LMFDB - Embedded newform 9801.2.a.cp.1.4

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:	$0$
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			1.4
Root			$-1.54614$ 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)

$f(q)$	$=$	$q-1.54614 q^{2} +0.390538 q^{4} -0.592032 q^{5} +0.721951 q^{7} +2.48845 q^{8} +0.915362 q^{10} -1.57496 q^{13} -1.11623 q^{14} -4.62856 q^{16} -4.59200 q^{17} +2.50860 q^{19} -0.231211 q^{20} +4.45200 q^{23} -4.64950 q^{25} +2.43511 q^{26} +0.281949 q^{28} -6.97862 q^{29} -9.12824 q^{31} +2.17948 q^{32} +7.09986 q^{34} -0.427418 q^{35} +2.89170 q^{37} -3.87864 q^{38} -1.47324 q^{40} +1.15412 q^{41} -4.21448 q^{43} -6.88340 q^{46} +0.227555 q^{47} -6.47879 q^{49} +7.18876 q^{50} -0.615084 q^{52} +5.70359 q^{53} +1.79654 q^{56} +10.7899 q^{58} -7.13355 q^{59} +4.55896 q^{61} +14.1135 q^{62} +5.88733 q^{64} +0.932429 q^{65} -8.09142 q^{67} -1.79335 q^{68} +0.660846 q^{70} +12.3094 q^{71} +15.4833 q^{73} -4.47096 q^{74} +0.979704 q^{76} -14.6196 q^{79} +2.74025 q^{80} -1.78443 q^{82} +6.95300 q^{83} +2.71861 q^{85} +6.51616 q^{86} -12.4803 q^{89} -1.13705 q^{91} +1.73868 q^{92} -0.351832 q^{94} -1.48517 q^{95} +14.1805 q^{97} +10.0171 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.54614	−1.09328	−0.546642	−	0.837367i	$-0.684094\pi$
$2$	−1.54614	−1.09328	−0.546642	+	0.837367i	$0.684094\pi$
$3$	0	0
$3$	0	0
$4$	0.390538	0.195269
$5$	−0.592032	−0.264765	−0.132382	−	0.991199i	$-0.542263\pi$
$5$	−0.592032	−0.264765	−0.132382	+	0.991199i	$0.542263\pi$
$6$	0	0
$7$	0.721951	0.272872	0.136436	−	0.990649i	$-0.456435\pi$
$7$	0.721951	0.272872	0.136436	+	0.990649i	$0.456435\pi$
$8$	2.48845	0.879799
$9$	0	0
$10$	0.915362	0.289463
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.57496	−0.436817	−0.218408	−	0.975857i	$-0.570086\pi$
$13$	−1.57496	−0.436817	−0.218408	+	0.975857i	$0.570086\pi$
$14$	−1.11623	−0.298326
$15$	0	0
$16$	−4.62856	−1.15714
$17$	−4.59200	−1.11372	−0.556862	−	0.830605i	$-0.687995\pi$
$17$	−4.59200	−1.11372	−0.556862	+	0.830605i	$0.687995\pi$
$18$	0	0
$19$	2.50860	0.575512	0.287756	−	0.957704i	$-0.407091\pi$
$19$	2.50860	0.575512	0.287756	+	0.957704i	$0.407091\pi$
$20$	−0.231211	−0.0517004
$21$	0	0
$22$	0	0
$23$	4.45200	0.928306	0.464153	−	0.885755i	$-0.346359\pi$
$23$	4.45200	0.928306	0.464153	+	0.885755i	$0.346359\pi$
$24$	0	0
$25$	−4.64950	−0.929900
$26$	2.43511	0.477564
$27$	0	0
$28$	0.281949	0.0532834
$29$	−6.97862	−1.29590	−0.647949	−	0.761684i	$-0.724373\pi$
$29$	−6.97862	−1.29590	−0.647949	+	0.761684i	$0.724373\pi$
$30$	0	0
$31$	−9.12824	−1.63948	−0.819740	−	0.572735i	$-0.805882\pi$
$31$	−9.12824	−1.63948	−0.819740	+	0.572735i	$0.805882\pi$
$32$	2.17948	0.385282
$33$	0	0
$34$	7.09986	1.21762
$35$	−0.427418	−0.0722468
$36$	0	0
$37$	2.89170	0.475392	0.237696	−	0.971340i	$-0.423608\pi$
$37$	2.89170	0.475392	0.237696	+	0.971340i	$0.423608\pi$
$38$	−3.87864	−0.629198
$39$	0	0
$40$	−1.47324	−0.232940
$41$	1.15412	0.180243	0.0901216	−	0.995931i	$-0.471274\pi$
$41$	1.15412	0.180243	0.0901216	+	0.995931i	$0.471274\pi$
$42$	0	0
$43$	−4.21448	−0.642702	−0.321351	−	0.946960i	$-0.604137\pi$
$43$	−4.21448	−0.642702	−0.321351	+	0.946960i	$0.604137\pi$
$44$	0	0
$45$	0	0
$46$	−6.88340	−1.01490
$47$	0.227555	0.0331924	0.0165962	−	0.999862i	$-0.494717\pi$
$47$	0.227555	0.0331924	0.0165962	+	0.999862i	$0.494717\pi$
$48$	0	0
$49$	−6.47879	−0.925541
$50$	7.18876	1.01664
$51$	0	0
$52$	−0.615084	−0.0852967
$53$	5.70359	0.783449	0.391724	−	0.920083i	$-0.371879\pi$
$53$	5.70359	0.783449	0.391724	+	0.920083i	$0.371879\pi$
$54$	0	0
$55$	0	0
$56$	1.79654	0.240072
$57$	0	0
$58$	10.7899	1.41678
$59$	−7.13355	−0.928710	−0.464355	−	0.885649i	$-0.653714\pi$
$59$	−7.13355	−0.928710	−0.464355	+	0.885649i	$0.653714\pi$
$60$	0	0
$61$	4.55896	0.583715	0.291857	−	0.956462i	$-0.405727\pi$
$61$	4.55896	0.583715	0.291857	+	0.956462i	$0.405727\pi$
$62$	14.1135	1.79242
$63$	0	0
$64$	5.88733	0.735917
$65$	0.932429	0.115654
$66$	0	0
$67$	−8.09142	−0.988524	−0.494262	−	0.869313i	$-0.664562\pi$
$67$	−8.09142	−0.988524	−0.494262	+	0.869313i	$0.664562\pi$
$68$	−1.79335	−0.217476
$69$	0	0
$70$	0.660846	0.0789862
$71$	12.3094	1.46086	0.730429	−	0.682989i	$-0.239320\pi$
$71$	12.3094	1.46086	0.730429	+	0.682989i	$0.239320\pi$
$72$	0	0
$73$	15.4833	1.81219	0.906093	−	0.423078i	$-0.139050\pi$
$73$	15.4833	1.81219	0.906093	+	0.423078i	$0.139050\pi$
$74$	−4.47096	−0.519738
$75$	0	0
$76$	0.979704	0.112380
$77$	0	0
$78$	0	0
$79$	−14.6196	−1.64483	−0.822414	−	0.568889i	$-0.807373\pi$
$79$	−14.6196	−1.64483	−0.822414	+	0.568889i	$0.807373\pi$
$80$	2.74025	0.306370
$81$	0	0
$82$	−1.78443	−0.197057
$83$	6.95300	0.763191	0.381596	−	0.924329i	$-0.375375\pi$
$83$	6.95300	0.763191	0.381596	+	0.924329i	$0.375375\pi$
$84$	0	0
$85$	2.71861	0.294875
$86$	6.51616	0.702656
$87$	0	0
$88$	0	0
