LMFDB - Embedded newform 363.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.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:	$2.89856959337$
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	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} -8.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -3.00000 q^{19} +8.00000 q^{20} -1.00000 q^{21} +2.00000 q^{23} +11.0000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{30} -5.00000 q^{31} +8.00000 q^{32} -8.00000 q^{34} +4.00000 q^{35} +2.00000 q^{36} +3.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -2.00000 q^{41} +2.00000 q^{42} +12.0000 q^{43} +4.00000 q^{45} -4.00000 q^{46} +2.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} -22.0000 q^{50} -4.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +2.00000 q^{54} +3.00000 q^{57} -12.0000 q^{58} -10.0000 q^{59} -8.00000 q^{60} +3.00000 q^{61} +10.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} -8.00000 q^{65} -1.00000 q^{67} +8.00000 q^{68} -2.00000 q^{69} -8.00000 q^{70} -11.0000 q^{73} -6.00000 q^{74} -11.0000 q^{75} -6.00000 q^{76} -4.00000 q^{78} +11.0000 q^{79} -16.0000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} +16.0000 q^{85} -24.0000 q^{86} -6.00000 q^{87} +12.0000 q^{89} -8.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +5.00000 q^{93} -4.00000 q^{94} -12.0000 q^{95} -8.00000 q^{96} +5.00000 q^{97} +12.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	2.00000	0.816497
$7$	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
$9$	1.00000	0.333333
$10$	−8.00000	−2.52982
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	−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
$15$	−4.00000	−1.03280
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−2.00000	−0.471405
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	8.00000	1.78885
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	8.00000	1.46059
$31$	−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
$34$	−8.00000	−1.37199
$35$	4.00000	0.676123
$36$	2.00000	0.333333
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	6.00000	0.973329
$39$	2.00000	0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	2.00000	0.308607
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	−4.00000	−0.589768
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	4.00000	0.577350
$49$	−6.00000	−0.857143
$50$	−22.0000	−3.11127
$51$	−4.00000	−0.560112
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	2.00000	0.272166
$55$	0	0
$56$	0	0
$57$	3.00000	0.397360
$58$	−12.0000	−1.57568
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−8.00000	−1.03280
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	10.0000	1.27000
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	8.00000	0.970143
$69$	−2.00000	−0.240772
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−6.00000	−0.697486
$75$	−11.0000	−1.27017
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−4.00000	−0.452911
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−16.0000	−1.78885
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−2.00000	−0.218218
$85$	16.0000	1.73544
$86$	−24.0000	−2.58799
$87$	−6.00000	−0.643268
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−8.00000	−0.843274
$91$	−2.00000	−0.209657
$92$	4.00000	0.417029
$93$	5.00000	0.518476
$94$	−4.00000	−0.412568
$95$	−12.0000	−1.23117
$96$	−8.00000	−0.816497
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	12.0000	1.21218
$99$	0	0
$100$	22.0000	2.20000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	8.00000	0.792118
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	−12.0000	−1.16554
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−2.00000	−0.192450
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	−4.00000	−0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−6.00000	−0.561951
$115$	8.00000	0.746004
$116$	12.0000	1.11417
