LMFDB - Embedded newform 4012.2.b.b.237.23

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			237.23
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.123544i q^{3} -0.454269i q^{5} -3.56076i q^{7} +2.98474 q^{9} +O(q^{10})$	$q-0.123544i q^{3} -0.454269i q^{5} -3.56076i q^{7} +2.98474 q^{9} -2.55914i q^{11} +1.05907 q^{13} -0.0561222 q^{15} +(2.55663 - 3.23476i) q^{17} -5.45088 q^{19} -0.439910 q^{21} -5.82839i q^{23} +4.79364 q^{25} -0.739378i q^{27} -10.2975i q^{29} +0.560481i q^{31} -0.316166 q^{33} -1.61754 q^{35} +10.4994i q^{37} -0.130842i q^{39} +6.36520i q^{41} -9.85998 q^{43} -1.35587i q^{45} -4.05578 q^{47} -5.67898 q^{49} +(-0.399634 - 0.315856i) q^{51} +10.3974 q^{53} -1.16254 q^{55} +0.673423i q^{57} +1.00000 q^{59} +6.58255i q^{61} -10.6279i q^{63} -0.481103i q^{65} -2.95294 q^{67} -0.720062 q^{69} -4.41564i q^{71} -7.04827i q^{73} -0.592225i q^{75} -9.11246 q^{77} +7.96911i q^{79} +8.86287 q^{81} +11.8663 q^{83} +(-1.46945 - 1.16140i) q^{85} -1.27220 q^{87} -18.3374 q^{89} -3.77109i q^{91} +0.0692441 q^{93} +2.47617i q^{95} -9.68685i q^{97} -7.63835i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.123544i		−	0.0713281i	−0.999364	−	0.0356641i	$-0.988645\pi$
$3$		−	0.123544i		−	0.0713281i	0.999364	−	0.0356641i	$-0.0113546\pi$
$4$		0			0
$5$		−	0.454269i		−	0.203155i	−0.994828	−	0.101578i	$-0.967611\pi$
$5$		−	0.454269i		−	0.203155i	0.994828	−	0.101578i	$-0.0323891\pi$
$6$		0			0
$7$		−	3.56076i		−	1.34584i	−0.739716	−	0.672920i	$-0.765040\pi$
$7$		−	3.56076i		−	1.34584i	0.739716	−	0.672920i	$-0.234960\pi$
$8$		0			0
$9$	2.98474			0.994912
$10$		0			0
$11$		−	2.55914i		−	0.771609i	−0.922581	−	0.385805i	$-0.873924\pi$
$11$		−	2.55914i		−	0.771609i	0.922581	−	0.385805i	$-0.126076\pi$
$12$		0			0
$13$	1.05907			0.293733			0.146867	−	0.989156i	$-0.453081\pi$
$13$	1.05907			0.293733			0.146867	+	0.989156i	$0.453081\pi$
$14$		0			0
$15$	−0.0561222			−0.0144907
$16$		0			0
$17$	2.55663	−	3.23476i	0.620074	−	0.784543i
$17$	2.55663	−	3.23476i	0.620074	−	0.784543i
$18$		0			0
$19$	−5.45088			−1.25052			−0.625258	−	0.780418i	$-0.715006\pi$
$19$	−5.45088			−1.25052			−0.625258	+	0.780418i	$0.715006\pi$
$20$		0			0
$21$	−0.439910			−0.0959962
$22$		0			0
$23$		−	5.82839i		−	1.21530i	−0.794204	−	0.607652i	$-0.792112\pi$
$23$		−	5.82839i		−	1.21530i	0.794204	−	0.607652i	$-0.207888\pi$
$24$		0			0
$25$	4.79364			0.958728
$26$		0			0
$27$		−	0.739378i		−	0.142293i
$28$		0			0
$29$		−	10.2975i		−	1.91220i	−0.293034	−	0.956102i	$-0.594665\pi$
$29$		−	10.2975i		−	1.91220i	0.293034	−	0.956102i	$-0.405335\pi$
$30$		0			0
$31$			0.560481i			0.100665i	0.998733	+	0.0503327i	$0.0160282\pi$
$31$			0.560481i			0.100665i	−0.998733	+	0.0503327i	$0.983972\pi$
$32$		0			0
$33$	−0.316166			−0.0550374
$34$		0			0
$35$	−1.61754			−0.273415
$36$		0			0
$37$			10.4994i			1.72610i	0.505122	+	0.863048i	$0.331448\pi$
$37$			10.4994i			1.72610i	−0.505122	+	0.863048i	$0.668552\pi$
$38$		0			0
$39$		−	0.130842i		−	0.0209514i
$40$		0			0
$41$			6.36520i			0.994078i	0.867728	+	0.497039i	$0.165579\pi$
$41$			6.36520i			0.994078i	−0.867728	+	0.497039i	$0.834421\pi$
$42$		0			0
$43$	−9.85998			−1.50363			−0.751816	−	0.659372i	$-0.770822\pi$
$43$	−9.85998			−1.50363			−0.751816	+	0.659372i	$0.770822\pi$
$44$		0			0
$45$		−	1.35587i		−	0.202122i
$46$		0			0
$47$	−4.05578			−0.591597			−0.295798	−	0.955250i	$-0.595586\pi$
$47$	−4.05578			−0.591597			−0.295798	+	0.955250i	$0.595586\pi$
$48$		0			0
$49$	−5.67898			−0.811283
$50$		0			0
$51$	−0.399634	−	0.315856i	−0.0559600	−	0.0442287i
$52$		0			0
$53$	10.3974			1.42820			0.714099	−	0.700044i	$-0.246836\pi$
$53$	10.3974			1.42820			0.714099	+	0.700044i	$0.246836\pi$
$54$		0			0
$55$	−1.16254			−0.156757
$56$		0			0
$57$			0.673423i			0.0891970i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			6.58255i			0.842809i	0.906873	+	0.421404i	$0.138463\pi$
$61$			6.58255i			0.842809i	−0.906873	+	0.421404i	$0.861537\pi$
$62$		0			0
$63$		−	10.6279i		−	1.33899i
$64$		0			0
$65$		−	0.481103i		−	0.0596735i
$66$		0			0
$67$	−2.95294			−0.360759			−0.180379	−	0.983597i	$-0.557733\pi$
$67$	−2.95294			−0.360759			−0.180379	+	0.983597i	$0.557733\pi$
$68$		0			0
$69$	−0.720062			−0.0866853
$70$		0			0
$71$		−	4.41564i		−	0.524040i	−0.965062	−	0.262020i	$-0.915611\pi$
$71$		−	4.41564i		−	0.524040i	0.965062	−	0.262020i	$-0.0843887\pi$
$72$		0			0
$73$		−	7.04827i		−	0.824938i	−0.910972	−	0.412469i	$-0.864667\pi$
$73$		−	7.04827i		−	0.824938i	0.910972	−	0.412469i	$-0.135333\pi$
$74$		0			0
$75$		−	0.592225i		−	0.0683842i
$76$		0			0
$77$	−9.11246			−1.03846
$78$		0			0
$79$			7.96911i			0.896595i	0.893884	+	0.448297i	$0.147969\pi$
$79$			7.96911i			0.896595i	−0.893884	+	0.448297i	$0.852031\pi$
$80$		0			0
$81$	8.86287			0.984763
$82$		0			0
$83$	11.8663			1.30249			0.651247	−	0.758866i	$-0.274246\pi$
$83$	11.8663			1.30249			0.651247	+	0.758866i	$0.274246\pi$
$84$		0			0
$85$	−1.46945	−	1.16140i	−0.159384	−	0.125971i
$86$		0			0
$87$	−1.27220			−0.136394
$88$		0			0
$89$	−18.3374			−1.94376			−0.971880	−	0.235475i	$-0.924335\pi$
