LMFDB - Embedded newform 312.6.q.a.217.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,6,Mod(217,312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312.217"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	312.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$50.0397517816$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 27696 x^{14} - 39172 x^{13} + 293413850 x^{12} + 1900944112 x^{11} + \cdots + 23\!\cdots\!33 $ x^16 - 4x^15 - 27696x^14 - 39172x^13 + 293413850x^12 + 1900944112x^11 - 1510277632076x^10 - 16835242841988x^9 + 4008698065423927x^8 + 62734380515262720x^7 - 5146471931709401916x^6 - 106028394330081758436x^5 + 2190548955462688552302x^4 + 65968620997679959890708x^3 + 556341743981408114123676x^2 + 1887766052104227281681808*x + 2307595861690755488364933
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.4
Root			$-3.19246 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	312.217
Dual form			312.6.q.a.289.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.50000 + 7.79423i) q^{3} -8.30754 q^{5} +(-49.9909 - 86.5868i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(188.412 - 326.339i) q^{11} +(525.236 + 308.902i) q^{13} +(37.3839 - 64.7509i) q^{15} +(-532.492 - 922.304i) q^{17} +(1374.49 + 2380.68i) q^{19} +899.836 q^{21} +(-778.742 + 1348.82i) q^{23} -3055.98 q^{25} +729.000 q^{27} +(-312.678 + 541.573i) q^{29} -3224.93 q^{31} +(1695.71 + 2937.05i) q^{33} +(415.302 + 719.324i) q^{35} +(2134.48 - 3697.03i) q^{37} +(-4771.21 + 2703.75i) q^{39} +(-6909.63 + 11967.8i) q^{41} +(-7108.66 - 12312.6i) q^{43} +(336.456 + 582.758i) q^{45} -14147.7 q^{47} +(3405.32 - 5898.18i) q^{49} +9584.86 q^{51} +36417.9 q^{53} +(-1565.24 + 2711.07i) q^{55} -24740.8 q^{57} +(-8314.37 - 14400.9i) q^{59} +(-16929.7 - 29323.2i) q^{61} +(-4049.26 + 7013.53i) q^{63} +(-4363.42 - 2566.22i) q^{65} +(-29001.5 + 50232.1i) q^{67} +(-7008.68 - 12139.4i) q^{69} +(-24974.9 - 43257.8i) q^{71} -25626.4 q^{73} +(13751.9 - 23819.0i) q^{75} -37675.5 q^{77} -50862.8 q^{79} +(-3280.50 + 5681.99i) q^{81} -75917.6 q^{83} +(4423.70 + 7662.08i) q^{85} +(-2814.10 - 4874.16i) q^{87} +(46625.4 - 80757.6i) q^{89} +(489.852 - 60920.8i) q^{91} +(14512.2 - 25135.8i) q^{93} +(-11418.6 - 19777.6i) q^{95} +(40034.5 + 69341.9i) q^{97} -30522.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 72 q^{3} - 188 q^{5} + 138 q^{7} - 648 q^{9} + 612 q^{11} - 228 q^{13} + 846 q^{15} + 1450 q^{17} + 1064 q^{19} - 2484 q^{21} + 1768 q^{23} + 7628 q^{25} + 11664 q^{27} - 3338 q^{29} - 3724 q^{31}+ \cdots - 99144 q^{99}+O(q^{100}) $ 16 * q - 72 * q^3 - 188 * q^5 + 138 * q^7 - 648 * q^9 + 612 * q^11 - 228 * q^13 + 846 * q^15 + 1450 * q^17 + 1064 * q^19 - 2484 * q^21 + 1768 * q^23 + 7628 * q^25 + 11664 * q^27 - 3338 * q^29 - 3724 * q^31 + 5508 * q^33 - 4704 * q^35 - 10462 * q^37 - 8586 * q^39 + 18486 * q^41 + 1530 * q^43 + 7614 * q^45 + 12312 * q^47 - 22226 * q^49 - 26100 * q^51 - 44972 * q^53 - 58080 * q^55 - 19152 * q^57 + 1664 * q^59 - 37208 * q^61 + 11178 * q^63 - 44010 * q^65 - 36198 * q^67 + 15912 * q^69 + 73856 * q^71 + 21768 * q^73 - 34326 * q^75 + 192808 * q^77 - 6468 * q^79 - 52488 * q^81 - 284712 * q^83 - 81558 * q^85 - 30042 * q^87 - 71976 * q^89 + 177606 * q^91 + 16758 * q^93 + 25216 * q^95 + 101682 * q^97 - 99144 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−4.50000	+	7.79423i	−0.288675	+	0.500000i
$3$	−4.50000	+	7.79423i	−0.288675	+	0.500000i
$4$		0			0
$5$	−8.30754			−0.148610			−0.0743049	−	0.997236i	$-0.523674\pi$
$5$	−8.30754			−0.148610			−0.0743049	+	0.997236i	$0.523674\pi$
$6$		0			0
$7$	−49.9909	−	86.5868i	−0.385608	−	0.667892i	0.606245	−	0.795278i	$-0.292675\pi$
$7$	−49.9909	−	86.5868i	−0.385608	−	0.667892i	−0.991853	+	0.127385i	$0.959342\pi$
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	188.412	−	326.339i	0.469490	−	0.813180i	−0.529902	−	0.848059i	$-0.677771\pi$
$11$	188.412	−	326.339i	0.469490	−	0.813180i	0.999392	+	0.0348789i	$0.0111046\pi$
$12$		0			0
$13$	525.236	+	308.902i	0.861977	+	0.506947i
$13$	525.236	+	308.902i	0.861977	+	0.506947i
$14$		0			0
$15$	37.3839	−	64.7509i	0.0429000	−	0.0743049i
$16$		0			0
$17$	−532.492	−	922.304i	−0.446880	−	0.774019i	0.551301	−	0.834306i	$-0.314132\pi$
$17$	−532.492	−	922.304i	−0.446880	−	0.774019i	−0.998181	+	0.0602875i	$0.980798\pi$
$18$		0			0
$19$	1374.49	+	2380.68i	0.873489	+	1.51293i	0.858364	+	0.513041i	$0.171481\pi$
$19$	1374.49	+	2380.68i	0.873489	+	1.51293i	0.0151247	+	0.999886i	$0.495185\pi$
$20$		0			0
$21$	899.836			0.445262
$22$		0			0
$23$	−778.742	+	1348.82i	−0.306955	+	0.531661i	−0.977695	−	0.210032i	$-0.932643\pi$
$23$	−778.742	+	1348.82i	−0.306955	+	0.531661i	0.670740	+	0.741693i	$0.265977\pi$
$24$		0			0
$25$	−3055.98			−0.977915
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	−312.678	+	541.573i	−0.0690401	+	0.119581i	−0.898479	−	0.439016i	$-0.855327\pi$
$29$	−312.678	+	541.573i	−0.0690401	+	0.119581i	0.829439	+	0.558597i	$0.188660\pi$
$30$		0			0
$31$	−3224.93			−0.602720			−0.301360	−	0.953510i	$-0.597441\pi$
$31$	−3224.93			−0.602720			−0.301360	+	0.953510i	$0.597441\pi$
$32$		0			0
$33$	1695.71	+	2937.05i	0.271060	+	0.469490i
$34$		0			0
$35$	415.302	+	719.324i	0.0573051	+	0.0992554i
$36$		0			0
$37$	2134.48	−	3697.03i	0.256324	−	0.443965i	−0.708931	−	0.705278i	$-0.750822\pi$
$37$	2134.48	−	3697.03i	0.256324	−	0.443965i	0.965254	+	0.261313i	$0.0841554\pi$
$38$		0			0
$39$	−4771.21	+	2703.75i	−0.502305	+	0.284646i
$40$		0			0
$41$	−6909.63	+	11967.8i	−0.641941	+	1.11188i	0.343057	+	0.939314i	$0.388537\pi$
$41$	−6909.63	+	11967.8i	−0.641941	+	1.11188i	−0.984999	+	0.172561i	$0.944796\pi$
$42$		0			0
$43$	−7108.66	−	12312.6i	−0.586295	−	1.01549i	−0.994713	−	0.102698i	$-0.967253\pi$
$43$	−7108.66	−	12312.6i	−0.586295	−	1.01549i	0.408417	−	0.912795i	$-0.366081\pi$
$44$		0			0
$45$	336.456	+	582.758i	0.0247683	+	0.0429000i
$46$		0			0
$47$	−14147.7			−0.934203			−0.467101	−	0.884204i	$-0.654702\pi$
$47$	−14147.7			−0.934203			−0.467101	+	0.884204i	$0.654702\pi$
$48$		0			0
$49$	3405.32	−	5898.18i	0.202613	−	0.350936i
$50$		0			0
$51$	9584.86			0.516013
$52$		0			0
$53$	36417.9			1.78084			0.890422	−	0.455136i	$-0.150409\pi$
$53$	36417.9			1.78084			0.890422	+	0.455136i	$0.150409\pi$
$54$		0			0
$55$	−1565.24	+	2711.07i	−0.0697708	+	0.120847i
$56$		0			0
$57$	−24740.8			−1.00862
$58$		0			0
$59$	−8314.37	−	14400.9i	−0.310956	−	0.538592i	0.667613	−	0.744508i	$-0.267316\pi$
$59$	−8314.37	−	14400.9i	−0.310956	−	0.538592i	−0.978570	+	0.205916i	$0.933983\pi$
$60$		0			0
$61$	−16929.7	−	29323.2i	−0.582540	−	1.00899i	−0.995177	−	0.0980932i	$-0.968726\pi$
$61$	−16929.7	−	29323.2i	−0.582540	−	1.00899i	0.412637	−	0.910895i	$-0.364608\pi$
