LMFDB - Embedded newform 9801.2.a.cn.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} - 22 x^{16} + 42 x^{15} + 198 x^{14} - 357 x^{13} - 944 x^{12} + 1579 x^{11} + \cdots - 55 $ x^18 - 2x^17 - 22x^16 + 42x^15 + 198x^14 - 357x^13 - 944x^12 + 1579x^11 + 2594x^10 - 3897x^9 - 4219x^8 + 5363x^7 + 4035x^6 - 3886x^5 - 2173x^4 + 1303x^3 + 576x^2 - 155*x - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.490494$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.490494 q^{2} -1.75942 q^{4} -1.70818 q^{5} -3.41052 q^{7} +1.84397 q^{8} +0.837851 q^{10} +3.99792 q^{13} +1.67284 q^{14} +2.61437 q^{16} -3.30458 q^{17} +4.41765 q^{19} +3.00539 q^{20} -7.49201 q^{23} -2.08213 q^{25} -1.96096 q^{26} +6.00053 q^{28} +1.69129 q^{29} -3.89852 q^{31} -4.97028 q^{32} +1.62087 q^{34} +5.82578 q^{35} +3.06553 q^{37} -2.16683 q^{38} -3.14983 q^{40} +0.390875 q^{41} -9.61268 q^{43} +3.67479 q^{46} +1.08026 q^{47} +4.63167 q^{49} +1.02127 q^{50} -7.03401 q^{52} +12.5638 q^{53} -6.28890 q^{56} -0.829567 q^{58} +3.65654 q^{59} +3.85318 q^{61} +1.91220 q^{62} -2.79086 q^{64} -6.82916 q^{65} +3.10120 q^{67} +5.81412 q^{68} -2.85751 q^{70} -7.01715 q^{71} +15.0426 q^{73} -1.50362 q^{74} -7.77248 q^{76} +3.99913 q^{79} -4.46581 q^{80} -0.191722 q^{82} -0.124966 q^{83} +5.64480 q^{85} +4.71496 q^{86} +7.93327 q^{89} -13.6350 q^{91} +13.1816 q^{92} -0.529860 q^{94} -7.54613 q^{95} +0.343160 q^{97} -2.27180 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{2} + 12 q^{4} + q^{5} - q^{7} - 6 q^{8} + 2 q^{10} - 3 q^{13} - 8 q^{16} - 20 q^{17} + 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} - 19 q^{28} - 21 q^{29} + 6 q^{31} - 9 q^{32} - 4 q^{34}+ \cdots - 82 q^{98}+O(q^{100}) $ 18 * q - 2 * q^2 + 12 * q^4 + q^5 - q^7 - 6 * q^8 + 2 * q^10 - 3 * q^13 - 8 * q^16 - 20 * q^17 + 3 * q^19 + 5 * q^20 + 10 * q^23 + 7 * q^25 - 2 * q^26 - 19 * q^28 - 21 * q^29 + 6 * q^31 - 9 * q^32 - 4 * q^34 - 38 * q^35 + 7 * q^37 - 13 * q^38 - 20 * q^41 - 4 * q^43 - 8 * q^46 - 7 * q^47 + 7 * q^49 - 25 * q^50 + 19 * q^52 - 31 * q^53 + 57 * q^56 + 12 * q^58 + 12 * q^59 + 16 * q^61 - 19 * q^62 - 16 * q^64 - 42 * q^65 - 5 * q^67 - 51 * q^68 + 8 * q^70 - 13 * q^71 - 9 * q^74 + 8 * q^76 + 2 * q^79 - 46 * q^80 - 34 * q^82 - 36 * q^83 - 25 * q^85 - 26 * q^86 + 14 * q^89 - 15 * q^91 + 15 * q^92 - 4 * q^94 - 64 * q^95 - 16 * q^97 - 82 * q^98

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 $

$2$	−0.490494	−0.346832	−0.173416	−	0.984849i	$-0.555480\pi$
$2$	−0.490494	−0.346832	−0.173416	+	0.984849i	$0.555480\pi$
$3$	0	0
$3$	0	0
$4$	−1.75942	−0.879708
$5$	−1.70818	−0.763920	−0.381960	−	0.924179i	$-0.624751\pi$
$5$	−1.70818	−0.763920	−0.381960	+	0.924179i	$0.624751\pi$
$6$	0	0
$7$	−3.41052	−1.28906	−0.644528	−	0.764581i	$-0.722946\pi$
$7$	−3.41052	−1.28906	−0.644528	+	0.764581i	$0.722946\pi$
$8$	1.84397	0.651942
$9$	0	0
$10$	0.837851	0.264952
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.99792	1.10882	0.554412	−	0.832242i	$-0.312943\pi$
$13$	3.99792	1.10882	0.554412	+	0.832242i	$0.312943\pi$
$14$	1.67284	0.447086
$15$	0	0
$16$	2.61437	0.653594
$17$	−3.30458	−0.801477	−0.400739	−	0.916192i	$-0.631246\pi$
$17$	−3.30458	−0.801477	−0.400739	+	0.916192i	$0.631246\pi$
$18$	0	0
$19$	4.41765	1.01348	0.506739	−	0.862099i	$-0.330851\pi$
$19$	4.41765	1.01348	0.506739	+	0.862099i	$0.330851\pi$
$20$	3.00539	0.672026
$21$	0	0
$22$	0	0
$23$	−7.49201	−1.56219	−0.781096	−	0.624411i	$-0.785339\pi$
$23$	−7.49201	−1.56219	−0.781096	+	0.624411i	$0.785339\pi$
$24$	0	0
$25$	−2.08213	−0.416426
$26$	−1.96096	−0.384575
$27$	0	0
$28$	6.00053	1.13399
$29$	1.69129	0.314064	0.157032	−	0.987593i	$-0.449807\pi$
$29$	1.69129	0.314064	0.157032	+	0.987593i	$0.449807\pi$
$30$	0	0
$31$	−3.89852	−0.700194	−0.350097	−	0.936713i	$-0.613851\pi$
$31$	−3.89852	−0.700194	−0.350097	+	0.936713i	$0.613851\pi$
$32$	−4.97028	−0.878629
$33$	0	0
$34$	1.62087	0.277978
$35$	5.82578	0.984736
$36$	0	0
$37$	3.06553	0.503970	0.251985	−	0.967731i	$-0.418917\pi$
$37$	3.06553	0.503970	0.251985	+	0.967731i	$0.418917\pi$
$38$	−2.16683	−0.351506
$39$	0	0
$40$	−3.14983	−0.498032
$41$	0.390875	0.0610444	0.0305222	−	0.999534i	$-0.490283\pi$
$41$	0.390875	0.0610444	0.0305222	+	0.999534i	$0.490283\pi$
$42$	0	0
$43$	−9.61268	−1.46592	−0.732960	−	0.680272i	$-0.761862\pi$
$43$	−9.61268	−1.46592	−0.732960	+	0.680272i	$0.761862\pi$
$44$	0	0
$45$	0	0
$46$	3.67479	0.541818
$47$	1.08026	0.157572	0.0787859	−	0.996892i	$-0.474896\pi$
$47$	1.08026	0.157572	0.0787859	+	0.996892i	$0.474896\pi$
$48$	0	0
$49$	4.63167	0.661666
$50$	1.02127	0.144430
$51$	0	0
$52$	−7.03401	−0.975442
$53$	12.5638	1.72577	0.862884	−	0.505402i	$-0.168656\pi$
$53$	12.5638	1.72577	0.862884	+	0.505402i	$0.168656\pi$
$54$	0	0
$55$	0	0
$56$	−6.28890	−0.840390
$57$	0	0
$58$	−0.829567	−0.108927
$59$	3.65654	0.476041	0.238021	−	0.971260i	$-0.423501\pi$
$59$	3.65654	0.476041	0.238021	+	0.971260i	$0.423501\pi$
$60$	0	0
$61$	3.85318	0.493349	0.246674	−	0.969098i	$-0.420662\pi$
$61$	3.85318	0.493349	0.246674	+	0.969098i	$0.420662\pi$
$62$	1.91220	0.242849
$63$	0	0
