LMFDB - Embedded newform 115.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("115.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.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:	$0.918279623245$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +3.00000 q^{17} -6.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +2.00000 q^{28} +7.00000 q^{29} -5.00000 q^{31} -8.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} -6.00000 q^{36} +11.0000 q^{37} -4.00000 q^{38} +1.00000 q^{41} +4.00000 q^{44} +3.00000 q^{45} +2.00000 q^{46} -6.00000 q^{49} +2.00000 q^{50} -4.00000 q^{52} +11.0000 q^{53} -2.00000 q^{55} +14.0000 q^{58} -13.0000 q^{59} -8.00000 q^{61} -10.0000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} +5.00000 q^{67} +6.00000 q^{68} -2.00000 q^{70} +5.00000 q^{71} +6.00000 q^{73} +22.0000 q^{74} -4.00000 q^{76} +2.00000 q^{77} -12.0000 q^{79} +4.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +9.00000 q^{83} -3.00000 q^{85} +4.00000 q^{89} +6.00000 q^{90} -2.00000 q^{91} +2.00000 q^{92} +2.00000 q^{95} -14.0000 q^{97} -12.0000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	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	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	−2.00000	−0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−6.00000	−1.41421
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	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
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	6.00000	1.02899
$35$	−1.00000	−0.169031
$36$	−6.00000	−1.00000
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	4.00000	0.603023
$45$	3.00000	0.447214
$46$	2.00000	0.294884
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	2.00000	0.282843
$51$	0	0
$52$	−4.00000	−0.554700
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	14.0000	1.83829
$59$	−13.0000	−1.69246	−0.846228	−	0.532821i	$-0.821132\pi$
$59$	−13.0000	−1.69246	−0.846228	+	0.532821i	$0.821132\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−10.0000	−1.27000
$63$	−3.00000	−0.377964
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	−2.00000	−0.239046
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	22.0000	2.55745
$75$	0	0
$76$	−4.00000	−0.458831
$77$	2.00000	0.227921
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	4.00000	0.447214
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	6.00000	0.632456
$91$	−2.00000	−0.209657
$92$	2.00000	0.208514
$93$	0	0
$94$	0	0
$95$	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	$0.751641\pi$
$98$	−12.0000	−1.21218
$99$	−6.00000	−0.603023
$100$	2.00000	0.200000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	22.0000	2.13683
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	14.0000	1.29987
$117$	6.00000	0.554700
$118$	−26.0000	−2.39349
$119$	3.00000	0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−16.0000	−1.44857
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−1.00000	−0.0894427
$126$	−6.00000	−0.534522
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	10.0000	0.863868
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	10.0000	0.839181
$143$	−4.00000	−0.334497
$144$	12.0000	1.00000
$145$	−7.00000	−0.581318
$146$	12.0000	0.993127
$147$	0	0
$148$	22.0000	1.80839
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−9.00000	−0.727607
$154$	4.00000	0.322329
$155$	5.00000	0.401610
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	−24.0000	−1.90934
$159$	0	0
$160$	8.00000	0.632456
$161$	1.00000	0.0788110
$162$	18.0000	1.41421
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	18.0000	1.39707
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	6.00000	0.458831
$172$	0	0
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−8.00000	−0.603023
$177$	0	0
$178$	8.00000	0.599625
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	6.00000	0.447214
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	0	0
$185$	−11.0000	−0.808736
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	0	0
$190$	4.00000	0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	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
$196$	−12.0000	−0.857143
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−12.0000	−0.852803
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	0	0
$201$	0	0
$202$	−10.0000	−0.703598
$203$	7.00000	0.491304
