LMFDB - Embedded newform 3328.2.a.bi.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,-2,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.27743$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27743 q^{3} +2.27743 q^{5} +0.121816 q^{7} +2.18667 q^{9} -2.39924 q^{11} +1.00000 q^{13} +5.18667 q^{15} -0.943042 q^{17} +7.86334 q^{19} +0.277427 q^{21} +0.665615 q^{23} +0.186674 q^{25} -1.85229 q^{27} +7.22047 q^{29} +7.49000 q^{31} -5.46410 q^{33} +0.277427 q^{35} -6.52106 q^{37} +2.27743 q^{39} -2.66562 q^{41} +6.03380 q^{43} +4.97999 q^{45} -0.676670 q^{47} -6.98516 q^{49} -2.14771 q^{51} +10.2626 q^{53} -5.46410 q^{55} +17.9082 q^{57} -2.15561 q^{59} +7.88924 q^{61} +0.266372 q^{63} +2.27743 q^{65} -1.17877 q^{67} +1.51589 q^{69} +12.6767 q^{71} -6.13287 q^{73} +0.425137 q^{75} -0.292266 q^{77} -2.68457 q^{79} -10.7785 q^{81} +5.17877 q^{83} -2.14771 q^{85} +16.4441 q^{87} +1.75637 q^{89} +0.121816 q^{91} +17.0579 q^{93} +17.9082 q^{95} -2.90925 q^{97} -5.24636 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} + 18 q^{15} - 2 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{25} - 14 q^{27} + 16 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{35} - 14 q^{37} - 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 18 * q^15 - 2 * q^17 + 6 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^25 - 14 * q^27 + 16 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^35 - 14 * q^37 - 2 * q^39 - 20 * q^41 + 14 * q^43 - 20 * q^45 + 20 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^57 + 2 * q^59 + 22 * q^63 - 2 * q^65 - 6 * q^67 - 20 * q^69 + 28 * q^71 + 8 * q^73 - 16 * q^75 - 16 * q^77 + 16 * q^79 + 20 * q^81 + 22 * q^83 - 2 * q^85 + 12 * q^87 + 8 * q^89 + 36 * q^93 + 4 * q^95 - 20 * q^97 - 2 * 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
$3$	2.27743	1.31487	0.657437	−	0.753510i	$-0.271641\pi$
$3$	2.27743	1.31487	0.657437	+	0.753510i	$0.271641\pi$
$4$	0	0
$5$	2.27743	1.01850	0.509248	−	0.860620i	$-0.329924\pi$
$5$	2.27743	1.01850	0.509248	+	0.860620i	$0.329924\pi$
$6$	0	0
$7$	0.121816	0.0460421	0.0230211	−	0.999735i	$-0.492672\pi$
$7$	0.121816	0.0460421	0.0230211	+	0.999735i	$0.492672\pi$
$8$	0	0
$9$	2.18667	0.728891
$10$	0	0
$11$	−2.39924	−0.723399	−0.361700	−	0.932295i	$-0.617803\pi$
$11$	−2.39924	−0.723399	−0.361700	+	0.932295i	$0.617803\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	5.18667	1.33919
$16$	0	0
$17$	−0.943042	−0.228721	−0.114361	−	0.993439i	$-0.536482\pi$
$17$	−0.943042	−0.228721	−0.114361	+	0.993439i	$0.536482\pi$
$18$	0	0
$19$	7.86334	1.80398	0.901988	−	0.431762i	$-0.142108\pi$
$19$	7.86334	1.80398	0.901988	+	0.431762i	$0.142108\pi$
$20$	0	0
$21$	0.277427	0.0605396
$22$	0	0
$23$	0.665615	0.138790	0.0693952	−	0.997589i	$-0.477893\pi$
$23$	0.665615	0.138790	0.0693952	+	0.997589i	$0.477893\pi$
$24$	0	0
$25$	0.186674	0.0373349
$26$	0	0
$27$	−1.85229	−0.356473
$28$	0	0
$29$	7.22047	1.34081	0.670404	−	0.741996i	$-0.266121\pi$
$29$	7.22047	1.34081	0.670404	+	0.741996i	$0.266121\pi$
$30$	0	0
$31$	7.49000	1.34524	0.672621	−	0.739987i	$-0.265168\pi$
$31$	7.49000	1.34524	0.672621	+	0.739987i	$0.265168\pi$
$32$	0	0
$33$	−5.46410	−0.951178
$34$	0	0
$35$	0.277427	0.0468937
$36$	0	0
$37$	−6.52106	−1.07206	−0.536028	−	0.844200i	$-0.680076\pi$
$37$	−6.52106	−1.07206	−0.536028	+	0.844200i	$0.680076\pi$
$38$	0	0
$39$	2.27743	0.364680
$40$	0	0
$41$	−2.66562	−0.416299	−0.208150	−	0.978097i	$-0.566744\pi$
$41$	−2.66562	−0.416299	−0.208150	+	0.978097i	$0.566744\pi$
$42$	0	0
$43$	6.03380	0.920145	0.460073	−	0.887881i	$-0.347823\pi$
$43$	6.03380	0.920145	0.460073	+	0.887881i	$0.347823\pi$
$44$	0	0
$45$	4.97999	0.742373
$46$	0	0
$47$	−0.676670	−0.0987025	−0.0493513	−	0.998781i	$-0.515715\pi$
$47$	−0.676670	−0.0987025	−0.0493513	+	0.998781i	$0.515715\pi$
$48$	0	0
$49$	−6.98516	−0.997880
$50$	0	0
$51$	−2.14771	−0.300740
$52$	0	0
$53$	10.2626	1.40967	0.704837	−	0.709369i	$-0.251020\pi$
$53$	10.2626	1.40967	0.704837	+	0.709369i	$0.251020\pi$
$54$	0	0
$55$	−5.46410	−0.736779
$56$	0	0
$57$	17.9082	2.37200
$58$	0	0
$59$	−2.15561	−0.280637	−0.140318	−	0.990106i	$-0.544813\pi$
$59$	−2.15561	−0.280637	−0.140318	+	0.990106i	$0.544813\pi$
$60$	0	0
$61$	7.88924	1.01011	0.505057	−	0.863086i	$-0.331472\pi$
$61$	7.88924	1.01011	0.505057	+	0.863086i	$0.331472\pi$
$62$	0	0
$63$	0.266372	0.0335597
$64$	0	0
$65$	2.27743	0.282480
$66$	0	0
$67$	−1.17877	−0.144010	−0.0720051	−	0.997404i	$-0.522940\pi$
$67$	−1.17877	−0.144010	−0.0720051	+	0.997404i	$0.522940\pi$
$68$	0	0
$69$	1.51589	0.182492
$70$	0	0
$71$	12.6767	1.50444	0.752222	−	0.658910i	$-0.228982\pi$
$71$	12.6767	1.50444	0.752222	+	0.658910i	$0.228982\pi$
