LMFDB - Embedded newform 4356.2.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4356,2,Mod(1,4356)] 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("4356.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4356, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4356.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,-1,0,0,0,0,0,7,0,0,0,-7,0,1,0,0,0,-4,0,-7,0,0,0, -15,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.7828351205$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4356.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{5} -3.85410 q^{7} +2.38197 q^{13} -2.38197 q^{17} +3.85410 q^{19} +2.47214 q^{23} -2.38197 q^{25} -8.61803 q^{29} +0.854102 q^{31} -6.23607 q^{35} -1.85410 q^{37} -8.61803 q^{41} -1.38197 q^{47} +7.85410 q^{49} +4.09017 q^{53} -1.09017 q^{59} +2.38197 q^{61} +3.85410 q^{65} +12.9443 q^{67} +6.38197 q^{71} -0.909830 q^{73} -7.14590 q^{79} -13.0344 q^{83} -3.85410 q^{85} -0.472136 q^{89} -9.18034 q^{91} +6.23607 q^{95} -14.5623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - q^{7} + 7 q^{13} - 7 q^{17} + q^{19} - 4 q^{23} - 7 q^{25} - 15 q^{29} - 5 q^{31} - 8 q^{35} + 3 q^{37} - 15 q^{41} - 5 q^{47} + 9 q^{49} - 3 q^{53} + 9 q^{59} + 7 q^{61} + q^{65} + 8 q^{67}+ \cdots - 9 q^{97}+O(q^{100}) $ 2 * q + q^5 - q^7 + 7 * q^13 - 7 * q^17 + q^19 - 4 * q^23 - 7 * q^25 - 15 * q^29 - 5 * q^31 - 8 * q^35 + 3 * q^37 - 15 * q^41 - 5 * q^47 + 9 * q^49 - 3 * q^53 + 9 * q^59 + 7 * q^61 + q^65 + 8 * q^67 + 15 * q^71 - 13 * q^73 - 21 * q^79 + 3 * q^83 - q^85 + 8 * q^89 + 4 * q^91 + 8 * q^95 - 9 * 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
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.38197	0.660639	0.330319	−	0.943869i	$-0.392844\pi$
$13$	2.38197	0.660639	0.330319	+	0.943869i	$0.392844\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.38197	−0.577712	−0.288856	−	0.957373i	$-0.593275\pi$
$17$	−2.38197	−0.577712	−0.288856	+	0.957373i	$0.593275\pi$
$18$	0	0
$19$	3.85410	0.884192	0.442096	−	0.896968i	$-0.354235\pi$
$19$	3.85410	0.884192	0.442096	+	0.896968i	$0.354235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.61803	−1.60033	−0.800164	−	0.599781i	$-0.795254\pi$
$29$	−8.61803	−1.60033	−0.800164	+	0.599781i	$0.795254\pi$
$30$	0	0
$31$	0.854102	0.153401	0.0767006	−	0.997054i	$-0.475561\pi$
$31$	0.854102	0.153401	0.0767006	+	0.997054i	$0.475561\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.23607	−1.05409
$36$	0	0
$37$	−1.85410	−0.304812	−0.152406	−	0.988318i	$-0.548702\pi$
$37$	−1.85410	−0.304812	−0.152406	+	0.988318i	$0.548702\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.61803	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$41$	−8.61803	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.38197	−0.201580	−0.100790	−	0.994908i	$-0.532137\pi$
$47$	−1.38197	−0.201580	−0.100790	+	0.994908i	$0.532137\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.09017	0.561828	0.280914	−	0.959733i	$-0.409362\pi$
$53$	4.09017	0.561828	0.280914	+	0.959733i	$0.409362\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.09017	−0.141928	−0.0709640	−	0.997479i	$-0.522608\pi$
$59$	−1.09017	−0.141928	−0.0709640	+	0.997479i	$0.522608\pi$
$60$	0	0
$61$	2.38197	0.304979	0.152490	−	0.988305i	$-0.451271\pi$
$61$	2.38197	0.304979	0.152490	+	0.988305i	$0.451271\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.85410	0.478043
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.38197	0.757400	0.378700	−	0.925519i	$-0.376371\pi$
$71$	6.38197	0.757400	0.378700	+	0.925519i	$0.376371\pi$
$72$	0	0
$73$	−0.909830	−0.106488	−0.0532438	−	0.998582i	$-0.516956\pi$
$73$	−0.909830	−0.106488	−0.0532438	+	0.998582i	$0.516956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.14590	−0.803976	−0.401988	−	0.915645i	$-0.631681\pi$
$79$	−7.14590	−0.803976	−0.401988	+	0.915645i	$0.631681\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.0344	−1.43072	−0.715358	−	0.698758i	$-0.753736\pi$
$83$	−13.0344	−1.43072	−0.715358	+	0.698758i	$0.753736\pi$
$84$	0	0
$85$	−3.85410	−0.418036
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.472136	−0.0500463	−0.0250232	−	0.999687i	$-0.507966\pi$
$89$	−0.472136	−0.0500463	−0.0250232	+	0.999687i	$0.507966\pi$
$90$	0	0
$91$	−9.18034	−0.962361
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.23607	0.639807
$96$	0	0
