LMFDB - Embedded newform 768.3.b.b.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("768.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.b (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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.b.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -6.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} -6.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +20.7846 q^{11} -14.0000i q^{13} -10.3923i q^{15} -6.00000 q^{17} -6.92820 q^{19} +12.0000i q^{21} -11.0000 q^{25} +5.19615 q^{27} +30.0000i q^{29} -20.7846i q^{31} +36.0000 q^{33} +41.5692 q^{35} -26.0000i q^{37} -24.2487i q^{39} +54.0000 q^{41} +20.7846 q^{43} -18.0000i q^{45} -41.5692i q^{47} +1.00000 q^{49} -10.3923 q^{51} +18.0000i q^{53} -124.708i q^{55} -12.0000 q^{57} -20.7846 q^{59} -70.0000i q^{61} +20.7846i q^{63} -84.0000 q^{65} +117.779 q^{67} -83.1384i q^{71} -82.0000 q^{73} -19.0526 q^{75} +144.000i q^{77} -76.2102i q^{79} +9.00000 q^{81} +20.7846 q^{83} +36.0000i q^{85} +51.9615i q^{87} -114.000 q^{89} +96.9948 q^{91} -36.0000i q^{93} +41.5692i q^{95} +34.0000 q^{97} +62.3538 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 24 q^{17} - 44 q^{25} + 144 q^{33} + 216 q^{41} + 4 q^{49} - 48 q^{57} - 336 q^{65} - 328 q^{73} + 36 q^{81} - 456 q^{89} + 136 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 - 24 * q^17 - 44 * q^25 + 144 * q^33 + 216 * q^41 + 4 * q^49 - 48 * q^57 - 336 * q^65 - 328 * q^73 + 36 * q^81 - 456 * q^89 + 136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	− 6.00000i	− 1.20000i	−0.800000	−	0.600000i	$-0.795167\pi$
$5$	− 6.00000i	− 1.20000i	0.800000	−	0.600000i	$-0.204833\pi$
$6$	0	0
$7$	6.92820i	0.989743i	0.868966	+	0.494872i	$0.164785\pi$
$7$	6.92820i	0.989743i	−0.868966	+	0.494872i	$0.835215\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	20.7846	1.88951	0.944755	−	0.327777i	$-0.106300\pi$
$11$	20.7846	1.88951	0.944755	+	0.327777i	$0.106300\pi$
$12$	0	0
$13$	− 14.0000i	− 1.07692i	−0.842650	−	0.538462i	$-0.819006\pi$
$13$	− 14.0000i	− 1.07692i	0.842650	−	0.538462i	$-0.180994\pi$
$14$	0	0
$15$	− 10.3923i	− 0.692820i
$16$	0	0
$17$	−6.00000	−0.352941	−0.176471	−	0.984306i	$-0.556468\pi$
$17$	−6.00000	−0.352941	−0.176471	+	0.984306i	$0.556468\pi$
$18$	0	0
$19$	−6.92820	−0.364642	−0.182321	−	0.983239i	$-0.558361\pi$
$19$	−6.92820	−0.364642	−0.182321	+	0.983239i	$0.558361\pi$
$20$	0	0
$21$	12.0000i	0.571429i
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−11.0000	−0.440000
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	30.0000i	1.03448i	0.855840	+	0.517241i	$0.173041\pi$
$29$	30.0000i	1.03448i	−0.855840	+	0.517241i	$0.826959\pi$
$30$	0	0
$31$	− 20.7846i	− 0.670471i	−0.942134	−	0.335236i	$-0.891184\pi$
$31$	− 20.7846i	− 0.670471i	0.942134	−	0.335236i	$-0.108816\pi$
$32$	0	0
$33$	36.0000	1.09091
$34$	0	0
$35$	41.5692	1.18769
$36$	0	0
$37$	− 26.0000i	− 0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$	− 26.0000i	− 0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	0	0
$39$	− 24.2487i	− 0.621762i
$40$	0	0
$41$	54.0000	1.31707	0.658537	−	0.752549i	$-0.271176\pi$
$41$	54.0000	1.31707	0.658537	+	0.752549i	$0.271176\pi$
$42$	0	0
$43$	20.7846	0.483363	0.241682	−	0.970356i	$-0.422301\pi$
$43$	20.7846	0.483363	0.241682	+	0.970356i	$0.422301\pi$
$44$	0	0
$45$	− 18.0000i	− 0.400000i
$46$	0	0
$47$	− 41.5692i	− 0.884451i	−0.896904	−	0.442226i	$-0.854189\pi$
$47$	− 41.5692i	− 0.884451i	0.896904	−	0.442226i	$-0.145811\pi$
$48$	0	0
$49$	1.00000	0.0204082
$50$	0	0
$51$	−10.3923	−0.203771
$52$	0	0
$53$	18.0000i	0.339623i	0.985477	+	0.169811i	$0.0543158\pi$
$53$	18.0000i	0.339623i	−0.985477	+	0.169811i	$0.945684\pi$
$54$	0	0
$55$	− 124.708i	− 2.26741i
$56$	0	0
$57$	−12.0000	−0.210526
$58$	0	0
$59$	−20.7846	−0.352282	−0.176141	−	0.984365i	$-0.556361\pi$
$59$	−20.7846	−0.352282	−0.176141	+	0.984365i	$0.556361\pi$
$60$	0	0
$61$	− 70.0000i	− 1.14754i	−0.819016	−	0.573770i	$-0.805480\pi$
$61$	− 70.0000i	− 1.14754i	0.819016	−	0.573770i	$-0.194520\pi$
$62$	0	0
$63$	20.7846i	0.329914i
$64$	0	0
$65$	−84.0000	−1.29231
$66$	0	0
$67$	117.779	1.75790	0.878951	−	0.476912i	$-0.158244\pi$
$67$	117.779	1.75790	0.878951	+	0.476912i	$0.158244\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 83.1384i	− 1.17096i	−0.810685	−	0.585482i	$-0.800905\pi$
$71$	− 83.1384i	− 1.17096i	0.810685	−	0.585482i	$-0.199095\pi$
$72$	0	0
$73$	−82.0000	−1.12329	−0.561644	−	0.827379i	$-0.689831\pi$
$73$	−82.0000	−1.12329	−0.561644	+	0.827379i	$0.689831\pi$
$74$	0	0
$75$	−19.0526	−0.254034
$76$	0	0
$77$	144.000i	1.87013i
$78$	0	0
$79$	− 76.2102i	− 0.964687i	−0.875982	−	0.482343i	$-0.839786\pi$
$79$	− 76.2102i	− 0.964687i	0.875982	−	0.482343i	$-0.160214\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	20.7846	0.250417	0.125208	−	0.992130i	$-0.460040\pi$
$83$	20.7846	0.250417	0.125208	+	0.992130i	$0.460040\pi$
$84$	0	0
$85$	36.0000i	0.423529i
$86$	0	0
