LMFDB - Embedded newform 1680.3.n.c.1471.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1471,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1471"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0,0,48,0,0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7 x^{15} + 6 x^{14} + 115 x^{13} - 535 x^{12} + 605 x^{11} + 2453 x^{10} - 10997 x^{9} + \cdots + 92164 $ x^16 - 7x^15 + 6x^14 + 115x^13 - 535x^12 + 605x^11 + 2453x^10 - 10997x^9 + 23723x^8 - 49514x^7 + 98500x^6 - 121640x^5 + 87351x^4 - 81896x^3 + 155185x^2 - 191418*x + 92164
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.13
Root			$-3.78650 - 0.542965i$ of defining polynomial
Character	$\chi$	$=$	1680.1471
Dual form			1680.3.n.c.1471.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607 q^{5} -2.64575i q^{7} -3.00000 q^{9} -20.0100i q^{11} +10.9121 q^{13} +3.87298i q^{15} -31.5852 q^{17} -0.112511i q^{19} +4.58258 q^{21} -13.6919i q^{23} +5.00000 q^{25} -5.19615i q^{27} -57.8012 q^{29} +59.4742i q^{31} +34.6584 q^{33} -5.91608i q^{35} -50.3894 q^{37} +18.9003i q^{39} +35.7441 q^{41} +60.1583i q^{43} -6.70820 q^{45} +21.4131i q^{47} -7.00000 q^{49} -54.7072i q^{51} -30.4859 q^{53} -44.7438i q^{55} +0.194874 q^{57} +91.0394i q^{59} -66.0447 q^{61} +7.93725i q^{63} +24.4002 q^{65} -30.7105i q^{67} +23.7151 q^{69} +67.8742i q^{71} -0.812277 q^{73} +8.66025i q^{75} -52.9415 q^{77} +70.9483i q^{79} +9.00000 q^{81} +67.1123i q^{83} -70.6267 q^{85} -100.115i q^{87} -128.939 q^{89} -28.8707i q^{91} -103.012 q^{93} -0.251581i q^{95} +74.2913 q^{97} +60.0300i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) $ 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 20.0100i	− 1.81909i	−0.415603	−	0.909546i	$-0.636429\pi$
$11$	− 20.0100i	− 1.81909i	0.415603	−	0.909546i	$-0.363571\pi$
$12$	0	0
$13$	10.9121	0.839391	0.419695	−	0.907665i	$-0.362137\pi$
$13$	10.9121	0.839391	0.419695	+	0.907665i	$0.362137\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	−31.5852	−1.85795	−0.928977	−	0.370139i	$-0.879310\pi$
$17$	−31.5852	−1.85795	−0.928977	+	0.370139i	$0.879310\pi$
$18$	0	0
$19$	− 0.112511i	− 0.00592161i	−0.999996	−	0.00296081i	$-0.999058\pi$
$19$	− 0.112511i	− 0.00592161i	0.999996	−	0.00296081i	$-0.000942455\pi$
$20$	0	0
$21$	4.58258	0.218218
$22$	0	0
$23$	− 13.6919i	− 0.595302i	−0.954675	−	0.297651i	$-0.903797\pi$
$23$	− 13.6919i	− 0.595302i	0.954675	−	0.297651i	$-0.0962031\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−57.8012	−1.99314	−0.996572	−	0.0827280i	$-0.973637\pi$
$29$	−57.8012	−1.99314	−0.996572	+	0.0827280i	$0.973637\pi$
$30$	0	0
$31$	59.4742i	1.91852i	0.282519	+	0.959262i	$0.408830\pi$
$31$	59.4742i	1.91852i	−0.282519	+	0.959262i	$0.591170\pi$
$32$	0	0
$33$	34.6584	1.05025
$34$	0	0
$35$	− 5.91608i	− 0.169031i
$36$	0	0
$37$	−50.3894	−1.36188	−0.680938	−	0.732341i	$-0.738428\pi$
$37$	−50.3894	−1.36188	−0.680938	+	0.732341i	$0.738428\pi$
$38$	0	0
$39$	18.9003i	0.484623i
$40$	0	0
$41$	35.7441	0.871807	0.435904	−	0.899993i	$-0.356429\pi$
$41$	35.7441	0.871807	0.435904	+	0.899993i	$0.356429\pi$
$42$	0	0
$43$	60.1583i	1.39903i	0.714617	+	0.699515i	$0.246601\pi$
$43$	60.1583i	1.39903i	−0.714617	+	0.699515i	$0.753399\pi$
$44$	0	0
$45$	−6.70820	−0.149071
$46$	0	0
$47$	21.4131i	0.455597i	0.973708	+	0.227799i	$0.0731528\pi$
$47$	21.4131i	0.455597i	−0.973708	+	0.227799i	$0.926847\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 54.7072i	− 1.07269i
$52$	0	0
$53$	−30.4859	−0.575206	−0.287603	−	0.957750i	$-0.592858\pi$
$53$	−30.4859	−0.575206	−0.287603	+	0.957750i	$0.592858\pi$
$54$	0	0
$55$	− 44.7438i	− 0.813523i
$56$	0	0
$57$	0.194874	0.00341884
$58$	0	0
$59$	91.0394i	1.54304i	0.636205	+	0.771520i	$0.280503\pi$
$59$	91.0394i	1.54304i	−0.636205	+	0.771520i	$0.719497\pi$
$60$	0	0
$61$	−66.0447	−1.08270	−0.541350	−	0.840797i	$-0.682087\pi$
$61$	−66.0447	−1.08270	−0.541350	+	0.840797i	$0.682087\pi$
$62$	0	0
$63$	7.93725i	0.125988i
$64$	0	0
$65$	24.4002	0.375387
$66$	0	0
$67$	− 30.7105i	− 0.458366i	−0.973383	−	0.229183i	$-0.926395\pi$
$67$	− 30.7105i	− 0.458366i	0.973383	−	0.229183i	$-0.0736055\pi$
$68$	0	0
$69$	23.7151	0.343698
$70$	0	0
$71$	67.8742i	0.955975i	0.878367	+	0.477988i	$0.158634\pi$
$71$	67.8742i	0.955975i	−0.878367	+	0.477988i	$0.841366\pi$
$72$	0	0
$73$	−0.812277	−0.0111271	−0.00556354	−	0.999985i	$-0.501771\pi$
$73$	−0.812277	−0.0111271	−0.00556354	+	0.999985i	$0.501771\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	−52.9415	−0.687552
$78$	0	0
$79$	70.9483i	0.898080i	0.893512	+	0.449040i	$0.148234\pi$
$79$	70.9483i	0.898080i	−0.893512	+	0.449040i	$0.851766\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	67.1123i	0.808582i	0.914630	+	0.404291i	$0.132482\pi$
