LMFDB - Embedded newform 1664.2.b.k.833.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.6
Root			$1.19252 - 0.760198i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.k.833.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.611393i q^{3} +1.62620i q^{5} -3.10261 q^{7} +2.62620 q^{9} -5.31965i q^{11} +1.00000i q^{13} +0.994247 q^{15} -1.89692 q^{17} +0.885578i q^{19} +1.89692i q^{21} +1.22279 q^{23} +2.35548 q^{25} -3.43982i q^{27} +2.00000i q^{29} -3.71400 q^{31} -3.25240 q^{33} -5.04546i q^{35} -11.1493i q^{37} +0.611393 q^{39} -5.79383 q^{41} -3.43982i q^{43} +4.27072i q^{45} +0.274184 q^{47} +2.62620 q^{49} +1.15976i q^{51} -7.79383i q^{53} +8.65080 q^{55} +0.541436 q^{57} -1.56000i q^{59} -7.04623i q^{61} -8.14807 q^{63} -1.62620 q^{65} -12.7477i q^{67} -0.747604i q^{69} +8.08505 q^{71} +16.2986 q^{73} -1.44012i q^{75} +16.5048i q^{77} +9.41650 q^{79} +5.77551 q^{81} -10.9765i q^{83} -3.08476i q^{85} +1.22279 q^{87} -5.25240 q^{89} -3.10261i q^{91} +2.27072i q^{93} -1.44012 q^{95} -8.50479 q^{97} -13.9704i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 8 * q^17 - 28 * q^25 + 32 * q^33 - 40 * q^41 - 4 * q^49 + 48 * q^57 + 16 * q^65 + 24 * q^73 - 52 * q^81 + 8 * q^89 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 0.611393i	− 0.352988i	−0.984302	−	0.176494i	$-0.943524\pi$
$3$	− 0.611393i	− 0.352988i	0.984302	−	0.176494i	$-0.0564756\pi$
$4$	0	0
$5$	1.62620i	0.727258i	0.931544	+	0.363629i	$0.118462\pi$
$5$	1.62620i	0.727258i	−0.931544	+	0.363629i	$0.881538\pi$
$6$	0	0
$7$	−3.10261	−1.17268	−0.586338	−	0.810066i	$-0.699431\pi$
$7$	−3.10261	−1.17268	−0.586338	+	0.810066i	$0.699431\pi$
$8$	0	0
$9$	2.62620	0.875399
$10$	0	0
$11$	− 5.31965i	− 1.60393i	−0.597369	−	0.801967i	$-0.703787\pi$
$11$	− 5.31965i	− 1.60393i	0.597369	−	0.801967i	$-0.296213\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0.994247	0.256713
$16$	0	0
$17$	−1.89692	−0.460070	−0.230035	−	0.973182i	$-0.573884\pi$
$17$	−1.89692	−0.460070	−0.230035	+	0.973182i	$0.573884\pi$
$18$	0	0
$19$	0.885578i	0.203165i	0.994827	+	0.101583i	$0.0323907\pi$
$19$	0.885578i	0.203165i	−0.994827	+	0.101583i	$0.967609\pi$
$20$	0	0
$21$	1.89692i	0.413941i
$22$	0	0
$23$	1.22279	0.254969	0.127484	−	0.991841i	$-0.459310\pi$
$23$	1.22279	0.254969	0.127484	+	0.991841i	$0.459310\pi$
$24$	0	0
$25$	2.35548	0.471096
$26$	0	0
$27$	− 3.43982i	− 0.661994i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	−3.71400	−0.667055	−0.333527	−	0.942740i	$-0.608239\pi$
$31$	−3.71400	−0.667055	−0.333527	+	0.942740i	$0.608239\pi$
$32$	0	0
$33$	−3.25240	−0.566169
$34$	0	0
$35$	− 5.04546i	− 0.852839i
$36$	0	0
$37$	− 11.1493i	− 1.83294i	−0.400108	−	0.916468i	$-0.631028\pi$
$37$	− 11.1493i	− 1.83294i	0.400108	−	0.916468i	$-0.368972\pi$
$38$	0	0
$39$	0.611393	0.0979013
$40$	0	0
$41$	−5.79383	−0.904845	−0.452422	−	0.891804i	$-0.649440\pi$
$41$	−5.79383	−0.904845	−0.452422	+	0.891804i	$0.649440\pi$
$42$	0	0
$43$	− 3.43982i	− 0.524568i	−0.964991	−	0.262284i	$-0.915524\pi$
$43$	− 3.43982i	− 0.524568i	0.964991	−	0.262284i	$-0.0844757\pi$
$44$	0	0
$45$	4.27072i	0.636641i
$46$	0	0
$47$	0.274184	0.0399939	0.0199970	−	0.999800i	$-0.493634\pi$
$47$	0.274184	0.0399939	0.0199970	+	0.999800i	$0.493634\pi$
$48$	0	0
$49$	2.62620	0.375171
$50$	0	0
$51$	1.15976i	0.162399i
$52$	0	0
$53$	− 7.79383i	− 1.07057i	−0.844673	−	0.535283i	$-0.820205\pi$
$53$	− 7.79383i	− 1.07057i	0.844673	−	0.535283i	$-0.179795\pi$
$54$	0	0
$55$	8.65080	1.16647
$56$	0	0
$57$	0.541436	0.0717150
$58$	0	0
$59$	− 1.56000i	− 0.203094i	−0.994831	−	0.101547i	$-0.967621\pi$
$59$	− 1.56000i	− 0.203094i	0.994831	−	0.101547i	$-0.0323793\pi$
$60$	0	0
$61$	− 7.04623i	− 0.902177i	−0.892479	−	0.451089i	$-0.851036\pi$
$61$	− 7.04623i	− 0.902177i	0.892479	−	0.451089i	$-0.148964\pi$
$62$	0	0
$63$	−8.14807	−1.02656
$64$	0	0
$65$	−1.62620	−0.201705
$66$	0	0
$67$	− 12.7477i	− 1.55737i	−0.627413	−	0.778687i	$-0.715886\pi$
$67$	− 12.7477i	− 1.55737i	0.627413	−	0.778687i	$-0.284114\pi$
$68$	0	0
$69$	− 0.747604i	− 0.0900009i
$70$	0	0
$71$	8.08505	0.959519	0.479759	−	0.877400i	$-0.340724\pi$
$71$	8.08505	0.959519	0.479759	+	0.877400i	$0.340724\pi$
$72$	0	0
$73$	16.2986	1.90761	0.953805	−	0.300427i	$-0.0971291\pi$
$73$	16.2986	1.90761	0.953805	+	0.300427i	$0.0971291\pi$
$74$	0	0
$75$	− 1.44012i	− 0.166291i
$76$	0	0
$77$	16.5048i	1.88090i
$78$	0	0
$79$	9.41650	1.05944	0.529720	−	0.848173i	$-0.322297\pi$
$79$	9.41650	1.05944	0.529720	+	0.848173i	$0.322297\pi$
$80$	0	0
$81$	5.77551	0.641723
$82$	0	0
$83$	− 10.9765i	− 1.20483i	−0.798184	−	0.602414i	$-0.794206\pi$
$83$	− 10.9765i	− 1.20483i	0.798184	−	0.602414i	$-0.205794\pi$
