LMFDB - Embedded newform 9522.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,-2,0,-1,-1,0,2,-1,0,-2,1,0,1,4,0,-2,-2,0,1,0,0,-1,2, 0,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.0335528047$
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	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} -1.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +2.00000 q^{35} +2.00000 q^{37} +2.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} -2.00000 q^{43} -1.00000 q^{44} -6.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -1.00000 q^{53} +2.00000 q^{55} +1.00000 q^{56} +1.00000 q^{58} +15.0000 q^{59} -8.00000 q^{61} +9.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +4.00000 q^{68} -2.00000 q^{70} +9.00000 q^{73} -2.00000 q^{74} -2.00000 q^{76} +1.00000 q^{77} +9.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -7.00000 q^{83} -8.00000 q^{85} +2.00000 q^{86} +1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} +6.00000 q^{94} +4.00000 q^{95} -9.00000 q^{97} +6.00000 q^{98} +O(q^{100})$

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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	2.00000	0.338062
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	1.00000	0.133631
$57$	0	0
$58$	1.00000	0.131306
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	9.00000	1.14300
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−2.00000	−0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−2.00000	−0.229416
$77$	1.00000	0.113961
$78$	0	0
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	2.00000	0.215666
$87$	0	0
$88$	1.00000	0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	6.00000	0.618853
$95$	4.00000	0.410391
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	−1.00000	−0.100000
$101$	11.0000	1.09454	0.547270	−	0.836956i	$-0.315667\pi$
$101$	11.0000	1.09454	0.547270	+	0.836956i	$0.315667\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	1.00000	0.0971286
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−15.0000	−1.38086
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	8.00000	0.724286
$123$	0	0
$124$	−9.00000	−0.808224
$125$	12.0000	1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	10.0000	0.863868
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	2.00000	0.166091
$146$	−9.00000	−0.744845
$147$	0	0
$148$	2.00000	0.164399
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	18.0000	1.44579
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−9.00000	−0.716002
$159$	0	0
$160$	2.00000	0.158114
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	7.00000	0.543305
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	8.00000	0.613572
$171$	0	0
$172$	−2.00000	−0.152499
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	9.00000	0.646162
$195$	0	0
$196$	−6.00000	−0.428571
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−11.0000	−0.773957
$203$	1.00000	0.0701862
$204$	0	0
$205$	−12.0000	−0.838116
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	−2.00000	−0.138675
$209$	2.00000	0.138343
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−1.00000	−0.0686803
$213$	0	0
$214$	−8.00000	−0.546869
$215$	4.00000	0.272798
$216$	0	0
$217$	9.00000	0.610960
$218$	6.00000	0.406371
$219$	0	0
$220$	2.00000	0.134840
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	0	0
$232$	1.00000	0.0656532
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	15.0000	0.976417
$237$	0	0
$238$	4.00000	0.259281
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	−8.00000	−0.512148
$245$	12.0000	0.766652
$246$	0	0
$247$	4.00000	0.254514
$248$	9.00000	0.571501
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	4.00000	0.248069
$261$	0	0
$262$	−3.00000	−0.185341
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−2.00000	−0.122628
$267$	0	0
$268$	−10.0000	−0.610847
$269$	19.0000	1.15845	0.579225	−	0.815168i	$-0.303355\pi$
$269$	19.0000	1.15845	0.579225	+	0.815168i	$0.303355\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	1.00000	0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	10.0000	0.599760
$279$	0	0
$280$	−2.00000	−0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−2.00000	−0.117444
$291$	0	0
$292$	9.00000	0.526685
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	−30.0000	−1.74667
$296$	−2.00000	−0.116248
$297$	0	0
$298$	22.0000	1.27443
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	−7.00000	−0.402805
$303$	0	0
$304$	−2.00000	−0.114708
$305$	16.0000	0.916157
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−18.0000	−1.02233
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	9.00000	0.506290
