LMFDB - Embedded newform 4761.2.a.bq.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4761,2,Mod(1,4761)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4761.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4761, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,2,0,4,-3,0,4,-9,0,1,-13,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.478891 q^{2} -1.77066 q^{4} -3.51334 q^{5} +0.805738 q^{7} -1.80574 q^{8} -1.68251 q^{10} -3.54620 q^{11} +6.61167 q^{13} +0.385861 q^{14} +2.67657 q^{16} +0.933899 q^{17} -3.74982 q^{19} +6.22094 q^{20} -1.69824 q^{22} +7.34354 q^{25} +3.16627 q^{26} -1.42669 q^{28} -1.58233 q^{29} +0.565015 q^{31} +4.89326 q^{32} +0.447236 q^{34} -2.83083 q^{35} +8.17703 q^{37} -1.79575 q^{38} +6.34417 q^{40} +3.81288 q^{41} -6.85630 q^{43} +6.27913 q^{44} +5.69427 q^{47} -6.35079 q^{49} +3.51676 q^{50} -11.7070 q^{52} +6.98011 q^{53} +12.4590 q^{55} -1.45495 q^{56} -0.757767 q^{58} -1.01360 q^{59} +12.2754 q^{61} +0.270581 q^{62} -3.00980 q^{64} -23.2290 q^{65} -2.48807 q^{67} -1.65362 q^{68} -1.35566 q^{70} -13.2550 q^{71} +6.70272 q^{73} +3.91591 q^{74} +6.63966 q^{76} -2.85731 q^{77} -1.43648 q^{79} -9.40370 q^{80} +1.82596 q^{82} -1.75113 q^{83} -3.28110 q^{85} -3.28342 q^{86} +6.40351 q^{88} -4.68976 q^{89} +5.32728 q^{91} +2.72694 q^{94} +13.1744 q^{95} -0.794619 q^{97} -3.04134 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 4 q^{7} - 9 q^{8} + q^{10} - 13 q^{11} + 7 q^{13} - 16 q^{14} + 6 q^{16} - q^{17} - 5 q^{19} + 2 q^{20} - 3 q^{22} - 10 q^{25} + 27 q^{26} + 12 q^{28} + 7 q^{29} + 8 q^{31}+ \cdots - 51 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 4 * q^7 - 9 * q^8 + q^10 - 13 * q^11 + 7 * q^13 - 16 * q^14 + 6 * q^16 - q^17 - 5 * q^19 + 2 * q^20 - 3 * q^22 - 10 * q^25 + 27 * q^26 + 12 * q^28 + 7 * q^29 + 8 * q^31 - 6 * q^32 - 18 * q^34 - 9 * q^35 + 7 * q^37 + 9 * q^38 + 12 * q^40 + 7 * q^41 - 10 * q^43 - 17 * q^44 - 13 * q^47 - q^49 - 4 * q^50 - q^52 - 17 * q^53 + 10 * q^55 - 38 * q^56 - 17 * q^58 + 10 * q^59 + 18 * q^61 + q^62 - 21 * q^64 - 24 * q^65 + 9 * q^67 - 25 * q^68 + 3 * q^70 - 37 * q^71 - 2 * q^73 + 5 * q^74 - 4 * q^76 - 6 * q^77 - 32 * q^79 - 8 * q^80 - 17 * q^82 - 13 * q^83 + 5 * q^85 - 26 * q^86 + 19 * q^88 - 30 * q^89 - 23 * q^91 + 8 * q^94 + 14 * q^95 + 20 * q^97 - 51 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.478891	0.338627	0.169314	−	0.985562i	$-0.445845\pi$
$2$	0.478891	0.338627	0.169314	+	0.985562i	$0.445845\pi$
$3$	0	0
$3$	0	0
$4$	−1.77066	−0.885331
$5$	−3.51334	−1.57121	−0.785606	−	0.618727i	$-0.787649\pi$
$5$	−3.51334	−1.57121	−0.785606	+	0.618727i	$0.787649\pi$
$6$	0	0
$7$	0.805738	0.304540	0.152270	−	0.988339i	$-0.451342\pi$
$7$	0.805738	0.304540	0.152270	+	0.988339i	$0.451342\pi$
$8$	−1.80574	−0.638425
$9$	0	0
$10$	−1.68251	−0.532055
$11$	−3.54620	−1.06922	−0.534610	−	0.845099i	$-0.679541\pi$
$11$	−3.54620	−1.06922	−0.534610	+	0.845099i	$0.679541\pi$
$12$	0	0
$13$	6.61167	1.83375	0.916874	−	0.399177i	$-0.130704\pi$
$13$	6.61167	1.83375	0.916874	+	0.399177i	$0.130704\pi$
$14$	0.385861	0.103126
$15$	0	0
$16$	2.67657	0.669143
$17$	0.933899	0.226504	0.113252	−	0.993566i	$-0.463873\pi$
$17$	0.933899	0.226504	0.113252	+	0.993566i	$0.463873\pi$
$18$	0	0
$19$	−3.74982	−0.860267	−0.430133	−	0.902765i	$-0.641533\pi$
$19$	−3.74982	−0.860267	−0.430133	+	0.902765i	$0.641533\pi$
$20$	6.22094	1.39104
$21$	0	0
$22$	−1.69824	−0.362067
$23$	0	0
$23$	0	0
$24$	0	0
$25$	7.34354	1.46871
$26$	3.16627	0.620957
$27$	0	0
$28$	−1.42669	−0.269619
$29$	−1.58233	−0.293832	−0.146916	−	0.989149i	$-0.546935\pi$
$29$	−1.58233	−0.293832	−0.146916	+	0.989149i	$0.546935\pi$
$30$	0	0
$31$	0.565015	0.101480	0.0507398	−	0.998712i	$-0.483842\pi$
$31$	0.565015	0.101480	0.0507398	+	0.998712i	$0.483842\pi$
$32$	4.89326	0.865015
$33$	0	0
$34$	0.447236	0.0767004
$35$	−2.83083	−0.478498
$36$	0	0
$37$	8.17703	1.34430	0.672148	−	0.740417i	$-0.265372\pi$
$37$	8.17703	1.34430	0.672148	+	0.740417i	$0.265372\pi$
$38$	−1.79575	−0.291310
$39$	0	0
$40$	6.34417	1.00310
$41$	3.81288	0.595472	0.297736	−	0.954648i	$-0.403769\pi$
$41$	3.81288	0.595472	0.297736	+	0.954648i	$0.403769\pi$
$42$	0	0
$43$	−6.85630	−1.04558	−0.522788	−	0.852463i	$-0.675108\pi$
$43$	−6.85630	−1.04558	−0.522788	+	0.852463i	$0.675108\pi$
$44$	6.27913	0.946614
$45$	0	0
$46$	0	0
$47$	5.69427	0.830594	0.415297	−	0.909686i	$-0.363678\pi$
$47$	5.69427	0.830594	0.415297	+	0.909686i	$0.363678\pi$
$48$	0	0
$49$	−6.35079	−0.907255
$50$	3.51676	0.497345
$51$	0	0
$52$	−11.7070	−1.62347
$53$	6.98011	0.958791	0.479396	−	0.877599i	$-0.340856\pi$
$53$	6.98011	0.958791	0.479396	+	0.877599i	$0.340856\pi$
$54$	0	0
$55$	12.4590	1.67997
$56$	−1.45495	−0.194426
$57$	0	0
$58$	−0.757767	−0.0994996
$59$	−1.01360	−0.131959	−0.0659797	−	0.997821i	$-0.521017\pi$
$59$	−1.01360	−0.131959	−0.0659797	+	0.997821i	$0.521017\pi$
$60$	0	0
$61$	12.2754	1.57170	0.785852	−	0.618414i	$-0.212225\pi$
$61$	12.2754	1.57170	0.785852	+	0.618414i	$0.212225\pi$
$62$	0.270581	0.0343638
$63$	0	0
$64$	−3.00980	−0.376226
