LMFDB - Embedded newform 4840.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} +1.00000 q^{5} +2.41421 q^{7} +2.82843 q^{9} +2.00000 q^{13} +2.41421 q^{15} +5.65685 q^{17} +4.82843 q^{19} +5.82843 q^{21} +3.65685 q^{23} +1.00000 q^{25} -0.414214 q^{27} +2.00000 q^{29} -5.65685 q^{31} +2.41421 q^{35} -4.00000 q^{37} +4.82843 q^{39} -9.48528 q^{41} +3.58579 q^{43} +2.82843 q^{45} +7.58579 q^{47} -1.17157 q^{49} +13.6569 q^{51} -7.65685 q^{53} +11.6569 q^{57} -11.3137 q^{59} +1.00000 q^{61} +6.82843 q^{63} +2.00000 q^{65} -6.41421 q^{67} +8.82843 q^{69} -4.00000 q^{71} +4.00000 q^{73} +2.41421 q^{75} -14.4853 q^{79} -9.48528 q^{81} -13.3137 q^{83} +5.65685 q^{85} +4.82843 q^{87} +2.65685 q^{89} +4.82843 q^{91} -13.6569 q^{93} +4.82843 q^{95} -17.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{13} + 2 q^{15} + 4 q^{19} + 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{35} - 8 q^{37} + 4 q^{39} - 2 q^{41} + 10 q^{43} + 18 q^{47} - 8 q^{49} + 16 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^13 + 2 * q^15 + 4 * q^19 + 6 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^35 - 8 * q^37 + 4 * q^39 - 2 * q^41 + 10 * q^43 + 18 * q^47 - 8 * q^49 + 16 * q^51 - 4 * q^53 + 12 * q^57 + 2 * q^61 + 8 * q^63 + 4 * q^65 - 10 * q^67 + 12 * q^69 - 8 * q^71 + 8 * q^73 + 2 * q^75 - 12 * q^79 - 2 * q^81 - 4 * q^83 + 4 * q^87 - 6 * q^89 + 4 * q^91 - 16 * q^93 + 4 * q^95 - 12 * 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 $

$2$	0	0
$2$	0	0
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.41421	0.912487	0.456243	−	0.889855i	$-0.349195\pi$
$7$	2.41421	0.912487	0.456243	+	0.889855i	$0.349195\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.41421	0.623347
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	4.82843	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$19$	4.82843	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$20$	0	0
$21$	5.82843	1.27187
$22$	0	0
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.41421	0.408077
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	−9.48528	−1.48135	−0.740676	−	0.671862i	$-0.765495\pi$
$41$	−9.48528	−1.48135	−0.740676	+	0.671862i	$0.765495\pi$
$42$	0	0
$43$	3.58579	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$43$	3.58579	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	7.58579	1.10650	0.553250	−	0.833015i	$-0.313387\pi$
$47$	7.58579	1.10650	0.553250	+	0.833015i	$0.313387\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	13.6569	1.91234
$52$	0	0
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.6569	1.54399
$58$	0	0
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	6.82843	0.860301
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−6.41421	−0.783621	−0.391810	−	0.920046i	$-0.628151\pi$
$67$	−6.41421	−0.783621	−0.391810	+	0.920046i	$0.628151\pi$
$68$	0	0
$69$	8.82843	1.06282
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	2.41421	0.278769
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−13.3137	−1.46137	−0.730685	−	0.682715i	$-0.760799\pi$
$83$	−13.3137	−1.46137	−0.730685	+	0.682715i	$0.760799\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	4.82843	0.517662
$88$	0	0
$89$	2.65685	0.281626	0.140813	−	0.990036i	$-0.455028\pi$
$89$	2.65685	0.281626	0.140813	+	0.990036i	$0.455028\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	0	0
$93$	−13.6569	−1.41615
$94$	0	0
$95$	4.82843	0.495386
$96$	0	0
$97$	−17.3137	−1.75794	−0.878970	−	0.476876i	$-0.841769\pi$
$97$	−17.3137	−1.75794	−0.878970	+	0.476876i	$0.841769\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.1716	1.01211	0.506055	−	0.862501i	$-0.331103\pi$
$101$	10.1716	1.01211	0.506055	+	0.862501i	$0.331103\pi$
$102$	0	0
$103$	13.3137	1.31184	0.655919	−	0.754831i	$-0.272281\pi$
$103$	13.3137	1.31184	0.655919	+	0.754831i	$0.272281\pi$
$104$	0	0
$105$	5.82843	0.568796
$106$	0	0
$107$	15.2426	1.47356	0.736781	−	0.676132i	$-0.236345\pi$
$107$	15.2426	1.47356	0.736781	+	0.676132i	$0.236345\pi$
$108$	0	0
$109$	16.6569	1.59544	0.797719	−	0.603030i	$-0.206040\pi$
$109$	16.6569	1.59544	0.797719	+	0.603030i	$0.206040\pi$
$110$	0	0
