LMFDB - Embedded newform 9996.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9996,2,Mod(1,9996)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9996, 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("9996.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9996 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9996.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:	$79.8184618605$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7053.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 3x + 3 $ x^4 - 2x^3 - 4x^2 + 3*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1428)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.89962$ of defining polynomial
Character	$\chi$	$=$	9996.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.29105 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.29105 q^{5} +1.00000 q^{9} +1.47358 q^{11} +4.89962 q^{13} +2.29105 q^{15} +1.00000 q^{17} +5.86033 q^{19} -7.83387 q^{23} +0.248918 q^{25} -1.00000 q^{27} +3.08215 q^{29} +9.62964 q^{31} -1.47358 q^{33} +1.70428 q^{37} -4.89962 q^{39} -7.79458 q^{41} +10.9324 q^{43} -2.29105 q^{45} -6.69887 q^{47} -1.00000 q^{51} +10.7017 q^{53} -3.37605 q^{55} -5.86033 q^{57} +5.62680 q^{59} +4.29390 q^{61} -11.2253 q^{65} -4.96823 q^{67} +7.83387 q^{69} -2.74357 q^{71} +8.90714 q^{73} -0.248918 q^{75} -4.00888 q^{79} +1.00000 q^{81} +4.57743 q^{83} -2.29105 q^{85} -3.08215 q^{87} -0.0793088 q^{89} -9.62964 q^{93} -13.4263 q^{95} +4.56388 q^{97} +1.47358 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} + 4 q^{17} + 9 q^{19} - 5 q^{23} + 18 q^{25} - 4 q^{27} + 5 q^{29} + 24 q^{31} - q^{33} + 2 q^{37} - 10 q^{39} - 14 q^{43} - 2 q^{45} + 2 q^{47} - 4 q^{51} - 15 q^{53} + 30 q^{55} - 9 q^{57} + 37 q^{59} - 19 q^{61} - 21 q^{65} + 2 q^{67} + 5 q^{69} - 11 q^{71} + 9 q^{73} - 18 q^{75} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 2 q^{85} - 5 q^{87} - 22 q^{89} - 24 q^{93} + 33 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9 + q^11 + 10 * q^13 + 2 * q^15 + 4 * q^17 + 9 * q^19 - 5 * q^23 + 18 * q^25 - 4 * q^27 + 5 * q^29 + 24 * q^31 - q^33 + 2 * q^37 - 10 * q^39 - 14 * q^43 - 2 * q^45 + 2 * q^47 - 4 * q^51 - 15 * q^53 + 30 * q^55 - 9 * q^57 + 37 * q^59 - 19 * q^61 - 21 * q^65 + 2 * q^67 + 5 * q^69 - 11 * q^71 + 9 * q^73 - 18 * q^75 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 2 * q^85 - 5 * q^87 - 22 * q^89 - 24 * q^93 + 33 * q^95 - 13 * q^97 + 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
$5$	−2.29105	−1.02459	−0.512295	−	0.858810i	$-0.671204\pi$
$5$	−2.29105	−1.02459	−0.512295	+	0.858810i	$0.671204\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.47358	0.444301	0.222151	−	0.975012i	$-0.428692\pi$
$11$	1.47358	0.444301	0.222151	+	0.975012i	$0.428692\pi$
$12$	0	0
$13$	4.89962	1.35891	0.679456	−	0.733717i	$-0.262216\pi$
$13$	4.89962	1.35891	0.679456	+	0.733717i	$0.262216\pi$
$14$	0	0
$15$	2.29105	0.591547
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.86033	1.34445	0.672226	−	0.740346i	$-0.265338\pi$
$19$	5.86033	1.34445	0.672226	+	0.740346i	$0.265338\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.83387	−1.63347	−0.816737	−	0.577010i	$-0.804219\pi$
$23$	−7.83387	−1.63347	−0.816737	+	0.577010i	$0.804219\pi$
$24$	0	0
$25$	0.248918	0.0497836
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.08215	0.572341	0.286171	−	0.958179i	$-0.407618\pi$
$29$	3.08215	0.572341	0.286171	+	0.958179i	$0.407618\pi$
$30$	0	0
$31$	9.62964	1.72953	0.864767	−	0.502173i	$-0.167466\pi$
$31$	9.62964	1.72953	0.864767	+	0.502173i	$0.167466\pi$
$32$	0	0
$33$	−1.47358	−0.256517
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.70428	0.280181	0.140091	−	0.990139i	$-0.455261\pi$
$37$	1.70428	0.280181	0.140091	+	0.990139i	$0.455261\pi$
$38$	0	0
$39$	−4.89962	−0.784568
$40$	0	0
$41$	−7.79458	−1.21731	−0.608654	−	0.793436i	$-0.708290\pi$
$41$	−7.79458	−1.21731	−0.608654	+	0.793436i	$0.708290\pi$
$42$	0	0
$43$	10.9324	1.66718	0.833589	−	0.552386i	$-0.186282\pi$
$43$	10.9324	1.66718	0.833589	+	0.552386i	$0.186282\pi$
$44$	0	0
$45$	−2.29105	−0.341530
$46$	0	0
$47$	−6.69887	−0.977131	−0.488566	−	0.872527i	$-0.662480\pi$
$47$	−6.69887	−0.977131	−0.488566	+	0.872527i	$0.662480\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	10.7017	1.46999	0.734997	−	0.678070i	$-0.237184\pi$
$53$	10.7017	1.46999	0.734997	+	0.678070i	$0.237184\pi$
$54$	0	0
$55$	−3.37605	−0.455226
$56$	0	0
$57$	−5.86033	−0.776220
$58$	0	0
$59$	5.62680	0.732546	0.366273	−	0.930507i	$-0.380634\pi$
$59$	5.62680	0.732546	0.366273	+	0.930507i	$0.380634\pi$
$60$	0	0
$61$	4.29390	0.549777	0.274888	−	0.961476i	$-0.411359\pi$
