LMFDB - Embedded newform 1152.2.c.c.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(1151,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("1152.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.c (of order $2$, degree $1$, 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:	$9.19876631285$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.1151
Dual form			1152.2.c.c.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421i q^{5} -4.82843i q^{7} +O(q^{10})$	$q-3.41421i q^{5} -4.82843i q^{7} +2.82843 q^{11} -2.82843 q^{13} -5.41421i q^{17} +5.65685i q^{19} +1.17157 q^{23} -6.65685 q^{25} -0.585786i q^{29} +3.17157i q^{31} -16.4853 q^{35} -3.65685 q^{37} -2.58579i q^{41} +9.65685i q^{43} +12.4853 q^{47} -16.3137 q^{49} +5.07107i q^{53} -9.65685i q^{55} -2.34315 q^{59} -7.65685 q^{61} +9.65685i q^{65} -12.0000i q^{67} -4.48528 q^{71} +4.00000 q^{73} -13.6569i q^{77} -6.48528i q^{79} +5.17157 q^{83} -18.4853 q^{85} -12.2426i q^{89} +13.6569i q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{23} - 4 q^{25} - 32 q^{35} + 8 q^{37} + 16 q^{47} - 20 q^{49} - 32 q^{59} - 8 q^{61} + 16 q^{71} + 16 q^{73} + 32 q^{83} - 40 q^{85} + 32 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q + 16 * q^23 - 4 * q^25 - 32 * q^35 + 8 * q^37 + 16 * q^47 - 20 * q^49 - 32 * q^59 - 8 * q^61 + 16 * q^71 + 16 * q^73 + 32 * q^83 - 40 * q^85 + 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$	− 3.41421i	− 1.52688i	−0.645877	−	0.763441i	$-0.723508\pi$
$5$	− 3.41421i	− 1.52688i	0.645877	−	0.763441i	$-0.276492\pi$
$6$	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
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.41421i	− 1.31314i	−0.754265	−	0.656570i	$-0.772007\pi$
$17$	− 5.41421i	− 1.31314i	0.754265	−	0.656570i	$-0.227993\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	0	0
$25$	−6.65685	−1.33137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.585786i	− 0.108778i	−0.998520	−	0.0543889i	$-0.982679\pi$
$29$	− 0.585786i	− 0.108778i	0.998520	−	0.0543889i	$-0.0173211\pi$
$30$	0	0
$31$	3.17157i	0.569631i	0.958582	+	0.284816i	$0.0919324\pi$
$31$	3.17157i	0.569631i	−0.958582	+	0.284816i	$0.908068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16.4853	−2.78652
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	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
$43$	9.65685i	1.47266i	0.676625	+	0.736328i	$0.263442\pi$
$43$	9.65685i	1.47266i	−0.676625	+	0.736328i	$0.736558\pi$
$44$	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
$53$	5.07107i	0.696565i	0.937390	+	0.348282i	$0.113235\pi$
$53$	5.07107i	0.696565i	−0.937390	+	0.348282i	$0.886765\pi$
$54$	0	0
$55$	− 9.65685i	− 1.30213i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.65685i	1.19779i
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 13.6569i	− 1.55634i
$78$	0	0
$79$	− 6.48528i	− 0.729651i	−0.931076	−	0.364826i	$-0.881129\pi$
$79$	− 6.48528i	− 0.729651i	0.931076	−	0.364826i	$-0.118871\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.17157	0.567654	0.283827	−	0.958876i	$-0.408396\pi$
$83$	5.17157	0.567654	0.283827	+	0.958876i	$0.408396\pi$
$84$	0	0
$85$	−18.4853	−2.00501
$86$	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
$91$	13.6569i	1.43163i
$92$	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
$101$	− 10.7279i	− 1.06747i	−0.845652	−	0.533734i	$-0.820788\pi$
$101$	− 10.7279i	− 1.06747i	0.845652	−	0.533734i	$-0.179212\pi$
