LMFDB - Embedded newform 8820.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8820,2,Mod(1,8820)] 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("8820.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,4,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$70.4280545828$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2940)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8820.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.00000 q^{11} +2.82843 q^{13} -1.17157 q^{17} -5.41421 q^{19} +7.41421 q^{23} +1.00000 q^{25} -3.65685 q^{29} -4.24264 q^{31} -10.4853 q^{37} -2.00000 q^{41} +1.65685 q^{43} -10.4853 q^{47} +4.58579 q^{53} -2.00000 q^{55} +2.82843 q^{59} +9.89949 q^{61} -2.82843 q^{65} -6.82843 q^{67} +10.4853 q^{71} -4.48528 q^{73} -10.0000 q^{79} -4.82843 q^{83} +1.17157 q^{85} -4.82843 q^{89} +5.41421 q^{95} -4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{11} - 8 q^{17} - 8 q^{19} + 12 q^{23} + 2 q^{25} + 4 q^{29} - 4 q^{37} - 4 q^{41} - 8 q^{43} - 4 q^{47} + 12 q^{53} - 4 q^{55} - 8 q^{67} + 4 q^{71} + 8 q^{73} - 20 q^{79} - 4 q^{83}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^11 - 8 * q^17 - 8 * q^19 + 12 * q^23 + 2 * q^25 + 4 * q^29 - 4 * q^37 - 4 * q^41 - 8 * q^43 - 4 * q^47 + 12 * q^53 - 4 * q^55 - 8 * q^67 + 4 * q^71 + 8 * q^73 - 20 * q^79 - 4 * q^83 + 8 * q^85 - 4 * q^89 + 8 * q^95 - 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.41421	1.54597	0.772985	−	0.634424i	$-0.218763\pi$
$23$	7.41421	1.54597	0.772985	+	0.634424i	$0.218763\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	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
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.4853	−1.52944	−0.764718	−	0.644365i	$-0.777122\pi$
$47$	−10.4853	−1.52944	−0.764718	+	0.644365i	$0.777122\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.58579	0.629906	0.314953	−	0.949107i	$-0.398011\pi$
$53$	4.58579	0.629906	0.314953	+	0.949107i	$0.398011\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	9.89949	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$61$	9.89949	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.82843	−0.350823
$66$	0	0
$67$	−6.82843	−0.834225	−0.417113	−	0.908855i	$-0.636958\pi$
$67$	−6.82843	−0.834225	−0.417113	+	0.908855i	$0.636958\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4853	1.24437	0.622187	−	0.782869i	$-0.286244\pi$
$71$	10.4853	1.24437	0.622187	+	0.782869i	$0.286244\pi$
$72$	0	0
$73$	−4.48528	−0.524962	−0.262481	−	0.964937i	$-0.584541\pi$
$73$	−4.48528	−0.524962	−0.262481	+	0.964937i	$0.584541\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.82843	−0.529989	−0.264994	−	0.964250i	$-0.585370\pi$
$83$	−4.82843	−0.529989	−0.264994	+	0.964250i	$0.585370\pi$
$84$	0	0
$85$	1.17157	0.127075
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.82843	−0.511812	−0.255906	−	0.966702i	$-0.582374\pi$
$89$	−4.82843	−0.511812	−0.255906	+	0.966702i	$0.582374\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.41421	0.555487
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.6569	1.15990	0.579950	−	0.814652i	$-0.303072\pi$
$101$	11.6569	1.15990	0.579950	+	0.814652i	$0.303072\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.07107	−0.490239	−0.245119	−	0.969493i	$-0.578827\pi$
$107$	−5.07107	−0.490239	−0.245119	+	0.969493i	$0.578827\pi$
$108$	0	0
$109$	−4.34315	−0.415998	−0.207999	−	0.978129i	$-0.566695\pi$
$109$	−4.34315	−0.415998	−0.207999	+	0.978129i	$0.566695\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.24264	0.587258	0.293629	−	0.955919i	$-0.405137\pi$
$113$	6.24264	0.587258	0.293629	+	0.955919i	$0.405137\pi$
$114$	0	0
$115$	−7.41421	−0.691379
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.1421	1.25491	0.627456	−	0.778652i	$-0.284096\pi$