$84$	0	0
$85$	− 3.08476i	− 0.334589i
$86$	0	0
$87$	1.22279	0.131097
$88$	0	0
$89$	−5.25240	−0.556753	−0.278376	−	0.960472i	$-0.589796\pi$
$89$	−5.25240	−0.556753	−0.278376	+	0.960472i	$0.589796\pi$
$90$	0	0
$91$	− 3.10261i	− 0.325242i
$92$	0	0
$93$	2.27072i	0.235463i
$94$	0	0
$95$	−1.44012	−0.147754
$96$	0	0
$97$	−8.50479	−0.863531	−0.431765	−	0.901986i	$-0.642109\pi$
$97$	−8.50479	−0.863531	−0.431765	+	0.901986i	$0.642109\pi$
$98$	0	0
$99$	− 13.9704i	− 1.40408i
$100$	0	0
$101$	− 11.4586i	− 1.14017i	−0.821586	−	0.570085i	$-0.806910\pi$
$101$	− 11.4586i	− 1.14017i	0.821586	−	0.570085i	$-0.193090\pi$
$102$	0	0
$103$	−13.6332	−1.34332	−0.671661	−	0.740858i	$-0.734419\pi$
$103$	−13.6332	−1.34332	−0.671661	+	0.740858i	$0.734419\pi$
$104$	0	0
$105$	−3.08476	−0.301042
$106$	0	0
$107$	− 17.6844i	− 1.70962i	−0.518941	−	0.854810i	$-0.673674\pi$
$107$	− 17.6844i	− 1.70962i	0.518941	−	0.854810i	$-0.326326\pi$
$108$	0	0
$109$	14.6079i	1.39918i	0.714544	+	0.699590i	$0.246634\pi$
$109$	14.6079i	1.39918i	−0.714544	+	0.699590i	$0.753366\pi$
$110$	0	0
$111$	−6.81662	−0.647005
$112$	0	0
$113$	10.2986	0.968813	0.484407	−	0.874843i	$-0.339036\pi$
$113$	10.2986	0.968813	0.484407	+	0.874843i	$0.339036\pi$
$114$	0	0
$115$	1.98849i	0.185428i
$116$	0	0
$117$	2.62620i	0.242792i
$118$	0	0
$119$	5.88539	0.539513
$120$	0	0
$121$	−17.2986	−1.57260
$122$	0	0
$123$	3.54231i	0.319399i
$124$	0	0
$125$	11.9615i	1.06987i
$126$	0	0
$127$	10.0909	0.895424	0.447712	−	0.894178i	$-0.352239\pi$
$127$	10.0909	0.895424	0.447712	+	0.894178i	$0.352239\pi$
$128$	0	0
$129$	−2.10308	−0.185166
$130$	0	0
$131$	5.21098i	0.455285i	0.973745	+	0.227643i	$0.0731018\pi$
$131$	5.21098i	0.455285i	−0.973745	+	0.227643i	$0.926898\pi$
$132$	0	0
$133$	− 2.74760i	− 0.238247i
$134$	0	0
$135$	5.59383	0.481440
$136$	0	0
$137$	−12.5048	−1.06836	−0.534178	−	0.845372i	$-0.679379\pi$
$137$	−12.5048	−1.06836	−0.534178	+	0.845372i	$0.679379\pi$
$138$	0	0
$139$	17.4559i	1.48059i	0.672282	+	0.740295i	$0.265314\pi$
$139$	17.4559i	1.48059i	−0.672282	+	0.740295i	$0.734686\pi$
$140$	0	0
$141$	− 0.167635i	− 0.0141174i
$142$	0	0
$143$	5.31965	0.444851
$144$	0	0
$145$	−3.25240	−0.270097
$146$	0	0
$147$	− 1.60564i	− 0.132431i
$148$	0	0
$149$	4.50479i	0.369047i	0.982828	+	0.184523i	$0.0590742\pi$
$149$	4.50479i	0.369047i	−0.982828	+	0.184523i	$0.940926\pi$
$150$	0	0
$151$	−14.5818	−1.18665	−0.593326	−	0.804962i	$-0.702186\pi$
$151$	−14.5818	−1.18665	−0.593326	+	0.804962i	$0.702186\pi$
$152$	0	0
$153$	−4.98168	−0.402745
$154$	0	0
$155$	− 6.03971i	− 0.485121i
$156$	0	0
$157$	− 1.79383i	− 0.143163i	−0.997435	−	0.0715817i	$-0.977195\pi$
$157$	− 1.79383i	− 0.143163i	0.997435	−	0.0715817i	$-0.0228047\pi$
$158$	0	0
$159$	−4.76510	−0.377897
$160$	0	0
$161$	−3.79383	−0.298996
$162$	0	0
$163$	− 7.85651i	− 0.615369i	−0.951488	−	0.307685i	$-0.900446\pi$
$163$	− 7.85651i	− 0.615369i	0.951488	−	0.307685i	$-0.0995542\pi$
$164$	0	0
$165$	− 5.28904i	− 0.411751i
$166$	0	0
$167$	−14.2620	−1.10363	−0.551814	−	0.833967i	$-0.686064\pi$
$167$	−14.2620	−1.10363	−0.551814	+	0.833967i	$0.686064\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.32570i	0.177851i
$172$	0	0
$173$	− 2.29862i	− 0.174761i	−0.996175	−	0.0873806i	$-0.972150\pi$
$173$	− 2.29862i	− 0.174761i	0.996175	−	0.0873806i	$-0.0278496\pi$
$174$	0	0
$175$	−7.30814	−0.552443
$176$	0	0
$177$	−0.953771	−0.0716898
$178$	0	0
$179$	15.5587i	1.16291i	0.813578	+	0.581456i	$0.197517\pi$
$179$	15.5587i	1.16291i	−0.813578	+	0.581456i	$0.802483\pi$
$180$	0	0
$181$	− 1.04623i	− 0.0777656i	−0.999244	−	0.0388828i	$-0.987620\pi$
$181$	− 1.04623i	− 0.0777656i	0.999244	−	0.0388828i	$-0.0123799\pi$
$182$	0	0
$183$	−4.30802	−0.318458
$184$	0	0
$185$	18.1310	1.33302
$186$	0	0
$187$	10.0909i	0.737921i
$188$	0	0
$189$	10.6724i	0.776305i
$190$	0	0
$191$	8.65080	0.625950	0.312975	−	0.949761i	$-0.398674\pi$
$191$	8.65080	0.625950	0.312975	+	0.949761i	$0.398674\pi$
$192$	0	0
$193$	12.5048	0.900115	0.450057	−	0.893000i	$-0.351404\pi$
$193$	12.5048	0.900115	0.450057	+	0.893000i	$0.351404\pi$
$194$	0	0
$195$	0.994247i	0.0711995i
$196$	0	0
$197$	− 15.3555i	− 1.09403i	−0.837122	−	0.547016i	$-0.815764\pi$
$197$	− 15.3555i	− 1.09403i	0.837122	−	0.547016i	$-0.184236\pi$
$198$	0	0
$199$	−14.9821	−1.06205	−0.531025	−	0.847356i	$-0.678193\pi$
$199$	−14.9821	−1.06205	−0.531025	+	0.847356i	$0.678193\pi$
$200$	0	0
$201$	−7.79383	−0.549735
$202$	0	0
$203$	− 6.20522i	− 0.435521i
$204$	0	0
$205$	− 9.42192i	− 0.658055i
$206$	0	0
$207$	3.21128	0.223199
$208$	0	0
$209$	4.71096	0.325864
$210$	0	0
$211$	9.77109i	0.672670i	0.941742	+	0.336335i	$0.109187\pi$