$64$	−2.79086	−0.348857
$65$	−6.82916	−0.847053
$66$	0	0
$67$	3.10120	0.378872	0.189436	−	0.981893i	$-0.439334\pi$
$67$	3.10120	0.378872	0.189436	+	0.981893i	$0.439334\pi$
$68$	5.81412	0.705066
$69$	0	0
$70$	−2.85751	−0.341538
$71$	−7.01715	−0.832783	−0.416391	−	0.909186i	$-0.636705\pi$
$71$	−7.01715	−0.832783	−0.416391	+	0.909186i	$0.636705\pi$
$72$	0	0
$73$	15.0426	1.76060	0.880302	−	0.474413i	$-0.157340\pi$
$73$	15.0426	1.76060	0.880302	+	0.474413i	$0.157340\pi$
$74$	−1.50362	−0.174793
$75$	0	0
$76$	−7.77248	−0.891565
$77$	0	0
$78$	0	0
$79$	3.99913	0.449937	0.224969	−	0.974366i	$-0.427772\pi$
$79$	3.99913	0.449937	0.224969	+	0.974366i	$0.427772\pi$
$80$	−4.46581	−0.499293
$81$	0	0
$82$	−0.191722	−0.0211721
$83$	−0.124966	−0.0137168	−0.00685839	−	0.999976i	$-0.502183\pi$
$83$	−0.124966	−0.0137168	−0.00685839	+	0.999976i	$0.502183\pi$
$84$	0	0
$85$	5.64480	0.612265
$86$	4.71496	0.508428
$87$	0	0
$88$	0	0
$89$	7.93327	0.840925	0.420462	−	0.907310i	$-0.361868\pi$
$89$	7.93327	0.840925	0.420462	+	0.907310i	$0.361868\pi$
$90$	0	0
$91$	−13.6350	−1.42934
$92$	13.1816	1.37427
$93$	0	0
$94$	−0.529860	−0.0546509
$95$	−7.54613	−0.774217
$96$	0	0
$97$	0.343160	0.0348426	0.0174213	−	0.999848i	$-0.494454\pi$
$97$	0.343160	0.0348426	0.0174213	+	0.999848i	$0.494454\pi$
$98$	−2.27180	−0.229487
$99$	0	0
$100$	3.66333	0.366333
$101$	7.11558	0.708027	0.354014	−	0.935240i	$-0.384817\pi$
$101$	7.11558	0.708027	0.354014	+	0.935240i	$0.384817\pi$
$102$	0	0
$103$	−1.65429	−0.163002	−0.0815009	−	0.996673i	$-0.525971\pi$
$103$	−1.65429	−0.163002	−0.0815009	+	0.996673i	$0.525971\pi$
$104$	7.37205	0.722889
$105$	0	0
$106$	−6.16246	−0.598551
$107$	3.30683	0.319683	0.159842	−	0.987143i	$-0.448902\pi$
$107$	3.30683	0.319683	0.159842	+	0.987143i	$0.448902\pi$
$108$	0	0
$109$	5.50709	0.527484	0.263742	−	0.964593i	$-0.415043\pi$
$109$	5.50709	0.527484	0.263742	+	0.964593i	$0.415043\pi$
$110$	0	0
$111$	0	0
$112$	−8.91638	−0.842519
$113$	16.4092	1.54365	0.771825	−	0.635835i	$-0.219344\pi$
$113$	16.4092	1.54365	0.771825	+	0.635835i	$0.219344\pi$
$114$	0	0
$115$	12.7977	1.19339
$116$	−2.97568	−0.276285
$117$	0	0
$118$	−1.79351	−0.165106
$119$	11.2703	1.03315
$120$	0	0
$121$	0	0
$122$	−1.88996	−0.171109
$123$	0	0
$124$	6.85911	0.615966
$125$	12.0975	1.08204
$126$	0	0
$127$	−10.7949	−0.957894	−0.478947	−	0.877844i	$-0.658981\pi$
$127$	−10.7949	−0.957894	−0.478947	+	0.877844i	$0.658981\pi$
$128$	11.3095	0.999624
$129$	0	0
$130$	3.34966	0.293785
$131$	−21.1557	−1.84838	−0.924189	−	0.381935i	$-0.875258\pi$
$131$	−21.1557	−1.84838	−0.924189	+	0.381935i	$0.875258\pi$
$132$	0	0
$133$	−15.0665	−1.30643
$134$	−1.52112	−0.131405
$135$	0	0
$136$	−6.09354	−0.522517
$137$	−3.20052	−0.273439	−0.136720	−	0.990610i	$-0.543656\pi$
$137$	−3.20052	−0.273439	−0.136720	+	0.990610i	$0.543656\pi$
$138$	0	0
$139$	14.9936	1.27174	0.635872	−	0.771794i	$-0.280640\pi$
$139$	14.9936	1.27174	0.635872	+	0.771794i	$0.280640\pi$
$140$	−10.2500	−0.866280
$141$	0	0
$142$	3.44187	0.288835
$143$	0	0
$144$	0	0
$145$	−2.88902	−0.239920
$146$	−7.37831	−0.610633
$147$	0	0
$148$	−5.39354	−0.443346
$149$	4.05112	0.331881	0.165940	−	0.986136i	$-0.446934\pi$
$149$	4.05112	0.331881	0.165940	+	0.986136i	$0.446934\pi$
$150$	0	0
$151$	7.79399	0.634266	0.317133	−	0.948381i	$-0.397280\pi$
$151$	7.79399	0.634266	0.317133	+	0.948381i	$0.397280\pi$
$152$	8.14602	0.660729
$153$	0	0
$154$	0	0
$155$	6.65936	0.534892
$156$	0	0
$157$	0.309152	0.0246730	0.0123365	−	0.999924i	$-0.496073\pi$
$157$	0.309152	0.0246730	0.0123365	+	0.999924i	$0.496073\pi$
$158$	−1.96155	−0.156052
$159$	0	0
$160$	8.49011	0.671202
$161$	25.5517	2.01375
$162$	0	0
$163$	10.9941	0.861125	0.430563	−	0.902561i	$-0.358315\pi$
$163$	10.9941	0.861125	0.430563	+	0.902561i	$0.358315\pi$
$164$	−0.687711	−0.0537012
$165$	0	0
$166$	0.0612950	0.00475741
$167$	−17.5739	−1.35991	−0.679953	−	0.733255i	$-0.738000\pi$
$167$	−17.5739	−1.35991	−0.679953	+	0.733255i	$0.738000\pi$
$168$	0	0
$169$	2.98339	0.229492
$170$	−2.76874	−0.212353
$171$	0	0
$172$	16.9127	1.28958
$173$	12.3657	0.940150	0.470075	−	0.882627i	$-0.344227\pi$
$173$	12.3657	0.940150	0.470075	+	0.882627i	$0.344227\pi$
$174$	0	0
$175$	7.10116	0.536797
$176$	0	0
$177$	0	0
$178$	−3.89122	−0.291659
$179$	−21.8823	−1.63556	−0.817780	−	0.575531i	$-0.804795\pi$
$179$	−21.8823	−1.63556	−0.817780	+	0.575531i	$0.804795\pi$
$180$	0	0
$181$	−4.38018	−0.325576	−0.162788	−	0.986661i	$-0.552049\pi$
$181$	−4.38018	−0.325576	−0.162788	+	0.986661i	$0.552049\pi$
$182$	6.68789	0.495739
$183$	0	0
$184$	−13.8150	−1.01846
$185$	−5.23647	−0.384993
$186$	0	0
$187$	0	0
$188$	−1.90062	−0.138617
$189$	0	0
$190$	3.70133	0.268523
$191$	−12.5147	−0.905534	−0.452767	−	0.891629i	$-0.649563\pi$
$191$	−12.5147	−0.905534	−0.452767	+	0.891629i	$0.649563\pi$
$192$	0	0
$193$	12.5657	0.904500	0.452250	−	0.891891i	$-0.350622\pi$
$193$	12.5657	0.904500	0.452250	+	0.891891i	$0.350622\pi$
$194$	−0.168318	−0.0120845
$195$	0	0
$196$	−8.14902	−0.582073
$197$	−10.3453	−0.737075	−0.368538	−	0.929613i	$-0.620141\pi$
