LMFDB - Embedded newform 9408.2.a.ef.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$75.1232582216$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +2.00000 q^{11} +2.58579 q^{13} +3.41421 q^{15} +2.24264 q^{17} -2.82843 q^{19} -7.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +6.82843 q^{29} +1.17157 q^{31} +2.00000 q^{33} +4.00000 q^{37} +2.58579 q^{39} -6.24264 q^{41} -5.65685 q^{43} +3.41421 q^{45} +2.82843 q^{47} +2.24264 q^{51} +2.00000 q^{53} +6.82843 q^{55} -2.82843 q^{57} -1.17157 q^{59} +12.2426 q^{61} +8.82843 q^{65} +5.65685 q^{67} -7.65685 q^{69} +9.31371 q^{71} -13.8995 q^{73} +6.65685 q^{75} +13.6569 q^{79} +1.00000 q^{81} +7.31371 q^{83} +7.65685 q^{85} +6.82843 q^{87} +14.2426 q^{89} +1.17157 q^{93} -9.65685 q^{95} -2.58579 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} + 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{39} - 4 q^{41} + 4 q^{45} - 4 q^{51} + 4 q^{53} + 8 q^{55} - 8 q^{59} + 16 q^{61} + 12 q^{65} - 4 q^{69} - 4 q^{71} - 8 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 8 q^{87} + 20 q^{89} + 8 q^{93} - 8 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 + 8 * q^31 + 4 * q^33 + 8 * q^37 + 8 * q^39 - 4 * q^41 + 4 * q^45 - 4 * q^51 + 4 * q^53 + 8 * q^55 - 8 * q^59 + 16 * q^61 + 12 * q^65 - 4 * q^69 - 4 * q^71 - 8 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 8 * q^87 + 20 * q^89 + 8 * q^93 - 8 * q^95 - 8 * q^97 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	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
$13$	2.58579	0.717168	0.358584	−	0.933497i	$-0.383260\pi$
$13$	2.58579	0.717168	0.358584	+	0.933497i	$0.383260\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.82843	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$29$	6.82843	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	2.58579	0.414057
$40$	0	0
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	0	0
$45$	3.41421	0.508961
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.24264	0.314033
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	6.82843	0.920745
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	−1.17157	−0.152526	−0.0762629	−	0.997088i	$-0.524299\pi$
$59$	−1.17157	−0.152526	−0.0762629	+	0.997088i	$0.524299\pi$
$60$	0	0
$61$	12.2426	1.56751	0.783755	−	0.621070i	$-0.213302\pi$
$61$	12.2426	1.56751	0.783755	+	0.621070i	$0.213302\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.82843	1.09503
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	−7.65685	−0.921777
$70$	0	0
$71$	9.31371	1.10533	0.552667	−	0.833402i	$-0.313610\pi$
$71$	9.31371	1.10533	0.552667	+	0.833402i	$0.313610\pi$
$72$	0	0
$73$	−13.8995	−1.62681	−0.813406	−	0.581696i	$-0.802389\pi$
$73$	−13.8995	−1.62681	−0.813406	+	0.581696i	$0.802389\pi$
$74$	0	0
$75$	6.65685	0.768667
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.31371	0.802784	0.401392	−	0.915906i	$-0.368527\pi$
$83$	7.31371	0.802784	0.401392	+	0.915906i	$0.368527\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	0	0
$87$	6.82843	0.732084
$88$	0	0
$89$	14.2426	1.50972	0.754858	−	0.655888i	$-0.227706\pi$
$89$	14.2426	1.50972	0.754858	+	0.655888i	$0.227706\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.17157	0.121486
$94$	0	0
$95$	−9.65685	−0.990772
$96$	0	0
$97$	−2.58579	−0.262547	−0.131273	−	0.991346i	$-0.541907\pi$
$97$	−2.58579	−0.262547	−0.131273	+	0.991346i	$0.541907\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	2.92893	0.291440	0.145720	−	0.989326i	$-0.453450\pi$
$101$	2.92893	0.291440	0.145720	+	0.989326i	$0.453450\pi$
$102$	0	0
$103$	−4.48528	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$103$	−4.48528	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.343146	0.0331732	0.0165866	−	0.999862i	$-0.494720\pi$
$107$	0.343146	0.0331732	0.0165866	+	0.999862i	$0.494720\pi$
$108$	0	0
$109$	5.65685	0.541828	0.270914	−	0.962604i	$-0.412674\pi$
$109$	5.65685	0.541828	0.270914	+	0.962604i	$0.412674\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−5.31371	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$114$	0	0
$115$	−26.1421	−2.43777
$116$	0	0
$117$	2.58579	0.239056
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−6.24264	−0.562880