$62$		0			0
$63$	−4049.26	+	7013.53i	−0.128536	+	0.222631i
$64$		0			0
$65$	−4363.42	−	2566.22i	−0.128098	−	0.0753373i
$66$		0			0
$67$	−29001.5	+	50232.1i	−0.789285	+	1.36708i	0.137121	+	0.990554i	$0.456215\pi$
$67$	−29001.5	+	50232.1i	−0.789285	+	1.36708i	−0.926406	+	0.376527i	$0.877118\pi$
$68$		0			0
$69$	−7008.68	−	12139.4i	−0.177220	−	0.306955i
$70$		0			0
$71$	−24974.9	−	43257.8i	−0.587973	−	1.01840i	−0.994498	−	0.104760i	$-0.966593\pi$
$71$	−24974.9	−	43257.8i	−0.587973	−	1.01840i	0.406524	−	0.913640i	$-0.366741\pi$
$72$		0			0
$73$	−25626.4			−0.562834			−0.281417	−	0.959586i	$-0.590804\pi$
$73$	−25626.4			−0.562834			−0.281417	+	0.959586i	$0.590804\pi$
$74$		0			0
$75$	13751.9	−	23819.0i	0.282300	−	0.488958i
$76$		0			0
$77$	−37675.5			−0.724156
$78$		0			0
$79$	−50862.8			−0.916923			−0.458461	−	0.888714i	$-0.651599\pi$
$79$	−50862.8			−0.916923			−0.458461	+	0.888714i	$0.651599\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−75917.6			−1.20961			−0.604807	−	0.796372i	$-0.706750\pi$
$83$	−75917.6			−1.20961			−0.604807	+	0.796372i	$0.706750\pi$
$84$		0			0
$85$	4423.70	+	7662.08i	0.0664108	+	0.115027i
$86$		0			0
$87$	−2814.10	−	4874.16i	−0.0398603	−	0.0690401i
$88$		0			0
$89$	46625.4	−	80757.6i	0.623947	−	1.08071i	−0.364797	−	0.931087i	$-0.618862\pi$
$89$	46625.4	−	80757.6i	0.623947	−	1.08071i	0.988744	−	0.149620i	$-0.0478051\pi$
$90$		0			0
$91$	489.852	−	60920.8i	0.00620100	−	0.771191i
$92$		0			0
$93$	14512.2	−	25135.8i	0.173990	−	0.301360i
$94$		0			0
$95$	−11418.6	−	19777.6i	−0.129809	−	0.224836i
$96$		0			0
$97$	40034.5	+	69341.9i	0.432022	+	0.748284i	0.997047	−	0.0767898i	$-0.0244671\pi$
$97$	40034.5	+	69341.9i	0.432022	+	0.748284i	−0.565026	+	0.825073i	$0.691134\pi$
$98$		0			0
$99$	−30522.7			−0.312993
$100$		0			0
$101$	−23603.8	+	40882.9i	−0.230239	+	0.398785i	−0.957878	−	0.287175i	$-0.907284\pi$
$101$	−23603.8	+	40882.9i	−0.230239	+	0.398785i	0.727640	+	0.685960i	$0.240617\pi$
$102$		0			0
$103$	−91933.6			−0.853849			−0.426925	−	0.904287i	$-0.640403\pi$
$103$	−91933.6			−0.853849			−0.426925	+	0.904287i	$0.640403\pi$
$104$		0			0
$105$	−7475.43			−0.0661703
$106$		0			0
$107$	−60480.3	+	104755.i	−0.510687	+	0.884536i	0.489236	+	0.872151i	$0.337276\pi$
$107$	−60480.3	+	104755.i	−0.510687	+	0.884536i	−0.999923	+	0.0123846i	$0.996058\pi$
$108$		0			0
$109$	−188986.			−1.52357			−0.761787	−	0.647827i	$-0.775678\pi$
$109$	−188986.			−1.52357			−0.761787	+	0.647827i	$0.775678\pi$
$110$		0			0
$111$	19210.4	+	33273.3i	0.147988	+	0.256324i
$112$		0			0
$113$	−79649.4	−	137957.i	−0.586795	−	1.01636i	−0.994649	−	0.103312i	$-0.967056\pi$
$113$	−79649.4	−	137957.i	−0.586795	−	1.01636i	0.407854	−	0.913047i	$-0.366277\pi$
$114$		0			0
$115$	6469.43	−	11205.4i	0.0456165	−	0.0790101i
$116$		0			0
$117$	396.852	−	49354.8i	0.00268018	−	0.333323i
$118$		0			0
$119$	−53239.5	+	92213.6i	−0.344641	+	0.596936i
$120$		0			0
$121$	9527.59	+	16502.3i	0.0591588	+	0.102466i
$122$		0			0
$123$	−62186.7	−	107711.i	−0.370625	−	0.641941i
$124$		0			0
$125$	51348.8			0.293938
$126$		0			0
$127$	−143235.	+	248090.i	−0.788024	+	1.36490i	0.139151	+	0.990271i	$0.455563\pi$
$127$	−143235.	+	248090.i	−0.788024	+	1.36490i	−0.927176	+	0.374627i	$0.877771\pi$
$128$		0			0
$129$	127956.			0.676995
$130$		0			0
$131$	−90405.3			−0.460273			−0.230136	−	0.973158i	$-0.573917\pi$
$131$	−90405.3			−0.460273			−0.230136	+	0.973158i	$0.573917\pi$
$132$		0			0
$133$	137424.	−	238025.i	0.673648	−	1.16679i
$134$		0			0
$135$	−6056.20			−0.0286000
$136$		0			0
$137$	−138697.	−	240230.i	−0.631343	−	1.09352i	−0.987277	−	0.159008i	$-0.949170\pi$
$137$	−138697.	−	240230.i	−0.631343	−	1.09352i	0.355934	−	0.934511i	$-0.384163\pi$
$138$		0			0
$139$	160749.	+	278426.i	0.705687	+	1.22229i	0.966443	+	0.256881i	$0.0826948\pi$
$139$	160749.	+	278426.i	0.705687	+	1.22229i	−0.260756	+	0.965405i	$0.583972\pi$
$140$		0			0
$141$	63664.7	−	110270.i	0.269681	−	0.467101i
$142$		0			0
$143$	199767.	−	113204.i	0.816929	−	0.462936i
$144$		0			0
$145$	2597.58	−	4499.14i	0.0102600	−	0.0177709i
$146$		0			0
$147$	30647.9	+	53083.7i	0.116979	+	0.202613i
$148$		0			0
$149$	38971.4	+	67500.4i	0.143807	+	0.249081i	0.928927	−	0.370262i	$-0.120732\pi$
$149$	38971.4	+	67500.4i	0.143807	+	0.249081i	−0.785120	+	0.619343i	$0.787399\pi$
$150$		0			0
$151$	−112154.			−0.400288			−0.200144	−	0.979766i	$-0.564141\pi$
$151$	−112154.			−0.400288			−0.200144	+	0.979766i	$0.564141\pi$
$152$		0			0
$153$	−43131.9	+	74706.6i	−0.148960	+	0.258006i
$154$		0			0
$155$	26791.2			0.0895701
$156$		0			0
$157$	−486043.			−1.57371			−0.786856	−	0.617137i	$-0.788293\pi$
$157$	−486043.			−1.57371			−0.786856	+	0.617137i	$0.788293\pi$
$158$		0			0
$159$	−163881.	+	283850.i	−0.514085	+	0.890422i
$160$		0			0
$161$	155720.			0.473456
$162$		0			0
$163$	−186993.	−	323882.i	−0.551261	−	0.954811i	−0.998184	−	0.0602391i	$-0.980814\pi$
$163$	−186993.	−	323882.i	−0.551261	−	0.954811i	0.446923	−	0.894572i	$-0.352520\pi$
$164$		0			0
$165$	−14087.1	−	24399.6i	−0.0402822	−	0.0697708i
$166$		0			0
$167$	−241984.	+	419129.i	−0.671423	+	1.16294i	0.306078	+	0.952007i	$0.400983\pi$
$167$	−241984.	+	419129.i	−0.671423	+	1.16294i	−0.977501	+	0.210932i	$0.932350\pi$
$168$		0			0
$169$	180452.	+	324493.i	0.486009	+	0.873954i
$170$		0			0
$171$	111334.	−	192835.i	0.291163	−	0.504309i
$172$		0			0
$173$	342555.	+	593322.i	0.870191	+	1.50722i	0.861798	+	0.507251i	$0.169338\pi$
$173$	342555.	+	593322.i	0.870191	+	1.50722i	0.00839283	+	0.999965i	$0.497328\pi$
$174$		0			0
$175$	152771.	+	264608.i	0.377092	+	0.653142i
$176$		0			0
$177$	149659.			0.359061
$178$		0			0
$179$	304510.	−	527426.i	0.710343	−	1.23035i	−0.254385	−	0.967103i	$-0.581873\pi$
$179$	304510.	−	527426.i	0.710343	−	1.23035i	0.964728	−	0.263248i	$-0.0847937\pi$
$180$		0			0
$181$	−195779.			−0.444190			−0.222095	−	0.975025i	$-0.571290\pi$
$181$	−195779.			−0.444190			−0.222095	+	0.975025i	$0.571290\pi$
$182$		0			0
$183$	304735.			0.672659
$184$		0			0
$185$	−17732.3	+	30713.3i	−0.0380922	+	0.0659776i
$186$		0			0
$187$	−401311.			−0.839222
$188$		0			0
$189$	−36443.4	−	63121.8i	−0.0742103	−	0.128536i
$190$		0			0
$191$	11582.8	+	20062.0i	0.0229737	+	0.0397915i	0.877284	−	0.479972i	$-0.159353\pi$
$191$	11582.8	+	20062.0i	0.0229737	+	0.0397915i	−0.854310	+	0.519764i	$0.826020\pi$
$192$		0			0
$193$	412798.	−	714987.i	0.797709	−	1.38167i	−0.123396	−	0.992358i	$-0.539378\pi$
$193$	412798.	−	714987.i	0.797709	−	1.38167i	0.921105	−	0.389315i	$-0.127288\pi$
$194$		0			0
$195$	39637.1	−	22461.5i	0.0746475	−	0.0423011i
$196$		0			0
