LMFDB - Embedded newform 4600.2.e.o.4049.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.o.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} -3.56155 q^{9} +5.12311 q^{11} -4.56155i q^{13} -3.12311i q^{17} -5.12311 q^{19} +1.00000i q^{23} -1.43845i q^{27} +0.561553 q^{29} -6.56155 q^{31} +13.1231i q^{33} -8.24621i q^{37} +11.6847 q^{39} +10.8078 q^{41} +8.00000i q^{43} +11.6847i q^{47} +7.00000 q^{49} +8.00000 q^{51} -2.00000i q^{53} -13.1231i q^{57} +6.24621 q^{59} +12.2462 q^{61} -5.12311i q^{67} -2.56155 q^{69} +9.43845 q^{71} +2.31534i q^{73} +5.12311 q^{79} -7.00000 q^{81} +2.24621i q^{83} +1.43845i q^{87} +13.3693 q^{89} -16.8078i q^{93} -13.3693i q^{97} -18.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} + 4 q^{11} - 4 q^{19} - 6 q^{29} - 18 q^{31} + 22 q^{39} + 2 q^{41} + 28 q^{49} + 32 q^{51} - 8 q^{59} + 16 q^{61} - 2 q^{69} + 46 q^{71} + 4 q^{79} - 28 q^{81} + 4 q^{89} - 40 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 + 4 * q^11 - 4 * q^19 - 6 * q^29 - 18 * q^31 + 22 * q^39 + 2 * q^41 + 28 * q^49 + 32 * q^51 - 8 * q^59 + 16 * q^61 - 2 * q^69 + 46 * q^71 + 4 * q^79 - 28 * q^81 + 4 * q^89 - 40 * q^99

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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	− 4.56155i	− 1.26515i	−0.774500	−	0.632574i	$-0.781999\pi$
$13$	− 4.56155i	− 1.26515i	0.774500	−	0.632574i	$-0.218001\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.12311i	− 0.757464i	−0.925506	−	0.378732i	$-0.876360\pi$
$17$	− 3.12311i	− 0.757464i	0.925506	−	0.378732i	$-0.123640\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	0	0
$29$	0.561553	0.104278	0.0521389	−	0.998640i	$-0.483396\pi$
$29$	0.561553	0.104278	0.0521389	+	0.998640i	$0.483396\pi$
$30$	0	0
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	0	0
$33$	13.1231i	2.28444i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.24621i	− 1.35567i	−0.735215	−	0.677834i	$-0.762919\pi$
$37$	− 8.24621i	− 1.35567i	0.735215	−	0.677834i	$-0.237081\pi$
$38$	0	0
$39$	11.6847	1.87104
$40$	0	0
$41$	10.8078	1.68789	0.843945	−	0.536430i	$-0.180228\pi$
$41$	10.8078	1.68789	0.843945	+	0.536430i	$0.180228\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.6847i	1.70438i	0.523230	+	0.852191i	$0.324727\pi$
$47$	11.6847i	1.70438i	−0.523230	+	0.852191i	$0.675273\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 13.1231i	− 1.73820i
$58$	0	0
$59$	6.24621	0.813187	0.406594	−	0.913609i	$-0.366716\pi$
$59$	6.24621	0.813187	0.406594	+	0.913609i	$0.366716\pi$
$60$	0	0
$61$	12.2462	1.56797	0.783983	−	0.620782i	$-0.213185\pi$
$61$	12.2462	1.56797	0.783983	+	0.620782i	$0.213185\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.12311i	− 0.625887i	−0.949772	−	0.312943i	$-0.898685\pi$
$67$	− 5.12311i	− 0.625887i	0.949772	−	0.312943i	$-0.101315\pi$
$68$	0	0
$69$	−2.56155	−0.308375
$70$	0	0
$71$	9.43845	1.12014	0.560069	−	0.828446i	$-0.310775\pi$
$71$	9.43845	1.12014	0.560069	+	0.828446i	$0.310775\pi$
$72$	0	0
$73$	2.31534i	0.270990i	0.990778	+	0.135495i	$0.0432625\pi$
$73$	2.31534i	0.270990i	−0.990778	+	0.135495i	$0.956737\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	2.24621i	0.246554i	0.992372	+	0.123277i	$0.0393403\pi$
$83$	2.24621i	0.246554i	−0.992372	+	0.123277i	$0.960660\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.43845i	0.154218i
$88$	0	0
$89$	13.3693	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$89$	13.3693	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 16.8078i	− 1.74288i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.3693i	− 1.35745i	−0.734393	−	0.678724i	$-0.762533\pi$
$97$	− 13.3693i	− 1.35745i	0.734393	−	0.678724i	$-0.237467\pi$
$98$	0	0
$99$	−18.2462	−1.83381
$100$	0	0
$101$	−4.24621	−0.422514	−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621	−0.422514	−0.211257	+	0.977431i	$0.567756\pi$
$102$	0	0
$103$	− 2.24621i	− 0.221326i	−0.993858	−	0.110663i	$-0.964703\pi$
$103$	− 2.24621i	− 0.221326i	0.993858	−	0.110663i	$-0.0352974\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.87689i	0.278120i	0.990284	+	0.139060i	$0.0444081\pi$
$107$	2.87689i	0.278120i	−0.990284	+	0.139060i	$0.955592\pi$
$108$	0	0
$109$	−4.87689	−0.467122	−0.233561	−	0.972342i	$-0.575038\pi$
