LMFDB - Embedded newform 4600.2.e.p.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not 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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.431642176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 19x^{4} + 97x^{2} + 64 $ x^6 + 19x^4 + 97x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$-0.878468i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.p.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.878468i q^{3} +0.121532i q^{7} +2.22829 q^{9} +O(q^{10})$	$q-0.878468i q^{3} +0.121532i q^{7} +2.22829 q^{9} +2.87847 q^{11} -5.22829i q^{13} -2.22829i q^{17} -1.22829 q^{19} +0.106762 q^{21} -1.00000i q^{23} -4.59289i q^{27} -9.34983 q^{29} -2.12153 q^{31} -2.52864i q^{33} +5.59289i q^{37} -4.59289 q^{39} +8.22829 q^{41} -8.00000i q^{43} -10.4566i q^{47} +6.98523 q^{49} -1.95749 q^{51} +3.59289i q^{53} +1.07902i q^{57} +0.650174 q^{59} -7.33506 q^{61} +0.270809i q^{63} +5.59289i q^{67} -0.878468 q^{69} -13.9852 q^{71} -12.9427i q^{73} +0.349826i q^{77} +3.51387 q^{79} +2.65017 q^{81} +11.1068i q^{83} +8.21352i q^{87} +0.486128 q^{89} +0.635404 q^{91} +1.86370i q^{93} +0.635404i q^{97} +6.41407 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 20 q^{9}+O(q^{10}) $ 6 * q - 20 * q^9	$ 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) $ 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 0.878468i	− 0.507184i	−0.967311	−	0.253592i	$-0.918388\pi$
$3$	− 0.878468i	− 0.507184i	0.967311	−	0.253592i	$-0.0816120\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.121532i	0.0459347i	0.999736	+	0.0229674i	$0.00731138\pi$
$7$	0.121532i	0.0459347i	−0.999736	+	0.0229674i	$0.992689\pi$
$8$	0	0
$9$	2.22829	0.742765
$10$	0	0
$11$	2.87847	0.867891	0.433945	−	0.900939i	$-0.357121\pi$
$11$	2.87847	0.867891	0.433945	+	0.900939i	$0.357121\pi$
$12$	0	0
$13$	− 5.22829i	− 1.45007i	−0.688713	−	0.725034i	$-0.741824\pi$
$13$	− 5.22829i	− 1.45007i	0.688713	−	0.725034i	$-0.258176\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.22829i	− 0.540441i	−0.962799	−	0.270220i	$-0.912903\pi$
$17$	− 2.22829i	− 0.540441i	0.962799	−	0.270220i	$-0.0870965\pi$
$18$	0	0
$19$	−1.22829	−0.281790	−0.140895	−	0.990025i	$-0.544998\pi$
$19$	−1.22829	−0.281790	−0.140895	+	0.990025i	$0.544998\pi$
$20$	0	0
$21$	0.106762	0.0232974
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.59289i	− 0.883902i
$28$	0	0
$29$	−9.34983	−1.73622	−0.868110	−	0.496373i	$-0.834665\pi$
$29$	−9.34983	−1.73622	−0.868110	+	0.496373i	$0.834665\pi$
$30$	0	0
$31$	−2.12153	−0.381038	−0.190519	−	0.981683i	$-0.561017\pi$
$31$	−2.12153	−0.381038	−0.190519	+	0.981683i	$0.561017\pi$
$32$	0	0
$33$	− 2.52864i	− 0.440180i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.59289i	0.919465i	0.888057	+	0.459733i	$0.152055\pi$
$37$	5.59289i	0.919465i	−0.888057	+	0.459733i	$0.847945\pi$
$38$	0	0
$39$	−4.59289	−0.735451
$40$	0	0
$41$	8.22829	1.28504	0.642522	−	0.766267i	$-0.277888\pi$
$41$	8.22829	1.28504	0.642522	+	0.766267i	$0.277888\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.4566i	− 1.52525i	−0.646841	−	0.762625i	$-0.723910\pi$
$47$	− 10.4566i	− 1.52525i	0.646841	−	0.762625i	$-0.276090\pi$
$48$	0	0
$49$	6.98523	0.997890
$50$	0	0
$51$	−1.95749	−0.274103
$52$	0	0
$53$	3.59289i	0.493521i	0.969076	+	0.246761i	$0.0793661\pi$
$53$	3.59289i	0.493521i	−0.969076	+	0.246761i	$0.920634\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.07902i	0.142919i
$58$	0	0
$59$	0.650174	0.0846455	0.0423227	−	0.999104i	$-0.486524\pi$
$59$	0.650174	0.0846455	0.0423227	+	0.999104i	$0.486524\pi$
$60$	0	0
$61$	−7.33506	−0.939158	−0.469579	−	0.882891i	$-0.655594\pi$
$61$	−7.33506	−0.939158	−0.469579	+	0.882891i	$0.655594\pi$
$62$	0	0
$63$	0.270809i	0.0341187i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.59289i	0.683280i	0.939831	+	0.341640i	$0.110982\pi$
$67$	5.59289i	0.683280i	−0.939831	+	0.341640i	$0.889018\pi$
$68$	0	0
$69$	−0.878468	−0.105755
$70$	0	0
$71$	−13.9852	−1.65974	−0.829871	−	0.557956i	$-0.811586\pi$
$71$	−13.9852	−1.65974	−0.829871	+	0.557956i	$0.811586\pi$
$72$	0	0
$73$	− 12.9427i	− 1.51483i	−0.652934	−	0.757415i	$-0.726462\pi$
$73$	− 12.9427i	− 1.51483i	0.652934	−	0.757415i	$-0.273538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.349826i	0.0398663i
$78$	0	0
$79$	3.51387	0.395342	0.197671	−	0.980268i	$-0.436662\pi$
$79$	3.51387	0.395342	0.197671	+	0.980268i	$0.436662\pi$
$80$	0	0
$81$	2.65017	0.294464
$82$	0	0
$83$	11.1068i	1.21913i	0.792738	+	0.609563i	$0.208655\pi$
