LMFDB - Embedded newform 672.3.g.a.463.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,3,Mod(463,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 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("672.463"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	672.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$18.3106737650$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			463.10
Character	$\chi$	$=$	672.463
Dual form			672.3.g.a.463.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +6.16432i q^{5} +2.64575i q^{7} +3.00000 q^{9} +18.3516 q^{11} -17.5694i q^{13} -10.6769i q^{15} +17.3798 q^{17} -9.17365 q^{19} -4.58258i q^{21} +12.5593i q^{23} -12.9988 q^{25} -5.19615 q^{27} -11.8380i q^{29} +52.4097i q^{31} -31.7859 q^{33} -16.3093 q^{35} +50.5280i q^{37} +30.4311i q^{39} -31.1053 q^{41} +54.6464 q^{43} +18.4930i q^{45} -9.54776i q^{47} -7.00000 q^{49} -30.1027 q^{51} +58.4627i q^{53} +113.125i q^{55} +15.8892 q^{57} +37.2670 q^{59} -40.0382i q^{61} +7.93725i q^{63} +108.303 q^{65} +60.0932 q^{67} -21.7534i q^{69} -65.0830i q^{71} -71.2799 q^{73} +22.5147 q^{75} +48.5538i q^{77} +130.677i q^{79} +9.00000 q^{81} -151.032 q^{83} +107.135i q^{85} +20.5041i q^{87} +104.920 q^{89} +46.4842 q^{91} -90.7763i q^{93} -56.5493i q^{95} +30.9691 q^{97} +55.0548 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{9} - 32 q^{11} + 16 q^{17} + 64 q^{19} - 72 q^{25} - 80 q^{41} - 32 q^{43} - 168 q^{49} - 192 q^{51} - 192 q^{65} + 32 q^{67} - 240 q^{73} + 384 q^{75} + 216 q^{81} + 320 q^{83} + 400 q^{89}+ \cdots - 96 q^{99}+O(q^{100}) $ 24 * q + 72 * q^9 - 32 * q^11 + 16 * q^17 + 64 * q^19 - 72 * q^25 - 80 * q^41 - 32 * q^43 - 168 * q^49 - 192 * q^51 - 192 * q^65 + 32 * q^67 - 240 * q^73 + 384 * q^75 + 216 * q^81 + 320 * q^83 + 400 * q^89 + 144 * q^97 - 96 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	6.16432i	1.23286i	0.787408	+	0.616432i	$0.211422\pi$
$5$	6.16432i	1.23286i	−0.787408	+	0.616432i	$0.788578\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	18.3516	1.66833	0.834164	−	0.551517i	$-0.185951\pi$
$11$	18.3516	1.66833	0.834164	+	0.551517i	$0.185951\pi$
$12$	0	0
$13$	− 17.5694i	− 1.35149i	−0.737135	−	0.675746i	$-0.763822\pi$
$13$	− 17.5694i	− 1.35149i	0.737135	−	0.675746i	$-0.236178\pi$
$14$	0	0
$15$	− 10.6769i	− 0.711794i
$16$	0	0
$17$	17.3798	1.02234	0.511170	−	0.859480i	$-0.329212\pi$
$17$	17.3798	1.02234	0.511170	+	0.859480i	$0.329212\pi$
$18$	0	0
$19$	−9.17365	−0.482824	−0.241412	−	0.970423i	$-0.577610\pi$
$19$	−9.17365	−0.482824	−0.241412	+	0.970423i	$0.577610\pi$
$20$	0	0
$21$	− 4.58258i	− 0.218218i
$22$	0	0
$23$	12.5593i	0.546058i	0.962006	+	0.273029i	$0.0880255\pi$
$23$	12.5593i	0.546058i	−0.962006	+	0.273029i	$0.911974\pi$
$24$	0	0
$25$	−12.9988	−0.519954
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	− 11.8380i	− 0.408208i	−0.978949	−	0.204104i	$-0.934572\pi$
$29$	− 11.8380i	− 0.408208i	0.978949	−	0.204104i	$-0.0654280\pi$
$30$	0	0
$31$	52.4097i	1.69064i	0.534264	+	0.845318i	$0.320589\pi$
$31$	52.4097i	1.69064i	−0.534264	+	0.845318i	$0.679411\pi$
$32$	0	0
$33$	−31.7859	−0.963209
$34$	0	0
$35$	−16.3093	−0.465979
$36$	0	0
$37$	50.5280i	1.36562i	0.730596	+	0.682810i	$0.239242\pi$
$37$	50.5280i	1.36562i	−0.730596	+	0.682810i	$0.760758\pi$
$38$	0	0
$39$	30.4311i	0.780284i
$40$	0	0
$41$	−31.1053	−0.758667	−0.379333	−	0.925260i	$-0.623847\pi$
$41$	−31.1053	−0.758667	−0.379333	+	0.925260i	$0.623847\pi$
$42$	0	0
$43$	54.6464	1.27085	0.635423	−	0.772164i	$-0.280826\pi$
$43$	54.6464	1.27085	0.635423	+	0.772164i	$0.280826\pi$
$44$	0	0
$45$	18.4930i	0.410955i
$46$	0	0
$47$	− 9.54776i	− 0.203144i	−0.994828	−	0.101572i	$-0.967613\pi$
$47$	− 9.54776i	− 0.203144i	0.994828	−	0.101572i	$-0.0323872\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−30.1027	−0.590248
$52$	0	0
$53$	58.4627i	1.10307i	0.834152	+	0.551535i	$0.185958\pi$
$53$	58.4627i	1.10307i	−0.834152	+	0.551535i	$0.814042\pi$
$54$	0	0
$55$	113.125i	2.05682i
$56$	0	0
$57$	15.8892	0.278758
$58$	0	0
$59$	37.2670	0.631645	0.315822	−	0.948818i	$-0.397720\pi$
$59$	37.2670	0.631645	0.315822	+	0.948818i	$0.397720\pi$
$60$	0	0
$61$	− 40.0382i	− 0.656364i	−0.944615	−	0.328182i	$-0.893564\pi$
$61$	− 40.0382i	− 0.656364i	0.944615	−	0.328182i	$-0.106436\pi$
$62$	0	0
$63$	7.93725i	0.125988i
$64$	0	0
$65$	108.303	1.66620
$66$	0	0
$67$	60.0932	0.896914	0.448457	−	0.893805i	$-0.351974\pi$
$67$	60.0932	0.896914	0.448457	+	0.893805i	$0.351974\pi$
$68$	0	0
$69$	− 21.7534i	− 0.315267i
$70$	0	0
$71$	− 65.0830i	− 0.916662i	−0.888782	−	0.458331i	$-0.848448\pi$
$71$	− 65.0830i	− 0.916662i	0.888782	−	0.458331i	$-0.151552\pi$
$72$	0	0
$73$	−71.2799	−0.976437	−0.488219	−	0.872721i	$-0.662353\pi$
$73$	−71.2799	−0.976437	−0.488219	+	0.872721i	$0.662353\pi$