$109$	−4.87689	−0.467122	−0.233561	+	0.972342i	$0.575038\pi$
$110$	0	0
$111$	21.1231	2.00492
$112$	0	0
$113$	11.1231i	1.04637i	0.852218	+	0.523187i	$0.175257\pi$
$113$	11.1231i	1.04637i	−0.852218	+	0.523187i	$0.824743\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.2462i	1.50196i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	27.6847i	2.49624i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.6847i	− 1.03685i	−0.855124	−	0.518423i	$-0.826519\pi$
$127$	− 11.6847i	− 1.03685i	0.855124	−	0.518423i	$-0.173481\pi$
$128$	0	0
$129$	−20.4924	−1.80426
$130$	0	0
$131$	15.6847	1.37037	0.685187	−	0.728367i	$-0.259720\pi$
$131$	15.6847	1.37037	0.685187	+	0.728367i	$0.259720\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.1231i	1.29205i	0.763315	+	0.646027i	$0.223571\pi$
$137$	15.1231i	1.29205i	−0.763315	+	0.646027i	$0.776429\pi$
$138$	0	0
$139$	15.6847	1.33036	0.665178	−	0.746685i	$-0.268356\pi$
$139$	15.6847	1.33036	0.665178	+	0.746685i	$0.268356\pi$
$140$	0	0
$141$	−29.9309	−2.52063
$142$	0	0
$143$	− 23.3693i	− 1.95424i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.9309i	1.47891i
$148$	0	0
$149$	3.75379	0.307522	0.153761	−	0.988108i	$-0.450861\pi$
$149$	3.75379	0.307522	0.153761	+	0.988108i	$0.450861\pi$
$150$	0	0
$151$	−14.5616	−1.18500	−0.592501	−	0.805570i	$-0.701859\pi$
$151$	−14.5616	−1.18500	−0.592501	+	0.805570i	$0.701859\pi$
$152$	0	0
$153$	11.1231i	0.899250i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.8769i	1.02769i	0.857884	+	0.513844i	$0.171779\pi$
$157$	12.8769i	1.02769i	−0.857884	+	0.513844i	$0.828221\pi$
$158$	0	0
$159$	5.12311	0.406289
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.93087i	0.777846i	0.921270	+	0.388923i	$0.127153\pi$
$163$	9.93087i	0.777846i	−0.921270	+	0.388923i	$0.872847\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	−7.80776	−0.600597
$170$	0	0
$171$	18.2462	1.39532
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.0000i	1.20263i
$178$	0	0
$179$	15.6847	1.17233	0.586163	−	0.810193i	$-0.300638\pi$
$179$	15.6847	1.17233	0.586163	+	0.810193i	$0.300638\pi$
$180$	0	0
$181$	20.2462	1.50489	0.752445	−	0.658656i	$-0.228875\pi$
$181$	20.2462	1.50489	0.752445	+	0.658656i	$0.228875\pi$
$182$	0	0
$183$	31.3693i	2.31889i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.0000i	− 1.17004i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.630683	0.0456346	0.0228173	−	0.999740i	$-0.492736\pi$
$191$	0.630683	0.0456346	0.0228173	+	0.999740i	$0.492736\pi$
$192$	0	0
$193$	4.56155i	0.328348i	0.986431	+	0.164174i	$0.0524958\pi$
$193$	4.56155i	0.328348i	−0.986431	+	0.164174i	$0.947504\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.9309i	0.850039i	0.905184	+	0.425020i	$0.139733\pi$
$197$	11.9309i	0.850039i	−0.905184	+	0.425020i	$0.860267\pi$
$198$	0	0
$199$	−2.87689	−0.203938	−0.101969	−	0.994788i	$-0.532514\pi$
$199$	−2.87689	−0.203938	−0.101969	+	0.994788i	$0.532514\pi$
$200$	0	0
$201$	13.1231	0.925633
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.56155i	− 0.247545i
$208$	0	0
$209$	−26.2462	−1.81549
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	24.1771i	1.65659i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.93087	−0.400771
$220$	0	0
$221$	−14.2462	−0.958304
$222$	0	0
$223$	− 4.49242i	− 0.300835i	−0.988623	−	0.150417i	$-0.951938\pi$
$223$	− 4.49242i	− 0.300835i	0.988623	−	0.150417i	$-0.0480618\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.2462i	0.680065i	0.940414	+	0.340032i	$0.110438\pi$
$227$	10.2462i	0.680065i	−0.940414	+	0.340032i	$0.889562\pi$
$228$	0	0
$229$	−23.1231	−1.52802	−0.764009	−	0.645206i	$-0.776772\pi$
$229$	−23.1231	−1.52802	−0.764009	+	0.645206i	$0.776772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.0540i	− 1.37929i	−0.724147	−	0.689646i	$-0.757766\pi$
$233$	− 21.0540i	− 1.37929i	0.724147	−	0.689646i	$-0.242234\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.1231i	0.852437i
$238$	0	0
$239$	11.0540	0.715022	0.357511	−	0.933909i	$-0.383625\pi$
$239$	11.0540	0.715022	0.357511	+	0.933909i	$0.383625\pi$
$240$	0	0
$241$	−26.4924	−1.70653	−0.853263	−	0.521480i	$-0.825380\pi$
$241$	−26.4924	−1.70653	−0.853263	+	0.521480i	$0.825380\pi$
$242$	0	0
