LMFDB - Embedded newform 4012.2.b.b.237.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.28
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.593296i q^{3} -3.87533i q^{5} -2.92807i q^{7} +2.64800 q^{9} +O(q^{10})$	$q+0.593296i q^{3} -3.87533i q^{5} -2.92807i q^{7} +2.64800 q^{9} +4.95085i q^{11} -3.80221 q^{13} +2.29922 q^{15} +(2.88587 - 2.94479i) q^{17} +1.82016 q^{19} +1.73721 q^{21} -8.40097i q^{23} -10.0182 q^{25} +3.35094i q^{27} +1.64389i q^{29} -7.62019i q^{31} -2.93732 q^{33} -11.3473 q^{35} -0.0884485i q^{37} -2.25584i q^{39} -2.48344i q^{41} +10.2661 q^{43} -10.2619i q^{45} -1.02570 q^{47} -1.57361 q^{49} +(1.74713 + 1.71217i) q^{51} +0.471127 q^{53} +19.1862 q^{55} +1.07989i q^{57} +1.00000 q^{59} +6.85661i q^{61} -7.75354i q^{63} +14.7348i q^{65} -12.9279 q^{67} +4.98426 q^{69} -8.36046i q^{71} +7.16000i q^{73} -5.94375i q^{75} +14.4964 q^{77} -8.40731i q^{79} +5.95590 q^{81} +2.81612 q^{83} +(-11.4120 - 11.1837i) q^{85} -0.975310 q^{87} -12.6865 q^{89} +11.1331i q^{91} +4.52103 q^{93} -7.05372i q^{95} -2.93724i q^{97} +13.1098i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			0.593296i			0.342540i	0.985224	+	0.171270i	$0.0547870\pi$
$3$			0.593296i			0.342540i	−0.985224	+	0.171270i	$0.945213\pi$
$4$		0			0
$5$		−	3.87533i		−	1.73310i	−0.499090	−	0.866550i	$-0.666332\pi$
$5$		−	3.87533i		−	1.73310i	0.499090	−	0.866550i	$-0.333668\pi$
$6$		0			0
$7$		−	2.92807i		−	1.10671i	−0.832946	−	0.553354i	$-0.813348\pi$
$7$		−	2.92807i		−	1.10671i	0.832946	−	0.553354i	$-0.186652\pi$
$8$		0			0
$9$	2.64800			0.882667
$10$		0			0
$11$			4.95085i			1.49274i	0.665533	+	0.746368i	$0.268204\pi$
$11$			4.95085i			1.49274i	−0.665533	+	0.746368i	$0.731796\pi$
$12$		0			0
$13$	−3.80221			−1.05454			−0.527272	−	0.849697i	$-0.676785\pi$
$13$	−3.80221			−1.05454			−0.527272	+	0.849697i	$0.676785\pi$
$14$		0			0
$15$	2.29922			0.593656
$16$		0			0
$17$	2.88587	−	2.94479i	0.699926	−	0.714216i
$17$	2.88587	−	2.94479i	0.699926	−	0.714216i
$18$		0			0
$19$	1.82016			0.417573			0.208787	−	0.977961i	$-0.433049\pi$
$19$	1.82016			0.417573			0.208787	+	0.977961i	$0.433049\pi$
$20$		0			0
$21$	1.73721			0.379091
$22$		0			0
$23$		−	8.40097i		−	1.75172i	−0.482562	−	0.875862i	$-0.660294\pi$
$23$		−	8.40097i		−	1.75172i	0.482562	−	0.875862i	$-0.339706\pi$
$24$		0			0
$25$	−10.0182			−2.00364
$26$		0			0
$27$			3.35094i			0.644888i
$28$		0			0
$29$			1.64389i			0.305262i	0.988283	+	0.152631i	$0.0487746\pi$
$29$			1.64389i			0.305262i	−0.988283	+	0.152631i	$0.951225\pi$
$30$		0			0
$31$		−	7.62019i		−	1.36863i	−0.729188	−	0.684313i	$-0.760102\pi$
$31$		−	7.62019i		−	1.36863i	0.729188	−	0.684313i	$-0.239898\pi$
$32$		0			0
$33$	−2.93732			−0.511321
$34$		0			0
$35$	−11.3473			−1.91804
$36$		0			0
$37$		−	0.0884485i		−	0.0145408i	−0.999974	−	0.00727042i	$-0.997686\pi$
$37$		−	0.0884485i		−	0.0145408i	0.999974	−	0.00727042i	$-0.00231427\pi$
$38$		0			0
$39$		−	2.25584i		−	0.361223i
$40$		0			0
$41$		−	2.48344i		−	0.387849i	−0.981016	−	0.193924i	$-0.937878\pi$
$41$		−	2.48344i		−	0.387849i	0.981016	−	0.193924i	$-0.0621217\pi$
$42$		0			0
$43$	10.2661			1.56557			0.782785	−	0.622292i	$-0.213798\pi$
$43$	10.2661			1.56557			0.782785	+	0.622292i	$0.213798\pi$
$44$		0			0
$45$		−	10.2619i		−	1.52975i
$46$		0			0
$47$	−1.02570			−0.149614			−0.0748070	−	0.997198i	$-0.523834\pi$
$47$	−1.02570			−0.149614			−0.0748070	+	0.997198i	$0.523834\pi$
$48$		0			0
$49$	−1.57361			−0.224802
$50$		0			0
$51$	1.74713	+	1.71217i	0.244647	+	0.239752i
$52$		0			0
$53$	0.471127			0.0647143			0.0323571	−	0.999476i	$-0.489699\pi$
$53$	0.471127			0.0647143			0.0323571	+	0.999476i	$0.489699\pi$
$54$		0			0
$55$	19.1862			2.58706
$56$		0			0
$57$			1.07989i			0.143035i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			6.85661i			0.877898i	0.898512	+	0.438949i	$0.144649\pi$
$61$			6.85661i			0.877898i	−0.898512	+	0.438949i	$0.855351\pi$
$62$		0			0
$63$		−	7.75354i		−	0.976854i
$64$		0			0
$65$			14.7348i			1.82763i
$66$		0			0
$67$	−12.9279			−1.57939			−0.789696	−	0.613499i	$-0.789762\pi$
$67$	−12.9279			−1.57939			−0.789696	+	0.613499i	$0.789762\pi$
$68$		0			0
$69$	4.98426			0.600035
$70$		0			0
$71$		−	8.36046i		−	0.992204i	−0.868264	−	0.496102i	$-0.834764\pi$
$71$		−	8.36046i		−	0.992204i	0.868264	−	0.496102i	$-0.165236\pi$
$72$		0			0
$73$			7.16000i			0.838014i	0.907983	+	0.419007i	$0.137622\pi$
$73$			7.16000i			0.838014i	−0.907983	+	0.419007i	$0.862378\pi$
$74$		0			0
$75$		−	5.94375i		−	0.686325i
$76$		0			0
$77$	14.4964			1.65202
$78$		0			0
$79$		−	8.40731i		−	0.945896i	−0.881090	−	0.472948i	$-0.843190\pi$
$79$		−	8.40731i		−	0.945896i	0.881090	−	0.472948i	$-0.156810\pi$
$80$		0			0
$81$	5.95590			0.661767
$82$		0			0
$83$	2.81612			0.309110			0.154555	−	0.987984i	$-0.450606\pi$
$83$	2.81612			0.309110			0.154555	+	0.987984i	$0.450606\pi$
$84$		0			0
$85$	−11.4120	−	11.1837i	−1.23781	−	1.21304i
$86$		0			0
$87$	−0.975310			−0.104564
$88$		0			0
$89$	−12.6865			−1.34476			−0.672381	−	0.740206i	$-0.734728\pi$
$89$	−12.6865			−1.34476			−0.672381	+	0.740206i	$0.734728\pi$