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−1.65685	−0.147022	−0.0735110	−	0.997294i	$-0.523420\pi$
$127$	−1.65685	−0.147022	−0.0735110	+	0.997294i	$0.523420\pi$
$128$	0	0
$129$	−5.65685	−0.498058
$130$	0	0
$131$	−15.3137	−1.33796	−0.668982	−	0.743278i	$-0.733270\pi$
$131$	−15.3137	−1.33796	−0.668982	+	0.743278i	$0.733270\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.41421	0.293849
$136$	0	0
$137$	14.1421	1.20824	0.604122	−	0.796892i	$-0.293524\pi$
$137$	14.1421	1.20824	0.604122	+	0.796892i	$0.293524\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0	0
$141$	2.82843	0.238197
$142$	0	0
$143$	5.17157	0.432469
$144$	0	0
$145$	23.3137	1.93610
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.3137	−1.41839	−0.709197	−	0.705010i	$-0.750942\pi$
$149$	−17.3137	−1.41839	−0.709197	+	0.705010i	$0.750942\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	2.24264	0.181307
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	11.7574	0.938339	0.469170	−	0.883108i	$-0.344553\pi$
$157$	11.7574	0.938339	0.469170	+	0.883108i	$0.344553\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	0	0
$165$	6.82843	0.531592
$166$	0	0
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	−6.31371	−0.485670
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	21.0711	1.60200	0.801002	−	0.598662i	$-0.204301\pi$
$173$	21.0711	1.60200	0.801002	+	0.598662i	$0.204301\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.17157	−0.0880608
$178$	0	0
$179$	19.6569	1.46922	0.734611	−	0.678488i	$-0.237365\pi$
$179$	19.6569	1.46922	0.734611	+	0.678488i	$0.237365\pi$
$180$	0	0
$181$	−2.58579	−0.192200	−0.0961000	−	0.995372i	$-0.530637\pi$
$181$	−2.58579	−0.192200	−0.0961000	+	0.995372i	$0.530637\pi$
$182$	0	0
$183$	12.2426	0.905002
$184$	0	0
$185$	13.6569	1.00407
$186$	0	0
$187$	4.48528	0.327996
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	5.31371	0.382489	0.191245	−	0.981542i	$-0.438748\pi$
$193$	5.31371	0.382489	0.191245	+	0.981542i	$0.438748\pi$
$194$	0	0
$195$	8.82843	0.632217
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	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
$201$	5.65685	0.399004
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−21.3137	−1.48861
$206$	0	0
$207$	−7.65685	−0.532188
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	−12.9706	−0.892930	−0.446465	−	0.894801i	$-0.647317\pi$
$211$	−12.9706	−0.892930	−0.446465	+	0.894801i	$0.647317\pi$
$212$	0	0
$213$	9.31371	0.638165
$214$	0	0
$215$	−19.3137	−1.31718
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.8995	−0.939241
$220$	0	0
$221$	5.79899	0.390082
$222$	0	0
$223$	−24.9706	−1.67215	−0.836076	−	0.548613i	$-0.815156\pi$
$223$	−24.9706	−1.67215	−0.836076	+	0.548613i	$0.815156\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	23.7990	1.57959	0.789797	−	0.613368i	$-0.210186\pi$
$227$	23.7990	1.57959	0.789797	+	0.613368i	$0.210186\pi$
$228$	0	0
$229$	−0.242641	−0.0160341	−0.00801707	−	0.999968i	$-0.502552\pi$
$229$	−0.242641	−0.0160341	−0.00801707	+	0.999968i	$0.502552\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.14214	−0.402385	−0.201192	−	0.979552i	$-0.564482\pi$
$233$	−6.14214	−0.402385	−0.201192	+	0.979552i	$0.564482\pi$
$234$	0	0
$235$	9.65685	0.629944
$236$	0	0
$237$	13.6569	0.887108
$238$	0	0
$239$	−15.6569	−1.01276	−0.506379	−	0.862311i	$-0.669016\pi$
$239$	−15.6569	−1.01276	−0.506379	+	0.862311i	$0.669016\pi$
$240$	0	0
$241$	16.2426	1.04628	0.523140	−	0.852247i	$-0.324760\pi$
$241$	16.2426	1.04628	0.523140	+	0.852247i	$0.324760\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.31371	−0.465360
$248$	0	0
$249$	7.31371	0.463487
$250$	0	0
$251$	−12.4853	−0.788064	−0.394032	−	0.919097i	$-0.628920\pi$
$251$	−12.4853	−0.788064	−0.394032	+	0.919097i	$0.628920\pi$
$252$	0	0
$253$	−15.3137	−0.962765
$254$	0	0
$255$	7.65685	0.479491
$256$	0	0
$257$	−23.2132	−1.44800	−0.724000	−	0.689800i	$-0.757698\pi$
$257$	−23.2132	−1.44800	−0.724000	+	0.689800i	$0.757698\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.82843	0.422669
$262$	0	0
$263$	5.31371	0.327657	0.163829	−	0.986489i	$-0.447616\pi$
$263$	5.31371	0.327657	0.163829	+	0.986489i	$0.447616\pi$
$264$	0	0
$265$	6.82843	0.419467
$266$	0	0
$267$	14.2426	0.871635