$74$	0	0
$75$	22.5147	0.300195
$76$	0	0
$77$	48.5538i	0.630569i
$78$	0	0
$79$	130.677i	1.65414i	0.562097	+	0.827071i	$0.309995\pi$
$79$	130.677i	1.65414i	−0.562097	+	0.827071i	$0.690005\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−151.032	−1.81967	−0.909834	−	0.414973i	$-0.863791\pi$
$83$	−151.032	−1.81967	−0.909834	+	0.414973i	$0.863791\pi$
$84$	0	0
$85$	107.135i	1.26041i
$86$	0	0
$87$	20.5041i	0.235679i
$88$	0	0
$89$	104.920	1.17888	0.589438	−	0.807813i	$-0.299349\pi$
$89$	104.920	1.17888	0.589438	+	0.807813i	$0.299349\pi$
$90$	0	0
$91$	46.4842	0.510816
$92$	0	0
$93$	− 90.7763i	− 0.976089i
$94$	0	0
$95$	− 56.5493i	− 0.595256i
$96$	0	0
$97$	30.9691	0.319269	0.159634	−	0.987176i	$-0.448969\pi$
$97$	30.9691	0.319269	0.159634	+	0.987176i	$0.448969\pi$
$98$	0	0
$99$	55.0548	0.556109
$100$	0	0
$101$	125.478i	1.24236i	0.783669	+	0.621178i	$0.213346\pi$
$101$	125.478i	1.24236i	−0.783669	+	0.621178i	$0.786654\pi$
$102$	0	0
$103$	− 110.356i	− 1.07142i	−0.844402	−	0.535710i	$-0.820044\pi$
$103$	− 110.356i	− 1.07142i	0.844402	−	0.535710i	$-0.179956\pi$
$104$	0	0
$105$	28.2485	0.269033
$106$	0	0
$107$	23.8889	0.223261	0.111631	−	0.993750i	$-0.464393\pi$
$107$	23.8889	0.223261	0.111631	+	0.993750i	$0.464393\pi$
$108$	0	0
$109$	196.446i	1.80225i	0.433556	+	0.901126i	$0.357258\pi$
$109$	196.446i	1.80225i	−0.433556	+	0.901126i	$0.642742\pi$
$110$	0	0
$111$	− 87.5170i	− 0.788441i
$112$	0	0
$113$	−107.844	−0.954371	−0.477186	−	0.878803i	$-0.658343\pi$
$113$	−107.844	−0.954371	−0.477186	+	0.878803i	$0.658343\pi$
$114$	0	0
$115$	−77.4198	−0.673215
$116$	0	0
$117$	− 52.7082i	− 0.450497i
$118$	0	0
$119$	45.9826i	0.386408i
$120$	0	0
$121$	215.781	1.78332
$122$	0	0
$123$	53.8760	0.438017
$124$	0	0
$125$	73.9790i	0.591832i
$126$	0	0
$127$	31.7684i	0.250145i	0.992148	+	0.125073i	$0.0399164\pi$
$127$	31.7684i	0.250145i	−0.992148	+	0.125073i	$0.960084\pi$
$128$	0	0
$129$	−94.6503	−0.733724
$130$	0	0
$131$	24.7991	0.189306	0.0946532	−	0.995510i	$-0.469826\pi$
$131$	24.7991	0.189306	0.0946532	+	0.995510i	$0.469826\pi$
$132$	0	0
$133$	− 24.2712i	− 0.182490i
$134$	0	0
$135$	− 32.0307i	− 0.237265i
$136$	0	0
$137$	−88.3273	−0.644725	−0.322362	−	0.946616i	$-0.604477\pi$
$137$	−88.3273	−0.644725	−0.322362	+	0.946616i	$0.604477\pi$
$138$	0	0
$139$	−126.221	−0.908062	−0.454031	−	0.890986i	$-0.650014\pi$
$139$	−126.221	−0.908062	−0.454031	+	0.890986i	$0.650014\pi$
$140$	0	0
$141$	16.5372i	0.117285i
$142$	0	0
$143$	− 322.426i	− 2.25473i
$144$	0	0
$145$	72.9734	0.503265
$146$	0	0
$147$	12.1244	0.0824786
$148$	0	0
$149$	− 95.4419i	− 0.640550i	−0.947325	−	0.320275i	$-0.896225\pi$
$149$	− 95.4419i	− 0.640550i	0.947325	−	0.320275i	$-0.103775\pi$
$150$	0	0
$151$	− 18.6965i	− 0.123818i	−0.998082	−	0.0619091i	$-0.980281\pi$
$151$	− 18.6965i	− 0.123818i	0.998082	−	0.0619091i	$-0.0197189\pi$
$152$	0	0
$153$	52.1393	0.340780
$154$	0	0
$155$	−323.070	−2.08432
$156$	0	0
$157$	− 4.06821i	− 0.0259122i	−0.999916	−	0.0129561i	$-0.995876\pi$
$157$	− 4.06821i	− 0.0259122i	0.999916	−	0.0129561i	$-0.00412417\pi$
$158$	0	0
$159$	− 101.260i	− 0.636858i
$160$	0	0
$161$	−33.2289	−0.206391
$162$	0	0
$163$	−194.179	−1.19128	−0.595642	−	0.803250i	$-0.703102\pi$
$163$	−194.179	−1.19128	−0.595642	+	0.803250i	$0.703102\pi$
$164$	0	0
$165$	− 195.939i	− 1.18751i
$166$	0	0
$167$	46.3885i	0.277775i	0.990308	+	0.138888i	$0.0443527\pi$
$167$	46.3885i	0.277775i	−0.990308	+	0.138888i	$0.955647\pi$
$168$	0	0
$169$	−139.683	−0.826529
$170$	0	0
$171$	−27.5209	−0.160941
$172$	0	0
$173$	− 256.754i	− 1.48412i	−0.670331	−	0.742062i	$-0.733848\pi$
$173$	− 256.754i	− 1.48412i	0.670331	−	0.742062i	$-0.266152\pi$
$174$	0	0
$175$	− 34.3917i	− 0.196524i
$176$	0	0
$177$	−64.5484	−0.364680
$178$	0	0
$179$	186.571	1.04229	0.521147	−	0.853467i	$-0.325504\pi$
$179$	186.571	1.04229	0.521147	+	0.853467i	$0.325504\pi$
$180$	0	0
$181$	12.2636i	0.0677546i	0.999426	+	0.0338773i	$0.0107855\pi$
$181$	12.2636i	0.0677546i	−0.999426	+	0.0338773i	$0.989214\pi$
$182$	0	0
$183$	69.3482i	0.378952i
$184$	0	0
$185$	−311.470	−1.68362
$186$	0	0
$187$	318.947	1.70560
$188$	0	0
$189$	− 13.7477i	− 0.0727393i
$190$	0	0
$191$	− 31.2212i	− 0.163462i	−0.996654	−	0.0817310i	$-0.973955\pi$
$191$	− 31.2212i	− 0.163462i	0.996654	−	0.0817310i	$-0.0260448\pi$
$192$	0	0
$193$	272.816	1.41355	0.706777	−	0.707437i	$-0.250149\pi$
$193$	272.816	1.41355	0.706777	+	0.707437i	$0.250149\pi$
$194$	0	0
$195$	−187.587	−0.961984
$196$	0	0
$197$	− 367.742i	− 1.86671i	−0.358952	−	0.933356i	$-0.616866\pi$
$197$	− 367.742i	− 1.86671i	0.358952	−	0.933356i	$-0.383134\pi$
$198$	0	0
$199$	220.530i	1.10819i	0.832453	+	0.554095i	$0.186936\pi$
$199$	220.530i	1.10819i	−0.832453	+	0.554095i	$0.813064\pi$
$200$	0	0
