LMFDB - Embedded newform 9801.2.a.cr.1.7

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:	$24$
Twist minimal:	no (minimal twist has level 891)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.36405 q^{2} -0.139360 q^{4} +2.07738 q^{5} -0.103732 q^{7} +2.91820 q^{8} +O(q^{10})$	$q-1.36405 q^{2} -0.139360 q^{4} +2.07738 q^{5} -0.103732 q^{7} +2.91820 q^{8} -2.83366 q^{10} -6.62590 q^{13} +0.141496 q^{14} -3.70186 q^{16} +3.79348 q^{17} +6.60020 q^{19} -0.289503 q^{20} +2.58108 q^{23} -0.684493 q^{25} +9.03808 q^{26} +0.0144561 q^{28} +9.95456 q^{29} -2.20217 q^{31} -0.786868 q^{32} -5.17450 q^{34} -0.215491 q^{35} +3.26024 q^{37} -9.00303 q^{38} +6.06221 q^{40} -1.43428 q^{41} +3.82125 q^{43} -3.52073 q^{46} +3.43538 q^{47} -6.98924 q^{49} +0.933685 q^{50} +0.923385 q^{52} -0.0632616 q^{53} -0.302710 q^{56} -13.5785 q^{58} +6.96732 q^{59} +4.24299 q^{61} +3.00388 q^{62} +8.47705 q^{64} -13.7645 q^{65} -5.43311 q^{67} -0.528658 q^{68} +0.293940 q^{70} -11.8294 q^{71} -12.2358 q^{73} -4.44715 q^{74} -0.919803 q^{76} +15.5747 q^{79} -7.69017 q^{80} +1.95644 q^{82} +4.92298 q^{83} +7.88049 q^{85} -5.21238 q^{86} -7.99493 q^{89} +0.687317 q^{91} -0.359699 q^{92} -4.68604 q^{94} +13.7111 q^{95} +3.09027 q^{97} +9.53369 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 26 q^{4} + 18 q^{7}+O(q^{10}) $ 24 * q + 26 * q^4 + 18 * q^7	$ 24 q + 26 q^{4} + 18 q^{7} + 16 q^{10} + 18 q^{13} + 18 q^{16} + 36 q^{19} + 20 q^{25} + 46 q^{28} - 12 q^{31} + 42 q^{34} + 14 q^{37} + 42 q^{40} + 42 q^{43} + 26 q^{46} + 38 q^{49} + 36 q^{52} - 40 q^{58} + 62 q^{61} - 16 q^{64} + 38 q^{67} + 6 q^{70} + 2 q^{73} + 102 q^{76} + 94 q^{79} - 24 q^{82} + 42 q^{85} - 34 q^{91} + 66 q^{94} - 18 q^{97}+O(q^{100}) $ 24 * q + 26 * q^4 + 18 * q^7 + 16 * q^10 + 18 * q^13 + 18 * q^16 + 36 * q^19 + 20 * q^25 + 46 * q^28 - 12 * q^31 + 42 * q^34 + 14 * q^37 + 42 * q^40 + 42 * q^43 + 26 * q^46 + 38 * q^49 + 36 * q^52 - 40 * q^58 + 62 * q^61 - 16 * q^64 + 38 * q^67 + 6 * q^70 + 2 * q^73 + 102 * q^76 + 94 * q^79 - 24 * q^82 + 42 * q^85 - 34 * q^91 + 66 * q^94 - 18 * q^97

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 $

$2$	−1.36405	−0.964531	−0.482266	−	0.876025i	$-0.660186\pi$
$2$	−1.36405	−0.964531	−0.482266	+	0.876025i	$0.660186\pi$
$3$	0	0
$3$	0	0
$4$	−0.139360	−0.0696799
$5$	2.07738	0.929032	0.464516	−	0.885565i	$-0.346228\pi$
$5$	2.07738	0.929032	0.464516	+	0.885565i	$0.346228\pi$
$6$	0	0
$7$	−0.103732	−0.0392070	−0.0196035	−	0.999808i	$-0.506240\pi$
$7$	−0.103732	−0.0392070	−0.0196035	+	0.999808i	$0.506240\pi$
$8$	2.91820	1.03174
$9$	0	0
$10$	−2.83366	−0.896081
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.62590	−1.83769	−0.918847	−	0.394613i	$-0.870879\pi$
$13$	−6.62590	−1.83769	−0.918847	+	0.394613i	$0.870879\pi$
$14$	0.141496	0.0378163
$15$	0	0
$16$	−3.70186	−0.925465
$17$	3.79348	0.920053	0.460027	−	0.887905i	$-0.347840\pi$
$17$	3.79348	0.920053	0.460027	+	0.887905i	$0.347840\pi$
$18$	0	0
$19$	6.60020	1.51419	0.757095	−	0.653304i	$-0.226618\pi$
$19$	6.60020	1.51419	0.757095	+	0.653304i	$0.226618\pi$
$20$	−0.289503	−0.0647349
$21$	0	0
$22$	0	0
$23$	2.58108	0.538193	0.269096	−	0.963113i	$-0.413275\pi$
$23$	2.58108	0.538193	0.269096	+	0.963113i	$0.413275\pi$
$24$	0	0
$25$	−0.684493	−0.136899
$26$	9.03808	1.77251
$27$	0	0
$28$	0.0144561	0.00273194
$29$	9.95456	1.84852	0.924258	−	0.381770i	$-0.124685\pi$
$29$	9.95456	1.84852	0.924258	+	0.381770i	$0.124685\pi$
$30$	0	0
$31$	−2.20217	−0.395522	−0.197761	−	0.980250i	$-0.563367\pi$
$31$	−2.20217	−0.395522	−0.197761	+	0.980250i	$0.563367\pi$
$32$	−0.786868	−0.139100
$33$	0	0
$34$	−5.17450	−0.887420
$35$	−0.215491	−0.0364245
$36$	0	0
$37$	3.26024	0.535981	0.267990	−	0.963422i	$-0.413640\pi$
$37$	3.26024	0.535981	0.267990	+	0.963422i	$0.413640\pi$
$38$	−9.00303	−1.46048
$39$	0	0
$40$	6.06221	0.958519
$41$	−1.43428	−0.223998	−0.111999	−	0.993708i	$-0.535725\pi$
$41$	−1.43428	−0.223998	−0.111999	+	0.993708i	$0.535725\pi$
$42$	0	0
$43$	3.82125	0.582735	0.291367	−	0.956611i	$-0.405890\pi$
$43$	3.82125	0.582735	0.291367	+	0.956611i	$0.405890\pi$
$44$	0	0
$45$	0	0
$46$	−3.52073	−0.519104
$47$	3.43538	0.501101	0.250551	−	0.968103i	$-0.419388\pi$
$47$	3.43538	0.501101	0.250551	+	0.968103i	$0.419388\pi$
$48$	0	0
$49$	−6.98924	−0.998463
$50$	0.933685	0.132043
$51$	0	0
$52$	0.923385	0.128050
$53$	−0.0632616	−0.00868965	−0.00434482	−	0.999991i	$-0.501383\pi$
$53$	−0.0632616	−0.00868965	−0.00434482	+	0.999991i	$0.501383\pi$
$54$	0	0
$55$	0	0
$56$	−0.302710	−0.0404514
$57$	0	0
$58$	−13.5785	−1.78295
$59$	6.96732	0.907068	0.453534	−	0.891239i	$-0.350163\pi$
$59$	6.96732	0.907068	0.453534	+	0.891239i	$0.350163\pi$
$60$	0	0
$61$	4.24299	0.543260	0.271630	−	0.962402i	$-0.412437\pi$
$61$	4.24299	0.543260	0.271630	+	0.962402i	$0.412437\pi$
$62$	3.00388	0.381493
$63$	0	0
$64$	8.47705	1.05963
$65$	−13.7645	−1.70728
$66$	0	0
$67$	−5.43311	−0.663761	−0.331880	−	0.943322i	$-0.607683\pi$
$67$	−5.43311	−0.663761	−0.331880	+	0.943322i	$0.607683\pi$
$68$	−0.528658	−0.0641092