$197$	−10.3453	−0.737075	−0.368538	+	0.929613i	$0.620141\pi$
$198$	0	0
$199$	26.4773	1.87693	0.938464	−	0.345376i	$-0.112249\pi$
$199$	26.4773	1.87693	0.938464	+	0.345376i	$0.112249\pi$
$200$	−3.83939	−0.271486
$201$	0	0
$202$	−3.49015	−0.245566
$203$	−5.76818	−0.404847
$204$	0	0
$205$	−0.667683	−0.0466330
$206$	0.811418	0.0565342
$207$	0	0
$208$	10.4521	0.724721
$209$	0	0
$210$	0	0
$211$	7.28390	0.501444	0.250722	−	0.968059i	$-0.419332\pi$
$211$	7.28390	0.501444	0.250722	+	0.968059i	$0.419332\pi$
$212$	−22.1049	−1.51817
$213$	0	0
$214$	−1.62198	−0.110876
$215$	16.4202	1.11985
$216$	0	0
$217$	13.2960	0.902590
$218$	−2.70120	−0.182948
$219$	0	0
$220$	0	0
$221$	−13.2114	−0.888698
$222$	0	0
$223$	23.5101	1.57435	0.787175	−	0.616730i	$-0.211543\pi$
$223$	23.5101	1.57435	0.787175	+	0.616730i	$0.211543\pi$
$224$	16.9512	1.13260
$225$	0	0
$226$	−8.04863	−0.535387
$227$	−11.9222	−0.791303	−0.395652	−	0.918401i	$-0.629481\pi$
$227$	−11.9222	−0.791303	−0.395652	+	0.918401i	$0.629481\pi$
$228$	0	0
$229$	3.05757	0.202050	0.101025	−	0.994884i	$-0.467788\pi$
$229$	3.05757	0.202050	0.101025	+	0.994884i	$0.467788\pi$
$230$	−6.27719	−0.413905
$231$	0	0
$232$	3.11869	0.204752
$233$	−24.5314	−1.60710	−0.803552	−	0.595235i	$-0.797059\pi$
$233$	−24.5314	−1.60710	−0.803552	+	0.595235i	$0.797059\pi$
$234$	0	0
$235$	−1.84527	−0.120372
$236$	−6.43338	−0.418777
$237$	0	0
$238$	−5.52803	−0.358329
$239$	−6.95063	−0.449599	−0.224799	−	0.974405i	$-0.572173\pi$
$239$	−6.95063	−0.449599	−0.224799	+	0.974405i	$0.572173\pi$
$240$	0	0
$241$	7.77059	0.500547	0.250274	−	0.968175i	$-0.419479\pi$
$241$	7.77059	0.500547	0.250274	+	0.968175i	$0.419479\pi$
$242$	0	0
$243$	0	0
$244$	−6.77934	−0.434003
$245$	−7.91170	−0.505460
$246$	0	0
$247$	17.6614	1.12377
$248$	−7.18875	−0.456486
$249$	0	0
$250$	−5.93377	−0.375284
$251$	−0.327168	−0.0206506	−0.0103253	−	0.999947i	$-0.503287\pi$
$251$	−0.327168	−0.0206506	−0.0103253	+	0.999947i	$0.503287\pi$
$252$	0	0
$253$	0	0
$254$	5.29484	0.332228
$255$	0	0
$256$	0.0344986	0.00215616
$257$	13.7292	0.856402	0.428201	−	0.903684i	$-0.359148\pi$
$257$	13.7292	0.856402	0.428201	+	0.903684i	$0.359148\pi$
$258$	0	0
$259$	−10.4551	−0.649645
$260$	12.0153	0.745159
$261$	0	0
$262$	10.3767	0.641076
$263$	−16.8542	−1.03927	−0.519637	−	0.854387i	$-0.673933\pi$
$263$	−16.8542	−1.03927	−0.519637	+	0.854387i	$0.673933\pi$
$264$	0	0
$265$	−21.4612	−1.31835
$266$	7.39003	0.453112
$267$	0	0
$268$	−5.45630	−0.333297
$269$	−12.3974	−0.755882	−0.377941	−	0.925830i	$-0.623368\pi$
$269$	−12.3974	−0.755882	−0.377941	+	0.925830i	$0.623368\pi$
$270$	0	0
$271$	−19.2695	−1.17054	−0.585269	−	0.810839i	$-0.699011\pi$
$271$	−19.2695	−1.17054	−0.585269	+	0.810839i	$0.699011\pi$
$272$	−8.63940	−0.523840
$273$	0	0
$274$	1.56984	0.0948374
$275$	0	0
$276$	0	0
$277$	−10.3511	−0.621936	−0.310968	−	0.950420i	$-0.600653\pi$
$277$	−10.3511	−0.621936	−0.310968	+	0.950420i	$0.600653\pi$
$278$	−7.35429	−0.441081
$279$	0	0
$280$	10.7426	0.641991
$281$	−16.4358	−0.980478	−0.490239	−	0.871588i	$-0.663090\pi$
$281$	−16.4358	−0.980478	−0.490239	+	0.871588i	$0.663090\pi$
$282$	0	0
$283$	1.26495	0.0751937	0.0375968	−	0.999293i	$-0.488030\pi$
$283$	1.26495	0.0751937	0.0375968	+	0.999293i	$0.488030\pi$
$284$	12.3461	0.732605
$285$	0	0
$286$	0	0
$287$	−1.33309	−0.0786896
$288$	0	0
$289$	−6.07978	−0.357634
$290$	1.41705	0.0832119
$291$	0	0
$292$	−26.4662	−1.54882
$293$	−24.0874	−1.40720	−0.703601	−	0.710596i	$-0.748426\pi$
$293$	−24.0874	−1.40720	−0.703601	+	0.710596i	$0.748426\pi$
$294$	0	0
$295$	−6.24602	−0.363658
$296$	5.65274	0.328559
$297$	0	0
$298$	−1.98705	−0.115107
$299$	−29.9525	−1.73220
$300$	0	0
$301$	32.7843	1.88965
$302$	−3.82290	−0.219983
$303$	0	0
$304$	11.5494	0.662403
$305$	−6.58191	−0.376879
$306$	0	0
$307$	−16.2949	−0.930001	−0.465001	−	0.885310i	$-0.653946\pi$
$307$	−16.2949	−0.930001	−0.465001	+	0.885310i	$0.653946\pi$
$308$	0	0
$309$	0	0
$310$	−3.26637	−0.185518
$311$	−21.3713	−1.21185	−0.605926	−	0.795521i	$-0.707197\pi$
$311$	−21.3713	−1.21185	−0.605926	+	0.795521i	$0.707197\pi$
$312$	0	0
$313$	3.11613	0.176134	0.0880669	−	0.996115i	$-0.471931\pi$
$313$	3.11613	0.176134	0.0880669	+	0.996115i	$0.471931\pi$
$314$	−0.151637	−0.00855738
$315$	0	0
$316$	−7.03613	−0.395813
$317$	18.2648	1.02586	0.512928	−	0.858432i	$-0.328561\pi$
$317$	18.2648	1.02586	0.512928	+	0.858432i	$0.328561\pi$
$318$	0	0
$319$	0	0
$320$	4.76728	0.266499
$321$	0	0
$322$	−12.5329	−0.698434
$323$	−14.5985	−0.812280
$324$	0	0
$325$	−8.32420	−0.461744
$326$	−5.39254	−0.298665
$327$	0	0
$328$	0.720761	0.0397974
$329$	−3.68425	−0.203119
$330$	0	0
$331$	3.76616	0.207007	0.103503	−	0.994629i	$-0.466995\pi$
$331$	3.76616	0.207007	0.103503	+	0.994629i	$0.466995\pi$
$332$	0.219867	0.0120668
$333$	0	0
$334$	8.61988	0.471659
$335$	−5.29740	−0.289428
$336$	0	0
$337$	−5.96618	−0.324998	−0.162499	−	0.986709i	$-0.551955\pi$
$337$	−5.96618	−0.324998	−0.162499	+	0.986709i	$0.551955\pi$
$338$	−1.46334	−0.0795950
$339$	0	0
$340$	−9.93155	−0.538614
$341$	0	0
$342$	0	0
$343$	8.07726	0.436131
$344$	−17.7255	−0.955695
$345$	0	0