$61$	4.29390	0.549777	0.274888	+	0.961476i	$0.411359\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.2253	−1.39233
$66$	0	0
$67$	−4.96823	−0.606966	−0.303483	−	0.952837i	$-0.598149\pi$
$67$	−4.96823	−0.606966	−0.303483	+	0.952837i	$0.598149\pi$
$68$	0	0
$69$	7.83387	0.943087
$70$	0	0
$71$	−2.74357	−0.325601	−0.162801	−	0.986659i	$-0.552053\pi$
$71$	−2.74357	−0.325601	−0.162801	+	0.986659i	$0.552053\pi$
$72$	0	0
$73$	8.90714	1.04250	0.521251	−	0.853403i	$-0.325466\pi$
$73$	8.90714	1.04250	0.521251	+	0.853403i	$0.325466\pi$
$74$	0	0
$75$	−0.248918	−0.0287426
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00888	−0.451034	−0.225517	−	0.974239i	$-0.572407\pi$
$79$	−4.00888	−0.451034	−0.225517	+	0.974239i	$0.572407\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.57743	0.502438	0.251219	−	0.967930i	$-0.419169\pi$
$83$	4.57743	0.502438	0.251219	+	0.967930i	$0.419169\pi$
$84$	0	0
$85$	−2.29105	−0.248499
$86$	0	0
$87$	−3.08215	−0.330441
$88$	0	0
$89$	−0.0793088	−0.00840672	−0.00420336	−	0.999991i	$-0.501338\pi$
$89$	−0.0793088	−0.00840672	−0.00420336	+	0.999991i	$0.501338\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.62964	−0.998547
$94$	0	0
$95$	−13.4263	−1.37751
$96$	0	0
$97$	4.56388	0.463392	0.231696	−	0.972788i	$-0.425573\pi$
$97$	4.56388	0.463392	0.231696	+	0.972788i	$0.425573\pi$
$98$	0	0
$99$	1.47358	0.148100
$100$	0	0
$101$	−9.84742	−0.979854	−0.489927	−	0.871763i	$-0.662977\pi$
$101$	−9.84742	−0.979854	−0.489927	+	0.871763i	$0.662977\pi$
$102$	0	0
$103$	13.5199	1.33216	0.666079	−	0.745882i	$-0.267972\pi$
$103$	13.5199	1.33216	0.666079	+	0.745882i	$0.267972\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.2464	−1.28057	−0.640287	−	0.768136i	$-0.721185\pi$
$107$	−13.2464	−1.28057	−0.640287	+	0.768136i	$0.721185\pi$
$108$	0	0
$109$	−16.6888	−1.59850	−0.799248	−	0.601001i	$-0.794769\pi$
$109$	−16.6888	−1.59850	−0.799248	+	0.601001i	$0.794769\pi$
$110$	0	0
$111$	−1.70428	−0.161763
$112$	0	0
$113$	−6.70491	−0.630745	−0.315372	−	0.948968i	$-0.602129\pi$
$113$	−6.70491	−0.630745	−0.315372	+	0.948968i	$0.602129\pi$
$114$	0	0
$115$	17.9478	1.67364
$116$	0	0
$117$	4.89962	0.452970
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.82856	−0.802597
$122$	0	0
$123$	7.79458	0.702813
$124$	0	0
$125$	10.8850	0.973582
$126$	0	0
$127$	−15.0539	−1.33581	−0.667907	−	0.744245i	$-0.732810\pi$
$127$	−15.0539	−1.33581	−0.667907	+	0.744245i	$0.732810\pi$
$128$	0	0
$129$	−10.9324	−0.962545
$130$	0	0
$131$	13.1631	1.15007	0.575033	−	0.818130i	$-0.304989\pi$
$131$	13.1631	1.15007	0.575033	+	0.818130i	$0.304989\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.29105	0.197182
$136$	0	0
$137$	15.1095	1.29089	0.645447	−	0.763805i	$-0.276671\pi$
$137$	15.1095	1.29089	0.645447	+	0.763805i	$0.276671\pi$
$138$	0	0
$139$	−1.49465	−0.126774	−0.0633871	−	0.997989i	$-0.520190\pi$
$139$	−1.49465	−0.126774	−0.0633871	+	0.997989i	$0.520190\pi$
$140$	0	0
$141$	6.69887	0.564147
$142$	0	0
$143$	7.21999	0.603766
$144$	0	0
$145$	−7.06137	−0.586415
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.4946	−1.59706	−0.798532	−	0.601953i	$-0.794390\pi$
$149$	−19.4946	−1.59706	−0.798532	+	0.601953i	$0.794390\pi$
$150$	0	0
$151$	2.52112	0.205166	0.102583	−	0.994724i	$-0.467289\pi$
$151$	2.52112	0.205166	0.102583	+	0.994724i	$0.467289\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−22.0620	−1.77206
$156$	0	0
$157$	−3.19535	−0.255017	−0.127508	−	0.991838i	$-0.540698\pi$
$157$	−3.19535	−0.255017	−0.127508	+	0.991838i	$0.540698\pi$
$158$	0	0
$159$	−10.7017	−0.848701
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0763	−1.57250	−0.786248	−	0.617911i	$-0.787979\pi$
$163$	−20.0763	−1.57250	−0.786248	+	0.617911i	$0.787979\pi$
$164$	0	0
$165$	3.37605	0.262825
$166$	0	0
$167$	−12.8056	−0.990925	−0.495462	−	0.868629i	$-0.665001\pi$
$167$	−12.8056	−0.990925	−0.495462	+	0.868629i	$0.665001\pi$
$168$	0	0
$169$	11.0063	0.846640
$170$	0	0
$171$	5.86033	0.448151
$172$	0	0
$173$	−11.5381	−0.877229	−0.438614	−	0.898675i	$-0.644531\pi$
$173$	−11.5381	−0.877229	−0.438614	+	0.898675i	$0.644531\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.62680	−0.422936
$178$	0	0
$179$	−6.90850	−0.516366	−0.258183	−	0.966096i	$-0.583124\pi$
$179$	−6.90850	−0.516366	−0.258183	+	0.966096i	$0.583124\pi$
$180$	0	0
$181$	6.80677	0.505943	0.252971	−	0.967474i	$-0.418592\pi$
$181$	6.80677	0.505943	0.252971	+	0.967474i	$0.418592\pi$
$182$	0	0
$183$	−4.29390	−0.317414
$184$	0	0
$185$	−3.90458	−0.287071
$186$	0	0
$187$	1.47358	0.107759
$188$	0	0
$189$	0	0
$190$	0	0
$191$	27.3912	1.98196	0.990980	−	0.134007i	$-0.0427844\pi$
$191$	27.3912	1.98196	0.990980	+	0.134007i	$0.0427844\pi$