$87$	51.9615i	0.597259i
$88$	0	0
$89$	−114.000	−1.28090	−0.640449	−	0.768000i	$-0.721252\pi$
$89$	−114.000	−1.28090	−0.640449	+	0.768000i	$0.721252\pi$
$90$	0	0
$91$	96.9948	1.06588
$92$	0	0
$93$	− 36.0000i	− 0.387097i
$94$	0	0
$95$	41.5692i	0.437571i
$96$	0	0
$97$	34.0000	0.350515	0.175258	−	0.984523i	$-0.443924\pi$
$97$	34.0000	0.350515	0.175258	+	0.984523i	$0.443924\pi$
$98$	0	0
$99$	62.3538	0.629837
$100$	0	0
$101$	18.0000i	0.178218i	0.996022	+	0.0891089i	$0.0284019\pi$
$101$	18.0000i	0.178218i	−0.996022	+	0.0891089i	$0.971598\pi$
$102$	0	0
$103$	131.636i	1.27802i	0.769199	+	0.639009i	$0.220655\pi$
$103$	131.636i	1.27802i	−0.769199	+	0.639009i	$0.779345\pi$
$104$	0	0
$105$	72.0000	0.685714
$106$	0	0
$107$	145.492	1.35974	0.679870	−	0.733332i	$-0.262036\pi$
$107$	145.492	1.35974	0.679870	+	0.733332i	$0.262036\pi$
$108$	0	0
$109$	34.0000i	0.311927i	0.987763	+	0.155963i	$0.0498482\pi$
$109$	34.0000i	0.311927i	−0.987763	+	0.155963i	$0.950152\pi$
$110$	0	0
$111$	− 45.0333i	− 0.405706i
$112$	0	0
$113$	−78.0000	−0.690265	−0.345133	−	0.938554i	$-0.612166\pi$
$113$	−78.0000	−0.690265	−0.345133	+	0.938554i	$0.612166\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 42.0000i	− 0.358974i
$118$	0	0
$119$	− 41.5692i	− 0.349321i
$120$	0	0
$121$	311.000	2.57025
$122$	0	0
$123$	93.5307	0.760413
$124$	0	0
$125$	− 84.0000i	− 0.672000i
$126$	0	0
$127$	103.923i	0.818292i	0.912469	+	0.409146i	$0.134173\pi$
$127$	103.923i	0.818292i	−0.912469	+	0.409146i	$0.865827\pi$
$128$	0	0
$129$	36.0000	0.279070
$130$	0	0
$131$	−103.923	−0.793306	−0.396653	−	0.917969i	$-0.629828\pi$
$131$	−103.923	−0.793306	−0.396653	+	0.917969i	$0.629828\pi$
$132$	0	0
$133$	− 48.0000i	− 0.360902i
$134$	0	0
$135$	− 31.1769i	− 0.230940i
$136$	0	0
$137$	−186.000	−1.35766	−0.678832	−	0.734294i	$-0.737514\pi$
$137$	−186.000	−1.35766	−0.678832	+	0.734294i	$0.737514\pi$
$138$	0	0
$139$	−48.4974	−0.348902	−0.174451	−	0.984666i	$-0.555815\pi$
$139$	−48.4974	−0.348902	−0.174451	+	0.984666i	$0.555815\pi$
$140$	0	0
$141$	− 72.0000i	− 0.510638i
$142$	0	0
$143$	− 290.985i	− 2.03486i
$144$	0	0
$145$	180.000	1.24138
$146$	0	0
$147$	1.73205	0.0117827
$148$	0	0
$149$	186.000i	1.24832i	0.781296	+	0.624161i	$0.214559\pi$
$149$	186.000i	1.24832i	−0.781296	+	0.624161i	$0.785441\pi$
$150$	0	0
$151$	34.6410i	0.229411i	0.993400	+	0.114705i	$0.0365924\pi$
$151$	34.6410i	0.229411i	−0.993400	+	0.114705i	$0.963408\pi$
$152$	0	0
$153$	−18.0000	−0.117647
$154$	0	0
$155$	−124.708	−0.804566
$156$	0	0
$157$	170.000i	1.08280i	0.840764	+	0.541401i	$0.182106\pi$
$157$	170.000i	1.08280i	−0.840764	+	0.541401i	$0.817894\pi$
$158$	0	0
$159$	31.1769i	0.196081i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−284.056	−1.74268	−0.871338	−	0.490683i	$-0.836747\pi$
$163$	−284.056	−1.74268	−0.871338	+	0.490683i	$0.836747\pi$
$164$	0	0
$165$	− 216.000i	− 1.30909i
$166$	0	0
$167$	207.846i	1.24459i	0.782784	+	0.622294i	$0.213799\pi$
$167$	207.846i	1.24459i	−0.782784	+	0.622294i	$0.786201\pi$
$168$	0	0
$169$	−27.0000	−0.159763
$170$	0	0
$171$	−20.7846	−0.121547
$172$	0	0
$173$	− 42.0000i	− 0.242775i	−0.992605	−	0.121387i	$-0.961266\pi$
$173$	− 42.0000i	− 0.242775i	0.992605	−	0.121387i	$-0.0387343\pi$
$174$	0	0
$175$	− 76.2102i	− 0.435487i
$176$	0	0
$177$	−36.0000	−0.203390
$178$	0	0
$179$	145.492	0.812806	0.406403	−	0.913694i	$-0.366783\pi$
$179$	145.492	0.812806	0.406403	+	0.913694i	$0.366783\pi$
$180$	0	0
$181$	− 82.0000i	− 0.453039i	−0.974007	−	0.226519i	$-0.927265\pi$
$181$	− 82.0000i	− 0.453039i	0.974007	−	0.226519i	$-0.0727346\pi$
$182$	0	0
$183$	− 121.244i	− 0.662533i
$184$	0	0
$185$	−156.000	−0.843243
$186$	0	0
$187$	−124.708	−0.666886
$188$	0	0
$189$	36.0000i	0.190476i
$190$	0	0
$191$	332.554i	1.74112i	0.492063	+	0.870560i	$0.336243\pi$
$191$	332.554i	1.74112i	−0.492063	+	0.870560i	$0.663757\pi$
$192$	0	0
$193$	−94.0000	−0.487047	−0.243523	−	0.969895i	$-0.578303\pi$
$193$	−94.0000	−0.487047	−0.243523	+	0.969895i	$0.578303\pi$
$194$	0	0
$195$	−145.492	−0.746114
$196$	0	0
$197$	258.000i	1.30964i	0.755783	+	0.654822i	$0.227257\pi$
$197$	258.000i	1.30964i	−0.755783	+	0.654822i	$0.772743\pi$
$198$	0	0
$199$	− 117.779i	− 0.591857i	−0.955210	−	0.295928i	$-0.904371\pi$
$199$	− 117.779i	− 0.591857i	0.955210	−	0.295928i	$-0.0956289\pi$
$200$	0	0
$201$	204.000	1.01493
$202$	0	0
$203$	−207.846	−1.02387
$204$	0	0
$205$	− 324.000i	− 1.58049i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−144.000	−0.688995
$210$	0	0
$211$	90.0666	0.426856	0.213428	−	0.976959i	$-0.431537\pi$
$211$	90.0666	0.426856	0.213428	+	0.976959i	$0.431537\pi$
$212$	0	0
$213$	− 144.000i	− 0.676056i
$214$	0	0
$215$	− 124.708i	− 0.580036i
$216$	0	0
$217$	144.000	0.663594
$218$	0	0
$219$	−142.028	−0.648530
$220$	0	0
$221$	84.0000i	0.380090i
$222$	0	0
$223$	353.338i	1.58448i	0.610212	+	0.792238i	$0.291084\pi$
$223$	353.338i	1.58448i	−0.610212	+	0.792238i	$0.708916\pi$
$224$	0	0
$225$	−33.0000	−0.146667
$226$	0	0