$102$	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
$107$	11.3137	1.09374	0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137	1.09374	0.546869	+	0.837218i	$0.315820\pi$
$108$	0	0
$109$	10.8284	1.03718	0.518588	−	0.855024i	$-0.326458\pi$
$109$	10.8284	1.03718	0.518588	+	0.855024i	$0.326458\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.2426i	− 1.15169i	−0.817559	−	0.575845i	$-0.804673\pi$
$113$	− 12.2426i	− 1.15169i	0.817559	−	0.575845i	$-0.195327\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	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
$125$	5.65685i	0.505964i
$126$	0	0
$127$	− 4.82843i	− 0.428454i	−0.976784	−	0.214227i	$-0.931277\pi$
$127$	− 4.82843i	− 0.428454i	0.976784	−	0.214227i	$-0.0687232\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	27.3137	2.36840
$134$	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
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	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
$149$	6.72792i	0.551173i	0.961276	+	0.275586i	$0.0888720\pi$
$149$	6.72792i	0.551173i	−0.961276	+	0.275586i	$0.911128\pi$
$150$	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
$155$	10.8284	0.869760
$156$	0	0
$157$	−0.343146	−0.0273860	−0.0136930	−	0.999906i	$-0.504359\pi$
$157$	−0.343146	−0.0273860	−0.0136930	+	0.999906i	$0.504359\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 5.65685i	− 0.445823i
$162$	0	0
$163$	− 1.65685i	− 0.129775i	−0.997893	−	0.0648874i	$-0.979331\pi$
$163$	− 1.65685i	− 0.129775i	0.997893	−	0.0648874i	$-0.0206688\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.5858i	1.56511i	0.622582	+	0.782554i	$0.286084\pi$
$173$	20.5858i	1.56511i	−0.622582	+	0.782554i	$0.713916\pi$
$174$	0	0
$175$	32.1421i	2.42972i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.3137	−1.44357	−0.721787	−	0.692115i	$-0.756679\pi$
$179$	−19.3137	−1.44357	−0.721787	+	0.692115i	$0.756679\pi$
$180$	0	0
$181$	6.14214	0.456541	0.228271	−	0.973598i	$-0.426693\pi$
$181$	6.14214	0.456541	0.228271	+	0.973598i	$0.426693\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.4853i	0.917936i
$186$	0	0
$187$	− 15.3137i	− 1.11985i
$188$	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
$197$	− 16.3848i	− 1.16737i	−0.811981	−	0.583683i	$-0.801611\pi$
$197$	− 16.3848i	− 1.16737i	0.811981	−	0.583683i	$-0.198389\pi$
$198$	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
$203$	−2.82843	−0.198517
$204$	0	0
$205$	−8.82843	−0.616604
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.0000i	1.10674i
$210$	0	0
$211$	− 23.3137i	− 1.60498i	−0.596664	−	0.802491i	$-0.703508\pi$
$211$	− 23.3137i	− 1.60498i	0.596664	−	0.802491i	$-0.296492\pi$
$212$	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
$221$	15.3137i	1.03011i
$222$	0	0
$223$	17.7990i	1.19191i	0.803018	+	0.595954i	$0.203226\pi$
$223$	17.7990i	1.19191i	−0.803018	+	0.595954i	$0.796774\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.8284	1.78067	0.890333	−	0.455311i	$-0.150472\pi$
$227$	26.8284	1.78067	0.890333	+	0.455311i	$0.150472\pi$
$228$	0	0
$229$	5.17157	0.341747	0.170874	−	0.985293i	$-0.445341\pi$
$229$	5.17157	0.341747	0.170874	+	0.985293i	$0.445341\pi$
$230$	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
$235$	− 42.6274i	− 2.78071i
$236$	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
$245$	55.6985i	3.55845i
$246$	0	0
$247$	− 16.0000i	− 1.01806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.4853	1.54550	0.772749	−	0.634712i	$-0.218881\pi$