$197$	418929.	−	725607.i	0.769087	−	1.33210i	−0.168972	−	0.985621i	$-0.554045\pi$
$197$	418929.	−	725607.i	0.769087	−	1.33210i	0.938059	−	0.346477i	$-0.112622\pi$
$198$		0			0
$199$	35950.1	+	62267.3i	0.0643527	+	0.111462i	0.896407	−	0.443233i	$-0.146168\pi$
$199$	35950.1	+	62267.3i	0.0643527	+	0.111462i	−0.832054	+	0.554695i	$0.812835\pi$
$200$		0			0
$201$	−261014.	−	452089.i	−0.455694	−	0.789285i
$202$		0			0
$203$	62524.1			0.106490
$204$		0			0
$205$	57402.1	−	99423.3i	0.0953988	−	0.165236i
$206$		0			0
$207$	126156.			0.204636
$208$		0			0
$209$	1.03588e6			1.64038
$210$		0			0
$211$	−511148.	+	885334.i	−0.790388	+	1.36899i	0.135339	+	0.990799i	$0.456788\pi$
$211$	−511148.	+	885334.i	−0.790388	+	1.36899i	−0.925727	+	0.378193i	$0.876546\pi$
$212$		0			0
$213$	449548.			0.678933
$214$		0			0
$215$	59055.5	+	102287.i	0.0871292	+	0.150912i
$216$		0			0
$217$	161217.	+	279236.i	0.232413	+	0.402552i
$218$		0			0
$219$	115319.	−	199738.i	0.162476	−	0.281417i
$220$		0			0
$221$	5217.80	−	648915.i	0.00718632	−	0.893731i
$222$		0			0
$223$	−317534.	+	549985.i	−0.427591	+	0.740609i	−0.996658	−	0.0816816i	$-0.973971\pi$
$223$	−317534.	+	549985.i	−0.427591	+	0.740609i	0.569068	+	0.822291i	$0.307304\pi$
$224$		0			0
$225$	123767.	+	214371.i	0.162986	+	0.282300i
$226$		0			0
$227$	−99025.7	−	171518.i	−0.127551	−	0.220925i	0.795176	−	0.606378i	$-0.207378\pi$
$227$	−99025.7	−	171518.i	−0.127551	−	0.220925i	−0.922727	+	0.385454i	$0.874045\pi$
$228$		0			0
$229$	631542.			0.795818			0.397909	−	0.917425i	$-0.369736\pi$
$229$	631542.			0.795818			0.397909	+	0.917425i	$0.369736\pi$
$230$		0			0
$231$	169540.	−	293651.i	0.209046	−	0.362078i
$232$		0			0
$233$	−322934.			−0.389694			−0.194847	−	0.980834i	$-0.562421\pi$
$233$	−322934.			−0.389694			−0.194847	+	0.980834i	$0.562421\pi$
$234$		0			0
$235$	117533.			0.138832
$236$		0			0
$237$	228883.	−	396436.i	0.264693	−	0.458461i
$238$		0			0
$239$	58666.5			0.0664347			0.0332174	−	0.999448i	$-0.489425\pi$
$239$	58666.5			0.0664347			0.0332174	+	0.999448i	$0.489425\pi$
$240$		0			0
$241$	476024.	+	824497.i	0.527942	+	0.914422i	0.999469	+	0.0325704i	$0.0103693\pi$
$241$	476024.	+	824497.i	0.527942	+	0.914422i	−0.471528	+	0.881851i	$0.656297\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−28289.8	+	48999.4i	−0.0301103	+	0.0521526i
$246$		0			0
$247$	−13468.4	+	1.67500e6i	−0.0140467	+	1.74692i
$248$		0			0
$249$	341629.	−	591719.i	0.349186	−	0.604807i
$250$		0			0
$251$	380656.	+	659316.i	0.381372	+	0.660555i	0.991259	−	0.131933i	$-0.0421185\pi$
$251$	380656.	+	659316.i	0.381372	+	0.660555i	−0.609887	+	0.792488i	$0.708785\pi$
$252$		0			0
$253$	293448.	+	508267.i	0.288224	+	0.499219i
$254$		0			0
$255$	−79626.6			−0.0766846
$256$		0			0
$257$	213274.	−	369401.i	0.201421	−	0.348871i	−0.747566	−	0.664188i	$-0.768777\pi$
$257$	213274.	−	369401.i	0.201421	−	0.348871i	0.948986	+	0.315317i	$0.102111\pi$
$258$		0			0
$259$	−426819.			−0.395362
$260$		0			0
$261$	50653.8			0.0460268
$262$		0			0
$263$	1.03918e6	−	1.79991e6i	0.926407	−	1.60458i	0.137124	−	0.990554i	$-0.456214\pi$
$263$	1.03918e6	−	1.79991e6i	0.926407	−	1.60458i	0.789283	−	0.614030i	$-0.210453\pi$
$264$		0			0
$265$	−302544.			−0.264651
$266$		0			0
$267$	419629.	+	726818.i	0.360236	+	0.623947i
$268$		0			0
$269$	−155737.	−	269744.i	−0.131223	−	0.227285i	0.792925	−	0.609319i	$-0.208557\pi$
$269$	−155737.	−	269744.i	−0.131223	−	0.227285i	−0.924148	+	0.382034i	$0.875224\pi$
$270$		0			0
$271$	117293.	−	203158.i	0.0970176	−	0.168039i	−0.813431	−	0.581661i	$-0.802403\pi$
$271$	117293.	−	203158.i	0.0970176	−	0.168039i	0.910449	+	0.413622i	$0.135736\pi$
$272$		0			0
$273$	472626.	+	277961.i	0.383805	+	0.225724i
$274$		0			0
$275$	−575783.	+	997286.i	−0.459121	+	0.795221i
$276$		0			0
$277$	−71004.3	−	122983.i	−0.0556013	−	0.0963043i	0.836885	−	0.547379i	$-0.184374\pi$
$277$	−71004.3	−	122983.i	−0.0556013	−	0.0963043i	−0.892486	+	0.451074i	$0.851041\pi$
$278$		0			0
$279$	130609.	+	226222.i	0.100453	+	0.173990i
$280$		0			0
$281$	876257.			0.662012			0.331006	−	0.943629i	$-0.392612\pi$
$281$	876257.			0.662012			0.331006	+	0.943629i	$0.392612\pi$
$282$		0			0
$283$	−484527.	+	839226.i	−0.359627	+	0.622892i	−0.987898	−	0.155102i	$-0.950429\pi$
$283$	−484527.	+	839226.i	−0.359627	+	0.622892i	0.628272	+	0.777994i	$0.283763\pi$
$284$		0			0
$285$	205535.			0.149891
$286$		0			0
$287$	1.38168e6			0.990151
$288$		0			0
$289$	142833.	−	247393.i	0.100596	−	0.174238i
$290$		0			0
$291$	−720622.			−0.498856
$292$		0			0
$293$	674250.	+	1.16784e6i	0.458830	+	0.794717i	0.998899	−	0.0469033i	$-0.0149353\pi$
$293$	674250.	+	1.16784e6i	0.458830	+	0.794717i	−0.540069	+	0.841621i	$0.681602\pi$
$294$		0			0
$295$	69072.0	+	119636.i	0.0462112	+	0.0800401i
$296$		0			0
$297$	137352.	−	237901.i	0.0903533	−	0.156497i
$298$		0			0
$299$	−825677.	+	467894.i	−0.534112	+	0.302670i
$300$		0			0
$301$	−710736.	+	1.23103e6i	−0.452160	+	0.783164i
$302$		0			0
$303$	−212434.	−	367947.i	−0.132928	−	0.230239i
$304$		0			0
$305$	140645.	+	243603.i	0.0865712	+	0.149946i
$306$		0			0
$307$	−750919.			−0.454723			−0.227362	−	0.973810i	$-0.573010\pi$
$307$	−750919.			−0.454723			−0.227362	+	0.973810i	$0.573010\pi$
$308$		0			0
$309$	413701.	−	716551.i	0.246485	−	0.426925i
$310$		0			0
$311$	−819434.			−0.480411			−0.240205	−	0.970722i	$-0.577215\pi$
$311$	−819434.			−0.480411			−0.240205	+	0.970722i	$0.577215\pi$
$312$		0			0
$313$	−702427.			−0.405266			−0.202633	−	0.979255i	$-0.564950\pi$
$313$	−702427.			−0.405266			−0.202633	+	0.979255i	$0.564950\pi$
$314$		0			0
$315$	33639.4	−	58265.2i	0.0191017	−	0.0330851i
$316$		0			0
$317$	2.12416e6			1.18724			0.593621	−	0.804744i	$-0.297698\pi$
$317$	2.12416e6			1.18724			0.593621	+	0.804744i	$0.297698\pi$
$318$		0			0
$319$	117824.	+	204077.i	0.0648273	+	0.112284i
$320$		0			0
$321$	−544323.	−	942795.i	−0.294845	−	0.510687i
$322$		0			0
$323$	1.46381e6	−	2.53539e6i	0.780689	−	1.35219i
$324$		0			0
$325$	−1.60511e6	−	944001.i	−0.842940	−	0.495751i
$326$		0			0
$327$	850438.	−	1.47300e6i	0.439818	−	0.761787i
$328$		0			0
$329$	707256.	+	1.22500e6i	0.360236	+	0.623947i
$330$		0			0
$331$	516569.	+	894723.i	0.259154	+	0.448868i	0.966016	−	0.258484i	$-0.0832229\pi$
$331$	516569.	+	894723.i	0.259154	+	0.448868i	−0.706861	+	0.707352i	$0.749890\pi$
$332$		0			0
$333$	−345786.			−0.170882
$334$		0			0
$335$	240931.	−	417305.i	0.117295	−	0.203162i
$336$		0			0
$337$	376704.			0.180687			0.0903433	−	0.995911i	$-0.471204\pi$
$337$	376704.			0.180687			0.0903433	+	0.995911i	$0.471204\pi$
$338$		0			0
$339$	1.43369e6			0.677573
$340$		0			0
$341$	−607614.	+	1.05242e6i	−0.282971	+	0.490120i
$342$		0			0
$343$	−2.36133e6			−1.08373
