LMFDB - Embedded newform 112.6.i.c.65.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-8] 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:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{37})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 10x^{2} + 9x + 81 $ x^4 - x^3 + 10x^2 + 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.1
Root			$1.77069 + 3.06693i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.6.i.c.81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.04138 - 8.73193i) q^{3} +(39.9138 - 69.1328i) q^{5} +(-43.1587 - 122.247i) q^{7} +(70.6689 - 122.402i) q^{9} +(175.952 + 304.757i) q^{11} -291.683 q^{13} -804.883 q^{15} +(185.038 + 320.495i) q^{17} +(752.463 - 1303.30i) q^{19} +(-849.873 + 993.152i) q^{21} +(-212.855 + 368.676i) q^{23} +(-1623.72 - 2812.37i) q^{25} -3875.19 q^{27} -7783.93 q^{29} +(-1287.59 - 2230.17i) q^{31} +(1774.08 - 3072.80i) q^{33} +(-10173.9 - 1895.67i) q^{35} +(-369.809 + 640.528i) q^{37} +(1470.48 + 2546.95i) q^{39} +7029.84 q^{41} -1835.23 q^{43} +(-5641.33 - 9771.08i) q^{45} +(-766.342 + 1327.34i) q^{47} +(-13081.7 + 10552.0i) q^{49} +(1865.69 - 3231.47i) q^{51} +(4768.73 + 8259.68i) q^{53} +28091.6 q^{55} -15173.8 q^{57} +(-14837.0 - 25698.5i) q^{59} +(23255.4 - 40279.5i) q^{61} +(-18013.3 - 3356.35i) q^{63} +(-11642.2 + 20164.8i) q^{65} +(13373.0 + 23162.8i) q^{67} +4292.34 q^{69} +14388.8 q^{71} +(35047.6 + 60704.2i) q^{73} +(-16371.6 + 28356.5i) q^{75} +(29661.8 - 34662.5i) q^{77} +(-13542.9 + 23457.0i) q^{79} +(2363.74 + 4094.13i) q^{81} +79755.4 q^{83} +29542.2 q^{85} +(39241.8 + 67968.8i) q^{87} +(-21788.7 + 37739.1i) q^{89} +(12588.6 + 35657.3i) q^{91} +(-12982.4 + 22486.2i) q^{93} +(-60067.3 - 104040. i) q^{95} +103374. q^{97} +49737.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{3} + 38 q^{5} + 168 q^{7} + 380 q^{9} + 424 q^{11} - 1848 q^{13} - 1784 q^{15} + 2346 q^{17} - 360 q^{19} - 1526 q^{21} - 12 q^{23} - 1872 q^{25} - 5744 q^{27} - 14104 q^{29} + 3548 q^{31} + 3398 q^{33}+ \cdots + 133888 q^{99}+O(q^{100}) $ 4 * q - 8 * q^3 + 38 * q^5 + 168 * q^7 + 380 * q^9 + 424 * q^11 - 1848 * q^13 - 1784 * q^15 + 2346 * q^17 - 360 * q^19 - 1526 * q^21 - 12 * q^23 - 1872 * q^25 - 5744 * q^27 - 14104 * q^29 + 3548 * q^31 + 3398 * q^33 - 27496 * q^35 - 11090 * q^37 + 1624 * q^39 + 7000 * q^41 + 25360 * q^43 - 1300 * q^45 - 22956 * q^47 + 4900 * q^49 - 384 * q^51 - 3042 * q^53 + 50152 * q^55 - 38116 * q^57 - 65808 * q^59 + 42486 * q^61 + 4760 * q^63 + 3164 * q^65 + 42312 * q^67 + 10308 * q^69 + 4416 * q^71 + 50506 * q^73 - 35608 * q^75 + 65338 * q^77 + 9004 * q^79 - 51178 * q^81 + 208656 * q^83 - 106212 * q^85 + 80008 * q^87 + 26666 * q^89 - 135632 * q^91 - 38718 * q^93 - 198140 * q^95 + 418264 * q^97 + 133888 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	−5.04138	−	8.73193i	−0.323405	−	0.560153i	0.657783	−	0.753207i	$-0.271494\pi$
$3$	−5.04138	−	8.73193i	−0.323405	−	0.560153i	−0.981188	+	0.193054i	$0.938161\pi$
$4$		0			0
$5$	39.9138	−	69.1328i	0.714000	−	1.23668i	−0.249344	−	0.968415i	$-0.580215\pi$
$5$	39.9138	−	69.1328i	0.714000	−	1.23668i	0.963344	−	0.268269i	$-0.0864517\pi$
$6$		0			0
$7$	−43.1587	−	122.247i	−0.332907	−	0.942960i
$7$	−43.1587	−	122.247i	−0.332907	−	0.942960i
$8$		0			0
$9$	70.6689	−	122.402i	0.290819	−	0.503713i
$10$		0			0
$11$	175.952	+	304.757i	0.438442	+	0.759403i	0.997570	−	0.0696781i	$-0.0221972\pi$
$11$	175.952	+	304.757i	0.438442	+	0.759403i	−0.559128	+	0.829082i	$0.688864\pi$
$12$		0			0
$13$	−291.683			−0.478688			−0.239344	−	0.970935i	$-0.576932\pi$
$13$	−291.683			−0.478688			−0.239344	+	0.970935i	$0.576932\pi$
$14$		0			0
$15$	−804.883			−0.923644
$16$		0			0
$17$	185.038	+	320.495i	0.155288	+	0.268967i	0.933164	−	0.359451i	$-0.117036\pi$
$17$	185.038	+	320.495i	0.155288	+	0.268967i	−0.777876	+	0.628418i	$0.783703\pi$
$18$		0			0
$19$	752.463	−	1303.30i	0.478190	−	0.828250i	−0.521497	−	0.853253i	$-0.674626\pi$
$19$	752.463	−	1303.30i	0.478190	−	0.828250i	0.999687	+	0.0250030i	$0.00795952\pi$
$20$		0			0
$21$	−849.873	+	993.152i	−0.420538	+	0.491437i
$22$		0			0
$23$	−212.855	+	368.676i	−0.0839006	+	0.145320i	−0.904922	−	0.425577i	$-0.860071\pi$
$23$	−212.855	+	368.676i	−0.0839006	+	0.145320i	0.821022	+	0.570897i	$0.193404\pi$
$24$		0			0
$25$	−1623.72	−	2812.37i	−0.519592	−	0.899960i
$26$		0			0
$27$	−3875.19			−1.02302
$28$		0			0
$29$	−7783.93			−1.71872			−0.859358	−	0.511374i	$-0.829137\pi$
$29$	−7783.93			−1.71872			−0.859358	+	0.511374i	$0.829137\pi$
$30$		0			0
$31$	−1287.59	−	2230.17i	−0.240643	−	0.416805i	0.720255	−	0.693710i	$-0.244025\pi$
$31$	−1287.59	−	2230.17i	−0.240643	−	0.416805i	−0.960898	+	0.276904i	$0.910692\pi$
$32$		0			0
$33$	1774.08	−	3072.80i	0.283588	−	0.491189i
$34$		0			0
$35$	−10173.9	−	1895.67i	−1.40384	−	0.261572i
$36$		0			0
$37$	−369.809	+	640.528i	−0.0444092	+	0.0769190i	−0.887376	−	0.461047i	$-0.847474\pi$
$37$	−369.809	+	640.528i	−0.0444092	+	0.0769190i	0.842966	+	0.537966i	$0.180807\pi$
$38$		0			0
$39$	1470.48	+	2546.95i	0.154810	+	0.268139i
$40$		0			0
$41$	7029.84			0.653109			0.326554	−	0.945178i	$-0.394112\pi$
$41$	7029.84			0.653109			0.326554	+	0.945178i	$0.394112\pi$
$42$		0			0
$43$	−1835.23			−0.151363			−0.0756816	−	0.997132i	$-0.524113\pi$
$43$	−1835.23			−0.151363			−0.0756816	+	0.997132i	$0.524113\pi$
$44$		0			0
$45$	−5641.33	−	9771.08i	−0.415289	−	0.719302i
$46$		0			0
$47$	−766.342	+	1327.34i	−0.0506032	+	0.0876473i	−0.890217	−	0.455536i	$-0.849448\pi$
$47$	−766.342	+	1327.34i	−0.0506032	+	0.0876473i	0.839614	+	0.543183i	$0.182781\pi$
$48$		0			0
$49$	−13081.7	+	10552.0i	−0.778346	+	0.627836i
$50$		0			0
$51$	1865.69	−	3231.47i	0.100442	−	0.173970i
$52$		0			0
$53$	4768.73	+	8259.68i	0.233192	+	0.403900i	0.958746	−	0.284265i	$-0.0917497\pi$
$53$	4768.73	+	8259.68i	0.233192	+	0.403900i	−0.725554	+	0.688165i	$0.758416\pi$
$54$		0			0
$55$	28091.6			1.25219
$56$		0			0
$57$	−15173.8			−0.618596
$58$		0			0
$59$	−14837.0	−	25698.5i	−0.554903	−	0.961120i	−0.997911	−	0.0646022i	$-0.979422\pi$
$59$	−14837.0	−	25698.5i	−0.554903	−	0.961120i	0.443008	−	0.896517i	$-0.353911\pi$
$60$		0			0
$61$	23255.4	−	40279.5i	0.800201	−	1.38599i	−0.119282	−	0.992860i	$-0.538059\pi$
$61$	23255.4	−	40279.5i	0.800201	−	1.38599i	0.919483	−	0.393129i	$-0.128607\pi$
$62$		0			0
$63$	−18013.3	−	3356.35i	−0.571796	−	0.106541i
$64$		0			0
$65$	−11642.2	+	20164.8i	−0.341783	+	0.591985i
$66$		0			0
$67$	13373.0	+	23162.8i	0.363951	+	0.630381i	0.988607	−	0.150518i	$-0.0480943\pi$
$67$	13373.0	+	23162.8i	0.363951	+	0.630381i	−0.624656	+	0.780900i	$0.714761\pi$
$68$		0			0
$69$	4292.34			0.108535
$70$		0			0
$71$	14388.8			0.338748			0.169374	−	0.985552i	$-0.445825\pi$
