LMFDB - Embedded newform 67.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [67,2,Mod(1,67)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(67, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("67.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	67.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:	$0.534997693543$
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	$\chi$	$=$	67.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +2.00000 q^{5} -4.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +2.00000 q^{5} -4.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{10} -4.00000 q^{11} -4.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} -4.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} +7.00000 q^{19} +4.00000 q^{20} +4.00000 q^{21} -8.00000 q^{22} +9.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} -5.00000 q^{29} -8.00000 q^{30} -10.0000 q^{31} -8.00000 q^{32} +8.00000 q^{33} +6.00000 q^{34} -4.00000 q^{35} +2.00000 q^{36} -1.00000 q^{37} +14.0000 q^{38} -4.00000 q^{39} +8.00000 q^{42} -2.00000 q^{43} -8.00000 q^{44} +2.00000 q^{45} +18.0000 q^{46} -1.00000 q^{47} +8.00000 q^{48} -3.00000 q^{49} -2.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} +10.0000 q^{53} +8.00000 q^{54} -8.00000 q^{55} -14.0000 q^{57} -10.0000 q^{58} +9.00000 q^{59} -8.00000 q^{60} -2.00000 q^{61} -20.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} +16.0000 q^{66} +1.00000 q^{67} +6.00000 q^{68} -18.0000 q^{69} -8.00000 q^{70} -7.00000 q^{73} -2.00000 q^{74} +2.00000 q^{75} +14.0000 q^{76} +8.00000 q^{77} -8.00000 q^{78} -8.00000 q^{79} -8.00000 q^{80} -11.0000 q^{81} +4.00000 q^{83} +8.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} +10.0000 q^{87} +7.00000 q^{89} +4.00000 q^{90} -4.00000 q^{91} +18.0000 q^{92} +20.0000 q^{93} -2.00000 q^{94} +14.0000 q^{95} +16.0000 q^{96} -6.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$0.250000\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	2.00000	1.00000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−4.00000	−1.63299
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−4.00000	−1.15470
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	−4.00000	−1.03280
$16$	−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	$0.381478\pi$
$18$	2.00000	0.471405
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	4.00000	0.894427
$21$	4.00000	0.872872
$22$	−8.00000	−1.70561
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	−4.00000	−0.755929
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	−8.00000	−1.46059
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−8.00000	−1.41421
$33$	8.00000	1.39262
$34$	6.00000	1.02899
$35$	−4.00000	−0.676123
$36$	2.00000	0.333333
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	14.0000	2.27110
$39$	−4.00000	−0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	8.00000	1.23443
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−8.00000	−1.20605
$45$	2.00000	0.298142
$46$	18.0000	2.65396
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	8.00000	1.15470
$49$	−3.00000	−0.428571
$50$	−2.00000	−0.282843
$51$	−6.00000	−0.840168
$52$	4.00000	0.554700
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	8.00000	1.08866
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−14.0000	−1.85435
$58$	−10.0000	−1.31306
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	−8.00000	−1.03280
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−20.0000	−2.54000
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	4.00000	0.496139
$66$	16.0000	1.96946
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	6.00000	0.727607
$69$	−18.0000	−2.16695
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−2.00000	−0.232495