$97$	−14.5623	−1.47858	−0.739289	−	0.673388i	$-0.764838\pi$
$97$	−14.5623	−1.47858	−0.739289	+	0.673388i	$0.764838\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.7984	−1.77100	−0.885502	−	0.464635i	$-0.846185\pi$
$101$	−17.7984	−1.77100	−0.885502	+	0.464635i	$0.846185\pi$
$102$	0	0
$103$	−7.85410	−0.773888	−0.386944	−	0.922103i	$-0.626469\pi$
$103$	−7.85410	−0.773888	−0.386944	+	0.922103i	$0.626469\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.5623	1.11777	0.558885	−	0.829245i	$-0.311229\pi$
$107$	11.5623	1.11777	0.558885	+	0.829245i	$0.311229\pi$
$108$	0	0
$109$	−12.4721	−1.19461	−0.597307	−	0.802013i	$-0.703763\pi$
$109$	−12.4721	−1.19461	−0.597307	+	0.802013i	$0.703763\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.5623	−1.65212	−0.826061	−	0.563580i	$-0.809424\pi$
$113$	−17.5623	−1.65212	−0.826061	+	0.563580i	$0.809424\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.18034	0.841560
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	−2.38197	−0.211365	−0.105683	−	0.994400i	$-0.533703\pi$
$127$	−2.38197	−0.211365	−0.105683	+	0.994400i	$0.533703\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.52786	−0.832453	−0.416227	−	0.909261i	$-0.636648\pi$
$131$	−9.52786	−0.832453	−0.416227	+	0.909261i	$0.636648\pi$
$132$	0	0
$133$	−14.8541	−1.28801
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.7984	−1.17887	−0.589437	−	0.807814i	$-0.700650\pi$
$137$	−13.7984	−1.17887	−0.589437	+	0.807814i	$0.700650\pi$
$138$	0	0
$139$	11.5623	0.980702	0.490351	−	0.871525i	$-0.336869\pi$
$139$	11.5623	0.980702	0.490351	+	0.871525i	$0.336869\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−13.9443	−1.15801
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.38197	−0.195138	−0.0975691	−	0.995229i	$-0.531107\pi$
$149$	−2.38197	−0.195138	−0.0975691	+	0.995229i	$0.531107\pi$
$150$	0	0
$151$	19.2705	1.56821	0.784106	−	0.620627i	$-0.213122\pi$
$151$	19.2705	1.56821	0.784106	+	0.620627i	$0.213122\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.38197	0.111002
$156$	0	0
$157$	17.5623	1.40162	0.700812	−	0.713346i	$-0.252821\pi$
$157$	17.5623	1.40162	0.700812	+	0.713346i	$0.252821\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.52786	−0.750901
$162$	0	0
$163$	11.3262	0.887139	0.443570	−	0.896240i	$-0.353712\pi$
$163$	11.3262	0.887139	0.443570	+	0.896240i	$0.353712\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.7984	−1.37728	−0.688640	−	0.725104i	$-0.741792\pi$
$167$	−17.7984	−1.37728	−0.688640	+	0.725104i	$0.741792\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.79837	0.516871	0.258435	−	0.966029i	$-0.416793\pi$
$173$	6.79837	0.516871	0.258435	+	0.966029i	$0.416793\pi$
$174$	0	0
$175$	9.18034	0.693968
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.61803	0.195681	0.0978405	−	0.995202i	$-0.468806\pi$
$179$	2.61803	0.195681	0.0978405	+	0.995202i	$0.468806\pi$
$180$	0	0
$181$	−1.61803	−0.120268	−0.0601338	−	0.998190i	$-0.519153\pi$
$181$	−1.61803	−0.120268	−0.0601338	+	0.998190i	$0.519153\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.0344	−1.88379	−0.941893	−	0.335913i	$-0.890955\pi$
$191$	−26.0344	−1.88379	−0.941893	+	0.335913i	$0.890955\pi$
$192$	0	0
$193$	−22.5623	−1.62407	−0.812035	−	0.583609i	$-0.801640\pi$
$193$	−22.5623	−1.62407	−0.812035	+	0.583609i	$0.801640\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	0	0
$199$	−0.944272	−0.0669377	−0.0334688	−	0.999440i	$-0.510655\pi$
$199$	−0.944272	−0.0669377	−0.0334688	+	0.999440i	$0.510655\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	33.2148	2.33122
$204$	0	0
$205$	−13.9443	−0.973910
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	11.9098	0.819907	0.409953	−	0.912107i	$-0.365545\pi$
$211$	11.9098	0.819907	0.409953	+	0.912107i	$0.365545\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.29180	−0.223462
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.67376	−0.381659
$222$	0	0
$223$	−12.1459	−0.813349	−0.406675	−	0.913573i	$-0.633312\pi$
$223$	−12.1459	−0.813349	−0.406675	+	0.913573i	$0.633312\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.67376	−0.376581	−0.188290	−	0.982113i	$-0.560295\pi$
$227$	−5.67376	−0.376581	−0.188290	+	0.982113i	$0.560295\pi$
$228$	0	0
$229$	16.2705	1.07519	0.537593	−	0.843205i	$-0.319334\pi$