$89$	−12.4803	−1.32291	−0.661453	−	0.749986i	$-0.730060\pi$
$89$	−12.4803	−1.32291	−0.661453	+	0.749986i	$0.730060\pi$
$90$	0	0
$91$	−1.13705	−0.119195
$92$	1.73868	0.181269
$93$	0	0
$94$	−0.351832	−0.0362887
$95$	−1.48517	−0.152375
$96$	0	0
$97$	14.1805	1.43981	0.719904	−	0.694073i	$-0.244186\pi$
$97$	14.1805	1.43981	0.719904	+	0.694073i	$0.244186\pi$
$98$	10.0171	1.01188
$99$	0	0
$100$	−1.81581	−0.181581
$101$	14.3685	1.42972	0.714861	−	0.699266i	$-0.246490\pi$
$101$	14.3685	1.42972	0.714861	+	0.699266i	$0.246490\pi$
$102$	0	0
$103$	−4.42885	−0.436388	−0.218194	−	0.975905i	$-0.570017\pi$
$103$	−4.42885	−0.436388	−0.218194	+	0.975905i	$0.570017\pi$
$104$	−3.91922	−0.384311
$105$	0	0
$106$	−8.81853	−0.856532
$107$	15.7444	1.52207	0.761035	−	0.648711i	$-0.224692\pi$
$107$	15.7444	1.52207	0.761035	+	0.648711i	$0.224692\pi$
$108$	0	0
$109$	−13.6970	−1.31194	−0.655969	−	0.754788i	$-0.727740\pi$
$109$	−13.6970	−1.31194	−0.655969	+	0.754788i	$0.727740\pi$
$110$	0	0
$111$	0	0
$112$	−3.34159	−0.315750
$113$	−8.03013	−0.755411	−0.377706	−	0.925926i	$-0.623287\pi$
$113$	−8.03013	−0.755411	−0.377706	+	0.925926i	$0.623287\pi$
$114$	0	0
$115$	−2.63573	−0.245783
$116$	−2.72542	−0.253049
$117$	0	0
$118$	11.0294	1.01534
$119$	−3.31520	−0.303904
$120$	0	0
$121$	0	0
$122$	−7.04877	−0.638166
$123$	0	0
$124$	−3.56493	−0.320140
$125$	5.71281	0.510969
$126$	0	0
$127$	−11.9963	−1.06450	−0.532250	−	0.846587i	$-0.678653\pi$
$127$	−11.9963	−1.06450	−0.532250	+	0.846587i	$0.678653\pi$
$128$	−13.4616	−1.18985
$129$	0	0
$130$	−1.44166	−0.126442
$131$	6.89109	0.602077	0.301039	−	0.953612i	$-0.402667\pi$
$131$	6.89109	0.602077	0.301039	+	0.953612i	$0.402667\pi$
$132$	0	0
$133$	1.81109	0.157041
$134$	12.5104	1.08074
$135$	0	0
$136$	−11.4270	−0.979854
$137$	−9.24655	−0.789986	−0.394993	−	0.918684i	$-0.629253\pi$
$137$	−9.24655	−0.789986	−0.394993	+	0.918684i	$0.629253\pi$
$138$	0	0
$139$	21.6113	1.83305	0.916523	−	0.399982i	$-0.130984\pi$
$139$	21.6113	1.83305	0.916523	+	0.399982i	$0.130984\pi$
$140$	−0.166923	−0.0141076
$141$	0	0
$142$	−19.0320	−1.59713
$143$	0	0
$144$	0	0
$145$	4.13157	0.343108
$146$	−23.9393	−1.98123
$147$	0	0
$148$	1.12932	0.0928293
$149$	1.14643	0.0939194	0.0469597	−	0.998897i	$-0.485047\pi$
$149$	1.14643	0.0939194	0.0469597	+	0.998897i	$0.485047\pi$
$150$	0	0
$151$	0.880640	0.0716655	0.0358327	−	0.999358i	$-0.488592\pi$
$151$	0.880640	0.0716655	0.0358327	+	0.999358i	$0.488592\pi$
$152$	6.24252	0.506335
$153$	0	0
$154$	0	0
$155$	5.40421	0.434077
$156$	0	0
$157$	−9.78621	−0.781025	−0.390512	−	0.920598i	$-0.627702\pi$
$157$	−9.78621	−0.781025	−0.390512	+	0.920598i	$0.627702\pi$
$158$	22.6038	1.79826
$159$	0	0
$160$	−1.29032	−0.102009
$161$	3.21412	0.253308
$162$	0	0
$163$	4.62976	0.362631	0.181315	−	0.983425i	$-0.441965\pi$
$163$	4.62976	0.362631	0.181315	+	0.983425i	$0.441965\pi$
$164$	0.450728	0.0351959
$165$	0	0
$166$	−10.7503	−0.834384
$167$	16.9874	1.31452	0.657262	−	0.753663i	$-0.271715\pi$
$167$	16.9874	1.31452	0.657262	+	0.753663i	$0.271715\pi$
$168$	0	0
$169$	−10.5195	−0.809191
$170$	−4.20335	−0.322382
$171$	0	0
$172$	−1.64592	−0.125500
$173$	0.591056	0.0449371	0.0224686	−	0.999748i	$-0.492847\pi$
$173$	0.591056	0.0449371	0.0224686	+	0.999748i	$0.492847\pi$
$174$	0	0
$175$	−3.35671	−0.253743
$176$	0	0
$177$	0	0
$178$	19.2962	1.44631
$179$	9.64093	0.720597	0.360298	−	0.932837i	$-0.382675\pi$
$179$	9.64093	0.720597	0.360298	+	0.932837i	$0.382675\pi$
$180$	0	0
$181$	−4.76187	−0.353947	−0.176973	−	0.984216i	$-0.556631\pi$
$181$	−4.76187	−0.353947	−0.176973	+	0.984216i	$0.556631\pi$
$182$	1.75803	0.130314
$183$	0	0
$184$	11.0786	0.816723
$185$	−1.71198	−0.125867
$186$	0	0
$187$	0	0
$188$	0.0888691	0.00648144
$189$	0	0
$190$	2.29628	0.166590
$191$	5.19802	0.376116	0.188058	−	0.982158i	$-0.439781\pi$
$191$	5.19802	0.376116	0.188058	+	0.982158i	$0.439781\pi$
$192$	0	0
$193$	3.26805	0.235239	0.117620	−	0.993059i	$-0.462474\pi$
$193$	3.26805	0.235239	0.117620	+	0.993059i	$0.462474\pi$
$194$	−21.9249	−1.57412
$195$	0	0
$196$	−2.53021	−0.180730
$197$	22.4626	1.60039	0.800197	−	0.599737i	$-0.204728\pi$
$197$	22.4626	1.60039	0.800197	+	0.599737i	$0.204728\pi$
$198$	0	0
$199$	−20.8291	−1.47654	−0.738269	−	0.674507i	$-0.764356\pi$
$199$	−20.8291	−1.47654	−0.738269	+	0.674507i	$0.764356\pi$
$200$	−11.5700	−0.818125
$201$	0	0
$202$	−22.2157	−1.56309
$203$	−5.03822	−0.353614
$204$	0	0
$205$	−0.683276	−0.0477220
$206$	6.84761	0.477096
$207$	0	0
$208$	7.28981	0.505457
$209$	0	0
$210$	0	0
$211$	3.39202	0.233517	0.116758	−	0.993160i	$-0.462750\pi$
$211$	3.39202	0.233517	0.116758	+	0.993160i	$0.462750\pi$
$212$	2.22747	0.152983
$213$	0	0
$214$	−24.3430	−1.66405
$215$	2.49511	0.170165
$216$	0	0
$217$	−6.59014	−0.447368
$218$	21.1775	1.43432
$219$	0	0
$220$	0	0
$221$	7.23224	0.486493
$222$	0	0
$223$	−11.0264	−0.738382	−0.369191	−	0.929354i	$-0.620365\pi$