$71$	14388.8			0.338748			0.169374	+	0.985552i	$0.445825\pi$
$72$		0			0
$73$	35047.6	+	60704.2i	0.769752	+	1.33325i	0.937697	+	0.347453i	$0.112953\pi$
$73$	35047.6	+	60704.2i	0.769752	+	1.33325i	−0.167946	+	0.985796i	$0.553713\pi$
$74$		0			0
$75$	−16371.6	+	28356.5i	−0.336077	+	0.582102i
$76$		0			0
$77$	29661.8	−	34662.5i	0.570126	−	0.666244i
$78$		0			0
$79$	−13542.9	+	23457.0i	−0.244143	+	0.422868i	−0.961890	−	0.273436i	$-0.911840\pi$
$79$	−13542.9	+	23457.0i	−0.244143	+	0.422868i	0.717748	+	0.696303i	$0.245173\pi$
$80$		0			0
$81$	2363.74	+	4094.13i	0.0400302	+	0.0693344i
$82$		0			0
$83$	79755.4			1.27076			0.635382	−	0.772198i	$-0.280843\pi$
$83$	79755.4			1.27076			0.635382	+	0.772198i	$0.280843\pi$
$84$		0			0
$85$	29542.2			0.443502
$86$		0			0
$87$	39241.8	+	67968.8i	0.555841	+	0.962745i
$88$		0			0
$89$	−21788.7	+	37739.1i	−0.291579	+	0.505029i	−0.974183	−	0.225759i	$-0.927514\pi$
$89$	−21788.7	+	37739.1i	−0.291579	+	0.505029i	0.682605	+	0.730788i	$0.260847\pi$
$90$		0			0
$91$	12588.6	+	35657.3i	0.159358	+	0.451383i
$92$		0			0
$93$	−12982.4	+	22486.2i	−0.155650	+	0.269594i
$94$		0			0
$95$	−60067.3	−	104040.i	−0.682856	−	1.18274i
$96$		0			0
$97$	103374.			1.11553			0.557765	−	0.829999i	$-0.311659\pi$
$97$	103374.			1.11553			0.557765	+	0.829999i	$0.311659\pi$
$98$		0			0
$99$	49737.3			0.510028
$100$		0			0
$101$	−14350.1	−	24855.1i	−0.139975	−	0.242444i	0.787512	−	0.616300i	$-0.211369\pi$
$101$	−14350.1	−	24855.1i	−0.139975	−	0.242444i	−0.927487	+	0.373855i	$0.878036\pi$
$102$		0			0
$103$	14614.0	−	25312.1i	0.135730	−	0.235091i	−0.790146	−	0.612918i	$-0.789995\pi$
$103$	14614.0	−	25312.1i	0.135730	−	0.235091i	0.925876	+	0.377828i	$0.123329\pi$
$104$		0			0
$105$	34737.7	+	98394.5i	0.307488	+	0.870959i
$106$		0			0
$107$	43929.2	−	76087.6i	0.370932	−	0.642472i	−0.618778	−	0.785566i	$-0.712372\pi$
$107$	43929.2	−	76087.6i	0.370932	−	0.642472i	0.989709	+	0.143094i	$0.0457051\pi$
$108$		0			0
$109$	−110314.	−	191069.i	−0.889333	−	1.54037i	−0.840665	−	0.541555i	$-0.817836\pi$
$109$	−110314.	−	191069.i	−0.889333	−	1.54037i	−0.0486678	−	0.998815i	$-0.515498\pi$
$110$		0			0
$111$	7457.39			0.0574486
$112$		0			0
$113$	39665.6			0.292225			0.146113	−	0.989268i	$-0.453324\pi$
$113$	39665.6			0.292225			0.146113	+	0.989268i	$0.453324\pi$
$114$		0			0
$115$	16991.7	+	29430.5i	0.119810	+	0.207517i
$116$		0			0
$117$	−20612.9	+	35702.6i	−0.139211	+	0.241121i
$118$		0			0
$119$	31193.5	−	36452.4i	0.201928	−	0.235971i
$120$		0			0
$121$	18607.4	−	32229.0i	0.115538	−	0.200117i
$122$		0			0
$123$	−35440.1	−	61384.0i	−0.211219	−	0.365841i
$124$		0			0
$125$	−9774.87			−0.0559546
$126$		0			0
$127$	−51740.3			−0.284655			−0.142328	−	0.989820i	$-0.545459\pi$
$127$	−51740.3			−0.284655			−0.142328	+	0.989820i	$0.545459\pi$
$128$		0			0
$129$	9252.11	+	16025.1i	0.0489515	+	0.0847866i
$130$		0			0
$131$	−83336.9	+	144344.i	−0.424286	+	0.734885i	−0.996353	−	0.0853215i	$-0.972808\pi$
$131$	−83336.9	+	144344.i	−0.424286	+	0.734885i	0.572067	+	0.820207i	$0.306142\pi$
$132$		0			0
$133$	−191800.	−	35737.4i	−0.940200	−	0.175184i
$134$		0			0
$135$	−154674.	+	267902.i	−0.730435	+	1.26515i
$136$		0			0
$137$	−14129.7	−	24473.4i	−0.0643178	−	0.111402i	0.832073	−	0.554666i	$-0.187154\pi$
$137$	−14129.7	−	24473.4i	−0.0643178	−	0.111402i	−0.896391	+	0.443264i	$0.853820\pi$
$138$		0			0
$139$	−336393.			−1.47676			−0.738380	−	0.674384i	$-0.764409\pi$
$139$	−336393.			−1.47676			−0.738380	+	0.674384i	$0.764409\pi$
$140$		0			0
$141$	15453.7			0.0654612
$142$		0			0
$143$	−51322.1	−	88892.4i	−0.209877	−	0.363517i
$144$		0			0
$145$	−310686.	+	538125.i	−1.22716	+	2.12551i
$146$		0			0
$147$	158089.	+	61031.3i	0.603405	+	0.232948i
$148$		0			0
$149$	177691.	−	307769.i	0.655691	−	1.13569i	−0.326030	−	0.945360i	$-0.605711\pi$
$149$	177691.	−	307769.i	0.655691	−	1.13569i	0.981720	−	0.190330i	$-0.0609557\pi$
$150$		0			0
$151$	−179399.	−	310727.i	−0.640290	−	1.10901i	−0.985368	−	0.170441i	$-0.945481\pi$
$151$	−179399.	−	310727.i	−0.640290	−	1.10901i	0.345078	−	0.938574i	$-0.387852\pi$
$152$		0			0
$153$	52305.7			0.180643
$154$		0			0
$155$	−205570.			−0.687275
$156$		0			0
$157$	−229455.	−	397428.i	−0.742932	−	1.28680i	−0.951155	−	0.308714i	$-0.900101\pi$
$157$	−229455.	−	397428.i	−0.742932	−	1.28680i	0.208223	−	0.978081i	$-0.433232\pi$
$158$		0			0
$159$	48082.0	−	83280.4i	0.150831	−	0.261246i
$160$		0			0
$161$	54256.1	+	10109.3i	0.164962	+	0.0307368i
$162$		0			0
$163$	251220.	−	435126.i	0.740603	−	1.28276i	−0.211618	−	0.977353i	$-0.567873\pi$
$163$	251220.	−	435126.i	0.740603	−	1.28276i	0.952221	−	0.305410i	$-0.0987935\pi$
$164$		0			0
$165$	−141621.	−	245294.i	−0.404964	−	0.701418i
$166$		0			0
$167$	676652.			1.87748			0.938738	−	0.344632i	$-0.111996\pi$
$167$	676652.			1.87748			0.938738	+	0.344632i	$0.111996\pi$
$168$		0			0
$169$	−286214.			−0.770858
$170$		0			0
$171$	−106351.	−	184206.i	−0.278133	−	0.481741i
$172$		0			0
$173$	124580.	−	215779.i	0.316470	−	0.548143i	−0.663279	−	0.748373i	$-0.730836\pi$
$173$	124580.	−	215779.i	0.316470	−	0.548143i	0.979749	+	0.200230i	$0.0641689\pi$
$174$		0			0
$175$	−273726.	+	319874.i	−0.675650	+	0.789557i
$176$		0			0
$177$	−149598.	+	259112.i	−0.358916	+	0.621661i
$178$		0			0
$179$	69628.9	+	120601.i	0.162427	+	0.281331i	0.935738	−	0.352695i	$-0.114735\pi$
$179$	69628.9	+	120601.i	0.162427	+	0.281331i	−0.773312	+	0.634026i	$0.781401\pi$
$180$		0			0
$181$	306246.			0.694823			0.347412	−	0.937713i	$-0.387061\pi$
$181$	306246.			0.694823			0.347412	+	0.937713i	$0.387061\pi$
$182$		0			0
$183$	−468957.			−1.03516
$184$		0			0
$185$	29521.0	+	51131.8i	0.0634163	+	0.109840i
$186$		0			0
$187$	−65115.4	+	112783.i	−0.136169	+	0.235852i
$188$		0			0
$189$	167248.	+	473730.i	0.340570	+	0.964665i
$190$		0			0
$191$	113747.	−	197015.i	0.225609	−	0.390766i	−0.730893	−	0.682492i	$-0.760896\pi$
$191$	113747.	−	197015.i	0.225609	−	0.390766i	0.956502	+	0.291726i	$0.0942295\pi$
$192$		0			0
$193$	336187.	+	582293.i	0.649663	+	1.12525i	0.983203	+	0.182513i	$0.0584231\pi$
$193$	336187.	+	582293.i	0.649663	+	1.12525i	−0.333541	+	0.942736i	$0.608244\pi$
$194$		0			0
$195$	234770.			0.442137
$196$		0			0
$197$	−1282.76			−0.00235493			−0.00117747	−	0.999999i	$-0.500375\pi$
$197$	−1282.76			−0.00235493			−0.00117747	+	0.999999i	$0.500375\pi$
$198$		0			0
$199$	184449.	+	319475.i	0.330175	+	0.571879i	0.982546	−	0.186020i	$-0.0595591\pi$
$199$	184449.	+	319475.i	0.330175	+	0.571879i	−0.652371	+	0.757900i	$0.726226\pi$
$200$		0			0
$201$	134837.	−	233545.i	0.235407	−	0.407737i
$202$		0			0
$203$	335944.	+	951563.i	0.572173	+	1.62068i
$204$		0			0
$205$	280588.	−	485992.i	0.466320	−	0.807690i
$206$		0			0
$207$	30084.5	+	52107.9i	0.0487997	+	0.0845236i
