LMFDB - Embedded newform 77.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, 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("77.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.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:	$0.614848095564$
Analytic rank:	$1$
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	$\chi$	$=$	77.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

\(q-3.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +6.00000 q^{12} -4.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} -5.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} +2.00000 q^{28} +10.0000 q^{29} +1.00000 q^{31} +3.00000 q^{33} +1.00000 q^{35} -12.0000 q^{36} -5.00000 q^{37} +12.0000 q^{39} -2.00000 q^{41} -8.00000 q^{43} +2.00000 q^{44} -6.00000 q^{45} +8.00000 q^{47} -12.0000 q^{48} +1.00000 q^{49} -6.00000 q^{51} +8.00000 q^{52} -6.00000 q^{53} +1.00000 q^{55} +18.0000 q^{57} +3.00000 q^{59} -6.00000 q^{60} -2.00000 q^{61} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -3.00000 q^{67} -4.00000 q^{68} +15.0000 q^{69} +1.00000 q^{71} +10.0000 q^{73} +12.0000 q^{75} +12.0000 q^{76} +1.00000 q^{77} +6.00000 q^{79} -4.00000 q^{80} +9.00000 q^{81} +12.0000 q^{83} -6.00000 q^{84} -2.00000 q^{85} -30.0000 q^{87} -15.0000 q^{89} +4.00000 q^{91} +10.0000 q^{92} -3.00000 q^{93} +6.00000 q^{95} -5.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	−2.00000	−1.00000
$5$	−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
$9$	6.00000	2.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	6.00000	1.73205
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	4.00000	1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	2.00000	0.447214
$21$	3.00000	0.654654
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	2.00000	0.377964
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	−12.0000	−2.00000
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	12.0000	1.92154
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	2.00000	0.301511
$45$	−6.00000	−0.894427
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−12.0000	−1.73205
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.00000	−0.840168
$52$	8.00000	1.10940
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	18.0000	2.38416
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−6.00000	−0.774597
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	−8.00000	−1.00000
$65$	4.00000	0.496139
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−4.00000	−0.485071
$69$	15.0000	1.80579
$70$	0	0
$71$	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
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	12.0000	1.37649
$77$	1.00000	0.113961
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−4.00000	−0.447214
$81$	9.00000	1.00000
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−6.00000	−0.654654
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−30.0000	−3.21634
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	10.0000	1.04257
$93$	−3.00000	−0.311086
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	8.00000	0.800000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	18.0000	1.73205
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	15.0000	1.42374
$112$	−4.00000	−0.377964
$113$	−19.0000	−1.78737	−0.893685	−	0.448695i	$-0.851889\pi$
$113$	−19.0000	−1.78737	−0.893685	+	0.448695i	$0.851889\pi$
$114$	0	0
$115$	5.00000	0.466252
$116$	−20.0000	−1.85695
$117$	−24.0000	−2.21880
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	−2.00000	−0.179605
$125$	9.00000	0.804984
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	24.0000	2.11308
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−6.00000	−0.522233
$133$	6.00000	0.520266
$134$	0	0
$135$	9.00000	0.774597
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−2.00000	−0.169031
$141$	−24.0000	−2.02116
$142$	0	0
$143$	4.00000	0.334497
