LMFDB - Embedded newform 9075.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
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	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} +5.00000 q^{32} -3.00000 q^{34} -1.00000 q^{36} -3.00000 q^{37} -3.00000 q^{38} +2.00000 q^{39} +3.00000 q^{41} -1.00000 q^{42} +12.0000 q^{43} +1.00000 q^{46} -1.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -3.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +3.00000 q^{56} -3.00000 q^{57} +6.00000 q^{58} -3.00000 q^{59} -10.0000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -6.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} -7.00000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -3.00000 q^{74} +3.00000 q^{76} +2.00000 q^{78} +7.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} -6.00000 q^{83} +1.00000 q^{84} +12.0000 q^{86} +6.00000 q^{87} -14.0000 q^{89} -2.00000 q^{91} -1.00000 q^{92} +2.00000 q^{93} -1.00000 q^{94} +5.00000 q^{96} +3.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$34$	−3.00000	−0.514496
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−3.00000	−0.486664
$39$	2.00000	0.320256
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−1.00000	−0.154303
$43$	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
$45$	0	0
$46$	1.00000	0.147442
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	−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	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	−3.00000	−0.397360
$58$	6.00000	0.787839
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	2.00000	0.254000
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	3.00000	0.363803
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	3.00000	0.344124
$77$	0	0
$78$	2.00000	0.226455
$79$	7.00000	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	12.0000	1.29399
$87$	6.00000	0.643268
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−1.00000	−0.104257
$93$	2.00000	0.207390
$94$	−1.00000	−0.103142
$95$	0	0
$96$	5.00000	0.510310
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	0	0
$101$	−11.0000	−1.09454	−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000	−1.09454	−0.547270	+	0.836956i	$0.684333\pi$
$102$	−3.00000	−0.297044
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−3.00000	−0.276172
$119$	3.00000	0.275010
$120$	0	0
$121$	0	0
$122$	−10.0000	−0.905357
$123$	3.00000	0.270501
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	−3.00000	−0.265165
$129$	12.0000	1.05654
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	−6.00000	−0.518321
$135$	0	0
$136$	9.00000	0.771744
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	1.00000	0.0851257
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	−7.00000	−0.587427
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	−6.00000	−0.494872
$148$	3.00000	0.246598
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	9.00000	0.729996
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	7.00000	0.556890
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	1.00000	0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−3.00000	−0.229416
$172$	−12.0000	−0.914991
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	−3.00000	−0.225494
$178$	−14.0000	−1.04934
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	−2.00000	−0.148250
$183$	−10.0000	−0.739221
$184$	−3.00000	−0.221163
$185$	0	0
$186$	2.00000	0.146647
$187$	0	0
$188$	1.00000	0.0729325
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	7.00000	0.505181
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	3.00000	0.215387
$195$	0	0
$196$	6.00000	0.428571
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	−11.0000	−0.773957
$203$	−6.00000	−0.421117
$204$	3.00000	0.210042
$205$	0	0
$206$	−8.00000	−0.557386
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	6.00000	0.412082
$213$	−7.00000	−0.479632
$214$	−8.00000	−0.546869
$215$	0	0
$216$	−3.00000	−0.204124
$217$	−2.00000	−0.135769
$218$	−12.0000	−0.812743
$219$	2.00000	0.135147
$220$	0	0
$221$	−6.00000	−0.403604
$222$	−3.00000	−0.201347
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	18.0000	1.19734
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	3.00000	0.198680
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	0	0
$231$	0	0
$232$	−18.0000	−1.18176
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	3.00000	0.195283
$237$	7.00000	0.454699
$238$	3.00000	0.194461
