LMFDB - Embedded newform 5445.2.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5445,2,Mod(1,5445)] 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("5445.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,-1,4,0,3,-3,0,-1,0,0,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.477260$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.477260 q^{2} -1.77222 q^{4} +1.00000 q^{5} -2.68522 q^{7} -1.80033 q^{8} +0.477260 q^{10} +4.66785 q^{13} -1.28155 q^{14} +2.68522 q^{16} -4.62632 q^{17} +4.34044 q^{19} -1.77222 q^{20} -2.77222 q^{23} +1.00000 q^{25} +2.22778 q^{26} +4.75881 q^{28} -3.01341 q^{29} +2.38630 q^{31} +4.88221 q^{32} -2.20796 q^{34} -2.68522 q^{35} -10.6429 q^{37} +2.07152 q^{38} -1.80033 q^{40} -2.21041 q^{41} +7.06719 q^{43} -1.32307 q^{46} -4.36215 q^{47} +0.210405 q^{49} +0.477260 q^{50} -8.27247 q^{52} +6.33404 q^{53} +4.83428 q^{56} -1.43818 q^{58} -11.7473 q^{59} +3.98263 q^{61} +1.13889 q^{62} -3.04036 q^{64} +4.66785 q^{65} +7.31984 q^{67} +8.19888 q^{68} -1.28155 q^{70} -1.19571 q^{71} -1.02171 q^{73} -5.07943 q^{74} -7.69223 q^{76} -3.50213 q^{79} +2.68522 q^{80} -1.05494 q^{82} +11.1158 q^{83} -4.62632 q^{85} +3.37289 q^{86} -2.76978 q^{89} -12.5342 q^{91} +4.91300 q^{92} -2.08188 q^{94} +4.34044 q^{95} +18.5342 q^{97} +0.100418 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8} - q^{10} + q^{13} - 2 q^{14} - 3 q^{16} + q^{17} + 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} + 13 q^{28} - 12 q^{29} - 5 q^{31} + 8 q^{32}+ \cdots + 8 q^{98}+O(q^{100}) $ 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8 - q^10 + q^13 - 2 * q^14 - 3 * q^16 + q^17 + 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 + 13 * q^28 - 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^37 - 20 * q^38 - 3 * q^40 - 11 * q^41 + 19 * q^43 - 4 * q^46 - 5 * q^47 + 3 * q^49 - q^50 - 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 + 12 * q^61 + 35 * q^62 - 3 * q^64 + q^65 - 19 * q^67 + 3 * q^68 - 2 * q^70 - 5 * q^71 + 11 * q^73 + 34 * q^79 - 3 * q^80 - 6 * q^82 + 11 * q^83 + q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 - q^94 + 20 * q^95 + 32 * q^97 + 8 * q^98

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 $

$2$	0.477260	0.337474	0.168737	−	0.985661i	$-0.446031\pi$
$2$	0.477260	0.337474	0.168737	+	0.985661i	$0.446031\pi$
$3$	0	0
$3$	0	0
$4$	−1.77222	−0.886111
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.68522	−1.01492	−0.507459	−	0.861676i	$-0.669415\pi$
$7$	−2.68522	−1.01492	−0.507459	+	0.861676i	$0.669415\pi$
$8$	−1.80033	−0.636513
$9$	0	0
$10$	0.477260	0.150923
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.66785	1.29463	0.647314	−	0.762223i	$-0.275892\pi$
$13$	4.66785	1.29463	0.647314	+	0.762223i	$0.275892\pi$
$14$	−1.28155	−0.342508
$15$	0	0
$16$	2.68522	0.671305
$17$	−4.62632	−1.12205	−0.561024	−	0.827799i	$-0.689593\pi$
$17$	−4.62632	−1.12205	−0.561024	+	0.827799i	$0.689593\pi$
$18$	0	0
$19$	4.34044	0.995766	0.497883	−	0.867244i	$-0.334111\pi$
$19$	4.34044	0.995766	0.497883	+	0.867244i	$0.334111\pi$
$20$	−1.77222	−0.396281
$21$	0	0
$22$	0	0
$23$	−2.77222	−0.578048	−0.289024	−	0.957322i	$-0.593331\pi$
$23$	−2.77222	−0.578048	−0.289024	+	0.957322i	$0.593331\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.22778	0.436903
$27$	0	0
$28$	4.75881	0.899330
$29$	−3.01341	−0.559577	−0.279789	−	0.960062i	$-0.590264\pi$
$29$	−3.01341	−0.559577	−0.279789	+	0.960062i	$0.590264\pi$
$30$	0	0
$31$	2.38630	0.428592	0.214296	−	0.976769i	$-0.431254\pi$
$31$	2.38630	0.428592	0.214296	+	0.976769i	$0.431254\pi$
$32$	4.88221	0.863061
$33$	0	0
$34$	−2.20796	−0.378662
$35$	−2.68522	−0.453885
$36$	0	0
$37$	−10.6429	−1.74968	−0.874842	−	0.484409i	$-0.839035\pi$
$37$	−10.6429	−1.74968	−0.874842	+	0.484409i	$0.839035\pi$
$38$	2.07152	0.336045
$39$	0	0
$40$	−1.80033	−0.284657
$41$	−2.21041	−0.345207	−0.172604	−	0.984991i	$-0.555218\pi$
$41$	−2.21041	−0.345207	−0.172604	+	0.984991i	$0.555218\pi$
$42$	0	0
$43$	7.06719	1.07774	0.538868	−	0.842390i	$-0.318852\pi$
$43$	7.06719	1.07774	0.538868	+	0.842390i	$0.318852\pi$
$44$	0	0
$45$	0	0
$46$	−1.32307	−0.195076
$47$	−4.36215	−0.636285	−0.318142	−	0.948043i	$-0.603059\pi$
$47$	−4.36215	−0.636285	−0.318142	+	0.948043i	$0.603059\pi$
$48$	0	0
$49$	0.210405	0.0300579
$50$	0.477260	0.0674948
$51$	0	0
$52$	−8.27247	−1.14718
$53$	6.33404	0.870047	0.435024	−	0.900419i	$-0.356740\pi$
$53$	6.33404	0.870047	0.435024	+	0.900419i	$0.356740\pi$
$54$	0	0
$55$	0	0
$56$	4.83428	0.646008
$57$	0	0
$58$	−1.43818	−0.188843
$59$	−11.7473	−1.52937	−0.764683	−	0.644407i	$-0.777104\pi$
$59$	−11.7473	−1.52937	−0.764683	+	0.644407i	$0.777104\pi$
$60$	0	0
$61$	3.98263	0.509923	0.254962	−	0.966951i	$-0.417937\pi$
$61$	3.98263	0.509923	0.254962	+	0.966951i	$0.417937\pi$
$62$	1.13889	0.144639
$63$	0	0
$64$	−3.04036	−0.380044
$65$	4.66785	0.578975
$66$	0	0
$67$	7.31984	0.894260	0.447130	−	0.894469i	$-0.352446\pi$
$67$	7.31984	0.894260	0.447130	+	0.894469i	$0.352446\pi$
$68$	8.19888	0.994260
$69$	0	0
$70$	−1.28155	−0.153174
$71$	−1.19571	−0.141905	−0.0709525	−	0.997480i	$-0.522604\pi$