$90$		0			0
$91$			11.1331i			1.16707i
$92$		0			0
$93$	4.52103			0.468809
$94$		0			0
$95$		−	7.05372i		−	0.723697i
$96$		0			0
$97$		−	2.93724i		−	0.298232i	−0.988820	−	0.149116i	$-0.952357\pi$
$97$		−	2.93724i		−	0.298232i	0.988820	−	0.149116i	$-0.0476427\pi$
$98$		0			0
$99$			13.1098i			1.31759i
$100$		0			0
$101$	−12.5736			−1.25112			−0.625562	−	0.780174i	$-0.715130\pi$
$101$	−12.5736			−1.25112			−0.625562	+	0.780174i	$0.715130\pi$
$102$		0			0
$103$	−0.918766			−0.0905287			−0.0452644	−	0.998975i	$-0.514413\pi$
$103$	−0.918766			−0.0905287			−0.0452644	+	0.998975i	$0.514413\pi$
$104$		0			0
$105$		−	6.73228i		−	0.657003i
$106$		0			0
$107$		−	7.47339i		−	0.722480i	−0.932473	−	0.361240i	$-0.882354\pi$
$107$		−	7.47339i		−	0.722480i	0.932473	−	0.361240i	$-0.117646\pi$
$108$		0			0
$109$			3.87296i			0.370963i	0.982648	+	0.185481i	$0.0593844\pi$
$109$			3.87296i			0.370963i	−0.982648	+	0.185481i	$0.940616\pi$
$110$		0			0
$111$	0.0524761			0.00498082
$112$		0			0
$113$			5.26006i			0.494825i	0.968910	+	0.247413i	$0.0795803\pi$
$113$			5.26006i			0.494825i	−0.968910	+	0.247413i	$0.920420\pi$
$114$		0			0
$115$	−32.5566			−3.03591
$116$		0			0
$117$	−10.0683			−0.930810
$118$		0			0
$119$	−8.62255	−	8.45003i	−0.790428	−	0.774613i
$120$		0			0
$121$	−13.5109			−1.22826
$122$		0			0
$123$	1.47342			0.132854
$124$		0			0
$125$			19.4472i			1.73941i
$126$		0			0
$127$	8.46239			0.750915			0.375458	−	0.926840i	$-0.377486\pi$
$127$	8.46239			0.750915			0.375458	+	0.926840i	$0.377486\pi$
$128$		0			0
$129$			6.09085i			0.536270i
$130$		0			0
$131$		−	17.4727i		−	1.52660i	−0.646046	−	0.763298i	$-0.723579\pi$
$131$		−	17.4727i		−	1.52660i	0.646046	−	0.763298i	$-0.276421\pi$
$132$		0			0
$133$		−	5.32956i		−	0.462132i
$134$		0			0
$135$	12.9860			1.11766
$136$		0			0
$137$	2.30666			0.197071			0.0985357	−	0.995134i	$-0.468584\pi$
$137$	2.30666			0.197071			0.0985357	+	0.995134i	$0.468584\pi$
$138$		0			0
$139$		−	10.7926i		−	0.915413i	−0.889103	−	0.457707i	$-0.848671\pi$
$139$		−	10.7926i		−	0.915413i	0.889103	−	0.457707i	$-0.151329\pi$
$140$		0			0
$141$		−	0.608545i		−	0.0512487i
$142$		0			0
$143$		−	18.8242i		−	1.57415i
$144$		0			0
$145$	6.37060			0.529049
$146$		0			0
$147$		−	0.933619i		−	0.0770036i
$148$		0			0
$149$	−13.8742			−1.13662			−0.568311	−	0.822814i	$-0.692403\pi$
$149$	−13.8742			−1.13662			−0.568311	+	0.822814i	$0.692403\pi$
$150$		0			0
$151$	−11.1809			−0.909891			−0.454945	−	0.890519i	$-0.650341\pi$
$151$	−11.1809			−0.909891			−0.454945	+	0.890519i	$0.650341\pi$
$152$		0			0
$153$	7.64178	−	7.79780i	0.617801	−	0.630415i
$154$		0			0
$155$	−29.5308			−2.37197
$156$		0			0
$157$	2.01139			0.160527			0.0802633	−	0.996774i	$-0.474424\pi$
$157$	2.01139			0.160527			0.0802633	+	0.996774i	$0.474424\pi$
$158$		0			0
$159$			0.279518i			0.0221672i
$160$		0			0
$161$	−24.5987			−1.93865
$162$		0			0
$163$			12.6552i			0.991235i	0.868541	+	0.495618i	$0.165058\pi$
$163$			12.6552i			0.991235i	−0.868541	+	0.495618i	$0.834942\pi$
$164$		0			0
$165$			11.3831i			0.886171i
$166$		0			0
$167$			18.5213i			1.43322i	0.697475	+	0.716609i	$0.254307\pi$
$167$			18.5213i			1.43322i	−0.697475	+	0.716609i	$0.745693\pi$
$168$		0			0
$169$	1.45680			0.112061
$170$		0			0
$171$	4.81978			0.368578
$172$		0			0
$173$			22.2040i			1.68814i	0.536236	+	0.844068i	$0.319846\pi$
$173$			22.2040i			1.68814i	−0.536236	+	0.844068i	$0.680154\pi$
$174$		0			0
$175$			29.3340i			2.21744i
$176$		0			0
$177$			0.593296i			0.0445949i
$178$		0			0
$179$	−18.3235			−1.36957			−0.684783	−	0.728747i	$-0.740103\pi$
$179$	−18.3235			−1.36957			−0.684783	+	0.728747i	$0.740103\pi$
$180$		0			0
$181$		−	5.00327i		−	0.371890i	−0.982560	−	0.185945i	$-0.940465\pi$
$181$		−	5.00327i		−	0.371890i	0.982560	−	0.185945i	$-0.0595347\pi$
$182$		0			0
$183$	−4.06800			−0.300715
$184$		0			0
$185$	−0.342767			−0.0252008
$186$		0			0
$187$	14.5792	+	14.2875i	1.06614	+	1.04480i
$188$		0			0
$189$	9.81179			0.713702
$190$		0			0
$191$	3.52982			0.255409			0.127704	−	0.991812i	$-0.459239\pi$
$191$	3.52982			0.255409			0.127704	+	0.991812i	$0.459239\pi$
$192$		0			0
$193$			17.0812i			1.22953i	0.788711	+	0.614764i	$0.210749\pi$
$193$			17.0812i			1.22953i	−0.788711	+	0.614764i	$0.789251\pi$
$194$		0			0
$195$	−8.74211			−0.626035
$196$		0			0
$197$			9.16663i			0.653095i	0.945181	+	0.326548i	$0.105885\pi$
$197$			9.16663i			0.653095i	−0.945181	+	0.326548i	$0.894115\pi$
$198$		0			0
$199$		−	4.96742i		−	0.352131i	−0.984378	−	0.176066i	$-0.943663\pi$
$199$		−	4.96742i		−	0.352131i	0.984378	−	0.176066i	$-0.0563371\pi$
$200$		0			0
$201$		−	7.67005i		−	0.541004i
$202$		0			0
$203$	4.81342			0.337836
$204$		0			0
$205$	−9.62417			−0.672181
$206$		0			0
$207$		−	22.2458i		−	1.54619i
$208$		0			0
$209$			9.01133i			0.623327i
$210$		0			0
$211$			16.3353i			1.12457i	0.826944	+	0.562284i	$0.190077\pi$
$211$			16.3353i			1.12457i	−0.826944	+	0.562284i	$0.809923\pi$
$212$		0			0
$213$	4.96023			0.339869
$214$		0			0
$215$		−	39.7847i		−	2.71329i