$111$	−9.65685	−0.916588
$112$	0	0
$113$	17.3137	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$114$	0	0
$115$	3.65685	0.341003
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−22.8995	−2.06478
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.41421	−0.391698	−0.195849	−	0.980634i	$-0.562746\pi$
$127$	−4.41421	−0.391698	−0.195849	+	0.980634i	$0.562746\pi$
$128$	0	0
$129$	8.65685	0.762194
$130$	0	0
$131$	−2.48528	−0.217140	−0.108570	−	0.994089i	$-0.534627\pi$
$131$	−2.48528	−0.217140	−0.108570	+	0.994089i	$0.534627\pi$
$132$	0	0
$133$	11.6569	1.01078
$134$	0	0
$135$	−0.414214	−0.0356498
$136$	0	0
$137$	13.3137	1.13747	0.568733	−	0.822522i	$-0.307434\pi$
$137$	13.3137	1.13747	0.568733	+	0.822522i	$0.307434\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	18.3137	1.54229
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−2.82843	−0.233285
$148$	0	0
$149$	−17.0000	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$150$	0	0
$151$	5.65685	0.460348	0.230174	−	0.973149i	$-0.426070\pi$
$151$	5.65685	0.460348	0.230174	+	0.973149i	$0.426070\pi$
$152$	0	0
$153$	16.0000	1.29352
$154$	0	0
$155$	−5.65685	−0.454369
$156$	0	0
$157$	17.6569	1.40917	0.704585	−	0.709619i	$-0.251133\pi$
$157$	17.6569	1.40917	0.704585	+	0.709619i	$0.251133\pi$
$158$	0	0
$159$	−18.4853	−1.46598
$160$	0	0
$161$	8.82843	0.695778
$162$	0	0
$163$	7.58579	0.594165	0.297082	−	0.954852i	$-0.403986\pi$
$163$	7.58579	0.594165	0.297082	+	0.954852i	$0.403986\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.41421	0.496347	0.248173	−	0.968716i	$-0.420170\pi$
$167$	6.41421	0.496347	0.248173	+	0.968716i	$0.420170\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	13.6569	1.04437
$172$	0	0
$173$	−3.65685	−0.278025	−0.139013	−	0.990291i	$-0.544393\pi$
$173$	−3.65685	−0.278025	−0.139013	+	0.990291i	$0.544393\pi$
$174$	0	0
$175$	2.41421	0.182497
$176$	0	0
$177$	−27.3137	−2.05302
$178$	0	0
$179$	−17.7990	−1.33036	−0.665179	−	0.746684i	$-0.731645\pi$
$179$	−17.7990	−1.33036	−0.665179	+	0.746684i	$0.731645\pi$
$180$	0	0
$181$	−25.8284	−1.91981	−0.959906	−	0.280322i	$-0.909559\pi$
$181$	−25.8284	−1.91981	−0.959906	+	0.280322i	$0.909559\pi$
$182$	0	0
$183$	2.41421	0.178464
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	16.8284	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$191$	16.8284	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$192$	0	0
$193$	1.31371	0.0945628	0.0472814	−	0.998882i	$-0.484944\pi$
$193$	1.31371	0.0945628	0.0472814	+	0.998882i	$0.484944\pi$
$194$	0	0
$195$	4.82843	0.345771
$196$	0	0
$197$	−6.97056	−0.496632	−0.248316	−	0.968679i	$-0.579877\pi$
$197$	−6.97056	−0.496632	−0.248316	+	0.968679i	$0.579877\pi$
$198$	0	0
$199$	−7.17157	−0.508379	−0.254190	−	0.967154i	$-0.581809\pi$
$199$	−7.17157	−0.508379	−0.254190	+	0.967154i	$0.581809\pi$
$200$	0	0
$201$	−15.4853	−1.09225
$202$	0	0
$203$	4.82843	0.338889
$204$	0	0
$205$	−9.48528	−0.662481
$206$	0	0
$207$	10.3431	0.718898
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	−9.65685	−0.661677
$214$	0	0
$215$	3.58579	0.244549
$216$	0	0
$217$	−13.6569	−0.927088
$218$	0	0
$219$	9.65685	0.652550
$220$	0	0
$221$	11.3137	0.761042
$222$	0	0
$223$	7.92893	0.530961	0.265480	−	0.964116i	$-0.414469\pi$
$223$	7.92893	0.530961	0.265480	+	0.964116i	$0.414469\pi$
$224$	0	0
$225$	2.82843	0.188562
$226$	0	0
$227$	26.0711	1.73040	0.865199	−	0.501429i	$-0.167192\pi$
$227$	26.0711	1.73040	0.865199	+	0.501429i	$0.167192\pi$
$228$	0	0
$229$	11.4853	0.758969	0.379484	−	0.925198i	$-0.376101\pi$
$229$	11.4853	0.758969	0.379484	+	0.925198i	$0.376101\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.34315	0.284529	0.142264	−	0.989829i	$-0.454562\pi$
$233$	4.34315	0.284529	0.142264	+	0.989829i	$0.454562\pi$
$234$	0	0
$235$	7.58579	0.494842
$236$	0	0
$237$	−34.9706	−2.27158
$238$	0	0
$239$	−24.1421	−1.56162	−0.780812	−	0.624765i	$-0.785195\pi$
$239$	−24.1421	−1.56162	−0.780812	+	0.624765i	$0.785195\pi$
$240$	0	0
$241$	16.1716	1.04170	0.520851	−	0.853647i	$-0.325615\pi$
$241$	16.1716	1.04170	0.520851	+	0.853647i	$0.325615\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	−1.17157	−0.0748490
