LMFDB - Embedded newform 3648.2.g.h.1825.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3648,2,Mod(1825,3648)] 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("3648.1825"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3648.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,-20,0,0,0,0,0,0,0,12,0,0,0, 0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)

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

Self dual:	no
Analytic conductor:	$29.1294266574$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1825.4
Root			$-1.26217 + 1.18614i$ of defining polynomial
Character	$\chi$	$=$	3648.1825
Dual form			3648.2.g.h.1825.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.52434i q^{5} +2.52434 q^{7} -1.00000 q^{9} -2.37228i q^{11} +2.52434 q^{15} +0.372281 q^{17} -1.00000i q^{19} -2.52434i q^{21} +5.04868 q^{23} -1.37228 q^{25} +1.00000i q^{27} -1.58457i q^{29} +3.46410 q^{31} -2.37228 q^{33} +6.37228i q^{35} +3.16915i q^{37} -2.74456 q^{41} -11.1168i q^{43} -2.52434i q^{45} +2.81929 q^{47} -0.627719 q^{49} -0.372281i q^{51} -6.63325i q^{53} +5.98844 q^{55} -1.00000 q^{57} +8.74456i q^{59} -2.81929i q^{61} -2.52434 q^{63} -0.744563i q^{67} -5.04868i q^{69} +8.80773 q^{71} +7.62772 q^{73} +1.37228i q^{75} -5.98844i q^{77} -6.63325 q^{79} +1.00000 q^{81} +8.00000i q^{83} +0.939764i q^{85} -1.58457 q^{87} +7.48913 q^{89} -3.46410i q^{93} +2.52434 q^{95} +2.74456 q^{97} +2.37228i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 20 q^{17} + 12 q^{25} + 4 q^{33} + 24 q^{41} - 28 q^{49} - 8 q^{57} + 84 q^{73} + 8 q^{81} - 32 q^{89} - 24 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 20 * q^17 + 12 * q^25 + 4 * q^33 + 24 * q^41 - 28 * q^49 - 8 * q^57 + 84 * q^73 + 8 * q^81 - 32 * q^89 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$.

$n$	$1217$	$1921$	$2053$	$2623$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	2.52434i	1.12892i	0.825461	+	0.564459i	$0.190915\pi$
$5$	2.52434i	1.12892i	−0.825461	+	0.564459i	$0.809085\pi$
$6$	0	0
$7$	2.52434	0.954110	0.477055	−	0.878873i	$-0.341704\pi$
$7$	2.52434	0.954110	0.477055	+	0.878873i	$0.341704\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.37228i	− 0.715270i	−0.933862	−	0.357635i	$-0.883583\pi$
$11$	− 2.37228i	− 0.715270i	0.933862	−	0.357635i	$-0.116417\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	2.52434	0.651781
$16$	0	0
$17$	0.372281	0.0902915	0.0451457	−	0.998980i	$-0.485625\pi$
$17$	0.372281	0.0902915	0.0451457	+	0.998980i	$0.485625\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 2.52434i	− 0.550856i
$22$	0	0
$23$	5.04868	1.05272	0.526361	−	0.850261i	$-0.323556\pi$
$23$	5.04868	1.05272	0.526361	+	0.850261i	$0.323556\pi$
$24$	0	0
$25$	−1.37228	−0.274456
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 1.58457i	− 0.294248i	−0.989118	−	0.147124i	$-0.952998\pi$
$29$	− 1.58457i	− 0.294248i	0.989118	−	0.147124i	$-0.0470016\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	−2.37228	−0.412961
$34$	0	0
$35$	6.37228i	1.07711i
$36$	0	0
$37$	3.16915i	0.521005i	0.965473	+	0.260502i	$0.0838882\pi$
$37$	3.16915i	0.521005i	−0.965473	+	0.260502i	$0.916112\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.74456	−0.428629	−0.214314	−	0.976765i	$-0.568752\pi$
$41$	−2.74456	−0.428629	−0.214314	+	0.976765i	$0.568752\pi$
$42$	0	0
$43$	− 11.1168i	− 1.69530i	−0.530554	−	0.847651i	$-0.678016\pi$
$43$	− 11.1168i	− 1.69530i	0.530554	−	0.847651i	$-0.321984\pi$
$44$	0	0
$45$	− 2.52434i	− 0.376306i
$46$	0	0
$47$	2.81929	0.411236	0.205618	−	0.978632i	$-0.434080\pi$
$47$	2.81929	0.411236	0.205618	+	0.978632i	$0.434080\pi$
$48$	0	0
$49$	−0.627719	−0.0896741
$50$	0	0
$51$	− 0.372281i	− 0.0521298i
$52$	0	0
$53$	− 6.63325i	− 0.911147i	−0.890198	−	0.455573i	$-0.849434\pi$
$53$	− 6.63325i	− 0.911147i	0.890198	−	0.455573i	$-0.150566\pi$
$54$	0	0
$55$	5.98844	0.807481
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	8.74456i	1.13845i	0.822183	+	0.569223i	$0.192756\pi$
$59$	8.74456i	1.13845i	−0.822183	+	0.569223i	$0.807244\pi$
$60$	0	0
$61$	− 2.81929i	− 0.360973i	−0.983577	−	0.180487i	$-0.942233\pi$
$61$	− 2.81929i	− 0.360973i	0.983577	−	0.180487i	$-0.0577673\pi$
$62$	0	0
$63$	−2.52434	−0.318037
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.744563i	− 0.0909628i	−0.998965	−	0.0454814i	$-0.985518\pi$
$67$	− 0.744563i	− 0.0909628i	0.998965	−	0.0454814i	$-0.0144822\pi$
$68$	0	0
$69$	− 5.04868i	− 0.607789i
$70$	0	0
$71$	8.80773	1.04529	0.522643	−	0.852552i	$-0.324946\pi$
$71$	8.80773	1.04529	0.522643	+	0.852552i	$0.324946\pi$
$72$	0	0
$73$	7.62772	0.892757	0.446378	−	0.894844i	$-0.352714\pi$