$216$		0			0
$217$	−22.3125			−1.51467
$218$		0			0
$219$	−4.24800			−0.287053
$220$		0			0
$221$	−10.9727	+	11.1967i	−0.738102	+	0.753171i
$222$		0			0
$223$	−6.02994			−0.403795			−0.201897	−	0.979407i	$-0.564711\pi$
$223$	−6.02994			−0.403795			−0.201897	+	0.979407i	$0.564711\pi$
$224$		0			0
$225$	−26.5282			−1.76854
$226$		0			0
$227$		−	19.0807i		−	1.26643i	−0.773975	−	0.633217i	$-0.781734\pi$
$227$		−	19.0807i		−	1.26643i	0.773975	−	0.633217i	$-0.218266\pi$
$228$		0			0
$229$	−9.92975			−0.656176			−0.328088	−	0.944647i	$-0.606404\pi$
$229$	−9.92975			−0.656176			−0.328088	+	0.944647i	$0.606404\pi$
$230$		0			0
$231$			8.60068i			0.565883i
$232$		0			0
$233$		−	17.1706i		−	1.12488i	−0.826838	−	0.562441i	$-0.809862\pi$
$233$		−	17.1706i		−	1.12488i	0.826838	−	0.562441i	$-0.190138\pi$
$234$		0			0
$235$			3.97493i			0.259296i
$236$		0			0
$237$	4.98802			0.324007
$238$		0			0
$239$	4.30844			0.278690			0.139345	−	0.990244i	$-0.455500\pi$
$239$	4.30844			0.278690			0.139345	+	0.990244i	$0.455500\pi$
$240$		0			0
$241$		−	27.4140i		−	1.76589i	−0.469473	−	0.882947i	$-0.655556\pi$
$241$		−	27.4140i		−	1.76589i	0.469473	−	0.882947i	$-0.344444\pi$
$242$		0			0
$243$			13.5864i			0.871569i
$244$		0			0
$245$			6.09828i			0.389605i
$246$		0			0
$247$	−6.92063			−0.440349
$248$		0			0
$249$			1.67080i			0.105882i
$250$		0			0
$251$	6.84664			0.432156			0.216078	−	0.976376i	$-0.430673\pi$
$251$	6.84664			0.432156			0.216078	+	0.976376i	$0.430673\pi$
$252$		0			0
$253$	41.5919			2.61486
$254$		0			0
$255$	6.63524	−	6.77071i	0.415515	−	0.423998i
$256$		0			0
$257$	20.2578			1.26364			0.631822	−	0.775113i	$-0.282307\pi$
$257$	20.2578			1.26364			0.631822	+	0.775113i	$0.282307\pi$
$258$		0			0
$259$	−0.258984			−0.0160925
$260$		0			0
$261$			4.35301i			0.269444i
$262$		0			0
$263$	−1.14819			−0.0708001			−0.0354001	−	0.999373i	$-0.511271\pi$
$263$	−1.14819			−0.0708001			−0.0354001	+	0.999373i	$0.511271\pi$
$264$		0			0
$265$		−	1.82577i		−	0.112156i
$266$		0			0
$267$		−	7.52682i		−	0.460634i
$268$		0			0
$269$			17.4737i			1.06539i	0.846307	+	0.532695i	$0.178821\pi$
$269$			17.4737i			1.06539i	−0.846307	+	0.532695i	$0.821179\pi$
$270$		0			0
$271$	−0.871087			−0.0529148			−0.0264574	−	0.999650i	$-0.508423\pi$
$271$	−0.871087			−0.0529148			−0.0264574	+	0.999650i	$0.508423\pi$
$272$		0			0
$273$	−6.60525			−0.399768
$274$		0			0
$275$		−	49.5985i		−	2.99090i
$276$		0			0
$277$		−	13.8936i		−	0.834787i	−0.908726	−	0.417394i	$-0.862944\pi$
$277$		−	13.8936i		−	0.834787i	0.908726	−	0.417394i	$-0.137056\pi$
$278$		0			0
$279$		−	20.1783i		−	1.20804i
$280$		0			0
$281$	6.08777			0.363166			0.181583	−	0.983376i	$-0.441878\pi$
$281$	6.08777			0.363166			0.181583	+	0.983376i	$0.441878\pi$
$282$		0			0
$283$		−	27.5681i		−	1.63876i	−0.573254	−	0.819378i	$-0.694319\pi$
$283$		−	27.5681i		−	1.63876i	0.573254	−	0.819378i	$-0.305681\pi$
$284$		0			0
$285$	4.18495			0.247895
$286$		0			0
$287$	−7.27171			−0.429235
$288$		0			0
$289$	−0.343546	−	16.9965i	−0.0202086	−	0.999796i
$290$		0			0
$291$	1.74265			0.102156
$292$		0			0
$293$	−5.36853			−0.313633			−0.156816	−	0.987628i	$-0.550123\pi$
$293$	−5.36853			−0.313633			−0.156816	+	0.987628i	$0.550123\pi$
$294$		0			0
$295$		−	3.87533i		−	0.225631i
$296$		0			0
$297$	−16.5900			−0.962647
$298$		0			0
$299$			31.9423i			1.84727i
$300$		0			0
$301$		−	30.0600i		−	1.73263i
$302$		0			0
$303$		−	7.45989i		−	0.428560i
$304$		0			0
$305$	26.5716			1.52149
$306$		0			0
$307$	15.5224			0.885910			0.442955	−	0.896544i	$-0.353930\pi$
$307$	15.5224			0.885910			0.442955	+	0.896544i	$0.353930\pi$
$308$		0			0
$309$		−	0.545100i		−	0.0310097i
$310$		0			0
$311$		−	27.5865i		−	1.56428i	−0.623100	−	0.782142i	$-0.714127\pi$
$311$		−	27.5865i		−	1.56428i	0.623100	−	0.782142i	$-0.285873\pi$
$312$		0			0
$313$			26.2054i			1.48122i	0.671935	+	0.740610i	$0.265463\pi$
$313$			26.2054i			1.48122i	−0.671935	+	0.740610i	$0.734537\pi$
$314$		0			0
$315$	−30.0475			−1.69299
$316$		0			0
$317$		−	11.8975i		−	0.668227i	−0.942533	−	0.334114i	$-0.891563\pi$
$317$		−	11.8975i		−	0.668227i	0.942533	−	0.334114i	$-0.108437\pi$
$318$		0			0
$319$	−8.13862			−0.455675
$320$		0			0
$321$	4.43393			0.247478
$322$		0			0
$323$	5.25274	−	5.35999i	0.292270	−	0.298238i
$324$		0			0
$325$	38.0913			2.11292
$326$		0			0
$327$	−2.29781			−0.127069
$328$		0			0
$329$			3.00333i			0.165579i
$330$		0			0
$331$	−21.4502			−1.17901			−0.589505	−	0.807764i	$-0.700677\pi$
$331$	−21.4502			−1.17901			−0.589505	+	0.807764i	$0.700677\pi$
$332$		0			0
$333$		−	0.234212i		−	0.0128347i
$334$		0			0
$335$			50.0998i			2.73724i
$336$		0			0
$337$		−	35.6109i		−	1.93985i	−0.243408	−	0.969924i	$-0.578265\pi$
$337$		−	35.6109i		−	1.93985i	0.243408	−	0.969924i	$-0.421735\pi$
$338$		0			0
$339$	−3.12077			−0.169497
$340$		0			0
$341$	37.7264			2.04300
$342$		0			0
$343$		−	15.8889i		−	0.857918i
$344$		0			0
$345$		−	19.3157i		−	1.03992i
$346$		0			0
$347$			13.8045i			0.741064i	0.928820	+	0.370532i	$0.120825\pi$
$347$			13.8045i			0.741064i	−0.928820	+	0.370532i	$0.879175\pi$
