LMFDB - Embedded newform 3420.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$27.3088374913$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +0.828427 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.828427 q^{7} +2.00000 q^{11} -6.24264 q^{13} -0.828427 q^{17} -1.00000 q^{19} +6.00000 q^{23} +1.00000 q^{25} +6.48528 q^{29} -6.82843 q^{31} -0.828427 q^{35} -1.75736 q^{37} -3.65685 q^{41} +4.82843 q^{43} +4.82843 q^{47} -6.31371 q^{49} -9.07107 q^{53} -2.00000 q^{55} -13.6569 q^{59} -13.6569 q^{61} +6.24264 q^{65} -3.41421 q^{67} -5.17157 q^{71} -2.48528 q^{73} +1.65685 q^{77} +1.65685 q^{79} +13.3137 q^{83} +0.828427 q^{85} +6.48528 q^{89} -5.17157 q^{91} +1.00000 q^{95} -10.2426 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{59} - 16 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{77} - 8 q^{79} + 4 q^{83} - 4 q^{85} - 4 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 - 16 * q^59 - 16 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^77 - 8 * q^79 + 4 * q^83 - 4 * q^85 - 4 * q^89 - 16 * q^91 + 2 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.48528	1.20429	0.602143	−	0.798388i	$-0.294314\pi$
$29$	6.48528	1.20429	0.602143	+	0.798388i	$0.294314\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.828427	−0.140030
$36$	0	0
$37$	−1.75736	−0.288908	−0.144454	−	0.989512i	$-0.546143\pi$
$37$	−1.75736	−0.288908	−0.144454	+	0.989512i	$0.546143\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	4.82843	0.736328	0.368164	−	0.929761i	$-0.379986\pi$
$43$	4.82843	0.736328	0.368164	+	0.929761i	$0.379986\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.07107	−1.24601	−0.623003	−	0.782219i	$-0.714088\pi$
$53$	−9.07107	−1.24601	−0.623003	+	0.782219i	$0.714088\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	0	0
$61$	−13.6569	−1.74858	−0.874291	−	0.485403i	$-0.838673\pi$
$61$	−13.6569	−1.74858	−0.874291	+	0.485403i	$0.838673\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.24264	0.774304
$66$	0	0
$67$	−3.41421	−0.417113	−0.208556	−	0.978010i	$-0.566876\pi$
$67$	−3.41421	−0.417113	−0.208556	+	0.978010i	$0.566876\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.17157	−0.613753	−0.306876	−	0.951749i	$-0.599284\pi$
$71$	−5.17157	−0.613753	−0.306876	+	0.951749i	$0.599284\pi$
$72$	0	0
$73$	−2.48528	−0.290880	−0.145440	−	0.989367i	$-0.546460\pi$
$73$	−2.48528	−0.290880	−0.145440	+	0.989367i	$0.546460\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.65685	0.188816
$78$	0	0
$79$	1.65685	0.186411	0.0932053	−	0.995647i	$-0.470289\pi$
$79$	1.65685	0.186411	0.0932053	+	0.995647i	$0.470289\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.3137	1.46137	0.730685	−	0.682715i	$-0.239201\pi$
$83$	13.3137	1.46137	0.730685	+	0.682715i	$0.239201\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.48528	0.687438	0.343719	−	0.939072i	$-0.388313\pi$
$89$	6.48528	0.687438	0.343719	+	0.939072i	$0.388313\pi$
$90$	0	0
$91$	−5.17157	−0.542128
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−10.2426	−1.03998	−0.519991	−	0.854172i	$-0.674065\pi$
$97$	−10.2426	−1.03998	−0.519991	+	0.854172i	$0.674065\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	3.89949	0.384229	0.192114	−	0.981373i	$-0.438466\pi$
$103$	3.89949	0.384229	0.192114	+	0.981373i	$0.438466\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.41421	−0.716759	−0.358380	−	0.933576i	$-0.616671\pi$
$107$	−7.41421	−0.716759	−0.358380	+	0.933576i	$0.616671\pi$
$108$	0	0
$109$	−3.17157	−0.303782	−0.151891	−	0.988397i	$-0.548536\pi$
$109$	−3.17157	−0.303782	−0.151891	+	0.988397i	$0.548536\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.07107	0.853334	0.426667	−	0.904409i	$-0.359688\pi$
$113$	9.07107	0.853334	0.426667	+	0.904409i	$0.359688\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.686292	−0.0629122
