LMFDB - Embedded newform 3808.2.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3808,2,Mod(1,3808)] 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("3808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17557 q^{3} +0.175571 q^{5} -1.00000 q^{7} -1.61803 q^{9} -1.28408 q^{11} +1.67853 q^{13} +0.206396 q^{15} +1.00000 q^{17} +3.53077 q^{19} -1.17557 q^{21} -2.07768 q^{23} -4.96917 q^{25} -5.42882 q^{27} -8.22835 q^{29} +1.42998 q^{31} -1.50953 q^{33} -0.175571 q^{35} +7.11798 q^{37} +1.97323 q^{39} -10.6150 q^{41} -2.23420 q^{43} -0.284079 q^{45} -3.43945 q^{47} +1.00000 q^{49} +1.17557 q^{51} +3.65833 q^{53} -0.225446 q^{55} +4.15067 q^{57} -4.45089 q^{59} -3.24482 q^{61} +1.61803 q^{63} +0.294701 q^{65} +3.16785 q^{67} -2.44246 q^{69} +4.61147 q^{71} +2.03373 q^{73} -5.84162 q^{75} +1.28408 q^{77} +0.932938 q^{79} -1.52786 q^{81} -5.38301 q^{83} +0.175571 q^{85} -9.67301 q^{87} -13.7775 q^{89} -1.67853 q^{91} +1.68104 q^{93} +0.619899 q^{95} -11.6668 q^{97} +2.07768 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{13} + 10 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{23} - 6 q^{25} + 6 q^{29} + 16 q^{31} + 4 q^{35} - 8 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^13 + 10 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^23 - 6 * q^25 + 6 * q^29 + 16 * q^31 + 4 * q^35 - 8 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 2 * q^45 + 14 * q^47 + 4 * q^49 - 14 * q^53 + 2 * q^55 - 10 * q^57 - 12 * q^59 - 2 * q^61 + 2 * q^63 - 8 * q^65 - 22 * q^67 - 10 * q^69 + 16 * q^71 - 14 * q^73 - 20 * q^75 + 2 * q^77 + 10 * q^79 - 24 * q^81 + 6 * q^83 - 4 * q^85 - 22 * q^89 + 2 * q^91 - 6 * q^95 - 12 * q^97 - 4 * q^99

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
$3$	1.17557	0.678716	0.339358	−	0.940657i	$-0.389790\pi$
$3$	1.17557	0.678716	0.339358	+	0.940657i	$0.389790\pi$
$4$	0	0
$5$	0.175571	0.0785175	0.0392588	−	0.999229i	$-0.487500\pi$
$5$	0.175571	0.0785175	0.0392588	+	0.999229i	$0.487500\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.61803	−0.539345
$10$	0	0
$11$	−1.28408	−0.387164	−0.193582	−	0.981084i	$-0.562011\pi$
$11$	−1.28408	−0.387164	−0.193582	+	0.981084i	$0.562011\pi$
$12$	0	0
$13$	1.67853	0.465541	0.232770	−	0.972532i	$-0.425221\pi$
$13$	1.67853	0.465541	0.232770	+	0.972532i	$0.425221\pi$
$14$	0	0
$15$	0.206396	0.0532911
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.53077	0.810014	0.405007	−	0.914314i	$-0.367269\pi$
$19$	3.53077	0.810014	0.405007	+	0.914314i	$0.367269\pi$
$20$	0	0
$21$	−1.17557	−0.256531
$22$	0	0
$23$	−2.07768	−0.433227	−0.216613	−	0.976257i	$-0.569501\pi$
$23$	−2.07768	−0.433227	−0.216613	+	0.976257i	$0.569501\pi$
$24$	0	0
$25$	−4.96917	−0.993835
$26$	0	0
$27$	−5.42882	−1.04478
$28$	0	0
$29$	−8.22835	−1.52797	−0.763983	−	0.645236i	$-0.776759\pi$
$29$	−8.22835	−1.52797	−0.763983	+	0.645236i	$0.776759\pi$
$30$	0	0
$31$	1.42998	0.256831	0.128416	−	0.991720i	$-0.459011\pi$
$31$	1.42998	0.256831	0.128416	+	0.991720i	$0.459011\pi$
$32$	0	0
$33$	−1.50953	−0.262775
$34$	0	0
$35$	−0.175571	−0.0296768
$36$	0	0
$37$	7.11798	1.17019	0.585094	−	0.810965i	$-0.301058\pi$
$37$	7.11798	1.17019	0.585094	+	0.810965i	$0.301058\pi$
$38$	0	0
$39$	1.97323	0.315970
$40$	0	0
$41$	−10.6150	−1.65779	−0.828894	−	0.559406i	$-0.811029\pi$
$41$	−10.6150	−1.65779	−0.828894	+	0.559406i	$0.811029\pi$
$42$	0	0
$43$	−2.23420	−0.340713	−0.170356	−	0.985383i	$-0.554492\pi$
$43$	−2.23420	−0.340713	−0.170356	+	0.985383i	$0.554492\pi$
$44$	0	0
$45$	−0.284079	−0.0423480
$46$	0	0
$47$	−3.43945	−0.501695	−0.250847	−	0.968027i	$-0.580709\pi$
$47$	−3.43945	−0.501695	−0.250847	+	0.968027i	$0.580709\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.17557	0.164613
$52$	0	0
$53$	3.65833	0.502510	0.251255	−	0.967921i	$-0.419157\pi$
$53$	3.65833	0.502510	0.251255	+	0.967921i	$0.419157\pi$
$54$	0	0
$55$	−0.225446	−0.0303992
$56$	0	0
$57$	4.15067	0.549769
$58$	0	0
$59$	−4.45089	−0.579457	−0.289728	−	0.957109i	$-0.593565\pi$
$59$	−4.45089	−0.579457	−0.289728	+	0.957109i	$0.593565\pi$
$60$	0	0
$61$	−3.24482	−0.415457	−0.207729	−	0.978186i	$-0.566607\pi$
$61$	−3.24482	−0.415457	−0.207729	+	0.978186i	$0.566607\pi$
$62$	0	0
$63$	1.61803	0.203853
$64$	0	0
$65$	0.294701	0.0365531
$66$	0	0
$67$	3.16785	0.387015	0.193507	−	0.981099i	$-0.438014\pi$
$67$	3.16785	0.387015	0.193507	+	0.981099i	$0.438014\pi$
$68$	0	0
$69$	−2.44246	−0.294038
$70$	0	0
$71$	4.61147	0.547281	0.273640	−	0.961832i	$-0.411772\pi$
$71$	4.61147	0.547281	0.273640	+	0.961832i	$0.411772\pi$
$72$	0	0
$73$	2.03373	0.238030	0.119015	−	0.992892i	$-0.462026\pi$
