LMFDB - Embedded newform 1521.4.a.bk.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 $ x^10 - 70x^8 + 1645x^6 - 14700x^4 + 44100x^2 - 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.917374$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.917374 q^{2} -7.15843 q^{4} +15.4704 q^{5} -20.5833 q^{7} +13.9059 q^{8} +O(q^{10})$	$q-0.917374 q^{2} -7.15843 q^{4} +15.4704 q^{5} -20.5833 q^{7} +13.9059 q^{8} -14.1922 q^{10} +65.8420 q^{11} +18.8826 q^{14} +44.5105 q^{16} -44.2956 q^{17} +147.053 q^{19} -110.744 q^{20} -60.4017 q^{22} -53.1586 q^{23} +114.334 q^{25} +147.344 q^{28} +38.6257 q^{29} +88.3894 q^{31} -152.080 q^{32} +40.6357 q^{34} -318.433 q^{35} +78.9587 q^{37} -134.903 q^{38} +215.131 q^{40} +354.966 q^{41} -407.846 q^{43} -471.325 q^{44} +48.7663 q^{46} +67.9674 q^{47} +80.6738 q^{49} -104.887 q^{50} -226.572 q^{53} +1018.60 q^{55} -286.231 q^{56} -35.4342 q^{58} -142.031 q^{59} +266.831 q^{61} -81.0862 q^{62} -216.569 q^{64} -411.187 q^{67} +317.087 q^{68} +292.122 q^{70} +91.5052 q^{71} +63.1328 q^{73} -72.4347 q^{74} -1052.67 q^{76} -1355.25 q^{77} -287.115 q^{79} +688.595 q^{80} -325.637 q^{82} +373.812 q^{83} -685.272 q^{85} +374.147 q^{86} +915.595 q^{88} -119.403 q^{89} +380.532 q^{92} -62.3515 q^{94} +2274.97 q^{95} +554.650 q^{97} -74.0080 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 60 q^{4}+O(q^{10}) $ 10 * q + 60 * q^4	$ 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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 $

$2$	−0.917374	−0.324341	−0.162170	−	0.986763i	$-0.551849\pi$
$2$	−0.917374	−0.324341	−0.162170	+	0.986763i	$0.551849\pi$
$3$	0	0
$3$	0	0
$4$	−7.15843	−0.894803
$5$	15.4704	1.38372	0.691858	−	0.722034i	$-0.256792\pi$
$5$	15.4704	1.38372	0.691858	+	0.722034i	$0.256792\pi$
$6$	0	0
$7$	−20.5833	−1.11140	−0.555698	−	0.831384i	$-0.687549\pi$
$7$	−20.5833	−1.11140	−0.555698	+	0.831384i	$0.687549\pi$
$8$	13.9059	0.614562
$9$	0	0
$10$	−14.1922	−0.448795
$11$	65.8420	1.80474	0.902369	−	0.430964i	$-0.141827\pi$
$11$	65.8420	1.80474	0.902369	+	0.430964i	$0.141827\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	18.8826	0.360471
$15$	0	0
$16$	44.5105	0.695476
$17$	−44.2956	−0.631957	−0.315979	−	0.948766i	$-0.602333\pi$
$17$	−44.2956	−0.631957	−0.315979	+	0.948766i	$0.602333\pi$
$18$	0	0
$19$	147.053	1.77560	0.887798	−	0.460234i	$-0.152234\pi$
$19$	147.053	1.77560	0.887798	+	0.460234i	$0.152234\pi$
$20$	−110.744	−1.23815
$21$	0	0
$22$	−60.4017	−0.585350
$23$	−53.1586	−0.481928	−0.240964	−	0.970534i	$-0.577464\pi$