$211$	9.77109i	0.672670i	−0.941742	+	0.336335i	$0.890813\pi$
$212$	0	0
$213$	− 4.94315i	− 0.338699i
$214$	0	0
$215$	5.59383	0.381496
$216$	0	0
$217$	11.5231	0.782240
$218$	0	0
$219$	− 9.96487i	− 0.673364i
$220$	0	0
$221$	− 1.89692i	− 0.127600i
$222$	0	0
$223$	18.4675	1.23668	0.618339	−	0.785912i	$-0.287806\pi$
$223$	18.4675	1.23668	0.618339	+	0.785912i	$0.287806\pi$
$224$	0	0
$225$	6.18596	0.412397
$226$	0	0
$227$	15.7416i	1.04481i	0.852699	+	0.522403i	$0.174964\pi$
$227$	15.7416i	1.04481i	−0.852699	+	0.522403i	$0.825036\pi$
$228$	0	0
$229$	− 17.0848i	− 1.12899i	−0.825436	−	0.564496i	$-0.809070\pi$
$229$	− 17.0848i	− 1.12899i	0.825436	−	0.564496i	$-0.190930\pi$
$230$	0	0
$231$	10.0909	0.663934
$232$	0	0
$233$	13.4200	0.879175	0.439588	−	0.898200i	$-0.355125\pi$
$233$	13.4200	0.879175	0.439588	+	0.898200i	$0.355125\pi$
$234$	0	0
$235$	0.445878i	0.0290859i
$236$	0	0
$237$	− 5.75719i	− 0.373970i
$238$	0	0
$239$	−17.5016	−1.13208	−0.566041	−	0.824377i	$-0.691525\pi$
$239$	−17.5016	−1.13208	−0.566041	+	0.824377i	$0.691525\pi$
$240$	0	0
$241$	−4.84006	−0.311776	−0.155888	−	0.987775i	$-0.549824\pi$
$241$	−4.84006	−0.311776	−0.155888	+	0.987775i	$0.549824\pi$
$242$	0	0
$243$	− 13.8506i	− 0.888515i
$244$	0	0
$245$	4.27072i	0.272846i
$246$	0	0
$247$	−0.885578	−0.0563480
$248$	0	0
$249$	−6.71096	−0.425290
$250$	0	0
$251$	1.05727i	0.0667344i	0.999443	+	0.0333672i	$0.0106231\pi$
$251$	1.05727i	0.0667344i	−0.999443	+	0.0333672i	$0.989377\pi$
$252$	0	0
$253$	− 6.50479i	− 0.408953i
$254$	0	0
$255$	−1.88600	−0.118106
$256$	0	0
$257$	−0.644520	−0.0402041	−0.0201020	−	0.999798i	$-0.506399\pi$
$257$	−0.644520	−0.0402041	−0.0201020	+	0.999798i	$0.506399\pi$
$258$	0	0
$259$	34.5920i	2.14944i
$260$	0	0
$261$	5.25240i	0.325115i
$262$	0	0
$263$	26.2610	1.61932	0.809662	−	0.586897i	$-0.199651\pi$
$263$	26.2610	1.61932	0.809662	+	0.586897i	$0.199651\pi$
$264$	0	0
$265$	12.6743	0.778577
$266$	0	0
$267$	3.21128i	0.196527i
$268$	0	0
$269$	27.7572i	1.69239i	0.532877	+	0.846193i	$0.321111\pi$
$269$	27.7572i	1.69239i	−0.532877	+	0.846193i	$0.678889\pi$
$270$	0	0
$271$	−26.6613	−1.61956	−0.809778	−	0.586737i	$-0.800412\pi$
$271$	−26.6613	−1.61956	−0.809778	+	0.586737i	$0.800412\pi$
$272$	0	0
$273$	−1.89692	−0.114807
$274$	0	0
$275$	− 12.5303i	− 0.755607i
$276$	0	0
$277$	14.5048i	0.871509i	0.900066	+	0.435754i	$0.143518\pi$
$277$	14.5048i	0.871509i	−0.900066	+	0.435754i	$0.856482\pi$
$278$	0	0
$279$	−9.75371	−0.583939
$280$	0	0
$281$	16.7110	0.996892	0.498446	−	0.866921i	$-0.333904\pi$
$281$	16.7110	0.996892	0.498446	+	0.866921i	$0.333904\pi$
$282$	0	0
$283$	18.3589i	1.09132i	0.838006	+	0.545661i	$0.183721\pi$
$283$	18.3589i	1.09132i	−0.838006	+	0.545661i	$0.816279\pi$
$284$	0	0
$285$	0.880483i	0.0521553i
$286$	0	0
$287$	17.9760	1.06109
$288$	0	0
$289$	−13.4017	−0.788336
$290$	0	0
$291$	5.19977i	0.304816i
$292$	0	0
$293$	− 33.1127i	− 1.93446i	−0.253897	−	0.967231i	$-0.581712\pi$
$293$	− 33.1127i	− 1.93446i	0.253897	−	0.967231i	$-0.418288\pi$
$294$	0	0
$295$	2.53686	0.147702
$296$	0	0
$297$	−18.2986	−1.06179
$298$	0	0
$299$	1.22279i	0.0707156i
$300$	0	0
$301$	10.6724i	0.615148i
$302$	0	0
$303$	−7.00569	−0.402466
$304$	0	0
$305$	11.4586	0.656115
$306$	0	0
$307$	26.7243i	1.52523i	0.646850	+	0.762617i	$0.276086\pi$
$307$	26.7243i	1.52523i	−0.646850	+	0.762617i	$0.723914\pi$
$308$	0	0
$309$	8.33527i	0.474177i
$310$	0	0
$311$	18.0673	1.02450	0.512251	−	0.858836i	$-0.328812\pi$
$311$	18.0673	1.02450	0.512251	+	0.858836i	$0.328812\pi$
$312$	0	0
$313$	−15.2139	−0.859938	−0.429969	−	0.902844i	$-0.641476\pi$
$313$	−15.2139	−0.859938	−0.429969	+	0.902844i	$0.641476\pi$
$314$	0	0
$315$	− 13.2504i	− 0.746574i
$316$	0	0
$317$	− 5.58767i	− 0.313835i	−0.987612	−	0.156917i	$-0.949844\pi$
$317$	− 5.58767i	− 0.313835i	0.987612	−	0.156917i	$-0.0501556\pi$
$318$	0	0
$319$	10.6393	0.595686
$320$	0	0
$321$	−10.8122	−0.603476
$322$	0	0
$323$	− 1.67987i	− 0.0934703i
$324$	0	0
$325$	2.35548i	0.130659i
$326$	0	0
$327$	8.93116	0.493894
$328$	0	0
$329$	−0.850688	−0.0468999
$330$	0	0
$331$	19.0789i	1.04867i	0.851511	+	0.524336i	$0.175687\pi$
$331$	19.0789i	1.04867i	−0.851511	+	0.524336i	$0.824313\pi$
$332$	0	0
$333$	− 29.2803i	− 1.60455i
$334$	0	0
$335$	20.7302	1.13261
$336$	0	0
$337$	−6.33716	−0.345207	−0.172603	−	0.984991i	$-0.555218\pi$
$337$	−6.33716	−0.345207	−0.172603	+	0.984991i	$0.555218\pi$
$338$	0	0
$339$	− 6.29651i	− 0.341980i
$340$	0	0
$341$	19.7572i	1.06991i
$342$	0	0
$343$	13.5702	0.732722
$344$	0	0
$345$	1.21575	0.0654539
$346$	0	0
$347$	5.04546i	0.270855i	0.990787	+	0.135427i	$0.0432407\pi$
$347$	5.04546i	0.270855i	−0.990787	+	0.135427i	$0.956759\pi$