$229$	16.2705	1.07519	0.537593	+	0.843205i	$0.319334\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.38197	−0.156048	−0.0780239	−	0.996951i	$-0.524861\pi$
$233$	−2.38197	−0.156048	−0.0780239	+	0.996951i	$0.524861\pi$
$234$	0	0
$235$	−2.23607	−0.145865
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.5623	0.747903	0.373952	−	0.927448i	$-0.378003\pi$
$239$	11.5623	0.747903	0.373952	+	0.927448i	$0.378003\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.7082	0.811897
$246$	0	0
$247$	9.18034	0.584131
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.7984	−1.62838	−0.814189	−	0.580599i	$-0.802818\pi$
$251$	−25.7984	−1.62838	−0.814189	+	0.580599i	$0.802818\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.90983	0.306267	0.153133	−	0.988206i	$-0.451064\pi$
$257$	4.90983	0.306267	0.153133	+	0.988206i	$0.451064\pi$
$258$	0	0
$259$	7.14590	0.444024
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.88854	−0.363103	−0.181552	−	0.983381i	$-0.558112\pi$
$263$	−5.88854	−0.363103	−0.181552	+	0.983381i	$0.558112\pi$
$264$	0	0
$265$	6.61803	0.406543
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.0902	1.46880	0.734402	−	0.678715i	$-0.237463\pi$
$269$	24.0902	1.46880	0.734402	+	0.678715i	$0.237463\pi$
$270$	0	0
$271$	−19.2705	−1.17060	−0.585300	−	0.810817i	$-0.699023\pi$
$271$	−19.2705	−1.17060	−0.585300	+	0.810817i	$0.699023\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.5623	−1.35564	−0.677819	−	0.735229i	$-0.737075\pi$
$277$	−22.5623	−1.35564	−0.677819	+	0.735229i	$0.737075\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−27.3262	−1.63015	−0.815073	−	0.579358i	$-0.803303\pi$
$281$	−27.3262	−1.63015	−0.815073	+	0.579358i	$0.803303\pi$
$282$	0	0
$283$	−11.5623	−0.687308	−0.343654	−	0.939096i	$-0.611665\pi$
$283$	−11.5623	−0.687308	−0.343654	+	0.939096i	$0.611665\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.2148	1.96061
$288$	0	0
$289$	−11.3262	−0.666249
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.61803	−0.503471	−0.251735	−	0.967796i	$-0.581001\pi$
$293$	−8.61803	−0.503471	−0.251735	+	0.967796i	$0.581001\pi$
$294$	0	0
$295$	−1.76393	−0.102700
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.88854	0.340543
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.85410	0.220685
$306$	0	0
$307$	9.52786	0.543784	0.271892	−	0.962328i	$-0.412351\pi$
$307$	9.52786	0.543784	0.271892	+	0.962328i	$0.412351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.7984	−0.952548	−0.476274	−	0.879297i	$-0.658013\pi$
$311$	−16.7984	−0.952548	−0.476274	+	0.879297i	$0.658013\pi$
$312$	0	0
$313$	−7.50658	−0.424297	−0.212148	−	0.977237i	$-0.568046\pi$
$313$	−7.50658	−0.424297	−0.212148	+	0.977237i	$0.568046\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.5623	0.817901	0.408950	−	0.912557i	$-0.365895\pi$
$317$	14.5623	0.817901	0.408950	+	0.912557i	$0.365895\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−9.18034	−0.510808
$324$	0	0
$325$	−5.67376	−0.314724
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.32624	0.293645
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	20.9443	1.14431
$336$	0	0
$337$	24.0344	1.30924	0.654620	−	0.755958i	$-0.272829\pi$
$337$	24.0344	1.30924	0.654620	+	0.755958i	$0.272829\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.7984	0.955467	0.477733	−	0.878505i	$-0.341458\pi$
$347$	17.7984	0.955467	0.477733	+	0.878505i	$0.341458\pi$
$348$	0	0
$349$	−16.3262	−0.873923	−0.436962	−	0.899480i	$-0.643945\pi$
$349$	−16.3262	−0.873923	−0.436962	+	0.899480i	$0.643945\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.9443	0.795403	0.397702	−	0.917515i	$-0.369808\pi$
$353$	14.9443	0.795403	0.397702	+	0.917515i	$0.369808\pi$
$354$	0	0
$355$	10.3262	0.548060
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.5623	−0.610235	−0.305118	−	0.952315i	$-0.598696\pi$
$359$	−11.5623	−0.610235	−0.305118	+	0.952315i	$0.598696\pi$
$360$	0	0
$361$	−4.14590	−0.218205
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.47214	−0.0770551
$366$	0	0
$367$	20.1459	1.05161	0.525804	−	0.850606i	$-0.323765\pi$
$367$	20.1459	1.05161	0.525804	+	0.850606i	$0.323765\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.7639	−0.818423
$372$	0	0
$373$	−6.58359	−0.340885	−0.170443	−	0.985368i	$-0.554520\pi$
$373$	−6.58359	−0.340885	−0.170443	+	0.985368i	$0.554520\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20.5279	−1.05724