$83$	67.1123i	0.808582i	−0.914630	+	0.404291i	$0.867518\pi$
$84$	0	0
$85$	−70.6267	−0.830902
$86$	0	0
$87$	− 100.115i	− 1.15074i
$88$	0	0
$89$	−128.939	−1.44876	−0.724378	−	0.689403i	$-0.757873\pi$
$89$	−128.939	−1.44876	−0.724378	+	0.689403i	$0.757873\pi$
$90$	0	0
$91$	− 28.8707i	− 0.317260i
$92$	0	0
$93$	−103.012	−1.10766
$94$	0	0
$95$	− 0.251581i	− 0.00264823i
$96$	0	0
$97$	74.2913	0.765890	0.382945	−	0.923771i	$-0.374910\pi$
$97$	74.2913	0.765890	0.382945	+	0.923771i	$0.374910\pi$
$98$	0	0
$99$	60.0300i	0.606364i
$100$	0	0
$101$	99.3982	0.984140	0.492070	−	0.870556i	$-0.336240\pi$
$101$	99.3982	0.984140	0.492070	+	0.870556i	$0.336240\pi$
$102$	0	0
$103$	− 110.380i	− 1.07165i	−0.844330	−	0.535824i	$-0.820001\pi$
$103$	− 110.380i	− 1.07165i	0.844330	−	0.535824i	$-0.179999\pi$
$104$	0	0
$105$	10.2470	0.0975900
$106$	0	0
$107$	− 118.521i	− 1.10767i	−0.832625	−	0.553837i	$-0.813163\pi$
$107$	− 118.521i	− 1.10767i	0.832625	−	0.553837i	$-0.186837\pi$
$108$	0	0
$109$	97.8209	0.897440	0.448720	−	0.893673i	$-0.351880\pi$
$109$	97.8209	0.897440	0.448720	+	0.893673i	$0.351880\pi$
$110$	0	0
$111$	− 87.2770i	− 0.786279i
$112$	0	0
$113$	−114.146	−1.01014	−0.505069	−	0.863079i	$-0.668533\pi$
$113$	−114.146	−1.01014	−0.505069	+	0.863079i	$0.668533\pi$
$114$	0	0
$115$	− 30.6161i	− 0.266227i
$116$	0	0
$117$	−32.7362	−0.279797
$118$	0	0
$119$	83.5666i	0.702240i
$120$	0	0
$121$	−279.401	−2.30910
$122$	0	0
$123$	61.9106i	0.503338i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	− 95.0784i	− 0.748649i	−0.927298	−	0.374324i	$-0.877875\pi$
$127$	− 95.0784i	− 0.748649i	0.927298	−	0.374324i	$-0.122125\pi$
$128$	0	0
$129$	−104.197	−0.807731
$130$	0	0
$131$	− 180.343i	− 1.37667i	−0.725395	−	0.688333i	$-0.758343\pi$
$131$	− 180.343i	− 1.37667i	0.725395	−	0.688333i	$-0.241657\pi$
$132$	0	0
$133$	−0.297675	−0.00223816
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	−120.582	−0.880160	−0.440080	−	0.897959i	$-0.645050\pi$
$137$	−120.582	−0.880160	−0.440080	+	0.897959i	$0.645050\pi$
$138$	0	0
$139$	113.931i	0.819648i	0.912165	+	0.409824i	$0.134410\pi$
$139$	113.931i	0.819648i	−0.912165	+	0.409824i	$0.865590\pi$
$140$	0	0
$141$	−37.0885	−0.263039
$142$	0	0
$143$	− 218.351i	− 1.52693i
$144$	0	0
$145$	−129.247	−0.891361
$146$	0	0
$147$	− 12.1244i	− 0.0824786i
$148$	0	0
$149$	266.069	1.78570	0.892848	−	0.450359i	$-0.148704\pi$
$149$	266.069	1.78570	0.892848	+	0.450359i	$0.148704\pi$
$150$	0	0
$151$	164.754i	1.09109i	0.838083	+	0.545543i	$0.183677\pi$
$151$	164.754i	1.09109i	−0.838083	+	0.545543i	$0.816323\pi$
$152$	0	0
$153$	94.7556	0.619318
$154$	0	0
$155$	132.988i	0.857990i
$156$	0	0
$157$	−65.3640	−0.416331	−0.208166	−	0.978094i	$-0.566749\pi$
$157$	−65.3640	−0.416331	−0.208166	+	0.978094i	$0.566749\pi$
$158$	0	0
$159$	− 52.8031i	− 0.332095i
$160$	0	0
$161$	−36.2255	−0.225003
$162$	0	0
$163$	− 100.406i	− 0.615988i	−0.951388	−	0.307994i	$-0.900342\pi$
$163$	− 100.406i	− 0.615988i	0.951388	−	0.307994i	$-0.0996576\pi$
$164$	0	0
$165$	77.4985	0.469688
$166$	0	0
$167$	185.759i	1.11233i	0.831071	+	0.556166i	$0.187728\pi$
$167$	185.759i	1.11233i	−0.831071	+	0.556166i	$0.812272\pi$
$168$	0	0
$169$	−49.9265	−0.295423
$170$	0	0
$171$	0.337532i	0.00197387i
$172$	0	0
$173$	244.698	1.41444	0.707219	−	0.706995i	$-0.249950\pi$
$173$	244.698	1.41444	0.707219	+	0.706995i	$0.249950\pi$
$174$	0	0
$175$	− 13.2288i	− 0.0755929i
$176$	0	0
$177$	−157.685	−0.890875
$178$	0	0
$179$	− 258.423i	− 1.44370i	−0.692049	−	0.721851i	$-0.743292\pi$
$179$	− 258.423i	− 1.44370i	0.692049	−	0.721851i	$-0.256708\pi$
$180$	0	0
$181$	−174.426	−0.963680	−0.481840	−	0.876259i	$-0.660031\pi$
$181$	−174.426	−0.963680	−0.481840	+	0.876259i	$0.660031\pi$
$182$	0	0
$183$	− 114.393i	− 0.625097i
$184$	0	0
$185$	−112.674	−0.609049
$186$	0	0
$187$	632.020i	3.37979i
$188$	0	0
$189$	−13.7477	−0.0727393
$190$	0	0
$191$	− 214.311i	− 1.12205i	−0.827799	−	0.561025i	$-0.810407\pi$
$191$	− 214.311i	− 1.12205i	0.827799	−	0.561025i	$-0.189593\pi$
$192$	0	0
$193$	11.8636	0.0614693	0.0307346	−	0.999528i	$-0.490215\pi$
$193$	11.8636	0.0614693	0.0307346	+	0.999528i	$0.490215\pi$
$194$	0	0
$195$	42.2623i	0.216730i
$196$	0	0
$197$	−158.345	−0.803780	−0.401890	−	0.915688i	$-0.631647\pi$
$197$	−158.345	−0.803780	−0.401890	+	0.915688i	$0.631647\pi$
$198$	0	0
$199$	− 26.9522i	− 0.135438i	−0.997704	−	0.0677192i	$-0.978428\pi$
$199$	− 26.9522i	− 0.135438i	0.997704	−	0.0677192i	$-0.0215722\pi$
$200$	0	0
$201$	53.1922	0.264638
$202$	0	0
$203$	152.928i	0.753338i
$204$	0	0
$205$	79.9262	0.389884
$206$	0	0
$207$	41.0758i	0.198434i
$208$	0	0
$209$	−2.25134	−0.0107720
$210$	0	0