$348$	0	0
$349$	25.4758i	1.36369i	0.731496	+	0.681845i	$0.238822\pi$
$349$	25.4758i	1.36369i	−0.731496	+	0.681845i	$0.761178\pi$
$350$	0	0
$351$	3.43982	0.183604
$352$	0	0
$353$	14.5414	0.773963	0.386981	−	0.922088i	$-0.373518\pi$
$353$	14.5414	0.773963	0.386981	+	0.922088i	$0.373518\pi$
$354$	0	0
$355$	13.1479i	0.697817i
$356$	0	0
$357$	− 3.59829i	− 0.190442i
$358$	0	0
$359$	−28.8783	−1.52414	−0.762069	−	0.647496i	$-0.775816\pi$
$359$	−28.8783	−1.52414	−0.762069	+	0.647496i	$0.775816\pi$
$360$	0	0
$361$	18.2158	0.958724
$362$	0	0
$363$	10.5763i	0.555110i
$364$	0	0
$365$	26.5048i	1.38732i
$366$	0	0
$367$	3.57707	0.186722	0.0933608	−	0.995632i	$-0.470239\pi$
$367$	3.57707	0.186722	0.0933608	+	0.995632i	$0.470239\pi$
$368$	0	0
$369$	−15.2158	−0.792100
$370$	0	0
$371$	24.1812i	1.25543i
$372$	0	0
$373$	20.7110i	1.07237i	0.844100	+	0.536186i	$0.180136\pi$
$373$	20.7110i	1.07237i	−0.844100	+	0.536186i	$0.819864\pi$
$374$	0	0
$375$	7.31316	0.377650
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	23.9701i	1.23126i	0.788035	+	0.615630i	$0.211098\pi$
$379$	23.9701i	1.23126i	−0.788035	+	0.615630i	$0.788902\pi$
$380$	0	0
$381$	− 6.16952i	− 0.316074i
$382$	0	0
$383$	−30.6606	−1.56669	−0.783343	−	0.621590i	$-0.786487\pi$
$383$	−30.6606	−1.56669	−0.783343	+	0.621590i	$0.786487\pi$
$384$	0	0
$385$	−26.8401	−1.36790
$386$	0	0
$387$	− 9.03365i	− 0.459206i
$388$	0	0
$389$	− 27.3449i	− 1.38644i	−0.720726	−	0.693220i	$-0.756192\pi$
$389$	− 27.3449i	− 1.38644i	0.720726	−	0.693220i	$-0.243808\pi$
$390$	0	0
$391$	−2.31952	−0.117303
$392$	0	0
$393$	3.18596	0.160710
$394$	0	0
$395$	15.3131i	0.770486i
$396$	0	0
$397$	8.09246i	0.406149i	0.979163	+	0.203074i	$0.0650933\pi$
$397$	8.09246i	0.406149i	−0.979163	+	0.203074i	$0.934907\pi$
$398$	0	0
$399$	−1.67987	−0.0840985
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	− 3.71400i	− 0.185008i
$404$	0	0
$405$	9.39212i	0.466698i
$406$	0	0
$407$	−59.3104	−2.93991
$408$	0	0
$409$	8.09246	0.400146	0.200073	−	0.979781i	$-0.435882\pi$
$409$	8.09246	0.400146	0.200073	+	0.979781i	$0.435882\pi$
$410$	0	0
$411$	7.64535i	0.377117i
$412$	0	0
$413$	4.84006i	0.238164i
$414$	0	0
$415$	17.8500	0.876220
$416$	0	0
$417$	10.6724	0.522631
$418$	0	0
$419$	9.93661i	0.485435i	0.970097	+	0.242718i	$0.0780388\pi$
$419$	9.93661i	0.485435i	−0.970097	+	0.242718i	$0.921961\pi$
$420$	0	0
$421$	− 9.96147i	− 0.485492i	−0.970090	−	0.242746i	$-0.921952\pi$
$421$	− 9.96147i	− 0.485492i	0.970090	−	0.242746i	$-0.0780482\pi$
$422$	0	0
$423$	0.720062	0.0350106
$424$	0	0
$425$	−4.46815	−0.216737
$426$	0	0
$427$	21.8617i	1.05796i
$428$	0	0
$429$	− 3.25240i	− 0.157027i
$430$	0	0
$431$	10.1083	0.486900	0.243450	−	0.969913i	$-0.421721\pi$
$431$	10.1083	0.486900	0.243450	+	0.969913i	$0.421721\pi$
$432$	0	0
$433$	9.83237	0.472513	0.236257	−	0.971691i	$-0.424079\pi$
$433$	9.83237	0.472513	0.236257	+	0.971691i	$0.424079\pi$
$434$	0	0
$435$	1.98849i	0.0953410i
$436$	0	0
$437$	1.08287i	0.0518008i
$438$	0	0
$439$	30.7298	1.46666	0.733328	−	0.679875i	$-0.237966\pi$
$439$	30.7298	1.46666	0.733328	+	0.679875i	$0.237966\pi$
$440$	0	0
$441$	6.89692	0.328425
$442$	0	0
$443$	4.62785i	0.219876i	0.993938	+	0.109938i	$0.0350652\pi$
$443$	4.62785i	0.219876i	−0.993938	+	0.109938i	$0.964935\pi$
$444$	0	0
$445$	− 8.54144i	− 0.404903i
$446$	0	0
$447$	2.75420	0.130269
$448$	0	0
$449$	35.1387	1.65830	0.829149	−	0.559028i	$-0.188826\pi$
$449$	35.1387	1.65830	0.829149	+	0.559028i	$0.188826\pi$
$450$	0	0
$451$	30.8211i	1.45131i
$452$	0	0
$453$	8.91524i	0.418874i
$454$	0	0
$455$	5.04546	0.236535
$456$	0	0
$457$	0.840061	0.0392964	0.0196482	−	0.999807i	$-0.493745\pi$
$457$	0.840061	0.0392964	0.0196482	+	0.999807i	$0.493745\pi$
$458$	0	0
$459$	6.52505i	0.304563i
$460$	0	0
$461$	− 32.4942i	− 1.51340i	−0.653760	−	0.756702i	$-0.726809\pi$
$461$	− 32.4942i	− 1.51340i	0.653760	−	0.756702i	$-0.273191\pi$
$462$	0	0
$463$	−6.06829	−0.282017	−0.141009	−	0.990008i	$-0.545035\pi$
$463$	−6.06829	−0.282017	−0.141009	+	0.990008i	$0.545035\pi$
$464$	0	0
$465$	−3.69264	−0.171242
$466$	0	0
$467$	− 35.0773i	− 1.62319i	−0.584224	−	0.811593i	$-0.698601\pi$
$467$	− 35.0773i	− 1.62319i	0.584224	−	0.811593i	$-0.301399\pi$
$468$	0	0
$469$	39.5510i	1.82630i
$470$	0	0
$471$	−1.09674	−0.0505350
$472$	0	0
$473$	−18.2986	−0.841372
$474$	0	0
$475$	2.08596i	0.0957104i
$476$	0	0
$477$	− 20.4681i	− 0.937172i
$478$	0	0
$479$	−8.50266	−0.388497	−0.194248	−	0.980952i	$-0.562227\pi$
$479$	−8.50266	−0.388497	−0.194248	+	0.980952i	$0.562227\pi$
$480$	0	0
$481$	11.1493	0.508365
$482$	0	0
$483$	2.31952i	0.105542i
$484$	0	0
$485$	− 13.8305i	− 0.628010i