$73$	2.03373	0.238030	0.119015	+	0.992892i	$0.462026\pi$
$74$	0	0
$75$	−5.84162	−0.674532
$76$	0	0
$77$	1.28408	0.146334
$78$	0	0
$79$	0.932938	0.104964	0.0524819	−	0.998622i	$-0.483287\pi$
$79$	0.932938	0.104964	0.0524819	+	0.998622i	$0.483287\pi$
$80$	0	0
$81$	−1.52786	−0.169763
$82$	0	0
$83$	−5.38301	−0.590862	−0.295431	−	0.955364i	$-0.595463\pi$
$83$	−5.38301	−0.590862	−0.295431	+	0.955364i	$0.595463\pi$
$84$	0	0
$85$	0.175571	0.0190433
$86$	0	0
$87$	−9.67301	−1.03706
$88$	0	0
$89$	−13.7775	−1.46041	−0.730204	−	0.683229i	$-0.760575\pi$
$89$	−13.7775	−1.46041	−0.730204	+	0.683229i	$0.760575\pi$
$90$	0	0
$91$	−1.67853	−0.175958
$92$	0	0
$93$	1.68104	0.174316
$94$	0	0
$95$	0.619899	0.0636003
$96$	0	0
$97$	−11.6668	−1.18458	−0.592290	−	0.805725i	$-0.701776\pi$
$97$	−11.6668	−1.18458	−0.592290	+	0.805725i	$0.701776\pi$
$98$	0	0
$99$	2.07768	0.208815
$100$	0	0
$101$	−9.96853	−0.991906	−0.495953	−	0.868349i	$-0.665181\pi$
$101$	−9.96853	−0.991906	−0.495953	+	0.868349i	$0.665181\pi$
$102$	0	0
$103$	−10.5362	−1.03816	−0.519080	−	0.854725i	$-0.673725\pi$
$103$	−10.5362	−1.03816	−0.519080	+	0.854725i	$0.673725\pi$
$104$	0	0
$105$	−0.206396	−0.0201421
$106$	0	0
$107$	1.71290	0.165593	0.0827963	−	0.996566i	$-0.473615\pi$
$107$	1.71290	0.165593	0.0827963	+	0.996566i	$0.473615\pi$
$108$	0	0
$109$	−5.56597	−0.533123	−0.266561	−	0.963818i	$-0.585888\pi$
$109$	−5.56597	−0.533123	−0.266561	+	0.963818i	$0.585888\pi$
$110$	0	0
$111$	8.36768	0.794225
$112$	0	0
$113$	7.24367	0.681427	0.340714	−	0.940167i	$-0.389331\pi$
$113$	7.24367	0.681427	0.340714	+	0.940167i	$0.389331\pi$
$114$	0	0
$115$	−0.364780	−0.0340159
$116$	0	0
$117$	−2.71592	−0.251087
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−9.35114	−0.850104
$122$	0	0
$123$	−12.4787	−1.12517
$124$	0	0
$125$	−1.75029	−0.156551
$126$	0	0
$127$	12.4868	1.10803	0.554013	−	0.832508i	$-0.313096\pi$
$127$	12.4868	1.10803	0.554013	+	0.832508i	$0.313096\pi$
$128$	0	0
$129$	−2.62646	−0.231247
$130$	0	0
$131$	−9.63342	−0.841676	−0.420838	−	0.907136i	$-0.638264\pi$
$131$	−9.63342	−0.841676	−0.420838	+	0.907136i	$0.638264\pi$
$132$	0	0
$133$	−3.53077	−0.306156
$134$	0	0
$135$	−0.953141	−0.0820334
$136$	0	0
$137$	−14.4282	−1.23268	−0.616341	−	0.787479i	$-0.711386\pi$
$137$	−14.4282	−1.23268	−0.616341	+	0.787479i	$0.711386\pi$
$138$	0	0
$139$	6.49151	0.550603	0.275302	−	0.961358i	$-0.411222\pi$
$139$	6.49151	0.550603	0.275302	+	0.961358i	$0.411222\pi$
$140$	0	0
$141$	−4.04331	−0.340508
$142$	0	0
$143$	−2.15537	−0.180241
$144$	0	0
$145$	−1.44466	−0.119972
$146$	0	0
$147$	1.17557	0.0969594
$148$	0	0
$149$	−5.25376	−0.430405	−0.215203	−	0.976569i	$-0.569041\pi$
$149$	−5.25376	−0.430405	−0.215203	+	0.976569i	$0.569041\pi$
$150$	0	0
$151$	−5.00998	−0.407706	−0.203853	−	0.979001i	$-0.565347\pi$
$151$	−5.00998	−0.407706	−0.203853	+	0.979001i	$0.565347\pi$
$152$	0	0
$153$	−1.61803	−0.130810
$154$	0	0
$155$	0.251062	0.0201658
$156$	0	0
$157$	−3.59273	−0.286731	−0.143366	−	0.989670i	$-0.545793\pi$
$157$	−3.59273	−0.286731	−0.143366	+	0.989670i	$0.545793\pi$
$158$	0	0
$159$	4.30062	0.341062
$160$	0	0
$161$	2.07768	0.163744
$162$	0	0
$163$	−13.6381	−1.06822	−0.534110	−	0.845415i	$-0.679353\pi$
$163$	−13.6381	−1.06822	−0.534110	+	0.845415i	$0.679353\pi$
$164$	0	0
$165$	−0.265028	−0.0206324
$166$	0	0
$167$	24.2752	1.87847	0.939236	−	0.343272i	$-0.111535\pi$
$167$	24.2752	1.87847	0.939236	+	0.343272i	$0.111535\pi$
$168$	0	0
$169$	−10.1825	−0.783272
$170$	0	0
$171$	−5.71290	−0.436877
$172$	0	0
$173$	1.63407	0.124236	0.0621179	−	0.998069i	$-0.480215\pi$
$173$	1.63407	0.124236	0.0621179	+	0.998069i	$0.480215\pi$
$174$	0	0
$175$	4.96917	0.375634
$176$	0	0
$177$	−5.23234	−0.393287
$178$	0	0
$179$	−19.9674	−1.49243	−0.746216	−	0.665704i	$-0.768131\pi$
$179$	−19.9674	−1.49243	−0.746216	+	0.665704i	$0.768131\pi$
$180$	0	0
$181$	−5.40456	−0.401718	−0.200859	−	0.979620i	$-0.564373\pi$
$181$	−5.40456	−0.401718	−0.200859	+	0.979620i	$0.564373\pi$
$182$	0	0
$183$	−3.81452	−0.281977
$184$	0	0
$185$	1.24971	0.0918803
$186$	0	0
$187$	−1.28408	−0.0939012
$188$	0	0
$189$	5.42882	0.394889
$190$	0	0
$191$	25.9295	1.87619	0.938096	−	0.346376i	$-0.112588\pi$
$191$	25.9295	1.87619	0.938096	+	0.346376i	$0.112588\pi$
$192$	0	0
$193$	6.15067	0.442735	0.221367	−	0.975190i	$-0.428948\pi$
$193$	6.15067	0.442735	0.221367	+	0.975190i	$0.428948\pi$
$194$	0	0
$195$	0.346441	0.0248092
$196$	0	0
$197$	4.55742	0.324703	0.162351	−	0.986733i	$-0.448092\pi$
$197$	4.55742	0.324703	0.162351	+	0.986733i	$0.448092\pi$
$198$	0	0
$199$	16.4710	1.16760	0.583799	−	0.811899i	$-0.301566\pi$
$199$	16.4710	1.16760	0.583799	+	0.811899i	$0.301566\pi$