$201$	−104.084	−0.517833
$202$	0	0
$203$	31.3205	0.154288
$204$	0	0
$205$	− 191.743i	− 0.935333i
$206$	0	0
$207$	37.6780i	0.182019i
$208$	0	0
$209$	−168.351	−0.805508
$210$	0	0
$211$	289.313	1.37115	0.685575	−	0.728002i	$-0.259551\pi$
$211$	289.313	1.37115	0.685575	+	0.728002i	$0.259551\pi$
$212$	0	0
$213$	112.727i	0.529235i
$214$	0	0
$215$	336.858i	1.56678i
$216$	0	0
$217$	−138.663	−0.639000
$218$	0	0
$219$	123.460	0.563746
$220$	0	0
$221$	− 305.352i	− 1.38168i
$222$	0	0
$223$	− 134.210i	− 0.601838i	−0.953650	−	0.300919i	$-0.902707\pi$
$223$	− 134.210i	− 0.601838i	0.953650	−	0.300919i	$-0.0972934\pi$
$224$	0	0
$225$	−38.9965	−0.173318
$226$	0	0
$227$	351.467	1.54831	0.774157	−	0.632994i	$-0.218174\pi$
$227$	351.467	1.54831	0.774157	+	0.632994i	$0.218174\pi$
$228$	0	0
$229$	271.439i	1.18532i	0.805452	+	0.592661i	$0.201922\pi$
$229$	271.439i	1.18532i	−0.805452	+	0.592661i	$0.798078\pi$
$230$	0	0
$231$	− 84.0976i	− 0.364059i
$232$	0	0
$233$	288.131	1.23661	0.618306	−	0.785938i	$-0.287819\pi$
$233$	288.131	1.23661	0.618306	+	0.785938i	$0.287819\pi$
$234$	0	0
$235$	58.8555	0.250449
$236$	0	0
$237$	− 226.340i	− 0.955020i
$238$	0	0
$239$	− 247.003i	− 1.03349i	−0.856141	−	0.516743i	$-0.827144\pi$
$239$	− 247.003i	− 1.03349i	0.856141	−	0.516743i	$-0.172856\pi$
$240$	0	0
$241$	−96.8368	−0.401813	−0.200906	−	0.979610i	$-0.564389\pi$
$241$	−96.8368	−0.401813	−0.200906	+	0.979610i	$0.564389\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	− 43.1502i	− 0.176123i
$246$	0	0
$247$	161.175i	0.652532i
$248$	0	0
$249$	261.596	1.05059
$250$	0	0
$251$	−47.0750	−0.187550	−0.0937748	−	0.995593i	$-0.529893\pi$
$251$	−47.0750	−0.187550	−0.0937748	+	0.995593i	$0.529893\pi$
$252$	0	0
$253$	230.484i	0.911004i
$254$	0	0
$255$	− 185.562i	− 0.727696i
$256$	0	0
$257$	−43.2628	−0.168338	−0.0841689	−	0.996452i	$-0.526824\pi$
$257$	−43.2628	−0.168338	−0.0841689	+	0.996452i	$0.526824\pi$
$258$	0	0
$259$	−133.684	−0.516156
$260$	0	0
$261$	− 35.5141i	− 0.136069i
$262$	0	0
$263$	− 234.603i	− 0.892028i	−0.895026	−	0.446014i	$-0.852843\pi$
$263$	− 234.603i	− 0.892028i	0.895026	−	0.446014i	$-0.147157\pi$
$264$	0	0
$265$	−360.383	−1.35993
$266$	0	0
$267$	−181.727	−0.680625
$268$	0	0
$269$	− 258.099i	− 0.959475i	−0.877412	−	0.479737i	$-0.840732\pi$
$269$	− 258.099i	− 0.959475i	0.877412	−	0.479737i	$-0.159268\pi$
$270$	0	0
$271$	− 33.9000i	− 0.125092i	−0.998042	−	0.0625462i	$-0.980078\pi$
$271$	− 33.9000i	− 0.125092i	0.998042	−	0.0625462i	$-0.0199221\pi$
$272$	0	0
$273$	−80.5130	−0.294920
$274$	0	0
$275$	−238.550	−0.867453
$276$	0	0
$277$	− 307.373i	− 1.10965i	−0.831967	−	0.554825i	$-0.812785\pi$
$277$	− 307.373i	− 1.10965i	0.831967	−	0.554825i	$-0.187215\pi$
$278$	0	0
$279$	157.229i	0.563545i
$280$	0	0
$281$	464.885	1.65439	0.827197	−	0.561911i	$-0.189934\pi$
$281$	464.885	1.65439	0.827197	+	0.561911i	$0.189934\pi$
$282$	0	0
$283$	−258.876	−0.914756	−0.457378	−	0.889272i	$-0.651211\pi$
$283$	−258.876	−0.914756	−0.457378	+	0.889272i	$0.651211\pi$
$284$	0	0
$285$	97.9463i	0.343671i
$286$	0	0
$287$	− 82.2970i	− 0.286749i
$288$	0	0
$289$	13.0566	0.0451787
$290$	0	0
$291$	−53.6400	−0.184330
$292$	0	0
$293$	56.2381i	0.191939i	0.995384	+	0.0959695i	$0.0305951\pi$
$293$	56.2381i	0.191939i	−0.995384	+	0.0959695i	$0.969405\pi$
$294$	0	0
$295$	229.726i	0.778732i
$296$	0	0
$297$	−95.3577	−0.321070
$298$	0	0
$299$	220.660	0.737993
$300$	0	0
$301$	144.581i	0.480335i
$302$	0	0
$303$	− 217.334i	− 0.717275i
$304$	0	0
$305$	246.808	0.809207
$306$	0	0
$307$	−322.183	−1.04946	−0.524728	−	0.851270i	$-0.675833\pi$
$307$	−322.183	−1.04946	−0.524728	+	0.851270i	$0.675833\pi$
$308$	0	0
$309$	191.142i	0.618584i
$310$	0	0
$311$	4.79256i	0.0154102i	0.999970	+	0.00770508i	$0.00245263\pi$
$311$	4.79256i	0.0154102i	−0.999970	+	0.00770508i	$0.997547\pi$
$312$	0	0
$313$	109.002	0.348248	0.174124	−	0.984724i	$-0.444291\pi$
$313$	109.002	0.348248	0.174124	+	0.984724i	$0.444291\pi$
$314$	0	0
$315$	−48.9278	−0.155326
$316$	0	0
$317$	38.2661i	0.120713i	0.998177	+	0.0603566i	$0.0192238\pi$
$317$	38.2661i	0.120713i	−0.998177	+	0.0603566i	$0.980776\pi$
$318$	0	0
$319$	− 217.247i	− 0.681024i
$320$	0	0
$321$	−41.3769	−0.128900
$322$	0	0
$323$	−159.436	−0.493610
$324$	0	0
$325$	228.382i	0.702713i
$326$	0	0
$327$	− 340.254i	− 1.04053i
$328$	0	0
$329$	25.2610	0.0767812
$330$	0	0
$331$	−94.2493	−0.284741	−0.142371	−	0.989813i	$-0.545472\pi$
$331$	−94.2493	−0.284741	−0.142371	+	0.989813i	$0.545472\pi$
$332$	0	0
$333$	151.584i	0.455207i
$334$	0	0
$335$	370.434i	1.10577i
$336$	0	0
$337$	329.007	0.976281	0.488140	−	0.872765i	$-0.337675\pi$
$337$	329.007	0.976281	0.488140	+	0.872765i	$0.337675\pi$
$338$	0	0
$339$	186.791	0.551006
$340$	0	0
$341$	961.802i	2.82053i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	134.095	0.388681