$208$		0			0
$209$	529589.			0.838635
$210$		0			0
$211$	−502168.			−0.776503			−0.388251	−	0.921553i	$-0.626921\pi$
$211$	−502168.			−0.776503			−0.388251	+	0.921553i	$0.626921\pi$
$212$		0			0
$213$	−72539.2	−	125642.i	−0.109553	−	0.189751i
$214$		0			0
$215$	−73251.1	+	126875.i	−0.108073	+	0.187188i
$216$		0			0
$217$	−217061.	+	253655.i	−0.312919	+	0.365674i
$218$		0			0
$219$	353376.	−	612066.i	0.497883	−	0.862358i
$220$		0			0
$221$	−53972.3	−	93482.7i	−0.0743344	−	0.128751i
$222$		0			0
$223$	1.17328e6			1.57993			0.789967	−	0.613149i	$-0.210098\pi$
$223$	1.17328e6			1.57993			0.789967	+	0.613149i	$0.210098\pi$
$224$		0			0
$225$	−458988.			−0.604428
$226$		0			0
$227$	−455079.	−	788220.i	−0.586168	−	1.01527i	−0.994729	−	0.102542i	$-0.967302\pi$
$227$	−455079.	−	788220.i	−0.586168	−	1.01527i	0.408560	−	0.912731i	$-0.366031\pi$
$228$		0			0
$229$	−260962.	+	452000.i	−0.328843	+	0.569573i	−0.982283	−	0.187406i	$-0.939992\pi$
$229$	−260962.	+	452000.i	−0.328843	+	0.569573i	0.653439	+	0.756979i	$0.273325\pi$
$230$		0			0
$231$	−452207.	−	84258.1i	−0.557580	−	0.103892i
$232$		0			0
$233$	521394.	−	903081.i	0.629182	−	1.08977i	−0.358535	−	0.933516i	$-0.616723\pi$
$233$	521394.	−	903081.i	0.629182	−	1.08977i	0.987716	−	0.156258i	$-0.0499432\pi$
$234$		0			0
$235$	61175.2	+	105959.i	0.0722613	+	0.125160i
$236$		0			0
$237$	273100.			0.315828
$238$		0			0
$239$	1.53447e6			1.73766			0.868830	−	0.495110i	$-0.164872\pi$
$239$	1.53447e6			1.73766			0.868830	+	0.495110i	$0.164872\pi$
$240$		0			0
$241$	503789.	+	872588.i	0.558735	+	0.967758i	0.997602	+	0.0692059i	$0.0220465\pi$
$241$	503789.	+	872588.i	0.558735	+	0.967758i	−0.438867	+	0.898552i	$0.644620\pi$
$242$		0			0
$243$	−447002.	+	774231.i	−0.485617	+	0.841114i
$244$		0			0
$245$	207353.	+	1.32554e6i	0.220696	+	1.41084i
$246$		0			0
$247$	−219480.	+	380151.i	−0.228904	+	0.396473i
$248$		0			0
$249$	−402077.	−	696419.i	−0.410971	−	0.711823i
$250$		0			0
$251$	−8511.89			−0.00852789			−0.00426394	−	0.999991i	$-0.501357\pi$
$251$	−8511.89			−0.00852789			−0.00426394	+	0.999991i	$0.501357\pi$
$252$		0			0
$253$	−149809.			−0.147142
$254$		0			0
$255$	−148934.	−	257961.i	−0.143431	−	0.248429i
$256$		0			0
$257$	263766.	−	456856.i	0.249107	−	0.431466i	−0.714171	−	0.699971i	$-0.753196\pi$
$257$	263766.	−	456856.i	0.249107	−	0.431466i	0.963278	+	0.268505i	$0.0865295\pi$
$258$		0			0
$259$	94263.0	+	17563.7i	0.0873156	+	0.0162692i
$260$		0			0
$261$	−550082.	+	952771.i	−0.499835	+	0.865739i
$262$		0			0
$263$	−176042.	−	304914.i	−0.156938	−	0.271824i	0.776825	−	0.629716i	$-0.216829\pi$
$263$	−176042.	−	304914.i	−0.156938	−	0.271824i	−0.933763	+	0.357892i	$0.883496\pi$
$264$		0			0
$265$	761353.			0.665996
$266$		0			0
$267$	439380.			0.377192
$268$		0			0
$269$	−239770.	−	415294.i	−0.202029	−	0.349925i	0.747153	−	0.664652i	$-0.231420\pi$
$269$	−239770.	−	415294.i	−0.202029	−	0.349925i	−0.949182	+	0.314727i	$0.898087\pi$
$270$		0			0
$271$	−488805.	+	846636.i	−0.404308	+	0.700283i	−0.994241	−	0.107170i	$-0.965821\pi$
$271$	−488805.	+	846636.i	−0.404308	+	0.700283i	0.589932	+	0.807453i	$0.299154\pi$
$272$		0			0
$273$	247893.	−	289685.i	0.201307	−	0.235245i
$274$		0			0
$275$	571395.	−	989684.i	0.455622	−	0.789160i
$276$		0			0
$277$	−484362.	−	838939.i	−0.379289	−	0.656948i	0.611670	−	0.791113i	$-0.290498\pi$
$277$	−484362.	−	838939.i	−0.379289	−	0.656948i	−0.990959	+	0.134165i	$0.957165\pi$
$278$		0			0
$279$	−363970.			−0.279934
$280$		0			0
$281$	−318333.			−0.240501			−0.120250	−	0.992744i	$-0.538370\pi$
$281$	−318333.			−0.240501			−0.120250	+	0.992744i	$0.538370\pi$
$282$		0			0
$283$	886051.	+	1.53468e6i	0.657646	+	1.13908i	0.981223	+	0.192875i	$0.0617812\pi$
$283$	886051.	+	1.53468e6i	0.657646	+	1.13908i	−0.323577	+	0.946202i	$0.604885\pi$
$284$		0			0
$285$	−605644.	+	1.04901e6i	−0.441678	+	0.765008i
$286$		0			0
$287$	−303398.	−	859377.i	−0.217425	−	0.615855i
$288$		0			0
$289$	641451.	−	1.11103e6i	0.451771	−	0.782491i
$290$		0			0
$291$	−521147.	−	902652.i	−0.360768	−	0.624868i
$292$		0			0
$293$	1.64148e6			1.11703			0.558516	−	0.829494i	$-0.311371\pi$
$293$	1.64148e6			1.11703			0.558516	+	0.829494i	$0.311371\pi$
$294$		0			0
$295$	−2.36881e6			−1.58480
$296$		0			0
$297$	−681846.	−	1.18099e6i	−0.448534	−	0.776883i
$298$		0			0
$299$	62086.2	−	107536.i	0.0401622	−	0.0695629i
$300$		0			0
$301$	79206.2	+	224352.i	0.0503898	+	0.142729i
$302$		0			0
$303$	−144689.	+	250608.i	−0.0905374	+	0.156815i
$304$		0			0
$305$	−1.85642e6	−	3.21542e6i	−1.14269	−	1.97919i
$306$		0			0
$307$	466930.			0.282752			0.141376	−	0.989956i	$-0.454847\pi$
$307$	466930.			0.282752			0.141376	+	0.989956i	$0.454847\pi$
$308$		0			0
$309$	−294698.			−0.175582
$310$		0			0
$311$	−1.21898e6	−	2.11134e6i	−0.714654	−	1.23782i	−0.963093	−	0.269169i	$-0.913251\pi$
$311$	−1.21898e6	−	2.11134e6i	−0.714654	−	1.23782i	0.248439	−	0.968647i	$-0.420082\pi$
$312$		0			0
$313$	−1.21047e6	+	2.09659e6i	−0.698381	+	1.20963i	0.270646	+	0.962679i	$0.412763\pi$
$313$	−1.21047e6	+	2.09659e6i	−0.698381	+	1.20963i	−0.969028	+	0.246953i	$0.920571\pi$
$314$		0			0
$315$	−951012.	+	1.11134e6i	−0.540020	+	0.631062i
$316$		0			0
$317$	−938057.	+	1.62476e6i	−0.524301	+	0.908116i	0.475298	+	0.879825i	$0.342340\pi$
$317$	−938057.	+	1.62476e6i	−0.524301	+	0.908116i	−0.999600	+	0.0282918i	$0.990993\pi$
$318$		0			0
$319$	−1.36960e6	−	2.37221e6i	−0.753557	−	1.30520i
$320$		0			0
$321$	−885855.			−0.479844
$322$		0			0
$323$	556936.			0.297029
$324$		0			0
$325$	473612.	+	820321.i	0.248722	+	0.430800i
$326$		0			0
$327$	−1.11227e6	+	1.92651e6i	−0.575229	+	0.996326i
$328$		0			0
$329$	195338.	+	36396.6i	0.0994940	+	0.0185384i
$330$		0			0
$331$	−541549.	+	937990.i	−0.271686	+	0.470575i	−0.969294	−	0.245906i	$-0.920915\pi$
$331$	−541549.	+	937990.i	−0.271686	+	0.470575i	0.697607	+	0.716480i	$0.254248\pi$
$332$		0			0
$333$	52268.0	+	90530.8i	0.0258301	+	0.0447390i
$334$		0			0
$335$	2.13507e6			1.03944
$336$		0			0
$337$	−2.59465e6			−1.24453			−0.622263	−	0.782809i	$-0.713786\pi$
$337$	−2.59465e6			−1.24453			−0.622263	+	0.782809i	$0.713786\pi$
$338$		0			0
$339$	−199969.	−	346357.i	−0.0945071	−	0.163691i
$340$		0			0
$341$	453107.	−	784804.i	0.211016	−	0.365490i
$342$		0			0
$343$	1.85454e6	+	1.14378e6i	0.851141	+	0.524938i
$344$		0			0
$345$	171324.	−	296741.i	0.0774943	−	0.134224i
$346$		0			0
$347$	−935255.	−	1.61991e6i	−0.416972	−	0.722216i	0.578662	−	0.815568i	$-0.303575\pi$
$347$	−935255.	−	1.61991e6i	−0.416972	−	0.722216i	−0.995633	+	0.0933518i	$0.970242\pi$
$348$		0			0
$349$	−1.61685e6			−0.710568			−0.355284	−	0.934758i	$-0.615616\pi$
$349$	−1.61685e6			−0.710568			−0.355284	+	0.934758i	$0.615616\pi$
$350$		0			0
