LMFDB - Embedded newform 9408.2.a.eh.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9408,2,Mod(1,9408)] 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("9408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,0,0,3,0,0,0,3,0,0,0,6,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.1232582216$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.523976$ of defining polynomial
Character	$\chi$	$=$	9408.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.20147 q^{5} +1.00000 q^{9} -4.24943 q^{11} +3.15352 q^{13} -3.20147 q^{15} -4.40294 q^{17} -3.15352 q^{19} +4.40294 q^{23} +5.24943 q^{25} -1.00000 q^{27} -7.20147 q^{29} +2.04795 q^{31} +4.24943 q^{33} +9.65237 q^{37} -3.15352 q^{39} -10.4989 q^{41} -0.750575 q^{43} +3.20147 q^{45} -2.40294 q^{47} +4.40294 q^{51} +3.29738 q^{53} -13.6044 q^{55} +3.15352 q^{57} -8.24943 q^{59} +8.09591 q^{61} +10.0959 q^{65} -5.15352 q^{67} -4.40294 q^{69} +6.40294 q^{71} +15.2494 q^{73} -5.24943 q^{75} -16.4509 q^{79} +1.00000 q^{81} -14.5565 q^{83} -14.0959 q^{85} +7.20147 q^{87} +2.40294 q^{89} -2.04795 q^{93} -10.0959 q^{95} +4.24943 q^{97} -4.24943 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} - 3 q^{37} - 3 q^{39} - 6 q^{41} - 15 q^{43} + 12 q^{47} - 6 q^{51} - 6 q^{53} - 12 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 - 3 * q^37 - 3 * q^39 - 6 * q^41 - 15 * q^43 + 12 * q^47 - 6 * q^51 - 6 * q^53 - 12 * q^55 + 3 * q^57 - 12 * q^59 + 18 * q^61 + 24 * q^65 - 9 * q^67 + 6 * q^69 + 33 * q^73 - 3 * q^75 - 27 * q^79 + 3 * q^81 - 18 * q^83 - 36 * q^85 + 12 * q^87 - 12 * q^89 - 3 * q^93 - 24 * q^95

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.20147	1.43174	0.715871	−	0.698233i	$-0.246030\pi$
$5$	3.20147	1.43174	0.715871	+	0.698233i	$0.246030\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.24943	−1.28125	−0.640625	−	0.767854i	$-0.721325\pi$
$11$	−4.24943	−1.28125	−0.640625	+	0.767854i	$0.721325\pi$
$12$	0	0
$13$	3.15352	0.874629	0.437314	−	0.899309i	$-0.355930\pi$
$13$	3.15352	0.874629	0.437314	+	0.899309i	$0.355930\pi$
$14$	0	0
$15$	−3.20147	−0.826617
$16$	0	0
$17$	−4.40294	−1.06787	−0.533935	−	0.845525i	$-0.679287\pi$
$17$	−4.40294	−1.06787	−0.533935	+	0.845525i	$0.679287\pi$
$18$	0	0
$19$	−3.15352	−0.723467	−0.361734	−	0.932282i	$-0.617815\pi$
$19$	−3.15352	−0.723467	−0.361734	+	0.932282i	$0.617815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.40294	0.918077	0.459039	−	0.888416i	$-0.348194\pi$
$23$	4.40294	0.918077	0.459039	+	0.888416i	$0.348194\pi$
$24$	0	0
$25$	5.24943	1.04989
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.20147	−1.33728	−0.668640	−	0.743586i	$-0.733123\pi$
$29$	−7.20147	−1.33728	−0.668640	+	0.743586i	$0.733123\pi$
$30$	0	0
$31$	2.04795	0.367823	0.183912	−	0.982943i	$-0.441124\pi$
$31$	2.04795	0.367823	0.183912	+	0.982943i	$0.441124\pi$
$32$	0	0
$33$	4.24943	0.739730
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.65237	1.58684	0.793420	−	0.608675i	$-0.208299\pi$
$37$	9.65237	1.58684	0.793420	+	0.608675i	$0.208299\pi$
$38$	0	0
$39$	−3.15352	−0.504967
$40$	0	0
$41$	−10.4989	−1.63964	−0.819822	−	0.572618i	$-0.805928\pi$
$41$	−10.4989	−1.63964	−0.819822	+	0.572618i	$0.805928\pi$
$42$	0	0
$43$	−0.750575	−0.114462	−0.0572308	−	0.998361i	$-0.518227\pi$
$43$	−0.750575	−0.114462	−0.0572308	+	0.998361i	$0.518227\pi$
$44$	0	0
$45$	3.20147	0.477247
$46$	0	0
$47$	−2.40294	−0.350506	−0.175253	−	0.984523i	$-0.556074\pi$
$47$	−2.40294	−0.350506	−0.175253	+	0.984523i	$0.556074\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.40294	0.616536
$52$	0	0
$53$	3.29738	0.452930	0.226465	−	0.974019i	$-0.427283\pi$
$53$	3.29738	0.452930	0.226465	+	0.974019i	$0.427283\pi$
$54$	0	0
$55$	−13.6044	−1.83442
$56$	0	0
$57$	3.15352	0.417694
$58$	0	0
$59$	−8.24943	−1.07398	−0.536992	−	0.843587i	$-0.680439\pi$
$59$	−8.24943	−1.07398	−0.536992	+	0.843587i	$0.680439\pi$
$60$	0	0
$61$	8.09591	1.03657	0.518287	−	0.855207i	$-0.326570\pi$
$61$	8.09591	1.03657	0.518287	+	0.855207i	$0.326570\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.0959	1.25224
$66$	0	0
$67$	−5.15352	−0.629603	−0.314801	−	0.949158i	$-0.601938\pi$
$67$	−5.15352	−0.629603	−0.314801	+	0.949158i	$0.601938\pi$
$68$	0	0
$69$	−4.40294	−0.530052
$70$	0	0
$71$	6.40294	0.759890	0.379945	−	0.925009i	$-0.375943\pi$
$71$	6.40294	0.759890	0.379945	+	0.925009i	$0.375943\pi$
$72$	0	0
$73$	15.2494	1.78481	0.892405	−	0.451235i	$-0.149016\pi$
$73$	15.2494	1.78481	0.892405	+	0.451235i	$0.149016\pi$
$74$	0	0
$75$	−5.24943	−0.606151
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.4509	−1.85087	−0.925435	−	0.378906i	$-0.876300\pi$