$89$	−18.3374			−1.94376			−0.971880	+	0.235475i	$0.924335\pi$
$90$		0			0
$91$		−	3.77109i		−	0.395318i
$92$		0			0
$93$	0.0692441			0.00718028
$94$		0			0
$95$			2.47617i			0.254049i
$96$		0			0
$97$		−	9.68685i		−	0.983551i	−0.870722	−	0.491775i	$-0.836348\pi$
$97$		−	9.68685i		−	0.983551i	0.870722	−	0.491775i	$-0.163652\pi$
$98$		0			0
$99$		−	7.63835i		−	0.767683i
$100$		0			0
$101$	10.7833			1.07297			0.536487	−	0.843909i	$-0.319751\pi$
$101$	10.7833			1.07297			0.536487	+	0.843909i	$0.319751\pi$
$102$		0			0
$103$	−2.61985			−0.258141			−0.129071	−	0.991635i	$-0.541199\pi$
$103$	−2.61985			−0.258141			−0.129071	+	0.991635i	$0.541199\pi$
$104$		0			0
$105$			0.199838i			0.0195021i
$106$		0			0
$107$			14.1782i			1.37066i	0.728234	+	0.685329i	$0.240342\pi$
$107$			14.1782i			1.37066i	−0.728234	+	0.685329i	$0.759658\pi$
$108$		0			0
$109$		−	2.82164i		−	0.270264i	−0.990828	−	0.135132i	$-0.956854\pi$
$109$		−	2.82164i		−	0.270264i	0.990828	−	0.135132i	$-0.0431459\pi$
$110$		0			0
$111$	1.29714			0.123119
$112$		0			0
$113$		−	7.43401i		−	0.699333i	−0.936874	−	0.349666i	$-0.886295\pi$
$113$		−	7.43401i		−	0.699333i	0.936874	−	0.349666i	$-0.113705\pi$
$114$		0			0
$115$	−2.64766			−0.246896
$116$		0			0
$117$	3.16104			0.292239
$118$		0			0
$119$	−11.5182	−	9.10354i	−1.05587	−	0.834520i
$120$		0			0
$121$	4.45081			0.404619
$122$		0			0
$123$	0.786382			0.0709057
$124$		0			0
$125$		−	4.44895i		−	0.397926i
$126$		0			0
$127$	−19.0541			−1.69078			−0.845388	−	0.534153i	$-0.820631\pi$
$127$	−19.0541			−1.69078			−0.845388	+	0.534153i	$0.820631\pi$
$128$		0			0
$129$			1.21814i			0.107251i
$130$		0			0
$131$			17.4469i			1.52434i	0.647377	+	0.762170i	$0.275866\pi$
$131$			17.4469i			1.52434i	−0.647377	+	0.762170i	$0.724134\pi$
$132$		0			0
$133$			19.4092i			1.68299i
$134$		0			0
$135$	−0.335877			−0.0289077
$136$		0			0
$137$	−15.2168			−1.30006			−0.650029	−	0.759910i	$-0.725243\pi$
$137$	−15.2168			−1.30006			−0.650029	+	0.759910i	$0.725243\pi$
$138$		0			0
$139$			12.3854i			1.05052i	0.850943	+	0.525258i	$0.176031\pi$
$139$			12.3854i			1.05052i	−0.850943	+	0.525258i	$0.823969\pi$
$140$		0			0
$141$			0.501067i			0.0421975i
$142$		0			0
$143$		−	2.71031i		−	0.226647i
$144$		0			0
$145$	−4.67786			−0.388475
$146$		0			0
$147$			0.701603i			0.0578673i
$148$		0			0
$149$	7.21156			0.590794			0.295397	−	0.955375i	$-0.404548\pi$
$149$	7.21156			0.590794			0.295397	+	0.955375i	$0.404548\pi$
$150$		0			0
$151$	−19.9609			−1.62440			−0.812199	−	0.583381i	$-0.801730\pi$
$151$	−19.9609			−1.62440			−0.812199	+	0.583381i	$0.801730\pi$
$152$		0			0
$153$	7.63087	−	9.65489i	0.616919	−	0.780552i
$154$		0			0
$155$	0.254610			0.0204507
$156$		0			0
$157$	15.9907			1.27619			0.638096	−	0.769956i	$-0.279722\pi$
$157$	15.9907			1.27619			0.638096	+	0.769956i	$0.279722\pi$
$158$		0			0
$159$		−	1.28454i		−	0.101871i
$160$		0			0
$161$	−20.7535			−1.63560
$162$		0			0
$163$			1.98112i			0.155173i	0.996986	+	0.0775867i	$0.0247215\pi$
$163$			1.98112i			0.155173i	−0.996986	+	0.0775867i	$0.975279\pi$
$164$		0			0
$165$			0.143625i			0.0111812i
$166$		0			0
$167$		−	3.16345i		−	0.244795i	−0.992481	−	0.122397i	$-0.960942\pi$
$167$		−	3.16345i		−	0.244795i	0.992481	−	0.122397i	$-0.0390583\pi$
$168$		0			0
$169$	−11.8784			−0.913721
$170$		0			0
$171$	−16.2694			−1.24415
$172$		0			0
$173$			17.3966i			1.32264i	0.750105	+	0.661319i	$0.230003\pi$
$173$			17.3966i			1.32264i	−0.750105	+	0.661319i	$0.769997\pi$
$174$		0			0
$175$		−	17.0690i		−	1.29029i
$176$		0			0
$177$		−	0.123544i		−	0.00928613i
$178$		0			0
$179$	12.5748			0.939887			0.469943	−	0.882697i	$-0.344274\pi$
$179$	12.5748			0.939887			0.469943	+	0.882697i	$0.344274\pi$
$180$		0			0
$181$		−	22.8834i		−	1.70091i	−0.526051	−	0.850453i	$-0.676328\pi$
$181$		−	22.8834i		−	1.70091i	0.526051	−	0.850453i	$-0.323672\pi$
$182$		0			0
$183$	0.813233			0.0601159
$184$		0			0
$185$	4.76957			0.350666
$186$		0			0
$187$	−8.27818	−	6.54277i	−0.605361	−	0.478455i
$188$		0			0
$189$	−2.63274			−0.191504
$190$		0			0
$191$	11.5865			0.838372			0.419186	−	0.907900i	$-0.362316\pi$
$191$	11.5865			0.838372			0.419186	+	0.907900i	$0.362316\pi$
$192$		0			0
$193$		−	12.8891i		−	0.927777i	−0.885893	−	0.463889i	$-0.846454\pi$
$193$		−	12.8891i		−	0.927777i	0.885893	−	0.463889i	$-0.153546\pi$
$194$		0			0
$195$	−0.0594374			−0.00425640
$196$		0			0
$197$		−	21.4768i		−	1.53016i	−0.643937	−	0.765078i	$-0.722700\pi$
$197$		−	21.4768i		−	1.53016i	0.643937	−	0.765078i	$-0.277300\pi$
$198$		0			0
$199$		−	8.87015i		−	0.628788i	−0.949293	−	0.314394i	$-0.898199\pi$
$199$		−	8.87015i		−	0.628788i	0.949293	−	0.314394i	$-0.101801\pi$
$200$		0			0
$201$			0.364817i			0.0257322i
$202$		0			0
$203$	−36.6670			−2.57352
$204$		0			0
$205$	2.89152			0.201952
$206$		0			0
$207$		−	17.3962i		−	1.20912i
$208$		0			0
$209$			13.9495i			0.964910i
$210$		0			0
$211$		−	23.9563i		−	1.64922i	−0.565701	−	0.824610i	$-0.691394\pi$
$211$		−	23.9563i		−	1.64922i	0.565701	−	0.824610i	$-0.308606\pi$
$212$		0			0
$213$	−0.545526			−0.0373788
$214$		0			0