$83$	11.1068i	1.21913i	−0.792738	+	0.609563i	$0.791345\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.21352i	0.880582i
$88$	0	0
$89$	0.486128	0.0515294	0.0257647	−	0.999668i	$-0.491798\pi$
$89$	0.486128	0.0515294	0.0257647	+	0.999668i	$0.491798\pi$
$90$	0	0
$91$	0.635404	0.0666085
$92$	0	0
$93$	1.86370i	0.193256i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.635404i	0.0645155i	0.999480	+	0.0322578i	$0.0102697\pi$
$97$	0.635404i	0.0645155i	−0.999480	+	0.0322578i	$0.989730\pi$
$98$	0	0
$99$	6.41407	0.644639
$100$	0	0
$101$	13.3498	1.32836	0.664179	−	0.747574i	$-0.268781\pi$
$101$	13.3498	1.32836	0.664179	+	0.747574i	$0.268781\pi$
$102$	0	0
$103$	7.33506i	0.722745i	0.932422	+	0.361372i	$0.117692\pi$
$103$	7.33506i	0.722745i	−0.932422	+	0.361372i	$0.882308\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0495i	1.55156i	0.631003	+	0.775781i	$0.282644\pi$
$107$	16.0495i	1.55156i	−0.631003	+	0.775781i	$0.717356\pi$
$108$	0	0
$109$	−1.01477	−0.0971973	−0.0485987	−	0.998818i	$-0.515476\pi$
$109$	−1.01477	−0.0971973	−0.0485987	+	0.998818i	$0.515476\pi$
$110$	0	0
$111$	4.91318	0.466338
$112$	0	0
$113$	− 17.3203i	− 1.62936i	−0.579914	−	0.814678i	$-0.696914\pi$
$113$	− 17.3203i	− 1.62936i	0.579914	−	0.814678i	$-0.303086\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 11.6502i	− 1.07706i
$118$	0	0
$119$	0.270809	0.0248250
$120$	0	0
$121$	−2.71442	−0.246766
$122$	0	0
$123$	− 7.22829i	− 0.651753i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.75694i	0.155903i	0.996957	+	0.0779514i	$0.0248379\pi$
$127$	1.75694i	0.155903i	−0.996957	+	0.0779514i	$0.975162\pi$
$128$	0	0
$129$	−7.02774	−0.618758
$130$	0	0
$131$	−20.2135	−1.76606	−0.883032	−	0.469313i	$-0.844502\pi$
$131$	−20.2135	−1.76606	−0.883032	+	0.469313i	$0.844502\pi$
$132$	0	0
$133$	− 0.149277i	− 0.0129439i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.9279i	− 1.53169i	−0.643027	−	0.765844i	$-0.722322\pi$
$137$	− 17.9279i	− 1.53169i	0.643027	−	0.765844i	$-0.277678\pi$
$138$	0	0
$139$	10.8637	0.921447	0.460723	−	0.887544i	$-0.347590\pi$
$139$	10.8637	0.921447	0.460723	+	0.887544i	$0.347590\pi$
$140$	0	0
$141$	−9.18578	−0.773582
$142$	0	0
$143$	− 15.0495i	− 1.25850i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.13630i	− 0.506114i
$148$	0	0
$149$	15.7144	1.28738	0.643688	−	0.765288i	$-0.277404\pi$
$149$	15.7144	1.28738	0.643688	+	0.765288i	$0.277404\pi$
$150$	0	0
$151$	−8.39234	−0.682959	−0.341479	−	0.939889i	$-0.610928\pi$
$151$	−8.39234	−0.682959	−0.341479	+	0.939889i	$0.610928\pi$
$152$	0	0
$153$	− 4.96529i	− 0.401420i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 19.5633i	− 1.56133i	−0.624953	−	0.780663i	$-0.714882\pi$
$157$	− 19.5633i	− 1.56133i	0.624953	−	0.780663i	$-0.285118\pi$
$158$	0	0
$159$	3.15624	0.250306
$160$	0	0
$161$	0.121532	0.00957805
$162$	0	0
$163$	− 1.01477i	− 0.0794829i	−0.999210	−	0.0397415i	$-0.987347\pi$
$163$	− 1.01477i	− 0.0794829i	0.999210	−	0.0397415i	$-0.0126534\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−14.3351	−1.10270
$170$	0	0
$171$	−2.73700	−0.209304
$172$	0	0
$173$	2.63540i	0.200366i	0.994969	+	0.100183i	$0.0319428\pi$
$173$	2.63540i	0.200366i	−0.994969	+	0.100183i	$0.968057\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 0.571157i	− 0.0429308i
$178$	0	0
$179$	14.4566	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$179$	14.4566	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$180$	0	0
$181$	10.7422	0.798459	0.399229	−	0.916851i	$-0.369278\pi$
$181$	10.7422	0.798459	0.399229	+	0.916851i	$0.369278\pi$
$182$	0	0
$183$	6.44361i	0.476326i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.41407i	− 0.469043i
$188$	0	0
$189$	0.558182	0.0406018
$190$	0	0
$191$	−22.6701	−1.64035	−0.820176	−	0.572112i	$-0.806124\pi$
$191$	−22.6701	−1.64035	−0.820176	+	0.572112i	$0.806124\pi$
$192$	0	0
$193$	− 2.24306i	− 0.161459i	−0.996736	−	0.0807296i	$-0.974275\pi$
$193$	− 2.24306i	− 0.161459i	0.996736	−	0.0807296i	$-0.0257250\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.57812i	− 0.539919i	−0.962872	−	0.269959i	$-0.912990\pi$
$197$	− 7.57812i	− 0.539919i	0.962872	−	0.269959i	$-0.0870103\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	4.91318	0.346549
$202$	0	0
$203$	− 1.13630i	− 0.0797528i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.22829i	− 0.154877i
$208$	0	0
$209$	−3.53560	−0.244563
$210$	0	0
$211$	−3.10676	−0.213878	−0.106939	−	0.994266i	$-0.534105\pi$
$211$	−3.10676	−0.213878	−0.106939	+	0.994266i	$0.534105\pi$
$212$	0	0
