LMFDB - Embedded newform 9801.2.a.bl.1.3

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:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
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			1.3
Root			$-2.27060$ 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)

$f(q)$	$=$	$q+2.15561 q^{2} +2.64667 q^{4} -3.62393 q^{5} +2.27060 q^{7} +1.39396 q^{8} +O(q^{10})$	$q+2.15561 q^{2} +2.64667 q^{4} -3.62393 q^{5} +2.27060 q^{7} +1.39396 q^{8} -7.81179 q^{10} -1.23835 q^{13} +4.89453 q^{14} -2.28849 q^{16} +5.69681 q^{17} +2.89453 q^{19} -9.59134 q^{20} -5.91726 q^{23} +8.13288 q^{25} -2.66940 q^{26} +6.00951 q^{28} -3.50895 q^{29} -2.50895 q^{31} -7.72102 q^{32} +12.2801 q^{34} -8.22849 q^{35} -0.333960 q^{37} +6.23948 q^{38} -5.05162 q^{40} -9.96741 q^{41} +7.15561 q^{43} -12.7553 q^{46} +8.69681 q^{47} -1.84439 q^{49} +17.5313 q^{50} -3.27750 q^{52} -6.16513 q^{53} +3.16513 q^{56} -7.56393 q^{58} -2.91726 q^{59} -6.27060 q^{61} -5.40832 q^{62} -12.0666 q^{64} +4.48769 q^{65} +9.36285 q^{67} +15.0775 q^{68} -17.7374 q^{70} +12.1230 q^{71} -4.31271 q^{73} -0.719889 q^{74} +7.66085 q^{76} -1.41670 q^{79} +8.29333 q^{80} -21.4859 q^{82} -2.75681 q^{83} -20.6448 q^{85} +15.4247 q^{86} -4.77116 q^{89} -2.81179 q^{91} -15.6610 q^{92} +18.7469 q^{94} -10.4896 q^{95} -2.55909 q^{97} -3.97578 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 11 q^{4} - 4 q^{5} - q^{7}+O(q^{10}) $ 4 * q + q^2 + 11 * q^4 - 4 * q^5 - q^7	$ 4 q + q^{2} + 11 q^{4} - 4 q^{5} - q^{7} - q^{10} - 7 q^{13} - q^{14} + 17 q^{16} - 5 q^{17} - 9 q^{19} + 10 q^{20} - 14 q^{23} + 14 q^{25} - 22 q^{26} + q^{28} - 6 q^{29} - 2 q^{31} - 34 q^{32} + 16 q^{34} - 8 q^{35} + 3 q^{37} - 3 q^{38} - 12 q^{40} - 2 q^{41} + 21 q^{43} - 2 q^{46} + 7 q^{47} - 15 q^{49} + 23 q^{50} + 10 q^{52} + 6 q^{53} - 18 q^{56} - 21 q^{58} - 2 q^{59} - 15 q^{61} - 20 q^{62} + 16 q^{64} + 19 q^{65} + 14 q^{67} - 7 q^{68} - 38 q^{70} + 3 q^{71} - 22 q^{73} - 36 q^{74} - 42 q^{76} - 11 q^{79} + 34 q^{80} - 17 q^{82} + 18 q^{83} - 13 q^{85} + 24 q^{86} + 6 q^{89} + 19 q^{91} - 67 q^{92} + 19 q^{94} - 30 q^{95} + 26 q^{97} + 15 q^{98}+O(q^{100}) $ 4 * q + q^2 + 11 * q^4 - 4 * q^5 - q^7 - q^10 - 7 * q^13 - q^14 + 17 * q^16 - 5 * q^17 - 9 * q^19 + 10 * q^20 - 14 * q^23 + 14 * q^25 - 22 * q^26 + q^28 - 6 * q^29 - 2 * q^31 - 34 * q^32 + 16 * q^34 - 8 * q^35 + 3 * q^37 - 3 * q^38 - 12 * q^40 - 2 * q^41 + 21 * q^43 - 2 * q^46 + 7 * q^47 - 15 * q^49 + 23 * q^50 + 10 * q^52 + 6 * q^53 - 18 * q^56 - 21 * q^58 - 2 * q^59 - 15 * q^61 - 20 * q^62 + 16 * q^64 + 19 * q^65 + 14 * q^67 - 7 * q^68 - 38 * q^70 + 3 * q^71 - 22 * q^73 - 36 * q^74 - 42 * q^76 - 11 * q^79 + 34 * q^80 - 17 * q^82 + 18 * q^83 - 13 * q^85 + 24 * q^86 + 6 * q^89 + 19 * q^91 - 67 * q^92 + 19 * q^94 - 30 * q^95 + 26 * q^97 + 15 * q^98

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 $

$2$	2.15561	1.52425	0.762124	−	0.647431i	$-0.224157\pi$
$2$	2.15561	1.52425	0.762124	+	0.647431i	$0.224157\pi$
$3$	0	0
$3$	0	0
$4$	2.64667	1.32333
$5$	−3.62393	−1.62067	−0.810336	−	0.585966i	$-0.800715\pi$
$5$	−3.62393	−1.62067	−0.810336	+	0.585966i	$0.800715\pi$
$6$	0	0
$7$	2.27060	0.858205	0.429103	−	0.903256i	$-0.358830\pi$
$7$	2.27060	0.858205	0.429103	+	0.903256i	$0.358830\pi$
$8$	1.39396	0.492840
$9$	0	0
$10$	−7.81179	−2.47031
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.23835	−0.343456	−0.171728	−	0.985144i	$-0.554935\pi$
$13$	−1.23835	−0.343456	−0.171728	+	0.985144i	$0.554935\pi$
$14$	4.89453	1.30812
$15$	0	0
$16$	−2.28849	−0.572123
$17$	5.69681	1.38168	0.690839	−	0.723008i	$-0.257241\pi$
$17$	5.69681	1.38168	0.690839	+	0.723008i	$0.257241\pi$
$18$	0	0
$19$	2.89453	0.664050	0.332025	−	0.943271i	$-0.392268\pi$
$19$	2.89453	0.664050	0.332025	+	0.943271i	$0.392268\pi$
$20$	−9.59134	−2.14469
$21$	0	0
$22$	0	0
$23$	−5.91726	−1.23383	−0.616917	−	0.787028i	$-0.711619\pi$
$23$	−5.91726	−1.23383	−0.616917	+	0.787028i	$0.711619\pi$
$24$	0	0
$25$	8.13288	1.62658
$26$	−2.66940	−0.523513
$27$	0	0
$28$	6.00951	1.13569
$29$	−3.50895	−0.651595	−0.325798	−	0.945440i	$-0.605633\pi$
$29$	−3.50895	−0.651595	−0.325798	+	0.945440i	$0.605633\pi$
$30$	0	0
$31$	−2.50895	−0.450620	−0.225310	−	0.974287i	$-0.572340\pi$
$31$	−2.50895	−0.450620	−0.225310	+	0.974287i	$0.572340\pi$
$32$	−7.72102	−1.36490
$33$	0	0
$34$	12.2801	2.10602
$35$	−8.22849	−1.39087
$36$	0	0
$37$	−0.333960	−0.0549027	−0.0274514	−	0.999623i	$-0.508739\pi$
$37$	−0.333960	−0.0549027	−0.0274514	+	0.999623i	$0.508739\pi$
$38$	6.23948	1.01218
$39$	0	0
$40$	−5.05162	−0.798732
$41$	−9.96741	−1.55665	−0.778324	−	0.627863i	$-0.783930\pi$
$41$	−9.96741	−1.55665	−0.778324	+	0.627863i	$0.783930\pi$
$42$	0	0
$43$	7.15561	1.09122	0.545610	−	0.838039i	$-0.316298\pi$
$43$	7.15561	1.09122	0.545610	+	0.838039i	$0.316298\pi$
$44$	0	0
$45$	0	0
$46$	−12.7553	−1.88067
$47$	8.69681	1.26856	0.634280	−	0.773104i	$-0.281297\pi$
$47$	8.69681	1.26856	0.634280	+	0.773104i	$0.281297\pi$
$48$	0	0
$49$	−1.84439	−0.263484
$50$	17.5313	2.47931
$51$	0	0
$52$	−3.27750	−0.454507
$53$	−6.16513	−0.846845	−0.423423	−	0.905932i	$-0.639171\pi$
