LMFDB - Embedded newform 9800.2.a.cv.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.239575536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 8x^{3} + 35x^{2} - 15x - 8 $ x^6 - x^5 - 12x^4 + 8x^3 + 35x^2 - 15x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.03837$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03837 q^{3} +1.15495 q^{9} +O(q^{10})$	$q+2.03837 q^{3} +1.15495 q^{9} +1.04807 q^{11} -1.15495 q^{13} -6.81986 q^{17} -4.75120 q^{19} +3.19331 q^{23} -3.76090 q^{27} +5.14130 q^{29} -5.57372 q^{31} +2.13636 q^{33} +4.91769 q^{37} -2.35420 q^{39} +9.68948 q^{41} +7.44559 q^{43} -9.29130 q^{47} -13.9014 q^{51} -5.60119 q^{53} -9.68469 q^{57} +10.8624 q^{59} -3.33443 q^{61} +5.45101 q^{67} +6.50915 q^{69} -4.10916 q^{71} -6.46879 q^{73} +0.612744 q^{79} -11.1309 q^{81} -0.275623 q^{83} +10.4799 q^{87} -17.2646 q^{89} -11.3613 q^{93} +7.59854 q^{97} +1.21047 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 7 q^{9}+O(q^{10}) $ 6 * q - q^3 + 7 * q^9	$ 6 q - q^{3} + 7 q^{9} + q^{11} - 7 q^{13} - 4 q^{17} - 5 q^{19} + 6 q^{23} - 7 q^{27} - 3 q^{29} - 2 q^{31} - 16 q^{33} + 9 q^{37} + 10 q^{39} + 6 q^{41} - 3 q^{43} - 27 q^{47} - 5 q^{53} - 26 q^{57} - 24 q^{59} + 9 q^{61} + 17 q^{67} + 15 q^{69} + 4 q^{71} - 18 q^{73} - 22 q^{79} - 6 q^{81} - 9 q^{83} - 39 q^{87} + 15 q^{89} + 10 q^{93} - 12 q^{97} + 11 q^{99}+O(q^{100}) $ 6 * q - q^3 + 7 * q^9 + q^11 - 7 * q^13 - 4 * q^17 - 5 * q^19 + 6 * q^23 - 7 * q^27 - 3 * q^29 - 2 * q^31 - 16 * q^33 + 9 * q^37 + 10 * q^39 + 6 * q^41 - 3 * q^43 - 27 * q^47 - 5 * q^53 - 26 * q^57 - 24 * q^59 + 9 * q^61 + 17 * q^67 + 15 * q^69 + 4 * q^71 - 18 * q^73 - 22 * q^79 - 6 * q^81 - 9 * q^83 - 39 * q^87 + 15 * q^89 + 10 * q^93 - 12 * q^97 + 11 * 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
$3$	2.03837	1.17685	0.588426	−	0.808551i	$-0.299748\pi$
$3$	2.03837	1.17685	0.588426	+	0.808551i	$0.299748\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.15495	0.384982
$10$	0	0
$11$	1.04807	0.316006	0.158003	−	0.987439i	$-0.449494\pi$
$11$	1.04807	0.316006	0.158003	+	0.987439i	$0.449494\pi$
$12$	0	0
$13$	−1.15495	−0.320324	−0.160162	−	0.987091i	$-0.551202\pi$
$13$	−1.15495	−0.320324	−0.160162	+	0.987091i	$0.551202\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.81986	−1.65406	−0.827030	−	0.562158i	$-0.809971\pi$
$17$	−6.81986	−1.65406	−0.827030	+	0.562158i	$0.809971\pi$
$18$	0	0
$19$	−4.75120	−1.09000	−0.545000	−	0.838436i	$-0.683470\pi$
$19$	−4.75120	−1.09000	−0.545000	+	0.838436i	$0.683470\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.19331	0.665852	0.332926	−	0.942953i	$-0.391964\pi$
$23$	3.19331	0.665852	0.332926	+	0.942953i	$0.391964\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.76090	−0.723786
$28$	0	0
$29$	5.14130	0.954716	0.477358	−	0.878709i	$-0.341595\pi$
$29$	5.14130	0.954716	0.477358	+	0.878709i	$0.341595\pi$
$30$	0	0
$31$	−5.57372	−1.00107	−0.500534	−	0.865717i	$-0.666863\pi$
$31$	−5.57372	−1.00107	−0.500534	+	0.865717i	$0.666863\pi$
$32$	0	0
$33$	2.13636	0.371892
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.91769	0.808463	0.404232	−	0.914657i	$-0.367539\pi$
$37$	4.91769	0.808463	0.404232	+	0.914657i	$0.367539\pi$
$38$	0	0
$39$	−2.35420	−0.376974
$40$	0	0
$41$	9.68948	1.51324	0.756621	−	0.653853i	$-0.226849\pi$
$41$	9.68948	1.51324	0.756621	+	0.653853i	$0.226849\pi$
$42$	0	0
$43$	7.44559	1.13544	0.567721	−	0.823221i	$-0.307825\pi$
$43$	7.44559	1.13544	0.567721	+	0.823221i	$0.307825\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.29130	−1.35528	−0.677638	−	0.735396i	$-0.736996\pi$
$47$	−9.29130	−1.35528	−0.677638	+	0.735396i	$0.736996\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−13.9014	−1.94658
$52$	0	0
$53$	−5.60119	−0.769383	−0.384692	−	0.923045i	$-0.625692\pi$
$53$	−5.60119	−0.769383	−0.384692	+	0.923045i	$0.625692\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.68469	−1.28277
$58$	0	0
$59$	10.8624	1.41416	0.707080	−	0.707134i	$-0.250012\pi$
$59$	10.8624	1.41416	0.707080	+	0.707134i	$0.250012\pi$
$60$	0	0
$61$	−3.33443	−0.426930	−0.213465	−	0.976951i	$-0.568475\pi$
$61$	−3.33443	−0.426930	−0.213465	+	0.976951i	$0.568475\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.45101	0.665947	0.332973	−	0.942936i	$-0.391948\pi$
$67$	5.45101	0.665947	0.332973	+	0.942936i	$0.391948\pi$
$68$	0	0
$69$	6.50915	0.783609
$70$	0	0
$71$	−4.10916	−0.487668	−0.243834	−	0.969817i	$-0.578405\pi$
$71$	−4.10916	−0.487668	−0.243834	+	0.969817i	$0.578405\pi$
$72$	0	0
$73$	−6.46879	−0.757114	−0.378557	−	0.925578i	$-0.623580\pi$