$223$	−11.0264	−0.738382	−0.369191	+	0.929354i	$0.620365\pi$
$224$	1.57348	0.105133
$225$	0	0
$226$	12.4157	0.825879
$227$	5.96689	0.396036	0.198018	−	0.980198i	$-0.436549\pi$
$227$	5.96689	0.396036	0.198018	+	0.980198i	$0.436549\pi$
$228$	0	0
$229$	24.7507	1.63557	0.817785	−	0.575523i	$-0.195202\pi$
$229$	24.7507	1.63557	0.817785	+	0.575523i	$0.195202\pi$
$230$	4.07519	0.268710
$231$	0	0
$232$	−17.3659	−1.14013
$233$	−4.18842	−0.274392	−0.137196	−	0.990544i	$-0.543809\pi$
$233$	−4.18842	−0.274392	−0.137196	+	0.990544i	$0.543809\pi$
$234$	0	0
$235$	−0.134720	−0.00878817
$236$	−2.78592	−0.181348
$237$	0	0
$238$	5.12575	0.332253
$239$	7.37284	0.476910	0.238455	−	0.971154i	$-0.423359\pi$
$239$	7.37284	0.476910	0.238455	+	0.971154i	$0.423359\pi$
$240$	0	0
$241$	−8.82924	−0.568741	−0.284371	−	0.958714i	$-0.591785\pi$
$241$	−8.82924	−0.568741	−0.284371	+	0.958714i	$0.591785\pi$
$242$	0	0
$243$	0	0
$244$	1.78045	0.113981
$245$	3.83565	0.245051
$246$	0	0
$247$	−3.95096	−0.251393
$248$	−22.7152	−1.44241
$249$	0	0
$250$	−8.83279	−0.558634
$251$	−20.5731	−1.29856	−0.649280	−	0.760550i	$-0.724930\pi$
$251$	−20.5731	−1.29856	−0.649280	+	0.760550i	$0.724930\pi$
$252$	0	0
$253$	0	0
$254$	18.5479	1.16380
$255$	0	0
$256$	9.03879	0.564924
$257$	3.38239	0.210987	0.105494	−	0.994420i	$-0.466358\pi$
$257$	3.38239	0.210987	0.105494	+	0.994420i	$0.466358\pi$
$258$	0	0
$259$	2.08766	0.129721
$260$	0.364149	0.0225836
$261$	0	0
$262$	−10.6546	−0.658241
$263$	14.2822	0.880681	0.440341	−	0.897831i	$-0.354858\pi$
$263$	14.2822	0.880681	0.440341	+	0.897831i	$0.354858\pi$
$264$	0	0
$265$	−3.37671	−0.207430
$266$	−2.80019	−0.171690
$267$	0	0
$268$	−3.16001	−0.193028
$269$	−6.72041	−0.409751	−0.204875	−	0.978788i	$-0.565679\pi$
$269$	−6.72041	−0.409751	−0.204875	+	0.978788i	$0.565679\pi$
$270$	0	0
$271$	−3.98035	−0.241789	−0.120895	−	0.992665i	$-0.538576\pi$
$271$	−3.98035	−0.241789	−0.120895	+	0.992665i	$0.538576\pi$
$272$	21.2543	1.28873
$273$	0	0
$274$	14.2964	0.863679
$275$	0	0
$276$	0	0
$277$	−13.1320	−0.789026	−0.394513	−	0.918890i	$-0.629087\pi$
$277$	−13.1320	−0.789026	−0.394513	+	0.918890i	$0.629087\pi$
$278$	−33.4140	−2.00404
$279$	0	0
$280$	−1.06361	−0.0635627
$281$	2.96787	0.177048	0.0885241	−	0.996074i	$-0.471785\pi$
$281$	2.96787	0.177048	0.0885241	+	0.996074i	$0.471785\pi$
$282$	0	0
$283$	8.09278	0.481066	0.240533	−	0.970641i	$-0.422678\pi$
$283$	8.09278	0.481066	0.240533	+	0.970641i	$0.422678\pi$
$284$	4.80729	0.285260
$285$	0	0
$286$	0	0
$287$	0.833217	0.0491833
$288$	0	0
$289$	4.08649	0.240382
$290$	−6.38797	−0.375114
$291$	0	0
$292$	6.04683	0.353864
$293$	15.7156	0.918114	0.459057	−	0.888407i	$-0.348187\pi$
$293$	15.7156	0.918114	0.459057	+	0.888407i	$0.348187\pi$
$294$	0	0
$295$	4.22329	0.245890
$296$	7.19584	0.418249
$297$	0	0
$298$	−1.77254	−0.102681
$299$	−7.01174	−0.405499
$300$	0	0
$301$	−3.04265	−0.175375
$302$	−1.36159	−0.0783507
$303$	0	0
$304$	−11.6112	−0.665948
$305$	−2.69905	−0.154547
$306$	0	0
$307$	−3.51315	−0.200506	−0.100253	−	0.994962i	$-0.531965\pi$
$307$	−3.51315	−0.200506	−0.100253	+	0.994962i	$0.531965\pi$
$308$	0	0
$309$	0	0
$310$	−8.35565	−0.474569
$311$	−26.3344	−1.49329	−0.746644	−	0.665224i	$-0.768336\pi$
$311$	−26.3344	−1.49329	−0.746644	+	0.665224i	$0.768336\pi$
$312$	0	0
$313$	−2.46114	−0.139112	−0.0695560	−	0.997578i	$-0.522158\pi$
$313$	−2.46114	−0.139112	−0.0695560	+	0.997578i	$0.522158\pi$
$314$	15.1308	0.853882
$315$	0	0
$316$	−5.70949	−0.321184
$317$	26.5071	1.48879	0.744395	−	0.667740i	$-0.232738\pi$
$317$	26.5071	1.48879	0.744395	+	0.667740i	$0.232738\pi$
$318$	0	0
$319$	0	0
$320$	−3.48549	−0.194845
$321$	0	0
$322$	−4.96947	−0.276938
$323$	−11.5195	−0.640962
$324$	0	0
$325$	7.32279	0.406196
$326$	−7.15824	−0.396458
$327$	0	0
$328$	2.87197	0.158578
$329$	0.164284	0.00905726
$330$	0	0
$331$	8.10397	0.445434	0.222717	−	0.974883i	$-0.428507\pi$
$331$	8.10397	0.445434	0.222717	+	0.974883i	$0.428507\pi$
$332$	2.71541	0.149028
$333$	0	0
$334$	−26.2648	−1.43715
$335$	4.79038	0.261726
$336$	0	0
$337$	24.7206	1.34662	0.673309	−	0.739361i	$-0.264872\pi$
$337$	24.7206	1.34662	0.673309	+	0.739361i	$0.264872\pi$
$338$	16.2646	0.884676
$339$	0	0
$340$	1.06172	0.0575799
$341$	0	0
$342$	0	0
$343$	−9.73102	−0.525426
$344$	−10.4875	−0.565449
$345$	0	0
$346$	−0.913853	−0.0491290
$347$	2.06964	0.111104	0.0555522	−	0.998456i	$-0.482308\pi$
$347$	2.06964	0.111104	0.0555522	+	0.998456i	$0.482308\pi$
$348$	0	0
$349$	−5.21427	−0.279113	−0.139557	−	0.990214i	$-0.544568\pi$
$349$	−5.21427	−0.279113	−0.139557	+	0.990214i	$0.544568\pi$
$350$	5.18993	0.277413
$351$	0	0
$352$	0	0
$353$	−21.6725	−1.15351	−0.576756	−	0.816917i	$-0.695681\pi$
$353$	−21.6725	−1.15351	−0.576756	+	0.816917i	$0.695681\pi$
$354$	0	0
$355$	−7.28756	−0.386784
$356$	−4.87402	−0.258323
$357$	0	0