$71$	−1.19571	−0.141905	−0.0709525	+	0.997480i	$0.522604\pi$
$72$	0	0
$73$	−1.02171	−0.119582	−0.0597908	−	0.998211i	$-0.519043\pi$
$73$	−1.02171	−0.119582	−0.0597908	+	0.998211i	$0.519043\pi$
$74$	−5.07943	−0.590472
$75$	0	0
$76$	−7.69223	−0.882360
$77$	0	0
$78$	0	0
$79$	−3.50213	−0.394021	−0.197010	−	0.980401i	$-0.563123\pi$
$79$	−3.50213	−0.394021	−0.197010	+	0.980401i	$0.563123\pi$
$80$	2.68522	0.300217
$81$	0	0
$82$	−1.05494	−0.116498
$83$	11.1158	1.22012	0.610061	−	0.792355i	$-0.291145\pi$
$83$	11.1158	1.22012	0.610061	+	0.792355i	$0.291145\pi$
$84$	0	0
$85$	−4.62632	−0.501795
$86$	3.37289	0.363708
$87$	0	0
$88$	0	0
$89$	−2.76978	−0.293596	−0.146798	−	0.989167i	$-0.546897\pi$
$89$	−2.76978	−0.293596	−0.146798	+	0.989167i	$0.546897\pi$
$90$	0	0
$91$	−12.5342	−1.31394
$92$	4.91300	0.512215
$93$	0	0
$94$	−2.08188	−0.214729
$95$	4.34044	0.445320
$96$	0	0
$97$	18.5342	1.88186	0.940931	−	0.338597i	$-0.109952\pi$
$97$	18.5342	1.88186	0.940931	+	0.338597i	$0.109952\pi$
$98$	0.100418	0.0101438
$99$	0	0
$100$	−1.77222	−0.177222
$101$	−7.11455	−0.707925	−0.353962	−	0.935260i	$-0.615166\pi$
$101$	−7.11455	−0.707925	−0.353962	+	0.935260i	$0.615166\pi$
$102$	0	0
$103$	7.52835	0.741791	0.370895	−	0.928675i	$-0.379051\pi$
$103$	7.52835	0.741791	0.370895	+	0.928675i	$0.379051\pi$
$104$	−8.40367	−0.824048
$105$	0	0
$106$	3.02298	0.293618
$107$	18.0292	1.74295	0.871475	−	0.490441i	$-0.163164\pi$
$107$	18.0292	1.74295	0.871475	+	0.490441i	$0.163164\pi$
$108$	0	0
$109$	16.3653	1.56751	0.783756	−	0.621068i	$-0.213301\pi$
$109$	16.3653	1.56751	0.783756	+	0.621068i	$0.213301\pi$
$110$	0	0
$111$	0	0
$112$	−7.21041	−0.681319
$113$	2.05377	0.193203	0.0966013	−	0.995323i	$-0.469203\pi$
$113$	2.05377	0.193203	0.0966013	+	0.995323i	$0.469203\pi$
$114$	0	0
$115$	−2.77222	−0.258511
$116$	5.34044	0.495848
$117$	0	0
$118$	−5.60651	−0.516121
$119$	12.4227	1.13879
$120$	0	0
$121$	0	0
$122$	1.90075	0.172086
$123$	0	0
$124$	−4.22906	−0.379780
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.0762667	0.00676758	0.00338379	−	0.999994i	$-0.498923\pi$
$127$	0.0762667	0.00676758	0.00338379	+	0.999994i	$0.498923\pi$
$128$	−11.2155	−0.991316
$129$	0	0
$130$	2.22778	0.195389
$131$	−11.4831	−1.00328	−0.501642	−	0.865075i	$-0.667270\pi$
$131$	−11.4831	−1.00328	−0.501642	+	0.865075i	$0.667270\pi$
$132$	0	0
$133$	−11.6550	−1.01062
$134$	3.49346	0.301789
$135$	0	0
$136$	8.32892	0.714199
$137$	18.3293	1.56598	0.782989	−	0.622036i	$-0.213694\pi$
$137$	18.3293	1.56598	0.782989	+	0.622036i	$0.213694\pi$
$138$	0	0
$139$	23.1874	1.96673	0.983363	−	0.181653i	$-0.0581446\pi$
$139$	23.1874	1.96673	0.983363	+	0.181653i	$0.0581446\pi$
$140$	4.75881	0.402193
$141$	0	0
$142$	−0.570666	−0.0478892
$143$	0	0
$144$	0	0
$145$	−3.01341	−0.250250
$146$	−0.487619	−0.0403557
$147$	0	0
$148$	18.8616	1.55041
$149$	14.6646	1.20137	0.600686	−	0.799485i	$-0.294894\pi$
$149$	14.6646	1.20137	0.600686	+	0.799485i	$0.294894\pi$
$150$	0	0
$151$	7.51902	0.611889	0.305944	−	0.952049i	$-0.401028\pi$
$151$	7.51902	0.611889	0.305944	+	0.952049i	$0.401028\pi$
$152$	−7.81423	−0.633818
$153$	0	0
$154$	0	0
$155$	2.38630	0.191672
$156$	0	0
$157$	−13.3819	−1.06799	−0.533996	−	0.845487i	$-0.679310\pi$
$157$	−13.3819	−1.06799	−0.533996	+	0.845487i	$0.679310\pi$
$158$	−1.67143	−0.132972
$159$	0	0
$160$	4.88221	0.385973
$161$	7.44403	0.586672
$162$	0	0
$163$	−0.771990	−0.0604669	−0.0302335	−	0.999543i	$-0.509625\pi$
$163$	−0.771990	−0.0604669	−0.0302335	+	0.999543i	$0.509625\pi$
$164$	3.91733	0.305892
$165$	0	0
$166$	5.30514	0.411759
$167$	−8.48232	−0.656381	−0.328191	−	0.944612i	$-0.606439\pi$
$167$	−8.48232	−0.656381	−0.328191	+	0.944612i	$0.606439\pi$
$168$	0	0
$169$	8.78880	0.676062
$170$	−2.20796	−0.169343
$171$	0	0
$172$	−12.5246	−0.954994
$173$	5.07871	0.386127	0.193064	−	0.981186i	$-0.438158\pi$
$173$	5.07871	0.386127	0.193064	+	0.981186i	$0.438158\pi$
$174$	0	0
$175$	−2.68522	−0.202984
$176$	0	0
$177$	0	0
$178$	−1.32190	−0.0990809
$179$	11.3170	0.845876	0.422938	−	0.906159i	$-0.360999\pi$
$179$	11.3170	0.845876	0.422938	+	0.906159i	$0.360999\pi$
$180$	0	0
$181$	7.40006	0.550042	0.275021	−	0.961438i	$-0.411315\pi$
$181$	7.40006	0.550042	0.275021	+	0.961438i	$0.411315\pi$
$182$	−5.98207	−0.443421
$183$	0	0
$184$	4.99092	0.367935
$185$	−10.6429	−0.782482
$186$	0	0
$187$	0	0
$188$	7.73070	0.563819
$189$	0	0
$190$	2.07152	0.150284
$191$	5.15608	0.373081	0.186540	−	0.982447i	$-0.440272\pi$
$191$	5.15608	0.373081	0.186540	+	0.982447i	$0.440272\pi$
$192$	0	0
$193$	4.03230	0.290251	0.145126	−	0.989413i	$-0.453641\pi$
$193$	4.03230	0.290251	0.145126	+	0.989413i	$0.453641\pi$
$194$	8.84563	0.635079
$195$	0	0
$196$	−0.372885	−0.0266346
$197$	11.4176	0.813469	0.406734	−	0.913547i	$-0.366667\pi$
$197$	11.4176	0.813469	0.406734	+	0.913547i	$0.366667\pi$
$198$	0	0
$199$	−7.16644	−0.508015	−0.254008	−	0.967202i	$-0.581749\pi$
$199$	−7.16644	−0.508015	−0.254008	+	0.967202i	$0.581749\pi$
$200$	−1.80033	−0.127303
$201$	0	0
$202$	−3.39549	−0.238906
$203$	8.09168	0.567925
$204$	0	0
$205$	−2.21041	−0.154381