$192$	0	0
$193$	−14.5517	−1.04745	−0.523727	−	0.851886i	$-0.675459\pi$
$193$	−14.5517	−1.04745	−0.523727	+	0.851886i	$0.675459\pi$
$194$	0	0
$195$	11.2253	0.803860
$196$	0	0
$197$	12.9356	0.921623	0.460812	−	0.887498i	$-0.347558\pi$
$197$	12.9356	0.921623	0.460812	+	0.887498i	$0.347558\pi$
$198$	0	0
$199$	−5.02527	−0.356232	−0.178116	−	0.984009i	$-0.557000\pi$
$199$	−5.02527	−0.356232	−0.178116	+	0.984009i	$0.557000\pi$
$200$	0	0
$201$	4.96823	0.350432
$202$	0	0
$203$	0	0
$204$	0	0
$205$	17.8578	1.24724
$206$	0	0
$207$	−7.83387	−0.544491
$208$	0	0
$209$	8.63567	0.597342
$210$	0	0
$211$	19.4352	1.33797	0.668987	−	0.743274i	$-0.266728\pi$
$211$	19.4352	1.33797	0.668987	+	0.743274i	$0.266728\pi$
$212$	0	0
$213$	2.74357	0.187986
$214$	0	0
$215$	−25.0467	−1.70817
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.90714	−0.601889
$220$	0	0
$221$	4.89962	0.329584
$222$	0	0
$223$	−23.0892	−1.54617	−0.773084	−	0.634304i	$-0.781287\pi$
$223$	−23.0892	−1.54617	−0.773084	+	0.634304i	$0.781287\pi$
$224$	0	0
$225$	0.248918	0.0165945
$226$	0	0
$227$	7.36029	0.488519	0.244260	−	0.969710i	$-0.421455\pi$
$227$	7.36029	0.488519	0.244260	+	0.969710i	$0.421455\pi$
$228$	0	0
$229$	9.03389	0.596976	0.298488	−	0.954413i	$-0.403518\pi$
$229$	9.03389	0.596976	0.298488	+	0.954413i	$0.403518\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.6636	−0.960646	−0.480323	−	0.877092i	$-0.659481\pi$
$233$	−14.6636	−0.960646	−0.480323	+	0.877092i	$0.659481\pi$
$234$	0	0
$235$	15.3475	1.00116
$236$	0	0
$237$	4.00888	0.260405
$238$	0	0
$239$	14.0768	0.910549	0.455275	−	0.890351i	$-0.349541\pi$
$239$	14.0768	0.910549	0.455275	+	0.890351i	$0.349541\pi$
$240$	0	0
$241$	11.6549	0.750759	0.375379	−	0.926871i	$-0.377512\pi$
$241$	11.6549	0.750759	0.375379	+	0.926871i	$0.377512\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.7134	1.82699
$248$	0	0
$249$	−4.57743	−0.290083
$250$	0	0
$251$	14.4043	0.909194	0.454597	−	0.890697i	$-0.349783\pi$
$251$	14.4043	0.909194	0.454597	+	0.890697i	$0.349783\pi$
$252$	0	0
$253$	−11.5438	−0.725754
$254$	0	0
$255$	2.29105	0.143471
$256$	0	0
$257$	6.67718	0.416511	0.208255	−	0.978074i	$-0.433221\pi$
$257$	6.67718	0.416511	0.208255	+	0.978074i	$0.433221\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.08215	0.190780
$262$	0	0
$263$	20.7328	1.27844	0.639218	−	0.769025i	$-0.279258\pi$
$263$	20.7328	1.27844	0.639218	+	0.769025i	$0.279258\pi$
$264$	0	0
$265$	−24.5182	−1.50614
$266$	0	0
$267$	0.0793088	0.00485362
$268$	0	0
$269$	−3.33382	−0.203266	−0.101633	−	0.994822i	$-0.532407\pi$
$269$	−3.33382	−0.203266	−0.101633	+	0.994822i	$0.532407\pi$
$270$	0	0
$271$	−20.1834	−1.22606	−0.613028	−	0.790061i	$-0.710049\pi$
$271$	−20.1834	−1.22606	−0.613028	+	0.790061i	$0.710049\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.366801	0.0221189
$276$	0	0
$277$	−6.36853	−0.382648	−0.191324	−	0.981527i	$-0.561278\pi$
$277$	−6.36853	−0.382648	−0.191324	+	0.981527i	$0.561278\pi$
$278$	0	0
$279$	9.62964	0.576511
$280$	0	0
$281$	−0.681848	−0.0406757	−0.0203378	−	0.999793i	$-0.506474\pi$
$281$	−0.681848	−0.0406757	−0.0203378	+	0.999793i	$0.506474\pi$
$282$	0	0
$283$	33.0221	1.96296	0.981479	−	0.191567i	$-0.0613570\pi$
$283$	33.0221	1.96296	0.981479	+	0.191567i	$0.0613570\pi$
$284$	0	0
$285$	13.4263	0.795307
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−4.56388	−0.267539
$292$	0	0
$293$	11.7974	0.689213	0.344606	−	0.938747i	$-0.388012\pi$
$293$	11.7974	0.689213	0.344606	+	0.938747i	$0.388012\pi$
$294$	0	0
$295$	−12.8913	−0.750559
$296$	0	0
$297$	−1.47358	−0.0855058
$298$	0	0
$299$	−38.3830	−2.21975
$300$	0	0
$301$	0	0
$302$	0	0
$303$	9.84742	0.565719
$304$	0	0
$305$	−9.83754	−0.563296
$306$	0	0
$307$	−0.370361	−0.0211376	−0.0105688	−	0.999944i	$-0.503364\pi$
$307$	−0.370361	−0.0211376	−0.0105688	+	0.999944i	$0.503364\pi$
$308$	0	0
$309$	−13.5199	−0.769121
$310$	0	0
$311$	30.7756	1.74513	0.872563	−	0.488502i	$-0.162457\pi$
$311$	30.7756	1.74513	0.872563	+	0.488502i	$0.162457\pi$
$312$	0	0
$313$	−18.2350	−1.03070	−0.515352	−	0.856978i	$-0.672339\pi$
$313$	−18.2350	−1.03070	−0.515352	+	0.856978i	$0.672339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.64723	−0.541843	−0.270921	−	0.962602i	$-0.587328\pi$
$317$	−9.64723	−0.541843	−0.270921	+	0.962602i	$0.587328\pi$
$318$	0	0
$319$	4.54180	0.254292
$320$	0	0
$321$	13.2464	0.739340
$322$	0	0
$323$	5.86033	0.326078
$324$	0	0
$325$	1.21961	0.0676515
$326$	0	0
$327$	16.6888	0.922893
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.82856	−0.540227	−0.270113	−	0.962829i	$-0.587061\pi$
$331$	−9.82856	−0.540227	−0.270113	+	0.962829i	$0.587061\pi$
$332$	0	0
$333$	1.70428	0.0933937