$358$	−14.9062	−0.787817
$359$	−8.75285	−0.461958	−0.230979	−	0.972959i	$-0.574193\pi$
$359$	−8.75285	−0.461958	−0.230979	+	0.972959i	$0.574193\pi$
$360$	0	0
$361$	−12.7069	−0.668786
$362$	7.36250	0.386964
$363$	0	0
$364$	−0.444060	−0.0232751
$365$	−9.16662	−0.479803
$366$	0	0
$367$	−6.76623	−0.353194	−0.176597	−	0.984283i	$-0.556509\pi$
$367$	−6.76623	−0.353194	−0.176597	+	0.984283i	$0.556509\pi$
$368$	−20.6063	−1.07418
$369$	0	0
$370$	2.64695	0.137608
$371$	4.11771	0.213781
$372$	0	0
$373$	−21.8933	−1.13359	−0.566797	−	0.823857i	$-0.691818\pi$
$373$	−21.8933	−1.13359	−0.566797	+	0.823857i	$0.691818\pi$
$374$	0	0
$375$	0	0
$376$	0.566260	0.0292026
$377$	10.9911	0.566069
$378$	0	0
$379$	−33.8236	−1.73740	−0.868701	−	0.495337i	$-0.835045\pi$
$379$	−33.8236	−1.73740	−0.868701	+	0.495337i	$0.835045\pi$
$380$	−0.580016	−0.0297542
$381$	0	0
$382$	−8.03686	−0.411201
$383$	18.3739	0.938862	0.469431	−	0.882969i	$-0.344459\pi$
$383$	18.3739	0.938862	0.469431	+	0.882969i	$0.344459\pi$
$384$	0	0
$385$	0	0
$386$	−5.05284	−0.257183
$387$	0	0
$388$	5.53801	0.281150
$389$	30.8065	1.56195	0.780977	−	0.624560i	$-0.214722\pi$
$389$	30.8065	1.56195	0.780977	+	0.624560i	$0.214722\pi$
$390$	0	0
$391$	−20.4436	−1.03388
$392$	−16.1221	−0.814290
$393$	0	0
$394$	−34.7302	−1.74968
$395$	8.65524	0.435493
$396$	0	0
$397$	−8.29578	−0.416353	−0.208177	−	0.978091i	$-0.566753\pi$
$397$	−8.29578	−0.416353	−0.208177	+	0.978091i	$0.566753\pi$
$398$	32.2047	1.61427
$399$	0	0
$400$	21.5205	1.07602
$401$	3.93362	0.196436	0.0982179	−	0.995165i	$-0.468686\pi$
$401$	3.93362	0.196436	0.0982179	+	0.995165i	$0.468686\pi$
$402$	0	0
$403$	14.3767	0.716152
$404$	5.61146	0.279181
$405$	0	0
$406$	7.78977	0.386600
$407$	0	0
$408$	0	0
$409$	−2.50425	−0.123827	−0.0619137	−	0.998082i	$-0.519720\pi$
$409$	−2.50425	−0.123827	−0.0619137	+	0.998082i	$0.519720\pi$
$410$	1.05644	0.0521737
$411$	0	0
$412$	−1.72964	−0.0852130
$413$	−5.15007	−0.253418
$414$	0	0
$415$	−4.11640	−0.202066
$416$	−3.43261	−0.168298
$417$	0	0
$418$	0	0
$419$	−10.7394	−0.524654	−0.262327	−	0.964979i	$-0.584490\pi$
$419$	−10.7394	−0.524654	−0.262327	+	0.964979i	$0.584490\pi$
$420$	0	0
$421$	22.1875	1.08135	0.540675	−	0.841232i	$-0.318169\pi$
$421$	22.1875	1.08135	0.540675	+	0.841232i	$0.318169\pi$
$422$	−5.24453	−0.255300
$423$	0	0
$424$	14.1931	0.689278
$425$	21.3505	1.03565
$426$	0	0
$427$	3.29134	0.159279
$428$	6.14879	0.297213
$429$	0	0
$430$	−3.85778	−0.186038
$431$	18.4582	0.889100	0.444550	−	0.895754i	$-0.353364\pi$
$431$	18.4582	0.889100	0.444550	+	0.895754i	$0.353364\pi$
$432$	0	0
$433$	15.1166	0.726459	0.363230	−	0.931700i	$-0.381674\pi$
$433$	15.1166	0.726459	0.363230	+	0.931700i	$0.381674\pi$
$434$	10.1893	0.489100
$435$	0	0
$436$	−5.34921	−0.256181
$437$	11.1683	0.534252
$438$	0	0
$439$	23.9553	1.14333	0.571663	−	0.820488i	$-0.306298\pi$
$439$	23.9553	1.14333	0.571663	+	0.820488i	$0.306298\pi$
$440$	0	0
$441$	0	0
$442$	−11.1820	−0.531875
$443$	−25.4977	−1.21143	−0.605716	−	0.795681i	$-0.707113\pi$
$443$	−25.4977	−1.21143	−0.605716	+	0.795681i	$0.707113\pi$
$444$	0	0
$445$	7.38872	0.350259
$446$	17.0483	0.807261
$447$	0	0
$448$	4.25036	0.200811
$449$	22.4824	1.06101	0.530505	−	0.847682i	$-0.322002\pi$
$449$	22.4824	1.06101	0.530505	+	0.847682i	$0.322002\pi$
$450$	0	0
$451$	0	0
$452$	−3.13607	−0.147508
$453$	0	0
$454$	−9.22562	−0.432980
$455$	0.673168	0.0315586
$456$	0	0
$457$	21.6896	1.01460	0.507298	−	0.861771i	$-0.330644\pi$
$457$	21.6896	1.01460	0.507298	+	0.861771i	$0.330644\pi$
$458$	−38.2679	−1.78814
$459$	0	0
$460$	−1.02935	−0.0479938
$461$	12.1036	0.563722	0.281861	−	0.959455i	$-0.409048\pi$
$461$	12.1036	0.563722	0.281861	+	0.959455i	$0.409048\pi$
$462$	0	0
$463$	20.0308	0.930910	0.465455	−	0.885071i	$-0.345891\pi$
$463$	20.0308	0.930910	0.465455	+	0.885071i	$0.345891\pi$
$464$	32.3009	1.49953
$465$	0	0
$466$	6.47586	0.299988
$467$	−6.06750	−0.280770	−0.140385	−	0.990097i	$-0.544834\pi$
$467$	−6.06750	−0.280770	−0.140385	+	0.990097i	$0.544834\pi$
$468$	0	0
$469$	−5.84161	−0.269740
$470$	0.208296	0.00960796
$471$	0	0
$472$	−17.7515	−0.817078
$473$	0	0
$474$	0	0
$475$	−11.6637	−0.535169
$476$	−1.29471	−0.0593430
$477$	0	0
$478$	−11.3994	−0.521397
$479$	−36.2728	−1.65735	−0.828673	−	0.559732i	$-0.810904\pi$
$479$	−36.2728	−1.65735	−0.828673	+	0.559732i	$0.810904\pi$
$480$	0	0
$481$	−4.55432	−0.207659
$482$	13.6512	0.621795
$483$	0	0
$484$	0	0
$485$	−8.39529	−0.381211
$486$	0	0
$487$	−8.68493	−0.393552	−0.196776	−	0.980449i	$-0.563047\pi$
$487$	−8.68493	−0.393552	−0.196776	+	0.980449i	$0.563047\pi$
$488$	11.3447	0.513552
$489$	0	0
$490$	−5.93044	−0.267910
$491$	33.1973	1.49817	0.749087	−	0.662472i	$-0.230492\pi$
$491$	33.1973	1.49817	0.749087	+	0.662472i	$0.230492\pi$
$492$	0	0
$493$	32.0458	1.44327
$494$	6.10872	0.274844
$495$	0	0
$496$	42.2506	1.89711