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	3.00000	0.191273
$247$	−6.00000	−0.381771
$248$	−6.00000	−0.381000
$249$	−6.00000	−0.380235
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−19.0000	−1.19217
$255$	0	0
$256$	−17.0000	−1.06250
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	12.0000	0.747087
$259$	3.00000	0.186411
$260$	0	0
$261$	6.00000	0.371391
$262$	2.00000	0.123560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	3.00000	0.183942
$267$	−14.0000	−0.856786
$268$	6.00000	0.366508
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	3.00000	0.181902
$273$	−2.00000	−0.121046
$274$	−16.0000	−0.966595
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	2.00000	0.119737
$280$	0	0
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	−1.00000	−0.0595491
$283$	−1.00000	−0.0594438	−0.0297219	−	0.999558i	$-0.509462\pi$
$283$	−1.00000	−0.0594438	−0.0297219	+	0.999558i	$0.509462\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	0	0
$287$	−3.00000	−0.177084
$288$	5.00000	0.294628
$289$	−8.00000	−0.470588
$290$	0	0
$291$	3.00000	0.175863
$292$	−2.00000	−0.117041
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	9.00000	0.523114
$297$	0	0
$298$	15.0000	0.868927
$299$	2.00000	0.115663
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−20.0000	−1.15087
$303$	−11.0000	−0.631933
$304$	3.00000	0.172062
$305$	0	0
$306$	−3.00000	−0.171499
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−6.00000	−0.339683
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−7.00000	−0.393781
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	−8.00000	−0.446516
$322$	−1.00000	−0.0557278
$323$	9.00000	0.500773
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−10.0000	−0.553849
$327$	−12.0000	−0.663602
$328$	−9.00000	−0.496942
$329$	1.00000	0.0551318
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	6.00000	0.329293
$333$	−3.00000	−0.164399
$334$	−6.00000	−0.328305
$335$	0	0
$336$	1.00000	0.0545545
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−9.00000	−0.489535
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	−3.00000	−0.162221
$343$	13.0000	0.701934
$344$	−36.0000	−1.94099
$345$	0	0
$346$	9.00000	0.483843
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	−6.00000	−0.321634
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−3.00000	−0.159448
$355$	0	0
$356$	14.0000	0.741999
$357$	3.00000	0.158777
$358$	−15.0000	−0.792775
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−5.00000	−0.262794
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−6.00000	−0.313197	−0.156599	−	0.987662i	$-0.550053\pi$
$367$	−6.00000	−0.313197	−0.156599	+	0.987662i	$0.550053\pi$
$368$	−1.00000	−0.0521286
$369$	3.00000	0.156174
$370$	0	0
$371$	6.00000	0.311504
$372$	−2.00000	−0.103695
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	12.0000	0.618031
$378$	−1.00000	−0.0514344
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	−19.0000	−0.973399
$382$	−3.00000	−0.153493
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	4.00000	0.203595
$387$	12.0000	0.609994
$388$	−3.00000	−0.152302
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	18.0000	0.909137
$393$	2.00000	0.100887
$394$	21.0000	1.05796
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−26.0000	−1.30326
$399$	3.00000	0.150188
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	−6.00000	−0.299253
$403$	4.00000	0.199254
$404$	11.0000	0.547270
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	9.00000	0.445566
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−16.0000	−0.789222
$412$	8.00000	0.394132
$413$	3.00000	0.147620
$414$	1.00000	0.0491473
$415$	0	0
$416$	10.0000	0.490290
$417$	4.00000	0.195881
$418$	0	0
$419$	−37.0000	−1.80757	−0.903784	−	0.427989i	$-0.859222\pi$
$419$	−37.0000	−1.80757	−0.903784	+	0.427989i	$0.859222\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	16.0000	0.778868
$423$	−1.00000	−0.0486217
$424$	18.0000	0.874157
$425$	0	0
$426$	−7.00000	−0.339151
$427$	10.0000	0.483934
$428$	8.00000	0.386695
$429$	0	0
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	−1.00000	−0.0481125
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	12.0000	0.574696
$437$	−3.00000	−0.143509
$438$	2.00000	0.0955637
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−6.00000	−0.285391
$443$	13.0000	0.617649	0.308824	−	0.951119i	$-0.400064\pi$