$79$	−16.4509	−1.85087	−0.925435	+	0.378906i	$0.876300\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.5565	−1.59778	−0.798890	−	0.601477i	$-0.794579\pi$
$83$	−14.5565	−1.59778	−0.798890	+	0.601477i	$0.794579\pi$
$84$	0	0
$85$	−14.0959	−1.52892
$86$	0	0
$87$	7.20147	0.772079
$88$	0	0
$89$	2.40294	0.254712	0.127356	−	0.991857i	$-0.459351\pi$
$89$	2.40294	0.254712	0.127356	+	0.991857i	$0.459351\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.04795	−0.212363
$94$	0	0
$95$	−10.0959	−1.03582
$96$	0	0
$97$	4.24943	0.431464	0.215732	−	0.976453i	$-0.430786\pi$
$97$	4.24943	0.431464	0.215732	+	0.976453i	$0.430786\pi$
$98$	0	0
$99$	−4.24943	−0.427083
$100$	0	0
$101$	−3.90409	−0.388472	−0.194236	−	0.980955i	$-0.562223\pi$
$101$	−3.90409	−0.388472	−0.194236	+	0.980955i	$0.562223\pi$
$102$	0	0
$103$	9.24943	0.911373	0.455686	−	0.890140i	$-0.349394\pi$
$103$	9.24943	0.911373	0.455686	+	0.890140i	$0.349394\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.5565	−1.60057	−0.800287	−	0.599617i	$-0.795319\pi$
$107$	−16.5565	−1.60057	−0.800287	+	0.599617i	$0.795319\pi$
$108$	0	0
$109$	−20.0553	−1.92095	−0.960475	−	0.278365i	$-0.910208\pi$
$109$	−20.0553	−1.92095	−0.960475	+	0.278365i	$0.910208\pi$
$110$	0	0
$111$	−9.65237	−0.916162
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	14.0959	1.31445
$116$	0	0
$117$	3.15352	0.291543
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.05761	0.641601
$122$	0	0
$123$	10.4989	0.946649
$124$	0	0
$125$	0.798528	0.0714225
$126$	0	0
$127$	12.4509	1.10484	0.552419	−	0.833566i	$-0.313705\pi$
$127$	12.4509	1.10484	0.552419	+	0.833566i	$0.313705\pi$
$128$	0	0
$129$	0.750575	0.0660844
$130$	0	0
$131$	−4.24943	−0.371274	−0.185637	−	0.982618i	$-0.559435\pi$
$131$	−4.24943	−0.371274	−0.185637	+	0.982618i	$0.559435\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.20147	−0.275539
$136$	0	0
$137$	−10.4029	−0.888784	−0.444392	−	0.895833i	$-0.646580\pi$
$137$	−10.4029	−0.888784	−0.444392	+	0.895833i	$0.646580\pi$
$138$	0	0
$139$	−11.6524	−0.988341	−0.494171	−	0.869365i	$-0.664528\pi$
$139$	−11.6524	−0.988341	−0.494171	+	0.869365i	$0.664528\pi$
$140$	0	0
$141$	2.40294	0.202364
$142$	0	0
$143$	−13.4006	−1.12062
$144$	0	0
$145$	−23.0553	−1.91464
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.4029	−0.852242	−0.426121	−	0.904666i	$-0.640120\pi$
$149$	−10.4029	−0.852242	−0.426121	+	0.904666i	$0.640120\pi$
$150$	0	0
$151$	−20.0074	−1.62818	−0.814088	−	0.580742i	$-0.802763\pi$
$151$	−20.0074	−1.62818	−0.814088	+	0.580742i	$0.802763\pi$
$152$	0	0
$153$	−4.40294	−0.355957
$154$	0	0
$155$	6.55646	0.526628
$156$	0	0
$157$	−14.4989	−1.15713	−0.578567	−	0.815635i	$-0.696388\pi$
$157$	−14.4989	−1.15713	−0.578567	+	0.815635i	$0.696388\pi$
$158$	0	0
$159$	−3.29738	−0.261499
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.7100	0.838871	0.419435	−	0.907785i	$-0.362228\pi$
$163$	10.7100	0.838871	0.419435	+	0.907785i	$0.362228\pi$
$164$	0	0
$165$	13.6044	1.05910
$166$	0	0
$167$	17.3047	1.33908	0.669540	−	0.742776i	$-0.266491\pi$
$167$	17.3047	1.33908	0.669540	+	0.742776i	$0.266491\pi$
$168$	0	0
$169$	−3.05531	−0.235024
$170$	0	0
$171$	−3.15352	−0.241156
$172$	0	0
$173$	3.90409	0.296823	0.148411	−	0.988926i	$-0.452584\pi$
$173$	3.90409	0.296823	0.148411	+	0.988926i	$0.452584\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.24943	0.620065
$178$	0	0
$179$	−20.8059	−1.55511	−0.777553	−	0.628818i	$-0.783539\pi$
$179$	−20.8059	−1.55511	−0.777553	+	0.628818i	$0.783539\pi$
$180$	0	0
$181$	13.2494	0.984822	0.492411	−	0.870363i	$-0.336116\pi$
$181$	13.2494	0.984822	0.492411	+	0.870363i	$0.336116\pi$
$182$	0	0
$183$	−8.09591	−0.598467
$184$	0	0
$185$	30.9018	2.27195
$186$	0	0
$187$	18.7100	1.36821
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.19181	0.303309	0.151654	−	0.988434i	$-0.451540\pi$
$191$	4.19181	0.303309	0.151654	+	0.988434i	$0.451540\pi$
$192$	0	0
$193$	−18.3047	−1.31760	−0.658802	−	0.752316i	$-0.728936\pi$
$193$	−18.3047	−1.31760	−0.658802	+	0.752316i	$0.728936\pi$
$194$	0	0
$195$	−10.0959	−0.722983
$196$	0	0
$197$	−11.9041	−0.848132	−0.424066	−	0.905631i	$-0.639397\pi$
$197$	−11.9041	−0.848132	−0.424066	+	0.905631i	$0.639397\pi$
$198$	0	0
$199$	0.805889	0.0571280	0.0285640	−	0.999592i	$-0.490907\pi$
$199$	0.805889	0.0571280	0.0285640	+	0.999592i	$0.490907\pi$
$200$	0	0
$201$	5.15352	0.363501
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−33.6118	−2.34755
$206$	0	0
$207$	4.40294	0.306026
$208$	0	0
$209$	13.4006	0.926942
$210$	0	0
$211$	−8.49885	−0.585085	−0.292542	−	0.956253i	$-0.594501\pi$
$211$	−8.49885	−0.585085	−0.292542	+	0.956253i	$0.594501\pi$
$212$	0	0
