LMFDB - Embedded newform 4368.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.a (trivial)

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:	yes
Analytic conductor:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} -3.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +9.00000 q^{29} -4.00000 q^{31} +5.00000 q^{33} +1.00000 q^{35} -11.0000 q^{37} +1.00000 q^{39} +5.00000 q^{43} -1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +3.00000 q^{51} -2.00000 q^{53} +5.00000 q^{55} -1.00000 q^{57} -4.00000 q^{59} -15.0000 q^{61} -1.00000 q^{63} +1.00000 q^{65} +2.00000 q^{67} +3.00000 q^{69} +12.0000 q^{71} +11.0000 q^{73} +4.00000 q^{75} +5.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +14.0000 q^{83} +3.00000 q^{85} -9.00000 q^{87} +6.00000 q^{89} +1.00000 q^{91} +4.00000 q^{93} -1.00000 q^{95} -14.0000 q^{97} -5.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−15.0000	−1.92055	−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000	−1.92055	−0.960277	+	0.279050i	$0.909981\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	15.0000	1.47799	0.738997	−	0.673709i	$-0.235300\pi$
$103$	15.0000	1.47799	0.738997	+	0.673709i	$0.235300\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	11.0000	1.04407
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	−13.0000	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$131$	−13.0000	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	15.0000	1.10883
$184$	0	0
$185$	11.0000	0.808736
$186$	0	0
$187$	15.0000	1.09691
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	−9.00000	−0.631676
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−5.00000	−0.340997
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−11.0000	−0.743311
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	11.0000	0.683507
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	20.0000	1.20605
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	15.0000	0.858898
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−15.0000	−0.853320
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	−45.0000	−2.51952
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	9.00000	0.497701
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−11.0000	−0.575766
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	29.0000	1.48183	0.740915	−	0.671598i	$-0.234392\pi$
$383$	29.0000	1.48183	0.740915	+	0.671598i	$0.234392\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	5.00000	0.254164
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	13.0000	0.655763
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	55.0000	2.72625
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	−19.0000	−0.937201
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−14.0000	−0.687233
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.00000	−0.341972	−0.170986	−	0.985273i	$-0.554695\pi$
$419$	−7.00000	−0.341972	−0.170986	+	0.985273i	$0.554695\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	15.0000	0.725901
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−23.0000	−1.08063
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−35.0000	−1.63011	−0.815056	−	0.579382i	$-0.803294\pi$
$461$	−35.0000	−1.63011	−0.815056	+	0.579382i	$0.803294\pi$
$462$	0	0
$463$	−9.00000	−0.418265	−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000	−0.418265	−0.209133	+	0.977887i	$0.567064\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	3.00000	0.138233
$472$	0	0
$473$	−25.0000	−1.14950
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	31.0000	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$479$	31.0000	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$480$	0	0
$481$	11.0000	0.501557
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	10.0000	0.452216
$490$	0	0
$491$	26.0000	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$492$	0	0
$493$	−27.0000	−1.21602
$494$	0	0
$495$	5.00000	0.224733
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	13.0000	0.576215	0.288107	−	0.957598i	$-0.406974\pi$
$509$	13.0000	0.576215	0.288107	+	0.957598i	$0.406974\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−15.0000	−0.660979
$516$	0	0
$517$	−40.0000	−1.75920
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.0000	0.432338
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	9.00000	0.385518
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−15.0000	−0.640184
$550$	0	0
$551$	9.00000	0.383413
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	−11.0000	−0.466924
$556$	0	0
$557$	−44.0000	−1.86434	−0.932170	−	0.362021i	$-0.882087\pi$
$557$	−44.0000	−1.86434	−0.932170	+	0.362021i	$0.882087\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−46.0000	−1.92842	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000	−1.92842	−0.964210	+	0.265139i	$0.914582\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	−12.0000	−0.498703
$580$	0	0
$581$	−14.0000	−0.580818
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	11.0000	0.450200
$598$	0	0
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	5.00000	0.199681
$628$	0	0
$629$	33.0000	1.31580
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	0	0
$633$	9.00000	0.357718
$634$	0	0
$635$	2.00000	0.0793676
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	5.00000	0.196875
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	0	0
$655$	13.0000	0.507952
$656$	0	0
$657$	11.0000	0.429151