$348$		0			0
$349$	−3.43255			−0.183740			−0.0918701	−	0.995771i	$-0.529284\pi$
$349$	−3.43255			−0.183740			−0.0918701	+	0.995771i	$0.529284\pi$
$350$		0			0
$351$		−	12.7410i		−	0.680062i
$352$		0			0
$353$	−4.17223			−0.222065			−0.111033	−	0.993817i	$-0.535416\pi$
$353$	−4.17223			−0.222065			−0.111033	+	0.993817i	$0.535416\pi$
$354$		0			0
$355$	−32.3996			−1.71959
$356$		0			0
$357$	5.01337	−	5.11573i	0.265336	−	0.270753i
$358$		0			0
$359$	26.5274			1.40006			0.700032	−	0.714112i	$-0.253169\pi$
$359$	26.5274			1.40006			0.700032	+	0.714112i	$0.253169\pi$
$360$		0			0
$361$	−15.6870			−0.825632
$362$		0			0
$363$		−	8.01595i		−	0.420728i
$364$		0			0
$365$	27.7474			1.45236
$366$		0			0
$367$			27.2314i			1.42147i	0.703462	+	0.710733i	$0.251637\pi$
$367$			27.2314i			1.42147i	−0.703462	+	0.710733i	$0.748363\pi$
$368$		0			0
$369$		−	6.57616i		−	0.342341i
$370$		0			0
$371$		−	1.37949i		−	0.0716198i
$372$		0			0
$373$	15.7377			0.814869			0.407434	−	0.913234i	$-0.366424\pi$
$373$	15.7377			0.814869			0.407434	+	0.913234i	$0.366424\pi$
$374$		0			0
$375$	−11.5379			−0.595815
$376$		0			0
$377$		−	6.25040i		−	0.321912i
$378$		0			0
$379$		−	12.7567i		−	0.655267i	−0.944805	−	0.327633i	$-0.893749\pi$
$379$		−	12.7567i		−	0.655267i	0.944805	−	0.327633i	$-0.106251\pi$
$380$		0			0
$381$			5.02070i			0.257218i
$382$		0			0
$383$	37.1510			1.89833			0.949164	−	0.314782i	$-0.101931\pi$
$383$	37.1510			1.89833			0.949164	+	0.314782i	$0.101931\pi$
$384$		0			0
$385$		−	56.1785i		−	2.86312i
$386$		0			0
$387$	27.1847			1.38188
$388$		0			0
$389$	31.1313			1.57842			0.789209	−	0.614125i	$-0.210491\pi$
$389$	31.1313			1.57842			0.789209	+	0.614125i	$0.210491\pi$
$390$		0			0
$391$	−24.7391	−	24.2441i	−1.25111	−	1.22608i
$392$		0			0
$393$	10.3665			0.522920
$394$		0			0
$395$	−32.5811			−1.63933
$396$		0			0
$397$		−	23.8999i		−	1.19950i	−0.800187	−	0.599750i	$-0.795267\pi$
$397$		−	23.8999i		−	1.19950i	0.800187	−	0.599750i	$-0.204733\pi$
$398$		0			0
$399$	3.16201			0.158298
$400$		0			0
$401$			16.5266i			0.825297i	0.910890	+	0.412648i	$0.135396\pi$
$401$			16.5266i			0.825297i	−0.910890	+	0.412648i	$0.864604\pi$
$402$		0			0
$403$			28.9736i			1.44328i
$404$		0			0
$405$		−	23.0811i		−	1.14691i
$406$		0			0
$407$	0.437895			0.0217057
$408$		0			0
$409$	−1.42032			−0.0702305			−0.0351152	−	0.999383i	$-0.511180\pi$
$409$	−1.42032			−0.0702305			−0.0351152	+	0.999383i	$0.511180\pi$
$410$		0			0
$411$			1.36853i			0.0675047i
$412$		0			0
$413$		−	2.92807i		−	0.144081i
$414$		0			0
$415$		−	10.9134i		−	0.535719i
$416$		0			0
$417$	6.40318			0.313565
$418$		0			0
$419$			20.5811i			1.00545i	0.864445	+	0.502727i	$0.167670\pi$
$419$			20.5811i			1.00545i	−0.864445	+	0.502727i	$0.832330\pi$
$420$		0			0
$421$	−26.0994			−1.27201			−0.636003	−	0.771687i	$-0.719413\pi$
$421$	−26.0994			−1.27201			−0.636003	+	0.771687i	$0.719413\pi$
$422$		0			0
$423$	−2.71606			−0.132059
$424$		0			0
$425$	−28.9112	+	29.5014i	−1.40240	+	1.43103i
$426$		0			0
$427$	20.0766			0.971577
$428$		0			0
$429$	11.1683			0.539210
$430$		0			0
$431$			8.97767i			0.432439i	0.976345	+	0.216219i	$0.0693727\pi$
$431$			8.97767i			0.432439i	−0.976345	+	0.216219i	$0.930627\pi$
$432$		0			0
$433$	6.45697			0.310302			0.155151	−	0.987891i	$-0.450414\pi$
$433$	6.45697			0.310302			0.155151	+	0.987891i	$0.450414\pi$
$434$		0			0
$435$			3.77965i			0.181220i
$436$		0			0
$437$		−	15.2911i		−	0.731473i
$438$		0			0
$439$			1.97734i			0.0943735i	0.998886	+	0.0471868i	$0.0150256\pi$
$439$			1.97734i			0.0943735i	−0.998886	+	0.0471868i	$0.984974\pi$
$440$		0			0
$441$	−4.16693			−0.198425
$442$		0			0
$443$	−30.5708			−1.45246			−0.726230	−	0.687452i	$-0.758729\pi$
$443$	−30.5708			−1.45246			−0.726230	+	0.687452i	$0.758729\pi$
$444$		0			0
$445$			49.1642i			2.33061i
$446$		0			0
$447$		−	8.23153i		−	0.389338i
$448$		0			0
$449$			39.4623i			1.86234i	0.364583	+	0.931171i	$0.381212\pi$
$449$			39.4623i			1.86234i	−0.364583	+	0.931171i	$0.618788\pi$
$450$		0			0
$451$	12.2952			0.578956
$452$		0			0
$453$		−	6.63360i		−	0.311674i
$454$		0			0
$455$	43.1446			2.02265
$456$		0			0
$457$	−15.5916			−0.729344			−0.364672	−	0.931136i	$-0.618819\pi$
$457$	−15.5916			−0.729344			−0.364672	+	0.931136i	$0.618819\pi$
$458$		0			0
$459$	9.86779	+	9.67035i	0.460589	+	0.451373i
$460$		0			0
$461$	23.7521			1.10625			0.553123	−	0.833100i	$-0.313436\pi$
$461$	23.7521			1.10625			0.553123	+	0.833100i	$0.313436\pi$
$462$		0			0
$463$	30.8449			1.43349			0.716743	−	0.697338i	$-0.245632\pi$
$463$	30.8449			1.43349			0.716743	+	0.697338i	$0.245632\pi$
$464$		0			0
$465$		−	17.5205i		−	0.812493i
$466$		0			0
$467$	42.2503			1.95511			0.977555	−	0.210683i	$-0.0675687\pi$
$467$	42.2503			1.95511			0.977555	+	0.210683i	$0.0675687\pi$
$468$		0			0
$469$			37.8538i			1.74792i
$470$		0			0
$471$			1.19335i			0.0549867i
$472$		0			0
$473$			50.8260i			2.33698i
$474$		0			0
$475$	−18.2347			−0.836666
$476$		0			0
$477$	1.24754			0.0571211
$478$		0			0