$73$	−6.46879	−0.757114	−0.378557	+	0.925578i	$0.623580\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.612744	0.0689391	0.0344695	−	0.999406i	$-0.489026\pi$
$79$	0.612744	0.0689391	0.0344695	+	0.999406i	$0.489026\pi$
$80$	0	0
$81$	−11.1309	−1.23677
$82$	0	0
$83$	−0.275623	−0.0302535	−0.0151268	−	0.999886i	$-0.504815\pi$
$83$	−0.275623	−0.0302535	−0.0151268	+	0.999886i	$0.504815\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.4799	1.12356
$88$	0	0
$89$	−17.2646	−1.83005	−0.915024	−	0.403399i	$-0.867829\pi$
$89$	−17.2646	−1.83005	−0.915024	+	0.403399i	$0.867829\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−11.3613	−1.17811
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.59854	0.771515	0.385757	−	0.922600i	$-0.373940\pi$
$97$	7.59854	0.771515	0.385757	+	0.922600i	$0.373940\pi$
$98$	0	0
$99$	1.21047	0.121657
$100$	0	0
$101$	−0.564671	−0.0561869	−0.0280934	−	0.999605i	$-0.508944\pi$
$101$	−0.564671	−0.0561869	−0.0280934	+	0.999605i	$0.508944\pi$
$102$	0	0
$103$	−15.5819	−1.53533	−0.767667	−	0.640848i	$-0.778583\pi$
$103$	−15.5819	−1.53533	−0.767667	+	0.640848i	$0.778583\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.0241	−1.35576	−0.677880	−	0.735172i	$-0.737101\pi$
$107$	−14.0241	−1.35576	−0.677880	+	0.735172i	$0.737101\pi$
$108$	0	0
$109$	−15.7978	−1.51315	−0.756577	−	0.653904i	$-0.773130\pi$
$109$	−15.7978	−1.51315	−0.756577	+	0.653904i	$0.773130\pi$
$110$	0	0
$111$	10.0241	0.951442
$112$	0	0
$113$	−20.3625	−1.91554	−0.957770	−	0.287536i	$-0.907164\pi$
$113$	−20.3625	−1.91554	−0.957770	+	0.287536i	$0.907164\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.33390	−0.123319
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.90154	−0.900140
$122$	0	0
$123$	19.7507	1.78086
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.358593	−0.0318199	−0.0159100	−	0.999873i	$-0.505065\pi$
$127$	−0.358593	−0.0318199	−0.0159100	+	0.999873i	$0.505065\pi$
$128$	0	0
$129$	15.1769	1.33625
$130$	0	0
$131$	−15.5114	−1.35524	−0.677620	−	0.735412i	$-0.736989\pi$
$131$	−15.5114	−1.35524	−0.677620	+	0.735412i	$0.736989\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.842399	−0.0719710	−0.0359855	−	0.999352i	$-0.511457\pi$
$137$	−0.842399	−0.0719710	−0.0359855	+	0.999352i	$0.511457\pi$
$138$	0	0
$139$	2.97876	0.252655	0.126327	−	0.991989i	$-0.459681\pi$
$139$	2.97876	0.252655	0.126327	+	0.991989i	$0.459681\pi$
$140$	0	0
$141$	−18.9391	−1.59496
$142$	0	0
$143$	−1.21047	−0.101224
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.10290	0.0903533	0.0451767	−	0.998979i	$-0.485615\pi$
$149$	1.10290	0.0903533	0.0451767	+	0.998979i	$0.485615\pi$
$150$	0	0
$151$	−12.5525	−1.02151	−0.510753	−	0.859728i	$-0.670633\pi$
$151$	−12.5525	−1.02151	−0.510753	+	0.859728i	$0.670633\pi$
$152$	0	0
$153$	−7.87657	−0.636783
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.0712	−1.68166	−0.840831	−	0.541298i	$-0.817933\pi$
$157$	−21.0712	−1.68166	−0.840831	+	0.541298i	$0.817933\pi$
$158$	0	0
$159$	−11.4173	−0.905451
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.8658	1.71266	0.856332	−	0.516425i	$-0.172738\pi$
$163$	21.8658	1.71266	0.856332	+	0.516425i	$0.172738\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.1680	−0.786821	−0.393411	−	0.919363i	$-0.628705\pi$
$167$	−10.1680	−0.786821	−0.393411	+	0.919363i	$0.628705\pi$
$168$	0	0
$169$	−11.6661	−0.897392
$170$	0	0
$171$	−5.48737	−0.419630
$172$	0	0
$173$	0.798446	0.0607048	0.0303524	−	0.999539i	$-0.490337\pi$
$173$	0.798446	0.0607048	0.0303524	+	0.999539i	$0.490337\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	22.1415	1.66426
$178$	0	0
$179$	−12.1755	−0.910042	−0.455021	−	0.890481i	$-0.650368\pi$
$179$	−12.1755	−0.910042	−0.455021	+	0.890481i	$0.650368\pi$
$180$	0	0
$181$	−4.00613	−0.297773	−0.148887	−	0.988854i	$-0.547569\pi$
$181$	−4.00613	−0.297773	−0.148887	+	0.988854i	$0.547569\pi$
$182$	0	0
$183$	−6.79680	−0.502434
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.14771	−0.522693
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.6616	1.20559	0.602797	−	0.797895i	$-0.294053\pi$
$191$	16.6616	1.20559	0.602797	+	0.797895i	$0.294053\pi$
$192$	0	0
$193$	−7.42501	−0.534464	−0.267232	−	0.963632i	$-0.586109\pi$
$193$	−7.42501	−0.534464	−0.267232	+	0.963632i	$0.586109\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.88883	−0.134574	−0.0672869	−	0.997734i	$-0.521434\pi$
$197$	−1.88883	−0.134574	−0.0672869	+	0.997734i	$0.521434\pi$
$198$	0	0
$199$	−17.2373	−1.22192	−0.610960	−	0.791662i	$-0.709216\pi$
$199$	−17.2373	−1.22192	−0.610960	+	0.791662i	$0.709216\pi$
$200$	0	0
$201$	11.1112	0.783721