$127$	14.1421	1.25491	0.627456	+	0.778652i	$0.284096\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.89949	0.674899	0.337450	−	0.941344i	$-0.390436\pi$
$137$	7.89949	0.674899	0.337450	+	0.941344i	$0.390436\pi$
$138$	0	0
$139$	7.55635	0.640921	0.320461	−	0.947262i	$-0.396162\pi$
$139$	7.55635	0.640921	0.320461	+	0.947262i	$0.396162\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	3.65685	0.303685
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.6569	1.61035	0.805176	−	0.593036i	$-0.202071\pi$
$149$	19.6569	1.61035	0.805176	+	0.593036i	$0.202071\pi$
$150$	0	0
$151$	5.31371	0.432423	0.216212	−	0.976346i	$-0.430630\pi$
$151$	5.31371	0.432423	0.216212	+	0.976346i	$0.430630\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.24264	0.340777
$156$	0	0
$157$	−10.1421	−0.809431	−0.404715	−	0.914443i	$-0.632629\pi$
$157$	−10.1421	−0.809431	−0.404715	+	0.914443i	$0.632629\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.1421	−1.42100	−0.710501	−	0.703696i	$-0.751532\pi$
$163$	−18.1421	−1.42100	−0.710501	+	0.703696i	$0.751532\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.1421	−1.07521	−0.537603	−	0.843198i	$-0.680670\pi$
$173$	−14.1421	−1.07521	−0.537603	+	0.843198i	$0.680670\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.4853	0.783707	0.391853	−	0.920028i	$-0.371834\pi$
$179$	10.4853	0.783707	0.391853	+	0.920028i	$0.371834\pi$
$180$	0	0
$181$	−24.2426	−1.80194	−0.900971	−	0.433880i	$-0.857144\pi$
$181$	−24.2426	−1.80194	−0.900971	+	0.433880i	$0.857144\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.4853	0.770893
$186$	0	0
$187$	−2.34315	−0.171348
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.9706	−1.08323	−0.541616	−	0.840626i	$-0.682187\pi$
$191$	−14.9706	−1.08323	−0.541616	+	0.840626i	$0.682187\pi$
$192$	0	0
$193$	12.8284	0.923410	0.461705	−	0.887033i	$-0.347238\pi$
$193$	12.8284	0.923410	0.461705	+	0.887033i	$0.347238\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.0711	−0.931275	−0.465638	−	0.884976i	$-0.654175\pi$
$197$	−13.0711	−0.931275	−0.465638	+	0.884976i	$0.654175\pi$
$198$	0	0
$199$	−8.24264	−0.584305	−0.292153	−	0.956372i	$-0.594372\pi$
$199$	−8.24264	−0.584305	−0.292153	+	0.956372i	$0.594372\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.8284	−0.749018
$210$	0	0
$211$	−17.6569	−1.21555	−0.607774	−	0.794110i	$-0.707937\pi$
$211$	−17.6569	−1.21555	−0.607774	+	0.794110i	$0.707937\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.65685	−0.112997
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.31371	−0.222904
$222$	0	0
$223$	−5.17157	−0.346314	−0.173157	−	0.984894i	$-0.555397\pi$
$223$	−5.17157	−0.346314	−0.173157	+	0.984894i	$0.555397\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	5.41421	0.357781	0.178891	−	0.983869i	$-0.442749\pi$
$229$	5.41421	0.357781	0.178891	+	0.983869i	$0.442749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.72792	−0.178712	−0.0893561	−	0.996000i	$-0.528481\pi$
$233$	−2.72792	−0.178712	−0.0893561	+	0.996000i	$0.528481\pi$
$234$	0	0
$235$	10.4853	0.683984
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.6569	1.27150	0.635748	−	0.771897i	$-0.280692\pi$
$239$	19.6569	1.27150	0.635748	+	0.771897i	$0.280692\pi$
$240$	0	0
$241$	−29.4142	−1.89474	−0.947368	−	0.320147i	$-0.896268\pi$
$241$	−29.4142	−1.89474	−0.947368	+	0.320147i	$0.896268\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.3137	−0.974388
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.5147	0.726803	0.363401	−	0.931633i	$-0.381615\pi$
$251$	11.5147	0.726803	0.363401	+	0.931633i	$0.381615\pi$
$252$	0	0
$253$	14.8284	0.932255
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.3137	−1.32951	−0.664756	−	0.747060i	$-0.731465\pi$
$257$	−21.3137	−1.32951	−0.664756	+	0.747060i	$0.731465\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.0711	1.05265	0.526324	−	0.850284i	$-0.323570\pi$