$206$	3.59298	0.250335
$207$	0	0
$208$	12.5342	0.869090
$209$	0	0
$210$	0	0
$211$	−3.48359	−0.239820	−0.119910	−	0.992785i	$-0.538261\pi$
$211$	−3.48359	−0.239820	−0.119910	+	0.992785i	$0.538261\pi$
$212$	−11.2253	−0.770959
$213$	0	0
$214$	8.60462	0.588200
$215$	7.06719	0.481978
$216$	0	0
$217$	−6.40774	−0.434986
$218$	7.81051	0.528995
$219$	0	0
$220$	0	0
$221$	−21.5950	−1.45264
$222$	0	0
$223$	10.5449	0.706141	0.353071	−	0.935597i	$-0.385138\pi$
$223$	10.5449	0.706141	0.353071	+	0.935597i	$0.385138\pi$
$224$	−13.1098	−0.875936
$225$	0	0
$226$	0.980183	0.0652008
$227$	0.216018	0.0143376	0.00716880	−	0.999974i	$-0.497718\pi$
$227$	0.216018	0.0143376	0.00716880	+	0.999974i	$0.497718\pi$
$228$	0	0
$229$	−0.0757097	−0.00500304	−0.00250152	−	0.999997i	$-0.500796\pi$
$229$	−0.0757097	−0.00500304	−0.00250152	+	0.999997i	$0.500796\pi$
$230$	−1.32307	−0.0872407
$231$	0	0
$232$	5.42514	0.356178
$233$	15.1201	0.990548	0.495274	−	0.868737i	$-0.335068\pi$
$233$	15.1201	0.990548	0.495274	+	0.868737i	$0.335068\pi$
$234$	0	0
$235$	−4.36215	−0.284555
$236$	20.8188	1.35519
$237$	0	0
$238$	5.92886	0.384311
$239$	−23.1646	−1.49839	−0.749197	−	0.662347i	$-0.769561\pi$
$239$	−23.1646	−1.49839	−0.749197	+	0.662347i	$0.769561\pi$
$240$	0	0
$241$	21.3349	1.37430	0.687151	−	0.726515i	$-0.258861\pi$
$241$	21.3349	1.37430	0.687151	+	0.726515i	$0.258861\pi$
$242$	0	0
$243$	0	0
$244$	−7.05810	−0.451849
$245$	0.210405	0.0134423
$246$	0	0
$247$	20.2605	1.28915
$248$	−4.29613	−0.272805
$249$	0	0
$250$	0.477260	0.0301846
$251$	6.22186	0.392721	0.196360	−	0.980532i	$-0.437088\pi$
$251$	6.22186	0.392721	0.196360	+	0.980532i	$0.437088\pi$
$252$	0	0
$253$	0	0
$254$	0.0363991	0.00228388
$255$	0	0
$256$	0.728021	0.0455013
$257$	14.2903	0.891406	0.445703	−	0.895181i	$-0.352954\pi$
$257$	14.2903	0.891406	0.445703	+	0.895181i	$0.352954\pi$
$258$	0	0
$259$	28.5785	1.77578
$260$	−8.27247	−0.513037
$261$	0	0
$262$	−5.48043	−0.338582
$263$	4.13132	0.254748	0.127374	−	0.991855i	$-0.459345\pi$
$263$	4.13132	0.254748	0.127374	+	0.991855i	$0.459345\pi$
$264$	0	0
$265$	6.33404	0.389097
$266$	−5.56249	−0.341058
$267$	0	0
$268$	−12.9724	−0.792414
$269$	1.68394	0.102672	0.0513359	−	0.998681i	$-0.483652\pi$
$269$	1.68394	0.102672	0.0513359	+	0.998681i	$0.483652\pi$
$270$	0	0
$271$	18.4310	1.11960	0.559801	−	0.828627i	$-0.310877\pi$
$271$	18.4310	1.11960	0.559801	+	0.828627i	$0.310877\pi$
$272$	−12.4227	−0.753237
$273$	0	0
$274$	8.74784	0.528476
$275$	0	0
$276$	0	0
$277$	3.42745	0.205935	0.102968	−	0.994685i	$-0.467166\pi$
$277$	3.42745	0.205935	0.102968	+	0.994685i	$0.467166\pi$
$278$	11.0664	0.663718
$279$	0	0
$280$	4.83428	0.288904
$281$	−22.8217	−1.36143	−0.680715	−	0.732548i	$-0.738331\pi$
$281$	−22.8217	−1.36143	−0.680715	+	0.732548i	$0.738331\pi$
$282$	0	0
$283$	29.0991	1.72976	0.864880	−	0.501978i	$-0.167394\pi$
$283$	29.0991	1.72976	0.864880	+	0.501978i	$0.167394\pi$
$284$	2.11907	0.125744
$285$	0	0
$286$	0	0
$287$	5.93542	0.350357
$288$	0	0
$289$	4.40288	0.258993
$290$	−1.43818	−0.0844530
$291$	0	0
$292$	1.81069	0.105963
$293$	21.2136	1.23931	0.619655	−	0.784874i	$-0.287273\pi$
$293$	21.2136	1.23931	0.619655	+	0.784874i	$0.287273\pi$
$294$	0	0
$295$	−11.7473	−0.683953
$296$	19.1608	1.11370
$297$	0	0
$298$	6.99883	0.405432
$299$	−12.9403	−0.748358
$300$	0	0
$301$	−18.9769	−1.09381
$302$	3.58853	0.206496
$303$	0	0
$304$	11.6550	0.668463
$305$	3.98263	0.228045
$306$	0	0
$307$	6.87520	0.392388	0.196194	−	0.980565i	$-0.437142\pi$
$307$	6.87520	0.392388	0.196194	+	0.980565i	$0.437142\pi$
$308$	0	0
$309$	0	0
$310$	1.13889	0.0646844
$311$	−25.1577	−1.42656	−0.713280	−	0.700879i	$-0.752791\pi$
$311$	−25.1577	−1.42656	−0.713280	+	0.700879i	$0.752791\pi$
$312$	0	0
$313$	−11.5793	−0.654503	−0.327251	−	0.944937i	$-0.606122\pi$
$313$	−11.5793	−0.654503	−0.327251	+	0.944937i	$0.606122\pi$
$314$	−6.38664	−0.360419
$315$	0	0
$316$	6.20656	0.349146
$317$	−20.7413	−1.16495	−0.582473	−	0.812850i	$-0.697915\pi$
$317$	−20.7413	−1.16495	−0.582473	+	0.812850i	$0.697915\pi$
$318$	0	0
$319$	0	0
$320$	−3.04036	−0.169961
$321$	0	0
$322$	3.55274	0.197986
$323$	−20.0803	−1.11730
$324$	0	0
$325$	4.66785	0.258926
$326$	−0.368440	−0.0204060
$327$	0	0
$328$	3.97946	0.219729
$329$	11.7133	0.645777
$330$	0	0
$331$	−32.1415	−1.76665	−0.883327	−	0.468757i	$-0.844702\pi$
$331$	−32.1415	−1.76665	−0.883327	+	0.468757i	$0.844702\pi$
$332$	−19.6997	−1.08116
$333$	0	0
$334$	−4.04827	−0.221511
$335$	7.31984	0.399925
$336$	0	0
$337$	17.9964	0.980326	0.490163	−	0.871631i	$-0.336937\pi$
$337$	17.9964	0.980326	0.490163	+	0.871631i	$0.336937\pi$
$338$	4.19454	0.228153
$339$	0	0
$340$	8.19888	0.444647
$341$	0	0
$342$	0	0
$343$	18.2316	0.984411
$344$	−12.7233	−0.685993
$345$	0	0
$346$	2.42387	0.130308
$347$	−8.04981	−0.432137	−0.216068	−	0.976378i	$-0.569323\pi$
$347$	−8.04981	−0.432137	−0.216068	+	0.976378i	$0.569323\pi$
$348$	0	0
$349$	−19.2294	−1.02932	−0.514662	−	0.857393i	$-0.672083\pi$
$349$	−19.2294	−1.02932	−0.514662	+	0.857393i	$0.672083\pi$
$350$	−1.28155	−0.0685016
$351$	0	0
$352$	0	0
