LMFDB - Embedded newform 145.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [145,2,Mod(1,145)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(145, 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("145.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 145 = 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	145.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:	$1.15783082931$
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	$\chi$	$=$	145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} +2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +6.00000 q^{22} +2.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -1.00000 q^{29} +2.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{35} +3.00000 q^{36} +10.0000 q^{37} +2.00000 q^{38} -3.00000 q^{40} +2.00000 q^{41} +8.00000 q^{43} +6.00000 q^{44} +3.00000 q^{45} -2.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +6.00000 q^{55} -6.00000 q^{56} +1.00000 q^{58} -8.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} -12.0000 q^{71} -9.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} +2.00000 q^{76} +12.0000 q^{77} -10.0000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} -14.0000 q^{83} +2.00000 q^{85} -8.00000 q^{86} -18.0000 q^{88} +18.0000 q^{89} -3.00000 q^{90} -4.00000 q^{91} -2.00000 q^{92} +12.0000 q^{94} +2.00000 q^{95} +2.00000 q^{97} +3.00000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	3.00000	1.06066
$9$	−3.00000	−1.00000
$10$	1.00000	0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	3.00000	0.707107
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	6.00000	1.27920
$23$	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
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	2.00000	0.377964
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	2.00000	0.342997
$35$	2.00000	0.338062
$36$	3.00000	0.500000
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−3.00000	−0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	6.00000	0.904534
$45$	3.00000	0.447214
$46$	−2.00000	−0.294884
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−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
$55$	6.00000	0.809040
$56$	−6.00000	−0.801784
$57$	0	0
$58$	1.00000	0.131306
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	6.00000	0.755929
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−9.00000	−1.06066
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	12.0000	1.36753
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	1.00000	0.111803
$81$	9.00000	1.00000
$82$	−2.00000	−0.220863
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−18.0000	−1.91881
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	−3.00000	−0.316228
$91$	−4.00000	−0.419314
$92$	−2.00000	−0.208514
$93$	0	0
$94$	12.0000	1.23771
$95$	2.00000	0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	3.00000	0.303046
$99$	18.0000	1.80907
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	2.00000	0.188982
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	1.00000	0.0928477
$117$	−6.00000	−0.554700
$118$	8.00000	0.736460
$119$	4.00000	0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	6.00000	0.543214
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	−6.00000	−0.534522
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	2.00000	0.175412
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	12.0000	1.00702
$143$	−12.0000	−1.00349
$144$	3.00000	0.250000
$145$	1.00000	0.0830455
$146$	6.00000	0.496564
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−6.00000	−0.486664
$153$	6.00000	0.485071
$154$	−12.0000	−0.966988
$155$	−2.00000	−0.160644
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	5.00000	0.395285
$161$	−4.00000	−0.315244
$162$	−9.00000	−0.707107
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	14.0000	1.08661
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	6.00000	0.458831
$172$	−8.00000	−0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	6.00000	0.452267
$177$	0	0
$178$	−18.0000	−1.34916
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−3.00000	−0.223607
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	6.00000	0.442326
