	0.882161i	$-0.656088\pi$
$911$	−7.12128e8			−0.941896			−0.470948	+	0.882161i	$0.656088\pi$
$912$		0			0
$913$	7.76957e8	−	7.76957e8i	1.02090	−	1.02090i
$914$		−	2.58732e8i		−	0.338853i
$915$		0			0
$916$	6.82834e8			0.888441
$917$	6.09249e8	+	6.09249e8i	0.790108	+	0.790108i
$918$		0			0
$919$			4.34345e8i			0.559613i	0.960056	+	0.279807i	$0.0902704\pi$
$919$			4.34345e8i			0.559613i	−0.960056	+	0.279807i	$0.909730\pi$
$920$	3.46345e8	−	2.62356e8i	0.444780	−	0.336920i
$921$		0			0
$922$	−3.50852e8	−	3.50852e8i	−0.447643	−	0.447643i
$923$	5.84627e8	−	5.84627e8i	0.743488	−	0.743488i
$924$		0			0
$925$	2.88245e8	−	1.61674e8i	0.364197	−	0.204275i
$926$	3.93360e8			0.495402
$927$		0			0
$928$	−3.47269e7	+	3.47269e7i	−0.0434532	+	0.0434532i
$929$			3.67090e8i			0.457852i	0.973444	+	0.228926i	$0.0735213\pi$
$929$			3.67090e8i			0.457852i	−0.973444	+	0.228926i	$0.926479\pi$
$930$		0			0
$931$	−6.92294e8			−0.857909
$932$	−3.82425e7	−	3.82425e7i	−0.0472388	−	0.0472388i
$933$		0			0
$934$			7.61836e8i			0.935020i
$935$	1.17040e8	−	8.48233e8i	0.143186	−	1.03772i
$936$		0			0
$937$	1.11262e8	+	1.11262e8i	0.135248	+	0.135248i	0.771490	−	0.636242i	$-0.219512\pi$
$937$	1.11262e8	+	1.11262e8i	0.135248	+	0.135248i	−0.636242	+	0.771490i	$0.719512\pi$
$938$	3.54137e8	−	3.54137e8i	0.429104	−	0.429104i
$939$		0			0
$940$	1.44507e8	+	1.99393e7i	0.173983	+	0.0240064i
$941$	7.63794e8			0.916659			0.458329	−	0.888782i	$-0.348448\pi$
$941$	7.63794e8			0.916659			0.458329	+	0.888782i	$0.348448\pi$
$942$		0			0
$943$	−6.88241e8	+	6.88241e8i	−0.820739	+	0.820739i
$944$		−	2.53120e8i		−	0.300892i
$945$		0			0
$946$	1.03599e8			0.122372
$947$	−1.13872e8	−	1.13872e8i	−0.134081	−	0.134081i	0.636881	−	0.770962i	$-0.280224\pi$
$947$	−1.13872e8	−	1.13872e8i	−0.134081	−	0.134081i	−0.770962	+	0.636881i	$0.780224\pi$
$948$		0			0
$949$			1.97807e9i			2.31443i
$950$	3.97339e8	+	7.08405e8i	0.463436	+	0.826248i
$951$		0			0
$952$	1.73908e8	+	1.73908e8i	0.201563	+	0.201563i
$953$	−9.79865e8	+	9.79865e8i	−1.13211	+	1.13211i	−0.142280	+	0.989826i	$0.545443\pi$
$953$	−9.79865e8	+	9.79865e8i	−1.13211	+	1.13211i	−0.989826	+	0.142280i	$0.954557\pi$
$954$		0			0
$955$	1.32554e8	+	1.74989e8i	0.152189	+	0.200910i
$956$	6.62422e7			0.0758160
$957$		0			0
$958$	8.16097e8	−	8.16097e8i	0.928208	−	0.928208i
$959$		−	7.29112e7i		−	0.0826682i
$960$		0			0
$961$	−5.95491e8			−0.670973
$962$	−2.42550e8	−	2.42550e8i	−0.272443	−	0.272443i
$963$		0			0
$964$			7.54690e8i			0.842437i
$965$	1.48662e9	+	2.05126e8i	1.65432	+	0.228265i
$966$		0			0
$967$	−2.18628e8	−	2.18628e8i	−0.241783	−	0.241783i	0.575804	−	0.817588i	$-0.304689\pi$