$334$	0	0
$335$	11.3825	0.621891
$336$	0	0
$337$	−2.77288	−0.151048	−0.0755241	−	0.997144i	$-0.524063\pi$
$337$	−2.77288	−0.151048	−0.0755241	+	0.997144i	$0.524063\pi$
$338$	0	0
$339$	6.70491	0.364161
$340$	0	0
$341$	14.1900	0.768434
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−17.9478	−0.966277
$346$	0	0
$347$	25.7819	1.38405	0.692024	−	0.721875i	$-0.256719\pi$
$347$	25.7819	1.38405	0.692024	+	0.721875i	$0.256719\pi$
$348$	0	0
$349$	25.0403	1.34038	0.670188	−	0.742191i	$-0.266213\pi$
$349$	25.0403	1.34038	0.670188	+	0.742191i	$0.266213\pi$
$350$	0	0
$351$	−4.89962	−0.261523
$352$	0	0
$353$	2.98781	0.159025	0.0795125	−	0.996834i	$-0.474664\pi$
$353$	2.98781	0.159025	0.0795125	+	0.996834i	$0.474664\pi$
$354$	0	0
$355$	6.28565	0.333608
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.5952	−0.770304	−0.385152	−	0.922853i	$-0.625851\pi$
$359$	−14.5952	−0.770304	−0.385152	+	0.922853i	$0.625851\pi$
$360$	0	0
$361$	15.3435	0.807554
$362$	0	0
$363$	8.82856	0.463379
$364$	0	0
$365$	−20.4067	−1.06814
$366$	0	0
$367$	34.4962	1.80069	0.900343	−	0.435180i	$-0.143315\pi$
$367$	34.4962	1.80069	0.900343	+	0.435180i	$0.143315\pi$
$368$	0	0
$369$	−7.79458	−0.405769
$370$	0	0
$371$	0	0
$372$	0	0
$373$	25.6334	1.32725	0.663624	−	0.748067i	$-0.269018\pi$
$373$	25.6334	1.32725	0.663624	+	0.748067i	$0.269018\pi$
$374$	0	0
$375$	−10.8850	−0.562098
$376$	0	0
$377$	15.1014	0.777761
$378$	0	0
$379$	19.1496	0.983647	0.491823	−	0.870695i	$-0.336331\pi$
$379$	19.1496	0.983647	0.491823	+	0.870695i	$0.336331\pi$
$380$	0	0
$381$	15.0539	0.771232
$382$	0	0
$383$	30.3173	1.54914	0.774572	−	0.632486i	$-0.217965\pi$
$383$	30.3173	1.54914	0.774572	+	0.632486i	$0.217965\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.9324	0.555726
$388$	0	0
$389$	36.3214	1.84157	0.920784	−	0.390074i	$-0.127551\pi$
$389$	36.3214	1.84157	0.920784	+	0.390074i	$0.127551\pi$
$390$	0	0
$391$	−7.83387	−0.396176
$392$	0	0
$393$	−13.1631	−0.663991
$394$	0	0
$395$	9.18454	0.462125
$396$	0	0
$397$	4.54281	0.227997	0.113999	−	0.993481i	$-0.463634\pi$
$397$	4.54281	0.227997	0.113999	+	0.993481i	$0.463634\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.0896	0.953288	0.476644	−	0.879096i	$-0.341853\pi$
$401$	19.0896	0.953288	0.476644	+	0.879096i	$0.341853\pi$
$402$	0	0
$403$	47.1816	2.35028
$404$	0	0
$405$	−2.29105	−0.113843
$406$	0	0
$407$	2.51139	0.124485
$408$	0	0
$409$	12.8351	0.634653	0.317327	−	0.948316i	$-0.397215\pi$
$409$	12.8351	0.634653	0.317327	+	0.948316i	$0.397215\pi$
$410$	0	0
$411$	−15.1095	−0.745299
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.4871	−0.514793
$416$	0	0
$417$	1.49465	0.0731932
$418$	0	0
$419$	−4.15303	−0.202889	−0.101444	−	0.994841i	$-0.532346\pi$
$419$	−4.15303	−0.202889	−0.101444	+	0.994841i	$0.532346\pi$
$420$	0	0
$421$	38.0557	1.85472	0.927360	−	0.374170i	$-0.122072\pi$
$421$	38.0557	1.85472	0.927360	+	0.374170i	$0.122072\pi$
$422$	0	0
$423$	−6.69887	−0.325710
$424$	0	0
$425$	0.248918	0.0120743
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7.21999	−0.348584
$430$	0	0
$431$	−6.65144	−0.320388	−0.160194	−	0.987086i	$-0.551212\pi$
$431$	−6.65144	−0.320388	−0.160194	+	0.987086i	$0.551212\pi$
$432$	0	0
$433$	13.9664	0.671182	0.335591	−	0.942008i	$-0.391064\pi$
$433$	13.9664	0.671182	0.335591	+	0.942008i	$0.391064\pi$
$434$	0	0
$435$	7.06137	0.338567
$436$	0	0
$437$	−45.9091	−2.19613
$438$	0	0
$439$	19.2674	0.919584	0.459792	−	0.888027i	$-0.347924\pi$
$439$	19.2674	0.919584	0.459792	+	0.888027i	$0.347924\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.1882	−0.864148	−0.432074	−	0.901838i	$-0.642218\pi$
$443$	−18.1882	−0.864148	−0.432074	+	0.901838i	$0.642218\pi$
$444$	0	0
$445$	0.181701	0.00861344
$446$	0	0
$447$	19.4946	0.922065
$448$	0	0
$449$	23.1290	1.09152	0.545762	−	0.837941i	$-0.316240\pi$
$449$	23.1290	1.09152	0.545762	+	0.837941i	$0.316240\pi$
$450$	0	0
$451$	−11.4859	−0.540851
$452$	0	0
$453$	−2.52112	−0.118452
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.43309	0.441261	0.220631	−	0.975357i	$-0.429188\pi$
$457$	9.43309	0.441261	0.220631	+	0.975357i	$0.429188\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−33.7016	−1.56964	−0.784820	−	0.619723i	$-0.787245\pi$
$461$	−33.7016	−1.56964	−0.784820	+	0.619723i	$0.787245\pi$
$462$	0	0
$463$	−7.72955	−0.359222	−0.179611	−	0.983738i	$-0.557484\pi$
$463$	−7.72955	−0.359222	−0.179611	+	0.983738i	$0.557484\pi$
$464$	0	0
$465$	22.0620	1.02310
$466$	0	0
$467$	19.5890	0.906470	0.453235	−	0.891391i	$-0.350270\pi$
$467$	19.5890	0.906470	0.453235	+	0.891391i	$0.350270\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.19535	0.147234
$472$	0	0
$473$	16.1098	0.740729