$53$	−6.16513	−0.846845	−0.423423	+	0.905932i	$0.639171\pi$
$54$	0	0
$55$	0	0
$56$	3.16513	0.422958
$57$	0	0
$58$	−7.56393	−0.993193
$59$	−2.91726	−0.379795	−0.189898	−	0.981804i	$-0.560816\pi$
$59$	−2.91726	−0.379795	−0.189898	+	0.981804i	$0.560816\pi$
$60$	0	0
$61$	−6.27060	−0.802868	−0.401434	−	0.915888i	$-0.631488\pi$
$61$	−6.27060	−0.802868	−0.401434	+	0.915888i	$0.631488\pi$
$62$	−5.40832	−0.686857
$63$	0	0
$64$	−12.0666	−1.50832
$65$	4.48769	0.556630
$66$	0	0
$67$	9.36285	1.14385	0.571927	−	0.820305i	$-0.306196\pi$
$67$	9.36285	1.14385	0.571927	+	0.820305i	$0.306196\pi$
$68$	15.0775	1.82842
$69$	0	0
$70$	−17.7374	−2.12003
$71$	12.1230	1.43874	0.719369	−	0.694628i	$-0.244431\pi$
$71$	12.1230	1.43874	0.719369	+	0.694628i	$0.244431\pi$
$72$	0	0
$73$	−4.31271	−0.504764	−0.252382	−	0.967628i	$-0.581214\pi$
$73$	−4.31271	−0.504764	−0.252382	+	0.967628i	$0.581214\pi$
$74$	−0.719889	−0.0836854
$75$	0	0
$76$	7.66085	0.878760
$77$	0	0
$78$	0	0
$79$	−1.41670	−0.159391	−0.0796954	−	0.996819i	$-0.525395\pi$
$79$	−1.41670	−0.159391	−0.0796954	+	0.996819i	$0.525395\pi$
$80$	8.29333	0.927223
$81$	0	0
$82$	−21.4859	−2.37272
$83$	−2.75681	−0.302599	−0.151300	−	0.988488i	$-0.548346\pi$
$83$	−2.75681	−0.302599	−0.151300	+	0.988488i	$0.548346\pi$
$84$	0	0
$85$	−20.6448	−2.23925
$86$	15.4247	1.66329
$87$	0	0
$88$	0	0
$89$	−4.77116	−0.505742	−0.252871	−	0.967500i	$-0.581375\pi$
$89$	−4.77116	−0.505742	−0.252871	+	0.967500i	$0.581375\pi$
$90$	0	0
$91$	−2.81179	−0.294756
$92$	−15.6610	−1.63277
$93$	0	0
$94$	18.7469	1.93360
$95$	−10.4896	−1.07621
$96$	0	0
$97$	−2.55909	−0.259836	−0.129918	−	0.991525i	$-0.541471\pi$
$97$	−2.55909	−0.259836	−0.129918	+	0.991525i	$0.541471\pi$
$98$	−3.97578	−0.401615
$99$	0	0
$100$	21.5250	2.15250
$101$	−18.3336	−1.82426	−0.912131	−	0.409898i	$-0.865564\pi$
$101$	−18.3336	−1.82426	−0.912131	+	0.409898i	$0.865564\pi$
$102$	0	0
$103$	−4.56245	−0.449551	−0.224776	−	0.974411i	$-0.572165\pi$
$103$	−4.56245	−0.449551	−0.224776	+	0.974411i	$0.572165\pi$
$104$	−1.72621	−0.169269
$105$	0	0
$106$	−13.2896	−1.29080
$107$	−5.11014	−0.494016	−0.247008	−	0.969013i	$-0.579447\pi$
$107$	−5.11014	−0.494016	−0.247008	+	0.969013i	$0.579447\pi$
$108$	0	0
$109$	−4.67891	−0.448159	−0.224079	−	0.974571i	$-0.571937\pi$
$109$	−4.67891	−0.448159	−0.224079	+	0.974571i	$0.571937\pi$
$110$	0	0
$111$	0	0
$112$	−5.19624	−0.490999
$113$	1.53168	0.144088	0.0720442	−	0.997401i	$-0.477048\pi$
$113$	1.53168	0.144088	0.0720442	+	0.997401i	$0.477048\pi$
$114$	0	0
$115$	21.4438	1.99964
$116$	−9.28701	−0.862277
$117$	0	0
$118$	−6.28849	−0.578902
$119$	12.9352	1.18576
$120$	0	0
$121$	0	0
$122$	−13.5170	−1.22377
$123$	0	0
$124$	−6.64034	−0.596320
$125$	−11.3533	−1.01547
$126$	0	0
$127$	−8.72758	−0.774447	−0.387224	−	0.921986i	$-0.626566\pi$
$127$	−8.72758	−0.774447	−0.387224	+	0.921986i	$0.626566\pi$
$128$	−10.5688	−0.934156
$129$	0	0
$130$	9.67373	0.848442
$131$	−3.68877	−0.322290	−0.161145	−	0.986931i	$-0.551519\pi$
$131$	−3.68877	−0.322290	−0.161145	+	0.986931i	$0.551519\pi$
$132$	0	0
$133$	6.57231	0.569892
$134$	20.1827	1.74352
$135$	0	0
$136$	7.94113	0.680946
$137$	−21.9685	−1.87690	−0.938449	−	0.345417i	$-0.887738\pi$
$137$	−21.9685	−1.87690	−0.938449	+	0.345417i	$0.887738\pi$
$138$	0	0
$139$	1.28997	0.109414	0.0547070	−	0.998502i	$-0.482578\pi$
$139$	1.28997	0.109414	0.0547070	+	0.998502i	$0.482578\pi$
$140$	−21.7781	−1.84058
$141$	0	0
$142$	26.1325	2.19299
$143$	0	0
$144$	0	0
$145$	12.7162	1.05602
$146$	−9.29652	−0.769386
$147$	0	0
$148$	−0.883882	−0.0726546
$149$	−10.9447	−0.896622	−0.448311	−	0.893878i	$-0.647974\pi$
$149$	−10.9447	−0.896622	−0.448311	+	0.893878i	$0.647974\pi$
$150$	0	0
$151$	−2.13772	−0.173965	−0.0869826	−	0.996210i	$-0.527722\pi$
$151$	−2.13772	−0.173965	−0.0869826	+	0.996210i	$0.527722\pi$
$152$	4.03486	0.327271
$153$	0	0
$154$	0	0
$155$	9.09225	0.730307
$156$	0	0
$157$	9.50057	0.758228	0.379114	−	0.925350i	$-0.376229\pi$
$157$	9.50057	0.758228	0.379114	+	0.925350i	$0.376229\pi$
$158$	−3.05385	−0.242951
$159$	0	0
$160$	27.9805	2.21205
$161$	−13.4357	−1.05888
$162$	0	0
$163$	−19.4713	−1.52511	−0.762556	−	0.646922i	$-0.776056\pi$
$163$	−19.4713	−1.52511	−0.762556	+	0.646922i	$0.776056\pi$
$164$	−26.3804	−2.05996
$165$	0	0
$166$	−5.94261	−0.461236
$167$	−15.2980	−1.18380	−0.591898	−	0.806013i	$-0.701621\pi$
$167$	−15.2980	−1.18380	−0.591898	+	0.806013i	$0.701621\pi$
$168$	0	0
$169$	−11.4665	−0.882038
$170$	−44.5023	−3.41317
$171$	0	0
$172$	18.9385	1.44405
$173$	14.0209	1.06599	0.532995	−	0.846118i	$-0.321066\pi$
$173$	14.0209	1.06599	0.532995	+	0.846118i	$0.321066\pi$
$174$	0	0
$175$	18.4665	1.39594
$176$	0	0
$177$	0	0
$178$	−10.2848	−0.770877
$179$	−3.94951	−0.295200	−0.147600	−	0.989047i	$-0.547155\pi$
$179$	−3.94951	−0.295200	−0.147600	+	0.989047i	$0.547155\pi$
$180$	0	0
$181$	13.3662	0.993502	0.496751	−	0.867893i	$-0.334526\pi$
$181$	13.3662	0.993502	0.496751	+	0.867893i	$0.334526\pi$
$182$	−6.06114	−0.449281
$183$	0	0
$184$	−8.24844	−0.608083
$185$	1.21025	0.0889793
$186$	0	0
$187$	0	0
$188$	23.0175	1.67873
$189$	0	0
$190$	−22.6115	−1.64041