$378$	0	0
$379$	3.14590	0.161594	0.0807970	−	0.996731i	$-0.474253\pi$
$379$	3.14590	0.161594	0.0807970	+	0.996731i	$0.474253\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.9098	−1.01734	−0.508672	−	0.860960i	$-0.669864\pi$
$383$	−19.9098	−1.01734	−0.508672	+	0.860960i	$0.669864\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.7984	1.35873	0.679366	−	0.733800i	$-0.262255\pi$
$389$	26.7984	1.35873	0.679366	+	0.733800i	$0.262255\pi$
$390$	0	0
$391$	−5.88854	−0.297796
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.5623	−0.581763
$396$	0	0
$397$	32.8328	1.64783	0.823916	−	0.566712i	$-0.191785\pi$
$397$	32.8328	1.64783	0.823916	+	0.566712i	$0.191785\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.74265	−0.336712	−0.168356	−	0.985726i	$-0.553846\pi$
$401$	−6.74265	−0.336712	−0.168356	+	0.985726i	$0.553846\pi$
$402$	0	0
$403$	2.03444	0.101343
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7.14590	−0.353342	−0.176671	−	0.984270i	$-0.556533\pi$
$409$	−7.14590	−0.353342	−0.176671	+	0.984270i	$0.556533\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.20163	0.206749
$414$	0	0
$415$	−21.0902	−1.03528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.58359	0.223923	0.111962	−	0.993713i	$-0.464287\pi$
$419$	4.58359	0.223923	0.111962	+	0.993713i	$0.464287\pi$
$420$	0	0
$421$	−9.27051	−0.451817	−0.225909	−	0.974149i	$-0.572535\pi$
$421$	−9.27051	−0.451817	−0.225909	+	0.974149i	$0.572535\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.67376	0.275218
$426$	0	0
$427$	−9.18034	−0.444268
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.6738	0.803147	0.401573	−	0.915827i	$-0.368463\pi$
$431$	16.6738	0.803147	0.401573	+	0.915827i	$0.368463\pi$
$432$	0	0
$433$	19.4508	0.934748	0.467374	−	0.884060i	$-0.345200\pi$
$433$	19.4508	0.934748	0.467374	+	0.884060i	$0.345200\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.52786	0.455780
$438$	0	0
$439$	−15.4164	−0.735785	−0.367893	−	0.929868i	$-0.619921\pi$
$439$	−15.4164	−0.735785	−0.367893	+	0.929868i	$0.619921\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.27051	−0.155387	−0.0776933	−	0.996977i	$-0.524755\pi$
$443$	−3.27051	−0.155387	−0.0776933	+	0.996977i	$0.524755\pi$
$444$	0	0
$445$	−0.763932	−0.0362139
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.5623	0.687238	0.343619	−	0.939109i	$-0.388347\pi$
$449$	14.5623	0.687238	0.343619	+	0.939109i	$0.388347\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.8541	−0.696371
$456$	0	0
$457$	−7.14590	−0.334271	−0.167136	−	0.985934i	$-0.553452\pi$
$457$	−7.14590	−0.334271	−0.167136	+	0.985934i	$0.553452\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.7771	1.57316	0.786578	−	0.617491i	$-0.211851\pi$
$461$	33.7771	1.57316	0.786578	+	0.617491i	$0.211851\pi$
$462$	0	0
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.6738	−0.586472	−0.293236	−	0.956040i	$-0.594732\pi$
$467$	−12.6738	−0.586472	−0.293236	+	0.956040i	$0.594732\pi$
$468$	0	0
$469$	−49.8885	−2.30364
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−9.18034	−0.421223
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.3262	1.24857	0.624284	−	0.781198i	$-0.285391\pi$
$479$	27.3262	1.24857	0.624284	+	0.781198i	$0.285391\pi$
$480$	0	0
$481$	−4.41641	−0.201371
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.5623	−1.06991
$486$	0	0
$487$	−18.0344	−0.817219	−0.408609	−	0.912709i	$-0.633986\pi$
$487$	−18.0344	−0.817219	−0.408609	+	0.912709i	$0.633986\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.5623	−0.521800	−0.260900	−	0.965366i	$-0.584019\pi$
$491$	−11.5623	−0.521800	−0.260900	+	0.965366i	$0.584019\pi$
$492$	0	0
$493$	20.5279	0.924528
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.5967	−1.10331
$498$	0	0
$499$	−23.2705	−1.04173	−0.520866	−	0.853639i	$-0.674391\pi$
$499$	−23.2705	−1.04173	−0.520866	+	0.853639i	$0.674391\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.5066	1.62775	0.813874	−	0.581042i	$-0.197355\pi$
$503$	36.5066	1.62775	0.813874	+	0.581042i	$0.197355\pi$
$504$	0	0
$505$	−28.7984	−1.28151
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.6869	1.27153	0.635763	−	0.771885i	$-0.280686\pi$
$509$	28.6869	1.27153	0.635763	+	0.771885i	$0.280686\pi$
$510$	0	0
$511$	3.50658	0.155122
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.7082	−0.559990