$474$	0	0
$475$	1.45874	0.0669317
$476$	0	0
$477$	10.7017	0.489998
$478$	0	0
$479$	−33.2867	−1.52091	−0.760453	−	0.649393i	$-0.775023\pi$
$479$	−33.2867	−1.52091	−0.760453	+	0.649393i	$0.775023\pi$
$480$	0	0
$481$	8.35031	0.380741
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.4561	−0.474786
$486$	0	0
$487$	12.4012	0.561950	0.280975	−	0.959715i	$-0.409342\pi$
$487$	12.4012	0.561950	0.280975	+	0.959715i	$0.409342\pi$
$488$	0	0
$489$	20.0763	0.907881
$490$	0	0
$491$	17.6678	0.797338	0.398669	−	0.917095i	$-0.369472\pi$
$491$	17.6678	0.797338	0.398669	+	0.917095i	$0.369472\pi$
$492$	0	0
$493$	3.08215	0.138813
$494$	0	0
$495$	−3.37605	−0.151742
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.6135	−0.609425	−0.304713	−	0.952444i	$-0.598560\pi$
$499$	−13.6135	−0.609425	−0.304713	+	0.952444i	$0.598560\pi$
$500$	0	0
$501$	12.8056	0.572111
$502$	0	0
$503$	−14.1267	−0.629877	−0.314938	−	0.949112i	$-0.601984\pi$
$503$	−14.1267	−0.629877	−0.314938	+	0.949112i	$0.601984\pi$
$504$	0	0
$505$	22.5609	1.00395
$506$	0	0
$507$	−11.0063	−0.488808
$508$	0	0
$509$	6.44345	0.285601	0.142801	−	0.989751i	$-0.454389\pi$
$509$	6.44345	0.285601	0.142801	+	0.989751i	$0.454389\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.86033	−0.258740
$514$	0	0
$515$	−30.9748	−1.36491
$516$	0	0
$517$	−9.87133	−0.434140
$518$	0	0
$519$	11.5381	0.506468
$520$	0	0
$521$	22.3806	0.980513	0.490256	−	0.871578i	$-0.336903\pi$
$521$	22.3806	0.980513	0.490256	+	0.871578i	$0.336903\pi$
$522$	0	0
$523$	−12.1480	−0.531194	−0.265597	−	0.964084i	$-0.585569\pi$
$523$	−12.1480	−0.531194	−0.265597	+	0.964084i	$0.585569\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.62964	0.419474
$528$	0	0
$529$	38.3694	1.66824
$530$	0	0
$531$	5.62680	0.244182
$532$	0	0
$533$	−38.1905	−1.65421
$534$	0	0
$535$	30.3481	1.31206
$536$	0	0
$537$	6.90850	0.298124
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.54970	0.367580	0.183790	−	0.982966i	$-0.441163\pi$
$541$	8.54970	0.367580	0.183790	+	0.982966i	$0.441163\pi$
$542$	0	0
$543$	−6.80677	−0.292106
$544$	0	0
$545$	38.2349	1.63780
$546$	0	0
$547$	−22.3676	−0.956370	−0.478185	−	0.878259i	$-0.658705\pi$
$547$	−22.3676	−0.956370	−0.478185	+	0.878259i	$0.658705\pi$
$548$	0	0
$549$	4.29390	0.183259
$550$	0	0
$551$	18.0624	0.769486
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.90458	0.165740
$556$	0	0
$557$	−38.0431	−1.61194	−0.805970	−	0.591956i	$-0.798356\pi$
$557$	−38.0431	−1.61194	−0.805970	+	0.591956i	$0.798356\pi$
$558$	0	0
$559$	53.5647	2.26555
$560$	0	0
$561$	−1.47358	−0.0622146
$562$	0	0
$563$	28.9025	1.21809	0.609047	−	0.793134i	$-0.291552\pi$
$563$	28.9025	1.21809	0.609047	+	0.793134i	$0.291552\pi$
$564$	0	0
$565$	15.3613	0.646254
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.5589	1.32302	0.661509	−	0.749937i	$-0.269916\pi$
$569$	31.5589	1.32302	0.661509	+	0.749937i	$0.269916\pi$
$570$	0	0
$571$	−1.66290	−0.0695900	−0.0347950	−	0.999394i	$-0.511078\pi$
$571$	−1.66290	−0.0695900	−0.0347950	+	0.999394i	$0.511078\pi$
$572$	0	0
$573$	−27.3912	−1.14429
$574$	0	0
$575$	−1.94999	−0.0813202
$576$	0	0
$577$	−6.49995	−0.270596	−0.135298	−	0.990805i	$-0.543199\pi$
$577$	−6.49995	−0.270596	−0.135298	+	0.990805i	$0.543199\pi$
$578$	0	0
$579$	14.5517	0.604748
$580$	0	0
$581$	0	0
$582$	0	0
$583$	15.7698	0.653120
$584$	0	0
$585$	−11.2253	−0.464109
$586$	0	0
$587$	18.7633	0.774446	0.387223	−	0.921986i	$-0.373434\pi$
$587$	18.7633	0.774446	0.387223	+	0.921986i	$0.373434\pi$
$588$	0	0
$589$	56.4329	2.32528
$590$	0	0
$591$	−12.9356	−0.532100
$592$	0	0
$593$	−18.9949	−0.780025	−0.390013	−	0.920809i	$-0.627529\pi$
$593$	−18.9949	−0.780025	−0.390013	+	0.920809i	$0.627529\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.02527	0.205671
$598$	0	0
$599$	−20.1298	−0.822479	−0.411240	−	0.911527i	$-0.634904\pi$
$599$	−20.1298	−0.822479	−0.411240	+	0.911527i	$0.634904\pi$
$600$	0	0
$601$	−31.6416	−1.29069	−0.645345	−	0.763891i	$-0.723286\pi$
$601$	−31.6416	−1.29069	−0.645345	+	0.763891i	$0.723286\pi$
$602$	0	0
$603$	−4.96823	−0.202322
$604$	0	0
$605$	20.2267	0.822332
$606$	0	0
$607$	−34.9986	−1.42055	−0.710275	−	0.703924i	$-0.751429\pi$
$607$	−34.9986	−1.42055	−0.710275	+	0.703924i	$0.751429\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.8220	−1.32783
$612$	0	0
$613$	−34.7761	−1.40459	−0.702296	−	0.711885i	$-0.747842\pi$
$613$	−34.7761	−1.40459	−0.702296	+	0.711885i	$0.747842\pi$
$614$	0	0
$615$	−17.8578	−0.720095
$616$	0	0
$617$	11.1428	0.448592	0.224296	−	0.974521i	$-0.427992\pi$
$617$	11.1428	0.448592	0.224296	+	0.974521i	$0.427992\pi$
$618$	0	0
$619$	−2.07161	−0.0832650	−0.0416325	−	0.999133i	$-0.513256\pi$