$75$	2.00000	0.230940
$76$	14.0000	1.60591
$77$	8.00000	0.911685
$78$	−8.00000	−0.905822
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−8.00000	−0.894427
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	8.00000	0.872872
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	10.0000	1.07211
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	4.00000	0.421637
$91$	−4.00000	−0.419314
$92$	18.0000	1.87663
$93$	20.0000	2.07390
$94$	−2.00000	−0.206284
$95$	14.0000	1.43637
$96$	16.0000	1.63299
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−6.00000	−0.606092
$99$	−4.00000	−0.402015
$100$	−2.00000	−0.200000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−12.0000	−1.18818
$103$	−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
$105$	8.00000	0.780720
$106$	20.0000	1.94257
$107$	−7.00000	−0.676716	−0.338358	−	0.941018i	$-0.609871\pi$
$107$	−7.00000	−0.676716	−0.338358	+	0.941018i	$0.609871\pi$
$108$	8.00000	0.769800
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−16.0000	−1.52554
$111$	2.00000	0.189832
$112$	8.00000	0.755929
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−28.0000	−2.62244
$115$	18.0000	1.67851
$116$	−10.0000	−0.928477
$117$	2.00000	0.184900
$118$	18.0000	1.65703
$119$	−6.00000	−0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	−4.00000	−0.362143
$123$	0	0
$124$	−20.0000	−1.79605
$125$	−12.0000	−1.07331
$126$	−4.00000	−0.356348
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	8.00000	0.701646
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	16.0000	1.39262
$133$	−14.0000	−1.21395
$134$	2.00000	0.172774
$135$	8.00000	0.688530
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−36.0000	−3.06452
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	−8.00000	−0.676123
$141$	2.00000	0.168430
$142$	0	0
$143$	−8.00000	−0.668994
$144$	−4.00000	−0.333333
$145$	−10.0000	−0.830455
$146$	−14.0000	−1.15865
$147$	6.00000	0.494872
$148$	−2.00000	−0.164399
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	4.00000	0.326599
$151$	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
$154$	16.0000	1.28932
$155$	−20.0000	−1.60644
$156$	−8.00000	−0.640513
$157$	9.00000	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$157$	9.00000	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$158$	−16.0000	−1.27289
$159$	−20.0000	−1.58610
$160$	−16.0000	−1.26491
$161$	−18.0000	−1.41860
$162$	−22.0000	−1.72848
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	0	0
$165$	16.0000	1.24560
$166$	8.00000	0.620920
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	7.00000	0.535303
$172$	−4.00000	−0.304997
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	20.0000	1.51620
$175$	2.00000	0.151186
$176$	16.0000	1.20605
$177$	−18.0000	−1.35296
$178$	14.0000	1.04934
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	4.00000	0.298142
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−8.00000	−0.592999
$183$	4.00000	0.295689
$184$	0	0
$185$	−2.00000	−0.147043
$186$	40.0000	2.93294
$187$	−12.0000	−0.877527
$188$	−2.00000	−0.145865
$189$	−8.00000	−0.581914
$190$	28.0000	2.03133
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	16.0000	1.15470
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	−6.00000	−0.428571
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−8.00000	−0.568535
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	4.00000	0.281439
$203$	10.0000	0.701862
$204$	−12.0000	−0.840168
$205$	0	0
$206$	−32.0000	−2.22955
$207$	9.00000	0.625543
$208$	−8.00000	−0.554700
$209$	−28.0000	−1.93680
$210$	16.0000	1.10410
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	20.0000	1.37361
$213$	0	0
$214$	−14.0000	−0.957020
$215$	−4.00000	−0.272798
$216$	0	0
$217$	20.0000	1.35769