$443$	13.0000	0.617649	0.308824	+	0.951119i	$0.400064\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	14.0000	0.662919
$447$	15.0000	0.709476
$448$	−7.00000	−0.330719
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−18.0000	−0.846649
$453$	−20.0000	−0.939682
$454$	−22.0000	−1.03251
$455$	0	0
$456$	9.00000	0.421464
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	23.0000	1.07472
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−9.00000	−0.416917
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	−2.00000	−0.0924500
$469$	6.00000	0.277054
$470$	0	0
$471$	−18.0000	−0.829396
$472$	9.00000	0.414259
$473$	0	0
$474$	7.00000	0.321521
$475$	0	0
$476$	−3.00000	−0.137505
$477$	−6.00000	−0.274721
$478$	24.0000	1.09773
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	22.0000	1.00207
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	0	0
$486$	1.00000	0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	30.0000	1.35804
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	−3.00000	−0.135250
$493$	−18.0000	−0.810679
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	7.00000	0.313993
$498$	−6.00000	−0.268866
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	20.0000	0.892644
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	19.0000	0.842989
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	−11.0000	−0.486136
$513$	−3.00000	−0.132453
$514$	6.00000	0.264649
$515$	0	0
$516$	−12.0000	−0.528271
$517$	0	0
$518$	3.00000	0.131812
$519$	9.00000	0.395056
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	6.00000	0.262613
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−3.00000	−0.130189
$532$	−3.00000	−0.130066
$533$	6.00000	0.259889
$534$	−14.0000	−0.605839
$535$	0	0
$536$	18.0000	0.777482
$537$	−15.0000	−0.647298
$538$	−6.00000	−0.258678
$539$	0	0
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	19.0000	0.816120
$543$	−5.00000	−0.214571
$544$	−15.0000	−0.643120
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	16.0000	0.683486
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−18.0000	−0.766826
$552$	−3.00000	−0.127688
$553$	−7.00000	−0.297670
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	2.00000	0.0846668
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	−17.0000	−0.717102
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	1.00000	0.0421076
$565$	0	0
$566$	−1.00000	−0.0420331
$567$	−1.00000	−0.0419961
$568$	21.0000	0.881140
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	−3.00000	−0.125218
$575$	0	0
$576$	7.00000	0.291667
$577$	5.00000	0.208153	0.104076	−	0.994569i	$-0.466811\pi$
$577$	5.00000	0.208153	0.104076	+	0.994569i	$0.466811\pi$
$578$	−8.00000	−0.332756
$579$	4.00000	0.166234
$580$	0	0
$581$	6.00000	0.248922
$582$	3.00000	0.124354
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−21.0000	−0.867502
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	6.00000	0.247436
$589$	−6.00000	−0.247226
$590$	0	0
$591$	21.0000	0.863825
$592$	3.00000	0.123299
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	−26.0000	−1.06411
$598$	2.00000	0.0817861
$599$	23.0000	0.939755	0.469877	−	0.882732i	$-0.344298\pi$
$599$	23.0000	0.939755	0.469877	+	0.882732i	$0.344298\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	−12.0000	−0.489083
$603$	−6.00000	−0.244339
$604$	20.0000	0.813788
$605$	0	0
$606$	−11.0000	−0.446844
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−15.0000	−0.608330
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	3.00000	0.121268
$613$	−32.0000	−1.29247	−0.646234	−	0.763139i	$-0.723657\pi$
$613$	−32.0000	−1.29247	−0.646234	+	0.763139i	$0.723657\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−8.00000	−0.321807
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	8.00000	0.320771
$623$	14.0000	0.560898
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	19.0000	0.759393
$627$	0	0
$628$	18.0000	0.718278
$629$	9.00000	0.358854
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−21.0000	−0.835335
$633$	16.0000	0.635943
$634$	30.0000	1.19145
$635$	0	0
$636$	6.00000	0.237915
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−7.00000	−0.276916
$640$	0	0
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	−8.00000	−0.315735
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	9.00000	0.354100
$647$	43.0000	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$647$	43.0000	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	10.0000	0.391630