$72$	0	0
$73$	−6.13287	−0.717798	−0.358899	−	0.933376i	$-0.616848\pi$
$73$	−6.13287	−0.717798	−0.358899	+	0.933376i	$0.616848\pi$
$74$	0	0
$75$	0.425137	0.0490906
$76$	0	0
$77$	−0.292266	−0.0333068
$78$	0	0
$79$	−2.68457	−0.302038	−0.151019	−	0.988531i	$-0.548255\pi$
$79$	−2.68457	−0.302038	−0.151019	+	0.988531i	$0.548255\pi$
$80$	0	0
$81$	−10.7785	−1.19761
$82$	0	0
$83$	5.17877	0.568444	0.284222	−	0.958758i	$-0.408265\pi$
$83$	5.17877	0.568444	0.284222	+	0.958758i	$0.408265\pi$
$84$	0	0
$85$	−2.14771	−0.232952
$86$	0	0
$87$	16.4441	1.76299
$88$	0	0
$89$	1.75637	0.186175	0.0930873	−	0.995658i	$-0.470326\pi$
$89$	1.75637	0.186175	0.0930873	+	0.995658i	$0.470326\pi$
$90$	0	0
$91$	0.121816	0.0127698
$92$	0	0
$93$	17.0579	1.76882
$94$	0	0
$95$	17.9082	1.83734
$96$	0	0
$97$	−2.90925	−0.295389	−0.147695	−	0.989033i	$-0.547185\pi$
$97$	−2.90925	−0.295389	−0.147695	+	0.989033i	$0.547185\pi$
$98$	0	0
$99$	−5.24636	−0.527279
$100$	0	0
$101$	17.5970	1.75096	0.875482	−	0.483251i	$-0.160544\pi$
$101$	17.5970	1.75096	0.875482	+	0.483251i	$0.160544\pi$
$102$	0	0
$103$	−7.48306	−0.737328	−0.368664	−	0.929563i	$-0.620185\pi$
$103$	−7.48306	−0.737328	−0.368664	+	0.929563i	$0.620185\pi$
$104$	0	0
$105$	0.631820	0.0616593
$106$	0	0
$107$	−4.42514	−0.427794	−0.213897	−	0.976856i	$-0.568616\pi$
$107$	−4.42514	−0.427794	−0.213897	+	0.976856i	$0.568616\pi$
$108$	0	0
$109$	−16.1180	−1.54383	−0.771914	−	0.635727i	$-0.780700\pi$
$109$	−16.1180	−1.54383	−0.771914	+	0.635727i	$0.780700\pi$
$110$	0	0
$111$	−14.8512	−1.40962
$112$	0	0
$113$	20.2815	1.90793	0.953964	−	0.299922i	$-0.0969608\pi$
$113$	20.2815	1.90793	0.953964	+	0.299922i	$0.0969608\pi$
$114$	0	0
$115$	1.51589	0.141357
$116$	0	0
$117$	2.18667	0.202158
$118$	0	0
$119$	−0.114878	−0.0105308
$120$	0	0
$121$	−5.24363	−0.476694
$122$	0	0
$123$	−6.07074	−0.547381
$124$	0	0
$125$	−10.9620	−0.980471
$126$	0	0
$127$	16.4441	1.45918	0.729589	−	0.683886i	$-0.239712\pi$
$127$	16.4441	1.45918	0.729589	+	0.683886i	$0.239712\pi$
$128$	0	0
$129$	13.7415	1.20987
$130$	0	0
$131$	−16.1338	−1.40962	−0.704810	−	0.709396i	$-0.748968\pi$
$131$	−16.1338	−1.40962	−0.704810	+	0.709396i	$0.748968\pi$
$132$	0	0
$133$	0.957882	0.0830589
$134$	0	0
$135$	−4.21845	−0.363067
$136$	0	0
$137$	−19.2584	−1.64535	−0.822677	−	0.568509i	$-0.807521\pi$
$137$	−19.2584	−1.64535	−0.822677	+	0.568509i	$0.807521\pi$
$138$	0	0
$139$	−17.2056	−1.45936	−0.729681	−	0.683787i	$-0.760332\pi$
$139$	−17.2056	−1.45936	−0.729681	+	0.683787i	$0.760332\pi$
$140$	0	0
$141$	−1.54107	−0.129781
$142$	0	0
$143$	−2.39924	−0.200635
$144$	0	0
$145$	16.4441	1.36561
$146$	0	0
$147$	−15.9082	−1.31209
$148$	0	0
$149$	−15.5970	−1.27775	−0.638877	−	0.769309i	$-0.720601\pi$
$149$	−15.5970	−1.27775	−0.638877	+	0.769309i	$0.720601\pi$
$150$	0	0
$151$	19.6049	1.59542	0.797711	−	0.603040i	$-0.206044\pi$
$151$	19.6049	1.59542	0.797711	+	0.603040i	$0.206044\pi$
$152$	0	0
$153$	−2.06213	−0.166713
$154$	0	0
$155$	17.0579	1.37013
$156$	0	0
$157$	−12.8850	−1.02834	−0.514169	−	0.857689i	$-0.671900\pi$
$157$	−12.8850	−1.02834	−0.514169	+	0.857689i	$0.671900\pi$
$158$	0	0
$159$	23.3723	1.85354
$160$	0	0
$161$	0.0810826	0.00639020
$162$	0	0
$163$	19.1977	1.50368	0.751841	−	0.659344i	$-0.229166\pi$
$163$	19.1977	1.50368	0.751841	+	0.659344i	$0.229166\pi$
$164$	0	0
$165$	−12.4441	−0.968771
$166$	0	0
$167$	11.1788	0.865039	0.432520	−	0.901624i	$-0.357625\pi$
$167$	11.1788	0.865039	0.432520	+	0.901624i	$0.357625\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.1946	1.31490
$172$	0	0
$173$	−22.7364	−1.72861	−0.864307	−	0.502965i	$-0.832242\pi$
$173$	−22.7364	−1.72861	−0.864307	+	0.502965i	$0.832242\pi$
$174$	0	0
$175$	0.0227399	0.00171898
$176$	0	0
$177$	−4.90925	−0.369002
$178$	0	0
$179$	−14.0041	−1.04672	−0.523358	−	0.852113i	$-0.675321\pi$
$179$	−14.0041	−1.04672	−0.523358	+	0.852113i	$0.675321\pi$
$180$	0	0
$181$	−15.3691	−1.14238	−0.571190	−	0.820818i	$-0.693518\pi$
$181$	−15.3691	−1.14238	−0.571190	+	0.820818i	$0.693518\pi$
$182$	0	0
$183$	17.9672	1.32817
$184$	0	0
$185$	−14.8512	−1.09188
$186$	0	0
$187$	2.26259	0.165457
$188$	0	0
$189$	−0.225639	−0.0164128
$190$	0	0
$191$	2.79533	0.202263	0.101132	−	0.994873i	$-0.467754\pi$
$191$	2.79533	0.202263	0.101132	+	0.994873i	$0.467754\pi$
$192$	0	0
$193$	11.4209	0.822097	0.411048	−	0.911614i	$-0.365163\pi$
$193$	11.4209	0.822097	0.411048	+	0.911614i	$0.365163\pi$
$194$	0	0
$195$	5.18667	0.371425
$196$	0	0
$197$	−11.1138	−0.791827	−0.395914	−	0.918288i	$-0.629572\pi$
$197$	−11.1138	−0.791827	−0.395914	+	0.918288i	$0.629572\pi$
$198$	0	0
$199$	−11.9703	−0.848554	−0.424277	−	0.905533i	$-0.639472\pi$
$199$	−11.9703	−0.848554	−0.424277	+	0.905533i	$0.639472\pi$