$344$		0			0
$345$	58224.9	+	100849.i	0.0263367	+	0.0456165i
$346$		0			0
$347$	−822897.	−	1.42530e6i	−0.366878	−	0.635452i	0.622197	−	0.782860i	$-0.286240\pi$
$347$	−822897.	−	1.42530e6i	−0.366878	−	0.635452i	−0.989076	+	0.147409i	$0.952907\pi$
$348$		0			0
$349$	−1.55893e6	+	2.70014e6i	−0.685113	+	1.18665i	0.288288	+	0.957544i	$0.406914\pi$
$349$	−1.55893e6	+	2.70014e6i	−0.685113	+	1.18665i	−0.973401	+	0.229107i	$0.926419\pi$
$350$		0			0
$351$	382897.	+	225190.i	0.165888	+	0.0975620i
$352$		0			0
$353$	1.21763e6	−	2.10900e6i	0.520091	−	0.900824i	−0.479636	−	0.877467i	$-0.659231\pi$
$353$	1.21763e6	−	2.10900e6i	0.520091	−	0.900824i	0.999727	−	0.0233565i	$-0.00743527\pi$
$354$		0			0
$355$	207480.	+	359366.i	0.0873787	+	0.151344i
$356$		0			0
$357$	−479156.	−	829922.i	−0.198979	−	0.344641i
$358$		0			0
$359$	−4.01124e6			−1.64264			−0.821320	−	0.570468i	$-0.806762\pi$
$359$	−4.01124e6			−1.64264			−0.821320	+	0.570468i	$0.806762\pi$
$360$		0			0
$361$	−2.54039e6	+	4.40009e6i	−1.02597	+	1.77702i
$362$		0			0
$363$	−171497.			−0.0683107
$364$		0			0
$365$	212892.			0.0836426
$366$		0			0
$367$	−995892.	+	1.72494e6i	−0.385964	+	0.668510i	−0.991902	−	0.127002i	$-0.959465\pi$
$367$	−995892.	+	1.72494e6i	−0.385964	+	0.668510i	0.605938	+	0.795512i	$0.292798\pi$
$368$		0			0
$369$	1.11936e6			0.427961
$370$		0			0
$371$	−1.82057e6	−	3.15331e6i	−0.686707	−	1.18941i
$372$		0			0
$373$	−1.82220e6	−	3.15614e6i	−0.678146	−	1.17458i	−0.975539	−	0.219828i	$-0.929450\pi$
$373$	−1.82220e6	−	3.15614e6i	−0.678146	−	1.17458i	0.297392	−	0.954755i	$-0.403883\pi$
$374$		0			0
$375$	−231070.	+	400224.i	−0.0848525	+	0.146969i
$376$		0			0
$377$	−331523.	+	187867.i	−0.120132	+	0.0680764i
$378$		0			0
$379$	2.30901e6	−	3.99931e6i	0.825709	−	1.43017i	−0.0756678	−	0.997133i	$-0.524109\pi$
$379$	2.30901e6	−	3.99931e6i	0.825709	−	1.43017i	0.901376	−	0.433036i	$-0.142558\pi$
$380$		0			0
$381$	−1.28911e6	−	2.23281e6i	−0.454966	−	0.788024i
$382$		0			0
$383$	563056.	+	975241.i	0.196135	+	0.339715i	0.947272	−	0.320431i	$-0.103828\pi$
$383$	563056.	+	975241.i	0.196135	+	0.339715i	−0.751137	+	0.660146i	$0.770494\pi$
$384$		0			0
$385$	312991.			0.107617
$386$		0			0
$387$	−575801.	+	997317.i	−0.195432	+	0.338498i
$388$		0			0
$389$	−5.15193e6			−1.72622			−0.863110	−	0.505016i	$-0.831487\pi$
$389$	−5.15193e6			−1.72622			−0.863110	+	0.505016i	$0.831487\pi$
$390$		0			0
$391$	1.65870e6			0.548688
$392$		0			0
$393$	406824.	−	704639.i	0.132869	−	0.230136i
$394$		0			0
$395$	422545.			0.136264
$396$		0			0
$397$	−1.89873e6	−	3.28870e6i	−0.604627	−	1.04725i	−0.992110	−	0.125368i	$-0.959989\pi$
$397$	−1.89873e6	−	3.28870e6i	−0.604627	−	1.04725i	0.387483	−	0.921877i	$-0.373345\pi$
$398$		0			0
$399$	1.23682e6	+	2.14223e6i	0.388931	+	0.673648i
$400$		0			0
$401$	−944546.	+	1.63600e6i	−0.293334	+	0.508069i	−0.974596	−	0.223971i	$-0.928098\pi$
$401$	−944546.	+	1.63600e6i	−0.293334	+	0.508069i	0.681262	+	0.732040i	$0.261431\pi$
$402$		0			0
$403$	−1.69385e6	−	996187.i	−0.519531	−	0.305547i
$404$		0			0
$405$	27252.9	−	47203.4i	0.00825610	−	0.0143000i
$406$		0			0
$407$	−804323.	−	1.39313e6i	−0.240683	−	0.416874i
$408$		0			0
$409$	−1.92001e6	−	3.32556e6i	−0.567540	−	0.983007i	−0.996808	−	0.0798306i	$-0.974562\pi$
$409$	−1.92001e6	−	3.32556e6i	−0.567540	−	0.983007i	0.429269	−	0.903177i	$-0.358771\pi$
$410$		0			0
$411$	2.49655e6			0.729013
$412$		0			0
$413$	−831286.	+	1.43983e6i	−0.239814	+	0.415371i
$414$		0			0
$415$	630688.			0.179761
$416$		0			0
$417$	−2.89349e6			−0.814857
$418$		0			0
$419$	2.14644e6	−	3.71775e6i	0.597289	−	1.03453i	−0.395931	−	0.918280i	$-0.629578\pi$
$419$	2.14644e6	−	3.71775e6i	0.597289	−	1.03453i	0.993219	−	0.116254i	$-0.0370887\pi$
$420$		0			0
$421$	−6.06716e6			−1.66832			−0.834161	−	0.551521i	$-0.814048\pi$
$421$	−6.06716e6			−1.66832			−0.834161	+	0.551521i	$0.814048\pi$
$422$		0			0
$423$	572982.	+	992434.i	0.155700	+	0.269681i
$424$		0			0
$425$	1.62729e6	+	2.81855e6i	0.437011	+	0.756925i
$426$		0			0
$427$	−1.69267e6	+	2.93178e6i	−0.449264	+	0.778148i
$428$		0			0
$429$	−16615.9	+	2.06645e6i	−0.00435894	+	0.542103i
$430$		0			0
$431$	1.93348e6	−	3.34888e6i	0.501356	−	0.868374i	−0.498643	−	0.866808i	$-0.666168\pi$
$431$	1.93348e6	−	3.34888e6i	0.501356	−	0.868374i	0.999999	−	0.00156664i	$-0.000498676\pi$
$432$		0			0
$433$	47573.8	+	82400.2i	0.0121940	+	0.0211207i	0.872058	−	0.489402i	$-0.162785\pi$
$433$	47573.8	+	82400.2i	0.0121940	+	0.0211207i	−0.859864	+	0.510523i	$0.829452\pi$
$434$		0			0
$435$	23378.2	+	40492.3i	0.00592364	+	0.0102600i
$436$		0			0
$437$	−4.28149e6			−1.07249
$438$		0			0
$439$	54806.8	−	94928.2i	0.0135729	−	0.0235090i	−0.859159	−	0.511708i	$-0.829013\pi$
$439$	54806.8	−	94928.2i	0.0135729	−	0.0235090i	0.872732	+	0.488199i	$0.162346\pi$
$440$		0			0
$441$	−551662.			−0.135075
$442$		0			0
$443$	−5.92396e6			−1.43418			−0.717088	−	0.696982i	$-0.754526\pi$
$443$	−5.92396e6			−1.43418			−0.717088	+	0.696982i	$0.754526\pi$
$444$		0			0
$445$	−387343.	+	670897.i	−0.0927246	+	0.160604i
$446$		0			0
$447$	−701485.			−0.166054
$448$		0			0
$449$	−164091.	−	284215.i	−0.0384123	−	0.0665320i	0.846180	−	0.532897i	$-0.178897\pi$
$449$	−164091.	−	284215.i	−0.0384123	−	0.0665320i	−0.884592	+	0.466365i	$0.845563\pi$
$450$		0			0
$451$	2.60371e6	+	4.50976e6i	0.602770	+	1.04403i
$452$		0			0
$453$	504694.	−	874155.i	0.115553	−	0.200144i
$454$		0			0
$455$	−4069.47	+	506102.i	−0.000921529	+	0.114607i
$456$		0			0
$457$	−562990.	+	975127.i	−0.126098	+	0.218409i	−0.922162	−	0.386804i	$-0.873579\pi$
$457$	−562990.	+	975127.i	−0.126098	+	0.218409i	0.796063	+	0.605213i	$0.206912\pi$
$458$		0			0
$459$	−388187.	−	672359.i	−0.0860021	−	0.148960i
$460$		0			0
$461$	3.81440e6	+	6.60674e6i	0.835939	+	1.44789i	0.893264	+	0.449533i	$0.148409\pi$
$461$	3.81440e6	+	6.60674e6i	0.835939	+	1.44789i	−0.0573252	+	0.998356i	$0.518257\pi$
$462$		0			0
$463$	−2.59151e6			−0.561823			−0.280912	−	0.959734i	$-0.590637\pi$
$463$	−2.59151e6			−0.561823			−0.280912	+	0.959734i	$0.590637\pi$
$464$		0			0
$465$	−120560.	+	208817.i	−0.0258567	+	0.0447850i
$466$		0			0
$467$	7.27838e6			1.54434			0.772169	−	0.635417i	$-0.219172\pi$
$467$	7.27838e6			1.54434			0.772169	+	0.635417i	$0.219172\pi$
$468$		0			0
$469$	5.79925e6			1.21742
$470$		0			0
$471$	2.18719e6	−	3.78833e6i	0.454291	−	0.786856i
$472$		0			0
$473$	−5.35741e6			−1.10104
$474$		0			0
$475$	−4.20042e6	−	7.27534e6i	−0.854198	−	1.47951i
$476$		0			0
$477$	−1.47493e6	−	2.55465e6i	−0.296807	−	0.514085i
$478$		0			0
$479$	1.29053e6	−	2.23526e6i	0.256998	−	0.445133i	−0.708439	−	0.705772i	$-0.750600\pi$