$200$	0	0
$201$	3.72404	0.262673
$202$	0	0
$203$	8.22835	0.577517
$204$	0	0
$205$	−1.86368	−0.130165
$206$	0	0
$207$	3.36176	0.233659
$208$	0	0
$209$	−4.53379	−0.313609
$210$	0	0
$211$	7.18253	0.494466	0.247233	−	0.968956i	$-0.420479\pi$
$211$	7.18253	0.494466	0.247233	+	0.968956i	$0.420479\pi$
$212$	0	0
$213$	5.42111	0.371448
$214$	0	0
$215$	−0.392260	−0.0267519
$216$	0	0
$217$	−1.42998	−0.0970732
$218$	0	0
$219$	2.39079	0.161555
$220$	0	0
$221$	1.67853	0.112910
$222$	0	0
$223$	8.28659	0.554911	0.277455	−	0.960739i	$-0.410509\pi$
$223$	8.28659	0.554911	0.277455	+	0.960739i	$0.410509\pi$
$224$	0	0
$225$	8.04029	0.536020
$226$	0	0
$227$	1.46435	0.0971923	0.0485961	−	0.998819i	$-0.484525\pi$
$227$	1.46435	0.0971923	0.0485961	+	0.998819i	$0.484525\pi$
$228$	0	0
$229$	9.00168	0.594848	0.297424	−	0.954746i	$-0.403873\pi$
$229$	9.00168	0.594848	0.297424	+	0.954746i	$0.403873\pi$
$230$	0	0
$231$	1.50953	0.0993195
$232$	0	0
$233$	−5.05863	−0.331402	−0.165701	−	0.986176i	$-0.552989\pi$
$233$	−5.05863	−0.331402	−0.165701	+	0.986176i	$0.552989\pi$
$234$	0	0
$235$	−0.603865	−0.0393918
$236$	0	0
$237$	1.09673	0.0712406
$238$	0	0
$239$	6.94252	0.449074	0.224537	−	0.974466i	$-0.427913\pi$
$239$	6.94252	0.449074	0.224537	+	0.974466i	$0.427913\pi$
$240$	0	0
$241$	−21.0009	−1.35279	−0.676394	−	0.736540i	$-0.736458\pi$
$241$	−21.0009	−1.35279	−0.676394	+	0.736540i	$0.736458\pi$
$242$	0	0
$243$	14.4904	0.929557
$244$	0	0
$245$	0.175571	0.0112168
$246$	0	0
$247$	5.92651	0.377095
$248$	0	0
$249$	−6.32810	−0.401027
$250$	0	0
$251$	12.7800	0.806664	0.403332	−	0.915054i	$-0.367852\pi$
$251$	12.7800	0.806664	0.403332	+	0.915054i	$0.367852\pi$
$252$	0	0
$253$	2.66791	0.167730
$254$	0	0
$255$	0.206396	0.0129250
$256$	0	0
$257$	−1.86297	−0.116209	−0.0581045	−	0.998311i	$-0.518506\pi$
$257$	−1.86297	−0.116209	−0.0581045	+	0.998311i	$0.518506\pi$
$258$	0	0
$259$	−7.11798	−0.442290
$260$	0	0
$261$	13.3138	0.824101
$262$	0	0
$263$	−15.4377	−0.951926	−0.475963	−	0.879465i	$-0.657900\pi$
$263$	−15.4377	−0.951926	−0.475963	+	0.879465i	$0.657900\pi$
$264$	0	0
$265$	0.642294	0.0394558
$266$	0	0
$267$	−16.1964	−0.991202
$268$	0	0
$269$	−14.1516	−0.862841	−0.431420	−	0.902151i	$-0.641987\pi$
$269$	−14.1516	−0.862841	−0.431420	+	0.902151i	$0.641987\pi$
$270$	0	0
$271$	−16.7514	−1.01758	−0.508788	−	0.860892i	$-0.669906\pi$
$271$	−16.7514	−1.01758	−0.508788	+	0.860892i	$0.669906\pi$
$272$	0	0
$273$	−1.97323	−0.119425
$274$	0	0
$275$	6.38081	0.384778
$276$	0	0
$277$	−3.58638	−0.215485	−0.107742	−	0.994179i	$-0.534362\pi$
$277$	−3.58638	−0.215485	−0.107742	+	0.994179i	$0.534362\pi$
$278$	0	0
$279$	−2.31375	−0.138521
$280$	0	0
$281$	17.6604	1.05353	0.526766	−	0.850010i	$-0.323404\pi$
$281$	17.6604	1.05353	0.526766	+	0.850010i	$0.323404\pi$
$282$	0	0
$283$	9.73131	0.578466	0.289233	−	0.957259i	$-0.406600\pi$
$283$	9.73131	0.578466	0.289233	+	0.957259i	$0.406600\pi$
$284$	0	0
$285$	0.728735	0.0431665
$286$	0	0
$287$	10.6150	0.626585
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−13.7151	−0.803993
$292$	0	0
$293$	−13.7397	−0.802680	−0.401340	−	0.915929i	$-0.631455\pi$
$293$	−13.7397	−0.802680	−0.401340	+	0.915929i	$0.631455\pi$
$294$	0	0
$295$	−0.781445	−0.0454975
$296$	0	0
$297$	6.97104	0.404501
$298$	0	0
$299$	−3.48746	−0.201685
$300$	0	0
$301$	2.23420	0.128777
$302$	0	0
$303$	−11.7187	−0.673222
$304$	0	0
$305$	−0.569696	−0.0326207
$306$	0	0
$307$	9.31928	0.531879	0.265940	−	0.963990i	$-0.414318\pi$
$307$	9.31928	0.531879	0.265940	+	0.963990i	$0.414318\pi$
$308$	0	0
$309$	−12.3860	−0.704616
$310$	0	0
$311$	20.2260	1.14691	0.573455	−	0.819237i	$-0.305603\pi$
$311$	20.2260	1.14691	0.573455	+	0.819237i	$0.305603\pi$
$312$	0	0
$313$	5.12423	0.289638	0.144819	−	0.989458i	$-0.453740\pi$
$313$	5.12423	0.289638	0.144819	+	0.989458i	$0.453740\pi$
$314$	0	0
$315$	0.284079	0.0160060
$316$	0	0
$317$	−27.2923	−1.53289	−0.766444	−	0.642312i	$-0.777976\pi$
$317$	−27.2923	−1.53289	−0.766444	+	0.642312i	$0.777976\pi$
$318$	0	0
$319$	10.5659	0.591574
$320$	0	0
$321$	2.01364	0.112390
$322$	0	0
$323$	3.53077	0.196457
$324$	0	0
$325$	−8.34092	−0.462671
$326$	0	0
$327$	−6.54319	−0.361839
$328$	0	0
$329$	3.43945	0.189623
$330$	0	0
$331$	−30.5556	−1.67949	−0.839743	−	0.542985i	$-0.817294\pi$
$331$	−30.5556	−1.67949	−0.839743	+	0.542985i	$0.817294\pi$
$332$	0	0
$333$	−11.5171	−0.631135
$334$	0	0
$335$	0.556182	0.0303874
$336$	0	0
$337$	0.504429	0.0274780	0.0137390	−	0.999906i	$-0.495627\pi$
$337$	0.504429	0.0274780	0.0137390	+	0.999906i	$0.495627\pi$
$338$	0	0
$339$	8.51545	0.462496
$340$	0	0
$341$	−1.83620	−0.0994360