$218$	4.00000	0.270914
$219$	14.0000	0.946032
$220$	−16.0000	−1.07872
$221$	6.00000	0.403604
$222$	4.00000	0.268462
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	16.0000	1.06904
$225$	−1.00000	−0.0666667
$226$	−24.0000	−1.59646
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−28.0000	−1.85435
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	36.0000	2.37377
$231$	−16.0000	−1.05272
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	4.00000	0.261488
$235$	−2.00000	−0.130466
$236$	18.0000	1.17170
$237$	16.0000	1.03931
$238$	−12.0000	−0.777844
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	16.0000	1.03280
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	10.0000	0.642824
$243$	10.0000	0.641500
$244$	−4.00000	−0.256074
$245$	−6.00000	−0.383326
$246$	0	0
$247$	14.0000	0.890799
$248$	0	0
$249$	−8.00000	−0.506979
$250$	−24.0000	−1.51789
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−4.00000	−0.251976
$253$	−36.0000	−2.26330
$254$	14.0000	0.878438
$255$	−12.0000	−0.751469
$256$	16.0000	1.00000
$257$	−1.00000	−0.0623783	−0.0311891	−	0.999514i	$-0.509929\pi$
$257$	−1.00000	−0.0623783	−0.0311891	+	0.999514i	$0.509929\pi$
$258$	8.00000	0.498058
$259$	2.00000	0.124274
$260$	8.00000	0.496139
$261$	−5.00000	−0.309492
$262$	−24.0000	−1.48272
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	−28.0000	−1.71679
$267$	−14.0000	−0.856786
$268$	2.00000	0.122169
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	16.0000	0.973729
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−12.0000	−0.727607
$273$	8.00000	0.484182
$274$	24.0000	1.44989
$275$	4.00000	0.241209
$276$	−36.0000	−2.16695
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	44.0000	2.63894
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	4.00000	0.238197
$283$	−3.00000	−0.178331	−0.0891657	−	0.996017i	$-0.528420\pi$
$283$	−3.00000	−0.178331	−0.0891657	+	0.996017i	$0.528420\pi$
$284$	0	0
$285$	−28.0000	−1.65858
$286$	−16.0000	−0.946100
$287$	0	0
$288$	−8.00000	−0.471405
$289$	−8.00000	−0.470588
$290$	−20.0000	−1.17444
$291$	0	0
$292$	−14.0000	−0.819288
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	12.0000	0.699854
$295$	18.0000	1.04800
$296$	0	0
$297$	−16.0000	−0.928414
$298$	42.0000	2.43299
$299$	18.0000	1.04097
$300$	4.00000	0.230940
$301$	4.00000	0.230556
$302$	6.00000	0.345261
$303$	−4.00000	−0.229794
$304$	−28.0000	−1.60591
$305$	−4.00000	−0.229039
$306$	6.00000	0.342997
$307$	−25.0000	−1.42683	−0.713413	−	0.700744i	$-0.752851\pi$
$307$	−25.0000	−1.42683	−0.713413	+	0.700744i	$0.752851\pi$
$308$	16.0000	0.911685
$309$	32.0000	1.82042
$310$	−40.0000	−2.27185
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	18.0000	1.01580
$315$	−4.00000	−0.225374
$316$	−16.0000	−0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−40.0000	−2.24309
$319$	20.0000	1.11979
$320$	−16.0000	−0.894427
$321$	14.0000	0.781404
$322$	−36.0000	−2.00620
$323$	21.0000	1.16847
$324$	−22.0000	−1.22222
$325$	−2.00000	−0.110940
$326$	38.0000	2.10463
$327$	−4.00000	−0.221201
$328$	0	0
$329$	2.00000	0.110264
$330$	32.0000	1.76154
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	8.00000	0.439057
$333$	−1.00000	−0.0547997
$334$	48.0000	2.62644
$335$	2.00000	0.109272
$336$	−16.0000	−0.872872
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−18.0000	−0.979071
$339$	24.0000	1.30350
$340$	12.0000	0.650791
$341$	40.0000	2.16612
$342$	14.0000	0.757033
$343$	20.0000	1.07990
$344$	0	0
$345$	−36.0000	−1.93817
$346$	22.0000	1.18273
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	20.0000	1.07211
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	4.00000	0.213809
$351$	8.00000	0.427008
$352$	32.0000	1.70561
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	−36.0000	−1.91338
