LMFDB - Embedded newform 7623.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
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	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{10} +4.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{20} -4.00000 q^{23} -4.00000 q^{25} -8.00000 q^{26} -2.00000 q^{28} -2.00000 q^{31} +8.00000 q^{32} -2.00000 q^{34} -1.00000 q^{35} +6.00000 q^{37} -2.00000 q^{41} +3.00000 q^{43} +8.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} +8.00000 q^{52} -12.0000 q^{53} +5.00000 q^{59} +12.0000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +4.00000 q^{65} +5.00000 q^{67} +2.00000 q^{68} +2.00000 q^{70} +6.00000 q^{71} -2.00000 q^{73} -12.0000 q^{74} +8.00000 q^{79} -4.00000 q^{80} +4.00000 q^{82} +15.0000 q^{83} +1.00000 q^{85} -6.00000 q^{86} -9.00000 q^{89} -4.00000 q^{91} -8.00000 q^{92} +14.0000 q^{94} +2.00000 q^{97} -2.00000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	−2.00000	−0.632456
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−8.00000	−1.56893
$27$	0	0
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	0	0
$45$	0	0
$46$	8.00000	1.17954
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.00000	1.13137
$51$	0	0
$52$	8.00000	1.10940
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	4.00000	0.496139
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−12.0000	−1.39497
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	4.00000	0.441726
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−6.00000	−0.646997
$87$	0	0
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	−8.00000	−0.834058
$93$	0	0
$94$	14.0000	1.44399
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−13.0000	−1.29355	−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000	−1.29355	−0.646774	+	0.762682i	$0.723882\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	24.0000	2.33109
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	−10.0000	−0.920575
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	0	0
$122$	−24.0000	−2.17286
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	0	0
$130$	−8.00000	−0.701646
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	0	0
$134$	−10.0000	−0.863868
$135$	0	0
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	12.0000	0.986394
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	8.00000	0.632456
$161$	4.00000	0.315244
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−30.0000	−2.32845
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−2.00000	−0.153393
$171$	0	0
$172$	6.00000	0.457496
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	18.0000	1.34916
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	8.00000	0.592999
$183$	0	0
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	−14.0000	−1.02105
$189$	0	0
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	2.00000	0.142857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	26.0000	1.82935
$203$	0	0
$204$	0	0
$205$	−2.00000	−0.139686
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−16.0000	−1.10940
$209$	0	0
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	−24.0000	−1.64833
$213$	0	0
$214$	24.0000	1.64061
$215$	3.00000	0.204598
$216$	0	0
$217$	2.00000	0.135769
$218$	−14.0000	−0.948200
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	−12.0000	−0.798228
$227$	−27.0000	−1.79205	−0.896026	−	0.444001i	$-0.853559\pi$
$227$	−27.0000	−1.79205	−0.896026	+	0.444001i	$0.853559\pi$
$228$	0	0
$229$	30.0000	1.98246	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000	1.98246	0.991228	+	0.132164i	$0.0421925\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−7.00000	−0.456630
$236$	10.0000	0.650945
$237$	0	0
$238$	2.00000	0.129641
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	0	0
$244$	24.0000	1.53644
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	18.0000	1.13842
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	−22.0000	−1.38040
$255$	0	0
$256$	16.0000	1.00000
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	8.00000	0.496139
$261$	0	0
$262$	−30.0000	−1.85341
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	10.0000	0.610847
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−16.0000	−0.966595