$215$			4.47909i			0.305471i
$216$		0			0
$217$	1.99574			0.135479
$218$		0			0
$219$	−0.870771			−0.0588413
$220$		0			0
$221$	2.70765	−	3.42583i	0.182136	−	0.230446i
$222$		0			0
$223$	16.6350			1.11396			0.556981	−	0.830525i	$-0.311960\pi$
$223$	16.6350			1.11396			0.556981	+	0.830525i	$0.311960\pi$
$224$		0			0
$225$	14.3078			0.953850
$226$		0			0
$227$		−	10.8491i		−	0.720079i	−0.932937	−	0.360039i	$-0.882763\pi$
$227$		−	10.8491i		−	0.720079i	0.932937	−	0.360039i	$-0.117237\pi$
$228$		0			0
$229$	−8.90060			−0.588168			−0.294084	−	0.955780i	$-0.595015\pi$
$229$	−8.90060			−0.588168			−0.294084	+	0.955780i	$0.595015\pi$
$230$		0			0
$231$			1.12579i			0.0740715i
$232$		0			0
$233$			4.73114i			0.309947i	0.987919	+	0.154974i	$0.0495292\pi$
$233$			4.73114i			0.309947i	−0.987919	+	0.154974i	$0.950471\pi$
$234$		0			0
$235$			1.84242i			0.120186i
$236$		0			0
$237$	0.984535			0.0639524
$238$		0			0
$239$	−0.214811			−0.0138950			−0.00694748	−	0.999976i	$-0.502211\pi$
$239$	−0.214811			−0.0138950			−0.00694748	+	0.999976i	$0.502211\pi$
$240$		0			0
$241$		−	17.5712i		−	1.13186i	−0.824454	−	0.565929i	$-0.808518\pi$
$241$		−	17.5712i		−	1.13186i	0.824454	−	0.565929i	$-0.191482\pi$
$242$		0			0
$243$		−	3.31309i		−	0.212535i
$244$		0			0
$245$			2.57979i			0.164817i
$246$		0			0
$247$	−5.77286			−0.367318
$248$		0			0
$249$		−	1.46601i		−	0.0929044i
$250$		0			0
$251$	−22.3649			−1.41166			−0.705830	−	0.708382i	$-0.749426\pi$
$251$	−22.3649			−1.41166			−0.705830	+	0.708382i	$0.749426\pi$
$252$		0			0
$253$	−14.9157			−0.937739
$254$		0			0
$255$	−0.143484	+	0.181542i	−0.00898530	+	0.0113686i
$256$		0			0
$257$	6.34853			0.396010			0.198005	−	0.980201i	$-0.436554\pi$
$257$	6.34853			0.396010			0.198005	+	0.980201i	$0.436554\pi$
$258$		0			0
$259$	37.3859			2.32305
$260$		0			0
$261$		−	30.7354i		−	1.90248i
$262$		0			0
$263$	1.36608			0.0842362			0.0421181	−	0.999113i	$-0.486589\pi$
$263$	1.36608			0.0842362			0.0421181	+	0.999113i	$0.486589\pi$
$264$		0			0
$265$		−	4.72324i		−	0.290146i
$266$		0			0
$267$			2.26547i			0.138645i
$268$		0			0
$269$			30.0617i			1.83289i	0.400159	+	0.916446i	$0.368955\pi$
$269$			30.0617i			1.83289i	−0.400159	+	0.916446i	$0.631045\pi$
$270$		0			0
$271$	26.7633			1.62575			0.812876	−	0.582436i	$-0.197900\pi$
$271$	26.7633			1.62575			0.812876	+	0.582436i	$0.197900\pi$
$272$		0			0
$273$	−0.465895			−0.0281973
$274$		0			0
$275$		−	12.2676i		−	0.739763i
$276$		0			0
$277$			3.74897i			0.225254i	0.993637	+	0.112627i	$0.0359265\pi$
$277$			3.74897i			0.225254i	−0.993637	+	0.112627i	$0.964073\pi$
$278$		0			0
$279$			1.67289i			0.100153i
$280$		0			0
$281$	−23.3772			−1.39457			−0.697283	−	0.716796i	$-0.745608\pi$
$281$	−23.3772			−1.39457			−0.697283	+	0.716796i	$0.745608\pi$
$282$		0			0
$283$			13.9107i			0.826906i	0.910525	+	0.413453i	$0.135677\pi$
$283$			13.9107i			0.826906i	−0.910525	+	0.413453i	$0.864323\pi$
$284$		0			0
$285$	0.305915			0.0181209
$286$		0			0
$287$	22.6649			1.33787
$288$		0			0
$289$	−3.92728	−	16.5401i	−0.231017	−	0.972950i
$290$		0			0
$291$	−1.19675			−0.0701548
$292$		0			0
$293$	−14.5996			−0.852916			−0.426458	−	0.904507i	$-0.640239\pi$
$293$	−14.5996			−0.852916			−0.426458	+	0.904507i	$0.640239\pi$
$294$		0			0
$295$		−	0.454269i		−	0.0264486i
$296$		0			0
$297$	−1.89217			−0.109795
$298$		0			0
$299$		−	6.17267i		−	0.356975i
$300$		0			0
$301$			35.1090i			2.02365i
$302$		0			0
$303$		−	1.33221i		−	0.0765332i
$304$		0			0
$305$	2.99025			0.171221
$306$		0			0
$307$	14.6278			0.834854			0.417427	−	0.908710i	$-0.362932\pi$
$307$	14.6278			0.834854			0.417427	+	0.908710i	$0.362932\pi$
$308$		0			0
$309$			0.323666i			0.0184127i
$310$		0			0
$311$			23.1670i			1.31368i	0.754030	+	0.656840i	$0.228108\pi$
$311$			23.1670i			1.31368i	−0.754030	+	0.656840i	$0.771892\pi$
$312$		0			0
$313$		−	30.7854i		−	1.74009i	−0.492968	−	0.870047i	$-0.664088\pi$
$313$		−	30.7854i		−	1.74009i	0.492968	−	0.870047i	$-0.335912\pi$
$314$		0			0
$315$	−4.82794			−0.272024
$316$		0			0
$317$		−	4.28361i		−	0.240591i	−0.992738	−	0.120296i	$-0.961616\pi$
$317$		−	4.28361i		−	0.240591i	0.992738	−	0.120296i	$-0.0383843\pi$
$318$		0			0
$319$	−26.3528			−1.47547
$320$		0			0
$321$	1.75163			0.0977664
$322$		0			0
$323$	−13.9359	+	17.6322i	−0.775413	+	0.981085i
$324$		0			0
$325$	5.07680			0.281610
$326$		0			0
$327$	−0.348597			−0.0192774
$328$		0			0
$329$			14.4417i			0.796194i
$330$		0			0
$331$	4.61534			0.253682			0.126841	−	0.991923i	$-0.459516\pi$
$331$	4.61534			0.253682			0.126841	+	0.991923i	$0.459516\pi$
$332$		0			0
$333$			31.3380i			1.71731i
$334$		0			0
$335$			1.34143i			0.0732901i
$336$		0			0
$337$		−	4.81381i		−	0.262225i	−0.991368	−	0.131112i	$-0.958145\pi$
$337$		−	4.81381i		−	0.262225i	0.991368	−	0.131112i	$-0.0418549\pi$
$338$		0			0
$339$	−0.918426			−0.0498821
$340$		0			0
$341$	1.43435			0.0776744
$342$		0			0
$343$		−	4.70383i		−	0.253983i
$344$		0			0
$345$			0.327102i			0.0176106i
$346$		0			0
$347$		−	28.7989i		−	1.54600i	−0.634404	−	0.773002i	$-0.718754\pi$