$486$	0	0
$487$	−0.171694	−0.00778019	−0.00389009	−	0.999992i	$-0.501238\pi$
$487$	−0.171694	−0.00778019	−0.00389009	+	0.999992i	$0.501238\pi$
$488$	0	0
$489$	−4.80342	−0.217218
$490$	0	0
$491$	− 2.85669i	− 0.128921i	−0.997920	−	0.0644603i	$-0.979467\pi$
$491$	− 2.85669i	− 0.128921i	0.997920	−	0.0644603i	$-0.0205326\pi$
$492$	0	0
$493$	− 3.79383i	− 0.170866i
$494$	0	0
$495$	22.7187	1.02113
$496$	0	0
$497$	−25.0848	−1.12521
$498$	0	0
$499$	− 18.2785i	− 0.818256i	−0.912477	−	0.409128i	$-0.865833\pi$
$499$	− 18.2785i	− 0.818256i	0.912477	−	0.409128i	$-0.134167\pi$
$500$	0	0
$501$	8.71970i	0.389567i
$502$	0	0
$503$	−24.1465	−1.07664	−0.538319	−	0.842741i	$-0.680940\pi$
$503$	−24.1465	−1.07664	−0.538319	+	0.842741i	$0.680940\pi$
$504$	0	0
$505$	18.6339	0.829197
$506$	0	0
$507$	0.611393i	0.0271529i
$508$	0	0
$509$	− 34.5972i	− 1.53350i	−0.641948	−	0.766748i	$-0.721874\pi$
$509$	− 34.5972i	− 1.53350i	0.641948	−	0.766748i	$-0.278126\pi$
$510$	0	0
$511$	−50.5683	−2.23701
$512$	0	0
$513$	3.04623	0.134494
$514$	0	0
$515$	− 22.1703i	− 0.976942i
$516$	0	0
$517$	− 1.45856i	− 0.0641476i
$518$	0	0
$519$	−1.40536	−0.0616886
$520$	0	0
$521$	10.3372	0.452879	0.226440	−	0.974025i	$-0.427291\pi$
$521$	10.3372	0.452879	0.226440	+	0.974025i	$0.427291\pi$
$522$	0	0
$523$	23.5239i	1.02863i	0.857602	+	0.514314i	$0.171953\pi$
$523$	23.5239i	1.02863i	−0.857602	+	0.514314i	$0.828047\pi$
$524$	0	0
$525$	4.46815i	0.195006i
$526$	0	0
$527$	7.04516	0.306892
$528$	0	0
$529$	−21.5048	−0.934991
$530$	0	0
$531$	− 4.09686i	− 0.177789i
$532$	0	0
$533$	− 5.79383i	− 0.250959i
$534$	0	0
$535$	28.7584	1.24333
$536$	0	0
$537$	9.51249	0.410494
$538$	0	0
$539$	− 13.9704i	− 0.601750i
$540$	0	0
$541$	− 35.1493i	− 1.51119i	−0.655041	−	0.755593i	$-0.727349\pi$
$541$	− 35.1493i	− 1.51119i	0.655041	−	0.755593i	$-0.272651\pi$
$542$	0	0
$543$	−0.639657	−0.0274503
$544$	0	0
$545$	−23.7553	−1.01757
$546$	0	0
$547$	37.6378i	1.60927i	0.593767	+	0.804637i	$0.297640\pi$
$547$	37.6378i	1.60927i	−0.593767	+	0.804637i	$0.702360\pi$
$548$	0	0
$549$	− 18.5048i	− 0.789765i
$550$	0	0
$551$	−1.77116	−0.0754538
$552$	0	0
$553$	−29.2158	−1.24238
$554$	0	0
$555$	− 11.0852i	− 0.470539i
$556$	0	0
$557$	9.39212i	0.397957i	0.980004	+	0.198979i	$0.0637624\pi$
$557$	9.39212i	0.397957i	−0.980004	+	0.198979i	$0.936238\pi$
$558$	0	0
$559$	3.43982	0.145489
$560$	0	0
$561$	6.16952	0.260477
$562$	0	0
$563$	− 12.4735i	− 0.525694i	−0.964838	−	0.262847i	$-0.915339\pi$
$563$	− 12.4735i	− 0.525694i	0.964838	−	0.262847i	$-0.0846615\pi$
$564$	0	0
$565$	16.7476i	0.704577i
$566$	0	0
$567$	−17.9192	−0.752534
$568$	0	0
$569$	−32.1589	−1.34817	−0.674086	−	0.738653i	$-0.735462\pi$
$569$	−32.1589	−1.34817	−0.674086	+	0.738653i	$0.735462\pi$
$570$	0	0
$571$	3.73139i	0.156154i	0.996947	+	0.0780768i	$0.0248779\pi$
$571$	3.73139i	0.156154i	−0.996947	+	0.0780768i	$0.975122\pi$
$572$	0	0
$573$	− 5.28904i	− 0.220953i
$574$	0	0
$575$	2.88025	0.120115
$576$	0	0
$577$	30.3911	1.26520	0.632599	−	0.774480i	$-0.281988\pi$
$577$	30.3911	1.26520	0.632599	+	0.774480i	$0.281988\pi$
$578$	0	0
$579$	− 7.64535i	− 0.317730i
$580$	0	0
$581$	34.0558i	1.41287i
$582$	0	0
$583$	−41.4604	−1.71712
$584$	0	0
$585$	−4.27072	−0.176572
$586$	0	0
$587$	− 30.8150i	− 1.27187i	−0.771743	−	0.635935i	$-0.780615\pi$
$587$	− 30.8150i	− 1.27187i	0.771743	−	0.635935i	$-0.219385\pi$
$588$	0	0
$589$	− 3.28904i	− 0.135523i
$590$	0	0
$591$	−9.38824	−0.386181
$592$	0	0
$593$	−42.4681	−1.74396	−0.871979	−	0.489543i	$-0.837163\pi$
$593$	−42.4681	−1.74396	−0.871979	+	0.489543i	$0.837163\pi$
$594$	0	0
$595$	9.57082i	0.392365i
$596$	0	0
$597$	9.15994i	0.374891i
$598$	0	0
$599$	40.7860	1.66647	0.833236	−	0.552918i	$-0.186486\pi$
$599$	40.7860	1.66647	0.833236	+	0.552918i	$0.186486\pi$
$600$	0	0
$601$	−17.7466	−0.723897	−0.361949	−	0.932198i	$-0.617888\pi$
$601$	−17.7466	−0.723897	−0.361949	+	0.932198i	$0.617888\pi$
$602$	0	0
$603$	− 33.4779i	− 1.36332i
$604$	0	0
$605$	− 28.1310i	− 1.14369i
$606$	0	0
$607$	−23.0497	−0.935560	−0.467780	−	0.883845i	$-0.654946\pi$
$607$	−23.0497	−0.935560	−0.467780	+	0.883845i	$0.654946\pi$
$608$	0	0
$609$	−3.79383	−0.153734
$610$	0	0
$611$	0.274184i	0.0110923i
$612$	0	0
$613$	22.5972i	0.912694i	0.889802	+	0.456347i	$0.150842\pi$
$613$	22.5972i	0.912694i	−0.889802	+	0.456347i	$0.849158\pi$
$614$	0	0
$615$	−5.76050	−0.232286
$616$	0	0
$617$	−19.2158	−0.773597	−0.386799	−	0.922164i	$-0.626419\pi$
$617$	−19.2158	−0.773597	−0.386799	+	0.922164i	$0.626419\pi$
$618$	0	0
$619$	− 11.6162i	− 0.466893i	−0.972370	−	0.233446i	$-0.925000\pi$
$619$	− 11.6162i	− 0.466893i	0.972370	−	0.233446i	$-0.0750003\pi$
$620$	0	0