$144$	24.0000	2.00000
$145$	−10.0000	−0.830455
$146$	0	0
$147$	−3.00000	−0.247436
$148$	10.0000	0.821995
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	−24.0000	−1.92154
$157$	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
$159$	18.0000	1.42749
$160$	0	0
$161$	5.00000	0.394055
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	4.00000	0.312348
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−36.0000	−2.75299
$172$	16.0000	1.21999
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	−9.00000	−0.676481
$178$	0	0
$179$	1.00000	0.0747435	0.0373718	−	0.999301i	$-0.488101\pi$
$179$	1.00000	0.0747435	0.0373718	+	0.999301i	$0.488101\pi$
$180$	12.0000	0.894427
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	5.00000	0.367607
$186$	0	0
$187$	−2.00000	−0.146254
$188$	−16.0000	−1.16692
$189$	9.00000	0.654654
$190$	0	0
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	24.0000	1.73205
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	−2.00000	−0.142857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	−10.0000	−0.701862
$204$	12.0000	0.840168
$205$	2.00000	0.139686
$206$	0	0
$207$	−30.0000	−2.08514
$208$	−16.0000	−1.10940
$209$	6.00000	0.415029
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	12.0000	0.824163
$213$	−3.00000	−0.205557
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	−30.0000	−2.02721
$220$	−2.00000	−0.134840
$221$	−8.00000	−0.538138
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−36.0000	−2.38416
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	−6.00000	−0.390567
$237$	−18.0000	−1.16923
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	12.0000	0.774597
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	4.00000	0.256074
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	−36.0000	−2.28141
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	12.0000	0.755929
$253$	5.00000	0.314347
$254$	0	0
$255$	6.00000	0.375735
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	5.00000	0.310685
$260$	−8.00000	−0.496139
$261$	60.0000	3.71391
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	45.0000	2.75396
$268$	6.00000	0.366508
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	8.00000	0.485071
$273$	−12.0000	−0.726273
$274$	0	0
$275$	4.00000	0.241209
$276$	−30.0000	−1.80579
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−2.00000	−0.118678
$285$	−18.0000	−1.06623
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	15.0000	0.879316
$292$	−20.0000	−1.17041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	20.0000	1.15663
$300$	−24.0000	−1.38564
$301$	8.00000	0.461112
$302$	0	0
$303$	36.0000	2.06815
$304$	−24.0000	−1.37649
$305$	2.00000	0.114520
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	−2.00000	−0.113961
$309$	36.0000	2.04797
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−23.0000	−1.30004	−0.650018	−	0.759918i	$-0.725239\pi$
$313$	−23.0000	−1.30004	−0.650018	+	0.759918i	$0.725239\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	−12.0000	−0.675053
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	0	0
$319$	−10.0000	−0.559893
$320$	8.00000	0.447214
$321$	30.0000	1.67444
$322$	0	0
$323$	−12.0000	−0.667698
$324$	−18.0000	−1.00000
$325$	16.0000	0.887520
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	−24.0000	−1.31717
$333$	−30.0000	−1.64399
$334$	0	0
$335$	3.00000	0.163908
$336$	12.0000	0.654654
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	57.0000	3.09582
$340$	4.00000	0.216930
$341$	−1.00000	−0.0541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−15.0000	−0.807573
$346$	0	0
$347$	14.0000	0.751559	0.375780	−	0.926709i	$-0.377375\pi$
$347$	14.0000	0.751559	0.375780	+	0.926709i	$0.377375\pi$
$348$	60.0000	3.21634
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	36.0000	1.92154
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	−1.00000	−0.0530745