$191$	−18.9205	−1.36904	−0.684518	−	0.728996i	$-0.739987\pi$
$191$	−18.9205	−1.36904	−0.684518	+	0.728996i	$0.739987\pi$
$192$	0	0
$193$	2.06450	0.148606	0.0743029	−	0.997236i	$-0.476327\pi$
$193$	2.06450	0.148606	0.0743029	+	0.997236i	$0.476327\pi$
$194$	−5.51640	−0.396055
$195$	0	0
$196$	−4.88148	−0.348677
$197$	−6.00148	−0.427588	−0.213794	−	0.976879i	$-0.568582\pi$
$197$	−6.00148	−0.427588	−0.213794	+	0.976879i	$0.568582\pi$
$198$	0	0
$199$	19.7132	1.39743	0.698717	−	0.715399i	$-0.253755\pi$
$199$	19.7132	1.39743	0.698717	+	0.715399i	$0.253755\pi$
$200$	11.3369	0.801641
$201$	0	0
$202$	−39.5202	−2.78063
$203$	−7.96741	−0.559202
$204$	0	0
$205$	36.1212	2.52281
$206$	−9.83487	−0.685228
$207$	0	0
$208$	2.83395	0.196499
$209$	0	0
$210$	0	0
$211$	11.6825	0.804253	0.402127	−	0.915584i	$-0.368271\pi$
$211$	11.6825	0.804253	0.402127	+	0.915584i	$0.368271\pi$
$212$	−16.3170	−1.12066
$213$	0	0
$214$	−11.0155	−0.753003
$215$	−25.9314	−1.76851
$216$	0	0
$217$	−5.69681	−0.386724
$218$	−10.0859	−0.683105
$219$	0	0
$220$	0	0
$221$	−7.05464	−0.474546
$222$	0	0
$223$	−18.1234	−1.21363	−0.606815	−	0.794843i	$-0.707553\pi$
$223$	−18.1234	−1.21363	−0.606815	+	0.794843i	$0.707553\pi$
$224$	−17.5313	−1.17136
$225$	0	0
$226$	3.30171	0.219627
$227$	17.5075	1.16201	0.581006	−	0.813900i	$-0.302659\pi$
$227$	17.5075	1.16201	0.581006	+	0.813900i	$0.302659\pi$
$228$	0	0
$229$	4.54268	0.300188	0.150094	−	0.988672i	$-0.452042\pi$
$229$	4.54268	0.300188	0.150094	+	0.988672i	$0.452042\pi$
$230$	46.2244	3.04795
$231$	0	0
$232$	−4.89134	−0.321132
$233$	−8.16962	−0.535210	−0.267605	−	0.963529i	$-0.586232\pi$
$233$	−8.16962	−0.535210	−0.267605	+	0.963529i	$0.586232\pi$
$234$	0	0
$235$	−31.5166	−2.05592
$236$	−7.72102	−0.502596
$237$	0	0
$238$	27.8832	1.80740
$239$	18.6008	1.20319	0.601594	−	0.798802i	$-0.294532\pi$
$239$	18.6008	1.20319	0.601594	+	0.798802i	$0.294532\pi$
$240$	0	0
$241$	−28.3526	−1.82635	−0.913177	−	0.407563i	$-0.866379\pi$
$241$	−28.3526	−1.82635	−0.913177	+	0.407563i	$0.866379\pi$
$242$	0	0
$243$	0	0
$244$	−16.5962	−1.06246
$245$	6.68393	0.427021
$246$	0	0
$247$	−3.58444	−0.228072
$248$	−3.49738	−0.222084
$249$	0	0
$250$	−24.4734	−1.54783
$251$	−4.93812	−0.311691	−0.155846	−	0.987781i	$-0.549810\pi$
$251$	−4.93812	−0.311691	−0.155846	+	0.987781i	$0.549810\pi$
$252$	0	0
$253$	0	0
$254$	−18.8133	−1.18045
$255$	0	0
$256$	1.35093	0.0844331
$257$	−17.1135	−1.06751	−0.533756	−	0.845639i	$-0.679220\pi$
$257$	−17.1135	−1.06751	−0.533756	+	0.845639i	$0.679220\pi$
$258$	0	0
$259$	−0.758289	−0.0471178
$260$	11.8774	0.736606
$261$	0	0
$262$	−7.95157	−0.491250
$263$	−4.44558	−0.274126	−0.137063	−	0.990562i	$-0.543766\pi$
$263$	−4.44558	−0.274126	−0.137063	+	0.990562i	$0.543766\pi$
$264$	0	0
$265$	22.3420	1.37246
$266$	14.1674	0.868656
$267$	0	0
$268$	24.7803	1.51370
$269$	−19.5899	−1.19441	−0.597207	−	0.802087i	$-0.703723\pi$
$269$	−19.5899	−1.19441	−0.597207	+	0.802087i	$0.703723\pi$
$270$	0	0
$271$	−6.77601	−0.411613	−0.205807	−	0.978593i	$-0.565982\pi$
$271$	−6.77601	−0.411613	−0.205807	+	0.978593i	$0.565982\pi$
$272$	−13.0371	−0.790490
$273$	0	0
$274$	−47.3557	−2.86086
$275$	0	0
$276$	0	0
$277$	−1.47652	−0.0887156	−0.0443578	−	0.999016i	$-0.514124\pi$
$277$	−1.47652	−0.0887156	−0.0443578	+	0.999016i	$0.514124\pi$
$278$	2.78068	0.166774
$279$	0	0
$280$	−11.4702	−0.685476
$281$	−7.34684	−0.438275	−0.219138	−	0.975694i	$-0.570324\pi$
$281$	−7.34684	−0.438275	−0.219138	+	0.975694i	$0.570324\pi$
$282$	0	0
$283$	14.7162	0.874786	0.437393	−	0.899270i	$-0.355902\pi$
$283$	14.7162	0.874786	0.437393	+	0.899270i	$0.355902\pi$
$284$	32.0856	1.90393
$285$	0	0
$286$	0	0
$287$	−22.6320	−1.33592
$288$	0	0
$289$	15.4536	0.909036
$290$	27.4112	1.60964
$291$	0	0
$292$	−11.4143	−0.667971
$293$	−1.97613	−0.115447	−0.0577234	−	0.998333i	$-0.518384\pi$
$293$	−1.97613	−0.115447	−0.0577234	+	0.998333i	$0.518384\pi$
$294$	0	0
$295$	10.5720	0.615523
$296$	−0.465528	−0.0270583
$297$	0	0
$298$	−23.5925	−1.36668
$299$	7.32764	0.423768
$300$	0	0
$301$	16.2475	0.936491
$302$	−4.60810	−0.265166
$303$	0	0
$304$	−6.62410	−0.379918
$305$	22.7242	1.30118
$306$	0	0
$307$	−11.9708	−0.683208	−0.341604	−	0.939844i	$-0.610970\pi$
$307$	−11.9708	−0.683208	−0.341604	+	0.939844i	$0.610970\pi$
$308$	0	0
$309$	0	0
$310$	19.5994	1.11317
$311$	−3.39732	−0.192645	−0.0963223	−	0.995350i	$-0.530708\pi$
$311$	−3.39732	−0.192645	−0.0963223	+	0.995350i	$0.530708\pi$
$312$	0	0
$313$	−22.6965	−1.28288	−0.641440	−	0.767173i	$-0.721663\pi$
$313$	−22.6965	−1.28288	−0.641440	+	0.767173i	$0.721663\pi$
$314$	20.4795	1.15573
$315$	0	0
$316$	−3.74952	−0.210927
$317$	−14.8509	−0.834112	−0.417056	−	0.908881i	$-0.636938\pi$
$317$	−14.8509	−0.834112	−0.417056	+	0.908881i	$0.636938\pi$
$318$	0	0
$319$	0	0
$320$	43.7284	2.44449
$321$	0	0
$322$	−28.9622	−1.61400
$323$	16.4896	0.917504
$324$	0	0
$325$	−10.0713	−0.558658
$326$	−41.9727	−2.32465
$327$	0	0
$328$	−13.8942	−0.767178
$329$	19.7469	1.08868
$330$	0	0
$331$	16.2028	0.890586	0.445293	−	0.895385i	$-0.353100\pi$
$331$	16.2028	0.890586	0.445293	+	0.895385i	$0.353100\pi$
$332$	−7.29635	−0.400439
$333$	0	0