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	−16.0000	−1.11477
$207$	−3.00000	−0.208514
$208$	8.00000	0.554700
$209$	−4.00000	−0.276686
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	22.0000	1.51097
$213$	0	0
$214$	−30.0000	−2.05076
$215$	0	0
$216$	0	0
$217$	−5.00000	−0.339422
$218$	−20.0000	−1.35457
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−6.00000	−0.403604
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−8.00000	−0.534522
$225$	−3.00000	−0.200000
$226$	−18.0000	−1.19734
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	12.0000	0.784465
$235$	0	0
$236$	−26.0000	−1.69246
$237$	0	0
$238$	6.00000	0.388922
$239$	−29.0000	−1.87585	−0.937927	−	0.346833i	$-0.887257\pi$
$239$	−29.0000	−1.87585	−0.937927	+	0.346833i	$0.887257\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	−16.0000	−1.02430
$245$	6.00000	0.383326
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	−2.00000	−0.126491
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	−6.00000	−0.377964
$253$	2.00000	0.125739
$254$	−8.00000	−0.501965
$255$	0	0
$256$	16.0000	1.00000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	11.0000	0.683507
$260$	4.00000	0.248069
$261$	−21.0000	−1.29987
$262$	24.0000	1.48272
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	−4.00000	−0.245256
$267$	0	0
$268$	10.0000	0.610847
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	12.0000	0.724947
$275$	2.00000	0.120605
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	−38.0000	−2.27909
$279$	15.0000	0.898027
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	−8.00000	−0.473050
$287$	1.00000	0.0590281
$288$	24.0000	1.41421
$289$	−8.00000	−0.470588
$290$	−14.0000	−0.822108
$291$	0	0
$292$	12.0000	0.702247
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	13.0000	0.756889
$296$	0	0
$297$	0	0
$298$	32.0000	1.85371
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	24.0000	1.38104
$303$	0	0
$304$	8.00000	0.458831
$305$	8.00000	0.458079
$306$	−18.0000	−1.02899
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	10.0000	0.567962
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−3.00000	−0.169570	−0.0847850	−	0.996399i	$-0.527020\pi$
$313$	−3.00000	−0.169570	−0.0847850	+	0.996399i	$0.527020\pi$
$314$	34.0000	1.91873
$315$	3.00000	0.169031
$316$	−24.0000	−1.35011
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	14.0000	0.783850
$320$	8.00000	0.447214
$321$	0	0
$322$	2.00000	0.111456
$323$	−6.00000	−0.333849
$324$	18.0000	1.00000
$325$	−2.00000	−0.110940
$326$	−36.0000	−1.99386
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	18.0000	0.987878
$333$	−33.0000	−1.80839
$334$	−48.0000	−2.62644
$335$	−5.00000	−0.273179
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−10.0000	−0.541530
$342$	12.0000	0.648886
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	48.0000	2.58050
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−16.0000	−0.852803
$353$	−8.00000	−0.425797	−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000	−0.425797	−0.212899	+	0.977074i	$0.568290\pi$
$354$	0	0
$355$	−5.00000	−0.265372
$356$	8.00000	0.423999
$357$	0	0
$358$	48.0000	2.53688
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	32.0000	1.68188
$363$	0	0
$364$	−4.00000	−0.209657
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	−4.00000	−0.208514
$369$	−3.00000	−0.156174
$370$	−22.0000	−1.14373
$371$	11.0000	0.571092
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	0	0
$377$	−14.0000	−0.721037
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	0	0
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	32.0000	1.62876
$387$	0	0
$388$	−28.0000	−1.42148
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	0	0
$394$	−4.00000	−0.201517
$395$	12.0000	0.603786
$396$	−12.0000	−0.603023
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	52.0000	2.60652
$399$	0	0
$400$	−4.00000	−0.200000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	−10.0000	−0.497519
$405$	−9.00000	−0.447214
$406$	14.0000	0.694808
$407$	22.0000	1.09050
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−16.0000	−0.788263
$413$	−13.0000	−0.639688
$414$	−6.00000	−0.294884
$415$	−9.00000	−0.441793
$416$	16.0000	0.784465
$417$	0	0
$418$	−8.00000	−0.391293
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	30.0000	1.46038
$423$	0	0
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−30.0000	−1.45010