$346$	−6.06532	−0.326074
$347$	24.3493	1.30714	0.653570	−	0.756866i	$-0.273270\pi$
$347$	24.3493	1.30714	0.653570	+	0.756866i	$0.273270\pi$
$348$	0	0
$349$	−16.7418	−0.896166	−0.448083	−	0.893992i	$-0.647893\pi$
$349$	−16.7418	−0.896166	−0.448083	+	0.893992i	$0.647893\pi$
$350$	−3.48307	−0.186178
$351$	0	0
$352$	0	0
$353$	8.03132	0.427464	0.213732	−	0.976892i	$-0.431438\pi$
$353$	8.03132	0.427464	0.213732	+	0.976892i	$0.431438\pi$
$354$	0	0
$355$	11.9865	0.636179
$356$	−13.9579	−0.739768
$357$	0	0
$358$	10.7331	0.567264
$359$	−28.8250	−1.52133	−0.760663	−	0.649147i	$-0.775126\pi$
$359$	−28.8250	−1.52133	−0.760663	+	0.649147i	$0.775126\pi$
$360$	0	0
$361$	0.515640	0.0271390
$362$	2.14845	0.112920
$363$	0	0
$364$	23.9896	1.25740
$365$	−25.6954	−1.34496
$366$	0	0
$367$	22.3434	1.16632	0.583158	−	0.812359i	$-0.301817\pi$
$367$	22.3434	1.16632	0.583158	+	0.812359i	$0.301817\pi$
$368$	−19.5869	−1.02104
$369$	0	0
$370$	2.56845	0.133528
$371$	−42.8491	−2.22461
$372$	0	0
$373$	−7.07872	−0.366522	−0.183261	−	0.983064i	$-0.558665\pi$
$373$	−7.07872	−0.366522	−0.183261	+	0.983064i	$0.558665\pi$
$374$	0	0
$375$	0	0
$376$	1.99196	0.102728
$377$	6.76164	0.348242
$378$	0	0
$379$	−24.3461	−1.25057	−0.625287	−	0.780394i	$-0.715018\pi$
$379$	−24.3461	−1.25057	−0.625287	+	0.780394i	$0.715018\pi$
$380$	13.2768	0.681084
$381$	0	0
$382$	6.13840	0.314068
$383$	17.6379	0.901254	0.450627	−	0.892712i	$-0.351200\pi$
$383$	17.6379	0.901254	0.450627	+	0.892712i	$0.351200\pi$
$384$	0	0
$385$	0	0
$386$	−6.16341	−0.313709
$387$	0	0
$388$	−0.603760	−0.0306513
$389$	17.8830	0.906704	0.453352	−	0.891332i	$-0.350228\pi$
$389$	17.8830	0.906704	0.453352	+	0.891332i	$0.350228\pi$
$390$	0	0
$391$	24.7579	1.25206
$392$	8.54066	0.431368
$393$	0	0
$394$	5.07433	0.255641
$395$	−6.83122	−0.343716
$396$	0	0
$397$	−21.9395	−1.10111	−0.550556	−	0.834798i	$-0.685584\pi$
$397$	−21.9395	−1.10111	−0.550556	+	0.834798i	$0.685584\pi$
$398$	−12.9870	−0.650978
$399$	0	0
$400$	−5.44347	−0.272174
$401$	−5.40756	−0.270041	−0.135020	−	0.990843i	$-0.543110\pi$
$401$	−5.40756	−0.270041	−0.135020	+	0.990843i	$0.543110\pi$
$402$	0	0
$403$	−15.5860	−0.776392
$404$	−12.5193	−0.622857
$405$	0	0
$406$	2.82926	0.140414
$407$	0	0
$408$	0	0
$409$	−23.6201	−1.16794	−0.583970	−	0.811775i	$-0.698502\pi$
$409$	−23.6201	−1.16794	−0.583970	+	0.811775i	$0.698502\pi$
$410$	0.327495	0.0161738
$411$	0	0
$412$	2.91058	0.143394
$413$	−12.4707	−0.613644
$414$	0	0
$415$	0.213464	0.0104785
$416$	−19.8708	−0.974245
$417$	0	0
$418$	0	0
$419$	18.1313	0.885773	0.442886	−	0.896578i	$-0.353955\pi$
$419$	18.1313	0.885773	0.442886	+	0.896578i	$0.353955\pi$
$420$	0	0
$421$	−12.8122	−0.624428	−0.312214	−	0.950012i	$-0.601071\pi$
$421$	−12.8122	−0.624428	−0.312214	+	0.950012i	$0.601071\pi$
$422$	−3.57271	−0.173917
$423$	0	0
$424$	23.1672	1.12510
$425$	6.88056	0.333756
$426$	0	0
$427$	−13.1413	−0.635954
$428$	−5.81809	−0.281228
$429$	0	0
$430$	−8.05399	−0.388398
$431$	−6.57916	−0.316907	−0.158453	−	0.987366i	$-0.550651\pi$
$431$	−6.57916	−0.316907	−0.158453	+	0.987366i	$0.550651\pi$
$432$	0	0
$433$	13.9978	0.672690	0.336345	−	0.941739i	$-0.390809\pi$
$433$	13.9978	0.672690	0.336345	+	0.941739i	$0.390809\pi$
$434$	−6.52160	−0.313047
$435$	0	0
$436$	−9.68927	−0.464032
$437$	−33.0971	−1.58325
$438$	0	0
$439$	−25.1782	−1.20169	−0.600846	−	0.799365i	$-0.705169\pi$
$439$	−25.1782	−1.20169	−0.600846	+	0.799365i	$0.705169\pi$
$440$	0	0
$441$	0	0
$442$	6.48013	0.308228
$443$	−26.8225	−1.27437	−0.637187	−	0.770709i	$-0.719902\pi$
$443$	−26.8225	−1.27437	−0.637187	+	0.770709i	$0.719902\pi$
$444$	0	0
$445$	−13.5514	−0.642399
$446$	−11.5315	−0.546034
$447$	0	0
$448$	9.51829	0.449697
$449$	−14.1758	−0.668999	−0.334499	−	0.942396i	$-0.608567\pi$
$449$	−14.1758	−0.668999	−0.334499	+	0.942396i	$0.608567\pi$
$450$	0	0
$451$	0	0
$452$	−28.8706	−1.35796
$453$	0	0
$454$	5.84776	0.274449
$455$	23.2910	1.09190
$456$	0	0
$457$	−24.6413	−1.15267	−0.576335	−	0.817213i	$-0.695518\pi$
$457$	−24.6413	−1.15267	−0.576335	+	0.817213i	$0.695518\pi$
$458$	−1.49972	−0.0700773
$459$	0	0
$460$	−22.5164	−1.04983
$461$	−26.2454	−1.22237	−0.611185	−	0.791488i	$-0.709307\pi$
$461$	−26.2454	−1.22237	−0.611185	+	0.791488i	$0.709307\pi$
$462$	0	0
$463$	12.6922	0.589855	0.294928	−	0.955520i	$-0.404704\pi$
$463$	12.6922	0.589855	0.294928	+	0.955520i	$0.404704\pi$
$464$	4.42166	0.205270
$465$	0	0
$466$	12.0325	0.557394
$467$	27.7040	1.28199	0.640994	−	0.767546i	$-0.278522\pi$
$467$	27.7040	1.28199	0.640994	+	0.767546i	$0.278522\pi$
$468$	0	0
$469$	−10.5767	−0.488387
$470$	0.905095	0.0417489
$471$	0	0
$472$	6.74256	0.310351
$473$	0	0
$474$	0	0
$475$	−9.19813	−0.422039
$476$	−19.8292	−0.908870
$477$	0	0
$478$	3.40924	0.155935
$479$	−15.6222	−0.713797	−0.356898	−	0.934143i	$-0.616166\pi$
$479$	−15.6222	−0.713797	−0.356898	+	0.934143i	$0.616166\pi$
$480$	0	0
$481$	12.2557	0.558814
$482$	−3.81143	−0.173606
$483$	0	0
$484$	0	0
$485$	−0.586177	−0.0266169
$486$	0	0
$487$	26.9270	1.22018	0.610090	−	0.792332i	$-0.291133\pi$
$487$	26.9270	1.22018	0.610090	+	0.792332i	$0.291133\pi$