$351$	1.13033e6			0.489706
$352$		0			0
$353$	289153.	+	500827.i	0.123507	+	0.213920i	0.921148	−	0.389212i	$-0.127253\pi$
$353$	289153.	+	500827.i	0.123507	+	0.213920i	−0.797642	+	0.603132i	$0.793919\pi$
$354$		0			0
$355$	574310.	−	994734.i	0.241866	−	0.418925i
$356$		0			0
$357$	−475558.	−	88609.1i	−0.197485	−	0.0367966i
$358$		0			0
$359$	−984090.	+	1.70449e6i	−0.402994	+	0.698006i	−0.994086	−	0.108598i	$-0.965364\pi$
$359$	−984090.	+	1.70449e6i	−0.402994	+	0.698006i	0.591092	+	0.806604i	$0.298697\pi$
$360$		0			0
$361$	105650.	+	182990.i	0.0426677	+	0.0739027i
$362$		0			0
$363$	−375229.			−0.149462
$364$		0			0
$365$	5.59553e6			2.19841
$366$		0			0
$367$	1.08726e6	+	1.88319e6i	0.421375	+	0.729842i	0.996074	−	0.0885223i	$-0.0282144\pi$
$367$	1.08726e6	+	1.88319e6i	0.421375	+	0.729842i	−0.574700	+	0.818364i	$0.694881\pi$
$368$		0			0
$369$	496791.	−	860468.i	0.189936	−	0.328979i
$370$		0			0
$371$	803910.	−	939440.i	0.303230	−	0.354352i
$372$		0			0
$373$	−692379.	+	1.19924e6i	−0.257675	+	0.446306i	−0.965619	−	0.259963i	$-0.916290\pi$
$373$	−692379.	+	1.19924e6i	−0.257675	+	0.446306i	0.707944	+	0.706269i	$0.249623\pi$
$374$		0			0
$375$	49278.9	+	85353.5i	0.0180960	+	0.0313432i
$376$		0			0
$377$	2.27044e6			0.822728
$378$		0			0
$379$	−3.37190e6			−1.20580			−0.602902	−	0.797815i	$-0.705989\pi$
$379$	−3.37190e6			−1.20580			−0.602902	+	0.797815i	$0.705989\pi$
$380$		0			0
$381$	260842.	+	451792.i	0.0920589	+	0.159451i
$382$		0			0
$383$	1.64030e6	−	2.84108e6i	0.571382	−	0.989662i	−0.425043	−	0.905173i	$-0.639741\pi$
$383$	1.64030e6	−	2.84108e6i	0.571382	−	0.989662i	0.996424	−	0.0844886i	$-0.0269257\pi$
$384$		0			0
$385$	−1.21240e6	−	3.43412e6i	−0.416863	−	1.18076i
$386$		0			0
$387$	−129694.	+	224637.i	−0.0440192	+	0.0762435i
$388$		0			0
$389$	1.47405e6	+	2.55313e6i	0.493899	+	0.855457i	0.999975	−	0.00703108i	$-0.00223808\pi$
$389$	1.47405e6	+	2.55313e6i	0.493899	+	0.855457i	−0.506077	+	0.862488i	$0.668905\pi$
$390$		0			0
$391$	−157545.			−0.0521150
$392$		0			0
$393$	1.68053e6			0.548865
$394$		0			0
$395$	1.08110e6	+	1.87251e6i	0.348636	+	0.603855i
$396$		0			0
$397$	34635.1	−	59989.7i	0.0110291	−	0.0191030i	−0.860458	−	0.509521i	$-0.829823\pi$
$397$	34635.1	−	59989.7i	0.0110291	−	0.0191030i	0.871487	+	0.490418i	$0.163156\pi$
$398$		0			0
$399$	654881.	+	1.85495e6i	0.205935	+	0.583311i
$400$		0			0
$401$	1.67393e6	−	2.89933e6i	0.519848	−	0.900404i	−0.479886	−	0.877331i	$-0.659322\pi$
$401$	1.67393e6	−	2.89933e6i	0.519848	−	0.900404i	0.999734	−	0.0230725i	$-0.00734485\pi$
$402$		0			0
$403$	375567.	+	650501.i	0.115193	+	0.199520i
$404$		0			0
$405$	377384.			0.114326
$406$		0			0
$407$	−260274.			−0.0778834
$408$		0			0
$409$	−1.45608e6	−	2.52201e6i	−0.430406	−	0.745485i	0.566502	−	0.824060i	$-0.308296\pi$
$409$	−1.45608e6	−	2.52201e6i	−0.430406	−	0.745485i	−0.996908	+	0.0785754i	$0.974963\pi$
$410$		0			0
$411$	−142466.	+	246759.i	−0.0416014	+	0.0720557i
$412$		0			0
$413$	−2.50122e6	+	2.92289e6i	−0.721566	+	0.843214i
$414$		0			0
$415$	3.18334e6	−	5.51371e6i	0.907326	−	1.57153i
$416$		0			0
$417$	1.69589e6	+	2.93736e6i	0.477592	+	0.827213i
$418$		0			0
$419$	4.62361e6			1.28661			0.643304	−	0.765611i	$-0.277563\pi$
$419$	4.62361e6			1.28661			0.643304	+	0.765611i	$0.277563\pi$
$420$		0			0
$421$	−2.63042e6			−0.723303			−0.361652	−	0.932313i	$-0.617787\pi$
$421$	−2.63042e6			−0.723303			−0.361652	+	0.932313i	$0.617787\pi$
$422$		0			0
$423$	108313.	+	187604.i	0.0294327	+	0.0509789i
$424$		0			0
$425$	600901.	−	1.04079e6i	0.161373	−	0.279506i
$426$		0			0
$427$	−5.92772e6	−	1.10449e6i	−1.57332	−	0.293152i
$428$		0			0
$429$	−517468.	+	896281.i	−0.135750	+	0.235126i
$430$		0			0
$431$	3.77064e6	+	6.53094e6i	0.977736	+	1.69349i	0.670592	+	0.741826i	$0.266040\pi$
$431$	3.77064e6	+	6.53094e6i	0.977736	+	1.69349i	0.307144	+	0.951663i	$0.400627\pi$
$432$		0			0
$433$	5.83558e6			1.49577			0.747883	−	0.663830i	$-0.231070\pi$
$433$	5.83558e6			1.49577			0.747883	+	0.663830i	$0.231070\pi$
$434$		0			0
$435$	6.26516e6			1.58748
$436$		0			0
$437$	320331.	+	554830.i	0.0802409	+	0.138981i
$438$		0			0
$439$	84051.9	−	145582.i	0.0208155	−	0.0360535i	−0.855430	−	0.517918i	$-0.826707\pi$
$439$	84051.9	−	145582.i	0.0208155	−	0.0360535i	0.876246	+	0.481865i	$0.160040\pi$
$440$		0			0
$441$	367126.	+	2.34693e6i	0.0898914	+	0.574649i
$442$		0			0
$443$	−1.42076e6	+	2.46082e6i	−0.343962	+	0.595760i	−0.985165	−	0.171612i	$-0.945102\pi$
$443$	−1.42076e6	+	2.46082e6i	−0.343962	+	0.595760i	0.641203	+	0.767372i	$0.278436\pi$
$444$		0			0
$445$	1.73934e6	+	3.01262e6i	0.416374	+	0.721181i
$446$		0			0
$447$	−3.58323e6			−0.848214
$448$		0			0
$449$	−1.41567e6			−0.331396			−0.165698	−	0.986177i	$-0.552988\pi$
$449$	−1.41567e6			−0.331396			−0.165698	+	0.986177i	$0.552988\pi$
$450$		0			0
$451$	1.23691e6	+	2.14240e6i	0.286350	+	0.495973i
$452$		0			0
$453$	−1.80883e6	+	3.13299e6i	−0.414146	+	0.717321i
$454$		0			0
$455$	2.96755e6	+	552933.i	0.672000	+	0.125211i
$456$		0			0
$457$	−778637.	+	1.34864e6i	−0.174399	+	0.302068i	−0.939953	−	0.341303i	$-0.889132\pi$
$457$	−778637.	+	1.34864e6i	−0.174399	+	0.302068i	0.765554	+	0.643372i	$0.222465\pi$
$458$		0			0
$459$	−717056.	−	1.24198e6i	−0.158862	−	0.275158i
$460$		0			0
$461$	−4.45345e6			−0.975987			−0.487994	−	0.872847i	$-0.662271\pi$
$461$	−4.45345e6			−0.975987			−0.487994	+	0.872847i	$0.662271\pi$
$462$		0			0
$463$	−4.92263e6			−1.06720			−0.533599	−	0.845738i	$-0.679161\pi$
$463$	−4.92263e6			−1.06720			−0.533599	+	0.845738i	$0.679161\pi$
$464$		0			0
$465$	1.03636e6	+	1.79502e6i	0.222268	+	0.384980i
$466$		0			0
$467$	2.54545e6	−	4.40885e6i	0.540098	−	0.935477i	−0.458800	−	0.888540i	$-0.651720\pi$
$467$	2.54545e6	−	4.40885e6i	0.540098	−	0.935477i	0.998898	−	0.0469376i	$-0.0149462\pi$
$468$		0			0
$469$	2.25442e6	−	2.63449e6i	0.473262	−	0.553049i
$470$		0			0
$471$	−2.31354e6	+	4.00717e6i	−0.480535	+	0.832312i
$472$		0			0
$473$	−322912.	−	559301.i	−0.0663639	−	0.114946i
$474$		0			0
$475$	−4.88717e6			−0.993856
$476$		0			0
$477$	1.34800e6			0.271266
$478$		0			0
$479$	−4.15042e6	−	7.18874e6i	−0.826521	−	1.43158i	−0.900752	−	0.434334i	$-0.856984\pi$
$479$	−4.15042e6	−	7.18874e6i	−0.826521	−	1.43158i	0.0742312	−	0.997241i	$-0.476350\pi$
$480$		0			0
$481$	107867.	−	186831.i	0.0212581	−	0.0368202i
$482$		0			0
$483$	−185252.	−	524726.i	−0.0361322	−	0.102344i
$484$		0			0
$485$	4.12604e6	−	7.14651e6i	0.796488	−	1.37956i
$486$		0			0
$487$	4.31701e6	+	7.47727e6i	0.824822	+	1.42863i	0.902055	+	0.431621i	$0.142058\pi$
$487$	4.31701e6	+	7.47727e6i	0.824822	+	1.42863i	−0.0772330	+	0.997013i	$0.524609\pi$
$488$		0			0
$489$	−5.06599e6			−0.958059
$490$		0			0