$355$	0	0
$356$	14.0000	0.741999
$357$	12.0000	0.635107
$358$	−24.0000	−1.26844
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	14.0000	0.735824
$363$	−10.0000	−0.524864
$364$	−8.00000	−0.419314
$365$	−14.0000	−0.732793
$366$	8.00000	0.418167
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−36.0000	−1.87663
$369$	0	0
$370$	−4.00000	−0.207950
$371$	−20.0000	−1.03835
$372$	40.0000	2.07390
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	−24.0000	−1.24101
$375$	24.0000	1.23935
$376$	0	0
$377$	−10.0000	−0.515026
$378$	−16.0000	−0.822951
$379$	−18.0000	−0.924598	−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000	−0.924598	−0.462299	+	0.886724i	$0.652975\pi$
$380$	28.0000	1.43637
$381$	−14.0000	−0.717242
$382$	−12.0000	−0.613973
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	−46.0000	−2.34134
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	−16.0000	−0.810191
$391$	27.0000	1.36545
$392$	0	0
$393$	24.0000	1.21064
$394$	−4.00000	−0.201517
$395$	−16.0000	−0.805047
$396$	−8.00000	−0.402015
$397$	−31.0000	−1.55585	−0.777923	−	0.628360i	$-0.783727\pi$
$397$	−31.0000	−1.55585	−0.777923	+	0.628360i	$0.783727\pi$
$398$	14.0000	0.701757
$399$	28.0000	1.40175
$400$	4.00000	0.200000
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	−4.00000	−0.199502
$403$	−20.0000	−0.996271
$404$	4.00000	0.199007
$405$	−22.0000	−1.09319
$406$	20.0000	0.992583
$407$	4.00000	0.198273
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	−32.0000	−1.57653
$413$	−18.0000	−0.885722
$414$	18.0000	0.884652
$415$	8.00000	0.392705
$416$	−16.0000	−0.784465
$417$	−44.0000	−2.15469
$418$	−56.0000	−2.73905
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	16.0000	0.780720
$421$	−21.0000	−1.02348	−0.511739	−	0.859141i	$-0.670998\pi$
$421$	−21.0000	−1.02348	−0.511739	+	0.859141i	$0.670998\pi$
$422$	−24.0000	−1.16830
$423$	−1.00000	−0.0486217
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	4.00000	0.193574
$428$	−14.0000	−0.676716
$429$	16.0000	0.772487
$430$	−8.00000	−0.385794
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	−16.0000	−0.769800
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	40.0000	1.92006
$435$	20.0000	0.958927
$436$	4.00000	0.191565
$437$	63.0000	3.01370
$438$	28.0000	1.33789
$439$	23.0000	1.09773	0.548865	−	0.835911i	$-0.315060\pi$
$439$	23.0000	1.09773	0.548865	+	0.835911i	$0.315060\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	4.00000	0.189832
$445$	14.0000	0.663664
$446$	22.0000	1.04173
$447$	−42.0000	−1.98653
$448$	16.0000	0.755929
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	−24.0000	−1.12887
$453$	−6.00000	−0.281905
$454$	6.00000	0.281594
$455$	−8.00000	−0.375046
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	8.00000	0.373815
$459$	12.0000	0.560112
$460$	36.0000	1.67851
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	−32.0000	−1.48877
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	20.0000	0.928477
$465$	40.0000	1.85496
$466$	20.0000	0.926482
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	4.00000	0.184900
$469$	−2.00000	−0.0923514
$470$	−4.00000	−0.184506
$471$	−18.0000	−0.829396
$472$	0	0
$473$	8.00000	0.367840
$474$	32.0000	1.46981
$475$	−7.00000	−0.321182
$476$	−12.0000	−0.550019
$477$	10.0000	0.457869
$478$	−40.0000	−1.82956
$479$	−29.0000	−1.32504	−0.662522	−	0.749043i	$-0.730514\pi$
$479$	−29.0000	−1.32504	−0.662522	+	0.749043i	$0.730514\pi$
$480$	32.0000	1.46059
$481$	−2.00000	−0.0911922
$482$	−38.0000	−1.73085
$483$	36.0000	1.63806
$484$	10.0000	0.454545
$485$	0	0
$486$	20.0000	0.907218
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−38.0000	−1.71842
$490$	−12.0000	−0.542105
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	28.0000	1.25978