$69$	0	0
$70$	0.293940	0.0351326
$71$	−11.8294	−1.40389	−0.701944	−	0.712232i	$-0.747684\pi$
$71$	−11.8294	−1.40389	−0.701944	+	0.712232i	$0.747684\pi$
$72$	0	0
$73$	−12.2358	−1.43209	−0.716045	−	0.698054i	$-0.754050\pi$
$73$	−12.2358	−1.43209	−0.716045	+	0.698054i	$0.754050\pi$
$74$	−4.44715	−0.516970
$75$	0	0
$76$	−0.919803	−0.105509
$77$	0	0
$78$	0	0
$79$	15.5747	1.75230	0.876148	−	0.482043i	$-0.160105\pi$
$79$	15.5747	1.75230	0.876148	+	0.482043i	$0.160105\pi$
$80$	−7.69017	−0.859787
$81$	0	0
$82$	1.95644	0.216053
$83$	4.92298	0.540368	0.270184	−	0.962809i	$-0.412916\pi$
$83$	4.92298	0.540368	0.270184	+	0.962809i	$0.412916\pi$
$84$	0	0
$85$	7.88049	0.854759
$86$	−5.21238	−0.562066
$87$	0	0
$88$	0	0
$89$	−7.99493	−0.847461	−0.423731	−	0.905788i	$-0.639280\pi$
$89$	−7.99493	−0.847461	−0.423731	+	0.905788i	$0.639280\pi$
$90$	0	0
$91$	0.687317	0.0720504
$92$	−0.359699	−0.0375012
$93$	0	0
$94$	−4.68604	−0.483328
$95$	13.7111	1.40673
$96$	0	0
$97$	3.09027	0.313769	0.156885	−	0.987617i	$-0.449855\pi$
$97$	3.09027	0.313769	0.156885	+	0.987617i	$0.449855\pi$
$98$	9.53369	0.963048
$99$	0	0
$100$	0.0953908	0.00953908
$101$	−2.49785	−0.248545	−0.124273	−	0.992248i	$-0.539660\pi$
$101$	−2.49785	−0.248545	−0.124273	+	0.992248i	$0.539660\pi$
$102$	0	0
$103$	−14.1509	−1.39433	−0.697167	−	0.716909i	$-0.745556\pi$
$103$	−14.1509	−1.39433	−0.697167	+	0.716909i	$0.745556\pi$
$104$	−19.3357	−1.89602
$105$	0	0
$106$	0.0862922	0.00838143
$107$	5.05455	0.488642	0.244321	−	0.969694i	$-0.421435\pi$
$107$	5.05455	0.488642	0.244321	+	0.969694i	$0.421435\pi$
$108$	0	0
$109$	16.0865	1.54081	0.770403	−	0.637557i	$-0.220055\pi$
$109$	16.0865	1.54081	0.770403	+	0.637557i	$0.220055\pi$
$110$	0	0
$111$	0	0
$112$	0.384001	0.0362847
$113$	10.2015	0.959673	0.479837	−	0.877358i	$-0.340696\pi$
$113$	10.2015	0.959673	0.479837	+	0.877358i	$0.340696\pi$
$114$	0	0
$115$	5.36189	0.499999
$116$	−1.38727	−0.128804
$117$	0	0
$118$	−9.50380	−0.874895
$119$	−0.393505	−0.0360725
$120$	0	0
$121$	0	0
$122$	−5.78767	−0.523991
$123$	0	0
$124$	0.306895	0.0275599
$125$	−11.8089	−1.05622
$126$	0	0
$127$	3.42113	0.303576	0.151788	−	0.988413i	$-0.451497\pi$
$127$	3.42113	0.303576	0.151788	+	0.988413i	$0.451497\pi$
$128$	−9.98941	−0.882947
$129$	0	0
$130$	18.7755	1.64672
$131$	−17.7504	−1.55086	−0.775432	−	0.631431i	$-0.782468\pi$
$131$	−17.7504	−1.55086	−0.775432	+	0.631431i	$0.782468\pi$
$132$	0	0
$133$	−0.684652	−0.0593668
$134$	7.41106	0.640218
$135$	0	0
$136$	11.0701	0.949255
$137$	8.71168	0.744289	0.372144	−	0.928175i	$-0.378623\pi$
$137$	8.71168	0.744289	0.372144	+	0.928175i	$0.378623\pi$
$138$	0	0
$139$	−3.82825	−0.324708	−0.162354	−	0.986733i	$-0.551909\pi$
$139$	−3.82825	−0.324708	−0.162354	+	0.986733i	$0.551909\pi$
$140$	0.0300307	0.00253806
$141$	0	0
$142$	16.1359	1.35409
$143$	0	0
$144$	0	0
$145$	20.6794	1.71733
$146$	16.6903	1.38130
$147$	0	0
$148$	−0.454347	−0.0373471
$149$	10.3566	0.848445	0.424223	−	0.905558i	$-0.360547\pi$
$149$	10.3566	0.848445	0.424223	+	0.905558i	$0.360547\pi$
$150$	0	0
$151$	−10.0155	−0.815051	−0.407526	−	0.913194i	$-0.633608\pi$
$151$	−10.0155	−0.815051	−0.407526	+	0.913194i	$0.633608\pi$
$152$	19.2607	1.56225
$153$	0	0
$154$	0	0
$155$	−4.57475	−0.367453
$156$	0	0
$157$	11.6002	0.925793	0.462897	−	0.886412i	$-0.346810\pi$
$157$	11.6002	0.925793	0.462897	+	0.886412i	$0.346810\pi$
$158$	−21.2448	−1.69014
$159$	0	0
$160$	−1.63462	−0.129228
$161$	−0.267741	−0.0211009
$162$	0	0
$163$	−22.7972	−1.78562	−0.892808	−	0.450437i	$-0.851268\pi$
$163$	−22.7972	−1.78562	−0.892808	+	0.450437i	$0.851268\pi$
$164$	0.199882	0.0156081
$165$	0	0
$166$	−6.71521	−0.521201
$167$	−4.56982	−0.353623	−0.176811	−	0.984245i	$-0.556578\pi$
$167$	−4.56982	−0.353623	−0.176811	+	0.984245i	$0.556578\pi$
$168$	0	0
$169$	30.9026	2.37712
$170$	−10.7494	−0.824442
$171$	0	0
$172$	−0.532528	−0.0406049
$173$	1.50022	0.114060	0.0570300	−	0.998372i	$-0.481837\pi$
$173$	1.50022	0.114060	0.0570300	+	0.998372i	$0.481837\pi$
$174$	0	0
$175$	0.0710038	0.00536738
$176$	0	0
$177$	0	0
$178$	10.9055	0.817402
$179$	−8.93750	−0.668020	−0.334010	−	0.942569i	$-0.608402\pi$
$179$	−8.93750	−0.668020	−0.334010	+	0.942569i	$0.608402\pi$
$180$	0	0
$181$	20.9653	1.55834	0.779168	−	0.626815i	$-0.215642\pi$
$181$	20.9653	1.55834	0.779168	+	0.626815i	$0.215642\pi$
$182$	−0.937537	−0.0694949
$183$	0	0
$184$	7.53211	0.555275
$185$	6.77276	0.497944
$186$	0	0
$187$	0	0
$188$	−0.478754	−0.0349167
$189$	0	0
$190$	−18.7027	−1.35684
$191$	−8.94242	−0.647051	−0.323526	−	0.946219i	$-0.604868\pi$
$191$	−8.94242	−0.647051	−0.323526	+	0.946219i	$0.604868\pi$
$192$	0	0
$193$	5.71872	0.411643	0.205821	−	0.978590i	$-0.434013\pi$
$193$	5.71872	0.411643	0.205821	+	0.978590i	$0.434013\pi$
$194$	−4.21529	−0.302640
$195$	0	0
$196$	0.974019	0.0695728
$197$	6.94318	0.494681	0.247341	−	0.968929i	$-0.420443\pi$
$197$	6.94318	0.494681	0.247341	+	0.968929i	$0.420443\pi$
$198$	0	0
$199$	24.0199	1.70273	0.851363	−	0.524576i	$-0.175776\pi$
$199$	24.0199	1.70273	0.851363	+	0.524576i	$0.175776\pi$