$213$	12.2856i	0.841794i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.257834i	− 0.0175029i
$218$	0	0
$219$	−11.3698	−0.768297
$220$	0	0
$221$	−11.6502	−0.783676
$222$	0	0
$223$	− 18.2135i	− 1.21967i	−0.792529	−	0.609834i	$-0.791236\pi$
$223$	− 18.2135i	− 1.21967i	0.792529	−	0.609834i	$-0.208764\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.9705i	0.927252i	0.886031	+	0.463626i	$0.153452\pi$
$227$	13.9705i	0.927252i	−0.886031	+	0.463626i	$0.846548\pi$
$228$	0	0
$229$	−12.2135	−0.807092	−0.403546	−	0.914959i	$-0.632223\pi$
$229$	−12.2135	−0.807092	−0.403546	+	0.914959i	$0.632223\pi$
$230$	0	0
$231$	0.307311	0.0202196
$232$	0	0
$233$	− 9.75694i	− 0.639198i	−0.947553	−	0.319599i	$-0.896452\pi$
$233$	− 9.75694i	− 0.639198i	0.947553	−	0.319599i	$-0.103548\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.08682i	− 0.200511i
$238$	0	0
$239$	−8.62063	−0.557622	−0.278811	−	0.960346i	$-0.589940\pi$
$239$	−8.62063	−0.557622	−0.278811	+	0.960346i	$0.589940\pi$
$240$	0	0
$241$	3.18578	0.205214	0.102607	−	0.994722i	$-0.467282\pi$
$241$	3.18578	0.205214	0.102607	+	0.994722i	$0.467282\pi$
$242$	0	0
$243$	− 16.1068i	− 1.03325i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.42188i	0.408614i
$248$	0	0
$249$	9.75694	0.618321
$250$	0	0
$251$	−25.2283	−1.59240	−0.796198	−	0.605036i	$-0.793159\pi$
$251$	−25.2283	−1.59240	−0.796198	+	0.605036i	$0.793159\pi$
$252$	0	0
$253$	− 2.87847i	− 0.180968i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.6701i	− 0.665583i	−0.943000	−	0.332792i	$-0.892009\pi$
$257$	− 10.6701i	− 0.665583i	0.943000	−	0.332792i	$-0.107991\pi$
$258$	0	0
$259$	−0.679714	−0.0422354
$260$	0	0
$261$	−20.8342	−1.28960
$262$	0	0
$263$	18.6849i	1.15216i	0.817394	+	0.576080i	$0.195418\pi$
$263$	18.6849i	1.15216i	−0.817394	+	0.576080i	$0.804582\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 0.427048i	− 0.0261349i
$268$	0	0
$269$	24.8637	1.51597	0.757983	−	0.652274i	$-0.226185\pi$
$269$	24.8637	1.51597	0.757983	+	0.652274i	$0.226185\pi$
$270$	0	0
$271$	0.578119	0.0351183	0.0175591	−	0.999846i	$-0.494410\pi$
$271$	0.578119	0.0351183	0.0175591	+	0.999846i	$0.494410\pi$
$272$	0	0
$273$	− 0.558182i	− 0.0337827i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 9.51387i	− 0.571633i	−0.958284	−	0.285817i	$-0.907735\pi$
$277$	− 9.51387i	− 0.571633i	0.958284	−	0.285817i	$-0.0922648\pi$
$278$	0	0
$279$	−4.72740	−0.283022
$280$	0	0
$281$	−13.1562	−0.784835	−0.392418	−	0.919787i	$-0.628361\pi$
$281$	−13.1562	−0.784835	−0.392418	+	0.919787i	$0.628361\pi$
$282$	0	0
$283$	− 24.5061i	− 1.45673i	−0.685187	−	0.728367i	$-0.740280\pi$
$283$	− 24.5061i	− 1.45673i	0.685187	−	0.728367i	$-0.259720\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00000i	0.0590281i
$288$	0	0
$289$	12.0347	0.707924
$290$	0	0
$291$	0.558182	0.0327212
$292$	0	0
$293$	6.29254i	0.367614i	0.982962	+	0.183807i	$0.0588422\pi$
$293$	6.29254i	0.367614i	−0.982962	+	0.183807i	$0.941158\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 13.2205i	− 0.767130i
$298$	0	0
$299$	−5.22829	−0.302360
$300$	0	0
$301$	0.972255	0.0560398
$302$	0	0
$303$	− 11.7274i	− 0.673721i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.3351i	0.761072i	0.924766	+	0.380536i	$0.124260\pi$
$307$	13.3351i	0.761072i	−0.924766	+	0.380536i	$0.875740\pi$
$308$	0	0
$309$	6.44361	0.366564
$310$	0	0
$311$	−24.4270	−1.38513	−0.692565	−	0.721355i	$-0.743520\pi$
$311$	−24.4270	−1.38513	−0.692565	+	0.721355i	$0.743520\pi$
$312$	0	0
$313$	− 26.5486i	− 1.50061i	−0.661089	−	0.750307i	$-0.729906\pi$
$313$	− 26.5486i	− 1.50061i	0.661089	−	0.750307i	$-0.270094\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.9852i	− 0.953986i	−0.878907	−	0.476993i	$-0.841727\pi$
$317$	− 16.9852i	− 0.953986i	0.878907	−	0.476993i	$-0.158273\pi$
$318$	0	0
$319$	−26.9132	−1.50685
$320$	0	0
$321$	14.0990	0.786927
$322$	0	0
$323$	2.73700i	0.152291i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.891443i	0.0492969i
$328$	0	0
$329$	1.27081	0.0700620
$330$	0	0
$331$	17.8914	0.983403	0.491701	−	0.870764i	$-0.336375\pi$
$331$	17.8914	0.983403	0.491701	+	0.870764i	$0.336375\pi$
$332$	0	0
$333$	12.4626i	0.682946i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.52864i	− 0.246691i	−0.992364	−	0.123345i	$-0.960638\pi$
$337$	− 4.52864i	− 0.246691i	0.992364	−	0.123345i	$-0.0393623\pi$
$338$	0	0
$339$	−15.2153	−0.826383
$340$	0	0
$341$	−6.10676	−0.330700
$342$	0	0
$343$	1.69965i	0.0917725i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 0.0937869i	− 0.00503475i	−0.999997	−	0.00251737i	$-0.999199\pi$
