LMFDB - Embedded newform 8379.2.a.bd.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] 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("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-10,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2793)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -1.73205 q^{8} -3.46410 q^{11} +3.46410 q^{13} -5.00000 q^{16} +4.00000 q^{17} -1.00000 q^{19} -6.00000 q^{22} +7.46410 q^{23} -5.00000 q^{25} +6.00000 q^{26} +4.00000 q^{29} -10.9282 q^{31} -5.19615 q^{32} +6.92820 q^{34} -8.92820 q^{37} -1.73205 q^{38} +2.92820 q^{41} +2.92820 q^{43} -3.46410 q^{44} +12.9282 q^{46} -8.92820 q^{47} -8.66025 q^{50} +3.46410 q^{52} +4.00000 q^{53} +6.92820 q^{58} -8.00000 q^{59} -14.9282 q^{61} -18.9282 q^{62} +1.00000 q^{64} +14.3923 q^{67} +4.00000 q^{68} -3.46410 q^{71} +8.00000 q^{73} -15.4641 q^{74} -1.00000 q^{76} -3.46410 q^{79} +5.07180 q^{82} +4.92820 q^{83} +5.07180 q^{86} +6.00000 q^{88} -12.0000 q^{89} +7.46410 q^{92} -15.4641 q^{94} -12.5359 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 10 q^{16} + 8 q^{17} - 2 q^{19} - 12 q^{22} + 8 q^{23} - 10 q^{25} + 12 q^{26} + 8 q^{29} - 8 q^{31} - 4 q^{37} - 8 q^{41} - 8 q^{43} + 12 q^{46} - 4 q^{47} + 8 q^{53} - 16 q^{59} - 16 q^{61}+ \cdots - 32 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 10 * q^16 + 8 * q^17 - 2 * q^19 - 12 * q^22 + 8 * q^23 - 10 * q^25 + 12 * q^26 + 8 * q^29 - 8 * q^31 - 4 * q^37 - 8 * q^41 - 8 * q^43 + 12 * q^46 - 4 * q^47 + 8 * q^53 - 16 * q^59 - 16 * q^61 - 24 * q^62 + 2 * q^64 + 8 * q^67 + 8 * q^68 + 16 * q^73 - 24 * q^74 - 2 * q^76 + 24 * q^82 - 4 * q^83 + 24 * q^86 + 12 * q^88 - 24 * q^89 + 8 * q^92 - 24 * q^94 - 32 * q^97

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	7.46410	1.55637	0.778186	−	0.628033i	$-0.216140\pi$
$23$	7.46410	1.55637	0.778186	+	0.628033i	$0.216140\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	6.00000	1.17670
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−10.9282	−1.96276	−0.981382	−	0.192068i	$-0.938481\pi$
$31$	−10.9282	−1.96276	−0.981382	+	0.192068i	$0.938481\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	6.92820	1.18818
$35$	0	0
$36$	0	0
$37$	−8.92820	−1.46779	−0.733894	−	0.679264i	$-0.762299\pi$
$37$	−8.92820	−1.46779	−0.733894	+	0.679264i	$0.762299\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	0	0
$41$	2.92820	0.457309	0.228654	−	0.973508i	$-0.426567\pi$
$41$	2.92820	0.457309	0.228654	+	0.973508i	$0.426567\pi$
$42$	0	0
$43$	2.92820	0.446547	0.223273	−	0.974756i	$-0.428326\pi$
$43$	2.92820	0.446547	0.223273	+	0.974756i	$0.428326\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	12.9282	1.90616
$47$	−8.92820	−1.30231	−0.651156	−	0.758944i	$-0.725716\pi$
$47$	−8.92820	−1.30231	−0.651156	+	0.758944i	$0.725716\pi$
$48$	0	0
$49$	0	0
$50$	−8.66025	−1.22474
$51$	0	0
$52$	3.46410	0.480384
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	6.92820	0.909718
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−14.9282	−1.91136	−0.955680	−	0.294407i	$-0.904878\pi$
$61$	−14.9282	−1.91136	−0.955680	+	0.294407i	$0.904878\pi$
$62$	−18.9282	−2.40388
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	14.3923	1.75830	0.879150	−	0.476545i	$-0.158111\pi$
$67$	14.3923	1.75830	0.879150	+	0.476545i	$0.158111\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−15.4641	−1.79767
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	−3.46410	−0.389742	−0.194871	−	0.980829i	$-0.562429\pi$
$79$	−3.46410	−0.389742	−0.194871	+	0.980829i	$0.562429\pi$
$80$	0	0
$81$	0	0
$82$	5.07180	0.560086
$83$	4.92820	0.540941	0.270470	−	0.962728i	$-0.412821\pi$
$83$	4.92820	0.540941	0.270470	+	0.962728i	$0.412821\pi$
$84$	0	0
$85$	0	0
$86$	5.07180	0.546906
$87$	0	0
$88$	6.00000	0.639602
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	7.46410	0.778186
$93$	0	0
$94$	−15.4641	−1.59500
$95$	0	0
$96$	0	0
$97$	−12.5359	−1.27283	−0.636414	−	0.771348i	$-0.719583\pi$
$97$	−12.5359	−1.27283	−0.636414	+	0.771348i	$0.719583\pi$
$98$	0	0
$99$	0	0