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.68810	0.256341
$208$	0	0
$209$	−4.97960	−0.344446
$210$	0	0
$211$	−5.25478	−0.361754	−0.180877	−	0.983506i	$-0.557894\pi$
$211$	−5.25478	−0.361754	−0.180877	+	0.983506i	$0.557894\pi$
$212$	0	0
$213$	−8.37598	−0.573913
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.1858	−0.891012
$220$	0	0
$221$	7.87657	0.529835
$222$	0	0
$223$	13.2760	0.889027	0.444513	−	0.895772i	$-0.353377\pi$
$223$	13.2760	0.889027	0.444513	+	0.895772i	$0.353377\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.8039	−0.783455	−0.391727	−	0.920081i	$-0.628122\pi$
$227$	−11.8039	−0.783455	−0.391727	+	0.920081i	$0.628122\pi$
$228$	0	0
$229$	13.4922	0.891592	0.445796	−	0.895135i	$-0.352921\pi$
$229$	13.4922	0.891592	0.445796	+	0.895135i	$0.352921\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.14060	0.0747230	0.0373615	−	0.999302i	$-0.488105\pi$
$233$	1.14060	0.0747230	0.0373615	+	0.999302i	$0.488105\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.24900	0.0811311
$238$	0	0
$239$	2.94796	0.190688	0.0953438	−	0.995444i	$-0.469605\pi$
$239$	2.94796	0.190688	0.0953438	+	0.995444i	$0.469605\pi$
$240$	0	0
$241$	10.6131	0.683650	0.341825	−	0.939764i	$-0.388955\pi$
$241$	10.6131	0.683650	0.341825	+	0.939764i	$0.388955\pi$
$242$	0	0
$243$	−11.4062	−0.731711
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.48737	0.349153
$248$	0	0
$249$	−0.561821	−0.0356039
$250$	0	0
$251$	−12.3119	−0.777120	−0.388560	−	0.921423i	$-0.627027\pi$
$251$	−12.3119	−0.777120	−0.388560	+	0.921423i	$0.627027\pi$
$252$	0	0
$253$	3.34683	0.210413
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.7484	−1.10712	−0.553558	−	0.832811i	$-0.686730\pi$
$257$	−17.7484	−1.10712	−0.553558	+	0.832811i	$0.686730\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.93792	0.367548
$262$	0	0
$263$	−4.17570	−0.257485	−0.128742	−	0.991678i	$-0.541094\pi$
$263$	−4.17570	−0.257485	−0.128742	+	0.991678i	$0.541094\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−35.1917	−2.15370
$268$	0	0
$269$	26.2681	1.60160	0.800798	−	0.598935i	$-0.204409\pi$
$269$	26.2681	1.60160	0.800798	+	0.598935i	$0.204409\pi$
$270$	0	0
$271$	26.3074	1.59806	0.799031	−	0.601290i	$-0.205346\pi$
$271$	26.3074	1.59806	0.799031	+	0.601290i	$0.205346\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.9321	1.25769	0.628845	−	0.777530i	$-0.283528\pi$
$277$	20.9321	1.25769	0.628845	+	0.777530i	$0.283528\pi$
$278$	0	0
$279$	−6.43734	−0.385393
$280$	0	0
$281$	15.7097	0.937162	0.468581	−	0.883420i	$-0.344765\pi$
$281$	15.7097	0.937162	0.468581	+	0.883420i	$0.344765\pi$
$282$	0	0
$283$	10.8087	0.642512	0.321256	−	0.946992i	$-0.395895\pi$
$283$	10.8087	0.642512	0.321256	+	0.946992i	$0.395895\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	29.5105	1.73591
$290$	0	0
$291$	15.4886	0.907959
$292$	0	0
$293$	9.91887	0.579467	0.289733	−	0.957107i	$-0.406433\pi$
$293$	9.91887	0.579467	0.289733	+	0.957107i	$0.406433\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.94170	−0.228721
$298$	0	0
$299$	−3.68810	−0.213288
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.15101	−0.0661236
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.64632	−0.322252	−0.161126	−	0.986934i	$-0.551513\pi$
$307$	−5.64632	−0.322252	−0.161126	+	0.986934i	$0.551513\pi$
$308$	0	0
$309$	−31.7617	−1.80686
$310$	0	0
$311$	12.0190	0.681534	0.340767	−	0.940148i	$-0.389313\pi$
$311$	12.0190	0.681534	0.340767	+	0.940148i	$0.389313\pi$
$312$	0	0
$313$	18.9785	1.07273	0.536363	−	0.843987i	$-0.319798\pi$
$313$	18.9785	1.07273	0.536363	+	0.843987i	$0.319798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.7418	0.715651	0.357826	−	0.933788i	$-0.383518\pi$
$317$	12.7418	0.715651	0.357826	+	0.933788i	$0.383518\pi$
$318$	0	0
$319$	5.38846	0.301696
$320$	0	0
$321$	−28.5863	−1.59553
$322$	0	0
$323$	32.4025	1.80292
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−32.2017	−1.78076
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.8384	1.31027	0.655137	−	0.755510i	$-0.272611\pi$
$331$	23.8384	1.31027	0.655137	+	0.755510i	$0.272611\pi$
$332$	0	0
$333$	5.67966	0.311244
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−23.7427	−1.29335	−0.646673	−	0.762767i	$-0.723840\pi$
$337$	−23.7427	−1.29335	−0.646673	+	0.762767i	$0.723840\pi$
$338$	0	0
$339$	−41.5062	−2.25431
$340$	0	0
$341$	−5.84166	−0.316344
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.53548	−0.136112	−0.0680558	−	0.997682i	$-0.521680\pi$
$347$	−2.53548	−0.136112	−0.0680558	+	0.997682i	$0.521680\pi$
$348$	0	0
$349$	24.9321	1.33459	0.667293	−	0.744795i	$-0.267453\pi$
