LMFDB - Embedded newform 2304.2.f.a.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(1151,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.2.f.a.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421 q^{5} -4.82843i q^{7} +O(q^{10})$	$q-3.41421 q^{5} -4.82843i q^{7} +2.82843i q^{11} +2.82843i q^{13} +5.41421i q^{17} -5.65685 q^{19} -1.17157 q^{23} +6.65685 q^{25} +0.585786 q^{29} -3.17157i q^{31} +16.4853i q^{35} -3.65685i q^{37} -2.58579i q^{41} +9.65685 q^{43} +12.4853 q^{47} -16.3137 q^{49} +5.07107 q^{53} -9.65685i q^{55} -2.34315i q^{59} +7.65685i q^{61} -9.65685i q^{65} +12.0000 q^{67} +4.48528 q^{71} -4.00000 q^{73} +13.6569 q^{77} +6.48528i q^{79} -5.17157i q^{83} -18.4853i q^{85} -12.2426i q^{89} +13.6569 q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5}+O(q^{10}) $ 4 * q - 8 * q^5	$ 4 q - 8 q^{5} - 16 q^{23} + 4 q^{25} + 8 q^{29} + 16 q^{43} + 16 q^{47} - 20 q^{49} - 8 q^{53} + 48 q^{67} - 16 q^{71} - 16 q^{73} + 32 q^{77} + 32 q^{91} + 32 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 - 16 * q^23 + 4 * q^25 + 8 * q^29 + 16 * q^43 + 16 * q^47 - 20 * q^49 - 8 * q^53 + 48 * q^67 - 16 * q^71 - 16 * q^73 + 32 * q^77 + 32 * q^91 + 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	− 4.82843i	− 1.82497i	−0.409106	−	0.912487i	$-0.634159\pi$
$7$	− 4.82843i	− 1.82497i	0.409106	−	0.912487i	$-0.365841\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.82843i	0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$	2.82843i	0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.82843i	0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$	2.82843i	0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.41421i	1.31314i	0.754265	+	0.656570i	$0.227993\pi$
$17$	5.41421i	1.31314i	−0.754265	+	0.656570i	$0.772007\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.17157	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$23$	−1.17157	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.585786	0.108778	0.0543889	−	0.998520i	$-0.482679\pi$
$29$	0.585786	0.108778	0.0543889	+	0.998520i	$0.482679\pi$
$30$	0	0
$31$	− 3.17157i	− 0.569631i	−0.958582	−	0.284816i	$-0.908068\pi$
$31$	− 3.17157i	− 0.569631i	0.958582	−	0.284816i	$-0.0919324\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.4853i	2.78652i
$36$	0	0
$37$	− 3.65685i	− 0.601183i	−0.953753	−	0.300592i	$-0.902816\pi$
$37$	− 3.65685i	− 0.601183i	0.953753	−	0.300592i	$-0.0971841\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.58579i	− 0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$41$	− 2.58579i	− 0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.4853	1.82117	0.910583	−	0.413327i	$-0.135633\pi$
$47$	12.4853	1.82117	0.910583	+	0.413327i	$0.135633\pi$
$48$	0	0
$49$	−16.3137	−2.33053
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.07107	0.696565	0.348282	−	0.937390i	$-0.386765\pi$
$53$	5.07107	0.696565	0.348282	+	0.937390i	$0.386765\pi$
$54$	0	0
$55$	− 9.65685i	− 1.30213i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.34315i	− 0.305052i	−0.988299	−	0.152526i	$-0.951259\pi$
$59$	− 2.34315i	− 0.305052i	0.988299	−	0.152526i	$-0.0487407\pi$
$60$	0	0
$61$	7.65685i	0.980360i	0.871621	+	0.490180i	$0.163069\pi$
$61$	7.65685i	0.980360i	−0.871621	+	0.490180i	$0.836931\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 9.65685i	− 1.19779i
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.48528	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$71$	4.48528	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.6569	1.55634
$78$	0	0
$79$	6.48528i	0.729651i	0.931076	+	0.364826i	$0.118871\pi$
$79$	6.48528i	0.729651i	−0.931076	+	0.364826i	$0.881129\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.17157i	− 0.567654i	−0.958876	−	0.283827i	$-0.908396\pi$
$83$	− 5.17157i	− 0.567654i	0.958876	−	0.283827i	$-0.0916041\pi$
$84$	0	0
$85$	− 18.4853i	− 2.00501i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.2426i	− 1.29772i	−0.760909	−	0.648859i	$-0.775247\pi$
$89$	− 12.2426i	− 1.29772i	0.760909	−	0.648859i	$-0.224753\pi$
$90$	0	0
$91$	13.6569	1.43163
$92$	0	0
$93$	0	0
$94$	0	0
$95$	19.3137	1.98154
$96$	0	0
$97$	13.6569	1.38664	0.693322	−	0.720628i	$-0.256146\pi$
$97$	13.6569	1.38664	0.693322	+	0.720628i	$0.256146\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.7279	−1.06747	−0.533734	−	0.845652i	$-0.679212\pi$