$100$	−5.00000	−0.500000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−10.9282	−1.07679	−0.538394	−	0.842693i	$-0.680969\pi$
$103$	−10.9282	−1.07679	−0.538394	+	0.842693i	$0.680969\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	6.92820	0.672927
$107$	18.3923	1.77805	0.889026	−	0.457857i	$-0.151383\pi$
$107$	18.3923	1.77805	0.889026	+	0.457857i	$0.151383\pi$
$108$	0	0
$109$	−8.92820	−0.855167	−0.427583	−	0.903976i	$-0.640635\pi$
$109$	−8.92820	−0.855167	−0.427583	+	0.903976i	$0.640635\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.07180	0.477115	0.238557	−	0.971128i	$-0.423326\pi$
$113$	5.07180	0.477115	0.238557	+	0.971128i	$0.423326\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	−13.8564	−1.27559
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−25.8564	−2.34093
$123$	0	0
$124$	−10.9282	−0.981382
$125$	0	0
$126$	0	0
$127$	−12.5359	−1.11238	−0.556191	−	0.831055i	$-0.687738\pi$
$127$	−12.5359	−1.11238	−0.556191	+	0.831055i	$0.687738\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	8.92820	0.780061	0.390030	−	0.920802i	$-0.372465\pi$
$131$	8.92820	0.780061	0.390030	+	0.920802i	$0.372465\pi$
$132$	0	0
$133$	0	0
$134$	24.9282	2.15347
$135$	0	0
$136$	−6.92820	−0.594089
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−12.0000	−1.00349
$144$	0	0
$145$	0	0
$146$	13.8564	1.14676
$147$	0	0
$148$	−8.92820	−0.733894
$149$	−20.9282	−1.71451	−0.857253	−	0.514896i	$-0.827831\pi$
$149$	−20.9282	−1.71451	−0.857253	+	0.514896i	$0.827831\pi$
$150$	0	0
$151$	−10.3923	−0.845714	−0.422857	−	0.906196i	$-0.638973\pi$
$151$	−10.3923	−0.845714	−0.422857	+	0.906196i	$0.638973\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.92820	−0.229355	−0.114677	−	0.993403i	$-0.536583\pi$
$163$	−2.92820	−0.229355	−0.114677	+	0.993403i	$0.536583\pi$
$164$	2.92820	0.228654
$165$	0	0
$166$	8.53590	0.662514
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	2.92820	0.223273
$173$	−9.85641	−0.749369	−0.374684	−	0.927152i	$-0.622249\pi$
$173$	−9.85641	−0.749369	−0.374684	+	0.927152i	$0.622249\pi$
$174$	0	0
$175$	0	0
$176$	17.3205	1.30558
$177$	0	0
$178$	−20.7846	−1.55787
$179$	11.4641	0.856867	0.428434	−	0.903573i	$-0.359066\pi$
$179$	11.4641	0.856867	0.428434	+	0.903573i	$0.359066\pi$
$180$	0	0
$181$	25.3205	1.88206	0.941029	−	0.338325i	$-0.109860\pi$
$181$	25.3205	1.88206	0.941029	+	0.338325i	$0.109860\pi$
$182$	0	0
$183$	0	0
$184$	−12.9282	−0.953080
$185$	0	0
$186$	0	0
$187$	−13.8564	−1.01328
$188$	−8.92820	−0.651156
$189$	0	0
$190$	0	0
$191$	0.535898	0.0387762	0.0193881	−	0.999812i	$-0.493828\pi$
$191$	0.535898	0.0387762	0.0193881	+	0.999812i	$0.493828\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−21.7128	−1.55889
$195$	0	0
$196$	0	0
$197$	4.92820	0.351120	0.175560	−	0.984469i	$-0.443826\pi$
$197$	4.92820	0.351120	0.175560	+	0.984469i	$0.443826\pi$
$198$	0	0
$199$	−5.85641	−0.415150	−0.207575	−	0.978219i	$-0.566557\pi$
$199$	−5.85641	−0.415150	−0.207575	+	0.978219i	$0.566557\pi$
$200$	8.66025	0.612372
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−18.9282	−1.31879
$207$	0	0
$208$	−17.3205	−1.20096
$209$	3.46410	0.239617
$210$	0	0
$211$	−0.535898	−0.0368928	−0.0184464	−	0.999830i	$-0.505872\pi$
$211$	−0.535898	−0.0368928	−0.0184464	+	0.999830i	$0.505872\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	31.8564	2.17766
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−15.4641	−1.04736
$219$	0	0
$220$	0	0
$221$	13.8564	0.932083
$222$	0	0
$223$	−13.0718	−0.875352	−0.437676	−	0.899133i	$-0.644198\pi$
$223$	−13.0718	−0.875352	−0.437676	+	0.899133i	$0.644198\pi$
$224$	0	0
$225$	0	0
$226$	8.78461	0.584344
$227$	−25.8564	−1.71615	−0.858075	−	0.513524i	$-0.828340\pi$
$227$	−25.8564	−1.71615	−0.858075	+	0.513524i	$0.828340\pi$
$228$	0	0
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	0	0
$231$	0	0
$232$	−6.92820	−0.454859
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	0	0
$239$	15.4641	1.00029	0.500145	−	0.865942i	$-0.333280\pi$
$239$	15.4641	1.00029	0.500145	+	0.865942i	$0.333280\pi$
$240$	0	0
$241$	−3.46410	−0.223142	−0.111571	−	0.993756i	$-0.535588\pi$