$495$	−8.00000	−0.359573
$496$	40.0000	1.79605
$497$	0	0
$498$	−16.0000	−0.716977
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−24.0000	−1.07331
$501$	−48.0000	−2.14448
$502$	−4.00000	−0.178529
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	−72.0000	−3.20079
$507$	18.0000	0.799408
$508$	14.0000	0.621150
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	−24.0000	−1.06274
$511$	14.0000	0.619324
$512$	32.0000	1.41421
$513$	28.0000	1.23623
$514$	−2.00000	−0.0882162
$515$	−32.0000	−1.41009
$516$	8.00000	0.352180
$517$	4.00000	0.175920
$518$	4.00000	0.175750
$519$	−22.0000	−0.965693
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	−10.0000	−0.437688
$523$	−19.0000	−0.830812	−0.415406	−	0.909636i	$-0.636360\pi$
$523$	−19.0000	−0.830812	−0.415406	+	0.909636i	$0.636360\pi$
$524$	−24.0000	−1.04844
$525$	−4.00000	−0.174574
$526$	32.0000	1.39527
$527$	−30.0000	−1.30682
$528$	−32.0000	−1.39262
$529$	58.0000	2.52174
$530$	40.0000	1.73749
$531$	9.00000	0.390567
$532$	−28.0000	−1.21395
$533$	0	0
$534$	−28.0000	−1.21168
$535$	−14.0000	−0.605273
$536$	0	0
$537$	24.0000	1.03568
$538$	4.00000	0.172452
$539$	12.0000	0.516877
$540$	16.0000	0.688530
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	16.0000	0.687259
$543$	−14.0000	−0.600798
$544$	−24.0000	−1.02899
$545$	4.00000	0.171341
$546$	16.0000	0.684737
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	24.0000	1.02523
$549$	−2.00000	−0.0853579
$550$	8.00000	0.341121
$551$	−35.0000	−1.49105
$552$	0	0
$553$	16.0000	0.680389
$554$	−4.00000	−0.169944
$555$	4.00000	0.169791
$556$	44.0000	1.86602
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−20.0000	−0.846668
$559$	−4.00000	−0.169182
$560$	16.0000	0.676123
$561$	24.0000	1.01328
$562$	−36.0000	−1.51857
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	4.00000	0.168430
$565$	−24.0000	−1.00969
$566$	−6.00000	−0.252199
$567$	22.0000	0.923913
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	−56.0000	−2.34558
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	−16.0000	−0.668994
$573$	12.0000	0.501307
$574$	0	0
$575$	−9.00000	−0.375326
$576$	−8.00000	−0.333333
$577$	−24.0000	−0.999133	−0.499567	−	0.866276i	$-0.666507\pi$
$577$	−24.0000	−0.999133	−0.499567	+	0.866276i	$0.666507\pi$
$578$	−16.0000	−0.665512
$579$	46.0000	1.91169
$580$	−20.0000	−0.830455
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−40.0000	−1.65663
$584$	0	0
$585$	4.00000	0.165380
$586$	36.0000	1.48715
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	12.0000	0.494872
$589$	−70.0000	−2.88430
$590$	36.0000	1.48210
$591$	4.00000	0.164538
$592$	4.00000	0.164399
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−32.0000	−1.31298
$595$	−12.0000	−0.491952
$596$	42.0000	1.72039
$597$	−14.0000	−0.572982
$598$	36.0000	1.47215
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	8.00000	0.326056
$603$	1.00000	0.0407231
$604$	6.00000	0.244137
$605$	10.0000	0.406558
$606$	−8.00000	−0.324978
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−56.0000	−2.27110
$609$	−20.0000	−0.810441
$610$	−8.00000	−0.323911
$611$	−2.00000	−0.0809113
$612$	6.00000	0.242536
$613$	39.0000	1.57520	0.787598	−	0.616190i	$-0.211325\pi$
$613$	39.0000	1.57520	0.787598	+	0.616190i	$0.211325\pi$
$614$	−50.0000	−2.01784
$615$	0	0
$616$	0	0
$617$	−1.00000	−0.0402585	−0.0201292	−	0.999797i	$-0.506408\pi$
$617$	−1.00000	−0.0402585	−0.0201292	+	0.999797i	$0.506408\pi$
$618$	64.0000	2.57446
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−40.0000	−1.60644
$621$	36.0000	1.44463
$622$	36.0000	1.44347
$623$	−14.0000	−0.560898
$624$	16.0000	0.640513
$625$	−19.0000	−0.760000
$626$	−32.0000	−1.27898
$627$	56.0000	2.23642
$628$	18.0000	0.718278
$629$	−3.00000	−0.119618
$630$	−8.00000	−0.318728