$65$	−23.2290	−2.88121
$66$	0	0
$67$	−2.48807	−0.303966	−0.151983	−	0.988383i	$-0.548566\pi$
$67$	−2.48807	−0.303966	−0.151983	+	0.988383i	$0.548566\pi$
$68$	−1.65362	−0.200531
$69$	0	0
$70$	−1.35566	−0.162032
$71$	−13.2550	−1.57308	−0.786540	−	0.617539i	$-0.788130\pi$
$71$	−13.2550	−1.57308	−0.786540	+	0.617539i	$0.788130\pi$
$72$	0	0
$73$	6.70272	0.784495	0.392247	−	0.919860i	$-0.371698\pi$
$73$	6.70272	0.784495	0.392247	+	0.919860i	$0.371698\pi$
$74$	3.91591	0.455215
$75$	0	0
$76$	6.63966	0.761621
$77$	−2.85731	−0.325621
$78$	0	0
$79$	−1.43648	−0.161617	−0.0808083	−	0.996730i	$-0.525750\pi$
$79$	−1.43648	−0.161617	−0.0808083	+	0.996730i	$0.525750\pi$
$80$	−9.40370	−1.05137
$81$	0	0
$82$	1.82596	0.201643
$83$	−1.75113	−0.192212	−0.0961058	−	0.995371i	$-0.530639\pi$
$83$	−1.75113	−0.192212	−0.0961058	+	0.995371i	$0.530639\pi$
$84$	0	0
$85$	−3.28110	−0.355886
$86$	−3.28342	−0.354060
$87$	0	0
$88$	6.40351	0.682616
$89$	−4.68976	−0.497113	−0.248557	−	0.968617i	$-0.579956\pi$
$89$	−4.68976	−0.497113	−0.248557	+	0.968617i	$0.579956\pi$
$90$	0	0
$91$	5.32728	0.558450
$92$	0	0
$93$	0	0
$94$	2.72694	0.281262
$95$	13.1744	1.35166
$96$	0	0
$97$	−0.794619	−0.0806814	−0.0403407	−	0.999186i	$-0.512844\pi$
$97$	−0.794619	−0.0806814	−0.0403407	+	0.999186i	$0.512844\pi$
$98$	−3.04134	−0.307221
$99$	0	0
$100$	−13.0029	−1.30029
$101$	12.5949	1.25324	0.626621	−	0.779324i	$-0.284437\pi$
$101$	12.5949	1.25324	0.626621	+	0.779324i	$0.284437\pi$
$102$	0	0
$103$	−10.5942	−1.04388	−0.521941	−	0.852982i	$-0.674792\pi$
$103$	−10.5942	−1.04388	−0.521941	+	0.852982i	$0.674792\pi$
$104$	−11.9389	−1.17071
$105$	0	0
$106$	3.34271	0.324673
$107$	−6.41935	−0.620582	−0.310291	−	0.950642i	$-0.600427\pi$
$107$	−6.41935	−0.620582	−0.310291	+	0.950642i	$0.600427\pi$
$108$	0	0
$109$	1.17341	0.112393	0.0561964	−	0.998420i	$-0.482103\pi$
$109$	1.17341	0.112393	0.0561964	+	0.998420i	$0.482103\pi$
$110$	5.96651	0.568884
$111$	0	0
$112$	2.15662	0.203781
$113$	−17.2605	−1.62373	−0.811865	−	0.583845i	$-0.801548\pi$
$113$	−17.2605	−1.62373	−0.811865	+	0.583845i	$0.801548\pi$
$114$	0	0
$115$	0	0
$116$	2.80178	0.260139
$117$	0	0
$118$	−0.485404	−0.0446850
$119$	0.752478	0.0689796
$120$	0	0
$121$	1.57554	0.143231
$122$	5.87859	0.532222
$123$	0	0
$124$	−1.00045	−0.0898431
$125$	−8.23364	−0.736439
$126$	0	0
$127$	9.46369	0.839767	0.419883	−	0.907578i	$-0.362071\pi$
$127$	9.46369	0.839767	0.419883	+	0.907578i	$0.362071\pi$
$128$	−11.2279	−0.992415
$129$	0	0
$130$	−11.1242	−0.975655
$131$	−13.4896	−1.17859	−0.589295	−	0.807918i	$-0.700594\pi$
$131$	−13.4896	−1.17859	−0.589295	+	0.807918i	$0.700594\pi$
$132$	0	0
$133$	−3.02137	−0.261986
$134$	−1.19151	−0.102931
$135$	0	0
$136$	−1.68638	−0.144606
$137$	−0.768502	−0.0656576	−0.0328288	−	0.999461i	$-0.510452\pi$
$137$	−0.768502	−0.0656576	−0.0328288	+	0.999461i	$0.510452\pi$
$138$	0	0
$139$	−14.8410	−1.25879	−0.629397	−	0.777084i	$-0.716698\pi$
$139$	−14.8410	−1.25879	−0.629397	+	0.777084i	$0.716698\pi$
$140$	5.01245	0.423629
$141$	0	0
$142$	−6.34771	−0.532688
$143$	−23.4463	−1.96068
$144$	0	0
$145$	5.55928	0.461673
$146$	3.20988	0.265651
$147$	0	0
$148$	−14.4788	−1.19015
$149$	−9.29327	−0.761334	−0.380667	−	0.924712i	$-0.624306\pi$
$149$	−9.29327	−0.761334	−0.380667	+	0.924712i	$0.624306\pi$
$150$	0	0
$151$	−8.43643	−0.686547	−0.343273	−	0.939236i	$-0.611536\pi$
$151$	−8.43643	−0.686547	−0.343273	+	0.939236i	$0.611536\pi$
$152$	6.77119	0.549216
$153$	0	0
$154$	−1.36834	−0.110264
$155$	−1.98509	−0.159446
$156$	0	0
$157$	−9.38633	−0.749111	−0.374555	−	0.927205i	$-0.622205\pi$
$157$	−9.38633	−0.749111	−0.374555	+	0.927205i	$0.622205\pi$
$158$	−0.687918	−0.0547278
$159$	0	0
$160$	−17.1917	−1.35912
$161$	0	0
$162$	0	0
$163$	−17.1171	−1.34071	−0.670356	−	0.742040i	$-0.733858\pi$
$163$	−17.1171	−1.34071	−0.670356	+	0.742040i	$0.733858\pi$
$164$	−6.75133	−0.527190
$165$	0	0
$166$	−0.838601	−0.0650881
$167$	−10.6935	−0.827485	−0.413742	−	0.910394i	$-0.635779\pi$
$167$	−10.6935	−0.827485	−0.413742	+	0.910394i	$0.635779\pi$
$168$	0	0
$169$	30.7142	2.36263
$170$	−1.57129	−0.120513
$171$	0	0
$172$	12.1402	0.925681
$173$	24.1400	1.83533	0.917666	−	0.397354i	$-0.130071\pi$
$173$	24.1400	1.83533	0.917666	+	0.397354i	$0.130071\pi$
$174$	0	0
$175$	5.91697	0.447281
$176$	−9.49167	−0.715461
$177$	0	0
$178$	−2.24588	−0.168336
$179$	−2.60662	−0.194828	−0.0974141	−	0.995244i	$-0.531057\pi$
$179$	−2.60662	−0.194828	−0.0974141	+	0.995244i	$0.531057\pi$
$180$	0	0
$181$	10.0296	0.745495	0.372747	−	0.927933i	$-0.378416\pi$
$181$	10.0296	0.745495	0.372747	+	0.927933i	$0.378416\pi$
$182$	2.55119	0.189107
$183$	0	0
$184$	0	0
$185$	−28.7287	−2.11217
$186$	0	0
$187$	−3.31179	−0.242182
$188$	−10.0826	−0.735351
$189$	0	0
$190$	6.30909	0.457710
$191$	−25.0497	−1.81253	−0.906267	−	0.422706i	$-0.861080\pi$
$191$	−25.0497	−1.81253	−0.906267	+	0.422706i	$0.861080\pi$
$192$	0	0
$193$	−0.0998613	−0.00718817	−0.00359409	−	0.999994i	$-0.501144\pi$
$193$	−0.0998613	−0.00718817	−0.00359409	+	0.999994i	$0.501144\pi$
$194$	−0.380536	−0.0273209