$73$	7.62772	0.892757	0.446378	+	0.894844i	$0.352714\pi$
$74$	0	0
$75$	1.37228i	0.158457i
$76$	0	0
$77$	− 5.98844i	− 0.682446i
$78$	0	0
$79$	−6.63325	−0.746299	−0.373149	−	0.927771i	$-0.621722\pi$
$79$	−6.63325	−0.746299	−0.373149	+	0.927771i	$0.621722\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0.939764i	0.101932i
$86$	0	0
$87$	−1.58457	−0.169884
$88$	0	0
$89$	7.48913	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$89$	7.48913	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 3.46410i	− 0.359211i
$94$	0	0
$95$	2.52434	0.258992
$96$	0	0
$97$	2.74456	0.278668	0.139334	−	0.990245i	$-0.455504\pi$
$97$	2.74456	0.278668	0.139334	+	0.990245i	$0.455504\pi$
$98$	0	0
$99$	2.37228i	0.238423i
$100$	0	0
$101$	11.6819i	1.16240i	0.813763	+	0.581198i	$0.197416\pi$
$101$	11.6819i	1.16240i	−0.813763	+	0.581198i	$0.802584\pi$
$102$	0	0
$103$	13.5615	1.33625	0.668125	−	0.744049i	$-0.267097\pi$
$103$	13.5615	1.33625	0.668125	+	0.744049i	$0.267097\pi$
$104$	0	0
$105$	6.37228	0.621871
$106$	0	0
$107$	8.74456i	0.845369i	0.906277	+	0.422684i	$0.138912\pi$
$107$	8.74456i	0.845369i	−0.906277	+	0.422684i	$0.861088\pi$
$108$	0	0
$109$	− 18.3152i	− 1.75428i	−0.480239	−	0.877138i	$-0.659450\pi$
$109$	− 18.3152i	− 1.75428i	0.480239	−	0.877138i	$-0.340550\pi$
$110$	0	0
$111$	3.16915	0.300802
$112$	0	0
$113$	−2.74456	−0.258187	−0.129093	−	0.991632i	$-0.541207\pi$
$113$	−2.74456	−0.258187	−0.129093	+	0.991632i	$0.541207\pi$
$114$	0	0
$115$	12.7446i	1.18844i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.939764	0.0861480
$120$	0	0
$121$	5.37228	0.488389
$122$	0	0
$123$	2.74456i	0.247469i
$124$	0	0
$125$	9.15759i	0.819080i
$126$	0	0
$127$	8.51278	0.755387	0.377693	−	0.925931i	$-0.376717\pi$
$127$	8.51278	0.755387	0.377693	+	0.925931i	$0.376717\pi$
$128$	0	0
$129$	−11.1168	−0.978784
$130$	0	0
$131$	19.1168i	1.67025i	0.550063	+	0.835123i	$0.314604\pi$
$131$	19.1168i	1.67025i	−0.550063	+	0.835123i	$0.685396\pi$
$132$	0	0
$133$	− 2.52434i	− 0.218888i
$134$	0	0
$135$	−2.52434	−0.217260
$136$	0	0
$137$	−5.11684	−0.437161	−0.218581	−	0.975819i	$-0.570143\pi$
$137$	−5.11684	−0.437161	−0.218581	+	0.975819i	$0.570143\pi$
$138$	0	0
$139$	− 11.1168i	− 0.942918i	−0.881888	−	0.471459i	$-0.843727\pi$
$139$	− 11.1168i	− 0.942918i	0.881888	−	0.471459i	$-0.156273\pi$
$140$	0	0
$141$	− 2.81929i	− 0.237427i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0.627719i	0.0517734i
$148$	0	0
$149$	− 11.3321i	− 0.928359i	−0.885741	−	0.464180i	$-0.846349\pi$
$149$	− 11.3321i	− 0.928359i	0.885741	−	0.464180i	$-0.153651\pi$
$150$	0	0
$151$	0.294954	0.0240030	0.0120015	−	0.999928i	$-0.496180\pi$
$151$	0.294954	0.0240030	0.0120015	+	0.999928i	$0.496180\pi$
$152$	0	0
$153$	−0.372281	−0.0300972
$154$	0	0
$155$	8.74456i	0.702380i
$156$	0	0
$157$	− 6.92820i	− 0.552931i	−0.961024	−	0.276465i	$-0.910837\pi$
$157$	− 6.92820i	− 0.552931i	0.961024	−	0.276465i	$-0.0891631\pi$
$158$	0	0
$159$	−6.63325	−0.526051
$160$	0	0
$161$	12.7446	1.00441
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	− 5.98844i	− 0.466199i
$166$	0	0
$167$	10.0974	0.781356	0.390678	−	0.920527i	$-0.372241\pi$
$167$	10.0974	0.781356	0.390678	+	0.920527i	$0.372241\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000i	0.0764719i
$172$	0	0
$173$	3.46410i	0.263371i	0.991292	+	0.131685i	$0.0420389\pi$
$173$	3.46410i	0.263371i	−0.991292	+	0.131685i	$0.957961\pi$
$174$	0	0
$175$	−3.46410	−0.261861
$176$	0	0
$177$	8.74456	0.657282
$178$	0	0
$179$	− 5.48913i	− 0.410276i	−0.978733	−	0.205138i	$-0.934236\pi$
$179$	− 5.48913i	− 0.410276i	0.978733	−	0.205138i	$-0.0657644\pi$
$180$	0	0
$181$	− 5.63858i	− 0.419113i	−0.977797	−	0.209556i	$-0.932798\pi$
$181$	− 5.63858i	− 0.419113i	0.977797	−	0.209556i	$-0.0672019\pi$
$182$	0	0
$183$	−2.81929	−0.208408
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	− 0.883156i	− 0.0645828i
$188$	0	0
$189$	2.52434i	0.183619i
$190$	0	0
$191$	−10.4472	−0.755933	−0.377967	−	0.925819i	$-0.623377\pi$
$191$	−10.4472	−0.755933	−0.377967	+	0.925819i	$0.623377\pi$
$192$	0	0
$193$	11.4891	0.827005	0.413503	−	0.910503i	$-0.364305\pi$
$193$	11.4891	0.827005	0.413503	+	0.910503i	$0.364305\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.34363i	0.380718i	0.981715	+	0.190359i	$0.0609652\pi$
$197$	5.34363i	0.380718i	−0.981715	+	0.190359i	$0.939035\pi$
$198$	0	0
$199$	3.81396	0.270364	0.135182	−	0.990821i	$-0.456838\pi$
$199$	3.81396	0.270364	0.135182	+	0.990821i	$0.456838\pi$
$200$	0	0
$201$	−0.744563	−0.0525174
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	− 6.92820i	− 0.483887i