$429$	0	0
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	−20.0000	−0.957826
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	−12.0000	−0.570782
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	28.0000	1.32584
$447$	0	0
$448$	−8.00000	−0.377964
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	−6.00000	−0.282843
$451$	2.00000	0.0941763
$452$	−18.0000	−0.846649
$453$	0	0
$454$	8.00000	0.375459
$455$	2.00000	0.0937614
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	16.0000	0.747631
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	−28.0000	−1.29987
$465$	0	0
$466$	8.00000	0.370593
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	12.0000	0.554700
$469$	5.00000	0.230879
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	6.00000	0.275010
$477$	−33.0000	−1.51097
$478$	−58.0000	−2.65286
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−22.0000	−1.00311
$482$	−36.0000	−1.63976
$483$	0	0
$484$	−14.0000	−0.636364
$485$	14.0000	0.635707
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	0	0
$490$	12.0000	0.542105
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	21.0000	0.945792
$494$	8.00000	0.359937
$495$	6.00000	0.269680
$496$	20.0000	0.898027
$497$	5.00000	0.224281
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	−2.00000	−0.0894427
$501$	0	0
$502$	−52.0000	−2.32087
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	0	0
$505$	5.00000	0.222497
$506$	4.00000	0.177822
$507$	0	0
$508$	−8.00000	−0.354943
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	32.0000	1.41421
$513$	0	0
$514$	−16.0000	−0.705730
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	22.0000	0.966625
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−42.0000	−1.83829
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	24.0000	1.04844
$525$	0	0
$526$	38.0000	1.65688
$527$	−15.0000	−0.653410
$528$	0	0
$529$	1.00000	0.0434783
$530$	−22.0000	−0.955619
$531$	39.0000	1.69246
$532$	−4.00000	−0.173422
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	15.0000	0.648507
$536$	0	0
$537$	0	0
$538$	42.0000	1.81075
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	26.0000	1.11680
$543$	0	0
$544$	−24.0000	−1.02899
$545$	10.0000	0.428353
$546$	0	0
$547$	24.0000	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$548$	12.0000	0.512615
$549$	24.0000	1.02430
$550$	4.00000	0.170561
$551$	−14.0000	−0.596420
$552$	0	0
$553$	−12.0000	−0.510292
$554$	12.0000	0.509831
$555$	0	0
$556$	−38.0000	−1.61156
$557$	11.0000	0.466085	0.233042	−	0.972467i	$-0.425132\pi$
$557$	11.0000	0.466085	0.233042	+	0.972467i	$0.425132\pi$
$558$	30.0000	1.27000
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	−12.0000	−0.506189
$563$	31.0000	1.30649	0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000	1.30649	0.653247	+	0.757145i	$0.273406\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	22.0000	0.924729
$567$	9.00000	0.377964
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	2.00000	0.0834784
$575$	1.00000	0.0417029
$576$	24.0000	1.00000
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	−14.0000	−0.581318
$581$	9.00000	0.373383
$582$	0	0
$583$	22.0000	0.911147
$584$	0	0
$585$	−6.00000	−0.248069
$586$	18.0000	0.743573
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	26.0000	1.07040
$591$	0	0
$592$	−44.0000	−1.80839
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	32.0000	1.31077
$597$	0	0
$598$	−4.00000	−0.163572
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	0	0
$603$	−15.0000	−0.610847
$604$	24.0000	0.976546
$605$	7.00000	0.284590
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	16.0000	0.648886
$609$	0	0
$610$	16.0000	0.647821
$611$	0	0
$612$	−18.0000	−0.727607
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−52.0000	−2.09855
$615$	0	0
$616$	0	0
$617$	13.0000	0.523360	0.261680	−	0.965155i	$-0.415723\pi$
$617$	13.0000	0.523360	0.261680	+	0.965155i	$0.415723\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	−16.0000	−0.641542
$623$	4.00000	0.160257
$624$	0	0
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	34.0000	1.35675
$629$	33.0000	1.31580
$630$	6.00000	0.239046
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	0	0
$634$	24.0000	0.953162
$635$	4.00000	0.158735
$636$	0	0
$637$	12.0000	0.475457
$638$	28.0000	1.10853
$639$	−15.0000	−0.593391