$213$	−6.40294	−0.438723
$214$	0	0
$215$	−2.40294	−0.163879
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−15.2494	−1.03046
$220$	0	0
$221$	−13.8848	−0.933991
$222$	0	0
$223$	−16.7985	−1.12491	−0.562456	−	0.826827i	$-0.690144\pi$
$223$	−16.7985	−1.12491	−0.562456	+	0.826827i	$0.690144\pi$
$224$	0	0
$225$	5.24943	0.349962
$226$	0	0
$227$	16.8635	1.11927	0.559635	−	0.828739i	$-0.310941\pi$
$227$	16.8635	1.11927	0.559635	+	0.828739i	$0.310941\pi$
$228$	0	0
$229$	−20.5542	−1.35826	−0.679129	−	0.734019i	$-0.737642\pi$
$229$	−20.5542	−1.35826	−0.679129	+	0.734019i	$0.737642\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.90409	0.648839	0.324419	−	0.945913i	$-0.394831\pi$
$233$	9.90409	0.648839	0.324419	+	0.945913i	$0.394831\pi$
$234$	0	0
$235$	−7.69296	−0.501833
$236$	0	0
$237$	16.4509	1.06860
$238$	0	0
$239$	4.40294	0.284803	0.142401	−	0.989809i	$-0.454518\pi$
$239$	4.40294	0.284803	0.142401	+	0.989809i	$0.454518\pi$
$240$	0	0
$241$	20.8635	1.34394	0.671968	−	0.740580i	$-0.265449\pi$
$241$	20.8635	1.34394	0.671968	+	0.740580i	$0.265449\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.94469	−0.632765
$248$	0	0
$249$	14.5565	0.922478
$250$	0	0
$251$	−3.44354	−0.217354	−0.108677	−	0.994077i	$-0.534661\pi$
$251$	−3.44354	−0.217354	−0.108677	+	0.994077i	$0.534661\pi$
$252$	0	0
$253$	−18.7100	−1.17629
$254$	0	0
$255$	14.0959	0.882720
$256$	0	0
$257$	−5.59706	−0.349135	−0.174567	−	0.984645i	$-0.555853\pi$
$257$	−5.59706	−0.349135	−0.174567	+	0.984645i	$0.555853\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.20147	−0.445760
$262$	0	0
$263$	−1.59706	−0.0984787	−0.0492393	−	0.998787i	$-0.515680\pi$
$263$	−1.59706	−0.0984787	−0.0492393	+	0.998787i	$0.515680\pi$
$264$	0	0
$265$	10.5565	0.648478
$266$	0	0
$267$	−2.40294	−0.147058
$268$	0	0
$269$	−17.7003	−1.07921	−0.539604	−	0.841919i	$-0.681426\pi$
$269$	−17.7003	−1.07921	−0.539604	+	0.841919i	$0.681426\pi$
$270$	0	0
$271$	−2.39558	−0.145521	−0.0727607	−	0.997349i	$-0.523181\pi$
$271$	−2.39558	−0.145521	−0.0727607	+	0.997349i	$0.523181\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.3070	−1.34517
$276$	0	0
$277$	5.24943	0.315407	0.157704	−	0.987486i	$-0.449591\pi$
$277$	5.24943	0.315407	0.157704	+	0.987486i	$0.449591\pi$
$278$	0	0
$279$	2.04795	0.122608
$280$	0	0
$281$	20.0959	1.19882	0.599411	−	0.800442i	$-0.295402\pi$
$281$	20.0959	1.19882	0.599411	+	0.800442i	$0.295402\pi$
$282$	0	0
$283$	23.2494	1.38203	0.691017	−	0.722838i	$-0.257163\pi$
$283$	23.2494	1.38203	0.691017	+	0.722838i	$0.257163\pi$
$284$	0	0
$285$	10.0959	0.598030
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.38592	0.140348
$290$	0	0
$291$	−4.24943	−0.249106
$292$	0	0
$293$	−11.2015	−0.654397	−0.327199	−	0.944956i	$-0.606105\pi$
$293$	−11.2015	−0.654397	−0.327199	+	0.944956i	$0.606105\pi$
$294$	0	0
$295$	−26.4103	−1.53767
$296$	0	0
$297$	4.24943	0.246577
$298$	0	0
$299$	13.8848	0.802977
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.90409	0.224284
$304$	0	0
$305$	25.9188	1.48411
$306$	0	0
$307$	28.9571	1.65267	0.826335	−	0.563179i	$-0.190422\pi$
$307$	28.9571	1.65267	0.826335	+	0.563179i	$0.190422\pi$
$308$	0	0
$309$	−9.24943	−0.526181
$310$	0	0
$311$	27.3047	1.54831	0.774155	−	0.632996i	$-0.218175\pi$
$311$	27.3047	1.54831	0.774155	+	0.632996i	$0.218175\pi$
$312$	0	0
$313$	−5.49885	−0.310813	−0.155407	−	0.987851i	$-0.549669\pi$
$313$	−5.49885	−0.310813	−0.155407	+	0.987851i	$0.549669\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.60442	0.539438	0.269719	−	0.962939i	$-0.413069\pi$
$317$	9.60442	0.539438	0.269719	+	0.962939i	$0.413069\pi$
$318$	0	0
$319$	30.6021	1.71339
$320$	0	0
$321$	16.5565	0.924092
$322$	0	0
$323$	13.8848	0.772569
$324$	0	0
$325$	16.5542	0.918260
$326$	0	0
$327$	20.0553	1.10906
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.5565	−0.855061	−0.427530	−	0.904001i	$-0.640616\pi$
$331$	−15.5565	−0.855061	−0.427530	+	0.904001i	$0.640616\pi$
$332$	0	0
$333$	9.65237	0.528947
$334$	0	0
$335$	−16.4989	−0.901428
$336$	0	0
$337$	17.9977	0.980397	0.490199	−	0.871611i	$-0.336924\pi$
$337$	17.9977	0.980397	0.490199	+	0.871611i	$0.336924\pi$
$338$	0	0
$339$	−8.00000	−0.434500
$340$	0	0
$341$	−8.70262	−0.471273
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−14.0959	−0.758898
$346$	0	0
$347$	−5.69296	−0.305614	−0.152807	−	0.988256i	$-0.548831\pi$
$347$	−5.69296	−0.305614	−0.152807	+	0.988256i	$0.548831\pi$
$348$	0	0
$349$	18.1918	0.973785	0.486893	−	0.873462i	$-0.338130\pi$
$349$	18.1918	0.973785	0.486893	+	0.873462i	$0.338130\pi$
$350$	0	0
$351$	−3.15352	−0.168322
$352$	0	0
$353$	−20.9977	−1.11759	−0.558797	−	0.829304i	$-0.688737\pi$