$200$	0	0
$201$	−2.68457	−0.189355
$202$	0	0
$203$	0.879569	0.0617336
$204$	0	0
$205$	−6.07074	−0.423999
$206$	0	0
$207$	1.45548	0.101163
$208$	0	0
$209$	−18.8661	−1.30499
$210$	0	0
$211$	−10.2774	−0.707527	−0.353764	−	0.935335i	$-0.615098\pi$
$211$	−10.2774	−0.707527	−0.353764	+	0.935335i	$0.615098\pi$
$212$	0	0
$213$	28.8702	1.97815
$214$	0	0
$215$	13.7415	0.937164
$216$	0	0
$217$	0.912402	0.0619379
$218$	0	0
$219$	−13.9672	−0.943814
$220$	0	0
$221$	−0.943042	−0.0634359
$222$	0	0
$223$	9.73459	0.651876	0.325938	−	0.945391i	$-0.394320\pi$
$223$	9.73459	0.651876	0.325938	+	0.945391i	$0.394320\pi$
$224$	0	0
$225$	0.408196	0.0272131
$226$	0	0
$227$	−0.772592	−0.0512787	−0.0256394	−	0.999671i	$-0.508162\pi$
$227$	−0.772592	−0.0512787	−0.0256394	+	0.999671i	$0.508162\pi$
$228$	0	0
$229$	5.13804	0.339531	0.169766	−	0.985484i	$-0.445699\pi$
$229$	5.13804	0.339531	0.169766	+	0.985484i	$0.445699\pi$
$230$	0	0
$231$	−0.665615	−0.0437943
$232$	0	0
$233$	−16.7836	−1.09953	−0.549767	−	0.835318i	$-0.685283\pi$
$233$	−16.7836	−1.09953	−0.549767	+	0.835318i	$0.685283\pi$
$234$	0	0
$235$	−1.54107	−0.100528
$236$	0	0
$237$	−6.11392	−0.397141
$238$	0	0
$239$	−11.8643	−0.767438	−0.383719	−	0.923450i	$-0.625357\pi$
$239$	−11.8643	−0.767438	−0.383719	+	0.923450i	$0.625357\pi$
$240$	0	0
$241$	−17.0476	−1.09813	−0.549066	−	0.835779i	$-0.685016\pi$
$241$	−17.0476	−1.09813	−0.549066	+	0.835779i	$0.685016\pi$
$242$	0	0
$243$	−18.9903	−1.21823
$244$	0	0
$245$	−15.9082	−1.01634
$246$	0	0
$247$	7.86334	0.500333
$248$	0	0
$249$	11.7943	0.747432
$250$	0	0
$251$	24.1518	1.52445	0.762225	−	0.647312i	$-0.224107\pi$
$251$	24.1518	1.52445	0.762225	+	0.647312i	$0.224107\pi$
$252$	0	0
$253$	−1.59697	−0.100401
$254$	0	0
$255$	−4.89125	−0.306302
$256$	0	0
$257$	10.6697	0.665560	0.332780	−	0.943005i	$-0.392013\pi$
$257$	10.6697	0.665560	0.332780	+	0.943005i	$0.392013\pi$
$258$	0	0
$259$	−0.794370	−0.0493597
$260$	0	0
$261$	15.7888	0.977303
$262$	0	0
$263$	30.0190	1.85105	0.925524	−	0.378689i	$-0.123625\pi$
$263$	30.0190	1.85105	0.925524	+	0.378689i	$0.123625\pi$
$264$	0	0
$265$	23.3723	1.43575
$266$	0	0
$267$	4.00000	0.244796
$268$	0	0
$269$	−4.84712	−0.295534	−0.147767	−	0.989022i	$-0.547209\pi$
$269$	−4.84712	−0.295534	−0.147767	+	0.989022i	$0.547209\pi$
$270$	0	0
$271$	−27.7401	−1.68509	−0.842544	−	0.538627i	$-0.818943\pi$
$271$	−27.7401	−1.68509	−0.842544	+	0.538627i	$0.818943\pi$
$272$	0	0
$273$	0.277427	0.0167907
$274$	0	0
$275$	−0.447877	−0.0270080
$276$	0	0
$277$	−18.5896	−1.11694	−0.558471	−	0.829524i	$-0.688612\pi$
$277$	−18.5896	−1.11694	−0.558471	+	0.829524i	$0.688612\pi$
$278$	0	0
$279$	16.3782	0.980536
$280$	0	0
$281$	11.5812	0.690875	0.345437	−	0.938442i	$-0.387731\pi$
$281$	11.5812	0.690875	0.345437	+	0.938442i	$0.387731\pi$
$282$	0	0
$283$	1.21732	0.0723619	0.0361809	−	0.999345i	$-0.488481\pi$
$283$	1.21732	0.0723619	0.0361809	+	0.999345i	$0.488481\pi$
$284$	0	0
$285$	40.7846	2.41587
$286$	0	0
$287$	−0.324715	−0.0191673
$288$	0	0
$289$	−16.1107	−0.947687
$290$	0	0
$291$	−6.62560	−0.388399
$292$	0	0
$293$	8.89441	0.519617	0.259808	−	0.965660i	$-0.416341\pi$
$293$	8.89441	0.519617	0.259808	+	0.965660i	$0.416341\pi$
$294$	0	0
$295$	−4.90925	−0.285827
$296$	0	0
$297$	4.44409	0.257872
$298$	0	0
$299$	0.665615	0.0384935
$300$	0	0
$301$	0.735013	0.0423654
$302$	0	0
$303$	40.0758	2.30230
$304$	0	0
$305$	17.9672	1.02880
$306$	0	0
$307$	16.0120	0.913854	0.456927	−	0.889504i	$-0.348950\pi$
$307$	16.0120	0.913854	0.456927	+	0.889504i	$0.348950\pi$
$308$	0	0
$309$	−17.0421	−0.969492
$310$	0	0
$311$	−28.0758	−1.59203	−0.796017	−	0.605274i	$-0.793063\pi$
$311$	−28.0758	−1.59203	−0.796017	+	0.605274i	$0.793063\pi$
$312$	0	0
$313$	−11.0949	−0.627119	−0.313560	−	0.949569i	$-0.601522\pi$
$313$	−11.0949	−0.627119	−0.313560	+	0.949569i	$0.601522\pi$
$314$	0	0
$315$	0.606643	0.0341805
$316$	0	0
$317$	−23.5831	−1.32456	−0.662279	−	0.749257i	$-0.730411\pi$
$317$	−23.5831	−1.32456	−0.662279	+	0.749257i	$0.730411\pi$
$318$	0	0
$319$	−17.3237	−0.969939
$320$	0	0
$321$	−10.0779	−0.562495
$322$	0	0
$323$	−7.41547	−0.412608
$324$	0	0
$325$	0.186674	0.0103548
$326$	0	0
$327$	−36.7076	−2.02994
$328$	0	0
$329$	−0.0824293	−0.00454447
$330$	0	0
$331$	16.9573	0.932055	0.466027	−	0.884770i	$-0.345685\pi$
$331$	16.9573	0.932055	0.466027	+	0.884770i	$0.345685\pi$
$332$	0	0
$333$	−14.2594	−0.781412
$334$	0	0
$335$	−2.68457	−0.146674
$336$	0	0
$337$	−7.09487	−0.386482	−0.193241	−	0.981151i	$-0.561900\pi$
$337$	−7.09487	−0.386482	−0.193241	+	0.981151i	$0.561900\pi$
$338$	0	0
$339$	46.1897	2.50868
$340$	0	0
$341$	−17.9703	−0.973147
$342$	0	0
$343$	−1.70362	−0.0919867