$488$	7.10514	0.321635
$489$	0	0
$490$	3.88064	0.175310
$491$	38.9997	1.76003	0.880017	−	0.474942i	$-0.157531\pi$
$491$	38.9997	1.76003	0.880017	+	0.474942i	$0.157531\pi$
$492$	0	0
$493$	−5.58899	−0.251715
$494$	−8.66283	−0.389759
$495$	0	0
$496$	−10.1922	−0.457642
$497$	23.9322	1.07350
$498$	0	0
$499$	−24.0351	−1.07596	−0.537980	−	0.842958i	$-0.680812\pi$
$499$	−24.0351	−1.07596	−0.537980	+	0.842958i	$0.680812\pi$
$500$	−21.2846	−0.951876
$501$	0	0
$502$	0.160474	0.00716229
$503$	−23.9865	−1.06950	−0.534752	−	0.845009i	$-0.679595\pi$
$503$	−23.9865	−1.06950	−0.534752	+	0.845009i	$0.679595\pi$
$504$	0	0
$505$	−12.1547	−0.540876
$506$	0	0
$507$	0	0
$508$	18.9927	0.842667
$509$	−3.69886	−0.163949	−0.0819746	−	0.996634i	$-0.526123\pi$
$509$	−3.69886	−0.163949	−0.0819746	+	0.996634i	$0.526123\pi$
$510$	0	0
$511$	−51.3032	−2.26952
$512$	−22.6358	−1.00037
$513$	0	0
$514$	−6.73407	−0.297027
$515$	2.82582	0.124520
$516$	0	0
$517$	0	0
$518$	5.12814	0.225318
$519$	0	0
$520$	−12.5928	−0.552230
$521$	6.41318	0.280966	0.140483	−	0.990083i	$-0.455134\pi$
$521$	6.41318	0.280966	0.140483	+	0.990083i	$0.455134\pi$
$522$	0	0
$523$	2.73856	0.119749	0.0598745	−	0.998206i	$-0.480930\pi$
$523$	2.73856	0.119749	0.0598745	+	0.998206i	$0.480930\pi$
$524$	37.2216	1.62603
$525$	0	0
$526$	8.26688	0.360453
$527$	12.8829	0.561190
$528$	0	0
$529$	33.1302	1.44044
$530$	10.5266	0.457245
$531$	0	0
$532$	26.5082	1.14928
$533$	1.56269	0.0676875
$534$	0	0
$535$	−5.64865	−0.244213
$536$	5.71852	0.247003
$537$	0	0
$538$	6.08085	0.262164
$539$	0	0
$540$	0	0
$541$	−20.9037	−0.898719	−0.449359	−	0.893351i	$-0.648348\pi$
$541$	−20.9037	−0.898719	−0.449359	+	0.893351i	$0.648348\pi$
$542$	9.45157	0.405980
$543$	0	0
$544$	16.4247	0.704201
$545$	−9.40709	−0.402956
$546$	0	0
$547$	41.8764	1.79050	0.895252	−	0.445560i	$-0.146996\pi$
$547$	41.8764	1.79050	0.895252	+	0.445560i	$0.146996\pi$
$548$	5.63105	0.240547
$549$	0	0
$550$	0	0
$551$	7.47152	0.318298
$552$	0	0
$553$	−13.6391	−0.579995
$554$	5.07714	0.215707
$555$	0	0
$556$	−26.3801	−1.11876
$557$	−12.5244	−0.530675	−0.265337	−	0.964156i	$-0.585483\pi$
$557$	−12.5244	−0.530675	−0.265337	+	0.964156i	$0.585483\pi$
$558$	0	0
$559$	−38.4308	−1.62545
$560$	15.2308	0.643617
$561$	0	0
$562$	8.06166	0.340061
$563$	−14.5087	−0.611470	−0.305735	−	0.952117i	$-0.598902\pi$
$563$	−14.5087	−0.611470	−0.305735	+	0.952117i	$0.598902\pi$
$564$	0	0
$565$	−28.0299	−1.17923
$566$	−0.620452	−0.0260796
$567$	0	0
$568$	−12.9394	−0.542926
$569$	−27.9724	−1.17266	−0.586332	−	0.810071i	$-0.699429\pi$
$569$	−27.9724	−1.17266	−0.586332	+	0.810071i	$0.699429\pi$
$570$	0	0
$571$	−9.04886	−0.378683	−0.189341	−	0.981911i	$-0.560635\pi$
$571$	−9.04886	−0.378683	−0.189341	+	0.981911i	$0.560635\pi$
$572$	0	0
$573$	0	0
$574$	0.653871	0.0272921
$575$	15.5993	0.650538
$576$	0	0
$577$	38.3471	1.59641	0.798204	−	0.602387i	$-0.205783\pi$
$577$	38.3471	1.59641	0.798204	+	0.602387i	$0.205783\pi$
$578$	2.98210	0.124039
$579$	0	0
$580$	5.08299	0.211060
$581$	0.426199	0.0176817
$582$	0	0
$583$	0	0
$584$	27.7381	1.14781
$585$	0	0
$586$	11.8147	0.488062
$587$	0.722617	0.0298256	0.0149128	−	0.999889i	$-0.495253\pi$
$587$	0.722617	0.0298256	0.0149128	+	0.999889i	$0.495253\pi$
$588$	0	0
$589$	−17.2223	−0.709632
$590$	3.06364	0.126128
$591$	0	0
$592$	8.01444	0.329391
$593$	−39.6596	−1.62863	−0.814313	−	0.580426i	$-0.802886\pi$
$593$	−39.6596	−1.62863	−0.814313	+	0.580426i	$0.802886\pi$
$594$	0	0
$595$	−19.2517	−0.789244
$596$	−7.12761	−0.291958
$597$	0	0
$598$	14.6915	0.600781
$599$	20.3768	0.832573	0.416287	−	0.909233i	$-0.363331\pi$
$599$	20.3768	0.832573	0.416287	+	0.909233i	$0.363331\pi$
$600$	0	0
$601$	48.3717	1.97312	0.986562	−	0.163389i	$-0.0522426\pi$
$601$	48.3717	1.97312	0.986562	+	0.163389i	$0.0522426\pi$
$602$	−16.0805	−0.655392
$603$	0	0
$604$	−13.7129	−0.557968
$605$	0	0
$606$	0	0
$607$	19.7023	0.799691	0.399845	−	0.916583i	$-0.369064\pi$
$607$	19.7023	0.799691	0.399845	+	0.916583i	$0.369064\pi$
$608$	−21.9569	−0.890472
$609$	0	0
$610$	3.22839	0.130714
$611$	4.31879	0.174720
$612$	0	0
$613$	−5.71781	−0.230940	−0.115470	−	0.993311i	$-0.536837\pi$
$613$	−5.71781	−0.230940	−0.115470	+	0.993311i	$0.536837\pi$
$614$	7.99257	0.322554
$615$	0	0
$616$	0	0
$617$	−6.10692	−0.245855	−0.122928	−	0.992416i	$-0.539228\pi$
$617$	−6.10692	−0.245855	−0.122928	+	0.992416i	$0.539228\pi$
$618$	0	0
$619$	−1.77973	−0.0715335	−0.0357668	−	0.999360i	$-0.511387\pi$
$619$	−1.77973	−0.0715335	−0.0357668	+	0.999360i	$0.511387\pi$
$620$	−11.7166	−0.470549
$621$	0	0
$622$	10.4825	0.420309
$623$	−27.0566	−1.08400
$624$	0	0
$625$	−10.2541	−0.410163
$626$	−1.52844	−0.0610888
$627$	0	0
$628$	−0.543927	−0.0217050
$629$	−10.1303	−0.403920
$630$	0	0
$631$	−44.8407	−1.78508	−0.892541	−	0.450966i	$-0.851079\pi$
$631$	−44.8407	−1.78508	−0.892541	+	0.450966i	$0.851079\pi$
$632$	7.37428	0.293333
$633$	0	0
$634$	−8.95880	−0.355799
$635$	18.4396	0.731754
$636$	0	0
$637$	18.5170	0.733672
$638$	0	0
$639$	0	0
$640$	−19.3185	−0.763633
$641$	27.0096	1.06681	0.533407	−	0.845859i	$-0.320912\pi$