$195$	0	0
$196$	11.2451	0.803222
$197$	−16.5794	−1.18123	−0.590616	−	0.806953i	$-0.701115\pi$
$197$	−16.5794	−1.18123	−0.590616	+	0.806953i	$0.701115\pi$
$198$	0	0
$199$	−20.0452	−1.42097	−0.710485	−	0.703713i	$-0.751524\pi$
$199$	−20.0452	−1.42097	−0.710485	+	0.703713i	$0.751524\pi$
$200$	−13.2605	−0.937659
$201$	0	0
$202$	6.03160	0.424382
$203$	−1.27495	−0.0894838
$204$	0	0
$205$	−13.3959	−0.935613
$206$	−5.07349	−0.353487
$207$	0	0
$208$	17.6966	1.22704
$209$	13.2976	0.919814
$210$	0	0
$211$	−2.32229	−0.159873	−0.0799366	−	0.996800i	$-0.525472\pi$
$211$	−2.32229	−0.159873	−0.0799366	+	0.996800i	$0.525472\pi$
$212$	−12.3594	−0.848848
$213$	0	0
$214$	−3.07417	−0.210146
$215$	24.0885	1.64282
$216$	0	0
$217$	0.455254	0.0309046
$218$	0.561938	0.0380593
$219$	0	0
$220$	−22.0607	−1.48733
$221$	6.17464	0.415351
$222$	0	0
$223$	−4.60115	−0.308116	−0.154058	−	0.988062i	$-0.549234\pi$
$223$	−4.60115	−0.308116	−0.154058	+	0.988062i	$0.549234\pi$
$224$	3.94269	0.263432
$225$	0	0
$226$	−8.26590	−0.549840
$227$	8.72329	0.578985	0.289492	−	0.957180i	$-0.406514\pi$
$227$	8.72329	0.578985	0.289492	+	0.957180i	$0.406514\pi$
$228$	0	0
$229$	9.31081	0.615276	0.307638	−	0.951503i	$-0.400461\pi$
$229$	9.31081	0.615276	0.307638	+	0.951503i	$0.400461\pi$
$230$	0	0
$231$	0	0
$232$	2.85728	0.187590
$233$	2.95114	0.193336	0.0966679	−	0.995317i	$-0.469182\pi$
$233$	2.95114	0.193336	0.0966679	+	0.995317i	$0.469182\pi$
$234$	0	0
$235$	−20.0059	−1.30504
$236$	1.79474	0.116828
$237$	0	0
$238$	0.360355	0.0233584
$239$	16.4327	1.06294	0.531471	−	0.847076i	$-0.321639\pi$
$239$	16.4327	1.06294	0.531471	+	0.847076i	$0.321639\pi$
$240$	0	0
$241$	1.32184	0.0851475	0.0425737	−	0.999093i	$-0.486444\pi$
$241$	1.32184	0.0851475	0.0425737	+	0.999093i	$0.486444\pi$
$242$	0.754511	0.0485018
$243$	0	0
$244$	−21.7356	−1.39148
$245$	22.3125	1.42549
$246$	0	0
$247$	−24.7926	−1.57751
$248$	−1.02027	−0.0647871
$249$	0	0
$250$	−3.94302	−0.249378
$251$	−18.6993	−1.18029	−0.590143	−	0.807299i	$-0.700929\pi$
$251$	−18.6993	−1.18029	−0.590143	+	0.807299i	$0.700929\pi$
$252$	0	0
$253$	0	0
$254$	4.53208	0.284368
$255$	0	0
$256$	0.642664	0.0401665
$257$	−6.32456	−0.394515	−0.197258	−	0.980352i	$-0.563204\pi$
$257$	−6.32456	−0.394515	−0.197258	+	0.980352i	$0.563204\pi$
$258$	0	0
$259$	6.58855	0.409392
$260$	41.1308	2.55082
$261$	0	0
$262$	−6.46004	−0.399103
$263$	−24.3804	−1.50336	−0.751680	−	0.659528i	$-0.770756\pi$
$263$	−24.3804	−1.50336	−0.751680	+	0.659528i	$0.770756\pi$
$264$	0	0
$265$	−24.5235	−1.50646
$266$	−1.44691	−0.0887156
$267$	0	0
$268$	4.40553	0.269110
$269$	13.5475	0.826003	0.413002	−	0.910730i	$-0.364480\pi$
$269$	13.5475	0.826003	0.413002	+	0.910730i	$0.364480\pi$
$270$	0	0
$271$	11.1231	0.675682	0.337841	−	0.941203i	$-0.390303\pi$
$271$	11.1231	0.675682	0.337841	+	0.941203i	$0.390303\pi$
$272$	2.49965	0.151564
$273$	0	0
$274$	−0.368029	−0.0222334
$275$	−26.0417	−1.57037
$276$	0	0
$277$	−14.7239	−0.884673	−0.442337	−	0.896849i	$-0.645850\pi$
$277$	−14.7239	−0.884673	−0.442337	+	0.896849i	$0.645850\pi$
$278$	−7.10721	−0.426262
$279$	0	0
$280$	5.11174	0.305485
$281$	21.0752	1.25724	0.628621	−	0.777712i	$-0.283620\pi$
$281$	21.0752	1.25724	0.628621	+	0.777712i	$0.283620\pi$
$282$	0	0
$283$	5.88718	0.349957	0.174978	−	0.984572i	$-0.444014\pi$
$283$	5.88718	0.349957	0.174978	+	0.984572i	$0.444014\pi$
$284$	23.4702	1.39270
$285$	0	0
$286$	−11.2282	−0.663940
$287$	3.07218	0.181345
$288$	0	0
$289$	−16.1278	−0.948696
$290$	2.66229	0.156335
$291$	0	0
$292$	−11.8683	−0.694538
$293$	−0.194666	−0.0113725	−0.00568626	−	0.999984i	$-0.501810\pi$
$293$	−0.194666	−0.0113725	−0.00568626	+	0.999984i	$0.501810\pi$
$294$	0	0
$295$	3.56111	0.207336
$296$	−14.7656	−0.858232
$297$	0	0
$298$	−4.45047	−0.257809
$299$	0	0
$300$	0	0
$301$	−5.52438	−0.318420
$302$	−4.04013	−0.232484
$303$	0	0
$304$	−10.0367	−0.575642
$305$	−43.1276	−2.46948
$306$	0	0
$307$	−12.9309	−0.738004	−0.369002	−	0.929429i	$-0.620300\pi$
$307$	−12.9309	−0.738004	−0.369002	+	0.929429i	$0.620300\pi$
$308$	5.05933	0.288282
$309$	0	0
$310$	−0.950641	−0.0539928
$311$	29.9897	1.70056	0.850279	−	0.526332i	$-0.176433\pi$
$311$	29.9897	1.70056	0.850279	+	0.526332i	$0.176433\pi$
$312$	0	0
$313$	1.53124	0.0865511	0.0432755	−	0.999063i	$-0.486221\pi$
$313$	1.53124	0.0865511	0.0432755	+	0.999063i	$0.486221\pi$
$314$	−4.49503	−0.253669
$315$	0	0
$316$	2.54352	0.143084
$317$	21.2348	1.19267	0.596333	−	0.802737i	$-0.296624\pi$
$317$	21.2348	1.19267	0.596333	+	0.802737i	$0.296624\pi$
$318$	0	0
$319$	5.61128	0.314171
$320$	10.5745	0.591130
$321$	0	0
$322$	0	0
$323$	−3.50195	−0.194854
$324$	0	0
$325$	48.5531	2.69324
$326$	−8.19721	−0.454002
$327$	0	0
$328$	−6.88506	−0.380164
$329$	4.58809	0.252950
$330$	0	0
$331$	−0.811437	−0.0446006	−0.0223003	−	0.999751i	$-0.507099\pi$
$331$	−0.811437	−0.0446006	−0.0223003	+	0.999751i	$0.507099\pi$
$332$	3.10066	0.170171
$333$	0	0
$334$	−5.12101	−0.280209
$335$	8.74141	0.477595
$336$	0	0
$337$	17.9618	0.978442	0.489221	−	0.872160i	$-0.337281\pi$