$619$	−2.07161	−0.0832650	−0.0416325	+	0.999133i	$0.513256\pi$
$620$	0	0
$621$	7.83387	0.314362
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.1826	−1.04731
$626$	0	0
$627$	−8.63567	−0.344876
$628$	0	0
$629$	1.70428	0.0679539
$630$	0	0
$631$	29.4945	1.17416	0.587078	−	0.809530i	$-0.300278\pi$
$631$	29.4945	1.17416	0.587078	+	0.809530i	$0.300278\pi$
$632$	0	0
$633$	−19.4352	−0.772480
$634$	0	0
$635$	34.4892	1.36866
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.74357	−0.108534
$640$	0	0
$641$	−16.3529	−0.645899	−0.322950	−	0.946416i	$-0.604674\pi$
$641$	−16.3529	−0.645899	−0.322950	+	0.946416i	$0.604674\pi$
$642$	0	0
$643$	−7.59072	−0.299349	−0.149674	−	0.988735i	$-0.547823\pi$
$643$	−7.59072	−0.299349	−0.149674	+	0.988735i	$0.547823\pi$
$644$	0	0
$645$	25.0467	0.986214
$646$	0	0
$647$	11.8628	0.466375	0.233187	−	0.972432i	$-0.425084\pi$
$647$	11.8628	0.466375	0.233187	+	0.972432i	$0.425084\pi$
$648$	0	0
$649$	8.29153	0.325471
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.0679	−1.68538	−0.842688	−	0.538403i	$-0.819028\pi$
$653$	−43.0679	−1.68538	−0.842688	+	0.538403i	$0.819028\pi$
$654$	0	0
$655$	−30.1574	−1.17835
$656$	0	0
$657$	8.90714	0.347501
$658$	0	0
$659$	−41.4674	−1.61534	−0.807670	−	0.589635i	$-0.799271\pi$
$659$	−41.4674	−1.61534	−0.807670	+	0.589635i	$0.799271\pi$
$660$	0	0
$661$	−34.7698	−1.35239	−0.676193	−	0.736724i	$-0.736372\pi$
$661$	−34.7698	−1.35239	−0.676193	+	0.736724i	$0.736372\pi$
$662$	0	0
$663$	−4.89962	−0.190286
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.1452	−0.934905
$668$	0	0
$669$	23.0892	0.892680
$670$	0	0
$671$	6.32740	0.244267
$672$	0	0
$673$	22.4767	0.866413	0.433207	−	0.901295i	$-0.357382\pi$
$673$	22.4767	0.866413	0.433207	+	0.901295i	$0.357382\pi$
$674$	0	0
$675$	−0.248918	−0.00958086
$676$	0	0
$677$	−30.1685	−1.15947	−0.579735	−	0.814805i	$-0.696844\pi$
$677$	−30.1685	−1.15947	−0.579735	+	0.814805i	$0.696844\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.36029	−0.282047
$682$	0	0
$683$	43.0090	1.64569	0.822846	−	0.568264i	$-0.192385\pi$
$683$	43.0090	1.64569	0.822846	+	0.568264i	$0.192385\pi$
$684$	0	0
$685$	−34.6167	−1.32264
$686$	0	0
$687$	−9.03389	−0.344664
$688$	0	0
$689$	52.4344	1.99759
$690$	0	0
$691$	34.6958	1.31989	0.659946	−	0.751313i	$-0.270579\pi$
$691$	34.6958	1.31989	0.659946	+	0.751313i	$0.270579\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.42431	0.129892
$696$	0	0
$697$	−7.79458	−0.295241
$698$	0	0
$699$	14.6636	0.554629
$700$	0	0
$701$	19.2181	0.725856	0.362928	−	0.931817i	$-0.381777\pi$
$701$	19.2181	0.725856	0.362928	+	0.931817i	$0.381777\pi$
$702$	0	0
$703$	9.98762	0.376690
$704$	0	0
$705$	−15.3475	−0.578019
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.4249	1.29286	0.646428	−	0.762975i	$-0.276262\pi$
$709$	34.4249	1.29286	0.646428	+	0.762975i	$0.276262\pi$
$710$	0	0
$711$	−4.00888	−0.150345
$712$	0	0
$713$	−75.4373	−2.82515
$714$	0	0
$715$	−16.5414	−0.618612
$716$	0	0
$717$	−14.0768	−0.525706
$718$	0	0
$719$	34.1944	1.27524	0.637618	−	0.770353i	$-0.279920\pi$
$719$	34.1944	1.27524	0.637618	+	0.770353i	$0.279920\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−11.6549	−0.433451
$724$	0	0
$725$	0.767204	0.0284932
$726$	0	0
$727$	−25.8650	−0.959280	−0.479640	−	0.877465i	$-0.659233\pi$
$727$	−25.8650	−0.959280	−0.479640	+	0.877465i	$0.659233\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.9324	0.404350
$732$	0	0
$733$	−17.6530	−0.652028	−0.326014	−	0.945365i	$-0.605706\pi$
$733$	−17.6530	−0.652028	−0.326014	+	0.945365i	$0.605706\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.32108	−0.269675
$738$	0	0
$739$	23.2868	0.856617	0.428309	−	0.903632i	$-0.359110\pi$
$739$	23.2868	0.856617	0.428309	+	0.903632i	$0.359110\pi$
$740$	0	0
$741$	−28.7134	−1.05481
$742$	0	0
$743$	24.9103	0.913869	0.456935	−	0.889500i	$-0.348947\pi$
$743$	24.9103	0.913869	0.456935	+	0.889500i	$0.348947\pi$
$744$	0	0
$745$	44.6632	1.63633
$746$	0	0
$747$	4.57743	0.167479
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.4799	−1.00276	−0.501378	−	0.865228i	$-0.667173\pi$
$751$	−27.4799	−1.00276	−0.501378	+	0.865228i	$0.667173\pi$
$752$	0	0
$753$	−14.4043	−0.524924
$754$	0	0
$755$	−5.77601	−0.210210
$756$	0	0
$757$	−10.2235	−0.371578	−0.185789	−	0.982590i	$-0.559484\pi$
$757$	−10.2235	−0.371578	−0.185789	+	0.982590i	$0.559484\pi$
$758$	0	0
$759$	11.5438	0.419014
$760$	0	0
$761$	−7.82647	−0.283709	−0.141855	−	0.989887i	$-0.545307\pi$
$761$	−7.82647	−0.283709	−0.141855	+	0.989887i	$0.545307\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.29105	−0.0828332
$766$	0	0
$767$	27.5692	0.995466
$768$	0	0