$347$		−	28.7989i		−	1.54600i	0.634404	−	0.773002i	$-0.281246\pi$
$348$		0			0
$349$	1.14168			0.0611125			0.0305563	−	0.999533i	$-0.490272\pi$
$349$	1.14168			0.0611125			0.0305563	+	0.999533i	$0.490272\pi$
$350$		0			0
$351$		−	0.783053i		−	0.0417963i
$352$		0			0
$353$	−6.61788			−0.352234			−0.176117	−	0.984369i	$-0.556354\pi$
$353$	−6.61788			−0.352234			−0.176117	+	0.984369i	$0.556354\pi$
$354$		0			0
$355$	−2.00589			−0.106462
$356$		0			0
$357$	−1.12469	+	1.42300i	−0.0595247	+	0.0753131i
$358$		0			0
$359$	22.4815			1.18653			0.593263	−	0.805009i	$-0.297839\pi$
$359$	22.4815			1.18653			0.593263	+	0.805009i	$0.297839\pi$
$360$		0			0
$361$	10.7121			0.563792
$362$		0			0
$363$		−	0.549871i		−	0.0288607i
$364$		0			0
$365$	−3.20182			−0.167591
$366$		0			0
$367$		−	0.731676i		−	0.0381932i	−0.999818	−	0.0190966i	$-0.993921\pi$
$367$		−	0.731676i		−	0.0381932i	0.999818	−	0.0190966i	$-0.00607900\pi$
$368$		0			0
$369$			18.9985i			0.989020i
$370$		0			0
$371$		−	37.0228i		−	1.92213i
$372$		0			0
$373$	18.2319			0.944013			0.472006	−	0.881595i	$-0.343530\pi$
$373$	18.2319			0.944013			0.472006	+	0.881595i	$0.343530\pi$
$374$		0			0
$375$	−0.549641			−0.0283833
$376$		0			0
$377$		−	10.9058i		−	0.561678i
$378$		0			0
$379$		−	17.1721i		−	0.882071i	−0.897490	−	0.441036i	$-0.854611\pi$
$379$		−	17.1721i		−	0.882071i	0.897490	−	0.441036i	$-0.145389\pi$
$380$		0			0
$381$			2.35401i			0.120600i
$382$		0			0
$383$	−8.19399			−0.418693			−0.209347	−	0.977841i	$-0.567134\pi$
$383$	−8.19399			−0.418693			−0.209347	+	0.977841i	$0.567134\pi$
$384$		0			0
$385$			4.13951i			0.210969i
$386$		0			0
$387$	−29.4294			−1.49598
$388$		0			0
$389$	−5.87389			−0.297818			−0.148909	−	0.988851i	$-0.547576\pi$
$389$	−5.87389			−0.297818			−0.148909	+	0.988851i	$0.547576\pi$
$390$		0			0
$391$	−18.8534	−	14.9010i	−0.953458	−	0.753578i
$392$		0			0
$393$	2.15546			0.108728
$394$		0			0
$395$	3.62012			0.182148
$396$		0			0
$397$			14.9831i			0.751982i	0.926623	+	0.375991i	$0.122698\pi$
$397$			14.9831i			0.751982i	−0.926623	+	0.375991i	$0.877302\pi$
$398$		0			0
$399$	2.39789			0.120045
$400$		0			0
$401$			11.5382i			0.576188i	0.957602	+	0.288094i	$0.0930216\pi$
$401$			11.5382i			0.576188i	−0.957602	+	0.288094i	$0.906978\pi$
$402$		0			0
$403$			0.593589i			0.0295688i
$404$		0			0
$405$		−	4.02613i		−	0.200060i
$406$		0			0
$407$	26.8695			1.33187
$408$		0			0
$409$	−15.2899			−0.756039			−0.378020	−	0.925798i	$-0.623395\pi$
$409$	−15.2899			−0.756039			−0.378020	+	0.925798i	$0.623395\pi$
$410$		0			0
$411$			1.87994i			0.0927306i
$412$		0			0
$413$		−	3.56076i		−	0.175213i
$414$		0			0
$415$		−	5.39049i		−	0.264609i
$416$		0			0
$417$	1.53014			0.0749313
$418$		0			0
$419$			18.7295i			0.914997i	0.889210	+	0.457499i	$0.151255\pi$
$419$			18.7295i			0.914997i	−0.889210	+	0.457499i	$0.848745\pi$
$420$		0			0
$421$	37.8151			1.84300			0.921498	−	0.388384i	$-0.126967\pi$
$421$	37.8151			1.84300			0.921498	+	0.388384i	$0.126967\pi$
$422$		0			0
$423$	−12.1054			−0.588587
$424$		0			0
$425$	12.2556	−	15.5062i	0.594482	−	0.752164i
$426$		0			0
$427$	23.4388			1.13428
$428$		0			0
$429$	−0.334842			−0.0161663
$430$		0			0
$431$		−	1.06536i		−	0.0513164i	−0.999671	−	0.0256582i	$-0.991832\pi$
$431$		−	1.06536i		−	0.0513164i	0.999671	−	0.0256582i	$-0.00816815\pi$
$432$		0			0
$433$	−0.599855			−0.0288272			−0.0144136	−	0.999896i	$-0.504588\pi$
$433$	−0.599855			−0.0288272			−0.0144136	+	0.999896i	$0.504588\pi$
$434$		0			0
$435$			0.577921i			0.0277092i
$436$		0			0
$437$			31.7698i			1.51976i
$438$		0			0
$439$		−	5.56664i		−	0.265681i	−0.991137	−	0.132841i	$-0.957590\pi$
$439$		−	5.56664i		−	0.265681i	0.991137	−	0.132841i	$-0.0424098\pi$
$440$		0			0
$441$	−16.9503			−0.807155
$442$		0			0
$443$	33.0999			1.57262			0.786311	−	0.617831i	$-0.211988\pi$
$443$	33.0999			1.57262			0.786311	+	0.617831i	$0.211988\pi$
$444$		0			0
$445$			8.33012i			0.394886i
$446$		0			0
$447$		−	0.890944i		−	0.0421402i
$448$		0			0
$449$		−	6.85503i		−	0.323509i	−0.986831	−	0.161754i	$-0.948285\pi$
$449$		−	6.85503i		−	0.323509i	0.986831	−	0.161754i	$-0.0517152\pi$
$450$		0			0
$451$	16.2894			0.767040
$452$		0			0
$453$			2.46605i			0.115865i
$454$		0			0
$455$	−1.71309			−0.0803109
$456$		0			0
$457$	4.38315			0.205035			0.102518	−	0.994731i	$-0.467310\pi$
$457$	4.38315			0.205035			0.102518	+	0.994731i	$0.467310\pi$
$458$		0			0
$459$	−2.39171	−	1.89032i	−0.111635	−	0.0882324i
$460$		0			0
$461$	32.2461			1.50185			0.750925	−	0.660388i	$-0.229608\pi$
$461$	32.2461			1.50185			0.750925	+	0.660388i	$0.229608\pi$
$462$		0			0
$463$	−20.5847			−0.956651			−0.478325	−	0.878183i	$-0.658756\pi$
$463$	−20.5847			−0.956651			−0.478325	+	0.878183i	$0.658756\pi$
$464$		0			0
$465$		−	0.0314555i		−	0.00145871i
$466$		0			0
$467$	10.4467			0.483416			0.241708	−	0.970349i	$-0.422292\pi$
$467$	10.4467			0.483416			0.241708	+	0.970349i	$0.422292\pi$
$468$		0			0
$469$			10.5147i			0.485523i
$470$		0			0
$471$		−	1.97555i		−	0.0910284i
$472$		0			0
$473$			25.2330i			1.16022i
$474$		0			0
$475$	−26.1295			−1.19891
$476$		0			0