$334$	−32.9766	−1.80440
$335$	−33.9303	−1.85381
$336$	0	0
$337$	11.2622	0.613492	0.306746	−	0.951791i	$-0.400760\pi$
$337$	11.2622	0.613492	0.306746	+	0.951791i	$0.400760\pi$
$338$	−24.7173	−1.34444
$339$	0	0
$340$	−54.6400	−2.96327
$341$	0	0
$342$	0	0
$343$	−20.0820	−1.08433
$344$	9.97465	0.537797
$345$	0	0
$346$	30.2236	1.62483
$347$	−17.0744	−0.916599	−0.458300	−	0.888798i	$-0.651541\pi$
$347$	−17.0744	−0.916599	−0.458300	+	0.888798i	$0.651541\pi$
$348$	0	0
$349$	−22.9724	−1.22969	−0.614843	−	0.788650i	$-0.710781\pi$
$349$	−22.9724	−1.22969	−0.614843	+	0.788650i	$0.710781\pi$
$350$	39.8066	2.12775
$351$	0	0
$352$	0	0
$353$	20.5134	1.09182	0.545910	−	0.837844i	$-0.316184\pi$
$353$	20.5134	1.09182	0.545910	+	0.837844i	$0.316184\pi$
$354$	0	0
$355$	−43.9330	−2.33172
$356$	−12.6277	−0.669266
$357$	0	0
$358$	−8.51362	−0.449959
$359$	−2.91840	−0.154027	−0.0770136	−	0.997030i	$-0.524538\pi$
$359$	−2.91840	−0.154027	−0.0770136	+	0.997030i	$0.524538\pi$
$360$	0	0
$361$	−10.6217	−0.559037
$362$	28.8124	1.51434
$363$	0	0
$364$	−7.44188	−0.390060
$365$	15.6289	0.818057
$366$	0	0
$367$	6.75139	0.352420	0.176210	−	0.984353i	$-0.443616\pi$
$367$	6.75139	0.352420	0.176210	+	0.984353i	$0.443616\pi$
$368$	13.5416	0.705905
$369$	0	0
$370$	2.60883	0.135627
$371$	−13.9985	−0.726767
$372$	0	0
$373$	22.3774	1.15866	0.579330	−	0.815093i	$-0.303314\pi$
$373$	22.3774	1.15866	0.579330	+	0.815093i	$0.303314\pi$
$374$	0	0
$375$	0	0
$376$	12.1230	0.625197
$377$	4.34530	0.223794
$378$	0	0
$379$	33.0321	1.69674	0.848372	−	0.529401i	$-0.177583\pi$
$379$	33.0321	1.69674	0.848372	+	0.529401i	$0.177583\pi$
$380$	−27.7624	−1.42418
$381$	0	0
$382$	−40.7852	−2.08675
$383$	−23.4842	−1.19999	−0.599993	−	0.800005i	$-0.704830\pi$
$383$	−23.4842	−1.19999	−0.599993	+	0.800005i	$0.704830\pi$
$384$	0	0
$385$	0	0
$386$	4.45026	0.226512
$387$	0	0
$388$	−6.77305	−0.343850
$389$	31.0414	1.57386	0.786931	−	0.617041i	$-0.211669\pi$
$389$	31.0414	1.57386	0.786931	+	0.617041i	$0.211669\pi$
$390$	0	0
$391$	−33.7095	−1.70476
$392$	−2.57101	−0.129855
$393$	0	0
$394$	−12.9369	−0.651750
$395$	5.13401	0.258320
$396$	0	0
$397$	17.3804	0.872297	0.436148	−	0.899875i	$-0.356342\pi$
$397$	17.3804	0.872297	0.436148	+	0.899875i	$0.356342\pi$
$398$	42.4941	2.13004
$399$	0	0
$400$	−18.6120	−0.930601
$401$	11.5479	0.576675	0.288338	−	0.957529i	$-0.406897\pi$
$401$	11.5479	0.576675	0.288338	+	0.957529i	$0.406897\pi$
$402$	0	0
$403$	3.10695	0.154768
$404$	−48.5230	−2.41411
$405$	0	0
$406$	−17.1746	−0.852363
$407$	0	0
$408$	0	0
$409$	−34.7356	−1.71757	−0.858783	−	0.512340i	$-0.828779\pi$
$409$	−34.7356	−1.71757	−0.858783	+	0.512340i	$0.828779\pi$
$410$	77.8633	3.84539
$411$	0	0
$412$	−12.0753	−0.594906
$413$	−6.62393	−0.325942
$414$	0	0
$415$	9.99049	0.490414
$416$	9.56132	0.468782
$417$	0	0
$418$	0	0
$419$	7.43071	0.363014	0.181507	−	0.983390i	$-0.441903\pi$
$419$	7.43071	0.363014	0.181507	+	0.983390i	$0.441903\pi$
$420$	0	0
$421$	11.8558	0.577815	0.288908	−	0.957357i	$-0.406708\pi$
$421$	11.8558	0.577815	0.288908	+	0.957357i	$0.406708\pi$
$422$	25.1828	1.22588
$423$	0	0
$424$	−8.59395	−0.417359
$425$	46.3314	2.24740
$426$	0	0
$427$	−14.2380	−0.689025
$428$	−13.5248	−0.653748
$429$	0	0
$430$	−55.8982	−2.69565
$431$	24.7496	1.19214	0.596072	−	0.802931i	$-0.296727\pi$
$431$	24.7496	1.19214	0.596072	+	0.802931i	$0.296727\pi$
$432$	0	0
$433$	19.0608	0.916003	0.458002	−	0.888951i	$-0.348565\pi$
$433$	19.0608	0.916003	0.458002	+	0.888951i	$0.348565\pi$
$434$	−12.2801	−0.589464
$435$	0	0
$436$	−12.3835	−0.593063
$437$	−17.1277	−0.819328
$438$	0	0
$439$	34.7485	1.65845	0.829227	−	0.558911i	$-0.188781\pi$
$439$	34.7485	1.65845	0.829227	+	0.558911i	$0.188781\pi$
$440$	0	0
$441$	0	0
$442$	−15.2071	−0.723326
$443$	1.35989	0.0646102	0.0323051	−	0.999478i	$-0.489715\pi$
$443$	1.35989	0.0646102	0.0323051	+	0.999478i	$0.489715\pi$
$444$	0	0
$445$	17.2904	0.819642
$446$	−39.0670	−1.84987
$447$	0	0
$448$	−27.3983	−1.29445
$449$	−31.7261	−1.49725	−0.748623	−	0.662995i	$-0.769285\pi$
$449$	−31.7261	−1.49725	−0.748623	+	0.662995i	$0.769285\pi$
$450$	0	0
$451$	0	0
$452$	4.05385	0.190677
$453$	0	0
$454$	37.7393	1.77119
$455$	10.1897	0.477702
$456$	0	0
$457$	−35.3825	−1.65512	−0.827561	−	0.561376i	$-0.810272\pi$
$457$	−35.3825	−1.65512	−0.827561	+	0.561376i	$0.810272\pi$
$458$	9.79225	0.457562
$459$	0	0
$460$	56.7545	2.64619
$461$	−31.2367	−1.45484	−0.727419	−	0.686194i	$-0.759280\pi$
$461$	−31.2367	−1.45484	−0.727419	+	0.686194i	$0.759280\pi$
$462$	0	0
$463$	−0.238696	−0.0110931	−0.00554657	−	0.999985i	$-0.501766\pi$
$463$	−0.238696	−0.0110931	−0.00554657	+	0.999985i	$0.501766\pi$
$464$	8.03019	0.372792
$465$	0	0
$466$	−17.6105	−0.815792
$467$	−1.40269	−0.0649087	−0.0324543	−	0.999473i	$-0.510332\pi$
$467$	−1.40269	−0.0649087	−0.0324543	+	0.999473i	$0.510332\pi$
$468$	0	0
$469$	21.2593	0.981661
$470$	−67.9377	−3.13373
$471$	0	0
$472$	−4.06655	−0.187178
$473$	0	0
$474$	0	0
$475$	23.5408	1.08013
$476$	34.2350	1.56916
$477$	0	0
$478$	40.0962	1.83396
$479$	24.9330	1.13922	0.569609	−	0.821916i	$-0.307095\pi$
$479$	24.9330	1.13922	0.569609	+	0.821916i	$0.307095\pi$