$497$	8.88678	0.398627
$498$	0	0
$499$	20.8769	0.934578	0.467289	−	0.884105i	$-0.345231\pi$
$499$	20.8769	0.934578	0.467289	+	0.884105i	$0.345231\pi$
$500$	2.23107	0.0997765
$501$	0	0
$502$	31.8087	1.41969
$503$	−23.3189	−1.03974	−0.519869	−	0.854246i	$-0.674019\pi$
$503$	−23.3189	−1.03974	−0.519869	+	0.854246i	$0.674019\pi$
$504$	0	0
$505$	−8.50663	−0.378540
$506$	0	0
$507$	0	0
$508$	−4.68502	−0.207864
$509$	−6.48121	−0.287275	−0.143637	−	0.989630i	$-0.545880\pi$
$509$	−6.48121	−0.287275	−0.143637	+	0.989630i	$0.545880\pi$
$510$	0	0
$511$	11.1782	0.494494
$512$	12.9480	0.572225
$513$	0	0
$514$	−5.22963	−0.230669
$515$	2.62202	0.115540
$516$	0	0
$517$	0	0
$518$	−3.22781	−0.141822
$519$	0	0
$520$	2.32030	0.101752
$521$	33.2103	1.45497	0.727484	−	0.686124i	$-0.240689\pi$
$521$	33.2103	1.45497	0.727484	+	0.686124i	$0.240689\pi$
$522$	0	0
$523$	−40.1386	−1.75514	−0.877569	−	0.479451i	$-0.840836\pi$
$523$	−40.1386	−1.75514	−0.877569	+	0.479451i	$0.840836\pi$
$524$	2.69123	0.117567
$525$	0	0
$526$	−22.0823	−0.962834
$527$	41.9169	1.82593
$528$	0	0
$529$	−3.17970	−0.138248
$530$	5.22085	0.226779
$531$	0	0
$532$	0.707298	0.0306652
$533$	−1.81770	−0.0787332
$534$	0	0
$535$	−9.32119	−0.402990
$536$	−20.1351	−0.869703
$537$	0	0
$538$	10.3907	0.447974
$539$	0	0
$540$	0	0
$541$	24.6485	1.05972	0.529860	−	0.848085i	$-0.322244\pi$
$541$	24.6485	1.05972	0.529860	+	0.848085i	$0.322244\pi$
$542$	6.15417	0.264344
$543$	0	0
$544$	−10.0082	−0.429098
$545$	8.10908	0.347355
$546$	0	0
$547$	25.7204	1.09972	0.549861	−	0.835256i	$-0.314681\pi$
$547$	25.7204	1.09972	0.549861	+	0.835256i	$0.314681\pi$
$548$	−3.61113	−0.154260
$549$	0	0
$550$	0	0
$551$	−17.5066	−0.745805
$552$	0	0
$553$	−10.5546	−0.448827
$554$	20.3039	0.862629
$555$	0	0
$556$	8.44003	0.357937
$557$	−11.6815	−0.494962	−0.247481	−	0.968893i	$-0.579603\pi$
$557$	−11.6815	−0.494962	−0.247481	+	0.968893i	$0.579603\pi$
$558$	0	0
$559$	6.63766	0.280743
$560$	1.97833	0.0835996
$561$	0	0
$562$	−4.58873	−0.193564
$563$	−8.12210	−0.342306	−0.171153	−	0.985244i	$-0.554749\pi$
$563$	−8.12210	−0.342306	−0.171153	+	0.985244i	$0.554749\pi$
$564$	0	0
$565$	4.75409	0.200006
$566$	−12.5125	−0.525941
$567$	0	0
$568$	30.6313	1.28526
$569$	18.0066	0.754877	0.377438	−	0.926035i	$-0.376805\pi$
$569$	18.0066	0.754877	0.377438	+	0.926035i	$0.376805\pi$
$570$	0	0
$571$	−2.60645	−0.109076	−0.0545382	−	0.998512i	$-0.517369\pi$
$571$	−2.60645	−0.109076	−0.0545382	+	0.998512i	$0.517369\pi$
$572$	0	0
$573$	0	0
$574$	−1.28827	−0.0537712
$575$	−20.6996	−0.863231
$576$	0	0
$577$	35.1888	1.46493	0.732465	−	0.680805i	$-0.238370\pi$
$577$	35.1888	1.46493	0.732465	+	0.680805i	$0.238370\pi$
$578$	−6.31827	−0.262805
$579$	0	0
$580$	1.61353	0.0669984
$581$	5.01972	0.208253
$582$	0	0
$583$	0	0
$584$	38.5294	1.59436
$585$	0	0
$586$	−24.2984	−1.00376
$587$	−23.5524	−0.972112	−0.486056	−	0.873928i	$-0.661565\pi$
$587$	−23.5524	−0.972112	−0.486056	+	0.873928i	$0.661565\pi$
$588$	0	0
$589$	−22.8991	−0.943541
$590$	−6.52979	−0.268827
$591$	0	0
$592$	−13.3844	−0.550095
$593$	−13.1828	−0.541354	−0.270677	−	0.962670i	$-0.587248\pi$
$593$	−13.1828	−0.541354	−0.270677	+	0.962670i	$0.587248\pi$
$594$	0	0
$595$	1.96270	0.0804630
$596$	0.447726	0.0183396
$597$	0	0
$598$	10.8411	0.443326
$599$	−34.2164	−1.39804	−0.699022	−	0.715100i	$-0.746381\pi$
$599$	−34.2164	−1.39804	−0.699022	+	0.715100i	$0.746381\pi$
$600$	0	0
$601$	29.0285	1.18410	0.592049	−	0.805902i	$-0.298319\pi$
$601$	29.0285	1.18410	0.592049	+	0.805902i	$0.298319\pi$
$602$	4.70435	0.191735
$603$	0	0
$604$	0.343924	0.0139941
$605$	0	0
$606$	0	0
$607$	−3.15635	−0.128112	−0.0640562	−	0.997946i	$-0.520404\pi$
$607$	−3.15635	−0.128112	−0.0640562	+	0.997946i	$0.520404\pi$
$608$	5.46745	0.221735
$609$	0	0
$610$	4.17310	0.168964
$611$	−0.358392	−0.0144990
$612$	0	0
$613$	33.9410	1.37086	0.685432	−	0.728136i	$-0.259613\pi$
$613$	33.9410	1.37086	0.685432	+	0.728136i	$0.259613\pi$
$614$	5.43181	0.219210
$615$	0	0
$616$	0	0
$617$	21.6077	0.869895	0.434948	−	0.900456i	$-0.356767\pi$
$617$	21.6077	0.869895	0.434948	+	0.900456i	$0.356767\pi$
$618$	0	0
$619$	20.5126	0.824471	0.412235	−	0.911077i	$-0.364748\pi$
$619$	20.5126	0.824471	0.412235	+	0.911077i	$0.364748\pi$
$620$	2.11055	0.0847617
$621$	0	0
$622$	40.7166	1.63259
$623$	−9.01014	−0.360984
$624$	0	0
$625$	19.8653	0.794613
$626$	3.80526	0.152089
$627$	0	0
$628$	−3.82189	−0.152510
$629$	−13.2787	−0.529456
$630$	0	0
$631$	29.8645	1.18889	0.594443	−	0.804138i	$-0.297372\pi$
$631$	29.8645	1.18889	0.594443	+	0.804138i	$0.297372\pi$
$632$	−36.3800	−1.44712
$633$	0	0
$634$	−40.9837	−1.62767
$635$	7.10220	0.281842
$636$	0	0
$637$	10.2039	0.404292
$638$	0	0
$639$	0	0
$640$	7.96969	0.315030
$641$	38.1201	1.50565	0.752826	−	0.658220i	$-0.228690\pi$
$641$	38.1201	1.50565	0.752826	+	0.658220i	$0.228690\pi$
$642$	0	0