$200$	−1.99749	−0.141244
$201$	0	0
$202$	3.40719	0.239729
$203$	−1.03261	−0.0724747
$204$	0	0
$205$	−2.97955	−0.208101
$206$	19.3026	1.34488
$207$	0	0
$208$	24.5282	1.70072
$209$	0	0
$210$	0	0
$211$	4.05355	0.279058	0.139529	−	0.990218i	$-0.455441\pi$
$211$	4.05355	0.279058	0.139529	+	0.990218i	$0.455441\pi$
$212$	0.00881612	0.000605494 0
$213$	0	0
$214$	−6.89467	−0.471310
$215$	7.93818	0.541379
$216$	0	0
$217$	0.228436	0.0155072
$218$	−21.9428	−1.48616
$219$	0	0
$220$	0	0
$221$	−25.1352	−1.69078
$222$	0	0
$223$	−10.1265	−0.678117	−0.339059	−	0.940765i	$-0.610109\pi$
$223$	−10.1265	−0.678117	−0.339059	+	0.940765i	$0.610109\pi$
$224$	0.0816233	0.00545369
$225$	0	0
$226$	−13.9153	−0.925634
$227$	6.54450	0.434374	0.217187	−	0.976130i	$-0.430312\pi$
$227$	6.54450	0.434374	0.217187	+	0.976130i	$0.430312\pi$
$228$	0	0
$229$	15.4522	1.02111	0.510555	−	0.859845i	$-0.329440\pi$
$229$	15.4522	1.02111	0.510555	+	0.859845i	$0.329440\pi$
$230$	−7.31390	−0.482264
$231$	0	0
$232$	29.0494	1.90719
$233$	21.2711	1.39351	0.696757	−	0.717307i	$-0.254625\pi$
$233$	21.2711	1.39351	0.696757	+	0.717307i	$0.254625\pi$
$234$	0	0
$235$	7.13658	0.465539
$236$	−0.970965	−0.0632044
$237$	0	0
$238$	0.536761	0.0347931
$239$	17.3437	1.12187	0.560936	−	0.827859i	$-0.310441\pi$
$239$	17.3437	1.12187	0.560936	+	0.827859i	$0.310441\pi$
$240$	0	0
$241$	10.6429	0.685572	0.342786	−	0.939414i	$-0.388629\pi$
$241$	10.6429	0.685572	0.342786	+	0.939414i	$0.388629\pi$
$242$	0	0
$243$	0	0
$244$	−0.591303	−0.0378543
$245$	−14.5193	−0.927604
$246$	0	0
$247$	−43.7323	−2.78262
$248$	−6.42638	−0.408076
$249$	0	0
$250$	16.1079	1.01875
$251$	27.9977	1.76720	0.883599	−	0.468245i	$-0.155113\pi$
$251$	27.9977	1.76720	0.883599	+	0.468245i	$0.155113\pi$
$252$	0	0
$253$	0	0
$254$	−4.66660	−0.292809
$255$	0	0
$256$	−3.32802	−0.208001
$257$	−17.9721	−1.12107	−0.560536	−	0.828130i	$-0.689405\pi$
$257$	−17.9721	−1.12107	−0.560536	+	0.828130i	$0.689405\pi$
$258$	0	0
$259$	−0.338191	−0.0210142
$260$	1.91822	0.118963
$261$	0	0
$262$	24.2125	1.49586
$263$	−13.7602	−0.848493	−0.424247	−	0.905547i	$-0.639461\pi$
$263$	−13.7602	−0.848493	−0.424247	+	0.905547i	$0.639461\pi$
$264$	0	0
$265$	−0.131418	−0.00807296
$266$	0.933901	0.0572612
$267$	0	0
$268$	0.757158	0.0462508
$269$	−14.3122	−0.872628	−0.436314	−	0.899794i	$-0.643716\pi$
$269$	−14.3122	−0.872628	−0.436314	+	0.899794i	$0.643716\pi$
$270$	0	0
$271$	−1.53931	−0.0935061	−0.0467531	−	0.998906i	$-0.514887\pi$
$271$	−1.53931	−0.0935061	−0.0467531	+	0.998906i	$0.514887\pi$
$272$	−14.0429	−0.851477
$273$	0	0
$274$	−11.8832	−0.717889
$275$	0	0
$276$	0	0
$277$	−23.9125	−1.43676	−0.718381	−	0.695650i	$-0.755116\pi$
$277$	−23.9125	−1.43676	−0.718381	+	0.695650i	$0.755116\pi$
$278$	5.22193	0.313191
$279$	0	0
$280$	−0.628844	−0.0375806
$281$	−15.2911	−0.912193	−0.456097	−	0.889930i	$-0.650753\pi$
$281$	−15.2911	−0.912193	−0.456097	+	0.889930i	$0.650753\pi$
$282$	0	0
$283$	20.4839	1.21764	0.608821	−	0.793307i	$-0.291643\pi$
$283$	20.4839	1.21764	0.608821	+	0.793307i	$0.291643\pi$
$284$	1.64854	0.0978228
$285$	0	0
$286$	0	0
$287$	0.148781	0.00878227
$288$	0	0
$289$	−2.60953	−0.153502
$290$	−28.2078	−1.65642
$291$	0	0
$292$	1.70518	0.0997879
$293$	10.8918	0.636307	0.318154	−	0.948039i	$-0.396937\pi$
$293$	10.8918	0.636307	0.318154	+	0.948039i	$0.396937\pi$
$294$	0	0
$295$	14.4738	0.842696
$296$	9.51404	0.552993
$297$	0	0
$298$	−14.1269	−0.818352
$299$	−17.1020	−0.989034
$300$	0	0
$301$	−0.396385	−0.0228473
$302$	13.6617	0.786142
$303$	0	0
$304$	−24.4330	−1.40133
$305$	8.81431	0.504706
$306$	0	0
$307$	16.2173	0.925568	0.462784	−	0.886471i	$-0.346850\pi$
$307$	16.2173	0.925568	0.462784	+	0.886471i	$0.346850\pi$
$308$	0	0
$309$	0	0
$310$	6.24020	0.354420
$311$	−12.8007	−0.725860	−0.362930	−	0.931816i	$-0.618224\pi$
$311$	−12.8007	−0.725860	−0.362930	+	0.931816i	$0.618224\pi$
$312$	0	0
$313$	12.1238	0.685275	0.342638	−	0.939468i	$-0.388680\pi$
$313$	12.1238	0.685275	0.342638	+	0.939468i	$0.388680\pi$
$314$	−15.8232	−0.892956
$315$	0	0
$316$	−2.17049	−0.122100
$317$	8.38371	0.470876	0.235438	−	0.971889i	$-0.424348\pi$
$317$	8.38371	0.470876	0.235438	+	0.971889i	$0.424348\pi$
$318$	0	0
$319$	0	0
$320$	17.6100	0.984432
$321$	0	0
$322$	0.365212	0.0203525
$323$	25.0377	1.39314
$324$	0	0
$325$	4.53539	0.251578
$326$	31.0966	1.72228
$327$	0	0
$328$	−4.18553	−0.231107
$329$	−0.356358	−0.0196467
$330$	0	0
$331$	−8.15077	−0.448007	−0.224003	−	0.974588i	$-0.571913\pi$
$331$	−8.15077	−0.448007	−0.224003	+	0.974588i	$0.571913\pi$
$332$	−0.686066	−0.0376528
$333$	0	0
$334$	6.23347	0.341080
$335$	−11.2866	−0.616655
$336$	0	0
$337$	30.0309	1.63589	0.817943	−	0.575299i	$-0.195114\pi$
$337$	30.0309	1.63589	0.817943	+	0.575299i	$0.195114\pi$
$338$	−42.1528	−2.29281
$339$	0	0
$340$	−1.09822	−0.0595596
$341$	0	0
$342$	0	0
$343$	1.45113	0.0783537
$344$	11.1512	0.601230
$345$	0	0
$346$	−2.04638	−0.110014
$347$	−34.0155	−1.82605	−0.913025	−	0.407905i	$-0.866260\pi$
$347$	−34.0155	−1.82605	−0.913025	+	0.407905i	$0.866260\pi$