$479$	1.29053e6	−	2.23526e6i	0.256998	−	0.445133i	0.965436	+	0.260640i	$0.0839334\pi$
$480$		0			0
$481$	2.26313e6	−	1.28247e6i	0.446012	−	0.252746i
$482$		0			0
$483$	−700741.	+	1.21372e6i	−0.136675	+	0.236728i
$484$		0			0
$485$	−332589.	−	576061.i	−0.0642027	−	0.111202i
$486$		0			0
$487$	2.55808e6	+	4.43073e6i	0.488756	+	0.846550i	0.999916	−	0.0129351i	$-0.00411750\pi$
$487$	2.55808e6	+	4.43073e6i	0.488756	+	0.846550i	−0.511160	+	0.859485i	$0.670784\pi$
$488$		0			0
$489$	3.36588e6			0.636541
$490$		0			0
$491$	3.43600e6	−	5.95132e6i	0.643204	−	1.11406i	−0.341509	−	0.939879i	$-0.610938\pi$
$491$	3.43600e6	−	5.95132e6i	0.643204	−	1.11406i	0.984713	−	0.174184i	$-0.0557287\pi$
$492$		0			0
$493$	665993.			0.123411
$494$		0			0
$495$	253569.			0.0465139
$496$		0			0
$497$	−2.49703e6	+	4.32499e6i	−0.453454	+	0.785406i
$498$		0			0
$499$	2.51797e6			0.452688			0.226344	−	0.974047i	$-0.427323\pi$
$499$	2.51797e6			0.452688			0.226344	+	0.974047i	$0.427323\pi$
$500$		0			0
$501$	−2.17786e6	−	3.77216e6i	−0.387646	−	0.671423i
$502$		0			0
$503$	−1.59007e6	−	2.75408e6i	−0.280218	−	0.485351i	0.691221	−	0.722644i	$-0.257073\pi$
$503$	−1.59007e6	−	2.75408e6i	−0.280218	−	0.485351i	−0.971438	+	0.237293i	$0.923740\pi$
$504$		0			0
$505$	196089.	−	339637.i	0.0342157	−	0.0592634i
$506$		0			0
$507$	−3.34120e6	−	53735.4i	−0.577276	−	0.00928412i
$508$		0			0
$509$	280620.	−	486048.i	0.0480092	−	0.0831543i	−0.841022	−	0.541001i	$-0.818046\pi$
$509$	280620.	−	486048.i	0.0480092	−	0.0831543i	0.889031	+	0.457846i	$0.151379\pi$
$510$		0			0
$511$	1.28109e6	+	2.21891e6i	0.217033	+	0.375912i
$512$		0			0
$513$	1.00200e6	+	1.73552e6i	0.168103	+	0.291163i
$514$		0			0
$515$	763742.			0.126890
$516$		0			0
$517$	−2.66559e6	+	4.61694e6i	−0.438599	+	0.759675i
$518$		0			0
$519$	−6.16599e6			−1.00481
$520$		0			0
$521$	6.78351e6			1.09486			0.547432	−	0.836850i	$-0.315606\pi$
$521$	6.78351e6			1.09486			0.547432	+	0.836850i	$0.315606\pi$
$522$		0			0
$523$	−5.09937e6	+	8.83236e6i	−0.815196	+	1.41196i	0.0939915	+	0.995573i	$0.470037\pi$
$523$	−5.09937e6	+	8.83236e6i	−0.815196	+	1.41196i	−0.909187	+	0.416387i	$0.863296\pi$
$524$		0			0
$525$	−2.74989e6			−0.435428
$526$		0			0
$527$	1.71725e6	+	2.97436e6i	0.269343	+	0.466517i
$528$		0			0
$529$	2.00529e6	+	3.47327e6i	0.311558	+	0.539634i
$530$		0			0
$531$	−673464.	+	1.16647e6i	−0.103652	+	0.179531i
$532$		0			0
$533$	−7.32608e6	+	4.15153e6i	−1.11700	+	0.632981i
$534$		0			0
$535$	502443.	−	870257.i	0.0758931	−	0.131451i
$536$		0			0
$537$	2.74059e6	+	4.74683e6i	0.410117	+	0.710343i
$538$		0			0
$539$	−1.28320e6	−	2.22257e6i	−0.190250	−	0.329522i
$540$		0			0
$541$	−3.76162e6			−0.552563			−0.276282	−	0.961077i	$-0.589102\pi$
$541$	−3.76162e6			−0.552563			−0.276282	+	0.961077i	$0.589102\pi$
$542$		0			0
$543$	881004.	−	1.52594e6i	0.128227	−	0.222095i
$544$		0			0
$545$	1.57001e6			0.226418
$546$		0			0
$547$	6.27984e6			0.897388			0.448694	−	0.893685i	$-0.351889\pi$
$547$	6.27984e6			0.897388			0.448694	+	0.893685i	$0.351889\pi$
$548$		0			0
$549$	−1.37131e6	+	2.37518e6i	−0.194180	+	0.336330i
$550$		0			0
$551$	−1.71909e6			−0.241223
$552$		0			0
$553$	2.54268e6	+	4.40405e6i	0.353573	+	0.612406i
$554$		0			0
$555$	−159591.	−	276419.i	−0.0219925	−	0.0380922i
$556$		0			0
$557$	−1.72957e6	+	2.99571e6i	−0.236211	+	0.409130i	−0.959624	−	0.281286i	$-0.909239\pi$
$557$	−1.72957e6	+	2.99571e6i	−0.236211	+	0.409130i	0.723413	+	0.690416i	$0.242572\pi$
$558$		0			0
$559$	69656.5	−	8.66287e6i	0.00942827	−	1.17255i
$560$		0			0
$561$	1.80590e6	−	3.12791e6i	0.242263	−	0.419611i
$562$		0			0
$563$	391261.	+	677683.i	0.0520230	+	0.0901064i	0.890864	−	0.454270i	$-0.150100\pi$
$563$	391261.	+	677683.i	0.0520230	+	0.0901064i	−0.838841	+	0.544376i	$0.816766\pi$
$564$		0			0
$565$	661691.	+	1.14608e6i	0.0872035	+	0.151041i
$566$		0			0
$567$	655981.			0.0856906
$568$		0			0
$569$	−822631.	+	1.42484e6i	−0.106518	+	0.184495i	−0.914358	−	0.404908i	$-0.867304\pi$
$569$	−822631.	+	1.42484e6i	−0.106518	+	0.184495i	0.807839	+	0.589403i	$0.200637\pi$
$570$		0			0
$571$	1.14533e7			1.47007			0.735037	−	0.678026i	$-0.237165\pi$
$571$	1.14533e7			1.47007			0.735037	+	0.678026i	$0.237165\pi$
$572$		0			0
$573$	−208490.			−0.0265277
$574$		0			0
$575$	2.37982e6	−	4.12198e6i	0.300176	−	0.519919i
$576$		0			0
$577$	−205235.			−0.0256632			−0.0128316	−	0.999918i	$-0.504085\pi$
$577$	−205235.			−0.0256632			−0.0128316	+	0.999918i	$0.504085\pi$
$578$		0			0
$579$	3.71518e6	+	6.43489e6i	0.460557	+	0.797709i
$580$		0			0
$581$	3.79519e6	+	6.57346e6i	0.466437	+	0.807893i
$582$		0			0
$583$	6.86157e6	−	1.18846e7i	0.836088	−	1.44815i
$584$		0			0
$585$	−3296.87	+	410017.i	−0.000398301	+	0.0495350i
$586$		0			0
$587$	1.04789e6	−	1.81500e6i	0.125522	−	0.217410i	−0.796415	−	0.604751i	$-0.793273\pi$
$587$	1.04789e6	−	1.81500e6i	0.125522	−	0.217410i	0.921937	+	0.387340i	$0.126606\pi$
$588$		0			0
$589$	−4.43262e6	−	7.67753e6i	−0.526469	−	0.911871i
$590$		0			0
$591$	3.77037e6	+	6.53046e6i	0.444032	+	0.769087i
$592$		0			0
$593$	1.52323e7			1.77880			0.889402	−	0.457126i	$-0.151121\pi$
$593$	1.52323e7			1.77880			0.889402	+	0.457126i	$0.151121\pi$
$594$		0			0
$595$	442290.	−	766068.i	0.0512170	−	0.0887105i
$596$		0			0
$597$	−647101.			−0.0743081
$598$		0			0
$599$	3.59165e6			0.409004			0.204502	−	0.978866i	$-0.434443\pi$
$599$	3.59165e6			0.409004			0.204502	+	0.978866i	$0.434443\pi$
$600$		0			0
$601$	6.32328e6	−	1.09522e7i	0.714095	−	1.23685i	−0.249212	−	0.968449i	$-0.580172\pi$
$601$	6.32328e6	−	1.09522e7i	0.714095	−	1.23685i	0.963307	−	0.268400i	$-0.0864950\pi$
$602$		0			0
$603$	4.69825e6			0.526190
$604$		0			0
$605$	−79150.8	−	137093.i	−0.00879158	−	0.0152275i
$606$		0			0
$607$	−5.57200e6	−	9.65099e6i	−0.613818	−	1.06316i	−0.990591	−	0.136858i	$-0.956299\pi$
$607$	−5.57200e6	−	9.65099e6i	−0.613818	−	1.06316i	0.376773	−	0.926306i	$-0.377034\pi$
$608$		0			0
$609$	−281359.	+	487327.i	−0.0307409	+	0.0532448i
$610$		0			0
$611$	−7.43088e6	−	4.37026e6i	−0.805262	−	0.473592i
$612$		0			0
$613$	6.97128e6	−	1.20746e7i	0.749310	−	1.29784i	−0.198844	−	0.980031i	$-0.563719\pi$
$613$	6.97128e6	−	1.20746e7i	0.749310	−	1.29784i	0.948154	−	0.317812i	$-0.102948\pi$
$614$		0			0
$615$	516619.	+	894810.i	0.0550785	+	0.0953988i
$616$		0			0
$617$	3.68600e6	+	6.38435e6i	0.389801	+	0.675155i	0.992423	−	0.122872i	$-0.0392105\pi$
$617$	3.68600e6	+	6.38435e6i	0.389801	+	0.675155i	−0.602622	+	0.798027i	$0.705877\pi$
$618$		0			0
$619$	8.41512e6			0.882742			0.441371	−	0.897325i	$-0.354492\pi$
$619$	8.41512e6			0.882742			0.441371	+	0.897325i	$0.354492\pi$
$620$		0			0
$621$	−567703.	+	983291.i	−0.0590734	+	0.102318i
$622$		0			0
$623$	−9.32338e6			−0.962395