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.67376	−0.336194	−0.168097	−	0.985770i	$-0.553762\pi$
$521$	−7.67376	−0.336194	−0.168097	+	0.985770i	$0.553762\pi$
$522$	0	0
$523$	28.4508	1.24407	0.622034	−	0.782990i	$-0.286306\pi$
$523$	28.4508	1.24407	0.622034	+	0.782990i	$0.286306\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.03444	−0.0886217
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−20.5279	−0.889160
$534$	0	0
$535$	18.7082	0.808826
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.27051	0.355577	0.177788	−	0.984069i	$-0.443106\pi$
$541$	8.27051	0.355577	0.177788	+	0.984069i	$0.443106\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−20.1803	−0.864431
$546$	0	0
$547$	38.3262	1.63871	0.819356	−	0.573285i	$-0.194331\pi$
$547$	38.3262	1.63871	0.819356	+	0.573285i	$0.194331\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−33.2148	−1.41500
$552$	0	0
$553$	27.5410	1.17116
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.7426	1.34498	0.672490	−	0.740107i	$-0.265225\pi$
$557$	31.7426	1.34498	0.672490	+	0.740107i	$0.265225\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.0344	0.549336	0.274668	−	0.961539i	$-0.411432\pi$
$563$	13.0344	0.549336	0.274668	+	0.961539i	$0.411432\pi$
$564$	0	0
$565$	−28.4164	−1.19549
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.9230	−1.25444	−0.627218	−	0.778843i	$-0.715807\pi$
$569$	−29.9230	−1.25444	−0.627218	+	0.778843i	$0.715807\pi$
$570$	0	0
$571$	9.52786	0.398729	0.199364	−	0.979925i	$-0.436112\pi$
$571$	9.52786	0.398729	0.199364	+	0.979925i	$0.436112\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.88854	−0.245569
$576$	0	0
$577$	−36.0902	−1.50245	−0.751227	−	0.660044i	$-0.770538\pi$
$577$	−36.0902	−1.50245	−0.751227	+	0.660044i	$0.770538\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	50.2361	2.08414
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.85410	0.324173	0.162087	−	0.986777i	$-0.448178\pi$
$587$	7.85410	0.324173	0.162087	+	0.986777i	$0.448178\pi$
$588$	0	0
$589$	3.29180	0.135636
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.4164	−1.53651	−0.768254	−	0.640145i	$-0.778874\pi$
$593$	−37.4164	−1.53651	−0.768254	+	0.640145i	$0.778874\pi$
$594$	0	0
$595$	14.8541	0.608959
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.9787	−1.87864	−0.939320	−	0.343043i	$-0.888542\pi$
$599$	−45.9787	−1.87864	−0.939320	+	0.343043i	$0.888542\pi$
$600$	0	0
$601$	33.5623	1.36904	0.684518	−	0.728996i	$-0.260013\pi$
$601$	33.5623	1.36904	0.684518	+	0.728996i	$0.260013\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.4377	0.870129	0.435065	−	0.900399i	$-0.356726\pi$
$607$	21.4377	0.870129	0.435065	+	0.900399i	$0.356726\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.29180	−0.133172
$612$	0	0
$613$	−10.4377	−0.421574	−0.210787	−	0.977532i	$-0.567603\pi$
$613$	−10.4377	−0.421574	−0.210787	+	0.977532i	$0.567603\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.52786	0.303060	0.151530	−	0.988453i	$-0.451580\pi$
$617$	7.52786	0.303060	0.151530	+	0.988453i	$0.451580\pi$
$618$	0	0
$619$	22.3262	0.897367	0.448684	−	0.893691i	$-0.351893\pi$
$619$	22.3262	0.897367	0.448684	+	0.893691i	$0.351893\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.81966	0.0729031
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.41641	0.176094
$630$	0	0
$631$	−23.8541	−0.949617	−0.474808	−	0.880089i	$-0.657483\pi$
$631$	−23.8541	−0.949617	−0.474808	+	0.880089i	$0.657483\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.85410	−0.152945
$636$	0	0
$637$	18.7082	0.741246
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.0902	−1.07000	−0.534999	−	0.844853i	$-0.679688\pi$
$641$	−27.0902	−1.07000	−0.534999	+	0.844853i	$0.679688\pi$
$642$	0	0
$643$	−12.2705	−0.483902	−0.241951	−	0.970289i	$-0.577787\pi$
$643$	−12.2705	−0.483902	−0.241951	+	0.970289i	$0.577787\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.74265	−0.265081	−0.132540	−	0.991178i	$-0.542313\pi$
$647$	−6.74265	−0.265081	−0.132540	+	0.991178i	$0.542313\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.67376	−0.300298	−0.150149	−	0.988663i	$-0.547975\pi$
$653$	−7.67376	−0.300298	−0.150149	+	0.988663i	$0.547975\pi$
$654$	0	0
$655$	−15.4164	−0.602369
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.3607	−1.57223	−0.786114	−	0.618081i	$-0.787910\pi$
$659$	−40.3607	−1.57223	−0.786114	+	0.618081i	$0.787910\pi$