$356$	30.0000	1.59000
$357$	6.00000	0.317554
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−3.00000	−0.157459
$364$	−8.00000	−0.419314
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	−20.0000	−1.04257
$369$	−12.0000	−0.624695
$370$	0	0
$371$	6.00000	0.311504
$372$	6.00000	0.311086
$373$	−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
$375$	−27.0000	−1.39427
$376$	0	0
$377$	−40.0000	−2.06010
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	−12.0000	−0.615587
$381$	−6.00000	−0.307389
$382$	0	0
$383$	17.0000	0.868659	0.434330	−	0.900754i	$-0.356985\pi$
$383$	17.0000	0.868659	0.434330	+	0.900754i	$0.356985\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−48.0000	−2.43998
$388$	10.0000	0.507673
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	−10.0000	−0.505722
$392$	0	0
$393$	−54.0000	−2.72394
$394$	0	0
$395$	−6.00000	−0.301893
$396$	12.0000	0.603023
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	−16.0000	−0.800000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	24.0000	1.19404
$405$	−9.00000	−0.447214
$406$	0	0
$407$	5.00000	0.247841
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	24.0000	1.18240
$413$	−3.00000	−0.147620
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	30.0000	1.46911
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	6.00000	0.292770
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	48.0000	2.33384
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	2.00000	0.0967868
$428$	20.0000	0.966736
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−36.0000	−1.73205
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	30.0000	1.43839
$436$	−8.00000	−0.383131
$437$	30.0000	1.43509
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	−30.0000	−1.42374
$445$	15.0000	0.711068
$446$	0	0
$447$	66.0000	3.12169
$448$	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	$0.384839\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	38.0000	1.78737
$453$	−18.0000	−0.845714
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	−18.0000	−0.840168
$460$	−10.0000	−0.466252
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	40.0000	1.85695
$465$	3.00000	0.139122
$466$	0	0
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	48.0000	2.21880
$469$	3.00000	0.138527
$470$	0	0
$471$	−21.0000	−0.967629
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	24.0000	1.10120
$476$	4.00000	0.183340
$477$	−36.0000	−1.64833
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	−15.0000	−0.682524
$484$	−2.00000	−0.0909091
$485$	5.00000	0.227038
$486$	0	0
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	−12.0000	−0.541002
$493$	20.0000	0.900755
$494$	0	0
$495$	6.00000	0.269680
$496$	4.00000	0.179605
$497$	−1.00000	−0.0448561
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	−18.0000	−0.804984
$501$	6.00000	0.268060
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	−9.00000	−0.399704
$508$	−4.00000	−0.177471
$509$	−31.0000	−1.37405	−0.687025	−	0.726633i	$-0.741084\pi$
$509$	−31.0000	−1.37405	−0.687025	+	0.726633i	$0.741084\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	54.0000	2.38416
$514$	0	0
$515$	12.0000	0.528783
$516$	−48.0000	−2.11308
$517$	−8.00000	−0.351840
$518$	0	0
$519$	−48.0000	−2.10697
$520$	0	0
$521$	7.00000	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$521$	7.00000	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$522$	0	0
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	−36.0000	−1.57267
$525$	−12.0000	−0.523723
$526$	0	0
$527$	2.00000	0.0871214
$528$	12.0000	0.522233
$529$	2.00000	0.0869565
$530$	0	0
$531$	18.0000	0.781133
$532$	−12.0000	−0.520266
$533$	8.00000	0.346518
$534$	0	0
$535$	10.0000	0.432338
$536$	0	0
$537$	−3.00000	−0.129460
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	−18.0000	−0.774597
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	−15.0000	−0.643712