$241$	−3.46410	−0.223142	−0.111571	+	0.993756i	$0.535588\pi$
$242$	1.73205	0.111340
$243$	0	0
$244$	−14.9282	−0.955680
$245$	0	0
$246$	0	0
$247$	−3.46410	−0.220416
$248$	18.9282	1.20194
$249$	0	0
$250$	0	0
$251$	−12.9282	−0.816021	−0.408010	−	0.912977i	$-0.633777\pi$
$251$	−12.9282	−0.816021	−0.408010	+	0.912977i	$0.633777\pi$
$252$	0	0
$253$	−25.8564	−1.62558
$254$	−21.7128	−1.36238
$255$	0	0
$256$	19.0000	1.18750
$257$	−9.85641	−0.614826	−0.307413	−	0.951576i	$-0.599463\pi$
$257$	−9.85641	−0.614826	−0.307413	+	0.951576i	$0.599463\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	15.4641	0.955375
$263$	7.46410	0.460256	0.230128	−	0.973160i	$-0.426085\pi$
$263$	7.46410	0.460256	0.230128	+	0.973160i	$0.426085\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	14.3923	0.879150
$269$	−10.9282	−0.666304	−0.333152	−	0.942873i	$-0.608112\pi$
$269$	−10.9282	−0.666304	−0.333152	+	0.942873i	$0.608112\pi$
$270$	0	0
$271$	25.8564	1.57066	0.785332	−	0.619074i	$-0.212492\pi$
$271$	25.8564	1.57066	0.785332	+	0.619074i	$0.212492\pi$
$272$	−20.0000	−1.21268
$273$	0	0
$274$	−24.2487	−1.46492
$275$	17.3205	1.04447
$276$	0	0
$277$	−25.7128	−1.54493	−0.772467	−	0.635055i	$-0.780977\pi$
$277$	−25.7128	−1.54493	−0.772467	+	0.635055i	$0.780977\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	−6.92820	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$281$	−6.92820	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	−20.7846	−1.22902
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	8.00000	0.468165
$293$	8.78461	0.513202	0.256601	−	0.966517i	$-0.417397\pi$
$293$	8.78461	0.513202	0.256601	+	0.966517i	$0.417397\pi$
$294$	0	0
$295$	0	0
$296$	15.4641	0.898833
$297$	0	0
$298$	−36.2487	−2.09983
$299$	25.8564	1.49531
$300$	0	0
$301$	0	0
$302$	−18.0000	−1.03578
$303$	0	0
$304$	5.00000	0.286770
$305$	0	0
$306$	0	0
$307$	25.8564	1.47570	0.737852	−	0.674963i	$-0.235840\pi$
$307$	25.8564	1.47570	0.737852	+	0.674963i	$0.235840\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.0718	−0.627824	−0.313912	−	0.949452i	$-0.601640\pi$
$311$	−11.0718	−0.627824	−0.313912	+	0.949452i	$0.601640\pi$
$312$	0	0
$313$	22.9282	1.29598	0.647989	−	0.761649i	$-0.275610\pi$
$313$	22.9282	1.29598	0.647989	+	0.761649i	$0.275610\pi$
$314$	−27.7128	−1.56392
$315$	0	0
$316$	−3.46410	−0.194871
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	−13.8564	−0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−17.3205	−0.960769
$326$	−5.07180	−0.280901
$327$	0	0
$328$	−5.07180	−0.280043
$329$	0	0
$330$	0	0
$331$	12.2487	0.673250	0.336625	−	0.941639i	$-0.390715\pi$
$331$	12.2487	0.673250	0.336625	+	0.941639i	$0.390715\pi$
$332$	4.92820	0.270470
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−1.73205	−0.0942111
$339$	0	0
$340$	0	0
$341$	37.8564	2.05004
$342$	0	0
$343$	0	0
$344$	−5.07180	−0.273453
$345$	0	0
$346$	−17.0718	−0.917785
$347$	27.4641	1.47435	0.737175	−	0.675702i	$-0.236159\pi$
$347$	27.4641	1.47435	0.737175	+	0.675702i	$0.236159\pi$
$348$	0	0
$349$	−9.07180	−0.485602	−0.242801	−	0.970076i	$-0.578066\pi$
$349$	−9.07180	−0.485602	−0.242801	+	0.970076i	$0.578066\pi$
$350$	0	0
$351$	0	0
$352$	18.0000	0.959403
$353$	−9.85641	−0.524604	−0.262302	−	0.964986i	$-0.584482\pi$
$353$	−9.85641	−0.524604	−0.262302	+	0.964986i	$0.584482\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	19.8564	1.04944
$359$	6.39230	0.337373	0.168686	−	0.985670i	$-0.446047\pi$
$359$	6.39230	0.337373	0.168686	+	0.985670i	$0.446047\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	43.8564	2.30504
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−37.3205	−1.94547
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.7846	−1.17974	−0.589871	−	0.807497i	$-0.700821\pi$
$373$	−22.7846	−1.17974	−0.589871	+	0.807497i	$0.700821\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	15.4641	0.797500
$377$	13.8564	0.713641
$378$	0	0
$379$	30.3923	1.56115	0.780574	−	0.625063i	$-0.214927\pi$
$379$	30.3923	1.56115	0.780574	+	0.625063i	$0.214927\pi$
$380$	0	0
$381$	0	0
$382$	0.928203	0.0474910