$348$	0	0
$349$	23.5467	1.26043	0.630213	−	0.776422i	$-0.282967\pi$
$349$	23.5467	1.26043	0.630213	+	0.776422i	$0.282967\pi$
$350$	−0.0968529	−0.00517701
$351$	0	0
$352$	0	0
$353$	−15.8273	−0.842404	−0.421202	−	0.906967i	$-0.638392\pi$
$353$	−15.8273	−0.842404	−0.421202	+	0.906967i	$0.638392\pi$
$354$	0	0
$355$	−24.5741	−1.30426
$356$	1.11417	0.0590510
$357$	0	0
$358$	12.1912	0.644326
$359$	−0.438445	−0.0231403	−0.0115701	−	0.999933i	$-0.503683\pi$
$359$	−0.438445	−0.0231403	−0.0115701	+	0.999933i	$0.503683\pi$
$360$	0	0
$361$	24.5627	1.29277
$362$	−28.5977	−1.50306
$363$	0	0
$364$	−0.0957844	−0.00502047
$365$	−25.4184	−1.33046
$366$	0	0
$367$	−13.8560	−0.723276	−0.361638	−	0.932319i	$-0.617782\pi$
$367$	−13.8560	−0.723276	−0.361638	+	0.932319i	$0.617782\pi$
$368$	−9.55480	−0.498078
$369$	0	0
$370$	−9.23841	−0.480282
$371$	0.00656224	0.000340695 0
$372$	0	0
$373$	7.01569	0.363259	0.181629	−	0.983367i	$-0.441863\pi$
$373$	7.01569	0.363259	0.181629	+	0.983367i	$0.441863\pi$
$374$	0	0
$375$	0	0
$376$	10.0251	0.517006
$377$	−65.9579	−3.39701
$378$	0	0
$379$	−9.97473	−0.512368	−0.256184	−	0.966628i	$-0.582465\pi$
$379$	−9.97473	−0.512368	−0.256184	+	0.966628i	$0.582465\pi$
$380$	−1.91078	−0.0980210
$381$	0	0
$382$	12.1979	0.624101
$383$	−18.8601	−0.963704	−0.481852	−	0.876253i	$-0.660036\pi$
$383$	−18.8601	−0.963704	−0.481852	+	0.876253i	$0.660036\pi$
$384$	0	0
$385$	0	0
$386$	−7.80064	−0.397042
$387$	0	0
$388$	−0.430659	−0.0218634
$389$	−23.3474	−1.18376	−0.591880	−	0.806026i	$-0.701614\pi$
$389$	−23.3474	−1.18376	−0.591880	+	0.806026i	$0.701614\pi$
$390$	0	0
$391$	9.79128	0.495166
$392$	−20.3960	−1.03015
$393$	0	0
$394$	−9.47086	−0.477135
$395$	32.3547	1.62794
$396$	0	0
$397$	−27.6991	−1.39018	−0.695090	−	0.718923i	$-0.744636\pi$
$397$	−27.6991	−1.39018	−0.695090	+	0.718923i	$0.744636\pi$
$398$	−32.7644	−1.64233
$399$	0	0
$400$	2.53390	0.126695
$401$	15.1059	0.754351	0.377176	−	0.926142i	$-0.376895\pi$
$401$	15.1059	0.754351	0.377176	+	0.926142i	$0.376895\pi$
$402$	0	0
$403$	14.5914	0.726849
$404$	0.348099	0.0173186
$405$	0	0
$406$	1.40853	0.0699041
$407$	0	0
$408$	0	0
$409$	−9.40802	−0.465197	−0.232598	−	0.972573i	$-0.574723\pi$
$409$	−9.40802	−0.465197	−0.232598	+	0.972573i	$0.574723\pi$
$410$	4.06427	0.200720
$411$	0	0
$412$	1.97207	0.0971570
$413$	−0.722733	−0.0355634
$414$	0	0
$415$	10.2269	0.502019
$416$	5.21371	0.255623
$417$	0	0
$418$	0	0
$419$	20.1038	0.982137	0.491069	−	0.871121i	$-0.336606\pi$
$419$	20.1038	0.982137	0.491069	+	0.871121i	$0.336606\pi$
$420$	0	0
$421$	−12.8452	−0.626039	−0.313019	−	0.949747i	$-0.601341\pi$
$421$	−12.8452	−0.626039	−0.313019	+	0.949747i	$0.601341\pi$
$422$	−5.52926	−0.269160
$423$	0	0
$424$	−0.184610	−0.00896545
$425$	−2.59661	−0.125954
$426$	0	0
$427$	−0.440134	−0.0212996
$428$	−0.704401	−0.0340485
$429$	0	0
$430$	−10.8281	−0.522177
$431$	−15.3007	−0.737008	−0.368504	−	0.929626i	$-0.620130\pi$
$431$	−15.3007	−0.737008	−0.368504	+	0.929626i	$0.620130\pi$
$432$	0	0
$433$	−12.0460	−0.578892	−0.289446	−	0.957194i	$-0.593471\pi$
$433$	−12.0460	−0.578892	−0.289446	+	0.957194i	$0.593471\pi$
$434$	−0.311598	−0.0149572
$435$	0	0
$436$	−2.24181	−0.107363
$437$	17.0357	0.814927
$438$	0	0
$439$	−4.14815	−0.197980	−0.0989902	−	0.995088i	$-0.531561\pi$
$439$	−4.14815	−0.197980	−0.0989902	+	0.995088i	$0.531561\pi$
$440$	0	0
$441$	0	0
$442$	34.2858	1.63081
$443$	19.0918	0.907080	0.453540	−	0.891236i	$-0.350161\pi$
$443$	19.0918	0.907080	0.453540	+	0.891236i	$0.350161\pi$
$444$	0	0
$445$	−16.6085	−0.787319
$446$	13.8130	0.654065
$447$	0	0
$448$	−0.879340	−0.0415449
$449$	3.41431	0.161131	0.0805655	−	0.996749i	$-0.474327\pi$
$449$	3.41431	0.161131	0.0805655	+	0.996749i	$0.474327\pi$
$450$	0	0
$451$	0	0
$452$	−1.42167	−0.0668699
$453$	0	0
$454$	−8.92705	−0.418967
$455$	1.42782	0.0669372
$456$	0	0
$457$	−29.5442	−1.38202	−0.691011	−	0.722845i	$-0.742834\pi$
$457$	−29.5442	−1.38202	−0.691011	+	0.722845i	$0.742834\pi$
$458$	−21.0776	−0.984892
$459$	0	0
$460$	−0.747232	−0.0348399
$461$	−40.7758	−1.89912	−0.949560	−	0.313586i	$-0.898470\pi$
$461$	−40.7758	−1.89912	−0.949560	+	0.313586i	$0.898470\pi$
$462$	0	0
$463$	−38.7501	−1.80087	−0.900435	−	0.434990i	$-0.856752\pi$
$463$	−38.7501	−1.80087	−0.900435	+	0.434990i	$0.856752\pi$
$464$	−36.8504	−1.71074
$465$	0	0
$466$	−29.0149	−1.34409
$467$	28.0641	1.29865	0.649325	−	0.760511i	$-0.275051\pi$
$467$	28.0641	1.29865	0.649325	+	0.760511i	$0.275051\pi$
$468$	0	0
$469$	0.563587	0.0260240
$470$	−9.73468	−0.449027
$471$	0	0
$472$	20.3320	0.935858
$473$	0	0
$474$	0	0
$475$	−4.51780	−0.207291
$476$	0.0548387	0.00251353
$477$	0	0
$478$	−23.6577	−1.08208
$479$	15.8245	0.723042	0.361521	−	0.932364i	$-0.382258\pi$
$479$	15.8245	0.723042	0.361521	+	0.932364i	$0.382258\pi$
$480$	0	0
$481$	−21.6021	−0.984969
$482$	−14.5175	−0.661255
$483$	0	0
$484$	0	0
$485$	6.41966	0.291502
$486$	0	0
$487$	−5.74666	−0.260406	−0.130203	−	0.991487i	$-0.541563\pi$
$487$	−5.74666	−0.260406	−0.130203	+	0.991487i	$0.541563\pi$