$643$	−16.8251	−0.663519	−0.331759	−	0.943364i	$-0.607642\pi$
$643$	−16.8251	−0.663519	−0.331759	+	0.943364i	$0.607642\pi$
$644$	1.25524	0.0494633
$645$	0	0
$646$	17.8107	0.700753
$647$	20.9745	0.824592	0.412296	−	0.911050i	$-0.364727\pi$
$647$	20.9745	0.824592	0.412296	+	0.911050i	$0.364727\pi$
$648$	0	0
$649$	0	0
$650$	−11.3220	−0.444087
$651$	0	0
$652$	1.80810	0.0708106
$653$	−19.7665	−0.773522	−0.386761	−	0.922180i	$-0.626406\pi$
$653$	−19.7665	−0.773522	−0.386761	+	0.922180i	$0.626406\pi$
$654$	0	0
$655$	−4.07975	−0.159409
$656$	−5.34191	−0.208566
$657$	0	0
$658$	−0.254005	−0.00990215
$659$	11.8055	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$659$	11.8055	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$660$	0	0
$661$	17.8076	0.692637	0.346318	−	0.938117i	$-0.387432\pi$
$661$	17.8076	0.692637	0.346318	+	0.938117i	$0.387432\pi$
$662$	−12.5298	−0.486986
$663$	0	0
$664$	17.3022	0.671455
$665$	−1.07222	−0.0415789
$666$	0	0
$667$	−31.0688	−1.20299
$668$	6.63422	0.256686
$669$	0	0
$670$	−7.40658	−0.286141
$671$	0	0
$672$	0	0
$673$	19.7053	0.759583	0.379791	−	0.925072i	$-0.375996\pi$
$673$	19.7053	0.759583	0.379791	+	0.925072i	$0.375996\pi$
$674$	−38.2215	−1.47224
$675$	0	0
$676$	−4.10826	−0.158010
$677$	12.4139	0.477105	0.238553	−	0.971130i	$-0.423327\pi$
$677$	12.4139	0.477105	0.238553	+	0.971130i	$0.423327\pi$
$678$	0	0
$679$	10.2376	0.392883
$680$	6.76512	0.259431
$681$	0	0
$682$	0	0
$683$	−2.02837	−0.0776135	−0.0388068	−	0.999247i	$-0.512356\pi$
$683$	−2.02837	−0.0776135	−0.0388068	+	0.999247i	$0.512356\pi$
$684$	0	0
$685$	5.47425	0.209160
$686$	15.0455	0.574439
$687$	0	0
$688$	19.5070	0.743696
$689$	−8.98296	−0.342223
$690$	0	0
$691$	2.34287	0.0891270	0.0445635	−	0.999007i	$-0.485810\pi$
$691$	2.34287	0.0891270	0.0445635	+	0.999007i	$0.485810\pi$
$692$	0.230830	0.00877483
$693$	0	0
$694$	−3.19995	−0.121469
$695$	−12.7946	−0.485326
$696$	0	0
$697$	−5.29972	−0.200741
$698$	8.06197	0.305150
$699$	0	0
$700$	−1.31092	−0.0495482
$701$	4.43368	0.167458	0.0837289	−	0.996489i	$-0.473317\pi$
$701$	4.43368	0.167458	0.0837289	+	0.996489i	$0.473317\pi$
$702$	0	0
$703$	7.25411	0.273594
$704$	0	0
$705$	0	0
$706$	33.5087	1.26111
$707$	10.3734	0.390131
$708$	0	0
$709$	29.8825	1.12226	0.561131	−	0.827727i	$-0.310366\pi$
$709$	29.8825	1.12226	0.561131	+	0.827727i	$0.310366\pi$
$710$	11.2676	0.422864
$711$	0	0
$712$	−31.0565	−1.16389
$713$	−40.6389	−1.52194
$714$	0	0
$715$	0	0
$716$	3.76515	0.140710
$717$	0	0
$718$	13.5331	0.505051
$719$	21.6321	0.806742	0.403371	−	0.915037i	$-0.367838\pi$
$719$	21.6321	0.806742	0.403371	+	0.915037i	$0.367838\pi$
$720$	0	0
$721$	−3.19741	−0.119078
$722$	19.6466	0.731172
$723$	0	0
$724$	−1.85969	−0.0691149
$725$	32.4471	1.20505
$726$	0	0
$727$	−12.4422	−0.461455	−0.230727	−	0.973018i	$-0.574111\pi$
$727$	−12.4422	−0.461455	−0.230727	+	0.973018i	$0.574111\pi$
$728$	−2.82948	−0.104868
$729$	0	0
$730$	14.1729	0.524561
$731$	19.3529	0.715793
$732$	0	0
$733$	−52.3522	−1.93367	−0.966836	−	0.255396i	$-0.917794\pi$
$733$	−52.3522	−1.93367	−0.966836	+	0.255396i	$0.917794\pi$
$734$	10.4615	0.386141
$735$	0	0
$736$	9.70306	0.357660
$737$	0	0
$738$	0	0
$739$	−46.6760	−1.71701	−0.858503	−	0.512809i	$-0.828605\pi$
$739$	−46.6760	−1.71701	−0.858503	+	0.512809i	$0.828605\pi$
$740$	−0.668592	−0.0245779
$741$	0	0
$742$	−6.36655	−0.233723
$743$	21.8719	0.802401	0.401201	−	0.915990i	$-0.368593\pi$
$743$	21.8719	0.802401	0.401201	+	0.915990i	$0.368593\pi$
$744$	0	0
$745$	−0.678725	−0.0248665
$746$	33.8501	1.23934
$747$	0	0
$748$	0	0
$749$	11.3667	0.415330
$750$	0	0
$751$	39.5921	1.44474	0.722369	−	0.691508i	$-0.243053\pi$
$751$	39.5921	1.44474	0.722369	+	0.691508i	$0.243053\pi$
$752$	−1.05325	−0.0384082
$753$	0	0
$754$	−16.9937	−0.618874
$755$	−0.521367	−0.0189745
$756$	0	0
$757$	23.1457	0.841246	0.420623	−	0.907235i	$-0.361811\pi$
$757$	23.1457	0.841246	0.420623	+	0.907235i	$0.361811\pi$
$758$	52.2959	1.89947
$759$	0	0
$760$	−3.69577	−0.134060
$761$	14.8929	0.539866	0.269933	−	0.962879i	$-0.412998\pi$
$761$	14.8929	0.539866	0.269933	+	0.962879i	$0.412998\pi$
$762$	0	0
$763$	−9.88858	−0.357991
$764$	2.03003	0.0734438
$765$	0	0
$766$	−28.4085	−1.02644
$767$	11.2351	0.405676
$768$	0	0
$769$	21.6233	0.779757	0.389879	−	0.920866i	$-0.372517\pi$
$769$	21.6233	0.779757	0.389879	+	0.920866i	$0.372517\pi$
$770$	0	0
$771$	0	0
$772$	1.27630	0.0459349
$773$	6.07292	0.218428	0.109214	−	0.994018i	$-0.465167\pi$
$773$	6.07292	0.218428	0.109214	+	0.994018i	$0.465167\pi$
$774$	0	0
$775$	42.4417	1.52455
$776$	35.2874	1.26674
$777$	0	0
$778$	−47.6311	−1.70766
$779$	2.89522	0.103732
$780$	0	0
$781$	0	0
$782$	31.6086	1.13032
$783$	0	0
$784$	29.9874	1.07098
$785$	5.79375	0.206788
$786$	0	0
$787$	51.1554	1.82349	0.911746	−	0.410754i	$-0.134735\pi$
$787$	51.1554	1.82349	0.911746	+	0.410754i	$0.134735\pi$