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.428825	−0.0230871
$346$	0	0
$347$	−18.7023	−1.00399	−0.501996	−	0.864870i	$-0.667401\pi$
$347$	−18.7023	−1.00399	−0.501996	+	0.864870i	$0.667401\pi$
$348$	0	0
$349$	−24.8600	−1.33073	−0.665364	−	0.746519i	$-0.731724\pi$
$349$	−24.8600	−1.33073	−0.665364	+	0.746519i	$0.731724\pi$
$350$	0	0
$351$	−9.11245	−0.486387
$352$	0	0
$353$	−21.2070	−1.12873	−0.564367	−	0.825524i	$-0.690880\pi$
$353$	−21.2070	−1.12873	−0.564367	+	0.825524i	$0.690880\pi$
$354$	0	0
$355$	0.809638	0.0429711
$356$	0	0
$357$	−1.17557	−0.0622178
$358$	0	0
$359$	17.1618	0.905766	0.452883	−	0.891570i	$-0.350395\pi$
$359$	17.1618	0.905766	0.452883	+	0.891570i	$0.350395\pi$
$360$	0	0
$361$	−6.53367	−0.343878
$362$	0	0
$363$	−10.9929	−0.576979
$364$	0	0
$365$	0.357063	0.0186895
$366$	0	0
$367$	28.5135	1.48839	0.744196	−	0.667962i	$-0.232833\pi$
$367$	28.5135	1.48839	0.744196	+	0.667962i	$0.232833\pi$
$368$	0	0
$369$	17.1755	0.894119
$370$	0	0
$371$	−3.65833	−0.189931
$372$	0	0
$373$	9.74843	0.504754	0.252377	−	0.967629i	$-0.418788\pi$
$373$	9.74843	0.504754	0.252377	+	0.967629i	$0.418788\pi$
$374$	0	0
$375$	−2.05759	−0.106254
$376$	0	0
$377$	−13.8115	−0.711331
$378$	0	0
$379$	−1.25782	−0.0646099	−0.0323050	−	0.999478i	$-0.510285\pi$
$379$	−1.25782	−0.0646099	−0.0323050	+	0.999478i	$0.510285\pi$
$380$	0	0
$381$	14.6791	0.752035
$382$	0	0
$383$	20.5428	1.04969	0.524844	−	0.851198i	$-0.324124\pi$
$383$	20.5428	1.04969	0.524844	+	0.851198i	$0.324124\pi$
$384$	0	0
$385$	0.225446	0.0114898
$386$	0	0
$387$	3.61502	0.183762
$388$	0	0
$389$	−18.2255	−0.924068	−0.462034	−	0.886862i	$-0.652880\pi$
$389$	−18.2255	−0.924068	−0.462034	+	0.886862i	$0.652880\pi$
$390$	0	0
$391$	−2.07768	−0.105073
$392$	0	0
$393$	−11.3248	−0.571259
$394$	0	0
$395$	0.163796	0.00824149
$396$	0	0
$397$	23.2775	1.16826	0.584132	−	0.811658i	$-0.301435\pi$
$397$	23.2775	1.16826	0.584132	+	0.811658i	$0.301435\pi$
$398$	0	0
$399$	−4.15067	−0.207793
$400$	0	0
$401$	22.1845	1.10784	0.553921	−	0.832569i	$-0.313131\pi$
$401$	22.1845	1.10784	0.553921	+	0.832569i	$0.313131\pi$
$402$	0	0
$403$	2.40026	0.119566
$404$	0	0
$405$	−0.268248	−0.0133293
$406$	0	0
$407$	−9.14005	−0.453055
$408$	0	0
$409$	17.0211	0.841641	0.420820	−	0.907144i	$-0.361742\pi$
$409$	17.0211	0.841641	0.420820	+	0.907144i	$0.361742\pi$
$410$	0	0
$411$	−16.9613	−0.836641
$412$	0	0
$413$	4.45089	0.219014
$414$	0	0
$415$	−0.945097	−0.0463930
$416$	0	0
$417$	7.63123	0.373703
$418$	0	0
$419$	18.5603	0.906728	0.453364	−	0.891325i	$-0.350224\pi$
$419$	18.5603	0.906728	0.453364	+	0.891325i	$0.350224\pi$
$420$	0	0
$421$	13.0307	0.635078	0.317539	−	0.948245i	$-0.397144\pi$
$421$	13.0307	0.635078	0.317539	+	0.948245i	$0.397144\pi$
$422$	0	0
$423$	5.56514	0.270586
$424$	0	0
$425$	−4.96917	−0.241040
$426$	0	0
$427$	3.24482	0.157028
$428$	0	0
$429$	−2.53379	−0.122332
$430$	0	0
$431$	−6.52097	−0.314104	−0.157052	−	0.987590i	$-0.550199\pi$
$431$	−6.52097	−0.314104	−0.157052	+	0.987590i	$0.550199\pi$
$432$	0	0
$433$	17.1820	0.825715	0.412858	−	0.910796i	$-0.364531\pi$
$433$	17.1820	0.825715	0.412858	+	0.910796i	$0.364531\pi$
$434$	0	0
$435$	−1.69829	−0.0814270
$436$	0	0
$437$	−7.33582	−0.350920
$438$	0	0
$439$	18.6382	0.889552	0.444776	−	0.895642i	$-0.353283\pi$
$439$	18.6382	0.889552	0.444776	+	0.895642i	$0.353283\pi$
$440$	0	0
$441$	−1.61803	−0.0770492
$442$	0	0
$443$	−7.22795	−0.343410	−0.171705	−	0.985148i	$-0.554928\pi$
$443$	−7.22795	−0.343410	−0.171705	+	0.985148i	$0.554928\pi$
$444$	0	0
$445$	−2.41892	−0.114668
$446$	0	0
$447$	−6.17617	−0.292123
$448$	0	0
$449$	1.70479	0.0804540	0.0402270	−	0.999191i	$-0.487192\pi$
$449$	1.70479	0.0804540	0.0402270	+	0.999191i	$0.487192\pi$
$450$	0	0
$451$	13.6305	0.641836
$452$	0	0
$453$	−5.88958	−0.276717
$454$	0	0
$455$	−0.294701	−0.0138158
$456$	0	0
$457$	26.9713	1.26166	0.630831	−	0.775920i	$-0.282714\pi$
$457$	26.9713	1.26166	0.630831	+	0.775920i	$0.282714\pi$
$458$	0	0
$459$	−5.42882	−0.253396
$460$	0	0
$461$	−9.22616	−0.429705	−0.214853	−	0.976647i	$-0.568927\pi$
$461$	−9.22616	−0.429705	−0.214853	+	0.976647i	$0.568927\pi$
$462$	0	0
$463$	11.3836	0.529043	0.264522	−	0.964380i	$-0.414786\pi$
$463$	11.3836	0.529043	0.264522	+	0.964380i	$0.414786\pi$
$464$	0	0
$465$	0.295141	0.0136868
$466$	0	0
$467$	23.8214	1.10232	0.551161	−	0.834399i	$-0.314185\pi$
$467$	23.8214	1.10232	0.551161	+	0.834399i	$0.314185\pi$
$468$	0	0
$469$	−3.16785	−0.146278
$470$	0	0
$471$	−4.22351	−0.194609
$472$	0	0
$473$	2.86889	0.131912
$474$	0	0
$475$	−17.5450	−0.805020
$476$	0	0
$477$	−5.91930	−0.271026
$478$	0	0