$275$	0	0
$276$	0	0
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−4.00000	−0.234082
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	5.00000	0.291111
$296$	0	0
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−10.0000	−0.575435
$303$	0	0
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	16.0000	0.900070
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	0	0
$319$	0	0
$320$	−8.00000	−0.447214
$321$	0	0
$322$	−8.00000	−0.445823
$323$	0	0
$324$	0	0
$325$	−16.0000	−0.887520
$326$	40.0000	2.21540
$327$	0	0
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	30.0000	1.64646
$333$	0	0
$334$	18.0000	0.984916
$335$	5.00000	0.273179
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−6.00000	−0.326357
$339$	0	0
$340$	2.00000	0.108465
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	18.0000	0.967686
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	−18.0000	−0.953998
$357$	0	0
$358$	0	0
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−32.0000	−1.68188
$363$	0	0
$364$	−8.00000	−0.419314
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	16.0000	0.834058
$369$	0	0
$370$	−12.0000	−0.623850
$371$	12.0000	0.623009
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	0	0
$381$	0	0
$382$	−4.00000	−0.204658
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	4.00000	0.203069
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	0	0
$394$	−44.0000	−2.21669
$395$	8.00000	0.402524
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	16.0000	0.800000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−26.0000	−1.29355
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	16.0000	0.788263
$413$	−5.00000	−0.246034
$414$	0	0
$415$	15.0000	0.736321
$416$	32.0000	1.56893
$417$	0	0
$418$	0	0
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	0	0
$421$	25.0000	1.21843	0.609213	−	0.793007i	$-0.291486\pi$
$421$	25.0000	1.21843	0.609213	+	0.793007i	$0.291486\pi$
$422$	6.00000	0.292075
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−12.0000	−0.580721
$428$	−24.0000	−1.16008
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	−32.0000	−1.53782	−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000	−1.53782	−0.768911	+	0.639356i	$0.779201\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	0	0
$442$	−8.00000	−0.380521
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	−20.0000	−0.947027
$447$	0	0
$448$	8.00000	0.377964
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	0	0
$452$	12.0000	0.564433
$453$	0	0
$454$	54.0000	2.53435
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−27.0000	−1.26301	−0.631503	−	0.775373i	$-0.717562\pi$
$457$	−27.0000	−1.26301	−0.631503	+	0.775373i	$0.717562\pi$
$458$	−60.0000	−2.80362
$459$	0	0
$460$	−8.00000	−0.373002
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	0	0
$466$	−12.0000	−0.555889
$467$	−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
$469$	−5.00000	−0.230879
$470$	14.0000	0.645772
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	12.0000	0.548867
$479$	−37.0000	−1.69057	−0.845287	−	0.534313i	$-0.820570\pi$
$479$	−37.0000	−1.69057	−0.845287	+	0.534313i	$0.820570\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	−20.0000	−0.910975
$483$	0	0
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	−31.0000	−1.40474	−0.702372	−	0.711810i	$-0.747876\pi$
$487$	−31.0000	−1.40474	−0.702372	+	0.711810i	$0.747876\pi$
$488$	0	0
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	34.0000	1.53440	0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000	1.53440	0.767199	+	0.641409i	$0.221650\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	−18.0000	−0.804984
$501$	0	0
$502$	40.0000	1.78529
$503$	13.0000	0.579641	0.289821	−	0.957081i	$-0.406404\pi$
$503$	13.0000	0.579641	0.289821	+	0.957081i	$0.406404\pi$
$504$	0	0
$505$	−13.0000	−0.578492
$506$	0	0
$507$	0	0
$508$	22.0000	0.976092
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−32.0000	−1.41421
$513$	0	0
$514$	−38.0000	−1.67611
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	12.0000	0.527250
$519$	0	0
$520$	0	0
$521$	45.0000	1.97149	0.985743	−	0.168259i	$-0.0538144\pi$
$521$	45.0000	1.97149	0.985743	+	0.168259i	$0.0538144\pi$
$522$	0	0