$353$	−20.9977	−1.11759	−0.558797	+	0.829304i	$0.688737\pi$
$354$	0	0
$355$	20.4989	1.08797
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.6907	−1.30312	−0.651562	−	0.758596i	$-0.725886\pi$
$359$	−24.6907	−1.30312	−0.651562	+	0.758596i	$0.725886\pi$
$360$	0	0
$361$	−9.05531	−0.476595
$362$	0	0
$363$	−7.05761	−0.370429
$364$	0	0
$365$	48.8206	2.55539
$366$	0	0
$367$	−5.85614	−0.305688	−0.152844	−	0.988250i	$-0.548843\pi$
$367$	−5.85614	−0.305688	−0.152844	+	0.988250i	$0.548843\pi$
$368$	0	0
$369$	−10.4989	−0.546548
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.635347	−0.0328970	−0.0164485	−	0.999865i	$-0.505236\pi$
$373$	−0.635347	−0.0328970	−0.0164485	+	0.999865i	$0.505236\pi$
$374$	0	0
$375$	−0.798528	−0.0412358
$376$	0	0
$377$	−22.7100	−1.16962
$378$	0	0
$379$	−10.7506	−0.552220	−0.276110	−	0.961126i	$-0.589045\pi$
$379$	−10.7506	−0.552220	−0.276110	+	0.961126i	$0.589045\pi$
$380$	0	0
$381$	−12.4509	−0.637879
$382$	0	0
$383$	14.9977	0.766347	0.383173	−	0.923676i	$-0.374831\pi$
$383$	14.9977	0.766347	0.383173	+	0.923676i	$0.374831\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.750575	−0.0381539
$388$	0	0
$389$	−6.21113	−0.314917	−0.157458	−	0.987526i	$-0.550330\pi$
$389$	−6.21113	−0.314917	−0.157458	+	0.987526i	$0.550330\pi$
$390$	0	0
$391$	−19.3859	−0.980388
$392$	0	0
$393$	4.24943	0.214355
$394$	0	0
$395$	−52.6671	−2.64997
$396$	0	0
$397$	−1.24943	−0.0627068	−0.0313534	−	0.999508i	$-0.509982\pi$
$397$	−1.24943	−0.0627068	−0.0313534	+	0.999508i	$0.509982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.11523	0.105629	0.0528147	−	0.998604i	$-0.483181\pi$
$401$	2.11523	0.105629	0.0528147	+	0.998604i	$0.483181\pi$
$402$	0	0
$403$	6.45826	0.321709
$404$	0	0
$405$	3.20147	0.159082
$406$	0	0
$407$	−41.0170	−2.03314
$408$	0	0
$409$	−16.1129	−0.796733	−0.398367	−	0.917226i	$-0.630423\pi$
$409$	−16.1129	−0.796733	−0.398367	+	0.917226i	$0.630423\pi$
$410$	0	0
$411$	10.4029	0.513139
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−46.6021	−2.28761
$416$	0	0
$417$	11.6524	0.570619
$418$	0	0
$419$	20.1106	0.982469	0.491234	−	0.871027i	$-0.336546\pi$
$419$	20.1106	0.982469	0.491234	+	0.871027i	$0.336546\pi$
$420$	0	0
$421$	−4.96171	−0.241819	−0.120909	−	0.992664i	$-0.538581\pi$
$421$	−4.96171	−0.241819	−0.120909	+	0.992664i	$0.538581\pi$
$422$	0	0
$423$	−2.40294	−0.116835
$424$	0	0
$425$	−23.1129	−1.12114
$426$	0	0
$427$	0	0
$428$	0	0
$429$	13.4006	0.646989
$430$	0	0
$431$	−20.8059	−1.00218	−0.501092	−	0.865394i	$-0.667068\pi$
$431$	−20.8059	−1.00218	−0.501092	+	0.865394i	$0.667068\pi$
$432$	0	0
$433$	−8.05531	−0.387114	−0.193557	−	0.981089i	$-0.562002\pi$
$433$	−8.05531	−0.387114	−0.193557	+	0.981089i	$0.562002\pi$
$434$	0	0
$435$	23.0553	1.10542
$436$	0	0
$437$	−13.8848	−0.664199
$438$	0	0
$439$	−16.1992	−0.773144	−0.386572	−	0.922259i	$-0.626341\pi$
$439$	−16.1992	−0.773144	−0.386572	+	0.922259i	$0.626341\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.4435	−0.733745	−0.366872	−	0.930271i	$-0.619571\pi$
$443$	−15.4435	−0.733745	−0.366872	+	0.930271i	$0.619571\pi$
$444$	0	0
$445$	7.69296	0.364681
$446$	0	0
$447$	10.4029	0.492042
$448$	0	0
$449$	31.5159	1.48733	0.743663	−	0.668555i	$-0.233087\pi$
$449$	31.5159	1.48733	0.743663	+	0.668555i	$0.233087\pi$
$450$	0	0
$451$	44.6141	2.10079
$452$	0	0
$453$	20.0074	0.940028
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.1152	−0.519948	−0.259974	−	0.965616i	$-0.583714\pi$
$457$	−11.1152	−0.519948	−0.259974	+	0.965616i	$0.583714\pi$
$458$	0	0
$459$	4.40294	0.205512
$460$	0	0
$461$	−13.5971	−0.633278	−0.316639	−	0.948546i	$-0.602554\pi$
$461$	−13.5971	−0.633278	−0.316639	+	0.948546i	$0.602554\pi$
$462$	0	0
$463$	2.55876	0.118916	0.0594579	−	0.998231i	$-0.481063\pi$
$463$	2.55876	0.118916	0.0594579	+	0.998231i	$0.481063\pi$
$464$	0	0
$465$	−6.55646	−0.304049
$466$	0	0
$467$	0.614078	0.0284161	0.0142081	−	0.999899i	$-0.495477\pi$
$467$	0.614078	0.0284161	0.0142081	+	0.999899i	$0.495477\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.4989	0.668072
$472$	0	0
$473$	3.18951	0.146654
$474$	0	0
$475$	−16.5542	−0.759557
$476$	0	0
$477$	3.29738	0.150977
$478$	0	0
$479$	20.0959	0.918205	0.459103	−	0.888383i	$-0.348171\pi$
$479$	20.0959	0.918205	0.459103	+	0.888383i	$0.348171\pi$
$480$	0	0
$481$	30.4389	1.38790
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.6044	0.617745
$486$	0	0
$487$	23.3720	1.05909	0.529544	−	0.848283i	$-0.322363\pi$
$487$	23.3720	1.05909	0.529544	+	0.848283i	$0.322363\pi$
$488$	0	0
$489$	−10.7100	−0.484322
$490$	0	0
$491$	−22.7483	−1.02662	−0.513308	−	0.858205i	$-0.671580\pi$