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	6.00000	0.256307
$549$	−12.0000	−0.512148
$550$	0	0
$551$	−60.0000	−2.55609
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	−15.0000	−0.636715
$556$	20.0000	0.848189
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	4.00000	0.169031
$561$	6.00000	0.253320
$562$	0	0
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	48.0000	2.02116
$565$	19.0000	0.799336
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−8.00000	−0.334497
$573$	−15.0000	−0.626634
$574$	0	0
$575$	20.0000	0.834058
$576$	−48.0000	−2.00000
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	0	0
$579$	−42.0000	−1.74546
$580$	20.0000	0.830455
$581$	−12.0000	−0.497844
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	6.00000	0.247436
$589$	−6.00000	−0.247226
$590$	0	0
$591$	−54.0000	−2.22126
$592$	−20.0000	−0.821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	44.0000	1.80231
$597$	24.0000	0.982255
$598$	0	0
$599$	−48.0000	−1.96123	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000	−1.96123	−0.980613	+	0.195952i	$0.937220\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	−18.0000	−0.733017
$604$	−12.0000	−0.488273
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	30.0000	1.21566
$610$	0	0
$611$	−32.0000	−1.29458
$612$	−24.0000	−0.970143
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	2.00000	0.0803219
$621$	45.0000	1.80579
$622$	0	0
$623$	15.0000	0.600962
$624$	48.0000	1.92154
$625$	11.0000	0.440000
$626$	0	0
$627$	−18.0000	−0.718851
$628$	−14.0000	−0.558661
$629$	−10.0000	−0.398726
$630$	0	0
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	0	0
$633$	6.00000	0.238479
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	−36.0000	−1.42749
$637$	−4.00000	−0.158486
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−29.0000	−1.14365	−0.571824	−	0.820376i	$-0.693764\pi$
$643$	−29.0000	−1.14365	−0.571824	+	0.820376i	$0.693764\pi$
$644$	−10.0000	−0.394055
$645$	−24.0000	−0.944999
$646$	0	0
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	3.00000	0.117579
$652$	−8.00000	−0.313304
$653$	−17.0000	−0.665261	−0.332631	−	0.943057i	$-0.607936\pi$
$653$	−17.0000	−0.665261	−0.332631	+	0.943057i	$0.607936\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	−8.00000	−0.312348
$657$	60.0000	2.34082
$658$	0	0
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	6.00000	0.233550
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	−50.0000	−1.93601
$668$	4.00000	0.154765
$669$	−3.00000	−0.115987
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	0	0
$675$	36.0000	1.38564
$676$	−6.00000	−0.230769
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	72.0000	2.75299
$685$	3.00000	0.114624
$686$	0	0
$687$	21.0000	0.801200
$688$	−32.0000	−1.21999
$689$	24.0000	0.914327
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	−32.0000	−1.21646
$693$	6.00000	0.227921
$694$	0	0
$695$	10.0000	0.379322
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	−18.0000	−0.680823
$700$	−8.00000	−0.302372
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	30.0000	1.13147
$704$	8.00000	0.301511
$705$	24.0000	0.903892
$706$	0	0
$707$	12.0000	0.451306
$708$	18.0000	0.676481
$709$	39.0000	1.46468	0.732338	−	0.680941i	$-0.238429\pi$
$709$	39.0000	1.46468	0.732338	+	0.680941i	$0.238429\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	−5.00000	−0.187251
$714$	0	0
$715$	−4.00000	−0.149592
$716$	−2.00000	−0.0747435
$717$	−12.0000	−0.448148
$718$	0	0
$719$	−11.0000	−0.410231	−0.205115	−	0.978738i	$-0.565757\pi$
$719$	−11.0000	−0.410231	−0.205115	+	0.978738i	$0.565757\pi$
$720$	−24.0000	−0.894427
$721$	12.0000	0.446903
$722$	0	0
$723$	36.0000	1.33885
$724$	−10.0000	−0.371647
$725$	−40.0000	−1.48556
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−16.0000	−0.591781
$732$	−12.0000	−0.443533
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	3.00000	0.110506