$346$	0	0
$347$	552.515	1.59226	0.796131	−	0.605124i	$-0.206876\pi$
$347$	552.515	1.59226	0.796131	+	0.605124i	$0.206876\pi$
$348$	0	0
$349$	178.048i	0.510165i	0.966919	+	0.255082i	$0.0821026\pi$
$349$	178.048i	0.510165i	−0.966919	+	0.255082i	$0.917897\pi$
$350$	0	0
$351$	91.2932i	0.260095i
$352$	0	0
$353$	−601.918	−1.70515	−0.852575	−	0.522604i	$-0.824960\pi$
$353$	−601.918	−1.70515	−0.852575	+	0.522604i	$0.824960\pi$
$354$	0	0
$355$	401.192	1.13012
$356$	0	0
$357$	− 79.6441i	− 0.223093i
$358$	0	0
$359$	132.354i	0.368674i	0.982863	+	0.184337i	$0.0590138\pi$
$359$	132.354i	0.368674i	−0.982863	+	0.184337i	$0.940986\pi$
$360$	0	0
$361$	−276.844	−0.766881
$362$	0	0
$363$	−373.744	−1.02960
$364$	0	0
$365$	− 439.392i	− 1.20381i
$366$	0	0
$367$	214.090i	0.583352i	0.956517	+	0.291676i	$0.0942128\pi$
$367$	214.090i	0.583352i	−0.956517	+	0.291676i	$0.905787\pi$
$368$	0	0
$369$	−93.3160	−0.252889
$370$	0	0
$371$	−154.678	−0.416921
$372$	0	0
$373$	− 460.986i	− 1.23589i	−0.786223	−	0.617943i	$-0.787966\pi$
$373$	− 460.986i	− 1.23589i	0.786223	−	0.617943i	$-0.212034\pi$
$374$	0	0
$375$	− 128.135i	− 0.341694i
$376$	0	0
$377$	−207.987	−0.551689
$378$	0	0
$379$	−539.587	−1.42371	−0.711856	−	0.702325i	$-0.752145\pi$
$379$	−539.587	−1.42371	−0.711856	+	0.702325i	$0.752145\pi$
$380$	0	0
$381$	− 55.0246i	− 0.144421i
$382$	0	0
$383$	− 298.687i	− 0.779863i	−0.920844	−	0.389931i	$-0.872499\pi$
$383$	− 298.687i	− 0.779863i	0.920844	−	0.389931i	$-0.127501\pi$
$384$	0	0
$385$	−299.301	−0.777405
$386$	0	0
$387$	163.939	0.423615
$388$	0	0
$389$	− 158.045i	− 0.406286i	−0.979149	−	0.203143i	$-0.934884\pi$
$389$	− 158.045i	− 0.406286i	0.979149	−	0.203143i	$-0.0651157\pi$
$390$	0	0
$391$	218.278i	0.558257i
$392$	0	0
$393$	−42.9534	−0.109296
$394$	0	0
$395$	−805.537	−2.03933
$396$	0	0
$397$	− 290.058i	− 0.730625i	−0.930885	−	0.365313i	$-0.880962\pi$
$397$	− 290.058i	− 0.730625i	0.930885	−	0.365313i	$-0.119038\pi$
$398$	0	0
$399$	42.0389i	0.105361i
$400$	0	0
$401$	111.817	0.278846	0.139423	−	0.990233i	$-0.455475\pi$
$401$	111.817	0.278846	0.139423	+	0.990233i	$0.455475\pi$
$402$	0	0
$403$	920.806	2.28488
$404$	0	0
$405$	55.4789i	0.136985i
$406$	0	0
$407$	927.269i	2.27830i
$408$	0	0
$409$	481.749	1.17787	0.588935	−	0.808180i	$-0.299547\pi$
$409$	481.749	1.17787	0.588935	+	0.808180i	$0.299547\pi$
$410$	0	0
$411$	152.987	0.372232
$412$	0	0
$413$	98.5993i	0.238739i
$414$	0	0
$415$	− 931.012i	− 2.24340i
$416$	0	0
$417$	218.621	0.524270
$418$	0	0
$419$	656.386	1.56655	0.783277	−	0.621673i	$-0.213547\pi$
$419$	656.386	1.56655	0.783277	+	0.621673i	$0.213547\pi$
$420$	0	0
$421$	238.193i	0.565779i	0.959153	+	0.282889i	$0.0912929\pi$
$421$	238.193i	0.565779i	−0.959153	+	0.282889i	$0.908707\pi$
$422$	0	0
$423$	− 28.6433i	− 0.0677146i
$424$	0	0
$425$	−225.917	−0.531569
$426$	0	0
$427$	105.931	0.248082
$428$	0	0
$429$	558.459i	1.30177i
$430$	0	0
$431$	− 329.786i	− 0.765166i	−0.923921	−	0.382583i	$-0.875035\pi$
$431$	− 329.786i	− 0.765166i	0.923921	−	0.382583i	$-0.124965\pi$
$432$	0	0
$433$	454.294	1.04918	0.524588	−	0.851356i	$-0.324219\pi$
$433$	454.294	1.04918	0.524588	+	0.851356i	$0.324219\pi$
$434$	0	0
$435$	−126.394	−0.290560
$436$	0	0
$437$	− 115.215i	− 0.263650i
$438$	0	0
$439$	358.082i	0.815676i	0.913054	+	0.407838i	$0.133717\pi$
$439$	358.082i	0.815676i	−0.913054	+	0.407838i	$0.866283\pi$
$440$	0	0
$441$	−21.0000	−0.0476190
$442$	0	0
$443$	158.477	0.357737	0.178868	−	0.983873i	$-0.442756\pi$
$443$	158.477	0.357737	0.178868	+	0.983873i	$0.442756\pi$
$444$	0	0
$445$	646.761i	1.45339i
$446$	0	0
$447$	165.310i	0.369821i
$448$	0	0
$449$	−433.724	−0.965978	−0.482989	−	0.875626i	$-0.660449\pi$
$449$	−433.724	−0.965978	−0.482989	+	0.875626i	$0.660449\pi$
$450$	0	0
$451$	−570.833	−1.26570
$452$	0	0
$453$	32.3834i	0.0714865i
$454$	0	0
$455$	286.544i	0.629766i
$456$	0	0
$457$	−352.377	−0.771066	−0.385533	−	0.922694i	$-0.625982\pi$
$457$	−352.377	−0.771066	−0.385533	+	0.922694i	$0.625982\pi$
$458$	0	0
$459$	−90.3080	−0.196749
$460$	0	0
$461$	407.443i	0.883824i	0.897058	+	0.441912i	$0.145700\pi$
$461$	407.443i	0.883824i	−0.897058	+	0.441912i	$0.854300\pi$
$462$	0	0
$463$	− 159.028i	− 0.343474i	−0.985143	−	0.171737i	$-0.945062\pi$
$463$	− 159.028i	− 0.343474i	0.985143	−	0.171737i	$-0.0549379\pi$
$464$	0	0
$465$	559.574	1.20338
$466$	0	0
$467$	−306.786	−0.656930	−0.328465	−	0.944516i	$-0.606531\pi$
$467$	−306.786	−0.656930	−0.328465	+	0.944516i	$0.606531\pi$
$468$	0	0
$469$	158.992i	0.339001i
$470$	0	0
$471$	7.04635i	0.0149604i
$472$	0	0
$473$	1002.85	2.12019
$474$	0	0
$475$	119.247	0.251046
$476$	0	0
$477$	175.388i	0.367690i
$478$	0	0
$479$	74.7230i	0.155998i	0.996953	+	0.0779990i	$0.0248531\pi$