$480$	0	0
$481$	0.413559	0.0188567
$482$	−61.1173	−2.78382
$483$	0	0
$484$	0	0
$485$	9.27396	0.421109
$486$	0	0
$487$	−16.0432	−0.726989	−0.363494	−	0.931596i	$-0.618416\pi$
$487$	−16.0432	−0.726989	−0.363494	+	0.931596i	$0.618416\pi$
$488$	−8.74097	−0.395685
$489$	0	0
$490$	14.4080	0.650886
$491$	10.0798	0.454894	0.227447	−	0.973790i	$-0.426962\pi$
$491$	10.0798	0.454894	0.227447	+	0.973790i	$0.426962\pi$
$492$	0	0
$493$	−19.9898	−0.900295
$494$	−7.72666	−0.347639
$495$	0	0
$496$	5.74170	0.257810
$497$	27.5265	1.23473
$498$	0	0
$499$	33.2537	1.48864	0.744319	−	0.667824i	$-0.232774\pi$
$499$	33.2537	1.48864	0.744319	+	0.667824i	$0.232774\pi$
$500$	−30.0485	−1.34381
$501$	0	0
$502$	−10.6447	−0.475095
$503$	−23.0498	−1.02774	−0.513870	−	0.857868i	$-0.671789\pi$
$503$	−23.0498	−1.02774	−0.513870	+	0.857868i	$0.671789\pi$
$504$	0	0
$505$	66.4398	2.95653
$506$	0	0
$507$	0	0
$508$	−23.0990	−1.02485
$509$	15.8923	0.704414	0.352207	−	0.935922i	$-0.385431\pi$
$509$	15.8923	0.704414	0.352207	+	0.935922i	$0.385431\pi$
$510$	0	0
$511$	−9.79242	−0.433191
$512$	24.0496	1.06285
$513$	0	0
$514$	−36.8901	−1.62715
$515$	16.5340	0.728575
$516$	0	0
$517$	0	0
$518$	−1.63458	−0.0718193
$519$	0	0
$520$	6.25567	0.274329
$521$	4.07140	0.178371	0.0891855	−	0.996015i	$-0.471574\pi$
$521$	4.07140	0.178371	0.0891855	+	0.996015i	$0.471574\pi$
$522$	0	0
$523$	−1.94690	−0.0851319	−0.0425659	−	0.999094i	$-0.513553\pi$
$523$	−1.94690	−0.0851319	−0.0425659	+	0.999094i	$0.513553\pi$
$524$	−9.76295	−0.426497
$525$	0	0
$526$	−9.58296	−0.417837
$527$	−14.2930	−0.622612
$528$	0	0
$529$	12.0140	0.522348
$530$	48.1607	2.09197
$531$	0	0
$532$	17.3947	0.754156
$533$	12.3431	0.534640
$534$	0	0
$535$	18.5188	0.800638
$536$	13.0515	0.563737
$537$	0	0
$538$	−42.2281	−1.82058
$539$	0	0
$540$	0	0
$541$	39.4039	1.69411	0.847053	−	0.531508i	$-0.178374\pi$
$541$	39.4039	1.69411	0.847053	+	0.531508i	$0.178374\pi$
$542$	−14.6064	−0.627401
$543$	0	0
$544$	−43.9852	−1.88585
$545$	16.9561	0.726318
$546$	0	0
$547$	25.1554	1.07557	0.537785	−	0.843082i	$-0.319261\pi$
$547$	25.1554	1.07557	0.537785	+	0.843082i	$0.319261\pi$
$548$	−58.1434	−2.48376
$549$	0	0
$550$	0	0
$551$	−10.1567	−0.432692
$552$	0	0
$553$	−3.21675	−0.136790
$554$	−3.18281	−0.135225
$555$	0	0
$556$	3.41412	0.144791
$557$	−30.0269	−1.27228	−0.636140	−	0.771574i	$-0.719470\pi$
$557$	−30.0269	−1.27228	−0.636140	+	0.771574i	$0.719470\pi$
$558$	0	0
$559$	−8.86115	−0.374787
$560$	18.8308	0.795747
$561$	0	0
$562$	−15.8369	−0.668041
$563$	18.3398	0.772929	0.386464	−	0.922304i	$-0.373696\pi$
$563$	18.3398	0.772929	0.386464	+	0.922304i	$0.373696\pi$
$564$	0	0
$565$	−5.55071	−0.233520
$566$	31.7224	1.33339
$567$	0	0
$568$	16.8990	0.709067
$569$	32.2270	1.35103	0.675513	−	0.737348i	$-0.263922\pi$
$569$	32.2270	1.35103	0.675513	+	0.737348i	$0.263922\pi$
$570$	0	0
$571$	37.2822	1.56021	0.780105	−	0.625648i	$-0.215166\pi$
$571$	37.2822	1.56021	0.780105	+	0.625648i	$0.215166\pi$
$572$	0	0
$573$	0	0
$574$	−48.7857	−2.03628
$575$	−48.1244	−2.00693
$576$	0	0
$577$	−2.91538	−0.121369	−0.0606845	−	0.998157i	$-0.519328\pi$
$577$	−2.91538	−0.121369	−0.0606845	+	0.998157i	$0.519328\pi$
$578$	33.3120	1.38560
$579$	0	0
$580$	33.6555	1.39747
$581$	−6.25960	−0.259692
$582$	0	0
$583$	0	0
$584$	−6.01175	−0.248768
$585$	0	0
$586$	−4.25977	−0.175970
$587$	19.2419	0.794198	0.397099	−	0.917776i	$-0.370017\pi$
$587$	19.2419	0.794198	0.397099	+	0.917776i	$0.370017\pi$
$588$	0	0
$589$	−7.26222	−0.299234
$590$	22.7891	0.938211
$591$	0	0
$592$	0.764265	0.0314111
$593$	14.7811	0.606986	0.303493	−	0.952834i	$-0.401847\pi$
$593$	14.7811	0.606986	0.303493	+	0.952834i	$0.401847\pi$
$594$	0	0
$595$	−46.8761	−1.92173
$596$	−28.9669	−1.18653
$597$	0	0
$598$	15.7955	0.645928
$599$	17.4805	0.714234	0.357117	−	0.934060i	$-0.383760\pi$
$599$	17.4805	0.714234	0.357117	+	0.934060i	$0.383760\pi$
$600$	0	0
$601$	−23.7844	−0.970185	−0.485093	−	0.874463i	$-0.661214\pi$
$601$	−23.7844	−0.970185	−0.485093	+	0.874463i	$0.661214\pi$
$602$	35.0234	1.42745
$603$	0	0
$604$	−5.65783	−0.230214
$605$	0	0
$606$	0	0
$607$	−39.1973	−1.59097	−0.795484	−	0.605975i	$-0.792783\pi$
$607$	−39.1973	−1.59097	−0.795484	+	0.605975i	$0.792783\pi$
$608$	−22.3487	−0.906360
$609$	0	0
$610$	48.9846	1.98333
$611$	−10.7697	−0.435695
$612$	0	0
$613$	−9.34643	−0.377499	−0.188749	−	0.982025i	$-0.560443\pi$
$613$	−9.34643	−0.377499	−0.188749	+	0.982025i	$0.560443\pi$
$614$	−25.8043	−1.04138
$615$	0	0
$616$	0	0
$617$	3.70963	0.149344	0.0746720	−	0.997208i	$-0.476209\pi$
$617$	3.70963	0.149344	0.0746720	+	0.997208i	$0.476209\pi$
$618$	0	0
$619$	30.2840	1.21722	0.608609	−	0.793470i	$-0.291728\pi$
$619$	30.2840	1.21722	0.608609	+	0.793470i	$0.291728\pi$
$620$	24.0642	0.966440
$621$	0	0
$622$	−7.32331	−0.293638
$623$	−10.8334	−0.434031
$624$	0	0
$625$	0.479312	0.0191725
$626$	−48.9248	−1.95543
$627$	0	0
$628$	25.1448	1.00339
$629$	−1.90251	−0.0758580
$630$	0	0
$631$	−10.7233	−0.426886	−0.213443	−	0.976956i	$-0.568468\pi$
$631$	−10.7233	−0.426886	−0.213443	+	0.976956i	$0.568468\pi$
$632$	−1.97482	−0.0785542
$633$	0	0