$101$	−10.7279	−1.06747	−0.533734	+	0.845652i	$0.679212\pi$
$102$	0	0
$103$	− 3.17157i	− 0.312504i	−0.987717	−	0.156252i	$-0.950059\pi$
$103$	− 3.17157i	− 0.312504i	0.987717	−	0.156252i	$-0.0499413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3137i	1.09374i	0.837218	+	0.546869i	$0.184180\pi$
$107$	11.3137i	1.09374i	−0.837218	+	0.546869i	$0.815820\pi$
$108$	0	0
$109$	− 10.8284i	− 1.03718i	−0.855024	−	0.518588i	$-0.826458\pi$
$109$	− 10.8284i	− 1.03718i	0.855024	−	0.518588i	$-0.173542\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.2426i	1.15169i	0.817559	+	0.575845i	$0.195327\pi$
$113$	12.2426i	1.15169i	−0.817559	+	0.575845i	$0.804673\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	26.1421	2.39645
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	4.82843i	0.428454i	0.976784	+	0.214227i	$0.0687232\pi$
$127$	4.82843i	0.428454i	−0.976784	+	0.214227i	$0.931277\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685i	0.494242i	0.968985	+	0.247121i	$0.0794845\pi$
$131$	5.65685i	0.494242i	−0.968985	+	0.247121i	$0.920516\pi$
$132$	0	0
$133$	27.3137i	2.36840i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.5858i	0.904405i	0.891915	+	0.452202i	$0.149362\pi$
$137$	10.5858i	0.904405i	−0.891915	+	0.452202i	$0.850638\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.72792	0.551173	0.275586	−	0.961276i	$-0.411128\pi$
$149$	6.72792	0.551173	0.275586	+	0.961276i	$0.411128\pi$
$150$	0	0
$151$	6.48528i	0.527765i	0.964555	+	0.263882i	$0.0850031\pi$
$151$	6.48528i	0.527765i	−0.964555	+	0.263882i	$0.914997\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.8284i	0.869760i
$156$	0	0
$157$	0.343146i	0.0273860i	0.999906	+	0.0136930i	$0.00435876\pi$
$157$	0.343146i	0.0273860i	−0.999906	+	0.0136930i	$0.995641\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.65685i	0.445823i
$162$	0	0
$163$	1.65685	0.129775	0.0648874	−	0.997893i	$-0.479331\pi$
$163$	1.65685	0.129775	0.0648874	+	0.997893i	$0.479331\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.5858	−1.56511	−0.782554	−	0.622582i	$-0.786084\pi$
$173$	−20.5858	−1.56511	−0.782554	+	0.622582i	$0.786084\pi$
$174$	0	0
$175$	− 32.1421i	− 2.42972i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.3137i	1.44357i	0.692115	+	0.721787i	$0.256679\pi$
$179$	19.3137i	1.44357i	−0.692115	+	0.721787i	$0.743321\pi$
$180$	0	0
$181$	6.14214i	0.456541i	0.973598	+	0.228271i	$0.0733071\pi$
$181$	6.14214i	0.456541i	−0.973598	+	0.228271i	$0.926693\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.4853i	0.917936i
$186$	0	0
$187$	−15.3137	−1.11985
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.3431	−0.748404	−0.374202	−	0.927347i	$-0.622083\pi$
$191$	−10.3431	−0.748404	−0.374202	+	0.927347i	$0.622083\pi$
$192$	0	0
$193$	5.31371	0.382489	0.191245	−	0.981542i	$-0.438748\pi$
$193$	5.31371	0.382489	0.191245	+	0.981542i	$0.438748\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.3848	−1.16737	−0.583683	−	0.811981i	$-0.698389\pi$
$197$	−16.3848	−1.16737	−0.583683	+	0.811981i	$0.698389\pi$
$198$	0	0
$199$	4.82843i	0.342278i	0.985247	+	0.171139i	$0.0547447\pi$
$199$	4.82843i	0.342278i	−0.985247	+	0.171139i	$0.945255\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.82843i	− 0.198517i
$204$	0	0
$205$	8.82843i	0.616604i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 16.0000i	− 1.10674i
$210$	0	0
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−32.9706	−2.24857
$216$	0	0
$217$	−15.3137	−1.03956
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.3137	−1.03011
$222$	0	0
$223$	− 17.7990i	− 1.19191i	−0.803018	−	0.595954i	$-0.796774\pi$
$223$	− 17.7990i	− 1.19191i	0.803018	−	0.595954i	$-0.203226\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26.8284i	− 1.78067i	−0.455311	−	0.890333i	$-0.650472\pi$
$227$	− 26.8284i	− 1.78067i	0.455311	−	0.890333i	$-0.349528\pi$
$228$	0	0
$229$	5.17157i	0.341747i	0.985293	+	0.170874i	$0.0546590\pi$
$229$	5.17157i	0.341747i	−0.985293	+	0.170874i	$0.945341\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.8995i	1.17263i	0.810081	+	0.586317i	$0.199423\pi$
$233$	17.8995i	1.17263i	−0.810081	+	0.586317i	$0.800577\pi$
$234$	0	0
$235$	−42.6274	−2.78071
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	20.9706	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$241$	20.9706	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	55.6985	3.55845
