LMFDB - Embedded newform 5915.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5915,2,Mod(1,5915)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5915, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5915.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5915 = 5 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5915.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:	$47.2315127956$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5915.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$

$q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} -2.00000 q^{10} -3.00000 q^{11} +2.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} +6.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} +6.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} -2.00000 q^{28} +7.00000 q^{29} +8.00000 q^{32} -2.00000 q^{34} -1.00000 q^{35} -6.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} -12.0000 q^{41} -4.00000 q^{43} -6.00000 q^{44} -3.00000 q^{45} -12.0000 q^{46} -11.0000 q^{47} +1.00000 q^{49} -2.00000 q^{50} +6.00000 q^{53} -3.00000 q^{55} -14.0000 q^{58} -4.00000 q^{59} -4.00000 q^{61} +3.00000 q^{63} -8.00000 q^{64} -6.00000 q^{67} +2.00000 q^{68} +2.00000 q^{70} +1.00000 q^{71} -1.00000 q^{73} -16.0000 q^{74} +4.00000 q^{76} +3.00000 q^{77} -4.00000 q^{80} +9.00000 q^{81} +24.0000 q^{82} -15.0000 q^{83} +1.00000 q^{85} +8.00000 q^{86} +6.00000 q^{89} +6.00000 q^{90} +12.0000 q^{92} +22.0000 q^{94} +2.00000 q^{95} -14.0000 q^{97} -2.00000 q^{98} +9.00000 q^{99} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−3.00000	−1.00000
$10$	−2.00000	−0.632456
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	6.00000	1.41421
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	6.00000	1.27920
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−1.00000	−0.169031
$36$	−6.00000	−1.00000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−6.00000	−0.904534
$45$	−3.00000	−0.447214
$46$	−12.0000	−1.76930
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.00000	−0.282843
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	−14.0000	−1.83829
$59$	−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
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−16.0000	−1.85996
$75$	0	0
$76$	4.00000	0.458831
$77$	3.00000	0.341882
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.00000	−0.447214
$81$	9.00000	1.00000
$82$	24.0000	2.65036
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	22.0000	2.26913
$95$	2.00000	0.205196
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−2.00000	−0.202031
$99$	9.00000	0.904534
$100$	2.00000	0.200000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	4.00000	0.377964
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	14.0000	1.29987
$117$	0	0
$118$	8.00000	0.736460
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000	0.724286
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	−6.00000	−0.534522
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	12.0000	1.03664
$135$	0	0
$136$	0	0
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−2.00000	−0.167836
$143$	0	0
$144$	12.0000	1.00000
$145$	7.00000	0.581318
$146$	2.00000	0.165521
$147$	0	0
$148$	16.0000	1.31519
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	0	0
$160$	8.00000	0.632456
$161$	−6.00000	−0.472866
$162$	−18.0000	−1.41421
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−24.0000	−1.87409
$165$	0	0
$166$	30.0000	2.32845
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	0	0
$170$	−2.00000	−0.153393
$171$	−6.00000	−0.458831
$172$	−8.00000	−0.609994
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	12.0000	0.904534
$177$	0	0
$178$	−12.0000	−0.899438
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	−6.00000	−0.447214
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	−3.00000	−0.219382
$188$	−22.0000	−1.60451
$189$	0	0
$190$	−4.00000	−0.290191
$191$	19.0000	1.37479	0.687396	−	0.726283i	$-0.258754\pi$
$191$	19.0000	1.37479	0.687396	+	0.726283i	$0.258754\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	28.0000	2.01028
$195$	0	0
$196$	2.00000	0.142857
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−18.0000	−1.27920
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	0	0
$202$	−20.0000	−1.40720
$203$	−7.00000	−0.491304