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	23.3693i	1.48695i
$248$	0	0
$249$	−5.75379	−0.364632
$250$	0	0
$251$	10.2462	0.646735	0.323368	−	0.946273i	$-0.395185\pi$
$251$	10.2462	0.646735	0.323368	+	0.946273i	$0.395185\pi$
$252$	0	0
$253$	5.12311i	0.322087i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 17.6847i	− 1.10314i	−0.834129	−	0.551569i	$-0.814029\pi$
$257$	− 17.6847i	− 1.10314i	0.834129	−	0.551569i	$-0.185971\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	10.8769i	0.670698i	0.942094	+	0.335349i	$0.108854\pi$
$263$	10.8769i	0.670698i	−0.942094	+	0.335349i	$0.891146\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	34.2462i	2.09583i
$268$	0	0
$269$	−25.6847	−1.56602	−0.783011	−	0.622008i	$-0.786317\pi$
$269$	−25.6847	−1.56602	−0.783011	+	0.622008i	$0.786317\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.80776i	− 0.168702i	−0.996436	−	0.0843511i	$-0.973118\pi$
$277$	− 2.80776i	− 0.168702i	0.996436	−	0.0843511i	$-0.0268817\pi$
$278$	0	0
$279$	23.3693	1.39908
$280$	0	0
$281$	−3.12311	−0.186309	−0.0931544	−	0.995652i	$-0.529695\pi$
$281$	−3.12311	−0.186309	−0.0931544	+	0.995652i	$0.529695\pi$
$282$	0	0
$283$	5.12311i	0.304537i	0.988339	+	0.152269i	$0.0486578\pi$
$283$	5.12311i	0.304537i	−0.988339	+	0.152269i	$0.951342\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.24621	0.426248
$290$	0	0
$291$	34.2462	2.00755
$292$	0	0
$293$	− 9.36932i	− 0.547361i	−0.961821	−	0.273681i	$-0.911759\pi$
$293$	− 9.36932i	− 0.547361i	0.961821	−	0.273681i	$-0.0882411\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 7.36932i	− 0.427611i
$298$	0	0
$299$	4.56155	0.263801
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 10.8769i	− 0.624861i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.75379i	− 0.100094i	−0.998747	−	0.0500470i	$-0.984063\pi$
$307$	− 1.75379i	− 0.100094i	0.998747	−	0.0500470i	$-0.0159371\pi$
$308$	0	0
$309$	5.75379	0.327322
$310$	0	0
$311$	1.43845	0.0815669	0.0407834	−	0.999168i	$-0.487015\pi$
$311$	1.43845	0.0815669	0.0407834	+	0.999168i	$0.487015\pi$
$312$	0	0
$313$	− 9.36932i	− 0.529585i	−0.964305	−	0.264793i	$-0.914697\pi$
$313$	− 9.36932i	− 0.529585i	0.964305	−	0.264793i	$-0.0853035\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.2462i	− 1.13714i	−0.822635	−	0.568570i	$-0.807497\pi$
$317$	− 20.2462i	− 1.13714i	0.822635	−	0.568570i	$-0.192503\pi$
$318$	0	0
$319$	2.87689	0.161075
$320$	0	0
$321$	−7.36932	−0.411315
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 12.4924i	− 0.690833i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−30.4233	−1.67222	−0.836108	−	0.548565i	$-0.815174\pi$
$331$	−30.4233	−1.67222	−0.836108	+	0.548565i	$0.815174\pi$
$332$	0	0
$333$	29.3693i	1.60943i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.6155i	1.06853i	0.845318	+	0.534263i	$0.179411\pi$
$337$	19.6155i	1.06853i	−0.845318	+	0.534263i	$0.820589\pi$
$338$	0	0
$339$	−28.4924	−1.54750
$340$	0	0
$341$	−33.6155	−1.82038
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4924i	0.885360i	0.896680	+	0.442680i	$0.145972\pi$
$347$	16.4924i	0.885360i	−0.896680	+	0.442680i	$0.854028\pi$
$348$	0	0
$349$	22.3153	1.19451	0.597256	−	0.802050i	$-0.296257\pi$
$349$	22.3153	1.19451	0.597256	+	0.802050i	$0.296257\pi$
$350$	0	0
$351$	−6.56155	−0.350230
$352$	0	0
$353$	− 11.4384i	− 0.608807i	−0.952543	−	0.304404i	$-0.901543\pi$
$353$	− 11.4384i	− 0.608807i	0.952543	−	0.304404i	$-0.0984571\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.6155	−0.929712	−0.464856	−	0.885386i	$-0.653894\pi$
$359$	−17.6155	−0.929712	−0.464856	+	0.885386i	$0.653894\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	39.0540i	2.04980i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.24621i	0.117251i	0.998280	+	0.0586256i	$0.0186718\pi$
$367$	2.24621i	0.117251i	−0.998280	+	0.0586256i	$0.981328\pi$
$368$	0	0
$369$	−38.4924	−2.00384
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 22.4924i	− 1.16461i	−0.812969	−	0.582307i	$-0.802150\pi$
$373$	− 22.4924i	− 1.16461i	0.812969	−	0.582307i	$-0.197850\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.56155i	− 0.131927i
$378$	0	0
$379$	−20.4924	−1.05263	−0.526313	−	0.850291i	$-0.676426\pi$
$379$	−20.4924	−1.05263	−0.526313	+	0.850291i	$0.676426\pi$