$227$	−145.492	−0.640935	−0.320468	−	0.947259i	$-0.603840\pi$
$227$	−145.492	−0.640935	−0.320468	+	0.947259i	$0.603840\pi$
$228$	0	0
$229$	− 226.000i	− 0.986900i	−0.869774	−	0.493450i	$-0.835736\pi$
$229$	− 226.000i	− 0.986900i	0.869774	−	0.493450i	$-0.164264\pi$
$230$	0	0
$231$	249.415i	1.07972i
$232$	0	0
$233$	−114.000	−0.489270	−0.244635	−	0.969615i	$-0.578668\pi$
$233$	−114.000	−0.489270	−0.244635	+	0.969615i	$0.578668\pi$
$234$	0	0
$235$	−249.415	−1.06134
$236$	0	0
$237$	− 132.000i	− 0.556962i
$238$	0	0
$239$	332.554i	1.39144i	0.718314	+	0.695719i	$0.244914\pi$
$239$	332.554i	1.39144i	−0.718314	+	0.695719i	$0.755086\pi$
$240$	0	0
$241$	178.000	0.738589	0.369295	−	0.929312i	$-0.379599\pi$
$241$	178.000	0.738589	0.369295	+	0.929312i	$0.379599\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	− 6.00000i	− 0.0244898i
$246$	0	0
$247$	96.9948i	0.392692i
$248$	0	0
$249$	36.0000	0.144578
$250$	0	0
$251$	103.923	0.414036	0.207018	−	0.978337i	$-0.433624\pi$
$251$	103.923	0.414036	0.207018	+	0.978337i	$0.433624\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	62.3538i	0.244525i
$256$	0	0
$257$	258.000	1.00389	0.501946	−	0.864899i	$-0.332618\pi$
$257$	258.000	1.00389	0.501946	+	0.864899i	$0.332618\pi$
$258$	0	0
$259$	180.133	0.695495
$260$	0	0
$261$	90.0000i	0.344828i
$262$	0	0
$263$	374.123i	1.42252i	0.702929	+	0.711260i	$0.251875\pi$
$263$	374.123i	1.42252i	−0.702929	+	0.711260i	$0.748125\pi$
$264$	0	0
$265$	108.000	0.407547
$266$	0	0
$267$	−197.454	−0.739527
$268$	0	0
$269$	510.000i	1.89591i	0.318403	+	0.947955i	$0.396853\pi$
$269$	510.000i	1.89591i	−0.318403	+	0.947955i	$0.603147\pi$
$270$	0	0
$271$	− 450.333i	− 1.66175i	−0.556462	−	0.830873i	$-0.687842\pi$
$271$	− 450.333i	− 1.66175i	0.556462	−	0.830873i	$-0.312158\pi$
$272$	0	0
$273$	168.000	0.615385
$274$	0	0
$275$	−228.631	−0.831384
$276$	0	0
$277$	14.0000i	0.0505415i	0.999681	+	0.0252708i	$0.00804479\pi$
$277$	14.0000i	0.0505415i	−0.999681	+	0.0252708i	$0.991955\pi$
$278$	0	0
$279$	− 62.3538i	− 0.223490i
$280$	0	0
$281$	−354.000	−1.25979	−0.629893	−	0.776682i	$-0.716901\pi$
$281$	−354.000	−1.25979	−0.629893	+	0.776682i	$0.716901\pi$
$282$	0	0
$283$	145.492	0.514107	0.257053	−	0.966397i	$-0.417248\pi$
$283$	145.492	0.514107	0.257053	+	0.966397i	$0.417248\pi$
$284$	0	0
$285$	72.0000i	0.252632i
$286$	0	0
$287$	374.123i	1.30356i
$288$	0	0
$289$	−253.000	−0.875433
$290$	0	0
$291$	58.8897	0.202370
$292$	0	0
$293$	498.000i	1.69966i	0.527058	+	0.849829i	$0.323295\pi$
$293$	498.000i	1.69966i	−0.527058	+	0.849829i	$0.676705\pi$
$294$	0	0
$295$	124.708i	0.422738i
$296$	0	0
$297$	108.000	0.363636
$298$	0	0
$299$	0	0
$300$	0	0
$301$	144.000i	0.478405i
$302$	0	0
$303$	31.1769i	0.102894i
$304$	0	0
$305$	−420.000	−1.37705
$306$	0	0
$307$	−187.061	−0.609321	−0.304660	−	0.952461i	$-0.598543\pi$
$307$	−187.061	−0.609321	−0.304660	+	0.952461i	$0.598543\pi$
$308$	0	0
$309$	228.000i	0.737864i
$310$	0	0
$311$	41.5692i	0.133663i	0.997764	+	0.0668315i	$0.0212890\pi$
$311$	41.5692i	0.133663i	−0.997764	+	0.0668315i	$0.978711\pi$
$312$	0	0
$313$	−290.000	−0.926518	−0.463259	−	0.886223i	$-0.653320\pi$
$313$	−290.000	−0.926518	−0.463259	+	0.886223i	$0.653320\pi$
$314$	0	0
$315$	124.708	0.395897
$316$	0	0
$317$	− 210.000i	− 0.662461i	−0.943550	−	0.331230i	$-0.892536\pi$
$317$	− 210.000i	− 0.662461i	0.943550	−	0.331230i	$-0.107464\pi$
$318$	0	0
$319$	623.538i	1.95467i
$320$	0	0
$321$	252.000	0.785047
$322$	0	0
$323$	41.5692	0.128697
$324$	0	0
$325$	154.000i	0.473846i
$326$	0	0
$327$	58.8897i	0.180091i
$328$	0	0
$329$	288.000	0.875380
$330$	0	0
$331$	200.918	0.607003	0.303501	−	0.952831i	$-0.401844\pi$
$331$	200.918	0.607003	0.303501	+	0.952831i	$0.401844\pi$
$332$	0	0
$333$	− 78.0000i	− 0.234234i
$334$	0	0
$335$	− 706.677i	− 2.10948i
$336$	0	0
$337$	−302.000	−0.896142	−0.448071	−	0.893998i	$-0.647889\pi$
$337$	−302.000	−0.896142	−0.448071	+	0.893998i	$0.647889\pi$
$338$	0	0
$339$	−135.100	−0.398525
$340$	0	0
$341$	− 432.000i	− 1.26686i
$342$	0	0
$343$	346.410i	1.00994i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−62.3538	−0.179694	−0.0898470	−	0.995956i	$-0.528638\pi$
$347$	−62.3538	−0.179694	−0.0898470	+	0.995956i	$0.528638\pi$
$348$	0	0
$349$	− 358.000i	− 1.02579i	−0.858452	−	0.512894i	$-0.828573\pi$
$349$	− 358.000i	− 1.02579i	0.858452	−	0.512894i	$-0.171427\pi$
$350$	0	0
$351$	− 72.7461i	− 0.207254i
$352$	0	0
$353$	−558.000	−1.58074	−0.790368	−	0.612632i	$-0.790111\pi$
$353$	−558.000	−1.58074	−0.790368	+	0.612632i	$0.790111\pi$
$354$	0	0
$355$	−498.831	−1.40516
$356$	0	0
$357$	− 72.0000i	− 0.201681i
$358$	0	0
$359$	83.1384i	0.231583i	0.993274	+	0.115792i	$0.0369405\pi$
$359$	83.1384i	0.231583i	−0.993274	+	0.115792i	$0.963059\pi$
$360$	0	0
$361$	−313.000	−0.867036
$362$	0	0
$363$	538.668	1.48393
$364$	0	0
$365$	492.000i	1.34795i
$366$	0	0
$367$	− 214.774i	− 0.585216i	−0.956232	−	0.292608i	$-0.905477\pi$
$367$	− 214.774i	− 0.585216i	0.956232	−	0.292608i	$-0.0945231\pi$