$251$	24.4853	1.54550	0.772749	+	0.634712i	$0.218881\pi$
$252$	0	0
$253$	3.31371	0.208331
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.89949i	0.617514i	0.951141	+	0.308757i	$0.0999129\pi$
$257$	9.89949i	0.617514i	−0.951141	+	0.308757i	$0.900087\pi$
$258$	0	0
$259$	17.6569i	1.09714i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.3431	−1.13109	−0.565543	−	0.824719i	$-0.691334\pi$
$263$	−18.3431	−1.13109	−0.565543	+	0.824719i	$0.691334\pi$
$264$	0	0
$265$	17.3137	1.06357
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 32.3848i	− 1.97453i	−0.159070	−	0.987267i	$-0.550850\pi$
$269$	− 32.3848i	− 1.97453i	0.159070	−	0.987267i	$-0.449150\pi$
$270$	0	0
$271$	− 1.51472i	− 0.0920126i	−0.998941	−	0.0460063i	$-0.985351\pi$
$271$	− 1.51472i	− 0.0920126i	0.998941	−	0.0460063i	$-0.0146494\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−18.8284	−1.13540
$276$	0	0
$277$	−21.1716	−1.27208	−0.636038	−	0.771658i	$-0.719428\pi$
$277$	−21.1716	−1.27208	−0.636038	+	0.771658i	$0.719428\pi$
$278$	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
$283$	− 3.31371i	− 0.196980i	−0.995138	−	0.0984898i	$-0.968599\pi$
$283$	− 3.31371i	− 0.196980i	0.995138	−	0.0984898i	$-0.0314012\pi$
$284$	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
$293$	− 28.8701i	− 1.68661i	−0.537438	−	0.843303i	$-0.680608\pi$
$293$	− 28.8701i	− 1.68661i	0.537438	−	0.843303i	$-0.319392\pi$
$294$	0	0
$295$	8.00000i	0.465778i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.31371	−0.191637
$300$	0	0
$301$	46.6274	2.68756
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.1421i	1.49689i
$306$	0	0
$307$	0.686292i	0.0391687i	0.999808	+	0.0195844i	$0.00623429\pi$
$307$	0.686292i	0.0391687i	−0.999808	+	0.0195844i	$0.993766\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.6569	−0.774409	−0.387205	−	0.921994i	$-0.626559\pi$
$311$	−13.6569	−0.774409	−0.387205	+	0.921994i	$0.626559\pi$
$312$	0	0
$313$	9.31371	0.526442	0.263221	−	0.964736i	$-0.415215\pi$
$313$	9.31371	0.526442	0.263221	+	0.964736i	$0.415215\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.2426i	0.799946i	0.916527	+	0.399973i	$0.130981\pi$
$317$	14.2426i	0.799946i	−0.916527	+	0.399973i	$0.869019\pi$
$318$	0	0
$319$	− 1.65685i	− 0.0927660i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	30.6274	1.70416
$324$	0	0
$325$	18.8284	1.04441
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 60.2843i	− 3.32358i
$330$	0	0
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	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
$341$	8.97056i	0.485783i
$342$	0	0
$343$	44.9706i	2.42818i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.4853	−0.884976	−0.442488	−	0.896774i	$-0.645904\pi$
$347$	−16.4853	−0.884976	−0.442488	+	0.896774i	$0.645904\pi$
$348$	0	0
$349$	11.6569	0.623977	0.311989	−	0.950086i	$-0.399005\pi$
$349$	11.6569	0.623977	0.311989	+	0.950086i	$0.399005\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.8701i	− 1.00435i	−0.864765	−	0.502176i	$-0.832533\pi$
$353$	− 18.8701i	− 1.00435i	0.864765	−	0.502176i	$-0.167467\pi$
$354$	0	0
$355$	15.3137i	0.812767i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.4853	1.08117	0.540586	−	0.841289i	$-0.318203\pi$
$359$	20.4853	1.08117	0.540586	+	0.841289i	$0.318203\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 13.6569i	− 0.714832i
$366$	0	0
$367$	− 20.8284i	− 1.08724i	−0.839333	−	0.543618i	$-0.817054\pi$