$337$	17.9618	0.978442	0.489221	+	0.872160i	$0.337281\pi$
$338$	14.7088	0.800051
$339$	0	0
$340$	5.80973	0.315077
$341$	−2.00365	−0.108504
$342$	0	0
$343$	−10.7572	−0.580836
$344$	12.3807	0.667521
$345$	0	0
$346$	11.5604	0.621493
$347$	4.66153	0.250244	0.125122	−	0.992141i	$-0.460068\pi$
$347$	4.66153	0.250244	0.125122	+	0.992141i	$0.460068\pi$
$348$	0	0
$349$	−15.9513	−0.853855	−0.426927	−	0.904286i	$-0.640404\pi$
$349$	−15.9513	−0.853855	−0.426927	+	0.904286i	$0.640404\pi$
$350$	2.83359	0.151462
$351$	0	0
$352$	−17.3525	−0.924891
$353$	−27.9148	−1.48576	−0.742878	−	0.669427i	$-0.766540\pi$
$353$	−27.9148	−1.48576	−0.742878	+	0.669427i	$0.766540\pi$
$354$	0	0
$355$	46.5693	2.47164
$356$	8.30398	0.440110
$357$	0	0
$358$	−1.24829	−0.0659741
$359$	33.0316	1.74334	0.871670	−	0.490093i	$-0.163037\pi$
$359$	33.0316	1.74334	0.871670	+	0.490093i	$0.163037\pi$
$360$	0	0
$361$	−4.93888	−0.259941
$362$	4.80309	0.252445
$363$	0	0
$364$	−9.43281	−0.494414
$365$	−23.5489	−1.23261
$366$	0	0
$367$	13.3155	0.695066	0.347533	−	0.937668i	$-0.387019\pi$
$367$	13.3155	0.695066	0.347533	+	0.937668i	$0.387019\pi$
$368$	0	0
$369$	0	0
$370$	−13.7579	−0.715240
$371$	5.62414	0.291991
$372$	0	0
$373$	24.7355	1.28075	0.640377	−	0.768060i	$-0.278778\pi$
$373$	24.7355	1.28075	0.640377	+	0.768060i	$0.278778\pi$
$374$	−1.58599	−0.0820096
$375$	0	0
$376$	−10.2824	−0.530272
$377$	−10.4619	−0.538814
$378$	0	0
$379$	−38.2939	−1.96703	−0.983513	−	0.180837i	$-0.942119\pi$
$379$	−38.2939	−1.96703	−0.983513	+	0.180837i	$0.942119\pi$
$380$	−23.3274	−1.19667
$381$	0	0
$382$	−11.9961	−0.613774
$383$	−16.7739	−0.857107	−0.428553	−	0.903516i	$-0.640977\pi$
$383$	−16.7739	−0.857107	−0.428553	+	0.903516i	$0.640977\pi$
$384$	0	0
$385$	10.0387	0.511619
$386$	−0.0478227	−0.00243411
$387$	0	0
$388$	1.40700	0.0714298
$389$	1.97171	0.0999698	0.0499849	−	0.998750i	$-0.484083\pi$
$389$	1.97171	0.0999698	0.0499849	+	0.998750i	$0.484083\pi$
$390$	0	0
$391$	0	0
$392$	11.4679	0.579214
$393$	0	0
$394$	−7.93973	−0.399998
$395$	5.04684	0.253934
$396$	0	0
$397$	−29.0865	−1.45981	−0.729904	−	0.683549i	$-0.760435\pi$
$397$	−29.0865	−1.45981	−0.729904	+	0.683549i	$0.760435\pi$
$398$	−9.59950	−0.481179
$399$	0	0
$400$	19.6555	0.982776
$401$	−8.80074	−0.439488	−0.219744	−	0.975558i	$-0.570522\pi$
$401$	−8.80074	−0.439488	−0.219744	+	0.975558i	$0.570522\pi$
$402$	0	0
$403$	3.73569	0.186088
$404$	−22.3014	−1.10953
$405$	0	0
$406$	−0.610562	−0.0303017
$407$	−28.9974	−1.43735
$408$	0	0
$409$	6.50603	0.321702	0.160851	−	0.986979i	$-0.448576\pi$
$409$	6.50603	0.321702	0.160851	+	0.986979i	$0.448576\pi$
$410$	−6.41520	−0.316824
$411$	0	0
$412$	18.7588	0.924182
$413$	−0.816695	−0.0401869
$414$	0	0
$415$	6.15231	0.302005
$416$	32.3527	1.58622
$417$	0	0
$418$	6.36811	0.311474
$419$	9.30261	0.454462	0.227231	−	0.973841i	$-0.427033\pi$
$419$	9.30261	0.454462	0.227231	+	0.973841i	$0.427033\pi$
$420$	0	0
$421$	4.86533	0.237122	0.118561	−	0.992947i	$-0.462172\pi$
$421$	4.86533	0.237122	0.118561	+	0.992947i	$0.462172\pi$
$422$	−1.11213	−0.0541374
$423$	0	0
$424$	−12.6042	−0.612116
$425$	6.85812	0.332668
$426$	0	0
$427$	9.89076	0.478648
$428$	11.3665	0.549421
$429$	0	0
$430$	11.5358	0.556304
$431$	33.1454	1.59656	0.798279	−	0.602288i	$-0.205744\pi$
$431$	33.1454	1.59656	0.798279	+	0.602288i	$0.205744\pi$
$432$	0	0
$433$	−30.7633	−1.47839	−0.739195	−	0.673492i	$-0.764794\pi$
$433$	−30.7633	−1.47839	−0.739195	+	0.673492i	$0.764794\pi$
$434$	0.218017	0.0104652
$435$	0	0
$436$	−2.07772	−0.0995048
$437$	0	0
$438$	0	0
$439$	31.0568	1.48226	0.741131	−	0.671361i	$-0.234290\pi$
$439$	31.0568	1.48226	0.741131	+	0.671361i	$0.234290\pi$
$440$	−22.4977	−1.07254
$441$	0	0
$442$	2.95698	0.140649
$443$	3.22899	0.153414	0.0767070	−	0.997054i	$-0.475559\pi$
$443$	3.22899	0.153414	0.0767070	+	0.997054i	$0.475559\pi$
$444$	0	0
$445$	16.4767	0.781070
$446$	−2.20345	−0.104336
$447$	0	0
$448$	−2.42511	−0.114576
$449$	−20.9309	−0.987791	−0.493895	−	0.869521i	$-0.664427\pi$
$449$	−20.9309	−0.987791	−0.493895	+	0.869521i	$0.664427\pi$
$450$	0	0
$451$	−13.5212	−0.636690
$452$	30.5625	1.43754
$453$	0	0
$454$	4.17751	0.196060
$455$	−18.7165	−0.877444
$456$	0	0
$457$	0.542266	0.0253662	0.0126831	−	0.999920i	$-0.495963\pi$
$457$	0.542266	0.0253662	0.0126831	+	0.999920i	$0.495963\pi$
$458$	4.45887	0.208349
$459$	0	0
$460$	0	0
$461$	−2.74045	−0.127636	−0.0638178	−	0.997962i	$-0.520328\pi$
$461$	−2.74045	−0.127636	−0.0638178	+	0.997962i	$0.520328\pi$
$462$	0	0
$463$	11.2918	0.524774	0.262387	−	0.964963i	$-0.415490\pi$
$463$	11.2918	0.524774	0.262387	+	0.964963i	$0.415490\pi$
$464$	−4.23524	−0.196616
$465$	0	0
$466$	1.41328	0.0654688
$467$	−5.78264	−0.267589	−0.133794	−	0.991009i	$-0.542716\pi$
$467$	−5.78264	−0.267589	−0.133794	+	0.991009i	$0.542716\pi$
$468$	0	0
$469$	−2.00473	−0.0925698
$470$	−9.58065	−0.441922
$471$	0	0
$472$	1.83029	0.0842461
$473$	24.3138	1.11795
$474$	0	0
$475$	−27.5369	−1.26348
$476$	−1.33239	−0.0610698
$477$	0	0
$478$	7.86948	0.359942
$479$	8.35747	0.381862	0.190931	−	0.981603i	$-0.438849\pi$