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	−17.3205	−0.881591
$387$	0	0
$388$	−12.5359	−0.636414
$389$	32.9282	1.66953	0.834763	−	0.550609i	$-0.185605\pi$
$389$	32.9282	1.66953	0.834763	+	0.550609i	$0.185605\pi$
$390$	0	0
$391$	29.8564	1.50990
$392$	0	0
$393$	0	0
$394$	8.53590	0.430032
$395$	0	0
$396$	0	0
$397$	−12.7846	−0.641641	−0.320821	−	0.947140i	$-0.603959\pi$
$397$	−12.7846	−0.641641	−0.320821	+	0.947140i	$0.603959\pi$
$398$	−10.1436	−0.508452
$399$	0	0
$400$	25.0000	1.25000
$401$	14.9282	0.745479	0.372739	−	0.927936i	$-0.378419\pi$
$401$	14.9282	0.745479	0.372739	+	0.927936i	$0.378419\pi$
$402$	0	0
$403$	−37.8564	−1.88576
$404$	0	0
$405$	0	0
$406$	0	0
$407$	30.9282	1.53305
$408$	0	0
$409$	−25.3205	−1.25202	−0.626009	−	0.779816i	$-0.715313\pi$
$409$	−25.3205	−1.25202	−0.626009	+	0.779816i	$0.715313\pi$
$410$	0	0
$411$	0	0
$412$	−10.9282	−0.538394
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−18.0000	−0.882523
$417$	0	0
$418$	6.00000	0.293470
$419$	32.6410	1.59462	0.797309	−	0.603571i	$-0.206256\pi$
$419$	32.6410	1.59462	0.797309	+	0.603571i	$0.206256\pi$
$420$	0	0
$421$	−8.92820	−0.435134	−0.217567	−	0.976045i	$-0.569812\pi$
$421$	−8.92820	−0.435134	−0.217567	+	0.976045i	$0.569812\pi$
$422$	−0.928203	−0.0451842
$423$	0	0
$424$	−6.92820	−0.336463
$425$	−20.0000	−0.970143
$426$	0	0
$427$	0	0
$428$	18.3923	0.889026
$429$	0	0
$430$	0	0
$431$	26.3923	1.27127	0.635636	−	0.771989i	$-0.280738\pi$
$431$	26.3923	1.27127	0.635636	+	0.771989i	$0.280738\pi$
$432$	0	0
$433$	9.32051	0.447915	0.223958	−	0.974599i	$-0.428102\pi$
$433$	9.32051	0.447915	0.223958	+	0.974599i	$0.428102\pi$
$434$	0	0
$435$	0	0
$436$	−8.92820	−0.427583
$437$	−7.46410	−0.357056
$438$	0	0
$439$	38.6410	1.84424	0.922118	−	0.386910i	$-0.126458\pi$
$439$	38.6410	1.84424	0.922118	+	0.386910i	$0.126458\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	33.3205	1.58311	0.791553	−	0.611101i	$-0.209273\pi$
$443$	33.3205	1.58311	0.791553	+	0.611101i	$0.209273\pi$
$444$	0	0
$445$	0	0
$446$	−22.6410	−1.07208
$447$	0	0
$448$	0	0
$449$	−17.0718	−0.805668	−0.402834	−	0.915273i	$-0.631975\pi$
$449$	−17.0718	−0.805668	−0.402834	+	0.915273i	$0.631975\pi$
$450$	0	0
$451$	−10.1436	−0.477643
$452$	5.07180	0.238557
$453$	0	0
$454$	−44.7846	−2.10185
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	−13.8564	−0.643962	−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564	−0.643962	−0.321981	+	0.946746i	$0.604349\pi$
$464$	−20.0000	−0.928477
$465$	0	0
$466$	−13.6077	−0.630364
$467$	8.92820	0.413148	0.206574	−	0.978431i	$-0.433769\pi$
$467$	8.92820	0.413148	0.206574	+	0.978431i	$0.433769\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	13.8564	0.637793
$473$	−10.1436	−0.466403
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	0	0
$478$	26.7846	1.22510
$479$	−36.9282	−1.68729	−0.843646	−	0.536899i	$-0.819596\pi$
$479$	−36.9282	−1.68729	−0.843646	+	0.536899i	$0.819596\pi$
$480$	0	0
$481$	−30.9282	−1.41020
$482$	−6.00000	−0.273293
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	2.39230	0.108406	0.0542028	−	0.998530i	$-0.482738\pi$
$487$	2.39230	0.108406	0.0542028	+	0.998530i	$0.482738\pi$
$488$	25.8564	1.17046
$489$	0	0
$490$	0	0
$491$	1.32051	0.0595937	0.0297968	−	0.999556i	$-0.490514\pi$
$491$	1.32051	0.0595937	0.0297968	+	0.999556i	$0.490514\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	−6.00000	−0.269953
$495$	0	0
$496$	54.6410	2.45345
$497$	0	0
$498$	0	0
$499$	−2.92820	−0.131084	−0.0655422	−	0.997850i	$-0.520878\pi$
$499$	−2.92820	−0.131084	−0.0655422	+	0.997850i	$0.520878\pi$
$500$	0	0
$501$	0	0
$502$	−22.3923	−0.999417
$503$	−36.6410	−1.63374	−0.816871	−	0.576820i	$-0.804293\pi$
$503$	−36.6410	−1.63374	−0.816871	+	0.576820i	$0.804293\pi$
$504$	0	0
$505$	0	0
$506$	−44.7846	−1.99092
$507$	0	0
$508$	−12.5359	−0.556191
$509$	−25.8564	−1.14607	−0.573033	−	0.819533i	$-0.694233\pi$
$509$	−25.8564	−1.14607	−0.573033	+	0.819533i	$0.694233\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	0	0
$514$	−17.0718	−0.753005
$515$	0	0
$516$	0	0
$517$	30.9282	1.36022