$347$	− 0.0937869i	− 0.00503475i	0.999997	−	0.00251737i	$-0.000801305\pi$
$348$	0	0
$349$	−8.07902	−0.432460	−0.216230	−	0.976342i	$-0.569376\pi$
$349$	−8.07902	−0.432460	−0.216230	+	0.976342i	$0.569376\pi$
$350$	0	0
$351$	−24.0130	−1.28172
$352$	0	0
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.237897i	− 0.0125908i
$358$	0	0
$359$	−8.69965	−0.459150	−0.229575	−	0.973291i	$-0.573734\pi$
$359$	−8.69965	−0.459150	−0.229575	+	0.973291i	$0.573734\pi$
$360$	0	0
$361$	−17.4913	−0.920594
$362$	0	0
$363$	2.38453i	0.125156i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.8064i	1.66028i	0.557554	+	0.830141i	$0.311740\pi$
$367$	31.8064i	1.66028i	−0.557554	+	0.830141i	$0.688260\pi$
$368$	0	0
$369$	18.3351	0.954485
$370$	0	0
$371$	−0.436651	−0.0226698
$372$	0	0
$373$	13.4288i	0.695319i	0.937621	+	0.347660i	$0.113023\pi$
$373$	13.4288i	0.695319i	−0.937621	+	0.347660i	$0.886977\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.8836i	2.51764i
$378$	0	0
$379$	−31.8212	−1.63454	−0.817272	−	0.576252i	$-0.804515\pi$
$379$	−31.8212	−1.63454	−0.817272	+	0.576252i	$0.804515\pi$
$380$	0	0
$381$	1.54341	0.0790714
$382$	0	0
$383$	− 34.3480i	− 1.75510i	−0.479483	−	0.877551i	$-0.659176\pi$
$383$	− 34.3480i	− 1.75510i	0.479483	−	0.877551i	$-0.340824\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 17.8264i	− 0.906164i
$388$	0	0
$389$	−8.14928	−0.413185	−0.206592	−	0.978427i	$-0.566237\pi$
$389$	−8.14928	−0.413185	−0.206592	+	0.978427i	$0.566237\pi$
$390$	0	0
$391$	−2.22829	−0.112690
$392$	0	0
$393$	17.7569i	0.895719i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.0625i	0.956717i	0.878165	+	0.478359i	$0.158768\pi$
$397$	19.0625i	0.956717i	−0.878165	+	0.478359i	$0.841232\pi$
$398$	0	0
$399$	−0.131135	−0.00656496
$400$	0	0
$401$	19.9705	0.997277	0.498639	−	0.866810i	$-0.333833\pi$
$401$	19.9705	0.997277	0.498639	+	0.866810i	$0.333833\pi$
$402$	0	0
$403$	11.0920i	0.552531i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.0990i	0.797996i
$408$	0	0
$409$	10.3351	0.511036	0.255518	−	0.966804i	$-0.417754\pi$
$409$	10.3351	0.511036	0.255518	+	0.966804i	$0.417754\pi$
$410$	0	0
$411$	−15.7491	−0.776847
$412$	0	0
$413$	0.0790169i	0.00388817i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.54341i	− 0.467343i
$418$	0	0
$419$	−5.72740	−0.279802	−0.139901	−	0.990166i	$-0.544678\pi$
$419$	−5.72740	−0.279802	−0.139901	+	0.990166i	$0.544678\pi$
$420$	0	0
$421$	−8.44361	−0.411516	−0.205758	−	0.978603i	$-0.565966\pi$
$421$	−8.44361	−0.411516	−0.205758	+	0.978603i	$0.565966\pi$
$422$	0	0
$423$	− 23.3003i	− 1.13290i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.891443i	− 0.0431400i
$428$	0	0
$429$	−13.2205	−0.638291
$430$	0	0
$431$	35.3698	1.70370	0.851851	−	0.523785i	$-0.175480\pi$
$431$	35.3698	1.70370	0.851851	+	0.523785i	$0.175480\pi$
$432$	0	0
$433$	− 13.3923i	− 0.643595i	−0.946809	−	0.321797i	$-0.895713\pi$
$433$	− 13.3923i	− 0.643595i	0.946809	−	0.321797i	$-0.104287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.22829i	0.0587573i
$438$	0	0
$439$	21.1137	1.00770	0.503852	−	0.863790i	$-0.331916\pi$
$439$	21.1137	1.00770	0.503852	+	0.863790i	$0.331916\pi$
$440$	0	0
$441$	15.5651	0.741197
$442$	0	0
$443$	31.1692i	1.48089i	0.672115	+	0.740447i	$0.265386\pi$
$443$	31.1692i	1.48089i	−0.672115	+	0.740447i	$0.734614\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 13.8046i	− 0.652936i
$448$	0	0
$449$	6.09199	0.287499	0.143749	−	0.989614i	$-0.454084\pi$
$449$	6.09199	0.287499	0.143749	+	0.989614i	$0.454084\pi$
$450$	0	0
$451$	23.6849	1.11528
$452$	0	0
$453$	7.37240i	0.346386i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.1345i	1.03541i	0.855560	+	0.517704i	$0.173213\pi$
$457$	22.1345i	1.03541i	−0.855560	+	0.517704i	$0.826787\pi$
$458$	0	0
$459$	−10.2343	−0.477697
$460$	0	0
$461$	−11.5434	−0.537630	−0.268815	−	0.963192i	$-0.586632\pi$
$461$	−11.5434	−0.537630	−0.268815	+	0.963192i	$0.586632\pi$
$462$	0	0
$463$	− 30.4270i	− 1.41406i	−0.707181	−	0.707032i	$-0.750033\pi$
$463$	− 30.4270i	− 1.41406i	0.707181	−	0.707032i	$-0.249967\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.3776i	0.480217i	0.970746	+	0.240108i	$0.0771830\pi$
$467$	10.3776i	0.480217i	−0.970746	+	0.240108i	$0.922817\pi$
$468$	0	0
$469$	−0.679714	−0.0313863
$470$	0	0
$471$	−17.1858	−0.791879
$472$	0	0
$473$	− 23.0277i	− 1.05882i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.00601i	0.366570i
$478$	0	0
$479$	29.6128	1.35304	0.676522	−	0.736422i	$-0.263486\pi$
$479$	29.6128	1.35304	0.676522	+	0.736422i	$0.263486\pi$