$263$	17.0711	1.05265	0.526324	+	0.850284i	$0.323570\pi$
$264$	0	0
$265$	−4.58579	−0.281703
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.1716	−0.925027	−0.462514	−	0.886612i	$-0.653052\pi$
$269$	−15.1716	−0.925027	−0.462514	+	0.886612i	$0.653052\pi$
$270$	0	0
$271$	−20.2426	−1.22965	−0.614826	−	0.788662i	$-0.710774\pi$
$271$	−20.2426	−1.22965	−0.614826	+	0.788662i	$0.710774\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	5.31371	0.319270	0.159635	−	0.987176i	$-0.448968\pi$
$277$	5.31371	0.319270	0.159635	+	0.987176i	$0.448968\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.97056	0.177209	0.0886045	−	0.996067i	$-0.471759\pi$
$281$	2.97056	0.177209	0.0886045	+	0.996067i	$0.471759\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.65685	−0.213636	−0.106818	−	0.994279i	$-0.534066\pi$
$293$	−3.65685	−0.213636	−0.106818	+	0.994279i	$0.534066\pi$
$294$	0	0
$295$	−2.82843	−0.164677
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.9706	1.21276
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.89949	−0.566843
$306$	0	0
$307$	−1.65685	−0.0945617	−0.0472808	−	0.998882i	$-0.515056\pi$
$307$	−1.65685	−0.0945617	−0.0472808	+	0.998882i	$0.515056\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.68629	0.492554	0.246277	−	0.969199i	$-0.420793\pi$
$311$	8.68629	0.492554	0.246277	+	0.969199i	$0.420793\pi$
$312$	0	0
$313$	−28.4853	−1.61008	−0.805042	−	0.593218i	$-0.797857\pi$
$313$	−28.4853	−1.61008	−0.805042	+	0.593218i	$0.797857\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.100505	−0.00564493	−0.00282246	−	0.999996i	$-0.500898\pi$
$317$	−0.100505	−0.00564493	−0.00282246	+	0.999996i	$0.500898\pi$
$318$	0	0
$319$	−7.31371	−0.409489
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.34315	0.352942
$324$	0	0
$325$	2.82843	0.156893
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−34.6274	−1.90329	−0.951647	−	0.307192i	$-0.900611\pi$
$331$	−34.6274	−1.90329	−0.951647	+	0.307192i	$0.900611\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.82843	0.373077
$336$	0	0
$337$	33.7990	1.84115	0.920574	−	0.390568i	$-0.127721\pi$
$337$	33.7990	1.84115	0.920574	+	0.390568i	$0.127721\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.48528	−0.459504
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.5858	−0.675640	−0.337820	−	0.941211i	$-0.609690\pi$
$347$	−12.5858	−0.675640	−0.337820	+	0.941211i	$0.609690\pi$
$348$	0	0
$349$	−29.2132	−1.56375	−0.781873	−	0.623437i	$-0.785736\pi$
$349$	−29.2132	−1.56375	−0.781873	+	0.623437i	$0.785736\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.2843	1.61187	0.805935	−	0.592005i	$-0.201663\pi$
$353$	30.2843	1.61187	0.805935	+	0.592005i	$0.201663\pi$
$354$	0	0
$355$	−10.4853	−0.556501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.9706	−1.42345	−0.711726	−	0.702457i	$-0.752086\pi$
$359$	−26.9706	−1.42345	−0.711726	+	0.702457i	$0.752086\pi$
$360$	0	0
$361$	10.3137	0.542827
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.48528	0.234770
$366$	0	0
$367$	30.8284	1.60923	0.804615	−	0.593796i	$-0.202372\pi$
$367$	30.8284	1.60923	0.804615	+	0.593796i	$0.202372\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.68629	0.139091	0.0695455	−	0.997579i	$-0.477845\pi$
$373$	2.68629	0.139091	0.0695455	+	0.997579i	$0.477845\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.3431	−0.532699
$378$	0	0
$379$	31.9411	1.64071	0.820353	−	0.571858i	$-0.193777\pi$
$379$	31.9411	1.64071	0.820353	+	0.571858i	$0.193777\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.85786	−0.197128	−0.0985638	−	0.995131i	$-0.531425\pi$
$383$	−3.85786	−0.197128	−0.0985638	+	0.995131i	$0.531425\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.17157	−0.363613	−0.181807	−	0.983334i	$-0.558194\pi$