$204$	0	0
$205$	−12.0000	−0.838116
$206$	32.0000	2.22955
$207$	−18.0000	−1.25109
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	−26.0000	−1.76094
$219$	0	0
$220$	−6.00000	−0.404520
$221$	0	0
$222$	0	0
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	−8.00000	−0.534522
$225$	−3.00000	−0.200000
$226$	−24.0000	−1.59646
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	−12.0000	−0.791257
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−11.0000	−0.717561
$236$	−8.00000	−0.520756
$237$	0	0
$238$	2.00000	0.129641
$239$	17.0000	1.09964	0.549819	−	0.835284i	$-0.314697\pi$
$239$	17.0000	1.09964	0.549819	+	0.835284i	$0.314697\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	4.00000	0.257130
$243$	0	0
$244$	−8.00000	−0.512148
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−2.00000	−0.126491
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	6.00000	0.377964
$253$	−18.0000	−1.13165
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−21.0000	−1.29987
$262$	−16.0000	−0.988483
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	4.00000	0.245256
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−44.0000	−2.65814
$275$	−3.00000	−0.180907
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	−24.0000	−1.41421
$289$	−16.0000	−0.941176
$290$	−14.0000	−0.822108
$291$	0	0
$292$	−2.00000	−0.117041
$293$	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
$297$	0	0
$298$	42.0000	2.43299
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	−6.00000	−0.345261
$303$	0	0
$304$	−8.00000	−0.458831
$305$	−4.00000	−0.229039
$306$	6.00000	0.342997
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	31.0000	1.75222	0.876112	−	0.482108i	$-0.160129\pi$
$313$	31.0000	1.75222	0.876112	+	0.482108i	$0.160129\pi$
$314$	−14.0000	−0.790066
$315$	3.00000	0.169031
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−21.0000	−1.17577
$320$	−8.00000	−0.447214
$321$	0	0
$322$	12.0000	0.668734
$323$	2.00000	0.111283
$324$	18.0000	1.00000
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	0	0
$329$	11.0000	0.606450
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−30.0000	−1.64646
$333$	−24.0000	−1.31519
$334$	32.0000	1.75096
$335$	−6.00000	−0.327815
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	0	0
$339$	0	0
$340$	2.00000	0.108465
$341$	0	0
$342$	12.0000	0.648886
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	22.0000	1.18273
$347$	34.0000	1.82522	0.912608	−	0.408836i	$-0.134065\pi$
$347$	34.0000	1.82522	0.912608	+	0.408836i	$0.134065\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−24.0000	−1.27920
$353$	5.00000	0.266123	0.133062	−	0.991108i	$-0.457519\pi$
$353$	5.00000	0.266123	0.133062	+	0.991108i	$0.457519\pi$
$354$	0	0
$355$	1.00000	0.0530745
$356$	12.0000	0.635999
$357$	0	0
$358$	2.00000	0.105703
$359$	5.00000	0.263890	0.131945	−	0.991257i	$-0.457878\pi$
$359$	5.00000	0.263890	0.131945	+	0.991257i	$0.457878\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	8.00000	0.420471
$363$	0	0
$364$	0	0
$365$	−1.00000	−0.0523424
$366$	0	0
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	−24.0000	−1.25109
$369$	36.0000	1.87409
$370$	−16.0000	−0.831800
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	27.0000	1.38690	0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000	1.38690	0.693448	+	0.720506i	$0.256091\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−38.0000	−1.94425
$383$	−33.0000	−1.68622	−0.843111	−	0.537740i	$-0.819278\pi$
$383$	−33.0000	−1.68622	−0.843111	+	0.537740i	$0.819278\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	−32.0000	−1.62876
$387$	12.0000	0.609994
$388$	−28.0000	−1.42148
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	0	0
$394$	44.0000	2.21669
$395$	0	0
$396$	18.0000	0.904534
$397$	−31.0000	−1.55585	−0.777923	−	0.628360i	$-0.783727\pi$
$397$	−31.0000	−1.55585	−0.777923	+	0.628360i	$0.783727\pi$
$398$	−44.0000	−2.20552
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	0	0
$404$	20.0000	0.995037
$405$	9.00000	0.447214
$406$	14.0000	0.694808
$407$	−24.0000	−1.18964
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	24.0000	1.18528
$411$	0	0
$412$	−32.0000	−1.57653
$413$	4.00000	0.196827
$414$	36.0000	1.76930
$415$	−15.0000	−0.736321
$416$	0	0
$417$	0	0
$418$	12.0000	0.586939