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.55635	0.847989	0.423994	−	0.905665i	$-0.360628\pi$
$127$	9.55635	0.847989	0.423994	+	0.905665i	$0.360628\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	−0.828427	−0.0718337
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.14214	−0.695630	−0.347815	−	0.937563i	$-0.613076\pi$
$137$	−8.14214	−0.695630	−0.347815	+	0.937563i	$0.613076\pi$
$138$	0	0
$139$	5.31371	0.450703	0.225351	−	0.974278i	$-0.427647\pi$
$139$	5.31371	0.450703	0.225351	+	0.974278i	$0.427647\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.4853	−1.04407
$144$	0	0
$145$	−6.48528	−0.538573
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−10.1421	−0.825355	−0.412678	−	0.910877i	$-0.635406\pi$
$151$	−10.1421	−0.825355	−0.412678	+	0.910877i	$0.635406\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.82843	0.548472
$156$	0	0
$157$	8.82843	0.704585	0.352293	−	0.935890i	$-0.385402\pi$
$157$	8.82843	0.704585	0.352293	+	0.935890i	$0.385402\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.97056	0.391735
$162$	0	0
$163$	14.4853	1.13457	0.567287	−	0.823520i	$-0.307993\pi$
$163$	14.4853	1.13457	0.567287	+	0.823520i	$0.307993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.7279	−1.13968	−0.569840	−	0.821755i	$-0.692995\pi$
$167$	−14.7279	−1.13968	−0.569840	+	0.821755i	$0.692995\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.8995	−1.20882	−0.604408	−	0.796675i	$-0.706590\pi$
$173$	−15.8995	−1.20882	−0.604408	+	0.796675i	$0.706590\pi$
$174$	0	0
$175$	0.828427	0.0626232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.3431	−0.773083	−0.386542	−	0.922272i	$-0.626330\pi$
$179$	−10.3431	−0.773083	−0.386542	+	0.922272i	$0.626330\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.75736	0.129204
$186$	0	0
$187$	−1.65685	−0.121161
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	3.41421	0.245760	0.122880	−	0.992422i	$-0.460787\pi$
$193$	3.41421	0.245760	0.122880	+	0.992422i	$0.460787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.3137	−1.23355	−0.616775	−	0.787139i	$-0.711561\pi$
$197$	−17.3137	−1.23355	−0.616775	+	0.787139i	$0.711561\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.37258	0.377081
$204$	0	0
$205$	3.65685	0.255406
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−28.4853	−1.96101	−0.980504	−	0.196500i	$-0.937042\pi$
$211$	−28.4853	−1.96101	−0.980504	+	0.196500i	$0.937042\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.82843	−0.329296
$216$	0	0
$217$	−5.65685	−0.384012
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.17157	0.347878
$222$	0	0
$223$	−23.2132	−1.55447	−0.777236	−	0.629210i	$-0.783379\pi$
$223$	−23.2132	−1.55447	−0.777236	+	0.629210i	$0.783379\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.4142	−1.55406	−0.777028	−	0.629466i	$-0.783274\pi$
$227$	−23.4142	−1.55406	−0.777028	+	0.629466i	$0.783274\pi$
$228$	0	0
$229$	−10.3431	−0.683494	−0.341747	−	0.939792i	$-0.611019\pi$
$229$	−10.3431	−0.683494	−0.341747	+	0.939792i	$0.611019\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−4.82843	−0.314972
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.6569	−1.40087	−0.700433	−	0.713718i	$-0.747010\pi$
$239$	−21.6569	−1.40087	−0.700433	+	0.713718i	$0.747010\pi$
$240$	0	0
$241$	26.2843	1.69312	0.846559	−	0.532294i	$-0.178670\pi$
$241$	26.2843	1.69312	0.846559	+	0.532294i	$0.178670\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.31371	0.403368
$246$	0	0
$247$	6.24264	0.397210
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.34315	0.147898	0.0739490	−	0.997262i	$-0.476440\pi$
$251$	2.34315	0.147898	0.0739490	+	0.997262i	$0.476440\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.24264	−0.389405	−0.194703	−	0.980862i	$-0.562374\pi$
$257$	−6.24264	−0.389405	−0.194703	+	0.980862i	$0.562374\pi$
