LMFDB - Embedded newform 1664.2.b.k.833.5

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			833.5
Root			$-0.760198 - 1.19252i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.k.833.8

$q$-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	$-0.881538\pi$
$5$	− 1.62620i	− 0.727258i	0.931544	−	0.363629i	$-0.118462\pi$
$6$	0	0
$7$	3.10261	1.17268	0.586338	−	0.810066i	$-0.300569\pi$
$7$	3.10261	1.17268	0.586338	+	0.810066i	$0.300569\pi$
$8$	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	$-0.540690\pi$
$23$	−1.22279	−0.254969	−0.127484	+	0.991841i	$0.540690\pi$
$24$	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	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	3.71400	0.667055	0.333527	−	0.942740i	$-0.391761\pi$
$31$	3.71400	0.667055	0.333527	+	0.942740i	$0.391761\pi$
$32$	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	$0.368972\pi$
$37$	11.1493i	1.83294i	−0.400108	+	0.916468i	$0.631028\pi$
$38$	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	$-0.506366\pi$
$47$	−0.274184	−0.0399939	−0.0199970	+	0.999800i	$0.506366\pi$
$48$	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	$0.179795\pi$
$53$	7.79383i	1.07057i	−0.844673	+	0.535283i	$0.820205\pi$
$54$	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	$0.148964\pi$
$61$	7.04623i	0.902177i	−0.892479	+	0.451089i	$0.851036\pi$
$62$	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	$-0.659276\pi$
$71$	−8.08505	−0.959519	−0.479759	+	0.877400i	$0.659276\pi$
$72$	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	$-0.677703\pi$
$79$	−9.41650	−1.05944	−0.529720	+	0.848173i	$0.677703\pi$
$80$	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	$0.193090\pi$
$101$	11.4586i	1.14017i	−0.821586	+	0.570085i	$0.806910\pi$
$102$	0	0
$103$	13.6332	1.34332	0.671661	−	0.740858i	$-0.265581\pi$
$103$	13.6332	1.34332	0.671661	+	0.740858i	$0.265581\pi$
$104$	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	$-0.753366\pi$
$109$	− 14.6079i	− 1.39918i	0.714544	−	0.699590i	$-0.246634\pi$
$110$	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	$-0.647761\pi$
$127$	−10.0909	−0.895424	−0.447712	+	0.894178i	$0.647761\pi$
$128$	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	$-0.940926\pi$
$149$	− 4.50479i	− 0.369047i	0.982828	−	0.184523i	$-0.0590742\pi$
$150$	0	0
$151$	14.5818	1.18665	0.593326	−	0.804962i	$-0.297814\pi$
$151$	14.5818	1.18665	0.593326	+	0.804962i	$0.297814\pi$
$152$	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	$0.0228047\pi$
$157$	1.79383i	0.143163i	−0.997435	+	0.0715817i	$0.977195\pi$
$158$	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	$-0.313936\pi$
$167$	14.2620	1.10363	0.551814	+	0.833967i	$0.313936\pi$
$168$	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	$0.0278496\pi$
$173$	2.29862i	0.174761i	−0.996175	+	0.0873806i	$0.972150\pi$
$174$	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	$0.0123799\pi$
$181$	1.04623i	0.0777656i	−0.999244	+	0.0388828i	$0.987620\pi$
$182$	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	$-0.601326\pi$
$191$	−8.65080	−0.625950	−0.312975	+	0.949761i	$0.601326\pi$
$192$	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	$0.184236\pi$
$197$	15.3555i	1.09403i	−0.837122	+	0.547016i	$0.815764\pi$
$198$	0	0
$199$	14.9821	1.06205	0.531025	−	0.847356i	$-0.321807\pi$
$199$	14.9821	1.06205	0.531025	+	0.847356i	$0.321807\pi$
$200$	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	$-0.712194\pi$
$223$	−18.4675	−1.23668	−0.618339	+	0.785912i	$0.712194\pi$
$224$	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	$0.190930\pi$
$229$	17.0848i	1.12899i	−0.825436	+	0.564496i	$0.809070\pi$
$230$	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	$-0.308475\pi$
$239$	17.5016	1.13208	0.566041	+	0.824377i	$0.308475\pi$
$240$	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	$-0.800349\pi$
$263$	−26.2610	−1.61932	−0.809662	+	0.586897i	$0.800349\pi$
$264$	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	$-0.678889\pi$
$269$	− 27.7572i	− 1.69239i	0.532877	−	0.846193i	$-0.321111\pi$