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	−3.00000	−0.117130
$657$	2.00000	0.0780274
$658$	1.00000	0.0389841
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	−29.0000	−1.12797	−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000	−1.12797	−0.563985	+	0.825785i	$0.690732\pi$
$662$	−16.0000	−0.621858
$663$	−6.00000	−0.233021
$664$	18.0000	0.698535
$665$	0	0
$666$	−3.00000	−0.116248
$667$	6.00000	0.232321
$668$	6.00000	0.232147
$669$	14.0000	0.541271
$670$	0	0
$671$	0	0
$672$	−5.00000	−0.192879
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	9.00000	0.346154
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	18.0000	0.691286
$679$	−3.00000	−0.115129
$680$	0	0
$681$	−22.0000	−0.843042
$682$	0	0
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	3.00000	0.114708
$685$	0	0
$686$	13.0000	0.496342
$687$	23.0000	0.877505
$688$	−12.0000	−0.457496
$689$	−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	$0.624220\pi$
$692$	−9.00000	−0.342129
$693$	0	0
$694$	2.00000	0.0759190
$695$	0	0
$696$	−18.0000	−0.682288
$697$	−9.00000	−0.340899
$698$	−6.00000	−0.227103
$699$	−9.00000	−0.340411
$700$	0	0
$701$	23.0000	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$702$	2.00000	0.0754851
$703$	9.00000	0.339441
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	11.0000	0.413698
$708$	3.00000	0.112747
$709$	−9.00000	−0.338002	−0.169001	−	0.985616i	$-0.554054\pi$
$709$	−9.00000	−0.338002	−0.169001	+	0.985616i	$0.554054\pi$
$710$	0	0
$711$	7.00000	0.262521
$712$	42.0000	1.57402
$713$	2.00000	0.0749006
$714$	3.00000	0.112272
$715$	0	0
$716$	15.0000	0.560576
$717$	24.0000	0.896296
$718$	−14.0000	−0.522475
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−10.0000	−0.372161
$723$	22.0000	0.818189
$724$	5.00000	0.185824
$725$	0	0
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.0000	−1.33151
$732$	10.0000	0.369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−6.00000	−0.221464
$735$	0	0
$736$	5.00000	0.184302
$737$	0	0
$738$	3.00000	0.110432
$739$	−41.0000	−1.50821	−0.754105	−	0.656754i	$-0.771929\pi$
$739$	−41.0000	−1.50821	−0.754105	+	0.656754i	$0.771929\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	6.00000	0.220267
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	12.0000	0.439351
$747$	−6.00000	−0.219529
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	1.00000	0.0364662
$753$	20.0000	0.728841
$754$	12.0000	0.437014
$755$	0	0
$756$	1.00000	0.0363696
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−14.0000	−0.508503
$759$	0	0
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	−19.0000	−0.688297
$763$	12.0000	0.434429
$764$	3.00000	0.108536
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−6.00000	−0.216647
$768$	−17.0000	−0.613435
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−4.00000	−0.143963
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−9.00000	−0.323081
$777$	3.00000	0.107624
$778$	−12.0000	−0.430221
$779$	−9.00000	−0.322458
$780$	0	0
$781$	0	0
$782$	−3.00000	−0.107280
$783$	6.00000	0.214423
$784$	6.00000	0.214286
$785$	0	0
$786$	2.00000	0.0713376
$787$	37.0000	1.31891	0.659454	−	0.751745i	$-0.270788\pi$
$787$	37.0000	1.31891	0.659454	+	0.751745i	$0.270788\pi$
$788$	−21.0000	−0.748094
$789$	0	0
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−20.0000	−0.710221
$794$	2.00000	0.0709773
$795$	0	0
$796$	26.0000	0.921546
$797$	−52.0000	−1.84193	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000	−1.84193	−0.920967	+	0.389640i	$0.872599\pi$
$798$	3.00000	0.106199
$799$	3.00000	0.106132
$800$	0	0
$801$	−14.0000	−0.494666
$802$	−22.0000	−0.776847
$803$	0	0
$804$	6.00000	0.211604
$805$	0	0
$806$	4.00000	0.140894
$807$	−6.00000	−0.211210
$808$	33.0000	1.16094
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	0	0
$811$	−47.0000	−1.65039	−0.825197	−	0.564846i	$-0.808936\pi$
$811$	−47.0000	−1.65039	−0.825197	+	0.564846i	$0.808936\pi$
$812$	6.00000	0.210559
$813$	19.0000	0.666359
$814$	0	0
$815$	0	0
$816$	3.00000	0.105021
$817$	−36.0000	−1.25948
$818$	10.0000	0.349642
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−16.0000	−0.558064
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	3.00000	0.104383
$827$	40.0000	1.39094	0.695468	−	0.718557i	$-0.255197\pi$
$827$	40.0000	1.39094	0.695468	+	0.718557i	$0.255197\pi$
$828$	−1.00000	−0.0347524
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	14.0000	0.485363
$833$	18.0000	0.623663