$258$	0	0
$259$	−1.45584	−0.0904618
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.65685	0.472142	0.236071	−	0.971736i	$-0.424140\pi$
$263$	7.65685	0.472142	0.236071	+	0.971736i	$0.424140\pi$
$264$	0	0
$265$	9.07107	0.557231
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.51472	0.336238	0.168119	−	0.985767i	$-0.446231\pi$
$269$	5.51472	0.336238	0.168119	+	0.985767i	$0.446231\pi$
$270$	0	0
$271$	12.3431	0.749793	0.374896	−	0.927067i	$-0.377678\pi$
$271$	12.3431	0.749793	0.374896	+	0.927067i	$0.377678\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	11.1716	0.671235	0.335617	−	0.941998i	$-0.391055\pi$
$277$	11.1716	0.671235	0.335617	+	0.941998i	$0.391055\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.9706	0.893069	0.446534	−	0.894766i	$-0.352658\pi$
$281$	14.9706	0.893069	0.446534	+	0.894766i	$0.352658\pi$
$282$	0	0
$283$	−25.3137	−1.50474	−0.752372	−	0.658739i	$-0.771090\pi$
$283$	−25.3137	−1.50474	−0.752372	+	0.658739i	$0.771090\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.02944	−0.178822
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.5858	1.20263	0.601317	−	0.799010i	$-0.294643\pi$
$293$	20.5858	1.20263	0.601317	+	0.799010i	$0.294643\pi$
$294$	0	0
$295$	13.6569	0.795133
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−37.4558	−2.16613
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.6569	0.781989
$306$	0	0
$307$	−11.8995	−0.679140	−0.339570	−	0.940581i	$-0.610282\pi$
$307$	−11.8995	−0.679140	−0.339570	+	0.940581i	$0.610282\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.2426	0.799946	0.399973	−	0.916527i	$-0.369019\pi$
$317$	14.2426	0.799946	0.399973	+	0.916527i	$0.369019\pi$
$318$	0	0
$319$	12.9706	0.726212
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.828427	0.0460949
$324$	0	0
$325$	−6.24264	−0.346279
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	6.82843	0.375324	0.187662	−	0.982234i	$-0.439909\pi$
$331$	6.82843	0.375324	0.187662	+	0.982234i	$0.439909\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.41421	0.186538
$336$	0	0
$337$	26.2426	1.42953	0.714764	−	0.699366i	$-0.246534\pi$
$337$	26.2426	1.42953	0.714764	+	0.699366i	$0.246534\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.6569	−0.739560
$342$	0	0
$343$	−11.0294	−0.595534
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.9706	0.588931	0.294465	−	0.955662i	$-0.404858\pi$
$347$	10.9706	0.588931	0.294465	+	0.955662i	$0.404858\pi$
$348$	0	0
$349$	−11.6569	−0.623977	−0.311989	−	0.950086i	$-0.600995\pi$
$349$	−11.6569	−0.623977	−0.311989	+	0.950086i	$0.600995\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.1421	1.71075	0.855377	−	0.518007i	$-0.173326\pi$
$353$	32.1421	1.71075	0.855377	+	0.518007i	$0.173326\pi$
$354$	0	0
$355$	5.17157	0.274479
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.31371	−0.0693349	−0.0346674	−	0.999399i	$-0.511037\pi$
$359$	−1.31371	−0.0693349	−0.0346674	+	0.999399i	$0.511037\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.48528	0.130086
$366$	0	0
$367$	0.142136	0.00741942	0.00370971	−	0.999993i	$-0.498819\pi$
$367$	0.142136	0.00741942	0.00370971	+	0.999993i	$0.498819\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.51472	−0.390145
$372$	0	0
$373$	−21.5563	−1.11615	−0.558073	−	0.829792i	$-0.688459\pi$
$373$	−21.5563	−1.11615	−0.558073	+	0.829792i	$0.688459\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−40.4853	−2.08510
$378$	0	0
$379$	33.6569	1.72884	0.864418	−	0.502773i	$-0.167687\pi$
$379$	33.6569	1.72884	0.864418	+	0.502773i	$0.167687\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.0711	−0.872291	−0.436145	−	0.899876i	$-0.643657\pi$
$383$	−17.0711	−0.872291	−0.436145	+	0.899876i	$0.643657\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	−4.97056	−0.251372
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.65685	−0.0833654