$185$	−10.0000	−0.735215
$186$	0	0
$187$	12.0000	0.877527
$188$	12.0000	0.875190
$189$	0	0
$190$	−2.00000	−0.145095
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−18.0000	−1.27920
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−10.0000	−0.703598
$203$	2.00000	0.140372
$204$	0	0
$205$	−2.00000	−0.139686
$206$	6.00000	0.418040
$207$	−6.00000	−0.417029
$208$	−2.00000	−0.138675
$209$	12.0000	0.830057
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−4.00000	−0.271538
$218$	14.0000	0.948200
$219$	0	0
$220$	−6.00000	−0.404520
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	10.0000	0.668153
$225$	−3.00000	−0.200000
$226$	−2.00000	−0.133038
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−3.00000	−0.196960
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	6.00000	0.392232
$235$	12.0000	0.782794
$236$	8.00000	0.520756
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−25.0000	−1.60706
$243$	0	0
$244$	6.00000	0.384111
$245$	3.00000	0.191663
$246$	0	0
$247$	−4.00000	−0.254514
$248$	6.00000	0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	−6.00000	−0.377964
$253$	−12.0000	−0.754434
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	2.00000	0.124035
$261$	3.00000	0.185695
$262$	−14.0000	−0.864923
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−4.00000	−0.245256
$267$	0	0
$268$	−2.00000	−0.122169
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−6.00000	−0.361814
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	6.00000	0.358569
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	12.0000	0.709575
$287$	−4.00000	−0.236113
$288$	15.0000	0.883883
$289$	−13.0000	−0.764706
$290$	−1.00000	−0.0587220
$291$	0	0
$292$	6.00000	0.351123
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	30.0000	1.74371
$297$	0	0
$298$	10.0000	0.579284
$299$	4.00000	0.231326
$300$	0	0
$301$	−16.0000	−0.922225
$302$	4.00000	0.230174
$303$	0	0
$304$	2.00000	0.114708
$305$	6.00000	0.343559
$306$	−6.00000	−0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	−12.0000	−0.683763
$309$	0	0
$310$	2.00000	0.113592
$311$	−22.0000	−1.24751	−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000	−1.24751	−0.623753	+	0.781622i	$0.714393\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−22.0000	−1.24153
$315$	−6.00000	−0.338062
$316$	10.0000	0.562544
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	−7.00000	−0.391312
$321$	0	0
$322$	4.00000	0.222911
$323$	4.00000	0.222566
$324$	−9.00000	−0.500000
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	0	0
$328$	6.00000	0.331295
$329$	24.0000	1.32316
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	14.0000	0.768350
$333$	−30.0000	−1.64399
$334$	−18.0000	−0.984916
$335$	−2.00000	−0.109272
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−2.00000	−0.108465
$341$	−12.0000	−0.649836
$342$	−6.00000	−0.324443
$343$	20.0000	1.07990
$344$	24.0000	1.29399
$345$	0	0
$346$	14.0000	0.752645
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	30.0000	1.59901
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−18.0000	−0.953998
$357$	0	0
$358$	12.0000	0.634220
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	9.00000	0.474342
$361$	−15.0000	−0.789474
$362$	−6.00000	−0.315353
$363$	0	0
$364$	4.00000	0.209657
$365$	6.00000	0.314054
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	−2.00000	−0.104257
$369$	−6.00000	−0.312348
$370$	10.0000	0.519875
$371$	12.0000	0.623009
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	−36.0000	−1.85656
$377$	−2.00000	−0.103005
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	22.0000	1.12562
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	10.0000	0.508987
$387$	−24.0000	−1.21999
$388$	−2.00000	−0.101535
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−2.00000	−0.100759
$395$	10.0000	0.503155
$396$	−18.0000	−0.904534
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	−10.0000	−0.497519
$405$	−9.00000	−0.447214
$406$	−2.00000	−0.0992583
$407$	−60.0000	−2.97409
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	6.00000	0.295599
$413$	16.0000	0.787309
$414$	6.00000	0.294884
$415$	14.0000	0.687233
$416$	−10.0000	−0.490290
$417$	0	0
$418$	−12.0000	−0.586939