$380$	0	0
$381$	29.9309	1.53340
$382$	0	0
$383$	− 26.2462i	− 1.34112i	−0.741856	−	0.670559i	$-0.766054\pi$
$383$	− 26.2462i	− 1.34112i	0.741856	−	0.670559i	$-0.233946\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 28.4924i	− 1.44835i
$388$	0	0
$389$	−1.36932	−0.0694271	−0.0347136	−	0.999397i	$-0.511052\pi$
$389$	−1.36932	−0.0694271	−0.0347136	+	0.999397i	$0.511052\pi$
$390$	0	0
$391$	3.12311	0.157942
$392$	0	0
$393$	40.1771i	2.02667i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.5616i	1.03195i	0.856602	+	0.515977i	$0.172571\pi$
$397$	20.5616i	1.03195i	−0.856602	+	0.515977i	$0.827429\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.7538	−0.586956	−0.293478	−	0.955966i	$-0.594813\pi$
$401$	−11.7538	−0.586956	−0.293478	+	0.955966i	$0.594813\pi$
$402$	0	0
$403$	29.9309i	1.49096i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 42.2462i	− 2.09407i
$408$	0	0
$409$	27.3002	1.34991	0.674954	−	0.737860i	$-0.264164\pi$
$409$	27.3002	1.34991	0.674954	+	0.737860i	$0.264164\pi$
$410$	0	0
$411$	−38.7386	−1.91084
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	40.1771i	1.96748i
$418$	0	0
$419$	12.4924	0.610295	0.305147	−	0.952305i	$-0.401294\pi$
$419$	12.4924	0.610295	0.305147	+	0.952305i	$0.401294\pi$
$420$	0	0
$421$	−27.1231	−1.32190	−0.660950	−	0.750430i	$-0.729846\pi$
$421$	−27.1231	−1.32190	−0.660950	+	0.750430i	$0.729846\pi$
$422$	0	0
$423$	− 41.6155i	− 2.02342i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	59.8617	2.89015
$430$	0	0
$431$	−28.4924	−1.37243	−0.686216	−	0.727398i	$-0.740729\pi$
$431$	−28.4924	−1.37243	−0.686216	+	0.727398i	$0.740729\pi$
$432$	0	0
$433$	− 24.7386i	− 1.18886i	−0.804146	−	0.594431i	$-0.797377\pi$
$433$	− 24.7386i	− 1.18886i	0.804146	−	0.594431i	$-0.202623\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.12311i	− 0.245071i
$438$	0	0
$439$	−11.0540	−0.527577	−0.263789	−	0.964580i	$-0.584972\pi$
$439$	−11.0540	−0.527577	−0.263789	+	0.964580i	$0.584972\pi$
$440$	0	0
$441$	−24.9309	−1.18718
$442$	0	0
$443$	6.06913i	0.288353i	0.989552	+	0.144177i	$0.0460533\pi$
$443$	6.06913i	0.288353i	−0.989552	+	0.144177i	$0.953947\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.61553i	0.454799i
$448$	0	0
$449$	8.24621	0.389163	0.194581	−	0.980886i	$-0.437665\pi$
$449$	8.24621	0.389163	0.194581	+	0.980886i	$0.437665\pi$
$450$	0	0
$451$	55.3693	2.60724
$452$	0	0
$453$	− 37.3002i	− 1.75252i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.36932i	− 0.251166i	−0.992083	−	0.125583i	$-0.959920\pi$
$457$	− 5.36932i	− 0.251166i	0.992083	−	0.125583i	$-0.0400801\pi$
$458$	0	0
$459$	−4.49242	−0.209688
$460$	0	0
$461$	−10.8078	−0.503368	−0.251684	−	0.967809i	$-0.580984\pi$
$461$	−10.8078	−0.503368	−0.251684	+	0.967809i	$0.580984\pi$
$462$	0	0
$463$	− 12.4924i	− 0.580572i	−0.956940	−	0.290286i	$-0.906250\pi$
$463$	− 12.4924i	− 0.580572i	0.956940	−	0.290286i	$-0.0937505\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 15.3693i	− 0.711207i	−0.934637	−	0.355604i	$-0.884275\pi$
$467$	− 15.3693i	− 0.711207i	0.934637	−	0.355604i	$-0.115725\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−32.9848	−1.51986
$472$	0	0
$473$	40.9848i	1.88449i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.12311i	0.326145i
$478$	0	0
$479$	−10.2462	−0.468161	−0.234081	−	0.972217i	$-0.575208\pi$
$479$	−10.2462	−0.468161	−0.234081	+	0.972217i	$0.575208\pi$
$480$	0	0
$481$	−37.6155	−1.71512
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.3153i	0.558061i	0.960282	+	0.279031i	$0.0900131\pi$
$487$	12.3153i	0.558061i	−0.960282	+	0.279031i	$0.909987\pi$
$488$	0	0
$489$	−25.4384	−1.15037
$490$	0	0
$491$	12.8078	0.578006	0.289003	−	0.957328i	$-0.406676\pi$
$491$	12.8078	0.578006	0.289003	+	0.957328i	$0.406676\pi$
$492$	0	0
$493$	− 1.75379i	− 0.0789867i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	23.0540	1.03204	0.516019	−	0.856577i	$-0.327413\pi$
$499$	23.0540	1.03204	0.516019	+	0.856577i	$0.327413\pi$
$500$	0	0
$501$	40.9848	1.83107
$502$	0	0
$503$	− 7.36932i	− 0.328582i	−0.986412	−	0.164291i	$-0.947466\pi$
$503$	− 7.36932i	− 0.328582i	0.986412	−	0.164291i	$-0.0525335\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 20.0000i	− 0.888231i