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	1.00000	0.0387783
$666$	0	0
$667$	−27.0000	−1.04544
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	75.0000	2.89534
$672$	0	0
$673$	−33.0000	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	−19.0000	−0.725953
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	5.00000	0.189934
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−11.0000	−0.414873
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	−5.00000	−0.186989
$716$	0	0
$717$	−12.0000	−0.448148
$718$	0	0
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	0	0
$721$	−15.0000	−0.558629
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	−36.0000	−1.33701
$726$	0	0
$727$	35.0000	1.29808	0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000	1.29808	0.649039	+	0.760755i	$0.275171\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.0000	−0.554795
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	0	0
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	0	0
$747$	14.0000	0.512233
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−7.00000	−0.255094
$754$	0	0
$755$	−23.0000	−0.837056
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	−15.0000	−0.544466
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	9.00000	0.325822
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	−3.00000	−0.107903	−0.0539513	−	0.998544i	$-0.517182\pi$
$773$	−3.00000	−0.107903	−0.0539513	+	0.998544i	$0.517182\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	−11.0000	−0.394623
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−60.0000	−2.14697
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	3.00000	0.107075
$786$	0	0
$787$	−39.0000	−1.39020	−0.695100	−	0.718913i	$-0.744640\pi$
$787$	−39.0000	−1.39020	−0.695100	+	0.718913i	$0.744640\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	15.0000	0.532666
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−55.0000	−1.94091
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	−22.0000	−0.771574
$814$	0	0
$815$	10.0000	0.350285
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−20.0000	−0.696311
$826$	0	0
$827$	19.0000	0.660695	0.330347	−	0.943859i	$-0.392834\pi$
$827$	19.0000	0.660695	0.330347	+	0.943859i	$0.392834\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−21.0000	−0.726735
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	28.0000	0.966667	0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000	0.966667	0.483334	+	0.875436i	$0.339426\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	26.0000	0.895488
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	33.0000	1.13123
$852$	0	0
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.0000	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$863$	−38.0000	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	50.0000	1.69613
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	−14.0000	−0.472208
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	−51.0000	−1.71629	−0.858143	−	0.513410i	$-0.828382\pi$
$883$	−51.0000	−1.71629	−0.858143	+	0.513410i	$0.828382\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−3.00000	−0.100167
$898$	0	0
$899$	−36.0000	−1.20067
$900$	0	0
$901$	6.00000	0.199889
$902$	0	0
$903$	5.00000	0.166390
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	19.0000	0.629498	0.314749	−	0.949175i	$-0.398080\pi$
$911$	19.0000	0.629498	0.314749	+	0.949175i	$0.398080\pi$
$912$	0	0
$913$	−70.0000	−2.31666
$914$	0	0
$915$	−15.0000	−0.495885
$916$	0	0
$917$	13.0000	0.429298
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	44.0000	1.44671
$926$	0	0
$927$	15.0000	0.492665
$928$	0	0
$929$	−8.00000	−0.262471	−0.131236	−	0.991351i	$-0.541894\pi$
$929$	−8.00000	−0.262471	−0.131236	+	0.991351i	$0.541894\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−10.0000	−0.327385
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	39.0000	1.26733	0.633665	−	0.773608i	$-0.281550\pi$
$947$	39.0000	1.26733	0.633665	+	0.773608i	$0.281550\pi$
$948$	0	0
$949$	−11.0000	−0.357075
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	0	0
$955$	3.00000	0.0970777
$956$	0	0
$957$	45.0000	1.45464
$958$	0	0
$959$	−19.0000	−0.613542
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−10.0000	−0.322245
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	5.00000	0.160789	0.0803946	−	0.996763i	$-0.474382\pi$
$967$	5.00000	0.160789	0.0803946	+	0.996763i	$0.474382\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−51.0000	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	0	0
$981$	−9.00000	−0.287348
$982$	0	0
$983$	−25.0000	−0.797376	−0.398688	−	0.917087i	$-0.630534\pi$
$983$	−25.0000	−0.797376	−0.398688	+	0.917087i	$0.630534\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−15.0000	−0.476972
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	11.0000	0.348723
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	11.0000	0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.e.1.1		1
4.3	odd	2		546.2.a.f.1.1	✓	1
12.11	even	2		1638.2.a.h.1.1		1
28.27	even	2		3822.2.a.w.1.1		1
52.51	odd	2		7098.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.f.1.1	✓	1	4.3	odd	2
1638.2.a.h.1.1		1	12.11	even	2
3822.2.a.w.1.1		1	28.27	even	2
4368.2.a.e.1.1		1	1.1	even	1	trivial
7098.2.a.n.1.1		1	52.51	odd	2

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

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L-functions

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