$317$	23.0000	1.29181	0.645904	−	0.763418i	$-0.276480\pi$
$317$	23.0000	1.29181	0.645904	+	0.763418i	$0.276480\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	−2.00000	−0.111803
$321$	0	0
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	6.00000	0.330791
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−7.00000	−0.384175
$333$	0	0
$334$	−16.0000	−0.875481
$335$	20.0000	1.09272
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−8.00000	−0.433861
$341$	9.00000	0.487377
$342$	0	0
$343$	13.0000	0.701934
$344$	2.00000	0.107833
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−1.00000	−0.0536828	−0.0268414	−	0.999640i	$-0.508545\pi$
$347$	−1.00000	−0.0536828	−0.0268414	+	0.999640i	$0.508545\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−28.0000	−1.49029	−0.745145	−	0.666903i	$-0.767620\pi$
$353$	−28.0000	−1.49029	−0.745145	+	0.666903i	$0.767620\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	4.00000	0.211407
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	12.0000	0.630706
$363$	0	0
$364$	2.00000	0.104828
$365$	−18.0000	−0.942163
$366$	0	0
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	0	0
$369$	0	0
$370$	4.00000	0.207950
$371$	1.00000	0.0519174
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	6.00000	0.309426
$377$	2.00000	0.103005
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	4.00000	0.204658
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	−5.00000	−0.254493
$387$	0	0
$388$	−9.00000	−0.456906
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	0	0
$392$	6.00000	0.303046
$393$	0	0
$394$	−15.0000	−0.755689
$395$	−18.0000	−0.905678
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−5.00000	−0.250627
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	18.0000	0.896644
$404$	11.0000	0.547270
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	1.00000	0.0492665
$413$	−15.0000	−0.738102
$414$	0	0
$415$	14.0000	0.687233
$416$	2.00000	0.0980581
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	−10.0000	−0.486792
$423$	0	0
$424$	1.00000	0.0485643
$425$	−4.00000	−0.194029
$426$	0	0
$427$	8.00000	0.387147
$428$	8.00000	0.386695
$429$	0	0
$430$	−4.00000	−0.192897
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	−6.00000	−0.287348
$437$	0	0
$438$	0	0
$439$	5.00000	0.238637	0.119318	−	0.992856i	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	8.00000	0.380521
$443$	35.0000	1.66290	0.831450	−	0.555599i	$-0.187511\pi$
$443$	35.0000	1.66290	0.831450	+	0.555599i	$0.187511\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	4.00000	0.189405
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	−14.0000	−0.658505
$453$	0	0
$454$	21.0000	0.985579
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	37.0000	1.71954	0.859768	−	0.510685i	$-0.170608\pi$
$463$	37.0000	1.71954	0.859768	+	0.510685i	$0.170608\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	12.0000	0.555889
$467$	−11.0000	−0.509019	−0.254510	−	0.967070i	$-0.581914\pi$
$467$	−11.0000	−0.509019	−0.254510	+	0.967070i	$0.581914\pi$
$468$	0	0
$469$	10.0000	0.461757
$470$	−12.0000	−0.553519
$471$	0	0
$472$	−15.0000	−0.690431
$473$	2.00000	0.0919601
$474$	0	0
$475$	2.00000	0.0917663
$476$	−4.00000	−0.183340
$477$	0	0
$478$	−30.0000	−1.37217
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−10.0000	−0.454545
$485$	18.0000	0.817338
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	−12.0000	−0.542105
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−9.00000	−0.404112
$497$	0	0
$498$	0	0
$499$	−26.0000	−1.16392	−0.581960	−	0.813217i	$-0.697714\pi$
$499$	−26.0000	−1.16392	−0.581960	+	0.813217i	$0.697714\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	28.0000	1.24970
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	−22.0000	−0.978987
$506$	0	0
$507$	0	0
$508$	16.0000	0.709885
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	6.00000	0.263880
$518$	2.00000	0.0878750
$519$	0	0
$520$	−4.00000	−0.175412
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	16.0000	0.697633
$527$	−36.0000	−1.56818
$528$	0	0
$529$	0	0
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	2.00000	0.0867110
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−16.0000	−0.691740
$536$	10.0000	0.431934
$537$	0	0
$538$	−19.0000	−0.819148
$539$	6.00000	0.258438
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	13.0000	0.558398
$543$	0	0
$544$	−4.00000	−0.171499
$545$	12.0000	0.514024
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	2.00000	0.0852029
$552$	0	0
$553$	−9.00000	−0.382719
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−10.0000	−0.424094
$557$	13.0000	0.550828	0.275414	−	0.961326i	$-0.411185\pi$