$738$	0	0
$739$	−18.0000	−0.662141	−0.331070	−	0.943606i	$-0.607410\pi$
$739$	−18.0000	−0.662141	−0.331070	+	0.943606i	$0.607410\pi$
$740$	−10.0000	−0.367607
$741$	−72.0000	−2.64499
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	22.0000	0.806018
$746$	0	0
$747$	72.0000	2.63434
$748$	4.00000	0.146254
$749$	10.0000	0.365392
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	32.0000	1.16692
$753$	63.0000	2.29585
$754$	0	0
$755$	−6.00000	−0.218362
$756$	−18.0000	−0.654654
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	−15.0000	−0.544466
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	−10.0000	−0.361787
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−12.0000	−0.433295
$768$	−48.0000	−1.73205
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−28.0000	−1.00774
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−15.0000	−0.538122
$778$	0	0
$779$	12.0000	0.429945
$780$	24.0000	0.859338
$781$	−1.00000	−0.0357828
$782$	0	0
$783$	−90.0000	−3.21634
$784$	4.00000	0.142857
$785$	−7.00000	−0.249841
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−36.0000	−1.28245
$789$	−54.0000	−1.92245
$790$	0	0
$791$	19.0000	0.675562
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	−18.0000	−0.638394
$796$	16.0000	0.567105
$797$	23.0000	0.814702	0.407351	−	0.913272i	$-0.366453\pi$
$797$	23.0000	0.814702	0.407351	+	0.913272i	$0.366453\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−90.0000	−3.17999
$802$	0	0
$803$	−10.0000	−0.352892
$804$	−18.0000	−0.634811
$805$	−5.00000	−0.176227
$806$	0	0
$807$	54.0000	1.90089
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	20.0000	0.701862
$813$	−48.0000	−1.68343
$814$	0	0
$815$	−4.00000	−0.140114
$816$	−24.0000	−0.840168
$817$	48.0000	1.67931
$818$	0	0
$819$	24.0000	0.838628
$820$	−4.00000	−0.139686
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−25.0000	−0.871445	−0.435723	−	0.900081i	$-0.643507\pi$
$823$	−25.0000	−0.871445	−0.435723	+	0.900081i	$0.643507\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	60.0000	2.08514
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	−72.0000	−2.49765
$832$	32.0000	1.10940
$833$	2.00000	0.0692959
$834$	0	0
$835$	2.00000	0.0692129
$836$	−12.0000	−0.415029
$837$	−9.00000	−0.311086
$838$	0	0
$839$	45.0000	1.55357	0.776786	−	0.629764i	$-0.216849\pi$
$839$	45.0000	1.55357	0.776786	+	0.629764i	$0.216849\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	12.0000	0.413302
$844$	4.00000	0.137686
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−24.0000	−0.824163
$849$	0	0
$850$	0	0
$851$	25.0000	0.856989
$852$	6.00000	0.205557
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	0	0
$857$	−28.0000	−0.956462	−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000	−0.956462	−0.478231	+	0.878234i	$0.658722\pi$
$858$	0	0
$859$	55.0000	1.87658	0.938288	−	0.345855i	$-0.112411\pi$
$859$	55.0000	1.87658	0.938288	+	0.345855i	$0.112411\pi$
$860$	−16.0000	−0.545595
$861$	−6.00000	−0.204479
$862$	0	0
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	0	0
$865$	−16.0000	−0.544016
$866$	0	0
$867$	39.0000	1.32451
$868$	2.00000	0.0678844
$869$	−6.00000	−0.203536
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	−30.0000	−1.01535
$874$	0	0
$875$	−9.00000	−0.304256
$876$	60.0000	2.02721
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	4.00000	0.134840
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	16.0000	0.538138
$885$	9.00000	0.302532
$886$	0	0
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−9.00000	−0.301511
$892$	−2.00000	−0.0669650
$893$	−48.0000	−1.60626
$894$	0	0
$895$	−1.00000	−0.0334263
$896$	0	0
$897$	−60.0000	−2.00334
$898$	0	0
$899$	10.0000	0.333519
$900$	48.0000	1.60000
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	−5.00000	−0.166206
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	−8.00000	−0.265489
$909$	−72.0000	−2.38809