$368$	0	0
$369$	162.000	0.439024
$370$	0	0
$371$	−124.708	−0.336139
$372$	0	0
$373$	− 554.000i	− 1.48525i	−0.669705	−	0.742627i	$-0.733579\pi$
$373$	− 554.000i	− 1.48525i	0.669705	−	0.742627i	$-0.266421\pi$
$374$	0	0
$375$	− 145.492i	− 0.387979i
$376$	0	0
$377$	420.000	1.11406
$378$	0	0
$379$	−533.472	−1.40758	−0.703788	−	0.710410i	$-0.748510\pi$
$379$	−533.472	−1.40758	−0.703788	+	0.710410i	$0.748510\pi$
$380$	0	0
$381$	180.000i	0.472441i
$382$	0	0
$383$	− 498.831i	− 1.30243i	−0.758893	−	0.651215i	$-0.774260\pi$
$383$	− 498.831i	− 1.30243i	0.758893	−	0.651215i	$-0.225740\pi$
$384$	0	0
$385$	864.000	2.24416
$386$	0	0
$387$	62.3538	0.161121
$388$	0	0
$389$	− 198.000i	− 0.508997i	−0.967073	−	0.254499i	$-0.918090\pi$
$389$	− 198.000i	− 0.508997i	0.967073	−	0.254499i	$-0.0819105\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−180.000	−0.458015
$394$	0	0
$395$	−457.261	−1.15762
$396$	0	0
$397$	− 646.000i	− 1.62720i	−0.581422	−	0.813602i	$-0.697504\pi$
$397$	− 646.000i	− 1.62720i	0.581422	−	0.813602i	$-0.302496\pi$
$398$	0	0
$399$	− 83.1384i	− 0.208367i
$400$	0	0
$401$	330.000	0.822943	0.411471	−	0.911423i	$-0.365015\pi$
$401$	330.000	0.822943	0.411471	+	0.911423i	$0.365015\pi$
$402$	0	0
$403$	−290.985	−0.722046
$404$	0	0
$405$	− 54.0000i	− 0.133333i
$406$	0	0
$407$	− 540.400i	− 1.32776i
$408$	0	0
$409$	−130.000	−0.317848	−0.158924	−	0.987291i	$-0.550803\pi$
$409$	−130.000	−0.317848	−0.158924	+	0.987291i	$0.550803\pi$
$410$	0	0
$411$	−322.161	−0.783848
$412$	0	0
$413$	− 144.000i	− 0.348668i
$414$	0	0
$415$	− 124.708i	− 0.300500i
$416$	0	0
$417$	−84.0000	−0.201439
$418$	0	0
$419$	353.338	0.843290	0.421645	−	0.906761i	$-0.361453\pi$
$419$	353.338	0.843290	0.421645	+	0.906761i	$0.361453\pi$
$420$	0	0
$421$	398.000i	0.945368i	0.881232	+	0.472684i	$0.156715\pi$
$421$	398.000i	0.945368i	−0.881232	+	0.472684i	$0.843285\pi$
$422$	0	0
$423$	− 124.708i	− 0.294817i
$424$	0	0
$425$	66.0000	0.155294
$426$	0	0
$427$	484.974	1.13577
$428$	0	0
$429$	− 504.000i	− 1.17483i
$430$	0	0
$431$	− 124.708i	− 0.289345i	−0.989480	−	0.144672i	$-0.953787\pi$
$431$	− 124.708i	− 0.289345i	0.989480	−	0.144672i	$-0.0462128\pi$
$432$	0	0
$433$	−142.000	−0.327945	−0.163972	−	0.986465i	$-0.552431\pi$
$433$	−142.000	−0.327945	−0.163972	+	0.986465i	$0.552431\pi$
$434$	0	0
$435$	311.769	0.716711
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 561.184i	− 1.27832i	−0.769072	−	0.639162i	$-0.779281\pi$
$439$	− 561.184i	− 1.27832i	0.769072	−	0.639162i	$-0.220719\pi$
$440$	0	0
$441$	3.00000	0.00680272
$442$	0	0
$443$	436.477	0.985275	0.492637	−	0.870235i	$-0.336033\pi$
$443$	436.477	0.985275	0.492637	+	0.870235i	$0.336033\pi$
$444$	0	0
$445$	684.000i	1.53708i
$446$	0	0
$447$	322.161i	0.720719i
$448$	0	0
$449$	−198.000	−0.440980	−0.220490	−	0.975389i	$-0.570766\pi$
$449$	−198.000	−0.440980	−0.220490	+	0.975389i	$0.570766\pi$
$450$	0	0
$451$	1122.37	2.48862
$452$	0	0
$453$	60.0000i	0.132450i
$454$	0	0
$455$	− 581.969i	− 1.27905i
$456$	0	0
$457$	446.000	0.975930	0.487965	−	0.872863i	$-0.337739\pi$
$457$	446.000	0.975930	0.487965	+	0.872863i	$0.337739\pi$
$458$	0	0
$459$	−31.1769	−0.0679236
$460$	0	0
$461$	342.000i	0.741866i	0.928660	+	0.370933i	$0.120962\pi$
$461$	342.000i	0.741866i	−0.928660	+	0.370933i	$0.879038\pi$
$462$	0	0
$463$	159.349i	0.344166i	0.985082	+	0.172083i	$0.0550497\pi$
$463$	159.349i	0.344166i	−0.985082	+	0.172083i	$0.944950\pi$
$464$	0	0
$465$	−216.000	−0.464516
$466$	0	0
$467$	−394.908	−0.845627	−0.422813	−	0.906217i	$-0.638957\pi$
$467$	−394.908	−0.845627	−0.422813	+	0.906217i	$0.638957\pi$
$468$	0	0
$469$	816.000i	1.73987i
$470$	0	0
$471$	294.449i	0.625156i
$472$	0	0
$473$	432.000	0.913319
$474$	0	0
$475$	76.2102	0.160443
$476$	0	0
$477$	54.0000i	0.113208i
$478$	0	0
$479$	789.815i	1.64888i	0.565947	+	0.824442i	$0.308511\pi$
$479$	789.815i	1.64888i	−0.565947	+	0.824442i	$0.691489\pi$
$480$	0	0
$481$	−364.000	−0.756757
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 204.000i	− 0.420619i
$486$	0	0
$487$	6.92820i	0.0142263i	0.999975	+	0.00711315i	$0.00226420\pi$
$487$	6.92820i	0.0142263i	−0.999975	+	0.00711315i	$0.997736\pi$
$488$	0	0
$489$	−492.000	−1.00613
$490$	0	0
$491$	644.323	1.31227	0.656133	−	0.754645i	$-0.272191\pi$
$491$	644.323	1.31227	0.656133	+	0.754645i	$0.272191\pi$
$492$	0	0
$493$	− 180.000i	− 0.365112i
$494$	0	0
$495$	− 374.123i	− 0.755804i
$496$	0	0
$497$	576.000	1.15895
$498$	0	0
$499$	810.600	1.62445	0.812224	−	0.583345i	$-0.198257\pi$
$499$	810.600	1.62445	0.812224	+	0.583345i	$0.198257\pi$
$500$	0	0
$501$	360.000i	0.718563i
$502$	0	0
$503$	332.554i	0.661141i	0.943781	+	0.330570i	$0.107241\pi$
$503$	332.554i	0.661141i	−0.943781	+	0.330570i	$0.892759\pi$
$504$	0	0
$505$	108.000	0.213861
$506$	0	0
$507$	−46.7654	−0.0922394
$508$	0	0
$509$	− 306.000i	− 0.601179i	−0.953754	−	0.300589i	$-0.902817\pi$
$509$	− 306.000i	− 0.601179i	0.953754	−	0.300589i	$-0.0971834\pi$
$510$	0	0