$491$	−95039.5			−0.0177910			−0.00889550	−	0.999960i	$-0.502832\pi$
$491$	−95039.5			−0.0177910			−0.00889550	+	0.999960i	$0.502832\pi$
$492$		0			0
$493$	−1.44032e6	−	2.49471e6i	−0.266896	−	0.462277i
$494$		0			0
$495$	1.98521e6	−	3.43848e6i	0.364160	−	0.630744i
$496$		0			0
$497$	−621000.	−	1.75898e6i	−0.112772	−	0.319426i
$498$		0			0
$499$	1.07102e6	−	1.85506e6i	0.192551	−	0.333507i	−0.753544	−	0.657397i	$-0.771657\pi$
$499$	1.07102e6	−	1.85506e6i	0.192551	−	0.333507i	0.946095	+	0.323890i	$0.104991\pi$
$500$		0			0
$501$	−3.41126e6	−	5.90848e6i	−0.607185	−	1.05167i
$502$		0			0
$503$	−5.24794e6			−0.924844			−0.462422	−	0.886660i	$-0.653019\pi$
$503$	−5.24794e6			−0.924844			−0.462422	+	0.886660i	$0.653019\pi$
$504$		0			0
$505$	−2.29107e6			−0.399769
$506$		0			0
$507$	1.44292e6	+	2.49920e6i	0.249299	+	0.431799i
$508$		0			0
$509$	5.29453e6	−	9.17040e6i	0.905802	−	1.56889i	0.0859643	−	0.996298i	$-0.472603\pi$
$509$	5.29453e6	−	9.17040e6i	0.905802	−	1.56889i	0.819837	−	0.572596i	$-0.194064\pi$
$510$		0			0
$511$	5.90829e6	−	6.90437e6i	1.00094	−	1.16969i
$512$		0			0
$513$	−2.91593e6	+	5.05055e6i	−0.489198	+	0.847315i
$514$		0			0
$515$	−1.16660e6	−	2.02061e6i	−0.193822	−	0.335709i
$516$		0			0
$517$	−539357.			−0.0887462
$518$		0			0
$519$	−2.51222e6			−0.409392
$520$		0			0
$521$	2.27232e6	+	3.93578e6i	0.366755	+	0.635238i	0.989056	−	0.147540i	$-0.0471354\pi$
$521$	2.27232e6	+	3.93578e6i	0.366755	+	0.635238i	−0.622301	+	0.782778i	$0.713802\pi$
$522$		0			0
$523$	−2.63599e6	+	4.56566e6i	−0.421394	+	0.729877i	−0.996076	−	0.0885004i	$-0.971793\pi$
$523$	−2.63599e6	+	4.56566e6i	−0.421394	+	0.729877i	0.574682	+	0.818377i	$0.305126\pi$
$524$		0			0
$525$	4.17307e6	+	777554.i	0.660782	+	0.123121i
$526$		0			0
$527$	476504.	−	825330.i	0.0747378	−	0.129450i
$528$		0			0
$529$	3.12756e6	+	5.41709e6i	0.485921	+	0.841641i
$530$		0			0
$531$	−4.19407e6			−0.645504
$532$		0			0
$533$	−2.05048e6			−0.312635
$534$		0			0
$535$	−3.50676e6	−	6.07389e6i	−0.529690	−	0.917451i
$536$		0			0
$537$	702052.	−	1.21599e6i	0.105059	−	0.181968i
$538$		0			0
$539$	−5.51755e6	−	2.13008e6i	−0.818040	−	0.315809i
$540$		0			0
$541$	−2.96723e6	+	5.13939e6i	−0.435871	+	0.754950i	−0.997366	−	0.0725298i	$-0.976893\pi$
$541$	−2.96723e6	+	5.13939e6i	−0.435871	+	0.754950i	0.561496	+	0.827480i	$0.310226\pi$
$542$		0			0
$543$	−1.54390e6	−	2.67412e6i	−0.224709	−	0.389208i
$544$		0			0
$545$	−1.76122e7			−2.53994
$546$		0			0
$547$	8.82017e6			1.26040			0.630200	−	0.776433i	$-0.282973\pi$
$547$	8.82017e6			1.26040			0.630200	+	0.776433i	$0.282973\pi$
$548$		0			0
$549$	−3.28687e6	−	5.69302e6i	−0.465427	−	0.806143i
$550$		0			0
$551$	−5.85712e6	+	1.01448e7i	−0.821874	+	1.42353i
$552$		0			0
$553$	3.45204e6	+	643206.i	0.480024	+	0.0894411i
$554$		0			0
$555$	297653.	−	515550.i	0.0410183	−	0.0710458i
$556$		0			0
$557$	591120.	+	1.02385e6i	0.0807304	+	0.139829i	0.903564	−	0.428453i	$-0.140941\pi$
$557$	591120.	+	1.02385e6i	0.0807304	+	0.139829i	−0.822833	+	0.568283i	$0.807608\pi$
$558$		0			0
$559$	535306.			0.0724556
$560$		0			0
$561$	1.31309e6			0.176151
$562$		0			0
$563$	2.03870e6	+	3.53114e6i	0.271071	+	0.469509i	0.969136	−	0.246525i	$-0.0792888\pi$
$563$	2.03870e6	+	3.53114e6i	0.271071	+	0.469509i	−0.698065	+	0.716034i	$0.745955\pi$
$564$		0			0
$565$	1.58321e6	−	2.74219e6i	0.208649	−	0.361391i
$566$		0			0
$567$	398478.	−	465658.i	0.0520532	−	0.0608288i
$568$		0			0
$569$	−4.08807e6	+	7.08075e6i	−0.529344	+	0.916851i	0.470070	+	0.882629i	$0.344229\pi$
$569$	−4.08807e6	+	7.08075e6i	−0.529344	+	0.916851i	−0.999414	+	0.0342216i	$0.989105\pi$
$570$		0			0
$571$	1.65307e6	+	2.86321e6i	0.212179	+	0.367504i	0.952396	−	0.304863i	$-0.0986108\pi$
$571$	1.65307e6	+	2.86321e6i	0.212179	+	0.367504i	−0.740217	+	0.672368i	$0.765277\pi$
$572$		0			0
$573$	−2.29377e6			−0.291852
$574$		0			0
$575$	1.38247e6			0.174376
$576$		0			0
$577$	3.57497e6	+	6.19203e6i	0.447026	+	0.774272i	0.998191	−	0.0601245i	$-0.0191498\pi$
$577$	3.57497e6	+	6.19203e6i	0.447026	+	0.774272i	−0.551165	+	0.834396i	$0.685816\pi$
$578$		0			0
$579$	3.38970e6	−	5.87112e6i	0.420208	−	0.727821i
$580$		0			0
$581$	−3.44214e6	−	9.74986e6i	−0.423046	−	1.19828i
$582$		0			0
$583$	−1.67813e6	+	2.90661e6i	−0.204482	+	0.354173i
$584$		0			0
$585$	1.64548e6	+	2.85005e6i	0.198794	+	0.344321i
$586$		0			0
$587$	−9.69191e6			−1.16095			−0.580476	−	0.814277i	$-0.697133\pi$
$587$	−9.69191e6			−1.16095			−0.580476	+	0.814277i	$0.697133\pi$
$588$		0			0
$589$	−3.87545e6			−0.460292
$590$		0			0
$591$	6466.87	+	11200.9i	0.000761597	+	0.00131912i
$592$		0			0
$593$	−3.31980e6	+	5.75006e6i	−0.387682	+	0.671484i	−0.992137	−	0.125154i	$-0.960057\pi$
$593$	−3.31980e6	+	5.75006e6i	−0.387682	+	0.671484i	0.604456	+	0.796639i	$0.293391\pi$
$594$		0			0
$595$	−1.27500e6	−	3.61145e6i	−0.147645	−	0.418205i
$596$		0			0
$597$	1.85976e6	−	3.22119e6i	0.213560	−	0.369897i
$598$		0			0
$599$	−1.62096e6	−	2.80758e6i	−0.184588	−	0.319716i	0.758849	−	0.651266i	$-0.225762\pi$
$599$	−1.62096e6	−	2.80758e6i	−0.184588	−	0.319716i	−0.943438	+	0.331550i	$0.892428\pi$
$600$		0			0
$601$	−5.65076e6			−0.638147			−0.319074	−	0.947730i	$-0.603372\pi$
$601$	−5.65076e6			−0.638147			−0.319074	+	0.947730i	$0.603372\pi$
$602$		0			0
$603$	3.78023e6			0.423375
$604$		0			0
$605$	−1.48539e6	−	2.57277e6i	−0.164988	−	0.285767i
$606$		0			0
$607$	117837.	−	204100.i	0.0129811	−	0.0224839i	−0.859462	−	0.511200i	$-0.829201\pi$
$607$	117837.	−	204100.i	0.0129811	−	0.0224839i	0.872443	+	0.488716i	$0.162535\pi$
$608$		0			0
$609$	6.61535e6	−	7.73063e6i	0.722786	−	0.844640i
$610$		0			0
$611$	223529.	−	387163.i	0.0242231	−	0.0419557i
$612$		0			0
$613$	−394438.	−	683187.i	−0.0423963	−	0.0734325i	0.844049	−	0.536267i	$-0.180166\pi$
$613$	−394438.	−	683187.i	−0.0423963	−	0.0734325i	−0.886445	+	0.462834i	$0.846833\pi$
$614$		0			0
$615$	−5.65820e6			−0.603240
$616$		0			0
$617$	1.67739e7			1.77387			0.886935	−	0.461894i	$-0.152830\pi$
$617$	1.67739e7			1.77387			0.886935	+	0.461894i	$0.152830\pi$
$618$		0			0
$619$	4.11150e6	+	7.12132e6i	0.431294	+	0.747023i	0.996985	−	0.0775941i	$-0.0247238\pi$
$619$	4.11150e6	+	7.12132e6i	0.431294	+	0.747023i	−0.565691	+	0.824617i	$0.691390\pi$
$620$		0			0
$621$	824854.	−	1.42869e6i	0.0858318	−	0.148665i
$622$		0			0
$623$	5.55386e6	+	1.03483e6i	0.573290	+	0.106819i
$624$		0			0
$625$	4.68399e6	−	8.11290e6i	0.479640	−	0.830761i
$626$		0			0
$627$	−2.66986e6	−	4.62433e6i	−0.271218	−	0.469764i
$628$		0			0
$629$	−273714.			−0.0275849
$630$		0			0
$631$	5.94507e6			0.594406			0.297203	−	0.954814i	$-0.403946\pi$
$631$	5.94507e6			0.594406			0.297203	+	0.954814i	$0.403946\pi$
$632$		0			0
$633$	2.53162e6	+	4.38490e6i	0.251125	+	0.434961i
$634$		0			0