$117$	−2.00000	−0.184900
$118$	20.0000	1.84115
$119$	4.00000	0.366679
$120$	0	0
$121$	0	0
$122$	−6.00000	−0.543214
$123$	2.00000	0.180334
$124$	−10.0000	−0.898027
$125$	24.0000	2.14663
$126$	−2.00000	−0.178174
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	16.0000	1.40329
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	2.00000	0.172774
$135$	−4.00000	−0.344265
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	4.00000	0.340503
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	8.00000	0.676123
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0	0
$144$	−4.00000	−0.333333
$145$	24.0000	1.99309
$146$	22.0000	1.82073
$147$	6.00000	0.494872
$148$	6.00000	0.493197
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	22.0000	1.79629
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−20.0000	−1.60644
$156$	4.00000	0.320256
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	−22.0000	−1.75023
$159$	−6.00000	−0.475831
$160$	32.0000	2.52982
$161$	2.00000	0.157622
$162$	−2.00000	−0.157135
$163$	25.0000	1.95815	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	25.0000	1.95815	0.979076	+	0.203497i	$0.0652307\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−32.0000	−2.45429
$171$	−3.00000	−0.229416
$172$	24.0000	1.82998
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	12.0000	0.909718
$175$	11.0000	0.831522
$176$	0	0
$177$	10.0000	0.751646
$178$	−24.0000	−1.79888
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	8.00000	0.596285
$181$	−23.0000	−1.70958	−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000	−1.70958	−0.854788	+	0.518977i	$0.826313\pi$
$182$	4.00000	0.296500
$183$	−3.00000	−0.221766
$184$	0	0
$185$	12.0000	0.882258
$186$	−10.0000	−0.733236
$187$	0	0
$188$	4.00000	0.291730
$189$	−1.00000	−0.0727393
$190$	24.0000	1.74114
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	8.00000	0.577350
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−10.0000	−0.717958
$195$	8.00000	0.572892
$196$	−12.0000	−0.857143
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	20.0000	1.40720
$203$	6.00000	0.421117
$204$	−8.00000	−0.560112
$205$	−8.00000	−0.558744
$206$	14.0000	0.975426
$207$	2.00000	0.139010
$208$	8.00000	0.554700
$209$	0	0
$210$	8.00000	0.552052
$211$	−21.0000	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	36.0000	2.46091
$215$	48.0000	3.27357
$216$	0	0
$217$	−5.00000	−0.339422
$218$	−2.00000	−0.135457
$219$	11.0000	0.743311
$220$	0	0
$221$	−8.00000	−0.538138
$222$	6.00000	0.402694
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	8.00000	0.534522
$225$	11.0000	0.733333
$226$	−12.0000	−0.798228
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	6.00000	0.397360
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	4.00000	0.261488
$235$	8.00000	0.521862
$236$	−20.0000	−1.30189
$237$	−11.0000	−0.714527
$238$	−8.00000	−0.518563
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	16.0000	1.03280
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	−24.0000	−1.53330
$246$	−4.00000	−0.255031
$247$	6.00000	0.381771
$248$	0	0
$249$	−6.00000	−0.380235
$250$	−48.0000	−3.03579
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	26.0000	1.63139
$255$	−16.0000	−1.00196
$256$	16.0000	1.00000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	24.0000	1.49417
$259$	3.00000	0.186411
$260$	−16.0000	−0.992278
$261$	6.00000	0.371391
$262$	12.0000	0.741362
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	6.00000	0.367884
$267$	−12.0000	−0.734388
$268$	−2.00000	−0.122169
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	8.00000	0.486864
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−16.0000	−0.970143
$273$	2.00000	0.121046
$274$	−16.0000	−0.966595
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	−32.0000	−1.91923
$279$	−5.00000	−0.299342
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	4.00000	0.238197