$477$	31.0336			1.42093
$478$		0			0
$479$		−	35.1744i		−	1.60716i	−0.595196	−	0.803581i	$-0.702926\pi$
$479$		−	35.1744i		−	1.60716i	0.595196	−	0.803581i	$-0.297074\pi$
$480$		0			0
$481$			11.1196i			0.507012i
$482$		0			0
$483$			2.56396i			0.116664i
$484$		0			0
$485$	−4.40044			−0.199814
$486$		0			0
$487$		−	16.9137i		−	0.766434i	−0.923658	−	0.383217i	$-0.874816\pi$
$487$		−	16.9137i		−	0.766434i	0.923658	−	0.383217i	$-0.125184\pi$
$488$		0			0
$489$	0.244756			0.0110682
$490$		0			0
$491$	−33.5236			−1.51290			−0.756449	−	0.654052i	$-0.773068\pi$
$491$	−33.5236			−1.51290			−0.756449	+	0.654052i	$0.773068\pi$
$492$		0			0
$493$	−33.3100	−	26.3270i	−1.50021	−	1.18571i
$494$		0			0
$495$	−3.46987			−0.155959
$496$		0			0
$497$	−15.7230			−0.705274
$498$		0			0
$499$		−	5.60167i		−	0.250765i	−0.992108	−	0.125383i	$-0.959984\pi$
$499$		−	5.60167i		−	0.250765i	0.992108	−	0.125383i	$-0.0400159\pi$
$500$		0			0
$501$	−0.390825			−0.0174608
$502$		0			0
$503$			33.9719i			1.51473i	0.652990	+	0.757367i	$0.273514\pi$
$503$			33.9719i			1.51473i	−0.652990	+	0.757367i	$0.726486\pi$
$504$		0			0
$505$		−	4.89850i		−	0.217981i
$506$		0			0
$507$			1.46750i			0.0651740i
$508$		0			0
$509$	−39.6811			−1.75883			−0.879417	−	0.476052i	$-0.842067\pi$
$509$	−39.6811			−1.75883			−0.879417	+	0.476052i	$0.842067\pi$
$510$		0			0
$511$	−25.0972			−1.11023
$512$		0			0
$513$			4.03026i			0.177940i
$514$		0			0
$515$			1.19012i			0.0524428i
$516$		0			0
$517$			10.3793i			0.456482i
$518$		0			0
$519$	2.14924			0.0943413
$520$		0			0
$521$			39.9053i			1.74828i	0.485671	+	0.874142i	$0.338575\pi$
$521$			39.9053i			1.74828i	−0.485671	+	0.874142i	$0.661425\pi$
$522$		0			0
$523$	−4.43348			−0.193863			−0.0969314	−	0.995291i	$-0.530903\pi$
$523$	−4.43348			−0.193863			−0.0969314	+	0.995291i	$0.530903\pi$
$524$		0			0
$525$	−2.10877			−0.0920342
$526$		0			0
$527$	1.81302	+	1.43294i	0.0789764	+	0.0624200i
$528$		0			0
$529$	−10.9701			−0.476962
$530$		0			0
$531$	2.98474			0.129527
$532$		0			0
$533$			6.74120i			0.291994i
$534$		0			0
$535$	6.44072			0.278457
$536$		0			0
$537$		−	1.55354i		−	0.0670404i
$538$		0			0
$539$			14.5333i			0.625993i
$540$		0			0
$541$			22.5883i			0.971149i	0.874195	+	0.485574i	$0.161389\pi$
$541$			22.5883i			0.971149i	−0.874195	+	0.485574i	$0.838611\pi$
$542$		0			0
$543$	−2.82710			−0.121322
$544$		0			0
$545$	−1.28179			−0.0549057
$546$		0			0
$547$		−	19.1095i		−	0.817062i	−0.912744	−	0.408531i	$-0.866041\pi$
$547$		−	19.1095i		−	0.817062i	0.912744	−	0.408531i	$-0.133959\pi$
$548$		0			0
$549$			19.6472i			0.838521i
$550$		0			0
$551$			56.1306i			2.39124i
$552$		0			0
$553$	28.3761			1.20667
$554$		0			0
$555$		−	0.589252i		−	0.0250123i
$556$		0			0
$557$	−6.81109			−0.288595			−0.144298	−	0.989534i	$-0.546092\pi$
$557$	−6.81109			−0.288595			−0.144298	+	0.989534i	$0.546092\pi$
$558$		0			0
$559$	−10.4424			−0.441667
$560$		0			0
$561$	−0.808319	+	1.02272i	−0.0341273	+	0.0431792i
$562$		0			0
$563$	−15.6517			−0.659640			−0.329820	−	0.944044i	$-0.606988\pi$
$563$	−15.6517			−0.659640			−0.329820	+	0.944044i	$0.606988\pi$
$564$		0			0
$565$	−3.37704			−0.142073
$566$		0			0
$567$		−	31.5585i		−	1.32533i
$568$		0			0
$569$	−34.9961			−1.46711			−0.733557	−	0.679628i	$-0.762141\pi$
$569$	−34.9961			−1.46711			−0.733557	+	0.679628i	$0.762141\pi$
$570$		0			0
$571$			24.3105i			1.01736i	0.860955	+	0.508682i	$0.169867\pi$
$571$			24.3105i			1.01736i	−0.860955	+	0.508682i	$0.830133\pi$
$572$		0			0
$573$		−	1.43145i		−	0.0597995i
$574$		0			0
$575$		−	27.9392i		−	1.16515i
$576$		0			0
$577$	32.1976			1.34040			0.670202	−	0.742178i	$-0.266207\pi$
$577$	32.1976			1.34040			0.670202	+	0.742178i	$0.266207\pi$
$578$		0			0
$579$	−1.59237			−0.0661766
$580$		0			0
$581$		−	42.2529i		−	1.75295i
$582$		0			0
$583$		−	26.6085i		−	1.10201i
$584$		0			0
$585$		−	1.43597i		−	0.0593699i
$586$		0			0
$587$	31.7249			1.30943			0.654713	−	0.755878i	$-0.272789\pi$
$587$	31.7249			1.30943			0.654713	+	0.755878i	$0.272789\pi$
$588$		0			0
$589$		−	3.05512i		−	0.125884i
$590$		0			0
$591$	−2.65332			−0.109143
$592$		0			0
$593$	30.6442			1.25841			0.629204	−	0.777240i	$-0.283381\pi$
$593$	30.6442			1.25841			0.629204	+	0.777240i	$0.283381\pi$
$594$		0			0
$595$	−4.13546	+	5.23235i	−0.169537	+	0.214506i
$596$		0			0
$597$	−1.09585			−0.0448503
$598$		0			0
$599$	40.4039			1.65086			0.825429	−	0.564506i	$-0.190933\pi$
$599$	40.4039			1.65086			0.825429	+	0.564506i	$0.190933\pi$
$600$		0			0
$601$			7.46970i			0.304695i	0.988327	+	0.152348i	$0.0486833\pi$
$601$			7.46970i			0.304695i	−0.988327	+	0.152348i	$0.951317\pi$
$602$		0			0
$603$	−8.81374			−0.358923
$604$		0			0
$605$		−	2.02187i		−	0.0822006i
$606$		0			0
$607$			21.4720i			0.871521i	0.900063	+	0.435761i	$0.143521\pi$
$607$			21.4720i			0.871521i	−0.900063	+	0.435761i	$0.856479\pi$
$608$		0			0
$609$			4.52999i			0.183564i
$610$		0			0
$611$	−4.29536			−0.173772
$612$		0			0
$613$	33.6928			1.36084			0.680421	−	0.732822i	$-0.261797\pi$
$613$	33.6928			1.36084			0.680421	+	0.732822i	$0.261797\pi$