$511$	− 568.113i	− 1.11177i
$512$	0	0
$513$	−36.0000	−0.0701754
$514$	0	0
$515$	789.815	1.53362
$516$	0	0
$517$	− 864.000i	− 1.67118i
$518$	0	0
$519$	− 72.7461i	− 0.140166i
$520$	0	0
$521$	−522.000	−1.00192	−0.500960	−	0.865471i	$-0.667020\pi$
$521$	−522.000	−1.00192	−0.500960	+	0.865471i	$0.667020\pi$
$522$	0	0
$523$	48.4974	0.0927293	0.0463646	−	0.998925i	$-0.485236\pi$
$523$	48.4974	0.0927293	0.0463646	+	0.998925i	$0.485236\pi$
$524$	0	0
$525$	− 132.000i	− 0.251429i
$526$	0	0
$527$	124.708i	0.236637i
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	−62.3538	−0.117427
$532$	0	0
$533$	− 756.000i	− 1.41839i
$534$	0	0
$535$	− 872.954i	− 1.63169i
$536$	0	0
$537$	252.000	0.469274
$538$	0	0
$539$	20.7846	0.0385614
$540$	0	0
$541$	802.000i	1.48244i	0.671262	+	0.741220i	$0.265752\pi$
$541$	802.000i	1.48244i	−0.671262	+	0.741220i	$0.734248\pi$
$542$	0	0
$543$	− 142.028i	− 0.261562i
$544$	0	0
$545$	204.000	0.374312
$546$	0	0
$547$	−34.6410	−0.0633291	−0.0316645	−	0.999499i	$-0.510081\pi$
$547$	−34.6410	−0.0633291	−0.0316645	+	0.999499i	$0.510081\pi$
$548$	0	0
$549$	− 210.000i	− 0.382514i
$550$	0	0
$551$	− 207.846i	− 0.377216i
$552$	0	0
$553$	528.000	0.954792
$554$	0	0
$555$	−270.200	−0.486847
$556$	0	0
$557$	− 474.000i	− 0.850987i	−0.904961	−	0.425494i	$-0.860100\pi$
$557$	− 474.000i	− 0.850987i	0.904961	−	0.425494i	$-0.139900\pi$
$558$	0	0
$559$	− 290.985i	− 0.520545i
$560$	0	0
$561$	−216.000	−0.385027
$562$	0	0
$563$	685.892	1.21828	0.609140	−	0.793062i	$-0.291515\pi$
$563$	685.892	1.21828	0.609140	+	0.793062i	$0.291515\pi$
$564$	0	0
$565$	468.000i	0.828319i
$566$	0	0
$567$	62.3538i	0.109971i
$568$	0	0
$569$	150.000	0.263620	0.131810	−	0.991275i	$-0.457921\pi$
$569$	150.000	0.263620	0.131810	+	0.991275i	$0.457921\pi$
$570$	0	0
$571$	672.036	1.17695	0.588473	−	0.808517i	$-0.299729\pi$
$571$	672.036	1.17695	0.588473	+	0.808517i	$0.299729\pi$
$572$	0	0
$573$	576.000i	1.00524i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−46.0000	−0.0797227	−0.0398614	−	0.999205i	$-0.512692\pi$
$577$	−46.0000	−0.0797227	−0.0398614	+	0.999205i	$0.512692\pi$
$578$	0	0
$579$	−162.813	−0.281197
$580$	0	0
$581$	144.000i	0.247849i
$582$	0	0
$583$	374.123i	0.641720i
$584$	0	0
$585$	−252.000	−0.430769
$586$	0	0
$587$	−353.338	−0.601939	−0.300970	−	0.953634i	$-0.597310\pi$
$587$	−353.338	−0.601939	−0.300970	+	0.953634i	$0.597310\pi$
$588$	0	0
$589$	144.000i	0.244482i
$590$	0	0
$591$	446.869i	0.756124i
$592$	0	0
$593$	114.000	0.192243	0.0961214	−	0.995370i	$-0.469356\pi$
$593$	114.000	0.192243	0.0961214	+	0.995370i	$0.469356\pi$
$594$	0	0
$595$	−249.415	−0.419185
$596$	0	0
$597$	− 204.000i	− 0.341709i
$598$	0	0
$599$	249.415i	0.416386i	0.978088	+	0.208193i	$0.0667582\pi$
$599$	249.415i	0.416386i	−0.978088	+	0.208193i	$0.933242\pi$
$600$	0	0
$601$	−626.000	−1.04160	−0.520799	−	0.853680i	$-0.674366\pi$
$601$	−626.000	−1.04160	−0.520799	+	0.853680i	$0.674366\pi$
$602$	0	0
$603$	353.338	0.585967
$604$	0	0
$605$	− 1866.00i	− 3.08430i
$606$	0	0
$607$	− 672.036i	− 1.10714i	−0.832802	−	0.553571i	$-0.813265\pi$
$607$	− 672.036i	− 1.10714i	0.832802	−	0.553571i	$-0.186735\pi$
$608$	0	0
$609$	−360.000	−0.591133
$610$	0	0
$611$	−581.969	−0.952486
$612$	0	0
$613$	694.000i	1.13214i	0.824358	+	0.566069i	$0.191536\pi$
$613$	694.000i	1.13214i	−0.824358	+	0.566069i	$0.808464\pi$
$614$	0	0
$615$	− 561.184i	− 0.912495i
$616$	0	0
$617$	30.0000	0.0486224	0.0243112	−	0.999704i	$-0.492261\pi$
$617$	30.0000	0.0486224	0.0243112	+	0.999704i	$0.492261\pi$
$618$	0	0
$619$	339.482	0.548436	0.274218	−	0.961668i	$-0.411581\pi$
$619$	339.482	0.548436	0.274218	+	0.961668i	$0.411581\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 789.815i	− 1.26776i
$624$	0	0
$625$	−779.000	−1.24640
$626$	0	0
$627$	−249.415	−0.397792
$628$	0	0
$629$	156.000i	0.248013i
$630$	0	0
$631$	− 464.190i	− 0.735641i	−0.929897	−	0.367821i	$-0.880104\pi$
$631$	− 464.190i	− 0.735641i	0.929897	−	0.367821i	$-0.119896\pi$
$632$	0	0
$633$	156.000	0.246445
$634$	0	0
$635$	623.538	0.981950
$636$	0	0
$637$	− 14.0000i	− 0.0219780i
$638$	0	0
$639$	− 249.415i	− 0.390321i
$640$	0	0
$641$	−390.000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$641$	−390.000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$642$	0	0
$643$	−810.600	−1.26065	−0.630326	−	0.776330i	$-0.717079\pi$
$643$	−810.600	−1.26065	−0.630326	+	0.776330i	$0.717079\pi$
$644$	0	0
$645$	− 216.000i	− 0.334884i
$646$	0	0
$647$	581.969i	0.899489i	0.893157	+	0.449744i	$0.148485\pi$
$647$	581.969i	0.899489i	−0.893157	+	0.449744i	$0.851515\pi$
$648$	0	0
$649$	−432.000	−0.665639
$650$	0	0
$651$	249.415	0.383126
$652$	0	0
$653$	774.000i	1.18530i	0.805461	+	0.592649i	$0.201918\pi$
$653$	774.000i	1.18530i	−0.805461	+	0.592649i	$0.798082\pi$
$654$	0	0
$655$	623.538i	0.951967i
$656$	0	0
$657$	−246.000	−0.374429
$658$	0	0
$659$	228.631	0.346936	0.173468	−	0.984840i	$-0.444503\pi$
$659$	228.631	0.346936	0.173468	+	0.984840i	$0.444503\pi$