$396$	0	0
$397$	−3.85786	−0.193621	−0.0968103	−	0.995303i	$-0.530864\pi$
$397$	−3.85786	−0.193621	−0.0968103	+	0.995303i	$0.530864\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.3137	1.46386	0.731928	−	0.681382i	$-0.238621\pi$
$401$	29.3137	1.46386	0.731928	+	0.681382i	$0.238621\pi$
$402$	0	0
$403$	42.6274	2.12342
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.51472	−0.174218
$408$	0	0
$409$	26.4853	1.30961	0.654806	−	0.755797i	$-0.272750\pi$
$409$	26.4853	1.30961	0.654806	+	0.755797i	$0.272750\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.3137	−0.556711
$414$	0	0
$415$	−13.3137	−0.653544
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	−19.6569	−0.958016	−0.479008	−	0.877810i	$-0.659004\pi$
$421$	−19.6569	−0.958016	−0.479008	+	0.877810i	$0.659004\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.828427	−0.0401846
$426$	0	0
$427$	−11.3137	−0.547509
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.4558	0.840818	0.420409	−	0.907335i	$-0.361887\pi$
$431$	17.4558	0.840818	0.420409	+	0.907335i	$0.361887\pi$
$432$	0	0
$433$	27.2132	1.30778	0.653892	−	0.756588i	$-0.273135\pi$
$433$	27.2132	1.30778	0.653892	+	0.756588i	$0.273135\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−18.3431	−0.875471	−0.437735	−	0.899104i	$-0.644219\pi$
$439$	−18.3431	−0.875471	−0.437735	+	0.899104i	$0.644219\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.2843	1.05876	0.529379	−	0.848386i	$-0.322425\pi$
$443$	22.2843	1.05876	0.529379	+	0.848386i	$0.322425\pi$
$444$	0	0
$445$	−6.48528	−0.307432
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.4853	1.43869	0.719345	−	0.694653i	$-0.244442\pi$
$449$	30.4853	1.43869	0.719345	+	0.694653i	$0.244442\pi$
$450$	0	0
$451$	−7.31371	−0.344389
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.17157	0.242447
$456$	0	0
$457$	−14.9706	−0.700293	−0.350147	−	0.936695i	$-0.613868\pi$
$457$	−14.9706	−0.700293	−0.350147	+	0.936695i	$0.613868\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.3137	−0.806380	−0.403190	−	0.915116i	$-0.632099\pi$
$461$	−17.3137	−0.806380	−0.403190	+	0.915116i	$0.632099\pi$
$462$	0	0
$463$	13.3137	0.618741	0.309370	−	0.950942i	$-0.399882\pi$
$463$	13.3137	0.618741	0.309370	+	0.950942i	$0.399882\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.3137	0.616085	0.308042	−	0.951373i	$-0.400326\pi$
$467$	13.3137	0.616085	0.308042	+	0.951373i	$0.400326\pi$
$468$	0	0
$469$	−2.82843	−0.130605
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.65685	0.444023
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	10.9706	0.500215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.2426	0.465094
$486$	0	0
$487$	−12.3848	−0.561208	−0.280604	−	0.959824i	$-0.590535\pi$
$487$	−12.3848	−0.561208	−0.280604	+	0.959824i	$0.590535\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.6863	−0.572524	−0.286262	−	0.958151i	$-0.592413\pi$
$491$	−12.6863	−0.572524	−0.286262	+	0.958151i	$0.592413\pi$
$492$	0	0
$493$	−5.37258	−0.241969
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.28427	−0.192176
$498$	0	0
$499$	6.97056	0.312045	0.156023	−	0.987753i	$-0.450133\pi$
$499$	6.97056	0.312045	0.156023	+	0.987753i	$0.450133\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.9706	1.38091	0.690455	−	0.723376i	$-0.257411\pi$
$503$	30.9706	1.38091	0.690455	+	0.723376i	$0.257411\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.79899	−0.257036	−0.128518	−	0.991707i	$-0.541022\pi$
$509$	−5.79899	−0.257036	−0.128518	+	0.991707i	$0.541022\pi$
$510$	0	0
$511$	−2.05887	−0.0910792
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.89949	−0.171832
$516$	0	0
$517$	9.65685	0.424708
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.6569	−1.73740	−0.868699	−	0.495340i	$-0.835043\pi$