$246$	0	0
$247$	9.65685	0.614451
$248$	0	0
$249$	−32.1421	−2.03693
$250$	0	0
$251$	0.686292	0.0433183	0.0216592	−	0.999765i	$-0.493105\pi$
$251$	0.686292	0.0433183	0.0216592	+	0.999765i	$0.493105\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	13.6569	0.855225
$256$	0	0
$257$	12.9706	0.809081	0.404541	−	0.914520i	$-0.367431\pi$
$257$	12.9706	0.809081	0.404541	+	0.914520i	$0.367431\pi$
$258$	0	0
$259$	−9.65685	−0.600048
$260$	0	0
$261$	5.65685	0.350150
$262$	0	0
$263$	24.6274	1.51859	0.759296	−	0.650746i	$-0.225544\pi$
$263$	24.6274	1.51859	0.759296	+	0.650746i	$0.225544\pi$
$264$	0	0
$265$	−7.65685	−0.470357
$266$	0	0
$267$	6.41421	0.392543
$268$	0	0
$269$	−16.3137	−0.994664	−0.497332	−	0.867560i	$-0.665687\pi$
$269$	−16.3137	−0.994664	−0.497332	+	0.867560i	$0.665687\pi$
$270$	0	0
$271$	−11.3137	−0.687259	−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137	−0.687259	−0.343629	+	0.939105i	$0.611656\pi$
$272$	0	0
$273$	11.6569	0.705505
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.6274	1.72005	0.860027	−	0.510248i	$-0.170446\pi$
$277$	28.6274	1.72005	0.860027	+	0.510248i	$0.170446\pi$
$278$	0	0
$279$	−16.0000	−0.957895
$280$	0	0
$281$	−5.31371	−0.316989	−0.158495	−	0.987360i	$-0.550664\pi$
$281$	−5.31371	−0.316989	−0.158495	+	0.987360i	$0.550664\pi$
$282$	0	0
$283$	−20.2132	−1.20155	−0.600775	−	0.799418i	$-0.705141\pi$
$283$	−20.2132	−1.20155	−0.600775	+	0.799418i	$0.705141\pi$
$284$	0	0
$285$	11.6569	0.690492
$286$	0	0
$287$	−22.8995	−1.35171
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	−41.7990	−2.45030
$292$	0	0
$293$	−19.3137	−1.12832	−0.564159	−	0.825666i	$-0.690800\pi$
$293$	−19.3137	−1.12832	−0.564159	+	0.825666i	$0.690800\pi$
$294$	0	0
$295$	−11.3137	−0.658710
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.31371	0.422963
$300$	0	0
$301$	8.65685	0.498973
$302$	0	0
$303$	24.5563	1.41073
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	−25.3137	−1.44473	−0.722365	−	0.691512i	$-0.756945\pi$
$307$	−25.3137	−1.44473	−0.722365	+	0.691512i	$0.756945\pi$
$308$	0	0
$309$	32.1421	1.82850
$310$	0	0
$311$	12.8284	0.727433	0.363717	−	0.931510i	$-0.381508\pi$
$311$	12.8284	0.727433	0.363717	+	0.931510i	$0.381508\pi$
$312$	0	0
$313$	−21.3137	−1.20472	−0.602361	−	0.798224i	$-0.705773\pi$
$313$	−21.3137	−1.20472	−0.602361	+	0.798224i	$0.705773\pi$
$314$	0	0
$315$	6.82843	0.384738
$316$	0	0
$317$	−2.68629	−0.150877	−0.0754386	−	0.997150i	$-0.524036\pi$
$317$	−2.68629	−0.150877	−0.0754386	+	0.997150i	$0.524036\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	36.7990	2.05392
$322$	0	0
$323$	27.3137	1.51978
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	40.2132	2.22380
$328$	0	0
$329$	18.3137	1.00967
$330$	0	0
$331$	4.82843	0.265394	0.132697	−	0.991157i	$-0.457636\pi$
$331$	4.82843	0.265394	0.132697	+	0.991157i	$0.457636\pi$
$332$	0	0
$333$	−11.3137	−0.619987
$334$	0	0
$335$	−6.41421	−0.350446
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	41.7990	2.27021
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.7279	−1.06521
$344$	0	0
$345$	8.82843	0.475307
$346$	0	0
$347$	1.38478	0.0743387	0.0371693	−	0.999309i	$-0.488166\pi$
$347$	1.38478	0.0743387	0.0371693	+	0.999309i	$0.488166\pi$
$348$	0	0
$349$	−5.31371	−0.284436	−0.142218	−	0.989835i	$-0.545423\pi$
$349$	−5.31371	−0.284436	−0.142218	+	0.989835i	$0.545423\pi$
$350$	0	0
$351$	−0.828427	−0.0442182
$352$	0	0
$353$	−34.6274	−1.84303	−0.921516	−	0.388341i	$-0.873048\pi$
$353$	−34.6274	−1.84303	−0.921516	+	0.388341i	$0.873048\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	32.9706	1.74499
$358$	0	0
$359$	−3.17157	−0.167389	−0.0836946	−	0.996491i	$-0.526672\pi$
$359$	−3.17157	−0.167389	−0.0836946	+	0.996491i	$0.526672\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	5.24264	0.273664	0.136832	−	0.990594i	$-0.456308\pi$
$367$	5.24264	0.273664	0.136832	+	0.990594i	$0.456308\pi$
$368$	0	0
$369$	−26.8284	−1.39663
$370$	0	0
$371$	−18.4853	−0.959708
$372$	0	0
$373$	−20.9706	−1.08581	−0.542907	−	0.839793i	$-0.682677\pi$
$373$	−20.9706	−1.08581	−0.542907	+	0.839793i	$0.682677\pi$