$660$	0	0
$661$	454.000i	0.686838i	0.939182	+	0.343419i	$0.111585\pi$
$661$	454.000i	0.686838i	−0.939182	+	0.343419i	$0.888415\pi$
$662$	0	0
$663$	145.492i	0.219445i
$664$	0	0
$665$	−288.000	−0.433083
$666$	0	0
$667$	0	0
$668$	0	0
$669$	612.000i	0.914798i
$670$	0	0
$671$	− 1454.92i	− 2.16829i
$672$	0	0
$673$	434.000	0.644874	0.322437	−	0.946591i	$-0.395498\pi$
$673$	434.000	0.644874	0.322437	+	0.946591i	$0.395498\pi$
$674$	0	0
$675$	−57.1577	−0.0846780
$676$	0	0
$677$	234.000i	0.345643i	0.984953	+	0.172821i	$0.0552883\pi$
$677$	234.000i	0.345643i	−0.984953	+	0.172821i	$0.944712\pi$
$678$	0	0
$679$	235.559i	0.346920i
$680$	0	0
$681$	−252.000	−0.370044
$682$	0	0
$683$	270.200	0.395608	0.197804	−	0.980242i	$-0.436619\pi$
$683$	270.200	0.395608	0.197804	+	0.980242i	$0.436619\pi$
$684$	0	0
$685$	1116.00i	1.62920i
$686$	0	0
$687$	− 391.443i	− 0.569787i
$688$	0	0
$689$	252.000	0.365747
$690$	0	0
$691$	20.7846	0.0300790	0.0150395	−	0.999887i	$-0.495213\pi$
$691$	20.7846	0.0300790	0.0150395	+	0.999887i	$0.495213\pi$
$692$	0	0
$693$	432.000i	0.623377i
$694$	0	0
$695$	290.985i	0.418683i
$696$	0	0
$697$	−324.000	−0.464849
$698$	0	0
$699$	−197.454	−0.282480
$700$	0	0
$701$	− 1074.00i	− 1.53210i	−0.642783	−	0.766049i	$-0.722220\pi$
$701$	− 1074.00i	− 1.53210i	0.642783	−	0.766049i	$-0.277780\pi$
$702$	0	0
$703$	180.133i	0.256235i
$704$	0	0
$705$	−432.000	−0.612766
$706$	0	0
$707$	−124.708	−0.176390
$708$	0	0
$709$	− 898.000i	− 1.26657i	−0.773918	−	0.633286i	$-0.781706\pi$
$709$	− 898.000i	− 1.26657i	0.773918	−	0.633286i	$-0.218294\pi$
$710$	0	0
$711$	− 228.631i	− 0.321562i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1745.91	−2.44183
$716$	0	0
$717$	576.000i	0.803347i
$718$	0	0
$719$	956.092i	1.32975i	0.746954	+	0.664876i	$0.231516\pi$
$719$	956.092i	1.32975i	−0.746954	+	0.664876i	$0.768484\pi$
$720$	0	0
$721$	−912.000	−1.26491
$722$	0	0
$723$	308.305	0.426425
$724$	0	0
$725$	− 330.000i	− 0.455172i
$726$	0	0
$727$	810.600i	1.11499i	0.830179	+	0.557496i	$0.188238\pi$
$727$	810.600i	1.11499i	−0.830179	+	0.557496i	$0.811762\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−124.708	−0.170599
$732$	0	0
$733$	370.000i	0.504775i	0.967626	+	0.252387i	$0.0812157\pi$
$733$	370.000i	0.504775i	−0.967626	+	0.252387i	$0.918784\pi$
$734$	0	0
$735$	− 10.3923i	− 0.0141392i
$736$	0	0
$737$	2448.00	3.32157
$738$	0	0
$739$	−852.169	−1.15314	−0.576569	−	0.817048i	$-0.695609\pi$
$739$	−852.169	−1.15314	−0.576569	+	0.817048i	$0.695609\pi$
$740$	0	0
$741$	168.000i	0.226721i
$742$	0	0
$743$	− 1371.78i	− 1.84628i	−0.384467	−	0.923139i	$-0.625615\pi$
$743$	− 1371.78i	− 1.84628i	0.384467	−	0.923139i	$-0.374385\pi$
$744$	0	0
$745$	1116.00	1.49799
$746$	0	0
$747$	62.3538	0.0834723
$748$	0	0
$749$	1008.00i	1.34579i
$750$	0	0
$751$	− 76.2102i	− 0.101478i	−0.998712	−	0.0507392i	$-0.983842\pi$
$751$	− 76.2102i	− 0.101478i	0.998712	−	0.0507392i	$-0.0161577\pi$
$752$	0	0
$753$	180.000	0.239044
$754$	0	0
$755$	207.846	0.275293
$756$	0	0
$757$	− 514.000i	− 0.678996i	−0.940607	−	0.339498i	$-0.889743\pi$
$757$	− 514.000i	− 0.678996i	0.940607	−	0.339498i	$-0.110257\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	966.000	1.26938	0.634691	−	0.772766i	$-0.281127\pi$
$761$	966.000	1.26938	0.634691	+	0.772766i	$0.281127\pi$
$762$	0	0
$763$	−235.559	−0.308727
$764$	0	0
$765$	108.000i	0.141176i
$766$	0	0
$767$	290.985i	0.379380i
$768$	0	0
$769$	−958.000	−1.24577	−0.622887	−	0.782312i	$-0.714040\pi$
$769$	−958.000	−1.24577	−0.622887	+	0.782312i	$0.714040\pi$
$770$	0	0
$771$	446.869	0.579597
$772$	0	0
$773$	546.000i	0.706339i	0.935559	+	0.353169i	$0.114896\pi$
$773$	546.000i	0.706339i	−0.935559	+	0.353169i	$0.885104\pi$
$774$	0	0
$775$	228.631i	0.295007i
$776$	0	0
$777$	312.000	0.401544
$778$	0	0
$779$	−374.123	−0.480261
$780$	0	0
$781$	− 1728.00i	− 2.21255i
$782$	0	0
$783$	155.885i	0.199086i
$784$	0	0
$785$	1020.00	1.29936
$786$	0	0
$787$	242.487	0.308116	0.154058	−	0.988062i	$-0.450766\pi$
$787$	242.487	0.308116	0.154058	+	0.988062i	$0.450766\pi$
$788$	0	0
$789$	648.000i	0.821293i
$790$	0	0
$791$	− 540.400i	− 0.683186i
$792$	0	0
$793$	−980.000	−1.23581
$794$	0	0
$795$	187.061	0.235297
$796$	0	0
$797$	− 1338.00i	− 1.67880i	−0.543518	−	0.839398i	$-0.682908\pi$
$797$	− 1338.00i	− 1.67880i	0.543518	−	0.839398i	$-0.317092\pi$
$798$	0	0
$799$	249.415i	0.312159i
$800$	0	0
$801$	−342.000	−0.426966
$802$	0	0
$803$	−1704.34	−2.12246
$804$	0	0
$805$	0	0
$806$	0	0
$807$	883.346i	1.09460i
$808$	0	0
$809$	966.000	1.19407	0.597033	−	0.802216i	$-0.296346\pi$
$809$	966.000	1.19407	0.597033	+	0.802216i	$0.296346\pi$
$810$	0	0
$811$	1517.28	1.87087	0.935436	−	0.353497i	$-0.115008\pi$
$811$	1517.28	1.87087	0.935436	+	0.353497i	$0.115008\pi$
$812$	0	0
$813$	− 780.000i	− 0.959410i
$814$	0	0
$815$	1704.34i	2.09121i
$816$	0	0
$817$	−144.000	−0.176255
$818$	0	0
$819$	290.985	0.355292
$820$	0	0
$821$	− 222.000i	− 0.270402i	−0.990818	−	0.135201i	$-0.956832\pi$