$634$	−32.0129	−1.27139
$635$	31.6281	1.25512
$636$	0	0
$637$	2.28400	0.0904952
$638$	0	0
$639$	0	0
$640$	38.3005	1.51396
$641$	43.8165	1.73065	0.865324	−	0.501213i	$-0.167113\pi$
$641$	43.8165	1.73065	0.865324	+	0.501213i	$0.167113\pi$
$642$	0	0
$643$	32.1790	1.26901	0.634507	−	0.772917i	$-0.281203\pi$
$643$	32.1790	1.26901	0.634507	+	0.772917i	$0.281203\pi$
$644$	−35.5599	−1.40126
$645$	0	0
$646$	35.5451	1.39850
$647$	−10.9108	−0.428946	−0.214473	−	0.976730i	$-0.568803\pi$
$647$	−10.9108	−0.428946	−0.214473	+	0.976730i	$0.568803\pi$
$648$	0	0
$649$	0	0
$650$	−21.7099	−0.851533
$651$	0	0
$652$	−51.5341	−2.01823
$653$	47.0662	1.84184	0.920922	−	0.389748i	$-0.127438\pi$
$653$	47.0662	1.84184	0.920922	+	0.389748i	$0.127438\pi$
$654$	0	0
$655$	13.3679	0.522326
$656$	22.8103	0.890593
$657$	0	0
$658$	42.5668	1.65943
$659$	26.9724	1.05070	0.525348	−	0.850887i	$-0.323935\pi$
$659$	26.9724	1.05070	0.525348	+	0.850887i	$0.323935\pi$
$660$	0	0
$661$	9.06690	0.352662	0.176331	−	0.984331i	$-0.443577\pi$
$661$	9.06690	0.352662	0.176331	+	0.984331i	$0.443577\pi$
$662$	34.9269	1.35747
$663$	0	0
$664$	−3.84289	−0.149133
$665$	−23.8176	−0.923607
$666$	0	0
$667$	20.7634	0.803961
$668$	−40.4887	−1.56656
$669$	0	0
$670$	−73.1406	−2.82567
$671$	0	0
$672$	0	0
$673$	8.73390	0.336667	0.168334	−	0.985730i	$-0.446161\pi$
$673$	8.73390	0.336667	0.168334	+	0.985730i	$0.446161\pi$
$674$	24.2770	0.935114
$675$	0	0
$676$	−30.3480	−1.16723
$677$	−35.7842	−1.37530	−0.687650	−	0.726043i	$-0.741357\pi$
$677$	−35.7842	−1.37530	−0.687650	+	0.726043i	$0.741357\pi$
$678$	0	0
$679$	−5.81066	−0.222993
$680$	−28.7781	−1.10359
$681$	0	0
$682$	0	0
$683$	40.9545	1.56708	0.783541	−	0.621340i	$-0.213412\pi$
$683$	40.9545	1.56708	0.783541	+	0.621340i	$0.213412\pi$
$684$	0	0
$685$	79.6125	3.04184
$686$	−43.2891	−1.65279
$687$	0	0
$688$	−16.3756	−0.624312
$689$	7.63458	0.290854
$690$	0	0
$691$	39.9926	1.52139	0.760696	−	0.649109i	$-0.224858\pi$
$691$	39.9926	1.52139	0.760696	+	0.649109i	$0.224858\pi$
$692$	37.1087	1.41066
$693$	0	0
$694$	−36.8057	−1.39713
$695$	−4.67477	−0.177324
$696$	0	0
$697$	−56.7824	−2.15079
$698$	−49.5196	−1.87435
$699$	0	0
$700$	48.8746	1.84729
$701$	−28.4894	−1.07603	−0.538015	−	0.842935i	$-0.680826\pi$
$701$	−28.4894	−1.07603	−0.538015	+	0.842935i	$0.680826\pi$
$702$	0	0
$703$	−0.966658	−0.0364582
$704$	0	0
$705$	0	0
$706$	44.2190	1.66421
$707$	−41.6283	−1.56559
$708$	0	0
$709$	38.4909	1.44556	0.722779	−	0.691080i	$-0.242865\pi$
$709$	38.4909	1.44556	0.722779	+	0.691080i	$0.242865\pi$
$710$	−94.7025	−3.55412
$711$	0	0
$712$	−6.65082	−0.249250
$713$	14.8461	0.555991
$714$	0	0
$715$	0	0
$716$	−10.4530	−0.390648
$717$	0	0
$718$	−6.29093	−0.234776
$719$	−19.0764	−0.711430	−0.355715	−	0.934594i	$-0.615763\pi$
$719$	−19.0764	−0.711430	−0.355715	+	0.934594i	$0.615763\pi$
$720$	0	0
$721$	−10.3595	−0.385807
$722$	−22.8963	−0.852111
$723$	0	0
$724$	35.3759	1.31473
$725$	−28.5378	−1.05987
$726$	0	0
$727$	−6.06336	−0.224878	−0.112439	−	0.993659i	$-0.535866\pi$
$727$	−6.06336	−0.224878	−0.112439	+	0.993659i	$0.535866\pi$
$728$	−3.91953	−0.145267
$729$	0	0
$730$	33.6900	1.24692
$731$	40.7641	1.50772
$732$	0	0
$733$	−37.7766	−1.39531	−0.697655	−	0.716434i	$-0.745773\pi$
$733$	−37.7766	−1.39531	−0.697655	+	0.716434i	$0.745773\pi$
$734$	14.5534	0.537175
$735$	0	0
$736$	45.6873	1.68406
$737$	0	0
$738$	0	0
$739$	11.0030	0.404752	0.202376	−	0.979308i	$-0.435134\pi$
$739$	11.0030	0.404752	0.202376	+	0.979308i	$0.435134\pi$
$740$	3.20313	0.117749
$741$	0	0
$742$	−30.1754	−1.10777
$743$	17.6653	0.648077	0.324038	−	0.946044i	$-0.394959\pi$
$743$	17.6653	0.648077	0.324038	+	0.946044i	$0.394959\pi$
$744$	0	0
$745$	39.6627	1.45313
$746$	48.2371	1.76608
$747$	0	0
$748$	0	0
$749$	−11.6031	−0.423967
$750$	0	0
$751$	−4.28513	−0.156367	−0.0781833	−	0.996939i	$-0.524912\pi$
$751$	−4.28513	−0.156367	−0.0781833	+	0.996939i	$0.524912\pi$
$752$	−19.9026	−0.725772
$753$	0	0
$754$	9.36679	0.341118
$755$	7.74695	0.281940
$756$	0	0
$757$	−8.95117	−0.325336	−0.162668	−	0.986681i	$-0.552010\pi$
$757$	−8.95117	−0.325336	−0.162668	+	0.986681i	$0.552010\pi$
$758$	71.2044	2.58626
$759$	0	0
$760$	−14.6221	−0.530398
$761$	−5.38074	−0.195052	−0.0975258	−	0.995233i	$-0.531093\pi$
$761$	−5.38074	−0.195052	−0.0975258	+	0.995233i	$0.531093\pi$
$762$	0	0
$763$	−10.6239	−0.384612
$764$	−50.0761	−1.81169
$765$	0	0
$766$	−50.6229	−1.82908
$767$	3.61259	0.130443
$768$	0	0
$769$	15.4491	0.557110	0.278555	−	0.960420i	$-0.410145\pi$
$769$	15.4491	0.557110	0.278555	+	0.960420i	$0.410145\pi$
$770$	0	0
$771$	0	0
$772$	5.46403	0.196655
$773$	32.7213	1.17690	0.588451	−	0.808533i	$-0.299738\pi$
$773$	32.7213	1.17690	0.588451	+	0.808533i	$0.299738\pi$
$774$	0	0
$775$	−20.4050	−0.732968
$776$	−3.56727	−0.128058
$777$	0	0
$778$	66.9133	2.39896
$779$	−28.8509	−1.03369
$780$	0	0
$781$	0	0
$782$	−72.6647	−2.59848
$783$	0	0
$784$	4.22086	0.150745
$785$	−34.4294	−1.22884
$786$	0	0
$787$	24.9150	0.888125	0.444063	−	0.895996i	$-0.353537\pi$
$787$	24.9150	0.888125	0.444063	+	0.895996i	$0.353537\pi$
$788$	−15.8839	−0.565841
$789$	0	0
$790$	11.0669	0.393744