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.0718	−0.572686	−0.286343	−	0.958127i	$-0.592440\pi$
$521$	−13.0718	−0.572686	−0.286343	+	0.958127i	$0.592440\pi$
$522$	0	0
$523$	−6.14359	−0.268641	−0.134320	−	0.990938i	$-0.542885\pi$
$523$	−6.14359	−0.268641	−0.134320	+	0.990938i	$0.542885\pi$
$524$	8.92820	0.390030
$525$	0	0
$526$	12.9282	0.563696
$527$	−43.7128	−1.90416
$528$	0	0
$529$	32.7128	1.42230
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.1436	0.439368
$534$	0	0
$535$	0	0
$536$	−24.9282	−1.07673
$537$	0	0
$538$	−18.9282	−0.816053
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	44.7846	1.92366
$543$	0	0
$544$	−20.7846	−0.891133
$545$	0	0
$546$	0	0
$547$	0.535898	0.0229134	0.0114567	−	0.999934i	$-0.496353\pi$
$547$	0.535898	0.0229134	0.0114567	+	0.999934i	$0.496353\pi$
$548$	−14.0000	−0.598050
$549$	0	0
$550$	30.0000	1.27920
$551$	−4.00000	−0.170406
$552$	0	0
$553$	0	0
$554$	−44.5359	−1.89215
$555$	0	0
$556$	4.00000	0.169638
$557$	−16.9282	−0.717271	−0.358635	−	0.933478i	$-0.616758\pi$
$557$	−16.9282	−0.717271	−0.358635	+	0.933478i	$0.616758\pi$
$558$	0	0
$559$	10.1436	0.429028
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	9.85641	0.415398	0.207699	−	0.978193i	$-0.433403\pi$
$563$	9.85641	0.415398	0.207699	+	0.978193i	$0.433403\pi$
$564$	0	0
$565$	0	0
$566$	−13.8564	−0.582428
$567$	0	0
$568$	6.00000	0.251754
$569$	16.7846	0.703647	0.351824	−	0.936066i	$-0.385562\pi$
$569$	16.7846	0.703647	0.351824	+	0.936066i	$0.385562\pi$
$570$	0	0
$571$	9.85641	0.412478	0.206239	−	0.978502i	$-0.433878\pi$
$571$	9.85641	0.412478	0.206239	+	0.978502i	$0.433878\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	0	0
$575$	−37.3205	−1.55637
$576$	0	0
$577$	12.7846	0.532230	0.266115	−	0.963941i	$-0.414260\pi$
$577$	12.7846	0.532230	0.266115	+	0.963941i	$0.414260\pi$
$578$	−1.73205	−0.0720438
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.8564	−0.573874
$584$	−13.8564	−0.573382
$585$	0	0
$586$	15.2154	0.628542
$587$	24.9282	1.02890	0.514449	−	0.857521i	$-0.327997\pi$
$587$	24.9282	1.02890	0.514449	+	0.857521i	$0.327997\pi$
$588$	0	0
$589$	10.9282	0.450289
$590$	0	0
$591$	0	0
$592$	44.6410	1.83473
$593$	39.7128	1.63081	0.815405	−	0.578891i	$-0.196514\pi$
$593$	39.7128	1.63081	0.815405	+	0.578891i	$0.196514\pi$
$594$	0	0
$595$	0	0
$596$	−20.9282	−0.857253
$597$	0	0
$598$	44.7846	1.83138
$599$	13.6077	0.555995	0.277998	−	0.960582i	$-0.410329\pi$
$599$	13.6077	0.555995	0.277998	+	0.960582i	$0.410329\pi$
$600$	0	0
$601$	−18.3923	−0.750238	−0.375119	−	0.926977i	$-0.622398\pi$
$601$	−18.3923	−0.750238	−0.375119	+	0.926977i	$0.622398\pi$
$602$	0	0
$603$	0	0
$604$	−10.3923	−0.422857
$605$	0	0
$606$	0	0
$607$	16.7846	0.681266	0.340633	−	0.940196i	$-0.389359\pi$
$607$	16.7846	0.681266	0.340633	+	0.940196i	$0.389359\pi$
$608$	5.19615	0.210732
$609$	0	0
$610$	0	0
$611$	−30.9282	−1.25122
$612$	0	0
$613$	19.8564	0.801993	0.400996	−	0.916080i	$-0.368664\pi$
$613$	19.8564	0.801993	0.400996	+	0.916080i	$0.368664\pi$
$614$	44.7846	1.80736
$615$	0	0
$616$	0	0
$617$	4.14359	0.166815	0.0834074	−	0.996516i	$-0.473420\pi$
$617$	4.14359	0.166815	0.0834074	+	0.996516i	$0.473420\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	−19.1769	−0.768924
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	39.7128	1.58724
$627$	0	0
$628$	−16.0000	−0.638470
$629$	−35.7128	−1.42396
$630$	0	0
$631$	28.7846	1.14590	0.572949	−	0.819591i	$-0.305799\pi$
$631$	28.7846	1.14590	0.572949	+	0.819591i	$0.305799\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	−48.4974	−1.92608
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−24.0000	−0.950169
$639$	0	0
$640$	0	0
$641$	13.0718	0.516305	0.258152	−	0.966104i	$-0.416886\pi$
$641$	13.0718	0.516305	0.258152	+	0.966104i	$0.416886\pi$
$642$	0	0
$643$	−45.5692	−1.79707	−0.898537	−	0.438897i	$-0.855369\pi$
$643$	−45.5692	−1.79707	−0.898537	+	0.438897i	$0.855369\pi$
$644$	0	0
$645$	0	0
$646$	−6.92820	−0.272587
$647$	1.21539	0.0477819	0.0238910	−	0.999715i	$-0.492395\pi$
$647$	1.21539	0.0477819	0.0238910	+	0.999715i	$0.492395\pi$
$648$	0	0
$649$	27.7128	1.08782