$769$	44.2965	1.59737	0.798686	−	0.601749i	$-0.205529\pi$
$769$	44.2965	1.59737	0.798686	+	0.601749i	$0.205529\pi$
$770$	0	0
$771$	−6.67718	−0.240473
$772$	0	0
$773$	−29.6525	−1.06652	−0.533262	−	0.845950i	$-0.679034\pi$
$773$	−29.6525	−1.06652	−0.533262	+	0.845950i	$0.679034\pi$
$774$	0	0
$775$	2.39699	0.0861025
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−45.6788	−1.63661
$780$	0	0
$781$	−4.04286	−0.144665
$782$	0	0
$783$	−3.08215	−0.110147
$784$	0	0
$785$	7.32071	0.261287
$786$	0	0
$787$	−44.0166	−1.56902	−0.784511	−	0.620115i	$-0.787086\pi$
$787$	−44.0166	−1.56902	−0.784511	+	0.620115i	$0.787086\pi$
$788$	0	0
$789$	−20.7328	−0.738106
$790$	0	0
$791$	0	0
$792$	0	0
$793$	21.0385	0.747098
$794$	0	0
$795$	24.5182	0.869570
$796$	0	0
$797$	−13.9946	−0.495714	−0.247857	−	0.968797i	$-0.579726\pi$
$797$	−13.9946	−0.495714	−0.247857	+	0.968797i	$0.579726\pi$
$798$	0	0
$799$	−6.69887	−0.236989
$800$	0	0
$801$	−0.0793088	−0.00280224
$802$	0	0
$803$	13.1254	0.463185
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.33382	0.117356
$808$	0	0
$809$	−45.4983	−1.59964	−0.799818	−	0.600243i	$-0.795071\pi$
$809$	−45.4983	−1.59964	−0.799818	+	0.600243i	$0.795071\pi$
$810$	0	0
$811$	−40.5999	−1.42565	−0.712827	−	0.701340i	$-0.752586\pi$
$811$	−40.5999	−1.42565	−0.712827	+	0.701340i	$0.752586\pi$
$812$	0	0
$813$	20.1834	0.707864
$814$	0	0
$815$	45.9958	1.61116
$816$	0	0
$817$	64.0676	2.24144
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.9189	0.799874	0.399937	−	0.916543i	$-0.369032\pi$
$821$	22.9189	0.799874	0.399937	+	0.916543i	$0.369032\pi$
$822$	0	0
$823$	21.5067	0.749677	0.374838	−	0.927090i	$-0.377698\pi$
$823$	21.5067	0.749677	0.374838	+	0.927090i	$0.377698\pi$
$824$	0	0
$825$	−0.366801	−0.0127704
$826$	0	0
$827$	3.07905	0.107069	0.0535345	−	0.998566i	$-0.482951\pi$
$827$	3.07905	0.107069	0.0535345	+	0.998566i	$0.482951\pi$
$828$	0	0
$829$	−45.1898	−1.56951	−0.784753	−	0.619809i	$-0.787210\pi$
$829$	−45.1898	−1.56951	−0.784753	+	0.619809i	$0.787210\pi$
$830$	0	0
$831$	6.36853	0.220922
$832$	0	0
$833$	0	0
$834$	0	0
$835$	29.3382	1.01529
$836$	0	0
$837$	−9.62964	−0.332849
$838$	0	0
$839$	46.4316	1.60300	0.801499	−	0.597996i	$-0.204036\pi$
$839$	46.4316	1.60300	0.801499	+	0.597996i	$0.204036\pi$
$840$	0	0
$841$	−19.5003	−0.672425
$842$	0	0
$843$	0.681848	0.0234841
$844$	0	0
$845$	−25.2160	−0.867458
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−33.0221	−1.13331
$850$	0	0
$851$	−13.3511	−0.457669
$852$	0	0
$853$	15.1781	0.519690	0.259845	−	0.965650i	$-0.416329\pi$
$853$	15.1781	0.519690	0.259845	+	0.965650i	$0.416329\pi$
$854$	0	0
$855$	−13.4263	−0.459171
$856$	0	0
$857$	−51.7157	−1.76658	−0.883288	−	0.468832i	$-0.844675\pi$
$857$	−51.7157	−1.76658	−0.883288	+	0.468832i	$0.844675\pi$
$858$	0	0
$859$	49.2339	1.67984	0.839919	−	0.542711i	$-0.182602\pi$
$859$	49.2339	1.67984	0.839919	+	0.542711i	$0.182602\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.0203	0.919780	0.459890	−	0.887976i	$-0.347889\pi$
$863$	27.0203	0.919780	0.459890	+	0.887976i	$0.347889\pi$
$864$	0	0
$865$	26.4345	0.898799
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−5.90740	−0.200395
$870$	0	0
$871$	−24.3424	−0.824812
$872$	0	0
$873$	4.56388	0.154464
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.54913	−0.0523105	−0.0261552	−	0.999658i	$-0.508326\pi$
$877$	−1.54913	−0.0523105	−0.0261552	+	0.999658i	$0.508326\pi$
$878$	0	0
$879$	−11.7974	−0.397917
$880$	0	0
$881$	8.30444	0.279784	0.139892	−	0.990167i	$-0.455325\pi$
$881$	8.30444	0.279784	0.139892	+	0.990167i	$0.455325\pi$
$882$	0	0
$883$	−13.3328	−0.448685	−0.224342	−	0.974510i	$-0.572023\pi$
$883$	−13.3328	−0.448685	−0.224342	+	0.974510i	$0.572023\pi$
$884$	0	0
$885$	12.8913	0.433336
$886$	0	0
$887$	−46.2026	−1.55133	−0.775665	−	0.631145i	$-0.782585\pi$
$887$	−46.2026	−1.55133	−0.775665	+	0.631145i	$0.782585\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.47358	0.0493668
$892$	0	0
$893$	−39.2576	−1.31371
$894$	0	0
$895$	15.8277	0.529063
$896$	0	0
$897$	38.3830	1.28157
$898$	0	0
$899$	29.6800	0.989884
$900$	0	0
$901$	10.7017	0.356526
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.5947	−0.518384
$906$	0	0
$907$	54.8338	1.82073	0.910363	−	0.413811i	$-0.135803\pi$
$907$	54.8338	1.82073	0.910363	+	0.413811i	$0.135803\pi$
$908$	0	0
$909$	−9.84742	−0.326618
$910$	0	0
$911$	22.7887	0.755023	0.377512	−	0.926005i	$-0.376780\pi$
$911$	22.7887	0.755023	0.377512	+	0.926005i	$0.376780\pi$
$912$	0	0
$913$	6.74521	0.223234
$914$	0	0
$915$	9.83754	0.325219
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36.9636	1.21932	0.609659	−	0.792664i	$-0.291307\pi$
$919$	36.9636	1.21932	0.609659	+	0.792664i	$0.291307\pi$