$791$	3.47783	0.123657
$792$	0	0
$793$	7.76519	0.275750
$794$	37.4654	1.32960
$795$	0	0
$796$	52.1743	1.84927
$797$	−9.09094	−0.322018	−0.161009	−	0.986953i	$-0.551475\pi$
$797$	−9.09094	−0.322018	−0.161009	+	0.986953i	$0.551475\pi$
$798$	0	0
$799$	49.5440	1.75274
$800$	−62.7941	−2.22011
$801$	0	0
$802$	24.8928	0.878997
$803$	0	0
$804$	0	0
$805$	48.6901	1.71610
$806$	6.69738	0.235905
$807$	0	0
$808$	−25.5564	−0.899070
$809$	10.0563	0.353560	0.176780	−	0.984250i	$-0.443432\pi$
$809$	10.0563	0.353560	0.176780	+	0.984250i	$0.443432\pi$
$810$	0	0
$811$	16.2533	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$811$	16.2533	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$812$	−21.0871	−0.740011
$813$	0	0
$814$	0	0
$815$	70.5628	2.47171
$816$	0	0
$817$	20.7121	0.724626
$818$	−74.8765	−2.61800
$819$	0	0
$820$	95.6007	3.33852
$821$	−20.8890	−0.729031	−0.364515	−	0.931197i	$-0.618765\pi$
$821$	−20.8890	−0.729031	−0.364515	+	0.931197i	$0.618765\pi$
$822$	0	0
$823$	−13.4793	−0.469859	−0.234930	−	0.972012i	$-0.575486\pi$
$823$	−13.4793	−0.469859	−0.234930	+	0.972012i	$0.575486\pi$
$824$	−6.35988	−0.221557
$825$	0	0
$826$	−14.2786	−0.496817
$827$	54.8435	1.90710	0.953548	−	0.301241i	$-0.0974008\pi$
$827$	54.8435	1.90710	0.953548	+	0.301241i	$0.0974008\pi$
$828$	0	0
$829$	1.57675	0.0547628	0.0273814	−	0.999625i	$-0.491283\pi$
$829$	1.57675	0.0547628	0.0273814	+	0.999625i	$0.491283\pi$
$830$	21.5356	0.747512
$831$	0	0
$832$	14.9426	0.518042
$833$	−10.5071	−0.364050
$834$	0	0
$835$	55.4389	1.91854
$836$	0	0
$837$	0	0
$838$	16.0177	0.553323
$839$	−29.0938	−1.00443	−0.502214	−	0.864743i	$-0.667481\pi$
$839$	−29.0938	−1.00443	−0.502214	+	0.864743i	$0.667481\pi$
$840$	0	0
$841$	−16.6873	−0.575424
$842$	25.5565	0.880734
$843$	0	0
$844$	30.9196	1.06429
$845$	41.5538	1.42949
$846$	0	0
$847$	0	0
$848$	14.1088	0.484499
$849$	0	0
$850$	99.8726	3.42560
$851$	1.97613	0.0677409
$852$	0	0
$853$	−25.7250	−0.880806	−0.440403	−	0.897800i	$-0.645164\pi$
$853$	−25.7250	−0.880806	−0.440403	+	0.897800i	$0.645164\pi$
$854$	−30.6916	−1.05025
$855$	0	0
$856$	−7.12334	−0.243471
$857$	−46.0114	−1.57172	−0.785860	−	0.618405i	$-0.787779\pi$
$857$	−46.0114	−1.57172	−0.785860	+	0.618405i	$0.787779\pi$
$858$	0	0
$859$	−44.0483	−1.50291	−0.751455	−	0.659785i	$-0.770647\pi$
$859$	−44.0483	−1.50291	−0.751455	+	0.659785i	$0.770647\pi$
$860$	−68.6319	−2.34033
$861$	0	0
$862$	53.3505	1.81712
$863$	13.4065	0.456362	0.228181	−	0.973619i	$-0.426722\pi$
$863$	13.4065	0.456362	0.228181	+	0.973619i	$0.426722\pi$
$864$	0	0
$865$	−50.8108	−1.72762
$866$	41.0877	1.39622
$867$	0	0
$868$	−15.0775	−0.511765
$869$	0	0
$870$	0	0
$871$	−11.5945	−0.392864
$872$	−6.52223	−0.220871
$873$	0	0
$874$	−36.9207	−1.24886
$875$	−25.7789	−0.871484
$876$	0	0
$877$	−12.3772	−0.417948	−0.208974	−	0.977921i	$-0.567012\pi$
$877$	−12.3772	−0.417948	−0.208974	+	0.977921i	$0.567012\pi$
$878$	74.9043	2.52790
$879$	0	0
$880$	0	0
$881$	31.1210	1.04849	0.524247	−	0.851566i	$-0.324347\pi$
$881$	31.1210	1.04849	0.524247	+	0.851566i	$0.324347\pi$
$882$	0	0
$883$	−49.1130	−1.65278	−0.826392	−	0.563096i	$-0.809610\pi$
$883$	−49.1130	−1.65278	−0.826392	+	0.563096i	$0.809610\pi$
$884$	−18.6713	−0.627983
$885$	0	0
$886$	2.93139	0.0984819
$887$	−46.3916	−1.55768	−0.778839	−	0.627224i	$-0.784191\pi$
$887$	−46.3916	−1.55768	−0.778839	+	0.627224i	$0.784191\pi$
$888$	0	0
$889$	−19.8168	−0.664634
$890$	37.2713	1.24934
$891$	0	0
$892$	−47.9665	−1.60604
$893$	25.1732	0.842388
$894$	0	0
$895$	14.3128	0.478423
$896$	−23.9974	−0.801698
$897$	0	0
$898$	−68.3892	−2.28218
$899$	8.80376	0.293622
$900$	0	0
$901$	−35.1215	−1.17007
$902$	0	0
$903$	0	0
$904$	2.13511	0.0710126
$905$	−48.4382	−1.61014
$906$	0	0
$907$	−7.42233	−0.246454	−0.123227	−	0.992378i	$-0.539324\pi$
$907$	−7.42233	−0.246454	−0.123227	+	0.992378i	$0.539324\pi$
$908$	46.3364	1.53773
$909$	0	0
$910$	21.9651	0.728137
$911$	−9.27060	−0.307149	−0.153574	−	0.988137i	$-0.549078\pi$
$911$	−9.27060	−0.307149	−0.153574	+	0.988137i	$0.549078\pi$
$912$	0	0
$913$	0	0
$914$	−76.2709	−2.52282
$915$	0	0
$916$	12.0229	0.397249
$917$	−8.37572	−0.276591
$918$	0	0
$919$	8.03556	0.265069	0.132534	−	0.991178i	$-0.457689\pi$
$919$	8.03556	0.265069	0.132534	+	0.991178i	$0.457689\pi$
$920$	29.8918	0.985503
$921$	0	0
$922$	−67.3342	−2.21753
$923$	−15.0125	−0.494143
$924$	0	0
$925$	−2.71606	−0.0893035
$926$	−0.514536	−0.0169087
$927$	0	0
$928$	27.0927	0.889360
$929$	9.53464	0.312821	0.156411	−	0.987692i	$-0.450008\pi$
$929$	9.53464	0.312821	0.156411	+	0.987692i	$0.450008\pi$
$930$	0	0
$931$	−5.33863	−0.174967
$932$	−21.6223	−0.708261
$933$	0	0
$934$	−3.02365	−0.0989369
$935$	0	0
$936$	0	0
$937$	17.8647	0.583614	0.291807	−	0.956477i	$-0.405744\pi$
$937$	17.8647	0.583614	0.291807	+	0.956477i	$0.405744\pi$
$938$	45.8267	1.49630
$939$	0	0
$940$	−83.4140	−2.72066
$941$	−48.8780	−1.59338	−0.796689	−	0.604390i	$-0.793417\pi$
$941$	−48.8780	−1.59338	−0.796689	+	0.604390i	$0.793417\pi$
$942$	0	0
$943$	58.9798	1.92065
$944$	6.67613	0.217290
$945$	0	0
$946$	0	0
$947$	46.1104	1.49839	0.749193	−	0.662352i	$-0.230442\pi$