$650$	−30.0000	−1.17670
$651$	0	0
$652$	−2.92820	−0.114677
$653$	36.6410	1.43387	0.716937	−	0.697138i	$-0.245544\pi$
$653$	36.6410	1.43387	0.716937	+	0.697138i	$0.245544\pi$
$654$	0	0
$655$	0	0
$656$	−14.6410	−0.571636
$657$	0	0
$658$	0	0
$659$	3.46410	0.134942	0.0674711	−	0.997721i	$-0.478507\pi$
$659$	3.46410	0.134942	0.0674711	+	0.997721i	$0.478507\pi$
$660$	0	0
$661$	−39.1769	−1.52381	−0.761903	−	0.647692i	$-0.775735\pi$
$661$	−39.1769	−1.52381	−0.761903	+	0.647692i	$0.775735\pi$
$662$	21.2154	0.824560
$663$	0	0
$664$	−8.53590	−0.331257
$665$	0	0
$666$	0	0
$667$	29.8564	1.15604
$668$	−13.8564	−0.536120
$669$	0	0
$670$	0	0
$671$	51.7128	1.99635
$672$	0	0
$673$	39.8564	1.53635	0.768176	−	0.640239i	$-0.221165\pi$
$673$	39.8564	1.53635	0.768176	+	0.640239i	$0.221165\pi$
$674$	−10.3923	−0.400297
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	65.5692	2.51078
$683$	40.2487	1.54007	0.770037	−	0.637999i	$-0.220238\pi$
$683$	40.2487	1.54007	0.770037	+	0.637999i	$0.220238\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−14.6410	−0.558184
$689$	13.8564	0.527887
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−9.85641	−0.374684
$693$	0	0
$694$	47.5692	1.80570
$695$	0	0
$696$	0	0
$697$	11.7128	0.443654
$698$	−15.7128	−0.594739
$699$	0	0
$700$	0	0
$701$	−4.92820	−0.186136	−0.0930678	−	0.995660i	$-0.529667\pi$
$701$	−4.92820	−0.186136	−0.0930678	+	0.995660i	$0.529667\pi$
$702$	0	0
$703$	8.92820	0.336734
$704$	−3.46410	−0.130558
$705$	0	0
$706$	−17.0718	−0.642506
$707$	0	0
$708$	0	0
$709$	39.8564	1.49684	0.748419	−	0.663226i	$-0.230813\pi$
$709$	39.8564	1.49684	0.748419	+	0.663226i	$0.230813\pi$
$710$	0	0
$711$	0	0
$712$	20.7846	0.778936
$713$	−81.5692	−3.05479
$714$	0	0
$715$	0	0
$716$	11.4641	0.428434
$717$	0	0
$718$	11.0718	0.413196
$719$	28.6410	1.06813	0.534065	−	0.845444i	$-0.320664\pi$
$719$	28.6410	1.06813	0.534065	+	0.845444i	$0.320664\pi$
$720$	0	0
$721$	0	0
$722$	1.73205	0.0644603
$723$	0	0
$724$	25.3205	0.941029
$725$	−20.0000	−0.742781
$726$	0	0
$727$	7.71281	0.286052	0.143026	−	0.989719i	$-0.454317\pi$
$727$	7.71281	0.286052	0.143026	+	0.989719i	$0.454317\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.7128	0.433214
$732$	0	0
$733$	18.6410	0.688522	0.344261	−	0.938874i	$-0.388130\pi$
$733$	18.6410	0.688522	0.344261	+	0.938874i	$0.388130\pi$
$734$	−6.92820	−0.255725
$735$	0	0
$736$	−38.7846	−1.42962
$737$	−49.8564	−1.83648
$738$	0	0
$739$	1.85641	0.0682890	0.0341445	−	0.999417i	$-0.489129\pi$
$739$	1.85641	0.0682890	0.0341445	+	0.999417i	$0.489129\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.2487	−0.889599	−0.444799	−	0.895630i	$-0.646725\pi$
$743$	−24.2487	−0.889599	−0.444799	+	0.895630i	$0.646725\pi$
$744$	0	0
$745$	0	0
$746$	−39.4641	−1.44488
$747$	0	0
$748$	−13.8564	−0.506640
$749$	0	0
$750$	0	0
$751$	4.53590	0.165517	0.0827586	−	0.996570i	$-0.473627\pi$
$751$	4.53590	0.165517	0.0827586	+	0.996570i	$0.473627\pi$
$752$	44.6410	1.62789
$753$	0	0
$754$	24.0000	0.874028
$755$	0	0
$756$	0	0
$757$	−24.1436	−0.877514	−0.438757	−	0.898606i	$-0.644581\pi$
$757$	−24.1436	−0.877514	−0.438757	+	0.898606i	$0.644581\pi$
$758$	52.6410	1.91201
$759$	0	0
$760$	0	0
$761$	23.7128	0.859589	0.429794	−	0.902927i	$-0.358586\pi$
$761$	23.7128	0.859589	0.429794	+	0.902927i	$0.358586\pi$
$762$	0	0
$763$	0	0
$764$	0.535898	0.0193881
$765$	0	0
$766$	6.92820	0.250326
$767$	−27.7128	−1.00065
$768$	0	0
$769$	34.6410	1.24919	0.624593	−	0.780950i	$-0.285265\pi$
$769$	34.6410	1.24919	0.624593	+	0.780950i	$0.285265\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−47.7128	−1.71611	−0.858055	−	0.513557i	$-0.828327\pi$
$773$	−47.7128	−1.71611	−0.858055	+	0.513557i	$0.828327\pi$
$774$	0	0
$775$	54.6410	1.96276
$776$	21.7128	0.779445
$777$	0	0
$778$	57.0333	2.04474
$779$	−2.92820	−0.104914
$780$	0	0
$781$	12.0000	0.429394
$782$	51.7128	1.84925
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.8564	1.20685	0.603425	−	0.797420i	$-0.293802\pi$
$787$	33.8564	1.20685	0.603425	+	0.797420i	$0.293802\pi$
$788$	4.92820	0.175560
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−51.7128	−1.83638