$479$		−	5.24436i		−	0.239621i	−0.992797	−	0.119810i	$-0.961771\pi$
$479$		−	5.24436i		−	0.239621i	0.992797	−	0.119810i	$-0.0382287\pi$
$480$		0			0
$481$			0.336300i			0.0153340i
$482$		0			0
$483$		−	14.5943i		−	0.664063i
$484$		0			0
$485$	−11.3828			−0.516866
$486$		0			0
$487$			32.3556i			1.46617i	0.680137	+	0.733085i	$0.261920\pi$
$487$			32.3556i			1.46617i	−0.680137	+	0.733085i	$0.738080\pi$
$488$		0			0
$489$	−7.50831			−0.339537
$490$		0			0
$491$	23.9506			1.08088			0.540438	−	0.841384i	$-0.318258\pi$
$491$	23.9506			1.08088			0.540438	+	0.841384i	$0.318258\pi$
$492$		0			0
$493$	4.84089	+	4.74403i	0.218023	+	0.213661i
$494$		0			0
$495$	50.8050			2.28351
$496$		0			0
$497$	−24.4800			−1.09808
$498$		0			0
$499$		−	12.4420i		−	0.556979i	−0.960439	−	0.278489i	$-0.910166\pi$
$499$		−	12.4420i		−	0.556979i	0.960439	−	0.278489i	$-0.0898337\pi$
$500$		0			0
$501$	−10.9886			−0.490934
$502$		0			0
$503$		−	31.5232i		−	1.40555i	−0.711413	−	0.702774i	$-0.751944\pi$
$503$		−	31.5232i		−	1.40555i	0.711413	−	0.702774i	$-0.248056\pi$
$504$		0			0
$505$			48.7270i			2.16833i
$506$		0			0
$507$			0.864311i			0.0383854i
$508$		0			0
$509$	−19.7227			−0.874195			−0.437097	−	0.899414i	$-0.643993\pi$
$509$	−19.7227			−0.874195			−0.437097	+	0.899414i	$0.643993\pi$
$510$		0			0
$511$	20.9650			0.927437
$512$		0			0
$513$			6.09924i			0.269288i
$514$		0			0
$515$			3.56052i			0.156895i
$516$		0			0
$517$		−	5.07809i		−	0.223334i
$518$		0			0
$519$	−13.1735			−0.578254
$520$		0			0
$521$		−	13.4265i		−	0.588228i	−0.955770	−	0.294114i	$-0.904975\pi$
$521$		−	13.4265i		−	0.588228i	0.955770	−	0.294114i	$-0.0950245\pi$
$522$		0			0
$523$	3.84227			0.168011			0.0840054	−	0.996465i	$-0.473229\pi$
$523$	3.84227			0.168011			0.0840054	+	0.996465i	$0.473229\pi$
$524$		0			0
$525$	−17.4037			−0.759562
$526$		0			0
$527$	−22.4398	−	21.9909i	−0.977495	−	0.957937i
$528$		0			0
$529$	−47.5764			−2.06854
$530$		0			0
$531$	2.64800			0.114913
$532$		0			0
$533$			9.44258i			0.409003i
$534$		0			0
$535$	−28.9618			−1.25213
$536$		0			0
$537$		−	10.8713i		−	0.469131i
$538$		0			0
$539$		−	7.79072i		−	0.335570i
$540$		0			0
$541$		−	19.6791i		−	0.846071i	−0.906113	−	0.423036i	$-0.860965\pi$
$541$		−	19.6791i		−	0.846071i	0.906113	−	0.423036i	$-0.139035\pi$
$542$		0			0
$543$	2.96842			0.127387
$544$		0			0
$545$	15.0090			0.642915
$546$		0			0
$547$		−	10.3727i		−	0.443505i	−0.975103	−	0.221753i	$-0.928822\pi$
$547$		−	10.3727i		−	0.443505i	0.975103	−	0.221753i	$-0.0711777\pi$
$548$		0			0
$549$			18.1563i			0.774892i
$550$		0			0
$551$			2.99213i			0.127469i
$552$		0			0
$553$	−24.6172			−1.04683
$554$		0			0
$555$		−	0.203362i		−	0.00863226i
$556$		0			0
$557$	30.8749			1.30821			0.654105	−	0.756404i	$-0.273045\pi$
$557$	30.8749			1.30821			0.654105	+	0.756404i	$0.273045\pi$
$558$		0			0
$559$	−39.0340			−1.65096
$560$		0			0
$561$	−8.47670	+	8.64977i	−0.357887	+	0.365194i
$562$		0			0
$563$	40.0945			1.68978			0.844890	−	0.534940i	$-0.179666\pi$
$563$	40.0945			1.68978			0.844890	+	0.534940i	$0.179666\pi$
$564$		0			0
$565$	20.3845			0.857582
$566$		0			0
$567$		−	17.4393i		−	0.732383i
$568$		0			0
$569$	−42.7366			−1.79161			−0.895806	−	0.444446i	$-0.853400\pi$
$569$	−42.7366			−1.79161			−0.895806	+	0.444446i	$0.853400\pi$
$570$		0			0
$571$			24.8026i			1.03796i	0.854788	+	0.518978i	$0.173687\pi$
$571$			24.8026i			1.03796i	−0.854788	+	0.518978i	$0.826313\pi$
$572$		0			0
$573$			2.09423i			0.0874876i
$574$		0			0
$575$			84.1626i			3.50982i
$576$		0			0
$577$	25.7144			1.07051			0.535253	−	0.844692i	$-0.320216\pi$
$577$	25.7144			1.07051			0.535253	+	0.844692i	$0.320216\pi$
$578$		0			0
$579$	−10.1342			−0.421162
$580$		0			0
$581$		−	8.24582i		−	0.342094i
$582$		0			0
$583$			2.33248i			0.0966014i
$584$		0			0
$585$			39.0178i			1.61319i
$586$		0			0
$587$	29.9337			1.23550			0.617748	−	0.786376i	$-0.288045\pi$
$587$	29.9337			1.23550			0.617748	+	0.786376i	$0.288045\pi$
$588$		0			0
$589$		−	13.8700i		−	0.571502i
$590$		0			0
$591$	−5.43852			−0.223711
$592$		0			0
$593$	13.7668			0.565336			0.282668	−	0.959218i	$-0.408781\pi$
$593$	13.7668			0.565336			0.282668	+	0.959218i	$0.408781\pi$
$594$		0			0
$595$	−32.7467	+	33.4153i	−1.34248	+	1.36989i
$596$		0			0
$597$	2.94715			0.120619
$598$		0			0
$599$	−0.413974			−0.0169145			−0.00845727	−	0.999964i	$-0.502692\pi$
$599$	−0.413974			−0.0169145			−0.00845727	+	0.999964i	$0.502692\pi$
$600$		0			0
$601$		−	9.76784i		−	0.398438i	−0.979955	−	0.199219i	$-0.936159\pi$
$601$		−	9.76784i		−	0.398438i	0.979955	−	0.199219i	$-0.0638406\pi$
$602$		0			0
$603$	−34.2330			−1.39408
$604$		0			0
$605$			52.3591i			2.12870i
$606$		0			0
$607$		−	20.5906i		−	0.835746i	−0.908505	−	0.417873i	$-0.862776\pi$
$607$		−	20.5906i		−	0.835746i	0.908505	−	0.417873i	$-0.137224\pi$
$608$		0			0
$609$			2.85578i			0.115722i
$610$		0			0
$611$	3.89993			0.157774
$612$		0			0
$613$	19.2067			0.775749			0.387875	−	0.921712i	$-0.373209\pi$
$613$	19.2067			0.775749			0.387875	+	0.921712i	$0.373209\pi$
$614$		0			0
$615$		−	5.70998i		−	0.230249i