$491$	−22.7483	−1.02662	−0.513308	+	0.858205i	$0.671580\pi$
$492$	0	0
$493$	31.7077	1.42804
$494$	0	0
$495$	−13.6044	−0.611473
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.53944	0.382278	0.191139	−	0.981563i	$-0.438782\pi$
$499$	8.53944	0.382278	0.191139	+	0.981563i	$0.438782\pi$
$500$	0	0
$501$	−17.3047	−0.773119
$502$	0	0
$503$	7.59706	0.338736	0.169368	−	0.985553i	$-0.445827\pi$
$503$	7.59706	0.338736	0.169368	+	0.985553i	$0.445827\pi$
$504$	0	0
$505$	−12.4989	−0.556192
$506$	0	0
$507$	3.05531	0.135691
$508$	0	0
$509$	4.99034	0.221193	0.110596	−	0.993865i	$-0.464724\pi$
$509$	4.99034	0.221193	0.110596	+	0.993865i	$0.464724\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.15352	0.139231
$514$	0	0
$515$	29.6118	1.30485
$516$	0	0
$517$	10.2111	0.449085
$518$	0	0
$519$	−3.90409	−0.171371
$520$	0	0
$521$	−12.4989	−0.547585	−0.273792	−	0.961789i	$-0.588278\pi$
$521$	−12.4989	−0.547585	−0.273792	+	0.961789i	$0.588278\pi$
$522$	0	0
$523$	−44.2618	−1.93544	−0.967718	−	0.252036i	$-0.918900\pi$
$523$	−44.2618	−1.93544	−0.967718	+	0.252036i	$0.918900\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.01702	−0.392788
$528$	0	0
$529$	−3.61408	−0.157134
$530$	0	0
$531$	−8.24943	−0.357995
$532$	0	0
$533$	−33.1083	−1.43408
$534$	0	0
$535$	−53.0051	−2.29161
$536$	0	0
$537$	20.8059	0.897840
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.9594	−1.11608	−0.558041	−	0.829813i	$-0.688447\pi$
$541$	−25.9594	−1.11608	−0.558041	+	0.829813i	$0.688447\pi$
$542$	0	0
$543$	−13.2494	−0.568587
$544$	0	0
$545$	−64.2065	−2.75031
$546$	0	0
$547$	−22.9018	−0.979210	−0.489605	−	0.871944i	$-0.662859\pi$
$547$	−22.9018	−0.979210	−0.489605	+	0.871944i	$0.662859\pi$
$548$	0	0
$549$	8.09591	0.345525
$550$	0	0
$551$	22.7100	0.967478
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−30.9018	−1.31171
$556$	0	0
$557$	6.50621	0.275677	0.137839	−	0.990455i	$-0.455984\pi$
$557$	6.50621	0.275677	0.137839	+	0.990455i	$0.455984\pi$
$558$	0	0
$559$	−2.36695	−0.100111
$560$	0	0
$561$	−18.7100	−0.789936
$562$	0	0
$563$	44.2448	1.86470	0.932349	−	0.361561i	$-0.117756\pi$
$563$	44.2448	1.86470	0.932349	+	0.361561i	$0.117756\pi$
$564$	0	0
$565$	25.6118	1.07750
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.0170	−0.545702	−0.272851	−	0.962056i	$-0.587967\pi$
$569$	−13.0170	−0.545702	−0.272851	+	0.962056i	$0.587967\pi$
$570$	0	0
$571$	−30.1705	−1.26260	−0.631299	−	0.775540i	$-0.717478\pi$
$571$	−30.1705	−1.26260	−0.631299	+	0.775540i	$0.717478\pi$
$572$	0	0
$573$	−4.19181	−0.175115
$574$	0	0
$575$	23.1129	0.963876
$576$	0	0
$577$	−29.8059	−1.24084	−0.620418	−	0.784272i	$-0.713037\pi$
$577$	−29.8059	−1.24084	−0.620418	+	0.784272i	$0.713037\pi$
$578$	0	0
$579$	18.3047	0.760719
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−14.0120	−0.580316
$584$	0	0
$585$	10.0959	0.417414
$586$	0	0
$587$	−18.2494	−0.753234	−0.376617	−	0.926369i	$-0.622913\pi$
$587$	−18.2494	−0.753234	−0.376617	+	0.926369i	$0.622913\pi$
$588$	0	0
$589$	−6.45826	−0.266108
$590$	0	0
$591$	11.9041	0.489669
$592$	0	0
$593$	−23.7889	−0.976892	−0.488446	−	0.872594i	$-0.662436\pi$
$593$	−23.7889	−0.976892	−0.488446	+	0.872594i	$0.662436\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.805889	−0.0329829
$598$	0	0
$599$	−15.5011	−0.633360	−0.316680	−	0.948532i	$-0.602568\pi$
$599$	−15.5011	−0.633360	−0.316680	+	0.948532i	$0.602568\pi$
$600$	0	0
$601$	15.8059	0.644736	0.322368	−	0.946614i	$-0.395521\pi$
$601$	15.8059	0.644736	0.322368	+	0.946614i	$0.395521\pi$
$602$	0	0
$603$	−5.15352	−0.209868
$604$	0	0
$605$	22.5948	0.918607
$606$	0	0
$607$	5.95205	0.241586	0.120793	−	0.992678i	$-0.461456\pi$
$607$	5.95205	0.241586	0.120793	+	0.992678i	$0.461456\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.57773	−0.306562
$612$	0	0
$613$	28.4029	1.14718	0.573592	−	0.819141i	$-0.305549\pi$
$613$	28.4029	1.14718	0.573592	+	0.819141i	$0.305549\pi$
$614$	0	0
$615$	33.6118	1.35536
$616$	0	0
$617$	−10.1918	−0.410307	−0.205153	−	0.978730i	$-0.565769\pi$
$617$	−10.1918	−0.410307	−0.205153	+	0.978730i	$0.565769\pi$
$618$	0	0
$619$	−20.8612	−0.838483	−0.419241	−	0.907875i	$-0.637704\pi$
$619$	−20.8612	−0.838483	−0.419241	+	0.907875i	$0.637704\pi$
$620$	0	0
$621$	−4.40294	−0.176684
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−23.6907	−0.947626
$626$	0	0
$627$	−13.4006	−0.535170
$628$	0	0
$629$	−42.4989	−1.69454
$630$	0	0
$631$	41.6191	1.65683	0.828416	−	0.560113i	$-0.189242\pi$
$631$	41.6191	1.65683	0.828416	+	0.560113i	$0.189242\pi$
$632$	0	0
$633$	8.49885	0.337799
$634$	0	0
$635$	39.8612	1.58184
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.40294	0.253297
$640$	0	0