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−14.0000	−0.681509
$423$	36.0000	1.75038
$424$	−18.0000	−0.874157
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	12.0000	0.580721
$428$	−6.00000	−0.290021
$429$	0	0
$430$	8.00000	0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	−4.00000	−0.191346
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	18.0000	0.858116
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	14.0000	0.662919
$447$	0	0
$448$	−14.0000	−0.661438
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	3.00000	0.141421
$451$	−12.0000	−0.565058
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	−22.0000	−1.03251
$455$	4.00000	0.187523
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	2.00000	0.0932505
$461$	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
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−18.0000	−0.833834
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	6.00000	0.277350
$469$	−4.00000	−0.184703
$470$	−12.0000	−0.553519
$471$	0	0
$472$	−24.0000	−1.10469
$473$	−48.0000	−2.20704
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	−4.00000	−0.183340
$477$	18.0000	0.824163
$478$	12.0000	0.548867
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	26.0000	1.18427
$483$	0	0
$484$	−25.0000	−1.13636
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	−18.0000	−0.814822
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	4.00000	0.179969
$495$	−18.0000	−0.809040
$496$	−2.00000	−0.0898027
$497$	24.0000	1.07655
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	6.00000	0.267793
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	18.0000	0.801784
$505$	−10.0000	−0.444994
$506$	12.0000	0.533465
$507$	0	0
$508$	16.0000	0.709885
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	11.0000	0.486136
$513$	0	0
$514$	30.0000	1.32324
$515$	6.00000	0.264392
$516$	0	0
$517$	72.0000	3.16656
$518$	20.0000	0.878750
$519$	0	0
$520$	−6.00000	−0.263117
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−3.00000	−0.131306
$523$	42.0000	1.83653	0.918266	−	0.395964i	$-0.129590\pi$
$523$	42.0000	1.83653	0.918266	+	0.395964i	$0.129590\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−6.00000	−0.260623
$531$	24.0000	1.04151
$532$	−4.00000	−0.173422
$533$	4.00000	0.173259
$534$	0	0
$535$	−6.00000	−0.259403
$536$	6.00000	0.259161
$537$	0	0
$538$	−26.0000	−1.12094
$539$	18.0000	0.775315
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	10.0000	0.428746
$545$	14.0000	0.599694
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−6.00000	−0.256307
$549$	18.0000	0.768221
$550$	6.00000	0.255841
$551$	2.00000	0.0852029
$552$	0	0
$553$	20.0000	0.850487
$554$	−18.0000	−0.764747
$555$	0	0
$556$	0	0
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	6.00000	0.254000
$559$	16.0000	0.676728
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−22.0000	−0.928014
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	−22.0000	−0.924729
$567$	−18.0000	−0.755929
$568$	−36.0000	−1.51053
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	4.00000	0.166957
$575$	2.00000	0.0834058
$576$	−21.0000	−0.875000
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−1.00000	−0.0415227
$581$	28.0000	1.16164
$582$	0	0
$583$	36.0000	1.49097
$584$	−18.0000	−0.744845
$585$	6.00000	0.248069
$586$	2.00000	0.0826192
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−8.00000	−0.329355
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	10.0000	0.409616
$597$	0	0
$598$	−4.00000	−0.163572
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	16.0000	0.652111
$603$	−6.00000	−0.244339
$604$	4.00000	0.162758
$605$	−25.0000	−1.01639
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	10.0000	0.405554
$609$	0	0
$610$	−6.00000	−0.242933
$611$	−24.0000	−0.970936
$612$	−6.00000	−0.242536
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	36.0000	1.45048
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	22.0000	0.882120
$623$	−36.0000	−1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−20.0000	−0.797452
$630$	6.00000	0.239046
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−30.0000	−1.19334
$633$	0	0
$634$	14.0000	0.556011
$635$	16.0000	0.634941