$344$	0	0
$345$	3.45233	0.185867
$346$	0	0
$347$	2.83228	0.152045	0.0760224	−	0.997106i	$-0.475778\pi$
$347$	2.83228	0.152045	0.0760224	+	0.997106i	$0.475778\pi$
$348$	0	0
$349$	7.34922	0.393395	0.196697	−	0.980464i	$-0.436978\pi$
$349$	7.34922	0.393395	0.196697	+	0.980464i	$0.436978\pi$
$350$	0	0
$351$	−1.85229	−0.0988679
$352$	0	0
$353$	30.0379	1.59876	0.799378	−	0.600828i	$-0.205162\pi$
$353$	30.0379	1.59876	0.799378	+	0.600828i	$0.205162\pi$
$354$	0	0
$355$	28.8702	1.53227
$356$	0	0
$357$	−0.261626	−0.0138467
$358$	0	0
$359$	−7.17877	−0.378881	−0.189441	−	0.981892i	$-0.560667\pi$
$359$	−7.17877	−0.378881	−0.189441	+	0.981892i	$0.560667\pi$
$360$	0	0
$361$	42.8322	2.25433
$362$	0	0
$363$	−11.9420	−0.626792
$364$	0	0
$365$	−13.9672	−0.731075
$366$	0	0
$367$	2.19731	0.114698	0.0573492	−	0.998354i	$-0.481735\pi$
$367$	2.19731	0.114698	0.0573492	+	0.998354i	$0.481735\pi$
$368$	0	0
$369$	−5.82883	−0.303437
$370$	0	0
$371$	1.25015	0.0649044
$372$	0	0
$373$	−1.44515	−0.0748268	−0.0374134	−	0.999300i	$-0.511912\pi$
$373$	−1.44515	−0.0748268	−0.0374134	+	0.999300i	$0.511912\pi$
$374$	0	0
$375$	−24.9652	−1.28919
$376$	0	0
$377$	7.22047	0.371873
$378$	0	0
$379$	6.72434	0.345406	0.172703	−	0.984974i	$-0.444750\pi$
$379$	6.72434	0.345406	0.172703	+	0.984974i	$0.444750\pi$
$380$	0	0
$381$	37.4502	1.91863
$382$	0	0
$383$	23.0721	1.17893	0.589465	−	0.807794i	$-0.299338\pi$
$383$	23.0721	1.17893	0.589465	+	0.807794i	$0.299338\pi$
$384$	0	0
$385$	−0.665615	−0.0339229
$386$	0	0
$387$	13.1939	0.670686
$388$	0	0
$389$	−5.00929	−0.253981	−0.126990	−	0.991904i	$-0.540532\pi$
$389$	−5.00929	−0.253981	−0.126990	+	0.991904i	$0.540532\pi$
$390$	0	0
$391$	−0.627703	−0.0317443
$392$	0	0
$393$	−36.7436	−1.85347
$394$	0	0
$395$	−6.11392	−0.307624
$396$	0	0
$397$	−9.72669	−0.488169	−0.244084	−	0.969754i	$-0.578487\pi$
$397$	−9.72669	−0.488169	−0.244084	+	0.969754i	$0.578487\pi$
$398$	0	0
$399$	2.18151	0.109212
$400$	0	0
$401$	20.6414	1.03078	0.515391	−	0.856955i	$-0.327647\pi$
$401$	20.6414	1.03078	0.515391	+	0.856955i	$0.327647\pi$
$402$	0	0
$403$	7.49000	0.373103
$404$	0	0
$405$	−24.5472	−1.21976
$406$	0	0
$407$	15.6456	0.775524
$408$	0	0
$409$	−9.62981	−0.476163	−0.238082	−	0.971245i	$-0.576519\pi$
$409$	−9.62981	−0.476163	−0.238082	+	0.971245i	$0.576519\pi$
$410$	0	0
$411$	−43.8596	−2.16343
$412$	0	0
$413$	−0.262588	−0.0129211
$414$	0	0
$415$	11.7943	0.578958
$416$	0	0
$417$	−39.1846	−1.91888
$418$	0	0
$419$	−19.9041	−0.972378	−0.486189	−	0.873854i	$-0.661613\pi$
$419$	−19.9041	−0.972378	−0.486189	+	0.873854i	$0.661613\pi$
$420$	0	0
$421$	2.53686	0.123639	0.0618195	−	0.998087i	$-0.480310\pi$
$421$	2.53686	0.123639	0.0618195	+	0.998087i	$0.480310\pi$
$422$	0	0
$423$	−1.47966	−0.0719434
$424$	0	0
$425$	−0.176042	−0.00853929
$426$	0	0
$427$	0.961036	0.0465078
$428$	0	0
$429$	−5.46410	−0.263809
$430$	0	0
$431$	16.9064	0.814354	0.407177	−	0.913349i	$-0.366513\pi$
$431$	16.9064	0.814354	0.407177	+	0.913349i	$0.366513\pi$
$432$	0	0
$433$	−20.2667	−0.973956	−0.486978	−	0.873414i	$-0.661901\pi$
$433$	−20.2667	−0.973956	−0.486978	+	0.873414i	$0.661901\pi$
$434$	0	0
$435$	37.4502	1.79560
$436$	0	0
$437$	5.23396	0.250374
$438$	0	0
$439$	−11.4020	−0.544187	−0.272093	−	0.962271i	$-0.587716\pi$
$439$	−11.4020	−0.544187	−0.272093	+	0.962271i	$0.587716\pi$
$440$	0	0
$441$	−15.2743	−0.727346
$442$	0	0
$443$	−35.0980	−1.66756	−0.833779	−	0.552099i	$-0.813827\pi$
$443$	−35.0980	−1.66756	−0.833779	+	0.552099i	$0.813827\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	−35.5210	−1.68008
$448$	0	0
$449$	−26.7256	−1.26126	−0.630630	−	0.776083i	$-0.717204\pi$
$449$	−26.7256	−1.26126	−0.630630	+	0.776083i	$0.717204\pi$
$450$	0	0
$451$	6.39546	0.301150
$452$	0	0
$453$	44.6487	2.09778
$454$	0	0
$455$	0.277427	0.0130060
$456$	0	0
$457$	−6.51694	−0.304850	−0.152425	−	0.988315i	$-0.548708\pi$
$457$	−6.51694	−0.304850	−0.152425	+	0.988315i	$0.548708\pi$
$458$	0	0
$459$	1.74679	0.0815331
$460$	0	0
$461$	−15.8586	−0.738609	−0.369304	−	0.929309i	$-0.620404\pi$
$461$	−15.8586	−0.738609	−0.369304	+	0.929309i	$0.620404\pi$
$462$	0	0
$463$	−9.08697	−0.422307	−0.211154	−	0.977453i	$-0.567722\pi$
$463$	−9.08697	−0.422307	−0.211154	+	0.977453i	$0.567722\pi$
$464$	0	0
$465$	38.8482	1.80154
$466$	0	0
$467$	−24.5085	−1.13412	−0.567060	−	0.823677i	$-0.691919\pi$
$467$	−24.5085	−1.13412	−0.567060	+	0.823677i	$0.691919\pi$
$468$	0	0
$469$	−0.143594	−0.00663053
$470$	0	0
$471$	−29.3447	−1.35213
$472$	0	0
$473$	−14.4765	−0.665632
$474$	0	0
$475$	1.46789	0.0673512
$476$	0	0
$477$	22.4409	1.02750
$478$	0	0
$479$	4.96578	0.226892	0.113446	−	0.993544i	$-0.463811\pi$
$479$	4.96578	0.226892	0.113446	+	0.993544i	$0.463811\pi$