$621$	− 4.20617i	− 0.168788i
$622$	0	0
$623$	16.2961	0.652891
$624$	0	0
$625$	−7.67432	−0.306973
$626$	0	0
$627$	− 2.88025i	− 0.115026i
$628$	0	0
$629$	21.1493i	0.843278i
$630$	0	0
$631$	38.0887	1.51629	0.758143	−	0.652089i	$-0.226107\pi$
$631$	38.0887	1.51629	0.758143	+	0.652089i	$0.226107\pi$
$632$	0	0
$633$	5.97398	0.237444
$634$	0	0
$635$	16.4098i	0.651205i
$636$	0	0
$637$	2.62620i	0.104054i
$638$	0	0
$639$	21.2329	0.839962
$640$	0	0
$641$	35.7205	1.41088	0.705438	−	0.708771i	$-0.250750\pi$
$641$	35.7205	1.41088	0.705438	+	0.708771i	$0.250750\pi$
$642$	0	0
$643$	− 30.1405i	− 1.18863i	−0.804234	−	0.594313i	$-0.797424\pi$
$643$	− 30.1405i	− 1.18863i	0.804234	−	0.594313i	$-0.202576\pi$
$644$	0	0
$645$	− 3.42003i	− 0.134664i
$646$	0	0
$647$	20.1818	0.793430	0.396715	−	0.917942i	$-0.370150\pi$
$647$	20.1818	0.793430	0.396715	+	0.917942i	$0.370150\pi$
$648$	0	0
$649$	−8.29862	−0.325750
$650$	0	0
$651$	− 7.04516i	− 0.276121i
$652$	0	0
$653$	3.83048i	0.149898i	0.997187	+	0.0749491i	$0.0238794\pi$
$653$	3.83048i	0.149898i	−0.997187	+	0.0749491i	$0.976121\pi$
$654$	0	0
$655$	−8.47408	−0.331110
$656$	0	0
$657$	42.8034	1.66992
$658$	0	0
$659$	− 7.71957i	− 0.300712i	−0.988632	−	0.150356i	$-0.951958\pi$
$659$	− 7.71957i	− 0.300712i	0.988632	−	0.150356i	$-0.0480420\pi$
$660$	0	0
$661$	11.4952i	0.447112i	0.974691	+	0.223556i	$0.0717665\pi$
$661$	11.4952i	0.447112i	−0.974691	+	0.223556i	$0.928233\pi$
$662$	0	0
$663$	−1.15976	−0.0450414
$664$	0	0
$665$	4.46815	0.173267
$666$	0	0
$667$	2.44557i	0.0946930i
$668$	0	0
$669$	− 11.2909i	− 0.436533i
$670$	0	0
$671$	−37.4834	−1.44703
$672$	0	0
$673$	−0.345895	−0.0133333	−0.00666664	−	0.999978i	$-0.502122\pi$
$673$	−0.345895	−0.0133333	−0.00666664	+	0.999978i	$0.502122\pi$
$674$	0	0
$675$	− 8.10243i	− 0.311863i
$676$	0	0
$677$	− 19.2524i	− 0.739930i	−0.929046	−	0.369965i	$-0.879370\pi$
$677$	− 19.2524i	− 0.739930i	0.929046	−	0.369965i	$-0.120630\pi$
$678$	0	0
$679$	26.3871	1.01264
$680$	0	0
$681$	9.62431	0.368804
$682$	0	0
$683$	− 32.5072i	− 1.24385i	−0.783076	−	0.621926i	$-0.786350\pi$
$683$	− 32.5072i	− 1.24385i	0.783076	−	0.621926i	$-0.213650\pi$
$684$	0	0
$685$	− 20.3353i	− 0.776971i
$686$	0	0
$687$	−10.4455	−0.398521
$688$	0	0
$689$	7.79383	0.296921
$690$	0	0
$691$	44.5395i	1.69436i	0.531305	+	0.847181i	$0.321702\pi$
$691$	44.5395i	1.69436i	−0.531305	+	0.847181i	$0.678298\pi$
$692$	0	0
$693$	43.3449i	1.64653i
$694$	0	0
$695$	−28.3868	−1.07677
$696$	0	0
$697$	10.9904	0.416292
$698$	0	0
$699$	− 8.20492i	− 0.310339i
$700$	0	0
$701$	29.6435i	1.11962i	0.828621	+	0.559809i	$0.189126\pi$
$701$	29.6435i	1.11962i	−0.828621	+	0.559809i	$0.810874\pi$
$702$	0	0
$703$	9.87358	0.372389
$704$	0	0
$705$	0.272607	0.0102670
$706$	0	0
$707$	35.5515i	1.33705i
$708$	0	0
$709$	21.5877i	0.810742i	0.914152	+	0.405371i	$0.132858\pi$
$709$	21.5877i	0.810742i	−0.914152	+	0.405371i	$0.867142\pi$
$710$	0	0
$711$	24.7296	0.927433
$712$	0	0
$713$	−4.54144	−0.170078
$714$	0	0
$715$	8.65080i	0.323521i
$716$	0	0
$717$	10.7003i	0.399611i
$718$	0	0
$719$	−27.9756	−1.04332	−0.521658	−	0.853155i	$-0.674686\pi$
$719$	−27.9756	−1.04332	−0.521658	+	0.853155i	$0.674686\pi$
$720$	0	0
$721$	42.2986	1.57528
$722$	0	0
$723$	2.95918i	0.110053i
$724$	0	0
$725$	4.71096i	0.174961i
$726$	0	0
$727$	19.4727	0.722201	0.361101	−	0.932527i	$-0.382401\pi$
$727$	19.4727	0.722201	0.361101	+	0.932527i	$0.382401\pi$
$728$	0	0
$729$	8.85838	0.328088
$730$	0	0
$731$	6.52505i	0.241338i
$732$	0	0
$733$	− 20.3372i	− 0.751170i	−0.926788	−	0.375585i	$-0.877442\pi$
$733$	− 20.3372i	− 0.751170i	0.926788	−	0.375585i	$-0.122558\pi$
$734$	0	0
$735$	2.61109	0.0963115
$736$	0	0
$737$	−67.8130	−2.49792
$738$	0	0
$739$	− 20.3017i	− 0.746811i	−0.927668	−	0.373405i	$-0.878190\pi$
$739$	− 20.3017i	− 0.746811i	0.927668	−	0.373405i	$-0.121810\pi$
$740$	0	0
$741$	0.541436i	0.0198902i
$742$	0	0
$743$	44.6373	1.63758	0.818791	−	0.574091i	$-0.194645\pi$
$743$	44.6373	1.63758	0.818791	+	0.574091i	$0.194645\pi$
$744$	0	0
$745$	−7.32568	−0.268392
$746$	0	0
$747$	− 28.8265i	− 1.05471i
$748$	0	0
$749$	54.8680i	2.00483i
$750$	0	0
$751$	38.8323	1.41701	0.708505	−	0.705706i	$-0.249370\pi$
$751$	38.8323	1.41701	0.708505	+	0.705706i	$0.249370\pi$
$752$	0	0
$753$	0.646409	0.0235564
$754$	0	0
$755$	− 23.7130i	− 0.863003i
$756$	0	0
$757$	18.1695i	0.660383i	0.943914	+	0.330191i	$0.107113\pi$
$757$	18.1695i	0.660383i	−0.943914	+	0.330191i	$0.892887\pi$
$758$	0	0
$759$	−3.97699	−0.144355
$760$	0	0
$761$	21.8496	0.792049	0.396025	−	0.918240i	$-0.370390\pi$
$761$	21.8496	0.792049	0.396025	+	0.918240i	$0.370390\pi$
$762$	0	0
$763$	− 45.3226i	− 1.64079i