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	25.0000	0.985904	0.492952	−	0.870057i	$-0.335918\pi$
$643$	25.0000	0.985904	0.492952	+	0.870057i	$0.335918\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−50.0000	−1.96570	−0.982851	−	0.184399i	$-0.940966\pi$
$647$	−50.0000	−1.96570	−0.982851	+	0.184399i	$0.940966\pi$
$648$	0	0
$649$	−26.0000	−1.02059
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−36.0000	−1.40987
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	−4.00000	−0.156174
$657$	−18.0000	−0.702247
$658$	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
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	22.0000	0.855054
$663$	0	0
$664$	0	0
$665$	2.00000	0.0775567
$666$	−66.0000	−2.55745
$667$	7.00000	0.271041
$668$	−48.0000	−1.85718
$669$	0	0
$670$	−10.0000	−0.386334
$671$	−16.0000	−0.617673
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	−18.0000	−0.692308
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	−20.0000	−0.765840
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	12.0000	0.458831
$685$	−6.00000	−0.229248
$686$	−26.0000	−0.992685
$687$	0	0
$688$	0	0
$689$	−22.0000	−0.838133
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	48.0000	1.82469
$693$	−6.00000	−0.227921
$694$	−24.0000	−0.911028
$695$	19.0000	0.720711
$696$	0	0
$697$	3.00000	0.113633
$698$	22.0000	0.832712
$699$	0	0
$700$	2.00000	0.0755929
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−22.0000	−0.829746
$704$	−16.0000	−0.603023
$705$	0	0
$706$	−16.0000	−0.602168
$707$	−5.00000	−0.188044
$708$	0	0
$709$	12.0000	0.450669	0.225335	−	0.974281i	$-0.427652\pi$
$709$	12.0000	0.450669	0.225335	+	0.974281i	$0.427652\pi$
$710$	−10.0000	−0.375293
$711$	36.0000	1.35011
$712$	0	0
$713$	−5.00000	−0.187251
$714$	0	0
$715$	4.00000	0.149592
$716$	48.0000	1.79384
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−37.0000	−1.37987	−0.689934	−	0.723873i	$-0.742360\pi$
$719$	−37.0000	−1.37987	−0.689934	+	0.723873i	$0.742360\pi$
$720$	−12.0000	−0.447214
$721$	−8.00000	−0.297936
$722$	−30.0000	−1.11648
$723$	0	0
$724$	32.0000	1.18927
$725$	7.00000	0.259973
$726$	0	0
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−12.0000	−0.444140
$731$	0	0
$732$	0	0
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	−26.0000	−0.959678
$735$	0	0
$736$	−8.00000	−0.294884
$737$	10.0000	0.368355
$738$	−6.00000	−0.220863
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	−22.0000	−0.808736
$741$	0	0
$742$	22.0000	0.807645
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−16.0000	−0.586195
$746$	−68.0000	−2.48966
$747$	−27.0000	−0.987878
$748$	12.0000	0.438763
$749$	−15.0000	−0.548088
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	0	0
$754$	−28.0000	−1.01970
$755$	−12.0000	−0.436725
$756$	0	0
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	0	0
$765$	9.00000	0.325396
$766$	22.0000	0.794892
$767$	26.0000	0.938806
$768$	0	0
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	32.0000	1.15171
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	0	0
$778$	20.0000	0.717035
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	10.0000	0.357828
$782$	6.00000	0.214560
$783$	0	0
$784$	24.0000	0.857143
$785$	−17.0000	−0.606756
$786$	0	0
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−4.00000	−0.142494
$789$	0	0
$790$	24.0000	0.853882
$791$	−9.00000	−0.320003
$792$	0	0
$793$	16.0000	0.568177
$794$	8.00000	0.283909
$795$	0	0
$796$	52.0000	1.84309
$797$	53.0000	1.87736	0.938678	−	0.344795i	$-0.112051\pi$
$797$	53.0000	1.87736	0.938678	+	0.344795i	$0.112051\pi$
$798$	0	0
$799$	0	0
$800$	−8.00000	−0.282843
$801$	−12.0000	−0.423999
$802$	4.00000	0.141245
$803$	12.0000	0.423471
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	20.0000	0.704470
$807$	0	0
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	−18.0000	−0.632456
$811$	15.0000	0.526721	0.263361	−	0.964697i	$-0.415169\pi$
$811$	15.0000	0.526721	0.263361	+	0.964697i	$0.415169\pi$
$812$	14.0000	0.491304
$813$	0	0
$814$	44.0000	1.54220
$815$	18.0000	0.630512
$816$	0	0
$817$	0	0
$818$	−22.0000	−0.769212
$819$	6.00000	0.209657
$820$	−2.00000	−0.0698430
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	−26.0000	−0.904656
$827$	−29.0000	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$827$	−29.0000	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$828$	−6.00000	−0.208514