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	72.0000	2.38416
$913$	−12.0000	−0.397142
$914$	0	0
$915$	−6.00000	−0.198354
$916$	14.0000	0.462573
$917$	−18.0000	−0.594412
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	84.0000	2.76789
$922$	0	0
$923$	−4.00000	−0.131662
$924$	6.00000	0.197386
$925$	20.0000	0.657596
$926$	0	0
$927$	−72.0000	−2.36479
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−12.0000	−0.393073
$933$	−24.0000	−0.785725
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	0	0
$939$	69.0000	2.25173
$940$	16.0000	0.521862
$941$	58.0000	1.89075	0.945373	−	0.325991i	$-0.105698\pi$
$941$	58.0000	1.89075	0.945373	+	0.325991i	$0.105698\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	12.0000	0.390567
$945$	−9.00000	−0.292770
$946$	0	0
$947$	5.00000	0.162478	0.0812391	−	0.996695i	$-0.474112\pi$
$947$	5.00000	0.162478	0.0812391	+	0.996695i	$0.474112\pi$
$948$	36.0000	1.16923
$949$	−40.0000	−1.29845
$950$	0	0
$951$	−27.0000	−0.875535
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	−5.00000	−0.161796
$956$	−8.00000	−0.258738
$957$	30.0000	0.969762
$958$	0	0
$959$	3.00000	0.0968751
$960$	−24.0000	−0.774597
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−60.0000	−1.93347
$964$	24.0000	0.772988
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	0	0
$969$	36.0000	1.15649
$970$	0	0
$971$	29.0000	0.930654	0.465327	−	0.885139i	$-0.345937\pi$
$971$	29.0000	0.930654	0.465327	+	0.885139i	$0.345937\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	0	0
$975$	−48.0000	−1.53723
$976$	−8.00000	−0.256074
$977$	−31.0000	−0.991778	−0.495889	−	0.868386i	$-0.665158\pi$
$977$	−31.0000	−0.991778	−0.495889	+	0.868386i	$0.665158\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	2.00000	0.0638877
$981$	24.0000	0.766261
$982$	0	0
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	24.0000	0.763928
$988$	−48.0000	−1.52708
$989$	40.0000	1.27193
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	51.0000	1.61844
$994$	0	0
$995$	8.00000	0.253617
$996$	72.0000	2.28141
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	0	0
$999$	45.0000	1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.2.a.a.1.1	✓	1
3.2	odd	2		693.2.a.c.1.1		1
4.3	odd	2		1232.2.a.l.1.1		1
5.2	odd	4		1925.2.b.e.1849.2		2
5.3	odd	4		1925.2.b.e.1849.1		2
5.4	even	2		1925.2.a.h.1.1		1
7.2	even	3		539.2.e.f.67.1		2
7.3	odd	6		539.2.e.c.177.1		2
7.4	even	3		539.2.e.f.177.1		2
7.5	odd	6		539.2.e.c.67.1		2
7.6	odd	2		539.2.a.c.1.1		1
8.3	odd	2		4928.2.a.a.1.1		1
8.5	even	2		4928.2.a.bj.1.1		1
11.2	odd	10		847.2.f.h.323.1		4
11.3	even	5		847.2.f.i.372.1		4
11.4	even	5		847.2.f.i.148.1		4
11.5	even	5		847.2.f.i.729.1		4
11.6	odd	10		847.2.f.h.729.1		4
11.7	odd	10		847.2.f.h.148.1		4
11.8	odd	10		847.2.f.h.372.1		4
11.9	even	5		847.2.f.i.323.1		4
11.10	odd	2		847.2.a.b.1.1		1
21.20	even	2		4851.2.a.j.1.1		1
28.27	even	2		8624.2.a.a.1.1		1
33.32	even	2		7623.2.a.j.1.1		1
77.76	even	2		5929.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.a.1.1	✓	1	1.1	even	1	trivial
539.2.a.c.1.1		1	7.6	odd	2
539.2.e.c.67.1		2	7.5	odd	6
539.2.e.c.177.1		2	7.3	odd	6
539.2.e.f.67.1		2	7.2	even	3
539.2.e.f.177.1		2	7.4	even	3
693.2.a.c.1.1		1	3.2	odd	2
847.2.a.b.1.1		1	11.10	odd	2
847.2.f.h.148.1		4	11.7	odd	10
847.2.f.h.323.1		4	11.2	odd	10
847.2.f.h.372.1		4	11.8	odd	10
847.2.f.h.729.1		4	11.6	odd	10
847.2.f.i.148.1		4	11.4	even	5
847.2.f.i.323.1		4	11.9	even	5
847.2.f.i.372.1		4	11.3	even	5
847.2.f.i.729.1		4	11.5	even	5
1232.2.a.l.1.1		1	4.3	odd	2
1925.2.a.h.1.1		1	5.4	even	2
1925.2.b.e.1849.1		2	5.3	odd	4
1925.2.b.e.1849.2		2	5.2	odd	4
4851.2.a.j.1.1		1	21.20	even	2
4928.2.a.a.1.1		1	8.3	odd	2
4928.2.a.bj.1.1		1	8.5	even	2
5929.2.a.f.1.1		1	77.76	even	2
7623.2.a.j.1.1		1	33.32	even	2
8624.2.a.a.1.1		1	28.27	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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