$283$	−11.0000	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$284$	0	0
$285$	12.0000	0.710819
$286$	0	0
$287$	−2.00000	−0.118056
$288$	8.00000	0.471405
$289$	−1.00000	−0.0588235
$290$	−48.0000	−2.81866
$291$	−5.00000	−0.293105
$292$	−22.0000	−1.28745
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	−12.0000	−0.699854
$295$	−40.0000	−2.32889
$296$	0	0
$297$	0	0
$298$	32.0000	1.85371
$299$	−4.00000	−0.231326
$300$	−22.0000	−1.27017
$301$	12.0000	0.691669
$302$	32.0000	1.84139
$303$	10.0000	0.574485
$304$	12.0000	0.688247
$305$	12.0000	0.687118
$306$	−8.00000	−0.457330
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	40.0000	2.27185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	2.00000	0.112867
$315$	4.00000	0.225374
$316$	22.0000	1.23760
$317$	−20.0000	−1.12331	−0.561656	−	0.827371i	$-0.689836\pi$
$317$	−20.0000	−1.12331	−0.561656	+	0.827371i	$0.689836\pi$
$318$	12.0000	0.672927
$319$	0	0
$320$	−32.0000	−1.78885
$321$	18.0000	1.00466
$322$	−4.00000	−0.222911
$323$	−12.0000	−0.667698
$324$	2.00000	0.111111
$325$	−22.0000	−1.22034
$326$	−50.0000	−2.76924
$327$	−1.00000	−0.0553001
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	12.0000	0.658586
$333$	3.00000	0.164399
$334$	36.0000	1.96983
$335$	−4.00000	−0.218543
$336$	4.00000	0.218218
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	18.0000	0.979071
$339$	−6.00000	−0.325875
$340$	32.0000	1.73544
$341$	0	0
$342$	6.00000	0.324443
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−8.00000	−0.430706
$346$	−48.0000	−2.58050
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	−12.0000	−0.643268
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	−22.0000	−1.17595
$351$	2.00000	0.106752
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	−20.0000	−1.06299
$355$	0	0
$356$	24.0000	1.27200
$357$	−4.00000	−0.211702
$358$	−12.0000	−0.634220
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	46.0000	2.41771
$363$	0	0
$364$	−4.00000	−0.209657
$365$	−44.0000	−2.30307
$366$	6.00000	0.313625
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−8.00000	−0.417029
$369$	−2.00000	−0.104116
$370$	−24.0000	−1.24770
$371$	6.00000	0.311504
$372$	10.0000	0.518476
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	−12.0000	−0.618031
$378$	2.00000	0.102869
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−24.0000	−1.23117
$381$	13.0000	0.666010
$382$	−16.0000	−0.818631
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	12.0000	0.609994
$388$	10.0000	0.507673
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−16.0000	−0.810191
$391$	8.00000	0.404577
$392$	0	0
$393$	6.00000	0.302660
$394$	16.0000	0.806068
$395$	44.0000	2.21388
$396$	0	0
$397$	31.0000	1.55585	0.777923	−	0.628360i	$-0.216273\pi$
$397$	31.0000	1.55585	0.777923	+	0.628360i	$0.216273\pi$
$398$	42.0000	2.10527
$399$	3.00000	0.150188
$400$	−44.0000	−2.20000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−2.00000	−0.0997509
$403$	10.0000	0.498135
$404$	−20.0000	−0.995037
$405$	4.00000	0.198762
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	21.0000	1.03838	0.519192	−	0.854658i	$-0.326233\pi$
$409$	21.0000	1.03838	0.519192	+	0.854658i	$0.326233\pi$
$410$	16.0000	0.790184
$411$	−8.00000	−0.394611
$412$	−14.0000	−0.689730
$413$	−10.0000	−0.492068
$414$	−4.00000	−0.196589
$415$	24.0000	1.17811
$416$	−16.0000	−0.784465
$417$	−16.0000	−0.783523
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	−8.00000	−0.390360
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	42.0000	2.04453
$423$	2.00000	0.0972433
$424$	0	0
$425$	44.0000	2.13431
$426$	0	0
$427$	3.00000	0.145180
$428$	−36.0000	−1.74013
$429$	0	0
$430$	−96.0000	−4.62953
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	4.00000	0.192450
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	10.0000	0.480015
$435$	−24.0000	−1.15071
$436$	2.00000	0.0957826
$437$	−6.00000	−0.287019
$438$	−22.0000	−1.05120
$439$	37.0000	1.76591	0.882957	−	0.469454i	$-0.155549\pi$