$488$	12.3819	0.560503
$489$	0	0
$490$	19.8051	0.894703
$491$	35.6797	1.61020	0.805101	−	0.593137i	$-0.202111\pi$
$491$	35.6797	1.61020	0.805101	+	0.593137i	$0.202111\pi$
$492$	0	0
$493$	37.7624	1.70073
$494$	59.6532	2.68392
$495$	0	0
$496$	8.15214	0.366042
$497$	1.22708	0.0550422
$498$	0	0
$499$	26.6029	1.19091	0.595454	−	0.803390i	$-0.296972\pi$
$499$	26.6029	1.19091	0.595454	+	0.803390i	$0.296972\pi$
$500$	1.64568	0.0735970
$501$	0	0
$502$	−38.1903	−1.70452
$503$	32.2906	1.43977	0.719883	−	0.694095i	$-0.244195\pi$
$503$	32.2906	1.43977	0.719883	+	0.694095i	$0.244195\pi$
$504$	0	0
$505$	−5.18898	−0.230906
$506$	0	0
$507$	0	0
$508$	−0.476768	−0.0211532
$509$	32.4304	1.43745	0.718725	−	0.695294i	$-0.244726\pi$
$509$	32.4304	1.43745	0.718725	+	0.695294i	$0.244726\pi$
$510$	0	0
$511$	1.26924	0.0561479
$512$	24.5184	1.08357
$513$	0	0
$514$	24.5150	1.08131
$515$	−29.3969	−1.29538
$516$	0	0
$517$	0	0
$518$	0.461311	0.0202688
$519$	0	0
$520$	−40.1676	−1.76147
$521$	13.4271	0.588251	0.294125	−	0.955767i	$-0.404972\pi$
$521$	13.4271	0.588251	0.294125	+	0.955767i	$0.404972\pi$
$522$	0	0
$523$	17.9310	0.784069	0.392034	−	0.919951i	$-0.371771\pi$
$523$	17.9310	0.784069	0.392034	+	0.919951i	$0.371771\pi$
$524$	2.47370	0.108064
$525$	0	0
$526$	18.7697	0.818398
$527$	−8.35390	−0.363901
$528$	0	0
$529$	−16.3380	−0.710349
$530$	0.179262	0.00778663
$531$	0	0
$532$	0.0954129	0.00413668
$533$	9.50343	0.411639
$534$	0	0
$535$	10.5002	0.453964
$536$	−15.8549	−0.684828
$537$	0	0
$538$	19.5226	0.841677
$539$	0	0
$540$	0	0
$541$	14.7959	0.636126	0.318063	−	0.948070i	$-0.396968\pi$
$541$	14.7959	0.636126	0.318063	+	0.948070i	$0.396968\pi$
$542$	2.09969	0.0901896
$543$	0	0
$544$	−2.98497	−0.127979
$545$	33.4178	1.43146
$546$	0	0
$547$	19.9452	0.852794	0.426397	−	0.904536i	$-0.359783\pi$
$547$	19.9452	0.852794	0.426397	+	0.904536i	$0.359783\pi$
$548$	−1.21406	−0.0518620
$549$	0	0
$550$	0	0
$551$	65.7021	2.79900
$552$	0	0
$553$	−1.61560	−0.0687022
$554$	32.6179	1.38580
$555$	0	0
$556$	0.533504	0.0226256
$557$	−13.0036	−0.550981	−0.275490	−	0.961304i	$-0.588840\pi$
$557$	−13.0036	−0.550981	−0.275490	+	0.961304i	$0.588840\pi$
$558$	0	0
$559$	−25.3192	−1.07089
$560$	0.797716	0.0337096
$561$	0	0
$562$	20.8579	0.879838
$563$	43.7343	1.84318	0.921590	−	0.388164i	$-0.126890\pi$
$563$	43.7343	1.84318	0.921590	+	0.388164i	$0.126890\pi$
$564$	0	0
$565$	21.1923	0.891567
$566$	−27.9411	−1.17445
$567$	0	0
$568$	−34.5205	−1.44845
$569$	−0.393421	−0.0164931	−0.00824654	−	0.999966i	$-0.502625\pi$
$569$	−0.393421	−0.0164931	−0.00824654	+	0.999966i	$0.502625\pi$
$570$	0	0
$571$	38.5003	1.61119	0.805594	−	0.592468i	$-0.201846\pi$
$571$	38.5003	1.61119	0.805594	+	0.592468i	$0.201846\pi$
$572$	0	0
$573$	0	0
$574$	−0.202945	−0.00847077
$575$	−1.76673	−0.0736779
$576$	0	0
$577$	0.0622460	0.00259134	0.00129567	−	0.999999i	$-0.499588\pi$
$577$	0.0622460	0.00259134	0.00129567	+	0.999999i	$0.499588\pi$
$578$	3.55954	0.148057
$579$	0	0
$580$	−2.88188	−0.119663
$581$	−0.510670	−0.0211862
$582$	0	0
$583$	0	0
$584$	−35.7065	−1.47754
$585$	0	0
$586$	−14.8570	−0.613738
$587$	11.9027	0.491278	0.245639	−	0.969361i	$-0.421002\pi$
$587$	11.9027	0.491278	0.245639	+	0.969361i	$0.421002\pi$
$588$	0	0
$589$	−14.5348	−0.598896
$590$	−19.7430	−0.812806
$591$	0	0
$592$	−12.0690	−0.496031
$593$	−29.3055	−1.20343	−0.601717	−	0.798709i	$-0.705516\pi$
$593$	−29.3055	−1.20343	−0.601717	+	0.798709i	$0.705516\pi$
$594$	0	0
$595$	−0.817458	−0.0335125
$596$	−1.44329	−0.0591196
$597$	0	0
$598$	23.3280	0.953954
$599$	−8.48712	−0.346774	−0.173387	−	0.984854i	$-0.555471\pi$
$599$	−8.48712	−0.346774	−0.173387	+	0.984854i	$0.555471\pi$
$600$	0	0
$601$	20.3588	0.830452	0.415226	−	0.909718i	$-0.363703\pi$
$601$	20.3588	0.830452	0.415226	+	0.909718i	$0.363703\pi$
$602$	0.540690	0.0220369
$603$	0	0
$604$	1.39576	0.0567927
$605$	0	0
$606$	0	0
$607$	30.5038	1.23811	0.619055	−	0.785348i	$-0.287516\pi$
$607$	30.5038	1.23811	0.619055	+	0.785348i	$0.287516\pi$
$608$	−5.19349	−0.210624
$609$	0	0
$610$	−12.0232	−0.486805
$611$	−22.7625	−0.920871
$612$	0	0
$613$	34.4613	1.39188	0.695939	−	0.718101i	$-0.254988\pi$
$613$	34.4613	1.39188	0.695939	+	0.718101i	$0.254988\pi$
$614$	−22.1212	−0.892740
$615$	0	0
$616$	0	0
$617$	−0.224835	−0.00905152	−0.00452576	−	0.999990i	$-0.501441\pi$
$617$	−0.224835	−0.00905152	−0.00452576	+	0.999990i	$0.501441\pi$
$618$	0	0
$619$	0.594526	0.0238960	0.0119480	−	0.999929i	$-0.496197\pi$
$619$	0.594526	0.0238960	0.0119480	+	0.999929i	$0.496197\pi$
$620$	0.637537	0.0256041
$621$	0	0
$622$	17.4608	0.700115
$623$	0.829329	0.0332264
$624$	0	0
$625$	−21.1090	−0.844360
$626$	−16.5374	−0.660969
$627$	0	0
$628$	−1.61660	−0.0645092
$629$	12.3677	0.493131
$630$	0	0
$631$	34.6701	1.38019	0.690097	−	0.723717i	$-0.257568\pi$
$631$	34.6701	1.38019	0.690097	+	0.723717i	$0.257568\pi$
$632$	45.4502	1.80791
$633$	0	0
$634$	−11.4358	−0.454175
$635$	7.10699	0.282032
$636$	0	0
$637$	46.3100	1.83487
$638$	0	0
$639$	0	0
$640$	−20.7518	−0.820286
$641$	21.8476	0.862930	0.431465	−	0.902130i	$-0.357997\pi$