$631$	−50.0000	−1.99047	−0.995234	−	0.0975126i	$-0.968911\pi$
$631$	−50.0000	−1.99047	−0.995234	+	0.0975126i	$0.968911\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	−36.0000	−1.42974
$635$	14.0000	0.555573
$636$	−40.0000	−1.58610
$637$	−6.00000	−0.237729
$638$	40.0000	1.58362
$639$	0	0
$640$	0	0
$641$	20.0000	0.789953	0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000	0.789953	0.394976	+	0.918691i	$0.370753\pi$
$642$	28.0000	1.10507
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−36.0000	−1.41860
$645$	8.00000	0.315000
$646$	42.0000	1.65247
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	−4.00000	−0.156893
$651$	−40.0000	−1.56772
$652$	38.0000	1.48819
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−8.00000	−0.312825
$655$	−24.0000	−0.937758
$656$	0	0
$657$	−7.00000	−0.273096
$658$	4.00000	0.155936
$659$	−9.00000	−0.350590	−0.175295	−	0.984516i	$-0.556088\pi$
$659$	−9.00000	−0.350590	−0.175295	+	0.984516i	$0.556088\pi$
$660$	32.0000	1.24560
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−12.0000	−0.466393
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−28.0000	−1.08579
$666$	−2.00000	−0.0774984
$667$	−45.0000	−1.74241
$668$	48.0000	1.85718
$669$	−22.0000	−0.850569
$670$	4.00000	0.154533
$671$	8.00000	0.308837
$672$	−32.0000	−1.23443
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	44.0000	1.69482
$675$	−4.00000	−0.153960
$676$	−18.0000	−0.692308
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	48.0000	1.84343
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	80.0000	3.06336
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	14.0000	0.535303
$685$	24.0000	0.916993
$686$	40.0000	1.52721
$687$	−8.00000	−0.305219
$688$	8.00000	0.304997
$689$	20.0000	0.761939
$690$	−72.0000	−2.74099
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	22.0000	0.836315
$693$	8.00000	0.303895
$694$	−60.0000	−2.27757
$695$	44.0000	1.66902
$696$	0	0
$697$	0	0
$698$	12.0000	0.454207
$699$	−20.0000	−0.756469
$700$	4.00000	0.151186
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	16.0000	0.603881
$703$	−7.00000	−0.264010
$704$	32.0000	1.20605
$705$	4.00000	0.150649
$706$	0	0
$707$	−4.00000	−0.150435
$708$	−36.0000	−1.35296
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−90.0000	−3.37053
$714$	24.0000	0.898177
$715$	−16.0000	−0.598366
$716$	−24.0000	−0.896922
$717$	40.0000	1.49383
$718$	38.0000	1.41815
$719$	25.0000	0.932343	0.466171	−	0.884694i	$-0.345633\pi$
$719$	25.0000	0.932343	0.466171	+	0.884694i	$0.345633\pi$
$720$	−8.00000	−0.298142
$721$	32.0000	1.19174
$722$	60.0000	2.23297
$723$	38.0000	1.41324
$724$	14.0000	0.520306
$725$	5.00000	0.185695
$726$	−20.0000	−0.742270
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−28.0000	−1.03633
$731$	−6.00000	−0.221918
$732$	8.00000	0.295689
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−16.0000	−0.590571
$735$	12.0000	0.442627
$736$	−72.0000	−2.65396
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−4.00000	−0.147043
$741$	−28.0000	−1.02861
$742$	−40.0000	−1.46845
$743$	−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
$745$	42.0000	1.53876
$746$	0	0
$747$	4.00000	0.146352
$748$	−24.0000	−0.877527
$749$	14.0000	0.511549
$750$	48.0000	1.75271
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	4.00000	0.145865
$753$	4.00000	0.145768
$754$	−20.0000	−0.728357
$755$	6.00000	0.218362
$756$	−16.0000	−0.581914
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	−36.0000	−1.30758
$759$	72.0000	2.61343
$760$	0	0
$761$	47.0000	1.70375	0.851874	−	0.523746i	$-0.175466\pi$
$761$	47.0000	1.70375	0.851874	+	0.523746i	$0.175466\pi$
$762$	−28.0000	−1.01433
$763$	−4.00000	−0.144810
$764$	−12.0000	−0.434145
$765$	6.00000	0.216930
$766$	−32.0000	−1.15621