$206$	0	0
$207$	−5.04868	−0.350907
$208$	0	0
$209$	−2.37228	−0.164094
$210$	0	0
$211$	18.2337i	1.25526i	0.778512	+	0.627629i	$0.215975\pi$
$211$	18.2337i	1.25526i	−0.778512	+	0.627629i	$0.784025\pi$
$212$	0	0
$213$	− 8.80773i	− 0.603496i
$214$	0	0
$215$	28.0627	1.91386
$216$	0	0
$217$	8.74456	0.593620
$218$	0	0
$219$	− 7.62772i	− 0.515433i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.6819	−0.782280	−0.391140	−	0.920331i	$-0.627919\pi$
$223$	−11.6819	−0.782280	−0.391140	+	0.920331i	$0.627919\pi$
$224$	0	0
$225$	1.37228	0.0914854
$226$	0	0
$227$	− 1.48913i	− 0.0988367i	−0.998778	−	0.0494184i	$-0.984263\pi$
$227$	− 1.48913i	− 0.0988367i	0.998778	−	0.0494184i	$-0.0157368\pi$
$228$	0	0
$229$	10.4472i	0.690371i	0.938534	+	0.345185i	$0.112184\pi$
$229$	10.4472i	0.690371i	−0.938534	+	0.345185i	$0.887816\pi$
$230$	0	0
$231$	−5.98844	−0.394010
$232$	0	0
$233$	13.1168	0.859313	0.429657	−	0.902992i	$-0.358635\pi$
$233$	13.1168	0.859313	0.429657	+	0.902992i	$0.358635\pi$
$234$	0	0
$235$	7.11684i	0.464252i
$236$	0	0
$237$	6.63325i	0.430876i
$238$	0	0
$239$	11.6270	0.752090	0.376045	−	0.926601i	$-0.377284\pi$
$239$	11.6270	0.752090	0.376045	+	0.926601i	$0.377284\pi$
$240$	0	0
$241$	5.25544	0.338532	0.169266	−	0.985570i	$-0.445860\pi$
$241$	5.25544	0.338532	0.169266	+	0.985570i	$0.445860\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 1.58457i	− 0.101235i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	− 2.37228i	− 0.149737i	−0.997193	−	0.0748685i	$-0.976146\pi$
$251$	− 2.37228i	− 0.149737i	0.997193	−	0.0748685i	$-0.0238537\pi$
$252$	0	0
$253$	− 11.9769i	− 0.752980i
$254$	0	0
$255$	0.939764	0.0588503
$256$	0	0
$257$	−10.7446	−0.670227	−0.335114	−	0.942178i	$-0.608775\pi$
$257$	−10.7446	−0.670227	−0.335114	+	0.942178i	$0.608775\pi$
$258$	0	0
$259$	8.00000i	0.497096i
$260$	0	0
$261$	1.58457i	0.0980827i
$262$	0	0
$263$	16.0858	0.991892	0.495946	−	0.868353i	$-0.334821\pi$
$263$	16.0858	0.991892	0.495946	+	0.868353i	$0.334821\pi$
$264$	0	0
$265$	16.7446	1.02861
$266$	0	0
$267$	− 7.48913i	− 0.458327i
$268$	0	0
$269$	6.63325i	0.404436i	0.979340	+	0.202218i	$0.0648150\pi$
$269$	6.63325i	0.404436i	−0.979340	+	0.202218i	$0.935185\pi$
$270$	0	0
$271$	12.9715	0.787965	0.393983	−	0.919118i	$-0.371097\pi$
$271$	12.9715	0.787965	0.393983	+	0.919118i	$0.371097\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.25544i	0.196310i
$276$	0	0
$277$	− 0.349857i	− 0.0210208i	−0.999945	−	0.0105104i	$-0.996654\pi$
$277$	− 0.349857i	− 0.0210208i	0.999945	−	0.0105104i	$-0.00334563\pi$
$278$	0	0
$279$	−3.46410	−0.207390
$280$	0	0
$281$	−19.4891	−1.16262	−0.581312	−	0.813681i	$-0.697460\pi$
$281$	−19.4891	−1.16262	−0.581312	+	0.813681i	$0.697460\pi$
$282$	0	0
$283$	− 28.6060i	− 1.70045i	−0.526421	−	0.850224i	$-0.676466\pi$
$283$	− 28.6060i	− 1.70045i	0.526421	−	0.850224i	$-0.323534\pi$
$284$	0	0
$285$	− 2.52434i	− 0.149529i
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−16.8614	−0.991847
$290$	0	0
$291$	− 2.74456i	− 0.160889i
$292$	0	0
$293$	− 14.8511i	− 0.867609i	−0.901007	−	0.433804i	$-0.857171\pi$
$293$	− 14.8511i	− 0.867609i	0.901007	−	0.433804i	$-0.142829\pi$
$294$	0	0
$295$	−22.0742	−1.28521
$296$	0	0
$297$	2.37228	0.137654
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 28.0627i	− 1.61751i
$302$	0	0
$303$	11.6819	0.671109
$304$	0	0
$305$	7.11684	0.407509
$306$	0	0
$307$	− 26.2337i	− 1.49724i	−0.663002	−	0.748618i	$-0.730718\pi$
$307$	− 26.2337i	− 1.49724i	0.663002	−	0.748618i	$-0.269282\pi$
$308$	0	0
$309$	− 13.5615i	− 0.771484i
$310$	0	0
$311$	5.98844	0.339573	0.169787	−	0.985481i	$-0.445692\pi$
$311$	5.98844	0.339573	0.169787	+	0.985481i	$0.445692\pi$
$312$	0	0
$313$	24.9783	1.41185	0.705927	−	0.708284i	$-0.250531\pi$
$313$	24.9783	1.41185	0.705927	+	0.708284i	$0.250531\pi$
$314$	0	0
$315$	− 6.37228i	− 0.359037i
$316$	0	0
$317$	− 9.80240i	− 0.550557i	−0.961364	−	0.275279i	$-0.911230\pi$
$317$	− 9.80240i	− 0.550557i	0.961364	−	0.275279i	$-0.0887701\pi$
$318$	0	0
$319$	−3.75906	−0.210467
$320$	0	0
$321$	8.74456	0.488074
$322$	0	0
$323$	− 0.372281i	− 0.0207143i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.3152	−1.01283
$328$	0	0
$329$	7.11684	0.392364
$330$	0	0
$331$	18.2337i	1.00221i	0.865385	+	0.501107i	$0.167074\pi$
$331$	18.2337i	1.00221i	−0.865385	+	0.501107i	$0.832926\pi$
$332$	0	0
$333$	− 3.16915i	− 0.173668i
$334$	0	0
$335$	1.87953	0.102690
$336$	0	0
$337$	−24.2337	−1.32009	−0.660047	−	0.751225i	$-0.729463\pi$
$337$	−24.2337	−1.32009	−0.660047	+	0.751225i	$0.729463\pi$
$338$	0	0
$339$	2.74456i	0.149064i
$340$	0	0
$341$	− 8.21782i	− 0.445020i