$479$	2.76881	0.126510	0.0632552	−	0.997997i	$-0.479852\pi$
$479$	2.76881	0.126510	0.0632552	+	0.997997i	$0.479852\pi$
$480$	0	0
$481$	11.9477	0.544770
$482$	0	0
$483$	2.44246	0.111136
$484$	0	0
$485$	−2.04834	−0.0930103
$486$	0	0
$487$	21.6146	0.979450	0.489725	−	0.871877i	$-0.337097\pi$
$487$	21.6146	0.979450	0.489725	+	0.871877i	$0.337097\pi$
$488$	0	0
$489$	−16.0326	−0.725018
$490$	0	0
$491$	25.0739	1.13157	0.565785	−	0.824553i	$-0.308573\pi$
$491$	25.0739	1.13157	0.565785	+	0.824553i	$0.308573\pi$
$492$	0	0
$493$	−8.22835	−0.370586
$494$	0	0
$495$	0.364780	0.0163956
$496$	0	0
$497$	−4.61147	−0.206853
$498$	0	0
$499$	−40.6855	−1.82133	−0.910667	−	0.413141i	$-0.864431\pi$
$499$	−40.6855	−1.82133	−0.910667	+	0.413141i	$0.864431\pi$
$500$	0	0
$501$	28.5372	1.27495
$502$	0	0
$503$	−20.1499	−0.898440	−0.449220	−	0.893421i	$-0.648298\pi$
$503$	−20.1499	−0.898440	−0.449220	+	0.893421i	$0.648298\pi$
$504$	0	0
$505$	−1.75018	−0.0778820
$506$	0	0
$507$	−11.9703	−0.531619
$508$	0	0
$509$	1.05103	0.0465860	0.0232930	−	0.999729i	$-0.492585\pi$
$509$	1.05103	0.0465860	0.0232930	+	0.999729i	$0.492585\pi$
$510$	0	0
$511$	−2.03373	−0.0899669
$512$	0	0
$513$	−19.1679	−0.846284
$514$	0	0
$515$	−1.84984	−0.0815138
$516$	0	0
$517$	4.41652	0.194238
$518$	0	0
$519$	1.92096	0.0843209
$520$	0	0
$521$	11.0103	0.482370	0.241185	−	0.970479i	$-0.422464\pi$
$521$	11.0103	0.482370	0.241185	+	0.970479i	$0.422464\pi$
$522$	0	0
$523$	−33.3849	−1.45982	−0.729911	−	0.683542i	$-0.760438\pi$
$523$	−33.3849	−1.45982	−0.729911	+	0.683542i	$0.760438\pi$
$524$	0	0
$525$	5.84162	0.254949
$526$	0	0
$527$	1.42998	0.0622908
$528$	0	0
$529$	−18.6832	−0.812314
$530$	0	0
$531$	7.20170	0.312527
$532$	0	0
$533$	−17.8176	−0.771768
$534$	0	0
$535$	0.300735	0.0130019
$536$	0	0
$537$	−23.4731	−1.01294
$538$	0	0
$539$	−1.28408	−0.0553092
$540$	0	0
$541$	42.9755	1.84766	0.923831	−	0.382801i	$-0.125041\pi$
$541$	42.9755	1.84766	0.923831	+	0.382801i	$0.125041\pi$
$542$	0	0
$543$	−6.35345	−0.272652
$544$	0	0
$545$	−0.977219	−0.0418595
$546$	0	0
$547$	−6.50098	−0.277962	−0.138981	−	0.990295i	$-0.544383\pi$
$547$	−6.50098	−0.277962	−0.138981	+	0.990295i	$0.544383\pi$
$548$	0	0
$549$	5.25024	0.224075
$550$	0	0
$551$	−29.0524	−1.23767
$552$	0	0
$553$	−0.932938	−0.0396726
$554$	0	0
$555$	1.46912	0.0623606
$556$	0	0
$557$	−41.4720	−1.75723	−0.878613	−	0.477534i	$-0.841531\pi$
$557$	−41.4720	−1.75723	−0.878613	+	0.477534i	$0.841531\pi$
$558$	0	0
$559$	−3.75018	−0.158616
$560$	0	0
$561$	−1.50953	−0.0637322
$562$	0	0
$563$	2.29000	0.0965120	0.0482560	−	0.998835i	$-0.484634\pi$
$563$	2.29000	0.0965120	0.0482560	+	0.998835i	$0.484634\pi$
$564$	0	0
$565$	1.27178	0.0535040
$566$	0	0
$567$	1.52786	0.0641643
$568$	0	0
$569$	12.1626	0.509885	0.254942	−	0.966956i	$-0.417944\pi$
$569$	12.1626	0.509885	0.254942	+	0.966956i	$0.417944\pi$
$570$	0	0
$571$	−2.46112	−0.102995	−0.0514973	−	0.998673i	$-0.516399\pi$
$571$	−2.46112	−0.102995	−0.0514973	+	0.998673i	$0.516399\pi$
$572$	0	0
$573$	30.4819	1.27340
$574$	0	0
$575$	10.3244	0.430556
$576$	0	0
$577$	28.3772	1.18136	0.590680	−	0.806906i	$-0.298860\pi$
$577$	28.3772	1.18136	0.590680	+	0.806906i	$0.298860\pi$
$578$	0	0
$579$	7.23054	0.300491
$580$	0	0
$581$	5.38301	0.223325
$582$	0	0
$583$	−4.69758	−0.194554
$584$	0	0
$585$	−0.476836	−0.0197147
$586$	0	0
$587$	13.7764	0.568614	0.284307	−	0.958733i	$-0.408236\pi$
$587$	13.7764	0.568614	0.284307	+	0.958733i	$0.408236\pi$
$588$	0	0
$589$	5.04892	0.208037
$590$	0	0
$591$	5.35757	0.220381
$592$	0	0
$593$	−6.41197	−0.263308	−0.131654	−	0.991296i	$-0.542029\pi$
$593$	−6.41197	−0.263308	−0.131654	+	0.991296i	$0.542029\pi$
$594$	0	0
$595$	−0.175571	−0.00719769
$596$	0	0
$597$	19.3628	0.792467
$598$	0	0
$599$	−42.3587	−1.73073	−0.865365	−	0.501142i	$-0.832913\pi$
$599$	−42.3587	−1.73073	−0.865365	+	0.501142i	$0.832913\pi$
$600$	0	0
$601$	−5.01302	−0.204485	−0.102243	−	0.994759i	$-0.532602\pi$
$601$	−5.01302	−0.204485	−0.102243	+	0.994759i	$0.532602\pi$
$602$	0	0
$603$	−5.12569	−0.208734
$604$	0	0
$605$	−1.64178	−0.0667480
$606$	0	0
$607$	−40.4563	−1.64207	−0.821035	−	0.570879i	$-0.806603\pi$
$607$	−40.4563	−1.64207	−0.821035	+	0.570879i	$0.806603\pi$
$608$	0	0
$609$	9.67301	0.391970
$610$	0	0
$611$	−5.77322	−0.233559
$612$	0	0
$613$	19.1708	0.774300	0.387150	−	0.922017i	$-0.373460\pi$
$613$	19.1708	0.774300	0.387150	+	0.922017i	$0.373460\pi$
$614$	0	0
$615$	−2.19089	−0.0883453
$616$	0	0
$617$	22.7692	0.916655	0.458327	−	0.888783i	$-0.348449\pi$
$617$	22.7692	0.916655	0.458327	+	0.888783i	$0.348449\pi$
$618$	0	0
$619$	−45.9644	−1.84746	−0.923732	−	0.383039i	$-0.874878\pi$