$967$	−2.18628e8	−	2.18628e8i	−0.241783	−	0.241783i	−0.817588	+	0.575804i	$0.804689\pi$
$968$	8.90846e7	−	8.90846e7i	0.0982147	−	0.0982147i
$969$		0			0
$970$	6.54392e8	−	4.95701e8i	0.717006	−	0.543131i
$971$	9.21253e8			1.00628			0.503142	−	0.864203i	$-0.332177\pi$
$971$	9.21253e8			1.00628			0.503142	+	0.864203i	$0.332177\pi$
$972$		0			0
$973$	7.14552e8	−	7.14552e8i	0.775703	−	0.775703i
$974$		−	8.03357e8i		−	0.869424i
$975$		0			0
$976$	2.87943e8			0.309711
$977$	1.10660e8	+	1.10660e8i	0.118661	+	0.118661i	0.763944	−	0.645283i	$-0.223260\pi$
$977$	1.10660e8	+	1.10660e8i	0.118661	+	0.118661i	−0.645283	+	0.763944i	$0.723260\pi$
$978$		0			0
$979$		−	7.15395e8i		−	0.762426i
$980$	−1.81959e8	−	2.40210e8i	−0.193328	−	0.255219i
$981$		0			0
$982$	2.91204e8	+	2.91204e8i	0.307512	+	0.307512i
$983$	5.81692e8	−	5.81692e8i	0.612397	−	0.612397i	−0.331173	−	0.943570i	$-0.607445\pi$
$983$	5.81692e8	−	5.81692e8i	0.612397	−	0.612397i	0.943570	+	0.331173i	$0.107445\pi$
$984$		0			0
$985$	1.71166e8	−	1.24050e9i	0.179106	−	1.29804i
$986$	−3.16780e8			−0.330467
$987$		0			0
$988$	5.96102e8	−	5.96102e8i	0.618087	−	0.618087i
$989$			3.39084e8i			0.350525i
$990$		0			0
$991$	−2.16687e8			−0.222644			−0.111322	−	0.993784i	$-0.535509\pi$
$991$	−2.16687e8			−0.222644			−0.111322	+	0.993784i	$0.535509\pi$
$992$	−6.99940e7	−	6.99940e7i	−0.0717011	−	0.0717011i
$993$		0			0
$994$			3.35583e8i			0.341696i
$995$	−2.33468e8	+	1.76852e8i	−0.237005	+	0.179531i
$996$		0			0
$997$	−3.07388e8	−	3.07388e8i	−0.310172	−	0.310172i	0.534804	−	0.844976i	$-0.320385\pi$
$997$	−3.07388e8	−	3.07388e8i	−0.310172	−	0.310172i	−0.844976	+	0.534804i	$0.820385\pi$
$998$	−7.19297e8	+	7.19297e8i	−0.723630	+	0.723630i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.7.g.f.37.2		8
3.2	odd	2		30.7.f.b.7.4	✓	8
5.2	odd	4		450.7.g.r.343.2		8
5.3	odd	4	inner	90.7.g.f.73.2		8
5.4	even	2		450.7.g.r.307.2		8
12.11	even	2		240.7.bg.b.97.2		8
15.2	even	4		150.7.f.h.43.1		8
15.8	even	4		30.7.f.b.13.4	yes	8
15.14	odd	2		150.7.f.h.7.1		8
60.23	odd	4		240.7.bg.b.193.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.7.f.b.7.4	✓	8	3.2	odd	2
30.7.f.b.13.4	yes	8	15.8	even	4
90.7.g.f.37.2		8	1.1	even	1	trivial
90.7.g.f.73.2		8	5.3	odd	4	inner
150.7.f.h.7.1		8	15.14	odd	2
150.7.f.h.43.1		8	15.2	even	4
240.7.bg.b.97.2		8	12.11	even	2
240.7.bg.b.193.2		8	60.23	odd	4
450.7.g.r.307.2		8	5.4	even	2