$764$	0	0
$765$	− 8.10119i	− 0.292899i
$766$	0	0
$767$	1.56000	0.0563282
$768$	0	0
$769$	41.7205	1.50448	0.752241	−	0.658888i	$-0.228973\pi$
$769$	41.7205	1.50448	0.752241	+	0.658888i	$0.228973\pi$
$770$	0	0
$771$	0.394055i	0.0141916i
$772$	0	0
$773$	− 2.91524i	− 0.104854i	−0.998625	−	0.0524269i	$-0.983304\pi$
$773$	− 2.91524i	− 0.104854i	0.998625	−	0.0524269i	$-0.0166957\pi$
$774$	0	0
$775$	−8.74826	−0.314247
$776$	0	0
$777$	21.1493	0.758727
$778$	0	0
$779$	− 5.13089i	− 0.183833i
$780$	0	0
$781$	− 43.0096i	− 1.53900i
$782$	0	0
$783$	6.87964	0.245858
$784$	0	0
$785$	2.91713	0.104117
$786$	0	0
$787$	37.6033i	1.34041i	0.742175	+	0.670207i	$0.233795\pi$
$787$	37.6033i	1.34041i	−0.742175	+	0.670207i	$0.766205\pi$
$788$	0	0
$789$	− 16.0558i	− 0.571602i
$790$	0	0
$791$	−31.9526	−1.13610
$792$	0	0
$793$	7.04623	0.250219
$794$	0	0
$795$	− 7.74899i	− 0.274828i
$796$	0	0
$797$	− 20.8401i	− 0.738193i	−0.929391	−	0.369096i	$-0.879667\pi$
$797$	− 20.8401i	− 0.738193i	0.929391	−	0.369096i	$-0.120333\pi$
$798$	0	0
$799$	−0.520105	−0.0184000
$800$	0	0
$801$	−13.7938	−0.487381
$802$	0	0
$803$	− 86.7029i	− 3.05968i
$804$	0	0
$805$	− 6.16952i	− 0.217447i
$806$	0	0
$807$	16.9706	0.597392
$808$	0	0
$809$	26.3372	0.925965	0.462983	−	0.886367i	$-0.346779\pi$
$809$	26.3372	0.925965	0.462983	+	0.886367i	$0.346779\pi$
$810$	0	0
$811$	20.9761i	0.736572i	0.929713	+	0.368286i	$0.120055\pi$
$811$	20.9761i	0.736572i	−0.929713	+	0.368286i	$0.879945\pi$
$812$	0	0
$813$	16.3005i	0.571684i
$814$	0	0
$815$	12.7762	0.447532
$816$	0	0
$817$	3.04623	0.106574
$818$	0	0
$819$	− 8.14807i	− 0.284717i
$820$	0	0
$821$	− 14.4296i	− 0.503597i	−0.967780	−	0.251799i	$-0.918978\pi$
$821$	− 14.4296i	− 0.503597i	0.967780	−	0.251799i	$-0.0810220\pi$
$822$	0	0
$823$	−48.7971	−1.70096	−0.850481	−	0.526006i	$-0.823689\pi$
$823$	−48.7971	−1.70096	−0.850481	+	0.526006i	$0.823689\pi$
$824$	0	0
$825$	−7.66095	−0.266720
$826$	0	0
$827$	32.4601i	1.12875i	0.825520	+	0.564373i	$0.190882\pi$
$827$	32.4601i	1.12875i	−0.825520	+	0.564373i	$0.809118\pi$
$828$	0	0
$829$	− 22.8401i	− 0.793268i	−0.917977	−	0.396634i	$-0.870178\pi$
$829$	− 22.8401i	− 0.793268i	0.917977	−	0.396634i	$-0.129822\pi$
$830$	0	0
$831$	8.86813	0.307632
$832$	0	0
$833$	−4.98168	−0.172605
$834$	0	0
$835$	− 23.1928i	− 0.802622i
$836$	0	0
$837$	12.7755i	0.441586i
$838$	0	0
$839$	−52.3503	−1.80733	−0.903667	−	0.428235i	$-0.859135\pi$
$839$	−52.3503	−1.80733	−0.903667	+	0.428235i	$0.859135\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 10.2170i	− 0.351891i
$844$	0	0
$845$	− 1.62620i	− 0.0559429i
$846$	0	0
$847$	53.6709	1.84415
$848$	0	0
$849$	11.2245	0.385223
$850$	0	0
$851$	− 13.6332i	− 0.467341i
$852$	0	0
$853$	21.2909i	0.728988i	0.931206	+	0.364494i	$0.118758\pi$
$853$	21.2909i	0.728988i	−0.931206	+	0.364494i	$0.881242\pi$
$854$	0	0
$855$	−3.78205	−0.129343
$856$	0	0
$857$	−8.50479	−0.290518	−0.145259	−	0.989394i	$-0.546402\pi$
$857$	−8.50479	−0.290518	−0.145259	+	0.989394i	$0.546402\pi$
$858$	0	0
$859$	− 43.8542i	− 1.49629i	−0.663538	−	0.748143i	$-0.730946\pi$
$859$	− 43.8542i	− 1.49629i	0.663538	−	0.748143i	$-0.269054\pi$
$860$	0	0
$861$	− 10.9904i	− 0.374552i
$862$	0	0
$863$	23.6155	0.803880	0.401940	−	0.915666i	$-0.368336\pi$
$863$	23.6155	0.803880	0.401940	+	0.915666i	$0.368336\pi$
$864$	0	0
$865$	3.73802	0.127096
$866$	0	0
$867$	8.19372i	0.278273i
$868$	0	0
$869$	− 50.0925i	− 1.69927i
$870$	0	0
$871$	12.7477	0.431938
$872$	0	0
$873$	−22.3353	−0.755934
$874$	0	0
$875$	− 37.1118i	− 1.25461i
$876$	0	0
$877$	35.1772i	1.18785i	0.804520	+	0.593925i	$0.202422\pi$
$877$	35.1772i	1.18785i	−0.804520	+	0.593925i	$0.797578\pi$
$878$	0	0
$879$	−20.2449	−0.682842
$880$	0	0
$881$	0.721586	0.0243108	0.0121554	−	0.999926i	$-0.496131\pi$
$881$	0.721586	0.0243108	0.0121554	+	0.999926i	$0.496131\pi$
$882$	0	0
$883$	− 27.9297i	− 0.939909i	−0.882691	−	0.469954i	$-0.844270\pi$
$883$	− 27.9297i	− 0.939909i	0.882691	−	0.469954i	$-0.155730\pi$
$884$	0	0
$885$	− 1.55102i	− 0.0521370i
$886$	0	0
$887$	−2.86789	−0.0962944	−0.0481472	−	0.998840i	$-0.515332\pi$
$887$	−2.86789	−0.0962944	−0.0481472	+	0.998840i	$0.515332\pi$
$888$	0	0
$889$	−31.3082	−1.05004
$890$	0	0
$891$	− 30.7237i	− 1.02928i
$892$	0	0
$893$	0.242812i	0.00812538i
$894$	0	0
$895$	−25.3015	−0.845737
$896$	0	0
$897$	0.747604	0.0249618
$898$	0	0
$899$	− 7.42801i	− 0.247738i
$900$	0	0
$901$	14.7842i	0.492535i
$902$	0	0
$903$	6.52505	0.217140
$904$	0	0
$905$	1.70138	0.0565556
$906$	0	0
$907$	− 4.53656i	− 0.150634i	−0.997160	−	0.0753170i	$-0.976003\pi$
$907$	− 4.53656i	− 0.150634i	0.997160	−	0.0753170i	$-0.0239969\pi$
$908$	0	0