$521$	−39.6569	−1.73740	−0.868699	+	0.495340i	$0.835043\pi$
$522$	0	0
$523$	18.7279	0.818915	0.409457	−	0.912329i	$-0.365718\pi$
$523$	18.7279	0.818915	0.409457	+	0.912329i	$0.365718\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.65685	0.246416
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22.8284	0.988809
$534$	0	0
$535$	7.41421	0.320544
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.6274	−0.543901
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.17157	0.135855
$546$	0	0
$547$	−0.384776	−0.0164518	−0.00822592	−	0.999966i	$-0.502618\pi$
$547$	−0.384776	−0.0164518	−0.00822592	+	0.999966i	$0.502618\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.48528	−0.276282
$552$	0	0
$553$	1.37258	0.0583682
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.20101	−0.0932598	−0.0466299	−	0.998912i	$-0.514848\pi$
$557$	−2.20101	−0.0932598	−0.0466299	+	0.998912i	$0.514848\pi$
$558$	0	0
$559$	−30.1421	−1.27488
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.58579	0.361848	0.180924	−	0.983497i	$-0.442091\pi$
$563$	8.58579	0.361848	0.180924	+	0.983497i	$0.442091\pi$
$564$	0	0
$565$	−9.07107	−0.381623
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.82843	0.202418	0.101209	−	0.994865i	$-0.467729\pi$
$569$	4.82843	0.202418	0.101209	+	0.994865i	$0.467729\pi$
$570$	0	0
$571$	−37.3137	−1.56153	−0.780765	−	0.624825i	$-0.785170\pi$
$571$	−37.3137	−1.56153	−0.780765	+	0.624825i	$0.785170\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.0294	0.457578
$582$	0	0
$583$	−18.1421	−0.751370
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.9706	−0.782999	−0.391499	−	0.920178i	$-0.628044\pi$
$587$	−18.9706	−0.782999	−0.391499	+	0.920178i	$0.628044\pi$
$588$	0	0
$589$	6.82843	0.281360
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.6274	1.50411	0.752054	−	0.659102i	$-0.229063\pi$
$593$	36.6274	1.50411	0.752054	+	0.659102i	$0.229063\pi$
$594$	0	0
$595$	0.686292	0.0281352
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−9.07107	−0.368183	−0.184092	−	0.982909i	$-0.558934\pi$
$607$	−9.07107	−0.368183	−0.184092	+	0.982909i	$0.558934\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.1421	−1.21942
$612$	0	0
$613$	34.4853	1.39285	0.696424	−	0.717631i	$-0.254773\pi$
$613$	34.4853	1.39285	0.696424	+	0.717631i	$0.254773\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.7990	1.84380	0.921899	−	0.387430i	$-0.126637\pi$
$617$	45.7990	1.84380	0.921899	+	0.387430i	$0.126637\pi$
$618$	0	0
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.37258	0.215248
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.45584	0.0580483
$630$	0	0
$631$	7.65685	0.304815	0.152407	−	0.988318i	$-0.451297\pi$
$631$	7.65685	0.304815	0.152407	+	0.988318i	$0.451297\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.55635	−0.379232
$636$	0	0
$637$	39.4142	1.56165
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.31371	−0.0518884	−0.0259442	−	0.999663i	$-0.508259\pi$
$641$	−1.31371	−0.0518884	−0.0259442	+	0.999663i	$0.508259\pi$
$642$	0	0
$643$	−20.6274	−0.813466	−0.406733	−	0.913547i	$-0.633332\pi$
$643$	−20.6274	−0.813466	−0.406733	+	0.913547i	$0.633332\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.3137	−1.15244	−0.576220	−	0.817294i	$-0.695473\pi$
$647$	−29.3137	−1.15244	−0.576220	+	0.817294i	$0.695473\pi$
$648$	0	0
$649$	−27.3137	−1.07216
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.7990	1.00959	0.504796	−	0.863239i	$-0.331568\pi$
$653$	25.7990	1.00959	0.504796	+	0.863239i	$0.331568\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.6274	−0.413985	−0.206993	−	0.978342i	$-0.566368\pi$
$659$	−10.6274	−0.413985	−0.206993	+	0.978342i	$0.566368\pi$
$660$	0	0
$661$	32.6274	1.26906	0.634530	−	0.772898i	$-0.281194\pi$
$661$	32.6274	1.26906	0.634530	+	0.772898i	$0.281194\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.828427	0.0321250