$660$	0	0
$661$	−30.3607	−1.18089	−0.590447	−	0.807077i	$-0.701048\pi$
$661$	−30.3607	−1.18089	−0.590447	+	0.807077i	$0.701048\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0344	−0.932016
$666$	0	0
$667$	−21.3050	−0.824931
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	36.8541	1.42062	0.710311	−	0.703888i	$-0.248554\pi$
$673$	36.8541	1.42062	0.710311	+	0.703888i	$0.248554\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.50658	0.134769	0.0673844	−	0.997727i	$-0.478535\pi$
$677$	3.50658	0.134769	0.0673844	+	0.997727i	$0.478535\pi$
$678$	0	0
$679$	56.1246	2.15386
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.944272	0.0361316	0.0180658	−	0.999837i	$-0.494249\pi$
$683$	0.944272	0.0361316	0.0180658	+	0.999837i	$0.494249\pi$
$684$	0	0
$685$	−22.3262	−0.853042
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.74265	0.371165
$690$	0	0
$691$	−24.0902	−0.916433	−0.458217	−	0.888841i	$-0.651512\pi$
$691$	−24.0902	−0.916433	−0.458217	+	0.888841i	$0.651512\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.7082	0.709643
$696$	0	0
$697$	20.5279	0.777548
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−48.9787	−1.84990	−0.924950	−	0.380088i	$-0.875894\pi$
$701$	−48.9787	−1.84990	−0.924950	+	0.380088i	$0.875894\pi$
$702$	0	0
$703$	−7.14590	−0.269513
$704$	0	0
$705$	0	0
$706$	0	0
$707$	68.5967	2.57985
$708$	0	0
$709$	23.3262	0.876035	0.438018	−	0.898966i	$-0.355681\pi$
$709$	23.3262	0.876035	0.438018	+	0.898966i	$0.355681\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.11146	0.0790747
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.2705	1.46454	0.732271	−	0.681013i	$-0.238460\pi$
$719$	39.2705	1.46454	0.732271	+	0.681013i	$0.238460\pi$
$720$	0	0
$721$	30.2705	1.12733
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.5279	0.762386
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.5623	1.23965	0.619826	−	0.784739i	$-0.287203\pi$
$733$	33.5623	1.23965	0.619826	+	0.784739i	$0.287203\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	23.6869	0.871337	0.435669	−	0.900107i	$-0.356512\pi$
$739$	23.6869	0.871337	0.435669	+	0.900107i	$0.356512\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.50658	−0.128644	−0.0643219	−	0.997929i	$-0.520488\pi$
$743$	−3.50658	−0.128644	−0.0643219	+	0.997929i	$0.520488\pi$
$744$	0	0
$745$	−3.85410	−0.141203
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−44.5623	−1.62827
$750$	0	0
$751$	6.90983	0.252143	0.126072	−	0.992021i	$-0.459763\pi$
$751$	6.90983	0.252143	0.126072	+	0.992021i	$0.459763\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	31.1803	1.13477
$756$	0	0
$757$	26.7426	0.971978	0.485989	−	0.873965i	$-0.338460\pi$
$757$	26.7426	0.971978	0.485989	+	0.873965i	$0.338460\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.9230	0.685958	0.342979	−	0.939343i	$-0.388564\pi$
$761$	18.9230	0.685958	0.342979	+	0.939343i	$0.388564\pi$
$762$	0	0
$763$	48.0689	1.74021
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.59675	−0.0937631
$768$	0	0
$769$	37.4164	1.34927	0.674635	−	0.738151i	$-0.264301\pi$
$769$	37.4164	1.34927	0.674635	+	0.738151i	$0.264301\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.3262	1.01882	0.509412	−	0.860523i	$-0.329863\pi$
$773$	28.3262	1.01882	0.509412	+	0.860523i	$0.329863\pi$
$774$	0	0
$775$	−2.03444	−0.0730793
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−33.2148	−1.19004
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	28.4164	1.01423
$786$	0	0
$787$	16.6738	0.594355	0.297178	−	0.954822i	$-0.403955\pi$
$787$	16.6738	0.594355	0.297178	+	0.954822i	$0.403955\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	67.6869	2.40667
$792$	0	0
$793$	5.67376	0.201481
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.7984	−0.488763	−0.244382	−	0.969679i	$-0.578585\pi$
$797$	−13.7984	−0.488763	−0.244382	+	0.969679i	$0.578585\pi$
$798$	0	0
$799$	3.29180	0.116455
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−15.4164	−0.543357
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.9230	0.665297	0.332648	−	0.943051i	$-0.392058\pi$
$809$	18.9230	0.665297	0.332648	+	0.943051i	$0.392058\pi$
$810$	0	0
$811$	−30.6180	−1.07514	−0.537572	−	0.843218i	$-0.680658\pi$
$811$	−30.6180	−1.07514	−0.537572	+	0.843218i	$0.680658\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.3262	0.641940