$636$	0	0
$637$	−6.00000	−0.237729
$638$	−6.00000	−0.237542
$639$	36.0000	1.42414
$640$	−3.00000	−0.118585
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	27.0000	1.06066
$649$	48.0000	1.88416
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	0	0
$655$	−14.0000	−0.547025
$656$	−2.00000	−0.0780869
$657$	18.0000	0.702247
$658$	−24.0000	−0.935617
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	−42.0000	−1.62992
$665$	−4.00000	−0.155113
$666$	30.0000	1.16248
$667$	−2.00000	−0.0774403
$668$	−18.0000	−0.696441
$669$	0	0
$670$	2.00000	0.0772667
$671$	36.0000	1.38976
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	9.00000	0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	6.00000	0.230089
$681$	0	0
$682$	12.0000	0.459504
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−6.00000	−0.229416
$685$	−6.00000	−0.229248
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000	0.532200
$693$	−36.0000	−1.36753
$694$	6.00000	0.227757
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	−34.0000	−1.28692
$699$	0	0
$700$	2.00000	0.0755929
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	−42.0000	−1.58293
$705$	0	0
$706$	14.0000	0.526897
$707$	−20.0000	−0.752177
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	−12.0000	−0.450352
$711$	30.0000	1.12509
$712$	54.0000	2.02374
$713$	4.00000	0.149801
$714$	0	0
$715$	12.0000	0.448775
$716$	12.0000	0.448461
$717$	0	0
$718$	−22.0000	−0.821033
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−3.00000	−0.111803
$721$	12.0000	0.446903
$722$	15.0000	0.558242
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−12.0000	−0.444750
$729$	−27.0000	−1.00000
$730$	−6.00000	−0.222070
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	−10.0000	−0.368605
$737$	−12.0000	−0.442026
$738$	6.00000	0.220863
$739$	46.0000	1.69214	0.846069	−	0.533074i	$-0.178963\pi$
$739$	46.0000	1.69214	0.846069	+	0.533074i	$0.178963\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	−12.0000	−0.440534
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	−18.0000	−0.659027
$747$	42.0000	1.53670
$748$	−12.0000	−0.438763
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	2.00000	0.0728357
$755$	4.00000	0.145575
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−6.00000	−0.217930
$759$	0	0
$760$	6.00000	0.217643
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	28.0000	1.01367
$764$	22.0000	0.795932
$765$	−6.00000	−0.216930
$766$	−14.0000	−0.505841
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	10.0000	0.359908
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	24.0000	0.862662
$775$	2.00000	0.0718421
$776$	6.00000	0.215387
$777$	0	0
$778$	14.0000	0.501924
$779$	−4.00000	−0.143315
$780$	0	0
$781$	72.0000	2.57636
$782$	4.00000	0.143040
$783$	0	0
$784$	3.00000	0.107143
$785$	−22.0000	−0.785214
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	−10.0000	−0.355784
$791$	−4.00000	−0.142224
$792$	54.0000	1.91881
$793$	−12.0000	−0.426132
$794$	30.0000	1.06466
$795$	0	0
$796$	4.00000	0.141776
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−5.00000	−0.176777
$801$	−54.0000	−1.90800
$802$	14.0000	0.494357
$803$	36.0000	1.27041
$804$	0	0
$805$	4.00000	0.140981
$806$	−4.00000	−0.140894
$807$	0	0
$808$	30.0000	1.05540
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	9.00000	0.316228
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	60.0000	2.10300
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−16.0000	−0.559769
$818$	22.0000	0.769212
$819$	12.0000	0.419314
$820$	2.00000	0.0698430
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−18.0000	−0.627060
$825$	0	0
$826$	−16.0000	−0.556711
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	6.00000	0.208514
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	−14.0000	−0.485947
$831$	0	0
$832$	14.0000	0.485363
$833$	6.00000	0.207888
$834$	0	0
$835$	−18.0000	−0.622916
$836$	−12.0000	−0.415029
$837$	0	0
$838$	−4.00000	−0.138178
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	−14.0000	−0.481900
$845$	9.00000	0.309609
$846$	−36.0000	−1.23771
$847$	−50.0000	−1.71802
$848$	6.00000	0.206041
$849$	0	0
$850$	2.00000	0.0685994
$851$	20.0000	0.685591