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	30.0000	1.31056
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−2.00000	−0.0871214
$528$	0	0
$529$	−7.00000	−0.304348
$530$	24.0000	1.04249
$531$	0	0
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	4.00000	0.172452
$539$	0	0
$540$	0	0
$541$	1.00000	0.0429934	0.0214967	−	0.999769i	$-0.493157\pi$
$541$	1.00000	0.0429934	0.0214967	+	0.999769i	$0.493157\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	8.00000	0.342997
$545$	7.00000	0.299847
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	16.0000	0.683486
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−50.0000	−2.12430
$555$	0	0
$556$	−4.00000	−0.169638
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	4.00000	0.169031
$561$	0	0
$562$	−48.0000	−2.02476
$563$	31.0000	1.30649	0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000	1.30649	0.653247	+	0.757145i	$0.273406\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−44.0000	−1.84946
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	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
$573$	0	0
$574$	−4.00000	−0.166957
$575$	16.0000	0.667246
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	32.0000	1.33102
$579$	0	0
$580$	0	0
$581$	−15.0000	−0.622305
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	18.0000	0.743573
$587$	31.0000	1.27951	0.639753	−	0.768580i	$-0.279036\pi$
$587$	31.0000	1.27951	0.639753	+	0.768580i	$0.279036\pi$
$588$	0	0
$589$	0	0
$590$	−10.0000	−0.411693
$591$	0	0
$592$	−24.0000	−0.986394
$593$	17.0000	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$593$	17.0000	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	4.00000	0.163846
$597$	0	0
$598$	32.0000	1.30858
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	0	0
$609$	0	0
$610$	−24.0000	−0.971732
$611$	−28.0000	−1.13276
$612$	0	0
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	44.0000	1.77570
$615$	0	0
$616$	0	0
$617$	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
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	42.0000	1.68405
$623$	9.00000	0.360577
$624$	0	0
$625$	11.0000	0.440000
$626$	−40.0000	−1.59872
$627$	0	0
$628$	4.00000	0.159617
$629$	6.00000	0.239236
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	0	0
$633$	0	0
$634$	−56.0000	−2.22404
$635$	11.0000	0.436522
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	0	0
$647$	11.0000	0.432455	0.216227	−	0.976343i	$-0.430625\pi$
$647$	11.0000	0.432455	0.216227	+	0.976343i	$0.430625\pi$
$648$	0	0
$649$	0	0
$650$	32.0000	1.25514
$651$	0	0
$652$	−40.0000	−1.56652
$653$	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
$655$	15.0000	0.586098
$656$	8.00000	0.312348
$657$	0	0
$658$	−14.0000	−0.545777
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−18.0000	−0.696441
$669$	0	0
$670$	−10.0000	−0.386334
$671$	0	0
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	6.00000	0.230769
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	2.00000	0.0763604
$687$	0	0
$688$	−12.0000	−0.457496
$689$	−48.0000	−1.82865
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	12.0000	0.454207
$699$	0	0
$700$	8.00000	0.302372
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	60.0000	2.25813
$707$	13.0000	0.488915
$708$	0	0
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	52.0000	1.94062
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	38.0000	1.41421
$723$	0	0
$724$	32.0000	1.18927
$725$	0	0
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	0	0
$730$	4.00000	0.148047
$731$	3.00000	0.110959
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	64.0000	2.36228
$735$	0	0
$736$	−32.0000	−1.17954
$737$	0	0
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	−24.0000	−0.881068
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	−38.0000	−1.39128
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−41.0000	−1.49611	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000	−1.49611	−0.748056	+	0.663636i	$0.769012\pi$
$752$	28.0000	1.02105
$753$	0	0
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	−38.0000	−1.38022
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	4.00000	0.144715
$765$	0	0
$766$	2.00000	0.0722629
$767$	20.0000	0.722158