$616$		0			0
$617$		−	19.2322i		−	0.774257i	−0.922026	−	0.387129i	$-0.873467\pi$
$617$		−	19.2322i		−	0.774257i	0.922026	−	0.387129i	$-0.126533\pi$
$618$		0			0
$619$		−	2.18242i		−	0.0877188i	−0.999038	−	0.0438594i	$-0.986035\pi$
$619$		−	2.18242i		−	0.0877188i	0.999038	−	0.0438594i	$-0.0139654\pi$
$620$		0			0
$621$	28.1511			1.12967
$622$		0			0
$623$			37.1469i			1.48826i
$624$		0			0
$625$	25.2732			1.01093
$626$		0			0
$627$	−5.34639			−0.213514
$628$		0			0
$629$	−0.260462	−	0.255251i	−0.0103853	−	0.0101775i
$630$		0			0
$631$	−49.7845			−1.98189			−0.990944	−	0.134277i	$-0.957129\pi$
$631$	−49.7845			−1.98189			−0.990944	+	0.134277i	$0.957129\pi$
$632$		0			0
$633$	−9.69167			−0.385209
$634$		0			0
$635$		−	32.7945i		−	1.30141i
$636$		0			0
$637$	5.98321			0.237064
$638$		0			0
$639$		−	22.1385i		−	0.875786i
$640$		0			0
$641$			28.1843i			1.11321i	0.830776	+	0.556607i	$0.187897\pi$
$641$			28.1843i			1.11321i	−0.830776	+	0.556607i	$0.812103\pi$
$642$		0			0
$643$			13.8432i			0.545924i	0.962025	+	0.272962i	$0.0880033\pi$
$643$			13.8432i			0.545924i	−0.962025	+	0.272962i	$0.911997\pi$
$644$		0			0
$645$	23.6041			0.929410
$646$		0			0
$647$	14.5492			0.571989			0.285995	−	0.958231i	$-0.407676\pi$
$647$	14.5492			0.571989			0.285995	+	0.958231i	$0.407676\pi$
$648$		0			0
$649$			4.95085i			0.194338i
$650$		0			0
$651$		−	13.2379i		−	0.518834i
$652$		0			0
$653$			5.49220i			0.214926i	0.994209	+	0.107463i	$0.0342728\pi$
$653$			5.49220i			0.214926i	−0.994209	+	0.107463i	$0.965727\pi$
$654$		0			0
$655$	−67.7125			−2.64575
$656$		0			0
$657$			18.9597i			0.739687i
$658$		0			0
$659$	−16.2140			−0.631608			−0.315804	−	0.948824i	$-0.602274\pi$
$659$	−16.2140			−0.631608			−0.315804	+	0.948824i	$0.602274\pi$
$660$		0			0
$661$	9.42581			0.366621			0.183311	−	0.983055i	$-0.441319\pi$
$661$	9.42581			0.366621			0.183311	+	0.983055i	$0.441319\pi$
$662$		0			0
$663$	−6.64296	−	6.51004i	−0.257991	−	0.252829i
$664$		0			0
$665$	−20.6538			−0.800921
$666$		0			0
$667$	13.8102			0.534734
$668$		0			0
$669$		−	3.57754i		−	0.138316i
$670$		0			0
$671$	−33.9460			−1.31047
$672$		0			0
$673$			4.71120i			0.181603i	0.995869	+	0.0908016i	$0.0289429\pi$
$673$			4.71120i			0.181603i	−0.995869	+	0.0908016i	$0.971057\pi$
$674$		0			0
$675$		−	33.5703i		−	1.29212i
$676$		0			0
$677$			1.39320i			0.0535450i	0.999642	+	0.0267725i	$0.00852297\pi$
$677$			1.39320i			0.0535450i	−0.999642	+	0.0267725i	$0.991477\pi$
$678$		0			0
$679$	−8.60046			−0.330055
$680$		0			0
$681$	11.3205			0.433803
$682$		0			0
$683$		−	37.0195i		−	1.41651i	−0.705956	−	0.708256i	$-0.749482\pi$
$683$		−	37.0195i		−	1.41651i	0.705956	−	0.708256i	$-0.250518\pi$
$684$		0			0
$685$		−	8.93908i		−	0.341545i
$686$		0			0
$687$		−	5.89128i		−	0.224766i
$688$		0			0
$689$	−1.79132			−0.0682440
$690$		0			0
$691$		−	34.7332i		−	1.32131i	−0.750688	−	0.660657i	$-0.770278\pi$
$691$		−	34.7332i		−	1.32131i	0.750688	−	0.660657i	$-0.229722\pi$
$692$		0			0
$693$	38.3866			1.45819
$694$		0			0
$695$	−41.8248			−1.58650
$696$		0			0
$697$	−7.31322	−	7.16689i	−0.277008	−	0.271465i
$698$		0			0
$699$	10.1872			0.385316
$700$		0			0
$701$	4.88392			0.184463			0.0922316	−	0.995738i	$-0.470600\pi$
$701$	4.88392			0.184463			0.0922316	+	0.995738i	$0.470600\pi$
$702$		0			0
$703$		−	0.160991i		−	0.00607187i
$704$		0			0
$705$	−2.35831			−0.0888192
$706$		0			0
$707$			36.8166i			1.38463i
$708$		0			0
$709$			48.3839i			1.81710i	0.417781	+	0.908548i	$0.362808\pi$
$709$			48.3839i			1.81710i	−0.417781	+	0.908548i	$0.637192\pi$
$710$		0			0
$711$		−	22.2626i		−	0.834911i
$712$		0			0
$713$	−64.0170			−2.39746
$714$		0			0
$715$	−72.9498			−2.72817
$716$		0			0
$717$			2.55618i			0.0954623i
$718$		0			0
$719$			9.80772i			0.365766i	0.983135	+	0.182883i	$0.0585430\pi$
$719$			9.80772i			0.365766i	−0.983135	+	0.182883i	$0.941457\pi$
$720$		0			0
$721$			2.69022i			0.100189i
$722$		0			0
$723$	16.2646			0.604888
$724$		0			0
$725$		−	16.4688i		−	0.611634i
$726$		0			0
$727$	14.5244			0.538680			0.269340	−	0.963045i	$-0.413194\pi$
$727$	14.5244			0.538680			0.269340	+	0.963045i	$0.413194\pi$
$728$		0			0
$729$	9.80694			0.363220
$730$		0			0
$731$	29.6267	−	30.2316i	1.09578	−	1.11816i
$732$		0			0
$733$	19.1472			0.707216			0.353608	−	0.935394i	$-0.384955\pi$
$733$	19.1472			0.707216			0.353608	+	0.935394i	$0.384955\pi$
$734$		0			0
$735$	−3.61808			−0.133455
$736$		0			0
$737$		−	64.0039i		−	2.35761i
$738$		0			0
$739$	−36.7662			−1.35247			−0.676233	−	0.736688i	$-0.736389\pi$
$739$	−36.7662			−1.35247			−0.676233	+	0.736688i	$0.736389\pi$
$740$		0			0
$741$		−	4.10598i		−	0.150837i
$742$		0			0
$743$			0.481977i			0.0176820i	0.999961	+	0.00884102i	$0.00281422\pi$
$743$			0.481977i			0.0176820i	−0.999961	+	0.00884102i	$0.997186\pi$
$744$		0			0
$745$			53.7673i			1.96988i
$746$		0			0
$747$	7.45710			0.272841
$748$		0			0
$749$	−21.8826			−0.799574
$750$		0			0
$751$		−	34.1238i		−	1.24519i	−0.782543	−	0.622597i	$-0.786078\pi$
$751$		−	34.1238i		−	1.24519i	0.782543	−	0.622597i	$-0.213922\pi$