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−26.0000	−1.26566
$423$	33.0000	1.60451
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	0	0
$430$	8.00000	0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	0	0
$436$	26.0000	1.24517
$437$	12.0000	0.574038
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−38.0000	−1.80543	−0.902717	−	0.430234i	$-0.858431\pi$
$443$	−38.0000	−1.80543	−0.902717	+	0.430234i	$0.858431\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−18.0000	−0.852325
$447$	0	0
$448$	8.00000	0.377964
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	6.00000	0.282843
$451$	36.0000	1.69517
$452$	24.0000	1.12887
$453$	0	0
$454$	−22.0000	−1.03251
$455$	0	0
$456$	0	0
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	−16.0000	−0.747631
$459$	0	0
$460$	12.0000	0.559503
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	−28.0000	−1.29987
$465$	0	0
$466$	36.0000	1.66767
$467$	5.00000	0.231372	0.115686	−	0.993286i	$-0.463093\pi$
$467$	5.00000	0.231372	0.115686	+	0.993286i	$0.463093\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	22.0000	1.01478
$471$	0	0
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	2.00000	0.0917663
$476$	−2.00000	−0.0916698
$477$	−18.0000	−0.824163
$478$	−34.0000	−1.55512
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	0	0
$482$	−20.0000	−0.910975
$483$	0	0
$484$	−4.00000	−0.181818
$485$	−14.0000	−0.635707
$486$	0	0
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	0	0
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	7.00000	0.315264
$494$	0	0
$495$	9.00000	0.404520
$496$	0	0
$497$	−1.00000	−0.0448561
$498$	0	0
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	−56.0000	−2.49940
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	36.0000	1.60040
$507$	0	0
$508$	16.0000	0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	−32.0000	−1.41421
$513$	0	0
$514$	4.00000	0.176432
$515$	−16.0000	−0.705044
$516$	0	0
$517$	33.0000	1.45134
$518$	16.0000	0.703000
$519$	0	0
$520$	0	0
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	42.0000	1.83829
$523$	7.00000	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	48.0000	2.09290
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	−12.0000	−0.521247
$531$	12.0000	0.520756
$532$	−4.00000	−0.173422
$533$	0	0
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	0	0
$538$	12.0000	0.517357
$539$	−3.00000	−0.129219
$540$	0	0
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	8.00000	0.342997
$545$	13.0000	0.556859
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	44.0000	1.87959
$549$	12.0000	0.512148
$550$	6.00000	0.255841
$551$	14.0000	0.596420
$552$	0	0
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	0	0
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	−54.0000	−2.27785
$563$	−41.0000	−1.72794	−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000	−1.72794	−0.863972	+	0.503540i	$0.832031\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−2.00000	−0.0840663
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	0	0
$571$	43.0000	1.79949	0.899747	−	0.436412i	$-0.143751\pi$
$571$	43.0000	1.79949	0.899747	+	0.436412i	$0.143751\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	6.00000	0.250217
$576$	24.0000	1.00000
$577$	35.0000	1.45707	0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000	1.45707	0.728535	+	0.685009i	$0.240202\pi$
$578$	32.0000	1.33102
$579$	0	0
$580$	14.0000	0.581318
$581$	15.0000	0.622305
$582$	0	0
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	−28.0000	−1.15667
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	−32.0000	−1.31519
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−42.0000	−1.72039
$597$	0	0
$598$	0	0
$599$	−19.0000	−0.776319	−0.388159	−	0.921592i	$-0.626889\pi$
$599$	−19.0000	−0.776319	−0.388159	+	0.921592i	$0.626889\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	−8.00000	−0.326056
$603$	18.0000	0.733017
$604$	6.00000	0.244137
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	11.0000	0.446476	0.223238	−	0.974764i	$-0.428337\pi$
$607$	11.0000	0.446476	0.223238	+	0.974764i	$0.428337\pi$
$608$	16.0000	0.648886
$609$	0	0
$610$	8.00000	0.323911
$611$	0	0
$612$	−6.00000	−0.242536
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−42.0000	−1.69498