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	2.00000	0.0691301
$838$	−37.0000	−1.27814
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−15.0000	−0.516934
$843$	−17.0000	−0.585511
$844$	−16.0000	−0.550743
$845$	0	0
$846$	−1.00000	−0.0343807
$847$	0	0
$848$	6.00000	0.206041
$849$	−1.00000	−0.0343199
$850$	0	0
$851$	−3.00000	−0.102839
$852$	7.00000	0.239816
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	24.0000	0.820303
$857$	−23.0000	−0.785665	−0.392833	−	0.919610i	$-0.628505\pi$
$857$	−23.0000	−0.785665	−0.392833	+	0.919610i	$0.628505\pi$
$858$	0	0
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	−6.00000	−0.204361
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−6.00000	−0.203888
$867$	−8.00000	−0.271694
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	36.0000	1.21911
$873$	3.00000	0.101535
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	35.0000	1.18119
$879$	−21.0000	−0.708312
$880$	0	0
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	−6.00000	−0.202031
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	13.0000	0.436744
$887$	54.0000	1.81314	0.906571	−	0.422053i	$-0.138690\pi$
$887$	54.0000	1.81314	0.906571	+	0.422053i	$0.138690\pi$
$888$	9.00000	0.302020
$889$	19.0000	0.637240
$890$	0	0
$891$	0	0
$892$	−14.0000	−0.468755
$893$	3.00000	0.100391
$894$	15.0000	0.501675
$895$	0	0
$896$	3.00000	0.100223
$897$	2.00000	0.0667781
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	−12.0000	−0.399335
$904$	−54.0000	−1.79601
$905$	0	0
$906$	−20.0000	−0.664455
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	22.0000	0.730096
$909$	−11.0000	−0.364847
$910$	0	0
$911$	29.0000	0.960813	0.480406	−	0.877046i	$-0.340489\pi$
$911$	29.0000	0.960813	0.480406	+	0.877046i	$0.340489\pi$
$912$	3.00000	0.0993399
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−23.0000	−0.759941
$917$	−2.00000	−0.0660458
$918$	−3.00000	−0.0990148
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−38.0000	−1.25146
$923$	−14.0000	−0.460816
$924$	0	0
$925$	0	0
$926$	22.0000	0.722965
$927$	−8.00000	−0.262754
$928$	30.0000	0.984798
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	9.00000	0.294805
$933$	8.00000	0.261908
$934$	−32.0000	−1.04707
$935$	0	0
$936$	−6.00000	−0.196116
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	6.00000	0.195907
$939$	19.0000	0.620042
$940$	0	0
$941$	49.0000	1.59735	0.798677	−	0.601760i	$-0.205534\pi$
$941$	49.0000	1.59735	0.798677	+	0.601760i	$0.205534\pi$
$942$	−18.0000	−0.586472
$943$	3.00000	0.0976934
$944$	3.00000	0.0976417
$945$	0	0
$946$	0	0
$947$	37.0000	1.20234	0.601169	−	0.799122i	$-0.294702\pi$
$947$	37.0000	1.20234	0.601169	+	0.799122i	$0.294702\pi$
$948$	−7.00000	−0.227349
$949$	4.00000	0.129845
$950$	0	0
$951$	30.0000	0.972817
$952$	−9.00000	−0.291692
$953$	−51.0000	−1.65205	−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000	−1.65205	−0.826026	+	0.563632i	$0.809404\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	14.0000	0.452319
$959$	16.0000	0.516667
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−6.00000	−0.193448
$963$	−8.00000	−0.257796
$964$	−22.0000	−0.708572
$965$	0	0
$966$	−1.00000	−0.0321745
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	9.00000	0.289122
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	−40.0000	−1.28168
$975$	0	0
$976$	10.0000	0.320092
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	−10.0000	−0.319765
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	−30.0000	−0.957338
$983$	−1.00000	−0.0318950	−0.0159475	−	0.999873i	$-0.505076\pi$
$983$	−1.00000	−0.0318950	−0.0159475	+	0.999873i	$0.505076\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	−18.0000	−0.573237
$987$	1.00000	0.0318304
$988$	6.00000	0.190885
$989$	12.0000	0.381578
$990$	0	0
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	10.0000	0.317500
$993$	−16.0000	−0.507745
$994$	7.00000	0.222027
$995$	0	0
$996$	6.00000	0.190117
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	8.00000	0.253236
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.p.1.1	yes	1
5.4	even	2		9075.2.a.e.1.1	✓	1
11.10	odd	2		9075.2.a.h.1.1	yes	1
55.54	odd	2		9075.2.a.m.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.e.1.1	✓	1	5.4	even	2
9075.2.a.h.1.1	yes	1	11.10	odd	2
9075.2.a.m.1.1	yes	1	55.54	odd	2
9075.2.a.p.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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