$367$	− 20.8284i	− 1.08724i	0.839333	−	0.543618i	$-0.182946\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.4853	1.27121
$372$	0	0
$373$	14.9706	0.775146	0.387573	−	0.921839i	$-0.373313\pi$
$373$	14.9706	0.775146	0.387573	+	0.921839i	$0.373313\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.65685i	0.0853323i
$378$	0	0
$379$	20.2843i	1.04193i	0.853577	+	0.520967i	$0.174428\pi$
$379$	20.2843i	1.04193i	−0.853577	+	0.520967i	$0.825572\pi$
$380$	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
$389$	2.44365i	0.123898i	0.998079	+	0.0619490i	$0.0197316\pi$
$389$	2.44365i	0.123898i	−0.998079	+	0.0619490i	$0.980268\pi$
$390$	0	0
$391$	− 6.34315i	− 0.320787i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.1421	−1.11409
$396$	0	0
$397$	−18.9706	−0.952105	−0.476053	−	0.879417i	$-0.657933\pi$
$397$	−18.9706	−0.952105	−0.476053	+	0.879417i	$0.657933\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 13.4142i	− 0.669874i	−0.942241	−	0.334937i	$-0.891285\pi$
$401$	− 13.4142i	− 0.669874i	0.942241	−	0.334937i	$-0.108715\pi$
$402$	0	0
$403$	− 8.97056i	− 0.446856i
$404$	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	$-0.417688\pi$
$409$	10.3431	0.511436	0.255718	+	0.966751i	$0.417688\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.3137i	0.556711i
$414$	0	0
$415$	− 17.6569i	− 0.866741i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.7990	0.576418	0.288209	−	0.957567i	$-0.406940\pi$
$419$	11.7990	0.576418	0.288209	+	0.957567i	$0.406940\pi$
$420$	0	0
$421$	7.51472	0.366245	0.183122	−	0.983090i	$-0.441380\pi$
$421$	7.51472	0.366245	0.183122	+	0.983090i	$0.441380\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	36.0416i	1.74828i
$426$	0	0
$427$	36.9706i	1.78913i
$428$	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
$437$	6.62742i	0.317032i
$438$	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
$443$	−17.4558	−0.829352	−0.414676	−	0.909969i	$-0.636105\pi$
$443$	−17.4558	−0.829352	−0.414676	+	0.909969i	$0.636105\pi$
$444$	0	0
$445$	−41.7990	−1.98146
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15.2721i	− 0.720734i	−0.932811	−	0.360367i	$-0.882651\pi$
$449$	− 15.2721i	− 0.720734i	0.932811	−	0.360367i	$-0.117349\pi$
$450$	0	0
$451$	− 7.31371i	− 0.344389i
$452$	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	$-0.567285\pi$
$457$	−8.97056	−0.419625	−0.209813	+	0.977742i	$0.567285\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.10051i	0.377278i	0.982046	+	0.188639i	$0.0604076\pi$
$461$	8.10051i	0.377278i	−0.982046	+	0.188639i	$0.939592\pi$
$462$	0	0
$463$	− 33.7990i	− 1.57077i	−0.619006	−	0.785386i	$-0.712464\pi$
$463$	− 33.7990i	− 1.57077i	0.619006	−	0.785386i	$-0.287536\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.14214	0.284224	0.142112	−	0.989851i	$-0.454611\pi$
$467$	6.14214	0.284224	0.142112	+	0.989851i	$0.454611\pi$
$468$	0	0
$469$	−57.9411	−2.67547
$470$	0	0
$471$	0	0
$472$	0	0
$473$	27.3137i	1.25589i
$474$	0	0
$475$	− 37.6569i	− 1.72781i
$476$	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
$485$	− 46.6274i	− 2.11724i
$486$	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
$491$	−35.3137	−1.59369	−0.796843	−	0.604187i	$-0.793498\pi$
$491$	−35.3137	−1.59369	−0.796843	+	0.604187i	$0.793498\pi$
$492$	0	0
$493$	−3.17157	−0.142840
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.6569i	0.971443i
$498$	0	0
$499$	6.62742i	0.296684i	0.988936	+	0.148342i	$0.0473936\pi$