$614$		0			0
$615$		−	0.357229i		−	0.0144049i
$616$		0			0
$617$			17.4231i			0.701428i	0.936483	+	0.350714i	$0.114061\pi$
$617$			17.4231i			0.701428i	−0.936483	+	0.350714i	$0.885939\pi$
$618$		0			0
$619$		−	35.5804i		−	1.43010i	−0.699075	−	0.715049i	$-0.746405\pi$
$619$		−	35.5804i		−	1.43010i	0.699075	−	0.715049i	$-0.253595\pi$
$620$		0			0
$621$	−4.30938			−0.172930
$622$		0			0
$623$			65.2950i			2.61599i
$624$		0			0
$625$	21.9472			0.877887
$626$		0			0
$627$	1.72338			0.0688252
$628$		0			0
$629$	33.9631	+	26.8432i	1.35420	+	1.07031i
$630$		0			0
$631$	13.4260			0.534479			0.267240	−	0.963630i	$-0.413889\pi$
$631$	13.4260			0.534479			0.267240	+	0.963630i	$0.413889\pi$
$632$		0			0
$633$	−2.95966			−0.117636
$634$		0			0
$635$			8.65568i			0.343490i
$636$		0			0
$637$	−6.01444			−0.238301
$638$		0			0
$639$		−	13.1795i		−	0.521374i
$640$		0			0
$641$		−	1.17967i		−	0.0465941i	−0.999729	−	0.0232971i	$-0.992584\pi$
$641$		−	1.17967i		−	0.0465941i	0.999729	−	0.0232971i	$-0.00741635\pi$
$642$		0			0
$643$			42.4219i			1.67296i	0.547999	+	0.836479i	$0.315390\pi$
$643$			42.4219i			1.67296i	−0.547999	+	0.836479i	$0.684610\pi$
$644$		0			0
$645$	0.553364			0.0217887
$646$		0			0
$647$	43.4812			1.70942			0.854711	−	0.519105i	$-0.173734\pi$
$647$	43.4812			1.70942			0.854711	+	0.519105i	$0.173734\pi$
$648$		0			0
$649$		−	2.55914i		−	0.100455i
$650$		0			0
$651$		−	0.246561i		−	0.00966350i
$652$		0			0
$653$			6.75483i			0.264337i	0.991227	+	0.132168i	$0.0421940\pi$
$653$			6.75483i			0.264337i	−0.991227	+	0.132168i	$0.957806\pi$
$654$		0			0
$655$	7.92558			0.309678
$656$		0			0
$657$		−	21.0372i		−	0.820741i
$658$		0			0
$659$	17.1645			0.668635			0.334317	−	0.942461i	$-0.391494\pi$
$659$	17.1645			0.668635			0.334317	+	0.942461i	$0.391494\pi$
$660$		0			0
$661$	−40.4195			−1.57214			−0.786068	−	0.618140i	$-0.787886\pi$
$661$	−40.4195			−1.57214			−0.786068	+	0.618140i	$0.787886\pi$
$662$		0			0
$663$	−0.423241	−	0.334514i	−0.0164373	−	0.0129914i
$664$		0			0
$665$	8.81702			0.341910
$666$		0			0
$667$	−60.0180			−2.32391
$668$		0			0
$669$		−	2.05515i		−	0.0794568i
$670$		0			0
$671$	16.8456			0.650319
$672$		0			0
$673$			18.7981i			0.724612i	0.932059	+	0.362306i	$0.118010\pi$
$673$			18.7981i			0.724612i	−0.932059	+	0.362306i	$0.881990\pi$
$674$		0			0
$675$		−	3.54431i		−	0.136421i
$676$		0			0
$677$			31.3773i			1.20593i	0.797769	+	0.602963i	$0.206013\pi$
$677$			31.3773i			1.20593i	−0.797769	+	0.602963i	$0.793987\pi$
$678$		0			0
$679$	−34.4925			−1.32370
$680$		0			0
$681$	−1.34034			−0.0513619
$682$		0			0
$683$			13.9333i			0.533141i	0.963815	+	0.266571i	$0.0858905\pi$
$683$			13.9333i			0.533141i	−0.963815	+	0.266571i	$0.914109\pi$
$684$		0			0
$685$			6.91252i			0.264114i
$686$		0			0
$687$			1.09962i			0.0419529i
$688$		0			0
$689$	11.0116			0.419509
$690$		0			0
$691$		−	27.0868i		−	1.03043i	−0.857061	−	0.515215i	$-0.827712\pi$
$691$		−	27.0868i		−	1.03043i	0.857061	−	0.515215i	$-0.172288\pi$
$692$		0			0
$693$	−27.1983			−1.03318
$694$		0			0
$695$	5.62631			0.213418
$696$		0			0
$697$	20.5899	+	16.2735i	0.779897	+	0.616402i
$698$		0			0
$699$	0.584503			0.0221079
$700$		0			0
$701$	7.57170			0.285979			0.142990	−	0.989724i	$-0.454328\pi$
$701$	7.57170			0.285979			0.142990	+	0.989724i	$0.454328\pi$
$702$		0			0
$703$		−	57.2311i		−	2.15851i
$704$		0			0
$705$	0.227620			0.00857265
$706$		0			0
$707$		−	38.3965i		−	1.44405i
$708$		0			0
$709$		−	2.50772i		−	0.0941793i	−0.998891	−	0.0470896i	$-0.985005\pi$
$709$		−	2.50772i		−	0.0941793i	0.998891	−	0.0470896i	$-0.0149946\pi$
$710$		0			0
$711$			23.7857i			0.892033i
$712$		0			0
$713$	3.26670			0.122339
$714$		0			0
$715$	−1.23121			−0.0460446
$716$		0			0
$717$			0.0265386i			0.000991102i
$718$		0			0
$719$		−	51.2654i		−	1.91188i	−0.293570	−	0.955938i	$-0.594843\pi$
$719$		−	51.2654i		−	1.91188i	0.293570	−	0.955938i	$-0.405157\pi$
$720$		0			0
$721$			9.32864i			0.347417i
$722$		0			0
$723$	−2.17081			−0.0807333
$724$		0			0
$725$		−	49.3627i		−	1.83328i
$726$		0			0
$727$	35.3839			1.31232			0.656159	−	0.754623i	$-0.272180\pi$
$727$	35.3839			1.31232			0.656159	+	0.754623i	$0.272180\pi$
$728$		0			0
$729$	26.1793			0.969603
$730$		0			0
$731$	−25.2083	+	31.8946i	−0.932364	+	1.17967i
$732$		0			0
$733$	−46.5913			−1.72089			−0.860444	−	0.509546i	$-0.829814\pi$
$733$	−46.5913			−1.72089			−0.860444	+	0.509546i	$0.829814\pi$
$734$		0			0
$735$	0.318717			0.0117561
$736$		0			0
$737$			7.55697i			0.278365i
$738$		0			0
$739$	−50.1732			−1.84565			−0.922826	−	0.385216i	$-0.874127\pi$
$739$	−50.1732			−1.84565			−0.922826	+	0.385216i	$0.874127\pi$
$740$		0			0
$741$			0.713202i			0.0262001i
$742$		0			0
$743$			21.6397i			0.793885i	0.917844	+	0.396942i	$0.129929\pi$
$743$			21.6397i			0.793885i	−0.917844	+	0.396942i	$0.870071\pi$
$744$		0			0
$745$		−	3.27599i		−	0.120023i
$746$		0			0
$747$	35.4177			1.29587
$748$		0			0
$749$	50.4851			1.84469
$750$		0			0
$751$			30.4336i			1.11054i	0.831671	+	0.555268i	$0.187384\pi$
$751$			30.4336i			1.11054i	−0.831671	+	0.555268i	$0.812616\pi$