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	0	0
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	−4.00000	−0.142857
$785$	2.00000	0.0713831
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	44.0000	1.56744
$789$	0	0
$790$	−16.0000	−0.569254
$791$	−6.00000	−0.213335
$792$	0	0
$793$	48.0000	1.70453
$794$	4.00000	0.141955
$795$	0	0
$796$	0	0
$797$	23.0000	0.814702	0.407351	−	0.913272i	$-0.366453\pi$
$797$	23.0000	0.814702	0.407351	+	0.913272i	$0.366453\pi$
$798$	0	0
$799$	−7.00000	−0.247642
$800$	−32.0000	−1.13137
$801$	0	0
$802$	−28.0000	−0.988714
$803$	0	0
$804$	0	0
$805$	4.00000	0.140981
$806$	16.0000	0.563576
$807$	0	0
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	0	0
$818$	−48.0000	−1.67828
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	10.0000	0.347945
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	−30.0000	−1.04132
$831$	0	0
$832$	−32.0000	−1.10940
$833$	1.00000	0.0346479
$834$	0	0
$835$	−9.00000	−0.311458
$836$	0	0
$837$	0	0
$838$	−46.0000	−1.58904
$839$	43.0000	1.48452	0.742262	−	0.670109i	$-0.233753\pi$
$839$	43.0000	1.48452	0.742262	+	0.670109i	$0.233753\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−50.0000	−1.72311
$843$	0	0
$844$	−6.00000	−0.206529
$845$	3.00000	0.103203
$846$	0	0
$847$	0	0
$848$	48.0000	1.64833
$849$	0	0
$850$	8.00000	0.274398
$851$	−24.0000	−0.822709
$852$	0	0
$853$	56.0000	1.91740	0.958702	−	0.284413i	$-0.0917988\pi$
$853$	56.0000	1.91740	0.958702	+	0.284413i	$0.0917988\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	0	0
$857$	−49.0000	−1.67381	−0.836904	−	0.547350i	$-0.815637\pi$
$857$	−49.0000	−1.67381	−0.836904	+	0.547350i	$0.815637\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	60.0000	2.04361
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−9.00000	−0.306009
$866$	64.0000	2.17481
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	64.0000	2.15990
$879$	0	0
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−56.0000	−1.88136
$887$	−37.0000	−1.24234	−0.621169	−	0.783676i	$-0.713342\pi$
$887$	−37.0000	−1.24234	−0.621169	+	0.783676i	$0.713342\pi$
$888$	0	0
$889$	−11.0000	−0.368928
$890$	18.0000	0.603361
$891$	0	0
$892$	20.0000	0.669650
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	68.0000	2.26919
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−54.0000	−1.79205
$909$	0	0
$910$	8.00000	0.265197
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	54.0000	1.78616
$915$	0	0
$916$	60.0000	1.98246
$917$	−15.0000	−0.495344
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	−6.00000	−0.197599
$923$	24.0000	0.789970
$924$	0	0
$925$	−24.0000	−0.789115
$926$	64.0000	2.10317
$927$	0	0
$928$	0	0
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	0	0
$931$	0	0
$932$	12.0000	0.393073
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	10.0000	0.326512
$939$	0	0
$940$	−14.0000	−0.456630
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	−20.0000	−0.650945
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	2.00000	0.0647185
$956$	−12.0000	−0.388108
$957$	0	0
$958$	74.0000	2.39083
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−48.0000	−1.54758
$963$	0	0
$964$	20.0000	0.644157
$965$	−5.00000	−0.160956
$966$	0	0
$967$	31.0000	0.996893	0.498446	−	0.866921i	$-0.333904\pi$
$967$	31.0000	0.996893	0.498446	+	0.866921i	$0.333904\pi$
$968$	0	0
$969$	0	0
$970$	−4.00000	−0.128432
$971$	−55.0000	−1.76503	−0.882517	−	0.470281i	$-0.844153\pi$
$971$	−55.0000	−1.76503	−0.882517	+	0.470281i	$0.844153\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	62.0000	1.98661
$975$	0	0
$976$	−48.0000	−1.53644
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	0	0
$980$	2.00000	0.0638877
$981$	0	0
$982$	−68.0000	−2.16997
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−16.0000	−0.508001
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.c.1.1	yes	1
3.2	odd	2		7623.2.a.p.1.1	yes	1
11.10	odd	2		7623.2.a.r.1.1	yes	1
33.32	even	2		7623.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.b.1.1	✓	1	33.32	even	2
7623.2.a.c.1.1	yes	1	1.1	even	1	trivial
7623.2.a.p.1.1	yes	1	3.2	odd	2
7623.2.a.r.1.1	yes	1	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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