$342$	0	0
$343$	−19.2549	−1.03967
$344$	0	0
$345$	12.7446	0.686144
$346$	0	0
$347$	− 31.1168i	− 1.67044i	−0.549916	−	0.835220i	$-0.685340\pi$
$347$	− 31.1168i	− 1.67044i	0.549916	−	0.835220i	$-0.314660\pi$
$348$	0	0
$349$	− 2.81929i	− 0.150913i	−0.997149	−	0.0754566i	$-0.975959\pi$
$349$	− 2.81929i	− 0.150913i	0.997149	−	0.0754566i	$-0.0240414\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.4891	−0.824403	−0.412201	−	0.911093i	$-0.635240\pi$
$353$	−15.4891	−0.824403	−0.412201	+	0.911093i	$0.635240\pi$
$354$	0	0
$355$	22.2337i	1.18004i
$356$	0	0
$357$	− 0.939764i	− 0.0497376i
$358$	0	0
$359$	12.9166	0.681714	0.340857	−	0.940115i	$-0.389283\pi$
$359$	12.9166	0.681714	0.340857	+	0.940115i	$0.389283\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 5.37228i	− 0.281972i
$364$	0	0
$365$	19.2549i	1.00785i
$366$	0	0
$367$	23.6588	1.23498	0.617490	−	0.786579i	$-0.288150\pi$
$367$	23.6588	1.23498	0.617490	+	0.786579i	$0.288150\pi$
$368$	0	0
$369$	2.74456	0.142876
$370$	0	0
$371$	− 16.7446i	− 0.869334i
$372$	0	0
$373$	− 6.92820i	− 0.358729i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	− 6.92820i	− 0.358729i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$	0	0
$375$	9.15759	0.472896
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 8.74456i	− 0.449178i	−0.974454	−	0.224589i	$-0.927896\pi$
$379$	− 8.74456i	− 0.449178i	0.974454	−	0.224589i	$-0.0721040\pi$
$380$	0	0
$381$	− 8.51278i	− 0.436123i
$382$	0	0
$383$	18.3152	0.935862	0.467931	−	0.883765i	$-0.345000\pi$
$383$	18.3152	0.935862	0.467931	+	0.883765i	$0.345000\pi$
$384$	0	0
$385$	15.1168	0.770426
$386$	0	0
$387$	11.1168i	0.565101i
$388$	0	0
$389$	26.4781i	1.34249i	0.741234	+	0.671246i	$0.234241\pi$
$389$	26.4781i	1.34249i	−0.741234	+	0.671246i	$0.765759\pi$
$390$	0	0
$391$	1.87953	0.0950518
$392$	0	0
$393$	19.1168	0.964317
$394$	0	0
$395$	− 16.7446i	− 0.842510i
$396$	0	0
$397$	− 11.0371i	− 0.553937i	−0.960879	−	0.276968i	$-0.910670\pi$
$397$	− 11.0371i	− 0.553937i	0.960879	−	0.276968i	$-0.0893298\pi$
$398$	0	0
$399$	−2.52434	−0.126375
$400$	0	0
$401$	−36.9783	−1.84661	−0.923303	−	0.384073i	$-0.874521\pi$
$401$	−36.9783	−1.84661	−0.923303	+	0.384073i	$0.874521\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.52434i	0.125435i
$406$	0	0
$407$	7.51811	0.372659
$408$	0	0
$409$	−34.4674	−1.70430	−0.852151	−	0.523296i	$-0.824702\pi$
$409$	−34.4674	−1.70430	−0.852151	+	0.523296i	$0.824702\pi$
$410$	0	0
$411$	5.11684i	0.252395i
$412$	0	0
$413$	22.0742i	1.08620i
$414$	0	0
$415$	−20.1947	−0.991319
$416$	0	0
$417$	−11.1168	−0.544394
$418$	0	0
$419$	25.4891i	1.24523i	0.782530	+	0.622613i	$0.213929\pi$
$419$	25.4891i	1.24523i	−0.782530	+	0.622613i	$0.786071\pi$
$420$	0	0
$421$	− 25.8333i	− 1.25904i	−0.776985	−	0.629519i	$-0.783252\pi$
$421$	− 25.8333i	− 1.25904i	0.776985	−	0.629519i	$-0.216748\pi$
$422$	0	0
$423$	−2.81929	−0.137079
$424$	0	0
$425$	−0.510875	−0.0247811
$426$	0	0
$427$	− 7.11684i	− 0.344408i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.4612	1.61177	0.805885	−	0.592073i	$-0.201690\pi$
$431$	33.4612	1.61177	0.805885	+	0.592073i	$0.201690\pi$
$432$	0	0
$433$	−22.7446	−1.09303	−0.546517	−	0.837448i	$-0.684047\pi$
$433$	−22.7446	−1.09303	−0.546517	+	0.837448i	$0.684047\pi$
$434$	0	0
$435$	− 4.00000i	− 0.191785i
$436$	0	0
$437$	− 5.04868i	− 0.241511i
$438$	0	0
$439$	−16.0309	−0.765113	−0.382556	−	0.923932i	$-0.624956\pi$
$439$	−16.0309	−0.765113	−0.382556	+	0.923932i	$0.624956\pi$
$440$	0	0
$441$	0.627719	0.0298914
$442$	0	0
$443$	0.883156i	0.0419600i	0.999780	+	0.0209800i	$0.00667863\pi$
$443$	0.883156i	0.0419600i	−0.999780	+	0.0209800i	$0.993321\pi$
$444$	0	0
$445$	18.9051i	0.896187i
$446$	0	0
$447$	−11.3321	−0.535988
$448$	0	0
$449$	−6.74456	−0.318296	−0.159148	−	0.987255i	$-0.550875\pi$
$449$	−6.74456	−0.318296	−0.159148	+	0.987255i	$0.550875\pi$
$450$	0	0
$451$	6.51087i	0.306585i
$452$	0	0
$453$	− 0.294954i	− 0.0138581i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.13859	0.287151	0.143576	−	0.989639i	$-0.454140\pi$
$457$	6.13859	0.287151	0.143576	+	0.989639i	$0.454140\pi$
$458$	0	0
$459$	0.372281i	0.0173766i
$460$	0	0
$461$	5.69349i	0.265172i	0.991172	+	0.132586i	$0.0423281\pi$
$461$	5.69349i	0.265172i	−0.991172	+	0.132586i	$0.957672\pi$
$462$	0	0
$463$	−3.81396	−0.177250	−0.0886248	−	0.996065i	$-0.528247\pi$
$463$	−3.81396	−0.177250	−0.0886248	+	0.996065i	$0.528247\pi$
$464$	0	0
$465$	8.74456	0.405519
$466$	0	0
$467$	16.8832i	0.781259i	0.920548	+	0.390630i	$0.127743\pi$
$467$	16.8832i	0.781259i	−0.920548	+	0.390630i	$0.872257\pi$
$468$	0	0
$469$	− 1.87953i	− 0.0867885i