$480$	0	0
$481$	−6.52106	−0.297335
$482$	0	0
$483$	0.184660	0.00840231
$484$	0	0
$485$	−6.62560	−0.300853
$486$	0	0
$487$	16.3201	0.739534	0.369767	−	0.929125i	$-0.379438\pi$
$487$	16.3201	0.739534	0.369767	+	0.929125i	$0.379438\pi$
$488$	0	0
$489$	43.7214	1.97715
$490$	0	0
$491$	−0.141402	−0.00638137	−0.00319068	−	0.999995i	$-0.501016\pi$
$491$	−0.141402	−0.00638137	−0.00319068	+	0.999995i	$0.501016\pi$
$492$	0	0
$493$	−6.80921	−0.306671
$494$	0	0
$495$	−11.9482	−0.537032
$496$	0	0
$497$	1.54422	0.0692678
$498$	0	0
$499$	25.9657	1.16238	0.581192	−	0.813766i	$-0.302587\pi$
$499$	25.9657	1.16238	0.581192	+	0.813766i	$0.302587\pi$
$500$	0	0
$501$	25.4588	1.13742
$502$	0	0
$503$	3.20151	0.142748	0.0713742	−	0.997450i	$-0.477262\pi$
$503$	3.20151	0.142748	0.0713742	+	0.997450i	$0.477262\pi$
$504$	0	0
$505$	40.0758	1.78335
$506$	0	0
$507$	2.27743	0.101144
$508$	0	0
$509$	42.3870	1.87877	0.939386	−	0.342860i	$-0.111396\pi$
$509$	42.3870	1.87877	0.939386	+	0.342860i	$0.111396\pi$
$510$	0	0
$511$	−0.747082	−0.0330490
$512$	0	0
$513$	−14.5652	−0.643069
$514$	0	0
$515$	−17.0421	−0.750965
$516$	0	0
$517$	1.62350	0.0714013
$518$	0	0
$519$	−51.7804	−2.27291
$520$	0	0
$521$	3.57390	0.156575	0.0782877	−	0.996931i	$-0.475055\pi$
$521$	3.57390	0.156575	0.0782877	+	0.996931i	$0.475055\pi$
$522$	0	0
$523$	−1.75637	−0.0768006	−0.0384003	−	0.999262i	$-0.512226\pi$
$523$	−1.75637	−0.0768006	−0.0384003	+	0.999262i	$0.512226\pi$
$524$	0	0
$525$	0.0517886	0.00226024
$526$	0	0
$527$	−7.06338	−0.307686
$528$	0	0
$529$	−22.5570	−0.980737
$530$	0	0
$531$	−4.71362	−0.204554
$532$	0	0
$533$	−2.66562	−0.115461
$534$	0	0
$535$	−10.0779	−0.435707
$536$	0	0
$537$	−31.8934	−1.37630
$538$	0	0
$539$	16.7591	0.721866
$540$	0	0
$541$	30.5211	1.31220	0.656101	−	0.754673i	$-0.272204\pi$
$541$	30.5211	1.31220	0.656101	+	0.754673i	$0.272204\pi$
$542$	0	0
$543$	−35.0021	−1.50208
$544$	0	0
$545$	−36.7076	−1.57238
$546$	0	0
$547$	−26.0262	−1.11280	−0.556401	−	0.830914i	$-0.687818\pi$
$547$	−26.0262	−1.11280	−0.556401	+	0.830914i	$0.687818\pi$
$548$	0	0
$549$	17.2512	0.736263
$550$	0	0
$551$	56.7770	2.41878
$552$	0	0
$553$	−0.327024	−0.0139065
$554$	0	0
$555$	−33.8226	−1.43569
$556$	0	0
$557$	−3.40101	−0.144106	−0.0720528	−	0.997401i	$-0.522955\pi$
$557$	−3.40101	−0.144106	−0.0720528	+	0.997401i	$0.522955\pi$
$558$	0	0
$559$	6.03380	0.255202
$560$	0	0
$561$	5.15288	0.217555
$562$	0	0
$563$	1.15192	0.0485475	0.0242738	−	0.999705i	$-0.492273\pi$
$563$	1.15192	0.0485475	0.0242738	+	0.999705i	$0.492273\pi$
$564$	0	0
$565$	46.1897	1.94322
$566$	0	0
$567$	−1.31299	−0.0551405
$568$	0	0
$569$	27.4903	1.15245	0.576227	−	0.817290i	$-0.304524\pi$
$569$	27.4903	1.15245	0.576227	+	0.817290i	$0.304524\pi$
$570$	0	0
$571$	8.41345	0.352092	0.176046	−	0.984382i	$-0.443669\pi$
$571$	8.41345	0.352092	0.176046	+	0.984382i	$0.443669\pi$
$572$	0	0
$573$	6.36617	0.265950
$574$	0	0
$575$	0.124253	0.00518172
$576$	0	0
$577$	34.7414	1.44631	0.723153	−	0.690688i	$-0.242692\pi$
$577$	34.7414	1.44631	0.723153	+	0.690688i	$0.242692\pi$
$578$	0	0
$579$	26.0103	1.08095
$580$	0	0
$581$	0.630858	0.0261724
$582$	0	0
$583$	−24.6224	−1.01976
$584$	0	0
$585$	4.97999	0.205897
$586$	0	0
$587$	14.2641	0.588741	0.294370	−	0.955691i	$-0.404890\pi$
$587$	14.2641	0.588741	0.294370	+	0.955691i	$0.404890\pi$
$588$	0	0
$589$	58.8964	2.42678
$590$	0	0
$591$	−25.3109	−1.04115
$592$	0	0
$593$	10.4081	0.427410	0.213705	−	0.976898i	$-0.431447\pi$
$593$	10.4081	0.427410	0.213705	+	0.976898i	$0.431447\pi$
$594$	0	0
$595$	−0.261626	−0.0107256
$596$	0	0
$597$	−27.2615	−1.11574
$598$	0	0
$599$	25.6939	1.04982	0.524911	−	0.851157i	$-0.324098\pi$
$599$	25.6939	1.04982	0.524911	+	0.851157i	$0.324098\pi$
$600$	0	0
$601$	−25.1161	−1.02451	−0.512254	−	0.858834i	$-0.671189\pi$
$601$	−25.1161	−1.02451	−0.512254	+	0.858834i	$0.671189\pi$
$602$	0	0
$603$	−2.57759	−0.104968
$604$	0	0
$605$	−11.9420	−0.485511
$606$	0	0
$607$	16.9535	0.688120	0.344060	−	0.938948i	$-0.388198\pi$
$607$	16.9535	0.688120	0.344060	+	0.938948i	$0.388198\pi$
$608$	0	0
$609$	2.00315	0.0811719
$610$	0	0
$611$	−0.676670	−0.0273752
$612$	0	0
$613$	−37.2255	−1.50353	−0.751763	−	0.659434i	$-0.770796\pi$
$613$	−37.2255	−1.50353	−0.751763	+	0.659434i	$0.770796\pi$
$614$	0	0
$615$	−13.8257	−0.557505
$616$	0	0
$617$	32.4630	1.30691	0.653457	−	0.756964i	$-0.273318\pi$
$617$	32.4630	1.30691	0.653457	+	0.756964i	$0.273318\pi$
$618$	0	0
$619$	−13.5850	−0.546025	−0.273013	−	0.962010i	$-0.588020\pi$
$619$	−13.5850	−0.546025	−0.273013	+	0.962010i	$0.588020\pi$
$620$	0	0
$621$	−1.23291	−0.0494751
$622$	0	0
$623$	0.213954	0.00857188
$624$	0	0
$625$	−25.8985	−1.03594