$624$		0			0
$625$	9.12337e6			0.934233
$626$		0			0
$627$	−4.66146e6	+	8.07388e6i	−0.473536	+	0.820188i
$628$		0			0
$629$	−4.54638e6			−0.458184
$630$		0			0
$631$	6.90214e6	+	1.19549e7i	0.690097	+	1.19528i	0.971806	+	0.235783i	$0.0757656\pi$
$631$	6.90214e6	+	1.19549e7i	0.690097	+	1.19528i	−0.281709	+	0.959500i	$0.590901\pi$
$632$		0			0
$633$	−4.60033e6	−	7.96800e6i	−0.456331	−	0.790388i
$634$		0			0
$635$	1.18993e6	−	2.06102e6i	0.117108	−	0.202837i
$636$		0			0
$637$	3.61056e6	−	2.04603e6i	0.352554	−	0.199785i
$638$		0			0
$639$	−2.02297e6	+	3.50388e6i	−0.195991	+	0.339467i
$640$		0			0
$641$	2.93781e6	+	5.08843e6i	0.282409	+	0.489146i	0.971977	−	0.235074i	$-0.0755333\pi$
$641$	2.93781e6	+	5.08843e6i	0.282409	+	0.489146i	−0.689569	+	0.724220i	$0.742200\pi$
$642$		0			0
$643$	2.41054e6	+	4.17518e6i	0.229926	+	0.398243i	0.957786	−	0.287483i	$-0.0928184\pi$
$643$	2.41054e6	+	4.17518e6i	0.229926	+	0.398243i	−0.727860	+	0.685725i	$0.759485\pi$
$644$		0			0
$645$	−1.06300e6			−0.100608
$646$		0			0
$647$	−430432.	+	745530.i	−0.0404244	+	0.0700172i	−0.885530	−	0.464583i	$-0.846204\pi$
$647$	−430432.	+	745530.i	−0.0404244	+	0.0700172i	0.845105	+	0.534600i	$0.179538\pi$
$648$		0			0
$649$	−6.26610e6			−0.583963
$650$		0			0
$651$	−2.90190e6			−0.268368
$652$		0			0
$653$	−2.06822e6	+	3.58227e6i	−0.189808	+	0.328757i	−0.945186	−	0.326532i	$-0.894120\pi$
$653$	−2.06822e6	+	3.58227e6i	−0.189808	+	0.328757i	0.755378	+	0.655289i	$0.227453\pi$
$654$		0			0
$655$	751046.			0.0684011
$656$		0			0
$657$	1.03787e6	+	1.79764e6i	0.0938056	+	0.162476i
$658$		0			0
$659$	1.75204e6	+	3.03463e6i	0.157156	+	0.272203i	0.933842	−	0.357686i	$-0.116434\pi$
$659$	1.75204e6	+	3.03463e6i	0.157156	+	0.272203i	−0.776686	+	0.629888i	$0.783101\pi$
$660$		0			0
$661$	5.81850e6	−	1.00779e7i	0.517974	−	0.897157i	−0.481808	−	0.876277i	$-0.660020\pi$
$661$	5.81850e6	−	1.00779e7i	0.517974	−	0.897157i	0.999782	−	0.0208801i	$-0.00664683\pi$
$662$		0			0
$663$	5.03431e6	+	2.96078e6i	0.444791	+	0.261591i
$664$		0			0
$665$	−1.14166e6	+	1.97740e6i	−0.100111	+	0.173397i
$666$		0			0
$667$	−486990.	−	843492.i	−0.0423844	−	0.0734119i
$668$		0			0
$669$	−2.85781e6	−	4.94987e6i	−0.246870	−	0.427591i
$670$		0			0
$671$	−1.27590e7			−1.09399
$672$		0			0
$673$	5.33070e6	−	9.23304e6i	0.453677	−	0.785791i	−0.544934	−	0.838479i	$-0.683445\pi$
$673$	5.33070e6	−	9.23304e6i	0.453677	−	0.785791i	0.998611	+	0.0526877i	$0.0167788\pi$
$674$		0			0
$675$	−2.22781e6			−0.188200
$676$		0			0
$677$	−5.00878e6			−0.420011			−0.210005	−	0.977700i	$-0.567348\pi$
$677$	−5.00878e6			−0.420011			−0.210005	+	0.977700i	$0.567348\pi$
$678$		0			0
$679$	4.00273e6	−	6.93293e6i	0.333182	−	0.577088i
$680$		0			0
$681$	1.78246e6			0.147283
$682$		0			0
$683$	6.51439e6	+	1.12833e7i	0.534346	+	0.925514i	0.999195	+	0.0401239i	$0.0127753\pi$
$683$	6.51439e6	+	1.12833e7i	0.534346	+	0.925514i	−0.464849	+	0.885390i	$0.653891\pi$
$684$		0			0
$685$	1.15223e6	+	1.99572e6i	0.0938239	+	0.162508i
$686$		0			0
$687$	−2.84194e6	+	4.92238e6i	−0.229733	+	0.397909i
$688$		0			0
$689$	1.91280e7	+	1.12496e7i	1.53505	+	0.902794i
$690$		0			0
$691$	−2.93345e6	+	5.08088e6i	−0.233713	+	0.404803i	−0.958898	−	0.283751i	$-0.908421\pi$
$691$	−2.93345e6	+	5.08088e6i	−0.233713	+	0.404803i	0.725185	+	0.688554i	$0.241754\pi$
$692$		0			0
$693$	1.52586e6	+	2.64286e6i	0.120693	+	0.209046i
$694$		0			0
$695$	−1.33543e6	−	2.31304e6i	−0.104872	−	0.181644i
$696$		0			0
$697$	1.47173e7			1.14748
$698$		0			0
$699$	1.45320e6	−	2.51702e6i	0.112495	−	0.194847i
$700$		0			0
$701$	−1.61845e7			−1.24396			−0.621978	−	0.783034i	$-0.713671\pi$
$701$	−1.61845e7			−1.24396			−0.621978	+	0.783034i	$0.713671\pi$
$702$		0			0
$703$	1.17353e7			0.895583
$704$		0			0
$705$	−528897.	+	916076.i	−0.0400773	+	0.0694159i
$706$		0			0
$707$	4.71990e6			0.355127
$708$		0			0
$709$	1.73222e6	+	3.00030e6i	0.129416	+	0.224155i	0.923451	−	0.383718i	$-0.125356\pi$
$709$	1.73222e6	+	3.00030e6i	0.129416	+	0.224155i	−0.794034	+	0.607873i	$0.792023\pi$
$710$		0			0
$711$	2.05994e6	+	3.56793e6i	0.152820	+	0.264693i
$712$		0			0
$713$	2.51139e6	−	4.34985e6i	0.185008	−	0.320443i
$714$		0			0
$715$	−1.65957e6	+	940446.i	−0.121404	+	0.0687969i
$716$		0			0
$717$	−263999.	+	457260.i	−0.0191781	+	0.0332174i
$718$		0			0
$719$	−379117.	−	656650.i	−0.0273496	−	0.0473709i	0.852027	−	0.523498i	$-0.175373\pi$
$719$	−379117.	−	656650.i	−0.0273496	−	0.0473709i	−0.879376	+	0.476128i	$0.842040\pi$
$720$		0			0
$721$	4.59584e6	+	7.96024e6i	0.329251	+	0.570280i
$722$		0			0
$723$	−8.56842e6			−0.609614
$724$		0			0
$725$	955538.	−	1.65504e6i	0.0675154	−	0.116940i
$726$		0			0
$727$	−1.56063e7			−1.09512			−0.547561	−	0.836766i	$-0.684444\pi$
$727$	−1.56063e7			−1.09512			−0.547561	+	0.836766i	$0.684444\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−7.57061e6	+	1.31127e7i	−0.524007	+	0.907607i
$732$		0			0
$733$	−1.53145e7			−1.05280			−0.526398	−	0.850238i	$-0.676458\pi$
$733$	−1.53145e7			−1.05280			−0.526398	+	0.850238i	$0.676458\pi$
$734$		0			0
$735$	−254608.	−	440995.i	−0.0173842	−	0.0301103i
$736$		0			0
$737$	1.09284e7	+	1.89286e7i	0.741122	+	1.28366i
$738$		0			0
$739$	−3.65859e6	+	6.33687e6i	−0.246435	+	0.426839i	−0.962534	−	0.271160i	$-0.912593\pi$
$739$	−3.65859e6	+	6.33687e6i	−0.246435	+	0.426839i	0.716099	+	0.697999i	$0.245926\pi$
$740$		0			0
$741$	−1.29948e7	−	7.64249e6i	−0.869406	−	0.511316i
$742$		0			0
$743$	1.34797e7	−	2.33475e7i	0.895792	−	1.55156i	0.0629707	−	0.998015i	$-0.479943\pi$
$743$	1.34797e7	−	2.33475e7i	0.895792	−	1.55156i	0.832821	−	0.553542i	$-0.186724\pi$
$744$		0			0
$745$	−323756.	−	560763.i	−0.0213711	−	0.0370159i
$746$		0			0
$747$	3.07466e6	+	5.32547e6i	0.201602	+	0.349186i
$748$		0			0
$749$	1.20939e7			0.787700
$750$		0			0
$751$	1.19385e6	−	2.06781e6i	0.0772413	−	0.133786i	−0.824817	−	0.565399i	$-0.808722\pi$
$751$	1.19385e6	−	2.06781e6i	0.0772413	−	0.133786i	0.902059	+	0.431613i	$0.142055\pi$
$752$		0			0
$753$	−6.85181e6			−0.440370
$754$		0			0
$755$	931725.			0.0594868
$756$		0			0
$757$	939670.	−	1.62756e6i	0.0595986	−	0.103228i	−0.834687	−	0.550725i	$-0.814351\pi$
$757$	939670.	−	1.62756e6i	0.0595986	−	0.103228i	0.894285	+	0.447497i	$0.147685\pi$
$758$		0			0
$759$	−5.28207e6			−0.332812
$760$		0			0
$761$	1.26795e6	+	2.19615e6i	0.0793668	+	0.137467i	0.902977	−	0.429689i	$-0.141377\pi$
$761$	1.26795e6	+	2.19615e6i	0.0793668	+	0.137467i	−0.823610	+	0.567156i	$0.808043\pi$
$762$		0			0
$763$	9.44759e6	+	1.63637e7i	0.587502	+	1.01758i
$764$		0			0
$765$	358320.	−	620628.i	0.0221369	−	0.0383423i
$766$		0			0
$767$	81471.0	−	1.01322e7i	0.00500051	−	0.621892i
$768$		0			0