450.7.g.r.343.2		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	90.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 + 4.00000i) q^{2} +32.0000i q^{4} +(-75.4773 - 99.6402i) q^{5} +(145.451 + 145.451i) q^{7} +(-128.000 + 128.000i) q^{8} +(96.6517 - 700.470i) q^{10} -1037.11 q^{11} +(2027.16 - 2027.16i) q^{13} +1163.61i q^{14} -1024.00 q^{16} +(-4670.49 - 4670.49i) q^{17} -9189.32i q^{19} +(3188.49 - 2415.27i) q^{20} +(-4148.42 - 4148.42i) q^{22} +(13577.9 - 13577.9i) q^{23} +(-4231.35 + 15041.2i) q^{25} +16217.3 q^{26} +(-4654.44 + 4654.44i) q^{28} -8478.24i q^{29} +17088.4 q^{31} +(-4096.00 - 4096.00i) q^{32} -37364.0i q^{34} +(3514.53 - 25471.1i) q^{35} +(-14956.3 - 14956.3i) q^{37} +(36757.3 - 36757.3i) q^{38} +(22415.0 + 3092.86i) q^{40} -50688.1 q^{41} +(-12486.6 + 12486.6i) q^{43} -33187.4i q^{44} +108624. q^{46} +(25787.6 + 25787.6i) q^{47} -75336.8i q^{49} +(-77090.0 + 43239.2i) q^{50} +(64869.0 + 64869.0i) q^{52} +(166649. - 166649. i) q^{53} +(78277.9 + 103337. i) q^{55} -37235.6 q^{56} +(33912.9 - 33912.9i) q^{58} +247188. i q^{59} -281194. q^{61} +(68353.5 + 68353.5i) q^{62} -32768.0i q^{64} +(-354991. - 48982.0i) q^{65} +(-304343. - 304343. i) q^{67} +(149456. - 149456. i) q^{68} +(115942. - 87826.2i) q^{70} +288398. q^{71} +(-487894. + 487894. i) q^{73} -119650. i q^{74} +294058. q^{76} +(-150848. - 150848. i) q^{77} -222009. i q^{79} +(77288.8 + 102032. i) q^{80} +(-202752. - 202752. i) q^{82} +(-749160. + 749160. i) q^{83} +(-112853. + 817885. i) q^{85} -99892.6 q^{86} +(132749. - 132749. i) q^{88} +689800. i q^{89} +589705. q^{91} +(434494. + 434494. i) q^{92} +206301. i q^{94} +(-915626. + 693585. i) q^{95} +(820944. + 820944. i) q^{97} +(301347. - 301347. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{2} - 60 q^{5} + 388 q^{7} - 1024 q^{8} - 976 q^{10} + 4192 q^{11} + 7884 q^{13} - 8192 q^{16} + 1372 q^{17} - 5888 q^{20} + 16768 q^{22} - 14888 q^{23} + 17932 q^{25} + 63072 q^{26} - 12416 q^{28}+ \cdots + 301888 q^{98}+O(q^{100}) \) 8 * q + 32 * q^2 - 60 * q^5 + 388 * q^7 - 1024 * q^8 - 976 * q^10 + 4192 * q^11 + 7884 * q^13 - 8192 * q^16 + 1372 * q^17 - 5888 * q^20 + 16768 * q^22 - 14888 * q^23 + 17932 * q^25 + 63072 * q^26 - 12416 * q^28 + 2240 * q^31 - 32768 * q^32 + 78232 * q^35 - 69996 * q^37 + 71008 * q^38 - 15872 * q^40 - 169064 * q^41 + 164136 * q^43 - 119104 * q^46 + 109248 * q^47 - 280832 * q^50 + 252288 * q^52 + 480740 * q^53 - 409068 * q^55 - 99328 * q^56 - 2272 * q^58 + 266088 * q^61 + 8960 * q^62 - 903396 * q^65 + 92488 * q^67 - 43904 * q^68 - 228352 * q^70 + 2526032 * q^71 + 324464 * q^73 + 568064 * q^76 - 2822104 * q^77 + 61440 * q^80 - 676256 * q^82 - 213264 * q^83 - 1537292 * q^85 + 1313088 * q^86 - 536576 * q^88 + 3340176 * q^91 - 476416 * q^92 - 3084784 * q^95 - 854736 * q^97 + 301888 * q^98

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