LMFDB - Embedded newform 8624.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +2.00000 q^{5} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +2.00000 q^{5} +6.00000 q^{9} +1.00000 q^{11} -7.00000 q^{13} -6.00000 q^{15} +2.00000 q^{17} +8.00000 q^{23} -1.00000 q^{25} -9.00000 q^{27} -5.00000 q^{29} -4.00000 q^{31} -3.00000 q^{33} +4.00000 q^{37} +21.0000 q^{39} +4.00000 q^{41} +8.00000 q^{43} +12.0000 q^{45} -2.00000 q^{47} -6.00000 q^{51} -6.00000 q^{53} +2.00000 q^{55} -3.00000 q^{59} +1.00000 q^{61} -14.0000 q^{65} -9.00000 q^{67} -24.0000 q^{69} +2.00000 q^{71} +4.00000 q^{73} +3.00000 q^{75} -9.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} +4.00000 q^{85} +15.0000 q^{87} +6.00000 q^{89} +12.0000 q^{93} +7.00000 q^{97} +6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	−6.00000	−1.54919
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	21.0000	3.36269
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	12.0000	1.78885
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−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
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−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
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.0000	−1.73649
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	−24.0000	−2.88926
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	15.0000	1.60817
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	−18.0000	−1.77359	−0.886796	−	0.462160i	$-0.847074\pi$
$103$	−18.0000	−1.77359	−0.886796	+	0.462160i	$0.847074\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	5.00000	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$114$	0	0
$115$	16.0000	1.49201
$116$	0	0
$117$	−42.0000	−3.88290
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	0	0
$129$	−24.0000	−2.11308
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−18.0000	−1.54919
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	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
$141$	6.00000	0.505291
$142$	0	0
$143$	−7.00000	−0.585369
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−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
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.0000	1.90071	0.950357	−	0.311160i	$-0.100718\pi$
$173$	25.0000	1.90071	0.950357	+	0.311160i	$0.100718\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.00000	0.676481
$178$	0	0
$179$	−19.0000	−1.42013	−0.710063	−	0.704138i	$-0.751334\pi$
$179$	−19.0000	−1.42013	−0.710063	+	0.704138i	$0.751334\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−3.00000	−0.221766
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	42.0000	3.00768
$196$	0	0
$197$	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
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	27.0000	1.90443
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	48.0000	3.33623
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.0000	−0.810885
$220$	0	0
$221$	−14.0000	−0.941742
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	27.0000	1.75384
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	−27.0000	−1.66489	−0.832446	−	0.554107i	$-0.813060\pi$
$263$	−27.0000	−1.66489	−0.832446	+	0.554107i	$0.813060\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−21.0000	−1.23104
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−9.00000	−0.522233
$298$	0	0
$299$	−56.0000	−3.23856
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.0000	1.55111
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	54.0000	3.07195
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	0	0
$324$	0	0
$325$	7.00000	0.388290
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	0	0
$333$	24.0000	1.31519
$334$	0	0
$335$	−18.0000	−0.983445
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−48.0000	−2.58423
$346$	0	0
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	63.0000	3.36269
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	0	0
$369$	24.0000	1.24939
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	36.0000	1.85903
$376$	0	0
$377$	35.0000	1.80259
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	0	0
$381$	−57.0000	−2.92020
$382$	0	0
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	48.0000	2.43998
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.0000	−0.905678
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	0	0
$403$	28.0000	1.39478
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	21.0000	1.01389
$430$	0	0
$431$	−25.0000	−1.20421	−0.602104	−	0.798418i	$-0.705671\pi$
$431$	−25.0000	−1.20421	−0.602104	+	0.798418i	$0.705671\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	30.0000	1.43839
$436$	0	0
$437$	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	$0.461929\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	30.0000	1.41895
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	−9.00000	−0.422857
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	−18.0000	−0.840168
$460$	0	0
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	0	0
$463$	2.00000	0.0929479	0.0464739	−	0.998920i	$-0.485202\pi$
$463$	2.00000	0.0929479	0.0464739	+	0.998920i	$0.485202\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−48.0000	−2.21172
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−36.0000	−1.64833
$478$	0	0
$479$	1.00000	0.0456912	0.0228456	−	0.999739i	$-0.492727\pi$
$479$	1.00000	0.0456912	0.0228456	+	0.999739i	$0.492727\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	0	0
$489$	−51.0000	−2.30630
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	57.0000	2.54657
$502$	0	0
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	−108.000	−4.79645
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−36.0000	−1.58635
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	0	0
$519$	−75.0000	−3.29213
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	−28.0000	−1.21281
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	57.0000	2.45973
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.00000	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$542$	0	0
$543$	66.0000	2.83233
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−30.0000	−1.28271	−0.641354	−	0.767245i	$-0.721627\pi$
$547$	−30.0000	−1.28271	−0.641354	+	0.767245i	$0.721627\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−24.0000	−1.01874
$556$	0	0
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	−56.0000	−2.36855
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−84.0000	−3.47297