$752$		0			0
$753$			2.76305i			0.100691i
$754$		0			0
$755$			9.06764i			0.330005i
$756$		0			0
$757$	34.3562			1.24870			0.624349	−	0.781145i	$-0.285364\pi$
$757$	34.3562			1.24870			0.624349	+	0.781145i	$0.285364\pi$
$758$		0			0
$759$			1.84274i			0.0668872i
$760$		0			0
$761$	25.6300			0.929089			0.464544	−	0.885550i	$-0.346218\pi$
$761$	25.6300			0.929089			0.464544	+	0.885550i	$0.346218\pi$
$762$		0			0
$763$	−10.0472			−0.363732
$764$		0			0
$765$	−4.38592	−	3.46647i	−0.158573	−	0.125331i
$766$		0			0
$767$	1.05907			0.0382408
$768$		0			0
$769$	−20.9528			−0.755579			−0.377789	−	0.925892i	$-0.623316\pi$
$769$	−20.9528			−0.755579			−0.377789	+	0.925892i	$0.623316\pi$
$770$		0			0
$771$		−	0.784322i		−	0.0282467i
$772$		0			0
$773$	31.1791			1.12143			0.560717	−	0.828008i	$-0.310526\pi$
$773$	31.1791			1.12143			0.560717	+	0.828008i	$0.310526\pi$
$774$		0			0
$775$			2.68675i			0.0965108i
$776$		0			0
$777$		−	4.61880i		−	0.165699i
$778$		0			0
$779$		−	34.6959i		−	1.24311i
$780$		0			0
$781$	−11.3002			−0.404354
$782$		0			0
$783$	−7.61377			−0.272094
$784$		0			0
$785$		−	7.26407i		−	0.259266i
$786$		0			0
$787$			4.50061i			0.160429i	0.996778	+	0.0802147i	$0.0255606\pi$
$787$			4.50061i			0.160429i	−0.996778	+	0.0802147i	$0.974439\pi$
$788$		0			0
$789$		−	0.168771i		−	0.00600841i
$790$		0			0
$791$	−26.4707			−0.941189
$792$		0			0
$793$			6.97138i			0.247561i
$794$		0			0
$795$	−0.583528			−0.0206956
$796$		0			0
$797$	24.4829			0.867229			0.433614	−	0.901098i	$-0.357238\pi$
$797$	24.4829			0.867229			0.433614	+	0.901098i	$0.357238\pi$
$798$		0			0
$799$	−10.3691	+	13.1195i	−0.366834	+	0.464133i
$800$		0			0
$801$	−54.7323			−1.93387
$802$		0			0
$803$	−18.0375			−0.636530
$804$		0			0
$805$			9.42767i			0.332282i
$806$		0			0
$807$	3.71394			0.130737
$808$		0			0
$809$			3.84717i			0.135259i	0.997710	+	0.0676296i	$0.0215436\pi$
$809$			3.84717i			0.135259i	−0.997710	+	0.0676296i	$0.978456\pi$
$810$		0			0
$811$			17.2547i			0.605896i	0.953007	+	0.302948i	$0.0979709\pi$
$811$			17.2547i			0.605896i	−0.953007	+	0.302948i	$0.902029\pi$
$812$		0			0
$813$		−	3.30644i		−	0.115962i
$814$		0			0
$815$	0.899963			0.0315243
$816$		0			0
$817$	53.7455			1.88032
$818$		0			0
$819$		−	11.2557i		−	0.393306i
$820$		0			0
$821$			10.2125i			0.356418i	0.983993	+	0.178209i	$0.0570304\pi$
$821$			10.2125i			0.356418i	−0.983993	+	0.178209i	$0.942970\pi$
$822$		0			0
$823$			0.129049i			0.00449837i	0.999997	+	0.00224918i	$0.000715938\pi$
$823$			0.129049i			0.00449837i	−0.999997	+	0.00224918i	$0.999284\pi$
$824$		0			0
$825$	−1.51559			−0.0527659
$826$		0			0
$827$			3.86396i			0.134363i	0.997741	+	0.0671816i	$0.0214007\pi$
$827$			3.86396i			0.134363i	−0.997741	+	0.0671816i	$0.978599\pi$
$828$		0			0
$829$	−20.8365			−0.723683			−0.361841	−	0.932240i	$-0.617852\pi$
$829$	−20.8365			−0.723683			−0.361841	+	0.932240i	$0.617852\pi$
$830$		0			0
$831$	0.463163			0.0160669
$832$		0			0
$833$	−14.5191	+	18.3701i	−0.503055	+	0.636487i
$834$		0			0
$835$	−1.43706			−0.0497314
$836$		0			0
$837$	0.414408			0.0143240
$838$		0			0
$839$			16.1733i			0.558365i	0.960238	+	0.279182i	$0.0900634\pi$
$839$			16.1733i			0.558365i	−0.960238	+	0.279182i	$0.909937\pi$
$840$		0			0
$841$	−77.0392			−2.65653
$842$		0			0
$843$			2.88811i			0.0994717i
$844$		0			0
$845$			5.39598i			0.185627i
$846$		0			0
$847$		−	15.8483i		−	0.544553i
$848$		0			0
$849$	1.71858			0.0589817
$850$		0			0
$851$	61.1948			2.09773
$852$		0			0
$853$		−	2.41135i		−	0.0825629i	−0.999148	−	0.0412815i	$-0.986856\pi$
$853$		−	2.41135i		−	0.0825629i	0.999148	−	0.0412815i	$-0.0131440\pi$
$854$		0			0
$855$			7.39071i			0.252757i
$856$		0			0
$857$		−	24.1069i		−	0.823475i	−0.911303	−	0.411737i	$-0.864922\pi$
$857$		−	24.1069i		−	0.823475i	0.911303	−	0.411737i	$-0.135078\pi$
$858$		0			0
$859$	32.4368			1.10673			0.553365	−	0.832939i	$-0.313344\pi$
$859$	32.4368			1.10673			0.553365	+	0.832939i	$0.313344\pi$
$860$		0			0
$861$		−	2.80011i		−	0.0954277i
$862$		0			0
$863$	−4.27350			−0.145472			−0.0727359	−	0.997351i	$-0.523173\pi$
$863$	−4.27350			−0.145472			−0.0727359	+	0.997351i	$0.523173\pi$
$864$		0			0
$865$	7.90274			0.268701
$866$		0			0
$867$	−2.04343	+	0.485192i	−0.0693987	+	0.0164780i
$868$		0			0
$869$	20.3941			0.691821
$870$		0			0
$871$	−3.12737			−0.105967
$872$		0			0
$873$		−	28.9127i		−	0.978547i
$874$		0			0
$875$	−15.8416			−0.535545
$876$		0			0
$877$			20.1177i			0.679326i	0.940547	+	0.339663i	$0.110313\pi$
$877$			20.1177i			0.679326i	−0.940547	+	0.339663i	$0.889687\pi$
$878$		0			0
$879$			1.80369i			0.0608369i
$880$		0			0
$881$			28.5492i			0.961846i	0.876763	+	0.480923i	$0.159698\pi$
$881$			28.5492i			0.961846i	−0.876763	+	0.480923i	$0.840302\pi$
$882$		0			0
$883$	27.3231			0.919496			0.459748	−	0.888049i	$-0.347940\pi$
$883$	27.3231			0.919496			0.459748	+	0.888049i	$0.347940\pi$
$884$		0			0
$885$	−0.0561222			−0.00188653
$886$		0			0
$887$			12.4943i			0.419518i	0.977753	+	0.209759i	$0.0672679\pi$
$887$			12.4943i			0.419518i	−0.977753	+	0.209759i	$0.932732\pi$