$439$	37.0000	1.76591	0.882957	+	0.469454i	$0.155549\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	16.0000	0.761042
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−6.00000	−0.284747
$445$	48.0000	2.27542
$446$	34.0000	1.60995
$447$	16.0000	0.756774
$448$	−8.00000	−0.377964
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	−22.0000	−1.03709
$451$	0	0
$452$	12.0000	0.564433
$453$	16.0000	0.751746
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	36.0000	1.68217
$459$	−4.00000	−0.186704
$460$	16.0000	0.746004
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−24.0000	−1.11417
$465$	20.0000	0.927478
$466$	−36.0000	−1.66767
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−4.00000	−0.184900
$469$	−1.00000	−0.0461757
$470$	−16.0000	−0.738025
$471$	1.00000	0.0460776
$472$	0	0
$473$	0	0
$474$	22.0000	1.01049
$475$	−33.0000	−1.51414
$476$	8.00000	0.366679
$477$	6.00000	0.274721
$478$	−12.0000	−0.548867
$479$	−22.0000	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$479$	−22.0000	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$480$	−32.0000	−1.46059
$481$	−6.00000	−0.273576
$482$	28.0000	1.27537
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	20.0000	0.908153
$486$	2.00000	0.0907218
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	0	0
$489$	−25.0000	−1.13054
$490$	48.0000	2.16842
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	4.00000	0.180334
$493$	24.0000	1.08091
$494$	−12.0000	−0.539906
$495$	0	0
$496$	20.0000	0.898027
$497$	0	0
$498$	12.0000	0.537733
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	48.0000	2.14663
$501$	18.0000	0.804181
$502$	4.00000	0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	9.00000	0.399704
$508$	−26.0000	−1.15356
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	32.0000	1.41698
$511$	−11.0000	−0.486611
$512$	−32.0000	−1.41421
$513$	3.00000	0.132453
$514$	28.0000	1.23503
$515$	−28.0000	−1.23383
$516$	−24.0000	−1.05654
$517$	0	0
$518$	−6.00000	−0.263625
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−12.0000	−0.525226
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	−12.0000	−0.524222
$525$	−11.0000	−0.480079
$526$	20.0000	0.872041
$527$	−20.0000	−0.871214
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−48.0000	−2.08499
$531$	−10.0000	−0.433963
$532$	−6.00000	−0.260133
$533$	4.00000	0.173259
$534$	24.0000	1.03858
$535$	−72.0000	−3.11283
$536$	0	0
$537$	−6.00000	−0.258919
$538$	28.0000	1.20717
$539$	0	0
$540$	−8.00000	−0.344265
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−16.0000	−0.687259
$543$	23.0000	0.987024
$544$	32.0000	1.37199
$545$	4.00000	0.171341
$546$	−4.00000	−0.171184
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	16.0000	0.683486
$549$	3.00000	0.128037
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	11.0000	0.467768
$554$	22.0000	0.934690
$555$	−12.0000	−0.509372
$556$	32.0000	1.35710
$557$	−8.00000	−0.338971	−0.169485	−	0.985533i	$-0.554211\pi$
$557$	−8.00000	−0.338971	−0.169485	+	0.985533i	$0.554211\pi$
$558$	10.0000	0.423334
$559$	−24.0000	−1.01509
$560$	−16.0000	−0.676123
$561$	0	0
$562$	24.0000	1.01238
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	−4.00000	−0.168430
$565$	24.0000	1.00969
$566$	22.0000	0.924729
$567$	1.00000	0.0419961
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	−24.0000	−1.00525
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	4.00000	0.166957
$575$	22.0000	0.917463
$576$	−8.00000	−0.333333
$577$	15.0000	0.624458	0.312229	−	0.950007i	$-0.398924\pi$
$577$	15.0000	0.624458	0.312229	+	0.950007i	$0.398924\pi$
$578$	2.00000	0.0831890
$579$	5.00000	0.207793
$580$	48.0000	1.99309
$581$	6.00000	0.248922
$582$	10.0000	0.414513
$583$	0	0
$584$	0	0
$585$	−8.00000	−0.330759
$586$	−24.0000	−0.991431
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	12.0000	0.494872
$589$	15.0000	0.618064
$590$	80.0000	3.29355