$920$	0	0
$921$	0.370361	0.0122038
$922$	0	0
$923$	−13.4424	−0.442463
$924$	0	0
$925$	0.424225	0.0139484
$926$	0	0
$927$	13.5199	0.444052
$928$	0	0
$929$	3.61132	0.118484	0.0592418	−	0.998244i	$-0.481132\pi$
$929$	3.61132	0.118484	0.0592418	+	0.998244i	$0.481132\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−30.7756	−1.00755
$934$	0	0
$935$	−3.37605	−0.110409
$936$	0	0
$937$	38.8014	1.26759	0.633794	−	0.773502i	$-0.281497\pi$
$937$	38.8014	1.26759	0.633794	+	0.773502i	$0.281497\pi$
$938$	0	0
$939$	18.2350	0.595077
$940$	0	0
$941$	−33.9637	−1.10719	−0.553593	−	0.832788i	$-0.686744\pi$
$941$	−33.9637	−1.10719	−0.553593	+	0.832788i	$0.686744\pi$
$942$	0	0
$943$	61.0617	1.98844
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.44977	0.0796068	0.0398034	−	0.999208i	$-0.487327\pi$
$947$	2.44977	0.0796068	0.0398034	+	0.999208i	$0.487327\pi$
$948$	0	0
$949$	43.6416	1.41667
$950$	0	0
$951$	9.64723	0.312833
$952$	0	0
$953$	37.0641	1.20062	0.600312	−	0.799766i	$-0.295043\pi$
$953$	37.0641	1.20062	0.600312	+	0.799766i	$0.295043\pi$
$954$	0	0
$955$	−62.7548	−2.03070
$956$	0	0
$957$	−4.54180	−0.146816
$958$	0	0
$959$	0	0
$960$	0	0
$961$	61.7300	1.99129
$962$	0	0
$963$	−13.2464	−0.426858
$964$	0	0
$965$	33.3387	1.07321
$966$	0	0
$967$	−19.1733	−0.616573	−0.308287	−	0.951294i	$-0.599756\pi$
$967$	−19.1733	−0.616573	−0.308287	+	0.951294i	$0.599756\pi$
$968$	0	0
$969$	−5.86033	−0.188261
$970$	0	0
$971$	16.4329	0.527358	0.263679	−	0.964611i	$-0.415064\pi$
$971$	16.4329	0.527358	0.263679	+	0.964611i	$0.415064\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.21961	−0.0390586
$976$	0	0
$977$	−17.5157	−0.560377	−0.280189	−	0.959945i	$-0.590397\pi$
$977$	−17.5157	−0.560377	−0.280189	+	0.959945i	$0.590397\pi$
$978$	0	0
$979$	−0.116868	−0.00373512
$980$	0	0
$981$	−16.6888	−0.532832
$982$	0	0
$983$	20.5810	0.656432	0.328216	−	0.944603i	$-0.393553\pi$
$983$	20.5810	0.656432	0.328216	+	0.944603i	$0.393553\pi$
$984$	0	0
$985$	−29.6361	−0.944286
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−85.6430	−2.72329
$990$	0	0
$991$	−13.5011	−0.428878	−0.214439	−	0.976737i	$-0.568792\pi$
$991$	−13.5011	−0.428878	−0.214439	+	0.976737i	$0.568792\pi$
$992$	0	0
$993$	9.82856	0.311900
$994$	0	0
$995$	11.5132	0.364992
$996$	0	0
$997$	15.7687	0.499401	0.249700	−	0.968323i	$-0.419668\pi$
$997$	15.7687	0.499401	0.249700	+	0.968323i	$0.419668\pi$
$998$	0	0
$999$	−1.70428	−0.0539209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9996.2.a.z.1.2		4
7.3	odd	6		1428.2.q.e.205.2	✓	8
7.5	odd	6		1428.2.q.e.613.2	yes	8
7.6	odd	2		9996.2.a.bd.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.q.e.205.2	✓	8	7.3	odd	6
1428.2.q.e.613.2	yes	8	7.5	odd	6
9996.2.a.z.1.2		4	1.1	even	1	trivial
9996.2.a.bd.1.3		4	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9996 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9996.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.29105 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.29105 q^{5} +1.00000 q^{9} +1.47358 q^{11} +4.89962 q^{13} +2.29105 q^{15} +1.00000 q^{17} +5.86033 q^{19} -7.83387 q^{23} +0.248918 q^{25} -1.00000 q^{27} +3.08215 q^{29} +9.62964 q^{31} -1.47358 q^{33} +1.70428 q^{37} -4.89962 q^{39} -7.79458 q^{41} +10.9324 q^{43} -2.29105 q^{45} -6.69887 q^{47} -1.00000 q^{51} +10.7017 q^{53} -3.37605 q^{55} -5.86033 q^{57} +5.62680 q^{59} +4.29390 q^{61} -11.2253 q^{65} -4.96823 q^{67} +7.83387 q^{69} -2.74357 q^{71} +8.90714 q^{73} -0.248918 q^{75} -4.00888 q^{79} +1.00000 q^{81} +4.57743 q^{83} -2.29105 q^{85} -3.08215 q^{87} -0.0793088 q^{89} -9.62964 q^{93} -13.4263 q^{95} +4.56388 q^{97} +1.47358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} + 4 q^{17} + 9 q^{19} - 5 q^{23} + 18 q^{25} - 4 q^{27} + 5 q^{29} + 24 q^{31} - q^{33} + 2 q^{37} - 10 q^{39} - 14 q^{43} - 2 q^{45} + 2 q^{47} - 4 q^{51} - 15 q^{53} + 30 q^{55} - 9 q^{57} + 37 q^{59} - 19 q^{61} - 21 q^{65} + 2 q^{67} + 5 q^{69} - 11 q^{71} + 9 q^{73} - 18 q^{75} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 2 q^{85} - 5 q^{87} - 22 q^{89} - 24 q^{93} + 33 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9 + q^11 + 10 * q^13 + 2 * q^15 + 4 * q^17 + 9 * q^19 - 5 * q^23 + 18 * q^25 - 4 * q^27 + 5 * q^29 + 24 * q^31 - q^33 + 2 * q^37 - 10 * q^39 - 14 * q^43 - 2 * q^45 + 2 * q^47 - 4 * q^51 - 15 * q^53 + 30 * q^55 - 9 * q^57 + 37 * q^59 - 19 * q^61 - 21 * q^65 + 2 * q^67 + 5 * q^69 - 11 * q^71 + 9 * q^73 - 18 * q^75 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 2 * q^85 - 5 * q^87 - 22 * q^89 - 24 * q^93 + 33 * q^95 - 13 * q^97 + q^99

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