$211$	2.10546i	0.00997850i	0.999988	+	0.00498925i	$0.00158813\pi$
$211$	2.10546i	0.00997850i	−0.999988	+	0.00498925i	$0.998412\pi$
$212$	0	0
$213$	−117.562	−0.551933
$214$	0	0
$215$	134.518i	0.625666i
$216$	0	0
$217$	157.354	0.725134
$218$	0	0
$219$	− 1.40691i	− 0.00642423i
$220$	0	0
$221$	−344.660	−1.55955
$222$	0	0
$223$	− 209.024i	− 0.937326i	−0.883377	−	0.468663i	$-0.844736\pi$
$223$	− 209.024i	− 0.937326i	0.883377	−	0.468663i	$-0.155264\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 60.4048i	− 0.266101i	−0.991109	−	0.133050i	$-0.957523\pi$
$227$	− 60.4048i	− 0.266101i	0.991109	−	0.133050i	$-0.0424772\pi$
$228$	0	0
$229$	−178.853	−0.781020	−0.390510	−	0.920599i	$-0.627701\pi$
$229$	−178.853	−0.781020	−0.390510	+	0.920599i	$0.627701\pi$
$230$	0	0
$231$	− 91.6974i	− 0.396958i
$232$	0	0
$233$	−276.663	−1.18739	−0.593697	−	0.804688i	$-0.702332\pi$
$233$	−276.663	−1.18739	−0.593697	+	0.804688i	$0.702332\pi$
$234$	0	0
$235$	47.8811i	0.203749i
$236$	0	0
$237$	−122.886	−0.518507
$238$	0	0
$239$	− 450.512i	− 1.88499i	−0.334225	−	0.942493i	$-0.608475\pi$
$239$	− 450.512i	− 1.88499i	0.334225	−	0.942493i	$-0.391525\pi$
$240$	0	0
$241$	−203.628	−0.844931	−0.422465	−	0.906379i	$-0.638835\pi$
$241$	−203.628	−0.844931	−0.422465	+	0.906379i	$0.638835\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−15.6525	−0.0638877
$246$	0	0
$247$	− 1.22773i	− 0.00497055i
$248$	0	0
$249$	−116.242	−0.466835
$250$	0	0
$251$	− 22.9426i	− 0.0914047i	−0.998955	−	0.0457023i	$-0.985447\pi$
$251$	− 22.9426i	− 0.0914047i	0.998955	−	0.0457023i	$-0.0145526\pi$
$252$	0	0
$253$	−273.976	−1.08291
$254$	0	0
$255$	− 122.329i	− 0.479721i
$256$	0	0
$257$	−244.168	−0.950070	−0.475035	−	0.879967i	$-0.657565\pi$
$257$	−244.168	−0.950070	−0.475035	+	0.879967i	$0.657565\pi$
$258$	0	0
$259$	133.318i	0.514741i
$260$	0	0
$261$	173.404	0.664381
$262$	0	0
$263$	− 247.769i	− 0.942087i	−0.882110	−	0.471043i	$-0.843877\pi$
$263$	− 247.769i	− 0.942087i	0.882110	−	0.471043i	$-0.156123\pi$
$264$	0	0
$265$	−68.1686	−0.257240
$266$	0	0
$267$	− 223.329i	− 0.836440i
$268$	0	0
$269$	145.210	0.539813	0.269907	−	0.962886i	$-0.413007\pi$
$269$	145.210	0.539813	0.269907	+	0.962886i	$0.413007\pi$
$270$	0	0
$271$	4.19355i	0.0154744i	0.999970	+	0.00773718i	$0.00246285\pi$
$271$	4.19355i	0.0154744i	−0.999970	+	0.00773718i	$0.997537\pi$
$272$	0	0
$273$	50.0054	0.183170
$274$	0	0
$275$	− 100.050i	− 0.363818i
$276$	0	0
$277$	−181.335	−0.654637	−0.327319	−	0.944914i	$-0.606145\pi$
$277$	−181.335	−0.654637	−0.327319	+	0.944914i	$0.606145\pi$
$278$	0	0
$279$	− 178.423i	− 0.639508i
$280$	0	0
$281$	−101.242	−0.360290	−0.180145	−	0.983640i	$-0.557657\pi$
$281$	−101.242	−0.360290	−0.180145	+	0.983640i	$0.557657\pi$
$282$	0	0
$283$	84.8156i	0.299702i	0.988709	+	0.149851i	$0.0478793\pi$
$283$	84.8156i	0.299702i	−0.988709	+	0.149851i	$0.952121\pi$
$284$	0	0
$285$	0.435752	0.00152895
$286$	0	0
$287$	− 94.5700i	− 0.329512i
$288$	0	0
$289$	708.625	2.45199
$290$	0	0
$291$	128.676i	0.442187i
$292$	0	0
$293$	−365.052	−1.24591	−0.622955	−	0.782258i	$-0.714068\pi$
$293$	−365.052	−1.24591	−0.622955	+	0.782258i	$0.714068\pi$
$294$	0	0
$295$	203.570i	0.690068i
$296$	0	0
$297$	−103.975	−0.350084
$298$	0	0
$299$	− 149.408i	− 0.499691i
$300$	0	0
$301$	159.164	0.528784
$302$	0	0
$303$	172.163i	0.568194i
$304$	0	0
$305$	−147.680	−0.484198
$306$	0	0
$307$	161.988i	0.527649i	0.964571	+	0.263825i	$0.0849840\pi$
$307$	161.988i	0.527649i	−0.964571	+	0.263825i	$0.915016\pi$
$308$	0	0
$309$	191.183	0.618716
$310$	0	0
$311$	142.246i	0.457384i	0.973499	+	0.228692i	$0.0734448\pi$
$311$	142.246i	0.457384i	−0.973499	+	0.228692i	$0.926555\pi$
$312$	0	0
$313$	500.375	1.59864	0.799321	−	0.600905i	$-0.205193\pi$
$313$	500.375	1.59864	0.799321	+	0.600905i	$0.205193\pi$
$314$	0	0
$315$	17.7482i	0.0563436i
$316$	0	0
$317$	178.879	0.564286	0.282143	−	0.959372i	$-0.408955\pi$
$317$	178.879	0.564286	0.282143	+	0.959372i	$0.408955\pi$
$318$	0	0
$319$	1156.60i	3.62571i
$320$	0	0
$321$	205.285	0.639516
$322$	0	0
$323$	3.55367i	0.0110021i
$324$	0	0
$325$	54.5604	0.167878
$326$	0	0
$327$	169.431i	0.518137i
$328$	0	0
$329$	56.6537	0.172200
$330$	0	0
$331$	0.490839i	0.00148290i	1.00000		0.000741449i	$0.000236010\pi$
$331$	0.490839i	0.00148290i	−1.00000		0.000741449i	$0.999764\pi$
$332$	0	0
$333$	151.168	0.453959
$334$	0	0
$335$	− 68.6709i	− 0.204988i
$336$	0	0
$337$	−447.909	−1.32911	−0.664553	−	0.747241i	$-0.731378\pi$
$337$	−447.909	−1.32911	−0.664553	+	0.747241i	$0.731378\pi$
$338$	0	0
$339$	− 197.706i	− 0.583204i
$340$	0	0
$341$	1190.08	3.48997
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	53.0286	0.153706
$346$	0	0