$641$	27.0096	1.06681	0.533407	+	0.845859i	$0.320912\pi$
$642$	0	0
$643$	−32.5574	−1.28394	−0.641969	−	0.766731i	$-0.721882\pi$
$643$	−32.5574	−1.28394	−0.641969	+	0.766731i	$0.721882\pi$
$644$	−44.9560	−1.77152
$645$	0	0
$646$	7.16046	0.281724
$647$	31.4833	1.23774	0.618868	−	0.785495i	$-0.287592\pi$
$647$	31.4833	1.23774	0.618868	+	0.785495i	$0.287592\pi$
$648$	0	0
$649$	0	0
$650$	4.08297	0.160147
$651$	0	0
$652$	−19.3432	−0.757538
$653$	4.65977	0.182351	0.0911755	−	0.995835i	$-0.470938\pi$
$653$	4.65977	0.182351	0.0911755	+	0.995835i	$0.470938\pi$
$654$	0	0
$655$	36.1376	1.41201
$656$	1.02189	0.0398982
$657$	0	0
$658$	1.80710	0.0704481
$659$	−32.0418	−1.24817	−0.624086	−	0.781355i	$-0.714529\pi$
$659$	−32.0418	−1.24817	−0.624086	+	0.781355i	$0.714529\pi$
$660$	0	0
$661$	−40.7054	−1.58325	−0.791627	−	0.611004i	$-0.790766\pi$
$661$	−40.7054	−1.58325	−0.791627	+	0.611004i	$0.790766\pi$
$662$	−1.84728	−0.0717965
$663$	0	0
$664$	−0.230433	−0.00894255
$665$	25.7362	0.998009
$666$	0	0
$667$	−12.6712	−0.490629
$668$	30.9197	1.19632
$669$	0	0
$670$	2.59834	0.100383
$671$	0	0
$672$	0	0
$673$	37.7635	1.45568	0.727838	−	0.685749i	$-0.240525\pi$
$673$	37.7635	1.45568	0.727838	+	0.685749i	$0.240525\pi$
$674$	2.92637	0.112720
$675$	0	0
$676$	−5.24903	−0.201886
$677$	18.9007	0.726412	0.363206	−	0.931709i	$-0.381682\pi$
$677$	18.9007	0.726412	0.363206	+	0.931709i	$0.381682\pi$
$678$	0	0
$679$	−1.17035	−0.0449140
$680$	10.4088	0.399161
$681$	0	0
$682$	0	0
$683$	−31.1053	−1.19021	−0.595105	−	0.803648i	$-0.702890\pi$
$683$	−31.1053	−1.19021	−0.595105	+	0.803648i	$0.702890\pi$
$684$	0	0
$685$	5.46706	0.208886
$686$	−3.96185	−0.151264
$687$	0	0
$688$	−25.1312	−0.958116
$689$	50.2290	1.91357
$690$	0	0
$691$	22.2114	0.844962	0.422481	−	0.906372i	$-0.361159\pi$
$691$	22.2114	0.844962	0.422481	+	0.906372i	$0.361159\pi$
$692$	−21.7565	−0.827057
$693$	0	0
$694$	−11.9432	−0.453358
$695$	−25.6118	−0.971511
$696$	0	0
$697$	−1.29167	−0.0489257
$698$	8.21174	0.310819
$699$	0	0
$700$	−12.4939	−0.472224
$701$	−29.9457	−1.13103	−0.565517	−	0.824736i	$-0.691323\pi$
$701$	−29.9457	−1.13103	−0.565517	+	0.824736i	$0.691323\pi$
$702$	0	0
$703$	13.5424	0.510763
$704$	0	0
$705$	0	0
$706$	−3.93931	−0.148258
$707$	−24.2679	−0.912687
$708$	0	0
$709$	17.2057	0.646175	0.323087	−	0.946369i	$-0.395279\pi$
$709$	17.2057	0.646175	0.323087	+	0.946369i	$0.395279\pi$
$710$	−5.87932	−0.220647
$711$	0	0
$712$	14.6287	0.548234
$713$	29.2077	1.09384
$714$	0	0
$715$	0	0
$716$	38.5001	1.43881
$717$	0	0
$718$	14.1385	0.527644
$719$	−4.72508	−0.176216	−0.0881079	−	0.996111i	$-0.528082\pi$
$719$	−4.72508	−0.176216	−0.0881079	+	0.996111i	$0.528082\pi$
$720$	0	0
$721$	5.64199	0.210119
$722$	−0.252919	−0.00941265
$723$	0	0
$724$	7.70655	0.286412
$725$	−3.52148	−0.130785
$726$	0	0
$727$	5.20088	0.192890	0.0964450	−	0.995338i	$-0.469253\pi$
$727$	5.20088	0.192890	0.0964450	+	0.995338i	$0.469253\pi$
$728$	−25.1426	−0.931845
$729$	0	0
$730$	12.6035	0.466475
$731$	31.7658	1.17490
$732$	0	0
$733$	49.0632	1.81219	0.906095	−	0.423075i	$-0.139049\pi$
$733$	49.0632	1.81219	0.906095	+	0.423075i	$0.139049\pi$
$734$	−10.9593	−0.404515
$735$	0	0
$736$	37.2374	1.37259
$737$	0	0
$738$	0	0
$739$	15.5942	0.573641	0.286821	−	0.957984i	$-0.407402\pi$
$739$	15.5942	0.573641	0.286821	+	0.957984i	$0.407402\pi$
$740$	9.21312	0.338681
$741$	0	0
$742$	21.0172	0.771566
$743$	11.1344	0.408483	0.204242	−	0.978921i	$-0.434527\pi$
$743$	11.1344	0.408483	0.204242	+	0.978921i	$0.434527\pi$
$744$	0	0
$745$	−6.92004	−0.253530
$746$	3.47207	0.127122
$747$	0	0
$748$	0	0
$749$	−11.2780	−0.412090
$750$	0	0
$751$	−8.08520	−0.295033	−0.147517	−	0.989060i	$-0.547128\pi$
$751$	−8.08520	−0.295033	−0.147517	+	0.989060i	$0.547128\pi$
$752$	2.82420	0.102988
$753$	0	0
$754$	−3.31654	−0.120781
$755$	−13.3135	−0.484528
$756$	0	0
$757$	29.8616	1.08534	0.542669	−	0.839947i	$-0.317414\pi$
$757$	29.8616	1.08534	0.542669	+	0.839947i	$0.317414\pi$
$758$	11.9416	0.433739
$759$	0	0
$760$	−13.9148	−0.504744
$761$	34.8926	1.26486	0.632428	−	0.774619i	$-0.282059\pi$
$761$	34.8926	1.26486	0.632428	+	0.774619i	$0.282059\pi$
$762$	0	0
$763$	−18.7821	−0.679957
$764$	22.0186	0.796606
$765$	0	0
$766$	−8.65128	−0.312583
$767$	14.6186	0.527846
$768$	0	0
$769$	5.60723	0.202202	0.101101	−	0.994876i	$-0.467763\pi$
$769$	5.60723	0.202202	0.101101	+	0.994876i	$0.467763\pi$
$770$	0	0
$771$	0	0
$772$	−22.1083	−0.795696
$773$	10.8401	0.389893	0.194946	−	0.980814i	$-0.437547\pi$
$773$	10.8401	0.389893	0.194946	+	0.980814i	$0.437547\pi$
$774$	0	0
$775$	8.11722	0.291579
$776$	0.632776	0.0227153
$777$	0	0
$778$	−8.77150	−0.314473
$779$	1.72675	0.0618672
$780$	0	0
$781$	0	0
$782$	−12.1436	−0.434255
$783$	0	0
$784$	12.1089	0.432461
$785$	−0.528086	−0.0188482
$786$	0	0
$787$	2.47643	0.0882752	0.0441376	−	0.999025i	$-0.485946\pi$
$787$	2.47643	0.0882752	0.0441376	+	0.999025i	$0.485946\pi$
$788$	18.2018	0.648411
$789$	0	0
$790$	3.35067	0.119212
$791$	−55.9640	−1.98985
$792$	0	0
$793$	15.4047	0.547037
$794$	10.7612	0.381900
$795$	0	0
$796$	−46.5846	−1.65115