$788$	8.77250	0.312507
$789$	0	0
$790$	−13.3822	−0.476117
$791$	−5.79736	−0.206130
$792$	0	0
$793$	−7.18019	−0.254976
$794$	12.8264	0.455192
$795$	0	0
$796$	−8.13457	−0.288322
$797$	25.2612	0.894798	0.447399	−	0.894334i	$-0.352350\pi$
$797$	25.2612	0.894798	0.447399	+	0.894334i	$0.352350\pi$
$798$	0	0
$799$	−1.04494	−0.0369672
$800$	−10.1335	−0.358274
$801$	0	0
$802$	−6.08192	−0.214760
$803$	0	0
$804$	0	0
$805$	−1.90286	−0.0670671
$806$	−22.2283	−0.782957
$807$	0	0
$808$	35.7554	1.25787
$809$	−51.3275	−1.80458	−0.902290	−	0.431130i	$-0.858115\pi$
$809$	−51.3275	−1.80458	−0.902290	+	0.431130i	$0.858115\pi$
$810$	0	0
$811$	4.54589	0.159628	0.0798138	−	0.996810i	$-0.474567\pi$
$811$	4.54589	0.159628	0.0798138	+	0.996810i	$0.474567\pi$
$812$	−1.96762	−0.0690498
$813$	0	0
$814$	0	0
$815$	−2.74097	−0.0960119
$816$	0	0
$817$	−10.5724	−0.369883
$818$	3.87192	0.135378
$819$	0	0
$820$	−0.266845	−0.00931864
$821$	−10.8738	−0.379498	−0.189749	−	0.981833i	$-0.560767\pi$
$821$	−10.8738	−0.379498	−0.189749	+	0.981833i	$0.560767\pi$
$822$	0	0
$823$	−23.6161	−0.823207	−0.411604	−	0.911363i	$-0.635031\pi$
$823$	−23.6161	−0.823207	−0.411604	+	0.911363i	$0.635031\pi$
$824$	−11.0210	−0.383934
$825$	0	0
$826$	7.96271	0.277058
$827$	27.6040	0.959886	0.479943	−	0.877300i	$-0.340657\pi$
$827$	27.6040	0.959886	0.479943	+	0.877300i	$0.340657\pi$
$828$	0	0
$829$	−16.1009	−0.559208	−0.279604	−	0.960115i	$-0.590203\pi$
$829$	−16.1009	−0.559208	−0.279604	+	0.960115i	$0.590203\pi$
$830$	6.36452	0.220916
$831$	0	0
$832$	−9.27234	−0.321461
$833$	29.7506	1.03080
$834$	0	0
$835$	−10.0571	−0.348039
$836$	0	0
$837$	0	0
$838$	16.6046	0.573596
$839$	19.4319	0.670865	0.335432	−	0.942064i	$-0.391118\pi$
$839$	19.4319	0.670865	0.335432	+	0.942064i	$0.391118\pi$
$840$	0	0
$841$	19.7012	0.679350
$842$	−34.3048	−1.18222
$843$	0	0
$844$	1.32471	0.0455986
$845$	6.22787	0.214245
$846$	0	0
$847$	0	0
$848$	−26.3994	−0.906559
$849$	0	0
$850$	−33.0108	−1.13226
$851$	12.8738	0.441309
$852$	0	0
$853$	−19.7855	−0.677443	−0.338722	−	0.940887i	$-0.609994\pi$
$853$	−19.7855	−0.677443	−0.338722	+	0.940887i	$0.609994\pi$
$854$	−5.08886	−0.174137
$855$	0	0
$856$	39.1791	1.33912
$857$	4.34065	0.148274	0.0741369	−	0.997248i	$-0.476380\pi$
$857$	4.34065	0.148274	0.0741369	+	0.997248i	$0.476380\pi$
$858$	0	0
$859$	−5.17935	−0.176717	−0.0883586	−	0.996089i	$-0.528162\pi$
$859$	−5.17935	−0.176717	−0.0883586	+	0.996089i	$0.528162\pi$
$860$	0.974434	0.0332279
$861$	0	0
$862$	−28.5389	−0.972039
$863$	2.69392	0.0917021	0.0458510	−	0.998948i	$-0.485400\pi$
$863$	2.69392	0.0917021	0.0458510	+	0.998948i	$0.485400\pi$
$864$	0	0
$865$	−0.349924	−0.0118978
$866$	−23.3724	−0.794226
$867$	0	0
$868$	−2.57370	−0.0873571
$869$	0	0
$870$	0	0
$871$	12.7437	0.431804
$872$	−34.0843	−1.15424
$873$	0	0
$874$	−17.2677	−0.584088
$875$	4.12437	0.139429
$876$	0	0
$877$	−20.8448	−0.703878	−0.351939	−	0.936023i	$-0.614478\pi$
$877$	−20.8448	−0.703878	−0.351939	+	0.936023i	$0.614478\pi$
$878$	−37.0382	−1.24998
$879$	0	0
$880$	0	0
$881$	−11.5843	−0.390286	−0.195143	−	0.980775i	$-0.562517\pi$
$881$	−11.5843	−0.390286	−0.195143	+	0.980775i	$0.562517\pi$
$882$	0	0
$883$	15.3973	0.518159	0.259079	−	0.965856i	$-0.416581\pi$
$883$	15.3973	0.518159	0.259079	+	0.965856i	$0.416581\pi$
$884$	2.82447	0.0949970
$885$	0	0
$886$	39.4229	1.32444
$887$	5.90996	0.198437	0.0992185	−	0.995066i	$-0.468366\pi$
$887$	5.90996	0.198437	0.0992185	+	0.995066i	$0.468366\pi$
$888$	0	0
$889$	−8.66074	−0.290472
$890$	−11.4240	−0.382932
$891$	0	0
$892$	−4.30623	−0.144183
$893$	0.570846	0.0191026
$894$	0	0
$895$	−5.70774	−0.190789
$896$	−9.71860	−0.324676
$897$	0	0
$898$	−34.7609	−1.15999
$899$	63.7025	2.12460
$900$	0	0
$901$	−26.1909	−0.872546
$902$	0	0
$903$	0	0
$904$	−19.9826	−0.664610
$905$	2.81918	0.0937127
$906$	0	0
$907$	−7.39332	−0.245491	−0.122746	−	0.992438i	$-0.539170\pi$
$907$	−7.39332	−0.245491	−0.122746	+	0.992438i	$0.539170\pi$
$908$	2.33030	0.0773336
$909$	0	0
$910$	−1.04081	−0.0345025
$911$	14.1988	0.470427	0.235214	−	0.971944i	$-0.424421\pi$
$911$	14.1988	0.470427	0.235214	+	0.971944i	$0.424421\pi$
$912$	0	0
$913$	0	0
$914$	−33.5350	−1.10924
$915$	0	0
$916$	9.66608	0.319376
$917$	4.97503	0.164290
$918$	0	0
$919$	−41.6426	−1.37366	−0.686831	−	0.726817i	$-0.740999\pi$
$919$	−41.6426	−1.37366	−0.686831	+	0.726817i	$0.740999\pi$
$920$	−6.55887	−0.216239
$921$	0	0
$922$	−18.7139	−0.616308
$923$	−19.3869	−0.638127
$924$	0	0
$925$	−13.4449	−0.442067
$926$	−30.9703	−1.01775
$927$	0	0
$928$	−15.2098	−0.499286
$929$	34.1328	1.11986	0.559930	−	0.828540i	$-0.310828\pi$
$929$	34.1328	1.11986	0.559930	+	0.828540i	$0.310828\pi$
$930$	0	0
$931$	−16.2527	−0.532660
$932$	−1.63574	−0.0535803
$933$	0	0
$934$	9.38118	0.306962
$935$	0	0
$936$	0	0
$937$	51.6822	1.68838	0.844192	−	0.536041i	$-0.180081\pi$