$268$	0	0
$269$	−14.7279	−0.897977	−0.448989	−	0.893537i	$-0.648216\pi$
$269$	−14.7279	−0.897977	−0.448989	+	0.893537i	$0.648216\pi$
$270$	0	0
$271$	10.1421	0.616091	0.308045	−	0.951372i	$-0.400325\pi$
$271$	10.1421	0.616091	0.308045	+	0.951372i	$0.400325\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.3137	0.802847
$276$	0	0
$277$	9.31371	0.559607	0.279803	−	0.960057i	$-0.409731\pi$
$277$	9.31371	0.559607	0.279803	+	0.960057i	$0.409731\pi$
$278$	0	0
$279$	1.17157	0.0701402
$280$	0	0
$281$	0.485281	0.0289495	0.0144747	−	0.999895i	$-0.495392\pi$
$281$	0.485281	0.0289495	0.0144747	+	0.999895i	$0.495392\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	−9.65685	−0.572023
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	−2.58579	−0.151581
$292$	0	0
$293$	16.5858	0.968952	0.484476	−	0.874805i	$-0.339010\pi$
$293$	16.5858	0.968952	0.484476	+	0.874805i	$0.339010\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−19.7990	−1.14501
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.92893	0.168263
$304$	0	0
$305$	41.7990	2.39340
$306$	0	0
$307$	30.1421	1.72030	0.860151	−	0.510039i	$-0.170369\pi$
$307$	30.1421	1.72030	0.860151	+	0.510039i	$0.170369\pi$
$308$	0	0
$309$	−4.48528	−0.255159
$310$	0	0
$311$	−6.14214	−0.348289	−0.174144	−	0.984720i	$-0.555716\pi$
$311$	−6.14214	−0.348289	−0.174144	+	0.984720i	$0.555716\pi$
$312$	0	0
$313$	−1.89949	−0.107366	−0.0536829	−	0.998558i	$-0.517096\pi$
$313$	−1.89949	−0.107366	−0.0536829	+	0.998558i	$0.517096\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	13.6569	0.764637
$320$	0	0
$321$	0.343146	0.0191525
$322$	0	0
$323$	−6.34315	−0.352942
$324$	0	0
$325$	17.2132	0.954817
$326$	0	0
$327$	5.65685	0.312825
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	19.3137	1.05522
$336$	0	0
$337$	−29.6569	−1.61551	−0.807756	−	0.589517i	$-0.799318\pi$
$337$	−29.6569	−1.61551	−0.807756	+	0.589517i	$0.799318\pi$
$338$	0	0
$339$	−5.31371	−0.288601
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−26.1421	−1.40745
$346$	0	0
$347$	−33.3137	−1.78837	−0.894187	−	0.447694i	$-0.852245\pi$
$347$	−33.3137	−1.78837	−0.894187	+	0.447694i	$0.852245\pi$
$348$	0	0
$349$	−9.89949	−0.529908	−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949	−0.529908	−0.264954	+	0.964261i	$0.585357\pi$
$350$	0	0
$351$	2.58579	0.138019
$352$	0	0
$353$	14.7279	0.783888	0.391944	−	0.919989i	$-0.371803\pi$
$353$	14.7279	0.783888	0.391944	+	0.919989i	$0.371803\pi$
$354$	0	0
$355$	31.7990	1.68772
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.343146	−0.0181105	−0.00905527	−	0.999959i	$-0.502882\pi$
$359$	−0.343146	−0.0181105	−0.00905527	+	0.999959i	$0.502882\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−47.4558	−2.48395
$366$	0	0
$367$	3.31371	0.172974	0.0864871	−	0.996253i	$-0.472436\pi$
$367$	3.31371	0.172974	0.0864871	+	0.996253i	$0.472436\pi$
$368$	0	0
$369$	−6.24264	−0.324979
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.6863	0.553315	0.276658	−	0.960969i	$-0.410773\pi$
$373$	10.6863	0.553315	0.276658	+	0.960969i	$0.410773\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	17.6569	0.909374
$378$	0	0
$379$	−8.68629	−0.446185	−0.223092	−	0.974797i	$-0.571615\pi$
$379$	−8.68629	−0.446185	−0.223092	+	0.974797i	$0.571615\pi$
$380$	0	0
$381$	−1.65685	−0.0848832
$382$	0	0
$383$	18.3431	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$383$	18.3431	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.65685	−0.287554
$388$	0	0
$389$	18.1421	0.919843	0.459921	−	0.887960i	$-0.347878\pi$
$389$	18.1421	0.919843	0.459921	+	0.887960i	$0.347878\pi$
$390$	0	0
$391$	−17.1716	−0.868404
$392$	0	0
$393$	−15.3137	−0.772474
$394$	0	0
$395$	46.6274	2.34608
$396$	0	0
$397$	−2.38478	−0.119688	−0.0598442	−	0.998208i	$-0.519060\pi$
$397$	−2.38478	−0.119688	−0.0598442	+	0.998208i	$0.519060\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.14214	−0.306724	−0.153362	−	0.988170i	$-0.549010\pi$
$401$	−6.14214	−0.306724	−0.153362	+	0.988170i	$0.549010\pi$
$402$	0	0
$403$	3.02944	0.150907
$404$	0	0
$405$	3.41421	0.169654
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	21.4142	1.05886	0.529432	−	0.848352i	$-0.322405\pi$
$409$	21.4142	1.05886	0.529432	+	0.848352i	$0.322405\pi$