$769$	7.57651e6	−	1.31229e7i	0.462012	−	0.800228i	−0.537049	−	0.843551i	$-0.680461\pi$
$769$	7.57651e6	−	1.31229e7i	0.462012	−	0.800228i	0.999061	+	0.0433227i	$0.0137944\pi$
$770$		0			0
$771$	1.91946e6	+	3.32461e6i	0.116290	+	0.201421i
$772$		0			0
$773$	−1.03514e7	−	1.79292e7i	−0.623091	−	1.07923i	−0.988907	−	0.148538i	$-0.952543\pi$
$773$	−1.03514e7	−	1.79292e7i	−0.623091	−	1.07923i	0.365815	−	0.930687i	$-0.380790\pi$
$774$		0			0
$775$	9.85532e6			0.589409
$776$		0			0
$777$	1.92069e6	−	3.32673e6i	0.114131	−	0.197681i
$778$		0			0
$779$	−3.79889e7			−2.24291
$780$		0			0
$781$	−1.88222e7			−1.10419
$782$		0			0
$783$	−227942.	+	394807.i	−0.0132868	+	0.0230134i
$784$		0			0
$785$	4.03782e6			0.233869
$786$		0			0
$787$	−1.53208e7	−	2.65364e7i	−0.881748	−	1.52723i	−0.849396	−	0.527756i	$-0.823034\pi$
$787$	−1.53208e7	−	2.65364e7i	−0.881748	−	1.52723i	−0.0323516	−	0.999477i	$-0.510300\pi$
$788$		0			0
$789$	9.35263e6	+	1.61992e7i	0.534861	+	0.926407i
$790$		0			0
$791$	−7.96349e6	+	1.37932e7i	−0.452546	+	0.783832i
$792$		0			0
$793$	165891.	−	2.06312e7i	0.00936788	−	1.16504i
$794$		0			0
$795$	1.36145e6	−	2.35809e6i	0.0763981	−	0.132325i
$796$		0			0
$797$	−9.75940e6	−	1.69038e7i	−0.544224	−	0.942623i	−0.998655	−	0.0518412i	$-0.983491\pi$
$797$	−9.75940e6	−	1.69038e7i	−0.544224	−	0.942623i	0.454432	−	0.890782i	$-0.349842\pi$
$798$		0			0
$799$	7.53354e6	+	1.30485e7i	0.417477	+	0.723091i
$800$		0			0
$801$	−7.55331e6			−0.415965
$802$		0			0
$803$	−4.82831e6	+	8.36287e6i	−0.264245	+	0.457685i
$804$		0			0
$805$	−1.29365e6			−0.0703603
$806$		0			0
$807$	2.80326e6			0.151523
$808$		0			0
$809$	−1.70191e7	+	2.94779e7i	−0.914250	+	1.58353i	−0.106253	+	0.994339i	$0.533885\pi$
$809$	−1.70191e7	+	2.94779e7i	−0.914250	+	1.58353i	−0.807996	+	0.589187i	$0.799448\pi$
$810$		0			0
$811$	1.17295e7			0.626219			0.313109	−	0.949717i	$-0.398629\pi$
$811$	1.17295e7			0.626219			0.313109	+	0.949717i	$0.398629\pi$
$812$		0			0
$813$	1.05564e6	+	1.82842e6i	0.0560131	+	0.0970176i
$814$		0			0
$815$	1.55345e6	+	2.69066e6i	0.0819228	+	0.141894i
$816$		0			0
$817$	1.95415e7	−	3.38469e7i	1.02424	−	1.77404i
$818$		0			0
$819$	−4.29331e6	+	2.43293e6i	−0.223657	+	0.126742i
$820$		0			0
$821$	7.30418e6	−	1.26512e7i	0.378193	−	0.655049i	−0.612606	−	0.790388i	$-0.709879\pi$
$821$	7.30418e6	−	1.26512e7i	0.378193	−	0.655049i	0.990799	+	0.135339i	$0.0432122\pi$
$822$		0			0
$823$	7.66251e6	+	1.32719e7i	0.394340	+	0.683018i	0.993017	−	0.117973i	$-0.0376398\pi$
$823$	7.66251e6	+	1.32719e7i	0.394340	+	0.683018i	−0.598676	+	0.800991i	$0.704306\pi$
$824$		0			0
$825$	−5.18205e6	−	8.97557e6i	−0.265074	−	0.459121i
$826$		0			0
$827$	−2.11573e6			−0.107571			−0.0537857	−	0.998552i	$-0.517129\pi$
$827$	−2.11573e6			−0.107571			−0.0537857	+	0.998552i	$0.517129\pi$
$828$		0			0
$829$	670755.	−	1.16178e6i	0.0338983	−	0.0587135i	−0.848579	−	0.529069i	$-0.822541\pi$
$829$	670755.	−	1.16178e6i	0.0338983	−	0.0587135i	0.882477	+	0.470356i	$0.155874\pi$
$830$		0			0
$831$	1.27808e6			0.0642029
$832$		0			0
$833$	−7.25322e6			−0.362175
$834$		0			0
$835$	2.01030e6	−	3.48193e6i	0.0997801	−	0.172824i
$836$		0			0
$837$	−2.35097e6			−0.115993
$838$		0			0
$839$	−3.73309e6	−	6.46589e6i	−0.183089	−	0.317120i	0.759842	−	0.650108i	$-0.225276\pi$
$839$	−3.73309e6	−	6.46589e6i	−0.183089	−	0.317120i	−0.942931	+	0.332988i	$0.891943\pi$
$840$		0			0
$841$	1.00600e7	+	1.74245e7i	0.490467	+	0.849514i
$842$		0			0
$843$	−3.94316e6	+	6.82975e6i	−0.191106	+	0.331006i
$844$		0			0
$845$	−1.49911e6	−	2.69574e6i	−0.0722258	−	0.129878i
$846$		0			0
$847$	952585.	−	1.64993e6i	0.0456242	−	0.0790234i
$848$		0			0
$849$	−4.36075e6	−	7.55303e6i	−0.207631	−	0.359627i
$850$		0			0
$851$	3.32443e6	+	5.75807e6i	0.157359	+	0.272554i
$852$		0			0
$853$	−6.24744e6			−0.293988			−0.146994	−	0.989137i	$-0.546960\pi$
$853$	−6.24744e6			−0.293988			−0.146994	+	0.989137i	$0.546960\pi$
$854$		0			0
$855$	−924909.	+	1.60199e6i	−0.0432697	+	0.0749453i
$856$		0			0
$857$	1.57515e7			0.732605			0.366303	−	0.930496i	$-0.380624\pi$
$857$	1.57515e7			0.732605			0.366303	+	0.930496i	$0.380624\pi$
$858$		0			0
$859$	−1.02670e7			−0.474746			−0.237373	−	0.971419i	$-0.576286\pi$
$859$	−1.02670e7			−0.474746			−0.237373	+	0.971419i	$0.576286\pi$
$860$		0			0
$861$	−6.21754e6	+	1.07691e7i	−0.285832	+	0.495075i
$862$		0			0
$863$	−3.21516e7			−1.46952			−0.734760	−	0.678327i	$-0.762705\pi$
$863$	−3.21516e7			−1.46952			−0.734760	+	0.678327i	$0.762705\pi$
$864$		0			0
$865$	−2.84579e6	−	4.92905e6i	−0.129319	−	0.223987i
$866$		0			0
$867$	1.28549e6	+	2.22654e6i	0.0580794	+	0.100596i
$868$		0			0
$869$	−9.58315e6	+	1.65985e7i	−0.430486	+	0.745623i
$870$		0			0
$871$	−3.07494e7	+	1.74251e7i	−1.37338	+	0.778267i
$872$		0			0
$873$	3.24280e6	−	5.61669e6i	0.144007	−	0.249428i
$874$		0			0
$875$	−2.56697e6	−	4.44613e6i	−0.113345	−	0.196319i
$876$		0			0
$877$	−5.22870e6	−	9.05637e6i	−0.229559	−	0.397608i	0.728118	−	0.685451i	$-0.240395\pi$
$877$	−5.22870e6	−	9.05637e6i	−0.229559	−	0.397608i	−0.957677	+	0.287843i	$0.907062\pi$
$878$		0			0
$879$	−1.21365e7			−0.529812
$880$		0			0
$881$	1.91590e6	−	3.31844e6i	0.0831637	−	0.144044i	−0.821444	−	0.570290i	$-0.806831\pi$
$881$	1.91590e6	−	3.31844e6i	0.0831637	−	0.144044i	0.904607	+	0.426246i	$0.140164\pi$
$882$		0			0
$883$	−4.25402e7			−1.83610			−0.918052	−	0.396459i	$-0.870239\pi$
$883$	−4.25402e7			−1.83610			−0.918052	+	0.396459i	$0.870239\pi$
$884$		0			0
$885$	−1.24330e6			−0.0533600
$886$		0			0
$887$	−9.80665e6	+	1.69856e7i	−0.418516	+	0.724890i	−0.995790	−	0.0916595i	$-0.970783\pi$
$887$	−9.80665e6	+	1.69856e7i	−0.418516	+	0.724890i	0.577275	+	0.816550i	$0.304116\pi$
$888$		0			0
$889$	2.86418e7			1.21547
$890$		0			0
$891$	1.23617e6	+	2.14111e6i	0.0521655	+	0.0903533i
$892$		0			0
$893$	−1.94459e7	−	3.36812e7i	−0.816016	−	1.41338i
$894$		0			0
$895$	−2.52973e6	+	4.38162e6i	−0.105564	+	0.182842i
$896$		0			0
$897$	68676.8	−	8.54104e6i	0.00284990	−	0.354429i
$898$		0			0
$899$	1.00836e6	−	1.74653e6i	0.0416119	−	0.0720738i
$900$		0			0
$901$	−1.93923e7	−	3.35884e7i	−0.795824	−	1.37841i
$902$		0			0
$903$	−6.39663e6	−	1.10793e7i	−0.261055	−	0.452160i
$904$		0			0
$905$	1.62644e6			0.0660110
$906$		0			0
$907$	1.14355e7	−	1.98069e7i	0.461571	−	0.799465i	−0.537468	−	0.843284i	$-0.680619\pi$
$907$	1.14355e7	−	1.98069e7i	0.461571	−	0.799465i	0.999039	+	0.0438194i	$0.0139526\pi$
$908$		0			0
$909$	3.82381e6			0.153492
$910$		0			0
$911$	−9.53579e6			−0.380681			−0.190340	−	0.981718i	$-0.560959\pi$
$911$	−9.53579e6			−0.380681			−0.190340	+	0.981718i	$0.560959\pi$
$912$		0			0
$913$	−1.43038e7	+	2.47748e7i	−0.567902	+	0.983635i