$374$	0	0
$375$	2.41421	0.124669
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−20.1421	−1.03463	−0.517316	−	0.855794i	$-0.673069\pi$
$379$	−20.1421	−1.03463	−0.517316	+	0.855794i	$0.673069\pi$
$380$	0	0
$381$	−10.6569	−0.545967
$382$	0	0
$383$	30.9706	1.58252	0.791261	−	0.611479i	$-0.209425\pi$
$383$	30.9706	1.58252	0.791261	+	0.611479i	$0.209425\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.1421	0.515554
$388$	0	0
$389$	26.3137	1.33416	0.667079	−	0.744987i	$-0.267544\pi$
$389$	26.3137	1.33416	0.667079	+	0.744987i	$0.267544\pi$
$390$	0	0
$391$	20.6863	1.04615
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	−14.4853	−0.728834
$396$	0	0
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	0	0
$399$	28.1421	1.40887
$400$	0	0
$401$	6.31371	0.315292	0.157646	−	0.987496i	$-0.449610\pi$
$401$	6.31371	0.315292	0.157646	+	0.987496i	$0.449610\pi$
$402$	0	0
$403$	−11.3137	−0.563576
$404$	0	0
$405$	−9.48528	−0.471327
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−39.7696	−1.96648	−0.983239	−	0.182322i	$-0.941639\pi$
$409$	−39.7696	−1.96648	−0.983239	+	0.182322i	$0.941639\pi$
$410$	0	0
$411$	32.1421	1.58545
$412$	0	0
$413$	−27.3137	−1.34402
$414$	0	0
$415$	−13.3137	−0.653544
$416$	0	0
$417$	28.9706	1.41869
$418$	0	0
$419$	−17.5147	−0.855650	−0.427825	−	0.903862i	$-0.640720\pi$
$419$	−17.5147	−0.855650	−0.427825	+	0.903862i	$0.640720\pi$
$420$	0	0
$421$	−7.68629	−0.374607	−0.187303	−	0.982302i	$-0.559975\pi$
$421$	−7.68629	−0.374607	−0.187303	+	0.982302i	$0.559975\pi$
$422$	0	0
$423$	21.4558	1.04322
$424$	0	0
$425$	5.65685	0.274398
$426$	0	0
$427$	2.41421	0.116832
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.5147	0.843654	0.421827	−	0.906676i	$-0.361389\pi$
$431$	17.5147	0.843654	0.421827	+	0.906676i	$0.361389\pi$
$432$	0	0
$433$	−28.9706	−1.39224	−0.696118	−	0.717927i	$-0.745091\pi$
$433$	−28.9706	−1.39224	−0.696118	+	0.717927i	$0.745091\pi$
$434$	0	0
$435$	4.82843	0.231505
$436$	0	0
$437$	17.6569	0.844642
$438$	0	0
$439$	−24.2843	−1.15903	−0.579513	−	0.814963i	$-0.696757\pi$
$439$	−24.2843	−1.15903	−0.579513	+	0.814963i	$0.696757\pi$
$440$	0	0
$441$	−3.31371	−0.157796
$442$	0	0
$443$	9.24264	0.439131	0.219566	−	0.975598i	$-0.429536\pi$
$443$	9.24264	0.439131	0.219566	+	0.975598i	$0.429536\pi$
$444$	0	0
$445$	2.65685	0.125947
$446$	0	0
$447$	−41.0416	−1.94120
$448$	0	0
$449$	−40.4558	−1.90923	−0.954615	−	0.297844i	$-0.903733\pi$
$449$	−40.4558	−1.90923	−0.954615	+	0.297844i	$0.903733\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.6569	0.641655
$454$	0	0
$455$	4.82843	0.226360
$456$	0	0
$457$	3.31371	0.155009	0.0775044	−	0.996992i	$-0.475305\pi$
$457$	3.31371	0.155009	0.0775044	+	0.996992i	$0.475305\pi$
$458$	0	0
$459$	−2.34315	−0.109369
$460$	0	0
$461$	−9.14214	−0.425792	−0.212896	−	0.977075i	$-0.568290\pi$
$461$	−9.14214	−0.425792	−0.212896	+	0.977075i	$0.568290\pi$
$462$	0	0
$463$	15.2426	0.708386	0.354193	−	0.935172i	$-0.384756\pi$
$463$	15.2426	0.708386	0.354193	+	0.935172i	$0.384756\pi$
$464$	0	0
$465$	−13.6569	−0.633321
$466$	0	0
$467$	42.5563	1.96927	0.984636	−	0.174617i	$-0.0558686\pi$
$467$	42.5563	1.96927	0.984636	+	0.174617i	$0.0558686\pi$
$468$	0	0
$469$	−15.4853	−0.715044
$470$	0	0
$471$	42.6274	1.96417
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.82843	0.221543
$476$	0	0
$477$	−21.6569	−0.991599
$478$	0	0
$479$	−1.79899	−0.0821979	−0.0410990	−	0.999155i	$-0.513086\pi$
$479$	−1.79899	−0.0821979	−0.0410990	+	0.999155i	$0.513086\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	21.3137	0.969807
$484$	0	0
$485$	−17.3137	−0.786175
$486$	0	0
$487$	18.6863	0.846757	0.423378	−	0.905953i	$-0.360844\pi$
$487$	18.6863	0.846757	0.423378	+	0.905953i	$0.360844\pi$
$488$	0	0
$489$	18.3137	0.828175
$490$	0	0
$491$	17.5147	0.790428	0.395214	−	0.918589i	$-0.370670\pi$
$491$	17.5147	0.790428	0.395214	+	0.918589i	$0.370670\pi$
$492$	0	0
$493$	11.3137	0.509544
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.65685	−0.433169
$498$	0	0
$499$	−37.7990	−1.69212	−0.846058	−	0.533092i	$-0.821030\pi$
$499$	−37.7990	−1.69212	−0.846058	+	0.533092i	$0.821030\pi$