$410$	0	0
$411$	14.1421	0.697580
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.9706	1.22576
$416$	0	0
$417$	−17.6569	−0.864660
$418$	0	0
$419$	33.1716	1.62054	0.810269	−	0.586059i	$-0.199321\pi$
$419$	33.1716	1.62054	0.810269	+	0.586059i	$0.199321\pi$
$420$	0	0
$421$	−16.6274	−0.810371	−0.405185	−	0.914235i	$-0.632793\pi$
$421$	−16.6274	−0.810371	−0.405185	+	0.914235i	$0.632793\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	14.9289	0.724160
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.17157	0.249686
$430$	0	0
$431$	−26.9706	−1.29913	−0.649563	−	0.760308i	$-0.725048\pi$
$431$	−26.9706	−1.29913	−0.649563	+	0.760308i	$0.725048\pi$
$432$	0	0
$433$	20.2426	0.972799	0.486400	−	0.873736i	$-0.338310\pi$
$433$	20.2426	0.972799	0.486400	+	0.873736i	$0.338310\pi$
$434$	0	0
$435$	23.3137	1.11781
$436$	0	0
$437$	21.6569	1.03599
$438$	0	0
$439$	−12.6863	−0.605484	−0.302742	−	0.953073i	$-0.597902\pi$
$439$	−12.6863	−0.605484	−0.302742	+	0.953073i	$0.597902\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.9706	−1.66150	−0.830751	−	0.556645i	$-0.812089\pi$
$443$	−34.9706	−1.66150	−0.830751	+	0.556645i	$0.812089\pi$
$444$	0	0
$445$	48.6274	2.30516
$446$	0	0
$447$	−17.3137	−0.818910
$448$	0	0
$449$	−5.31371	−0.250769	−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371	−0.250769	−0.125385	+	0.992108i	$0.540017\pi$
$450$	0	0
$451$	−12.4853	−0.587909
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	2.24264	0.104678
$460$	0	0
$461$	−16.5858	−0.772477	−0.386239	−	0.922399i	$-0.626226\pi$
$461$	−16.5858	−0.772477	−0.386239	+	0.922399i	$0.626226\pi$
$462$	0	0
$463$	−26.6274	−1.23748	−0.618741	−	0.785595i	$-0.712357\pi$
$463$	−26.6274	−1.23748	−0.618741	+	0.785595i	$0.712357\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	0.201010	0.00930164	0.00465082	−	0.999989i	$-0.498520\pi$
$467$	0.201010	0.00930164	0.00465082	+	0.999989i	$0.498520\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	11.7574	0.541751
$472$	0	0
$473$	−11.3137	−0.520205
$474$	0	0
$475$	−18.8284	−0.863907
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−1.85786	−0.0848880	−0.0424440	−	0.999099i	$-0.513514\pi$
$479$	−1.85786	−0.0848880	−0.0424440	+	0.999099i	$0.513514\pi$
$480$	0	0
$481$	10.3431	0.471607
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.82843	−0.400878
$486$	0	0
$487$	26.6274	1.20660	0.603302	−	0.797513i	$-0.293851\pi$
$487$	26.6274	1.20660	0.603302	+	0.797513i	$0.293851\pi$
$488$	0	0
$489$	11.3137	0.511624
$490$	0	0
$491$	−5.02944	−0.226975	−0.113488	−	0.993539i	$-0.536202\pi$
$491$	−5.02944	−0.226975	−0.113488	+	0.993539i	$0.536202\pi$
$492$	0	0
$493$	15.3137	0.689695
$494$	0	0
$495$	6.82843	0.306915
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.31371	−0.148342	−0.0741710	−	0.997246i	$-0.523631\pi$
$499$	−3.31371	−0.148342	−0.0741710	+	0.997246i	$0.523631\pi$
$500$	0	0
$501$	19.7990	0.884554
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	−6.31371	−0.280402
$508$	0	0
$509$	5.55635	0.246281	0.123140	−	0.992389i	$-0.460703\pi$
$509$	5.55635	0.246281	0.123140	+	0.992389i	$0.460703\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.82843	−0.124878
$514$	0	0
$515$	−15.3137	−0.674803
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	21.0711	0.924917
$520$	0	0
$521$	35.4142	1.55152	0.775762	−	0.631025i	$-0.217366\pi$
$521$	35.4142	1.55152	0.775762	+	0.631025i	$0.217366\pi$
$522$	0	0
$523$	−25.6569	−1.12190	−0.560948	−	0.827851i	$-0.689563\pi$
$523$	−25.6569	−1.12190	−0.560948	+	0.827851i	$0.689563\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.62742	0.114452
$528$	0	0
$529$	35.6274	1.54902
$530$	0	0
$531$	−1.17157	−0.0508419
$532$	0	0
$533$	−16.1421	−0.699194
$534$	0	0
$535$	1.17157	0.0506515
$536$	0	0
$537$	19.6569	0.848256
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.3137	−0.744374	−0.372187	−	0.928158i	$-0.621392\pi$
$541$	−17.3137	−0.744374	−0.372187	+	0.928158i	$0.621392\pi$
$542$	0	0
$543$	−2.58579	−0.110967
$544$	0	0
$545$	19.3137	0.827308
$546$	0	0
$547$	36.9706	1.58075	0.790374	−	0.612625i	$-0.209886\pi$
$547$	36.9706	1.58075	0.790374	+	0.612625i	$0.209886\pi$
$548$	0	0
$549$	12.2426	0.522503
$550$	0	0
$551$	−19.3137	−0.822792