$909$	− 30.0925i	− 0.998104i
$910$	0	0
$911$	−8.39870	−0.278261	−0.139131	−	0.990274i	$-0.544431\pi$
$911$	−8.39870	−0.278261	−0.139131	+	0.990274i	$0.544431\pi$
$912$	0	0
$913$	−58.3911	−1.93246
$914$	0	0
$915$	− 7.00569i	− 0.231601i
$916$	0	0
$917$	− 16.1676i	− 0.533902i
$918$	0	0
$919$	3.63360	0.119861	0.0599307	−	0.998203i	$-0.480912\pi$
$919$	3.63360	0.119861	0.0599307	+	0.998203i	$0.480912\pi$
$920$	0	0
$921$	16.3390	0.538390
$922$	0	0
$923$	8.08505i	0.266123i
$924$	0	0
$925$	− 26.2620i	− 0.863489i
$926$	0	0
$927$	−35.8036	−1.17594
$928$	0	0
$929$	23.3449	0.765920	0.382960	−	0.923765i	$-0.374905\pi$
$929$	23.3449	0.765920	0.382960	+	0.923765i	$0.374905\pi$
$930$	0	0
$931$	2.32570i	0.0762218i
$932$	0	0
$933$	− 11.0462i	− 0.361637i
$934$	0	0
$935$	−16.4098	−0.536659
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	9.30166i	0.303548i
$940$	0	0
$941$	10.1676i	0.331455i	0.986172	+	0.165728i	$0.0529973\pi$
$941$	10.1676i	0.331455i	−0.986172	+	0.165728i	$0.947003\pi$
$942$	0	0
$943$	−7.08462	−0.230707
$944$	0	0
$945$	−17.3555	−0.564574
$946$	0	0
$947$	− 29.6269i	− 0.962746i	−0.876516	−	0.481373i	$-0.840138\pi$
$947$	− 29.6269i	− 0.962746i	0.876516	−	0.481373i	$-0.159862\pi$
$948$	0	0
$949$	16.2986i	0.529076i
$950$	0	0
$951$	−3.41626	−0.110780
$952$	0	0
$953$	−41.1127	−1.33177	−0.665885	−	0.746054i	$-0.731946\pi$
$953$	−41.1127	−1.33177	−0.665885	+	0.746054i	$0.731946\pi$
$954$	0	0
$955$	14.0679i	0.455227i
$956$	0	0
$957$	− 6.50479i	− 0.210270i
$958$	0	0
$959$	38.7975	1.25284
$960$	0	0
$961$	−17.2062	−0.555038
$962$	0	0
$963$	− 46.4429i	− 1.49660i
$964$	0	0
$965$	20.3353i	0.654615i
$966$	0	0
$967$	−42.9056	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$967$	−42.9056	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$968$	0	0
$969$	−1.02706	−0.0329939
$970$	0	0
$971$	− 24.5011i	− 0.786277i	−0.919479	−	0.393138i	$-0.871389\pi$
$971$	− 24.5011i	− 0.786277i	0.919479	−	0.393138i	$-0.128611\pi$
$972$	0	0
$973$	− 54.1589i	− 1.73625i
$974$	0	0
$975$	1.44012	0.0461209
$976$	0	0
$977$	−15.0096	−0.480199	−0.240100	−	0.970748i	$-0.577180\pi$
$977$	−15.0096	−0.480199	−0.240100	+	0.970748i	$0.577180\pi$
$978$	0	0
$979$	27.9409i	0.892995i
$980$	0	0
$981$	38.3632i	1.22484i
$982$	0	0
$983$	−13.9940	−0.446339	−0.223170	−	0.974780i	$-0.571640\pi$
$983$	−13.9940	−0.446339	−0.223170	+	0.974780i	$0.571640\pi$
$984$	0	0
$985$	24.9711	0.795644
$986$	0	0
$987$	0.520105i	0.0165551i
$988$	0	0
$989$	− 4.20617i	− 0.133748i
$990$	0	0
$991$	46.7739	1.48582	0.742911	−	0.669390i	$-0.233445\pi$
$991$	46.7739	1.48582	0.742911	+	0.669390i	$0.233445\pi$
$992$	0	0
$993$	11.6647	0.370169
$994$	0	0
$995$	− 24.3638i	− 0.772385i
$996$	0	0
$997$	− 41.7572i	− 1.32246i	−0.750182	−	0.661232i	$-0.770034\pi$
$997$	− 41.7572i	− 1.32246i	0.750182	−	0.661232i	$-0.229966\pi$
$998$	0	0
$999$	−38.3516	−1.21339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.k.833.6	yes	12
4.3	odd	2	inner	1664.2.b.k.833.8	yes	12
8.3	odd	2	inner	1664.2.b.k.833.5	✓	12
8.5	even	2	inner	1664.2.b.k.833.7	yes	12
16.3	odd	4		3328.2.a.bo.1.3		6
16.5	even	4		3328.2.a.bp.1.3		6
16.11	odd	4		3328.2.a.bp.1.4		6
16.13	even	4		3328.2.a.bo.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.k.833.5	✓	12	8.3	odd	2	inner
1664.2.b.k.833.6	yes	12	1.1	even	1	trivial
1664.2.b.k.833.7	yes	12	8.5	even	2	inner
1664.2.b.k.833.8	yes	12	4.3	odd	2	inner
3328.2.a.bo.1.3		6	16.3	odd	4
3328.2.a.bo.1.4		6	16.13	even	4
3328.2.a.bp.1.3		6	16.5	even	4
3328.2.a.bp.1.4		6	16.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.611393i q^{3} +1.62620i q^{5} -3.10261 q^{7} +2.62620 q^{9} -5.31965i q^{11} +1.00000i q^{13} +0.994247 q^{15} -1.89692 q^{17} +0.885578i q^{19} +1.89692i q^{21} +1.22279 q^{23} +2.35548 q^{25} -3.43982i q^{27} +2.00000i q^{29} -3.71400 q^{31} -3.25240 q^{33} -5.04546i q^{35} -11.1493i q^{37} +0.611393 q^{39} -5.79383 q^{41} -3.43982i q^{43} +4.27072i q^{45} +0.274184 q^{47} +2.62620 q^{49} +1.15976i q^{51} -7.79383i q^{53} +8.65080 q^{55} +0.541436 q^{57} -1.56000i q^{59} -7.04623i q^{61} -8.14807 q^{63} -1.62620 q^{65} -12.7477i q^{67} -0.747604i q^{69} +8.08505 q^{71} +16.2986 q^{73} -1.44012i q^{75} +16.5048i q^{77} +9.41650 q^{79} +5.77551 q^{81} -10.9765i q^{83} -3.08476i q^{85} +1.22279 q^{87} -5.25240 q^{89} -3.10261i q^{91} +2.27072i q^{93} -1.44012 q^{95} -8.50479 q^{97} -13.9704i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 8 * q^17 - 28 * q^25 + 32 * q^33 - 40 * q^41 - 4 * q^49 + 48 * q^57 + 16 * q^65 + 24 * q^73 - 52 * q^81 + 8 * q^89 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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