$347$	− 397.388i	− 1.14521i	−0.819831	−	0.572606i	$-0.805933\pi$
$347$	− 397.388i	− 1.14521i	0.819831	−	0.572606i	$-0.194067\pi$
$348$	0	0
$349$	649.279	1.86040	0.930200	−	0.367054i	$-0.119634\pi$
$349$	649.279	1.86040	0.930200	+	0.367054i	$0.119634\pi$
$350$	0	0
$351$	− 56.7008i	− 0.161541i
$352$	0	0
$353$	9.28745	0.0263101	0.0131550	−	0.999913i	$-0.495813\pi$
$353$	9.28745	0.0263101	0.0131550	+	0.999913i	$0.495813\pi$
$354$	0	0
$355$	151.771i	0.427525i
$356$	0	0
$357$	−144.742	−0.405439
$358$	0	0
$359$	317.216i	0.883609i	0.897111	+	0.441805i	$0.145662\pi$
$359$	317.216i	0.883609i	−0.897111	+	0.441805i	$0.854338\pi$
$360$	0	0
$361$	360.987	0.999965
$362$	0	0
$363$	− 483.936i	− 1.33316i
$364$	0	0
$365$	−1.81631	−0.00497618
$366$	0	0
$367$	− 446.161i	− 1.21570i	−0.794053	−	0.607849i	$-0.792033\pi$
$367$	− 446.161i	− 1.21570i	0.794053	−	0.607849i	$-0.207967\pi$
$368$	0	0
$369$	−107.232	−0.290602
$370$	0	0
$371$	80.6581i	0.217407i
$372$	0	0
$373$	114.126	0.305969	0.152985	−	0.988229i	$-0.451112\pi$
$373$	114.126	0.305969	0.152985	+	0.988229i	$0.451112\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−630.731	−1.67303
$378$	0	0
$379$	228.631i	0.603249i	0.953427	+	0.301624i	$0.0975289\pi$
$379$	228.631i	0.603249i	−0.953427	+	0.301624i	$0.902471\pi$
$380$	0	0
$381$	164.681	0.432233
$382$	0	0
$383$	− 373.966i	− 0.976414i	−0.872728	−	0.488207i	$-0.837651\pi$
$383$	− 373.966i	− 0.976414i	0.872728	−	0.488207i	$-0.162349\pi$
$384$	0	0
$385$	−118.381	−0.307483
$386$	0	0
$387$	− 180.475i	− 0.466344i
$388$	0	0
$389$	−506.048	−1.30090	−0.650448	−	0.759551i	$-0.725419\pi$
$389$	−506.048	−1.30090	−0.650448	+	0.759551i	$0.725419\pi$
$390$	0	0
$391$	432.463i	1.10604i
$392$	0	0
$393$	312.364	0.794818
$394$	0	0
$395$	158.645i	0.401634i
$396$	0	0
$397$	314.380	0.791888	0.395944	−	0.918275i	$-0.370417\pi$
$397$	314.380	0.791888	0.395944	+	0.918275i	$0.370417\pi$
$398$	0	0
$399$	− 0.515588i	− 0.00129220i
$400$	0	0
$401$	−67.3041	−0.167841	−0.0839203	−	0.996472i	$-0.526744\pi$
$401$	−67.3041	−0.167841	−0.0839203	+	0.996472i	$0.526744\pi$
$402$	0	0
$403$	648.988i	1.61039i
$404$	0	0
$405$	20.1246	0.0496904
$406$	0	0
$407$	1008.29i	2.47738i
$408$	0	0
$409$	−703.321	−1.71961	−0.859805	−	0.510622i	$-0.829415\pi$
$409$	−703.321	−1.71961	−0.859805	+	0.510622i	$0.829415\pi$
$410$	0	0
$411$	− 208.854i	− 0.508161i
$412$	0	0
$413$	240.868	0.583214
$414$	0	0
$415$	150.068i	0.361609i
$416$	0	0
$417$	−197.334	−0.473224
$418$	0	0
$419$	396.598i	0.946535i	0.880919	+	0.473267i	$0.156926\pi$
$419$	396.598i	0.946535i	−0.880919	+	0.473267i	$0.843074\pi$
$420$	0	0
$421$	512.979	1.21848	0.609239	−	0.792987i	$-0.291475\pi$
$421$	512.979	1.21848	0.609239	+	0.792987i	$0.291475\pi$
$422$	0	0
$423$	− 64.2392i	− 0.151866i
$424$	0	0
$425$	−157.926	−0.371591
$426$	0	0
$427$	174.738i	0.409222i
$428$	0	0
$429$	378.195	0.881573
$430$	0	0
$431$	492.946i	1.14373i	0.820349	+	0.571863i	$0.193779\pi$
$431$	492.946i	1.14373i	−0.820349	+	0.571863i	$0.806221\pi$
$432$	0	0
$433$	−664.110	−1.53374	−0.766871	−	0.641801i	$-0.778187\pi$
$433$	−664.110	−1.53374	−0.766871	+	0.641801i	$0.778187\pi$
$434$	0	0
$435$	− 223.863i	− 0.514628i
$436$	0	0
$437$	−1.54049	−0.00352515
$438$	0	0
$439$	− 775.746i	− 1.76707i	−0.468361	−	0.883537i	$-0.655155\pi$
$439$	− 775.746i	− 1.76707i	0.468361	−	0.883537i	$-0.344845\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	− 110.881i	− 0.250295i	−0.992138	−	0.125148i	$-0.960060\pi$
$443$	− 110.881i	− 0.250295i	0.992138	−	0.125148i	$-0.0399405\pi$
$444$	0	0
$445$	−288.317	−0.647903
$446$	0	0
$447$	460.844i	1.03097i
$448$	0	0
$449$	786.476	1.75162	0.875808	−	0.482659i	$-0.160329\pi$
$449$	786.476	1.75162	0.875808	+	0.482659i	$0.160329\pi$
$450$	0	0
$451$	− 715.240i	− 1.58590i
$452$	0	0
$453$	−285.362	−0.629939
$454$	0	0
$455$	− 64.5567i	− 0.141883i
$456$	0	0
$457$	−539.266	−1.18001	−0.590007	−	0.807398i	$-0.700875\pi$
$457$	−539.266	−1.18001	−0.590007	+	0.807398i	$0.700875\pi$
$458$	0	0
$459$	164.122i	0.357563i
$460$	0	0
$461$	162.303	0.352067	0.176033	−	0.984384i	$-0.443673\pi$
$461$	162.303	0.352067	0.176033	+	0.984384i	$0.443673\pi$
$462$	0	0
$463$	813.003i	1.75595i	0.478710	+	0.877973i	$0.341104\pi$
$463$	813.003i	1.75595i	−0.478710	+	0.877973i	$0.658896\pi$
$464$	0	0
$465$	−230.343	−0.495361
$466$	0	0
$467$	− 922.032i	− 1.97437i	−0.159569	−	0.987187i	$-0.551010\pi$
$467$	− 922.032i	− 1.97437i	0.159569	−	0.987187i	$-0.448990\pi$
$468$	0	0
$469$	−81.2525	−0.173246
$470$	0	0
$471$	− 113.214i	− 0.240369i
$472$	0	0
$473$	1203.77	2.54497
$474$	0	0
$475$	− 0.562553i	− 0.00118432i
$476$	0	0
$477$	91.4577	0.191735
$478$	0	0
$479$	− 232.951i	− 0.486328i	−0.969985	−	0.243164i	$-0.921815\pi$