$829$	45.0000	1.56291	0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000	1.56291	0.781457	+	0.623959i	$0.214477\pi$
$830$	−18.0000	−0.624789
$831$	0	0
$832$	16.0000	0.554700
$833$	−18.0000	−0.623663
$834$	0	0
$835$	24.0000	0.830554
$836$	−8.00000	−0.276686
$837$	0	0
$838$	−36.0000	−1.24360
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−4.00000	−0.137849
$843$	0	0
$844$	30.0000	1.03264
$845$	9.00000	0.309609
$846$	0	0
$847$	−7.00000	−0.240523
$848$	−44.0000	−1.51097
$849$	0	0
$850$	6.00000	0.205798
$851$	11.0000	0.377075
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−16.0000	−0.547509
$855$	−6.00000	−0.205196
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	0	0
$861$	0	0
$862$	56.0000	1.90737
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	−10.0000	−0.339422
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	42.0000	1.42148
$874$	−4.00000	−0.135302
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	16.0000	0.539974
$879$	0	0
$880$	8.00000	0.269680
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	36.0000	1.21218
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−64.0000	−2.15012
$887$	−26.0000	−0.872995	−0.436497	−	0.899706i	$-0.643781\pi$
$887$	−26.0000	−0.872995	−0.436497	+	0.899706i	$0.643781\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	−8.00000	−0.268161
$891$	18.0000	0.603023
$892$	28.0000	0.937509
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	−42.0000	−1.40156
$899$	−35.0000	−1.16732
$900$	−6.00000	−0.200000
$901$	33.0000	1.09939
$902$	4.00000	0.133185
$903$	0	0
$904$	0	0
$905$	−16.0000	−0.531858
$906$	0	0
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	8.00000	0.265489
$909$	15.0000	0.497519
$910$	4.00000	0.132599
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	50.0000	1.65385
$915$	0	0
$916$	16.0000	0.528655
$917$	12.0000	0.396275
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	0	0
$922$	−4.00000	−0.131733
$923$	−10.0000	−0.329154
$924$	0	0
$925$	11.0000	0.361678
$926$	−20.0000	−0.657241
$927$	24.0000	0.788263
$928$	−56.0000	−1.83829
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	8.00000	0.262049
$933$	0	0
$934$	66.0000	2.15959
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	10.0000	0.326512
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	1.00000	0.0325645
$944$	52.0000	1.69246
$945$	0	0
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−66.0000	−2.13683
$955$	0	0
$956$	−58.0000	−1.87585
$957$	0	0
$958$	24.0000	0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−44.0000	−1.41862
$963$	45.0000	1.45010
$964$	−36.0000	−1.15948
$965$	−16.0000	−0.515058
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	0	0
$969$	0	0
$970$	28.0000	0.899026
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−19.0000	−0.609112
$974$	−56.0000	−1.79436
$975$	0	0
$976$	32.0000	1.02430
$977$	−39.0000	−1.24772	−0.623860	−	0.781536i	$-0.714437\pi$
$977$	−39.0000	−1.24772	−0.623860	+	0.781536i	$0.714437\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	12.0000	0.383326
$981$	30.0000	0.957826
$982$	18.0000	0.574403
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	42.0000	1.33755
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	12.0000	0.381385
$991$	−1.00000	−0.0317660	−0.0158830	−	0.999874i	$-0.505056\pi$
$991$	−1.00000	−0.0317660	−0.0158830	+	0.999874i	$0.505056\pi$
$992$	40.0000	1.27000
$993$	0	0
$994$	10.0000	0.317181
$995$	−26.0000	−0.824255
$996$	0	0
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.a.a.1.1	✓	1
3.2	odd	2		1035.2.a.b.1.1		1
4.3	odd	2		1840.2.a.d.1.1		1
5.2	odd	4		575.2.b.a.24.2		2
5.3	odd	4		575.2.b.a.24.1		2
5.4	even	2		575.2.a.b.1.1		1
7.6	odd	2		5635.2.a.j.1.1		1
8.3	odd	2		7360.2.a.n.1.1		1
8.5	even	2		7360.2.a.q.1.1		1
15.14	odd	2		5175.2.a.y.1.1		1
20.19	odd	2		9200.2.a.t.1.1		1
23.22	odd	2		2645.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.a.1.1	✓	1	1.1	even	1	trivial
575.2.a.b.1.1		1	5.4	even	2
575.2.b.a.24.1		2	5.3	odd	4
575.2.b.a.24.2		2	5.2	odd	4
1035.2.a.b.1.1		1	3.2	odd	2
1840.2.a.d.1.1		1	4.3	odd	2
2645.2.a.c.1.1		1	23.22	odd	2
5175.2.a.y.1.1		1	15.14	odd	2
5635.2.a.j.1.1		1	7.6	odd	2
7360.2.a.n.1.1		1	8.3	odd	2
7360.2.a.q.1.1		1	8.5	even	2
9200.2.a.t.1.1		1	20.19	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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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