$389$	−7.17157	−0.363613	−0.181807	+	0.983334i	$0.558194\pi$
$390$	0	0
$391$	−8.68629	−0.439285
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	28.2843	1.41955	0.709773	−	0.704430i	$-0.248797\pi$
$397$	28.2843	1.41955	0.709773	+	0.704430i	$0.248797\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.4853	−1.92186	−0.960932	−	0.276786i	$-0.910731\pi$
$401$	−38.4853	−1.92186	−0.960932	+	0.276786i	$0.910731\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.9706	−1.03947
$408$	0	0
$409$	24.7279	1.22272	0.611359	−	0.791354i	$-0.290623\pi$
$409$	24.7279	1.22272	0.611359	+	0.791354i	$0.290623\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.82843	0.237018
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.9706	−1.61072	−0.805359	−	0.592788i	$-0.798027\pi$
$419$	−32.9706	−1.61072	−0.805359	+	0.592788i	$0.798027\pi$
$420$	0	0
$421$	−32.2843	−1.57344	−0.786720	−	0.617311i	$-0.788222\pi$
$421$	−32.2843	−1.57344	−0.786720	+	0.617311i	$0.788222\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.17157	−0.0568296
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.4853	−1.85377	−0.926885	−	0.375344i	$-0.877524\pi$
$431$	−38.4853	−1.85377	−0.926885	+	0.375344i	$0.877524\pi$
$432$	0	0
$433$	−13.1716	−0.632985	−0.316493	−	0.948595i	$-0.602505\pi$
$433$	−13.1716	−0.632985	−0.316493	+	0.948595i	$0.602505\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−40.1421	−1.92026
$438$	0	0
$439$	−24.7279	−1.18020	−0.590100	−	0.807330i	$-0.700912\pi$
$439$	−24.7279	−1.18020	−0.590100	+	0.807330i	$0.700912\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.5858	0.978060	0.489030	−	0.872267i	$-0.337351\pi$
$443$	20.5858	0.978060	0.489030	+	0.872267i	$0.337351\pi$
$444$	0	0
$445$	4.82843	0.228889
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.51472	0.445080	0.222540	−	0.974924i	$-0.428565\pi$
$457$	9.51472	0.445080	0.222540	+	0.974924i	$0.428565\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.4558	0.533552	0.266776	−	0.963759i	$-0.414042\pi$
$461$	11.4558	0.533552	0.266776	+	0.963759i	$0.414042\pi$
$462$	0	0
$463$	−5.85786	−0.272238	−0.136119	−	0.990692i	$-0.543463\pi$
$463$	−5.85786	−0.272238	−0.136119	+	0.990692i	$0.543463\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.9411	1.38551	0.692755	−	0.721173i	$-0.256397\pi$
$467$	29.9411	1.38551	0.692755	+	0.721173i	$0.256397\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.31371	0.152364
$474$	0	0
$475$	−5.41421	−0.248421
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.9411	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$479$	−33.9411	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$480$	0	0
$481$	−29.6569	−1.35224
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	−20.2843	−0.919168	−0.459584	−	0.888134i	$-0.652001\pi$
$487$	−20.2843	−0.919168	−0.459584	+	0.888134i	$0.652001\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.1127	1.31384	0.656919	−	0.753961i	$-0.271859\pi$
$491$	29.1127	1.31384	0.656919	+	0.753961i	$0.271859\pi$
$492$	0	0
$493$	4.28427	0.192954
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−30.2843	−1.35571	−0.677855	−	0.735196i	$-0.737090\pi$
$499$	−30.2843	−1.35571	−0.677855	+	0.735196i	$0.737090\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.8284	−1.46375	−0.731874	−	0.681440i	$-0.761354\pi$
$503$	−32.8284	−1.46375	−0.731874	+	0.681440i	$0.761354\pi$
$504$	0	0
$505$	−11.6569	−0.518723
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.4853	−1.70583	−0.852915	−	0.522050i	$-0.825168\pi$
$509$	−38.4853	−1.70583	−0.852915	+	0.522050i	$0.825168\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.82843	0.124635
$516$	0	0
$517$	−20.9706	−0.922284
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.1127	−0.924964	−0.462482	−	0.886629i	$-0.653041\pi$