$914$		0			0
$915$	−2.53160e6			−0.0999638
$916$		0			0
$917$	4.51944e6	+	7.82790e6i	0.177485	+	0.307413i
$918$		0			0
$919$	1.05394e6	+	1.82547e6i	0.0411648	+	0.0712995i	0.885874	−	0.463926i	$-0.153560\pi$
$919$	1.05394e6	+	1.82547e6i	0.0411648	+	0.0712995i	−0.844709	+	0.535226i	$0.820226\pi$
$920$		0			0
$921$	3.37914e6	−	5.85284e6i	0.131267	−	0.227362i
$922$		0			0
$923$	244725.	−	3.04353e7i	0.00945525	−	1.17591i
$924$		0			0
$925$	−6.52295e6	+	1.12981e7i	−0.250663	+	0.434161i
$926$		0			0
$927$	3.72331e6	+	6.44896e6i	0.142308	+	0.246485i
$928$		0			0
$929$	−3.72642e6	−	6.45434e6i	−0.141662	−	0.245365i	0.786461	−	0.617640i	$-0.211911\pi$
$929$	−3.72642e6	−	6.45434e6i	−0.141662	−	0.245365i	−0.928122	+	0.372275i	$0.878578\pi$
$930$		0			0
$931$	1.87223e7			0.707921
$932$		0			0
$933$	3.68745e6	−	6.38685e6i	0.138683	−	0.240205i
$934$		0			0
$935$	3.33391e6			0.124717
$936$		0			0
$937$	1.38470e7			0.515238			0.257619	−	0.966247i	$-0.417062\pi$
$937$	1.38470e7			0.515238			0.257619	+	0.966247i	$0.417062\pi$
$938$		0			0
$939$	3.16092e6	−	5.47488e6i	0.116990	−	0.202633i
$940$		0			0
$941$	−4.47011e6			−0.164567			−0.0822837	−	0.996609i	$-0.526221\pi$
$941$	−4.47011e6			−0.164567			−0.0822837	+	0.996609i	$0.526221\pi$
$942$		0			0
$943$	−1.07616e7	−	1.86397e7i	−0.394094	−	0.682590i
$944$		0			0
$945$	302755.	+	524387.i	0.0110284	+	0.0191017i
$946$		0			0
$947$	1.23846e7	−	2.14508e7i	0.448754	−	0.777264i	−0.549551	−	0.835460i	$-0.685201\pi$
$947$	1.23846e7	−	2.14508e7i	0.448754	−	0.777264i	0.998305	+	0.0581955i	$0.0185347\pi$
$948$		0			0
$949$	−1.34599e7	−	7.91604e6i	−0.485150	−	0.285327i
$950$		0			0
$951$	−9.55873e6	+	1.65562e7i	−0.342727	+	0.593621i
$952$		0			0
$953$	−1.82023e6	−	3.15273e6i	−0.0649222	−	0.112449i	0.831737	−	0.555169i	$-0.187347\pi$
$953$	−1.82023e6	−	3.15273e6i	−0.0649222	−	0.112449i	−0.896659	+	0.442721i	$0.854013\pi$
$954$		0			0
$955$	−96224.6	−	166666.i	−0.00341411	−	0.00591341i
$956$		0			0
$957$	−2.12084e6			−0.0748561
$958$		0			0
$959$	−1.38672e7	+	2.40187e7i	−0.486902	+	0.843339i
$960$		0			0
$961$	−1.82290e7			−0.636729
$962$		0			0
$963$	9.79782e6			0.340458
$964$		0			0
$965$	−3.42934e6	+	5.93979e6i	−0.118547	+	0.205330i
$966$		0			0
$967$	−1.27353e7			−0.437970			−0.218985	−	0.975728i	$-0.570275\pi$
$967$	−1.27353e7			−0.437970			−0.218985	+	0.975728i	$0.570275\pi$
$968$		0			0
$969$	1.31743e7	+	2.28185e7i	0.450731	+	0.780689i
$970$		0			0
$971$	−2.80611e6	−	4.86033e6i	−0.0955119	−	0.165431i	0.814310	−	0.580430i	$-0.197115\pi$
$971$	−2.80611e6	−	4.86033e6i	−0.0955119	−	0.165431i	−0.909822	+	0.414998i	$0.863782\pi$
$972$		0			0
$973$	1.60720e7	−	2.78375e7i	0.544237	−	0.942646i
$974$		0			0
$975$	1.45808e7	−	8.26261e6i	0.491212	−	0.278359i
$976$		0			0
$977$	1.16834e7	−	2.02363e7i	0.391593	−	0.678258i	−0.601067	−	0.799199i	$-0.705258\pi$
$977$	1.16834e7	−	2.02363e7i	0.391593	−	0.678258i	0.992660	+	0.120940i	$0.0385909\pi$
$978$		0			0
$979$	−1.75695e7	−	3.04313e7i	−0.585873	−	1.01476i
$980$		0			0
$981$	7.65394e6	+	1.32570e7i	0.253929	+	0.439818i
$982$		0			0
$983$	1.02853e7			0.339496			0.169748	−	0.985488i	$-0.445705\pi$
$983$	1.02853e7			0.339496			0.169748	+	0.985488i	$0.445705\pi$
$984$		0			0
$985$	−3.48027e6	+	6.02801e6i	−0.114294	+	0.197963i
$986$		0			0
$987$	−1.27306e7			−0.415965
$988$		0			0
$989$	2.21432e7			0.719864
$990$		0			0
$991$	−9.12437e6	+	1.58039e7i	−0.295134	+	0.511187i	−0.975016	−	0.222135i	$-0.928698\pi$
$991$	−9.12437e6	+	1.58039e7i	−0.295134	+	0.511187i	0.679882	+	0.733321i	$0.262031\pi$
$992$		0			0
$993$	−9.29824e6			−0.299245
$994$		0			0
$995$	−298657.	−	517289.i	−0.00956345	−	0.0165644i
$996$		0			0
$997$	6.88342e6	+	1.19224e7i	0.219314	+	0.379863i	0.954598	−	0.297896i	$-0.0962848\pi$
$997$	6.88342e6	+	1.19224e7i	0.219314	+	0.379863i	−0.735284	+	0.677759i	$0.762951\pi$
$998$		0			0
$999$	1.55604e6	−	2.69514e6i	0.0493295	−	0.0854412i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.6.q.a.217.4	✓	16
13.3	even	3	inner	312.6.q.a.289.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.6.q.a.217.4	✓	16	1.1	even	1	trivial
312.6.q.a.289.4	yes	16	13.3	even	3	inner

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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	312.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} -8.30754 q^{5} +(-49.9909 - 86.5868i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(188.412 - 326.339i) q^{11} +(525.236 + 308.902i) q^{13} +(37.3839 - 64.7509i) q^{15} +(-532.492 - 922.304i) q^{17} +(1374.49 + 2380.68i) q^{19} +899.836 q^{21} +(-778.742 + 1348.82i) q^{23} -3055.98 q^{25} +729.000 q^{27} +(-312.678 + 541.573i) q^{29} -3224.93 q^{31} +(1695.71 + 2937.05i) q^{33} +(415.302 + 719.324i) q^{35} +(2134.48 - 3697.03i) q^{37} +(-4771.21 + 2703.75i) q^{39} +(-6909.63 + 11967.8i) q^{41} +(-7108.66 - 12312.6i) q^{43} +(336.456 + 582.758i) q^{45} -14147.7 q^{47} +(3405.32 - 5898.18i) q^{49} +9584.86 q^{51} +36417.9 q^{53} +(-1565.24 + 2711.07i) q^{55} -24740.8 q^{57} +(-8314.37 - 14400.9i) q^{59} +(-16929.7 - 29323.2i) q^{61} +(-4049.26 + 7013.53i) q^{63} +(-4363.42 - 2566.22i) q^{65} +(-29001.5 + 50232.1i) q^{67} +(-7008.68 - 12139.4i) q^{69} +(-24974.9 - 43257.8i) q^{71} -25626.4 q^{73} +(13751.9 - 23819.0i) q^{75} -37675.5 q^{77} -50862.8 q^{79} +(-3280.50 + 5681.99i) q^{81} -75917.6 q^{83} +(4423.70 + 7662.08i) q^{85} +(-2814.10 - 4874.16i) q^{87} +(46625.4 - 80757.6i) q^{89} +(489.852 - 60920.8i) q^{91} +(14512.2 - 25135.8i) q^{93} +(-11418.6 - 19777.6i) q^{95} +(40034.5 + 69341.9i) q^{97} -30522.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 72 q^{3} - 188 q^{5} + 138 q^{7} - 648 q^{9} + 612 q^{11} - 228 q^{13} + 846 q^{15} + 1450 q^{17} + 1064 q^{19} - 2484 q^{21} + 1768 q^{23} + 7628 q^{25} + 11664 q^{27} - 3338 q^{29} - 3724 q^{31}+ \cdots - 99144 q^{99}+O(q^{100}) \) 16 * q - 72 * q^3 - 188 * q^5 + 138 * q^7 - 648 * q^9 + 612 * q^11 - 228 * q^13 + 846 * q^15 + 1450 * q^17 + 1064 * q^19 - 2484 * q^21 + 1768 * q^23 + 7628 * q^25 + 11664 * q^27 - 3338 * q^29 - 3724 * q^31 + 5508 * q^33 - 4704 * q^35 - 10462 * q^37 - 8586 * q^39 + 18486 * q^41 + 1530 * q^43 + 7614 * q^45 + 12312 * q^47 - 22226 * q^49 - 26100 * q^51 - 44972 * q^53 - 58080 * q^55 - 19152 * q^57 + 1664 * q^59 - 37208 * q^61 + 11178 * q^63 - 44010 * q^65 - 36198 * q^67 + 15912 * q^69 + 73856 * q^71 + 21768 * q^73 - 34326 * q^75 + 192808 * q^77 - 6468 * q^79 - 52488 * q^81 - 284712 * q^83 - 81558 * q^85 - 30042 * q^87 - 71976 * q^89 + 177606 * q^91 + 16758 * q^93 + 25216 * q^95 + 101682 * q^97 - 99144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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