$635$	−2.06515e6	+	3.57695e6i	−0.203244	+	0.352029i
$636$		0			0
$637$	3.81569e6	−	3.07785e6i	0.372585	−	0.300537i
$638$		0			0
$639$	1.01684e6	−	1.76122e6i	0.0985144	−	0.170632i
$640$		0			0
$641$	5.33804e6	+	9.24576e6i	0.513141	+	0.888786i	0.999884	+	0.0152411i	$0.00485157\pi$
$641$	5.33804e6	+	9.24576e6i	0.513141	+	0.888786i	−0.486743	+	0.873545i	$0.661815\pi$
$642$		0			0
$643$	3.13159e6			0.298701			0.149351	−	0.988784i	$-0.452282\pi$
$643$	3.13159e6			0.298701			0.149351	+	0.988784i	$0.452282\pi$
$644$		0			0
$645$	1.47715e6			0.139806
$646$		0			0
$647$	−2.46728e6	−	4.27346e6i	−0.231717	−	0.401346i	0.726596	−	0.687065i	$-0.241101\pi$
$647$	−2.46728e6	−	4.27346e6i	−0.231717	−	0.401346i	−0.958314	+	0.285719i	$0.907768\pi$
$648$		0			0
$649$	5.22120e6	−	9.04339e6i	0.486585	−	0.842790i
$650$		0			0
$651$	3.30918e6	+	616588.i	0.306033	+	0.0570220i
$652$		0			0
$653$	−2.86112e6	+	4.95561e6i	−0.262575	+	0.454793i	−0.966925	−	0.255059i	$-0.917905\pi$
$653$	−2.86112e6	+	4.95561e6i	−0.262575	+	0.454793i	0.704351	+	0.709852i	$0.251238\pi$
$654$		0			0
$655$	6.65258e6	+	1.15226e7i	0.605881	+	1.04942i
$656$		0			0
$657$	9.90710e6			0.895433
$658$		0			0
$659$	−362477.			−0.0325137			−0.0162569	−	0.999868i	$-0.505175\pi$
$659$	−362477.			−0.0325137			−0.0162569	+	0.999868i	$0.505175\pi$
$660$		0			0
$661$	9.56053e6	+	1.65593e7i	0.851096	+	1.47414i	0.880220	+	0.474565i	$0.157395\pi$
$661$	9.56053e6	+	1.65593e7i	0.851096	+	1.47414i	−0.0291249	+	0.999576i	$0.509272\pi$
$662$		0			0
$663$	−544190.	+	942564.i	−0.0480802	+	0.0832774i
$664$		0			0
$665$	−1.01261e7	+	1.18333e7i	−0.887950	+	1.03765i
$666$		0			0
$667$	1.65685e6	−	2.86975e6i	0.144201	−	0.249764i
$668$		0			0
$669$	−5.91495e6	−	1.02450e7i	−0.510958	−	0.885006i
$670$		0			0
$671$	1.63673e7			1.40337
$672$		0			0
$673$	−573374.			−0.0487978			−0.0243989	−	0.999702i	$-0.507767\pi$
$673$	−573374.			−0.0487978			−0.0243989	+	0.999702i	$0.507767\pi$
$674$		0			0
$675$	6.29224e6	+	1.08985e7i	0.531552	+	0.920675i
$676$		0			0
$677$	−5.84516e6	+	1.01241e7i	−0.490146	+	0.848957i	−0.999936	−	0.0113419i	$-0.996390\pi$
$677$	−5.84516e6	+	1.01241e7i	−0.490146	+	0.848957i	0.509790	+	0.860299i	$0.329723\pi$
$678$		0			0
$679$	−4.46148e6	−	1.26371e7i	−0.371368	−	1.05190i
$680$		0			0
$681$	−4.58846e6	+	7.94744e6i	−0.379139	+	0.656689i
$682$		0			0
$683$	9.18370e6	+	1.59066e7i	0.753297	+	1.30475i	0.946217	+	0.323534i	$0.104871\pi$
$683$	9.18370e6	+	1.59066e7i	0.753297	+	1.30475i	−0.192920	+	0.981214i	$0.561796\pi$
$684$		0			0
$685$	−2.25588e6			−0.183692
$686$		0			0
$687$	5.26244e6			0.425398
$688$		0			0
$689$	−1.39096e6	−	2.40921e6i	−0.111626	−	0.193342i
$690$		0			0
$691$	1.17806e7	−	2.04045e7i	0.938579	−	1.62567i	0.170454	−	0.985366i	$-0.445477\pi$
$691$	1.17806e7	−	2.04045e7i	0.938579	−	1.62567i	0.768125	−	0.640300i	$-0.221190\pi$
$692$		0			0
$693$	−2.14660e6	−	6.08024e6i	−0.169792	−	0.480936i
$694$		0			0
$695$	−1.34267e7	+	2.32558e7i	−1.05441	+	1.82629i
$696$		0			0
$697$	1.30078e6	+	2.25303e6i	0.101420	+	0.175665i
$698$		0			0
$699$	−1.05142e7			−0.813921
$700$		0			0
$701$	1.32980e7			1.02210			0.511048	−	0.859552i	$-0.329257\pi$
$701$	1.32980e7			1.02210			0.511048	+	0.859552i	$0.329257\pi$
$702$		0			0
$703$	556535.	+	963946.i	0.0424721	+	0.0735639i
$704$		0			0
$705$	616815.	−	1.06836e6i	0.0467393	−	0.0809549i
$706$		0			0
$707$	−2.41913e6	+	2.82697e6i	−0.182016	+	0.212703i
$708$		0			0
$709$	3.42677e6	−	5.93533e6i	0.256017	−	0.443434i	−0.709154	−	0.705053i	$-0.750923\pi$
$709$	3.42677e6	−	5.93533e6i	0.256017	−	0.443434i	0.965171	+	0.261619i	$0.0842564\pi$
$710$		0			0
$711$	1.91412e6	+	3.31536e6i	0.142003	+	0.245956i
$712$		0			0
$713$	1.09628e6			0.0807602
$714$		0			0
$715$	−8.19384e6			−0.599408
$716$		0			0
$717$	−7.73587e6	−	1.33989e7i	−0.561968	−	0.973357i
$718$		0			0
$719$	1.32865e7	−	2.30128e7i	0.958490	−	1.66015i	0.232318	−	0.972640i	$-0.425369\pi$
$719$	1.32865e7	−	2.30128e7i	0.958490	−	1.66015i	0.726172	−	0.687513i	$-0.241298\pi$
$720$		0			0
$721$	−3.72505e6	−	694075.i	−0.266866	−	0.0497242i
$722$		0			0
$723$	5.07958e6	−	8.79810e6i	0.361395	−	0.625955i
$724$		0			0
$725$	1.26390e7	+	2.18913e7i	0.893031	+	1.54678i
$726$		0			0
$727$	−2.16991e6			−0.152267			−0.0761335	−	0.997098i	$-0.524258\pi$
$727$	−2.16991e6			−0.152267			−0.0761335	+	0.997098i	$0.524258\pi$
$728$		0			0
$729$	1.01628e7			0.708264
$730$		0			0
$731$	−339587.	−	588182.i	−0.0235049	−	0.0407116i
$732$		0			0
$733$	8.73265e6	−	1.51254e7i	0.600324	−	1.03979i	−0.392447	−	0.919774i	$-0.628372\pi$
$733$	8.73265e6	−	1.51254e7i	0.600324	−	1.03979i	0.992772	−	0.120018i	$-0.0382951\pi$
$734$		0			0
$735$	1.05292e7	−	8.49316e6i	0.718914	−	0.579897i
$736$		0			0
$737$	−4.70602e6	+	8.15106e6i	−0.319143	+	0.552771i
$738$		0			0
$739$	−6.82304e6	−	1.18178e7i	−0.459586	−	0.796026i	0.539353	−	0.842080i	$-0.318669\pi$
$739$	−6.82304e6	−	1.18178e7i	−0.459586	−	0.796026i	−0.998939	+	0.0460536i	$0.985335\pi$
$740$		0			0
$741$	4.42594e6			0.296114
$742$		0			0
$743$	−1.48965e7			−0.989944			−0.494972	−	0.868909i	$-0.664822\pi$
$743$	−1.48965e7			−0.989944			−0.494972	+	0.868909i	$0.664822\pi$
$744$		0			0
$745$	−1.41846e7	−	2.45685e7i	−0.936326	−	1.62176i
$746$		0			0
$747$	5.63623e6	−	9.76224e6i	0.369562	−	0.640100i
$748$		0			0
$749$	−1.11974e7	−	2.08637e6i	−0.729311	−	0.135890i
$750$		0			0
$751$	−1.26731e7	+	2.19505e7i	−0.819944	+	1.42019i	0.0857783	+	0.996314i	$0.472662\pi$
$751$	−1.26731e7	+	2.19505e7i	−0.819944	+	1.42019i	−0.905723	+	0.423871i	$0.860671\pi$
$752$		0			0
$753$	42911.7	+	74325.2i	0.00275796	+	0.00477693i
$754$		0			0
$755$	−2.86419e7			−1.82867
$756$		0			0
$757$	−2.66725e7			−1.69170			−0.845852	−	0.533417i	$-0.820908\pi$
$757$	−2.66725e7			−1.69170			−0.845852	+	0.533417i	$0.820908\pi$
$758$		0			0
$759$	755245.	+	1.30812e6i	0.0475864	+	0.0824221i
$760$		0			0
$761$	−289914.	+	502146.i	−0.0181471	+	0.0314318i	−0.874956	−	0.484202i	$-0.839110\pi$
$761$	−289914.	+	502146.i	−0.0181471	+	0.0314318i	0.856809	+	0.515634i	$0.172443\pi$
$762$		0			0
$763$	−1.85967e7	+	2.17319e7i	−1.15644	+	1.35141i
$764$		0			0
$765$	2.08772e6	−	3.61604e6i	0.128979	−	0.223398i
$766$		0			0
$767$	4.32770e6	+	7.49580e6i	0.265625	+	0.460076i
$768$		0			0
$769$	1.52438e7			0.929562			0.464781	−	0.885426i	$-0.346133\pi$
$769$	1.52438e7			0.929562			0.464781	+	0.885426i	$0.346133\pi$
$770$		0			0
$771$	−5.31898e6			−0.322250
$772$		0			0
$773$	9.74628e6	+	1.68810e7i	0.586665	+	1.01613i	0.994666	+	0.103152i	$0.0328927\pi$
$773$	9.74628e6	+	1.68810e7i	0.586665	+	1.01613i	−0.408001	+	0.912982i	$0.633774\pi$
$774$		0			0
$775$	−4.18138e6	+	7.24236e6i	−0.250072	+	0.433137i
$776$		0			0
$777$	−321851.	−	911643.i	−0.0191250	−	0.0541717i