$353$	−14.8497	−0.790371	−0.395186	−	0.918601i	$-0.629320\pi$
$353$	−14.8497	−0.790371	−0.395186	+	0.918601i	$0.629320\pi$
$354$	0	0
$355$	−1.19571	−0.0634618
$356$	4.90866	0.260159
$357$	0	0
$358$	5.40117	0.285461
$359$	10.2233	0.539563	0.269782	−	0.962922i	$-0.413048\pi$
$359$	10.2233	0.539563	0.269782	+	0.962922i	$0.413048\pi$
$360$	0	0
$361$	−0.160555	−0.00845028
$362$	3.53175	0.185625
$363$	0	0
$364$	22.2134	1.16430
$365$	−1.02171	−0.0534785
$366$	0	0
$367$	14.1434	0.738279	0.369139	−	0.929374i	$-0.379653\pi$
$367$	14.1434	0.738279	0.369139	+	0.929374i	$0.379653\pi$
$368$	−7.44403	−0.388047
$369$	0	0
$370$	−5.07943	−0.264067
$371$	−17.0083	−0.883026
$372$	0	0
$373$	−12.4600	−0.645154	−0.322577	−	0.946543i	$-0.604549\pi$
$373$	−12.4600	−0.645154	−0.322577	+	0.946543i	$0.604549\pi$
$374$	0	0
$375$	0	0
$376$	7.85331	0.405004
$377$	−14.0662	−0.724444
$378$	0	0
$379$	16.3370	0.839174	0.419587	−	0.907715i	$-0.362175\pi$
$379$	16.3370	0.839174	0.419587	+	0.907715i	$0.362175\pi$
$380$	−7.69223	−0.394603
$381$	0	0
$382$	2.46079	0.125905
$383$	0.812648	0.0415244	0.0207622	−	0.999784i	$-0.493391\pi$
$383$	0.812648	0.0415244	0.0207622	+	0.999784i	$0.493391\pi$
$384$	0	0
$385$	0	0
$386$	1.92445	0.0979522
$387$	0	0
$388$	−32.8467	−1.66754
$389$	30.3941	1.54104	0.770521	−	0.637414i	$-0.219996\pi$
$389$	30.3941	1.54104	0.770521	+	0.637414i	$0.219996\pi$
$390$	0	0
$391$	12.8252	0.648598
$392$	−0.378799	−0.0191322
$393$	0	0
$394$	5.44915	0.274524
$395$	−3.50213	−0.176211
$396$	0	0
$397$	−14.8996	−0.747789	−0.373894	−	0.927471i	$-0.621978\pi$
$397$	−14.8996	−0.747789	−0.373894	+	0.927471i	$0.621978\pi$
$398$	−3.42025	−0.171442
$399$	0	0
$400$	2.68522	0.134261
$401$	−12.1692	−0.607700	−0.303850	−	0.952720i	$-0.598272\pi$
$401$	−12.1692	−0.607700	−0.303850	+	0.952720i	$0.598272\pi$
$402$	0	0
$403$	11.1389	0.554867
$404$	12.6086	0.627300
$405$	0	0
$406$	3.86184	0.191660
$407$	0	0
$408$	0	0
$409$	0.262108	0.0129604	0.00648020	−	0.999979i	$-0.497937\pi$
$409$	0.262108	0.0129604	0.00648020	+	0.999979i	$0.497937\pi$
$410$	−1.05494	−0.0520997
$411$	0	0
$412$	−13.3419	−0.657309
$413$	31.5440	1.55218
$414$	0	0
$415$	11.1158	0.545655
$416$	22.7894	1.11734
$417$	0	0
$418$	0	0
$419$	1.26916	0.0620023	0.0310012	−	0.999519i	$-0.490130\pi$
$419$	1.26916	0.0620023	0.0310012	+	0.999519i	$0.490130\pi$
$420$	0	0
$421$	−29.6655	−1.44581	−0.722903	−	0.690949i	$-0.757193\pi$
$421$	−29.6655	−1.44581	−0.722903	+	0.690949i	$0.757193\pi$
$422$	−1.66258	−0.0809331
$423$	0	0
$424$	−11.4034	−0.553797
$425$	−4.62632	−0.224410
$426$	0	0
$427$	−10.6942	−0.517530
$428$	−31.9518	−1.54445
$429$	0	0
$430$	3.37289	0.162655
$431$	31.3549	1.51031	0.755156	−	0.655545i	$-0.227561\pi$
$431$	31.3549	1.51031	0.755156	+	0.655545i	$0.227561\pi$
$432$	0	0
$433$	−26.0325	−1.25104	−0.625521	−	0.780207i	$-0.715114\pi$
$433$	−26.0325	−1.25104	−0.625521	+	0.780207i	$0.715114\pi$
$434$	−3.05816	−0.146796
$435$	0	0
$436$	−29.0030	−1.38899
$437$	−12.0327	−0.575601
$438$	0	0
$439$	−14.4191	−0.688185	−0.344093	−	0.938936i	$-0.611813\pi$
$439$	−14.4191	−0.688185	−0.344093	+	0.938936i	$0.611813\pi$
$440$	0	0
$441$	0	0
$442$	−10.3064	−0.490226
$443$	0.330608	0.0157077	0.00785383	−	0.999969i	$-0.497500\pi$
$443$	0.330608	0.0157077	0.00785383	+	0.999969i	$0.497500\pi$
$444$	0	0
$445$	−2.76978	−0.131300
$446$	5.03268	0.238304
$447$	0	0
$448$	8.16402	0.385714
$449$	−8.50828	−0.401531	−0.200765	−	0.979639i	$-0.564343\pi$
$449$	−8.50828	−0.401531	−0.200765	+	0.979639i	$0.564343\pi$
$450$	0	0
$451$	0	0
$452$	−3.63974	−0.171199
$453$	0	0
$454$	0.103097	0.00483856
$455$	−12.5342	−0.587612
$456$	0	0
$457$	−1.13847	−0.0532555	−0.0266277	−	0.999645i	$-0.508477\pi$
$457$	−1.13847	−0.0532555	−0.0266277	+	0.999645i	$0.508477\pi$
$458$	−0.0361332	−0.00168839
$459$	0	0
$460$	4.91300	0.229070
$461$	14.5073	0.675670	0.337835	−	0.941205i	$-0.390305\pi$
$461$	14.5073	0.675670	0.337835	+	0.941205i	$0.390305\pi$
$462$	0	0
$463$	−4.89739	−0.227601	−0.113801	−	0.993504i	$-0.536302\pi$
$463$	−4.89739	−0.227601	−0.113801	+	0.993504i	$0.536302\pi$
$464$	−8.09168	−0.375647
$465$	0	0
$466$	7.21620	0.334284
$467$	−32.4230	−1.50036	−0.750180	−	0.661234i	$-0.770033\pi$
$467$	−32.4230	−1.50036	−0.750180	+	0.661234i	$0.770033\pi$
$468$	0	0
$469$	−19.6554	−0.907601
$470$	−2.08188	−0.0960299
$471$	0	0
$472$	21.1490	0.973461
$473$	0	0
$474$	0	0
$475$	4.34044	0.199153
$476$	−22.0158	−1.00909
$477$	0	0
$478$	−11.0555	−0.505669
$479$	17.7354	0.810352	0.405176	−	0.914239i	$-0.367210\pi$
$479$	17.7354	0.810352	0.405176	+	0.914239i	$0.367210\pi$
$480$	0	0
$481$	−49.6795	−2.26519
$482$	10.1823	0.463791
$483$	0	0
$484$	0	0
$485$	18.5342	0.841595
$486$	0	0
$487$	−18.4603	−0.836514	−0.418257	−	0.908329i	$-0.637359\pi$
$487$	−18.4603	−0.836514	−0.418257	+	0.908329i	$0.637359\pi$
$488$	−7.17005	−0.324573
$489$	0	0
$490$	0.100418	0.00453642
$491$	−11.4338	−0.516002	−0.258001	−	0.966145i	$-0.583064\pi$
$491$	−11.4338	−0.516002	−0.258001	+	0.966145i	$0.583064\pi$
$492$	0	0
$493$	13.9410	0.627873
$494$	9.66954	0.435053
$495$	0	0
$496$	6.40774	0.287716