$557$	13.0000	0.550828	0.275414	+	0.961326i	$0.411185\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	2.00000	0.0845154
$561$	0	0
$562$	−30.0000	−1.26547
$563$	19.0000	0.800755	0.400377	−	0.916350i	$-0.368879\pi$
$563$	19.0000	0.800755	0.400377	+	0.916350i	$0.368879\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	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	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	6.00000	0.250435
$575$	0	0
$576$	0	0
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	2.00000	0.0830455
$581$	7.00000	0.290409
$582$	0	0
$583$	1.00000	0.0414158
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−19.0000	−0.784883
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	30.0000	1.23508
$591$	0	0
$592$	2.00000	0.0821995
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	−22.0000	−0.901155
$597$	0	0
$598$	0	0
$599$	−26.0000	−1.06233	−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000	−1.06233	−0.531166	+	0.847268i	$0.678246\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	−2.00000	−0.0815139
$603$	0	0
$604$	7.00000	0.284826
$605$	20.0000	0.813116
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−16.0000	−0.647821
$611$	12.0000	0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	18.0000	0.722897
$621$	0	0
$622$	−8.00000	−0.320771
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	14.0000	0.559553
$627$	0	0
$628$	−18.0000	−0.718278
$629$	8.00000	0.318981
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−9.00000	−0.358001
$633$	0	0
$634$	−23.0000	−0.913447
$635$	−32.0000	−1.26988
$636$	0	0
$637$	12.0000	0.475457
$638$	−1.00000	−0.0395904
$639$	0	0
$640$	2.00000	0.0790569
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	8.00000	0.314756
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	0	0
$649$	−15.0000	−0.588802
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	6.00000	0.234261
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	7.00000	0.271653
$665$	−4.00000	−0.155113
$666$	0	0
$667$	0	0
$668$	16.0000	0.619059
$669$	0	0
$670$	−20.0000	−0.772667
$671$	8.00000	0.308837
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	9.00000	0.345388
$680$	8.00000	0.306786
$681$	0	0
$682$	−9.00000	−0.344628
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	−13.0000	−0.496342
$687$	0	0
$688$	−2.00000	−0.0762493
$689$	2.00000	0.0761939
$690$	0	0
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	1.00000	0.0379595
$695$	20.0000	0.758643
$696$	0	0
$697$	24.0000	0.909065
$698$	16.0000	0.605609
$699$	0	0
$700$	1.00000	0.0377964
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	28.0000	1.05379
$707$	−11.0000	−0.413698
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−6.00000	−0.223918
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	15.0000	0.558242
$723$	0	0
$724$	−12.0000	−0.445976
$725$	1.00000	0.0371391
$726$	0	0
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	18.0000	0.666210
$731$	−8.00000	−0.295891
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−3.00000	−0.110732
$735$	0	0
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	46.0000	1.69214	0.846069	−	0.533074i	$-0.178963\pi$
$739$	46.0000	1.69214	0.846069	+	0.533074i	$0.178963\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−1.00000	−0.0367112
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	44.0000	1.61204
$746$	10.0000	0.366126
$747$	0	0
$748$	−4.00000	−0.146254
$749$	−8.00000	−0.292314
$750$	0	0
$751$	15.0000	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−48.0000	−1.74459	−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000	−1.74459	−0.872295	+	0.488980i	$0.837369\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	−4.00000	−0.145095
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−36.0000	−1.30073
$767$	−30.0000	−1.08324
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	5.00000	0.179954
$773$	−1.00000	−0.0359675	−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000	−0.0359675	−0.0179838	+	0.999838i	$0.505725\pi$
$774$	0	0
$775$	9.00000	0.323290
$776$	9.00000	0.323081
$777$	0	0
$778$	15.0000	0.537776
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	36.0000	1.28490
$786$	0	0
$787$	42.0000	1.49714	0.748569	−	0.663057i	$-0.230741\pi$
$787$	42.0000	1.49714	0.748569	+	0.663057i	$0.230741\pi$
$788$	15.0000	0.534353
$789$	0	0
$790$	18.0000	0.640411
$791$	14.0000	0.497783
$792$	0	0
$793$	16.0000	0.568177
$794$	−8.00000	−0.283909
$795$	0	0
$796$	5.00000	0.177220
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	0	0
$802$	0	0
$803$	−9.00000	−0.317603
$804$	0	0
$805$	0	0
$806$	−18.0000	−0.634023
$807$	0	0