$591$	8.00000	0.329076
$592$	−12.0000	−0.493197
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	−32.0000	−1.31077
$597$	21.0000	0.859473
$598$	8.00000	0.327144
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	−24.0000	−0.978167
$603$	−1.00000	−0.0407231
$604$	−32.0000	−1.30206
$605$	0	0
$606$	−20.0000	−0.812444
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−24.0000	−0.973329
$609$	−6.00000	−0.243132
$610$	−24.0000	−0.971732
$611$	−4.00000	−0.161823
$612$	8.00000	0.323381
$613$	−13.0000	−0.525065	−0.262533	−	0.964923i	$-0.584558\pi$
$613$	−13.0000	−0.525065	−0.262533	+	0.964923i	$0.584558\pi$
$614$	38.0000	1.53356
$615$	8.00000	0.322591
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	−14.0000	−0.563163
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−40.0000	−1.60644
$621$	−2.00000	−0.0802572
$622$	−48.0000	−1.92462
$623$	12.0000	0.480770
$624$	−8.00000	−0.320256
$625$	41.0000	1.64000
$626$	20.0000	0.799361
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	12.0000	0.478471
$630$	−8.00000	−0.318728
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	21.0000	0.834675
$634$	40.0000	1.58860
$635$	−52.0000	−2.06356
$636$	−12.0000	−0.475831
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	−36.0000	−1.42081
$643$	−37.0000	−1.45914	−0.729569	−	0.683907i	$-0.760279\pi$
$643$	−37.0000	−1.45914	−0.729569	+	0.683907i	$0.760279\pi$
$644$	4.00000	0.157622
$645$	−48.0000	−1.89000
$646$	24.0000	0.944267
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	0	0
$649$	0	0
$650$	44.0000	1.72582
$651$	5.00000	0.195965
$652$	50.0000	1.95815
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	2.00000	0.0782062
$655$	−24.0000	−0.937758
$656$	8.00000	0.312348
$657$	−11.0000	−0.429151
$658$	−4.00000	−0.155936
$659$	46.0000	1.79191	0.895953	−	0.444149i	$-0.146494\pi$
$659$	46.0000	1.79191	0.895953	+	0.444149i	$0.146494\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	22.0000	0.855054
$663$	8.00000	0.310694
$664$	0	0
$665$	−12.0000	−0.465340
$666$	−6.00000	−0.232495
$667$	12.0000	0.464642
$668$	−36.0000	−1.39288
$669$	17.0000	0.657258
$670$	8.00000	0.309067
$671$	0	0
$672$	−8.00000	−0.308607
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−10.0000	−0.385186
$675$	−11.0000	−0.423390
$676$	−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	$0.425932\pi$
$678$	12.0000	0.460857
$679$	5.00000	0.191882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.0000	−1.30097	−0.650487	−	0.759517i	$-0.725435\pi$
$683$	−34.0000	−1.30097	−0.650487	+	0.759517i	$0.725435\pi$
$684$	−6.00000	−0.229416
$685$	32.0000	1.22266
$686$	26.0000	0.992685
$687$	18.0000	0.686743
$688$	−48.0000	−1.82998
$689$	−12.0000	−0.457164
$690$	16.0000	0.609110
$691$	11.0000	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$692$	48.0000	1.82469
$693$	0	0
$694$	4.00000	0.151838
$695$	64.0000	2.42766
$696$	0	0
$697$	−8.00000	−0.303022
$698$	−30.0000	−1.13552
$699$	−18.0000	−0.680823
$700$	22.0000	0.831522
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	−4.00000	−0.150970
$703$	−9.00000	−0.339441
$704$	0	0
$705$	−8.00000	−0.301297
$706$	24.0000	0.903252
$707$	−10.0000	−0.376089
$708$	20.0000	0.751646
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−10.0000	−0.374503
$714$	8.00000	0.299392
$715$	0	0
$716$	12.0000	0.448461
$717$	−6.00000	−0.224074
$718$	−8.00000	−0.298557
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−16.0000	−0.596285
$721$	−7.00000	−0.260694
$722$	20.0000	0.744323
$723$	14.0000	0.520666
$724$	−46.0000	−1.70958
$725$	66.0000	2.45118
$726$	0	0
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	88.0000	3.25703
$731$	48.0000	1.77534
$732$	−6.00000	−0.221766
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	16.0000	0.590571
$735$	24.0000	0.885253
$736$	16.0000	0.589768
$737$	0	0
$738$	4.00000	0.147242
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	24.0000	0.882258