$470$	0	0
$471$	−6.92820	−0.319235
$472$	0	0
$473$	−26.3723	−1.21260
$474$	0	0
$475$	1.37228i	0.0629646i
$476$	0	0
$477$	6.63325i	0.303716i
$478$	0	0
$479$	−11.3870	−0.520284	−0.260142	−	0.965570i	$-0.583769\pi$
$479$	−11.3870	−0.520284	−0.260142	+	0.965570i	$0.583769\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 12.7446i	− 0.579898i
$484$	0	0
$485$	6.92820i	0.314594i
$486$	0	0
$487$	−29.9971	−1.35930	−0.679649	−	0.733537i	$-0.737868\pi$
$487$	−29.9971	−1.35930	−0.679649	+	0.733537i	$0.737868\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 14.9783i	− 0.675959i	−0.941153	−	0.337979i	$-0.890257\pi$
$491$	− 14.9783i	− 0.675959i	0.941153	−	0.337979i	$-0.109743\pi$
$492$	0	0
$493$	− 0.589907i	− 0.0265681i
$494$	0	0
$495$	−5.98844	−0.269160
$496$	0	0
$497$	22.2337	0.997317
$498$	0	0
$499$	− 4.88316i	− 0.218600i	−0.994009	−	0.109300i	$-0.965139\pi$
$499$	− 4.88316i	− 0.218600i	0.994009	−	0.109300i	$-0.0348609\pi$
$500$	0	0
$501$	− 10.0974i	− 0.451116i
$502$	0	0
$503$	15.1460	0.675328	0.337664	−	0.941267i	$-0.390363\pi$
$503$	15.1460	0.675328	0.337664	+	0.941267i	$0.390363\pi$
$504$	0	0
$505$	−29.4891	−1.31225
$506$	0	0
$507$	− 13.0000i	− 0.577350i
$508$	0	0
$509$	5.93354i	0.262999i	0.991316	+	0.131500i	$0.0419792\pi$
$509$	5.93354i	0.262999i	−0.991316	+	0.131500i	$0.958021\pi$
$510$	0	0
$511$	19.2549	0.851788
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	34.2337i	1.50852i
$516$	0	0
$517$	− 6.68815i	− 0.294145i
$518$	0	0
$519$	3.46410	0.152057
$520$	0	0
$521$	−24.9783	−1.09432	−0.547159	−	0.837029i	$-0.684291\pi$
$521$	−24.9783	−1.09432	−0.547159	+	0.837029i	$0.684291\pi$
$522$	0	0
$523$	− 15.2554i	− 0.667074i	−0.942737	−	0.333537i	$-0.891758\pi$
$523$	− 15.2554i	− 0.667074i	0.942737	−	0.333537i	$-0.108242\pi$
$524$	0	0
$525$	3.46410i	0.151186i
$526$	0	0
$527$	1.28962	0.0561767
$528$	0	0
$529$	2.48913	0.108223
$530$	0	0
$531$	− 8.74456i	− 0.379482i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−22.0742	−0.954352
$536$	0	0
$537$	−5.48913	−0.236873
$538$	0	0
$539$	1.48913i	0.0641412i
$540$	0	0
$541$	31.8217i	1.36812i	0.729424	+	0.684061i	$0.239788\pi$
$541$	31.8217i	1.36812i	−0.729424	+	0.684061i	$0.760212\pi$
$542$	0	0
$543$	−5.63858	−0.241975
$544$	0	0
$545$	46.2337	1.98043
$546$	0	0
$547$	13.4891i	0.576753i	0.957517	+	0.288377i	$0.0931155\pi$
$547$	13.4891i	0.576753i	−0.957517	+	0.288377i	$0.906884\pi$
$548$	0	0
$549$	2.81929i	0.120324i
$550$	0	0
$551$	−1.58457	−0.0675051
$552$	0	0
$553$	−16.7446	−0.712051
$554$	0	0
$555$	8.00000i	0.339581i
$556$	0	0
$557$	39.0449i	1.65438i	0.561919	+	0.827192i	$0.310063\pi$
$557$	39.0449i	1.65438i	−0.561919	+	0.827192i	$0.689937\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.883156	−0.0372869
$562$	0	0
$563$	36.4674i	1.53692i	0.639900	+	0.768458i	$0.278976\pi$
$563$	36.4674i	1.53692i	−0.639900	+	0.768458i	$0.721024\pi$
$564$	0	0
$565$	− 6.92820i	− 0.291472i
$566$	0	0
$567$	2.52434	0.106012
$568$	0	0
$569$	−25.2554	−1.05876	−0.529382	−	0.848384i	$-0.677576\pi$
$569$	−25.2554	−1.05876	−0.529382	+	0.848384i	$0.677576\pi$
$570$	0	0
$571$	24.4674i	1.02393i	0.859007	+	0.511964i	$0.171082\pi$
$571$	24.4674i	1.02393i	−0.859007	+	0.511964i	$0.828918\pi$
$572$	0	0
$573$	10.4472i	0.436438i
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	−17.1168	−0.712584	−0.356292	−	0.934375i	$-0.615959\pi$
$577$	−17.1168	−0.712584	−0.356292	+	0.934375i	$0.615959\pi$
$578$	0	0
$579$	− 11.4891i	− 0.477472i
$580$	0	0
$581$	20.1947i	0.837817i
$582$	0	0
$583$	−15.7359	−0.651716
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.11684i	0.128646i	0.997929	+	0.0643230i	$0.0204888\pi$
$587$	3.11684i	0.128646i	−0.997929	+	0.0643230i	$0.979511\pi$
$588$	0	0
$589$	− 3.46410i	− 0.142736i
$590$	0	0
$591$	5.34363	0.219808
$592$	0	0
$593$	28.9783	1.18999	0.594997	−	0.803728i	$-0.297153\pi$
$593$	28.9783	1.18999	0.594997	+	0.803728i	$0.297153\pi$
$594$	0	0
$595$	2.37228i	0.0972541i
$596$	0	0
$597$	− 3.81396i	− 0.156095i
$598$	0	0
$599$	−22.6641	−0.926032	−0.463016	−	0.886350i	$-0.653233\pi$
$599$	−22.6641	−0.926032	−0.463016	+	0.886350i	$0.653233\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	0.744563i	0.0303209i
$604$	0	0
$605$	13.5615i	0.551351i
$606$	0	0
$607$	−7.92287	−0.321579	−0.160790	−	0.986989i	$-0.551404\pi$
$607$	−7.92287	−0.321579	−0.160790	+	0.986989i	$0.551404\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	0	0
$612$	0	0
$613$	23.0140i	0.929526i	0.885435	+	0.464763i	$0.153861\pi$
$613$	23.0140i	0.929526i	−0.885435	+	0.464763i	$0.846139\pi$
$614$	0	0
$615$	−6.92820	−0.279372
$616$	0	0