$479$	74.7230i	0.155998i	−0.996953	+	0.0779990i	$0.975147\pi$
$480$	0	0
$481$	887.745	1.84562
$482$	0	0
$483$	57.5541	0.119160
$484$	0	0
$485$	190.903i	0.393615i
$486$	0	0
$487$	459.351i	0.943225i	0.881806	+	0.471612i	$0.156328\pi$
$487$	459.351i	0.943225i	−0.881806	+	0.471612i	$0.843672\pi$
$488$	0	0
$489$	336.328	0.687788
$490$	0	0
$491$	−495.412	−1.00899	−0.504493	−	0.863416i	$-0.668320\pi$
$491$	−495.412	−1.00899	−0.504493	+	0.863416i	$0.668320\pi$
$492$	0	0
$493$	− 205.742i	− 0.417327i
$494$	0	0
$495$	339.375i	0.685607i
$496$	0	0
$497$	172.193	0.346466
$498$	0	0
$499$	277.598	0.556309	0.278155	−	0.960536i	$-0.410277\pi$
$499$	277.598	0.556309	0.278155	+	0.960536i	$0.410277\pi$
$500$	0	0
$501$	− 80.3472i	− 0.160374i
$502$	0	0
$503$	− 470.363i	− 0.935116i	−0.883962	−	0.467558i	$-0.845134\pi$
$503$	− 470.363i	− 0.935116i	0.883962	−	0.467558i	$-0.154866\pi$
$504$	0	0
$505$	−773.486	−1.53166
$506$	0	0
$507$	241.939	0.477196
$508$	0	0
$509$	− 431.842i	− 0.848413i	−0.905566	−	0.424206i	$-0.860553\pi$
$509$	− 431.842i	− 0.848413i	0.905566	−	0.424206i	$-0.139447\pi$
$510$	0	0
$511$	− 188.589i	− 0.369058i
$512$	0	0
$513$	47.6677	0.0929194
$514$	0	0
$515$	680.271	1.32091
$516$	0	0
$517$	− 175.217i	− 0.338911i
$518$	0	0
$519$	444.710i	0.856860i
$520$	0	0
$521$	−145.399	−0.279077	−0.139539	−	0.990217i	$-0.544562\pi$
$521$	−145.399	−0.279077	−0.139539	+	0.990217i	$0.544562\pi$
$522$	0	0
$523$	624.823	1.19469	0.597345	−	0.801984i	$-0.296222\pi$
$523$	624.823	1.19469	0.597345	+	0.801984i	$0.296222\pi$
$524$	0	0
$525$	59.5682i	0.113463i
$526$	0	0
$527$	910.869i	1.72840i
$528$	0	0
$529$	371.263	0.701821
$530$	0	0
$531$	111.801	0.210548
$532$	0	0
$533$	546.502i	1.02533i
$534$	0	0
$535$	147.259i	0.275251i
$536$	0	0
$537$	−323.150	−0.601769
$538$	0	0
$539$	−128.461	−0.238333
$540$	0	0
$541$	873.673i	1.61492i	0.589921	+	0.807461i	$0.299159\pi$
$541$	873.673i	1.61492i	−0.589921	+	0.807461i	$0.700841\pi$
$542$	0	0
$543$	− 21.2411i	− 0.0391181i
$544$	0	0
$545$	−1210.95	−2.22193
$546$	0	0
$547$	−63.9331	−0.116879	−0.0584397	−	0.998291i	$-0.518613\pi$
$547$	−63.9331	−0.116879	−0.0584397	+	0.998291i	$0.518613\pi$
$548$	0	0
$549$	− 120.115i	− 0.218788i
$550$	0	0
$551$	108.598i	0.197092i
$552$	0	0
$553$	−345.740	−0.625207
$554$	0	0
$555$	539.483	0.972041
$556$	0	0
$557$	442.096i	0.793710i	0.917881	+	0.396855i	$0.129898\pi$
$557$	442.096i	0.793710i	−0.917881	+	0.396855i	$0.870102\pi$
$558$	0	0
$559$	− 960.104i	− 1.71754i
$560$	0	0
$561$	−552.432	−0.984727
$562$	0	0
$563$	−207.884	−0.369243	−0.184621	−	0.982810i	$-0.559106\pi$
$563$	−207.884	−0.369243	−0.184621	+	0.982810i	$0.559106\pi$
$564$	0	0
$565$	− 664.785i	− 1.17661i
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	−36.9418	−0.0649241	−0.0324620	−	0.999473i	$-0.510335\pi$
$569$	−36.9418	−0.0649241	−0.0324620	+	0.999473i	$0.510335\pi$
$570$	0	0
$571$	−752.995	−1.31873	−0.659365	−	0.751823i	$-0.729175\pi$
$571$	−752.995	−1.31873	−0.659365	+	0.751823i	$0.729175\pi$
$572$	0	0
$573$	54.0768i	0.0943748i
$574$	0	0
$575$	− 163.257i	− 0.283925i
$576$	0	0
$577$	−645.055	−1.11795	−0.558973	−	0.829186i	$-0.688804\pi$
$577$	−645.055	−1.11795	−0.558973	+	0.829186i	$0.688804\pi$
$578$	0	0
$579$	−472.531	−0.816115
$580$	0	0
$581$	− 399.594i	− 0.687770i
$582$	0	0
$583$	1072.88i	1.84028i
$584$	0	0
$585$	324.910	0.555402
$586$	0	0
$587$	321.664	0.547980	0.273990	−	0.961733i	$-0.411656\pi$
$587$	321.664	0.547980	0.273990	+	0.961733i	$0.411656\pi$
$588$	0	0
$589$	− 480.788i	− 0.816279i
$590$	0	0
$591$	636.948i	1.07775i
$592$	0	0
$593$	563.214	0.949771	0.474886	−	0.880047i	$-0.342489\pi$
$593$	563.214	0.949771	0.474886	+	0.880047i	$0.342489\pi$
$594$	0	0
$595$	−283.451	−0.476389
$596$	0	0
$597$	− 381.969i	− 0.639814i
$598$	0	0
$599$	518.990i	0.866427i	0.901291	+	0.433213i	$0.142620\pi$
$599$	518.990i	0.866427i	−0.901291	+	0.433213i	$0.857380\pi$
$600$	0	0
$601$	662.908	1.10301	0.551504	−	0.834172i	$-0.314054\pi$
$601$	662.908	1.10301	0.551504	+	0.834172i	$0.314054\pi$
$602$	0	0
$603$	180.280	0.298971
$604$	0	0
$605$	1330.15i	2.19859i
$606$	0	0
$607$	− 787.794i	− 1.29785i	−0.760853	−	0.648925i	$-0.775219\pi$
$607$	− 787.794i	− 1.29785i	0.760853	−	0.648925i	$-0.224781\pi$
$608$	0	0
$609$	−54.2486	−0.0890782
$610$	0	0
$611$	−167.748	−0.274547
$612$	0	0
$613$	− 1056.68i	− 1.72378i	−0.507092	−	0.861892i	$-0.669279\pi$
$613$	− 1056.68i	− 1.72378i	0.507092	−	0.861892i	$-0.330721\pi$
$614$	0	0
$615$	332.109i	0.540015i
$616$	0	0
$617$	−39.6456	−0.0642554	−0.0321277	−	0.999484i	$-0.510228\pi$
$617$	−39.6456	−0.0642554	−0.0321277	+	0.999484i	$0.510228\pi$
$618$	0	0
$619$	−479.135	−0.774047	−0.387023	−	0.922070i	$-0.626497\pi$
$619$	−479.135	−0.774047	−0.387023	+	0.922070i	$0.626497\pi$
$620$	0	0
$621$	− 65.2602i	− 0.105089i
$622$	0	0