$246$	0	0
$247$	− 16.0000i	− 1.01806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.4853i	1.54550i	0.634712	+	0.772749i	$0.281119\pi$
$251$	24.4853i	1.54550i	−0.634712	+	0.772749i	$0.718881\pi$
$252$	0	0
$253$	− 3.31371i	− 0.208331i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.89949i	− 0.617514i	−0.951141	−	0.308757i	$-0.900087\pi$
$257$	− 9.89949i	− 0.617514i	0.951141	−	0.308757i	$-0.0999129\pi$
$258$	0	0
$259$	−17.6569	−1.09714
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.3431	1.13109	0.565543	−	0.824719i	$-0.308666\pi$
$263$	18.3431	1.13109	0.565543	+	0.824719i	$0.308666\pi$
$264$	0	0
$265$	−17.3137	−1.06357
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.3848	1.97453	0.987267	−	0.159070i	$-0.0508495\pi$
$269$	32.3848	1.97453	0.987267	+	0.159070i	$0.0508495\pi$
$270$	0	0
$271$	1.51472i	0.0920126i	0.998941	+	0.0460063i	$0.0146494\pi$
$271$	1.51472i	0.0920126i	−0.998941	+	0.0460063i	$0.985351\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.8284i	1.13540i
$276$	0	0
$277$	− 21.1716i	− 1.27208i	−0.771658	−	0.636038i	$-0.780572\pi$
$277$	− 21.1716i	− 1.27208i	0.771658	−	0.636038i	$-0.219428\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.10051i	0.363926i	0.983305	+	0.181963i	$0.0582450\pi$
$281$	6.10051i	0.363926i	−0.983305	+	0.181963i	$0.941755\pi$
$282$	0	0
$283$	−3.31371	−0.196980	−0.0984898	−	0.995138i	$-0.531401\pi$
$283$	−3.31371	−0.196980	−0.0984898	+	0.995138i	$0.531401\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.4853	−0.736983
$288$	0	0
$289$	−12.3137	−0.724336
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.8701	−1.68661	−0.843303	−	0.537438i	$-0.819392\pi$
$293$	−28.8701	−1.68661	−0.843303	+	0.537438i	$0.819392\pi$
$294$	0	0
$295$	8.00000i	0.465778i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.31371i	− 0.191637i
$300$	0	0
$301$	− 46.6274i	− 2.68756i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 26.1421i	− 1.49689i
$306$	0	0
$307$	−0.686292	−0.0391687	−0.0195844	−	0.999808i	$-0.506234\pi$
$307$	−0.686292	−0.0391687	−0.0195844	+	0.999808i	$0.506234\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.6569	0.774409	0.387205	−	0.921994i	$-0.373441\pi$
$311$	13.6569	0.774409	0.387205	+	0.921994i	$0.373441\pi$
$312$	0	0
$313$	−9.31371	−0.526442	−0.263221	−	0.964736i	$-0.584785\pi$
$313$	−9.31371	−0.526442	−0.263221	+	0.964736i	$0.584785\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.2426	−0.799946	−0.399973	−	0.916527i	$-0.630981\pi$
$317$	−14.2426	−0.799946	−0.399973	+	0.916527i	$0.630981\pi$
$318$	0	0
$319$	1.65685i	0.0927660i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 30.6274i	− 1.70416i
$324$	0	0
$325$	18.8284i	1.04441i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 60.2843i	− 3.32358i
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−40.9706	−2.23846
$336$	0	0
$337$	23.3137	1.26998	0.634989	−	0.772521i	$-0.281004\pi$
$337$	23.3137	1.26998	0.634989	+	0.772521i	$0.281004\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.97056	0.485783
$342$	0	0
$343$	44.9706i	2.42818i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.4853i	− 0.884976i	−0.896774	−	0.442488i	$-0.854096\pi$
$347$	− 16.4853i	− 0.884976i	0.896774	−	0.442488i	$-0.145904\pi$
$348$	0	0
$349$	− 11.6569i	− 0.623977i	−0.950086	−	0.311989i	$-0.899005\pi$
$349$	− 11.6569i	− 0.623977i	0.950086	−	0.311989i	$-0.100995\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.8701i	1.00435i	0.864765	+	0.502176i	$0.167467\pi$
$353$	18.8701i	1.00435i	−0.864765	+	0.502176i	$0.832533\pi$
$354$	0	0
$355$	−15.3137	−0.812767
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.4853	−1.08117	−0.540586	−	0.841289i	$-0.681797\pi$
$359$	−20.4853	−1.08117	−0.540586	+	0.841289i	$0.681797\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.6569	0.714832
$366$	0	0
$367$	20.8284i	1.08724i	0.839333	+	0.543618i	$0.182946\pi$
$367$	20.8284i	1.08724i	−0.839333	+	0.543618i	$0.817054\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 24.4853i	− 1.27121i
$372$	0	0
$373$	14.9706i	0.775146i	0.921839	+	0.387573i	$0.126687\pi$
$373$	14.9706i	0.775146i	−0.921839	+	0.387573i	$0.873313\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.65685i	0.0853323i
$378$	0	0
$379$	20.2843	1.04193	0.520967	−	0.853577i	$-0.325572\pi$