$641$	−48.8013	−1.92754	−0.963768	−	0.266744i	$-0.914052\pi$
$641$	−48.8013	−1.92754	−0.963768	+	0.266744i	$0.914052\pi$
$642$	0	0
$643$	−11.8442	−0.467089	−0.233544	−	0.972346i	$-0.575032\pi$
$643$	−11.8442	−0.467089	−0.233544	+	0.972346i	$0.575032\pi$
$644$	0	0
$645$	2.40294	0.0946159
$646$	0	0
$647$	33.5159	1.31764	0.658822	−	0.752298i	$-0.271055\pi$
$647$	33.5159	1.31764	0.658822	+	0.752298i	$0.271055\pi$
$648$	0	0
$649$	35.0553	1.37604
$650$	0	0
$651$	0	0
$652$	0	0
$653$	46.5062	1.81993	0.909964	−	0.414686i	$-0.136109\pi$
$653$	46.5062	1.81993	0.909964	+	0.414686i	$0.136109\pi$
$654$	0	0
$655$	−13.6044	−0.531569
$656$	0	0
$657$	15.2494	0.594937
$658$	0	0
$659$	10.1152	0.394033	0.197017	−	0.980400i	$-0.436875\pi$
$659$	10.1152	0.394033	0.197017	+	0.980400i	$0.436875\pi$
$660$	0	0
$661$	−33.2641	−1.29383	−0.646913	−	0.762564i	$-0.723940\pi$
$661$	−33.2641	−1.29383	−0.646913	+	0.762564i	$0.723940\pi$
$662$	0	0
$663$	13.8848	0.539240
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−31.7077	−1.22773
$668$	0	0
$669$	16.7985	0.649469
$670$	0	0
$671$	−34.4029	−1.32811
$672$	0	0
$673$	−50.4966	−1.94650	−0.973249	−	0.229751i	$-0.926209\pi$
$673$	−50.4966	−1.94650	−0.973249	+	0.229751i	$0.926209\pi$
$674$	0	0
$675$	−5.24943	−0.202050
$676$	0	0
$677$	−28.7985	−1.10682	−0.553409	−	0.832910i	$-0.686673\pi$
$677$	−28.7985	−1.10682	−0.553409	+	0.832910i	$0.686673\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−16.8635	−0.646211
$682$	0	0
$683$	13.2471	0.506887	0.253444	−	0.967350i	$-0.418437\pi$
$683$	13.2471	0.506887	0.253444	+	0.967350i	$0.418437\pi$
$684$	0	0
$685$	−33.3047	−1.27251
$686$	0	0
$687$	20.5542	0.784190
$688$	0	0
$689$	10.3983	0.396145
$690$	0	0
$691$	39.6671	1.50901	0.754504	−	0.656296i	$-0.227878\pi$
$691$	39.6671	1.50901	0.754504	+	0.656296i	$0.227878\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−37.3047	−1.41505
$696$	0	0
$697$	46.2259	1.75093
$698$	0	0
$699$	−9.90409	−0.374607
$700$	0	0
$701$	2.10787	0.0796130	0.0398065	−	0.999207i	$-0.487326\pi$
$701$	2.10787	0.0796130	0.0398065	+	0.999207i	$0.487326\pi$
$702$	0	0
$703$	−30.4389	−1.14803
$704$	0	0
$705$	7.69296	0.289734
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14.9977	0.563250	0.281625	−	0.959524i	$-0.409127\pi$
$709$	14.9977	0.563250	0.281625	+	0.959524i	$0.409127\pi$
$710$	0	0
$711$	−16.4509	−0.616957
$712$	0	0
$713$	9.01702	0.337690
$714$	0	0
$715$	−42.9018	−1.60444
$716$	0	0
$717$	−4.40294	−0.164431
$718$	0	0
$719$	22.8059	0.850516	0.425258	−	0.905072i	$-0.360183\pi$
$719$	22.8059	0.850516	0.425258	+	0.905072i	$0.360183\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−20.8635	−0.775922
$724$	0	0
$725$	−37.8036	−1.40399
$726$	0	0
$727$	16.9691	0.629348	0.314674	−	0.949200i	$-0.398105\pi$
$727$	16.9691	0.629348	0.314674	+	0.949200i	$0.398105\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.30474	0.122230
$732$	0	0
$733$	−1.34533	−0.0496909	−0.0248455	−	0.999691i	$-0.507909\pi$
$733$	−1.34533	−0.0496909	−0.0248455	+	0.999691i	$0.507909\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.8995	0.806678
$738$	0	0
$739$	−45.2641	−1.66507	−0.832534	−	0.553973i	$-0.813111\pi$
$739$	−45.2641	−1.66507	−0.832534	+	0.553973i	$0.813111\pi$
$740$	0	0
$741$	9.94469	0.365327
$742$	0	0
$743$	−27.8036	−1.02001	−0.510007	−	0.860170i	$-0.670357\pi$
$743$	−27.8036	−1.02001	−0.510007	+	0.860170i	$0.670357\pi$
$744$	0	0
$745$	−33.3047	−1.22019
$746$	0	0
$747$	−14.5565	−0.532593
$748$	0	0
$749$	0	0
$750$	0	0
$751$	30.0627	1.09700	0.548501	−	0.836150i	$-0.315199\pi$
$751$	30.0627	1.09700	0.548501	+	0.836150i	$0.315199\pi$
$752$	0	0
$753$	3.44354	0.125489
$754$	0	0
$755$	−64.0530	−2.33113
$756$	0	0
$757$	17.3241	0.629654	0.314827	−	0.949149i	$-0.398054\pi$
$757$	17.3241	0.629654	0.314827	+	0.949149i	$0.398054\pi$
$758$	0	0
$759$	18.7100	0.679129
$760$	0	0
$761$	12.2111	0.442653	0.221327	−	0.975200i	$-0.428961\pi$
$761$	12.2111	0.442653	0.221327	+	0.975200i	$0.428961\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−14.0959	−0.509639
$766$	0	0
$767$	−26.0147	−0.939337
$768$	0	0
$769$	11.1152	0.400825	0.200413	−	0.979712i	$-0.435772\pi$
$769$	11.1152	0.400825	0.200413	+	0.979712i	$0.435772\pi$
$770$	0	0
$771$	5.59706	0.201573
$772$	0	0
$773$	0.287717	0.0103485	0.00517423	−	0.999987i	$-0.498353\pi$
$773$	0.287717	0.0103485	0.00517423	+	0.999987i	$0.498353\pi$
$774$	0	0
$775$	10.7506	0.386172
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.1083	1.18623
$780$	0	0
$781$	−27.2088	−0.973609
$782$	0	0
$783$	7.20147	0.257360
$784$	0	0
$785$	−46.4177	−1.65672
$786$	0	0
$787$	11.5011	0.409972	0.204986	−	0.978765i	$-0.434285\pi$