$499$	6.62742i	0.296684i	−0.988936	+	0.148342i	$0.952606\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.7990	0.704442	0.352221	−	0.935917i	$-0.385427\pi$
$503$	15.7990	0.704442	0.352221	+	0.935917i	$0.385427\pi$
$504$	0	0
$505$	−36.6274	−1.62990
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 13.2721i	− 0.588275i	−0.955763	−	0.294137i	$-0.904968\pi$
$509$	− 13.2721i	− 0.588275i	0.955763	−	0.294137i	$-0.0950323\pi$
$510$	0	0
$511$	− 19.3137i	− 0.854388i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.8284	−0.477158
$516$	0	0
$517$	35.3137	1.55310
$518$	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
$523$	− 45.6569i	− 1.99643i	−0.0596823	−	0.998217i	$-0.519009\pi$
$523$	− 45.6569i	− 1.99643i	0.0596823	−	0.998217i	$-0.480991\pi$
$524$	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
$533$	7.31371i	0.316792i
$534$	0	0
$535$	− 38.6274i	− 1.67001i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−46.1421	−1.98748
$540$	0	0
$541$	−35.7990	−1.53912	−0.769559	−	0.638575i	$-0.779524\pi$
$541$	−35.7990	−1.53912	−0.769559	+	0.638575i	$0.779524\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 36.9706i	− 1.58364i
$546$	0	0
$547$	9.65685i	0.412897i	0.978457	+	0.206449i	$0.0661906\pi$
$547$	9.65685i	0.412897i	−0.978457	+	0.206449i	$0.933809\pi$
$548$	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
$557$	9.07107i	0.384353i	0.981360	+	0.192177i	$0.0615547\pi$
$557$	9.07107i	0.384353i	−0.981360	+	0.192177i	$0.938445\pi$
$558$	0	0
$559$	− 27.3137i	− 1.15525i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.4558	1.07284	0.536418	−	0.843952i	$-0.319777\pi$
$563$	25.4558	1.07284	0.536418	+	0.843952i	$0.319777\pi$
$564$	0	0
$565$	−41.7990	−1.75850
$566$	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
$571$	24.0000i	1.00437i	0.864761	+	0.502184i	$0.167470\pi$
$571$	24.0000i	1.00437i	−0.864761	+	0.502184i	$0.832530\pi$
$572$	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
$581$	− 24.9706i	− 1.03595i
$582$	0	0
$583$	14.3431i	0.594032i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.6569	1.22407	0.612035	−	0.790831i	$-0.290351\pi$
$587$	29.6569	1.22407	0.612035	+	0.790831i	$0.290351\pi$
$588$	0	0
$589$	−17.9411	−0.739251
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.0711i	− 0.618895i	−0.950917	−	0.309447i	$-0.899856\pi$
$593$	− 15.0711i	− 0.618895i	0.950917	−	0.309447i	$-0.100144\pi$
$594$	0	0
$595$	89.2548i	3.65909i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.4853	1.81762	0.908810	−	0.417211i	$-0.136992\pi$
$599$	44.4853	1.81762	0.908810	+	0.417211i	$0.136992\pi$
$600$	0	0
$601$	21.3137	0.869404	0.434702	−	0.900574i	$-0.356854\pi$
$601$	21.3137	0.869404	0.434702	+	0.900574i	$0.356854\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.2426i	0.416423i
$606$	0	0
$607$	19.1716i	0.778150i	0.921206	+	0.389075i	$0.127205\pi$
$607$	19.1716i	0.778150i	−0.921206	+	0.389075i	$0.872795\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.3137	−1.42864
$612$	0	0
$613$	11.6569	0.470816	0.235408	−	0.971897i	$-0.424357\pi$
$613$	11.6569	0.470816	0.235408	+	0.971897i	$0.424357\pi$
$614$	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
$619$	33.9411i	1.36421i	0.731255	+	0.682105i	$0.238935\pi$
$619$	33.9411i	1.36421i	−0.731255	+	0.682105i	$0.761065\pi$
$620$	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
$629$	19.7990i	0.789437i
$630$	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
$635$	−16.4853	−0.654198