$552$	0	0
$553$	0	0
$554$	0	0
$555$	13.6569	0.579701
$556$	0	0
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	−14.6274	−0.618674
$560$	0	0
$561$	4.48528	0.189369
$562$	0	0
$563$	−1.17157	−0.0493759	−0.0246880	−	0.999695i	$-0.507859\pi$
$563$	−1.17157	−0.0493759	−0.0246880	+	0.999695i	$0.507859\pi$
$564$	0	0
$565$	−18.1421	−0.763245
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.4853	−0.691099	−0.345549	−	0.938401i	$-0.612307\pi$
$569$	−16.4853	−0.691099	−0.345549	+	0.938401i	$0.612307\pi$
$570$	0	0
$571$	−22.3431	−0.935032	−0.467516	−	0.883985i	$-0.654851\pi$
$571$	−22.3431	−0.935032	−0.467516	+	0.883985i	$0.654851\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−50.9706	−2.12562
$576$	0	0
$577$	33.8995	1.41125	0.705627	−	0.708583i	$-0.250665\pi$
$577$	33.8995	1.41125	0.705627	+	0.708583i	$0.250665\pi$
$578$	0	0
$579$	5.31371	0.220830
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	8.82843	0.365011
$586$	0	0
$587$	−22.8284	−0.942230	−0.471115	−	0.882072i	$-0.656148\pi$
$587$	−22.8284	−0.942230	−0.471115	+	0.882072i	$0.656148\pi$
$588$	0	0
$589$	−3.31371	−0.136539
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	6.92893	0.284537	0.142269	−	0.989828i	$-0.454560\pi$
$593$	6.92893	0.284537	0.142269	+	0.989828i	$0.454560\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−21.6569	−0.886356
$598$	0	0
$599$	−2.00000	−0.0817178	−0.0408589	−	0.999165i	$-0.513009\pi$
$599$	−2.00000	−0.0817178	−0.0408589	+	0.999165i	$0.513009\pi$
$600$	0	0
$601$	15.0711	0.614762	0.307381	−	0.951587i	$-0.400547\pi$
$601$	15.0711	0.614762	0.307381	+	0.951587i	$0.400547\pi$
$602$	0	0
$603$	5.65685	0.230365
$604$	0	0
$605$	−23.8995	−0.971653
$606$	0	0
$607$	18.3431	0.744525	0.372263	−	0.928127i	$-0.378582\pi$
$607$	18.3431	0.744525	0.372263	+	0.928127i	$0.378582\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.31371	0.295881
$612$	0	0
$613$	−4.68629	−0.189278	−0.0946388	−	0.995512i	$-0.530170\pi$
$613$	−4.68629	−0.189278	−0.0946388	+	0.995512i	$0.530170\pi$
$614$	0	0
$615$	−21.3137	−0.859452
$616$	0	0
$617$	−24.4853	−0.985740	−0.492870	−	0.870103i	$-0.664052\pi$
$617$	−24.4853	−0.985740	−0.492870	+	0.870103i	$0.664052\pi$
$618$	0	0
$619$	28.9706	1.16443	0.582213	−	0.813037i	$-0.302187\pi$
$619$	28.9706	1.16443	0.582213	+	0.813037i	$0.302187\pi$
$620$	0	0
$621$	−7.65685	−0.307259
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	−5.65685	−0.225913
$628$	0	0
$629$	8.97056	0.357680
$630$	0	0
$631$	23.3137	0.928104	0.464052	−	0.885808i	$-0.346395\pi$
$631$	23.3137	0.928104	0.464052	+	0.885808i	$0.346395\pi$
$632$	0	0
$633$	−12.9706	−0.515534
$634$	0	0
$635$	−5.65685	−0.224485
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.31371	0.368445
$640$	0	0
$641$	−10.8284	−0.427697	−0.213849	−	0.976867i	$-0.568600\pi$
$641$	−10.8284	−0.427697	−0.213849	+	0.976867i	$0.568600\pi$
$642$	0	0
$643$	−34.4264	−1.35764	−0.678822	−	0.734302i	$-0.737509\pi$
$643$	−34.4264	−1.35764	−0.678822	+	0.734302i	$0.737509\pi$
$644$	0	0
$645$	−19.3137	−0.760477
$646$	0	0
$647$	−26.8284	−1.05473	−0.527367	−	0.849638i	$-0.676821\pi$
$647$	−26.8284	−1.05473	−0.527367	+	0.849638i	$0.676821\pi$
$648$	0	0
$649$	−2.34315	−0.0919765
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.4853	−1.42778	−0.713890	−	0.700258i	$-0.753068\pi$
$653$	−36.4853	−1.42778	−0.713890	+	0.700258i	$0.753068\pi$
$654$	0	0
$655$	−52.2843	−2.04292
$656$	0	0
$657$	−13.8995	−0.542271
$658$	0	0
$659$	−9.31371	−0.362811	−0.181405	−	0.983408i	$-0.558065\pi$
$659$	−9.31371	−0.362811	−0.181405	+	0.983408i	$0.558065\pi$
$660$	0	0
$661$	−23.5563	−0.916236	−0.458118	−	0.888891i	$-0.651476\pi$
$661$	−23.5563	−0.916236	−0.458118	+	0.888891i	$0.651476\pi$
$662$	0	0
$663$	5.79899	0.225214
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−52.2843	−2.02446
$668$	0	0
$669$	−24.9706	−0.965418
$670$	0	0
$671$	24.4853	0.945244
$672$	0	0
$673$	23.3137	0.898677	0.449339	−	0.893361i	$-0.351660\pi$
$673$	23.3137	0.898677	0.449339	+	0.893361i	$0.351660\pi$
$674$	0	0
$675$	6.65685	0.256222
$676$	0	0
$677$	31.4142	1.20735	0.603673	−	0.797232i	$-0.293703\pi$
$677$	31.4142	1.20735	0.603673	+	0.797232i	$0.293703\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	23.7990	0.911979