$666$	0	0
$667$	38.9117	1.50667
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27.3137	−1.05443
$672$	0	0
$673$	1.75736	0.0677412	0.0338706	−	0.999426i	$-0.489217\pi$
$673$	1.75736	0.0677412	0.0338706	+	0.999426i	$0.489217\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.55635	0.213548	0.106774	−	0.994283i	$-0.465948\pi$
$677$	5.55635	0.213548	0.106774	+	0.994283i	$0.465948\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.2721	0.507842	0.253921	−	0.967225i	$-0.418280\pi$
$683$	13.2721	0.507842	0.253921	+	0.967225i	$0.418280\pi$
$684$	0	0
$685$	8.14214	0.311095
$686$	0	0
$687$	0	0
$688$	0	0
$689$	56.6274	2.15733
$690$	0	0
$691$	−4.34315	−0.165221	−0.0826105	−	0.996582i	$-0.526326\pi$
$691$	−4.34315	−0.165221	−0.0826105	+	0.996582i	$0.526326\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.31371	−0.201560
$696$	0	0
$697$	3.02944	0.114748
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.3431	−1.44820	−0.724100	−	0.689695i	$-0.757745\pi$
$701$	−38.3431	−1.44820	−0.724100	+	0.689695i	$0.757745\pi$
$702$	0	0
$703$	1.75736	0.0662801
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.31371	−0.124625
$708$	0	0
$709$	−14.9706	−0.562231	−0.281116	−	0.959674i	$-0.590704\pi$
$709$	−14.9706	−0.562231	−0.281116	+	0.959674i	$0.590704\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−40.9706	−1.53436
$714$	0	0
$715$	12.4853	0.466923
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.2843	0.681888	0.340944	−	0.940084i	$-0.389253\pi$
$719$	18.2843	0.681888	0.340944	+	0.940084i	$0.389253\pi$
$720$	0	0
$721$	3.23045	0.120308
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.48528	0.240857
$726$	0	0
$727$	−46.4853	−1.72404	−0.862022	−	0.506871i	$-0.830802\pi$
$727$	−46.4853	−1.72404	−0.862022	+	0.506871i	$0.830802\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	28.3431	1.04688	0.523439	−	0.852063i	$-0.324649\pi$
$733$	28.3431	1.04688	0.523439	+	0.852063i	$0.324649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.82843	−0.251528
$738$	0	0
$739$	−10.6274	−0.390936	−0.195468	−	0.980710i	$-0.562623\pi$
$739$	−10.6274	−0.390936	−0.195468	+	0.980710i	$0.562623\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.0416	1.24887	0.624433	−	0.781078i	$-0.285330\pi$
$743$	34.0416	1.24887	0.624433	+	0.781078i	$0.285330\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.14214	−0.224429
$750$	0	0
$751$	−14.1421	−0.516054	−0.258027	−	0.966138i	$-0.583072\pi$
$751$	−14.1421	−0.516054	−0.258027	+	0.966138i	$0.583072\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.1421	0.369110
$756$	0	0
$757$	8.34315	0.303237	0.151618	−	0.988439i	$-0.451552\pi$
$757$	8.34315	0.303237	0.151618	+	0.988439i	$0.451552\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.6274	0.530243	0.265122	−	0.964215i	$-0.414588\pi$
$761$	14.6274	0.530243	0.265122	+	0.964215i	$0.414588\pi$
$762$	0	0
$763$	−2.62742	−0.0951189
$764$	0	0
$765$	0	0
$766$	0	0
$767$	85.2548	3.07837
$768$	0	0
$769$	−6.62742	−0.238991	−0.119495	−	0.992835i	$-0.538128\pi$
$769$	−6.62742	−0.238991	−0.119495	+	0.992835i	$0.538128\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.8995	0.715735	0.357868	−	0.933772i	$-0.383504\pi$
$773$	19.8995	0.715735	0.357868	+	0.933772i	$0.383504\pi$
$774$	0	0
$775$	−6.82843	−0.245284
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.65685	0.131020
$780$	0	0
$781$	−10.3431	−0.370107
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.82843	−0.315100
$786$	0	0
$787$	45.8406	1.63404	0.817021	−	0.576608i	$-0.195624\pi$
$787$	45.8406	1.63404	0.817021	+	0.576608i	$0.195624\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.51472	0.267193
$792$	0	0
$793$	85.2548	3.02749
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.89949	0.279814	0.139907	−	0.990165i	$-0.455320\pi$