$480$	0	0
$481$	29.2413	1.33329
$482$	0	0
$483$	− 0.106762i	− 0.00485783i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.1562i	0.505537i	0.967527	+	0.252769i	$0.0813412\pi$
$487$	11.1562i	0.505537i	−0.967527	+	0.252769i	$0.918659\pi$
$488$	0	0
$489$	−0.891443	−0.0403125
$490$	0	0
$491$	38.2630	1.72679	0.863393	−	0.504533i	$-0.168335\pi$
$491$	38.2630	1.72679	0.863393	+	0.504533i	$0.168335\pi$
$492$	0	0
$493$	20.8342i	0.938323i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.69965i	− 0.0762398i
$498$	0	0
$499$	−6.40711	−0.286822	−0.143411	−	0.989663i	$-0.545807\pi$
$499$	−6.40711	−0.286822	−0.143411	+	0.989663i	$0.545807\pi$
$500$	0	0
$501$	−7.02774	−0.313976
$502$	0	0
$503$	18.0920i	0.806682i	0.915050	+	0.403341i	$0.132151\pi$
$503$	18.0920i	0.806682i	−0.915050	+	0.403341i	$0.867849\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.5929i	0.559270i
$508$	0	0
$509$	34.1285	1.51272	0.756359	−	0.654156i	$-0.226976\pi$
$509$	34.1285	1.51272	0.756359	+	0.654156i	$0.226976\pi$
$510$	0	0
$511$	1.57295	0.0695833
$512$	0	0
$513$	5.64142i	0.249075i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 30.0990i	− 1.32375i
$518$	0	0
$519$	2.31512	0.101622
$520$	0	0
$521$	35.8854	1.57217	0.786085	−	0.618119i	$-0.212105\pi$
$521$	35.8854	1.57217	0.786085	+	0.618119i	$0.212105\pi$
$522$	0	0
$523$	− 11.7865i	− 0.515387i	−0.966227	−	0.257693i	$-0.917038\pi$
$523$	− 11.7865i	− 0.515387i	0.966227	−	0.257693i	$-0.0829624\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.72740i	0.205929i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	1.44878	0.0628717
$532$	0	0
$533$	− 43.0199i	− 1.86340i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.6997i	− 0.548030i
$538$	0	0
$539$	20.1068	0.866060
$540$	0	0
$541$	−17.2413	−0.741260	−0.370630	−	0.928781i	$-0.620858\pi$
$541$	−17.2413	−0.741260	−0.370630	+	0.928781i	$0.620858\pi$
$542$	0	0
$543$	− 9.43665i	− 0.404965i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.6571i	0.626694i	0.949639	+	0.313347i	$0.101450\pi$
$547$	14.6571i	0.626694i	−0.949639	+	0.313347i	$0.898550\pi$
$548$	0	0
$549$	−16.3447	−0.697573
$550$	0	0
$551$	11.4843	0.489249
$552$	0	0
$553$	0.427048i	0.0181599i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.07902i	0.0880908i	0.999030	+	0.0440454i	$0.0140246\pi$
$557$	2.07902i	0.0880908i	−0.999030	+	0.0440454i	$0.985975\pi$
$558$	0	0
$559$	−41.8264	−1.76907
$560$	0	0
$561$	−5.63456	−0.237891
$562$	0	0
$563$	5.10676i	0.215224i	0.994193	+	0.107612i	$0.0343205\pi$
$563$	5.10676i	0.215224i	−0.994193	+	0.107612i	$0.965680\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.322081i	0.0135261i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	10.0642	0.421176	0.210588	−	0.977575i	$-0.432462\pi$
$571$	10.0642	0.421176	0.210588	+	0.977575i	$0.432462\pi$
$572$	0	0
$573$	19.9150i	0.831960i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 15.7274i	− 0.654740i	−0.944896	−	0.327370i	$-0.893838\pi$
$577$	− 15.7274i	− 0.654740i	0.944896	−	0.327370i	$-0.106162\pi$
$578$	0	0
$579$	−1.97046	−0.0818895
$580$	0	0
$581$	−1.34983	−0.0560002
$582$	0	0
$583$	10.3420i	0.428323i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.7344i	− 0.773250i	−0.922237	−	0.386625i	$-0.873641\pi$
$587$	− 18.7344i	− 0.773250i	0.922237	−	0.386625i	$-0.126359\pi$
$588$	0	0
$589$	2.60586	0.107373
$590$	0	0
$591$	−6.65714	−0.273838
$592$	0	0
$593$	3.21532i	0.132037i	0.997818	+	0.0660187i	$0.0210297\pi$
$593$	3.21532i	0.132037i	−0.997818	+	0.0660187i	$0.978970\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 12.2986i	− 0.503346i
$598$	0	0
$599$	−22.7639	−0.930108	−0.465054	−	0.885282i	$-0.653965\pi$
$599$	−22.7639	−0.930108	−0.465054	+	0.885282i	$0.653965\pi$
$600$	0	0
$601$	4.30552	0.175626	0.0878128	−	0.996137i	$-0.472012\pi$
$601$	4.30552	0.175626	0.0878128	+	0.996137i	$0.472012\pi$
$602$	0	0
$603$	12.4626i	0.507516i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.0868i	0.450000i	0.974359	+	0.225000i	$0.0722383\pi$
$607$	11.0868i	0.450000i	−0.974359	+	0.225000i	$0.927762\pi$
$608$	0	0
$609$	−0.998205	−0.0404493
$610$	0	0
$611$	−54.6701	−2.21172
$612$	0	0
$613$	− 29.9409i	− 1.20930i	−0.796490	−	0.604651i	$-0.793313\pi$
$613$	− 29.9409i	− 1.20930i	0.796490	−	0.604651i	$-0.206687\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.0130i	1.40957i	0.709421	+	0.704785i	$0.248956\pi$
$617$	35.0130i	1.40957i	−0.709421	+	0.704785i	$0.751044\pi$
$618$	0	0
$619$	−4.98523	−0.200373	−0.100187	−	0.994969i	$-0.531944\pi$