$617$	−18.8832	−0.760207	−0.380104	−	0.924944i	$-0.624112\pi$
$617$	−18.8832	−0.760207	−0.380104	+	0.924944i	$0.624112\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	5.04868i	0.202596i
$622$	0	0
$623$	18.9051	0.757416
$624$	0	0
$625$	−29.9783	−1.19913
$626$	0	0
$627$	2.37228i	0.0947398i
$628$	0	0
$629$	1.17981i	0.0470423i
$630$	0	0
$631$	0.644810	0.0256695	0.0128347	−	0.999918i	$-0.495914\pi$
$631$	0.644810	0.0256695	0.0128347	+	0.999918i	$0.495914\pi$
$632$	0	0
$633$	18.2337	0.724724
$634$	0	0
$635$	21.4891i	0.852770i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.80773	−0.348428
$640$	0	0
$641$	−14.7446	−0.582375	−0.291188	−	0.956666i	$-0.594050\pi$
$641$	−14.7446	−0.582375	−0.291188	+	0.956666i	$0.594050\pi$
$642$	0	0
$643$	20.8832i	0.823551i	0.911285	+	0.411776i	$0.135091\pi$
$643$	20.8832i	0.823551i	−0.911285	+	0.411776i	$0.864909\pi$
$644$	0	0
$645$	− 28.0627i	− 1.10497i
$646$	0	0
$647$	−0.939764	−0.0369459	−0.0184730	−	0.999829i	$-0.505880\pi$
$647$	−0.939764	−0.0369459	−0.0184730	+	0.999829i	$0.505880\pi$
$648$	0	0
$649$	20.7446	0.814295
$650$	0	0
$651$	− 8.74456i	− 0.342726i
$652$	0	0
$653$	39.0449i	1.52794i	0.645249	+	0.763972i	$0.276754\pi$
$653$	39.0449i	1.52794i	−0.645249	+	0.763972i	$0.723246\pi$
$654$	0	0
$655$	−48.2574	−1.88557
$656$	0	0
$657$	−7.62772	−0.297586
$658$	0	0
$659$	− 27.7228i	− 1.07993i	−0.841688	−	0.539964i	$-0.818438\pi$
$659$	− 27.7228i	− 1.07993i	0.841688	−	0.539964i	$-0.181562\pi$
$660$	0	0
$661$	34.6410i	1.34738i	0.739014	+	0.673690i	$0.235292\pi$
$661$	34.6410i	1.34738i	−0.739014	+	0.673690i	$0.764708\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.37228	0.247106
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	11.6819i	0.451649i
$670$	0	0
$671$	−6.68815	−0.258193
$672$	0	0
$673$	7.48913	0.288685	0.144342	−	0.989528i	$-0.453893\pi$
$673$	7.48913	0.288685	0.144342	+	0.989528i	$0.453893\pi$
$674$	0	0
$675$	− 1.37228i	− 0.0528191i
$676$	0	0
$677$	18.6101i	0.715245i	0.933866	+	0.357623i	$0.116413\pi$
$677$	18.6101i	0.715245i	−0.933866	+	0.357623i	$0.883587\pi$
$678$	0	0
$679$	6.92820	0.265880
$680$	0	0
$681$	−1.48913	−0.0570634
$682$	0	0
$683$	− 41.4891i	− 1.58754i	−0.608220	−	0.793769i	$-0.708116\pi$
$683$	− 41.4891i	− 1.58754i	0.608220	−	0.793769i	$-0.291884\pi$
$684$	0	0
$685$	− 12.9166i	− 0.493520i
$686$	0	0
$687$	10.4472	0.398586
$688$	0	0
$689$	0	0
$690$	0	0
$691$	20.8832i	0.794433i	0.917725	+	0.397216i	$0.130024\pi$
$691$	20.8832i	0.794433i	−0.917725	+	0.397216i	$0.869976\pi$
$692$	0	0
$693$	5.98844i	0.227482i
$694$	0	0
$695$	28.0627	1.06448
$696$	0	0
$697$	−1.02175	−0.0387015
$698$	0	0
$699$	− 13.1168i	− 0.496125i
$700$	0	0
$701$	− 25.5383i	− 0.964569i	−0.876015	−	0.482285i	$-0.839807\pi$
$701$	− 25.5383i	− 0.964569i	0.876015	−	0.482285i	$-0.160193\pi$
$702$	0	0
$703$	3.16915	0.119527
$704$	0	0
$705$	7.11684	0.268036
$706$	0	0
$707$	29.4891i	1.10905i
$708$	0	0
$709$	40.3894i	1.51686i	0.651757	+	0.758428i	$0.274032\pi$
$709$	40.3894i	1.51686i	−0.651757	+	0.758428i	$0.725968\pi$
$710$	0	0
$711$	6.63325	0.248766
$712$	0	0
$713$	17.4891	0.654973
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 11.6270i	− 0.434219i
$718$	0	0
$719$	9.15759	0.341520	0.170760	−	0.985313i	$-0.445378\pi$
$719$	9.15759	0.341520	0.170760	+	0.985313i	$0.445378\pi$
$720$	0	0
$721$	34.2337	1.27493
$722$	0	0
$723$	− 5.25544i	− 0.195452i
$724$	0	0
$725$	2.17448i	0.0807582i
$726$	0	0
$727$	−47.2627	−1.75288	−0.876438	−	0.481514i	$-0.840087\pi$
$727$	−47.2627	−1.75288	−0.876438	+	0.481514i	$0.840087\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 4.13859i	− 0.153071i
$732$	0	0
$733$	27.7128i	1.02360i	0.859106	+	0.511798i	$0.171020\pi$
$733$	27.7128i	1.02360i	−0.859106	+	0.511798i	$0.828980\pi$
$734$	0	0
$735$	−1.58457	−0.0584479
$736$	0	0
$737$	−1.76631	−0.0650629
$738$	0	0
$739$	− 33.3505i	− 1.22682i	−0.789765	−	0.613410i	$-0.789798\pi$
$739$	− 33.3505i	− 1.22682i	0.789765	−	0.613410i	$-0.210202\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.8482	−1.64532	−0.822660	−	0.568534i	$-0.807511\pi$
$743$	−44.8482	−1.64532	−0.822660	+	0.568534i	$0.807511\pi$
$744$	0	0
$745$	28.6060	1.04804
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	22.0742i	0.806575i
$750$	0	0
$751$	−16.0309	−0.584975	−0.292488	−	0.956269i	$-0.594483\pi$
$751$	−16.0309	−0.584975	−0.292488	+	0.956269i	$0.594483\pi$
$752$	0	0
$753$	−2.37228	−0.0864507
$754$	0	0
$755$	0.744563i	0.0270974i
$756$	0	0
$757$	− 34.4010i	− 1.25032i	−0.780495	−	0.625162i	$-0.785033\pi$
$757$	− 34.4010i	− 1.25032i	0.780495	−	0.625162i	$-0.214967\pi$
$758$	0	0
$759$	−11.9769	−0.434733