$500$	0	0
$501$	15.4853	0.691831
$502$	0	0
$503$	9.38478	0.418446	0.209223	−	0.977868i	$-0.432906\pi$
$503$	9.38478	0.418446	0.209223	+	0.977868i	$0.432906\pi$
$504$	0	0
$505$	10.1716	0.452629
$506$	0	0
$507$	−21.7279	−0.964971
$508$	0	0
$509$	11.4853	0.509076	0.254538	−	0.967063i	$-0.418077\pi$
$509$	11.4853	0.509076	0.254538	+	0.967063i	$0.418077\pi$
$510$	0	0
$511$	9.65685	0.427194
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	13.3137	0.586672
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−8.82843	−0.387525
$520$	0	0
$521$	6.02944	0.264154	0.132077	−	0.991239i	$-0.457835\pi$
$521$	6.02944	0.264154	0.132077	+	0.991239i	$0.457835\pi$
$522$	0	0
$523$	−13.3137	−0.582168	−0.291084	−	0.956698i	$-0.594016\pi$
$523$	−13.3137	−0.582168	−0.291084	+	0.956698i	$0.594016\pi$
$524$	0	0
$525$	5.82843	0.254373
$526$	0	0
$527$	−32.0000	−1.39394
$528$	0	0
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−32.0000	−1.38868
$532$	0	0
$533$	−18.9706	−0.821706
$534$	0	0
$535$	15.2426	0.658997
$536$	0	0
$537$	−42.9706	−1.85432
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.14214	−0.221078	−0.110539	−	0.993872i	$-0.535258\pi$
$541$	−5.14214	−0.221078	−0.110539	+	0.993872i	$0.535258\pi$
$542$	0	0
$543$	−62.3553	−2.67592
$544$	0	0
$545$	16.6569	0.713501
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	0	0
$549$	2.82843	0.120714
$550$	0	0
$551$	9.65685	0.411396
$552$	0	0
$553$	−34.9706	−1.48710
$554$	0	0
$555$	−9.65685	−0.409911
$556$	0	0
$557$	22.6274	0.958754	0.479377	−	0.877609i	$-0.340863\pi$
$557$	22.6274	0.958754	0.479377	+	0.877609i	$0.340863\pi$
$558$	0	0
$559$	7.17157	0.303325
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.89949	0.122199	0.0610996	−	0.998132i	$-0.480539\pi$
$563$	2.89949	0.122199	0.0610996	+	0.998132i	$0.480539\pi$
$564$	0	0
$565$	17.3137	0.728393
$566$	0	0
$567$	−22.8995	−0.961688
$568$	0	0
$569$	−16.4558	−0.689865	−0.344932	−	0.938628i	$-0.612098\pi$
$569$	−16.4558	−0.689865	−0.344932	+	0.938628i	$0.612098\pi$
$570$	0	0
$571$	23.4558	0.981597	0.490798	−	0.871273i	$-0.336705\pi$
$571$	23.4558	0.981597	0.490798	+	0.871273i	$0.336705\pi$
$572$	0	0
$573$	40.6274	1.69723
$574$	0	0
$575$	3.65685	0.152501
$576$	0	0
$577$	−33.9411	−1.41299	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	−33.9411	−1.41299	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	3.17157	0.131806
$580$	0	0
$581$	−32.1421	−1.33348
$582$	0	0
$583$	0	0
$584$	0	0
$585$	5.65685	0.233882
$586$	0	0
$587$	36.6985	1.51471	0.757354	−	0.653004i	$-0.226492\pi$
$587$	36.6985	1.51471	0.757354	+	0.653004i	$0.226492\pi$
$588$	0	0
$589$	−27.3137	−1.12544
$590$	0	0
$591$	−16.8284	−0.692229
$592$	0	0
$593$	−23.6569	−0.971471	−0.485735	−	0.874106i	$-0.661448\pi$
$593$	−23.6569	−0.971471	−0.485735	+	0.874106i	$0.661448\pi$
$594$	0	0
$595$	13.6569	0.559876
$596$	0	0
$597$	−17.3137	−0.708603
$598$	0	0
$599$	15.8579	0.647935	0.323967	−	0.946068i	$-0.394983\pi$
$599$	15.8579	0.647935	0.323967	+	0.946068i	$0.394983\pi$
$600$	0	0
$601$	39.9411	1.62923	0.814616	−	0.580000i	$-0.196948\pi$
$601$	39.9411	1.62923	0.814616	+	0.580000i	$0.196948\pi$
$602$	0	0
$603$	−18.1421	−0.738805
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.5980	1.36370	0.681850	−	0.731492i	$-0.261176\pi$
$607$	33.5980	1.36370	0.681850	+	0.731492i	$0.261176\pi$
$608$	0	0
$609$	11.6569	0.472360
$610$	0	0
$611$	15.1716	0.613776
$612$	0	0
$613$	14.9706	0.604655	0.302328	−	0.953204i	$-0.402236\pi$
$613$	14.9706	0.604655	0.302328	+	0.953204i	$0.402236\pi$
$614$	0	0
$615$	−22.8995	−0.923397
$616$	0	0
$617$	−40.0000	−1.61034	−0.805170	−	0.593045i	$-0.797926\pi$
$617$	−40.0000	−1.61034	−0.805170	+	0.593045i	$0.797926\pi$
$618$	0	0
$619$	7.31371	0.293963	0.146981	−	0.989139i	$-0.453044\pi$
$619$	7.31371	0.293963	0.146981	+	0.989139i	$0.453044\pi$
$620$	0	0
$621$	−1.51472	−0.0607836
$622$	0	0
$623$	6.41421	0.256980
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.6274	−0.902214
$630$	0	0
$631$	34.3431	1.36718	0.683590	−	0.729867i	$-0.260418\pi$
$631$	34.3431	1.36718	0.683590	+	0.729867i	$0.260418\pi$
$632$	0	0