$752$		0			0
$753$			4.06208i			0.148030i
$754$		0			0
$755$			43.3298i			1.57693i
$756$		0			0
$757$	−15.2759			−0.555212			−0.277606	−	0.960695i	$-0.589541\pi$
$757$	−15.2759			−0.555212			−0.277606	+	0.960695i	$0.589541\pi$
$758$		0			0
$759$			24.6763i			0.895694i
$760$		0			0
$761$	38.2701			1.38729			0.693645	−	0.720317i	$-0.256004\pi$
$761$	38.2701			1.38729			0.693645	+	0.720317i	$0.256004\pi$
$762$		0			0
$763$	11.3403			0.410547
$764$		0			0
$765$	−30.2190	−	29.6144i	−1.09257	−	1.07071i
$766$		0			0
$767$	−3.80221			−0.137290
$768$		0			0
$769$	39.8202			1.43595			0.717976	−	0.696068i	$-0.245069\pi$
$769$	39.8202			1.43595			0.717976	+	0.696068i	$0.245069\pi$
$770$		0			0
$771$			12.0189i			0.432848i
$772$		0			0
$773$	21.8048			0.784265			0.392133	−	0.919909i	$-0.371737\pi$
$773$	21.8048			0.784265			0.392133	+	0.919909i	$0.371737\pi$
$774$		0			0
$775$			76.3405i			2.74223i
$776$		0			0
$777$		−	0.153654i		−	0.00551231i
$778$		0			0
$779$		−	4.52027i		−	0.161955i
$780$		0			0
$781$	41.3914			1.48110
$782$		0			0
$783$	−5.50855			−0.196860
$784$		0			0
$785$		−	7.79481i		−	0.278209i
$786$		0			0
$787$			36.9239i			1.31619i	0.752933	+	0.658097i	$0.228638\pi$
$787$			36.9239i			1.31619i	−0.752933	+	0.658097i	$0.771362\pi$
$788$		0			0
$789$		−	0.681214i		−	0.0242518i
$790$		0			0
$791$	15.4018			0.547627
$792$		0			0
$793$		−	26.0702i		−	0.925782i
$794$		0			0
$795$	1.08322			0.0384180
$796$		0			0
$797$	49.7539			1.76237			0.881187	−	0.472768i	$-0.156745\pi$
$797$	49.7539			1.76237			0.881187	+	0.472768i	$0.156745\pi$
$798$		0			0
$799$	−2.96004	+	3.02047i	−0.104719	+	0.106857i
$800$		0			0
$801$	−33.5937			−1.18698
$802$		0			0
$803$	−35.4480			−1.25093
$804$		0			0
$805$			95.3280i			3.35987i
$806$		0			0
$807$	−10.3671			−0.364938
$808$		0			0
$809$			3.35385i			0.117915i	0.998260	+	0.0589575i	$0.0187777\pi$
$809$			3.35385i			0.117915i	−0.998260	+	0.0589575i	$0.981222\pi$
$810$		0			0
$811$		−	29.9645i		−	1.05220i	−0.850424	−	0.526098i	$-0.823655\pi$
$811$		−	29.9645i		−	1.05220i	0.850424	−	0.526098i	$-0.176345\pi$
$812$		0			0
$813$		−	0.516812i		−	0.0181254i
$814$		0			0
$815$	49.0433			1.71791
$816$		0			0
$817$	18.6860			0.653741
$818$		0			0
$819$			29.4806i			1.03013i
$820$		0			0
$821$		−	50.6512i		−	1.76774i	−0.467733	−	0.883870i	$-0.654929\pi$
$821$		−	50.6512i		−	1.76774i	0.467733	−	0.883870i	$-0.345071\pi$
$822$		0			0
$823$			34.4677i			1.20147i	0.799449	+	0.600734i	$0.205125\pi$
$823$			34.4677i			1.20147i	−0.799449	+	0.600734i	$0.794875\pi$
$824$		0			0
$825$	29.4266			1.02450
$826$		0			0
$827$			52.8530i			1.83788i	0.394400	+	0.918939i	$0.370953\pi$
$827$			52.8530i			1.83788i	−0.394400	+	0.918939i	$0.629047\pi$
$828$		0			0
$829$	−45.8961			−1.59404			−0.797019	−	0.603954i	$-0.793591\pi$
$829$	−45.8961			−1.59404			−0.797019	+	0.603954i	$0.793591\pi$
$830$		0			0
$831$	8.24304			0.285948
$832$		0			0
$833$	−4.54124	+	4.63396i	−0.157345	+	0.160557i
$834$		0			0
$835$	71.7760			2.48391
$836$		0			0
$837$	25.5348			0.882611
$838$		0			0
$839$		−	22.5261i		−	0.777686i	−0.921304	−	0.388843i	$-0.872875\pi$
$839$		−	22.5261i		−	0.777686i	0.921304	−	0.388843i	$-0.127125\pi$
$840$		0			0
$841$	26.2976			0.906815
$842$		0			0
$843$			3.61185i			0.124399i
$844$		0			0
$845$		−	5.64557i		−	0.194213i
$846$		0			0
$847$			39.5608i			1.35933i
$848$		0			0
$849$	16.3561			0.561339
$850$		0			0
$851$	−0.743054			−0.0254716
$852$		0			0
$853$			42.2949i			1.44815i	0.689721	+	0.724075i	$0.257733\pi$
$853$			42.2949i			1.44815i	−0.689721	+	0.724075i	$0.742267\pi$
$854$		0			0
$855$		−	18.6783i		−	0.638783i
$856$		0			0
$857$			43.4900i			1.48559i	0.669520	+	0.742794i	$0.266500\pi$
$857$			43.4900i			1.48559i	−0.669520	+	0.742794i	$0.733500\pi$
$858$		0			0
$859$	−23.7871			−0.811606			−0.405803	−	0.913961i	$-0.633008\pi$
$859$	−23.7871			−0.811606			−0.405803	+	0.913961i	$0.633008\pi$
$860$		0			0
$861$		−	4.31427i		−	0.147030i
$862$		0			0
$863$	6.34212			0.215888			0.107944	−	0.994157i	$-0.465573\pi$
$863$	6.34212			0.215888			0.107944	+	0.994157i	$0.465573\pi$
$864$		0			0
$865$	86.0477			2.92571
$866$		0			0
$867$	10.0840	−	0.203824i	0.342470	−	0.00692223i
$868$		0			0
$869$	41.6233			1.41197
$870$		0			0
$871$	49.1545			1.66554
$872$		0			0
$873$		−	7.77782i		−	0.263239i
$874$		0			0
$875$	56.9427			1.92501
$876$		0			0
$877$		−	35.9426i		−	1.21369i	−0.794819	−	0.606847i	$-0.792434\pi$
$877$		−	35.9426i		−	1.21369i	0.794819	−	0.606847i	$-0.207566\pi$
$878$		0			0
$879$		−	3.18512i		−	0.107432i
$880$		0			0
$881$			35.4598i			1.19467i	0.801992	+	0.597335i	$0.203774\pi$
$881$			35.4598i			1.19467i	−0.801992	+	0.597335i	$0.796226\pi$
$882$		0			0
$883$	−33.0617			−1.11262			−0.556308	−	0.830976i	$-0.687782\pi$
$883$	−33.0617			−1.11262			−0.556308	+	0.830976i	$0.687782\pi$
$884$		0			0
$885$	2.29922			0.0772874
$886$		0			0
$887$		−	58.1191i		−	1.95145i	−0.219006	−	0.975723i	$-0.570281\pi$
$887$		−	58.1191i		−	1.95145i	0.219006	−	0.975723i	$-0.429719\pi$
$888$		0			0
$889$		−	24.7785i		−	0.831044i