$619$	−4.98523	−0.200373	−0.100187	+	0.994969i	$0.531944\pi$
$620$	0	0
$621$	−4.59289	−0.184306
$622$	0	0
$623$	0.0590800i	0.00236699i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.10592i	0.124038i
$628$	0	0
$629$	12.4626	0.496916
$630$	0	0
$631$	−3.64237	−0.145000	−0.0725002	−	0.997368i	$-0.523098\pi$
$631$	−3.64237	−0.145000	−0.0725002	+	0.997368i	$0.523098\pi$
$632$	0	0
$633$	2.72919i	0.108476i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 36.5208i	− 1.44701i
$638$	0	0
$639$	−31.1632	−1.23280
$640$	0	0
$641$	38.6997	1.52854	0.764272	−	0.644894i	$-0.223098\pi$
$641$	38.6997	1.52854	0.764272	+	0.644894i	$0.223098\pi$
$642$	0	0
$643$	28.8637i	1.13827i	0.822243	+	0.569137i	$0.192722\pi$
$643$	28.8637i	1.13827i	−0.822243	+	0.569137i	$0.807278\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.8854i	− 0.781777i	−0.920438	−	0.390888i	$-0.872168\pi$
$647$	− 19.8854i	− 0.781777i	0.920438	−	0.390888i	$-0.127832\pi$
$648$	0	0
$649$	1.87151	0.0734630
$650$	0	0
$651$	−0.226499	−0.00887719
$652$	0	0
$653$	27.1988i	1.06437i	0.846628	+	0.532185i	$0.178629\pi$
$653$	27.1988i	1.06437i	−0.846628	+	0.532185i	$0.821371\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 28.8402i	− 1.12516i
$658$	0	0
$659$	17.7274	0.690561	0.345281	−	0.938499i	$-0.387784\pi$
$659$	17.7274	0.690561	0.345281	+	0.938499i	$0.387784\pi$
$660$	0	0
$661$	40.7048	1.58323	0.791617	−	0.611018i	$-0.209240\pi$
$661$	40.7048	1.58323	0.791617	+	0.611018i	$0.209240\pi$
$662$	0	0
$663$	10.2343i	0.397468i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.34983i	0.362027i
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−21.1137	−0.815086
$672$	0	0
$673$	− 12.2135i	− 0.470797i	−0.971899	−	0.235398i	$-0.924361\pi$
$673$	− 12.2135i	− 0.470797i	0.971899	−	0.235398i	$-0.0756395\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.5061i	0.634380i	0.948362	+	0.317190i	$0.102739\pi$
$677$	16.5061i	0.634380i	−0.948362	+	0.317190i	$0.897261\pi$
$678$	0	0
$679$	−0.0772219	−0.00296350
$680$	0	0
$681$	12.2726	0.470287
$682$	0	0
$683$	29.1988i	1.11726i	0.829417	+	0.558630i	$0.188673\pi$
$683$	29.1988i	1.11726i	−0.829417	+	0.558630i	$0.811327\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.7292i	0.409344i
$688$	0	0
$689$	18.7847	0.715639
$690$	0	0
$691$	−15.9150	−0.605434	−0.302717	−	0.953080i	$-0.597894\pi$
$691$	−15.9150	−0.605434	−0.302717	+	0.953080i	$0.597894\pi$
$692$	0	0
$693$	0.779514i	0.0296113i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.3351i	− 0.694490i
$698$	0	0
$699$	−8.57116	−0.324191
$700$	0	0
$701$	34.5981	1.30675	0.653375	−	0.757034i	$-0.273352\pi$
$701$	34.5981	1.30675	0.653375	+	0.757034i	$0.273352\pi$
$702$	0	0
$703$	− 6.86971i	− 0.259096i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.62243i	0.0610177i
$708$	0	0
$709$	−35.8212	−1.34529	−0.672646	−	0.739964i	$-0.734842\pi$
$709$	−35.8212	−1.34529	−0.672646	+	0.739964i	$0.734842\pi$
$710$	0	0
$711$	7.82994	0.293646
$712$	0	0
$713$	2.12153i	0.0794520i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.57295i	0.282817i
$718$	0	0
$719$	32.5781	1.21496	0.607479	−	0.794335i	$-0.292181\pi$
$719$	32.5781	1.21496	0.607479	+	0.794335i	$0.292181\pi$
$720$	0	0
$721$	−0.891443	−0.0331991
$722$	0	0
$723$	− 2.79861i	− 0.104081i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 31.9557i	− 1.18517i	−0.805508	−	0.592585i	$-0.798107\pi$
$727$	− 31.9557i	− 1.18517i	0.805508	−	0.592585i	$-0.201893\pi$
$728$	0	0
$729$	−6.19875	−0.229583
$730$	0	0
$731$	−17.8264	−0.659331
$732$	0	0
$733$	2.19359i	0.0810220i	0.999179	+	0.0405110i	$0.0128986\pi$
$733$	2.19359i	0.0810220i	−0.999179	+	0.0405110i	$0.987101\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0990i	0.593013i
$738$	0	0
$739$	51.7473	1.90356	0.951778	−	0.306787i	$-0.0992539\pi$
$739$	51.7473	1.90356	0.951778	+	0.306787i	$0.0992539\pi$
$740$	0	0
$741$	5.64142	0.207243
$742$	0	0
$743$	14.4436i	0.529885i	0.964264	+	0.264942i	$0.0853529\pi$
$743$	14.4436i	0.529885i	−0.964264	+	0.264942i	$0.914647\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.7491i	0.905523i
$748$	0	0
$749$	−1.95052	−0.0712706
$750$	0	0
$751$	17.4288	0.635987	0.317994	−	0.948093i	$-0.396991\pi$
$751$	17.4288	0.635987	0.317994	+	0.948093i	$0.396991\pi$
$752$	0	0
$753$	22.1623i	0.807637i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 34.9331i	− 1.26967i	−0.772650	−	0.634833i	$-0.781069\pi$
$757$	− 34.9331i	− 1.26967i	0.772650	−	0.634833i	$-0.218931\pi$
$758$	0	0
$759$	−2.52864	−0.0917839
$760$	0	0