$619$	−45.9644	−1.84746	−0.923732	+	0.383039i	$0.874878\pi$
$620$	0	0
$621$	11.2794	0.452626
$622$	0	0
$623$	13.7775	0.551982
$624$	0	0
$625$	24.5386	0.981543
$626$	0	0
$627$	−5.32979	−0.212851
$628$	0	0
$629$	7.11798	0.283812
$630$	0	0
$631$	39.0466	1.55442	0.777211	−	0.629240i	$-0.216634\pi$
$631$	39.0466	1.55442	0.777211	+	0.629240i	$0.216634\pi$
$632$	0	0
$633$	8.44357	0.335602
$634$	0	0
$635$	2.19232	0.0869994
$636$	0	0
$637$	1.67853	0.0665058
$638$	0	0
$639$	−7.46151	−0.295173
$640$	0	0
$641$	10.0465	0.396814	0.198407	−	0.980120i	$-0.436423\pi$
$641$	10.0465	0.396814	0.198407	+	0.980120i	$0.436423\pi$
$642$	0	0
$643$	14.2560	0.562204	0.281102	−	0.959678i	$-0.409300\pi$
$643$	14.2560	0.562204	0.281102	+	0.959678i	$0.409300\pi$
$644$	0	0
$645$	−0.461129	−0.0181570
$646$	0	0
$647$	−3.14184	−0.123519	−0.0617593	−	0.998091i	$-0.519671\pi$
$647$	−3.14184	−0.123519	−0.0617593	+	0.998091i	$0.519671\pi$
$648$	0	0
$649$	5.71530	0.224345
$650$	0	0
$651$	−1.68104	−0.0658851
$652$	0	0
$653$	−11.7536	−0.459955	−0.229977	−	0.973196i	$-0.573865\pi$
$653$	−11.7536	−0.459955	−0.229977	+	0.973196i	$0.573865\pi$
$654$	0	0
$655$	−1.69135	−0.0660863
$656$	0	0
$657$	−3.29064	−0.128380
$658$	0	0
$659$	13.5857	0.529222	0.264611	−	0.964355i	$-0.414756\pi$
$659$	13.5857	0.529222	0.264611	+	0.964355i	$0.414756\pi$
$660$	0	0
$661$	10.1929	0.396457	0.198228	−	0.980156i	$-0.436481\pi$
$661$	10.1929	0.396457	0.198228	+	0.980156i	$0.436481\pi$
$662$	0	0
$663$	1.97323	0.0766340
$664$	0	0
$665$	−0.619899	−0.0240386
$666$	0	0
$667$	17.0959	0.661956
$668$	0	0
$669$	9.74147	0.376627
$670$	0	0
$671$	4.16661	0.160850
$672$	0	0
$673$	45.9612	1.77167	0.885836	−	0.463998i	$-0.153585\pi$
$673$	45.9612	1.77167	0.885836	+	0.463998i	$0.153585\pi$
$674$	0	0
$675$	26.9768	1.03834
$676$	0	0
$677$	44.1316	1.69611	0.848057	−	0.529904i	$-0.177772\pi$
$677$	44.1316	1.69611	0.848057	+	0.529904i	$0.177772\pi$
$678$	0	0
$679$	11.6668	0.447729
$680$	0	0
$681$	1.72145	0.0659659
$682$	0	0
$683$	−35.9370	−1.37509	−0.687546	−	0.726141i	$-0.741312\pi$
$683$	−35.9370	−1.37509	−0.687546	+	0.726141i	$0.741312\pi$
$684$	0	0
$685$	−2.53316	−0.0967872
$686$	0	0
$687$	10.5821	0.403733
$688$	0	0
$689$	6.14062	0.233939
$690$	0	0
$691$	−37.5275	−1.42761	−0.713807	−	0.700343i	$-0.753031\pi$
$691$	−37.5275	−1.42761	−0.713807	+	0.700343i	$0.753031\pi$
$692$	0	0
$693$	−2.07768	−0.0789247
$694$	0	0
$695$	1.13972	0.0432320
$696$	0	0
$697$	−10.6150	−0.402072
$698$	0	0
$699$	−5.94678	−0.224928
$700$	0	0
$701$	−11.6195	−0.438863	−0.219431	−	0.975628i	$-0.570420\pi$
$701$	−11.6195	−0.438863	−0.219431	+	0.975628i	$0.570420\pi$
$702$	0	0
$703$	25.1319	0.947869
$704$	0	0
$705$	−0.709886	−0.0267359
$706$	0	0
$707$	9.96853	0.374905
$708$	0	0
$709$	−3.14102	−0.117963	−0.0589817	−	0.998259i	$-0.518785\pi$
$709$	−3.14102	−0.117963	−0.0589817	+	0.998259i	$0.518785\pi$
$710$	0	0
$711$	−1.50953	−0.0566116
$712$	0	0
$713$	−2.97104	−0.111266
$714$	0	0
$715$	−0.378419	−0.0141521
$716$	0	0
$717$	8.16142	0.304794
$718$	0	0
$719$	−0.526599	−0.0196388	−0.00981941	−	0.999952i	$-0.503126\pi$
$719$	−0.526599	−0.0196388	−0.00981941	+	0.999952i	$0.503126\pi$
$720$	0	0
$721$	10.5362	0.392388
$722$	0	0
$723$	−24.6881	−0.918159
$724$	0	0
$725$	40.8881	1.51855
$726$	0	0
$727$	35.6204	1.32109	0.660543	−	0.750788i	$-0.270326\pi$
$727$	35.6204	1.32109	0.660543	+	0.750788i	$0.270326\pi$
$728$	0	0
$729$	21.6180	0.800668
$730$	0	0
$731$	−2.23420	−0.0826350
$732$	0	0
$733$	2.41601	0.0892374	0.0446187	−	0.999004i	$-0.485793\pi$
$733$	2.41601	0.0892374	0.0446187	+	0.999004i	$0.485793\pi$
$734$	0	0
$735$	0.206396	0.00761301
$736$	0	0
$737$	−4.06777	−0.149838
$738$	0	0
$739$	−46.5039	−1.71067	−0.855336	−	0.518073i	$-0.826650\pi$
$739$	−46.5039	−1.71067	−0.855336	+	0.518073i	$0.826650\pi$
$740$	0	0
$741$	6.96703	0.255940
$742$	0	0
$743$	−37.4235	−1.37294	−0.686468	−	0.727160i	$-0.740840\pi$
$743$	−37.4235	−1.37294	−0.686468	+	0.727160i	$0.740840\pi$
$744$	0	0
$745$	−0.922406	−0.0337943
$746$	0	0
$747$	8.70989	0.318678
$748$	0	0
$749$	−1.71290	−0.0625881
$750$	0	0
$751$	22.6554	0.826707	0.413353	−	0.910571i	$-0.364357\pi$
$751$	22.6554	0.826707	0.413353	+	0.910571i	$0.364357\pi$
$752$	0	0
$753$	15.0238	0.547496
$754$	0	0
$755$	−0.879605	−0.0320121
$756$	0	0
$757$	−21.8732	−0.794993	−0.397497	−	0.917604i	$-0.630121\pi$
$757$	−21.8732	−0.794993	−0.397497	+	0.917604i	$0.630121\pi$
$758$	0	0
$759$	3.13632	0.113841
$760$	0	0
$761$	−35.1992	−1.27597	−0.637984	−	0.770049i	$-0.720232\pi$
$761$	−35.1992	−1.27597	−0.637984	+	0.770049i	$0.720232\pi$
$762$	0	0
$763$	5.56597	0.201501
$764$	0	0
$765$	−0.284079	−0.0102709