$947$	46.1104	1.49839	0.749193	+	0.662352i	$0.230442\pi$
$948$	0	0
$949$	5.34064	0.173364
$950$	50.7450	1.64638
$951$	0	0
$952$	18.0311	0.584392
$953$	−44.3793	−1.43759	−0.718793	−	0.695224i	$-0.755305\pi$
$953$	−44.3793	−1.43759	−0.718793	+	0.695224i	$0.755305\pi$
$954$	0	0
$955$	68.5664	2.21876
$956$	49.2302	1.59222
$957$	0	0
$958$	53.7459	1.73645
$959$	−49.8817	−1.61076
$960$	0	0
$961$	−24.7052	−0.796942
$962$	0.891474	0.0287423
$963$	0	0
$964$	−75.0400	−2.41687
$965$	−7.48159	−0.240841
$966$	0	0
$967$	−22.1663	−0.712819	−0.356409	−	0.934330i	$-0.615999\pi$
$967$	−22.1663	−0.712819	−0.356409	+	0.934330i	$0.615999\pi$
$968$	0	0
$969$	0	0
$970$	19.9911	0.641874
$971$	−23.1985	−0.744476	−0.372238	−	0.928137i	$-0.621409\pi$
$971$	−23.1985	−0.744476	−0.372238	+	0.928137i	$0.621409\pi$
$972$	0	0
$973$	2.92900	0.0938996
$974$	−34.5830	−1.10811
$975$	0	0
$976$	14.3502	0.459339
$977$	−15.5682	−0.498071	−0.249035	−	0.968494i	$-0.580114\pi$
$977$	−15.5682	−0.498071	−0.249035	+	0.968494i	$0.580114\pi$
$978$	0	0
$979$	0	0
$980$	17.6901	0.565091
$981$	0	0
$982$	21.7281	0.693371
$983$	−29.5841	−0.943586	−0.471793	−	0.881709i	$-0.656393\pi$
$983$	−29.5841	−0.943586	−0.471793	+	0.881709i	$0.656393\pi$
$984$	0	0
$985$	21.7490	0.692979
$986$	−43.0903	−1.37227
$987$	0	0
$988$	−9.48681	−0.301816
$989$	−42.3416	−1.34639
$990$	0	0
$991$	−7.97155	−0.253225	−0.126612	−	0.991952i	$-0.540410\pi$
$991$	−7.97155	−0.253225	−0.126612	+	0.991952i	$0.540410\pi$
$992$	19.3716	0.615050
$993$	0	0
$994$	59.3365	1.88204
$995$	−71.4394	−2.26478
$996$	0	0
$997$	−47.9941	−1.51999	−0.759994	−	0.649931i	$-0.774798\pi$
$997$	−47.9941	−1.51999	−0.759994	+	0.649931i	$0.774798\pi$
$998$	71.6820	2.26905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.bl.1.3		4
3.2	odd	2		9801.2.a.bi.1.2		4
9.2	odd	6		1089.2.e.i.364.3		8
9.5	odd	6		1089.2.e.i.727.3		8
11.10	odd	2		891.2.a.p.1.2		4
33.32	even	2		891.2.a.q.1.3		4
99.32	even	6		99.2.e.e.34.2	✓	8
99.43	odd	6		297.2.e.e.199.3		8
99.65	even	6		99.2.e.e.67.2	yes	8
99.76	odd	6		297.2.e.e.100.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.e.34.2	✓	8	99.32	even	6
99.2.e.e.67.2	yes	8	99.65	even	6
297.2.e.e.100.3		8	99.76	odd	6
297.2.e.e.199.3		8	99.43	odd	6
891.2.a.p.1.2		4	11.10	odd	2
891.2.a.q.1.3		4	33.32	even	2
1089.2.e.i.364.3		8	9.2	odd	6
1089.2.e.i.727.3		8	9.5	odd	6
9801.2.a.bi.1.2		4	3.2	odd	2
9801.2.a.bl.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q+2.15561 q^{2} +2.64667 q^{4} -3.62393 q^{5} +2.27060 q^{7} +1.39396 q^{8} +O(q^{10})\)	\(q+2.15561 q^{2} +2.64667 q^{4} -3.62393 q^{5} +2.27060 q^{7} +1.39396 q^{8} -7.81179 q^{10} -1.23835 q^{13} +4.89453 q^{14} -2.28849 q^{16} +5.69681 q^{17} +2.89453 q^{19} -9.59134 q^{20} -5.91726 q^{23} +8.13288 q^{25} -2.66940 q^{26} +6.00951 q^{28} -3.50895 q^{29} -2.50895 q^{31} -7.72102 q^{32} +12.2801 q^{34} -8.22849 q^{35} -0.333960 q^{37} +6.23948 q^{38} -5.05162 q^{40} -9.96741 q^{41} +7.15561 q^{43} -12.7553 q^{46} +8.69681 q^{47} -1.84439 q^{49} +17.5313 q^{50} -3.27750 q^{52} -6.16513 q^{53} +3.16513 q^{56} -7.56393 q^{58} -2.91726 q^{59} -6.27060 q^{61} -5.40832 q^{62} -12.0666 q^{64} +4.48769 q^{65} +9.36285 q^{67} +15.0775 q^{68} -17.7374 q^{70} +12.1230 q^{71} -4.31271 q^{73} -0.719889 q^{74} +7.66085 q^{76} -1.41670 q^{79} +8.29333 q^{80} -21.4859 q^{82} -2.75681 q^{83} -20.6448 q^{85} +15.4247 q^{86} -4.77116 q^{89} -2.81179 q^{91} -15.6610 q^{92} +18.7469 q^{94} -10.4896 q^{95} -2.55909 q^{97} -3.97578 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 11 q^{4} - 4 q^{5} - q^{7}+O(q^{10}) \) 4 * q + q^2 + 11 * q^4 - 4 * q^5 - q^7	\( 4 q + q^{2} + 11 q^{4} - 4 q^{5} - q^{7} - q^{10} - 7 q^{13} - q^{14} + 17 q^{16} - 5 q^{17} - 9 q^{19} + 10 q^{20} - 14 q^{23} + 14 q^{25} - 22 q^{26} + q^{28} - 6 q^{29} - 2 q^{31} - 34 q^{32} + 16 q^{34} - 8 q^{35} + 3 q^{37} - 3 q^{38} - 12 q^{40} - 2 q^{41} + 21 q^{43} - 2 q^{46} + 7 q^{47} - 15 q^{49} + 23 q^{50} + 10 q^{52} + 6 q^{53} - 18 q^{56} - 21 q^{58} - 2 q^{59} - 15 q^{61} - 20 q^{62} + 16 q^{64} + 19 q^{65} + 14 q^{67} - 7 q^{68} - 38 q^{70} + 3 q^{71} - 22 q^{73} - 36 q^{74} - 42 q^{76} - 11 q^{79} + 34 q^{80} - 17 q^{82} + 18 q^{83} - 13 q^{85} + 24 q^{86} + 6 q^{89} + 19 q^{91} - 67 q^{92} + 19 q^{94} - 30 q^{95} + 26 q^{97} + 15 q^{98}+O(q^{100}) \) 4 * q + q^2 + 11 * q^4 - 4 * q^5 - q^7 - q^10 - 7 * q^13 - q^14 + 17 * q^16 - 5 * q^17 - 9 * q^19 + 10 * q^20 - 14 * q^23 + 14 * q^25 - 22 * q^26 + q^28 - 6 * q^29 - 2 * q^31 - 34 * q^32 + 16 * q^34 - 8 * q^35 + 3 * q^37 - 3 * q^38 - 12 * q^40 - 2 * q^41 + 21 * q^43 - 2 * q^46 + 7 * q^47 - 15 * q^49 + 23 * q^50 + 10 * q^52 + 6 * q^53 - 18 * q^56 - 21 * q^58 - 2 * q^59 - 15 * q^61 - 20 * q^62 + 16 * q^64 + 19 * q^65 + 14 * q^67 - 7 * q^68 - 38 * q^70 + 3 * q^71 - 22 * q^73 - 36 * q^74 - 42 * q^76 - 11 * q^79 + 34 * q^80 - 17 * q^82 + 18 * q^83 - 13 * q^85 + 24 * q^86 + 6 * q^89 + 19 * q^91 - 67 * q^92 + 19 * q^94 - 30 * q^95 + 26 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

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Database

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