$479$	8.35747	0.381862	0.190931	+	0.981603i	$0.438849\pi$
$480$	0	0
$481$	54.0638	2.46510
$482$	0.633020	0.0288333
$483$	0	0
$484$	−2.78974	−0.126807
$485$	2.79177	0.126768
$486$	0	0
$487$	−3.46845	−0.157170	−0.0785851	−	0.996907i	$-0.525040\pi$
$487$	−3.46845	−0.157170	−0.0785851	+	0.996907i	$0.525040\pi$
$488$	−22.1662	−1.00342
$489$	0	0
$490$	10.6852	0.482710
$491$	−0.367628	−0.0165908	−0.00829540	−	0.999966i	$-0.502641\pi$
$491$	−0.367628	−0.0165908	−0.00829540	+	0.999966i	$0.502641\pi$
$492$	0	0
$493$	−1.47774	−0.0665541
$494$	−11.8729	−0.534189
$495$	0	0
$496$	1.51230	0.0679044
$497$	−10.6801	−0.479066
$498$	0	0
$499$	13.5738	0.607645	0.303823	−	0.952729i	$-0.401737\pi$
$499$	13.5738	0.607645	0.303823	+	0.952729i	$0.401737\pi$
$500$	14.5790	0.651993
$501$	0	0
$502$	−8.95491	−0.399677
$503$	−16.7713	−0.747797	−0.373899	−	0.927470i	$-0.621979\pi$
$503$	−16.7713	−0.747797	−0.373899	+	0.927470i	$0.621979\pi$
$504$	0	0
$505$	−44.2502	−1.96911
$506$	0	0
$507$	0	0
$508$	−16.7570	−0.743472
$509$	−2.99973	−0.132961	−0.0664804	−	0.997788i	$-0.521177\pi$
$509$	−2.99973	−0.132961	−0.0664804	+	0.997788i	$0.521177\pi$
$510$	0	0
$511$	5.40064	0.238910
$512$	22.7636	1.00602
$513$	0	0
$514$	−3.02878	−0.133594
$515$	37.2212	1.64016
$516$	0	0
$517$	−20.1930	−0.888088
$518$	3.15520	0.138631
$519$	0	0
$520$	41.9455	1.83943
$521$	−9.18185	−0.402264	−0.201132	−	0.979564i	$-0.564462\pi$
$521$	−9.18185	−0.402264	−0.201132	+	0.979564i	$0.564462\pi$
$522$	0	0
$523$	7.31656	0.319931	0.159965	−	0.987123i	$-0.448862\pi$
$523$	7.31656	0.319931	0.159965	+	0.987123i	$0.448862\pi$
$524$	23.8855	1.04344
$525$	0	0
$526$	−11.6756	−0.509079
$527$	0.527667	0.0229855
$528$	0	0
$529$	0	0
$530$	−11.7441	−0.510130
$531$	0	0
$532$	5.34983	0.231944
$533$	25.2095	1.09195
$534$	0	0
$535$	22.5534	0.975067
$536$	4.49280	0.194059
$537$	0	0
$538$	6.48776	0.279707
$539$	22.5212	0.970055
$540$	0	0
$541$	0.663839	0.0285407	0.0142703	−	0.999898i	$-0.495457\pi$
$541$	0.663839	0.0285407	0.0142703	+	0.999898i	$0.495457\pi$
$542$	5.32677	0.228804
$543$	0	0
$544$	4.56982	0.195929
$545$	−4.12260	−0.176593
$546$	0	0
$547$	−18.2097	−0.778590	−0.389295	−	0.921113i	$-0.627281\pi$
$547$	−18.2097	−0.778590	−0.389295	+	0.921113i	$0.627281\pi$
$548$	1.36076	0.0581287
$549$	0	0
$550$	−12.4711	−0.531771
$551$	5.93346	0.252774
$552$	0	0
$553$	−1.15743	−0.0492188
$554$	−7.05115	−0.299575
$555$	0	0
$556$	26.2783	1.11445
$557$	−25.4783	−1.07955	−0.539774	−	0.841810i	$-0.681490\pi$
$557$	−25.4783	−1.07955	−0.539774	+	0.841810i	$0.681490\pi$
$558$	0	0
$559$	−45.3316	−1.91732
$560$	−7.57692	−0.320183
$561$	0	0
$562$	10.0927	0.425737
$563$	−21.2243	−0.894497	−0.447249	−	0.894410i	$-0.647596\pi$
$563$	−21.2243	−0.894497	−0.447249	+	0.894410i	$0.647596\pi$
$564$	0	0
$565$	60.6419	2.55122
$566$	2.81932	0.118505
$567$	0	0
$568$	23.9351	1.00429
$569$	−33.6394	−1.41024	−0.705118	−	0.709090i	$-0.749106\pi$
$569$	−33.6394	−1.41024	−0.705118	+	0.709090i	$0.749106\pi$
$570$	0	0
$571$	12.8568	0.538041	0.269020	−	0.963134i	$-0.413300\pi$
$571$	12.8568	0.538041	0.269020	+	0.963134i	$0.413300\pi$
$572$	41.5155	1.73585
$573$	0	0
$574$	1.47124	0.0614085
$575$	0	0
$576$	0	0
$577$	−10.9844	−0.457288	−0.228644	−	0.973510i	$-0.573429\pi$
$577$	−10.9844	−0.457288	−0.228644	+	0.973510i	$0.573429\pi$
$578$	−7.72348	−0.321254
$579$	0	0
$580$	−9.84360	−0.408733
$581$	−1.41095	−0.0585362
$582$	0	0
$583$	−24.7529	−1.02516
$584$	−12.1034	−0.500841
$585$	0	0
$586$	−0.0932239	−0.00385104
$587$	13.5995	0.561309	0.280655	−	0.959809i	$-0.409448\pi$
$587$	13.5995	0.561309	0.280655	+	0.959809i	$0.409448\pi$
$588$	0	0
$589$	−2.11870	−0.0872995
$590$	1.70539	0.0702097
$591$	0	0
$592$	21.8864	0.899526
$593$	−42.8777	−1.76078	−0.880388	−	0.474254i	$-0.842718\pi$
$593$	−42.8777	−1.76078	−0.880388	+	0.474254i	$0.842718\pi$
$594$	0	0
$595$	−2.64371	−0.108382
$596$	16.4552	0.674033
$597$	0	0
$598$	0	0
$599$	14.6628	0.599106	0.299553	−	0.954080i	$-0.403162\pi$
$599$	14.6628	0.599106	0.299553	+	0.954080i	$0.403162\pi$
$600$	0	0
$601$	−31.2518	−1.27479	−0.637393	−	0.770539i	$-0.719987\pi$
$601$	−31.2518	−1.27479	−0.637393	+	0.770539i	$0.719987\pi$
$602$	−2.64558	−0.107826
$603$	0	0
$604$	14.9381	0.607822
$605$	−5.53539	−0.225046
$606$	0	0
$607$	3.89830	0.158227	0.0791135	−	0.996866i	$-0.474791\pi$
$607$	3.89830	0.158227	0.0791135	+	0.996866i	$0.474791\pi$
$608$	−18.3488	−0.744144
$609$	0	0
$610$	−20.6535	−0.836234
$611$	37.6486	1.52310
$612$	0	0
$613$	34.9028	1.40971	0.704856	−	0.709350i	$-0.251012\pi$
$613$	34.9028	1.40971	0.704856	+	0.709350i	$0.251012\pi$
$614$	−6.19249	−0.249908
$615$	0	0
$616$	5.15955	0.207884
$617$	−15.1918	−0.611601	−0.305800	−	0.952096i	$-0.598924\pi$
$617$	−15.1918	−0.611601	−0.305800	+	0.952096i	$0.598924\pi$
$618$	0	0
$619$	4.96244	0.199457	0.0997286	−	0.995015i	$-0.468203\pi$
$619$	4.96244	0.199457	0.0997286	+	0.995015i	$0.468203\pi$
$620$	3.51492	0.141163
$621$	0	0
$622$	14.3618	0.575856
$623$	−3.77872	−0.151391
$624$	0	0
$625$	−7.79014	−0.311606
$626$	0.733300	0.0293086