$767$	18.0000	0.649942
$768$	−32.0000	−1.15470
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	32.0000	1.15320
$771$	2.00000	0.0720282
$772$	−46.0000	−1.65558
$773$	9.00000	0.323708	0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000	0.323708	0.161854	+	0.986815i	$0.448253\pi$
$774$	−4.00000	−0.143777
$775$	10.0000	0.359211
$776$	0	0
$777$	−4.00000	−0.143499
$778$	−20.0000	−0.717035
$779$	0	0
$780$	−16.0000	−0.572892
$781$	0	0
$782$	54.0000	1.93104
$783$	−20.0000	−0.714742
$784$	12.0000	0.428571
$785$	18.0000	0.642448
$786$	48.0000	1.71210
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−4.00000	−0.142494
$789$	−32.0000	−1.13923
$790$	−32.0000	−1.13851
$791$	24.0000	0.853342
$792$	0	0
$793$	−4.00000	−0.142044
$794$	−62.0000	−2.20030
$795$	−40.0000	−1.41865
$796$	14.0000	0.496217
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	56.0000	1.98238
$799$	−3.00000	−0.106132
$800$	8.00000	0.282843
$801$	7.00000	0.247333
$802$	−40.0000	−1.41245
$803$	28.0000	0.988099
$804$	−4.00000	−0.141069
$805$	−36.0000	−1.26883
$806$	−40.0000	−1.40894
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−44.0000	−1.54600
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	20.0000	0.701862
$813$	−16.0000	−0.561144
$814$	8.00000	0.280400
$815$	38.0000	1.33108
$816$	24.0000	0.840168
$817$	−14.0000	−0.489798
$818$	16.0000	0.559427
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	−48.0000	−1.67419
$823$	51.0000	1.77775	0.888874	−	0.458151i	$-0.151488\pi$
$823$	51.0000	1.77775	0.888874	+	0.458151i	$0.151488\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	−36.0000	−1.25260
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	18.0000	0.625543
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	16.0000	0.555368
$831$	4.00000	0.138758
$832$	−16.0000	−0.554700
$833$	−9.00000	−0.311832
$834$	−88.0000	−3.04719
$835$	48.0000	1.66111
$836$	−56.0000	−1.93680
$837$	−40.0000	−1.38260
$838$	−18.0000	−0.621800
$839$	−23.0000	−0.794048	−0.397024	−	0.917808i	$-0.629957\pi$
$839$	−23.0000	−0.794048	−0.397024	+	0.917808i	$0.629957\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−42.0000	−1.44742
$843$	36.0000	1.23991
$844$	−24.0000	−0.826114
$845$	−18.0000	−0.619219
$846$	−2.00000	−0.0687614
$847$	−10.0000	−0.343604
$848$	−40.0000	−1.37361
$849$	6.00000	0.205919
$850$	−6.00000	−0.205798
$851$	−9.00000	−0.308516
$852$	0	0
$853$	29.0000	0.992941	0.496471	−	0.868054i	$-0.334629\pi$
$853$	29.0000	0.992941	0.496471	+	0.868054i	$0.334629\pi$
$854$	8.00000	0.273754
$855$	14.0000	0.478790
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	32.0000	1.09246
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−18.0000	−0.613082
$863$	1.00000	0.0340404	0.0170202	−	0.999855i	$-0.494582\pi$
$863$	1.00000	0.0340404	0.0170202	+	0.999855i	$0.494582\pi$
$864$	−32.0000	−1.08866
$865$	22.0000	0.748022
$866$	−36.0000	−1.22333
$867$	16.0000	0.543388
$868$	40.0000	1.35769
$869$	32.0000	1.08553
$870$	40.0000	1.35613
$871$	2.00000	0.0677674
$872$	0	0
$873$	0	0
$874$	126.000	4.26201
$875$	24.0000	0.811348
$876$	28.0000	0.946032
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	46.0000	1.55242
$879$	−36.0000	−1.21425
$880$	32.0000	1.07872
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	−6.00000	−0.202031
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	12.0000	0.403604
$885$	−36.0000	−1.21013
$886$	4.00000	0.134383
$887$	−31.0000	−1.04088	−0.520439	−	0.853899i	$-0.674232\pi$
$887$	−31.0000	−1.04088	−0.520439	+	0.853899i	$0.674232\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	28.0000	0.938562
$891$	44.0000	1.47406
$892$	22.0000	0.736614
$893$	−7.00000	−0.234246
$894$	−84.0000	−2.80938
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−36.0000	−1.20201