$797$	9.67969	0.342872	0.171436	−	0.985195i	$-0.445159\pi$
$797$	9.67969	0.342872	0.171436	+	0.985195i	$0.445159\pi$
$798$	0	0
$799$	−3.56980	−0.126290
$800$	10.3488	0.365884
$801$	0	0
$802$	2.65237	0.0936586
$803$	0	0
$804$	0	0
$805$	−43.6468	−1.53835
$806$	7.64482	0.269277
$807$	0	0
$808$	13.1209	0.461593
$809$	−7.86532	−0.276530	−0.138265	−	0.990395i	$-0.544153\pi$
$809$	−7.86532	−0.276530	−0.138265	+	0.990395i	$0.544153\pi$
$810$	0	0
$811$	−39.5047	−1.38720	−0.693599	−	0.720361i	$-0.743976\pi$
$811$	−39.5047	−1.38720	−0.693599	+	0.720361i	$0.743976\pi$
$812$	10.1486	0.356147
$813$	0	0
$814$	0	0
$815$	−18.7799	−0.657831
$816$	0	0
$817$	−42.4655	−1.48568
$818$	11.5855	0.405079
$819$	0	0
$820$	1.17473	0.0410234
$821$	−3.35362	−0.117042	−0.0585211	−	0.998286i	$-0.518638\pi$
$821$	−3.35362	−0.117042	−0.0585211	+	0.998286i	$0.518638\pi$
$822$	0	0
$823$	16.1018	0.561274	0.280637	−	0.959814i	$-0.409454\pi$
$823$	16.1018	0.561274	0.280637	+	0.959814i	$0.409454\pi$
$824$	−3.05046	−0.106268
$825$	0	0
$826$	6.11681	0.212831
$827$	15.2905	0.531702	0.265851	−	0.964014i	$-0.414347\pi$
$827$	15.2905	0.531702	0.265851	+	0.964014i	$0.414347\pi$
$828$	0	0
$829$	−42.1463	−1.46380	−0.731901	−	0.681411i	$-0.761367\pi$
$829$	−42.1463	−1.46380	−0.731901	+	0.681411i	$0.761367\pi$
$830$	−0.104703	−0.00363428
$831$	0	0
$832$	−11.1576	−0.386822
$833$	−15.3057	−0.530311
$834$	0	0
$835$	30.0193	1.03886
$836$	0	0
$837$	0	0
$838$	−8.89330	−0.307214
$839$	−2.64739	−0.0913981	−0.0456990	−	0.998955i	$-0.514552\pi$
$839$	−2.64739	−0.0913981	−0.0456990	+	0.998955i	$0.514552\pi$
$840$	0	0
$841$	−26.1395	−0.901364
$842$	6.28431	0.216572
$843$	0	0
$844$	−12.8154	−0.441125
$845$	−5.09616	−0.175313
$846$	0	0
$847$	0	0
$848$	32.8464	1.12795
$849$	0	0
$850$	−3.37487	−0.115757
$851$	−22.9670	−0.787298
$852$	0	0
$853$	−9.27037	−0.317411	−0.158706	−	0.987326i	$-0.550732\pi$
$853$	−9.27037	−0.317411	−0.158706	+	0.987326i	$0.550732\pi$
$854$	6.44575	0.220569
$855$	0	0
$856$	6.09770	0.208415
$857$	−20.5356	−0.701483	−0.350742	−	0.936472i	$-0.614070\pi$
$857$	−20.5356	−0.701483	−0.350742	+	0.936472i	$0.614070\pi$
$858$	0	0
$859$	36.0278	1.22925	0.614626	−	0.788819i	$-0.289307\pi$
$859$	36.0278	1.22925	0.614626	+	0.788819i	$0.289307\pi$
$860$	−28.8899	−0.985137
$861$	0	0
$862$	3.22704	0.109913
$863$	−4.31646	−0.146934	−0.0734670	−	0.997298i	$-0.523406\pi$
$863$	−4.31646	−0.146934	−0.0734670	+	0.997298i	$0.523406\pi$
$864$	0	0
$865$	−21.1229	−0.718199
$866$	−6.86583	−0.233310
$867$	0	0
$868$	−23.3932	−0.794015
$869$	0	0
$870$	0	0
$871$	12.3984	0.420102
$872$	10.1549	0.343889
$873$	0	0
$874$	16.2339	0.549121
$875$	−41.2589	−1.39481
$876$	0	0
$877$	−31.1160	−1.05071	−0.525356	−	0.850882i	$-0.676068\pi$
$877$	−31.1160	−1.05071	−0.525356	+	0.850882i	$0.676068\pi$
$878$	12.3498	0.416785
$879$	0	0
$880$	0	0
$881$	−43.9267	−1.47993	−0.739964	−	0.672647i	$-0.765157\pi$
$881$	−43.9267	−1.47993	−0.739964	+	0.672647i	$0.765157\pi$
$882$	0	0
$883$	−5.87946	−0.197860	−0.0989298	−	0.995094i	$-0.531542\pi$
$883$	−5.87946	−0.197860	−0.0989298	+	0.995094i	$0.531542\pi$
$884$	23.2444	0.781794
$885$	0	0
$886$	13.1563	0.441993
$887$	−7.69510	−0.258376	−0.129188	−	0.991620i	$-0.541237\pi$
$887$	−7.69510	−0.258376	−0.129188	+	0.991620i	$0.541237\pi$
$888$	0	0
$889$	36.8163	1.23478
$890$	6.64689	0.222804
$891$	0	0
$892$	−41.3640	−1.38497
$893$	4.77220	0.159696
$894$	0	0
$895$	37.3788	1.24944
$896$	−38.5711	−1.28857
$897$	0	0
$898$	6.95316	0.232030
$899$	−6.59352	−0.219906
$900$	0	0
$901$	−41.5180	−1.38316
$902$	0	0
$903$	0	0
$904$	30.2581	1.00637
$905$	7.48212	0.248714
$906$	0	0
$907$	−18.9906	−0.630571	−0.315285	−	0.948997i	$-0.602100\pi$
$907$	−18.9906	−0.630571	−0.315285	+	0.948997i	$0.602100\pi$
$908$	20.9761	0.696116
$909$	0	0
$910$	−11.4241	−0.378705
$911$	−50.0353	−1.65774	−0.828871	−	0.559439i	$-0.811017\pi$
$911$	−50.0353	−1.65774	−0.828871	+	0.559439i	$0.811017\pi$
$912$	0	0
$913$	0	0
$914$	12.0864	0.399783
$915$	0	0
$916$	−5.37954	−0.177745
$917$	72.1519	2.38266
$918$	0	0
$919$	−3.07419	−0.101408	−0.0507042	−	0.998714i	$-0.516147\pi$
$919$	−3.07419	−0.101408	−0.0507042	+	0.998714i	$0.516147\pi$
$920$	23.5985	0.778021
$921$	0	0
$922$	12.8732	0.423956
$923$	−28.0540	−0.923410
$924$	0	0
$925$	−6.38283	−0.209866
$926$	−6.22544	−0.204581
$927$	0	0
$928$	−8.40617	−0.275946
$929$	0.905574	0.0297109	0.0148555	−	0.999890i	$-0.495271\pi$
$929$	0.905574	0.0297109	0.0148555	+	0.999890i	$0.495271\pi$
$930$	0	0
$931$	20.4611	0.670585
$932$	43.1609	1.41378
$933$	0	0
$934$	−13.5887	−0.444634
$935$	0	0
$936$	0	0
$937$	22.0542	0.720481	0.360240	−	0.932860i	$-0.382695\pi$
$937$	22.0542	0.720481	0.360240	+	0.932860i	$0.382695\pi$
$938$	5.18781	0.169388
$939$	0	0
$940$	3.24660	0.105892
$941$	5.46092	0.178021	0.0890105	−	0.996031i	$-0.471630\pi$
$941$	5.46092	0.178021	0.0890105	+	0.996031i	$0.471630\pi$
$942$	0	0
$943$	−2.92844	−0.0953630
$944$	9.55957	0.311138
$945$	0	0
$946$	0	0
$947$	−32.4406	−1.05418	−0.527089	−	0.849810i	$-0.676717\pi$
$947$	−32.4406	−1.05418	−0.527089	+	0.849810i	$0.676717\pi$