$778$		0			0
$779$	5.28969e6	−	9.16201e6i	0.312311	−	0.540938i
$780$		0			0
$781$	2.53173e6	+	4.38508e6i	0.148521	+	0.257247i
$782$		0			0
$783$	3.01642e7			1.75828
$784$		0			0
$785$	−3.66337e7			−2.12181
$786$		0			0
$787$	6.81384e6	+	1.18019e7i	0.392153	+	0.679228i	0.992733	−	0.120336i	$-0.0383971\pi$
$787$	6.81384e6	+	1.18019e7i	0.392153	+	0.679228i	−0.600580	+	0.799564i	$0.705064\pi$
$788$		0			0
$789$	−1.77499e6	+	3.07438e6i	−0.101509	+	0.175819i
$790$		0			0
$791$	−1.71192e6	−	4.84900e6i	−0.0972839	−	0.275557i
$792$		0			0
$793$	−6.78320e6	+	1.17488e7i	−0.383046	+	0.663456i
$794$		0			0
$795$	−3.83827e6	−	6.64808e6i	−0.215386	−	0.373060i
$796$		0			0
$797$	3.62853e6			0.202341			0.101171	−	0.994869i	$-0.467741\pi$
$797$	3.62853e6			0.202341			0.101171	+	0.994869i	$0.467741\pi$
$798$		0			0
$799$	−567208.			−0.0314323
$800$		0			0
$801$	3.07956e6	+	5.33396e6i	0.169593	+	0.293744i
$802$		0			0
$803$	−1.23334e7	+	2.13620e7i	−0.674983	+	1.16910i
$804$		0			0
$805$	2.86446e6	−	3.34737e6i	0.155795	−	0.182060i
$806$		0			0
$807$	−2.41754e6	+	4.18731e6i	−0.130674	+	0.226335i
$808$		0			0
$809$	−1.49390e7	−	2.58751e7i	−0.802509	−	1.38999i	−0.917960	−	0.396673i	$-0.870165\pi$
$809$	−1.49390e7	−	2.58751e7i	−0.802509	−	1.38999i	0.115451	−	0.993313i	$-0.463169\pi$
$810$		0			0
$811$	1.02643e6			0.0547995			0.0273998	−	0.999625i	$-0.491277\pi$
$811$	1.02643e6			0.0547995			0.0273998	+	0.999625i	$0.491277\pi$
$812$		0			0
$813$	9.85702e6			0.523021
$814$		0			0
$815$	−2.00543e7	−	3.47351e7i	−1.05758	−	1.83178i
$816$		0			0
$817$	−1.38094e6	+	2.39187e6i	−0.0723804	+	0.125367i
$818$		0			0
$819$	5.25416e6	+	978989.i	0.273712	+	0.0509997i
$820$		0			0
$821$	5.75312e6	−	9.96470e6i	0.297883	−	0.515948i	−0.677768	−	0.735275i	$-0.737053\pi$
$821$	5.75312e6	−	9.96470e6i	0.297883	−	0.515948i	0.975651	+	0.219327i	$0.0703861\pi$
$822$		0			0
$823$	−1.25605e7	−	2.17554e7i	−0.646408	−	1.11961i	−0.983974	−	0.178310i	$-0.942937\pi$
$823$	−1.25605e7	−	2.17554e7i	−0.646408	−	1.11961i	0.337566	−	0.941302i	$-0.390396\pi$
$824$		0			0
$825$	−1.15225e7			−0.589401
$826$		0			0
$827$	2.14447e6			0.109032			0.0545162	−	0.998513i	$-0.482638\pi$
$827$	2.14447e6			0.109032			0.0545162	+	0.998513i	$0.482638\pi$
$828$		0			0
$829$	−409033.	−	708465.i	−0.0206715	−	0.0358041i	0.855505	−	0.517795i	$-0.173247\pi$
$829$	−409033.	−	708465.i	−0.0206715	−	0.0358041i	−0.876176	+	0.481991i	$0.839914\pi$
$830$		0			0
$831$	−4.88370e6	+	8.45882e6i	−0.245328	+	0.424920i
$832$		0			0
$833$	−5.80247e6	−	2.24008e6i	−0.289735	−	0.111854i
$834$		0			0
$835$	2.70078e7	−	4.67788e7i	1.34052	−	2.32184i
$836$		0			0
$837$	4.98964e6	+	8.64232e6i	0.246182	+	0.426399i
$838$		0			0
$839$	2.11279e7			1.03622			0.518110	−	0.855314i	$-0.326636\pi$
$839$	2.11279e7			1.03622			0.518110	+	0.855314i	$0.326636\pi$
$840$		0			0
$841$	4.00785e7			1.95398
$842$		0			0
$843$	1.60484e6	+	2.77966e6i	0.0777790	+	0.134717i
$844$		0			0
$845$	−1.14239e7	+	1.97868e7i	−0.550393	+	0.953308i
$846$		0			0
$847$	−4.74298e6	−	883742.i	−0.227166	−	0.0423269i
$848$		0			0
$849$	8.93384e6	−	1.54739e7i	0.425372	−	0.736766i
$850$		0			0
$851$	−157432.	−	272679.i	−0.00745191	−	0.0129071i
$852$		0			0
$853$	−1.89000e7			−0.889386			−0.444693	−	0.895683i	$-0.646687\pi$
$853$	−1.89000e7			−0.889386			−0.444693	+	0.895683i	$0.646687\pi$
$854$		0			0
$855$	−1.69796e7			−0.794349
$856$		0			0
$857$	−1.86143e7	−	3.22409e7i	−0.865753	−	1.49953i	−0.866297	−	0.499529i	$-0.833507\pi$
$857$	−1.86143e7	−	3.22409e7i	−0.865753	−	1.49953i	0.000544054	−	1.00000i	$-0.499827\pi$
$858$		0			0
$859$	1.51032e6	−	2.61595e6i	0.0698371	−	0.120961i	−0.828992	−	0.559260i	$-0.811085\pi$
$859$	1.51032e6	−	2.61595e6i	0.0698371	−	0.120961i	0.898829	+	0.438298i	$0.144419\pi$
$860$		0			0
$861$	−5.97447e6	+	6.98170e6i	−0.274657	+	0.320962i
$862$		0			0
$863$	−1.35921e7	+	2.35423e7i	−0.621242	+	1.07602i	0.368013	+	0.929821i	$0.380038\pi$
$863$	−1.35921e7	+	2.35423e7i	−0.621242	+	1.07602i	−0.989255	+	0.146202i	$0.953295\pi$
$864$		0			0
$865$	−9.94493e6	−	1.72251e7i	−0.451920	−	0.782748i
$866$		0			0
$867$	−1.29352e7			−0.584420
$868$		0			0
$869$	−9.53158e6			−0.428169
$870$		0			0
$871$	−3.90068e6	−	6.75618e6i	−0.174219	−	0.301756i
$872$		0			0
$873$	7.30532e6	−	1.26532e7i	0.324417	−	0.561906i
$874$		0			0
$875$	421871.	+	1.19495e6i	0.0186277	+	0.0527630i
$876$		0			0
$877$	−8.69339e6	+	1.50574e7i	−0.381672	+	0.661075i	−0.991301	−	0.131611i	$-0.957985\pi$
$877$	−8.69339e6	+	1.50574e7i	−0.381672	+	0.661075i	0.609629	+	0.792687i	$0.291318\pi$
$878$		0			0
$879$	−8.27531e6	−	1.43333e7i	−0.361253	−	0.625709i
$880$		0			0
$881$	−8.14472e6			−0.353538			−0.176769	−	0.984252i	$-0.556565\pi$
$881$	−8.14472e6			−0.353538			−0.176769	+	0.984252i	$0.556565\pi$
$882$		0			0
$883$	3.10298e7			1.33930			0.669649	−	0.742678i	$-0.266445\pi$
$883$	3.10298e7			1.33930			0.669649	+	0.742678i	$0.266445\pi$
$884$		0			0
$885$	1.19421e7	+	2.06843e7i	0.512533	+	0.887732i
$886$		0			0
$887$	−7.35138e6	+	1.27330e7i	−0.313733	+	0.543401i	−0.979167	−	0.203055i	$-0.934913\pi$
$887$	−7.35138e6	+	1.27330e7i	−0.313733	+	0.543401i	0.665435	+	0.746456i	$0.268246\pi$
$888$		0			0
$889$	2.23304e6	+	6.32509e6i	0.0947638	+	0.268419i
$890$		0			0
$891$	−831810.	+	1.44074e6i	−0.0351018	+	0.0607982i
$892$		0			0
$893$	1.15329e6	+	1.99755e6i	0.0483959	+	0.0838242i
$894$		0			0
$895$	1.11166e7			0.463890
$896$		0			0
$897$	−1.25200e6			−0.0519545
$898$		0			0
$899$	1.00225e7	+	1.73595e7i	0.413596	+	0.716370i
$900$		0			0
$901$	−1.76479e6	+	3.05671e6i	−0.0724238	+	0.125442i
$902$		0			0
$903$	1.55971e6	−	1.82267e6i	0.0636540	−	0.0743854i
$904$		0			0
$905$	1.22235e7	−	2.11716e7i	0.496104	−	0.859277i
$906$		0			0
$907$	6.54701e6	+	1.13398e7i	0.264256	+	0.457705i	0.967368	−	0.253374i	$-0.0815403\pi$
$907$	6.54701e6	+	1.13398e7i	0.264256	+	0.457705i	−0.703112	+	0.711079i	$0.748207\pi$
$908$		0			0
$909$	−4.05643e6			−0.162830
$910$		0			0
$911$	−2.68695e7			−1.07266			−0.536332	−	0.844007i	$-0.680191\pi$
$911$	−2.68695e7			−1.07266			−0.536332	+	0.844007i	$0.680191\pi$
$912$		0			0
$913$	1.40331e7	+	2.43061e7i	0.557156	+	0.965023i
$914$		0			0
$915$	−1.87179e7	+	3.24203e7i	−0.739101	+	1.28016i
$916$		0			0
$917$	2.12423e7	+	3.95800e6i	0.834215	+	0.155436i
$918$		0			0
$919$	636586.	−	1.10260e6i	0.0248638	−	0.0430654i	−0.853326	−	0.521378i	$-0.825418\pi$
$919$	636586.	−	1.10260e6i	0.0248638	−	0.0430654i	0.878190	+	0.478313i	$0.158751\pi$
$920$		0			0
$921$	−2.35397e6	−	4.07720e6i	−0.0914435	−	0.158385i
$922$		0			0
$923$	−4.19695e6			−0.162155
$924$		0			0
$925$	2.40187e6			0.0922986
$926$		0			0
$927$	−2.06551e6	−	3.57756e6i	−0.0789454	−	0.136737i
$928$		0			0