$626$	0	0
$627$	−42.9661	−1.71590
$628$	0	0
$629$	6.14964	0.245202
$630$	0	0
$631$	−40.2115	−1.60080	−0.800398	−	0.599470i	$-0.795378\pi$
$631$	−40.2115	−1.60080	−0.800398	+	0.599470i	$0.795378\pi$
$632$	0	0
$633$	−23.4061	−0.930309
$634$	0	0
$635$	37.4502	1.48617
$636$	0	0
$637$	−6.98516	−0.276762
$638$	0	0
$639$	27.7198	1.09658
$640$	0	0
$641$	25.5155	1.00780	0.503901	−	0.863761i	$-0.331898\pi$
$641$	25.5155	1.00780	0.503901	+	0.863761i	$0.331898\pi$
$642$	0	0
$643$	−42.6755	−1.68296	−0.841479	−	0.540289i	$-0.818315\pi$
$643$	−42.6755	−1.68296	−0.841479	+	0.540289i	$0.818315\pi$
$644$	0	0
$645$	31.2953	1.23225
$646$	0	0
$647$	23.6614	0.930226	0.465113	−	0.885251i	$-0.346014\pi$
$647$	23.6614	0.930226	0.465113	+	0.885251i	$0.346014\pi$
$648$	0	0
$649$	5.17184	0.203012
$650$	0	0
$651$	2.07793	0.0814404
$652$	0	0
$653$	10.3333	0.404375	0.202187	−	0.979347i	$-0.435195\pi$
$653$	10.3333	0.404375	0.202187	+	0.979347i	$0.435195\pi$
$654$	0	0
$655$	−36.7436	−1.43569
$656$	0	0
$657$	−13.4106	−0.523197
$658$	0	0
$659$	−44.8261	−1.74618	−0.873088	−	0.487563i	$-0.837886\pi$
$659$	−44.8261	−1.74618	−0.873088	+	0.487563i	$0.837886\pi$
$660$	0	0
$661$	−0.492728	−0.0191649	−0.00958244	−	0.999954i	$-0.503050\pi$
$661$	−0.492728	−0.0191649	−0.00958244	+	0.999954i	$0.503050\pi$
$662$	0	0
$663$	−2.14771	−0.0834102
$664$	0	0
$665$	2.18151	0.0845952
$666$	0	0
$667$	4.80605	0.186091
$668$	0	0
$669$	22.1698	0.857135
$670$	0	0
$671$	−18.9282	−0.730715
$672$	0	0
$673$	37.2107	1.43437	0.717184	−	0.696884i	$-0.245431\pi$
$673$	37.2107	1.43437	0.717184	+	0.696884i	$0.245431\pi$
$674$	0	0
$675$	−0.345775	−0.0133089
$676$	0	0
$677$	−3.93241	−0.151135	−0.0755674	−	0.997141i	$-0.524077\pi$
$677$	−3.93241	−0.151135	−0.0755674	+	0.997141i	$0.524077\pi$
$678$	0	0
$679$	−0.354393	−0.0136004
$680$	0	0
$681$	−1.75952	−0.0674250
$682$	0	0
$683$	37.3064	1.42749	0.713745	−	0.700406i	$-0.246998\pi$
$683$	37.3064	1.42749	0.713745	+	0.700406i	$0.246998\pi$
$684$	0	0
$685$	−43.8596	−1.67579
$686$	0	0
$687$	11.7015	0.446441
$688$	0	0
$689$	10.2626	0.390973
$690$	0	0
$691$	−33.3222	−1.26764	−0.633818	−	0.773482i	$-0.718513\pi$
$691$	−33.3222	−1.26764	−0.633818	+	0.773482i	$0.718513\pi$
$692$	0	0
$693$	−0.639091	−0.0242771
$694$	0	0
$695$	−39.1846	−1.48636
$696$	0	0
$697$	2.51379	0.0952165
$698$	0	0
$699$	−38.2235	−1.44575
$700$	0	0
$701$	−10.9610	−0.413993	−0.206996	−	0.978342i	$-0.566369\pi$
$701$	−10.9610	−0.413993	−0.206996	+	0.978342i	$0.566369\pi$
$702$	0	0
$703$	−51.2773	−1.93396
$704$	0	0
$705$	−3.50967	−0.132182
$706$	0	0
$707$	2.14359	0.0806181
$708$	0	0
$709$	5.94821	0.223390	0.111695	−	0.993743i	$-0.464372\pi$
$709$	5.94821	0.223390	0.111695	+	0.993743i	$0.464372\pi$
$710$	0	0
$711$	−5.87028	−0.220153
$712$	0	0
$713$	4.98546	0.186707
$714$	0	0
$715$	−5.46410	−0.204346
$716$	0	0
$717$	−27.0201	−1.00908
$718$	0	0
$719$	−29.7951	−1.11117	−0.555585	−	0.831460i	$-0.687506\pi$
$719$	−29.7951	−1.11117	−0.555585	+	0.831460i	$0.687506\pi$
$720$	0	0
$721$	−0.911556	−0.0339481
$722$	0	0
$723$	−38.8246	−1.44390
$724$	0	0
$725$	1.34788	0.0500589
$726$	0	0
$727$	−7.60013	−0.281873	−0.140937	−	0.990019i	$-0.545011\pi$
$727$	−7.60013	−0.281873	−0.140937	+	0.990019i	$0.545011\pi$
$728$	0	0
$729$	−10.9137	−0.404210
$730$	0	0
$731$	−5.69012	−0.210457
$732$	0	0
$733$	34.6108	1.27838	0.639189	−	0.769050i	$-0.279270\pi$
$733$	34.6108	1.27838	0.639189	+	0.769050i	$0.279270\pi$
$734$	0	0
$735$	−36.2298	−1.33635
$736$	0	0
$737$	2.82816	0.104177
$738$	0	0
$739$	13.7471	0.505696	0.252848	−	0.967506i	$-0.418633\pi$
$739$	13.7471	0.505696	0.252848	+	0.967506i	$0.418633\pi$
$740$	0	0
$741$	17.9082	0.657874
$742$	0	0
$743$	−10.2219	−0.375003	−0.187502	−	0.982264i	$-0.560039\pi$
$743$	−10.2219	−0.375003	−0.187502	+	0.982264i	$0.560039\pi$
$744$	0	0
$745$	−35.5210	−1.30139
$746$	0	0
$747$	11.3243	0.414334
$748$	0	0
$749$	−0.539053	−0.0196966
$750$	0	0
$751$	3.43989	0.125523	0.0627616	−	0.998029i	$-0.480009\pi$
$751$	3.43989	0.125523	0.0627616	+	0.998029i	$0.480009\pi$
$752$	0	0
$753$	55.0040	2.00446
$754$	0	0
$755$	44.6487	1.62493
$756$	0	0
$757$	−46.9229	−1.70544	−0.852722	−	0.522365i	$-0.825050\pi$
$757$	−46.9229	−1.70544	−0.852722	+	0.522365i	$0.825050\pi$
$758$	0	0
$759$	−3.63699	−0.132014
$760$	0	0
$761$	−50.1569	−1.81819	−0.909093	−	0.416593i	$-0.863224\pi$
$761$	−50.1569	−1.81819	−0.909093	+	0.416593i	$0.863224\pi$
$762$	0	0
$763$	−1.96343	−0.0710811
$764$	0	0
$765$	−4.69634	−0.169797
$766$	0	0
$767$	−2.15561	−0.0778346
$768$	0	0
$769$	20.5603	0.741424	0.370712	−	0.928748i	$-0.379114\pi$
$769$	20.5603	0.741424	0.370712	+	0.928748i	$0.379114\pi$
$770$	0	0
$771$	24.2995	0.875126
$772$	0	0