$379$	20.2843	1.04193	0.520967	+	0.853577i	$0.325572\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.6274	−0.747426	−0.373713	−	0.927544i	$-0.621916\pi$
$383$	−14.6274	−0.747426	−0.373713	+	0.927544i	$0.621916\pi$
$384$	0	0
$385$	−46.6274	−2.37635
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.44365	0.123898	0.0619490	−	0.998079i	$-0.480268\pi$
$389$	2.44365	0.123898	0.0619490	+	0.998079i	$0.480268\pi$
$390$	0	0
$391$	− 6.34315i	− 0.320787i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 22.1421i	− 1.11409i
$396$	0	0
$397$	18.9706i	0.952105i	0.879417	+	0.476053i	$0.157933\pi$
$397$	18.9706i	0.952105i	−0.879417	+	0.476053i	$0.842067\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.4142i	0.669874i	0.942241	+	0.334937i	$0.108715\pi$
$401$	13.4142i	0.669874i	−0.942241	+	0.334937i	$0.891285\pi$
$402$	0	0
$403$	8.97056	0.446856
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.3431	0.512691
$408$	0	0
$409$	−10.3431	−0.511436	−0.255718	−	0.966751i	$-0.582312\pi$
$409$	−10.3431	−0.511436	−0.255718	+	0.966751i	$0.582312\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.3137	−0.556711
$414$	0	0
$415$	17.6569i	0.866741i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 11.7990i	− 0.576418i	−0.957567	−	0.288209i	$-0.906940\pi$
$419$	− 11.7990i	− 0.576418i	0.957567	−	0.288209i	$-0.0930599\pi$
$420$	0	0
$421$	7.51472i	0.366245i	0.983090	+	0.183122i	$0.0586205\pi$
$421$	7.51472i	0.366245i	−0.983090	+	0.183122i	$0.941380\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	36.0416i	1.74828i
$426$	0	0
$427$	36.9706	1.78913
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.4853	1.37209	0.686044	−	0.727560i	$-0.259346\pi$
$431$	28.4853	1.37209	0.686044	+	0.727560i	$0.259346\pi$
$432$	0	0
$433$	28.6274	1.37575	0.687873	−	0.725831i	$-0.258545\pi$
$433$	28.6274	1.37575	0.687873	+	0.725831i	$0.258545\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.62742	0.317032
$438$	0	0
$439$	7.85786i	0.375035i	0.982261	+	0.187518i	$0.0600442\pi$
$439$	7.85786i	0.375035i	−0.982261	+	0.187518i	$0.939956\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 17.4558i	− 0.829352i	−0.909969	−	0.414676i	$-0.863895\pi$
$443$	− 17.4558i	− 0.829352i	0.909969	−	0.414676i	$-0.136105\pi$
$444$	0	0
$445$	41.7990i	1.98146i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.2721i	0.720734i	0.932811	+	0.360367i	$0.117349\pi$
$449$	15.2721i	0.720734i	−0.932811	+	0.360367i	$0.882651\pi$
$450$	0	0
$451$	7.31371	0.344389
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−46.6274	−2.18593
$456$	0	0
$457$	8.97056	0.419625	0.209813	−	0.977742i	$-0.432715\pi$
$457$	8.97056	0.419625	0.209813	+	0.977742i	$0.432715\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.10051	−0.377278	−0.188639	−	0.982046i	$-0.560408\pi$
$461$	−8.10051	−0.377278	−0.188639	+	0.982046i	$0.560408\pi$
$462$	0	0
$463$	33.7990i	1.57077i	0.619006	+	0.785386i	$0.287536\pi$
$463$	33.7990i	1.57077i	−0.619006	+	0.785386i	$0.712464\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.14214i	− 0.284224i	−0.989851	−	0.142112i	$-0.954611\pi$
$467$	− 6.14214i	− 0.284224i	0.989851	−	0.142112i	$-0.0453893\pi$
$468$	0	0
$469$	− 57.9411i	− 2.67547i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	27.3137i	1.25589i
$474$	0	0
$475$	−37.6569	−1.72781
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.17157	0.419060	0.209530	−	0.977802i	$-0.432807\pi$
$479$	9.17157	0.419060	0.209530	+	0.977802i	$0.432807\pi$
$480$	0	0
$481$	10.3431	0.471607
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−46.6274	−2.11724
$486$	0	0
$487$	28.8284i	1.30634i	0.757211	+	0.653170i	$0.226561\pi$
$487$	28.8284i	1.30634i	−0.757211	+	0.653170i	$0.773439\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 35.3137i	− 1.59369i	−0.604187	−	0.796843i	$-0.706502\pi$
$491$	− 35.3137i	− 1.59369i	0.604187	−	0.796843i	$-0.293498\pi$
$492$	0	0
$493$	3.17157i	0.142840i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 21.6569i	− 0.971443i
$498$	0	0
$499$	−6.62742	−0.296684	−0.148342	−	0.988936i	$-0.547394\pi$
$499$	−6.62742	−0.296684	−0.148342	+	0.988936i	$0.547394\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.7990	−0.704442	−0.352221	−	0.935917i	$-0.614573\pi$
$503$	−15.7990	−0.704442	−0.352221	+	0.935917i	$0.614573\pi$
$504$	0	0
$505$	36.6274	1.62990
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.2721	0.588275	0.294137	−	0.955763i	$-0.404968\pi$