$479$	− 232.951i	− 0.486328i	0.969985	−	0.243164i	$-0.0781854\pi$
$480$	0	0
$481$	−549.853	−1.14315
$482$	0	0
$483$	− 62.7443i	− 0.129905i
$484$	0	0
$485$	166.120	0.342516
$486$	0	0
$487$	− 313.380i	− 0.643491i	−0.946826	−	0.321745i	$-0.895731\pi$
$487$	− 313.380i	− 0.643491i	0.946826	−	0.321745i	$-0.104269\pi$
$488$	0	0
$489$	173.908	0.355641
$490$	0	0
$491$	237.064i	0.482819i	0.970423	+	0.241410i	$0.0776098\pi$
$491$	237.064i	0.482819i	−0.970423	+	0.241410i	$0.922390\pi$
$492$	0	0
$493$	1825.66	3.70317
$494$	0	0
$495$	134.231i	0.271174i
$496$	0	0
$497$	179.578	0.361325
$498$	0	0
$499$	− 393.596i	− 0.788770i	−0.918945	−	0.394385i	$-0.870958\pi$
$499$	− 393.596i	− 0.788770i	0.918945	−	0.394385i	$-0.129042\pi$
$500$	0	0
$501$	−321.745	−0.642205
$502$	0	0
$503$	− 571.089i	− 1.13537i	−0.823248	−	0.567683i	$-0.807840\pi$
$503$	− 571.089i	− 1.13537i	0.823248	−	0.567683i	$-0.192160\pi$
$504$	0	0
$505$	222.261	0.440121
$506$	0	0
$507$	− 86.4752i	− 0.170563i
$508$	0	0
$509$	269.146	0.528774	0.264387	−	0.964417i	$-0.414830\pi$
$509$	269.146	0.528774	0.264387	+	0.964417i	$0.414830\pi$
$510$	0	0
$511$	2.14908i	0.00420564i
$512$	0	0
$513$	−0.584622	−0.00113961
$514$	0	0
$515$	− 246.816i	− 0.479255i
$516$	0	0
$517$	428.476	0.828774
$518$	0	0
$519$	423.829i	0.816626i
$520$	0	0
$521$	−212.098	−0.407098	−0.203549	−	0.979065i	$-0.565248\pi$
$521$	−212.098	−0.407098	−0.203549	+	0.979065i	$0.565248\pi$
$522$	0	0
$523$	441.249i	0.843688i	0.906668	+	0.421844i	$0.138617\pi$
$523$	441.249i	0.843688i	−0.906668	+	0.421844i	$0.861383\pi$
$524$	0	0
$525$	22.9129	0.0436436
$526$	0	0
$527$	− 1878.51i	− 3.56453i
$528$	0	0
$529$	341.531	0.645616
$530$	0	0
$531$	− 273.118i	− 0.514347i
$532$	0	0
$533$	390.042	0.731787
$534$	0	0
$535$	− 265.021i	− 0.495367i
$536$	0	0
$537$	447.601	0.833522
$538$	0	0
$539$	140.070i	0.259870i
$540$	0	0
$541$	−197.182	−0.364478	−0.182239	−	0.983254i	$-0.558334\pi$
$541$	−197.182	−0.364478	−0.182239	+	0.983254i	$0.558334\pi$
$542$	0	0
$543$	− 302.115i	− 0.556381i
$544$	0	0
$545$	218.734	0.401347
$546$	0	0
$547$	1026.06i	1.87580i	0.346908	+	0.937899i	$0.387232\pi$
$547$	1026.06i	1.87580i	−0.346908	+	0.937899i	$0.612768\pi$
$548$	0	0
$549$	198.134	0.360900
$550$	0	0
$551$	6.50325i	0.0118026i
$552$	0	0
$553$	187.712	0.339442
$554$	0	0
$555$	− 195.157i	− 0.351635i
$556$	0	0
$557$	314.481	0.564598	0.282299	−	0.959326i	$-0.408903\pi$
$557$	314.481	0.564598	0.282299	+	0.959326i	$0.408903\pi$
$558$	0	0
$559$	656.453i	1.17433i
$560$	0	0
$561$	−1094.69	−1.95132
$562$	0	0
$563$	− 552.861i	− 0.981991i	−0.871162	−	0.490996i	$-0.836633\pi$
$563$	− 552.861i	− 0.981991i	0.871162	−	0.490996i	$-0.163367\pi$
$564$	0	0
$565$	−255.237	−0.451748
$566$	0	0
$567$	− 23.8118i	− 0.0419961i
$568$	0	0
$569$	948.437	1.66685	0.833425	−	0.552633i	$-0.186377\pi$
$569$	948.437	1.66685	0.833425	+	0.552633i	$0.186377\pi$
$570$	0	0
$571$	404.335i	0.708118i	0.935223	+	0.354059i	$0.115199\pi$
$571$	404.335i	0.708118i	−0.935223	+	0.354059i	$0.884801\pi$
$572$	0	0
$573$	371.198	0.647816
$574$	0	0
$575$	− 68.4597i	− 0.119060i
$576$	0	0
$577$	−859.682	−1.48992	−0.744959	−	0.667111i	$-0.767531\pi$
$577$	−859.682	−1.48992	−0.744959	+	0.667111i	$0.767531\pi$
$578$	0	0
$579$	20.5483i	0.0354893i
$580$	0	0
$581$	177.562	0.305615
$582$	0	0
$583$	610.024i	1.04635i
$584$	0	0
$585$	−73.2005	−0.125129
$586$	0	0
$587$	− 142.567i	− 0.242874i	−0.992599	−	0.121437i	$-0.961250\pi$
$587$	− 142.567i	− 0.242874i	0.992599	−	0.121437i	$-0.0387502\pi$
$588$	0	0
$589$	6.69148	0.0113608
$590$	0	0
$591$	− 274.261i	− 0.464062i
$592$	0	0
$593$	−453.633	−0.764980	−0.382490	−	0.923960i	$-0.624933\pi$
$593$	−453.633	−0.764980	−0.382490	+	0.923960i	$0.624933\pi$
$594$	0	0
$595$	186.861i	0.314051i
$596$	0	0
$597$	46.6826	0.0781954
$598$	0	0
$599$	− 37.1329i	− 0.0619915i	−0.999520	−	0.0309957i	$-0.990132\pi$
$599$	− 37.1329i	− 0.0619915i	0.999520	−	0.0309957i	$-0.00986783\pi$
$600$	0	0
$601$	182.536	0.303721	0.151861	−	0.988402i	$-0.451474\pi$
$601$	182.536	0.303721	0.151861	+	0.988402i	$0.451474\pi$
$602$	0	0
$603$	92.1316i	0.152789i
$604$	0	0
$605$	−624.759	−1.03266
$606$	0	0
$607$	− 374.869i	− 0.617576i	−0.951131	−	0.308788i	$-0.900077\pi$
$607$	− 374.869i	− 0.617576i	0.951131	−	0.308788i	$-0.0999234\pi$
$608$	0	0
$609$	−264.878	−0.434940
$610$	0	0
$611$	233.661i	0.382424i
$612$	0	0
$613$	419.852	0.684913	0.342456	−	0.939534i	$-0.388741\pi$
$613$	419.852	0.684913	0.342456	+	0.939534i	$0.388741\pi$
$614$	0	0
$615$	138.436i	0.225100i
$616$	0	0
$617$	−212.859	−0.344990	−0.172495	−	0.985010i	$-0.555183\pi$
$617$	−212.859	−0.344990	−0.172495	+	0.985010i	$0.555183\pi$
$618$	0	0
$619$	307.647i	0.497007i	0.968631	+	0.248504i	$0.0799388\pi$