$521$	−21.1127	−0.924964	−0.462482	+	0.886629i	$0.653041\pi$
$522$	0	0
$523$	13.1716	0.575953	0.287976	−	0.957638i	$-0.407018\pi$
$523$	13.1716	0.575953	0.287976	+	0.957638i	$0.407018\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.97056	0.216521
$528$	0	0
$529$	31.9706	1.39002
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.65685	−0.245026
$534$	0	0
$535$	5.07107	0.219241
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	36.2843	1.55998	0.779991	−	0.625790i	$-0.215223\pi$
$541$	36.2843	1.55998	0.779991	+	0.625790i	$0.215223\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.34315	0.186040
$546$	0	0
$547$	27.1127	1.15926	0.579628	−	0.814881i	$-0.303198\pi$
$547$	27.1127	1.15926	0.579628	+	0.814881i	$0.303198\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.7990	0.843465
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.6985	1.34311	0.671554	−	0.740956i	$-0.265627\pi$
$557$	31.6985	1.34311	0.671554	+	0.740956i	$0.265627\pi$
$558$	0	0
$559$	4.68629	0.198209
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−38.4853	−1.62196	−0.810981	−	0.585073i	$-0.801066\pi$
$563$	−38.4853	−1.62196	−0.810981	+	0.585073i	$0.801066\pi$
$564$	0	0
$565$	−6.24264	−0.262630
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.4853	−1.44570	−0.722849	−	0.691006i	$-0.757168\pi$
$569$	−34.4853	−1.44570	−0.722849	+	0.691006i	$0.757168\pi$
$570$	0	0
$571$	20.2843	0.848870	0.424435	−	0.905458i	$-0.360473\pi$
$571$	20.2843	0.848870	0.424435	+	0.905458i	$0.360473\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.41421	0.309194
$576$	0	0
$577$	2.82843	0.117749	0.0588745	−	0.998265i	$-0.481249\pi$
$577$	2.82843	0.117749	0.0588745	+	0.998265i	$0.481249\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.17157	0.379848
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.8284	0.694584	0.347292	−	0.937757i	$-0.387101\pi$
$587$	16.8284	0.694584	0.347292	+	0.937757i	$0.387101\pi$
$588$	0	0
$589$	22.9706	0.946486
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.97056	−0.121986	−0.0609932	−	0.998138i	$-0.519427\pi$
$593$	−2.97056	−0.121986	−0.0609932	+	0.998138i	$0.519427\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−19.7574	−0.805919	−0.402960	−	0.915218i	$-0.632019\pi$
$601$	−19.7574	−0.805919	−0.402960	+	0.915218i	$0.632019\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−21.4558	−0.870866	−0.435433	−	0.900221i	$-0.643405\pi$
$607$	−21.4558	−0.870866	−0.435433	+	0.900221i	$0.643405\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−29.6569	−1.19979
$612$	0	0
$613$	37.3137	1.50709	0.753543	−	0.657398i	$-0.228343\pi$
$613$	37.3137	1.50709	0.753543	+	0.657398i	$0.228343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.6985	−1.43717	−0.718583	−	0.695441i	$-0.755209\pi$
$617$	−35.6985	−1.43717	−0.718583	+	0.695441i	$0.755209\pi$
$618$	0	0
$619$	14.1005	0.566747	0.283374	−	0.959010i	$-0.408546\pi$
$619$	14.1005	0.566747	0.283374	+	0.959010i	$0.408546\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.2843	0.489806
$630$	0	0
$631$	6.68629	0.266177	0.133089	−	0.991104i	$-0.457511\pi$
$631$	6.68629	0.266177	0.133089	+	0.991104i	$0.457511\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−14.1421	−0.561214
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.02944	−0.356641	−0.178321	−	0.983972i	$-0.557066\pi$
$641$	−9.02944	−0.356641	−0.178321	+	0.983972i	$0.557066\pi$
$642$	0	0
$643$	−32.2843	−1.27317	−0.636584	−	0.771208i	$-0.719653\pi$
$643$	−32.2843	−1.27317	−0.636584	+	0.771208i	$0.719653\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.3137	−1.38833	−0.694163	−	0.719818i	$-0.744225\pi$
$647$	−35.3137	−1.38833	−0.694163	+	0.719818i	$0.744225\pi$
$648$	0	0
$649$	5.65685	0.222051
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.5858	−0.649052	−0.324526	−	0.945877i	$-0.605205\pi$