$890$		0			0
$891$			29.4868i			0.987844i
$892$		0			0
$893$	−1.86694			−0.0624748
$894$		0			0
$895$			71.0098i			2.37360i
$896$		0			0
$897$	−18.9512			−0.632763
$898$		0			0
$899$	12.5267			0.417790
$900$		0			0
$901$	1.35961	−	1.38737i	0.0452952	−	0.0462200i
$902$		0			0
$903$	17.8345			0.593494
$904$		0			0
$905$	−19.3893			−0.644523
$906$		0			0
$907$		−	28.3702i		−	0.942016i	−0.882129	−	0.471008i	$-0.843890\pi$
$907$		−	28.3702i		−	0.942016i	0.882129	−	0.471008i	$-0.156110\pi$
$908$		0			0
$909$	−33.2950			−1.10433
$910$		0			0
$911$		−	7.65666i		−	0.253676i	−0.991923	−	0.126838i	$-0.959517\pi$
$911$		−	7.65666i		−	0.253676i	0.991923	−	0.126838i	$-0.0404829\pi$
$912$		0			0
$913$			13.9422i			0.461419i
$914$		0			0
$915$			15.7648i			0.521169i
$916$		0			0
$917$	−51.1613			−1.68950
$918$		0			0
$919$	41.8360			1.38004			0.690021	−	0.723790i	$-0.257601\pi$
$919$	41.8360			1.38004			0.690021	+	0.723790i	$0.257601\pi$
$920$		0			0
$921$			9.20938i			0.303459i
$922$		0			0
$923$			31.7882i			1.04632i
$924$		0			0
$925$			0.886094i			0.0291346i
$926$		0			0
$927$	−2.43289			−0.0799067
$928$		0			0
$929$			19.5569i			0.641640i	0.947140	+	0.320820i	$0.103958\pi$
$929$			19.5569i			0.641640i	−0.947140	+	0.320820i	$0.896042\pi$
$930$		0			0
$931$	−2.86423			−0.0938714
$932$		0			0
$933$	16.3669			0.535829
$934$		0			0
$935$	55.3687	−	56.4992i	1.81075	−	1.84772i
$936$		0			0
$937$	−32.9555			−1.07661			−0.538305	−	0.842750i	$-0.680935\pi$
$937$	−32.9555			−1.07661			−0.538305	+	0.842750i	$0.680935\pi$
$938$		0			0
$939$	−15.5476			−0.507376
$940$		0			0
$941$			23.6300i			0.770316i	0.922851	+	0.385158i	$0.125853\pi$
$941$			23.6300i			0.770316i	−0.922851	+	0.385158i	$0.874147\pi$
$942$		0			0
$943$	−20.8634			−0.679404
$944$		0			0
$945$		−	38.0239i		−	1.23692i
$946$		0			0
$947$			23.0768i			0.749896i	0.927046	+	0.374948i	$0.122339\pi$
$947$			23.0768i			0.749896i	−0.927046	+	0.374948i	$0.877661\pi$
$948$		0			0
$949$		−	27.2238i		−	0.883722i
$950$		0			0
$951$	7.05871			0.228894
$952$		0			0
$953$	28.2644			0.915575			0.457787	−	0.889062i	$-0.348642\pi$
$953$	28.2644			0.915575			0.457787	+	0.889062i	$0.348642\pi$
$954$		0			0
$955$		−	13.6792i		−	0.442649i
$956$		0			0
$957$		−	4.82861i		−	0.156087i
$958$		0			0
$959$		−	6.75407i		−	0.218100i
$960$		0			0
$961$	−27.0673			−0.873140
$962$		0			0
$963$		−	19.7895i		−	0.637709i
$964$		0			0
$965$	66.1952			2.13090
$966$		0			0
$967$	−41.8946			−1.34724			−0.673620	−	0.739078i	$-0.735262\pi$
$967$	−41.8946			−1.34724			−0.673620	+	0.739078i	$0.735262\pi$
$968$		0			0
$969$	3.18006	+	3.11643i	0.102158	+	0.100114i
$970$		0			0
$971$	35.1614			1.12838			0.564192	−	0.825643i	$-0.309188\pi$
$971$	35.1614			1.12838			0.564192	+	0.825643i	$0.309188\pi$
$972$		0			0
$973$	−31.6014			−1.01310
$974$		0			0
$975$			22.5994i			0.723760i
$976$		0			0
$977$	24.7177			0.790789			0.395395	−	0.918511i	$-0.370608\pi$
$977$	24.7177			0.790789			0.395395	+	0.918511i	$0.370608\pi$
$978$		0			0
$979$		−	62.8087i		−	2.00737i
$980$		0			0
$981$			10.2556i			0.327436i
$982$		0			0
$983$			24.5831i			0.784078i	0.919949	+	0.392039i	$0.128230\pi$
$983$			24.5831i			0.784078i	−0.919949	+	0.392039i	$0.871770\pi$
$984$		0			0
$985$	35.5237			1.13188
$986$		0			0
$987$	−1.78186			−0.0567174
$988$		0			0
$989$		−	86.2455i		−	2.74245i
$990$		0			0
$991$			19.2420i			0.611241i	0.952153	+	0.305621i	$0.0988639\pi$
$991$			19.2420i			0.611241i	−0.952153	+	0.305621i	$0.901136\pi$
$992$		0			0
$993$		−	12.7263i		−	0.403858i
$994$		0			0
$995$	−19.2504			−0.610279
$996$		0			0
$997$			1.95987i			0.0620696i	0.999518	+	0.0310348i	$0.00988028\pi$
$997$			1.95987i			0.0620696i	−0.999518	+	0.0310348i	$0.990120\pi$
$998$		0			0
$999$	0.296385			0.00937722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.28	yes	46
17.16	even	2	inner	4012.2.b.b.237.19	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.19	✓	46	17.16	even	2	inner
4012.2.b.b.237.28	yes	46	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.593296i q^{3} -3.87533i q^{5} -2.92807i q^{7} +2.64800 q^{9} +O(q^{10})\)	\(q+0.593296i q^{3} -3.87533i q^{5} -2.92807i q^{7} +2.64800 q^{9} +4.95085i q^{11} -3.80221 q^{13} +2.29922 q^{15} +(2.88587 - 2.94479i) q^{17} +1.82016 q^{19} +1.73721 q^{21} -8.40097i q^{23} -10.0182 q^{25} +3.35094i q^{27} +1.64389i q^{29} -7.62019i q^{31} -2.93732 q^{33} -11.3473 q^{35} -0.0884485i q^{37} -2.25584i q^{39} -2.48344i q^{41} +10.2661 q^{43} -10.2619i q^{45} -1.02570 q^{47} -1.57361 q^{49} +(1.74713 + 1.71217i) q^{51} +0.471127 q^{53} +19.1862 q^{55} +1.07989i q^{57} +1.00000 q^{59} +6.85661i q^{61} -7.75354i q^{63} +14.7348i q^{65} -12.9279 q^{67} +4.98426 q^{69} -8.36046i q^{71} +7.16000i q^{73} -5.94375i q^{75} +14.4964 q^{77} -8.40731i q^{79} +5.95590 q^{81} +2.81612 q^{83} +(-11.4120 - 11.1837i) q^{85} -0.975310 q^{87} -12.6865 q^{89} +11.1331i q^{91} +4.52103 q^{93} -7.05372i q^{95} -2.93724i q^{97} +13.1098i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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