$508$	0	0
$509$	15.3002	0.678169	0.339084	−	0.940756i	$-0.389883\pi$
$509$	15.3002	0.678169	0.339084	+	0.940756i	$0.389883\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.36932i	0.325363i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	59.8617i	2.63272i
$518$	0	0
$519$	−25.6155	−1.12440
$520$	0	0
$521$	19.6155	0.859372	0.429686	−	0.902978i	$-0.358624\pi$
$521$	19.6155	0.859372	0.429686	+	0.902978i	$0.358624\pi$
$522$	0	0
$523$	− 5.75379i	− 0.251596i	−0.992056	−	0.125798i	$-0.959851\pi$
$523$	− 5.75379i	− 0.251596i	0.992056	−	0.125798i	$-0.0401491\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.4924i	0.892664i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−22.2462	−0.965403
$532$	0	0
$533$	− 49.3002i	− 2.13543i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	40.1771i	1.73377i
$538$	0	0
$539$	35.8617	1.54467
$540$	0	0
$541$	35.9309	1.54479	0.772394	−	0.635143i	$-0.219059\pi$
$541$	35.9309	1.54479	0.772394	+	0.635143i	$0.219059\pi$
$542$	0	0
$543$	51.8617i	2.22560i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.5464i	1.86191i	0.365135	+	0.930955i	$0.381023\pi$
$547$	43.5464i	1.86191i	−0.365135	+	0.930955i	$0.618977\pi$
$548$	0	0
$549$	−43.6155	−1.86147
$550$	0	0
$551$	−2.87689	−0.122560
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 0.876894i	− 0.0371552i	−0.999827	−	0.0185776i	$-0.994086\pi$
$557$	− 0.876894i	− 0.0371552i	0.999827	−	0.0185776i	$-0.00591378\pi$
$558$	0	0
$559$	36.4924	1.54347
$560$	0	0
$561$	40.9848	1.73038
$562$	0	0
$563$	− 5.75379i	− 0.242493i	−0.992622	−	0.121247i	$-0.961311\pi$
$563$	− 5.75379i	− 0.242493i	0.992622	−	0.121247i	$-0.0386892\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.24621	0.345699	0.172850	−	0.984948i	$-0.444703\pi$
$569$	8.24621	0.345699	0.172850	+	0.984948i	$0.444703\pi$
$570$	0	0
$571$	−11.5076	−0.481577	−0.240789	−	0.970578i	$-0.577406\pi$
$571$	−11.5076	−0.481577	−0.240789	+	0.970578i	$0.577406\pi$
$572$	0	0
$573$	1.61553i	0.0674897i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.9309i	0.663211i	0.943418	+	0.331605i	$0.107590\pi$
$577$	15.9309i	0.663211i	−0.943418	+	0.331605i	$0.892410\pi$
$578$	0	0
$579$	−11.6847	−0.485598
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 10.2462i	− 0.424355i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 23.0540i	− 0.951539i	−0.879570	−	0.475770i	$-0.842170\pi$
$587$	− 23.0540i	− 0.951539i	0.879570	−	0.475770i	$-0.157830\pi$
$588$	0	0
$589$	33.6155	1.38510
$590$	0	0
$591$	−30.5616	−1.25713
$592$	0	0
$593$	44.7386i	1.83720i	0.395194	+	0.918598i	$0.370677\pi$
$593$	44.7386i	1.83720i	−0.395194	+	0.918598i	$0.629323\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7.36932i	− 0.301606i
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−38.8078	−1.58300	−0.791501	−	0.611168i	$-0.790700\pi$
$601$	−38.8078	−1.58300	−0.791501	+	0.611168i	$0.790700\pi$
$602$	0	0
$603$	18.2462i	0.743043i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 14.7386i	− 0.598223i	−0.954218	−	0.299111i	$-0.903310\pi$
$607$	− 14.7386i	− 0.598223i	0.954218	−	0.299111i	$-0.0966902\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	53.3002	2.15629
$612$	0	0
$613$	28.1080i	1.13527i	0.823281	+	0.567635i	$0.192141\pi$
$613$	28.1080i	1.13527i	−0.823281	+	0.567635i	$0.807859\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.9848i	1.40844i	0.709983	+	0.704218i	$0.248702\pi$
$617$	34.9848i	1.40844i	−0.709983	+	0.704218i	$0.751298\pi$
$618$	0	0
$619$	−28.4924	−1.14521	−0.572604	−	0.819832i	$-0.694067\pi$
$619$	−28.4924	−1.14521	−0.572604	+	0.819832i	$0.694067\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 67.2311i	− 2.68495i
$628$	0	0
$629$	−25.7538	−1.02687
$630$	0	0
$631$	−6.73863	−0.268261	−0.134130	−	0.990964i	$-0.542824\pi$
$631$	−6.73863	−0.268261	−0.134130	+	0.990964i	$0.542824\pi$
$632$	0	0
$633$	− 10.2462i	− 0.407250i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 31.9309i	− 1.26515i
$638$	0	0
$639$	−33.6155	−1.32981
$640$	0	0
$641$	−13.3693	−0.528056	−0.264028	−	0.964515i	$-0.585051\pi$
$641$	−13.3693	−0.528056	−0.264028	+	0.964515i	$0.585051\pi$
$642$	0	0
$643$	23.3693i	0.921596i	0.887505	+	0.460798i	$0.152437\pi$