$623$	277.592i	0.445574i
$624$	0	0
$625$	−781.001	−1.24960
$626$	0	0
$627$	291.593	0.465060
$628$	0	0
$629$	878.165i	1.39613i
$630$	0	0
$631$	− 1038.15i	− 1.64525i	−0.568583	−	0.822626i	$-0.692508\pi$
$631$	− 1038.15i	− 1.64525i	0.568583	−	0.822626i	$-0.307492\pi$
$632$	0	0
$633$	−501.104	−0.791634
$634$	0	0
$635$	−195.831	−0.308395
$636$	0	0
$637$	122.986i	0.193070i
$638$	0	0
$639$	− 195.249i	− 0.305554i
$640$	0	0
$641$	770.901	1.20265	0.601327	−	0.799003i	$-0.294639\pi$
$641$	770.901	1.20265	0.601327	+	0.799003i	$0.294639\pi$
$642$	0	0
$643$	−610.183	−0.948963	−0.474482	−	0.880265i	$-0.657364\pi$
$643$	−610.183	−0.948963	−0.474482	+	0.880265i	$0.657364\pi$
$644$	0	0
$645$	− 583.455i	− 0.904581i
$646$	0	0
$647$	1049.62i	1.62228i	0.584850	+	0.811142i	$0.301153\pi$
$647$	1049.62i	1.62228i	−0.584850	+	0.811142i	$0.698847\pi$
$648$	0	0
$649$	683.910	1.05379
$650$	0	0
$651$	240.171	0.368927
$652$	0	0
$653$	− 743.849i	− 1.13913i	−0.821948	−	0.569563i	$-0.807112\pi$
$653$	− 743.849i	− 1.13913i	0.821948	−	0.569563i	$-0.192888\pi$
$654$	0	0
$655$	152.870i	0.233389i
$656$	0	0
$657$	−213.840	−0.325479
$658$	0	0
$659$	−476.526	−0.723105	−0.361552	−	0.932352i	$-0.617753\pi$
$659$	−476.526	−0.723105	−0.361552	+	0.932352i	$0.617753\pi$
$660$	0	0
$661$	− 970.379i	− 1.46805i	−0.679124	−	0.734024i	$-0.737640\pi$
$661$	− 970.379i	− 1.46805i	0.679124	−	0.734024i	$-0.262360\pi$
$662$	0	0
$663$	528.885i	0.797715i
$664$	0	0
$665$	149.615	0.224986
$666$	0	0
$667$	148.678	0.222905
$668$	0	0
$669$	232.458i	0.347472i
$670$	0	0
$671$	− 734.765i	− 1.09503i
$672$	0	0
$673$	−399.194	−0.593156	−0.296578	−	0.955009i	$-0.595845\pi$
$673$	−399.194	−0.593156	−0.296578	+	0.955009i	$0.595845\pi$
$674$	0	0
$675$	67.5440	0.100065
$676$	0	0
$677$	− 371.606i	− 0.548900i	−0.961601	−	0.274450i	$-0.911504\pi$
$677$	− 371.606i	− 0.548900i	0.961601	−	0.274450i	$-0.0884958\pi$
$678$	0	0
$679$	81.9364i	0.120672i
$680$	0	0
$681$	−608.759	−0.893920
$682$	0	0
$683$	863.088	1.26367	0.631836	−	0.775102i	$-0.282302\pi$
$683$	863.088	1.26367	0.631836	+	0.775102i	$0.282302\pi$
$684$	0	0
$685$	− 544.478i	− 0.794858i
$686$	0	0
$687$	− 470.145i	− 0.684346i
$688$	0	0
$689$	1027.15	1.49079
$690$	0	0
$691$	128.866	0.186492	0.0932461	−	0.995643i	$-0.470276\pi$
$691$	128.866	0.186492	0.0932461	+	0.995643i	$0.470276\pi$
$692$	0	0
$693$	145.661i	0.210190i
$694$	0	0
$695$	− 778.064i	− 1.11952i
$696$	0	0
$697$	−540.604	−0.775615
$698$	0	0
$699$	−499.057	−0.713958
$700$	0	0
$701$	35.1038i	0.0500767i	0.999686	+	0.0250384i	$0.00797080\pi$
$701$	35.1038i	0.0500767i	−0.999686	+	0.0250384i	$0.992029\pi$
$702$	0	0
$703$	− 463.526i	− 0.659354i
$704$	0	0
$705$	−101.941	−0.144597
$706$	0	0
$707$	−331.984	−0.469567
$708$	0	0
$709$	355.246i	0.501053i	0.968110	+	0.250526i	$0.0806037\pi$
$709$	355.246i	0.501053i	−0.968110	+	0.250526i	$0.919396\pi$
$710$	0	0
$711$	392.032i	0.551381i
$712$	0	0
$713$	−658.231	−0.923185
$714$	0	0
$715$	1987.54	2.77978
$716$	0	0
$717$	427.822i	0.596683i
$718$	0	0
$719$	− 543.096i	− 0.755349i	−0.925938	−	0.377674i	$-0.876724\pi$
$719$	− 543.096i	− 0.755349i	0.925938	−	0.377674i	$-0.123276\pi$
$720$	0	0
$721$	291.975	0.404958
$722$	0	0
$723$	167.726	0.231987
$724$	0	0
$725$	153.881i	0.212249i
$726$	0	0
$727$	893.793i	1.22943i	0.788751	+	0.614713i	$0.210728\pi$
$727$	893.793i	1.22943i	−0.788751	+	0.614713i	$0.789272\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	949.742	1.29924
$732$	0	0
$733$	− 1282.23i	− 1.74929i	−0.484767	−	0.874643i	$-0.661096\pi$
$733$	− 1282.23i	− 1.74929i	0.484767	−	0.874643i	$-0.338904\pi$
$734$	0	0
$735$	74.7384i	0.101685i
$736$	0	0
$737$	1102.81	1.49635
$738$	0	0
$739$	−677.536	−0.916828	−0.458414	−	0.888739i	$-0.651582\pi$
$739$	−677.536	−0.916828	−0.458414	+	0.888739i	$0.651582\pi$
$740$	0	0
$741$	− 279.164i	− 0.376739i
$742$	0	0
$743$	− 1069.32i	− 1.43919i	−0.694393	−	0.719596i	$-0.744327\pi$
$743$	− 1069.32i	− 1.43919i	0.694393	−	0.719596i	$-0.255673\pi$
$744$	0	0
$745$	588.334	0.789711
$746$	0	0
$747$	−453.097	−0.606556
$748$	0	0
$749$	63.2042i	0.0843848i
$750$	0	0
$751$	− 634.789i	− 0.845259i	−0.906303	−	0.422629i	$-0.861107\pi$
$751$	− 634.789i	− 0.845259i	0.906303	−	0.422629i	$-0.138893\pi$
$752$	0	0
$753$	81.5362	0.108282
$754$	0	0
$755$	115.252	0.152651
$756$	0	0
$757$	− 216.902i	− 0.286528i	−0.989684	−	0.143264i	$-0.954240\pi$
$757$	− 216.902i	− 0.286528i	0.989684	−	0.143264i	$-0.0457599\pi$
$758$	0	0
$759$	− 399.210i	− 0.525968i
$760$	0	0
$761$	−471.719	−0.619868	−0.309934	−	0.950758i	$-0.600307\pi$
$761$	−471.719	−0.619868	−0.309934	+	0.950758i	$0.600307\pi$
$762$	0	0
$763$	−519.746	−0.681188
$764$	0	0
$765$	321.404i	0.420135i
$766$	0	0
$767$	− 654.759i	− 0.853662i
$768$	0	0
$769$	−1382.58	−1.79790	−0.898948	−	0.438056i	$-0.855667\pi$