$509$	13.2721	0.588275	0.294137	+	0.955763i	$0.404968\pi$
$510$	0	0
$511$	19.3137i	0.854388i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.8284i	0.477158i
$516$	0	0
$517$	35.3137i	1.55310i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3.07107i	− 0.134546i	−0.997735	−	0.0672730i	$-0.978570\pi$
$521$	− 3.07107i	− 0.134546i	0.997735	−	0.0672730i	$-0.0214298\pi$
$522$	0	0
$523$	−45.6569	−1.99643	−0.998217	−	0.0596823i	$-0.980991\pi$
$523$	−45.6569	−1.99643	−0.998217	+	0.0596823i	$0.980991\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.1716	0.748005
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.31371	0.316792
$534$	0	0
$535$	− 38.6274i	− 1.67001i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 46.1421i	− 1.98748i
$540$	0	0
$541$	35.7990i	1.53912i	0.638575	+	0.769559i	$0.279524\pi$
$541$	35.7990i	1.53912i	−0.638575	+	0.769559i	$0.720476\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	36.9706i	1.58364i
$546$	0	0
$547$	−9.65685	−0.412897	−0.206449	−	0.978457i	$-0.566191\pi$
$547$	−9.65685	−0.412897	−0.206449	+	0.978457i	$0.566191\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	31.3137	1.33159
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.07107	−0.384353	−0.192177	−	0.981360i	$-0.561555\pi$
$557$	−9.07107	−0.384353	−0.192177	+	0.981360i	$0.561555\pi$
$558$	0	0
$559$	27.3137i	1.15525i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 25.4558i	− 1.07284i	−0.843952	−	0.536418i	$-0.819777\pi$
$563$	− 25.4558i	− 1.07284i	0.843952	−	0.536418i	$-0.180223\pi$
$564$	0	0
$565$	− 41.7990i	− 1.75850i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 2.10051i	− 0.0880578i	−0.999030	−	0.0440289i	$-0.985981\pi$
$569$	− 2.10051i	− 0.0880578i	0.999030	−	0.0440289i	$-0.0140194\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.79899	−0.325240
$576$	0	0
$577$	−29.3137	−1.22035	−0.610173	−	0.792268i	$-0.708900\pi$
$577$	−29.3137	−1.22035	−0.610173	+	0.792268i	$0.708900\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−24.9706	−1.03595
$582$	0	0
$583$	14.3431i	0.594032i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.6569i	1.22407i	0.790831	+	0.612035i	$0.209649\pi$
$587$	29.6569i	1.22407i	−0.790831	+	0.612035i	$0.790351\pi$
$588$	0	0
$589$	17.9411i	0.739251i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.0711i	0.618895i	0.950917	+	0.309447i	$0.100144\pi$
$593$	15.0711i	0.618895i	−0.950917	+	0.309447i	$0.899856\pi$
$594$	0	0
$595$	−89.2548	−3.65909
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−44.4853	−1.81762	−0.908810	−	0.417211i	$-0.863008\pi$
$599$	−44.4853	−1.81762	−0.908810	+	0.417211i	$0.863008\pi$
$600$	0	0
$601$	−21.3137	−0.869404	−0.434702	−	0.900574i	$-0.643146\pi$
$601$	−21.3137	−0.869404	−0.434702	+	0.900574i	$0.643146\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.2426	−0.416423
$606$	0	0
$607$	− 19.1716i	− 0.778150i	−0.921206	−	0.389075i	$-0.872795\pi$
$607$	− 19.1716i	− 0.778150i	0.921206	−	0.389075i	$-0.127205\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.3137i	1.42864i
$612$	0	0
$613$	11.6569i	0.470816i	0.971897	+	0.235408i	$0.0756426\pi$
$613$	11.6569i	0.470816i	−0.971897	+	0.235408i	$0.924357\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.5858i	− 1.23134i	−0.788005	−	0.615669i	$-0.788886\pi$
$617$	− 30.5858i	− 1.23134i	0.788005	−	0.615669i	$-0.211114\pi$
$618$	0	0
$619$	33.9411	1.36421	0.682105	−	0.731255i	$-0.261065\pi$
$619$	33.9411	1.36421	0.682105	+	0.731255i	$0.261065\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−59.1127	−2.36830
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.7990	0.789437
$630$	0	0
$631$	1.51472i	0.0603000i	0.999545	+	0.0301500i	$0.00959850\pi$
$631$	1.51472i	0.0603000i	−0.999545	+	0.0301500i	$0.990402\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.4853i	− 0.654198i
$636$	0	0
$637$	− 46.1421i	− 1.82822i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 38.8701i	− 1.53527i	−0.640884	−	0.767637i	$-0.721432\pi$
$641$	− 38.8701i	− 1.53527i	0.640884	−	0.767637i	$-0.278568\pi$
$642$	0	0
$643$	1.65685	0.0653400	0.0326700	−	0.999466i	$-0.489599\pi$
$643$	1.65685	0.0653400	0.0326700	+	0.999466i	$0.489599\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.7990	0.935635	0.467817	−	0.883825i	$-0.345040\pi$
$647$	23.7990	0.935635	0.467817	+	0.883825i	$0.345040\pi$