$349$	24.9321	1.33459	0.667293	+	0.744795i	$0.267453\pi$
$350$	0	0
$351$	4.34363	0.231846
$352$	0	0
$353$	2.35067	0.125113	0.0625567	−	0.998041i	$-0.480075\pi$
$353$	2.35067	0.125113	0.0625567	+	0.998041i	$0.480075\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.3016	−1.70481	−0.852407	−	0.522880i	$-0.824858\pi$
$359$	−32.3016	−1.70481	−0.852407	+	0.522880i	$0.824858\pi$
$360$	0	0
$361$	3.57387	0.188098
$362$	0	0
$363$	−20.1830	−1.05933
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.55523	−0.394380	−0.197190	−	0.980365i	$-0.563182\pi$
$367$	−7.55523	−0.394380	−0.197190	+	0.980365i	$0.563182\pi$
$368$	0	0
$369$	11.1908	0.582571
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−29.0647	−1.50491	−0.752457	−	0.658641i	$-0.771132\pi$
$373$	−29.0647	−1.50491	−0.752457	+	0.658641i	$0.771132\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.93792	−0.305819
$378$	0	0
$379$	−9.36318	−0.480954	−0.240477	−	0.970655i	$-0.577304\pi$
$379$	−9.36318	−0.480954	−0.240477	+	0.970655i	$0.577304\pi$
$380$	0	0
$381$	−0.730944	−0.0374474
$382$	0	0
$383$	1.93471	0.0988589	0.0494295	−	0.998778i	$-0.484260\pi$
$383$	1.93471	0.0988589	0.0494295	+	0.998778i	$0.484260\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.59925	0.437124
$388$	0	0
$389$	0.757390	0.0384012	0.0192006	−	0.999816i	$-0.493888\pi$
$389$	0.757390	0.0384012	0.0192006	+	0.999816i	$0.493888\pi$
$390$	0	0
$391$	−21.7780	−1.10136
$392$	0	0
$393$	−31.6180	−1.59492
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−33.4580	−1.67921	−0.839604	−	0.543199i	$-0.817213\pi$
$397$	−33.4580	−1.67921	−0.839604	+	0.543199i	$0.817213\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.5011	1.52315	0.761577	−	0.648074i	$-0.224425\pi$
$401$	30.5011	1.52315	0.761577	+	0.648074i	$0.224425\pi$
$402$	0	0
$403$	6.43734	0.320667
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.15410	0.255479
$408$	0	0
$409$	−18.6737	−0.923353	−0.461676	−	0.887048i	$-0.652752\pi$
$409$	−18.6737	−0.923353	−0.461676	+	0.887048i	$0.652752\pi$
$410$	0	0
$411$	−1.71712	−0.0846992
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.07180	0.297338
$418$	0	0
$419$	−34.8449	−1.70228	−0.851142	−	0.524936i	$-0.824089\pi$
$419$	−34.8449	−1.70228	−0.851142	+	0.524936i	$0.824089\pi$
$420$	0	0
$421$	5.53106	0.269567	0.134784	−	0.990875i	$-0.456966\pi$
$421$	5.53106	0.269567	0.134784	+	0.990875i	$0.456966\pi$
$422$	0	0
$423$	−10.7309	−0.521757
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.46738	−0.119126
$430$	0	0
$431$	10.5655	0.508922	0.254461	−	0.967083i	$-0.418102\pi$
$431$	10.5655	0.508922	0.254461	+	0.967083i	$0.418102\pi$
$432$	0	0
$433$	−14.3654	−0.690358	−0.345179	−	0.938537i	$-0.612182\pi$
$433$	−14.3654	−0.690358	−0.345179	+	0.938537i	$0.612182\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.1721	−0.725778
$438$	0	0
$439$	31.6168	1.50899	0.754493	−	0.656308i	$-0.227883\pi$
$439$	31.6168	1.50899	0.754493	+	0.656308i	$0.227883\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.9930	0.569804	0.284902	−	0.958557i	$-0.408039\pi$
$443$	11.9930	0.569804	0.284902	+	0.958557i	$0.408039\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.24812	0.106333
$448$	0	0
$449$	−4.98107	−0.235071	−0.117536	−	0.993069i	$-0.537499\pi$
$449$	−4.98107	−0.235071	−0.117536	+	0.993069i	$0.537499\pi$
$450$	0	0
$451$	10.1553	0.478194
$452$	0	0
$453$	−25.5866	−1.20216
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.9785	−1.26200	−0.631000	−	0.775783i	$-0.717355\pi$
$457$	−26.9785	−1.26200	−0.631000	+	0.775783i	$0.717355\pi$
$458$	0	0
$459$	25.6488	1.19718
$460$	0	0
$461$	22.6886	1.05672	0.528358	−	0.849022i	$-0.322808\pi$
$461$	22.6886	1.05672	0.528358	+	0.849022i	$0.322808\pi$
$462$	0	0
$463$	−27.0921	−1.25908	−0.629539	−	0.776969i	$-0.716756\pi$
$463$	−27.0921	−1.25908	−0.629539	+	0.776969i	$0.716756\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.55482	−0.210772	−0.105386	−	0.994431i	$-0.533608\pi$
$467$	−4.55482	−0.210772	−0.105386	+	0.994431i	$0.533608\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−42.9508	−1.97907
$472$	0	0
$473$	7.80352	0.358806
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.46907	−0.296199
$478$	0	0
$479$	8.22937	0.376010	0.188005	−	0.982168i	$-0.439798\pi$
$479$	8.22937	0.376010	0.188005	+	0.982168i	$0.439798\pi$
$480$	0	0
$481$	−5.67966	−0.258970
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.0063	0.861259	0.430629	−	0.902529i	$-0.358292\pi$
$487$	19.0063	0.861259	0.430629	+	0.902529i	$0.358292\pi$
$488$	0	0
$489$	44.5706	2.01555
$490$	0	0
$491$	19.2359	0.868102	0.434051	−	0.900888i	$-0.357084\pi$