$497$	3.21075	0.144022
$498$	0	0
$499$	10.8815	0.487125	0.243562	−	0.969885i	$-0.421684\pi$
$499$	10.8815	0.487125	0.243562	+	0.969885i	$0.421684\pi$
$500$	−1.77222	−0.0792562
$501$	0	0
$502$	2.96945	0.132533
$503$	44.7247	1.99417	0.997087	−	0.0762683i	$-0.0243005\pi$
$503$	44.7247	1.99417	0.997087	+	0.0762683i	$0.0243005\pi$
$504$	0	0
$505$	−7.11455	−0.316594
$506$	0	0
$507$	0	0
$508$	−0.135162	−0.00599683
$509$	13.8093	0.612088	0.306044	−	0.952017i	$-0.400994\pi$
$509$	13.8093	0.612088	0.306044	+	0.952017i	$0.400994\pi$
$510$	0	0
$511$	2.74350	0.121365
$512$	22.7784	1.00667
$513$	0	0
$514$	6.82020	0.300826
$515$	7.52835	0.331739
$516$	0	0
$517$	0	0
$518$	13.6394	0.599281
$519$	0	0
$520$	−8.40367	−0.368525
$521$	3.64972	0.159897	0.0799486	−	0.996799i	$-0.474524\pi$
$521$	3.64972	0.159897	0.0799486	+	0.996799i	$0.474524\pi$
$522$	0	0
$523$	4.98707	0.218069	0.109035	−	0.994038i	$-0.465224\pi$
$523$	4.98707	0.218069	0.109035	+	0.994038i	$0.465224\pi$
$524$	20.3506	0.889021
$525$	0	0
$526$	1.97171	0.0859707
$527$	−11.0398	−0.480901
$528$	0	0
$529$	−15.3148	−0.665860
$530$	3.02298	0.131310
$531$	0	0
$532$	20.6553	0.895522
$533$	−10.3178	−0.446915
$534$	0	0
$535$	18.0292	0.779471
$536$	−13.1781	−0.569208
$537$	0	0
$538$	0.803678	0.0346490
$539$	0	0
$540$	0	0
$541$	−0.247594	−0.0106449	−0.00532246	−	0.999986i	$-0.501694\pi$
$541$	−0.247594	−0.0106449	−0.00532246	+	0.999986i	$0.501694\pi$
$542$	8.79637	0.377837
$543$	0	0
$544$	−22.5867	−0.968396
$545$	16.3653	0.701013
$546$	0	0
$547$	25.3120	1.08226	0.541132	−	0.840938i	$-0.317996\pi$
$547$	25.3120	1.08226	0.541132	+	0.840938i	$0.317996\pi$
$548$	−32.4836	−1.38763
$549$	0	0
$550$	0	0
$551$	−13.0796	−0.557208
$552$	0	0
$553$	9.40400	0.399899
$554$	1.63578	0.0694978
$555$	0	0
$556$	−41.0932	−1.74274
$557$	−38.6671	−1.63838	−0.819190	−	0.573523i	$-0.805576\pi$
$557$	−38.6671	−1.63838	−0.819190	+	0.573523i	$0.805576\pi$
$558$	0	0
$559$	32.9885	1.39527
$560$	−7.21041	−0.304695
$561$	0	0
$562$	−10.8919	−0.459447
$563$	13.9362	0.587341	0.293671	−	0.955907i	$-0.405123\pi$
$563$	13.9362	0.587341	0.293671	+	0.955907i	$0.405123\pi$
$564$	0	0
$565$	2.05377	0.0864028
$566$	13.8878	0.583749
$567$	0	0
$568$	2.15268	0.0903243
$569$	27.9125	1.17015	0.585076	−	0.810979i	$-0.301065\pi$
$569$	27.9125	1.17015	0.585076	+	0.810979i	$0.301065\pi$
$570$	0	0
$571$	31.4113	1.31452	0.657261	−	0.753663i	$-0.271715\pi$
$571$	31.4113	1.31452	0.657261	+	0.753663i	$0.271715\pi$
$572$	0	0
$573$	0	0
$574$	2.83274	0.118236
$575$	−2.77222	−0.115610
$576$	0	0
$577$	20.7349	0.863206	0.431603	−	0.902064i	$-0.357948\pi$
$577$	20.7349	0.863206	0.431603	+	0.902064i	$0.357948\pi$
$578$	2.10132	0.0874034
$579$	0	0
$580$	5.34044	0.221750
$581$	−29.8485	−1.23832
$582$	0	0
$583$	0	0
$584$	1.83941	0.0761153
$585$	0	0
$586$	10.1244	0.418235
$587$	−15.2367	−0.628886	−0.314443	−	0.949276i	$-0.601818\pi$
$587$	−15.2367	−0.628886	−0.314443	+	0.949276i	$0.601818\pi$
$588$	0	0
$589$	10.3576	0.426777
$590$	−5.60651	−0.230816
$591$	0	0
$592$	−28.5785	−1.17457
$593$	27.5413	1.13098	0.565492	−	0.824754i	$-0.308686\pi$
$593$	27.5413	1.13098	0.565492	+	0.824754i	$0.308686\pi$
$594$	0	0
$595$	12.4227	0.509281
$596$	−25.9890	−1.06455
$597$	0	0
$598$	−6.17589	−0.252551
$599$	26.7331	1.09228	0.546142	−	0.837693i	$-0.316096\pi$
$599$	26.7331	1.09228	0.546142	+	0.837693i	$0.316096\pi$
$600$	0	0
$601$	−2.40680	−0.0981753	−0.0490876	−	0.998794i	$-0.515631\pi$
$601$	−2.40680	−0.0981753	−0.0490876	+	0.998794i	$0.515631\pi$
$602$	−9.05694	−0.369133
$603$	0	0
$604$	−13.3254	−0.542202
$605$	0	0
$606$	0	0
$607$	−10.2929	−0.417774	−0.208887	−	0.977940i	$-0.566984\pi$
$607$	−10.2929	−0.417774	−0.208887	+	0.977940i	$0.566984\pi$
$608$	21.1910	0.859407
$609$	0	0
$610$	1.90075	0.0769591
$611$	−20.3618	−0.823752
$612$	0	0
$613$	−27.8756	−1.12589	−0.562943	−	0.826496i	$-0.690331\pi$
$613$	−27.8756	−1.12589	−0.562943	+	0.826496i	$0.690331\pi$
$614$	3.28126	0.132421
$615$	0	0
$616$	0	0
$617$	28.7216	1.15629	0.578143	−	0.815935i	$-0.303778\pi$
$617$	28.7216	1.15629	0.578143	+	0.815935i	$0.303778\pi$
$618$	0	0
$619$	22.6968	0.912261	0.456131	−	0.889913i	$-0.349235\pi$
$619$	22.6968	0.912261	0.456131	+	0.889913i	$0.349235\pi$
$620$	−4.22906	−0.169843
$621$	0	0
$622$	−12.0067	−0.481427
$623$	7.43746	0.297976
$624$	0	0
$625$	1.00000	0.0400000
$626$	−5.52635	−0.220877
$627$	0	0
$628$	23.7157	0.946360
$629$	49.2375	1.96323
$630$	0	0
$631$	28.5364	1.13602	0.568008	−	0.823023i	$-0.307714\pi$
$631$	28.5364	1.13602	0.568008	+	0.823023i	$0.307714\pi$
$632$	6.30500	0.250799
$633$	0	0
$634$	−9.89897	−0.393138
$635$	0.0762667	0.00302655
$636$	0	0
$637$	0.982140	0.0389138
$638$	0	0
$639$	0	0
$640$	−11.2155	−0.443330
$641$	−2.37375	−0.0937575	−0.0468788	−	0.998901i	$-0.514927\pi$
$641$	−2.37375	−0.0937575	−0.0468788	+	0.998901i	$0.514927\pi$
$642$	0	0
$643$	11.3603	0.448007	0.224004	−	0.974588i	$-0.428087\pi$
$643$	11.3603	0.448007	0.224004	+	0.974588i	$0.428087\pi$
$644$	−13.1925	−0.519856
$645$	0	0
$646$	−9.58352	−0.377059
$647$	48.4220	1.90366	0.951832	−	0.306621i	$-0.0991984\pi$