$643$	23.3693i	0.921596i	−0.887505	+	0.460798i	$0.847563\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 29.3002i	− 1.15191i	−0.817482	−	0.575955i	$-0.804630\pi$
$647$	− 29.3002i	− 1.15191i	0.817482	−	0.575955i	$-0.195370\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 25.0540i	− 0.980438i	−0.871599	−	0.490219i	$-0.836917\pi$
$653$	− 25.0540i	− 0.980438i	0.871599	−	0.490219i	$-0.163083\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8.24621i	− 0.321715i
$658$	0	0
$659$	−2.24621	−0.0875000	−0.0437500	−	0.999043i	$-0.513930\pi$
$659$	−2.24621	−0.0875000	−0.0437500	+	0.999043i	$0.513930\pi$
$660$	0	0
$661$	−0.246211	−0.00957651	−0.00478825	−	0.999989i	$-0.501524\pi$
$661$	−0.246211	−0.00957651	−0.00478825	+	0.999989i	$0.501524\pi$
$662$	0	0
$663$	− 36.4924i	− 1.41725i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.561553i	0.0217434i
$668$	0	0
$669$	11.5076	0.444909
$670$	0	0
$671$	62.7386	2.42200
$672$	0	0
$673$	2.94602i	0.113561i	0.998387	+	0.0567805i	$0.0180835\pi$
$673$	2.94602i	0.113561i	−0.998387	+	0.0567805i	$0.981916\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 38.9848i	− 1.49831i	−0.662395	−	0.749155i	$-0.730460\pi$
$677$	− 38.9848i	− 1.49831i	0.662395	−	0.749155i	$-0.269540\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−26.2462	−1.00576
$682$	0	0
$683$	− 16.3153i	− 0.624289i	−0.950035	−	0.312145i	$-0.898953\pi$
$683$	− 16.3153i	− 0.624289i	0.950035	−	0.312145i	$-0.101047\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 59.2311i	− 2.25981i
$688$	0	0
$689$	−9.12311	−0.347563
$690$	0	0
$691$	20.9848	0.798301	0.399151	−	0.916885i	$-0.369305\pi$
$691$	20.9848	0.798301	0.399151	+	0.916885i	$0.369305\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 33.7538i	− 1.27852i
$698$	0	0
$699$	53.9309	2.03985
$700$	0	0
$701$	29.8617	1.12786	0.563931	−	0.825822i	$-0.309288\pi$
$701$	29.8617	1.12786	0.563931	+	0.825822i	$0.309288\pi$
$702$	0	0
$703$	42.2462i	1.59335i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.3693	1.40343	0.701717	−	0.712456i	$-0.252417\pi$
$709$	37.3693	1.40343	0.701717	+	0.712456i	$0.252417\pi$
$710$	0	0
$711$	−18.2462	−0.684286
$712$	0	0
$713$	− 6.56155i	− 0.245732i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	28.3153i	1.05746i
$718$	0	0
$719$	−4.49242	−0.167539	−0.0837695	−	0.996485i	$-0.526696\pi$
$719$	−4.49242	−0.167539	−0.0837695	+	0.996485i	$0.526696\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 67.8617i	− 2.52381i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 39.3693i	− 1.46013i	−0.683379	−	0.730064i	$-0.739490\pi$
$727$	− 39.3693i	− 1.46013i	0.683379	−	0.730064i	$-0.260510\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	24.9848	0.924098
$732$	0	0
$733$	− 49.3693i	− 1.82350i	−0.410749	−	0.911749i	$-0.634733\pi$
$733$	− 49.3693i	− 1.82350i	0.410749	−	0.911749i	$-0.365267\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 26.2462i	− 0.966792i
$738$	0	0
$739$	−0.315342	−0.0116000	−0.00580001	−	0.999983i	$-0.501846\pi$
$739$	−0.315342	−0.0116000	−0.00580001	+	0.999983i	$0.501846\pi$
$740$	0	0
$741$	−59.8617	−2.19908
$742$	0	0
$743$	− 34.2462i	− 1.25637i	−0.778063	−	0.628186i	$-0.783798\pi$
$743$	− 34.2462i	− 1.25637i	0.778063	−	0.628186i	$-0.216202\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	40.9848	1.49556	0.747779	−	0.663948i	$-0.231120\pi$
$751$	40.9848	1.49556	0.747779	+	0.663948i	$0.231120\pi$
$752$	0	0
$753$	26.2462i	0.956465i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.50758i	− 0.0547938i	−0.999625	−	0.0273969i	$-0.991278\pi$
$757$	− 1.50758i	− 0.0547938i	0.999625	−	0.0273969i	$-0.00872180\pi$
$758$	0	0
$759$	−13.1231	−0.476339
$760$	0	0
$761$	−18.3153	−0.663931	−0.331965	−	0.943292i	$-0.607712\pi$
$761$	−18.3153	−0.663931	−0.331965	+	0.943292i	$0.607712\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 28.4924i	− 1.02880i
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	45.3002	1.63145
$772$	0	0
$773$	− 2.63068i	− 0.0946191i	−0.998880	−	0.0473095i	$-0.984935\pi$
$773$	− 2.63068i	− 0.0946191i	0.998880	−	0.0473095i	$-0.0150647\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−55.3693	−1.98381
$780$	0	0
$781$	48.3542	1.73025
$782$	0	0