$641$	21.8476	0.862930	0.431465	+	0.902130i	$0.357997\pi$
$642$	0	0
$643$	30.5370	1.20426	0.602130	−	0.798398i	$-0.294319\pi$
$643$	30.5370	1.20426	0.602130	+	0.798398i	$0.294319\pi$
$644$	0.0373123	0.00147031
$645$	0	0
$646$	−34.1528	−1.34372
$647$	26.7412	1.05131	0.525653	−	0.850699i	$-0.323821\pi$
$647$	26.7412	1.05131	0.525653	+	0.850699i	$0.323821\pi$
$648$	0	0
$649$	0	0
$650$	−6.18651	−0.242655
$651$	0	0
$652$	3.17702	0.124422
$653$	−29.3589	−1.14890	−0.574451	−	0.818539i	$-0.694784\pi$
$653$	−29.3589	−1.14890	−0.574451	+	0.818539i	$0.694784\pi$
$654$	0	0
$655$	−36.8744	−1.44080
$656$	5.30952	0.207302
$657$	0	0
$658$	0.486091	0.0189498
$659$	36.5480	1.42371	0.711854	−	0.702328i	$-0.247856\pi$
$659$	36.5480	1.42371	0.711854	+	0.702328i	$0.247856\pi$
$660$	0	0
$661$	−3.81296	−0.148307	−0.0741535	−	0.997247i	$-0.523625\pi$
$661$	−3.81296	−0.148307	−0.0741535	+	0.997247i	$0.523625\pi$
$662$	11.1181	0.432116
$663$	0	0
$664$	14.3662	0.557519
$665$	−1.42228	−0.0551537
$666$	0	0
$667$	25.6935	0.994857
$668$	0.636849	0.0246404
$669$	0	0
$670$	15.3956	0.594783
$671$	0	0
$672$	0	0
$673$	15.0359	0.579591	0.289795	−	0.957089i	$-0.406413\pi$
$673$	15.0359	0.579591	0.289795	+	0.957089i	$0.406413\pi$
$674$	−40.9637	−1.57786
$675$	0	0
$676$	−4.30658	−0.165638
$677$	−30.1500	−1.15876	−0.579379	−	0.815058i	$-0.696705\pi$
$677$	−30.1500	−1.15876	−0.579379	+	0.815058i	$0.696705\pi$
$678$	0	0
$679$	−0.320559	−0.0123019
$680$	22.9969	0.881889
$681$	0	0
$682$	0	0
$683$	45.7767	1.75160	0.875798	−	0.482677i	$-0.160336\pi$
$683$	45.7767	1.75160	0.875798	+	0.482677i	$0.160336\pi$
$684$	0	0
$685$	18.0975	0.691468
$686$	−1.97942	−0.0755745
$687$	0	0
$688$	−14.1457	−0.539300
$689$	0.419165	0.0159689
$690$	0	0
$691$	24.7085	0.939956	0.469978	−	0.882678i	$-0.344262\pi$
$691$	24.7085	0.939956	0.469978	+	0.882678i	$0.344262\pi$
$692$	−0.209071	−0.00794768
$693$	0	0
$694$	46.3990	1.76128
$695$	−7.95273	−0.301664
$696$	0	0
$697$	−5.44092	−0.206090
$698$	−32.1190	−1.21572
$699$	0	0
$700$	−0.00989507	−0.000373999 0
$701$	36.8337	1.39119	0.695595	−	0.718434i	$-0.255141\pi$
$701$	36.8337	1.39119	0.695595	+	0.718434i	$0.255141\pi$
$702$	0	0
$703$	21.5183	0.811577
$704$	0	0
$705$	0	0
$706$	21.5893	0.812525
$707$	0.259106	0.00974470
$708$	0	0
$709$	8.15030	0.306091	0.153045	−	0.988219i	$-0.451092\pi$
$709$	8.15030	0.306091	0.153045	+	0.988219i	$0.451092\pi$
$710$	33.5204	1.25800
$711$	0	0
$712$	−23.3308	−0.874359
$713$	−5.68399	−0.212867
$714$	0	0
$715$	0	0
$716$	1.24553	0.0465476
$717$	0	0
$718$	0.598063	0.0223195
$719$	35.0876	1.30855	0.654273	−	0.756258i	$-0.272975\pi$
$719$	35.0876	1.30855	0.654273	+	0.756258i	$0.272975\pi$
$720$	0	0
$721$	1.46790	0.0546676
$722$	−33.5048	−1.24692
$723$	0	0
$724$	−2.92171	−0.108585
$725$	−6.81383	−0.253059
$726$	0	0
$727$	22.5664	0.836940	0.418470	−	0.908231i	$-0.362566\pi$
$727$	22.5664	0.836940	0.418470	+	0.908231i	$0.362566\pi$
$728$	2.00573	0.0743373
$729$	0	0
$730$	34.6720	1.28327
$731$	14.4958	0.536147
$732$	0	0
$733$	−34.5100	−1.27466	−0.637328	−	0.770593i	$-0.719960\pi$
$733$	−34.5100	−1.27466	−0.637328	+	0.770593i	$0.719960\pi$
$734$	18.9003	0.697622
$735$	0	0
$736$	−2.03097	−0.0748626
$737$	0	0
$738$	0	0
$739$	−21.3031	−0.783646	−0.391823	−	0.920041i	$-0.628155\pi$
$739$	−21.3031	−0.783646	−0.391823	+	0.920041i	$0.628155\pi$
$740$	−0.943851	−0.0346967
$741$	0	0
$742$	−0.00895125	−0.000328611 0
$743$	−44.2579	−1.62367	−0.811833	−	0.583890i	$-0.801530\pi$
$743$	−44.2579	−1.62367	−0.811833	+	0.583890i	$0.801530\pi$
$744$	0	0
$745$	21.5146	0.788233
$746$	−9.56977	−0.350374
$747$	0	0
$748$	0	0
$749$	−0.524318	−0.0191582
$750$	0	0
$751$	−24.9595	−0.910786	−0.455393	−	0.890290i	$-0.650501\pi$
$751$	−24.9595	−0.910786	−0.455393	+	0.890290i	$0.650501\pi$
$752$	−12.7173	−0.463752
$753$	0	0
$754$	89.9701	3.27652
$755$	−20.8060	−0.757209
$756$	0	0
$757$	40.7314	1.48041	0.740204	−	0.672382i	$-0.234729\pi$
$757$	40.7314	1.48041	0.740204	+	0.672382i	$0.234729\pi$
$758$	13.6061	0.494194
$759$	0	0
$760$	40.0118	1.45138
$761$	−19.3865	−0.702762	−0.351381	−	0.936233i	$-0.614288\pi$
$761$	−19.3865	−0.702762	−0.351381	+	0.936233i	$0.614288\pi$
$762$	0	0
$763$	−1.66868	−0.0604103
$764$	1.24621	0.0450865
$765$	0	0
$766$	25.7261	0.929522
$767$	−46.1648	−1.66691
$768$	0	0
$769$	−10.7672	−0.388276	−0.194138	−	0.980974i	$-0.562191\pi$
$769$	−10.7672	−0.388276	−0.194138	+	0.980974i	$0.562191\pi$
$770$	0	0
$771$	0	0
$772$	−0.796960	−0.0286832
$773$	29.3838	1.05686	0.528432	−	0.848976i	$-0.322780\pi$
$773$	29.3838	1.05686	0.528432	+	0.848976i	$0.322780\pi$
$774$	0	0
$775$	1.50737	0.0541464
$776$	9.01802	0.323728
$777$	0	0
$778$	31.8471	1.14177
$779$	−9.46657	−0.339175
$780$	0	0
$781$	0	0
$782$	−13.3558	−0.477603
$783$	0	0
$784$	25.8732	0.924042
$785$	24.0979	0.860092
$786$	0	0
$787$	3.55720	0.126801	0.0634003	−	0.997988i	$-0.479806\pi$
$787$	3.55720	0.126801	0.0634003	+	0.997988i	$0.479806\pi$
$788$	−0.967600	−0.0344693
$789$	0	0
$790$	−44.1335	−1.57020