$491$	19.2359	0.868102	0.434051	+	0.900888i	$0.357084\pi$
$492$	0	0
$493$	−35.0630	−1.57916
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.2105	0.546618	0.273309	−	0.961926i	$-0.411882\pi$
$499$	12.2105	0.546618	0.273309	+	0.961926i	$0.411882\pi$
$500$	0	0
$501$	−20.7261	−0.925972
$502$	0	0
$503$	2.17917	0.0971646	0.0485823	−	0.998819i	$-0.484530\pi$
$503$	2.17917	0.0971646	0.0485823	+	0.998819i	$0.484530\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.7798	−1.05610
$508$	0	0
$509$	12.4401	0.551398	0.275699	−	0.961244i	$-0.411091\pi$
$509$	12.4401	0.551398	0.275699	+	0.961244i	$0.411091\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	17.8688	0.788926
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.73797	−0.428275
$518$	0	0
$519$	1.62753	0.0714405
$520$	0	0
$521$	18.8545	0.826030	0.413015	−	0.910724i	$-0.364476\pi$
$521$	18.8545	0.826030	0.413015	+	0.910724i	$0.364476\pi$
$522$	0	0
$523$	−19.0063	−0.831088	−0.415544	−	0.909573i	$-0.636409\pi$
$523$	−19.0063	−0.831088	−0.415544	+	0.909573i	$0.636409\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.0120	1.65583
$528$	0	0
$529$	−12.8027	−0.556641
$530$	0	0
$531$	12.5454	0.544426
$532$	0	0
$533$	−11.1908	−0.484728
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−24.8182	−1.07099
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.3621	1.04741	0.523703	−	0.851901i	$-0.324550\pi$
$541$	24.3621	1.04741	0.523703	+	0.851901i	$0.324550\pi$
$542$	0	0
$543$	−8.16596	−0.350435
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.44667	0.147369	0.0736845	−	0.997282i	$-0.476524\pi$
$547$	3.44667	0.147369	0.0736845	+	0.997282i	$0.476524\pi$
$548$	0	0
$549$	−3.85109	−0.164360
$550$	0	0
$551$	−24.4273	−1.04064
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.5215	1.63221	0.816104	−	0.577905i	$-0.196130\pi$
$557$	38.5215	1.63221	0.816104	+	0.577905i	$0.196130\pi$
$558$	0	0
$559$	−8.59925	−0.363709
$560$	0	0
$561$	−14.5697	−0.615132
$562$	0	0
$563$	10.7673	0.453788	0.226894	−	0.973919i	$-0.427143\pi$
$563$	10.7673	0.453788	0.226894	+	0.973919i	$0.427143\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.3706	−0.979747	−0.489873	−	0.871794i	$-0.662957\pi$
$569$	−23.3706	−0.979747	−0.489873	+	0.871794i	$0.662957\pi$
$570$	0	0
$571$	−11.5645	−0.483960	−0.241980	−	0.970281i	$-0.577797\pi$
$571$	−11.5645	−0.483960	−0.241980	+	0.970281i	$0.577797\pi$
$572$	0	0
$573$	33.9625	1.41881
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.16239	0.0900213	0.0450107	−	0.998987i	$-0.485668\pi$
$577$	2.16239	0.0900213	0.0450107	+	0.998987i	$0.485668\pi$
$578$	0	0
$579$	−15.1349	−0.628985
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.87046	−0.243130
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.65367	−0.0682544	−0.0341272	−	0.999417i	$-0.510865\pi$
$587$	−1.65367	−0.0682544	−0.0341272	+	0.999417i	$0.510865\pi$
$588$	0	0
$589$	26.4818	1.09116
$590$	0	0
$591$	−3.85014	−0.158374
$592$	0	0
$593$	−25.5873	−1.05074	−0.525372	−	0.850873i	$-0.676074\pi$
$593$	−25.5873	−1.05074	−0.525372	+	0.850873i	$0.676074\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−35.1360	−1.43802
$598$	0	0
$599$	11.7773	0.481209	0.240604	−	0.970623i	$-0.422654\pi$
$599$	11.7773	0.481209	0.240604	+	0.970623i	$0.422654\pi$
$600$	0	0
$601$	18.5454	0.756484	0.378242	−	0.925707i	$-0.376529\pi$
$601$	18.5454	0.756484	0.378242	+	0.925707i	$0.376529\pi$
$602$	0	0
$603$	6.29562	0.256377
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.14662	0.290072	0.145036	−	0.989426i	$-0.453670\pi$
$607$	7.14662	0.290072	0.145036	+	0.989426i	$0.453670\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.7309	0.434128
$612$	0	0
$613$	20.5746	0.831002	0.415501	−	0.909593i	$-0.363606\pi$
$613$	20.5746	0.831002	0.415501	+	0.909593i	$0.363606\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−44.4857	−1.79093	−0.895464	−	0.445134i	$-0.853156\pi$
$617$	−44.4857	−1.79093	−0.895464	+	0.445134i	$0.853156\pi$
$618$	0	0
$619$	−28.8136	−1.15812	−0.579058	−	0.815286i	$-0.696580\pi$
$619$	−28.8136	−1.15812	−0.579058	+	0.815286i	$0.696580\pi$
$620$	0	0
$621$	−12.0097	−0.481934
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10.1503	−0.405362
$628$	0	0
$629$	−33.5380	−1.33725
$630$	0	0
$631$	48.6854	1.93813	0.969067	−	0.246796i	$-0.0793777\pi$
$631$	48.6854	1.93813	0.969067	+	0.246796i	$0.0793777\pi$
$632$	0	0
$633$	−10.7112	−0.425731
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.74585	−0.187743
$640$	0	0
$641$	36.2497	1.43178	0.715888	−	0.698216i	$-0.246022\pi$
$641$	36.2497	1.43178	0.715888	+	0.698216i	$0.246022\pi$
$642$	0	0