$787$	11.5011	0.409972	0.204986	+	0.978765i	$0.434285\pi$
$788$	0	0
$789$	1.59706	0.0568567
$790$	0	0
$791$	0	0
$792$	0	0
$793$	25.5306	0.906618
$794$	0	0
$795$	−10.5565	−0.374399
$796$	0	0
$797$	18.6980	0.662318	0.331159	−	0.943575i	$-0.392560\pi$
$797$	18.6980	0.662318	0.331159	+	0.943575i	$0.392560\pi$
$798$	0	0
$799$	10.5800	0.374295
$800$	0	0
$801$	2.40294	0.0849039
$802$	0	0
$803$	−64.8013	−2.28679
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.7003	0.623081
$808$	0	0
$809$	−53.8036	−1.89163	−0.945817	−	0.324701i	$-0.894736\pi$
$809$	−53.8036	−1.89163	−0.945817	+	0.324701i	$0.894736\pi$
$810$	0	0
$811$	34.7866	1.22152	0.610761	−	0.791815i	$-0.290864\pi$
$811$	34.7866	1.22152	0.610761	+	0.791815i	$0.290864\pi$
$812$	0	0
$813$	2.39558	0.0840168
$814$	0	0
$815$	34.2877	1.20105
$816$	0	0
$817$	2.36695	0.0828092
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.8110	−1.31961	−0.659806	−	0.751436i	$-0.729361\pi$
$821$	−37.8110	−1.31961	−0.659806	+	0.751436i	$0.729361\pi$
$822$	0	0
$823$	−54.4177	−1.89688	−0.948440	−	0.316956i	$-0.897339\pi$
$823$	−54.4177	−1.89688	−0.948440	+	0.316956i	$0.897339\pi$
$824$	0	0
$825$	22.3070	0.776631
$826$	0	0
$827$	7.86120	0.273361	0.136680	−	0.990615i	$-0.456357\pi$
$827$	7.86120	0.273361	0.136680	+	0.990615i	$0.456357\pi$
$828$	0	0
$829$	−6.45826	−0.224305	−0.112152	−	0.993691i	$-0.535774\pi$
$829$	−6.45826	−0.224305	−0.112152	+	0.993691i	$0.535774\pi$
$830$	0	0
$831$	−5.24943	−0.182101
$832$	0	0
$833$	0	0
$834$	0	0
$835$	55.4006	1.91722
$836$	0	0
$837$	−2.04795	−0.0707876
$838$	0	0
$839$	−28.3218	−0.977776	−0.488888	−	0.872347i	$-0.662597\pi$
$839$	−28.3218	−0.977776	−0.488888	+	0.872347i	$0.662597\pi$
$840$	0	0
$841$	22.8612	0.788317
$842$	0	0
$843$	−20.0959	−0.692140
$844$	0	0
$845$	−9.78150	−0.336494
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−23.2494	−0.797918
$850$	0	0
$851$	42.4989	1.45684
$852$	0	0
$853$	−8.75057	−0.299614	−0.149807	−	0.988715i	$-0.547865\pi$
$853$	−8.75057	−0.299614	−0.149807	+	0.988715i	$0.547865\pi$
$854$	0	0
$855$	−10.0959	−0.345273
$856$	0	0
$857$	−24.6907	−0.843417	−0.421708	−	0.906731i	$-0.638569\pi$
$857$	−24.6907	−0.843417	−0.421708	+	0.906731i	$0.638569\pi$
$858$	0	0
$859$	8.49885	0.289977	0.144989	−	0.989433i	$-0.453685\pi$
$859$	8.49885	0.289977	0.144989	+	0.989433i	$0.453685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.6141	0.633631	0.316815	−	0.948487i	$-0.397386\pi$
$863$	18.6141	0.633631	0.316815	+	0.948487i	$0.397386\pi$
$864$	0	0
$865$	12.4989	0.424974
$866$	0	0
$867$	−2.38592	−0.0810302
$868$	0	0
$869$	69.9069	2.37143
$870$	0	0
$871$	−16.2517	−0.550669
$872$	0	0
$873$	4.24943	0.143821
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.1106	1.48951	0.744755	−	0.667338i	$-0.232566\pi$
$877$	44.1106	1.48951	0.744755	+	0.667338i	$0.232566\pi$
$878$	0	0
$879$	11.2015	0.377816
$880$	0	0
$881$	−11.2900	−0.380370	−0.190185	−	0.981748i	$-0.560909\pi$
$881$	−11.2900	−0.380370	−0.190185	+	0.981748i	$0.560909\pi$
$882$	0	0
$883$	42.2471	1.42173	0.710864	−	0.703329i	$-0.248304\pi$
$883$	42.2471	1.42173	0.710864	+	0.703329i	$0.248304\pi$
$884$	0	0
$885$	26.4103	0.887773
$886$	0	0
$887$	−3.98068	−0.133658	−0.0668290	−	0.997764i	$-0.521288\pi$
$887$	−3.98068	−0.133658	−0.0668290	+	0.997764i	$0.521288\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.24943	−0.142361
$892$	0	0
$893$	7.57773	0.253579
$894$	0	0
$895$	−66.6095	−2.22651
$896$	0	0
$897$	−13.8848	−0.463599
$898$	0	0
$899$	−14.7483	−0.491883
$900$	0	0
$901$	−14.5182	−0.483670
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.4177	1.41001
$906$	0	0
$907$	12.5735	0.417496	0.208748	−	0.977969i	$-0.433061\pi$
$907$	12.5735	0.417496	0.208748	+	0.977969i	$0.433061\pi$
$908$	0	0
$909$	−3.90409	−0.129491
$910$	0	0
$911$	7.09821	0.235174	0.117587	−	0.993063i	$-0.462484\pi$
$911$	7.09821	0.235174	0.117587	+	0.993063i	$0.462484\pi$
$912$	0	0
$913$	61.8566	2.04715
$914$	0	0
$915$	−25.9188	−0.856850
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.4412	−1.23507	−0.617536	−	0.786542i	$-0.711869\pi$
$919$	−37.4412	−1.23507	−0.617536	+	0.786542i	$0.711869\pi$
$920$	0	0
$921$	−28.9571	−0.954169
$922$	0	0
$923$	20.1918	0.664622
$924$	0	0
$925$	50.6694	1.66600
$926$	0	0
$927$	9.24943	0.303791
$928$	0	0
$929$	7.88477	0.258691	0.129345	−	0.991600i	$-0.458712\pi$
$929$	7.88477	0.258691	0.129345	+	0.991600i	$0.458712\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−27.3047	−0.893917
$934$	0	0
$935$	59.8995	1.95892
$936$	0	0
$937$	36.9954	1.20859	0.604294	−	0.796762i	$-0.293455\pi$
$937$	36.9954	1.20859	0.604294	+	0.796762i	$0.293455\pi$