$773$	35.9780	1.29404	0.647019	−	0.762474i	$-0.276015\pi$
$773$	35.9780	1.29404	0.647019	+	0.762474i	$0.276015\pi$
$774$	0	0
$775$	1.39819	0.0502245
$776$	0	0
$777$	−1.80912	−0.0649018
$778$	0	0
$779$	−20.9607	−0.750993
$780$	0	0
$781$	−30.4144	−1.08831
$782$	0	0
$783$	−13.3744	−0.477962
$784$	0	0
$785$	−29.3447	−1.04736
$786$	0	0
$787$	37.1874	1.32559	0.662794	−	0.748802i	$-0.269371\pi$
$787$	37.1874	1.32559	0.662794	+	0.748802i	$0.269371\pi$
$788$	0	0
$789$	68.3660	2.43389
$790$	0	0
$791$	2.47062	0.0878450
$792$	0	0
$793$	7.88924	0.280155
$794$	0	0
$795$	53.2287	1.88783
$796$	0	0
$797$	16.6012	0.588044	0.294022	−	0.955799i	$-0.405006\pi$
$797$	16.6012	0.588044	0.294022	+	0.955799i	$0.405006\pi$
$798$	0	0
$799$	0.638129	0.0225754
$800$	0	0
$801$	3.84060	0.135701
$802$	0	0
$803$	14.7142	0.519255
$804$	0	0
$805$	0.184660	0.00650840
$806$	0	0
$807$	−11.0390	−0.388590
$808$	0	0
$809$	−15.6794	−0.551258	−0.275629	−	0.961264i	$-0.588886\pi$
$809$	−15.6794	−0.551258	−0.275629	+	0.961264i	$0.588886\pi$
$810$	0	0
$811$	−15.8179	−0.555440	−0.277720	−	0.960662i	$-0.589579\pi$
$811$	−15.8179	−0.555440	−0.277720	+	0.960662i	$0.589579\pi$
$812$	0	0
$813$	−63.1760	−2.21568
$814$	0	0
$815$	43.7214	1.53149
$816$	0	0
$817$	47.4458	1.65992
$818$	0	0
$819$	0.266372	0.00930779
$820$	0	0
$821$	9.30290	0.324673	0.162337	−	0.986735i	$-0.448097\pi$
$821$	9.30290	0.324673	0.162337	+	0.986735i	$0.448097\pi$
$822$	0	0
$823$	−12.2839	−0.428188	−0.214094	−	0.976813i	$-0.568680\pi$
$823$	−12.2839	−0.428188	−0.214094	+	0.976813i	$0.568680\pi$
$824$	0	0
$825$	−1.02001	−0.0355121
$826$	0	0
$827$	−48.9731	−1.70296	−0.851480	−	0.524387i	$-0.824294\pi$
$827$	−48.9731	−1.70296	−0.851480	+	0.524387i	$0.824294\pi$
$828$	0	0
$829$	9.04212	0.314046	0.157023	−	0.987595i	$-0.449810\pi$
$829$	9.04212	0.314046	0.157023	+	0.987595i	$0.449810\pi$
$830$	0	0
$831$	−42.3365	−1.46864
$832$	0	0
$833$	6.58730	0.228237
$834$	0	0
$835$	25.4588	0.881040
$836$	0	0
$837$	−13.8736	−0.479543
$838$	0	0
$839$	−12.7694	−0.440850	−0.220425	−	0.975404i	$-0.570744\pi$
$839$	−12.7694	−0.440850	−0.220425	+	0.975404i	$0.570744\pi$
$840$	0	0
$841$	23.1352	0.797765
$842$	0	0
$843$	26.3753	0.908413
$844$	0	0
$845$	2.27743	0.0783459
$846$	0	0
$847$	−0.638758	−0.0219480
$848$	0	0
$849$	2.77235	0.0951467
$850$	0	0
$851$	−4.34052	−0.148791
$852$	0	0
$853$	10.8260	0.370674	0.185337	−	0.982675i	$-0.440662\pi$
$853$	10.8260	0.370674	0.185337	+	0.982675i	$0.440662\pi$
$854$	0	0
$855$	39.1594	1.33922
$856$	0	0
$857$	−33.1939	−1.13388	−0.566942	−	0.823758i	$-0.691874\pi$
$857$	−33.1939	−1.13388	−0.566942	+	0.823758i	$0.691874\pi$
$858$	0	0
$859$	29.9248	1.02102	0.510511	−	0.859871i	$-0.329456\pi$
$859$	29.9248	1.02102	0.510511	+	0.859871i	$0.329456\pi$
$860$	0	0
$861$	−0.739514	−0.0252026
$862$	0	0
$863$	−20.6691	−0.703584	−0.351792	−	0.936078i	$-0.614428\pi$
$863$	−20.6691	−0.703584	−0.351792	+	0.936078i	$0.614428\pi$
$864$	0	0
$865$	−51.7804	−1.76059
$866$	0	0
$867$	−36.6909	−1.24609
$868$	0	0
$869$	6.44094	0.218494
$870$	0	0
$871$	−1.17877	−0.0399412
$872$	0	0
$873$	−6.36158	−0.215307
$874$	0	0
$875$	−1.33535	−0.0451430
$876$	0	0
$877$	−54.2290	−1.83118	−0.915592	−	0.402110i	$-0.868277\pi$
$877$	−54.2290	−1.83118	−0.915592	+	0.402110i	$0.868277\pi$
$878$	0	0
$879$	20.2564	0.683230
$880$	0	0
$881$	11.3894	0.383720	0.191860	−	0.981422i	$-0.438548\pi$
$881$	11.3894	0.383720	0.191860	+	0.981422i	$0.438548\pi$
$882$	0	0
$883$	−52.7784	−1.77613	−0.888067	−	0.459714i	$-0.847952\pi$
$883$	−52.7784	−1.77613	−0.888067	+	0.459714i	$0.847952\pi$
$884$	0	0
$885$	−11.1805	−0.375827
$886$	0	0
$887$	−9.30386	−0.312393	−0.156197	−	0.987726i	$-0.549923\pi$
$887$	−9.30386	−0.312393	−0.156197	+	0.987726i	$0.549923\pi$
$888$	0	0
$889$	2.00315	0.0671836
$890$	0	0
$891$	25.8602	0.866349
$892$	0	0
$893$	−5.32089	−0.178057
$894$	0	0
$895$	−31.8934	−1.06608
$896$	0	0
$897$	1.51589	0.0506141
$898$	0	0
$899$	54.0813	1.80371
$900$	0	0
$901$	−9.67806	−0.322423
$902$	0	0
$903$	1.67394	0.0557052
$904$	0	0
$905$	−35.0021	−1.16351
$906$	0	0
$907$	15.0399	0.499393	0.249696	−	0.968324i	$-0.419669\pi$
$907$	15.0399	0.499393	0.249696	+	0.968324i	$0.419669\pi$
$908$	0	0
$909$	38.4789	1.27626
$910$	0	0
$911$	−14.3523	−0.475513	−0.237756	−	0.971325i	$-0.576412\pi$
$911$	−14.3523	−0.475513	−0.237756	+	0.971325i	$0.576412\pi$
$912$	0	0
$913$	−12.4251	−0.411212
$914$	0	0
$915$	40.9189	1.35274
$916$	0	0
$917$	−1.96536	−0.0649019
$918$	0	0
$919$	−52.4527	−1.73026	−0.865128	−	0.501552i	$-0.832763\pi$
$919$	−52.4527	−1.73026	−0.865128	+	0.501552i	$0.832763\pi$
$920$	0	0
$921$	36.4662	1.20160
$922$	0	0
$923$	12.6767	0.417258
$924$	0	0
$925$	−1.21732	−0.0400251