$821$	− 222.000i	− 0.270402i	0.990818	−	0.135201i	$-0.0431680\pi$
$822$	0	0
$823$	1281.72i	1.55737i	0.627413	+	0.778686i	$0.284114\pi$
$823$	1281.72i	1.55737i	−0.627413	+	0.778686i	$0.715886\pi$
$824$	0	0
$825$	−396.000	−0.480000
$826$	0	0
$827$	−1434.14	−1.73415	−0.867073	−	0.498182i	$-0.834001\pi$
$827$	−1434.14	−1.73415	−0.867073	+	0.498182i	$0.834001\pi$
$828$	0	0
$829$	226.000i	0.272618i	0.990666	+	0.136309i	$0.0435239\pi$
$829$	226.000i	0.272618i	−0.990666	+	0.136309i	$0.956476\pi$
$830$	0	0
$831$	24.2487i	0.0291802i
$832$	0	0
$833$	−6.00000	−0.00720288
$834$	0	0
$835$	1247.08	1.49350
$836$	0	0
$837$	− 108.000i	− 0.129032i
$838$	0	0
$839$	498.831i	0.594554i	0.954791	+	0.297277i	$0.0960784\pi$
$839$	498.831i	0.594554i	−0.954791	+	0.297277i	$0.903922\pi$
$840$	0	0
$841$	−59.0000	−0.0701546
$842$	0	0
$843$	−613.146	−0.727338
$844$	0	0
$845$	162.000i	0.191716i
$846$	0	0
$847$	2154.67i	2.54389i
$848$	0	0
$849$	252.000	0.296820
$850$	0	0
$851$	0	0
$852$	0	0
$853$	70.0000i	0.0820633i	0.999158	+	0.0410317i	$0.0130644\pi$
$853$	70.0000i	0.0820633i	−0.999158	+	0.0410317i	$0.986936\pi$
$854$	0	0
$855$	124.708i	0.145857i
$856$	0	0
$857$	−42.0000	−0.0490082	−0.0245041	−	0.999700i	$-0.507801\pi$
$857$	−42.0000	−0.0490082	−0.0245041	+	0.999700i	$0.507801\pi$
$858$	0	0
$859$	−921.451	−1.07270	−0.536351	−	0.843995i	$-0.680198\pi$
$859$	−921.451	−1.07270	−0.536351	+	0.843995i	$0.680198\pi$
$860$	0	0
$861$	648.000i	0.752613i
$862$	0	0
$863$	166.277i	0.192673i	0.995349	+	0.0963365i	$0.0307125\pi$
$863$	166.277i	0.192673i	−0.995349	+	0.0963365i	$0.969287\pi$
$864$	0	0
$865$	−252.000	−0.291329
$866$	0	0
$867$	−438.209	−0.505431
$868$	0	0
$869$	− 1584.00i	− 1.82278i
$870$	0	0
$871$	− 1648.91i	− 1.89313i
$872$	0	0
$873$	102.000	0.116838
$874$	0	0
$875$	581.969	0.665108
$876$	0	0
$877$	− 166.000i	− 0.189282i	−0.995511	−	0.0946408i	$-0.969830\pi$
$877$	− 166.000i	− 0.189282i	0.995511	−	0.0946408i	$-0.0301703\pi$
$878$	0	0
$879$	862.561i	0.981298i
$880$	0	0
$881$	−702.000	−0.796822	−0.398411	−	0.917207i	$-0.630438\pi$
$881$	−702.000	−0.796822	−0.398411	+	0.917207i	$0.630438\pi$
$882$	0	0
$883$	630.466	0.714005	0.357003	−	0.934103i	$-0.383799\pi$
$883$	630.466	0.714005	0.357003	+	0.934103i	$0.383799\pi$
$884$	0	0
$885$	216.000i	0.244068i
$886$	0	0
$887$	1288.65i	1.45281i	0.687265	+	0.726407i	$0.258811\pi$
$887$	1288.65i	1.45281i	−0.687265	+	0.726407i	$0.741189\pi$
$888$	0	0
$889$	−720.000	−0.809899
$890$	0	0
$891$	187.061	0.209946
$892$	0	0
$893$	288.000i	0.322508i
$894$	0	0
$895$	− 872.954i	− 0.975367i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	623.538	0.693591
$900$	0	0
$901$	− 108.000i	− 0.119867i
$902$	0	0
$903$	249.415i	0.276207i
$904$	0	0
$905$	−492.000	−0.543646
$906$	0	0
$907$	−1447.99	−1.59647	−0.798233	−	0.602349i	$-0.794232\pi$
$907$	−1447.99	−1.59647	−0.798233	+	0.602349i	$0.794232\pi$
$908$	0	0
$909$	54.0000i	0.0594059i
$910$	0	0
$911$	332.554i	0.365043i	0.983202	+	0.182521i	$0.0584258\pi$
$911$	332.554i	0.365043i	−0.983202	+	0.182521i	$0.941574\pi$
$912$	0	0
$913$	432.000	0.473165
$914$	0	0
$915$	−727.461	−0.795040
$916$	0	0
$917$	− 720.000i	− 0.785169i
$918$	0	0
$919$	561.184i	0.610647i	0.952249	+	0.305323i	$0.0987646\pi$
$919$	561.184i	0.610647i	−0.952249	+	0.305323i	$0.901235\pi$
$920$	0	0
$921$	−324.000	−0.351792
$922$	0	0
$923$	−1163.94	−1.26104
$924$	0	0
$925$	286.000i	0.309189i
$926$	0	0
$927$	394.908i	0.426006i
$928$	0	0
$929$	−438.000	−0.471475	−0.235737	−	0.971817i	$-0.575751\pi$
$929$	−438.000	−0.471475	−0.235737	+	0.971817i	$0.575751\pi$
$930$	0	0
$931$	−6.92820	−0.00744168
$932$	0	0
$933$	72.0000i	0.0771704i
$934$	0	0
$935$	748.246i	0.800263i
$936$	0	0
$937$	−1826.00	−1.94877	−0.974386	−	0.224881i	$-0.927801\pi$
$937$	−1826.00	−1.94877	−0.974386	+	0.224881i	$0.927801\pi$
$938$	0	0
$939$	−502.295	−0.534925
$940$	0	0
$941$	− 330.000i	− 0.350691i	−0.984507	−	0.175345i	$-0.943896\pi$
$941$	− 330.000i	− 0.350691i	0.984507	−	0.175345i	$-0.0561042\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	216.000	0.228571
$946$	0	0
$947$	−187.061	−0.197531	−0.0987653	−	0.995111i	$-0.531489\pi$
$947$	−187.061	−0.197531	−0.0987653	+	0.995111i	$0.531489\pi$
$948$	0	0
$949$	1148.00i	1.20969i
$950$	0	0
$951$	− 363.731i	− 0.382472i
$952$	0	0
$953$	1110.00	1.16474	0.582371	−	0.812923i	$-0.302125\pi$
$953$	1110.00	1.16474	0.582371	+	0.812923i	$0.302125\pi$
$954$	0	0
$955$	1995.32	2.08934
$956$	0	0
$957$	1080.00i	1.12853i
$958$	0	0
$959$	− 1288.65i	− 1.34374i
$960$	0	0
$961$	529.000	0.550468
$962$	0	0
$963$	436.477	0.453247
$964$	0	0
$965$	564.000i	0.584456i
$966$	0	0
$967$	− 755.174i	− 0.780945i	−0.920615	−	0.390473i	$-0.872312\pi$
$967$	− 755.174i	− 0.780945i	0.920615	−	0.390473i	$-0.127688\pi$
$968$	0	0
$969$	72.0000	0.0743034
$970$	0	0
$971$	−394.908	−0.406702	−0.203351	−	0.979106i	$-0.565183\pi$
$971$	−394.908	−0.406702	−0.203351	+	0.979106i	$0.565183\pi$
$972$	0	0