$586$	0	0
$587$	−27.0000	−1.11441	−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000	−1.11441	−0.557205	+	0.830375i	$0.688126\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−45.0000	−1.85105
$592$	0	0
$593$	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
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	−54.0000	−2.19905
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.0000	0.566379
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−72.0000	−2.88926
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	0	0
$633$	−60.0000	−2.38479
$634$	0	0
$635$	38.0000	1.50798
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	0	0
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	0	0
$645$	−48.0000	−1.89000
$646$	0	0
$647$	−30.0000	−1.17942	−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000	−1.17942	−0.589711	+	0.807614i	$0.700758\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.0000	1.56532	0.782660	−	0.622449i	$-0.213862\pi$
$653$	40.0000	1.56532	0.782660	+	0.622449i	$0.213862\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24.0000	0.936329
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	42.0000	1.63114
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−40.0000	−1.54881
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	1.00000	0.0386046
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	9.00000	0.346410
$676$	0	0
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	−21.0000	−0.803543	−0.401771	−	0.915740i	$-0.631605\pi$
$683$	−21.0000	−0.803543	−0.401771	+	0.915740i	$0.631605\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	84.0000	3.20480
$688$	0	0
$689$	42.0000	1.60007
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−48.0000	−1.80268	−0.901339	−	0.433114i	$-0.857415\pi$
$709$	−48.0000	−1.80268	−0.901339	+	0.433114i	$0.857415\pi$
$710$	0	0
$711$	−54.0000	−2.02516
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−14.0000	−0.523570
$716$	0	0
$717$	−15.0000	−0.560185
$718$	0	0
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.00000	0.185695
$726$	0	0
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.00000	−0.331519
$738$	0	0
$739$	−42.0000	−1.54499	−0.772497	−	0.635018i	$-0.780993\pi$
$739$	−42.0000	−1.54499	−0.772497	+	0.635018i	$0.780993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	0	0
$747$	−36.0000	−1.31717
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	−72.0000	−2.62383
$754$	0	0
$755$	6.00000	0.218362
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	21.0000	0.758266
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−9.00000	−0.324127
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	32.0000	1.14213
$786$	0	0
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	0	0
$789$	81.0000	2.88368
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7.00000	−0.248577
$794$	0	0
$795$	36.0000	1.27679
$796$	0	0
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	36.0000	1.27200
$802$	0	0
$803$	4.00000	0.141157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	72.0000	2.53452
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	0	0
$813$	−33.0000	−1.15736
$814$	0	0
$815$	34.0000	1.19097
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	27.0000	0.936620
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−38.0000	−1.31504
$836$	0	0
$837$	36.0000	1.24434
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	84.0000	2.89311
$844$	0	0
$845$	72.0000	2.47688
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.0000	1.50301	0.751506	−	0.659727i	$-0.229328\pi$
$857$	44.0000	1.50301	0.751506	+	0.659727i	$0.229328\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	50.0000	1.70005
$866$	0	0
$867$	39.0000	1.32451
$868$	0	0
$869$	−9.00000	−0.305304
$870$	0	0
$871$	63.0000	2.13467
$872$	0	0
$873$	42.0000	1.42148
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.0000	−0.979260	−0.489630	−	0.871930i	$-0.662868\pi$
$877$	−29.0000	−0.979260	−0.489630	+	0.871930i	$0.662868\pi$
$878$	0	0
$879$	54.0000	1.82137
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	41.0000	1.37665	0.688323	−	0.725405i	$-0.258347\pi$
$887$	41.0000	1.37665	0.688323	+	0.725405i	$0.258347\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−38.0000	−1.27020
$896$	0	0
$897$	168.000	5.60936
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−44.0000	−1.46261
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	−54.0000	−1.79107
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	6.00000	0.197707
$922$	0	0
$923$	−14.0000	−0.460816
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	−108.000	−3.54719
$928$	0	0
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−30.0000	−0.982156
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	−3.00000	−0.0979013
$940$	0	0
$941$	−11.0000	−0.358590	−0.179295	−	0.983795i	$-0.557382\pi$
$941$	−11.0000	−0.358590	−0.179295	+	0.983795i	$0.557382\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	0	0
$957$	15.0000	0.484881
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.0000	−1.31575	−0.657876	−	0.753126i	$-0.728545\pi$
$971$	−41.0000	−1.31575	−0.657876	+	0.753126i	$0.728545\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−21.0000	−0.672538
$976$	0	0
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	64.0000	2.03508
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	0	0
$993$	39.0000	1.23763
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	0	0
$999$	−36.0000	−1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.b.1.1		1
4.3	odd	2		1078.2.a.m.1.1		1
7.2	even	3		1232.2.q.e.529.1		2
7.4	even	3		1232.2.q.e.177.1		2
7.6	odd	2		8624.2.a.be.1.1		1
12.11	even	2		9702.2.a.i.1.1		1
28.3	even	6		1078.2.e.f.177.1		2
28.11	odd	6		154.2.e.a.23.1	✓	2
28.19	even	6		1078.2.e.f.67.1		2
28.23	odd	6		154.2.e.a.67.1	yes	2
28.27	even	2		1078.2.a.g.1.1		1
84.11	even	6		1386.2.k.o.793.1		2
84.23	even	6		1386.2.k.o.991.1		2
84.83	odd	2		9702.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.a.23.1	✓	2	28.11	odd	6
154.2.e.a.67.1	yes	2	28.23	odd	6
1078.2.a.g.1.1		1	28.27	even	2
1078.2.a.m.1.1		1	4.3	odd	2
1078.2.e.f.67.1		2	28.19	even	6
1078.2.e.f.177.1		2	28.3	even	6
1232.2.q.e.177.1		2	7.4	even	3
1232.2.q.e.529.1		2	7.2	even	3
1386.2.k.o.793.1		2	84.11	even	6
1386.2.k.o.991.1		2	84.23	even	6
8624.2.a.b.1.1		1	1.1	even	1	trivial
8624.2.a.be.1.1		1	7.6	odd	2
9702.2.a.i.1.1		1	12.11	even	2
9702.2.a.y.1.1		1	84.83	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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