$783$	− 0.807764i	− 0.0288671i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 26.2462i	− 0.935576i	−0.883841	−	0.467788i	$-0.845051\pi$
$787$	− 26.2462i	− 0.935576i	0.883841	−	0.467788i	$-0.154949\pi$
$788$	0	0
$789$	−27.8617	−0.991904
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 55.8617i	− 1.98371i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.2462i	0.433783i	0.976196	+	0.216892i	$0.0695918\pi$
$797$	12.2462i	0.433783i	−0.976196	+	0.216892i	$0.930408\pi$
$798$	0	0
$799$	36.4924	1.29101
$800$	0	0
$801$	−47.6155	−1.68241
$802$	0	0
$803$	11.8617i	0.418592i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 65.7926i	− 2.31601i
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	33.9309	1.19147	0.595737	−	0.803180i	$-0.296860\pi$
$811$	33.9309	1.19147	0.595737	+	0.803180i	$0.296860\pi$
$812$	0	0
$813$	61.4773i	2.15610i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 40.9848i	− 1.43388i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.7386	−0.863384	−0.431692	−	0.902021i	$-0.642083\pi$
$821$	−24.7386	−0.863384	−0.431692	+	0.902021i	$0.642083\pi$
$822$	0	0
$823$	− 49.4384i	− 1.72332i	−0.507489	−	0.861658i	$-0.669426\pi$
$823$	− 49.4384i	− 1.72332i	0.507489	−	0.861658i	$-0.330574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.49242i	0.156217i	0.996945	+	0.0781084i	$0.0248880\pi$
$827$	4.49242i	0.156217i	−0.996945	+	0.0781084i	$0.975112\pi$
$828$	0	0
$829$	20.2462	0.703180	0.351590	−	0.936154i	$-0.385641\pi$
$829$	20.2462	0.703180	0.351590	+	0.936154i	$0.385641\pi$
$830$	0	0
$831$	7.19224	0.249496
$832$	0	0
$833$	− 21.8617i	− 0.757464i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.43845i	0.326240i
$838$	0	0
$839$	−50.2462	−1.73469	−0.867346	−	0.497706i	$-0.834176\pi$
$839$	−50.2462	−1.73469	−0.867346	+	0.497706i	$0.834176\pi$
$840$	0	0
$841$	−28.6847	−0.989126
$842$	0	0
$843$	− 8.00000i	− 0.275535i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−13.1231	−0.450384
$850$	0	0
$851$	8.24621	0.282676
$852$	0	0
$853$	34.9848i	1.19786i	0.800802	+	0.598929i	$0.204407\pi$
$853$	34.9848i	1.19786i	−0.800802	+	0.598929i	$0.795593\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.5616i	0.565732i	0.959159	+	0.282866i	$0.0912852\pi$
$857$	16.5616i	0.565732i	−0.959159	+	0.282866i	$0.908715\pi$
$858$	0	0
$859$	−16.9460	−0.578191	−0.289095	−	0.957300i	$-0.593354\pi$
$859$	−16.9460	−0.578191	−0.289095	+	0.957300i	$0.593354\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.17708i	0.278351i	0.990268	+	0.139176i	$0.0444452\pi$
$863$	8.17708i	0.278351i	−0.990268	+	0.139176i	$0.955555\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18.5616i	0.630383i
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	−23.3693	−0.791839
$872$	0	0
$873$	47.6155i	1.61154i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.9848i	1.04628i	0.852246	+	0.523142i	$0.175240\pi$
$877$	30.9848i	1.04628i	−0.852246	+	0.523142i	$0.824760\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	4.87689	0.164307	0.0821534	−	0.996620i	$-0.473820\pi$
$881$	4.87689	0.164307	0.0821534	+	0.996620i	$0.473820\pi$
$882$	0	0
$883$	− 36.9848i	− 1.24464i	−0.782763	−	0.622320i	$-0.786190\pi$
$883$	− 36.9848i	− 1.24464i	0.782763	−	0.622320i	$-0.213810\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.6847i	0.660946i	0.943815	+	0.330473i	$0.107208\pi$
$887$	19.6847i	0.660946i	−0.943815	+	0.330473i	$0.892792\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−35.8617	−1.20141
$892$	0	0
$893$	− 59.8617i	− 2.00320i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.6847i	0.390139i
$898$	0	0
$899$	−3.68466	−0.122890
$900$	0	0
$901$	−6.24621	−0.208091
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	42.8769i	1.42370i	0.702330	+	0.711852i	$0.252143\pi$
$907$	42.8769i	1.42370i	−0.702330	+	0.711852i	$0.747857\pi$
$908$	0	0
$909$	15.1231	0.501602
$910$	0	0
$911$	−37.1231	−1.22994	−0.614972	−	0.788549i	$-0.710833\pi$
$911$	−37.1231	−1.22994	−0.614972	+	0.788549i	$0.710833\pi$
$912$	0	0
$913$	11.5076i	0.380845i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.7538	−1.24538	−0.622691	−	0.782468i	$-0.713961\pi$
$919$	−37.7538	−1.24538	−0.622691	+	0.782468i	$0.713961\pi$
$920$	0	0
$921$	4.49242	0.148030
$922$	0	0