$766$	0	0
$767$	−7.47096	−0.269761
$768$	0	0
$769$	−13.5342	−0.488055	−0.244028	−	0.969768i	$-0.578469\pi$
$769$	−13.5342	−0.488055	−0.244028	+	0.969768i	$0.578469\pi$
$770$	0	0
$771$	−2.19005	−0.0788729
$772$	0	0
$773$	−30.4262	−1.09435	−0.547177	−	0.837017i	$-0.684298\pi$
$773$	−30.4262	−1.09435	−0.547177	+	0.837017i	$0.684298\pi$
$774$	0	0
$775$	−7.10581	−0.255248
$776$	0	0
$777$	−8.36768	−0.300189
$778$	0	0
$779$	−37.4792	−1.34283
$780$	0	0
$781$	−5.92149	−0.211888
$782$	0	0
$783$	44.6703	1.59639
$784$	0	0
$785$	−0.630778	−0.0225134
$786$	0	0
$787$	34.2565	1.22111	0.610557	−	0.791972i	$-0.290946\pi$
$787$	34.2565	1.22111	0.610557	+	0.791972i	$0.290946\pi$
$788$	0	0
$789$	−18.1480	−0.646088
$790$	0	0
$791$	−7.24367	−0.257555
$792$	0	0
$793$	−5.44654	−0.193412
$794$	0	0
$795$	0.755062	0.0267793
$796$	0	0
$797$	21.4922	0.761291	0.380646	−	0.924721i	$-0.375702\pi$
$797$	21.4922	0.761291	0.380646	+	0.924721i	$0.375702\pi$
$798$	0	0
$799$	−3.43945	−0.121679
$800$	0	0
$801$	22.2924	0.787663
$802$	0	0
$803$	−2.61147	−0.0921568
$804$	0	0
$805$	0.364780	0.0128568
$806$	0	0
$807$	−16.6362	−0.585624
$808$	0	0
$809$	47.2001	1.65947	0.829734	−	0.558159i	$-0.188492\pi$
$809$	47.2001	1.65947	0.829734	+	0.558159i	$0.188492\pi$
$810$	0	0
$811$	14.3999	0.505650	0.252825	−	0.967512i	$-0.418640\pi$
$811$	14.3999	0.505650	0.252825	+	0.967512i	$0.418640\pi$
$812$	0	0
$813$	−19.6925	−0.690645
$814$	0	0
$815$	−2.39445	−0.0838740
$816$	0	0
$817$	−7.88845	−0.275982
$818$	0	0
$819$	2.71592	0.0949020
$820$	0	0
$821$	20.9566	0.731390	0.365695	−	0.930735i	$-0.380831\pi$
$821$	20.9566	0.731390	0.365695	+	0.930735i	$0.380831\pi$
$822$	0	0
$823$	−15.6310	−0.544862	−0.272431	−	0.962175i	$-0.587828\pi$
$823$	−15.6310	−0.544862	−0.272431	+	0.962175i	$0.587828\pi$
$824$	0	0
$825$	7.50110	0.261155
$826$	0	0
$827$	18.3808	0.639164	0.319582	−	0.947559i	$-0.396458\pi$
$827$	18.3808	0.639164	0.319582	+	0.947559i	$0.396458\pi$
$828$	0	0
$829$	−9.54196	−0.331406	−0.165703	−	0.986176i	$-0.552989\pi$
$829$	−9.54196	−0.331406	−0.165703	+	0.986176i	$0.552989\pi$
$830$	0	0
$831$	−4.21605	−0.146253
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	4.26201	0.147493
$836$	0	0
$837$	−7.76309	−0.268332
$838$	0	0
$839$	13.6166	0.470096	0.235048	−	0.971984i	$-0.424475\pi$
$839$	13.6166	0.470096	0.235048	+	0.971984i	$0.424475\pi$
$840$	0	0
$841$	38.7058	1.33468
$842$	0	0
$843$	20.7611	0.715049
$844$	0	0
$845$	−1.78775	−0.0615005
$846$	0	0
$847$	9.35114	0.321309
$848$	0	0
$849$	11.4398	0.392614
$850$	0	0
$851$	−14.7889	−0.506957
$852$	0	0
$853$	−2.58000	−0.0883376	−0.0441688	−	0.999024i	$-0.514064\pi$
$853$	−2.58000	−0.0883376	−0.0441688	+	0.999024i	$0.514064\pi$
$854$	0	0
$855$	−1.00302	−0.0343025
$856$	0	0
$857$	−46.9063	−1.60229	−0.801145	−	0.598470i	$-0.795775\pi$
$857$	−46.9063	−1.60229	−0.801145	+	0.598470i	$0.795775\pi$
$858$	0	0
$859$	26.7384	0.912301	0.456151	−	0.889903i	$-0.349228\pi$
$859$	26.7384	0.912301	0.456151	+	0.889903i	$0.349228\pi$
$860$	0	0
$861$	12.4787	0.425273
$862$	0	0
$863$	4.91043	0.167153	0.0835765	−	0.996501i	$-0.473366\pi$
$863$	4.91043	0.167153	0.0835765	+	0.996501i	$0.473366\pi$
$864$	0	0
$865$	0.286894	0.00975469
$866$	0	0
$867$	1.17557	0.0399245
$868$	0	0
$869$	−1.19797	−0.0406382
$870$	0	0
$871$	5.31734	0.180171
$872$	0	0
$873$	18.8772	0.638897
$874$	0	0
$875$	1.75029	0.0591707
$876$	0	0
$877$	−5.58086	−0.188452	−0.0942261	−	0.995551i	$-0.530038\pi$
$877$	−5.58086	−0.188452	−0.0942261	+	0.995551i	$0.530038\pi$
$878$	0	0
$879$	−16.1520	−0.544792
$880$	0	0
$881$	20.6060	0.694234	0.347117	−	0.937822i	$-0.387161\pi$
$881$	20.6060	0.694234	0.347117	+	0.937822i	$0.387161\pi$
$882$	0	0
$883$	33.0088	1.11083	0.555417	−	0.831572i	$-0.312559\pi$
$883$	33.0088	1.11083	0.555417	+	0.831572i	$0.312559\pi$
$884$	0	0
$885$	−0.918644	−0.0308799
$886$	0	0
$887$	−34.4323	−1.15612	−0.578062	−	0.815993i	$-0.696191\pi$
$887$	−34.4323	−1.15612	−0.578062	+	0.815993i	$0.696191\pi$
$888$	0	0
$889$	−12.4868	−0.418794
$890$	0	0
$891$	1.96190	0.0657261
$892$	0	0
$893$	−12.1439	−0.406380
$894$	0	0
$895$	−3.50568	−0.117182
$896$	0	0
$897$	−4.09975	−0.136887
$898$	0	0
$899$	−11.7664	−0.392430
$900$	0	0
$901$	3.65833	0.121877
$902$	0	0
$903$	2.62646	0.0874032
$904$	0	0
$905$	−0.948882	−0.0315419
$906$	0	0
$907$	−47.0094	−1.56092	−0.780461	−	0.625204i	$-0.785016\pi$
$907$	−47.0094	−1.56092	−0.780461	+	0.625204i	$0.785016\pi$
$908$	0	0
$909$	16.1294	0.534979
$910$	0	0
$911$	−24.4595	−0.810378	−0.405189	−	0.914233i	$-0.632794\pi$
$911$	−24.4595	−0.810378	−0.405189	+	0.914233i	$0.632794\pi$
$912$	0	0
$913$	6.91220	0.228761