$647$	48.4220	1.90366	0.951832	+	0.306621i	$0.0991984\pi$
$648$	0	0
$649$	0	0
$650$	2.22778	0.0873806
$651$	0	0
$652$	1.36814	0.0535804
$653$	−29.7548	−1.16440	−0.582198	−	0.813047i	$-0.697807\pi$
$653$	−29.7548	−1.16440	−0.582198	+	0.813047i	$0.697807\pi$
$654$	0	0
$655$	−11.4831	−0.448682
$656$	−5.93542	−0.231739
$657$	0	0
$658$	5.59030	0.217933
$659$	−28.4931	−1.10993	−0.554966	−	0.831873i	$-0.687269\pi$
$659$	−28.4931	−1.10993	−0.554966	+	0.831873i	$0.687269\pi$
$660$	0	0
$661$	−1.02875	−0.0400139	−0.0200070	−	0.999800i	$-0.506369\pi$
$661$	−1.02875	−0.0400139	−0.0200070	+	0.999800i	$0.506369\pi$
$662$	−15.3398	−0.596200
$663$	0	0
$664$	−20.0122	−0.776623
$665$	−11.6550	−0.451963
$666$	0	0
$667$	8.35386	0.323463
$668$	15.0326	0.581627
$669$	0	0
$670$	3.49346	0.134964
$671$	0	0
$672$	0	0
$673$	−13.0369	−0.502536	−0.251268	−	0.967918i	$-0.580848\pi$
$673$	−13.0369	−0.502536	−0.251268	+	0.967918i	$0.580848\pi$
$674$	8.58896	0.330834
$675$	0	0
$676$	−15.5757	−0.599066
$677$	−37.1064	−1.42612	−0.713058	−	0.701105i	$-0.752690\pi$
$677$	−37.1064	−1.42612	−0.713058	+	0.701105i	$0.752690\pi$
$678$	0	0
$679$	−49.7684	−1.90994
$680$	8.32892	0.319399
$681$	0	0
$682$	0	0
$683$	32.8992	1.25885	0.629426	−	0.777061i	$-0.283290\pi$
$683$	32.8992	1.25885	0.629426	+	0.777061i	$0.283290\pi$
$684$	0	0
$685$	18.3293	0.700326
$686$	8.70119	0.332213
$687$	0	0
$688$	18.9769	0.723489
$689$	29.5663	1.12639
$690$	0	0
$691$	36.6998	1.39613	0.698064	−	0.716035i	$-0.254045\pi$
$691$	36.6998	1.39613	0.698064	+	0.716035i	$0.254045\pi$
$692$	−9.00061	−0.342152
$693$	0	0
$694$	−3.84185	−0.145835
$695$	23.1874	0.879546
$696$	0	0
$697$	10.2261	0.387339
$698$	−9.17741	−0.347370
$699$	0	0
$700$	4.75881	0.179866
$701$	−36.2497	−1.36913	−0.684566	−	0.728951i	$-0.740008\pi$
$701$	−36.2497	−1.36913	−0.684566	+	0.728951i	$0.740008\pi$
$702$	0	0
$703$	−46.1949	−1.74227
$704$	0	0
$705$	0	0
$706$	−7.08718	−0.266730
$707$	19.1041	0.718485
$708$	0	0
$709$	−18.6013	−0.698585	−0.349293	−	0.937014i	$-0.613578\pi$
$709$	−18.6013	−0.698585	−0.349293	+	0.937014i	$0.613578\pi$
$710$	−0.570666	−0.0214167
$711$	0	0
$712$	4.98652	0.186878
$713$	−6.61536	−0.247747
$714$	0	0
$715$	0	0
$716$	−20.0563	−0.749540
$717$	0	0
$718$	4.87915	0.182088
$719$	37.4280	1.39583	0.697914	−	0.716181i	$-0.254112\pi$
$719$	37.4280	1.39583	0.697914	+	0.716181i	$0.254112\pi$
$720$	0	0
$721$	−20.2153	−0.752856
$722$	−0.0766267	−0.00285175
$723$	0	0
$724$	−13.1146	−0.487399
$725$	−3.01341	−0.111915
$726$	0	0
$727$	14.6011	0.541526	0.270763	−	0.962646i	$-0.412724\pi$
$727$	14.6011	0.541526	0.270763	+	0.962646i	$0.412724\pi$
$728$	22.5657	0.836341
$729$	0	0
$730$	−0.487619	−0.0180476
$731$	−32.6951	−1.20927
$732$	0	0
$733$	−41.8369	−1.54528	−0.772640	−	0.634844i	$-0.781064\pi$
$733$	−41.8369	−1.54528	−0.772640	+	0.634844i	$0.781064\pi$
$734$	6.75007	0.249150
$735$	0	0
$736$	−13.5346	−0.498891
$737$	0	0
$738$	0	0
$739$	11.9299	0.438849	0.219424	−	0.975630i	$-0.429582\pi$
$739$	11.9299	0.438849	0.219424	+	0.975630i	$0.429582\pi$
$740$	18.8616	0.693366
$741$	0	0
$742$	−8.11738	−0.297998
$743$	46.6803	1.71253	0.856267	−	0.516534i	$-0.172778\pi$
$743$	46.6803	1.71253	0.856267	+	0.516534i	$0.172778\pi$
$744$	0	0
$745$	14.6646	0.537270
$746$	−5.94666	−0.217723
$747$	0	0
$748$	0	0
$749$	−48.4124	−1.76895
$750$	0	0
$751$	14.4039	0.525607	0.262803	−	0.964849i	$-0.415353\pi$
$751$	14.4039	0.525607	0.262803	+	0.964849i	$0.415353\pi$
$752$	−11.7133	−0.427141
$753$	0	0
$754$	−6.71322	−0.244481
$755$	7.51902	0.273645
$756$	0	0
$757$	−16.0616	−0.583768	−0.291884	−	0.956454i	$-0.594282\pi$
$757$	−16.0616	−0.583768	−0.291884	+	0.956454i	$0.594282\pi$
$758$	7.79698	0.283199
$759$	0	0
$760$	−7.81423	−0.283452
$761$	−38.8840	−1.40954	−0.704772	−	0.709433i	$-0.748951\pi$
$761$	−38.8840	−1.40954	−0.704772	+	0.709433i	$0.748951\pi$
$762$	0	0
$763$	−43.9445	−1.59090
$764$	−9.13772	−0.330591
$765$	0	0
$766$	0.387844	0.0140134
$767$	−54.8345	−1.97996
$768$	0	0
$769$	−43.0017	−1.55068	−0.775341	−	0.631543i	$-0.782422\pi$
$769$	−43.0017	−1.55068	−0.775341	+	0.631543i	$0.782422\pi$
$770$	0	0
$771$	0	0
$772$	−7.14613	−0.257195
$773$	7.85462	0.282511	0.141256	−	0.989973i	$-0.454886\pi$
$773$	7.85462	0.282511	0.141256	+	0.989973i	$0.454886\pi$
$774$	0	0
$775$	2.38630	0.0857184
$776$	−33.3677	−1.19783
$777$	0	0
$778$	14.5059	0.520061
$779$	−9.59414	−0.343746
$780$	0	0
$781$	0	0
$782$	6.12096	0.218885
$783$	0	0
$784$	0.564984	0.0201780
$785$	−13.3819	−0.477620
$786$	0	0
$787$	11.4249	0.407253	0.203627	−	0.979049i	$-0.434727\pi$
$787$	11.4249	0.407253	0.203627	+	0.979049i	$0.434727\pi$
$788$	−20.2345	−0.720824
$789$	0	0
$790$	−1.67143	−0.0594667
$791$	−5.51483	−0.196085
$792$	0	0
$793$	18.5903	0.660161
$794$	−7.11097	−0.252359
$795$	0	0
$796$	12.7005	0.450158
$797$	4.28502	0.151783	0.0758915	−	0.997116i	$-0.475820\pi$
$797$	4.28502	0.151783	0.0758915	+	0.997116i	$0.475820\pi$
$798$	0	0
$799$	20.1807	0.713942
$800$	4.88221	0.172612
$801$	0	0
$802$	−5.80787	−0.205083
$803$	0	0
$804$	0	0
$805$	7.44403	0.262368
$806$	5.31614	0.187253
$807$	0	0