$760$	0	0
$761$	−32.0951	−1.16345	−0.581723	−	0.813387i	$-0.697621\pi$
$761$	−32.0951	−1.16345	−0.581723	+	0.813387i	$0.697621\pi$
$762$	0	0
$763$	− 46.2337i	− 1.67377i
$764$	0	0
$765$	− 0.939764i	− 0.0339772i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.8614	0.788342	0.394171	−	0.919037i	$-0.371032\pi$
$769$	21.8614	0.788342	0.394171	+	0.919037i	$0.371032\pi$
$770$	0	0
$771$	10.7446i	0.386956i
$772$	0	0
$773$	40.0945i	1.44210i	0.692885	+	0.721049i	$0.256340\pi$
$773$	40.0945i	1.44210i	−0.692885	+	0.721049i	$0.743660\pi$
$774$	0	0
$775$	−4.75372	−0.170759
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	2.74456i	0.0983342i
$780$	0	0
$781$	− 20.8944i	− 0.747661i
$782$	0	0
$783$	1.58457	0.0566281
$784$	0	0
$785$	17.4891	0.624214
$786$	0	0
$787$	− 6.97825i	− 0.248748i	−0.992235	−	0.124374i	$-0.960308\pi$
$787$	− 6.97825i	− 0.248748i	0.992235	−	0.124374i	$-0.0396922\pi$
$788$	0	0
$789$	− 16.0858i	− 0.572669i
$790$	0	0
$791$	−6.92820	−0.246339
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 16.7446i	− 0.593868i
$796$	0	0
$797$	37.5152i	1.32886i	0.747352	+	0.664428i	$0.231325\pi$
$797$	37.5152i	1.32886i	−0.747352	+	0.664428i	$0.768675\pi$
$798$	0	0
$799$	1.04957	0.0371311
$800$	0	0
$801$	−7.48913	−0.264615
$802$	0	0
$803$	− 18.0951i	− 0.638562i
$804$	0	0
$805$	32.1716i	1.13390i
$806$	0	0
$807$	6.63325	0.233501
$808$	0	0
$809$	−2.13859	−0.0751889	−0.0375945	−	0.999293i	$-0.511970\pi$
$809$	−2.13859	−0.0751889	−0.0375945	+	0.999293i	$0.511970\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	0	0
$813$	− 12.9715i	− 0.454932i
$814$	0	0
$815$	−10.0974	−0.353695
$816$	0	0
$817$	−11.1168	−0.388929
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 14.5012i	− 0.506096i	−0.967454	−	0.253048i	$-0.918567\pi$
$821$	− 14.5012i	− 0.506096i	0.967454	−	0.253048i	$-0.0814331\pi$
$822$	0	0
$823$	42.2140	1.47149	0.735744	−	0.677259i	$-0.236833\pi$
$823$	42.2140	1.47149	0.735744	+	0.677259i	$0.236833\pi$
$824$	0	0
$825$	3.25544	0.113340
$826$	0	0
$827$	− 16.7446i	− 0.582265i	−0.956683	−	0.291133i	$-0.905968\pi$
$827$	− 16.7446i	− 0.582265i	0.956683	−	0.291133i	$-0.0940321\pi$
$828$	0	0
$829$	− 30.2921i	− 1.05209i	−0.850458	−	0.526043i	$-0.823675\pi$
$829$	− 30.2921i	− 1.05209i	0.850458	−	0.526043i	$-0.176325\pi$
$830$	0	0
$831$	−0.349857	−0.0121364
$832$	0	0
$833$	−0.233688	−0.00809681
$834$	0	0
$835$	25.4891i	0.882088i
$836$	0	0
$837$	3.46410i	0.119737i
$838$	0	0
$839$	−52.9562	−1.82825	−0.914125	−	0.405432i	$-0.867121\pi$
$839$	−52.9562	−1.82825	−0.914125	+	0.405432i	$0.867121\pi$
$840$	0	0
$841$	26.4891	0.913418
$842$	0	0
$843$	19.4891i	0.671241i
$844$	0	0
$845$	32.8164i	1.12892i
$846$	0	0
$847$	13.5615	0.465977
$848$	0	0
$849$	−28.6060	−0.981754
$850$	0	0
$851$	16.0000i	0.548473i
$852$	0	0
$853$	15.8457i	0.542548i	0.962502	+	0.271274i	$0.0874449\pi$
$853$	15.8457i	0.542548i	−0.962502	+	0.271274i	$0.912555\pi$
$854$	0	0
$855$	−2.52434	−0.0863305
$856$	0	0
$857$	−46.0000	−1.57133	−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000	−1.57133	−0.785665	+	0.618652i	$0.787679\pi$
$858$	0	0
$859$	− 27.1168i	− 0.925215i	−0.886563	−	0.462607i	$-0.846914\pi$
$859$	− 27.1168i	− 0.925215i	0.886563	−	0.462607i	$-0.153086\pi$
$860$	0	0
$861$	6.92820i	0.236113i
$862$	0	0
$863$	34.0511	1.15911	0.579557	−	0.814932i	$-0.303226\pi$
$863$	34.0511	1.15911	0.579557	+	0.814932i	$0.303226\pi$
$864$	0	0
$865$	−8.74456	−0.297324
$866$	0	0
$867$	16.8614i	0.572643i
$868$	0	0
$869$	15.7359i	0.533805i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.74456	−0.0928894
$874$	0	0
$875$	23.1168i	0.781492i
$876$	0	0
$877$	22.6641i	0.765314i	0.923891	+	0.382657i	$0.124991\pi$
$877$	22.6641i	0.765314i	−0.923891	+	0.382657i	$0.875009\pi$
$878$	0	0
$879$	−14.8511	−0.500914
$880$	0	0
$881$	36.0951	1.21607	0.608037	−	0.793908i	$-0.291957\pi$
$881$	36.0951	1.21607	0.608037	+	0.793908i	$0.291957\pi$
$882$	0	0
$883$	− 3.39403i	− 0.114218i	−0.998368	−	0.0571091i	$-0.981812\pi$
$883$	− 3.39403i	− 0.114218i	0.998368	−	0.0571091i	$-0.0181883\pi$
$884$	0	0
$885$	22.0742i	0.742017i
$886$	0	0
$887$	−24.6535	−0.827783	−0.413891	−	0.910326i	$-0.635831\pi$
$887$	−24.6535	−0.827783	−0.413891	+	0.910326i	$0.635831\pi$
$888$	0	0
$889$	21.4891	0.720722
$890$	0	0
$891$	− 2.37228i	− 0.0794744i
$892$	0	0
$893$	− 2.81929i	− 0.0943440i
$894$	0	0
$895$	13.8564	0.463169
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.48913i	− 0.183073i
$900$	0	0
$901$	− 2.46943i	− 0.0822688i
$902$	0	0
$903$	−28.0627	−0.933867
$904$	0	0
$905$	14.2337	0.473144
$906$	0	0
$907$	21.2119i	0.704331i	0.935938	+	0.352165i	$0.114555\pi$