$633$	−19.3137	−0.767651
$634$	0	0
$635$	−4.41421	−0.175173
$636$	0	0
$637$	−2.34315	−0.0928388
$638$	0	0
$639$	−11.3137	−0.447563
$640$	0	0
$641$	−48.6274	−1.92067	−0.960334	−	0.278853i	$-0.910046\pi$
$641$	−48.6274	−1.92067	−0.960334	+	0.278853i	$0.910046\pi$
$642$	0	0
$643$	0.272078	0.0107297	0.00536485	−	0.999986i	$-0.498292\pi$
$643$	0.272078	0.0107297	0.00536485	+	0.999986i	$0.498292\pi$
$644$	0	0
$645$	8.65685	0.340863
$646$	0	0
$647$	−29.8701	−1.17431	−0.587157	−	0.809473i	$-0.699753\pi$
$647$	−29.8701	−1.17431	−0.587157	+	0.809473i	$0.699753\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−32.9706	−1.29222
$652$	0	0
$653$	−7.65685	−0.299636	−0.149818	−	0.988714i	$-0.547869\pi$
$653$	−7.65685	−0.299636	−0.149818	+	0.988714i	$0.547869\pi$
$654$	0	0
$655$	−2.48528	−0.0971080
$656$	0	0
$657$	11.3137	0.441390
$658$	0	0
$659$	−23.1716	−0.902636	−0.451318	−	0.892363i	$-0.649046\pi$
$659$	−23.1716	−0.902636	−0.451318	+	0.892363i	$0.649046\pi$
$660$	0	0
$661$	35.3431	1.37469	0.687345	−	0.726332i	$-0.258776\pi$
$661$	35.3431	1.37469	0.687345	+	0.726332i	$0.258776\pi$
$662$	0	0
$663$	27.3137	1.06078
$664$	0	0
$665$	11.6569	0.452033
$666$	0	0
$667$	7.31371	0.283188
$668$	0	0
$669$	19.1421	0.740078
$670$	0	0
$671$	0	0
$672$	0	0
$673$	16.9706	0.654167	0.327084	−	0.944995i	$-0.393934\pi$
$673$	16.9706	0.654167	0.327084	+	0.944995i	$0.393934\pi$
$674$	0	0
$675$	−0.414214	−0.0159431
$676$	0	0
$677$	−8.68629	−0.333841	−0.166921	−	0.985970i	$-0.553382\pi$
$677$	−8.68629	−0.333841	−0.166921	+	0.985970i	$0.553382\pi$
$678$	0	0
$679$	−41.7990	−1.60410
$680$	0	0
$681$	62.9411	2.41191
$682$	0	0
$683$	19.8701	0.760307	0.380153	−	0.924923i	$-0.375871\pi$
$683$	19.8701	0.760307	0.380153	+	0.924923i	$0.375871\pi$
$684$	0	0
$685$	13.3137	0.508691
$686$	0	0
$687$	27.7279	1.05789
$688$	0	0
$689$	−15.3137	−0.583406
$690$	0	0
$691$	−14.4853	−0.551046	−0.275523	−	0.961294i	$-0.588851\pi$
$691$	−14.4853	−0.551046	−0.275523	+	0.961294i	$0.588851\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	−53.6569	−2.03240
$698$	0	0
$699$	10.4853	0.396590
$700$	0	0
$701$	39.9411	1.50856	0.754278	−	0.656555i	$-0.227987\pi$
$701$	39.9411	1.50856	0.754278	+	0.656555i	$0.227987\pi$
$702$	0	0
$703$	−19.3137	−0.728430
$704$	0	0
$705$	18.3137	0.689734
$706$	0	0
$707$	24.5563	0.923537
$708$	0	0
$709$	−35.1421	−1.31979	−0.659895	−	0.751358i	$-0.729399\pi$
$709$	−35.1421	−1.31979	−0.659895	+	0.751358i	$0.729399\pi$
$710$	0	0
$711$	−40.9706	−1.53652
$712$	0	0
$713$	−20.6863	−0.774708
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−58.2843	−2.17667
$718$	0	0
$719$	−31.1716	−1.16250	−0.581252	−	0.813724i	$-0.697437\pi$
$719$	−31.1716	−1.16250	−0.581252	+	0.813724i	$0.697437\pi$
$720$	0	0
$721$	32.1421	1.19704
$722$	0	0
$723$	39.0416	1.45197
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	52.0711	1.93121	0.965605	−	0.260015i	$-0.0837276\pi$
$727$	52.0711	1.93121	0.965605	+	0.260015i	$0.0837276\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	20.2843	0.750241
$732$	0	0
$733$	40.2843	1.48793	0.743967	−	0.668217i	$-0.232942\pi$
$733$	40.2843	1.48793	0.743967	+	0.668217i	$0.232942\pi$
$734$	0	0
$735$	−2.82843	−0.104328
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−33.5147	−1.23286	−0.616429	−	0.787410i	$-0.711421\pi$
$739$	−33.5147	−1.23286	−0.616429	+	0.787410i	$0.711421\pi$
$740$	0	0
$741$	23.3137	0.856450
$742$	0	0
$743$	32.4142	1.18916	0.594581	−	0.804036i	$-0.297318\pi$
$743$	32.4142	1.18916	0.594581	+	0.804036i	$0.297318\pi$
$744$	0	0
$745$	−17.0000	−0.622832
$746$	0	0
$747$	−37.6569	−1.37779
$748$	0	0
$749$	36.7990	1.34461
$750$	0	0
$751$	4.82843	0.176192	0.0880959	−	0.996112i	$-0.471922\pi$
$751$	4.82843	0.176192	0.0880959	+	0.996112i	$0.471922\pi$
$752$	0	0
$753$	1.65685	0.0603791
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	−30.9706	−1.12564	−0.562822	−	0.826578i	$-0.690284\pi$
$757$	−30.9706	−1.12564	−0.562822	+	0.826578i	$0.690284\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	40.2132	1.45582
$764$	0	0
$765$	16.0000	0.578481
$766$	0	0
$767$	−22.6274	−0.817029
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	31.3137	1.12774