$808$	12.8086	0.450603
$809$	−19.8484	−0.697833	−0.348917	−	0.937154i	$-0.613450\pi$
$809$	−19.8484	−0.697833	−0.348917	+	0.937154i	$0.613450\pi$
$810$	0	0
$811$	3.13836	0.110203	0.0551014	−	0.998481i	$-0.482452\pi$
$811$	3.13836	0.110203	0.0551014	+	0.998481i	$0.482452\pi$
$812$	−14.3403	−0.503245
$813$	0	0
$814$	0	0
$815$	−0.771990	−0.0270416
$816$	0	0
$817$	30.6747	1.07317
$818$	0.125094	0.00437379
$819$	0	0
$820$	3.91733	0.136799
$821$	23.4999	0.820151	0.410076	−	0.912052i	$-0.365502\pi$
$821$	23.4999	0.820151	0.410076	+	0.912052i	$0.365502\pi$
$822$	0	0
$823$	−12.6564	−0.441173	−0.220586	−	0.975367i	$-0.570797\pi$
$823$	−12.6564	−0.441173	−0.220586	+	0.975367i	$0.570797\pi$
$824$	−13.5535	−0.472159
$825$	0	0
$826$	15.0547	0.523820
$827$	30.2236	1.05098	0.525488	−	0.850801i	$-0.323883\pi$
$827$	30.2236	1.05098	0.525488	+	0.850801i	$0.323883\pi$
$828$	0	0
$829$	2.00166	0.0695204	0.0347602	−	0.999396i	$-0.488933\pi$
$829$	2.00166	0.0695204	0.0347602	+	0.999396i	$0.488933\pi$
$830$	5.30514	0.184144
$831$	0	0
$832$	−14.1919	−0.492016
$833$	−0.973403	−0.0337264
$834$	0	0
$835$	−8.48232	−0.293543
$836$	0	0
$837$	0	0
$838$	0.605717	0.0209242
$839$	−35.3744	−1.22126	−0.610629	−	0.791917i	$-0.709083\pi$
$839$	−35.3744	−1.22126	−0.610629	+	0.791917i	$0.709083\pi$
$840$	0	0
$841$	−19.9193	−0.686873
$842$	−14.1581	−0.487922
$843$	0	0
$844$	6.17370	0.212508
$845$	8.78880	0.302344
$846$	0	0
$847$	0	0
$848$	17.0083	0.584067
$849$	0	0
$850$	−2.20796	−0.0757324
$851$	29.5045	1.01140
$852$	0	0
$853$	18.2034	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$853$	18.2034	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$854$	−5.10393	−0.174653
$855$	0	0
$856$	−32.4585	−1.10941
$857$	−29.2837	−1.00031	−0.500156	−	0.865935i	$-0.666724\pi$
$857$	−29.2837	−1.00031	−0.500156	+	0.865935i	$0.666724\pi$
$858$	0	0
$859$	8.44030	0.287979	0.143990	−	0.989579i	$-0.454007\pi$
$859$	8.44030	0.287979	0.143990	+	0.989579i	$0.454007\pi$
$860$	−12.5246	−0.427086
$861$	0	0
$862$	14.9644	0.509691
$863$	19.3487	0.658636	0.329318	−	0.944219i	$-0.393181\pi$
$863$	19.3487	0.658636	0.329318	+	0.944219i	$0.393181\pi$
$864$	0	0
$865$	5.07871	0.172681
$866$	−12.4243	−0.422194
$867$	0	0
$868$	11.3559	0.385446
$869$	0	0
$870$	0	0
$871$	34.1679	1.15773
$872$	−29.4630	−0.997743
$873$	0	0
$874$	−5.74271	−0.194250
$875$	−2.68522	−0.0907770
$876$	0	0
$877$	−17.5140	−0.591405	−0.295702	−	0.955280i	$-0.595554\pi$
$877$	−17.5140	−0.591405	−0.295702	+	0.955280i	$0.595554\pi$
$878$	−6.88165	−0.232245
$879$	0	0
$880$	0	0
$881$	20.0575	0.675754	0.337877	−	0.941190i	$-0.390291\pi$
$881$	20.0575	0.675754	0.337877	+	0.941190i	$0.390291\pi$
$882$	0	0
$883$	26.8980	0.905189	0.452594	−	0.891717i	$-0.350499\pi$
$883$	26.8980	0.905189	0.452594	+	0.891717i	$0.350499\pi$
$884$	38.2711	1.28720
$885$	0	0
$886$	0.157786	0.00530092
$887$	6.80568	0.228512	0.114256	−	0.993451i	$-0.463552\pi$
$887$	6.80568	0.228512	0.114256	+	0.993451i	$0.463552\pi$
$888$	0	0
$889$	−0.204793	−0.00686854
$890$	−1.32190	−0.0443103
$891$	0	0
$892$	−18.6880	−0.625720
$893$	−18.9337	−0.633591
$894$	0	0
$895$	11.3170	0.378287
$896$	30.1160	1.00610
$897$	0	0
$898$	−4.06066	−0.135506
$899$	−7.19091	−0.239830
$900$	0	0
$901$	−29.3033	−0.976235
$902$	0	0
$903$	0	0
$904$	−3.69747	−0.122976
$905$	7.40006	0.245986
$906$	0	0
$907$	−21.3313	−0.708295	−0.354148	−	0.935190i	$-0.615229\pi$
$907$	−21.3313	−0.708295	−0.354148	+	0.935190i	$0.615229\pi$
$908$	−0.382831	−0.0127047
$909$	0	0
$910$	−5.98207	−0.198304
$911$	27.5888	0.914059	0.457029	−	0.889452i	$-0.348913\pi$
$911$	27.5888	0.914059	0.457029	+	0.889452i	$0.348913\pi$
$912$	0	0
$913$	0	0
$914$	−0.543347	−0.0179723
$915$	0	0
$916$	0.134174	0.00443325
$917$	30.8347	1.01825
$918$	0	0
$919$	6.00889	0.198215	0.0991075	−	0.995077i	$-0.468401\pi$
$919$	6.00889	0.198215	0.0991075	+	0.995077i	$0.468401\pi$
$920$	4.99092	0.164546
$921$	0	0
$922$	6.92373	0.228021
$923$	−5.58140	−0.183714
$924$	0	0
$925$	−10.6429	−0.349937
$926$	−2.33733	−0.0768094
$927$	0	0
$928$	−14.7121	−0.482949
$929$	−9.59739	−0.314880	−0.157440	−	0.987529i	$-0.550324\pi$
$929$	−9.59739	−0.314880	−0.157440	+	0.987529i	$0.550324\pi$
$930$	0	0
$931$	0.913252	0.0299306
$932$	−26.7961	−0.877736
$933$	0	0
$934$	−15.4742	−0.506332
$935$	0	0
$936$	0	0
$937$	38.5917	1.26074	0.630369	−	0.776296i	$-0.282904\pi$
$937$	38.5917	1.26074	0.630369	+	0.776296i	$0.282904\pi$
$938$	−9.38072	−0.306291
$939$	0	0
$940$	7.73070	0.252148
$941$	−29.1846	−0.951391	−0.475695	−	0.879610i	$-0.657804\pi$
$941$	−29.1846	−0.951391	−0.475695	+	0.879610i	$0.657804\pi$
$942$	0	0
$943$	6.12774	0.199547
$944$	−31.5440	−1.02667
$945$	0	0
$946$	0	0
$947$	−46.7623	−1.51957	−0.759785	−	0.650174i	$-0.774696\pi$
$947$	−46.7623	−1.51957	−0.759785	+	0.650174i	$0.774696\pi$
$948$	0	0
$949$	−4.76917	−0.154814
$950$	2.07152	0.0672090
$951$	0	0
$952$	−22.3650	−0.724853
$953$	−6.15238	−0.199295	−0.0996475	−	0.995023i	$-0.531772\pi$
$953$	−6.15238	−0.199295	−0.0996475	+	0.995023i	$0.531772\pi$
$954$	0	0
$955$	5.15608	0.166847
$956$	41.0529	1.32774