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−6.00000	−0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	−62.0000	−2.47802
$627$	0	0
$628$	14.0000	0.558661
$629$	8.00000	0.318981
$630$	−6.00000	−0.239046
$631$	19.0000	0.756378	0.378189	−	0.925728i	$-0.376547\pi$
$631$	19.0000	0.756378	0.378189	+	0.925728i	$0.376547\pi$
$632$	0	0
$633$	0	0
$634$	−12.0000	−0.476581
$635$	8.00000	0.317470
$636$	0	0
$637$	0	0
$638$	42.0000	1.66280
$639$	−3.00000	−0.118678
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−4.00000	−0.157378
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	16.0000	0.626608
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	48.0000	1.87409
$657$	3.00000	0.117041
$658$	−22.0000	−0.857649
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	−24.0000	−0.933492	−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000	−0.933492	−0.466746	+	0.884391i	$0.654574\pi$
$662$	40.0000	1.55464
$663$	0	0
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	48.0000	1.85996
$667$	42.0000	1.62625
$668$	−32.0000	−1.23812
$669$	0	0
$670$	12.0000	0.463600
$671$	12.0000	0.463255
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	−40.0000	−1.54074
$675$	0	0
$676$	0	0
$677$	−5.00000	−0.192166	−0.0960828	−	0.995373i	$-0.530631\pi$
$677$	−5.00000	−0.192166	−0.0960828	+	0.995373i	$0.530631\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−12.0000	−0.458831
$685$	22.0000	0.840577
$686$	2.00000	0.0763604
$687$	0	0
$688$	16.0000	0.609994
$689$	0	0
$690$	0	0
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	−22.0000	−0.836315
$693$	−9.00000	−0.341882
$694$	−68.0000	−2.58124
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	32.0000	1.21122
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	24.0000	0.904534
$705$	0	0
$706$	−10.0000	−0.376355
$707$	−10.0000	−0.376089
$708$	0	0
$709$	23.0000	0.863783	0.431892	−	0.901926i	$-0.357846\pi$
$709$	23.0000	0.863783	0.431892	+	0.901926i	$0.357846\pi$
$710$	−2.00000	−0.0750587
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	−10.0000	−0.373197
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	12.0000	0.447214
$721$	16.0000	0.595871
$722$	30.0000	1.11648
$723$	0	0
$724$	−8.00000	−0.297318
$725$	7.00000	0.259973
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	2.00000	0.0740233
$731$	−4.00000	−0.147945
$732$	0	0
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	48.0000	1.76930
$737$	18.0000	0.663039
$738$	−72.0000	−2.65036
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	16.0000	0.588172
$741$	0	0
$742$	12.0000	0.440534
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	−21.0000	−0.769380
$746$	12.0000	0.439351
$747$	45.0000	1.64646
$748$	−6.00000	−0.219382
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	44.0000	1.60451
$753$	0	0
$754$	0	0
$755$	3.00000	0.109181
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	−54.0000	−1.96137
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	−13.0000	−0.470632
$764$	38.0000	1.37479
$765$	−3.00000	−0.108465
$766$	66.0000	2.38468
$767$	0	0
$768$	0	0
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	−6.00000	−0.216225
$771$	0	0
$772$	32.0000	1.15171
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	−24.0000	−0.862662
$775$	0	0
$776$	0	0
$777$	0	0
$778$	6.00000	0.215110
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−3.00000	−0.107348
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−4.00000	−0.142857
$785$	7.00000	0.249841
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	−44.0000	−1.56744
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	62.0000	2.20030
$795$	0	0
$796$	44.0000	1.55954
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−11.0000	−0.389152
$800$	8.00000	0.282843
$801$	−18.0000	−0.635999
$802$	28.0000	0.988714
$803$	3.00000	0.105868
$804$	0	0
$805$	−6.00000	−0.211472
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	−18.0000	−0.632456
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−14.0000	−0.491304
$813$	0	0
$814$	48.0000	1.68240
$815$	8.00000	0.280228
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−64.0000	−2.23771
$819$	0	0
$820$	−24.0000	−0.838116
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	−36.0000	−1.25109
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	30.0000	1.04132
$831$	0	0
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−16.0000	−0.553703