$791$	−1.05822	−0.0376259
$792$	0	0
$793$	−28.1137	−0.998346
$794$	37.7831	1.34087
$795$	0	0
$796$	−3.34741	−0.118646
$797$	35.0148	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$797$	35.0148	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$798$	0	0
$799$	13.0320	0.461040
$800$	0.538606	0.0190426
$801$	0	0
$802$	−20.6052	−0.727595
$803$	0	0
$804$	0	0
$805$	−0.556199	−0.0196034
$806$	−19.9034	−0.701068
$807$	0	0
$808$	−7.28922	−0.256434
$809$	−19.1540	−0.673419	−0.336709	−	0.941609i	$-0.609314\pi$
$809$	−19.1540	−0.673419	−0.336709	+	0.941609i	$0.609314\pi$
$810$	0	0
$811$	−6.20233	−0.217793	−0.108897	−	0.994053i	$-0.534732\pi$
$811$	−6.20233	−0.217793	−0.108897	+	0.994053i	$0.534732\pi$
$812$	0.143904	0.00505003
$813$	0	0
$814$	0	0
$815$	−47.3585	−1.65890
$816$	0	0
$817$	25.2210	0.882371
$818$	12.8330	0.448697
$819$	0	0
$820$	0.415230	0.0145005
$821$	−25.0661	−0.874811	−0.437406	−	0.899264i	$-0.644103\pi$
$821$	−25.0661	−0.874811	−0.437406	+	0.899264i	$0.644103\pi$
$822$	0	0
$823$	2.93966	0.102470	0.0512350	−	0.998687i	$-0.483684\pi$
$823$	2.93966	0.102470	0.0512350	+	0.998687i	$0.483684\pi$
$824$	−41.2953	−1.43859
$825$	0	0
$826$	0.985847	0.0343020
$827$	32.2686	1.12209	0.561044	−	0.827786i	$-0.310400\pi$
$827$	32.2686	1.12209	0.561044	+	0.827786i	$0.310400\pi$
$828$	0	0
$829$	1.05143	0.0365178	0.0182589	−	0.999833i	$-0.494188\pi$
$829$	1.05143	0.0365178	0.0182589	+	0.999833i	$0.494188\pi$
$830$	−13.9500	−0.484213
$831$	0	0
$832$	−56.1681	−1.94728
$833$	−26.5135	−0.918639
$834$	0	0
$835$	−9.49324	−0.328527
$836$	0	0
$837$	0	0
$838$	−27.4227	−0.947302
$839$	−2.35866	−0.0814299	−0.0407149	−	0.999171i	$-0.512964\pi$
$839$	−2.35866	−0.0814299	−0.0407149	+	0.999171i	$0.512964\pi$
$840$	0	0
$841$	70.0932	2.41701
$842$	17.5216	0.603834
$843$	0	0
$844$	−0.564902	−0.0194447
$845$	64.1964	2.20842
$846$	0	0
$847$	0	0
$848$	0.234185	0.00804196
$849$	0	0
$850$	3.54191	0.121487
$851$	8.41496	0.288461
$852$	0	0
$853$	−29.2448	−1.00132	−0.500661	−	0.865643i	$-0.666910\pi$
$853$	−29.2448	−1.00132	−0.500661	+	0.865643i	$0.666910\pi$
$854$	0.600366	0.0205441
$855$	0	0
$856$	14.7502	0.504151
$857$	−28.2043	−0.963439	−0.481720	−	0.876325i	$-0.659988\pi$
$857$	−28.2043	−0.963439	−0.481720	+	0.876325i	$0.659988\pi$
$858$	0	0
$859$	26.6444	0.909095	0.454548	−	0.890722i	$-0.349801\pi$
$859$	26.6444	0.909095	0.454548	+	0.890722i	$0.349801\pi$
$860$	−1.10626	−0.0377233
$861$	0	0
$862$	20.8709	0.710867
$863$	−24.2053	−0.823959	−0.411979	−	0.911193i	$-0.635162\pi$
$863$	−24.2053	−0.823959	−0.411979	+	0.911193i	$0.635162\pi$
$864$	0	0
$865$	3.11654	0.105965
$866$	16.4313	0.558359
$867$	0	0
$868$	−0.0318348	−0.00108054
$869$	0	0
$870$	0	0
$871$	35.9993	1.21979
$872$	46.9436	1.58971
$873$	0	0
$874$	−23.2376	−0.786022
$875$	1.22495	0.0414110
$876$	0	0
$877$	−9.16732	−0.309558	−0.154779	−	0.987949i	$-0.549467\pi$
$877$	−9.16732	−0.309558	−0.154779	+	0.987949i	$0.549467\pi$
$878$	5.65830	0.190958
$879$	0	0
$880$	0	0
$881$	32.2415	1.08624	0.543122	−	0.839654i	$-0.317242\pi$
$881$	32.2415	1.08624	0.543122	+	0.839654i	$0.317242\pi$
$882$	0	0
$883$	−43.6444	−1.46875	−0.734375	−	0.678744i	$-0.762525\pi$
$883$	−43.6444	−1.46875	−0.734375	+	0.678744i	$0.762525\pi$
$884$	3.50284	0.117813
$885$	0	0
$886$	−26.0423	−0.874907
$887$	8.62161	0.289485	0.144743	−	0.989469i	$-0.453765\pi$
$887$	8.62161	0.289485	0.144743	+	0.989469i	$0.453765\pi$
$888$	0	0
$889$	−0.354880	−0.0119023
$890$	22.6549	0.759393
$891$	0	0
$892$	1.41122	0.0472512
$893$	22.6742	0.758763
$894$	0	0
$895$	−18.5666	−0.620613
$896$	1.03622	0.0346177
$897$	0	0
$898$	−4.65729	−0.155416
$899$	−21.9217	−0.731129
$900$	0	0
$901$	−0.239981	−0.00799494
$902$	0	0
$903$	0	0
$904$	29.7699	0.990133
$905$	43.5528	1.44774
$906$	0	0
$907$	−8.39800	−0.278851	−0.139426	−	0.990233i	$-0.544526\pi$
$907$	−8.39800	−0.278851	−0.139426	+	0.990233i	$0.544526\pi$
$908$	−0.912040	−0.0302671
$909$	0	0
$910$	−1.94762	−0.0645630
$911$	31.5874	1.04654	0.523269	−	0.852168i	$-0.324712\pi$
$911$	31.5874	1.04654	0.523269	+	0.852168i	$0.324712\pi$
$912$	0	0
$913$	0	0
$914$	40.2999	1.33300
$915$	0	0
$916$	−2.15341	−0.0711508
$917$	1.84129	0.0608047
$918$	0	0
$919$	−40.9615	−1.35120	−0.675598	−	0.737270i	$-0.736115\pi$
$919$	−40.9615	−1.35120	−0.675598	+	0.737270i	$0.736115\pi$
$920$	15.6471	0.515868
$921$	0	0
$922$	55.6204	1.83176
$923$	78.3802	2.57992
$924$	0	0
$925$	−2.23161	−0.0733750
$926$	52.8572	1.73700
$927$	0	0
$928$	−7.83292	−0.257128
$929$	11.3720	0.373105	0.186552	−	0.982445i	$-0.440269\pi$
$929$	11.3720	0.373105	0.186552	+	0.982445i	$0.440269\pi$
$930$	0	0
$931$	−46.1304	−1.51186
$932$	−2.96433	−0.0971000
$933$	0	0
$934$	−38.2809	−1.25259
$935$	0	0
$936$	0	0
$937$	−0.357906	−0.0116923	−0.00584614	−	0.999983i	$-0.501861\pi$
$937$	−0.357906	−0.0116923	−0.00584614	+	0.999983i	$0.501861\pi$
$938$	−0.768763	−0.0251010
$939$	0	0
$940$	−0.994553	−0.0324387
$941$	−18.3556	−0.598375	−0.299187	−	0.954194i	$-0.596716\pi$
$941$	−18.3556	−0.598375	−0.299187	+	0.954194i	$0.596716\pi$
$942$	0	0