$643$	−17.4717	−0.689015	−0.344507	−	0.938784i	$-0.611954\pi$
$643$	−17.4717	−0.689015	−0.344507	+	0.938784i	$0.611954\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.67004	0.262226	0.131113	−	0.991367i	$-0.458145\pi$
$647$	6.67004	0.262226	0.131113	+	0.991367i	$0.458145\pi$
$648$	0	0
$649$	11.3845	0.446883
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.0250	0.627107	0.313554	−	0.949570i	$-0.398480\pi$
$653$	16.0250	0.627107	0.313554	+	0.949570i	$0.398480\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.47109	−0.291475
$658$	0	0
$659$	12.3934	0.482780	0.241390	−	0.970428i	$-0.422397\pi$
$659$	12.3934	0.482780	0.241390	+	0.970428i	$0.422397\pi$
$660$	0	0
$661$	12.3132	0.478928	0.239464	−	0.970905i	$-0.423028\pi$
$661$	12.3132	0.478928	0.239464	+	0.970905i	$0.423028\pi$
$662$	0	0
$663$	16.0553	0.623538
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.4178	0.635699
$668$	0	0
$669$	27.0614	1.04625
$670$	0	0
$671$	−3.49473	−0.134913
$672$	0	0
$673$	−10.3139	−0.397572	−0.198786	−	0.980043i	$-0.563700\pi$
$673$	−10.3139	−0.397572	−0.198786	+	0.980043i	$0.563700\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.3992	1.16834	0.584168	−	0.811633i	$-0.301421\pi$
$677$	30.3992	1.16834	0.584168	+	0.811633i	$0.301421\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−24.0608	−0.922010
$682$	0	0
$683$	−25.7960	−0.987057	−0.493529	−	0.869730i	$-0.664293\pi$
$683$	−25.7960	−0.987057	−0.493529	+	0.869730i	$0.664293\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	27.5021	1.04927
$688$	0	0
$689$	6.46907	0.246452
$690$	0	0
$691$	−15.6453	−0.595176	−0.297588	−	0.954694i	$-0.596182\pi$
$691$	−15.6453	−0.595176	−0.297588	+	0.954694i	$0.596182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−66.0809	−2.50299
$698$	0	0
$699$	2.32496	0.0879379
$700$	0	0
$701$	30.3083	1.14473	0.572365	−	0.819999i	$-0.306026\pi$
$701$	30.3083	1.14473	0.572365	+	0.819999i	$0.306026\pi$
$702$	0	0
$703$	−23.3649	−0.881224
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	40.7086	1.52885	0.764423	−	0.644715i	$-0.223024\pi$
$709$	40.7086	1.52885	0.764423	+	0.644715i	$0.223024\pi$
$710$	0	0
$711$	0.707686	0.0265403
$712$	0	0
$713$	−17.7986	−0.666564
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00902	0.224411
$718$	0	0
$719$	11.2692	0.420269	0.210134	−	0.977673i	$-0.432610\pi$
$719$	11.2692	0.420269	0.210134	+	0.977673i	$0.432610\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.6334	0.804555
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.5742	−0.948495	−0.474248	−	0.880392i	$-0.657280\pi$
$727$	−25.5742	−0.948495	−0.474248	+	0.880392i	$0.657280\pi$
$728$	0	0
$729$	10.1427	0.375655
$730$	0	0
$731$	−50.7779	−1.87809
$732$	0	0
$733$	−7.86641	−0.290552	−0.145276	−	0.989391i	$-0.546407\pi$
$733$	−7.86641	−0.290552	−0.145276	+	0.989391i	$0.546407\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.71306	0.210443
$738$	0	0
$739$	14.0515	0.516892	0.258446	−	0.966026i	$-0.416790\pi$
$739$	14.0515	0.516892	0.258446	+	0.966026i	$0.416790\pi$
$740$	0	0
$741$	11.1853	0.410902
$742$	0	0
$743$	−11.3464	−0.416258	−0.208129	−	0.978101i	$-0.566737\pi$
$743$	−11.3464	−0.416258	−0.208129	+	0.978101i	$0.566737\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.318329	−0.0116471
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.1936	−1.75861	−0.879306	−	0.476258i	$-0.841993\pi$
$751$	−48.1936	−1.75861	−0.879306	+	0.476258i	$0.841993\pi$
$752$	0	0
$753$	−25.0962	−0.914556
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.3236	0.738676	0.369338	−	0.929295i	$-0.379585\pi$
$757$	20.3236	0.738676	0.369338	+	0.929295i	$0.379585\pi$
$758$	0	0
$759$	6.82206	0.247625
$760$	0	0
$761$	6.92883	0.251170	0.125585	−	0.992083i	$-0.459919\pi$
$761$	6.92883	0.251170	0.125585	+	0.992083i	$0.459919\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.5454	−0.452989
$768$	0	0
$769$	−32.5653	−1.17434	−0.587168	−	0.809465i	$-0.699757\pi$
$769$	−32.5653	−1.17434	−0.587168	+	0.809465i	$0.699757\pi$
$770$	0	0
$771$	−36.1778	−1.30291
$772$	0	0
$773$	31.4728	1.13200	0.565998	−	0.824407i	$-0.308491\pi$
$773$	31.4728	1.13200	0.565998	+	0.824407i	$0.308491\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−46.0366	−1.64943
$780$	0	0
$781$	−4.30670	−0.154106
$782$	0	0
$783$	−19.3359	−0.691010
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−45.9267	−1.63711	−0.818555	−	0.574428i	$-0.805224\pi$
$787$	−45.9267	−1.63711	−0.818555	+	0.574428i	$0.805224\pi$
$788$	0	0
$789$	−8.51162	−0.303022
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.85109	0.136756