$769$	−1382.58	−1.79790	−0.898948	+	0.438056i	$0.855667\pi$
$770$	0	0
$771$	74.9334	0.0971899
$772$	0	0
$773$	− 402.193i	− 0.520302i	−0.965568	−	0.260151i	$-0.916228\pi$
$773$	− 402.193i	− 0.520302i	0.965568	−	0.260151i	$-0.0837723\pi$
$774$	0	0
$775$	− 681.265i	− 0.879052i
$776$	0	0
$777$	231.548	0.298003
$778$	0	0
$779$	285.349	0.366302
$780$	0	0
$781$	− 1194.38i	− 1.52929i
$782$	0	0
$783$	61.5122i	0.0785596i
$784$	0	0
$785$	25.0778	0.0319462
$786$	0	0
$787$	−593.635	−0.754301	−0.377151	−	0.926152i	$-0.623096\pi$
$787$	−593.635	−0.754301	−0.377151	+	0.926152i	$0.623096\pi$
$788$	0	0
$789$	406.345i	0.515013i
$790$	0	0
$791$	− 285.328i	− 0.360718i
$792$	0	0
$793$	−703.446	−0.887070
$794$	0	0
$795$	624.201	0.785159
$796$	0	0
$797$	− 1173.37i	− 1.47224i	−0.676853	−	0.736118i	$-0.736657\pi$
$797$	− 1173.37i	− 1.47224i	0.676853	−	0.736118i	$-0.263343\pi$
$798$	0	0
$799$	− 165.938i	− 0.207682i
$800$	0	0
$801$	314.760	0.392959
$802$	0	0
$803$	−1308.10	−1.62902
$804$	0	0
$805$	− 204.833i	− 0.254451i
$806$	0	0
$807$	447.040i	0.553953i
$808$	0	0
$809$	−583.712	−0.721523	−0.360762	−	0.932658i	$-0.617483\pi$
$809$	−583.712	−0.721523	−0.360762	+	0.932658i	$0.617483\pi$
$810$	0	0
$811$	124.565	0.153595	0.0767973	−	0.997047i	$-0.475531\pi$
$811$	124.565	0.153595	0.0767973	+	0.997047i	$0.475531\pi$
$812$	0	0
$813$	58.7166i	0.0722221i
$814$	0	0
$815$	− 1196.98i	− 1.46869i
$816$	0	0
$817$	−501.307	−0.613595
$818$	0	0
$819$	139.453	0.170272
$820$	0	0
$821$	− 53.7147i	− 0.0654260i	−0.999465	−	0.0327130i	$-0.989585\pi$
$821$	− 53.7147i	− 0.0654260i	0.999465	−	0.0327130i	$-0.0104147\pi$
$822$	0	0
$823$	− 1141.75i	− 1.38730i	−0.720312	−	0.693651i	$-0.756001\pi$
$823$	− 1141.75i	− 1.38730i	0.720312	−	0.693651i	$-0.243999\pi$
$824$	0	0
$825$	413.180	0.500824
$826$	0	0
$827$	795.993	0.962507	0.481253	−	0.876582i	$-0.340182\pi$
$827$	795.993	0.962507	0.481253	+	0.876582i	$0.340182\pi$
$828$	0	0
$829$	1171.16i	1.41274i	0.707844	+	0.706369i	$0.249668\pi$
$829$	1171.16i	1.41274i	−0.707844	+	0.706369i	$0.750332\pi$
$830$	0	0
$831$	532.386i	0.640657i
$832$	0	0
$833$	−121.658	−0.146049
$834$	0	0
$835$	−285.954	−0.342459
$836$	0	0
$837$	− 272.329i	− 0.325363i
$838$	0	0
$839$	− 440.914i	− 0.525524i	−0.964861	−	0.262762i	$-0.915367\pi$
$839$	− 440.914i	− 0.525524i	0.964861	−	0.262762i	$-0.0846334\pi$
$840$	0	0
$841$	700.861	0.833366
$842$	0	0
$843$	−805.204	−0.955165
$844$	0	0
$845$	− 861.053i	− 1.01900i
$846$	0	0
$847$	570.904i	0.674031i
$848$	0	0
$849$	448.386	0.528134
$850$	0	0
$851$	−634.598	−0.745708
$852$	0	0
$853$	− 631.048i	− 0.739798i	−0.929072	−	0.369899i	$-0.879392\pi$
$853$	− 631.048i	− 0.739798i	0.929072	−	0.369899i	$-0.120608\pi$
$854$	0	0
$855$	− 169.648i	− 0.198419i
$856$	0	0
$857$	432.516	0.504686	0.252343	−	0.967638i	$-0.418799\pi$
$857$	432.516	0.504686	0.252343	+	0.967638i	$0.418799\pi$
$858$	0	0
$859$	−171.236	−0.199343	−0.0996716	−	0.995020i	$-0.531779\pi$
$859$	−171.236	−0.199343	−0.0996716	+	0.995020i	$0.531779\pi$
$860$	0	0
$861$	142.543i	0.165555i
$862$	0	0
$863$	− 908.096i	− 1.05225i	−0.850406	−	0.526127i	$-0.823644\pi$
$863$	− 908.096i	− 1.05225i	0.850406	−	0.526127i	$-0.176356\pi$
$864$	0	0
$865$	1582.71	1.82972
$866$	0	0
$867$	−22.6148	−0.0260839
$868$	0	0
$869$	2398.14i	2.75965i
$870$	0	0
$871$	− 1055.80i	− 1.21217i
$872$	0	0
$873$	92.9072	0.106423
$874$	0	0
$875$	−195.730	−0.223691
$876$	0	0
$877$	596.521i	0.680183i	0.940392	+	0.340092i	$0.110458\pi$
$877$	596.521i	0.680183i	−0.940392	+	0.340092i	$0.889542\pi$
$878$	0	0
$879$	− 97.4073i	− 0.110816i
$880$	0	0
$881$	−268.205	−0.304433	−0.152216	−	0.988347i	$-0.548641\pi$
$881$	−268.205	−0.304433	−0.152216	+	0.988347i	$0.548641\pi$
$882$	0	0
$883$	−434.461	−0.492028	−0.246014	−	0.969266i	$-0.579121\pi$
$883$	−434.461	−0.492028	−0.246014	+	0.969266i	$0.579121\pi$
$884$	0	0
$885$	− 397.897i	− 0.449601i
$886$	0	0
$887$	357.593i	0.403148i	0.979473	+	0.201574i	$0.0646057\pi$
$887$	357.593i	0.403148i	−0.979473	+	0.201574i	$0.935394\pi$
$888$	0	0
$889$	−84.0514	−0.0945460
$890$	0	0
$891$	165.164	0.185370
$892$	0	0
$893$	87.5878i	0.0980827i
$894$	0	0
$895$	1150.08i	1.28501i
$896$	0	0
$897$	−382.194	−0.426080
$898$	0	0
$899$	620.427	0.690131
$900$	0	0
$901$	1016.07i	1.12771i
$902$	0	0
$903$	− 250.421i	− 0.277321i
$904$	0	0
$905$	−75.5966	−0.0835322
$906$	0	0
$907$	−1201.18	−1.32434	−0.662170	−	0.749353i	$-0.730365\pi$
$907$	−1201.18	−1.32434	−0.662170	+	0.749353i	$0.730365\pi$
$908$	0	0
$909$	376.434i	0.414119i
$910$	0	0
$911$	1432.03i	1.57193i	0.618270	+	0.785966i	$0.287834\pi$
$911$	1432.03i	1.57193i	−0.618270	+	0.785966i	$0.712166\pi$
$912$	0	0
$913$	−2771.69	−3.03580
$914$	0	0
$915$	−427.484	−0.467196
$916$	0	0
$917$	65.6124i	0.0715511i