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.8541	0.902314	0.451157	−	0.892445i	$-0.351011\pi$
$821$	25.8541	0.902314	0.451157	+	0.892445i	$0.351011\pi$
$822$	0	0
$823$	−39.5066	−1.37711	−0.688556	−	0.725183i	$-0.741755\pi$
$823$	−39.5066	−1.37711	−0.688556	+	0.725183i	$0.741755\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.0344	0.453252	0.226626	−	0.973982i	$-0.427231\pi$
$827$	13.0344	0.453252	0.226626	+	0.973982i	$0.427231\pi$
$828$	0	0
$829$	34.8673	1.21099	0.605495	−	0.795849i	$-0.292975\pi$
$829$	34.8673	1.21099	0.605495	+	0.795849i	$0.292975\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.7082	−0.648201
$834$	0	0
$835$	−28.7984	−0.996609
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.61803	0.0903846	0.0451923	−	0.998978i	$-0.485610\pi$
$839$	2.61803	0.0903846	0.0451923	+	0.998978i	$0.485610\pi$
$840$	0	0
$841$	45.2705	1.56105
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.8541	−0.407794
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.58359	−0.157124
$852$	0	0
$853$	17.7984	0.609405	0.304702	−	0.952448i	$-0.401443\pi$
$853$	17.7984	0.609405	0.304702	+	0.952448i	$0.401443\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.94427	−0.100574	−0.0502872	−	0.998735i	$-0.516014\pi$
$857$	−2.94427	−0.100574	−0.0502872	+	0.998735i	$0.516014\pi$
$858$	0	0
$859$	−31.0557	−1.05961	−0.529804	−	0.848120i	$-0.677734\pi$
$859$	−31.0557	−1.05961	−0.529804	+	0.848120i	$0.677734\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.2148	0.858321	0.429161	−	0.903228i	$-0.358809\pi$
$863$	25.2148	0.858321	0.429161	+	0.903228i	$0.358809\pi$
$864$	0	0
$865$	11.0000	0.374011
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	30.8328	1.04473
$872$	0	0
$873$	0	0
$874$	0	0
$875$	46.0344	1.55625
$876$	0	0
$877$	27.6738	0.934477	0.467238	−	0.884131i	$-0.345249\pi$
$877$	27.6738	0.934477	0.467238	+	0.884131i	$0.345249\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−15.8885	−0.535299	−0.267649	−	0.963516i	$-0.586247\pi$
$881$	−15.8885	−0.535299	−0.267649	+	0.963516i	$0.586247\pi$
$882$	0	0
$883$	5.02129	0.168980	0.0844899	−	0.996424i	$-0.473074\pi$
$883$	5.02129	0.168980	0.0844899	+	0.996424i	$0.473074\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.03444	−0.0683099	−0.0341549	−	0.999417i	$-0.510874\pi$
$887$	−2.03444	−0.0683099	−0.0341549	+	0.999417i	$0.510874\pi$
$888$	0	0
$889$	9.18034	0.307899
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.32624	−0.178236
$894$	0	0
$895$	4.23607	0.141596
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.36068	−0.245492
$900$	0	0
$901$	−9.74265	−0.324575
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.61803	−0.0870264
$906$	0	0
$907$	1.97871	0.0657021	0.0328511	−	0.999460i	$-0.489541\pi$
$907$	1.97871	0.0657021	0.0328511	+	0.999460i	$0.489541\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.5066	0.646282	0.323141	−	0.946351i	$-0.395261\pi$
$911$	19.5066	0.646282	0.323141	+	0.946351i	$0.395261\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.7214	1.21265
$918$	0	0
$919$	−21.4377	−0.707164	−0.353582	−	0.935403i	$-0.615037\pi$
$919$	−21.4377	−0.707164	−0.353582	+	0.935403i	$0.615037\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.2016	0.500368
$924$	0	0
$925$	4.41641	0.145211
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.9098	−0.653220	−0.326610	−	0.945159i	$-0.605906\pi$
$929$	−19.9098	−0.653220	−0.326610	+	0.945159i	$0.605906\pi$
$930$	0	0
$931$	30.2705	0.992076
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.14590	−0.233446	−0.116723	−	0.993164i	$-0.537239\pi$
$937$	−7.14590	−0.233446	−0.116723	+	0.993164i	$0.537239\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.9098	−0.388249	−0.194125	−	0.980977i	$-0.562187\pi$
$941$	−11.9098	−0.388249	−0.194125	+	0.980977i	$0.562187\pi$
$942$	0	0
$943$	−21.3050	−0.693785
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.7771	0.772652	0.386326	−	0.922362i	$-0.373744\pi$
$947$	23.7771	0.772652	0.386326	+	0.922362i	$0.373744\pi$
$948$	0	0
$949$	−2.16718	−0.0703498
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45.3394	−1.46869	−0.734344	−	0.678778i	$-0.762510\pi$
$953$	−45.3394	−1.46869	−0.734344	+	0.678778i	$0.762510\pi$
$954$	0	0
$955$	−42.1246	−1.36312
$956$	0	0
$957$	0	0
$958$	0	0
$959$	53.1803	1.71728
$960$	0	0