$888$		0			0
$889$			67.8469i			2.27551i
$890$		0			0
$891$		−	22.6813i		−	0.759852i
$892$		0			0
$893$	22.1076			0.739802
$894$		0			0
$895$		−	5.71236i		−	0.190943i
$896$		0			0
$897$	−0.762596			−0.0254623
$898$		0			0
$899$	5.77158			0.192493
$900$		0			0
$901$	26.5824	−	33.6332i	0.885589	−	1.12048i
$902$		0			0
$903$	4.33750			0.144343
$904$		0			0
$905$	−10.3952			−0.345549
$906$		0			0
$907$			42.6576i			1.41642i	0.706000	+	0.708212i	$0.250498\pi$
$907$			42.6576i			1.41642i	−0.706000	+	0.708212i	$0.749502\pi$
$908$		0			0
$909$	32.1852			1.06751
$910$		0			0
$911$		−	52.7572i		−	1.74792i	−0.485994	−	0.873962i	$-0.661542\pi$
$911$		−	52.7572i		−	1.74792i	0.485994	−	0.873962i	$-0.338458\pi$
$912$		0			0
$913$		−	30.3675i		−	1.00502i
$914$		0			0
$915$		−	0.369427i		−	0.0122129i
$916$		0			0
$917$	62.1241			2.05152
$918$		0			0
$919$	−39.1272			−1.29069			−0.645344	−	0.763892i	$-0.723286\pi$
$919$	−39.1272			−1.29069			−0.645344	+	0.763892i	$0.723286\pi$
$920$		0			0
$921$		−	1.80718i		−	0.0595486i
$922$		0			0
$923$		−	4.67647i		−	0.153928i
$924$		0			0
$925$			50.3305i			1.65486i
$926$		0			0
$927$	−7.81956			−0.256828
$928$		0			0
$929$			36.7996i			1.20736i	0.797228	+	0.603678i	$0.206299\pi$
$929$			36.7996i			1.20736i	−0.797228	+	0.603678i	$0.793701\pi$
$930$		0			0
$931$	30.9554			1.01452
$932$		0			0
$933$	2.86214			0.0937024
$934$		0			0
$935$	−2.97218	+	3.76053i	−0.0972007	+	0.122982i
$936$		0			0
$937$	14.1198			0.461274			0.230637	−	0.973040i	$-0.425919\pi$
$937$	14.1198			0.461274			0.230637	+	0.973040i	$0.425919\pi$
$938$		0			0
$939$	−3.80335			−0.124118
$940$		0			0
$941$			18.5654i			0.605215i	0.953115	+	0.302608i	$0.0978572\pi$
$941$			18.5654i			0.605215i	−0.953115	+	0.302608i	$0.902143\pi$
$942$		0			0
$943$	37.0989			1.20811
$944$		0			0
$945$			1.19598i			0.0389051i
$946$		0			0
$947$		−	22.6296i		−	0.735364i	−0.929952	−	0.367682i	$-0.880151\pi$
$947$		−	22.6296i		−	0.735364i	0.929952	−	0.367682i	$-0.119849\pi$
$948$		0			0
$949$		−	7.46461i		−	0.242312i
$950$		0			0
$951$	−0.529214			−0.0171609
$952$		0			0
$953$	−15.2691			−0.494614			−0.247307	−	0.968937i	$-0.579546\pi$
$953$	−15.2691			−0.494614			−0.247307	+	0.968937i	$0.579546\pi$
$954$		0			0
$955$		−	5.26341i		−	0.170320i
$956$		0			0
$957$			3.25573i			0.105243i
$958$		0			0
$959$			54.1832i			1.74967i
$960$		0			0
$961$	30.6859			0.989866
$962$		0			0
$963$			42.3182i			1.36368i
$964$		0			0
$965$	−5.85512			−0.188483
$966$		0			0
$967$	28.5811			0.919105			0.459552	−	0.888151i	$-0.348010\pi$
$967$	28.5811			0.919105			0.459552	+	0.888151i	$0.348010\pi$
$968$		0			0
$969$	2.17836	+	1.72169i	0.0699789	+	0.0553087i
$970$		0			0
$971$	52.1479			1.67351			0.836754	−	0.547580i	$-0.184451\pi$
$971$	52.1479			1.67351			0.836754	+	0.547580i	$0.184451\pi$
$972$		0			0
$973$	44.1014			1.41382
$974$		0			0
$975$		−	0.627208i		−	0.0200867i
$976$		0			0
$977$	4.83514			0.154690			0.0773449	−	0.997004i	$-0.475356\pi$
$977$	4.83514			0.154690			0.0773449	+	0.997004i	$0.475356\pi$
$978$		0			0
$979$			46.9279i			1.49982i
$980$		0			0
$981$		−	8.42186i		−	0.268889i
$982$		0			0
$983$		−	3.02318i		−	0.0964245i	−0.998837	−	0.0482123i	$-0.984648\pi$
$983$		−	3.02318i		−	0.0964245i	0.998837	−	0.0482123i	$-0.0153524\pi$
$984$		0			0
$985$	−9.75624			−0.310860
$986$		0			0
$987$	1.78418			0.0567910
$988$		0			0
$989$			57.4678i			1.82737i
$990$		0			0
$991$		−	3.84056i		−	0.121999i	−0.998138	−	0.0609997i	$-0.980571\pi$
$991$		−	3.84056i		−	0.121999i	0.998138	−	0.0609997i	$-0.0194289\pi$
$992$		0			0
$993$		−	0.570197i		−	0.0180947i
$994$		0			0
$995$	−4.02944			−0.127742
$996$		0			0
$997$		−	22.1552i		−	0.701663i	−0.936439	−	0.350832i	$-0.885899\pi$
$997$		−	22.1552i		−	0.701663i	0.936439	−	0.350832i	$-0.114101\pi$
$998$		0			0
$999$	7.76305			0.245612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.23	✓	46
17.16	even	2	inner	4012.2.b.b.237.24	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.23	✓	46	1.1	even	1	trivial
4012.2.b.b.237.24	yes	46	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.123544i q^{3} -0.454269i q^{5} -3.56076i q^{7} +2.98474 q^{9} +O(q^{10})\)	\(q-0.123544i q^{3} -0.454269i q^{5} -3.56076i q^{7} +2.98474 q^{9} -2.55914i q^{11} +1.05907 q^{13} -0.0561222 q^{15} +(2.55663 - 3.23476i) q^{17} -5.45088 q^{19} -0.439910 q^{21} -5.82839i q^{23} +4.79364 q^{25} -0.739378i q^{27} -10.2975i q^{29} +0.560481i q^{31} -0.316166 q^{33} -1.61754 q^{35} +10.4994i q^{37} -0.130842i q^{39} +6.36520i q^{41} -9.85998 q^{43} -1.35587i q^{45} -4.05578 q^{47} -5.67898 q^{49} +(-0.399634 - 0.315856i) q^{51} +10.3974 q^{53} -1.16254 q^{55} +0.673423i q^{57} +1.00000 q^{59} +6.58255i q^{61} -10.6279i q^{63} -0.481103i q^{65} -2.95294 q^{67} -0.720062 q^{69} -4.41564i q^{71} -7.04827i q^{73} -0.592225i q^{75} -9.11246 q^{77} +7.96911i q^{79} +8.86287 q^{81} +11.8663 q^{83} +(-1.46945 - 1.16140i) q^{85} -1.27220 q^{87} -18.3374 q^{89} -3.77109i q^{91} +0.0692441 q^{93} +2.47617i q^{95} -9.68685i q^{97} -7.63835i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more