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	−12.0000	−0.410632
$855$	−6.00000	−0.205196
$856$	18.0000	0.615227
$857$	58.0000	1.98124	0.990621	−	0.136637i	$-0.0436295\pi$
$857$	58.0000	1.98124	0.990621	+	0.136637i	$0.0436295\pi$
$858$	0	0
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	24.0000	0.817443
$863$	−42.0000	−1.42970	−0.714848	−	0.699280i	$-0.753504\pi$
$863$	−42.0000	−1.42970	−0.714848	+	0.699280i	$0.753504\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	34.0000	1.15537
$867$	0	0
$868$	4.00000	0.135769
$869$	60.0000	2.03536
$870$	0	0
$871$	4.00000	0.135535
$872$	−42.0000	−1.42230
$873$	−6.00000	−0.203069
$874$	4.00000	0.135302
$875$	2.00000	0.0676123
$876$	0	0
$877$	−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
$879$	0	0
$880$	−6.00000	−0.202260
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	−9.00000	−0.303046
$883$	−50.0000	−1.68263	−0.841317	−	0.540542i	$-0.818219\pi$
$883$	−50.0000	−1.68263	−0.841317	+	0.540542i	$0.818219\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	12.0000	0.403148
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	18.0000	0.603361
$891$	−54.0000	−1.80907
$892$	14.0000	0.468755
$893$	24.0000	0.803129
$894$	0	0
$895$	12.0000	0.401116
$896$	−6.00000	−0.200446
$897$	0	0
$898$	−10.0000	−0.333704
$899$	−2.00000	−0.0667037
$900$	3.00000	0.100000
$901$	12.0000	0.399778
$902$	12.0000	0.399556
$903$	0	0
$904$	6.00000	0.199557
$905$	−6.00000	−0.199447
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−22.0000	−0.730096
$909$	−30.0000	−0.995037
$910$	−4.00000	−0.132599
$911$	14.0000	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$912$	0	0
$913$	84.0000	2.77999
$914$	38.0000	1.25693
$915$	0	0
$916$	6.00000	0.198246
$917$	−28.0000	−0.924641
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−24.0000	−0.789970
$924$	0	0
$925$	10.0000	0.328798
$926$	−22.0000	−0.722965
$927$	18.0000	0.591198
$928$	5.00000	0.164133
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−18.0000	−0.589610
$933$	0	0
$934$	−36.0000	−1.17796
$935$	−12.0000	−0.392442
$936$	−18.0000	−0.588348
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	4.00000	0.130258
$944$	8.00000	0.260378
$945$	0	0
$946$	48.0000	1.56061
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	2.00000	0.0648886
$951$	0	0
$952$	12.0000	0.388922
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−18.0000	−0.582772
$955$	22.0000	0.711903
$956$	12.0000	0.388108
$957$	0	0
$958$	14.0000	0.452319
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−20.0000	−0.644826
$963$	−18.0000	−0.580042
$964$	26.0000	0.837404
$965$	10.0000	0.321911
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	75.0000	2.41059
$969$	0	0
$970$	2.00000	0.0642161
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	34.0000	1.08943
$975$	0	0
$976$	6.00000	0.192055
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−108.000	−3.45169
$980$	−3.00000	−0.0958315
$981$	42.0000	1.34096
$982$	2.00000	0.0638226
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	4.00000	0.127257
$989$	16.0000	0.508770
$990$	18.0000	0.572078
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	−24.0000	−0.761234
$995$	4.00000	0.126809
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	12.0000	0.379853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	145.2.a.a.1.1	✓	1
3.2	odd	2		1305.2.a.f.1.1		1
4.3	odd	2		2320.2.a.e.1.1		1
5.2	odd	4		725.2.b.a.349.1		2
5.3	odd	4		725.2.b.a.349.2		2
5.4	even	2		725.2.a.a.1.1		1
7.6	odd	2		7105.2.a.b.1.1		1
8.3	odd	2		9280.2.a.o.1.1		1
8.5	even	2		9280.2.a.l.1.1		1
15.14	odd	2		6525.2.a.d.1.1		1
29.28	even	2		4205.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.a.1.1	✓	1	1.1	even	1	trivial
725.2.a.a.1.1		1	5.4	even	2
725.2.b.a.349.1		2	5.2	odd	4
725.2.b.a.349.2		2	5.3	odd	4
1305.2.a.f.1.1		1	3.2	odd	2
2320.2.a.e.1.1		1	4.3	odd	2
4205.2.a.a.1.1		1	29.28	even	2
6525.2.a.d.1.1		1	15.14	odd	2
7105.2.a.b.1.1		1	7.6	odd	2
9280.2.a.l.1.1		1	8.5	even	2
9280.2.a.o.1.1		1	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 145 = 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	145.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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