$619$	307.647i	0.497007i	−0.968631	+	0.248504i	$0.920061\pi$
$620$	0	0
$621$	−71.1454	−0.114566
$622$	0	0
$623$	341.141i	0.547578i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 3.89943i	− 0.00621919i
$628$	0	0
$629$	1591.56	2.53030
$630$	0	0
$631$	− 194.008i	− 0.307461i	−0.988113	−	0.153731i	$-0.950871\pi$
$631$	− 194.008i	− 0.307461i	0.988113	−	0.153731i	$-0.0491288\pi$
$632$	0	0
$633$	−3.64677	−0.00576109
$634$	0	0
$635$	− 212.602i	− 0.334806i
$636$	0	0
$637$	−76.3846	−0.119913
$638$	0	0
$639$	− 203.623i	− 0.318658i
$640$	0	0
$641$	420.553	0.656089	0.328045	−	0.944662i	$-0.393610\pi$
$641$	420.553	0.656089	0.328045	+	0.944662i	$0.393610\pi$
$642$	0	0
$643$	544.105i	0.846197i	0.906084	+	0.423099i	$0.139058\pi$
$643$	544.105i	0.846197i	−0.906084	+	0.423099i	$0.860942\pi$
$644$	0	0
$645$	−232.992	−0.361228
$646$	0	0
$647$	− 109.115i	− 0.168648i	−0.996438	−	0.0843239i	$-0.973127\pi$
$647$	− 109.115i	− 0.168648i	0.996438	−	0.0843239i	$-0.0268730\pi$
$648$	0	0
$649$	1821.70	2.80693
$650$	0	0
$651$	272.545i	0.418656i
$652$	0	0
$653$	−84.5159	−0.129427	−0.0647135	−	0.997904i	$-0.520613\pi$
$653$	−84.5159	−0.129427	−0.0647135	+	0.997904i	$0.520613\pi$
$654$	0	0
$655$	− 403.260i	− 0.615664i
$656$	0	0
$657$	2.43683	0.00370903
$658$	0	0
$659$	− 68.3917i	− 0.103781i	−0.998653	−	0.0518905i	$-0.983475\pi$
$659$	− 68.3917i	− 0.103781i	0.998653	−	0.0518905i	$-0.0165247\pi$
$660$	0	0
$661$	925.213	1.39972	0.699858	−	0.714282i	$-0.253247\pi$
$661$	925.213	1.39972	0.699858	+	0.714282i	$0.253247\pi$
$662$	0	0
$663$	− 596.969i	− 0.900406i
$664$	0	0
$665$	−0.665622	−0.00100094
$666$	0	0
$667$	791.410i	1.18652i
$668$	0	0
$669$	362.040	0.541166
$670$	0	0
$671$	1321.56i	1.96953i
$672$	0	0
$673$	−1135.50	−1.68723	−0.843614	−	0.536951i	$-0.819576\pi$
$673$	−1135.50	−1.68723	−0.843614	+	0.536951i	$0.819576\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	−750.183	−1.10810	−0.554050	−	0.832484i	$-0.686918\pi$
$677$	−750.183	−1.10810	−0.554050	+	0.832484i	$0.686918\pi$
$678$	0	0
$679$	− 196.556i	− 0.289479i
$680$	0	0
$681$	104.624	0.153633
$682$	0	0
$683$	541.257i	0.792471i	0.918149	+	0.396235i	$0.129684\pi$
$683$	541.257i	0.792471i	−0.918149	+	0.396235i	$0.870316\pi$
$684$	0	0
$685$	−269.629	−0.393619
$686$	0	0
$687$	− 309.783i	− 0.450922i
$688$	0	0
$689$	−332.665	−0.482823
$690$	0	0
$691$	300.995i	0.435594i	0.975994	+	0.217797i	$0.0698871\pi$
$691$	300.995i	0.435594i	−0.975994	+	0.217797i	$0.930113\pi$
$692$	0	0
$693$	158.825	0.229184
$694$	0	0
$695$	254.758i	0.366558i
$696$	0	0
$697$	−1128.98	−1.61978
$698$	0	0
$699$	− 479.194i	− 0.685543i
$700$	0	0
$701$	−672.761	−0.959716	−0.479858	−	0.877346i	$-0.659312\pi$
$701$	−672.761	−0.959716	−0.479858	+	0.877346i	$0.659312\pi$
$702$	0	0
$703$	5.66934i	0.00806450i
$704$	0	0
$705$	−82.9325	−0.117635
$706$	0	0
$707$	− 262.983i	− 0.371970i
$708$	0	0
$709$	−65.4502	−0.0923135	−0.0461567	−	0.998934i	$-0.514697\pi$
$709$	−65.4502	−0.0923135	−0.0461567	+	0.998934i	$0.514697\pi$
$710$	0	0
$711$	− 212.845i	− 0.299360i
$712$	0	0
$713$	814.317	1.14210
$714$	0	0
$715$	− 488.247i	− 0.682864i
$716$	0	0
$717$	780.309	1.08830
$718$	0	0
$719$	854.235i	1.18809i	0.804433	+	0.594044i	$0.202469\pi$
$719$	854.235i	1.18809i	−0.804433	+	0.594044i	$0.797531\pi$
$720$	0	0
$721$	−292.037	−0.405045
$722$	0	0
$723$	− 352.695i	− 0.487821i
$724$	0	0
$725$	−289.006	−0.398629
$726$	0	0
$727$	− 117.161i	− 0.161157i	−0.996748	−	0.0805786i	$-0.974323\pi$
$727$	− 117.161i	− 0.161157i	0.996748	−	0.0805786i	$-0.0256768\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1900.11i	− 2.59933i
$732$	0	0
$733$	−71.3372	−0.0973222	−0.0486611	−	0.998815i	$-0.515495\pi$
$733$	−71.3372	−0.0973222	−0.0486611	+	0.998815i	$0.515495\pi$
$734$	0	0
$735$	− 27.1109i	− 0.0368856i
$736$	0	0
$737$	−614.518	−0.833811
$738$	0	0
$739$	− 528.835i	− 0.715609i	−0.933796	−	0.357805i	$-0.883525\pi$
$739$	− 528.835i	− 0.715609i	0.933796	−	0.357805i	$-0.116475\pi$
$740$	0	0
$741$	2.12648	0.00286975
$742$	0	0
$743$	762.107i	1.02572i	0.858473	+	0.512858i	$0.171413\pi$
$743$	762.107i	1.02572i	−0.858473	+	0.512858i	$0.828587\pi$
$744$	0	0
$745$	594.947	0.798587
$746$	0	0
$747$	− 201.337i	− 0.269527i
$748$	0	0
$749$	−313.577	−0.418661
$750$	0	0
$751$	− 632.536i	− 0.842258i	−0.907001	−	0.421129i	$-0.861634\pi$
$751$	− 632.536i	− 0.842258i	0.907001	−	0.421129i	$-0.138366\pi$
$752$	0	0
$753$	39.7377	0.0527725
$754$	0	0
$755$	368.401i	0.487948i
$756$	0	0
$757$	−1084.95	−1.43322	−0.716610	−	0.697474i	$-0.754307\pi$
$757$	−1084.95	−1.43322	−0.716610	+	0.697474i	$0.754307\pi$
$758$	0	0
$759$	− 474.540i	− 0.625218i
$760$	0	0