$636$	0	0
$637$	46.1421	1.82822
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.8701i	1.53527i	0.640884	+	0.767637i	$0.278568\pi$
$641$	38.8701i	1.53527i	−0.640884	+	0.767637i	$0.721432\pi$
$642$	0	0
$643$	− 1.65685i	− 0.0653400i	−0.999466	−	0.0326700i	$-0.989599\pi$
$643$	− 1.65685i	− 0.0653400i	0.999466	−	0.0326700i	$-0.0104010\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.7990	−0.935635	−0.467817	−	0.883825i	$-0.654960\pi$
$647$	−23.7990	−0.935635	−0.467817	+	0.883825i	$0.654960\pi$
$648$	0	0
$649$	−6.62742	−0.260149
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 30.7279i	− 1.20248i	−0.799070	−	0.601238i	$-0.794674\pi$
$653$	− 30.7279i	− 1.20248i	0.799070	−	0.601238i	$-0.205326\pi$
$654$	0	0
$655$	19.3137i	0.754649i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.9706	0.972715	0.486358	−	0.873760i	$-0.338325\pi$
$659$	24.9706	0.972715	0.486358	+	0.873760i	$0.338325\pi$
$660$	0	0
$661$	7.65685	0.297817	0.148909	−	0.988851i	$-0.452424\pi$
$661$	7.65685	0.297817	0.148909	+	0.988851i	$0.452424\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 93.2548i	− 3.61627i
$666$	0	0
$667$	− 0.686292i	− 0.0265733i
$668$	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
$677$	− 6.72792i	− 0.258575i	−0.991607	−	0.129288i	$-0.958731\pi$
$677$	− 6.72792i	− 0.258575i	0.991607	−	0.129288i	$-0.0412690\pi$
$678$	0	0
$679$	− 65.9411i	− 2.53059i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.14214	−0.235022	−0.117511	−	0.993072i	$-0.537492\pi$
$683$	−6.14214	−0.235022	−0.117511	+	0.993072i	$0.537492\pi$
$684$	0	0
$685$	36.1421	1.38092
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 14.3431i	− 0.546430i
$690$	0	0
$691$	36.9706i	1.40643i	0.710979	+	0.703213i	$0.248252\pi$
$691$	36.9706i	1.40643i	−0.710979	+	0.703213i	$0.751748\pi$
$692$	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
$701$	− 22.2426i	− 0.840093i	−0.907503	−	0.420046i	$-0.862014\pi$
$701$	− 22.2426i	− 0.840093i	0.907503	−	0.420046i	$-0.137986\pi$
$702$	0	0
$703$	− 20.6863i	− 0.780198i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−51.7990	−1.94810
$708$	0	0
$709$	−48.0833	−1.80580	−0.902902	−	0.429846i	$-0.858568\pi$
$709$	−48.0833	−1.80580	−0.902902	+	0.429846i	$0.858568\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.71573i	0.139155i
$714$	0	0
$715$	27.3137i	1.02147i
$716$	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
$725$	3.89949i	0.144824i
$726$	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
$731$	52.2843	1.93380
$732$	0	0
$733$	−7.51472	−0.277562	−0.138781	−	0.990323i	$-0.544318\pi$
$733$	−7.51472	−0.277562	−0.138781	+	0.990323i	$0.544318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 33.9411i	− 1.25024i
$738$	0	0
$739$	− 8.00000i	− 0.294285i	−0.989115	−	0.147142i	$-0.952992\pi$
$739$	− 8.00000i	− 0.294285i	0.989115	−	0.147142i	$-0.0470076\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.9706	0.622590	0.311295	−	0.950313i	$-0.399237\pi$
$743$	16.9706	0.622590	0.311295	+	0.950313i	$0.399237\pi$
$744$	0	0
$745$	22.9706	0.841576
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 54.6274i	− 1.99604i
$750$	0	0
$751$	35.1716i	1.28343i	0.766944	+	0.641714i	$0.221777\pi$
$751$	35.1716i	1.28343i	−0.766944	+	0.641714i	$0.778223\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.1421	0.805835
$756$	0	0
$757$	16.4853	0.599168	0.299584	−	0.954070i	$-0.403152\pi$
$757$	16.4853	0.599168	0.299584	+	0.954070i	$0.403152\pi$