$914$	0	0
$915$	−0.669717	−0.0221402
$916$	0	0
$917$	9.63342	0.318124
$918$	0	0
$919$	−27.3622	−0.902595	−0.451298	−	0.892374i	$-0.649039\pi$
$919$	−27.3622	−0.902595	−0.451298	+	0.892374i	$0.649039\pi$
$920$	0	0
$921$	10.9555	0.360995
$922$	0	0
$923$	7.74050	0.254782
$924$	0	0
$925$	−35.3705	−1.16297
$926$	0	0
$927$	17.0479	0.559926
$928$	0	0
$929$	21.0522	0.690699	0.345349	−	0.938474i	$-0.387760\pi$
$929$	21.0522	0.690699	0.345349	+	0.938474i	$0.387760\pi$
$930$	0	0
$931$	3.53077	0.115716
$932$	0	0
$933$	23.7771	0.778426
$934$	0	0
$935$	−0.225446	−0.00737289
$936$	0	0
$937$	25.8171	0.843409	0.421704	−	0.906733i	$-0.361432\pi$
$937$	25.8171	0.843409	0.421704	+	0.906733i	$0.361432\pi$
$938$	0	0
$939$	6.02389	0.196582
$940$	0	0
$941$	40.2603	1.31245	0.656224	−	0.754566i	$-0.272153\pi$
$941$	40.2603	1.31245	0.656224	+	0.754566i	$0.272153\pi$
$942$	0	0
$943$	22.0546	0.718198
$944$	0	0
$945$	0.953141	0.0310057
$946$	0	0
$947$	52.0040	1.68990	0.844952	−	0.534843i	$-0.179629\pi$
$947$	52.0040	1.68990	0.844952	+	0.534843i	$0.179629\pi$
$948$	0	0
$949$	3.41368	0.110813
$950$	0	0
$951$	−32.0840	−1.04040
$952$	0	0
$953$	16.6751	0.540159	0.270079	−	0.962838i	$-0.412950\pi$
$953$	16.6751	0.540159	0.270079	+	0.962838i	$0.412950\pi$
$954$	0	0
$955$	4.55245	0.147314
$956$	0	0
$957$	12.4209	0.401511
$958$	0	0
$959$	14.4282	0.465910
$960$	0	0
$961$	−28.9552	−0.934038
$962$	0	0
$963$	−2.77154	−0.0893115
$964$	0	0
$965$	1.07988	0.0347624
$966$	0	0
$967$	−10.9615	−0.352497	−0.176248	−	0.984346i	$-0.556396\pi$
$967$	−10.9615	−0.352497	−0.176248	+	0.984346i	$0.556396\pi$
$968$	0	0
$969$	4.15067	0.133339
$970$	0	0
$971$	33.2272	1.06631	0.533156	−	0.846017i	$-0.321006\pi$
$971$	33.2272	1.06631	0.533156	+	0.846017i	$0.321006\pi$
$972$	0	0
$973$	−6.49151	−0.208108
$974$	0	0
$975$	−9.80534	−0.314022
$976$	0	0
$977$	−19.8481	−0.634996	−0.317498	−	0.948259i	$-0.602843\pi$
$977$	−19.8481	−0.634996	−0.317498	+	0.948259i	$0.602843\pi$
$978$	0	0
$979$	17.6913	0.565418
$980$	0	0
$981$	9.00592	0.287537
$982$	0	0
$983$	34.8380	1.11116	0.555581	−	0.831463i	$-0.312496\pi$
$983$	34.8380	1.11116	0.555581	+	0.831463i	$0.312496\pi$
$984$	0	0
$985$	0.800149	0.0254949
$986$	0	0
$987$	4.04331	0.128700
$988$	0	0
$989$	4.64197	0.147606
$990$	0	0
$991$	−32.8254	−1.04273	−0.521366	−	0.853333i	$-0.674577\pi$
$991$	−32.8254	−1.04273	−0.521366	+	0.853333i	$0.674577\pi$
$992$	0	0
$993$	−35.9202	−1.13989
$994$	0	0
$995$	2.89182	0.0916768
$996$	0	0
$997$	4.47716	0.141793	0.0708966	−	0.997484i	$-0.477414\pi$
$997$	4.47716	0.141793	0.0708966	+	0.997484i	$0.477414\pi$
$998$	0	0
$999$	−38.6423	−1.22259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.c.1.3	✓	4
4.3	odd	2		3808.2.a.d.1.2	yes	4
8.3	odd	2		7616.2.a.bm.1.3		4
8.5	even	2		7616.2.a.bl.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.c.1.3	✓	4	1.1	even	1	trivial
3808.2.a.d.1.2	yes	4	4.3	odd	2
7616.2.a.bl.1.2		4	8.5	even	2
7616.2.a.bm.1.3		4	8.3	odd	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:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17557 q^{3} +0.175571 q^{5} -1.00000 q^{7} -1.61803 q^{9} -1.28408 q^{11} +1.67853 q^{13} +0.206396 q^{15} +1.00000 q^{17} +3.53077 q^{19} -1.17557 q^{21} -2.07768 q^{23} -4.96917 q^{25} -5.42882 q^{27} -8.22835 q^{29} +1.42998 q^{31} -1.50953 q^{33} -0.175571 q^{35} +7.11798 q^{37} +1.97323 q^{39} -10.6150 q^{41} -2.23420 q^{43} -0.284079 q^{45} -3.43945 q^{47} +1.00000 q^{49} +1.17557 q^{51} +3.65833 q^{53} -0.225446 q^{55} +4.15067 q^{57} -4.45089 q^{59} -3.24482 q^{61} +1.61803 q^{63} +0.294701 q^{65} +3.16785 q^{67} -2.44246 q^{69} +4.61147 q^{71} +2.03373 q^{73} -5.84162 q^{75} +1.28408 q^{77} +0.932938 q^{79} -1.52786 q^{81} -5.38301 q^{83} +0.175571 q^{85} -9.67301 q^{87} -13.7775 q^{89} -1.67853 q^{91} +1.68104 q^{93} +0.619899 q^{95} -11.6668 q^{97} +2.07768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{13} + 10 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{23} - 6 q^{25} + 6 q^{29} + 16 q^{31} + 4 q^{35} - 8 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^13 + 10 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^23 - 6 * q^25 + 6 * q^29 + 16 * q^31 + 4 * q^35 - 8 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 2 * q^45 + 14 * q^47 + 4 * q^49 - 14 * q^53 + 2 * q^55 - 10 * q^57 - 12 * q^59 - 2 * q^61 + 2 * q^63 - 8 * q^65 - 22 * q^67 - 10 * q^69 + 16 * q^71 - 14 * q^73 - 20 * q^75 + 2 * q^77 + 10 * q^79 - 24 * q^81 + 6 * q^83 - 4 * q^85 - 22 * q^89 + 2 * q^91 - 6 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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