$794$	−22.1436	−0.785847
$795$	0	0
$796$	−5.85641	−0.207575
$797$	16.7846	0.594541	0.297271	−	0.954793i	$-0.403924\pi$
$797$	16.7846	0.594541	0.297271	+	0.954793i	$0.403924\pi$
$798$	0	0
$799$	−35.7128	−1.26343
$800$	25.9808	0.918559
$801$	0	0
$802$	25.8564	0.913021
$803$	−27.7128	−0.977964
$804$	0	0
$805$	0	0
$806$	−65.5692	−2.30958
$807$	0	0
$808$	0	0
$809$	−17.7128	−0.622749	−0.311375	−	0.950287i	$-0.600789\pi$
$809$	−17.7128	−0.622749	−0.311375	+	0.950287i	$0.600789\pi$
$810$	0	0
$811$	−47.7128	−1.67542	−0.837712	−	0.546113i	$-0.816107\pi$
$811$	−47.7128	−1.67542	−0.837712	+	0.546113i	$0.816107\pi$
$812$	0	0
$813$	0	0
$814$	53.5692	1.87760
$815$	0	0
$816$	0	0
$817$	−2.92820	−0.102445
$818$	−43.8564	−1.53340
$819$	0	0
$820$	0	0
$821$	−38.7846	−1.35359	−0.676796	−	0.736171i	$-0.736632\pi$
$821$	−38.7846	−1.35359	−0.676796	+	0.736171i	$0.736632\pi$
$822$	0	0
$823$	−45.8564	−1.59845	−0.799227	−	0.601029i	$-0.794757\pi$
$823$	−45.8564	−1.59845	−0.799227	+	0.601029i	$0.794757\pi$
$824$	18.9282	0.659395
$825$	0	0
$826$	0	0
$827$	18.3923	0.639563	0.319782	−	0.947491i	$-0.396390\pi$
$827$	18.3923	0.639563	0.319782	+	0.947491i	$0.396390\pi$
$828$	0	0
$829$	16.2487	0.564341	0.282171	−	0.959364i	$-0.408946\pi$
$829$	16.2487	0.564341	0.282171	+	0.959364i	$0.408946\pi$
$830$	0	0
$831$	0	0
$832$	3.46410	0.120096
$833$	0	0
$834$	0	0
$835$	0	0
$836$	3.46410	0.119808
$837$	0	0
$838$	56.5359	1.95300
$839$	−6.14359	−0.212100	−0.106050	−	0.994361i	$-0.533820\pi$
$839$	−6.14359	−0.212100	−0.106050	+	0.994361i	$0.533820\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−15.4641	−0.532928
$843$	0	0
$844$	−0.535898	−0.0184464
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−20.0000	−0.686803
$849$	0	0
$850$	−34.6410	−1.18818
$851$	−66.6410	−2.28442
$852$	0	0
$853$	−19.7128	−0.674954	−0.337477	−	0.941334i	$-0.609573\pi$
$853$	−19.7128	−0.674954	−0.337477	+	0.941334i	$0.609573\pi$
$854$	0	0
$855$	0	0
$856$	−31.8564	−1.08883
$857$	1.85641	0.0634136	0.0317068	−	0.999497i	$-0.489906\pi$
$857$	1.85641	0.0634136	0.0317068	+	0.999497i	$0.489906\pi$
$858$	0	0
$859$	−26.1436	−0.892008	−0.446004	−	0.895031i	$-0.647153\pi$
$859$	−26.1436	−0.892008	−0.446004	+	0.895031i	$0.647153\pi$
$860$	0	0
$861$	0	0
$862$	45.7128	1.55698
$863$	26.3923	0.898405	0.449202	−	0.893430i	$-0.351708\pi$
$863$	26.3923	0.898405	0.449202	+	0.893430i	$0.351708\pi$
$864$	0	0
$865$	0	0
$866$	16.1436	0.548582
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	49.8564	1.68932
$872$	15.4641	0.523681
$873$	0	0
$874$	−12.9282	−0.437303
$875$	0	0
$876$	0	0
$877$	−17.2154	−0.581322	−0.290661	−	0.956826i	$-0.593875\pi$
$877$	−17.2154	−0.581322	−0.290661	+	0.956826i	$0.593875\pi$
$878$	66.9282	2.25872
$879$	0	0
$880$	0	0
$881$	−37.5692	−1.26574	−0.632870	−	0.774258i	$-0.718123\pi$
$881$	−37.5692	−1.26574	−0.632870	+	0.774258i	$0.718123\pi$
$882$	0	0
$883$	−16.7846	−0.564847	−0.282424	−	0.959290i	$-0.591138\pi$
$883$	−16.7846	−0.564847	−0.282424	+	0.959290i	$0.591138\pi$
$884$	13.8564	0.466041
$885$	0	0
$886$	57.7128	1.93890
$887$	−26.1436	−0.877816	−0.438908	−	0.898532i	$-0.644635\pi$
$887$	−26.1436	−0.877816	−0.438908	+	0.898532i	$0.644635\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−13.0718	−0.437676
$893$	8.92820	0.298771
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−29.5692	−0.986738
$899$	−43.7128	−1.45790
$900$	0	0
$901$	16.0000	0.533037
$902$	−17.5692	−0.584991
$903$	0	0
$904$	−8.78461	−0.292172
$905$	0	0
$906$	0	0
$907$	−45.3205	−1.50484	−0.752421	−	0.658682i	$-0.771114\pi$
$907$	−45.3205	−1.50484	−0.752421	+	0.658682i	$0.771114\pi$
$908$	−25.8564	−0.858075
$909$	0	0
$910$	0	0
$911$	7.17691	0.237782	0.118891	−	0.992907i	$-0.462066\pi$
$911$	7.17691	0.237782	0.118891	+	0.992907i	$0.462066\pi$
$912$	0	0
$913$	−17.0718	−0.564994
$914$	31.1769	1.03124
$915$	0	0
$916$	−6.92820	−0.228914
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	0	0
$922$	55.4256	1.82535
$923$	−12.0000	−0.394985
$924$	0	0
$925$	44.6410	1.46779
$926$	−24.0000	−0.788689
$927$	0	0