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.5988	0.481696	0.240848	−	0.970563i	$-0.422575\pi$
$797$	13.5988	0.481696	0.240848	+	0.970563i	$0.422575\pi$
$798$	0	0
$799$	63.3654	2.24171
$800$	0	0
$801$	−19.9397	−0.704535
$802$	0	0
$803$	−6.77976	−0.239253
$804$	0	0
$805$	0	0
$806$	0	0
$807$	53.5441	1.88484
$808$	0	0
$809$	17.0751	0.600329	0.300165	−	0.953887i	$-0.402958\pi$
$809$	17.0751	0.600329	0.300165	+	0.953887i	$0.402958\pi$
$810$	0	0
$811$	−35.7020	−1.25367	−0.626833	−	0.779153i	$-0.715649\pi$
$811$	−35.7020	−1.25367	−0.626833	+	0.779153i	$0.715649\pi$
$812$	0	0
$813$	53.6242	1.88068
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−35.3755	−1.23763
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.4670	−0.574701	−0.287351	−	0.957825i	$-0.592774\pi$
$821$	−16.4670	−0.574701	−0.287351	+	0.957825i	$0.592774\pi$
$822$	0	0
$823$	29.0400	1.01227	0.506135	−	0.862454i	$-0.331074\pi$
$823$	29.0400	1.01227	0.506135	+	0.862454i	$0.331074\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.8903	−1.66531	−0.832654	−	0.553794i	$-0.813179\pi$
$827$	−47.8903	−1.66531	−0.832654	+	0.553794i	$0.813179\pi$
$828$	0	0
$829$	24.1186	0.837672	0.418836	−	0.908062i	$-0.362438\pi$
$829$	24.1186	0.837672	0.418836	+	0.908062i	$0.362438\pi$
$830$	0	0
$831$	42.6674	1.48012
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.9622	0.724559
$838$	0	0
$839$	25.1230	0.867343	0.433672	−	0.901071i	$-0.357218\pi$
$839$	25.1230	0.867343	0.433672	+	0.901071i	$0.357218\pi$
$840$	0	0
$841$	−2.56701	−0.0885175
$842$	0	0
$843$	32.0222	1.10290
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.0321	0.756141
$850$	0	0
$851$	15.7037	0.538317
$852$	0	0
$853$	2.78647	0.0954069	0.0477035	−	0.998862i	$-0.484810\pi$
$853$	2.78647	0.0954069	0.0477035	+	0.998862i	$0.484810\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.7254	1.15204	0.576019	−	0.817436i	$-0.304605\pi$
$857$	33.7254	1.15204	0.576019	+	0.817436i	$0.304605\pi$
$858$	0	0
$859$	19.1171	0.652266	0.326133	−	0.945324i	$-0.394254\pi$
$859$	19.1171	0.652266	0.326133	+	0.945324i	$0.394254\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.0437	−1.12482	−0.562410	−	0.826859i	$-0.690126\pi$
$863$	−33.0437	−1.12482	−0.562410	+	0.826859i	$0.690126\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	60.1533	2.04291
$868$	0	0
$869$	0.642200	0.0217852
$870$	0	0
$871$	−6.29562	−0.213319
$872$	0	0
$873$	8.77590	0.297019
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.185101	−0.00625043	−0.00312521	−	0.999995i	$-0.500995\pi$
$877$	−0.185101	−0.00625043	−0.00312521	+	0.999995i	$0.500995\pi$
$878$	0	0
$879$	20.2183	0.681947
$880$	0	0
$881$	43.7063	1.47250	0.736252	−	0.676708i	$-0.236594\pi$
$881$	43.7063	1.47250	0.736252	+	0.676708i	$0.236594\pi$
$882$	0	0
$883$	−4.70129	−0.158211	−0.0791055	−	0.996866i	$-0.525206\pi$
$883$	−4.70129	−0.158211	−0.0791055	+	0.996866i	$0.525206\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.9752	−0.972893	−0.486447	−	0.873710i	$-0.661707\pi$
$887$	−28.9752	−0.972893	−0.486447	+	0.873710i	$0.661707\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.6660	−0.390827
$892$	0	0
$893$	44.1448	1.47725
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.51771	−0.251009
$898$	0	0
$899$	−28.6562	−0.955736
$900$	0	0
$901$	38.1994	1.27261
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.9661	1.19423	0.597117	−	0.802154i	$-0.296313\pi$
$907$	35.9661	1.19423	0.597117	+	0.802154i	$0.296313\pi$
$908$	0	0
$909$	−0.652164	−0.0216309
$910$	0	0
$911$	29.1414	0.965499	0.482749	−	0.875759i	$-0.339638\pi$
$911$	29.1414	0.965499	0.482749	+	0.875759i	$0.339638\pi$
$912$	0	0
$913$	−0.288873	−0.00956030
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	27.6014	0.910485	0.455243	−	0.890367i	$-0.349553\pi$
$919$	27.6014	0.910485	0.455243	+	0.890367i	$0.349553\pi$
$920$	0	0
$921$	−11.5093	−0.379244
$922$	0	0
$923$	4.74585	0.156212
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.9963	−0.591076
$928$	0	0
$929$	41.4287	1.35923	0.679616	−	0.733568i	$-0.262147\pi$
$929$	41.4287	1.35923	0.679616	+	0.733568i	$0.262147\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.4991	0.802065
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.6450	−0.576438	−0.288219	−	0.957564i	$-0.593063\pi$
$937$	−17.6450	−0.576438	−0.288219	+	0.957564i	$0.593063\pi$
$938$	0	0
$939$	38.6851	1.26244
$940$	0	0
$941$	24.1005	0.785653	0.392827	−	0.919613i	$-0.371497\pi$
$941$	24.1005	0.785653	0.392827	+	0.919613i	$0.371497\pi$
$942$	0	0
$943$	30.9415	1.00760
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.64168	−0.313312	−0.156656	−	0.987653i	$-0.550071\pi$