$761$	−316.765	−0.416248	−0.208124	−	0.978102i	$-0.566736\pi$
$761$	−316.765	−0.416248	−0.208124	+	0.978102i	$0.566736\pi$
$762$	0	0
$763$	− 258.810i	− 0.339200i
$764$	0	0
$765$	211.880	0.276967
$766$	0	0
$767$	993.429i	1.29521i
$768$	0	0
$769$	517.571	0.673044	0.336522	−	0.941676i	$-0.390749\pi$
$769$	517.571	0.673044	0.336522	+	0.941676i	$0.390749\pi$
$770$	0	0
$771$	− 422.911i	− 0.548523i
$772$	0	0
$773$	−243.065	−0.314444	−0.157222	−	0.987563i	$-0.550254\pi$
$773$	−243.065	−0.314444	−0.157222	+	0.987563i	$0.550254\pi$
$774$	0	0
$775$	297.371i	0.383705i
$776$	0	0
$777$	−230.913	−0.297186
$778$	0	0
$779$	− 4.02159i	− 0.00516250i
$780$	0	0
$781$	1358.16	1.73901
$782$	0	0
$783$	300.344i	0.383581i
$784$	0	0
$785$	−146.158	−0.186189
$786$	0	0
$787$	138.994i	0.176612i	0.996093	+	0.0883061i	$0.0281454\pi$
$787$	138.994i	0.176612i	−0.996093	+	0.0883061i	$0.971855\pi$
$788$	0	0
$789$	429.148	0.543914
$790$	0	0
$791$	302.001i	0.381797i
$792$	0	0
$793$	−720.685	−0.908808
$794$	0	0
$795$	− 118.071i	− 0.148518i
$796$	0	0
$797$	−100.299	−0.125845	−0.0629227	−	0.998018i	$-0.520042\pi$
$797$	−100.299	−0.125845	−0.0629227	+	0.998018i	$0.520042\pi$
$798$	0	0
$799$	− 676.336i	− 0.846479i
$800$	0	0
$801$	386.818	0.482919
$802$	0	0
$803$	16.2537i	0.0202412i
$804$	0	0
$805$	−81.0026	−0.100624
$806$	0	0
$807$	251.511i	0.311661i
$808$	0	0
$809$	−132.563	−0.163860	−0.0819302	−	0.996638i	$-0.526108\pi$
$809$	−132.563	−0.163860	−0.0819302	+	0.996638i	$0.526108\pi$
$810$	0	0
$811$	122.569i	0.151134i	0.997141	+	0.0755668i	$0.0240766\pi$
$811$	122.569i	0.151134i	−0.997141	+	0.0755668i	$0.975923\pi$
$812$	0	0
$813$	−7.26345	−0.00893413
$814$	0	0
$815$	− 224.515i	− 0.275478i
$816$	0	0
$817$	6.76845	0.00828452
$818$	0	0
$819$	86.6120i	0.105753i
$820$	0	0
$821$	479.566	0.584125	0.292062	−	0.956399i	$-0.405659\pi$
$821$	479.566	0.584125	0.292062	+	0.956399i	$0.405659\pi$
$822$	0	0
$823$	− 934.971i	− 1.13605i	−0.823011	−	0.568026i	$-0.807707\pi$
$823$	− 934.971i	− 1.13605i	0.823011	−	0.568026i	$-0.192293\pi$
$824$	0	0
$825$	173.292	0.210051
$826$	0	0
$827$	− 268.789i	− 0.325017i	−0.986707	−	0.162508i	$-0.948042\pi$
$827$	− 268.789i	− 0.325017i	0.986707	−	0.162508i	$-0.0519585\pi$
$828$	0	0
$829$	432.432	0.521631	0.260816	−	0.965389i	$-0.416009\pi$
$829$	432.432	0.521631	0.260816	+	0.965389i	$0.416009\pi$
$830$	0	0
$831$	− 314.081i	− 0.377955i
$832$	0	0
$833$	221.096	0.265422
$834$	0	0
$835$	415.370i	0.497450i
$836$	0	0
$837$	309.037	0.369220
$838$	0	0
$839$	− 1147.96i	− 1.36825i	−0.729365	−	0.684124i	$-0.760185\pi$
$839$	− 1147.96i	− 1.36825i	0.729365	−	0.684124i	$-0.239815\pi$
$840$	0	0
$841$	2499.98	2.97262
$842$	0	0
$843$	− 175.356i	− 0.208014i
$844$	0	0
$845$	−111.639	−0.132117
$846$	0	0
$847$	739.225i	0.872757i
$848$	0	0
$849$	−146.905	−0.173033
$850$	0	0
$851$	689.928i	0.810727i
$852$	0	0
$853$	800.937	0.938965	0.469482	−	0.882942i	$-0.344441\pi$
$853$	800.937	0.938965	0.469482	+	0.882942i	$0.344441\pi$
$854$	0	0
$855$	0.754744i	0 0.000882742i
$856$	0	0
$857$	−865.820	−1.01029	−0.505146	−	0.863034i	$-0.668561\pi$
$857$	−865.820	−1.01029	−0.505146	+	0.863034i	$0.668561\pi$
$858$	0	0
$859$	− 72.4322i	− 0.0843216i	−0.999111	−	0.0421608i	$-0.986576\pi$
$859$	− 72.4322i	− 0.0843216i	0.999111	−	0.0421608i	$-0.0134242\pi$
$860$	0	0
$861$	163.800	0.190244
$862$	0	0
$863$	356.880i	0.413534i	0.978390	+	0.206767i	$0.0662943\pi$
$863$	356.880i	0.413534i	−0.978390	+	0.206767i	$0.933706\pi$
$864$	0	0
$865$	547.161	0.632556
$866$	0	0
$867$	1227.37i	1.41566i
$868$	0	0
$869$	1419.68	1.63369
$870$	0	0
$871$	− 335.116i	− 0.384748i
$872$	0	0
$873$	−222.874	−0.255297
$874$	0	0
$875$	− 29.5804i	− 0.0338062i
$876$	0	0
$877$	−1268.05	−1.44590	−0.722950	−	0.690901i	$-0.757214\pi$
$877$	−1268.05	−1.44590	−0.722950	+	0.690901i	$0.757214\pi$
$878$	0	0
$879$	− 632.288i	− 0.719326i
$880$	0	0
$881$	−715.271	−0.811885	−0.405943	−	0.913899i	$-0.633057\pi$
$881$	−715.271	−0.811885	−0.405943	+	0.913899i	$0.633057\pi$
$882$	0	0
$883$	− 62.0769i	− 0.0703023i	−0.999382	−	0.0351512i	$-0.988809\pi$
$883$	− 62.0769i	− 0.0703023i	0.999382	−	0.0351512i	$-0.0111913\pi$
$884$	0	0
$885$	−352.594	−0.398411
$886$	0	0
$887$	− 1253.32i	− 1.41299i	−0.707720	−	0.706493i	$-0.750276\pi$
$887$	− 1253.32i	− 1.41299i	0.707720	−	0.706493i	$-0.249724\pi$
$888$	0	0
$889$	−251.554	−0.282963
$890$	0	0
$891$	− 180.090i	− 0.202121i
$892$	0	0
$893$	2.40920	0.00269787
$894$	0	0
$895$	− 577.851i	− 0.645643i
$896$	0	0
$897$	258.781	0.288497
$898$	0	0
$899$	− 3437.68i	− 3.82389i
$900$	0	0
$901$	962.904	1.06871
$902$	0	0
$903$	275.680i	0.305294i
$904$	0	0
$905$	−390.028	−0.430971
$906$	0	0
$907$	1013.00i	1.11687i	0.829547	+	0.558437i	$0.188599\pi$