$929$	4.65852e6	−	8.06880e6i	0.177096	−	0.306740i	−0.763789	−	0.645467i	$-0.776663\pi$
$929$	4.65852e6	−	8.06880e6i	0.177096	−	0.306740i	0.940885	+	0.338727i	$0.109996\pi$
$930$		0			0
$931$	3.90905e6	+	2.49894e7i	0.147808	+	0.944890i
$932$		0			0
$933$	−1.22907e7	+	2.12881e7i	−0.462245	+	0.800631i
$934$		0			0
$935$	5.19801e6	+	9.00322e6i	0.194450	+	0.336797i
$936$		0			0
$937$	1.18158e7			0.439657			0.219829	−	0.975539i	$-0.429450\pi$
$937$	1.18158e7			0.439657			0.219829	+	0.975539i	$0.429450\pi$
$938$		0			0
$939$	2.44097e7			0.903439
$940$		0			0
$941$	1.26765e7	+	2.19563e7i	0.466685	+	0.808323i	0.999276	−	0.0380504i	$-0.0121148\pi$
$941$	1.26765e7	+	2.19563e7i	0.466685	+	0.808323i	−0.532591	+	0.846373i	$0.678781\pi$
$942$		0			0
$943$	−1.49634e6	+	2.59173e6i	−0.0547962	+	0.0949098i
$944$		0			0
$945$	3.94258e7	+	7.34606e6i	1.43615	+	0.267593i
$946$		0			0
$947$	1.32471e7	−	2.29446e7i	0.480004	−	0.831391i	−0.519733	−	0.854329i	$-0.673969\pi$
$947$	1.32471e7	−	2.29446e7i	0.480004	−	0.831391i	0.999737	+	0.0229375i	$0.00730187\pi$
$948$		0			0
$949$	−1.02228e7	−	1.77063e7i	−0.368471	−	0.638210i
$950$		0			0
$951$	1.89164e7			0.678246
$952$		0			0
$953$	−1.88335e7			−0.671735			−0.335868	−	0.941909i	$-0.609030\pi$
$953$	−1.88335e7			−0.671735			−0.335868	+	0.941909i	$0.609030\pi$
$954$		0			0
$955$	−9.08015e6	−	1.57273e7i	−0.322170	−	0.558014i
$956$		0			0
$957$	−1.38093e7	+	2.39184e7i	−0.487408	+	0.844215i
$958$		0			0
$959$	−2.38198e6	+	2.78355e6i	−0.0836355	+	0.0977356i
$960$		0			0
$961$	1.09988e7	−	1.90505e7i	0.384182	−	0.665423i
$962$		0			0
$963$	−6.20886e6	−	1.07541e7i	−0.215748	−	0.373686i
$964$		0			0
$965$	5.36740e7			1.85544
$966$		0			0
$967$	3.14956e7			1.08314			0.541569	−	0.840656i	$-0.317831\pi$
$967$	3.14956e7			1.08314			0.541569	+	0.840656i	$0.317831\pi$
$968$		0			0
$969$	−2.80773e6	−	4.86312e6i	−0.0960606	−	0.166382i
$970$		0			0
$971$	−3.42834e6	+	5.93806e6i	−0.116691	+	0.202114i	−0.918454	−	0.395527i	$-0.870562\pi$
$971$	−3.42834e6	+	5.93806e6i	−0.116691	+	0.202114i	0.801764	+	0.597641i	$0.203895\pi$
$972$		0			0
$973$	1.45183e7	+	4.11231e7i	0.491624	+	1.39253i
$974$		0			0
$975$	4.77532e6	−	8.27110e6i	0.160876	−	0.278645i
$976$		0			0
$977$	−1.40735e7	−	2.43761e7i	−0.471701	−	0.817010i	0.527775	−	0.849384i	$-0.323027\pi$
$977$	−1.40735e7	−	2.43761e7i	−0.471701	−	0.817010i	−0.999476	+	0.0323741i	$0.989693\pi$
$978$		0			0
$979$	−1.53350e7			−0.511361
$980$		0			0
$981$	−3.11831e7			−1.03454
$982$		0			0
$983$	1.17458e7	+	2.03443e7i	0.387703	+	0.671521i	0.992140	−	0.125132i	$-0.0399354\pi$
$983$	1.17458e7	+	2.03443e7i	0.387703	+	0.671521i	−0.604437	+	0.796653i	$0.706602\pi$
$984$		0			0
$985$	−51199.7	+	88680.5i	−0.00168142	+	0.00291231i
$986$		0			0
$987$	−666960.	−	1.88917e6i	−0.0217925	−	0.0617273i
$988$		0			0
$989$	390639.	−	676607.i	0.0126995	−	0.0219961i
$990$		0			0
$991$	−1.07206e6	−	1.85686e6i	−0.0346765	−	0.0600615i	0.848166	−	0.529730i	$-0.177707\pi$
$991$	−1.07206e6	−	1.85686e6i	−0.0346765	−	0.0600615i	−0.882843	+	0.469669i	$0.844373\pi$
$992$		0			0
$993$	1.09206e7			0.351459
$994$		0			0
$995$	2.94483e7			0.942979
$996$		0			0
$997$	−1.25436e7	−	2.17261e7i	−0.399654	−	0.692220i	0.594029	−	0.804443i	$-0.297536\pi$
$997$	−1.25436e7	−	2.17261e7i	−0.399654	−	0.692220i	−0.993683	+	0.112223i	$0.964203\pi$
$998$		0			0
$999$	1.43308e6	−	2.48216e6i	0.0454314	−	0.0786895i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.i.c.65.1		4
4.3	odd	2		7.6.c.a.2.1	✓	4
7.2	even	3		784.6.a.ba.1.2		2
7.4	even	3	inner	112.6.i.c.81.1		4
7.5	odd	6		784.6.a.t.1.1		2
12.11	even	2		63.6.e.d.37.2		4
28.3	even	6		49.6.c.f.18.1		4
28.11	odd	6		7.6.c.a.4.1	yes	4
28.19	even	6		49.6.a.e.1.2		2
28.23	odd	6		49.6.a.d.1.2		2
28.27	even	2		49.6.c.f.30.1		4
84.11	even	6		63.6.e.d.46.2		4
84.23	even	6		441.6.a.n.1.1		2
84.47	odd	6		441.6.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.c.a.2.1	✓	4	4.3	odd	2
7.6.c.a.4.1	yes	4	28.11	odd	6
49.6.a.d.1.2		2	28.23	odd	6
49.6.a.e.1.2		2	28.19	even	6
49.6.c.f.18.1		4	28.3	even	6
49.6.c.f.30.1		4	28.27	even	2
63.6.e.d.37.2		4	12.11	even	2
63.6.e.d.46.2		4	84.11	even	6
112.6.i.c.65.1		4	1.1	even	1	trivial
112.6.i.c.81.1		4	7.4	even	3	inner
441.6.a.m.1.1		2	84.47	odd	6
441.6.a.n.1.1		2	84.23	even	6
784.6.a.t.1.1		2	7.5	odd	6
784.6.a.ba.1.2		2	7.2	even	3

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.04138 - 8.73193i) q^{3} +(39.9138 - 69.1328i) q^{5} +(-43.1587 - 122.247i) q^{7} +(70.6689 - 122.402i) q^{9} +(175.952 + 304.757i) q^{11} -291.683 q^{13} -804.883 q^{15} +(185.038 + 320.495i) q^{17} +(752.463 - 1303.30i) q^{19} +(-849.873 + 993.152i) q^{21} +(-212.855 + 368.676i) q^{23} +(-1623.72 - 2812.37i) q^{25} -3875.19 q^{27} -7783.93 q^{29} +(-1287.59 - 2230.17i) q^{31} +(1774.08 - 3072.80i) q^{33} +(-10173.9 - 1895.67i) q^{35} +(-369.809 + 640.528i) q^{37} +(1470.48 + 2546.95i) q^{39} +7029.84 q^{41} -1835.23 q^{43} +(-5641.33 - 9771.08i) q^{45} +(-766.342 + 1327.34i) q^{47} +(-13081.7 + 10552.0i) q^{49} +(1865.69 - 3231.47i) q^{51} +(4768.73 + 8259.68i) q^{53} +28091.6 q^{55} -15173.8 q^{57} +(-14837.0 - 25698.5i) q^{59} +(23255.4 - 40279.5i) q^{61} +(-18013.3 - 3356.35i) q^{63} +(-11642.2 + 20164.8i) q^{65} +(13373.0 + 23162.8i) q^{67} +4292.34 q^{69} +14388.8 q^{71} +(35047.6 + 60704.2i) q^{73} +(-16371.6 + 28356.5i) q^{75} +(29661.8 - 34662.5i) q^{77} +(-13542.9 + 23457.0i) q^{79} +(2363.74 + 4094.13i) q^{81} +79755.4 q^{83} +29542.2 q^{85} +(39241.8 + 67968.8i) q^{87} +(-21788.7 + 37739.1i) q^{89} +(12588.6 + 35657.3i) q^{91} +(-12982.4 + 22486.2i) q^{93} +(-60067.3 - 104040. i) q^{95} +103374. q^{97} +49737.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{3} + 38 q^{5} + 168 q^{7} + 380 q^{9} + 424 q^{11} - 1848 q^{13} - 1784 q^{15} + 2346 q^{17} - 360 q^{19} - 1526 q^{21} - 12 q^{23} - 1872 q^{25} - 5744 q^{27} - 14104 q^{29} + 3548 q^{31} + 3398 q^{33}+ \cdots + 133888 q^{99}+O(q^{100}) \) 4 * q - 8 * q^3 + 38 * q^5 + 168 * q^7 + 380 * q^9 + 424 * q^11 - 1848 * q^13 - 1784 * q^15 + 2346 * q^17 - 360 * q^19 - 1526 * q^21 - 12 * q^23 - 1872 * q^25 - 5744 * q^27 - 14104 * q^29 + 3548 * q^31 + 3398 * q^33 - 27496 * q^35 - 11090 * q^37 + 1624 * q^39 + 7000 * q^41 + 25360 * q^43 - 1300 * q^45 - 22956 * q^47 + 4900 * q^49 - 384 * q^51 - 3042 * q^53 + 50152 * q^55 - 38116 * q^57 - 65808 * q^59 + 42486 * q^61 + 4760 * q^63 + 3164 * q^65 + 42312 * q^67 + 10308 * q^69 + 4416 * q^71 + 50506 * q^73 - 35608 * q^75 + 65338 * q^77 + 9004 * q^79 - 51178 * q^81 + 208656 * q^83 - 106212 * q^85 + 80008 * q^87 + 26666 * q^89 - 135632 * q^91 - 38718 * q^93 - 198140 * q^95 + 418264 * q^97 + 133888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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