$270$	0	0
$271$	26.6613	1.61956	0.809778	−	0.586737i	$-0.199588\pi$
$271$	26.6613	1.61956	0.809778	+	0.586737i	$0.199588\pi$
$272$	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	$-0.856482\pi$
$277$	− 14.5048i	− 0.871509i	0.900066	−	0.435754i	$-0.143518\pi$
$278$	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	$0.418288\pi$
$293$	33.1127i	1.93446i	−0.253897	+	0.967231i	$0.581712\pi$
$294$	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	$-0.671188\pi$
$311$	−18.0673	−1.02450	−0.512251	+	0.858836i	$0.671188\pi$
$312$	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	$0.0501556\pi$
$317$	5.58767i	0.313835i	−0.987612	+	0.156917i	$0.949844\pi$
$318$	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	$-0.761178\pi$
$349$	− 25.4758i	− 1.36369i	0.731496	−	0.681845i	$-0.238822\pi$
$350$	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	$-0.224184\pi$
$359$	28.8783	1.52414	0.762069	+	0.647496i	$0.224184\pi$
$360$	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	$-0.529761\pi$
$367$	−3.57707	−0.186722	−0.0933608	+	0.995632i	$0.529761\pi$
$368$	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	$-0.819864\pi$
$373$	− 20.7110i	− 1.07237i	0.844100	−	0.536186i	$-0.180136\pi$
$374$	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	$-0.213513\pi$
$383$	30.6606	1.56669	0.783343	+	0.621590i	$0.213513\pi$
$384$	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	$0.243808\pi$
$389$	27.3449i	1.38644i	−0.720726	+	0.693220i	$0.756192\pi$
$390$	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	$-0.934907\pi$
$397$	− 8.09246i	− 0.406149i	0.979163	−	0.203074i	$-0.0650933\pi$
$398$	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	$0.0780482\pi$
$421$	9.96147i	0.485492i	−0.970090	+	0.242746i	$0.921952\pi$
$422$	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	$-0.578279\pi$
$431$	−10.1083	−0.486900	−0.243450	+	0.969913i	$0.578279\pi$
$432$	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	$-0.762034\pi$
$439$	−30.7298	−1.46666	−0.733328	+	0.679875i	$0.762034\pi$
$440$	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	$0.273191\pi$
$461$	32.4942i	1.51340i	−0.653760	+	0.756702i	$0.726809\pi$
$462$	0	0
$463$	6.06829	0.282017	0.141009	−	0.990008i	$-0.454965\pi$
$463$	6.06829	0.282017	0.141009	+	0.990008i	$0.454965\pi$
$464$	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	$-0.437773\pi$
$479$	8.50266	0.388497	0.194248	+	0.980952i	$0.437773\pi$
$480$	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	$-0.498762\pi$
$487$	0.171694	0.00778019	0.00389009	+	0.999992i	$0.498762\pi$
$488$	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	$-0.319060\pi$
$503$	24.1465	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$504$	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	$0.278126\pi$
$509$	34.5972i	1.53350i	−0.641948	+	0.766748i	$0.721874\pi$
$510$	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	$0.272651\pi$
$541$	35.1493i	1.51119i	−0.655041	+	0.755593i	$0.727349\pi$
$542$	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	$-0.936238\pi$
$557$	− 9.39212i	− 0.397957i	0.980004	−	0.198979i	$-0.0637624\pi$
$558$	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	$-0.813514\pi$
$599$	−40.7860	−1.66647	−0.833236	+	0.552918i	$0.813514\pi$
$600$	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	$-0.345054\pi$
$607$	23.0497	0.935560	0.467780	+	0.883845i	$0.345054\pi$
$608$	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	$-0.849158\pi$
$613$	− 22.5972i	− 0.912694i	0.889802	−	0.456347i	$-0.150842\pi$
$614$	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	$-0.773893\pi$
$631$	−38.0887	−1.51629	−0.758143	+	0.652089i	$0.773893\pi$
$632$	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	$-0.629850\pi$
$647$	−20.1818	−0.793430	−0.396715	+	0.917942i	$0.629850\pi$
$648$	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	$-0.976121\pi$
$653$	− 3.83048i	− 0.149898i	0.997187	−	0.0749491i	$-0.0238794\pi$
$654$	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	$-0.928233\pi$