$948$	0	0
$949$	60.1392	1.95220
$950$	4.51163	0.146377
$951$	0	0
$952$	20.7822	0.673554
$953$	−53.8088	−1.74304	−0.871518	−	0.490363i	$-0.836864\pi$
$953$	−53.8088	−1.74304	−0.871518	+	0.490363i	$0.836864\pi$
$954$	0	0
$955$	21.3774	0.691756
$956$	12.2290	0.395516
$957$	0	0
$958$	7.66260	0.247567
$959$	10.9155	0.352479
$960$	0	0
$961$	−15.8016	−0.509728
$962$	−6.01137	−0.193814
$963$	0	0
$964$	−13.6717	−0.440335
$965$	−21.4645	−0.690966
$966$	0	0
$967$	28.8151	0.926630	0.463315	−	0.886194i	$-0.346660\pi$
$967$	28.8151	0.926630	0.463315	+	0.886194i	$0.346660\pi$
$968$	0	0
$969$	0	0
$970$	0.287516	0.00923160
$971$	−5.39405	−0.173103	−0.0865517	−	0.996247i	$-0.527585\pi$
$971$	−5.39405	−0.173103	−0.0865517	+	0.996247i	$0.527585\pi$
$972$	0	0
$973$	−51.1362	−1.63935
$974$	−13.2075	−0.423197
$975$	0	0
$976$	10.0736	0.322450
$977$	3.31236	0.105972	0.0529859	−	0.998595i	$-0.483126\pi$
$977$	3.31236	0.105972	0.0529859	+	0.998595i	$0.483126\pi$
$978$	0	0
$979$	0	0
$980$	13.9200	0.444657
$981$	0	0
$982$	−19.1291	−0.610435
$983$	−33.8721	−1.08035	−0.540176	−	0.841552i	$-0.681643\pi$
$983$	−33.8721	−1.08035	−0.540176	+	0.841552i	$0.681643\pi$
$984$	0	0
$985$	17.6717	0.563067
$986$	2.74137	0.0873029
$987$	0	0
$988$	−31.0738	−0.988589
$989$	72.0183	2.29005
$990$	0	0
$991$	8.66640	0.275297	0.137649	−	0.990481i	$-0.456046\pi$
$991$	8.66640	0.275297	0.137649	+	0.990481i	$0.456046\pi$
$992$	19.3767	0.615211
$993$	0	0
$994$	−11.7386	−0.372325
$995$	−45.2280	−1.43382
$996$	0	0
$997$	30.2654	0.958515	0.479258	−	0.877674i	$-0.340906\pi$
$997$	30.2654	0.958515	0.479258	+	0.877674i	$0.340906\pi$
$998$	11.7891	0.373177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cn.1.9		18
3.2	odd	2		9801.2.a.co.1.10		18
9.2	odd	6		1089.2.e.o.364.9		36
9.5	odd	6		1089.2.e.o.727.9		36
11.2	odd	10		891.2.f.e.730.5		36
11.6	odd	10		891.2.f.e.487.5		36
11.10	odd	2		9801.2.a.cp.1.10		18
33.2	even	10		891.2.f.f.730.5		36
33.17	even	10		891.2.f.f.487.5		36
33.32	even	2		9801.2.a.cm.1.9		18
99.2	even	30		99.2.m.b.4.5	✓	72
99.13	odd	30		297.2.n.b.235.5		72
99.32	even	6		1089.2.e.p.727.10		36
99.50	even	30		99.2.m.b.25.5	yes	72
99.61	odd	30		297.2.n.b.91.5		72
99.65	even	6		1089.2.e.p.364.10		36
99.68	even	30		99.2.m.b.70.5	yes	72
99.79	odd	30		297.2.n.b.37.5		72
99.83	even	30		99.2.m.b.58.5	yes	72
99.94	odd	30		297.2.n.b.289.5		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.4.5	✓	72	99.2	even	30
99.2.m.b.25.5	yes	72	99.50	even	30
99.2.m.b.58.5	yes	72	99.83	even	30
99.2.m.b.70.5	yes	72	99.68	even	30
297.2.n.b.37.5		72	99.79	odd	30
297.2.n.b.91.5		72	99.61	odd	30
297.2.n.b.235.5		72	99.13	odd	30
297.2.n.b.289.5		72	99.94	odd	30
891.2.f.e.487.5		36	11.6	odd	10
891.2.f.e.730.5		36	11.2	odd	10
891.2.f.f.487.5		36	33.17	even	10
891.2.f.f.730.5		36	33.2	even	10
1089.2.e.o.364.9		36	9.2	odd	6
1089.2.e.o.727.9		36	9.5	odd	6
1089.2.e.p.364.10		36	99.65	even	6
1089.2.e.p.727.10		36	99.32	even	6
9801.2.a.cm.1.9		18	33.32	even	2
9801.2.a.cn.1.9		18	1.1	even	1	trivial
9801.2.a.co.1.10		18	3.2	odd	2
9801.2.a.cp.1.10		18	11.10	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q-0.490494 q^{2} -1.75942 q^{4} -1.70818 q^{5} -3.41052 q^{7} +1.84397 q^{8} +0.837851 q^{10} +3.99792 q^{13} +1.67284 q^{14} +2.61437 q^{16} -3.30458 q^{17} +4.41765 q^{19} +3.00539 q^{20} -7.49201 q^{23} -2.08213 q^{25} -1.96096 q^{26} +6.00053 q^{28} +1.69129 q^{29} -3.89852 q^{31} -4.97028 q^{32} +1.62087 q^{34} +5.82578 q^{35} +3.06553 q^{37} -2.16683 q^{38} -3.14983 q^{40} +0.390875 q^{41} -9.61268 q^{43} +3.67479 q^{46} +1.08026 q^{47} +4.63167 q^{49} +1.02127 q^{50} -7.03401 q^{52} +12.5638 q^{53} -6.28890 q^{56} -0.829567 q^{58} +3.65654 q^{59} +3.85318 q^{61} +1.91220 q^{62} -2.79086 q^{64} -6.82916 q^{65} +3.10120 q^{67} +5.81412 q^{68} -2.85751 q^{70} -7.01715 q^{71} +15.0426 q^{73} -1.50362 q^{74} -7.77248 q^{76} +3.99913 q^{79} -4.46581 q^{80} -0.191722 q^{82} -0.124966 q^{83} +5.64480 q^{85} +4.71496 q^{86} +7.93327 q^{89} -13.6350 q^{91} +13.1816 q^{92} -0.529860 q^{94} -7.54613 q^{95} +0.343160 q^{97} -2.27180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{2} + 12 q^{4} + q^{5} - q^{7} - 6 q^{8} + 2 q^{10} - 3 q^{13} - 8 q^{16} - 20 q^{17} + 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} - 19 q^{28} - 21 q^{29} + 6 q^{31} - 9 q^{32} - 4 q^{34}+ \cdots - 82 q^{98}+O(q^{100}) \) 18 * q - 2 * q^2 + 12 * q^4 + q^5 - q^7 - 6 * q^8 + 2 * q^10 - 3 * q^13 - 8 * q^16 - 20 * q^17 + 3 * q^19 + 5 * q^20 + 10 * q^23 + 7 * q^25 - 2 * q^26 - 19 * q^28 - 21 * q^29 + 6 * q^31 - 9 * q^32 - 4 * q^34 - 38 * q^35 + 7 * q^37 - 13 * q^38 - 20 * q^41 - 4 * q^43 - 8 * q^46 - 7 * q^47 + 7 * q^49 - 25 * q^50 + 19 * q^52 - 31 * q^53 + 57 * q^56 + 12 * q^58 + 12 * q^59 + 16 * q^61 - 19 * q^62 - 16 * q^64 - 42 * q^65 - 5 * q^67 - 51 * q^68 + 8 * q^70 - 13 * q^71 - 9 * q^74 + 8 * q^76 + 2 * q^79 - 46 * q^80 - 34 * q^82 - 36 * q^83 - 25 * q^85 - 26 * q^86 + 14 * q^89 - 15 * q^91 + 15 * q^92 - 4 * q^94 - 64 * q^95 - 16 * q^97 - 82 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more