$797$	7.89949	0.279814	0.139907	+	0.990165i	$0.455320\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.97056	−0.175407
$804$	0	0
$805$	−4.97056	−0.175189
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−51.2548	−1.80202	−0.901012	−	0.433794i	$-0.857175\pi$
$809$	−51.2548	−1.80202	−0.901012	+	0.433794i	$0.857175\pi$
$810$	0	0
$811$	−19.7990	−0.695237	−0.347618	−	0.937636i	$-0.613009\pi$
$811$	−19.7990	−0.695237	−0.347618	+	0.937636i	$0.613009\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.4853	−0.507397
$816$	0	0
$817$	−4.82843	−0.168925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−20.8284	−0.726033	−0.363017	−	0.931783i	$-0.618253\pi$
$823$	−20.8284	−0.726033	−0.363017	+	0.931783i	$0.618253\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.9289	0.519130	0.259565	−	0.965726i	$-0.416421\pi$
$827$	14.9289	0.519130	0.259565	+	0.965726i	$0.416421\pi$
$828$	0	0
$829$	44.4264	1.54299	0.771496	−	0.636234i	$-0.219509\pi$
$829$	44.4264	1.54299	0.771496	+	0.636234i	$0.219509\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.23045	0.181224
$834$	0	0
$835$	14.7279	0.509681
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.5980	1.22898	0.614489	−	0.788925i	$-0.289362\pi$
$839$	35.5980	1.22898	0.614489	+	0.788925i	$0.289362\pi$
$840$	0	0
$841$	13.0589	0.450306
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−25.9706	−0.893415
$846$	0	0
$847$	−5.79899	−0.199256
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.5442	−0.361449
$852$	0	0
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.2132	−0.519673	−0.259837	−	0.965653i	$-0.583669\pi$
$857$	−15.2132	−0.519673	−0.259837	+	0.965653i	$0.583669\pi$
$858$	0	0
$859$	32.2843	1.10153	0.550763	−	0.834662i	$-0.314337\pi$
$859$	32.2843	1.10153	0.550763	+	0.834662i	$0.314337\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.8995	−0.541225	−0.270613	−	0.962688i	$-0.587226\pi$
$863$	−15.8995	−0.541225	−0.270613	+	0.962688i	$0.587226\pi$
$864$	0	0
$865$	15.8995	0.540599
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.31371	0.112410
$870$	0	0
$871$	21.3137	0.722187
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.828427	−0.0280059
$876$	0	0
$877$	−22.9289	−0.774255	−0.387128	−	0.922026i	$-0.626533\pi$
$877$	−22.9289	−0.774255	−0.387128	+	0.922026i	$0.626533\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.2843	0.413868	0.206934	−	0.978355i	$-0.433652\pi$
$881$	12.2843	0.413868	0.206934	+	0.978355i	$0.433652\pi$
$882$	0	0
$883$	−44.8284	−1.50860	−0.754298	−	0.656532i	$-0.772023\pi$
$883$	−44.8284	−1.50860	−0.754298	+	0.656532i	$0.772023\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.9289	1.17280	0.586399	−	0.810022i	$-0.300545\pi$
$887$	34.9289	1.17280	0.586399	+	0.810022i	$0.300545\pi$
$888$	0	0
$889$	7.91674	0.265519
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.82843	−0.161577
$894$	0	0
$895$	10.3431	0.345733
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−44.2843	−1.47696
$900$	0	0
$901$	7.51472	0.250352
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	16.3848	0.544048	0.272024	−	0.962291i	$-0.412307\pi$
$907$	16.3848	0.544048	0.272024	+	0.962291i	$0.412307\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.7990	−1.45113	−0.725563	−	0.688156i	$-0.758420\pi$
$911$	−43.7990	−1.45113	−0.725563	+	0.688156i	$0.758420\pi$
$912$	0	0
$913$	26.6274	0.881239
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.68629	−0.154755
$918$	0	0
$919$	12.6863	0.418482	0.209241	−	0.977864i	$-0.432901\pi$
$919$	12.6863	0.418482	0.209241	+	0.977864i	$0.432901\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	32.2843	1.06265
$924$	0	0
$925$	−1.75736	−0.0577816
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.3137	−0.961752	−0.480876	−	0.876789i	$-0.659681\pi$
$929$	−29.3137	−0.961752	−0.480876	+	0.876789i	$0.659681\pi$