$653$	−16.5858	−0.649052	−0.324526	+	0.945877i	$0.605205\pi$
$654$	0	0
$655$	−11.3137	−0.442063
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.79899	−0.381714	−0.190857	−	0.981618i	$-0.561127\pi$
$659$	−9.79899	−0.381714	−0.190857	+	0.981618i	$0.561127\pi$
$660$	0	0
$661$	−35.7574	−1.39080	−0.695400	−	0.718623i	$-0.744773\pi$
$661$	−35.7574	−1.39080	−0.695400	+	0.718623i	$0.744773\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−27.1127	−1.04981
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.7990	0.764332
$672$	0	0
$673$	40.8284	1.57382	0.786910	−	0.617068i	$-0.211680\pi$
$673$	40.8284	1.57382	0.786910	+	0.617068i	$0.211680\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.9706	1.65149	0.825746	−	0.564041i	$-0.190754\pi$
$677$	42.9706	1.65149	0.825746	+	0.564041i	$0.190754\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.44365	0.0935037	0.0467518	−	0.998907i	$-0.485113\pi$
$683$	2.44365	0.0935037	0.0467518	+	0.998907i	$0.485113\pi$
$684$	0	0
$685$	−7.89949	−0.301824
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.9706	0.494139
$690$	0	0
$691$	19.2721	0.733144	0.366572	−	0.930390i	$-0.380531\pi$
$691$	19.2721	0.733144	0.366572	+	0.930390i	$0.380531\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.55635	−0.286629
$696$	0	0
$697$	2.34315	0.0887530
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.9706	1.16974	0.584871	−	0.811126i	$-0.301145\pi$
$701$	30.9706	1.16974	0.584871	+	0.811126i	$0.301145\pi$
$702$	0	0
$703$	56.7696	2.14110
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.68629	0.326221	0.163110	−	0.986608i	$-0.447847\pi$
$709$	8.68629	0.326221	0.163110	+	0.986608i	$0.447847\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.4558	−1.17803
$714$	0	0
$715$	−5.65685	−0.211554
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.4853	0.465622	0.232811	−	0.972522i	$-0.425208\pi$
$719$	12.4853	0.465622	0.232811	+	0.972522i	$0.425208\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.65685	−0.135812
$726$	0	0
$727$	43.5980	1.61696	0.808480	−	0.588524i	$-0.200291\pi$
$727$	43.5980	1.61696	0.808480	+	0.588524i	$0.200291\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.94113	−0.0717951
$732$	0	0
$733$	−24.4853	−0.904385	−0.452192	−	0.891920i	$-0.649358\pi$
$733$	−24.4853	−0.904385	−0.452192	+	0.891920i	$0.649358\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.6569	−0.503057
$738$	0	0
$739$	−6.62742	−0.243793	−0.121897	−	0.992543i	$-0.538898\pi$
$739$	−6.62742	−0.243793	−0.121897	+	0.992543i	$0.538898\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.0416	0.661883	0.330942	−	0.943651i	$-0.392634\pi$
$743$	18.0416	0.661883	0.330942	+	0.943651i	$0.392634\pi$
$744$	0	0
$745$	−19.6569	−0.720171
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.6569	0.498346	0.249173	−	0.968459i	$-0.419841\pi$
$751$	13.6569	0.498346	0.249173	+	0.968459i	$0.419841\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.31371	−0.193386
$756$	0	0
$757$	0.828427	0.0301097	0.0150548	−	0.999887i	$-0.495208\pi$
$757$	0.828427	0.0301097	0.0150548	+	0.999887i	$0.495208\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.48528	0.235091	0.117546	−	0.993067i	$-0.462497\pi$
$761$	6.48528	0.235091	0.117546	+	0.993067i	$0.462497\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	13.2132	0.476480	0.238240	−	0.971206i	$-0.423429\pi$
$769$	13.2132	0.476480	0.238240	+	0.971206i	$0.423429\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.4558	1.34719	0.673597	−	0.739099i	$-0.264749\pi$
$773$	37.4558	1.34719	0.673597	+	0.739099i	$0.264749\pi$
$774$	0	0
$775$	−4.24264	−0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.8284	0.387969
$780$	0	0
$781$	20.9706	0.750386
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.1421	0.361988