$648$	0	0
$649$	6.62742	0.260149
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.7279	1.20248	0.601238	−	0.799070i	$-0.294674\pi$
$653$	30.7279	1.20248	0.601238	+	0.799070i	$0.294674\pi$
$654$	0	0
$655$	− 19.3137i	− 0.754649i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 24.9706i	− 0.972715i	−0.873760	−	0.486358i	$-0.838325\pi$
$659$	− 24.9706i	− 0.972715i	0.873760	−	0.486358i	$-0.161675\pi$
$660$	0	0
$661$	7.65685i	0.297817i	0.988851	+	0.148909i	$0.0475760\pi$
$661$	7.65685i	0.297817i	−0.988851	+	0.148909i	$0.952424\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 93.2548i	− 3.61627i
$666$	0	0
$667$	−0.686292	−0.0265733
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−21.6569	−0.836054
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.72792	−0.258575	−0.129288	−	0.991607i	$-0.541269\pi$
$677$	−6.72792	−0.258575	−0.129288	+	0.991607i	$0.541269\pi$
$678$	0	0
$679$	− 65.9411i	− 2.53059i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.14214i	− 0.235022i	−0.993072	−	0.117511i	$-0.962508\pi$
$683$	− 6.14214i	− 0.235022i	0.993072	−	0.117511i	$-0.0374916\pi$
$684$	0	0
$685$	− 36.1421i	− 1.38092i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.3431i	0.546430i
$690$	0	0
$691$	−36.9706	−1.40643	−0.703213	−	0.710979i	$-0.748252\pi$
$691$	−36.9706	−1.40643	−0.703213	+	0.710979i	$0.748252\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−40.9706	−1.55410
$696$	0	0
$697$	14.0000	0.530288
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.2426	0.840093	0.420046	−	0.907503i	$-0.362014\pi$
$701$	22.2426	0.840093	0.420046	+	0.907503i	$0.362014\pi$
$702$	0	0
$703$	20.6863i	0.780198i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	51.7990i	1.94810i
$708$	0	0
$709$	− 48.0833i	− 1.80580i	−0.429846	−	0.902902i	$-0.641432\pi$
$709$	− 48.0833i	− 1.80580i	0.429846	−	0.902902i	$-0.358568\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.71573i	0.139155i
$714$	0	0
$715$	27.3137	1.02147
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.5147	0.429427	0.214713	−	0.976677i	$-0.431118\pi$
$719$	11.5147	0.429427	0.214713	+	0.976677i	$0.431118\pi$
$720$	0	0
$721$	−15.3137	−0.570312
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.89949	0.144824
$726$	0	0
$727$	− 3.17157i	− 0.117627i	−0.998269	−	0.0588136i	$-0.981268\pi$
$727$	− 3.17157i	− 0.117627i	0.998269	−	0.0588136i	$-0.0187317\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	52.2843i	1.93380i
$732$	0	0
$733$	7.51472i	0.277562i	0.990323	+	0.138781i	$0.0443185\pi$
$733$	7.51472i	0.277562i	−0.990323	+	0.138781i	$0.955682\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.9411i	1.25024i
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.9706	−0.622590	−0.311295	−	0.950313i	$-0.600763\pi$
$743$	−16.9706	−0.622590	−0.311295	+	0.950313i	$0.600763\pi$
$744$	0	0
$745$	−22.9706	−0.841576
$746$	0	0
$747$	0	0
$748$	0	0
$749$	54.6274	1.99604
$750$	0	0
$751$	− 35.1716i	− 1.28343i	−0.766944	−	0.641714i	$-0.778223\pi$
$751$	− 35.1716i	− 1.28343i	0.766944	−	0.641714i	$-0.221777\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 22.1421i	− 0.805835i
$756$	0	0
$757$	16.4853i	0.599168i	0.954070	+	0.299584i	$0.0968478\pi$
$757$	16.4853i	0.599168i	−0.954070	+	0.299584i	$0.903152\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.6985i	1.22157i	0.791797	+	0.610785i	$0.209146\pi$
$761$	33.6985i	1.22157i	−0.791797	+	0.610785i	$0.790854\pi$
$762$	0	0
$763$	−52.2843	−1.89282
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.62742	0.239302
$768$	0	0
$769$	1.31371	0.0473735	0.0236868	−	0.999719i	$-0.492460\pi$
$769$	1.31371	0.0473735	0.0236868	+	0.999719i	$0.492460\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.4142	0.554411	0.277205	−	0.960811i	$-0.410592\pi$
$773$	15.4142	0.554411	0.277205	+	0.960811i	$0.410592\pi$
$774$	0	0
$775$	− 21.1127i	− 0.758391i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.6274i	0.524082i
$780$	0	0
$781$	12.6863i	0.453951i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.17157i	− 0.0418152i
$786$	0	0
$787$	37.6569	1.34232	0.671161	−	0.741312i	$-0.265796\pi$
$787$	37.6569	1.34232	0.671161	+	0.741312i	$0.265796\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	59.1127	2.10181
$792$	0	0
$793$	−21.6569	−0.769057
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.07107	0.0379392	0.0189696	−	0.999820i	$-0.493961\pi$