$627$	0	0
$628$	16.6200	0.663212
$629$	7.63652	0.304488
$630$	0	0
$631$	−36.6674	−1.45971	−0.729854	−	0.683603i	$-0.760412\pi$
$631$	−36.6674	−1.45971	−0.729854	+	0.683603i	$0.760412\pi$
$632$	2.59391	0.103180
$633$	0	0
$634$	10.1692	0.403869
$635$	−33.2491	−1.31945
$636$	0	0
$637$	−41.9893	−1.66368
$638$	2.68719	0.106387
$639$	0	0
$640$	39.4474	1.55930
$641$	13.7339	0.542457	0.271229	−	0.962515i	$-0.412570\pi$
$641$	13.7339	0.542457	0.271229	+	0.962515i	$0.412570\pi$
$642$	0	0
$643$	34.2265	1.34976	0.674880	−	0.737927i	$-0.264195\pi$
$643$	34.2265	1.34976	0.674880	+	0.737927i	$0.264195\pi$
$644$	0	0
$645$	0	0
$646$	−1.67705	−0.0659828
$647$	38.5284	1.51471	0.757353	−	0.653005i	$-0.226492\pi$
$647$	38.5284	1.51471	0.757353	+	0.653005i	$0.226492\pi$
$648$	0	0
$649$	3.59442	0.141093
$650$	23.2516	0.912005
$651$	0	0
$652$	30.3085	1.18697
$653$	−20.7209	−0.810873	−0.405437	−	0.914123i	$-0.632881\pi$
$653$	−20.7209	−0.810873	−0.405437	+	0.914123i	$0.632881\pi$
$654$	0	0
$655$	47.3934	1.85181
$656$	10.2055	0.398456
$657$	0	0
$658$	2.19720	0.0856556
$659$	−15.8482	−0.617358	−0.308679	−	0.951166i	$-0.599887\pi$
$659$	−15.8482	−0.617358	−0.308679	+	0.951166i	$0.599887\pi$
$660$	0	0
$661$	0.136085	0.00529310	0.00264655	−	0.999996i	$-0.499158\pi$
$661$	0.136085	0.00529310	0.00264655	+	0.999996i	$0.499158\pi$
$662$	−0.388590	−0.0151030
$663$	0	0
$664$	3.16208	0.122713
$665$	10.6151	0.411636
$666$	0	0
$667$	0	0
$668$	18.9345	0.732598
$669$	0	0
$670$	4.18619	0.161727
$671$	−43.5310	−1.68050
$672$	0	0
$673$	−20.6423	−0.795701	−0.397851	−	0.917450i	$-0.630244\pi$
$673$	−20.6423	−0.795701	−0.397851	+	0.917450i	$0.630244\pi$
$674$	8.60175	0.331327
$675$	0	0
$676$	−54.3845	−2.09171
$677$	28.2982	1.08759	0.543795	−	0.839218i	$-0.316987\pi$
$677$	28.2982	1.08759	0.543795	+	0.839218i	$0.316987\pi$
$678$	0	0
$679$	−0.640255	−0.0245707
$680$	5.92481	0.227206
$681$	0	0
$682$	−0.959533	−0.0367424
$683$	19.5149	0.746718	0.373359	−	0.927687i	$-0.378206\pi$
$683$	19.5149	0.746718	0.373359	+	0.927687i	$0.378206\pi$
$684$	0	0
$685$	2.70001	0.103162
$686$	−5.15155	−0.196687
$687$	0	0
$688$	−18.3514	−0.699640
$689$	46.1502	1.75818
$690$	0	0
$691$	18.2405	0.693902	0.346951	−	0.937883i	$-0.387217\pi$
$691$	18.2405	0.693902	0.346951	+	0.937883i	$0.387217\pi$
$692$	−42.7438	−1.62488
$693$	0	0
$694$	2.23237	0.0847395
$695$	52.1413	1.97783
$696$	0	0
$697$	3.56085	0.134877
$698$	−7.63895	−0.289139
$699$	0	0
$700$	−10.4770	−0.395992
$701$	25.0197	0.944982	0.472491	−	0.881335i	$-0.343355\pi$
$701$	25.0197	0.944982	0.472491	+	0.881335i	$0.343355\pi$
$702$	0	0
$703$	−30.6624	−1.15645
$704$	10.6734	0.402268
$705$	0	0
$706$	−13.3682	−0.503118
$707$	10.1482	0.381663
$708$	0	0
$709$	31.8325	1.19549	0.597747	−	0.801685i	$-0.296063\pi$
$709$	31.8325	1.19549	0.597747	+	0.801685i	$0.296063\pi$
$710$	22.3016	0.836966
$711$	0	0
$712$	8.46847	0.317369
$713$	0	0
$714$	0	0
$715$	82.3748	3.08064
$716$	4.61545	0.172487
$717$	0	0
$718$	15.8185	0.590343
$719$	1.82665	0.0681225	0.0340612	−	0.999420i	$-0.489156\pi$
$719$	1.82665	0.0681225	0.0340612	+	0.999420i	$0.489156\pi$
$720$	0	0
$721$	−8.53619	−0.317904
$722$	−2.36519	−0.0880232
$723$	0	0
$724$	−17.7590	−0.660010
$725$	−11.6199	−0.431554
$726$	0	0
$727$	37.4929	1.39053	0.695267	−	0.718751i	$-0.255286\pi$
$727$	37.4929	1.39053	0.695267	+	0.718751i	$0.255286\pi$
$728$	−9.61967	−0.356529
$729$	0	0
$730$	−11.2774	−0.417395
$731$	−6.40309	−0.236827
$732$	0	0
$733$	3.31404	0.122407	0.0612035	−	0.998125i	$-0.480506\pi$
$733$	3.31404	0.122407	0.0612035	+	0.998125i	$0.480506\pi$
$734$	6.37670	0.235368
$735$	0	0
$736$	0	0
$737$	8.82318	0.325006
$738$	0	0
$739$	−5.35905	−0.197136	−0.0985680	−	0.995130i	$-0.531426\pi$
$739$	−5.35905	−0.197136	−0.0985680	+	0.995130i	$0.531426\pi$
$740$	50.8688	1.86997
$741$	0	0
$742$	2.69335	0.0988760
$743$	−25.7930	−0.946255	−0.473128	−	0.880994i	$-0.656875\pi$
$743$	−25.7930	−0.946255	−0.473128	+	0.880994i	$0.656875\pi$
$744$	0	0
$745$	32.6504	1.19622
$746$	11.8456	0.433699
$747$	0	0
$748$	5.86407	0.214412
$749$	−5.17232	−0.188992
$750$	0	0
$751$	−38.7211	−1.41295	−0.706476	−	0.707737i	$-0.749716\pi$
$751$	−38.7211	−1.41295	−0.706476	+	0.707737i	$0.749716\pi$
$752$	15.2411	0.555787
$753$	0	0
$754$	−5.01010	−0.182457
$755$	29.6400	1.07871
$756$	0	0
$757$	−25.8929	−0.941092	−0.470546	−	0.882375i	$-0.655943\pi$
$757$	−25.8929	−0.941092	−0.470546	+	0.882375i	$0.655943\pi$
$758$	−18.3386	−0.666089
$759$	0	0
$760$	−23.7895	−0.862934
$761$	−30.5806	−1.10855	−0.554273	−	0.832335i	$-0.687004\pi$
$761$	−30.5806	−1.10855	−0.554273	+	0.832335i	$0.687004\pi$
$762$	0	0
$763$	0.945465	0.0342281
$764$	44.3546	1.60469
$765$	0	0
$766$	−8.03288	−0.290240
$767$	−6.70158	−0.241980
$768$	0	0
$769$	15.1190	0.545205	0.272602	−	0.962127i	$-0.412116\pi$
$769$	15.1190	0.545205	0.272602	+	0.962127i	$0.412116\pi$
$770$	4.80744	0.173248
$771$	0	0
$772$	0.176821	0.00636391
$773$	−25.1640	−0.905085	−0.452542	−	0.891743i	$-0.649483\pi$
$773$	−25.1640	−0.905085	−0.452542	+	0.891743i	$0.649483\pi$