$808$	−11.0000	−0.386979
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	1.00000	0.0350931
$813$	0	0
$814$	2.00000	0.0701000
$815$	−8.00000	−0.280228
$816$	0	0
$817$	4.00000	0.139942
$818$	−7.00000	−0.244749
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−9.00000	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$821$	−9.00000	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$822$	0	0
$823$	48.0000	1.67317	0.836587	−	0.547833i	$-0.184547\pi$
$823$	48.0000	1.67317	0.836587	+	0.547833i	$0.184547\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	15.0000	0.521917
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	−14.0000	−0.485947
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	−24.0000	−0.831551
$834$	0	0
$835$	−32.0000	−1.10741
$836$	2.00000	0.0691714
$837$	0	0
$838$	21.0000	0.725433
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−28.0000	−0.964944
$843$	0	0
$844$	10.0000	0.344214
$845$	18.0000	0.619219
$846$	0	0
$847$	10.0000	0.343604
$848$	−1.00000	−0.0343401
$849$	0	0
$850$	4.00000	0.137199
$851$	0	0
$852$	0	0
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−8.00000	−0.273434
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−56.0000	−1.91070	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000	−1.91070	−0.955348	+	0.295484i	$0.904519\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	−36.0000	−1.22616
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−36.0000	−1.22404
$866$	−10.0000	−0.339814
$867$	0	0
$868$	9.00000	0.305480
$869$	−9.00000	−0.305304
$870$	0	0
$871$	20.0000	0.677674
$872$	6.00000	0.203186
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	−5.00000	−0.168742
$879$	0	0
$880$	2.00000	0.0674200
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	18.0000	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$883$	18.0000	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−35.0000	−1.17585
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−4.00000	−0.133930
$893$	12.0000	0.401565
$894$	0	0
$895$	8.00000	0.267411
$896$	1.00000	0.0334077
$897$	0	0
$898$	−20.0000	−0.667409
$899$	9.00000	0.300167
$900$	0	0
$901$	−4.00000	−0.133259
$902$	6.00000	0.199778
$903$	0	0
$904$	14.0000	0.465633
$905$	24.0000	0.797787
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−21.0000	−0.696909
$909$	0	0
$910$	4.00000	0.132599
$911$	44.0000	1.45779	0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000	1.45779	0.728893	+	0.684628i	$0.240035\pi$
$912$	0	0
$913$	7.00000	0.231666
$914$	25.0000	0.826927
$915$	0	0
$916$	20.0000	0.660819
$917$	−3.00000	−0.0990687
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	0	0
$922$	3.00000	0.0987997
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−37.0000	−1.21590
$927$	0	0
$928$	1.00000	0.0328266
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	−12.0000	−0.393073
$933$	0	0
$934$	11.0000	0.359931
$935$	8.00000	0.261628
$936$	0	0
$937$	−41.0000	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$938$	−10.0000	−0.326512
$939$	0	0
$940$	12.0000	0.391397
$941$	−3.00000	−0.0977972	−0.0488986	−	0.998804i	$-0.515571\pi$
$941$	−3.00000	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$942$	0	0
$943$	0	0
$944$	15.0000	0.488208
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	−2.00000	−0.0648886
$951$	0	0
$952$	4.00000	0.129641
$953$	40.0000	1.29573	0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000	1.29573	0.647864	+	0.761756i	$0.275663\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	30.0000	0.970269
$957$	0	0
$958$	−22.0000	−0.710788
$959$	12.0000	0.387500
$960$	0	0
$961$	50.0000	1.61290
$962$	4.00000	0.128965
$963$	0	0
$964$	22.0000	0.708572
$965$	−10.0000	−0.321911
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	−18.0000	−0.577945
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	−13.0000	−0.416547
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	12.0000	0.383326
$981$	0	0
$982$	33.0000	1.05307
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	0	0
$985$	−30.0000	−0.955879
$986$	4.00000	0.127386
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	9.00000	0.285750
$993$	0	0
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	26.0000	0.823016
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.a.1.1	✓	1
3.2	odd	2		9522.2.a.m.1.1	yes	1
23.22	odd	2		9522.2.a.e.1.1	yes	1
69.68	even	2		9522.2.a.h.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.a.1.1	✓	1	1.1	even	1	trivial
9522.2.a.e.1.1	yes	1	23.22	odd	2
9522.2.a.h.1.1	yes	1	69.68	even	2
9522.2.a.m.1.1	yes	1	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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