$661$	− 11.4952i	− 0.447112i	0.974691	−	0.223556i	$-0.0717665\pi$
$662$	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	$0.120630\pi$
$677$	19.2524i	0.739930i	−0.929046	+	0.369965i	$0.879370\pi$
$678$	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	$-0.810874\pi$
$701$	− 29.6435i	− 1.11962i	0.828621	−	0.559809i	$-0.189126\pi$
$702$	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	$-0.867142\pi$
$709$	− 21.5877i	− 0.810742i	0.914152	−	0.405371i	$-0.132858\pi$
$710$	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	$-0.325314\pi$
$719$	27.9756	1.04332	0.521658	+	0.853155i	$0.325314\pi$
$720$	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	$-0.617599\pi$
$727$	−19.4727	−0.722201	−0.361101	+	0.932527i	$0.617599\pi$
$728$	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	$0.122558\pi$
$733$	20.3372i	0.751170i	−0.926788	+	0.375585i	$0.877442\pi$
$734$	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	$-0.805355\pi$
$743$	−44.6373	−1.63758	−0.818791	+	0.574091i	$0.805355\pi$
$744$	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	$-0.750630\pi$
$751$	−38.8323	−1.41701	−0.708505	+	0.705706i	$0.750630\pi$
$752$	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	$-0.892887\pi$
$757$	− 18.1695i	− 0.660383i	0.943914	−	0.330191i	$-0.107113\pi$
$758$	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	$0.0166957\pi$
$773$	2.91524i	0.104854i	−0.998625	+	0.0524269i	$0.983304\pi$
$774$	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	$0.120333\pi$
$797$	20.8401i	0.738193i	−0.929391	+	0.369096i	$0.879667\pi$
$798$	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	$0.0810220\pi$
$821$	14.4296i	0.503597i	−0.967780	+	0.251799i	$0.918978\pi$
$822$	0	0
$823$	48.7971	1.70096	0.850481	−	0.526006i	$-0.176311\pi$
$823$	48.7971	1.70096	0.850481	+	0.526006i	$0.176311\pi$
$824$	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	$0.129822\pi$
$829$	22.8401i	0.793268i	−0.917977	+	0.396634i	$0.870178\pi$
$830$	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	$-0.140865\pi$
$839$	52.3503	1.80733	0.903667	+	0.428235i	$0.140865\pi$
$840$	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	$-0.881242\pi$
$853$	− 21.2909i	− 0.728988i	0.931206	−	0.364494i	$-0.118758\pi$
$854$	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	$-0.631664\pi$
$863$	−23.6155	−0.803880	−0.401940	+	0.915666i	$0.631664\pi$
$864$	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	$-0.797578\pi$
$877$	− 35.1772i	− 1.18785i	0.804520	−	0.593925i	$-0.202422\pi$
$878$	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	$-0.484668\pi$
$887$	2.86789	0.0962944	0.0481472	+	0.998840i	$0.484668\pi$
$888$	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	$-0.455569\pi$
$911$	8.39870	0.278261	0.139131	+	0.990274i	$0.455569\pi$
$912$	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	$-0.519088\pi$
$919$	−3.63360	−0.119861	−0.0599307	+	0.998203i	$0.519088\pi$
$920$	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	$-0.947003\pi$
$941$	− 10.1676i	− 0.331455i	0.986172	−	0.165728i	$-0.0529973\pi$
$942$	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	$-0.257665\pi$
$967$	42.9056	1.37975	0.689875	+	0.723928i	$0.257665\pi$
$968$	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	$-0.428360\pi$
$983$	13.9940	0.446339	0.223170	+	0.974780i	$0.428360\pi$
$984$	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	$-0.766555\pi$
$991$	−46.7739	−1.48582	−0.742911	+	0.669390i	$0.766555\pi$
$992$	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	$0.229966\pi$
$997$	41.7572i	1.32246i	−0.750182	+	0.661232i	$0.770034\pi$
$998$	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.833.5	✓	12
4.3	odd	2	inner	1664.2.b.k.833.7	yes	12
8.3	odd	2	inner	1664.2.b.k.833.6	yes	12
8.5	even	2	inner	1664.2.b.k.833.8	yes	12
16.3	odd	4		3328.2.a.bp.1.3		6
16.5	even	4		3328.2.a.bo.1.3		6
16.11	odd	4		3328.2.a.bo.1.4		6
16.13	even	4		3328.2.a.bp.1.4		6

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