$797$	1.07107	0.0379392	0.0189696	+	0.999820i	$0.493961\pi$
$798$	0	0
$799$	67.5980i	2.39144i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 11.3137i	− 0.399252i
$804$	0	0
$805$	− 19.3137i	− 0.680719i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	50.6690i	1.78143i	0.454563	+	0.890714i	$0.349795\pi$
$809$	50.6690i	1.78143i	−0.454563	+	0.890714i	$0.650205\pi$
$810$	0	0
$811$	−47.5980	−1.67139	−0.835696	−	0.549193i	$-0.814935\pi$
$811$	−47.5980	−1.67139	−0.835696	+	0.549193i	$0.814935\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.65685	−0.198151
$816$	0	0
$817$	−54.6274	−1.91117
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.58579	0.299646	0.149823	−	0.988713i	$-0.452130\pi$
$821$	8.58579	0.299646	0.149823	+	0.988713i	$0.452130\pi$
$822$	0	0
$823$	32.4264i	1.13031i	0.824984	+	0.565157i	$0.191184\pi$
$823$	32.4264i	1.13031i	−0.824984	+	0.565157i	$0.808816\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 48.9706i	− 1.70287i	−0.524457	−	0.851437i	$-0.675732\pi$
$827$	− 48.9706i	− 1.70287i	0.524457	−	0.851437i	$-0.324268\pi$
$828$	0	0
$829$	43.7990i	1.52120i	0.649220	+	0.760601i	$0.275096\pi$
$829$	43.7990i	1.52120i	−0.649220	+	0.760601i	$0.724904\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 88.3259i	− 3.06031i
$834$	0	0
$835$	−19.3137	−0.668378
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.4853	−0.707230	−0.353615	−	0.935391i	$-0.615048\pi$
$839$	−20.4853	−0.707230	−0.353615	+	0.935391i	$0.615048\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17.0711	−0.587263
$846$	0	0
$847$	− 14.4853i	− 0.497720i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.28427i	0.146863i
$852$	0	0
$853$	14.2843i	0.489084i	0.969639	+	0.244542i	$0.0786376\pi$
$853$	14.2843i	0.489084i	−0.969639	+	0.244542i	$0.921362\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.10051i	0.0717519i	0.999356	+	0.0358759i	$0.0114221\pi$
$857$	2.10051i	0.0717519i	−0.999356	+	0.0358759i	$0.988578\pi$
$858$	0	0
$859$	9.65685	0.329488	0.164744	−	0.986336i	$-0.447320\pi$
$859$	9.65685	0.329488	0.164744	+	0.986336i	$0.447320\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.9706	1.12233	0.561166	−	0.827704i	$-0.310353\pi$
$863$	32.9706	1.12233	0.561166	+	0.827704i	$0.310353\pi$
$864$	0	0
$865$	70.2843	2.38974
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18.3431	−0.622249
$870$	0	0
$871$	33.9411i	1.15005i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	27.3137i	0.923372i
$876$	0	0
$877$	26.9706i	0.910731i	0.890305	+	0.455366i	$0.150491\pi$
$877$	26.9706i	0.910731i	−0.890305	+	0.455366i	$0.849509\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7279i	0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$	12.7279i	0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$	0	0
$883$	16.2843	0.548009	0.274005	−	0.961728i	$-0.411652\pi$
$883$	16.2843	0.548009	0.274005	+	0.961728i	$0.411652\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.6569	−0.458552	−0.229276	−	0.973361i	$-0.573636\pi$
$887$	−13.6569	−0.458552	−0.229276	+	0.973361i	$0.573636\pi$
$888$	0	0
$889$	23.3137	0.781917
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−70.6274	−2.36346
$894$	0	0
$895$	− 65.9411i	− 2.20417i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.85786i	− 0.0619632i
$900$	0	0
$901$	27.4558i	0.914687i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 20.9706i	− 0.697085i
$906$	0	0
$907$	25.6569	0.851922	0.425961	−	0.904742i	$-0.359936\pi$
$907$	25.6569	0.851922	0.425961	+	0.904742i	$0.359936\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.5980	1.84204	0.921022	−	0.389511i	$-0.127356\pi$
$911$	55.5980	1.84204	0.921022	+	0.389511i	$0.127356\pi$
$912$	0	0
$913$	14.6274	0.484097
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.3137	0.901978
$918$	0	0
$919$	− 27.4558i	− 0.905685i	−0.891591	−	0.452842i	$-0.850410\pi$
$919$	− 27.4558i	− 0.905685i	0.891591	−	0.452842i	$-0.149590\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.6863i	0.417574i
$924$	0	0
$925$	− 24.3431i	− 0.800398i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.72792i	0.286354i	0.989697	+	0.143177i	$0.0457318\pi$
$929$	8.72792i	0.286354i	−0.989697	+	0.143177i	$0.954268\pi$
$930$	0	0
$931$	92.2843	3.02449
$932$	0	0
$933$	0	0
$934$	0	0
$935$	52.2843	1.70988
$936$	0	0