$947$	−9.64168	−0.313312	−0.156656	+	0.987653i	$0.550071\pi$
$948$	0	0
$949$	7.47109	0.242522
$950$	0	0
$951$	25.9725	0.842216
$952$	0	0
$953$	−12.3033	−0.398544	−0.199272	−	0.979944i	$-0.563858\pi$
$953$	−12.3033	−0.398544	−0.199272	+	0.979944i	$0.563858\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.9837	0.355052
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0.0662997	0.00213870
$962$	0	0
$963$	−16.1971	−0.521943
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.7098	−0.505193	−0.252597	−	0.967572i	$-0.581285\pi$
$967$	−15.7098	−0.505193	−0.252597	+	0.967572i	$0.581285\pi$
$968$	0	0
$969$	66.0482	2.12178
$970$	0	0
$971$	−4.71582	−0.151338	−0.0756689	−	0.997133i	$-0.524109\pi$
$971$	−4.71582	−0.151338	−0.0756689	+	0.997133i	$0.524109\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.0115	0.928159	0.464079	−	0.885794i	$-0.346385\pi$
$977$	29.0115	0.928159	0.464079	+	0.885794i	$0.346385\pi$
$978$	0	0
$979$	−18.0946	−0.578306
$980$	0	0
$981$	−18.2456	−0.582537
$982$	0	0
$983$	−29.6408	−0.945394	−0.472697	−	0.881225i	$-0.656720\pi$
$983$	−29.6408	−0.945394	−0.472697	+	0.881225i	$0.656720\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.7761	0.756036
$990$	0	0
$991$	39.0278	1.23976	0.619879	−	0.784697i	$-0.287182\pi$
$991$	39.0278	1.23976	0.619879	+	0.784697i	$0.287182\pi$
$992$	0	0
$993$	48.5913	1.54200
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.3710	0.391792	0.195896	−	0.980625i	$-0.437238\pi$
$997$	12.3710	0.391792	0.195896	+	0.980625i	$0.437238\pi$
$998$	0	0
$999$	−18.4949	−0.585154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cv.1.5		6
5.2	odd	4		1960.2.g.f.1569.4		12
5.3	odd	4		1960.2.g.f.1569.9		12
5.4	even	2		9800.2.a.cx.1.2		6
7.2	even	3		1400.2.q.o.1201.2		12
7.4	even	3		1400.2.q.o.401.2		12
7.6	odd	2		9800.2.a.cy.1.2		6
35.2	odd	12		280.2.bg.a.249.9	yes	24
35.4	even	6		1400.2.q.n.401.5		12
35.9	even	6		1400.2.q.n.1201.5		12
35.13	even	4		1960.2.g.e.1569.4		12
35.18	odd	12		280.2.bg.a.9.9	yes	24
35.23	odd	12		280.2.bg.a.249.4	yes	24
35.27	even	4		1960.2.g.e.1569.9		12
35.32	odd	12		280.2.bg.a.9.4	✓	24
35.34	odd	2		9800.2.a.cw.1.5		6
140.23	even	12		560.2.bw.f.529.9		24
140.67	even	12		560.2.bw.f.289.9		24
140.107	even	12		560.2.bw.f.529.4		24
140.123	even	12		560.2.bw.f.289.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bg.a.9.4	✓	24	35.32	odd	12
280.2.bg.a.9.9	yes	24	35.18	odd	12
280.2.bg.a.249.4	yes	24	35.23	odd	12
280.2.bg.a.249.9	yes	24	35.2	odd	12
560.2.bw.f.289.4		24	140.123	even	12
560.2.bw.f.289.9		24	140.67	even	12
560.2.bw.f.529.4		24	140.107	even	12
560.2.bw.f.529.9		24	140.23	even	12
1400.2.q.n.401.5		12	35.4	even	6
1400.2.q.n.1201.5		12	35.9	even	6
1400.2.q.o.401.2		12	7.4	even	3
1400.2.q.o.1201.2		12	7.2	even	3
1960.2.g.e.1569.4		12	35.13	even	4
1960.2.g.e.1569.9		12	35.27	even	4
1960.2.g.f.1569.4		12	5.2	odd	4
1960.2.g.f.1569.9		12	5.3	odd	4
9800.2.a.cv.1.5		6	1.1	even	1	trivial
9800.2.a.cw.1.5		6	35.34	odd	2
9800.2.a.cx.1.2		6	5.4	even	2
9800.2.a.cy.1.2		6	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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.03837 q^{3} +1.15495 q^{9} +O(q^{10})\)	\(q+2.03837 q^{3} +1.15495 q^{9} +1.04807 q^{11} -1.15495 q^{13} -6.81986 q^{17} -4.75120 q^{19} +3.19331 q^{23} -3.76090 q^{27} +5.14130 q^{29} -5.57372 q^{31} +2.13636 q^{33} +4.91769 q^{37} -2.35420 q^{39} +9.68948 q^{41} +7.44559 q^{43} -9.29130 q^{47} -13.9014 q^{51} -5.60119 q^{53} -9.68469 q^{57} +10.8624 q^{59} -3.33443 q^{61} +5.45101 q^{67} +6.50915 q^{69} -4.10916 q^{71} -6.46879 q^{73} +0.612744 q^{79} -11.1309 q^{81} -0.275623 q^{83} +10.4799 q^{87} -17.2646 q^{89} -11.3613 q^{93} +7.59854 q^{97} +1.21047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 7 q^{9}+O(q^{10}) \) 6 * q - q^3 + 7 * q^9	\( 6 q - q^{3} + 7 q^{9} + q^{11} - 7 q^{13} - 4 q^{17} - 5 q^{19} + 6 q^{23} - 7 q^{27} - 3 q^{29} - 2 q^{31} - 16 q^{33} + 9 q^{37} + 10 q^{39} + 6 q^{41} - 3 q^{43} - 27 q^{47} - 5 q^{53} - 26 q^{57} - 24 q^{59} + 9 q^{61} + 17 q^{67} + 15 q^{69} + 4 q^{71} - 18 q^{73} - 22 q^{79} - 6 q^{81} - 9 q^{83} - 39 q^{87} + 15 q^{89} + 10 q^{93} - 12 q^{97} + 11 q^{99}+O(q^{100}) \) 6 * q - q^3 + 7 * q^9 + q^11 - 7 * q^13 - 4 * q^17 - 5 * q^19 + 6 * q^23 - 7 * q^27 - 3 * q^29 - 2 * q^31 - 16 * q^33 + 9 * q^37 + 10 * q^39 + 6 * q^41 - 3 * q^43 - 27 * q^47 - 5 * q^53 - 26 * q^57 - 24 * q^59 + 9 * q^61 + 17 * q^67 + 15 * q^69 + 4 * q^71 - 18 * q^73 - 22 * q^79 - 6 * q^81 - 9 * q^83 - 39 * q^87 + 15 * q^89 + 10 * q^93 - 12 * q^97 + 11 * q^99

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