$938$	0	0
$939$	5.49885	0.179448
$940$	0	0
$941$	12.7026	0.414094	0.207047	−	0.978331i	$-0.433615\pi$
$941$	12.7026	0.414094	0.207047	+	0.978331i	$0.433615\pi$
$942$	0	0
$943$	−46.2259	−1.50532
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.1129	0.751069	0.375535	−	0.926808i	$-0.377459\pi$
$947$	23.1129	0.751069	0.375535	+	0.926808i	$0.377459\pi$
$948$	0	0
$949$	48.0894	1.56105
$950$	0	0
$951$	−9.60442	−0.311445
$952$	0	0
$953$	−43.4200	−1.40651	−0.703255	−	0.710937i	$-0.748271\pi$
$953$	−43.4200	−1.40651	−0.703255	+	0.710937i	$0.748271\pi$
$954$	0	0
$955$	13.4200	0.434260
$956$	0	0
$957$	−30.6021	−0.989226
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.8059	−0.864706
$962$	0	0
$963$	−16.5565	−0.533525
$964$	0	0
$965$	−58.6021	−1.88647
$966$	0	0
$967$	−11.6450	−0.374478	−0.187239	−	0.982314i	$-0.559954\pi$
$967$	−11.6450	−0.374478	−0.187239	+	0.982314i	$0.559954\pi$
$968$	0	0
$969$	−13.8848	−0.446043
$970$	0	0
$971$	−33.3624	−1.07065	−0.535324	−	0.844647i	$-0.679811\pi$
$971$	−33.3624	−1.07065	−0.535324	+	0.844647i	$0.679811\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−16.5542	−0.530158
$976$	0	0
$977$	19.1895	0.613927	0.306963	−	0.951721i	$-0.400687\pi$
$977$	19.1895	0.613927	0.306963	+	0.951721i	$0.400687\pi$
$978$	0	0
$979$	−10.2111	−0.326349
$980$	0	0
$981$	−20.0553	−0.640317
$982$	0	0
$983$	6.07658	0.193813	0.0969065	−	0.995293i	$-0.469105\pi$
$983$	6.07658	0.193813	0.0969065	+	0.995293i	$0.469105\pi$
$984$	0	0
$985$	−38.1106	−1.21431
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.30474	−0.105085
$990$	0	0
$991$	31.6597	1.00570	0.502852	−	0.864372i	$-0.332284\pi$
$991$	31.6597	1.00570	0.502852	+	0.864372i	$0.332284\pi$
$992$	0	0
$993$	15.5565	0.493669
$994$	0	0
$995$	2.58003	0.0817925
$996$	0	0
$997$	−15.3601	−0.486458	−0.243229	−	0.969969i	$-0.578207\pi$
$997$	−15.3601	−0.486458	−0.243229	+	0.969969i	$0.578207\pi$
$998$	0	0
$999$	−9.65237	−0.305387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.eh.1.3		3
4.3	odd	2		9408.2.a.ej.1.3		3
7.2	even	3		1344.2.q.z.193.1		6
7.4	even	3		1344.2.q.z.961.1		6
7.6	odd	2		9408.2.a.ei.1.1		3
8.3	odd	2		4704.2.a.bs.1.1		3
8.5	even	2		4704.2.a.bu.1.1		3
28.11	odd	6		1344.2.q.y.961.1		6
28.23	odd	6		1344.2.q.y.193.1		6
28.27	even	2		9408.2.a.eg.1.1		3
56.11	odd	6		672.2.q.l.289.3	yes	6
56.13	odd	2		4704.2.a.bt.1.3		3
56.27	even	2		4704.2.a.bv.1.3		3
56.37	even	6		672.2.q.k.193.3	✓	6
56.51	odd	6		672.2.q.l.193.3	yes	6
56.53	even	6		672.2.q.k.289.3	yes	6
168.11	even	6		2016.2.s.v.289.1		6
168.53	odd	6		2016.2.s.u.289.1		6
168.107	even	6		2016.2.s.v.865.1		6
168.149	odd	6		2016.2.s.u.865.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.k.193.3	✓	6	56.37	even	6
672.2.q.k.289.3	yes	6	56.53	even	6
672.2.q.l.193.3	yes	6	56.51	odd	6
672.2.q.l.289.3	yes	6	56.11	odd	6
1344.2.q.y.193.1		6	28.23	odd	6
1344.2.q.y.961.1		6	28.11	odd	6
1344.2.q.z.193.1		6	7.2	even	3
1344.2.q.z.961.1		6	7.4	even	3
2016.2.s.u.289.1		6	168.53	odd	6
2016.2.s.u.865.1		6	168.149	odd	6
2016.2.s.v.289.1		6	168.11	even	6
2016.2.s.v.865.1		6	168.107	even	6
4704.2.a.bs.1.1		3	8.3	odd	2
4704.2.a.bt.1.3		3	56.13	odd	2
4704.2.a.bu.1.1		3	8.5	even	2
4704.2.a.bv.1.3		3	56.27	even	2
9408.2.a.eg.1.1		3	28.27	even	2
9408.2.a.eh.1.3		3	1.1	even	1	trivial
9408.2.a.ei.1.1		3	7.6	odd	2
9408.2.a.ej.1.3		3	4.3	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:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.20147 q^{5} +1.00000 q^{9} -4.24943 q^{11} +3.15352 q^{13} -3.20147 q^{15} -4.40294 q^{17} -3.15352 q^{19} +4.40294 q^{23} +5.24943 q^{25} -1.00000 q^{27} -7.20147 q^{29} +2.04795 q^{31} +4.24943 q^{33} +9.65237 q^{37} -3.15352 q^{39} -10.4989 q^{41} -0.750575 q^{43} +3.20147 q^{45} -2.40294 q^{47} +4.40294 q^{51} +3.29738 q^{53} -13.6044 q^{55} +3.15352 q^{57} -8.24943 q^{59} +8.09591 q^{61} +10.0959 q^{65} -5.15352 q^{67} -4.40294 q^{69} +6.40294 q^{71} +15.2494 q^{73} -5.24943 q^{75} -16.4509 q^{79} +1.00000 q^{81} -14.5565 q^{83} -14.0959 q^{85} +7.20147 q^{87} +2.40294 q^{89} -2.04795 q^{93} -10.0959 q^{95} +4.24943 q^{97} -4.24943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} - 3 q^{37} - 3 q^{39} - 6 q^{41} - 15 q^{43} + 12 q^{47} - 6 q^{51} - 6 q^{53} - 12 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 - 3 * q^37 - 3 * q^39 - 6 * q^41 - 15 * q^43 + 12 * q^47 - 6 * q^51 - 6 * q^53 - 12 * q^55 + 3 * q^57 - 12 * q^59 + 18 * q^61 + 24 * q^65 - 9 * q^67 + 6 * q^69 + 33 * q^73 - 3 * q^75 - 27 * q^79 + 3 * q^81 - 18 * q^83 - 36 * q^85 + 12 * q^87 - 12 * q^89 - 3 * q^93 - 24 * q^95

Introduction

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