$923$	− 43.0540i	− 1.41714i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	−34.8078	−1.14201	−0.571003	−	0.820948i	$-0.693445\pi$
$929$	−34.8078	−1.14201	−0.571003	+	0.820948i	$0.693445\pi$
$930$	0	0
$931$	−35.8617	−1.17532
$932$	0	0
$933$	3.68466i	0.120630i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.7538i	0.514654i	0.966324	+	0.257327i	$0.0828417\pi$
$937$	15.7538i	0.514654i	−0.966324	+	0.257327i	$0.917158\pi$
$938$	0	0
$939$	24.0000	0.783210
$940$	0	0
$941$	39.1231	1.27538	0.637688	−	0.770294i	$-0.279891\pi$
$941$	39.1231	1.27538	0.637688	+	0.770294i	$0.279891\pi$
$942$	0	0
$943$	10.8078i	0.351949i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.56155i	− 0.0832393i	−0.999134	−	0.0416196i	$-0.986748\pi$
$947$	− 2.56155i	− 0.0832393i	0.999134	−	0.0416196i	$-0.0132518\pi$
$948$	0	0
$949$	10.5616	0.342843
$950$	0	0
$951$	51.8617	1.68173
$952$	0	0
$953$	− 44.2462i	− 1.43328i	−0.697446	−	0.716638i	$-0.745680\pi$
$953$	− 44.2462i	− 1.43328i	0.697446	−	0.716638i	$-0.254320\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.36932i	0.238216i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.0540	0.388838
$962$	0	0
$963$	− 10.2462i	− 0.330180i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35.0540i	1.12726i	0.826027	+	0.563630i	$0.190596\pi$
$967$	35.0540i	1.12726i	−0.826027	+	0.563630i	$0.809404\pi$
$968$	0	0
$969$	−40.9848	−1.31662
$970$	0	0
$971$	31.3693	1.00669	0.503345	−	0.864086i	$-0.332103\pi$
$971$	31.3693	1.00669	0.503345	+	0.864086i	$0.332103\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 57.2311i	− 1.83098i	−0.402337	−	0.915492i	$-0.631802\pi$
$977$	− 57.2311i	− 1.83098i	0.402337	−	0.915492i	$-0.368198\pi$
$978$	0	0
$979$	68.4924	2.18903
$980$	0	0
$981$	17.3693	0.554560
$982$	0	0
$983$	− 22.1080i	− 0.705134i	−0.935787	−	0.352567i	$-0.885309\pi$
$983$	− 22.1080i	− 0.705134i	0.935787	−	0.352567i	$-0.114691\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	8.98485	0.285413	0.142707	−	0.989765i	$-0.454419\pi$
$991$	8.98485	0.285413	0.142707	+	0.989765i	$0.454419\pi$
$992$	0	0
$993$	− 77.9309i	− 2.47306i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	0	0
$999$	−11.8617	−0.375289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.o.4049.4		4
5.2	odd	4		4600.2.a.s.1.2		2
5.3	odd	4		184.2.a.e.1.1	✓	2
5.4	even	2	inner	4600.2.e.o.4049.1		4
15.8	even	4		1656.2.a.j.1.1		2
20.3	even	4		368.2.a.i.1.2		2
20.7	even	4		9200.2.a.br.1.1		2
35.13	even	4		9016.2.a.w.1.2		2
40.3	even	4		1472.2.a.p.1.1		2
40.13	odd	4		1472.2.a.u.1.2		2
60.23	odd	4		3312.2.a.t.1.2		2
115.68	even	4		4232.2.a.o.1.1		2
460.183	odd	4		8464.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.e.1.1	✓	2	5.3	odd	4
368.2.a.i.1.2		2	20.3	even	4
1472.2.a.p.1.1		2	40.3	even	4
1472.2.a.u.1.2		2	40.13	odd	4
1656.2.a.j.1.1		2	15.8	even	4
3312.2.a.t.1.2		2	60.23	odd	4
4232.2.a.o.1.1		2	115.68	even	4
4600.2.a.s.1.2		2	5.2	odd	4
4600.2.e.o.4049.1		4	5.4	even	2	inner
4600.2.e.o.4049.4		4	1.1	even	1	trivial
8464.2.a.bd.1.2		2	460.183	odd	4
9016.2.a.w.1.2		2	35.13	even	4
9200.2.a.br.1.1		2	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)

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} -3.56155 q^{9} +5.12311 q^{11} -4.56155i q^{13} -3.12311i q^{17} -5.12311 q^{19} +1.00000i q^{23} -1.43845i q^{27} +0.561553 q^{29} -6.56155 q^{31} +13.1231i q^{33} -8.24621i q^{37} +11.6847 q^{39} +10.8078 q^{41} +8.00000i q^{43} +11.6847i q^{47} +7.00000 q^{49} +8.00000 q^{51} -2.00000i q^{53} -13.1231i q^{57} +6.24621 q^{59} +12.2462 q^{61} -5.12311i q^{67} -2.56155 q^{69} +9.43845 q^{71} +2.31534i q^{73} +5.12311 q^{79} -7.00000 q^{81} +2.24621i q^{83} +1.43845i q^{87} +13.3693 q^{89} -16.8078i q^{93} -13.3693i q^{97} -18.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} + 4 q^{11} - 4 q^{19} - 6 q^{29} - 18 q^{31} + 22 q^{39} + 2 q^{41} + 28 q^{49} + 32 q^{51} - 8 q^{59} + 16 q^{61} - 2 q^{69} + 46 q^{71} + 4 q^{79} - 28 q^{81} + 4 q^{89} - 40 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 + 4 * q^11 - 4 * q^19 - 6 * q^29 - 18 * q^31 + 22 * q^39 + 2 * q^41 + 28 * q^49 + 32 * q^51 - 8 * q^59 + 16 * q^61 - 2 * q^69 + 46 * q^71 + 4 * q^79 - 28 * q^81 + 4 * q^89 - 40 * q^99

Introduction

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