$961$	−30.2705	−0.976468
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−36.5066	−1.17519
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.90983	−0.221747	−0.110873	−	0.993835i	$-0.535365\pi$
$971$	−6.90983	−0.221747	−0.110873	+	0.993835i	$0.535365\pi$
$972$	0	0
$973$	−44.5623	−1.42860
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.4508	0.654281	0.327140	−	0.944976i	$-0.393915\pi$
$977$	20.4508	0.654281	0.327140	+	0.944976i	$0.393915\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.2705	1.25253	0.626267	−	0.779608i	$-0.284582\pi$
$983$	39.2705	1.25253	0.626267	+	0.779608i	$0.284582\pi$
$984$	0	0
$985$	4.76393	0.151791
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.52786	−0.0484365
$996$	0	0
$997$	29.9230	0.947670	0.473835	−	0.880614i	$-0.342869\pi$
$997$	29.9230	0.947670	0.473835	+	0.880614i	$0.342869\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.2.a.t.1.2		2
3.2	odd	2		484.2.a.b.1.1		2
11.5	even	5		396.2.j.a.289.1		4
11.9	even	5		396.2.j.a.37.1		4
11.10	odd	2		4356.2.a.u.1.2		2
12.11	even	2		1936.2.a.ba.1.2		2
24.5	odd	2		7744.2.a.da.1.2		2
24.11	even	2		7744.2.a.bp.1.1		2
33.2	even	10		484.2.e.c.81.1		4
33.5	odd	10		44.2.e.a.25.1	✓	4
33.8	even	10		484.2.e.d.9.1		4
33.14	odd	10		484.2.e.e.9.1		4
33.17	even	10		484.2.e.c.245.1		4
33.20	odd	10		44.2.e.a.37.1	yes	4
33.26	odd	10		484.2.e.e.269.1		4
33.29	even	10		484.2.e.d.269.1		4
33.32	even	2		484.2.a.c.1.1		2
132.71	even	10		176.2.m.b.113.1		4
132.119	even	10		176.2.m.b.81.1		4
132.131	odd	2		1936.2.a.z.1.2		2
165.38	even	20		1100.2.cb.a.949.2		8
165.53	even	20		1100.2.cb.a.1049.1		8
165.104	odd	10		1100.2.n.a.201.1		4
165.119	odd	10		1100.2.n.a.301.1		4
165.137	even	20		1100.2.cb.a.949.1		8
165.152	even	20		1100.2.cb.a.1049.2		8
264.5	odd	10		704.2.m.e.641.1		4
264.53	odd	10		704.2.m.e.257.1		4
264.131	odd	2		7744.2.a.bo.1.1		2
264.197	even	2		7744.2.a.db.1.2		2
264.203	even	10		704.2.m.d.641.1		4
264.251	even	10		704.2.m.d.257.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.e.a.25.1	✓	4	33.5	odd	10
44.2.e.a.37.1	yes	4	33.20	odd	10
176.2.m.b.81.1		4	132.119	even	10
176.2.m.b.113.1		4	132.71	even	10
396.2.j.a.37.1		4	11.9	even	5
396.2.j.a.289.1		4	11.5	even	5
484.2.a.b.1.1		2	3.2	odd	2
484.2.a.c.1.1		2	33.32	even	2
484.2.e.c.81.1		4	33.2	even	10
484.2.e.c.245.1		4	33.17	even	10
484.2.e.d.9.1		4	33.8	even	10
484.2.e.d.269.1		4	33.29	even	10
484.2.e.e.9.1		4	33.14	odd	10
484.2.e.e.269.1		4	33.26	odd	10
704.2.m.d.257.1		4	264.251	even	10
704.2.m.d.641.1		4	264.203	even	10
704.2.m.e.257.1		4	264.53	odd	10
704.2.m.e.641.1		4	264.5	odd	10
1100.2.n.a.201.1		4	165.104	odd	10
1100.2.n.a.301.1		4	165.119	odd	10
1100.2.cb.a.949.1		8	165.137	even	20
1100.2.cb.a.949.2		8	165.38	even	20
1100.2.cb.a.1049.1		8	165.53	even	20
1100.2.cb.a.1049.2		8	165.152	even	20
1936.2.a.z.1.2		2	132.131	odd	2
1936.2.a.ba.1.2		2	12.11	even	2
4356.2.a.t.1.2		2	1.1	even	1	trivial
4356.2.a.u.1.2		2	11.10	odd	2
7744.2.a.bo.1.1		2	264.131	odd	2
7744.2.a.bp.1.1		2	24.11	even	2
7744.2.a.da.1.2		2	24.5	odd	2
7744.2.a.db.1.2		2	264.197	even	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:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4356.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{5} -3.85410 q^{7} +2.38197 q^{13} -2.38197 q^{17} +3.85410 q^{19} +2.47214 q^{23} -2.38197 q^{25} -8.61803 q^{29} +0.854102 q^{31} -6.23607 q^{35} -1.85410 q^{37} -8.61803 q^{41} -1.38197 q^{47} +7.85410 q^{49} +4.09017 q^{53} -1.09017 q^{59} +2.38197 q^{61} +3.85410 q^{65} +12.9443 q^{67} +6.38197 q^{71} -0.909830 q^{73} -7.14590 q^{79} -13.0344 q^{83} -3.85410 q^{85} -0.472136 q^{89} -9.18034 q^{91} +6.23607 q^{95} -14.5623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - q^{7} + 7 q^{13} - 7 q^{17} + q^{19} - 4 q^{23} - 7 q^{25} - 15 q^{29} - 5 q^{31} - 8 q^{35} + 3 q^{37} - 15 q^{41} - 5 q^{47} + 9 q^{49} - 3 q^{53} + 9 q^{59} + 7 q^{61} + q^{65} + 8 q^{67}+ \cdots - 9 q^{97}+O(q^{100}) \) 2 * q + q^5 - q^7 + 7 * q^13 - 7 * q^17 + q^19 - 4 * q^23 - 7 * q^25 - 15 * q^29 - 5 * q^31 - 8 * q^35 + 3 * q^37 - 15 * q^41 - 5 * q^47 + 9 * q^49 - 3 * q^53 + 9 * q^59 + 7 * q^61 + q^65 + 8 * q^67 + 15 * q^71 - 13 * q^73 - 21 * q^79 + 3 * q^83 - q^85 + 8 * q^89 + 4 * q^91 + 8 * q^95 - 9 * q^97

Introduction

L-functions

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