$898$	78.0000	2.60289
$899$	50.0000	1.66759
$900$	−2.00000	−0.0666667
$901$	30.0000	0.999445
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	14.0000	0.465376
$906$	−12.0000	−0.398673
$907$	−43.0000	−1.42779	−0.713896	−	0.700252i	$-0.753071\pi$
$907$	−43.0000	−1.42779	−0.713896	+	0.700252i	$0.753071\pi$
$908$	6.00000	0.199117
$909$	2.00000	0.0663358
$910$	−16.0000	−0.530395
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	56.0000	1.85435
$913$	−16.0000	−0.529523
$914$	58.0000	1.91847
$915$	8.00000	0.264472
$916$	8.00000	0.264327
$917$	24.0000	0.792550
$918$	24.0000	0.792118
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	50.0000	1.64756
$922$	74.0000	2.43706
$923$	0	0
$924$	−32.0000	−1.05272
$925$	1.00000	0.0328798
$926$	−16.0000	−0.525793
$927$	−16.0000	−0.525509
$928$	40.0000	1.31306
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	80.0000	2.62330
$931$	−21.0000	−0.688247
$932$	20.0000	0.655122
$933$	−36.0000	−1.17859
$934$	−24.0000	−0.785304
$935$	−24.0000	−0.784884
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	−4.00000	−0.130605
$939$	32.0000	1.04428
$940$	−4.00000	−0.130466
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	−36.0000	−1.17294
$943$	0	0
$944$	−36.0000	−1.17170
$945$	−16.0000	−0.520480
$946$	16.0000	0.520205
$947$	9.00000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$948$	32.0000	1.03931
$949$	−14.0000	−0.454459
$950$	−14.0000	−0.454220
$951$	36.0000	1.16738
$952$	0	0
$953$	−7.00000	−0.226752	−0.113376	−	0.993552i	$-0.536167\pi$
$953$	−7.00000	−0.226752	−0.113376	+	0.993552i	$0.536167\pi$
$954$	20.0000	0.647524
$955$	−12.0000	−0.388311
$956$	−40.0000	−1.29369
$957$	−40.0000	−1.29302
$958$	−58.0000	−1.87389
$959$	−24.0000	−0.775000
$960$	32.0000	1.03280
$961$	69.0000	2.22581
$962$	−4.00000	−0.128965
$963$	−7.00000	−0.225572
$964$	−38.0000	−1.22390
$965$	−46.0000	−1.48079
$966$	72.0000	2.31656
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	−42.0000	−1.34923
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	20.0000	0.641500
$973$	−44.0000	−1.41058
$974$	4.00000	0.128168
$975$	4.00000	0.128103
$976$	8.00000	0.256074
$977$	−13.0000	−0.415907	−0.207953	−	0.978139i	$-0.566680\pi$
$977$	−13.0000	−0.415907	−0.207953	+	0.978139i	$0.566680\pi$
$978$	−76.0000	−2.43021
$979$	−28.0000	−0.894884
$980$	−12.0000	−0.383326
$981$	2.00000	0.0638551
$982$	−18.0000	−0.574403
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	−30.0000	−0.955395
$987$	−4.00000	−0.127321
$988$	28.0000	0.890799
$989$	−18.0000	−0.572367
$990$	−16.0000	−0.508513
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	80.0000	2.54000
$993$	12.0000	0.380808
$994$	0	0
$995$	14.0000	0.443830
$996$	−16.0000	−0.506979
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	48.0000	1.51941
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.a.a.1.1	✓	1
3.2	odd	2		603.2.a.a.1.1		1
4.3	odd	2		1072.2.a.b.1.1		1
5.2	odd	4		1675.2.c.a.1274.2		2
5.3	odd	4		1675.2.c.a.1274.1		2
5.4	even	2		1675.2.a.a.1.1		1
7.6	odd	2		3283.2.a.e.1.1		1
8.3	odd	2		4288.2.a.a.1.1		1
8.5	even	2		4288.2.a.e.1.1		1
11.10	odd	2		8107.2.a.a.1.1		1
12.11	even	2		9648.2.a.g.1.1		1
67.66	odd	2		4489.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.a.a.1.1	✓	1	1.1	even	1	trivial
603.2.a.a.1.1		1	3.2	odd	2
1072.2.a.b.1.1		1	4.3	odd	2
1675.2.a.a.1.1		1	5.4	even	2
1675.2.c.a.1274.1		2	5.3	odd	4
1675.2.c.a.1274.2		2	5.2	odd	4
3283.2.a.e.1.1		1	7.6	odd	2
4288.2.a.a.1.1		1	8.3	odd	2
4288.2.a.e.1.1		1	8.5	even	2
4489.2.a.a.1.1		1	67.66	odd	2
8107.2.a.a.1.1		1	11.10	odd	2
9648.2.a.g.1.1		1	12.11	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:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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