$761$	−23.2482	−0.842748	−0.421374	−	0.906887i	$-0.638452\pi$
$761$	−23.2482	−0.842748	−0.421374	+	0.906887i	$0.638452\pi$
$762$	0	0
$763$	− 0.123327i	− 0.00446473i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.39930i	− 0.122742i
$768$	0	0
$769$	48.6997	1.75615	0.878077	−	0.478519i	$-0.158826\pi$
$769$	48.6997	1.75615	0.878077	+	0.478519i	$0.158826\pi$
$770$	0	0
$771$	−9.37335	−0.337573
$772$	0	0
$773$	15.6128i	0.561554i	0.959773	+	0.280777i	$0.0905922\pi$
$773$	15.6128i	0.561554i	−0.959773	+	0.280777i	$0.909408\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.597107i	0.0214211i
$778$	0	0
$779$	−10.1068	−0.362112
$780$	0	0
$781$	−40.2560	−1.44047
$782$	0	0
$783$	42.9427i	1.53465i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.2630i	0.864883i	0.901662	+	0.432441i	$0.142348\pi$
$787$	24.2630i	0.864883i	−0.901662	+	0.432441i	$0.857652\pi$
$788$	0	0
$789$	16.4141	0.584356
$790$	0	0
$791$	2.10497	0.0748440
$792$	0	0
$793$	38.3498i	1.36184i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.5911i	1.57950i	0.613430	+	0.789749i	$0.289789\pi$
$797$	44.5911i	1.57950i	−0.613430	+	0.789749i	$0.710211\pi$
$798$	0	0
$799$	−23.3003	−0.824307
$800$	0	0
$801$	1.08323	0.0382742
$802$	0	0
$803$	− 37.2552i	− 1.31471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 21.8420i	− 0.768874i
$808$	0	0
$809$	44.7144	1.57208	0.786038	−	0.618179i	$-0.212129\pi$
$809$	44.7144	1.57208	0.786038	+	0.618179i	$0.212129\pi$
$810$	0	0
$811$	1.19179	0.0418495	0.0209247	−	0.999781i	$-0.493339\pi$
$811$	1.19179	0.0418495	0.0209247	+	0.999781i	$0.493339\pi$
$812$	0	0
$813$	− 0.507859i	− 0.0178114i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.82635i	0.343780i
$818$	0	0
$819$	1.41587	0.0494744
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 26.7551i	− 0.932626i	−0.884620	−	0.466313i	$-0.845582\pi$
$823$	− 26.7551i	− 0.932626i	0.884620	−	0.466313i	$-0.154418\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.2907i	− 0.948992i	−0.880258	−	0.474496i	$-0.842630\pi$
$827$	− 27.2907i	− 0.948992i	0.880258	−	0.474496i	$-0.157370\pi$
$828$	0	0
$829$	−34.2925	−1.19103	−0.595515	−	0.803344i	$-0.703052\pi$
$829$	−34.2925	−1.19103	−0.595515	+	0.803344i	$0.703052\pi$
$830$	0	0
$831$	−8.35763	−0.289923
$832$	0	0
$833$	− 15.5651i	− 0.539300i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.74396i	0.336801i
$838$	0	0
$839$	40.6701	1.40409	0.702044	−	0.712133i	$-0.252271\pi$
$839$	40.6701	1.40409	0.702044	+	0.712133i	$0.252271\pi$
$840$	0	0
$841$	58.4192	2.01446
$842$	0	0
$843$	11.5573i	0.398056i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.329889i	− 0.0113351i
$848$	0	0
$849$	−21.5278	−0.738832
$850$	0	0
$851$	5.59289	0.191722
$852$	0	0
$853$	34.6276i	1.18563i	0.805340	+	0.592813i	$0.201983\pi$
$853$	34.6276i	1.18563i	−0.805340	+	0.592813i	$0.798017\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.4861i	0.904749i	0.891828	+	0.452374i	$0.149423\pi$
$857$	26.4861i	0.904749i	−0.891828	+	0.452374i	$0.850577\pi$
$858$	0	0
$859$	−40.9627	−1.39763	−0.698814	−	0.715304i	$-0.746288\pi$
$859$	−40.9627	−1.39763	−0.698814	+	0.715304i	$0.746288\pi$
$860$	0	0
$861$	0.878468	0.0299381
$862$	0	0
$863$	18.5712i	0.632170i	0.948731	+	0.316085i	$0.102368\pi$
$863$	18.5712i	0.632170i	−0.948731	+	0.316085i	$0.897632\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 10.5721i	− 0.359048i
$868$	0	0
$869$	10.1146	0.343113
$870$	0	0
$871$	29.2413	0.990803
$872$	0	0
$873$	1.41587i	0.0479199i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 28.4913i	− 0.962083i	−0.876698	−	0.481041i	$-0.840259\pi$
$877$	− 28.4913i	− 0.962083i	0.876698	−	0.481041i	$-0.159741\pi$
$878$	0	0
$879$	5.52780	0.186448
$880$	0	0
$881$	−4.91318	−0.165529	−0.0827645	−	0.996569i	$-0.526375\pi$
$881$	−4.91318	−0.165529	−0.0827645	+	0.996569i	$0.526375\pi$
$882$	0	0
$883$	− 30.1710i	− 1.01534i	−0.861553	−	0.507668i	$-0.830508\pi$
$883$	− 30.1710i	− 1.01534i	0.861553	−	0.507668i	$-0.169492\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.7865i	0.865825i	0.901436	+	0.432913i	$0.142514\pi$
$887$	25.7865i	0.865825i	−0.901436	+	0.432913i	$0.857486\pi$
$888$	0	0
$889$	−0.213524	−0.00716136
$890$	0	0
$891$	7.62844	0.255562
$892$	0	0
$893$	12.8438i	0.429800i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.59289i	0.153352i
$898$	0	0
$899$	19.8360	0.661566
$900$	0	0
$901$	8.00601	0.266719
$902$	0	0
$903$	− 0.854095i	− 0.0284225i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.3776i	− 1.47353i	−0.676147	−	0.736767i	$-0.736352\pi$
$907$	− 44.3776i	− 1.47353i	0.676147	−	0.736767i	$-0.263648\pi$