$682$	0	0
$683$	19.6569	0.752149	0.376074	−	0.926590i	$-0.377274\pi$
$683$	19.6569	0.752149	0.376074	+	0.926590i	$0.377274\pi$
$684$	0	0
$685$	48.2843	1.84485
$686$	0	0
$687$	−0.242641	−0.00925732
$688$	0	0
$689$	5.17157	0.197021
$690$	0	0
$691$	−0.686292	−0.0261078	−0.0130539	−	0.999915i	$-0.504155\pi$
$691$	−0.686292	−0.0261078	−0.0130539	+	0.999915i	$0.504155\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−60.2843	−2.28671
$696$	0	0
$697$	−14.0000	−0.530288
$698$	0	0
$699$	−6.14214	−0.232317
$700$	0	0
$701$	17.1716	0.648561	0.324281	−	0.945961i	$-0.394878\pi$
$701$	17.1716	0.648561	0.324281	+	0.945961i	$0.394878\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	0	0
$705$	9.65685	0.363698
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.2843	1.36268	0.681342	−	0.731965i	$-0.261397\pi$
$709$	36.2843	1.36268	0.681342	+	0.731965i	$0.261397\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	0	0
$713$	−8.97056	−0.335950
$714$	0	0
$715$	17.6569	0.660329
$716$	0	0
$717$	−15.6569	−0.584716
$718$	0	0
$719$	−41.9411	−1.56414	−0.782070	−	0.623191i	$-0.785836\pi$
$719$	−41.9411	−1.56414	−0.782070	+	0.623191i	$0.785836\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	16.2426	0.604070
$724$	0	0
$725$	45.4558	1.68819
$726$	0	0
$727$	−12.4853	−0.463053	−0.231527	−	0.972829i	$-0.574372\pi$
$727$	−12.4853	−0.463053	−0.231527	+	0.972829i	$0.574372\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.6863	−0.469219
$732$	0	0
$733$	49.6985	1.83566	0.917828	−	0.396979i	$-0.129941\pi$
$733$	49.6985	1.83566	0.917828	+	0.396979i	$0.129941\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3137	0.416746
$738$	0	0
$739$	−4.68629	−0.172388	−0.0861940	−	0.996278i	$-0.527470\pi$
$739$	−4.68629	−0.172388	−0.0861940	+	0.996278i	$0.527470\pi$
$740$	0	0
$741$	−7.31371	−0.268676
$742$	0	0
$743$	−50.9706	−1.86993	−0.934964	−	0.354742i	$-0.884569\pi$
$743$	−50.9706	−1.86993	−0.934964	+	0.354742i	$0.884569\pi$
$744$	0	0
$745$	−59.1127	−2.16572
$746$	0	0
$747$	7.31371	0.267595
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.6569	0.498346	0.249173	−	0.968459i	$-0.419841\pi$
$751$	13.6569	0.498346	0.249173	+	0.968459i	$0.419841\pi$
$752$	0	0
$753$	−12.4853	−0.454989
$754$	0	0
$755$	40.9706	1.49107
$756$	0	0
$757$	−26.3431	−0.957458	−0.478729	−	0.877963i	$-0.658902\pi$
$757$	−26.3431	−0.957458	−0.478729	+	0.877963i	$0.658902\pi$
$758$	0	0
$759$	−15.3137	−0.555852
$760$	0	0
$761$	18.5269	0.671600	0.335800	−	0.941933i	$-0.390993\pi$
$761$	18.5269	0.671600	0.335800	+	0.941933i	$0.390993\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	7.65685	0.276834
$766$	0	0
$767$	−3.02944	−0.109387
$768$	0	0
$769$	−29.6985	−1.07095	−0.535477	−	0.844550i	$-0.679868\pi$
$769$	−29.6985	−1.07095	−0.535477	+	0.844550i	$0.679868\pi$
$770$	0	0
$771$	−23.2132	−0.836003
$772$	0	0
$773$	−9.55635	−0.343718	−0.171859	−	0.985122i	$-0.554977\pi$
$773$	−9.55635	−0.343718	−0.171859	+	0.985122i	$0.554977\pi$
$774$	0	0
$775$	7.79899	0.280148
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.6569	0.632622
$780$	0	0
$781$	18.6274	0.666541
$782$	0	0
$783$	6.82843	0.244028
$784$	0	0
$785$	40.1421	1.43273
$786$	0	0
$787$	24.6863	0.879971	0.439986	−	0.898005i	$-0.354984\pi$
$787$	24.6863	0.879971	0.439986	+	0.898005i	$0.354984\pi$
$788$	0	0
$789$	5.31371	0.189173
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31.6569	1.12417
$794$	0	0
$795$	6.82843	0.242179
$796$	0	0
$797$	−8.38478	−0.297004	−0.148502	−	0.988912i	$-0.547445\pi$
$797$	−8.38478	−0.297004	−0.148502	+	0.988912i	$0.547445\pi$
$798$	0	0
$799$	6.34315	0.224404
$800$	0	0
$801$	14.2426	0.503239
$802$	0	0
$803$	−27.7990	−0.981005
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.7279	−0.518447
$808$	0	0
$809$	19.9411	0.701093	0.350546	−	0.936545i	$-0.385996\pi$
$809$	19.9411	0.701093	0.350546	+	0.936545i	$0.385996\pi$
$810$	0	0
$811$	−17.6569	−0.620016	−0.310008	−	0.950734i	$-0.600332\pi$
$811$	−17.6569	−0.620016	−0.310008	+	0.950734i	$0.600332\pi$
$812$	0	0
$813$	10.1421	0.355700
$814$	0	0
$815$	38.6274	1.35306
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.6863	0.372954	0.186477	−	0.982459i	$-0.440293\pi$
$821$	10.6863	0.372954	0.186477	+	0.982459i	$0.440293\pi$