$918$	0	0
$919$	− 437.793i	− 0.476380i	−0.971219	−	0.238190i	$-0.923446\pi$
$919$	− 437.793i	− 0.476380i	0.971219	−	0.238190i	$-0.0765541\pi$
$920$	0	0
$921$	558.038	0.605904
$922$	0	0
$923$	−1143.47	−1.23886
$924$	0	0
$925$	− 656.805i	− 0.710059i
$926$	0	0
$927$	− 331.068i	− 0.357140i
$928$	0	0
$929$	1102.35	1.18660	0.593298	−	0.804983i	$-0.297826\pi$
$929$	1102.35	1.18660	0.593298	+	0.804983i	$0.297826\pi$
$930$	0	0
$931$	64.2155	0.0689748
$932$	0	0
$933$	− 8.30096i	− 0.00889706i
$934$	0	0
$935$	1966.09i	2.10277i
$936$	0	0
$937$	1227.69	1.31024	0.655118	−	0.755527i	$-0.272619\pi$
$937$	1227.69	1.31024	0.655118	+	0.755527i	$0.272619\pi$
$938$	0	0
$939$	−188.796	−0.201061
$940$	0	0
$941$	− 1235.97i	− 1.31347i	−0.754123	−	0.656733i	$-0.771938\pi$
$941$	− 1235.97i	− 1.31347i	0.754123	−	0.656733i	$-0.228062\pi$
$942$	0	0
$943$	− 390.662i	− 0.414276i
$944$	0	0
$945$	84.7454	0.0896777
$946$	0	0
$947$	353.554	0.373341	0.186671	−	0.982423i	$-0.440230\pi$
$947$	353.554	0.373341	0.186671	+	0.982423i	$0.440230\pi$
$948$	0	0
$949$	1252.34i	1.31965i
$950$	0	0
$951$	− 66.2788i	− 0.0696938i
$952$	0	0
$953$	−477.524	−0.501075	−0.250537	−	0.968107i	$-0.580607\pi$
$953$	−477.524	−0.501075	−0.250537	+	0.968107i	$0.580607\pi$
$954$	0	0
$955$	192.458	0.201526
$956$	0	0
$957$	376.282i	0.393190i
$958$	0	0
$959$	− 233.692i	− 0.243683i
$960$	0	0
$961$	−1785.78	−1.85825
$962$	0	0
$963$	71.6668	0.0744204
$964$	0	0
$965$	1681.72i	1.74272i
$966$	0	0
$967$	− 957.602i	− 0.990281i	−0.868813	−	0.495141i	$-0.835117\pi$
$967$	− 957.602i	− 0.990281i	0.868813	−	0.495141i	$-0.164883\pi$
$968$	0	0
$969$	276.151	0.284986
$970$	0	0
$971$	−1417.31	−1.45964	−0.729821	−	0.683639i	$-0.760396\pi$
$971$	−1417.31	−1.45964	−0.729821	+	0.683639i	$0.760396\pi$
$972$	0	0
$973$	− 333.948i	− 0.343215i
$974$	0	0
$975$	− 395.569i	− 0.405711i
$976$	0	0
$977$	−1820.53	−1.86339	−0.931693	−	0.363248i	$-0.881668\pi$
$977$	−1820.53	−1.86339	−0.931693	+	0.363248i	$0.881668\pi$
$978$	0	0
$979$	1925.45	1.96675
$980$	0	0
$981$	589.337i	0.600751i
$982$	0	0
$983$	789.256i	0.802905i	0.915880	+	0.401453i	$0.131495\pi$
$983$	789.256i	0.802905i	−0.915880	+	0.401453i	$0.868505\pi$
$984$	0	0
$985$	2266.88	2.30140
$986$	0	0
$987$	−43.7533	−0.0443296
$988$	0	0
$989$	686.322i	0.693956i
$990$	0	0
$991$	245.162i	0.247388i	0.992320	+	0.123694i	$0.0394742\pi$
$991$	245.162i	0.247388i	−0.992320	+	0.123694i	$0.960526\pi$
$992$	0	0
$993$	163.245	0.164395
$994$	0	0
$995$	−1359.42	−1.36625
$996$	0	0
$997$	− 1676.61i	− 1.68166i	−0.541302	−	0.840828i	$-0.682068\pi$
$997$	− 1676.61i	− 1.68166i	0.541302	−	0.840828i	$-0.317932\pi$
$998$	0	0
$999$	− 262.551i	− 0.262814i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.g.a.463.10		24
3.2	odd	2		2016.3.g.d.1135.4		24
4.3	odd	2		168.3.g.a.43.22	yes	24
8.3	odd	2	inner	672.3.g.a.463.3		24
8.5	even	2		168.3.g.a.43.21	✓	24
12.11	even	2		504.3.g.d.379.3		24
24.5	odd	2		504.3.g.d.379.4		24
24.11	even	2		2016.3.g.d.1135.21		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.g.a.43.21	✓	24	8.5	even	2
168.3.g.a.43.22	yes	24	4.3	odd	2
504.3.g.d.379.3		24	12.11	even	2
504.3.g.d.379.4		24	24.5	odd	2
672.3.g.a.463.3		24	8.3	odd	2	inner
672.3.g.a.463.10		24	1.1	even	1	trivial
2016.3.g.d.1135.4		24	3.2	odd	2
2016.3.g.d.1135.21		24	24.11	even	2

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +6.16432i q^{5} +2.64575i q^{7} +3.00000 q^{9} +18.3516 q^{11} -17.5694i q^{13} -10.6769i q^{15} +17.3798 q^{17} -9.17365 q^{19} -4.58258i q^{21} +12.5593i q^{23} -12.9988 q^{25} -5.19615 q^{27} -11.8380i q^{29} +52.4097i q^{31} -31.7859 q^{33} -16.3093 q^{35} +50.5280i q^{37} +30.4311i q^{39} -31.1053 q^{41} +54.6464 q^{43} +18.4930i q^{45} -9.54776i q^{47} -7.00000 q^{49} -30.1027 q^{51} +58.4627i q^{53} +113.125i q^{55} +15.8892 q^{57} +37.2670 q^{59} -40.0382i q^{61} +7.93725i q^{63} +108.303 q^{65} +60.0932 q^{67} -21.7534i q^{69} -65.0830i q^{71} -71.2799 q^{73} +22.5147 q^{75} +48.5538i q^{77} +130.677i q^{79} +9.00000 q^{81} -151.032 q^{83} +107.135i q^{85} +20.5041i q^{87} +104.920 q^{89} +46.4842 q^{91} -90.7763i q^{93} -56.5493i q^{95} +30.9691 q^{97} +55.0548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{9} - 32 q^{11} + 16 q^{17} + 64 q^{19} - 72 q^{25} - 80 q^{41} - 32 q^{43} - 168 q^{49} - 192 q^{51} - 192 q^{65} + 32 q^{67} - 240 q^{73} + 384 q^{75} + 216 q^{81} + 320 q^{83} + 400 q^{89}+ \cdots - 96 q^{99}+O(q^{100}) \) 24 * q + 72 * q^9 - 32 * q^11 + 16 * q^17 + 64 * q^19 - 72 * q^25 - 80 * q^41 - 32 * q^43 - 168 * q^49 - 192 * q^51 - 192 * q^65 + 32 * q^67 - 240 * q^73 + 384 * q^75 + 216 * q^81 + 320 * q^83 + 400 * q^89 + 144 * q^97 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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