$758$	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
$763$	− 52.2843i	− 1.89282i
$764$	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
$773$	15.4142i	0.554411i	0.960811	+	0.277205i	$0.0894082\pi$
$773$	15.4142i	0.554411i	−0.960811	+	0.277205i	$0.910592\pi$
$774$	0	0
$775$	− 21.1127i	− 0.758391i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.6274	0.524082
$780$	0	0
$781$	−12.6863	−0.453951
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.17157i	0.0418152i
$786$	0	0
$787$	− 37.6569i	− 1.34232i	−0.741312	−	0.671161i	$-0.765796\pi$
$787$	− 37.6569i	− 1.34232i	0.741312	−	0.671161i	$-0.234204\pi$
$788$	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
$797$	− 1.07107i	− 0.0379392i	−0.999820	−	0.0189696i	$-0.993961\pi$
$797$	− 1.07107i	− 0.0379392i	0.999820	−	0.0189696i	$-0.00603857\pi$
$798$	0	0
$799$	− 67.5980i	− 2.39144i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3137	0.399252
$804$	0	0
$805$	−19.3137	−0.680719
$806$	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
$811$	− 47.5980i	− 1.67139i	−0.549193	−	0.835696i	$-0.685065\pi$
$811$	− 47.5980i	− 1.67139i	0.549193	−	0.835696i	$-0.314935\pi$
$812$	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
$821$	8.58579i	0.299646i	0.988713	+	0.149823i	$0.0478704\pi$
$821$	8.58579i	0.299646i	−0.988713	+	0.149823i	$0.952130\pi$
$822$	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
$827$	−48.9706	−1.70287	−0.851437	−	0.524457i	$-0.824268\pi$
$827$	−48.9706	−1.70287	−0.851437	+	0.524457i	$0.824268\pi$
$828$	0	0
$829$	−43.7990	−1.52120	−0.760601	−	0.649220i	$-0.775096\pi$
$829$	−43.7990	−1.52120	−0.760601	+	0.649220i	$0.775096\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	88.3259i	3.06031i
$834$	0	0
$835$	19.3137i	0.668378i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.4853	0.707230	0.353615	−	0.935391i	$-0.384952\pi$
$839$	20.4853	0.707230	0.353615	+	0.935391i	$0.384952\pi$
$840$	0	0
$841$	28.6569	0.988167
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.0711i	0.587263i
$846$	0	0
$847$	14.4853i	0.497720i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.28427	−0.146863
$852$	0	0
$853$	14.2843	0.489084	0.244542	−	0.969639i	$-0.421362\pi$
$853$	14.2843	0.489084	0.244542	+	0.969639i	$0.421362\pi$
$854$	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
$859$	9.65685i	0.329488i	0.986336	+	0.164744i	$0.0526797\pi$
$859$	9.65685i	0.329488i	−0.986336	+	0.164744i	$0.947320\pi$
$860$	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
$869$	− 18.3431i	− 0.622249i
$870$	0	0
$871$	33.9411i	1.15005i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	27.3137	0.923372
$876$	0	0
$877$	−26.9706	−0.910731	−0.455366	−	0.890305i	$-0.650491\pi$
$877$	−26.9706	−0.910731	−0.455366	+	0.890305i	$0.650491\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 12.7279i	− 0.428815i	−0.976744	−	0.214407i	$-0.931218\pi$
$881$	− 12.7279i	− 0.428815i	0.976744	−	0.214407i	$-0.0687820\pi$
$882$	0	0
$883$	− 16.2843i	− 0.548009i	−0.961728	−	0.274005i	$-0.911652\pi$
$883$	− 16.2843i	− 0.548009i	0.961728	−	0.274005i	$-0.0883484\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.6569	0.458552	0.229276	−	0.973361i	$-0.426364\pi$
$887$	13.6569	0.458552	0.229276	+	0.973361i	$0.426364\pi$
$888$	0	0
$889$	−23.3137	−0.781917
$890$	0	0
$891$	0	0
$892$	0	0
$893$	70.6274i	2.36346i
$894$	0	0
$895$	65.9411i	2.20417i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.85786	0.0619632