$822$	0	0
$823$	8.97056	0.312694	0.156347	−	0.987702i	$-0.450028\pi$
$823$	8.97056	0.312694	0.156347	+	0.987702i	$0.450028\pi$
$824$	0	0
$825$	13.3137	0.463524
$826$	0	0
$827$	−47.6569	−1.65719	−0.828596	−	0.559848i	$-0.810860\pi$
$827$	−47.6569	−1.65719	−0.828596	+	0.559848i	$0.810860\pi$
$828$	0	0
$829$	−0.727922	−0.0252818	−0.0126409	−	0.999920i	$-0.504024\pi$
$829$	−0.727922	−0.0252818	−0.0126409	+	0.999920i	$0.504024\pi$
$830$	0	0
$831$	9.31371	0.323089
$832$	0	0
$833$	0	0
$834$	0	0
$835$	67.5980	2.33932
$836$	0	0
$837$	1.17157	0.0404955
$838$	0	0
$839$	−50.8284	−1.75479	−0.877396	−	0.479767i	$-0.840721\pi$
$839$	−50.8284	−1.75479	−0.877396	+	0.479767i	$0.840721\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	0	0
$843$	0.485281	0.0167140
$844$	0	0
$845$	−21.5563	−0.741561
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.48528	−0.291214
$850$	0	0
$851$	−30.6274	−1.04989
$852$	0	0
$853$	−49.4975	−1.69476	−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975	−1.69476	−0.847381	+	0.530986i	$0.821822\pi$
$854$	0	0
$855$	−9.65685	−0.330257
$856$	0	0
$857$	−15.4142	−0.526540	−0.263270	−	0.964722i	$-0.584801\pi$
$857$	−15.4142	−0.526540	−0.263270	+	0.964722i	$0.584801\pi$
$858$	0	0
$859$	57.4558	1.96037	0.980184	−	0.198089i	$-0.0634735\pi$
$859$	57.4558	1.96037	0.980184	+	0.198089i	$0.0634735\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.3137	0.589365	0.294683	−	0.955595i	$-0.404786\pi$
$863$	17.3137	0.589365	0.294683	+	0.955595i	$0.404786\pi$
$864$	0	0
$865$	71.9411	2.44607
$866$	0	0
$867$	−11.9706	−0.406542
$868$	0	0
$869$	27.3137	0.926554
$870$	0	0
$871$	14.6274	0.495631
$872$	0	0
$873$	−2.58579	−0.0875156
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.3137	0.382037	0.191018	−	0.981586i	$-0.438821\pi$
$877$	11.3137	0.382037	0.191018	+	0.981586i	$0.438821\pi$
$878$	0	0
$879$	16.5858	0.559425
$880$	0	0
$881$	−21.7574	−0.733024	−0.366512	−	0.930413i	$-0.619448\pi$
$881$	−21.7574	−0.733024	−0.366512	+	0.930413i	$0.619448\pi$
$882$	0	0
$883$	4.68629	0.157706	0.0788531	−	0.996886i	$-0.474874\pi$
$883$	4.68629	0.157706	0.0788531	+	0.996886i	$0.474874\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	2.82843	0.0949693	0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843	0.0949693	0.0474846	+	0.998872i	$0.484879\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	67.1127	2.24333
$896$	0	0
$897$	−19.7990	−0.661069
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	4.48528	0.149426
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.82843	−0.293467
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	2.92893	0.0971465
$910$	0	0
$911$	−1.02944	−0.0341068	−0.0170534	−	0.999855i	$-0.505429\pi$
$911$	−1.02944	−0.0341068	−0.0170534	+	0.999855i	$0.505429\pi$
$912$	0	0
$913$	14.6274	0.484097
$914$	0	0
$915$	41.7990	1.38183
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.28427	−0.273273	−0.136636	−	0.990621i	$-0.543629\pi$
$919$	−8.28427	−0.273273	−0.136636	+	0.990621i	$0.543629\pi$
$920$	0	0
$921$	30.1421	0.993217
$922$	0	0
$923$	24.0833	0.792710
$924$	0	0
$925$	26.6274	0.875504
$926$	0	0
$927$	−4.48528	−0.147316
$928$	0	0
$929$	39.2132	1.28654	0.643272	−	0.765638i	$-0.277577\pi$
$929$	39.2132	1.28654	0.643272	+	0.765638i	$0.277577\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.14214	−0.201084
$934$	0	0
$935$	15.3137	0.500812
$936$	0	0
$937$	−30.5858	−0.999194	−0.499597	−	0.866258i	$-0.666519\pi$
$937$	−30.5858	−0.999194	−0.499597	+	0.866258i	$0.666519\pi$
$938$	0	0
$939$	−1.89949	−0.0619877
$940$	0	0
$941$	35.2132	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$941$	35.2132	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$942$	0	0
$943$	47.7990	1.55655
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.6863	−0.997170	−0.498585	−	0.866841i	$-0.666147\pi$
$947$	−30.6863	−0.997170	−0.498585	+	0.866841i	$0.666147\pi$
$948$	0	0
$949$	−35.9411	−1.16670
$950$	0	0
$951$	−10.0000	−0.324272
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	−61.4558	−1.98866
$956$	0	0
$957$	13.6569	0.441463
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	0.343146	0.0110577
$964$	0	0
$965$	18.1421	0.584016
$966$	0	0