$957$	0	0
$958$	8.46440	0.273472
$959$	−49.2182	−1.58934
$960$	0	0
$961$	−25.3056	−0.816309
$962$	−23.7100	−0.764442
$963$	0	0
$964$	−37.8102	−1.21778
$965$	4.03230	0.129804
$966$	0	0
$967$	−3.39625	−0.109216	−0.0546080	−	0.998508i	$-0.517391\pi$
$967$	−3.39625	−0.109216	−0.0546080	+	0.998508i	$0.517391\pi$
$968$	0	0
$969$	0	0
$970$	8.84563	0.284016
$971$	9.39500	0.301500	0.150750	−	0.988572i	$-0.451831\pi$
$971$	9.39500	0.301500	0.150750	+	0.988572i	$0.451831\pi$
$972$	0	0
$973$	−62.2631	−1.99606
$974$	−8.81035	−0.282302
$975$	0	0
$976$	10.6942	0.342314
$977$	16.5552	0.529649	0.264824	−	0.964297i	$-0.414686\pi$
$977$	16.5552	0.529649	0.264824	+	0.964297i	$0.414686\pi$
$978$	0	0
$979$	0	0
$980$	−0.372885	−0.0119114
$981$	0	0
$982$	−5.45692	−0.174137
$983$	−50.8180	−1.62084	−0.810421	−	0.585848i	$-0.800762\pi$
$983$	−50.8180	−1.62084	−0.810421	+	0.585848i	$0.800762\pi$
$984$	0	0
$985$	11.4176	0.363794
$986$	6.65350	0.211891
$987$	0	0
$988$	−35.9062	−1.14233
$989$	−19.5918	−0.622983
$990$	0	0
$991$	11.3642	0.360996	0.180498	−	0.983575i	$-0.442229\pi$
$991$	11.3642	0.360996	0.180498	+	0.983575i	$0.442229\pi$
$992$	11.6504	0.369901
$993$	0	0
$994$	1.53236	0.0486036
$995$	−7.16644	−0.227191
$996$	0	0
$997$	29.1644	0.923647	0.461824	−	0.886972i	$-0.347195\pi$
$997$	29.1644	0.923647	0.461824	+	0.886972i	$0.347195\pi$
$998$	5.19332	0.164392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bi.1.3		4
3.2	odd	2		605.2.a.k.1.2		4
11.7	odd	10		495.2.n.e.181.1		8
11.8	odd	10		495.2.n.e.361.1		8
11.10	odd	2		5445.2.a.bp.1.2		4
12.11	even	2		9680.2.a.cm.1.3		4
15.14	odd	2		3025.2.a.w.1.3		4
33.2	even	10		605.2.g.m.81.1		8
33.5	odd	10		605.2.g.e.366.2		8
33.8	even	10		55.2.g.b.31.2	yes	8
33.14	odd	10		605.2.g.k.251.1		8
33.17	even	10		605.2.g.m.366.1		8
33.20	odd	10		605.2.g.e.81.2		8
33.26	odd	10		605.2.g.k.511.1		8
33.29	even	10		55.2.g.b.16.2	✓	8
33.32	even	2		605.2.a.j.1.3		4
132.95	odd	10		880.2.bo.h.401.1		8
132.107	odd	10		880.2.bo.h.801.1		8
132.131	odd	2		9680.2.a.cn.1.3		4
165.8	odd	20		275.2.z.a.174.2		16
165.29	even	10		275.2.h.a.126.1		8
165.62	odd	20		275.2.z.a.49.2		16
165.74	even	10		275.2.h.a.251.1		8
165.107	odd	20		275.2.z.a.174.3		16
165.128	odd	20		275.2.z.a.49.3		16
165.164	even	2		3025.2.a.bd.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.16.2	✓	8	33.29	even	10
55.2.g.b.31.2	yes	8	33.8	even	10
275.2.h.a.126.1		8	165.29	even	10
275.2.h.a.251.1		8	165.74	even	10
275.2.z.a.49.2		16	165.62	odd	20
275.2.z.a.49.3		16	165.128	odd	20
275.2.z.a.174.2		16	165.8	odd	20
275.2.z.a.174.3		16	165.107	odd	20
495.2.n.e.181.1		8	11.7	odd	10
495.2.n.e.361.1		8	11.8	odd	10
605.2.a.j.1.3		4	33.32	even	2
605.2.a.k.1.2		4	3.2	odd	2
605.2.g.e.81.2		8	33.20	odd	10
605.2.g.e.366.2		8	33.5	odd	10
605.2.g.k.251.1		8	33.14	odd	10
605.2.g.k.511.1		8	33.26	odd	10
605.2.g.m.81.1		8	33.2	even	10
605.2.g.m.366.1		8	33.17	even	10
880.2.bo.h.401.1		8	132.95	odd	10
880.2.bo.h.801.1		8	132.107	odd	10
3025.2.a.w.1.3		4	15.14	odd	2
3025.2.a.bd.1.2		4	165.164	even	2
5445.2.a.bi.1.3		4	1.1	even	1	trivial
5445.2.a.bp.1.2		4	11.10	odd	2
9680.2.a.cm.1.3		4	12.11	even	2
9680.2.a.cn.1.3		4	132.131	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.477260 q^{2} -1.77222 q^{4} +1.00000 q^{5} -2.68522 q^{7} -1.80033 q^{8} +0.477260 q^{10} +4.66785 q^{13} -1.28155 q^{14} +2.68522 q^{16} -4.62632 q^{17} +4.34044 q^{19} -1.77222 q^{20} -2.77222 q^{23} +1.00000 q^{25} +2.22778 q^{26} +4.75881 q^{28} -3.01341 q^{29} +2.38630 q^{31} +4.88221 q^{32} -2.20796 q^{34} -2.68522 q^{35} -10.6429 q^{37} +2.07152 q^{38} -1.80033 q^{40} -2.21041 q^{41} +7.06719 q^{43} -1.32307 q^{46} -4.36215 q^{47} +0.210405 q^{49} +0.477260 q^{50} -8.27247 q^{52} +6.33404 q^{53} +4.83428 q^{56} -1.43818 q^{58} -11.7473 q^{59} +3.98263 q^{61} +1.13889 q^{62} -3.04036 q^{64} +4.66785 q^{65} +7.31984 q^{67} +8.19888 q^{68} -1.28155 q^{70} -1.19571 q^{71} -1.02171 q^{73} -5.07943 q^{74} -7.69223 q^{76} -3.50213 q^{79} +2.68522 q^{80} -1.05494 q^{82} +11.1158 q^{83} -4.62632 q^{85} +3.37289 q^{86} -2.76978 q^{89} -12.5342 q^{91} +4.91300 q^{92} -2.08188 q^{94} +4.34044 q^{95} +18.5342 q^{97} +0.100418 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8} - q^{10} + q^{13} - 2 q^{14} - 3 q^{16} + q^{17} + 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} + 13 q^{28} - 12 q^{29} - 5 q^{31} + 8 q^{32}+ \cdots + 8 q^{98}+O(q^{100}) \) 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8 - q^10 + q^13 - 2 * q^14 - 3 * q^16 + q^17 + 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 + 13 * q^28 - 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^37 - 20 * q^38 - 3 * q^40 - 11 * q^41 + 19 * q^43 - 4 * q^46 - 5 * q^47 + 3 * q^49 - q^50 - 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 + 12 * q^61 + 35 * q^62 - 3 * q^64 + q^65 - 19 * q^67 + 3 * q^68 - 2 * q^70 - 5 * q^71 + 11 * q^73 + 34 * q^79 - 3 * q^80 - 6 * q^82 + 11 * q^83 + q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 - q^94 + 20 * q^95 + 32 * q^97 + 8 * q^98

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