$928$	−20.7846	−0.682288
$929$	−44.0000	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$929$	−44.0000	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$930$	0	0
$931$	0	0
$932$	−7.85641	−0.257345
$933$	0	0
$934$	15.4641	0.506001
$935$	0	0
$936$	0	0
$937$	−28.7846	−0.940352	−0.470176	−	0.882573i	$-0.655810\pi$
$937$	−28.7846	−0.940352	−0.470176	+	0.882573i	$0.655810\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15.7128	0.512223	0.256112	−	0.966647i	$-0.417559\pi$
$941$	15.7128	0.512223	0.256112	+	0.966647i	$0.417559\pi$
$942$	0	0
$943$	21.8564	0.711743
$944$	40.0000	1.30189
$945$	0	0
$946$	−17.5692	−0.571225
$947$	6.67949	0.217054	0.108527	−	0.994093i	$-0.465387\pi$
$947$	6.67949	0.217054	0.108527	+	0.994093i	$0.465387\pi$
$948$	0	0
$949$	27.7128	0.899596
$950$	8.66025	0.280976
$951$	0	0
$952$	0	0
$953$	30.9282	1.00186	0.500931	−	0.865487i	$-0.332991\pi$
$953$	30.9282	1.00186	0.500931	+	0.865487i	$0.332991\pi$
$954$	0	0
$955$	0	0
$956$	15.4641	0.500145
$957$	0	0
$958$	−63.9615	−2.06650
$959$	0	0
$960$	0	0
$961$	88.4256	2.85244
$962$	−53.5692	−1.72714
$963$	0	0
$964$	−3.46410	−0.111571
$965$	0	0
$966$	0	0
$967$	28.7846	0.925651	0.462825	−	0.886450i	$-0.346836\pi$
$967$	28.7846	0.925651	0.462825	+	0.886450i	$0.346836\pi$
$968$	−1.73205	−0.0556702
$969$	0	0
$970$	0	0
$971$	−49.8564	−1.59997	−0.799984	−	0.600021i	$-0.795159\pi$
$971$	−49.8564	−1.59997	−0.799984	+	0.600021i	$0.795159\pi$
$972$	0	0
$973$	0	0
$974$	4.14359	0.132769
$975$	0	0
$976$	74.6410	2.38920
$977$	−10.6410	−0.340436	−0.170218	−	0.985406i	$-0.554447\pi$
$977$	−10.6410	−0.340436	−0.170218	+	0.985406i	$0.554447\pi$
$978$	0	0
$979$	41.5692	1.32856
$980$	0	0
$981$	0	0
$982$	2.28719	0.0729871
$983$	51.7128	1.64938	0.824691	−	0.565583i	$-0.191349\pi$
$983$	51.7128	1.64938	0.824691	+	0.565583i	$0.191349\pi$
$984$	0	0
$985$	0	0
$986$	27.7128	0.882556
$987$	0	0
$988$	−3.46410	−0.110208
$989$	21.8564	0.694993
$990$	0	0
$991$	37.0333	1.17640	0.588201	−	0.808715i	$-0.299836\pi$
$991$	37.0333	1.17640	0.588201	+	0.808715i	$0.299836\pi$
$992$	56.7846	1.80291
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29.8564	−0.945562	−0.472781	−	0.881180i	$-0.656750\pi$
$997$	−29.8564	−0.945562	−0.472781	+	0.881180i	$0.656750\pi$
$998$	−5.07180	−0.160545
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bd.1.2		2
3.2	odd	2		2793.2.a.r.1.1	yes	2
7.6	odd	2		8379.2.a.bc.1.2		2
21.20	even	2		2793.2.a.q.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.2.a.q.1.1	✓	2	21.20	even	2
2793.2.a.r.1.1	yes	2	3.2	odd	2
8379.2.a.bc.1.2		2	7.6	odd	2
8379.2.a.bd.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.73205 q^{8} -3.46410 q^{11} +3.46410 q^{13} -5.00000 q^{16} +4.00000 q^{17} -1.00000 q^{19} -6.00000 q^{22} +7.46410 q^{23} -5.00000 q^{25} +6.00000 q^{26} +4.00000 q^{29} -10.9282 q^{31} -5.19615 q^{32} +6.92820 q^{34} -8.92820 q^{37} -1.73205 q^{38} +2.92820 q^{41} +2.92820 q^{43} -3.46410 q^{44} +12.9282 q^{46} -8.92820 q^{47} -8.66025 q^{50} +3.46410 q^{52} +4.00000 q^{53} +6.92820 q^{58} -8.00000 q^{59} -14.9282 q^{61} -18.9282 q^{62} +1.00000 q^{64} +14.3923 q^{67} +4.00000 q^{68} -3.46410 q^{71} +8.00000 q^{73} -15.4641 q^{74} -1.00000 q^{76} -3.46410 q^{79} +5.07180 q^{82} +4.92820 q^{83} +5.07180 q^{86} +6.00000 q^{88} -12.0000 q^{89} +7.46410 q^{92} -15.4641 q^{94} -12.5359 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 10 q^{16} + 8 q^{17} - 2 q^{19} - 12 q^{22} + 8 q^{23} - 10 q^{25} + 12 q^{26} + 8 q^{29} - 8 q^{31} - 4 q^{37} - 8 q^{41} - 8 q^{43} + 12 q^{46} - 4 q^{47} + 8 q^{53} - 16 q^{59} - 16 q^{61}+ \cdots - 32 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 10 * q^16 + 8 * q^17 - 2 * q^19 - 12 * q^22 + 8 * q^23 - 10 * q^25 + 12 * q^26 + 8 * q^29 - 8 * q^31 - 4 * q^37 - 8 * q^41 - 8 * q^43 + 12 * q^46 - 4 * q^47 + 8 * q^53 - 16 * q^59 - 16 * q^61 - 24 * q^62 + 2 * q^64 + 8 * q^67 + 8 * q^68 + 16 * q^73 - 24 * q^74 - 2 * q^76 + 24 * q^82 - 4 * q^83 + 24 * q^86 + 12 * q^88 - 24 * q^89 + 8 * q^92 - 24 * q^94 - 32 * q^97

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