$741$	−6.00000	−0.220416
$742$	−12.0000	−0.440534
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	0	0
$745$	−64.0000	−2.34478
$746$	−14.0000	−0.512576
$747$	6.00000	0.219529
$748$	0	0
$749$	−18.0000	−0.657706
$750$	48.0000	1.75271
$751$	19.0000	0.693320	0.346660	−	0.937991i	$-0.387316\pi$
$751$	19.0000	0.693320	0.346660	+	0.937991i	$0.387316\pi$
$752$	−8.00000	−0.291730
$753$	2.00000	0.0728841
$754$	24.0000	0.874028
$755$	−64.0000	−2.32920
$756$	−2.00000	−0.0727393
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	−26.0000	−0.941881
$763$	1.00000	0.0362024
$764$	16.0000	0.578860
$765$	16.0000	0.578481
$766$	−52.0000	−1.87884
$767$	20.0000	0.722158
$768$	−16.0000	−0.577350
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−10.0000	−0.359908
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	−24.0000	−0.862662
$775$	−55.0000	−1.97566
$776$	0	0
$777$	−3.00000	−0.107624
$778$	−36.0000	−1.29066
$779$	6.00000	0.214972
$780$	16.0000	0.572892
$781$	0	0
$782$	−16.0000	−0.572159
$783$	−6.00000	−0.214423
$784$	24.0000	0.857143
$785$	−4.00000	−0.142766
$786$	−12.0000	−0.428026
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−16.0000	−0.569976
$789$	10.0000	0.356009
$790$	−88.0000	−3.13090
$791$	6.00000	0.213335
$792$	0	0
$793$	−6.00000	−0.213066
$794$	−62.0000	−2.20030
$795$	−24.0000	−0.851192
$796$	−42.0000	−1.48865
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	−6.00000	−0.212398
$799$	8.00000	0.283020
$800$	88.0000	3.11127
$801$	12.0000	0.423999
$802$	56.0000	1.97743
$803$	0	0
$804$	2.00000	0.0705346
$805$	8.00000	0.281963
$806$	−20.0000	−0.704470
$807$	14.0000	0.492823
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	−8.00000	−0.281091
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	12.0000	0.421117
$813$	−8.00000	−0.280572
$814$	0	0
$815$	100.000	3.50285
$816$	16.0000	0.560112
$817$	−36.0000	−1.25948
$818$	−42.0000	−1.46850
$819$	−2.00000	−0.0698857
$820$	−16.0000	−0.558744
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	16.0000	0.558064
$823$	−27.0000	−0.941161	−0.470580	−	0.882357i	$-0.655955\pi$
$823$	−27.0000	−0.941161	−0.470580	+	0.882357i	$0.655955\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	4.00000	0.139010
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	−48.0000	−1.66610
$831$	11.0000	0.381586
$832$	16.0000	0.554700
$833$	−24.0000	−0.831551
$834$	32.0000	1.10807
$835$	−72.0000	−2.49166
$836$	0	0
$837$	5.00000	0.172825
$838$	−52.0000	−1.79631
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	4.00000	0.137849
$843$	12.0000	0.413302
$844$	−42.0000	−1.44570
$845$	−36.0000	−1.23844
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−24.0000	−0.824163
$849$	11.0000	0.377519
$850$	−88.0000	−3.01838
$851$	6.00000	0.205677
$852$	0	0
$853$	11.0000	0.376633	0.188316	−	0.982108i	$-0.439697\pi$
$853$	11.0000	0.376633	0.188316	+	0.982108i	$0.439697\pi$
$854$	−6.00000	−0.205316
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	−45.0000	−1.53538	−0.767690	−	0.640821i	$-0.778594\pi$
$859$	−45.0000	−1.53538	−0.767690	+	0.640821i	$0.778594\pi$
$860$	96.0000	3.27357
$861$	2.00000	0.0681598
$862$	−36.0000	−1.22616
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	−8.00000	−0.272166
$865$	96.0000	3.26410
$866$	34.0000	1.15537
$867$	1.00000	0.0339618
$868$	−10.0000	−0.339422
$869$	0	0
$870$	48.0000	1.62735
$871$	2.00000	0.0677674
$872$	0	0
$873$	5.00000	0.169224
$874$	12.0000	0.405906
$875$	24.0000	0.811348
$876$	22.0000	0.743311
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	−74.0000	−2.49738
$879$	−12.0000	−0.404750
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	12.0000	0.404061
$883$	49.0000	1.64898	0.824491	−	0.565876i	$-0.191462\pi$
$883$	49.0000	1.64898	0.824491	+	0.565876i	$0.191462\pi$
$884$	−16.0000	−0.538138
$885$	40.0000	1.34459
$886$	−8.00000	−0.268765
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	−96.0000	−3.21793