$907$	21.2119i	0.704331i	−0.935938	+	0.352165i	$0.885445\pi$
$908$	0	0
$909$	− 11.6819i	− 0.387465i
$910$	0	0
$911$	24.6535	0.816806	0.408403	−	0.912802i	$-0.366086\pi$
$911$	24.6535	0.816806	0.408403	+	0.912802i	$0.366086\pi$
$912$	0	0
$913$	18.9783	0.628088
$914$	0	0
$915$	− 7.11684i	− 0.235276i
$916$	0	0
$917$	48.2574i	1.59360i
$918$	0	0
$919$	16.7306	0.551892	0.275946	−	0.961173i	$-0.411009\pi$
$919$	16.7306	0.551892	0.275946	+	0.961173i	$0.411009\pi$
$920$	0	0
$921$	−26.2337	−0.864429
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.34896i	− 0.142993i
$926$	0	0
$927$	−13.5615	−0.445417
$928$	0	0
$929$	28.5109	0.935411	0.467706	−	0.883884i	$-0.345081\pi$
$929$	28.5109	0.935411	0.467706	+	0.883884i	$0.345081\pi$
$930$	0	0
$931$	0.627719i	0.0205726i
$932$	0	0
$933$	− 5.98844i	− 0.196053i
$934$	0	0
$935$	2.22938	0.0729087
$936$	0	0
$937$	−50.6060	−1.65322	−0.826612	−	0.562772i	$-0.809735\pi$
$937$	−50.6060	−1.65322	−0.826612	+	0.562772i	$0.809735\pi$
$938$	0	0
$939$	− 24.9783i	− 0.815134i
$940$	0	0
$941$	− 34.3461i	− 1.11965i	−0.828611	−	0.559825i	$-0.810868\pi$
$941$	− 34.3461i	− 1.11965i	0.828611	−	0.559825i	$-0.189132\pi$
$942$	0	0
$943$	−13.8564	−0.451227
$944$	0	0
$945$	−6.37228	−0.207290
$946$	0	0
$947$	33.9565i	1.10344i	0.834030	+	0.551719i	$0.186028\pi$
$947$	33.9565i	1.10344i	−0.834030	+	0.551719i	$0.813972\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−9.80240	−0.317865
$952$	0	0
$953$	−40.9783	−1.32742	−0.663708	−	0.747992i	$-0.731018\pi$
$953$	−40.9783	−1.32742	−0.663708	+	0.747992i	$0.731018\pi$
$954$	0	0
$955$	− 26.3723i	− 0.853387i
$956$	0	0
$957$	3.75906i	0.121513i
$958$	0	0
$959$	−12.9166	−0.417100
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	− 8.74456i	− 0.281790i
$964$	0	0
$965$	29.0024i	0.933621i
$966$	0	0
$967$	46.4327	1.49318	0.746588	−	0.665286i	$-0.231691\pi$
$967$	46.4327	1.49318	0.746588	+	0.665286i	$0.231691\pi$
$968$	0	0
$969$	−0.372281	−0.0119594
$970$	0	0
$971$	9.48913i	0.304521i	0.988340	+	0.152260i	$0.0486552\pi$
$971$	9.48913i	0.304521i	−0.988340	+	0.152260i	$0.951345\pi$
$972$	0	0
$973$	− 28.0627i	− 0.899648i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.4891	−1.39134	−0.695670	−	0.718361i	$-0.744892\pi$
$977$	−43.4891	−1.39134	−0.695670	+	0.718361i	$0.744892\pi$
$978$	0	0
$979$	− 17.7663i	− 0.567814i
$980$	0	0
$981$	18.3152i	0.584759i
$982$	0	0
$983$	8.91754	0.284425	0.142213	−	0.989836i	$-0.454578\pi$
$983$	8.91754	0.284425	0.142213	+	0.989836i	$0.454578\pi$
$984$	0	0
$985$	−13.4891	−0.429799
$986$	0	0
$987$	− 7.11684i	− 0.226532i
$988$	0	0
$989$	− 56.1253i	− 1.78468i
$990$	0	0
$991$	4.16381	0.132268	0.0661340	−	0.997811i	$-0.478934\pi$
$991$	4.16381	0.132268	0.0661340	+	0.997811i	$0.478934\pi$
$992$	0	0
$993$	18.2337	0.578629
$994$	0	0
$995$	9.62772i	0.305219i
$996$	0	0
$997$	6.57835i	0.208338i	0.994560	+	0.104169i	$0.0332183\pi$
$997$	6.57835i	0.208338i	−0.994560	+	0.104169i	$0.966782\pi$
$998$	0	0
$999$	−3.16915	−0.100267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3648.2.g.h.1825.4	yes	8
4.3	odd	2	inner	3648.2.g.h.1825.8	yes	8
8.3	odd	2	inner	3648.2.g.h.1825.1	✓	8
8.5	even	2	inner	3648.2.g.h.1825.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3648.2.g.h.1825.1	✓	8	8.3	odd	2	inner
3648.2.g.h.1825.4	yes	8	1.1	even	1	trivial
3648.2.g.h.1825.5	yes	8	8.5	even	2	inner
3648.2.g.h.1825.8	yes	8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.52434i q^{5} +2.52434 q^{7} -1.00000 q^{9} -2.37228i q^{11} +2.52434 q^{15} +0.372281 q^{17} -1.00000i q^{19} -2.52434i q^{21} +5.04868 q^{23} -1.37228 q^{25} +1.00000i q^{27} -1.58457i q^{29} +3.46410 q^{31} -2.37228 q^{33} +6.37228i q^{35} +3.16915i q^{37} -2.74456 q^{41} -11.1168i q^{43} -2.52434i q^{45} +2.81929 q^{47} -0.627719 q^{49} -0.372281i q^{51} -6.63325i q^{53} +5.98844 q^{55} -1.00000 q^{57} +8.74456i q^{59} -2.81929i q^{61} -2.52434 q^{63} -0.744563i q^{67} -5.04868i q^{69} +8.80773 q^{71} +7.62772 q^{73} +1.37228i q^{75} -5.98844i q^{77} -6.63325 q^{79} +1.00000 q^{81} +8.00000i q^{83} +0.939764i q^{85} -1.58457 q^{87} +7.48913 q^{89} -3.46410i q^{93} +2.52434 q^{95} +2.74456 q^{97} +2.37228i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 20 q^{17} + 12 q^{25} + 4 q^{33} + 24 q^{41} - 28 q^{49} - 8 q^{57} + 84 q^{73} + 8 q^{81} - 32 q^{89} - 24 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 20 * q^17 + 12 * q^25 + 4 * q^33 + 24 * q^41 - 28 * q^49 - 8 * q^57 + 84 * q^73 + 8 * q^81 - 32 * q^89 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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