$907$	1013.00i	1.11687i	−0.829547	+	0.558437i	$0.811401\pi$
$908$	0	0
$909$	−298.195	−0.328047
$910$	0	0
$911$	− 55.4511i	− 0.0608683i	−0.999537	−	0.0304342i	$-0.990311\pi$
$911$	− 55.4511i	− 0.0608683i	0.999537	−	0.0304342i	$-0.00968899\pi$
$912$	0	0
$913$	1342.92	1.47089
$914$	0	0
$915$	− 255.790i	− 0.279552i
$916$	0	0
$917$	−477.143	−0.520331
$918$	0	0
$919$	− 564.739i	− 0.614515i	−0.951626	−	0.307258i	$-0.900589\pi$
$919$	− 564.739i	− 0.614515i	0.951626	−	0.307258i	$-0.0994113\pi$
$920$	0	0
$921$	−280.572	−0.304638
$922$	0	0
$923$	740.649i	0.802437i
$924$	0	0
$925$	−251.947	−0.272375
$926$	0	0
$927$	331.139i	0.357216i
$928$	0	0
$929$	−1558.36	−1.67746	−0.838729	−	0.544549i	$-0.816701\pi$
$929$	−1558.36	−1.67746	−0.838729	+	0.544549i	$0.816701\pi$
$930$	0	0
$931$	0.787574i	0 0.000845945i
$932$	0	0
$933$	−246.378	−0.264071
$934$	0	0
$935$	1413.24i	1.51149i
$936$	0	0
$937$	−1296.05	−1.38319	−0.691597	−	0.722283i	$-0.743093\pi$
$937$	−1296.05	−1.38319	−0.691597	+	0.722283i	$0.743093\pi$
$938$	0	0
$939$	866.674i	0.922976i
$940$	0	0
$941$	−1124.14	−1.19462	−0.597311	−	0.802010i	$-0.703764\pi$
$941$	−1124.14	−1.19462	−0.597311	+	0.802010i	$0.703764\pi$
$942$	0	0
$943$	− 489.406i	− 0.518988i
$944$	0	0
$945$	−30.7409	−0.0325300
$946$	0	0
$947$	1400.34i	1.47871i	0.673318	+	0.739353i	$0.264869\pi$
$947$	1400.34i	1.47871i	−0.673318	+	0.739353i	$0.735131\pi$
$948$	0	0
$949$	−8.86364	−0.00933997
$950$	0	0
$951$	309.827i	0.325791i
$952$	0	0
$953$	−352.414	−0.369794	−0.184897	−	0.982758i	$-0.559195\pi$
$953$	−352.414	−0.369794	−0.184897	+	0.982758i	$0.559195\pi$
$954$	0	0
$955$	− 479.215i	− 0.501796i
$956$	0	0
$957$	−2003.29	−2.09331
$958$	0	0
$959$	319.030i	0.332669i
$960$	0	0
$961$	−2576.18	−2.68073
$962$	0	0
$963$	355.563i	0.369225i
$964$	0	0
$965$	26.5278	0.0274899
$966$	0	0
$967$	1867.80i	1.93154i	0.259402	+	0.965769i	$0.416474\pi$
$967$	1867.80i	1.93154i	−0.259402	+	0.965769i	$0.583526\pi$
$968$	0	0
$969$	−6.15514	−0.00635205
$970$	0	0
$971$	1801.04i	1.85483i	0.374030	+	0.927417i	$0.377976\pi$
$971$	1801.04i	1.85483i	−0.374030	+	0.927417i	$0.622024\pi$
$972$	0	0
$973$	301.433	0.309798
$974$	0	0
$975$	94.5014i	0.0969245i
$976$	0	0
$977$	−633.312	−0.648221	−0.324110	−	0.946019i	$-0.605065\pi$
$977$	−633.312	−0.648221	−0.324110	+	0.946019i	$0.605065\pi$
$978$	0	0
$979$	2580.08i	2.63542i
$980$	0	0
$981$	−293.463	−0.299147
$982$	0	0
$983$	− 696.054i	− 0.708092i	−0.935228	−	0.354046i	$-0.884806\pi$
$983$	− 696.054i	− 0.708092i	0.935228	−	0.354046i	$-0.115194\pi$
$984$	0	0
$985$	−354.069	−0.359461
$986$	0	0
$987$	98.1271i	0.0994195i
$988$	0	0
$989$	823.684	0.832845
$990$	0	0
$991$	470.828i	0.475104i	0.971375	+	0.237552i	$0.0763450\pi$
$991$	470.828i	0.475104i	−0.971375	+	0.237552i	$0.923655\pi$
$992$	0	0
$993$	−0.850158	−0.000856151 0
$994$	0	0
$995$	− 60.2670i	− 0.0605699i
$996$	0	0
$997$	−1148.19	−1.15164	−0.575822	−	0.817575i	$-0.695318\pi$
$997$	−1148.19	−1.15164	−0.575822	+	0.817575i	$0.695318\pi$
$998$	0	0
$999$	261.831i	0.262093i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.n.c.1471.13	yes	16
4.3	odd	2	inner	1680.3.n.c.1471.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.3.n.c.1471.8	✓	16	4.3	odd	2	inner
1680.3.n.c.1471.13	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607 q^{5} -2.64575i q^{7} -3.00000 q^{9} -20.0100i q^{11} +10.9121 q^{13} +3.87298i q^{15} -31.5852 q^{17} -0.112511i q^{19} +4.58258 q^{21} -13.6919i q^{23} +5.00000 q^{25} -5.19615i q^{27} -57.8012 q^{29} +59.4742i q^{31} +34.6584 q^{33} -5.91608i q^{35} -50.3894 q^{37} +18.9003i q^{39} +35.7441 q^{41} +60.1583i q^{43} -6.70820 q^{45} +21.4131i q^{47} -7.00000 q^{49} -54.7072i q^{51} -30.4859 q^{53} -44.7438i q^{55} +0.194874 q^{57} +91.0394i q^{59} -66.0447 q^{61} +7.93725i q^{63} +24.4002 q^{65} -30.7105i q^{67} +23.7151 q^{69} +67.8742i q^{71} -0.812277 q^{73} +8.66025i q^{75} -52.9415 q^{77} +70.9483i q^{79} +9.00000 q^{81} +67.1123i q^{83} -70.6267 q^{85} -100.115i q^{87} -128.939 q^{89} -28.8707i q^{91} -103.012 q^{93} -0.251581i q^{95} +74.2913 q^{97} +60.0300i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 48 q^{13} - 96 q^{17} + 80 q^{25} + 112 q^{29} + 48 q^{33} - 224 q^{37} + 160 q^{41} - 112 q^{49} - 96 q^{53} + 80 q^{61} - 80 q^{65} + 48 q^{69} - 48 q^{73} + 144 q^{81} - 160 q^{89}+ \cdots + 592 q^{97}+O(q^{100}) \) 16 * q - 48 * q^9 + 48 * q^13 - 96 * q^17 + 80 * q^25 + 112 * q^29 + 48 * q^33 - 224 * q^37 + 160 * q^41 - 112 * q^49 - 96 * q^53 + 80 * q^61 - 80 * q^65 + 48 * q^69 - 48 * q^73 + 144 * q^81 - 160 * q^89 + 592 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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