$774$	0	0
$775$	4.14921	0.149044
$776$	1.43487	0.0515090
$777$	0	0
$778$	0.944237	0.0338525
$779$	−14.2976	−0.512265
$780$	0	0
$781$	47.0049	1.68197
$782$	0	0
$783$	0	0
$784$	−16.9983	−0.607084
$785$	32.9774	1.17701
$786$	0	0
$787$	−21.9597	−0.782778	−0.391389	−	0.920225i	$-0.628005\pi$
$787$	−21.9597	−0.782778	−0.391389	+	0.920225i	$0.628005\pi$
$788$	29.3565	1.04578
$789$	0	0
$790$	2.41689	0.0859890
$791$	−13.9074	−0.494492
$792$	0	0
$793$	81.1609	2.88211
$794$	−13.9293	−0.494331
$795$	0	0
$796$	35.4934	1.25803
$797$	−4.80630	−0.170248	−0.0851239	−	0.996370i	$-0.527129\pi$
$797$	−4.80630	−0.170248	−0.0851239	+	0.996370i	$0.527129\pi$
$798$	0	0
$799$	5.31787	0.188133
$800$	35.9339	1.27045
$801$	0	0
$802$	−4.21460	−0.148823
$803$	−23.7692	−0.838797
$804$	0	0
$805$	0	0
$806$	1.78899	0.0630145
$807$	0	0
$808$	−22.7431	−0.800101
$809$	29.5278	1.03814	0.519071	−	0.854731i	$-0.326278\pi$
$809$	29.5278	1.03814	0.519071	+	0.854731i	$0.326278\pi$
$810$	0	0
$811$	−22.5010	−0.790118	−0.395059	−	0.918656i	$-0.629276\pi$
$811$	−22.5010	−0.790118	−0.395059	+	0.918656i	$0.629276\pi$
$812$	2.25750	0.0792228
$813$	0	0
$814$	−13.8866	−0.486725
$815$	60.1380	2.10654
$816$	0	0
$817$	25.7098	0.899474
$818$	3.11568	0.108937
$819$	0	0
$820$	23.7197	0.828327
$821$	18.6961	0.652498	0.326249	−	0.945284i	$-0.394215\pi$
$821$	18.6961	0.652498	0.326249	+	0.945284i	$0.394215\pi$
$822$	0	0
$823$	−40.7624	−1.42089	−0.710444	−	0.703754i	$-0.751506\pi$
$823$	−40.7624	−1.42089	−0.710444	+	0.703754i	$0.751506\pi$
$824$	19.1304	0.666440
$825$	0	0
$826$	−0.391108	−0.0136084
$827$	23.5429	0.818666	0.409333	−	0.912385i	$-0.365761\pi$
$827$	23.5429	0.818666	0.409333	+	0.912385i	$0.365761\pi$
$828$	0	0
$829$	−10.5885	−0.367752	−0.183876	−	0.982949i	$-0.558865\pi$
$829$	−10.5885	−0.367752	−0.183876	+	0.982949i	$0.558865\pi$
$830$	2.94629	0.102267
$831$	0	0
$832$	−19.8998	−0.689903
$833$	−5.93099	−0.205497
$834$	0	0
$835$	37.5697	1.30015
$836$	−23.5456	−0.814340
$837$	0	0
$838$	4.45494	0.153893
$839$	39.3110	1.35717	0.678583	−	0.734523i	$-0.262594\pi$
$839$	39.3110	1.35717	0.678583	+	0.734523i	$0.262594\pi$
$840$	0	0
$841$	−26.4962	−0.913663
$842$	2.32996	0.0802959
$843$	0	0
$844$	4.11199	0.141541
$845$	−107.909	−3.71219
$846$	0	0
$847$	1.26947	0.0436195
$848$	18.6828	0.641569
$849$	0	0
$850$	3.28430	0.112650
$851$	0	0
$852$	0	0
$853$	−41.5065	−1.42116	−0.710578	−	0.703618i	$-0.751567\pi$
$853$	−41.5065	−1.42116	−0.710578	+	0.703618i	$0.751567\pi$
$854$	4.73660	0.162083
$855$	0	0
$856$	11.5917	0.396195
$857$	−11.1992	−0.382557	−0.191278	−	0.981536i	$-0.561263\pi$
$857$	−11.1992	−0.382557	−0.191278	+	0.981536i	$0.561263\pi$
$858$	0	0
$859$	2.01412	0.0687209	0.0343605	−	0.999410i	$-0.489061\pi$
$859$	2.01412	0.0687209	0.0343605	+	0.999410i	$0.489061\pi$
$860$	−42.6526	−1.45444
$861$	0	0
$862$	15.8730	0.540638
$863$	−22.0663	−0.751146	−0.375573	−	0.926793i	$-0.622554\pi$
$863$	−22.0663	−0.751146	−0.375573	+	0.926793i	$0.622554\pi$
$864$	0	0
$865$	−84.8120	−2.88369
$866$	−14.7323	−0.500623
$867$	0	0
$868$	−0.806101	−0.0273609
$869$	5.09404	0.172804
$870$	0	0
$871$	−16.4503	−0.557396
$872$	−2.11888	−0.0717543
$873$	0	0
$874$	0	0
$875$	−6.63416	−0.224275
$876$	0	0
$877$	−56.1439	−1.89585	−0.947923	−	0.318501i	$-0.896821\pi$
$877$	−56.1439	−1.89585	−0.947923	+	0.318501i	$0.896821\pi$
$878$	14.8729	0.501934
$879$	0	0
$880$	33.3474	1.12414
$881$	−40.7594	−1.37322	−0.686609	−	0.727027i	$-0.740902\pi$
$881$	−40.7594	−1.37322	−0.686609	+	0.727027i	$0.740902\pi$
$882$	0	0
$883$	−7.62344	−0.256549	−0.128275	−	0.991739i	$-0.540944\pi$
$883$	−7.62344	−0.256549	−0.128275	+	0.991739i	$0.540944\pi$
$884$	−10.9332	−0.367723
$885$	0	0
$886$	1.54634	0.0519502
$887$	−40.3041	−1.35328	−0.676640	−	0.736314i	$-0.736565\pi$
$887$	−40.3041	−1.35328	−0.676640	+	0.736314i	$0.736565\pi$
$888$	0	0
$889$	7.62526	0.255743
$890$	7.89055	0.264492
$891$	0	0
$892$	8.14709	0.272785
$893$	−21.3525	−0.714533
$894$	0	0
$895$	9.15794	0.306116
$896$	−9.04675	−0.302231
$897$	0	0
$898$	−10.0236	−0.334493
$899$	−0.894042	−0.0298180
$900$	0	0
$901$	6.51872	0.217170
$902$	−6.47520	−0.215601
$903$	0	0
$904$	31.1679	1.03663
$905$	−35.2374	−1.17133
$906$	0	0
$907$	19.4348	0.645321	0.322660	−	0.946515i	$-0.395423\pi$
$907$	19.4348	0.645321	0.322660	+	0.946515i	$0.395423\pi$
$908$	−15.4460	−0.512593
$909$	0	0
$910$	−8.96318	−0.297127
$911$	−47.8408	−1.58504	−0.792518	−	0.609848i	$-0.791230\pi$
$911$	−47.8408	−1.58504	−0.792518	+	0.609848i	$0.791230\pi$
$912$	0	0
$913$	6.20986	0.205516
$914$	0.259687	0.00858967
$915$	0	0
$916$	−16.4863	−0.544723
$917$	−10.8691	−0.358928
$918$	0	0
$919$	−50.2660	−1.65812	−0.829061	−	0.559158i	$-0.811124\pi$
$919$	−50.2660	−1.65812	−0.829061	+	0.559158i	$0.811124\pi$
$920$	0	0
$921$	0	0
$922$	−1.31238	−0.0432209
$923$	−87.6378	−2.88463
$924$	0	0
$925$	60.0483	1.97438
$926$	5.40754	0.177703
$927$	0	0
$928$	−7.74278	−0.254169
$929$	−46.9029	−1.53883	−0.769417	−	0.638747i	$-0.779453\pi$
$929$	−46.9029	−1.53883	−0.769417	+	0.638747i	$0.779453\pi$