$930$	0	0
$931$	6.31371	0.206923
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.65685	0.0541849
$936$	0	0
$937$	−11.8579	−0.387380	−0.193690	−	0.981063i	$-0.562046\pi$
$937$	−11.8579	−0.387380	−0.193690	+	0.981063i	$0.562046\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.2843	−0.856843	−0.428421	−	0.903579i	$-0.640930\pi$
$941$	−26.2843	−0.856843	−0.428421	+	0.903579i	$0.640930\pi$
$942$	0	0
$943$	−21.9411	−0.714501
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.6569	0.638762	0.319381	−	0.947626i	$-0.396525\pi$
$947$	19.6569	0.638762	0.319381	+	0.947626i	$0.396525\pi$
$948$	0	0
$949$	15.5147	0.503629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.2426	−0.461364	−0.230682	−	0.973029i	$-0.574096\pi$
$953$	−14.2426	−0.461364	−0.230682	+	0.973029i	$0.574096\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.74517	−0.217813
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.41421	−0.109907
$966$	0	0
$967$	−24.6274	−0.791964	−0.395982	−	0.918258i	$-0.629596\pi$
$967$	−24.6274	−0.791964	−0.395982	+	0.918258i	$0.629596\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.1127	−0.870088	−0.435044	−	0.900409i	$-0.643267\pi$
$971$	−27.1127	−0.870088	−0.435044	+	0.900409i	$0.643267\pi$
$972$	0	0
$973$	4.40202	0.141122
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.7279	−1.11104	−0.555522	−	0.831502i	$-0.687482\pi$
$977$	−34.7279	−1.11104	−0.555522	+	0.831502i	$0.687482\pi$
$978$	0	0
$979$	12.9706	0.414541
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.5858	0.656585	0.328292	−	0.944576i	$-0.393527\pi$
$983$	20.5858	0.656585	0.328292	+	0.944576i	$0.393527\pi$
$984$	0	0
$985$	17.3137	0.551661
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.9706	0.921210
$990$	0	0
$991$	11.1127	0.353006	0.176503	−	0.984300i	$-0.443521\pi$
$991$	11.1127	0.353006	0.176503	+	0.984300i	$0.443521\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.6569	0.686568
$996$	0	0
$997$	−30.4853	−0.965479	−0.482739	−	0.875764i	$-0.660358\pi$
$997$	−30.4853	−0.965479	−0.482739	+	0.875764i	$0.660358\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.a.g.1.2		2
3.2	odd	2		380.2.a.c.1.1	✓	2
12.11	even	2		1520.2.a.o.1.2		2
15.2	even	4		1900.2.c.d.1749.4		4
15.8	even	4		1900.2.c.d.1749.1		4
15.14	odd	2		1900.2.a.e.1.2		2
24.5	odd	2		6080.2.a.bl.1.2		2
24.11	even	2		6080.2.a.y.1.1		2
57.56	even	2		7220.2.a.m.1.2		2
60.59	even	2		7600.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.c.1.1	✓	2	3.2	odd	2
1520.2.a.o.1.2		2	12.11	even	2
1900.2.a.e.1.2		2	15.14	odd	2
1900.2.c.d.1749.1		4	15.8	even	4
1900.2.c.d.1749.4		4	15.2	even	4
3420.2.a.g.1.2		2	1.1	even	1	trivial
6080.2.a.y.1.1		2	24.11	even	2
6080.2.a.bl.1.2		2	24.5	odd	2
7220.2.a.m.1.2		2	57.56	even	2
7600.2.a.u.1.1		2	60.59	even	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.828427 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.828427 q^{7} +2.00000 q^{11} -6.24264 q^{13} -0.828427 q^{17} -1.00000 q^{19} +6.00000 q^{23} +1.00000 q^{25} +6.48528 q^{29} -6.82843 q^{31} -0.828427 q^{35} -1.75736 q^{37} -3.65685 q^{41} +4.82843 q^{43} +4.82843 q^{47} -6.31371 q^{49} -9.07107 q^{53} -2.00000 q^{55} -13.6569 q^{59} -13.6569 q^{61} +6.24264 q^{65} -3.41421 q^{67} -5.17157 q^{71} -2.48528 q^{73} +1.65685 q^{77} +1.65685 q^{79} +13.3137 q^{83} +0.828427 q^{85} +6.48528 q^{89} -5.17157 q^{91} +1.00000 q^{95} -10.2426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{59} - 16 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{77} - 8 q^{79} + 4 q^{83} - 4 q^{85} - 4 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 - 16 * q^59 - 16 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^77 - 8 * q^79 + 4 * q^83 - 4 * q^85 - 4 * q^89 - 16 * q^91 + 2 * q^95 - 12 * q^97

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