$772$	0	0
$773$	−28.9706	−1.04200	−0.520999	−	0.853557i	$-0.674441\pi$
$773$	−28.9706	−1.04200	−0.520999	+	0.853557i	$0.674441\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	−23.3137	−0.836375
$778$	0	0
$779$	−45.7990	−1.64092
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.828427	−0.0296056
$784$	0	0
$785$	17.6569	0.630200
$786$	0	0
$787$	0.213203	0.00759988	0.00379994	−	0.999993i	$-0.498790\pi$
$787$	0.213203	0.00759988	0.00379994	+	0.999993i	$0.498790\pi$
$788$	0	0
$789$	59.4558	2.11668
$790$	0	0
$791$	41.7990	1.48620
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−18.4853	−0.655605
$796$	0	0
$797$	34.3431	1.21650	0.608248	−	0.793747i	$-0.291872\pi$
$797$	34.3431	1.21650	0.608248	+	0.793747i	$0.291872\pi$
$798$	0	0
$799$	42.9117	1.51811
$800$	0	0
$801$	7.51472	0.265520
$802$	0	0
$803$	0	0
$804$	0	0
$805$	8.82843	0.311161
$806$	0	0
$807$	−39.3848	−1.38641
$808$	0	0
$809$	24.6274	0.865854	0.432927	−	0.901429i	$-0.357481\pi$
$809$	24.6274	0.865854	0.432927	+	0.901429i	$0.357481\pi$
$810$	0	0
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	−27.3137	−0.957934
$814$	0	0
$815$	7.58579	0.265719
$816$	0	0
$817$	17.3137	0.605730
$818$	0	0
$819$	13.6569	0.477209
$820$	0	0
$821$	−3.97056	−0.138574	−0.0692868	−	0.997597i	$-0.522072\pi$
$821$	−3.97056	−0.138574	−0.0692868	+	0.997597i	$0.522072\pi$
$822$	0	0
$823$	−1.92893	−0.0672383	−0.0336192	−	0.999435i	$-0.510703\pi$
$823$	−1.92893	−0.0672383	−0.0336192	+	0.999435i	$0.510703\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.3553	−1.12511	−0.562553	−	0.826761i	$-0.690181\pi$
$827$	−32.3553	−1.12511	−0.562553	+	0.826761i	$0.690181\pi$
$828$	0	0
$829$	32.9411	1.14409	0.572046	−	0.820221i	$-0.306150\pi$
$829$	32.9411	1.14409	0.572046	+	0.820221i	$0.306150\pi$
$830$	0	0
$831$	69.1127	2.39749
$832$	0	0
$833$	−6.62742	−0.229626
$834$	0	0
$835$	6.41421	0.221973
$836$	0	0
$837$	2.34315	0.0809910
$838$	0	0
$839$	41.7990	1.44306	0.721531	−	0.692382i	$-0.243439\pi$
$839$	41.7990	1.44306	0.721531	+	0.692382i	$0.243439\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−12.8284	−0.441834
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−48.7990	−1.67478
$850$	0	0
$851$	−14.6274	−0.501421
$852$	0	0
$853$	−29.6569	−1.01543	−0.507716	−	0.861525i	$-0.669510\pi$
$853$	−29.6569	−1.01543	−0.507716	+	0.861525i	$0.669510\pi$
$854$	0	0
$855$	13.6569	0.467055
$856$	0	0
$857$	−10.9706	−0.374747	−0.187374	−	0.982289i	$-0.559998\pi$
$857$	−10.9706	−0.374747	−0.187374	+	0.982289i	$0.559998\pi$
$858$	0	0
$859$	−7.85786	−0.268107	−0.134053	−	0.990974i	$-0.542799\pi$
$859$	−7.85786	−0.268107	−0.134053	+	0.990974i	$0.542799\pi$
$860$	0	0
$861$	−55.2843	−1.88408
$862$	0	0
$863$	−7.87006	−0.267900	−0.133950	−	0.990988i	$-0.542766\pi$
$863$	−7.87006	−0.267900	−0.133950	+	0.990988i	$0.542766\pi$
$864$	0	0
$865$	−3.65685	−0.124337
$866$	0	0
$867$	36.2132	1.22986
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.8284	−0.434675
$872$	0	0
$873$	−48.9706	−1.65740
$874$	0	0
$875$	2.41421	0.0816153
$876$	0	0
$877$	−47.3137	−1.59767	−0.798835	−	0.601550i	$-0.794550\pi$
$877$	−47.3137	−1.59767	−0.798835	+	0.601550i	$0.794550\pi$
$878$	0	0
$879$	−46.6274	−1.57270
$880$	0	0
$881$	2.51472	0.0847230	0.0423615	−	0.999102i	$-0.486512\pi$
$881$	2.51472	0.0847230	0.0423615	+	0.999102i	$0.486512\pi$
$882$	0	0
$883$	−32.3431	−1.08843	−0.544217	−	0.838945i	$-0.683173\pi$
$883$	−32.3431	−1.08843	−0.544217	+	0.838945i	$0.683173\pi$
$884$	0	0
$885$	−27.3137	−0.918140
$886$	0	0
$887$	−0.615224	−0.0206572	−0.0103286	−	0.999947i	$-0.503288\pi$
$887$	−0.615224	−0.0206572	−0.0103286	+	0.999947i	$0.503288\pi$
$888$	0	0
$889$	−10.6569	−0.357419
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.6274	1.22569
$894$	0	0
$895$	−17.7990	−0.594955
$896$	0	0
$897$	17.6569	0.589545
$898$	0	0
$899$	−11.3137	−0.377333
$900$	0	0
$901$	−43.3137	−1.44299
$902$	0	0
$903$	20.8995	0.695492
$904$	0	0
$905$	−25.8284	−0.858566
$906$	0	0
$907$	40.6985	1.35137	0.675686	−	0.737190i	$-0.263848\pi$
$907$	40.6985	1.35137	0.675686	+	0.737190i	$0.263848\pi$
$908$	0	0