$930$	0	0
$931$	23.8143	0.780481
$932$	−5.22548	−0.171166
$933$	0	0
$934$	−2.76926	−0.0906128
$935$	11.6354	0.380520
$936$	0	0
$937$	−0.762958	−0.0249248	−0.0124624	−	0.999922i	$-0.503967\pi$
$937$	−0.762958	−0.0249248	−0.0124624	+	0.999922i	$0.503967\pi$
$938$	−0.960048	−0.0313467
$939$	0	0
$940$	35.4237	1.15539
$941$	32.4347	1.05734	0.528670	−	0.848827i	$-0.322691\pi$
$941$	32.4347	1.05734	0.528670	+	0.848827i	$0.322691\pi$
$942$	0	0
$943$	0	0
$944$	−2.71297	−0.0882997
$945$	0	0
$946$	11.6437	0.378568
$947$	−31.4609	−1.02234	−0.511171	−	0.859479i	$-0.670788\pi$
$947$	−31.4609	−1.02234	−0.511171	+	0.859479i	$0.670788\pi$
$948$	0	0
$949$	44.3162	1.43857
$950$	−13.1872	−0.427849
$951$	0	0
$952$	−1.35878	−0.0440383
$953$	−4.69904	−0.152217	−0.0761084	−	0.997100i	$-0.524249\pi$
$953$	−4.69904	−0.152217	−0.0761084	+	0.997100i	$0.524249\pi$
$954$	0	0
$955$	88.0081	2.84788
$956$	−29.0968	−0.941057
$957$	0	0
$958$	4.00232	0.129309
$959$	−0.619211	−0.0199954
$960$	0	0
$961$	−30.6808	−0.989702
$962$	25.8907	0.834750
$963$	0	0
$964$	−2.34054	−0.0753837
$965$	0.350846	0.0112941
$966$	0	0
$967$	−26.0335	−0.837180	−0.418590	−	0.908175i	$-0.637476\pi$
$967$	−26.0335	−0.837180	−0.418590	+	0.908175i	$0.637476\pi$
$968$	−2.84501	−0.0914420
$969$	0	0
$970$	1.33695	0.0429270
$971$	−1.34939	−0.0433041	−0.0216521	−	0.999766i	$-0.506893\pi$
$971$	−1.34939	−0.0433041	−0.0216521	+	0.999766i	$0.506893\pi$
$972$	0	0
$973$	−11.9579	−0.383354
$974$	−1.66101	−0.0532222
$975$	0	0
$976$	32.8560	1.05170
$977$	−12.9384	−0.413935	−0.206967	−	0.978348i	$-0.566359\pi$
$977$	−12.9384	−0.413935	−0.206967	+	0.978348i	$0.566359\pi$
$978$	0	0
$979$	16.6308	0.531523
$980$	−39.5078	−1.26203
$981$	0	0
$982$	−0.176054	−0.00561810
$983$	9.74749	0.310897	0.155448	−	0.987844i	$-0.450318\pi$
$983$	9.74749	0.310897	0.155448	+	0.987844i	$0.450318\pi$
$984$	0	0
$985$	58.2490	1.85597
$986$	−0.707678	−0.0225370
$987$	0	0
$988$	43.8993	1.39662
$989$	0	0
$990$	0	0
$991$	−20.9779	−0.666384	−0.333192	−	0.942859i	$-0.608126\pi$
$991$	−20.9779	−0.666384	−0.333192	+	0.942859i	$0.608126\pi$
$992$	2.76477	0.0877814
$993$	0	0
$994$	−5.11459	−0.162225
$995$	70.4257	2.23264
$996$	0	0
$997$	−9.07531	−0.287418	−0.143709	−	0.989620i	$-0.545903\pi$
$997$	−9.07531	−0.287418	−0.143709	+	0.989620i	$0.545903\pi$
$998$	6.50036	0.205765
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bq.1.3		5
3.2	odd	2		1587.2.a.p.1.3		5
23.5	odd	22		207.2.i.b.163.1		10
23.14	odd	22		207.2.i.b.127.1		10
23.22	odd	2		4761.2.a.br.1.3		5
69.5	even	22		69.2.e.a.25.1	✓	10
69.14	even	22		69.2.e.a.58.1	yes	10
69.68	even	2		1587.2.a.o.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.a.25.1	✓	10	69.5	even	22
69.2.e.a.58.1	yes	10	69.14	even	22
207.2.i.b.127.1		10	23.14	odd	22
207.2.i.b.163.1		10	23.5	odd	22
1587.2.a.o.1.3		5	69.68	even	2
1587.2.a.p.1.3		5	3.2	odd	2
4761.2.a.bq.1.3		5	1.1	even	1	trivial
4761.2.a.br.1.3		5	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q+0.478891 q^{2} -1.77066 q^{4} -3.51334 q^{5} +0.805738 q^{7} -1.80574 q^{8} -1.68251 q^{10} -3.54620 q^{11} +6.61167 q^{13} +0.385861 q^{14} +2.67657 q^{16} +0.933899 q^{17} -3.74982 q^{19} +6.22094 q^{20} -1.69824 q^{22} +7.34354 q^{25} +3.16627 q^{26} -1.42669 q^{28} -1.58233 q^{29} +0.565015 q^{31} +4.89326 q^{32} +0.447236 q^{34} -2.83083 q^{35} +8.17703 q^{37} -1.79575 q^{38} +6.34417 q^{40} +3.81288 q^{41} -6.85630 q^{43} +6.27913 q^{44} +5.69427 q^{47} -6.35079 q^{49} +3.51676 q^{50} -11.7070 q^{52} +6.98011 q^{53} +12.4590 q^{55} -1.45495 q^{56} -0.757767 q^{58} -1.01360 q^{59} +12.2754 q^{61} +0.270581 q^{62} -3.00980 q^{64} -23.2290 q^{65} -2.48807 q^{67} -1.65362 q^{68} -1.35566 q^{70} -13.2550 q^{71} +6.70272 q^{73} +3.91591 q^{74} +6.63966 q^{76} -2.85731 q^{77} -1.43648 q^{79} -9.40370 q^{80} +1.82596 q^{82} -1.75113 q^{83} -3.28110 q^{85} -3.28342 q^{86} +6.40351 q^{88} -4.68976 q^{89} +5.32728 q^{91} +2.72694 q^{94} +13.1744 q^{95} -0.794619 q^{97} -3.04134 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 4 q^{7} - 9 q^{8} + q^{10} - 13 q^{11} + 7 q^{13} - 16 q^{14} + 6 q^{16} - q^{17} - 5 q^{19} + 2 q^{20} - 3 q^{22} - 10 q^{25} + 27 q^{26} + 12 q^{28} + 7 q^{29} + 8 q^{31}+ \cdots - 51 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 4 * q^7 - 9 * q^8 + q^10 - 13 * q^11 + 7 * q^13 - 16 * q^14 + 6 * q^16 - q^17 - 5 * q^19 + 2 * q^20 - 3 * q^22 - 10 * q^25 + 27 * q^26 + 12 * q^28 + 7 * q^29 + 8 * q^31 - 6 * q^32 - 18 * q^34 - 9 * q^35 + 7 * q^37 + 9 * q^38 + 12 * q^40 + 7 * q^41 - 10 * q^43 - 17 * q^44 - 13 * q^47 - q^49 - 4 * q^50 - q^52 - 17 * q^53 + 10 * q^55 - 38 * q^56 - 17 * q^58 + 10 * q^59 + 18 * q^61 + q^62 - 21 * q^64 - 24 * q^65 + 9 * q^67 - 25 * q^68 + 3 * q^70 - 37 * q^71 - 2 * q^73 + 5 * q^74 - 4 * q^76 - 6 * q^77 - 32 * q^79 - 8 * q^80 - 17 * q^82 - 13 * q^83 + 5 * q^85 - 26 * q^86 + 19 * q^88 - 30 * q^89 - 23 * q^91 + 8 * q^94 + 14 * q^95 + 20 * q^97 - 51 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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