$908$	0	0
$909$	29.7473	0.986657
$910$	0	0
$911$	47.7968	1.58358	0.791789	−	0.610794i	$-0.209150\pi$
$911$	47.7968	1.58358	0.791789	+	0.610794i	$0.209150\pi$
$912$	0	0
$913$	31.9705i	1.05807i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.45659i	− 0.0811237i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	11.7144	0.386003
$922$	0	0
$923$	73.1189i	2.40674i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.3447i	0.536829i
$928$	0	0
$929$	−24.0199	−0.788069	−0.394034	−	0.919096i	$-0.628921\pi$
$929$	−24.0199	−0.788069	−0.394034	+	0.919096i	$0.628921\pi$
$930$	0	0
$931$	−8.57991	−0.281195
$932$	0	0
$933$	21.4584i	0.702516i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.384533i	0.0125621i	0.999980	+	0.00628107i	$0.00199934\pi$
$937$	0.384533i	0.0125621i	−0.999980	+	0.00628107i	$0.998001\pi$
$938$	0	0
$939$	−23.3221	−0.761087
$940$	0	0
$941$	25.0920	0.817976	0.408988	−	0.912540i	$-0.365882\pi$
$941$	25.0920	0.817976	0.408988	+	0.912540i	$0.365882\pi$
$942$	0	0
$943$	− 8.22829i	− 0.267950i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.7126i	1.45297i	0.687185	+	0.726483i	$0.258846\pi$
$947$	44.7126i	1.45297i	−0.687185	+	0.726483i	$0.741154\pi$
$948$	0	0
$949$	−67.6683	−2.19661
$950$	0	0
$951$	−14.9210	−0.483846
$952$	0	0
$953$	21.2500i	0.688356i	0.938904	+	0.344178i	$0.111842\pi$
$953$	21.2500i	0.688356i	−0.938904	+	0.344178i	$0.888158\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	23.6424i	0.764249i
$958$	0	0
$959$	2.17882	0.0703577
$960$	0	0
$961$	−26.4991	−0.854810
$962$	0	0
$963$	35.7629i	1.15244i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 41.0417i	− 1.31981i	−0.751349	−	0.659906i	$-0.770596\pi$
$967$	− 41.0417i	− 1.31981i	0.751349	−	0.659906i	$-0.229404\pi$
$968$	0	0
$969$	2.40437	0.0772394
$970$	0	0
$971$	35.2760	1.13206	0.566030	−	0.824385i	$-0.308479\pi$
$971$	35.2760	1.13206	0.566030	+	0.824385i	$0.308479\pi$
$972$	0	0
$973$	1.32029i	0.0423264i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.79164i	− 0.217284i	−0.994081	−	0.108642i	$-0.965350\pi$
$977$	− 6.79164i	− 0.217284i	0.994081	−	0.108642i	$-0.0346502\pi$
$978$	0	0
$979$	1.39930	0.0447219
$980$	0	0
$981$	−2.26121	−0.0721947
$982$	0	0
$983$	− 19.5981i	− 0.625081i	−0.949904	−	0.312540i	$-0.898820\pi$
$983$	− 19.5981i	− 0.625081i	0.949904	−	0.312540i	$-0.101180\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.11636i	− 0.0355343i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−37.2995	−1.18486	−0.592429	−	0.805623i	$-0.701831\pi$
$991$	−37.2995	−1.18486	−0.592429	+	0.805623i	$0.701831\pi$
$992$	0	0
$993$	− 15.7171i	− 0.498766i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 19.7274i	− 0.624773i	−0.949955	−	0.312386i	$-0.898872\pi$
$997$	− 19.7274i	− 0.624773i	0.949955	−	0.312386i	$-0.101128\pi$
$998$	0	0
$999$	25.6875	0.812717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.p.4049.3		6
5.2	odd	4		4600.2.a.x.1.2		3
5.3	odd	4		920.2.a.h.1.2	✓	3
5.4	even	2	inner	4600.2.e.p.4049.4		6
15.8	even	4		8280.2.a.bj.1.2		3
20.3	even	4		1840.2.a.s.1.2		3
20.7	even	4		9200.2.a.ce.1.2		3
40.3	even	4		7360.2.a.cc.1.2		3
40.13	odd	4		7360.2.a.by.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.2	✓	3	5.3	odd	4
1840.2.a.s.1.2		3	20.3	even	4
4600.2.a.x.1.2		3	5.2	odd	4
4600.2.e.p.4049.3		6	1.1	even	1	trivial
4600.2.e.p.4049.4		6	5.4	even	2	inner
7360.2.a.by.1.2		3	40.13	odd	4
7360.2.a.cc.1.2		3	40.3	even	4
8280.2.a.bj.1.2		3	15.8	even	4
9200.2.a.ce.1.2		3	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.878468i q^{3} +0.121532i q^{7} +2.22829 q^{9} +O(q^{10})\)	\(q-0.878468i q^{3} +0.121532i q^{7} +2.22829 q^{9} +2.87847 q^{11} -5.22829i q^{13} -2.22829i q^{17} -1.22829 q^{19} +0.106762 q^{21} -1.00000i q^{23} -4.59289i q^{27} -9.34983 q^{29} -2.12153 q^{31} -2.52864i q^{33} +5.59289i q^{37} -4.59289 q^{39} +8.22829 q^{41} -8.00000i q^{43} -10.4566i q^{47} +6.98523 q^{49} -1.95749 q^{51} +3.59289i q^{53} +1.07902i q^{57} +0.650174 q^{59} -7.33506 q^{61} +0.270809i q^{63} +5.59289i q^{67} -0.878468 q^{69} -13.9852 q^{71} -12.9427i q^{73} +0.349826i q^{77} +3.51387 q^{79} +2.65017 q^{81} +11.1068i q^{83} +8.21352i q^{87} +0.486128 q^{89} +0.635404 q^{91} +1.86370i q^{93} +0.635404i q^{97} +6.41407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 20 q^{9}+O(q^{10}) \) 6 * q - 20 * q^9	\( 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) \) 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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