$937$	51.6822	1.68838	0.844192	+	0.536041i	$0.180081\pi$
$938$	9.03192	0.294903
$939$	0	0
$940$	−0.0526133	−0.00171606
$941$	−18.0062	−0.586985	−0.293492	−	0.955961i	$-0.594818\pi$
$941$	−18.0062	−0.586985	−0.293492	+	0.955961i	$0.594818\pi$
$942$	0	0
$943$	5.13814	0.167321
$944$	33.0181	1.07465
$945$	0	0
$946$	0	0
$947$	−28.6303	−0.930360	−0.465180	−	0.885216i	$-0.654010\pi$
$947$	−28.6303	−0.930360	−0.465180	+	0.885216i	$0.654010\pi$
$948$	0	0
$949$	−24.3857	−0.791593
$950$	18.0337	0.585091
$951$	0	0
$952$	−8.24970	−0.267374
$953$	−16.6893	−0.540620	−0.270310	−	0.962773i	$-0.587126\pi$
$953$	−16.6893	−0.540620	−0.270310	+	0.962773i	$0.587126\pi$
$954$	0	0
$955$	−3.07740	−0.0995822
$956$	2.87938	0.0931257
$957$	0	0
$958$	56.0827	1.81195
$959$	−6.67555	−0.215565
$960$	0	0
$961$	52.3248	1.68790
$962$	7.04160	0.227030
$963$	0	0
$964$	−3.44815	−0.111058
$965$	−1.93479	−0.0622830
$966$	0	0
$967$	24.3193	0.782057	0.391029	−	0.920379i	$-0.372119\pi$
$967$	24.3193	0.782057	0.391029	+	0.920379i	$0.372119\pi$
$968$	0	0
$969$	0	0
$970$	12.9803	0.416771
$971$	−7.55191	−0.242352	−0.121176	−	0.992631i	$-0.538667\pi$
$971$	−7.55191	−0.242352	−0.121176	+	0.992631i	$0.538667\pi$
$972$	0	0
$973$	15.6023	0.500186
$974$	13.4281	0.430264
$975$	0	0
$976$	−21.1014	−0.675439
$977$	−2.03229	−0.0650186	−0.0325093	−	0.999471i	$-0.510350\pi$
$977$	−2.03229	−0.0650186	−0.0325093	+	0.999471i	$0.510350\pi$
$978$	0	0
$979$	0	0
$980$	1.49797	0.0478508
$981$	0	0
$982$	−51.3276	−1.63793
$983$	−33.4396	−1.06656	−0.533279	−	0.845940i	$-0.679040\pi$
$983$	−33.4396	−1.06656	−0.533279	+	0.845940i	$0.679040\pi$
$984$	0	0
$985$	−13.2986	−0.423728
$986$	−49.5473	−1.57791
$987$	0	0
$988$	−1.54300	−0.0490893
$989$	−18.7629	−0.596624
$990$	0	0
$991$	40.1009	1.27385	0.636923	−	0.770927i	$-0.280207\pi$
$991$	40.1009	1.27385	0.636923	+	0.770927i	$0.280207\pi$
$992$	−19.8949	−0.631662
$993$	0	0
$994$	−13.7402	−0.435812
$995$	12.3315	0.390935
$996$	0	0
$997$	16.1301	0.510845	0.255423	−	0.966829i	$-0.417785\pi$
$997$	16.1301	0.510845	0.255423	+	0.966829i	$0.417785\pi$
$998$	−32.2785	−1.02176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cp.1.4		18
3.2	odd	2		9801.2.a.cm.1.15		18
9.2	odd	6		1089.2.e.p.364.4		36
9.5	odd	6		1089.2.e.p.727.4		36
11.5	even	5		891.2.f.e.487.8		36
11.9	even	5		891.2.f.e.730.8		36
11.10	odd	2		9801.2.a.cn.1.15		18
33.5	odd	10		891.2.f.f.487.2		36
33.20	odd	10		891.2.f.f.730.2		36
33.32	even	2		9801.2.a.co.1.4		18
99.5	odd	30		99.2.m.b.25.2	yes	72
99.16	even	15		297.2.n.b.91.2		72
99.20	odd	30		99.2.m.b.4.2	✓	72
99.31	even	15		297.2.n.b.235.2		72
99.32	even	6		1089.2.e.o.727.15		36
99.38	odd	30		99.2.m.b.58.8	yes	72
99.49	even	15		297.2.n.b.289.8		72
99.65	even	6		1089.2.e.o.364.15		36
99.86	odd	30		99.2.m.b.70.8	yes	72
99.97	even	15		297.2.n.b.37.8		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.4.2	✓	72	99.20	odd	30
99.2.m.b.25.2	yes	72	99.5	odd	30
99.2.m.b.58.8	yes	72	99.38	odd	30
99.2.m.b.70.8	yes	72	99.86	odd	30
297.2.n.b.37.8		72	99.97	even	15
297.2.n.b.91.2		72	99.16	even	15
297.2.n.b.235.2		72	99.31	even	15
297.2.n.b.289.8		72	99.49	even	15
891.2.f.e.487.8		36	11.5	even	5
891.2.f.e.730.8		36	11.9	even	5
891.2.f.f.487.2		36	33.5	odd	10
891.2.f.f.730.2		36	33.20	odd	10
1089.2.e.o.364.15		36	99.65	even	6
1089.2.e.o.727.15		36	99.32	even	6
1089.2.e.p.364.4		36	9.2	odd	6
1089.2.e.p.727.4		36	9.5	odd	6
9801.2.a.cm.1.15		18	3.2	odd	2
9801.2.a.cn.1.15		18	11.10	odd	2
9801.2.a.co.1.4		18	33.32	even	2
9801.2.a.cp.1.4		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.54614 q^{2} +0.390538 q^{4} -0.592032 q^{5} +0.721951 q^{7} +2.48845 q^{8} +0.915362 q^{10} -1.57496 q^{13} -1.11623 q^{14} -4.62856 q^{16} -4.59200 q^{17} +2.50860 q^{19} -0.231211 q^{20} +4.45200 q^{23} -4.64950 q^{25} +2.43511 q^{26} +0.281949 q^{28} -6.97862 q^{29} -9.12824 q^{31} +2.17948 q^{32} +7.09986 q^{34} -0.427418 q^{35} +2.89170 q^{37} -3.87864 q^{38} -1.47324 q^{40} +1.15412 q^{41} -4.21448 q^{43} -6.88340 q^{46} +0.227555 q^{47} -6.47879 q^{49} +7.18876 q^{50} -0.615084 q^{52} +5.70359 q^{53} +1.79654 q^{56} +10.7899 q^{58} -7.13355 q^{59} +4.55896 q^{61} +14.1135 q^{62} +5.88733 q^{64} +0.932429 q^{65} -8.09142 q^{67} -1.79335 q^{68} +0.660846 q^{70} +12.3094 q^{71} +15.4833 q^{73} -4.47096 q^{74} +0.979704 q^{76} -14.6196 q^{79} +2.74025 q^{80} -1.78443 q^{82} +6.95300 q^{83} +2.71861 q^{85} +6.51616 q^{86} -12.4803 q^{89} -1.13705 q^{91} +1.73868 q^{92} -0.351832 q^{94} -1.48517 q^{95} +14.1805 q^{97} +10.0171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} + 12 q^{4} + q^{5} + q^{7} + 6 q^{8} - 2 q^{10} + 3 q^{13} - 8 q^{16} + 20 q^{17} - 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} + 19 q^{28} + 21 q^{29} + 6 q^{31} + 9 q^{32} - 4 q^{34}+ \cdots + 82 q^{98}+O(q^{100}) \) 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