$943$	−3.70201	−0.120554
$944$	−25.7920	−0.839460
$945$	0	0
$946$	0	0
$947$	−51.2153	−1.66427	−0.832137	−	0.554571i	$-0.812883\pi$
$947$	−51.2153	−1.66427	−0.832137	+	0.554571i	$0.812883\pi$
$948$	0	0
$949$	81.0731	2.63175
$950$	6.16251	0.199938
$951$	0	0
$952$	−1.14832	−0.0372174
$953$	16.2172	0.525327	0.262664	−	0.964887i	$-0.415399\pi$
$953$	16.2172	0.525327	0.262664	+	0.964887i	$0.415399\pi$
$954$	0	0
$955$	−18.5768	−0.601132
$956$	−2.41702	−0.0781719
$957$	0	0
$958$	−21.5855	−0.697396
$959$	−0.903679	−0.0291813
$960$	0	0
$961$	−26.1504	−0.843562
$962$	29.4664	0.950033
$963$	0	0
$964$	−1.48320	−0.0477706
$965$	11.8800	0.382429
$966$	0	0
$967$	32.5309	1.04612	0.523061	−	0.852295i	$-0.324790\pi$
$967$	32.5309	1.04612	0.523061	+	0.852295i	$0.324790\pi$
$968$	0	0
$969$	0	0
$970$	−8.75675	−0.281162
$971$	2.21917	0.0712166	0.0356083	−	0.999366i	$-0.488663\pi$
$971$	2.21917	0.0712166	0.0356083	+	0.999366i	$0.488663\pi$
$972$	0	0
$973$	0.397111	0.0127308
$974$	7.83875	0.251170
$975$	0	0
$976$	−15.7070	−0.502768
$977$	−21.3581	−0.683306	−0.341653	−	0.939826i	$-0.610987\pi$
$977$	−21.3581	−0.683306	−0.341653	+	0.939826i	$0.610987\pi$
$978$	0	0
$979$	0	0
$980$	2.02341	0.0646354
$981$	0	0
$982$	−48.6690	−1.55309
$983$	−14.4569	−0.461104	−0.230552	−	0.973060i	$-0.574053\pi$
$983$	−14.4569	−0.461104	−0.230552	+	0.973060i	$0.574053\pi$
$984$	0	0
$985$	14.4236	0.459575
$986$	−51.5099	−1.64041
$987$	0	0
$988$	6.09453	0.193893
$989$	9.86295	0.313624
$990$	0	0
$991$	4.68782	0.148914	0.0744568	−	0.997224i	$-0.476278\pi$
$991$	4.68782	0.148914	0.0744568	+	0.997224i	$0.476278\pi$
$992$	1.73282	0.0550171
$993$	0	0
$994$	−1.67381	−0.0530899
$995$	49.8985	1.58189
$996$	0	0
$997$	33.7831	1.06992	0.534961	−	0.844877i	$-0.320326\pi$
$997$	33.7831	1.06992	0.534961	+	0.844877i	$0.320326\pi$
$998$	−36.2877	−1.14867
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cr.1.7		24
3.2	odd	2	inner	9801.2.a.cr.1.18		24
11.5	even	5		891.2.f.g.487.9	yes	48
11.9	even	5		891.2.f.g.730.9	yes	48
11.10	odd	2		9801.2.a.cq.1.18		24
33.5	odd	10		891.2.f.g.487.4	✓	48
33.20	odd	10		891.2.f.g.730.4	yes	48
33.32	even	2		9801.2.a.cq.1.7		24
99.5	odd	30		891.2.n.l.784.4		96
99.16	even	15		891.2.n.l.190.4		96
99.20	odd	30		891.2.n.l.433.4		96
99.31	even	15		891.2.n.l.136.4		96
99.38	odd	30		891.2.n.l.190.9		96
99.49	even	15		891.2.n.l.784.9		96
99.86	odd	30		891.2.n.l.136.9		96
99.97	even	15		891.2.n.l.433.9		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
891.2.f.g.487.4	✓	48	33.5	odd	10
891.2.f.g.487.9	yes	48	11.5	even	5
891.2.f.g.730.4	yes	48	33.20	odd	10
891.2.f.g.730.9	yes	48	11.9	even	5
891.2.n.l.136.4		96	99.31	even	15
891.2.n.l.136.9		96	99.86	odd	30
891.2.n.l.190.4		96	99.16	even	15
891.2.n.l.190.9		96	99.38	odd	30
891.2.n.l.433.4		96	99.20	odd	30
891.2.n.l.433.9		96	99.97	even	15
891.2.n.l.784.4		96	99.5	odd	30
891.2.n.l.784.9		96	99.49	even	15
9801.2.a.cq.1.7		24	33.32	even	2
9801.2.a.cq.1.18		24	11.10	odd	2
9801.2.a.cr.1.7		24	1.1	even	1	trivial
9801.2.a.cr.1.18		24	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.36405 q^{2} -0.139360 q^{4} +2.07738 q^{5} -0.103732 q^{7} +2.91820 q^{8} +O(q^{10})\)	\(q-1.36405 q^{2} -0.139360 q^{4} +2.07738 q^{5} -0.103732 q^{7} +2.91820 q^{8} -2.83366 q^{10} -6.62590 q^{13} +0.141496 q^{14} -3.70186 q^{16} +3.79348 q^{17} +6.60020 q^{19} -0.289503 q^{20} +2.58108 q^{23} -0.684493 q^{25} +9.03808 q^{26} +0.0144561 q^{28} +9.95456 q^{29} -2.20217 q^{31} -0.786868 q^{32} -5.17450 q^{34} -0.215491 q^{35} +3.26024 q^{37} -9.00303 q^{38} +6.06221 q^{40} -1.43428 q^{41} +3.82125 q^{43} -3.52073 q^{46} +3.43538 q^{47} -6.98924 q^{49} +0.933685 q^{50} +0.923385 q^{52} -0.0632616 q^{53} -0.302710 q^{56} -13.5785 q^{58} +6.96732 q^{59} +4.24299 q^{61} +3.00388 q^{62} +8.47705 q^{64} -13.7645 q^{65} -5.43311 q^{67} -0.528658 q^{68} +0.293940 q^{70} -11.8294 q^{71} -12.2358 q^{73} -4.44715 q^{74} -0.919803 q^{76} +15.5747 q^{79} -7.69017 q^{80} +1.95644 q^{82} +4.92298 q^{83} +7.88049 q^{85} -5.21238 q^{86} -7.99493 q^{89} +0.687317 q^{91} -0.359699 q^{92} -4.68604 q^{94} +13.7111 q^{95} +3.09027 q^{97} +9.53369 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 26 q^{4} + 18 q^{7}+O(q^{10}) \) 24 * q + 26 * q^4 + 18 * q^7	\( 24 q + 26 q^{4} + 18 q^{7} + 16 q^{10} + 18 q^{13} + 18 q^{16} + 36 q^{19} + 20 q^{25} + 46 q^{28} - 12 q^{31} + 42 q^{34} + 14 q^{37} + 42 q^{40} + 42 q^{43} + 26 q^{46} + 38 q^{49} + 36 q^{52} - 40 q^{58} + 62 q^{61} - 16 q^{64} + 38 q^{67} + 6 q^{70} + 2 q^{73} + 102 q^{76} + 94 q^{79} - 24 q^{82} + 42 q^{85} - 34 q^{91} + 66 q^{94} - 18 q^{97}+O(q^{100}) \) 24 * q + 26 * q^4 + 18 * q^7 + 16 * q^10 + 18 * q^13 + 18 * q^16 + 36 * q^19 + 20 * q^25 + 46 * q^28 - 12 * q^31 + 42 * q^34 + 14 * q^37 + 42 * q^40 + 42 * q^43 + 26 * q^46 + 38 * q^49 + 36 * q^52 - 40 * q^58 + 62 * q^61 - 16 * q^64 + 38 * q^67 + 6 * q^70 + 2 * q^73 + 102 * q^76 + 94 * q^79 - 24 * q^82 + 42 * q^85 - 34 * q^91 + 66 * q^94 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more