$786$	0	0
$787$	39.1127	1.39422	0.697109	−	0.716966i	$-0.254470\pi$
$787$	39.1127	1.39422	0.697109	+	0.716966i	$0.254470\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	28.0000	0.994309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.1127	0.535319	0.267660	−	0.963514i	$-0.413750\pi$
$797$	15.1127	0.535319	0.267660	+	0.963514i	$0.413750\pi$
$798$	0	0
$799$	12.2843	0.434586
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.97056	−0.316564
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.51472	−0.193887	−0.0969436	−	0.995290i	$-0.530907\pi$
$809$	−5.51472	−0.193887	−0.0969436	+	0.995290i	$0.530907\pi$
$810$	0	0
$811$	1.89949	0.0667003	0.0333501	−	0.999444i	$-0.489382\pi$
$811$	1.89949	0.0667003	0.0333501	+	0.999444i	$0.489382\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.1421	0.635491
$816$	0	0
$817$	−8.97056	−0.313840
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40.4264	−1.41089	−0.705446	−	0.708764i	$-0.749253\pi$
$821$	−40.4264	−1.41089	−0.705446	+	0.708764i	$0.749253\pi$
$822$	0	0
$823$	25.4558	0.887335	0.443667	−	0.896191i	$-0.353677\pi$
$823$	25.4558	0.887335	0.443667	+	0.896191i	$0.353677\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.2426	0.495265	0.247633	−	0.968854i	$-0.420347\pi$
$827$	14.2426	0.495265	0.247633	+	0.968854i	$0.420347\pi$
$828$	0	0
$829$	−3.07107	−0.106663	−0.0533313	−	0.998577i	$-0.516984\pi$
$829$	−3.07107	−0.106663	−0.0533313	+	0.998577i	$0.516984\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.4853	1.12152	0.560758	−	0.827980i	$-0.310510\pi$
$839$	32.4853	1.12152	0.560758	+	0.827980i	$0.310510\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.00000	0.172005
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−77.7401	−2.66490
$852$	0	0
$853$	−40.2843	−1.37931	−0.689654	−	0.724139i	$-0.742237\pi$
$853$	−40.2843	−1.37931	−0.689654	+	0.724139i	$0.742237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.4558	−0.869555	−0.434778	−	0.900538i	$-0.643173\pi$
$857$	−25.4558	−0.869555	−0.434778	+	0.900538i	$0.643173\pi$
$858$	0	0
$859$	−10.1005	−0.344625	−0.172312	−	0.985042i	$-0.555124\pi$
$859$	−10.1005	−0.344625	−0.172312	+	0.985042i	$0.555124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.8995	1.22203	0.611017	−	0.791618i	$-0.290761\pi$
$863$	35.8995	1.22203	0.611017	+	0.791618i	$0.290761\pi$
$864$	0	0
$865$	14.1421	0.480847
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	−19.3137	−0.654420
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.9706	−1.58608	−0.793042	−	0.609167i	$-0.791504\pi$
$877$	−46.9706	−1.58608	−0.793042	+	0.609167i	$0.791504\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	0	0
$883$	−16.9706	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$883$	−16.9706	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.1716	−0.912332	−0.456166	−	0.889895i	$-0.650778\pi$
$887$	−27.1716	−0.912332	−0.456166	+	0.889895i	$0.650778\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	56.7696	1.89972
$894$	0	0
$895$	−10.4853	−0.350484
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.5147	0.517445
$900$	0	0
$901$	−5.37258	−0.178987
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.2426	0.805853
$906$	0	0
$907$	33.6569	1.11756	0.558779	−	0.829317i	$-0.311270\pi$
$907$	33.6569	1.11756	0.558779	+	0.829317i	$0.311270\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47.9411	1.58836	0.794180	−	0.607682i	$-0.207901\pi$
$911$	47.9411	1.58836	0.794180	+	0.607682i	$0.207901\pi$
$912$	0	0
$913$	−9.65685	−0.319595
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	28.2843	0.933012	0.466506	−	0.884518i	$-0.345513\pi$
$919$	28.2843	0.933012	0.466506	+	0.884518i	$0.345513\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.6569	0.976167
$924$	0	0