$926$	0	0
$927$	−16.3630	−0.537432
$928$	0	0
$929$	−30.0600	−0.986237	−0.493119	−	0.869962i	$-0.664143\pi$
$929$	−30.0600	−0.986237	−0.493119	+	0.869962i	$0.664143\pi$
$930$	0	0
$931$	−54.9267	−1.80015
$932$	0	0
$933$	−63.9406	−2.09332
$934$	0	0
$935$	5.15288	0.168517
$936$	0	0
$937$	−47.9979	−1.56802	−0.784011	−	0.620746i	$-0.786830\pi$
$937$	−47.9979	−1.56802	−0.784011	+	0.620746i	$0.786830\pi$
$938$	0	0
$939$	−25.2678	−0.824582
$940$	0	0
$941$	−32.7500	−1.06762	−0.533809	−	0.845605i	$-0.679240\pi$
$941$	−32.7500	−1.06762	−0.533809	+	0.845605i	$0.679240\pi$
$942$	0	0
$943$	−1.77427	−0.0577783
$944$	0	0
$945$	−0.513875	−0.0167164
$946$	0	0
$947$	28.1584	0.915024	0.457512	−	0.889203i	$-0.348741\pi$
$947$	28.1584	0.915024	0.457512	+	0.889203i	$0.348741\pi$
$948$	0	0
$949$	−6.13287	−0.199081
$950$	0	0
$951$	−53.7088	−1.74163
$952$	0	0
$953$	12.5179	0.405495	0.202747	−	0.979231i	$-0.435013\pi$
$953$	12.5179	0.405495	0.202747	+	0.979231i	$0.435013\pi$
$954$	0	0
$955$	6.36617	0.206004
$956$	0	0
$957$	−39.4534	−1.27535
$958$	0	0
$959$	−2.34598	−0.0757556
$960$	0	0
$961$	25.1000	0.809679
$962$	0	0
$963$	−9.67634	−0.311816
$964$	0	0
$965$	26.0103	0.837302
$966$	0	0
$967$	12.1894	0.391985	0.195992	−	0.980605i	$-0.437207\pi$
$967$	12.1894	0.391985	0.195992	+	0.980605i	$0.437207\pi$
$968$	0	0
$969$	−16.8882	−0.542527
$970$	0	0
$971$	28.1654	0.903872	0.451936	−	0.892050i	$-0.350734\pi$
$971$	28.1654	0.903872	0.451936	+	0.892050i	$0.350734\pi$
$972$	0	0
$973$	−2.09592	−0.0671922
$974$	0	0
$975$	0.425137	0.0136153
$976$	0	0
$977$	−8.26028	−0.264270	−0.132135	−	0.991232i	$-0.542183\pi$
$977$	−8.26028	−0.264270	−0.132135	+	0.991232i	$0.542183\pi$
$978$	0	0
$979$	−4.21395	−0.134679
$980$	0	0
$981$	−35.2449	−1.12528
$982$	0	0
$983$	36.2115	1.15497	0.577484	−	0.816402i	$-0.304035\pi$
$983$	36.2115	1.15497	0.577484	+	0.816402i	$0.304035\pi$
$984$	0	0
$985$	−25.3109	−0.806473
$986$	0	0
$987$	−0.187727	−0.00597541
$988$	0	0
$989$	4.01619	0.127707
$990$	0	0
$991$	−9.38302	−0.298061	−0.149031	−	0.988833i	$-0.547615\pi$
$991$	−9.38302	−0.298061	−0.149031	+	0.988833i	$0.547615\pi$
$992$	0	0
$993$	38.6189	1.22553
$994$	0	0
$995$	−27.2615	−0.864249
$996$	0	0
$997$	−29.3260	−0.928763	−0.464381	−	0.885635i	$-0.653723\pi$
$997$	−29.3260	−0.928763	−0.464381	+	0.885635i	$0.653723\pi$
$998$	0	0
$999$	12.0789	0.382159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bi.1.4		4
4.3	odd	2		3328.2.a.bm.1.1		4
8.3	odd	2		3328.2.a.bj.1.4		4
8.5	even	2		3328.2.a.bn.1.1		4
16.3	odd	4		832.2.b.c.417.2	✓	8
16.5	even	4		832.2.b.d.417.2	yes	8
16.11	odd	4		832.2.b.c.417.7	yes	8
16.13	even	4		832.2.b.d.417.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.2	✓	8	16.3	odd	4
832.2.b.c.417.7	yes	8	16.11	odd	4
832.2.b.d.417.2	yes	8	16.5	even	4
832.2.b.d.417.7	yes	8	16.13	even	4
3328.2.a.bi.1.4		4	1.1	even	1	trivial
3328.2.a.bj.1.4		4	8.3	odd	2
3328.2.a.bm.1.1		4	4.3	odd	2
3328.2.a.bn.1.1		4	8.5	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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27743 q^{3} +2.27743 q^{5} +0.121816 q^{7} +2.18667 q^{9} -2.39924 q^{11} +1.00000 q^{13} +5.18667 q^{15} -0.943042 q^{17} +7.86334 q^{19} +0.277427 q^{21} +0.665615 q^{23} +0.186674 q^{25} -1.85229 q^{27} +7.22047 q^{29} +7.49000 q^{31} -5.46410 q^{33} +0.277427 q^{35} -6.52106 q^{37} +2.27743 q^{39} -2.66562 q^{41} +6.03380 q^{43} +4.97999 q^{45} -0.676670 q^{47} -6.98516 q^{49} -2.14771 q^{51} +10.2626 q^{53} -5.46410 q^{55} +17.9082 q^{57} -2.15561 q^{59} +7.88924 q^{61} +0.266372 q^{63} +2.27743 q^{65} -1.17877 q^{67} +1.51589 q^{69} +12.6767 q^{71} -6.13287 q^{73} +0.425137 q^{75} -0.292266 q^{77} -2.68457 q^{79} -10.7785 q^{81} +5.17877 q^{83} -2.14771 q^{85} +16.4441 q^{87} +1.75637 q^{89} +0.121816 q^{91} +17.0579 q^{93} +17.9082 q^{95} -2.90925 q^{97} -5.24636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} + 18 q^{15} - 2 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{25} - 14 q^{27} + 16 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{35} - 14 q^{37} - 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 18 * q^15 - 2 * q^17 + 6 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^25 - 14 * q^27 + 16 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^35 - 14 * q^37 - 2 * q^39 - 20 * q^41 + 14 * q^43 - 20 * q^45 + 20 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^57 + 2 * q^59 + 22 * q^63 - 2 * q^65 - 6 * q^67 - 20 * q^69 + 28 * q^71 + 8 * q^73 - 16 * q^75 - 16 * q^77 + 16 * q^79 + 20 * q^81 + 22 * q^83 - 2 * q^85 + 12 * q^87 + 8 * q^89 + 36 * q^93 + 4 * q^95 - 20 * q^97 - 2 * q^99

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