$937$	11.3726	0.371526	0.185763	−	0.982595i	$-0.440524\pi$
$937$	11.3726	0.371526	0.185763	+	0.982595i	$0.440524\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.7279	−1.52329	−0.761643	−	0.647996i	$-0.775607\pi$
$941$	−46.7279	−1.52329	−0.761643	+	0.647996i	$0.775607\pi$
$942$	0	0
$943$	3.02944i	0.0986521i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.5980i	1.54673i	0.633963	+	0.773363i	$0.281427\pi$
$947$	47.5980i	1.54673i	−0.633963	+	0.773363i	$0.718573\pi$
$948$	0	0
$949$	− 11.3137i	− 0.367259i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.5563i	1.15178i	0.817526	+	0.575892i	$0.195345\pi$
$953$	35.5563i	1.15178i	−0.817526	+	0.575892i	$0.804655\pi$
$954$	0	0
$955$	35.3137	1.14272
$956$	0	0
$957$	0	0
$958$	0	0
$959$	51.1127	1.65052
$960$	0	0
$961$	20.9411	0.675520
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18.1421	−0.584016
$966$	0	0
$967$	50.0833i	1.61057i	0.592889	+	0.805285i	$0.297987\pi$
$967$	50.0833i	1.61057i	−0.592889	+	0.805285i	$0.702013\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 26.8284i	− 0.860965i	−0.902599	−	0.430483i	$-0.858343\pi$
$971$	− 26.8284i	− 0.860965i	0.902599	−	0.430483i	$-0.141657\pi$
$972$	0	0
$973$	− 57.9411i	− 1.85751i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.5269i	1.68048i	0.542211	+	0.840242i	$0.317587\pi$
$977$	52.5269i	1.68048i	−0.542211	+	0.840242i	$0.682413\pi$
$978$	0	0
$979$	34.6274	1.10670
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.6569	1.45623	0.728114	−	0.685456i	$-0.240397\pi$
$983$	45.6569	1.45623	0.728114	+	0.685456i	$0.240397\pi$
$984$	0	0
$985$	55.9411	1.78243
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.3137	−0.359755
$990$	0	0
$991$	− 25.7990i	− 0.819532i	−0.912191	−	0.409766i	$-0.865610\pi$
$991$	− 25.7990i	− 0.819532i	0.912191	−	0.409766i	$-0.134390\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 16.4853i	− 0.522619i
$996$	0	0
$997$	61.5980i	1.95083i	0.220381	+	0.975414i	$0.429270\pi$
$997$	61.5980i	1.95083i	−0.220381	+	0.975414i	$0.570730\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.f.a.1151.1		4
3.2	odd	2		2304.2.f.h.1151.3		4
4.3	odd	2		2304.2.f.b.1151.2		4
8.3	odd	2		2304.2.f.h.1151.4		4
8.5	even	2		2304.2.f.g.1151.3		4
12.11	even	2		2304.2.f.g.1151.4		4
16.3	odd	4		1152.2.c.b.1151.4	yes	4
16.5	even	4		1152.2.c.d.1151.1	yes	4
16.11	odd	4		1152.2.c.a.1151.1	✓	4
16.13	even	4		1152.2.c.c.1151.4	yes	4
24.5	odd	2		2304.2.f.b.1151.1		4
24.11	even	2	inner	2304.2.f.a.1151.2		4
48.5	odd	4		1152.2.c.a.1151.4	yes	4
48.11	even	4		1152.2.c.d.1151.4	yes	4
48.29	odd	4		1152.2.c.b.1151.1	yes	4
48.35	even	4		1152.2.c.c.1151.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.c.a.1151.1	✓	4	16.11	odd	4
1152.2.c.a.1151.4	yes	4	48.5	odd	4
1152.2.c.b.1151.1	yes	4	48.29	odd	4
1152.2.c.b.1151.4	yes	4	16.3	odd	4
1152.2.c.c.1151.1	yes	4	48.35	even	4
1152.2.c.c.1151.4	yes	4	16.13	even	4
1152.2.c.d.1151.1	yes	4	16.5	even	4
1152.2.c.d.1151.4	yes	4	48.11	even	4
2304.2.f.a.1151.1		4	1.1	even	1	trivial
2304.2.f.a.1151.2		4	24.11	even	2	inner
2304.2.f.b.1151.1		4	24.5	odd	2
2304.2.f.b.1151.2		4	4.3	odd	2
2304.2.f.g.1151.3		4	8.5	even	2
2304.2.f.g.1151.4		4	12.11	even	2
2304.2.f.h.1151.3		4	3.2	odd	2
2304.2.f.h.1151.4		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41421 q^{5} -4.82843i q^{7} +O(q^{10})\)	\(q-3.41421 q^{5} -4.82843i q^{7} +2.82843i q^{11} +2.82843i q^{13} +5.41421i q^{17} -5.65685 q^{19} -1.17157 q^{23} +6.65685 q^{25} +0.585786 q^{29} -3.17157i q^{31} +16.4853i q^{35} -3.65685i q^{37} -2.58579i q^{41} +9.65685 q^{43} +12.4853 q^{47} -16.3137 q^{49} +5.07107 q^{53} -9.65685i q^{55} -2.34315i q^{59} +7.65685i q^{61} -9.65685i q^{65} +12.0000 q^{67} +4.48528 q^{71} -4.00000 q^{73} +13.6569 q^{77} +6.48528i q^{79} -5.17157i q^{83} -18.4853i q^{85} -12.2426i q^{89} +13.6569 q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5}+O(q^{10}) \) 4 * q - 8 * q^5	\( 4 q - 8 q^{5} - 16 q^{23} + 4 q^{25} + 8 q^{29} + 16 q^{43} + 16 q^{47} - 20 q^{49} - 8 q^{53} + 48 q^{67} - 16 q^{71} - 16 q^{73} + 32 q^{77} + 32 q^{91} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 - 16 * q^23 + 4 * q^25 + 8 * q^29 + 16 * q^43 + 16 * q^47 - 20 * q^49 - 8 * q^53 + 48 * q^67 - 16 * q^71 - 16 * q^73 + 32 * q^77 + 32 * q^91 + 32 * q^95 + 32 * q^97

Introduction

L-functions

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