$925$	−10.4853	−0.344754
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.1716	0.628999	0.314499	−	0.949258i	$-0.398163\pi$
$929$	19.1716	0.628999	0.314499	+	0.949258i	$0.398163\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.34315	0.0766291
$936$	0	0
$937$	17.6569	0.576824	0.288412	−	0.957506i	$-0.406873\pi$
$937$	17.6569	0.576824	0.288412	+	0.957506i	$0.406873\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−51.2548	−1.67086	−0.835430	−	0.549597i	$-0.814781\pi$
$941$	−51.2548	−1.67086	−0.835430	+	0.549597i	$0.814781\pi$
$942$	0	0
$943$	−14.8284	−0.482880
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.9289	−0.745090	−0.372545	−	0.928014i	$-0.621515\pi$
$947$	−22.9289	−0.745090	−0.372545	+	0.928014i	$0.621515\pi$
$948$	0	0
$949$	−12.6863	−0.411814
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−46.0416	−1.49143	−0.745717	−	0.666262i	$-0.767893\pi$
$953$	−46.0416	−1.49143	−0.745717	+	0.666262i	$0.767893\pi$
$954$	0	0
$955$	14.9706	0.484436
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.8284	−0.412962
$966$	0	0
$967$	−20.2010	−0.649621	−0.324810	−	0.945779i	$-0.605300\pi$
$967$	−20.2010	−0.649621	−0.324810	+	0.945779i	$0.605300\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.4853	0.657404	0.328702	−	0.944434i	$-0.393389\pi$
$971$	20.4853	0.657404	0.328702	+	0.944434i	$0.393389\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.0416	−0.961117	−0.480558	−	0.876963i	$-0.659566\pi$
$977$	−30.0416	−0.961117	−0.480558	+	0.876963i	$0.659566\pi$
$978$	0	0
$979$	−9.65685	−0.308634
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.5980	1.77330	0.886650	−	0.462441i	$-0.153026\pi$
$983$	55.5980	1.77330	0.886650	+	0.462441i	$0.153026\pi$
$984$	0	0
$985$	13.0711	0.416479
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.2843	0.390617
$990$	0	0
$991$	−11.3137	−0.359392	−0.179696	−	0.983722i	$-0.557511\pi$
$991$	−11.3137	−0.359392	−0.179696	+	0.983722i	$0.557511\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.24264	0.261309
$996$	0	0
$997$	−3.31371	−0.104946	−0.0524731	−	0.998622i	$-0.516710\pi$
$997$	−3.31371	−0.104946	−0.0524731	+	0.998622i	$0.516710\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bh.1.2		2
3.2	odd	2		2940.2.a.o.1.2	✓	2
7.6	odd	2		8820.2.a.bm.1.1		2
21.2	odd	6		2940.2.q.r.361.2		4
21.5	even	6		2940.2.q.p.361.1		4
21.11	odd	6		2940.2.q.r.961.2		4
21.17	even	6		2940.2.q.p.961.1		4
21.20	even	2		2940.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.a.o.1.2	✓	2	3.2	odd	2
2940.2.a.q.1.1	yes	2	21.20	even	2
2940.2.q.p.361.1		4	21.5	even	6
2940.2.q.p.961.1		4	21.17	even	6
2940.2.q.r.361.2		4	21.2	odd	6
2940.2.q.r.961.2		4	21.11	odd	6
8820.2.a.bh.1.2		2	1.1	even	1	trivial
8820.2.a.bm.1.1		2	7.6	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:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.00000 q^{11} +2.82843 q^{13} -1.17157 q^{17} -5.41421 q^{19} +7.41421 q^{23} +1.00000 q^{25} -3.65685 q^{29} -4.24264 q^{31} -10.4853 q^{37} -2.00000 q^{41} +1.65685 q^{43} -10.4853 q^{47} +4.58579 q^{53} -2.00000 q^{55} +2.82843 q^{59} +9.89949 q^{61} -2.82843 q^{65} -6.82843 q^{67} +10.4853 q^{71} -4.48528 q^{73} -10.0000 q^{79} -4.82843 q^{83} +1.17157 q^{85} -4.82843 q^{89} +5.41421 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{11} - 8 q^{17} - 8 q^{19} + 12 q^{23} + 2 q^{25} + 4 q^{29} - 4 q^{37} - 4 q^{41} - 8 q^{43} - 4 q^{47} + 12 q^{53} - 4 q^{55} - 8 q^{67} + 4 q^{71} + 8 q^{73} - 20 q^{79} - 4 q^{83}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^11 - 8 * q^17 - 8 * q^19 + 12 * q^23 + 2 * q^25 + 4 * q^29 - 4 * q^37 - 4 * q^41 - 8 * q^43 - 4 * q^47 + 12 * q^53 - 4 * q^55 - 8 * q^67 + 4 * q^71 + 8 * q^73 - 20 * q^79 - 4 * q^83 + 8 * q^85 - 4 * q^89 + 8 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more