$909$	28.7696	0.954226
$910$	0	0
$911$	−27.1716	−0.900234	−0.450117	−	0.892969i	$-0.648618\pi$
$911$	−27.1716	−0.900234	−0.450117	+	0.892969i	$0.648618\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.41421	0.0798114
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	32.4264	1.06965	0.534824	−	0.844963i	$-0.320378\pi$
$919$	32.4264	1.06965	0.534824	+	0.844963i	$0.320378\pi$
$920$	0	0
$921$	−61.1127	−2.01373
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	37.6569	1.23681
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	30.9706	1.01393
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	−51.4558	−1.67920
$940$	0	0
$941$	51.8284	1.68956	0.844779	−	0.535115i	$-0.179732\pi$
$941$	51.8284	1.68956	0.844779	+	0.535115i	$0.179732\pi$
$942$	0	0
$943$	−34.6863	−1.12954
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	−6.48528	−0.210300
$952$	0	0
$953$	24.2843	0.786645	0.393322	−	0.919401i	$-0.371326\pi$
$953$	24.2843	0.786645	0.393322	+	0.919401i	$0.371326\pi$
$954$	0	0
$955$	16.8284	0.544555
$956$	0	0
$957$	0	0
$958$	0	0
$959$	32.1421	1.03792
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	43.1127	1.38929
$964$	0	0
$965$	1.31371	0.0422898
$966$	0	0
$967$	41.5980	1.33770	0.668850	−	0.743397i	$-0.266787\pi$
$967$	41.5980	1.33770	0.668850	+	0.743397i	$0.266787\pi$
$968$	0	0
$969$	65.9411	2.11833
$970$	0	0
$971$	−3.71573	−0.119243	−0.0596217	−	0.998221i	$-0.518989\pi$
$971$	−3.71573	−0.119243	−0.0596217	+	0.998221i	$0.518989\pi$
$972$	0	0
$973$	28.9706	0.928754
$974$	0	0
$975$	4.82843	0.154633
$976$	0	0
$977$	2.62742	0.0840585	0.0420293	−	0.999116i	$-0.486618\pi$
$977$	2.62742	0.0840585	0.0420293	+	0.999116i	$0.486618\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	47.1127	1.50419
$982$	0	0
$983$	−5.92893	−0.189104	−0.0945518	−	0.995520i	$-0.530142\pi$
$983$	−5.92893	−0.189104	−0.0945518	+	0.995520i	$0.530142\pi$
$984$	0	0
$985$	−6.97056	−0.222101
$986$	0	0
$987$	44.2132	1.40732
$988$	0	0
$989$	13.1127	0.416960
$990$	0	0
$991$	58.4853	1.85785	0.928923	−	0.370273i	$-0.120736\pi$
$991$	58.4853	1.85785	0.928923	+	0.370273i	$0.120736\pi$
$992$	0	0
$993$	11.6569	0.369919
$994$	0	0
$995$	−7.17157	−0.227354
$996$	0	0
$997$	−29.3137	−0.928374	−0.464187	−	0.885737i	$-0.653654\pi$
$997$	−29.3137	−0.928374	−0.464187	+	0.885737i	$0.653654\pi$
$998$	0	0
$999$	1.65685	0.0524205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.p.1.2	yes	2
4.3	odd	2		9680.2.a.bj.1.1		2
11.10	odd	2		4840.2.a.o.1.2	✓	2
44.43	even	2		9680.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.o.1.2	✓	2	11.10	odd	2
4840.2.a.p.1.2	yes	2	1.1	even	1	trivial
9680.2.a.bj.1.1		2	4.3	odd	2
9680.2.a.bk.1.1		2	44.43	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +1.00000 q^{5} +2.41421 q^{7} +2.82843 q^{9} +2.00000 q^{13} +2.41421 q^{15} +5.65685 q^{17} +4.82843 q^{19} +5.82843 q^{21} +3.65685 q^{23} +1.00000 q^{25} -0.414214 q^{27} +2.00000 q^{29} -5.65685 q^{31} +2.41421 q^{35} -4.00000 q^{37} +4.82843 q^{39} -9.48528 q^{41} +3.58579 q^{43} +2.82843 q^{45} +7.58579 q^{47} -1.17157 q^{49} +13.6569 q^{51} -7.65685 q^{53} +11.6569 q^{57} -11.3137 q^{59} +1.00000 q^{61} +6.82843 q^{63} +2.00000 q^{65} -6.41421 q^{67} +8.82843 q^{69} -4.00000 q^{71} +4.00000 q^{73} +2.41421 q^{75} -14.4853 q^{79} -9.48528 q^{81} -13.3137 q^{83} +5.65685 q^{85} +4.82843 q^{87} +2.65685 q^{89} +4.82843 q^{91} -13.6569 q^{93} +4.82843 q^{95} -17.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{13} + 2 q^{15} + 4 q^{19} + 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{35} - 8 q^{37} + 4 q^{39} - 2 q^{41} + 10 q^{43} + 18 q^{47} - 8 q^{49} + 16 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^13 + 2 * q^15 + 4 * q^19 + 6 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^35 - 8 * q^37 + 4 * q^39 - 2 * q^41 + 10 * q^43 + 18 * q^47 - 8 * q^49 + 16 * q^51 - 4 * q^53 + 12 * q^57 + 2 * q^61 + 8 * q^63 + 4 * q^65 - 10 * q^67 + 12 * q^69 - 8 * q^71 + 8 * q^73 + 2 * q^75 - 12 * q^79 - 2 * q^81 - 4 * q^83 + 4 * q^87 - 6 * q^89 + 4 * q^91 - 16 * q^93 + 4 * q^95 - 12 * q^97

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