$891$	0	0
$892$	−34.0000	−1.13840
$893$	−6.00000	−0.200782
$894$	−32.0000	−1.07024
$895$	24.0000	0.802232
$896$	0	0
$897$	4.00000	0.133556
$898$	−40.0000	−1.33482
$899$	−30.0000	−1.00056
$900$	22.0000	0.733333
$901$	24.0000	0.799556
$902$	0	0
$903$	−12.0000	−0.399335
$904$	0	0
$905$	−92.0000	−3.05818
$906$	−32.0000	−1.06313
$907$	33.0000	1.09575	0.547874	−	0.836561i	$-0.315438\pi$
$907$	33.0000	1.09575	0.547874	+	0.836561i	$0.315438\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	16.0000	0.530395
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	−12.0000	−0.397360
$913$	0	0
$914$	−36.0000	−1.19077
$915$	−12.0000	−0.396708
$916$	−36.0000	−1.18947
$917$	−6.00000	−0.198137
$918$	8.00000	0.264039
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	19.0000	0.626071
$922$	12.0000	0.395199
$923$	0	0
$924$	0	0
$925$	33.0000	1.08503
$926$	−32.0000	−1.05159
$927$	−7.00000	−0.229910
$928$	48.0000	1.57568
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	−40.0000	−1.31165
$931$	18.0000	0.589926
$932$	36.0000	1.17922
$933$	−24.0000	−0.785725
$934$	−48.0000	−1.57061
$935$	0	0
$936$	0	0
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	2.00000	0.0653023
$939$	10.0000	0.326338
$940$	16.0000	0.521862
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	−2.00000	−0.0651635
$943$	−4.00000	−0.130258
$944$	40.0000	1.30189
$945$	−4.00000	−0.130120
$946$	0	0
$947$	54.0000	1.75476	0.877382	−	0.479792i	$-0.159288\pi$
$947$	54.0000	1.75476	0.877382	+	0.479792i	$0.159288\pi$
$948$	−22.0000	−0.714527
$949$	22.0000	0.714150
$950$	66.0000	2.14132
$951$	20.0000	0.648544
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	−12.0000	−0.388514
$955$	32.0000	1.03550
$956$	12.0000	0.388108
$957$	0	0
$958$	44.0000	1.42158
$959$	8.00000	0.258333
$960$	32.0000	1.03280
$961$	−6.00000	−0.193548
$962$	12.0000	0.386896
$963$	−18.0000	−0.580042
$964$	−28.0000	−0.901819
$965$	−20.0000	−0.643823
$966$	4.00000	0.128698
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	−40.0000	−1.28432
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	−2.00000	−0.0641500
$973$	16.0000	0.512936
$974$	80.0000	2.56337
$975$	22.0000	0.704564
$976$	−12.0000	−0.384111
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	50.0000	1.59882
$979$	0	0
$980$	−48.0000	−1.53330
$981$	1.00000	0.0319275
$982$	28.0000	0.893516
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−32.0000	−1.01960
$986$	−48.0000	−1.52863
$987$	−2.00000	−0.0636607
$988$	12.0000	0.381771
$989$	24.0000	0.763156
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−40.0000	−1.27000
$993$	11.0000	0.349074
$994$	0	0
$995$	−84.0000	−2.66298
$996$	−12.0000	−0.380235
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	−46.0000	−1.45610
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.a.a.1.1	✓	1
3.2	odd	2		1089.2.a.k.1.1		1
4.3	odd	2		5808.2.a.bh.1.1		1
5.4	even	2		9075.2.a.t.1.1		1
11.2	odd	10		363.2.e.d.202.1		4
11.3	even	5		363.2.e.i.130.1		4
11.4	even	5		363.2.e.i.148.1		4
11.5	even	5		363.2.e.i.124.1		4
11.6	odd	10		363.2.e.d.124.1		4
11.7	odd	10		363.2.e.d.148.1		4
11.8	odd	10		363.2.e.d.130.1		4
11.9	even	5		363.2.e.i.202.1		4
11.10	odd	2		363.2.a.c.1.1	yes	1
33.32	even	2		1089.2.a.a.1.1		1
44.43	even	2		5808.2.a.bi.1.1		1
55.54	odd	2		9075.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.a.1.1	✓	1	1.1	even	1	trivial
363.2.a.c.1.1	yes	1	11.10	odd	2
363.2.e.d.124.1		4	11.6	odd	10
363.2.e.d.130.1		4	11.8	odd	10
363.2.e.d.148.1		4	11.7	odd	10
363.2.e.d.202.1		4	11.2	odd	10
363.2.e.i.124.1		4	11.5	even	5
363.2.e.i.130.1		4	11.3	even	5
363.2.e.i.148.1		4	11.4	even	5
363.2.e.i.202.1		4	11.9	even	5
1089.2.a.a.1.1		1	33.32	even	2
1089.2.a.k.1.1		1	3.2	odd	2
5808.2.a.bh.1.1		1	4.3	odd	2
5808.2.a.bi.1.1		1	44.43	even	2
9075.2.a.b.1.1		1	55.54	odd	2
9075.2.a.t.1.1		1	5.4	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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