$973$	− 336.000i	− 0.345324i
$974$	0	0
$975$	266.736i	0.273575i
$976$	0	0
$977$	−918.000	−0.939611	−0.469806	−	0.882770i	$-0.655676\pi$
$977$	−918.000	−0.939611	−0.469806	+	0.882770i	$0.655676\pi$
$978$	0	0
$979$	−2369.45	−2.42027
$980$	0	0
$981$	102.000i	0.103976i
$982$	0	0
$983$	− 41.5692i	− 0.0422881i	−0.999776	−	0.0211441i	$-0.993269\pi$
$983$	− 41.5692i	− 0.0422881i	0.999776	−	0.0211441i	$-0.00673086\pi$
$984$	0	0
$985$	1548.00	1.57157
$986$	0	0
$987$	498.831	0.505401
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 48.4974i	− 0.0489379i	−0.999701	−	0.0244689i	$-0.992211\pi$
$991$	− 48.4974i	− 0.0489379i	0.999701	−	0.0244689i	$-0.00778948\pi$
$992$	0	0
$993$	348.000	0.350453
$994$	0	0
$995$	−706.677	−0.710228
$996$	0	0
$997$	− 554.000i	− 0.555667i	−0.960629	−	0.277834i	$-0.910384\pi$
$997$	− 554.000i	− 0.555667i	0.960629	−	0.277834i	$-0.0896164\pi$
$998$	0	0
$999$	− 135.100i	− 0.135235i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.b.b.127.3		4
3.2	odd	2		2304.3.b.n.127.4		4
4.3	odd	2	inner	768.3.b.b.127.1		4
8.3	odd	2	inner	768.3.b.b.127.4		4
8.5	even	2	inner	768.3.b.b.127.2		4
12.11	even	2		2304.3.b.n.127.3		4
16.3	odd	4		192.3.g.a.127.1		2
16.5	even	4		48.3.g.a.31.1	✓	2
16.11	odd	4		48.3.g.a.31.2	yes	2
16.13	even	4		192.3.g.a.127.2		2
24.5	odd	2		2304.3.b.n.127.2		4
24.11	even	2		2304.3.b.n.127.1		4
48.5	odd	4		144.3.g.b.127.1		2
48.11	even	4		144.3.g.b.127.2		2
48.29	odd	4		576.3.g.i.127.1		2
48.35	even	4		576.3.g.i.127.2		2
80.27	even	4		1200.3.j.a.799.3		4
80.37	odd	4		1200.3.j.a.799.2		4
80.43	even	4		1200.3.j.a.799.1		4
80.53	odd	4		1200.3.j.a.799.4		4
80.59	odd	4		1200.3.e.h.751.1		2
80.69	even	4		1200.3.e.h.751.2		2
112.27	even	4		2352.3.m.a.1471.1		2
112.69	odd	4		2352.3.m.a.1471.2		2
144.5	odd	12		1296.3.o.p.703.1		2
144.11	even	12		1296.3.o.p.271.1		2
144.43	odd	12		1296.3.o.c.271.1		2
144.59	even	12		1296.3.o.n.703.1		2
144.85	even	12		1296.3.o.c.703.1		2
144.101	odd	12		1296.3.o.n.271.1		2
144.133	even	12		1296.3.o.a.271.1		2
144.139	odd	12		1296.3.o.a.703.1		2
240.53	even	4		3600.3.j.i.1999.1		4
240.59	even	4		3600.3.e.t.3151.1		2
240.107	odd	4		3600.3.j.i.1999.2		4
240.149	odd	4		3600.3.e.t.3151.2		2
240.197	even	4		3600.3.j.i.1999.3		4
240.203	odd	4		3600.3.j.i.1999.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.g.a.31.1	✓	2	16.5	even	4
48.3.g.a.31.2	yes	2	16.11	odd	4
144.3.g.b.127.1		2	48.5	odd	4
144.3.g.b.127.2		2	48.11	even	4
192.3.g.a.127.1		2	16.3	odd	4
192.3.g.a.127.2		2	16.13	even	4
576.3.g.i.127.1		2	48.29	odd	4
576.3.g.i.127.2		2	48.35	even	4
768.3.b.b.127.1		4	4.3	odd	2	inner
768.3.b.b.127.2		4	8.5	even	2	inner
768.3.b.b.127.3		4	1.1	even	1	trivial
768.3.b.b.127.4		4	8.3	odd	2	inner
1200.3.e.h.751.1		2	80.59	odd	4
1200.3.e.h.751.2		2	80.69	even	4
1200.3.j.a.799.1		4	80.43	even	4
1200.3.j.a.799.2		4	80.37	odd	4
1200.3.j.a.799.3		4	80.27	even	4
1200.3.j.a.799.4		4	80.53	odd	4
1296.3.o.a.271.1		2	144.133	even	12
1296.3.o.a.703.1		2	144.139	odd	12
1296.3.o.c.271.1		2	144.43	odd	12
1296.3.o.c.703.1		2	144.85	even	12
1296.3.o.n.271.1		2	144.101	odd	12
1296.3.o.n.703.1		2	144.59	even	12
1296.3.o.p.271.1		2	144.11	even	12
1296.3.o.p.703.1		2	144.5	odd	12
2304.3.b.n.127.1		4	24.11	even	2
2304.3.b.n.127.2		4	24.5	odd	2
2304.3.b.n.127.3		4	12.11	even	2
2304.3.b.n.127.4		4	3.2	odd	2
2352.3.m.a.1471.1		2	112.27	even	4
2352.3.m.a.1471.2		2	112.69	odd	4
3600.3.e.t.3151.1		2	240.59	even	4
3600.3.e.t.3151.2		2	240.149	odd	4
3600.3.j.i.1999.1		4	240.53	even	4
3600.3.j.i.1999.2		4	240.107	odd	4
3600.3.j.i.1999.3		4	240.197	even	4
3600.3.j.i.1999.4		4	240.203	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -6.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} -6.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +20.7846 q^{11} -14.0000i q^{13} -10.3923i q^{15} -6.00000 q^{17} -6.92820 q^{19} +12.0000i q^{21} -11.0000 q^{25} +5.19615 q^{27} +30.0000i q^{29} -20.7846i q^{31} +36.0000 q^{33} +41.5692 q^{35} -26.0000i q^{37} -24.2487i q^{39} +54.0000 q^{41} +20.7846 q^{43} -18.0000i q^{45} -41.5692i q^{47} +1.00000 q^{49} -10.3923 q^{51} +18.0000i q^{53} -124.708i q^{55} -12.0000 q^{57} -20.7846 q^{59} -70.0000i q^{61} +20.7846i q^{63} -84.0000 q^{65} +117.779 q^{67} -83.1384i q^{71} -82.0000 q^{73} -19.0526 q^{75} +144.000i q^{77} -76.2102i q^{79} +9.00000 q^{81} +20.7846 q^{83} +36.0000i q^{85} +51.9615i q^{87} -114.000 q^{89} +96.9948 q^{91} -36.0000i q^{93} +41.5692i q^{95} +34.0000 q^{97} +62.3538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 24 q^{17} - 44 q^{25} + 144 q^{33} + 216 q^{41} + 4 q^{49} - 48 q^{57} - 336 q^{65} - 328 q^{73} + 36 q^{81} - 456 q^{89} + 136 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 - 24 * q^17 - 44 * q^25 + 144 * q^33 + 216 * q^41 + 4 * q^49 - 48 * q^57 - 336 * q^65 - 328 * q^73 + 36 * q^81 - 456 * q^89 + 136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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