$967$	33.6569	1.08233	0.541166	−	0.840916i	$-0.317983\pi$
$967$	33.6569	1.08233	0.541166	+	0.840916i	$0.317983\pi$
$968$	0	0
$969$	−6.34315	−0.203771
$970$	0	0
$971$	−50.6274	−1.62471	−0.812356	−	0.583162i	$-0.801815\pi$
$971$	−50.6274	−1.62471	−0.812356	+	0.583162i	$0.801815\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	17.2132	0.551264
$976$	0	0
$977$	−21.1716	−0.677339	−0.338669	−	0.940905i	$-0.609977\pi$
$977$	−21.1716	−0.677339	−0.338669	+	0.940905i	$0.609977\pi$
$978$	0	0
$979$	28.4853	0.910394
$980$	0	0
$981$	5.65685	0.180609
$982$	0	0
$983$	53.2548	1.69857	0.849283	−	0.527938i	$-0.177035\pi$
$983$	53.2548	1.69857	0.849283	+	0.527938i	$0.177035\pi$
$984$	0	0
$985$	−6.82843	−0.217572
$986$	0	0
$987$	0	0
$988$	0	0
$989$	43.3137	1.37730
$990$	0	0
$991$	−12.9706	−0.412024	−0.206012	−	0.978550i	$-0.566049\pi$
$991$	−12.9706	−0.412024	−0.206012	+	0.978550i	$0.566049\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	−73.9411	−2.34409
$996$	0	0
$997$	26.3848	0.835614	0.417807	−	0.908536i	$-0.362799\pi$
$997$	26.3848	0.835614	0.417807	+	0.908536i	$0.362799\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.ef.1.2		2
4.3	odd	2		9408.2.a.dq.1.2		2
7.6	odd	2		9408.2.a.di.1.1		2
8.3	odd	2		2352.2.a.be.1.1		2
8.5	even	2		147.2.a.d.1.2	✓	2
24.5	odd	2		441.2.a.j.1.1		2
24.11	even	2		7056.2.a.cv.1.2		2
28.27	even	2		9408.2.a.dt.1.1		2
40.29	even	2		3675.2.a.bf.1.1		2
56.3	even	6		2352.2.q.bd.961.1		4
56.5	odd	6		147.2.e.d.67.1		4
56.11	odd	6		2352.2.q.bb.961.2		4
56.13	odd	2		147.2.a.e.1.2	yes	2
56.19	even	6		2352.2.q.bd.1537.1		4
56.27	even	2		2352.2.a.bc.1.2		2
56.37	even	6		147.2.e.e.67.1		4
56.45	odd	6		147.2.e.d.79.1		4
56.51	odd	6		2352.2.q.bb.1537.2		4
56.53	even	6		147.2.e.e.79.1		4
168.5	even	6		441.2.e.g.361.2		4
168.53	odd	6		441.2.e.f.226.2		4
168.83	odd	2		7056.2.a.cf.1.1		2
168.101	even	6		441.2.e.g.226.2		4
168.125	even	2		441.2.a.i.1.1		2
168.149	odd	6		441.2.e.f.361.2		4
280.69	odd	2		3675.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.a.d.1.2	✓	2	8.5	even	2
147.2.a.e.1.2	yes	2	56.13	odd	2
147.2.e.d.67.1		4	56.5	odd	6
147.2.e.d.79.1		4	56.45	odd	6
147.2.e.e.67.1		4	56.37	even	6
147.2.e.e.79.1		4	56.53	even	6
441.2.a.i.1.1		2	168.125	even	2
441.2.a.j.1.1		2	24.5	odd	2
441.2.e.f.226.2		4	168.53	odd	6
441.2.e.f.361.2		4	168.149	odd	6
441.2.e.g.226.2		4	168.101	even	6
441.2.e.g.361.2		4	168.5	even	6
2352.2.a.bc.1.2		2	56.27	even	2
2352.2.a.be.1.1		2	8.3	odd	2
2352.2.q.bb.961.2		4	56.11	odd	6
2352.2.q.bb.1537.2		4	56.51	odd	6
2352.2.q.bd.961.1		4	56.3	even	6
2352.2.q.bd.1537.1		4	56.19	even	6
3675.2.a.bd.1.1		2	280.69	odd	2
3675.2.a.bf.1.1		2	40.29	even	2
7056.2.a.cf.1.1		2	168.83	odd	2
7056.2.a.cv.1.2		2	24.11	even	2
9408.2.a.di.1.1		2	7.6	odd	2
9408.2.a.dq.1.2		2	4.3	odd	2
9408.2.a.dt.1.1		2	28.27	even	2
9408.2.a.ef.1.2		2	1.1	even	1	trivial

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +2.00000 q^{11} +2.58579 q^{13} +3.41421 q^{15} +2.24264 q^{17} -2.82843 q^{19} -7.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +6.82843 q^{29} +1.17157 q^{31} +2.00000 q^{33} +4.00000 q^{37} +2.58579 q^{39} -6.24264 q^{41} -5.65685 q^{43} +3.41421 q^{45} +2.82843 q^{47} +2.24264 q^{51} +2.00000 q^{53} +6.82843 q^{55} -2.82843 q^{57} -1.17157 q^{59} +12.2426 q^{61} +8.82843 q^{65} +5.65685 q^{67} -7.65685 q^{69} +9.31371 q^{71} -13.8995 q^{73} +6.65685 q^{75} +13.6569 q^{79} +1.00000 q^{81} +7.31371 q^{83} +7.65685 q^{85} +6.82843 q^{87} +14.2426 q^{89} +1.17157 q^{93} -9.65685 q^{95} -2.58579 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} + 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{39} - 4 q^{41} + 4 q^{45} - 4 q^{51} + 4 q^{53} + 8 q^{55} - 8 q^{59} + 16 q^{61} + 12 q^{65} - 4 q^{69} - 4 q^{71} - 8 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 8 q^{87} + 20 q^{89} + 8 q^{93} - 8 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 + 8 * q^31 + 4 * q^33 + 8 * q^37 + 8 * q^39 - 4 * q^41 + 4 * q^45 - 4 * q^51 + 4 * q^53 + 8 * q^55 - 8 * q^59 + 16 * q^61 + 12 * q^65 - 4 * q^69 - 4 * q^71 - 8 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 8 * q^87 + 20 * q^89 + 8 * q^93 - 8 * q^95 - 8 * q^97 + 4 * q^99

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