$900$	0	0
$901$	27.4558	0.914687
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 20.9706i	− 0.697085i
$906$	0	0
$907$	25.6569i	0.851922i	0.904742	+	0.425961i	$0.140064\pi$
$907$	25.6569i	0.851922i	−0.904742	+	0.425961i	$0.859936\pi$
$908$	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
$917$	27.3137i	0.901978i
$918$	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
$923$	12.6863	0.417574
$924$	0	0
$925$	24.3431	0.800398
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.72792i	− 0.286354i	−0.989697	−	0.143177i	$-0.954268\pi$
$929$	− 8.72792i	− 0.286354i	0.989697	−	0.143177i	$-0.0457318\pi$
$930$	0	0
$931$	− 92.2843i	− 3.02449i
$932$	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	$-0.559476\pi$
$937$	−11.3726	−0.371526	−0.185763	+	0.982595i	$0.559476\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.7279i	1.52329i	0.647996	+	0.761643i	$0.275607\pi$
$941$	46.7279i	1.52329i	−0.647996	+	0.761643i	$0.724393\pi$
$942$	0	0
$943$	− 3.02944i	− 0.0986521i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.5980	−1.54673	−0.773363	−	0.633963i	$-0.781427\pi$
$947$	−47.5980	−1.54673	−0.773363	+	0.633963i	$0.781427\pi$
$948$	0	0
$949$	−11.3137	−0.367259
$950$	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
$955$	35.3137i	1.14272i
$956$	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
$965$	− 18.1421i	− 0.584016i
$966$	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
$971$	−26.8284	−0.860965	−0.430483	−	0.902599i	$-0.641657\pi$
$971$	−26.8284	−0.860965	−0.430483	+	0.902599i	$0.641657\pi$
$972$	0	0
$973$	57.9411	1.85751
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 52.5269i	− 1.68048i	−0.542211	−	0.840242i	$-0.682413\pi$
$977$	− 52.5269i	− 1.68048i	0.542211	−	0.840242i	$-0.317587\pi$
$978$	0	0
$979$	− 34.6274i	− 1.10670i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.6569	−1.45623	−0.728114	−	0.685456i	$-0.759603\pi$
$983$	−45.6569	−1.45623	−0.728114	+	0.685456i	$0.759603\pi$
$984$	0	0
$985$	−55.9411	−1.78243
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.3137i	0.359755i
$990$	0	0
$991$	25.7990i	0.819532i	0.912191	+	0.409766i	$0.134390\pi$
$991$	25.7990i	0.819532i	−0.912191	+	0.409766i	$0.865610\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.4853	0.522619
$996$	0	0
$997$	61.5980	1.95083	0.975414	−	0.220381i	$-0.0707302\pi$
$997$	61.5980	1.95083	0.975414	+	0.220381i	$0.0707302\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.41421i q^{5} -4.82843i q^{7} +O(q^{10})\)	\(q-3.41421i q^{5} -4.82843i q^{7} +2.82843 q^{11} -2.82843 q^{13} -5.41421i q^{17} +5.65685i q^{19} +1.17157 q^{23} -6.65685 q^{25} -0.585786i q^{29} +3.17157i q^{31} -16.4853 q^{35} -3.65685 q^{37} -2.58579i q^{41} +9.65685i q^{43} +12.4853 q^{47} -16.3137 q^{49} +5.07107i q^{53} -9.65685i q^{55} -2.34315 q^{59} -7.65685 q^{61} +9.65685i q^{65} -12.0000i q^{67} -4.48528 q^{71} +4.00000 q^{73} -13.6569i q^{77} -6.48528i q^{79} +5.17157 q^{83} -18.4853 q^{85} -12.2426i q^{89} +13.6569i q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{23} - 4 q^{25} - 32 q^{35} + 8 q^{37} + 16 q^{47} - 20 q^{49} - 32 q^{59} - 8 q^{61} + 16 q^{71} + 16 q^{73} + 32 q^{83} - 40 q^{85} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q + 16 * q^23 - 4 * q^25 - 32 * q^35 + 8 * q^37 + 16 * q^47 - 20 * q^49 - 32 * q^59 - 8 * q^61 + 16 * q^71 + 16 * q^73 + 32 * q^83 - 40 * q^85 + 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more