$836$	−12.0000	−0.415029
$837$	0	0
$838$	56.0000	1.93449
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−12.0000	−0.413547
$843$	0	0
$844$	26.0000	0.894957
$845$	0	0
$846$	−66.0000	−2.26913
$847$	2.00000	0.0687208
$848$	−24.0000	−0.824163
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	48.0000	1.64542
$852$	0	0
$853$	21.0000	0.719026	0.359513	−	0.933140i	$-0.382943\pi$
$853$	21.0000	0.719026	0.359513	+	0.933140i	$0.382943\pi$
$854$	−8.00000	−0.273754
$855$	−6.00000	−0.205196
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	48.0000	1.63489
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	0	0
$865$	−11.0000	−0.374011
$866$	−12.0000	−0.407777
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	42.0000	1.42148
$874$	−24.0000	−0.811812
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	12.0000	0.404520
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	6.00000	0.202031
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	0	0
$885$	0	0
$886$	76.0000	2.55327
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	−12.0000	−0.402241
$891$	−27.0000	−0.904534
$892$	18.0000	0.602685
$893$	−22.0000	−0.736202
$894$	0	0
$895$	−1.00000	−0.0334263
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	0	0
$900$	−6.00000	−0.200000
$901$	6.00000	0.199889
$902$	−72.0000	−2.39734
$903$	0	0
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	22.0000	0.730096
$909$	−30.0000	−0.995037
$910$	0	0
$911$	−57.0000	−1.88849	−0.944247	−	0.329238i	$-0.893208\pi$
$911$	−57.0000	−1.88849	−0.944247	+	0.329238i	$0.893208\pi$
$912$	0	0
$913$	45.0000	1.48928
$914$	−60.0000	−1.98462
$915$	0	0
$916$	16.0000	0.528655
$917$	−8.00000	−0.264183
$918$	0	0
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	0	0
$921$	0	0
$922$	20.0000	0.658665
$923$	0	0
$924$	0	0
$925$	8.00000	0.263038
$926$	−20.0000	−0.657241
$927$	48.0000	1.57653
$928$	56.0000	1.83829
$929$	56.0000	1.83730	0.918650	−	0.395072i	$-0.129280\pi$
$929$	56.0000	1.83730	0.918650	+	0.395072i	$0.129280\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−36.0000	−1.17922
$933$	0	0
$934$	−10.0000	−0.327210
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	49.0000	1.60076	0.800380	−	0.599493i	$-0.204631\pi$
$937$	49.0000	1.60076	0.800380	+	0.599493i	$0.204631\pi$
$938$	−12.0000	−0.391814
$939$	0	0
$940$	−22.0000	−0.717561
$941$	−44.0000	−1.43436	−0.717180	−	0.696888i	$-0.754567\pi$
$941$	−44.0000	−1.43436	−0.717180	+	0.696888i	$0.754567\pi$
$942$	0	0
$943$	−72.0000	−2.34464
$944$	16.0000	0.520756
$945$	0	0
$946$	−24.0000	−0.780307
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	0	0
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	36.0000	1.16554
$955$	19.0000	0.614826
$956$	34.0000	1.09964
$957$	0	0
$958$	−44.0000	−1.42158
$959$	−22.0000	−0.710417
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−6.00000	−0.193347
$964$	20.0000	0.644157
$965$	16.0000	0.515058
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	0	0
$970$	28.0000	0.899026
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	0	0
$974$	−68.0000	−2.17886
$975$	0	0
$976$	16.0000	0.512148
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	2.00000	0.0638877
$981$	−39.0000	−1.24517
$982$	−24.0000	−0.765871
$983$	−43.0000	−1.37149	−0.685744	−	0.727843i	$-0.740523\pi$
$983$	−43.0000	−1.37149	−0.685744	+	0.727843i	$0.740523\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	−14.0000	−0.445851
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	−18.0000	−0.572078
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	0	0
$994$	2.00000	0.0634361
$995$	22.0000	0.697447
$996$	0	0
$997$	29.0000	0.918439	0.459220	−	0.888323i	$-0.348129\pi$
$997$	29.0000	0.918439	0.459220	+	0.888323i	$0.348129\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5915.2.a.a.1.1		1
13.3	even	3		455.2.i.d.386.1	yes	2
13.9	even	3		455.2.i.d.211.1	✓	2
13.12	even	2		5915.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.i.d.211.1	✓	2	13.9	even	3
455.2.i.d.386.1	yes	2	13.3	even	3
5915.2.a.a.1.1		1	1.1	even	1	trivial
5915.2.a.i.1.1		1	13.12	even	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:	\( N \)	\(=\)	\( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5915.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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