$23$	−53.1586	−0.481928	−0.240964	+	0.970534i	$0.577464\pi$
$24$	0	0
$25$	114.334	0.914669
$26$	0	0
$27$	0	0
$28$	147.344	0.994480
$29$	38.6257	0.247331	0.123666	−	0.992324i	$-0.460535\pi$
$29$	38.6257	0.247331	0.123666	+	0.992324i	$0.460535\pi$
$30$	0	0
$31$	88.3894	0.512104	0.256052	−	0.966663i	$-0.417578\pi$
$31$	88.3894	0.512104	0.256052	+	0.966663i	$0.417578\pi$
$32$	−152.080	−0.840133
$33$	0	0
$34$	40.6357	0.204969
$35$	−318.433	−1.53786
$36$	0	0
$37$	78.9587	0.350831	0.175415	−	0.984495i	$-0.443873\pi$
$37$	78.9587	0.350831	0.175415	+	0.984495i	$0.443873\pi$
$38$	−134.903	−0.575898
$39$	0	0
$40$	215.131	0.850379
$41$	354.966	1.35211	0.676054	−	0.736852i	$-0.263689\pi$
$41$	354.966	1.35211	0.676054	+	0.736852i	$0.263689\pi$
$42$	0	0
$43$	−407.846	−1.44642	−0.723208	−	0.690630i	$-0.757333\pi$
$43$	−407.846	−1.44642	−0.723208	+	0.690630i	$0.757333\pi$
$44$	−471.325	−1.61489
$45$	0	0
$46$	48.7663	0.156309
$47$	67.9674	0.210938	0.105469	−	0.994423i	$-0.466366\pi$
$47$	67.9674	0.210938	0.105469	+	0.994423i	$0.466366\pi$
$48$	0	0
$49$	80.6738	0.235201
$50$	−104.887	−0.296664
$51$	0	0
$52$	0	0
$53$	−226.572	−0.587209	−0.293604	−	0.955927i	$-0.594855\pi$
$53$	−226.572	−0.587209	−0.293604	+	0.955927i	$0.594855\pi$
$54$	0	0
$55$	1018.60	2.49724
$56$	−286.231	−0.683021
$57$	0	0
$58$	−35.4342	−0.0802196
$59$	−142.031	−0.313404	−0.156702	−	0.987646i	$-0.550086\pi$
$59$	−142.031	−0.313404	−0.156702	+	0.987646i	$0.550086\pi$
$60$	0	0
$61$	266.831	0.560069	0.280035	−	0.959990i	$-0.409654\pi$
$61$	266.831	0.560069	0.280035	+	0.959990i	$0.409654\pi$
$62$	−81.0862	−0.166096
$63$	0	0
$64$	−216.569	−0.422987
$65$	0	0
$66$	0	0
$67$	−411.187	−0.749768	−0.374884	−	0.927072i	$-0.622317\pi$
$67$	−411.187	−0.749768	−0.374884	+	0.927072i	$0.622317\pi$
$68$	317.087	0.565477
$69$	0	0
$70$	292.122	0.498789
$71$	91.5052	0.152953	0.0764765	−	0.997071i	$-0.475633\pi$
$71$	91.5052	0.152953	0.0764765	+	0.997071i	$0.475633\pi$
$72$	0	0
$73$	63.1328	0.101221	0.0506105	−	0.998718i	$-0.483883\pi$
$73$	63.1328	0.101221	0.0506105	+	0.998718i	$0.483883\pi$
$74$	−72.4347	−0.113789
$75$	0	0
$76$	−1052.67	−1.58881
$77$	−1355.25	−2.00578
$78$	0	0
$79$	−287.115	−0.408899	−0.204449	−	0.978877i	$-0.565540\pi$
$79$	−287.115	−0.408899	−0.204449	+	0.978877i	$0.565540\pi$
$80$	688.595	0.962341
$81$	0	0
$82$	−325.637	−0.438543
$83$	373.812	0.494352	0.247176	−	0.968971i	$-0.420497\pi$
$83$	373.812	0.494352	0.247176	+	0.968971i	$0.420497\pi$
$84$	0	0
$85$	−685.272	−0.874449
$86$	374.147	0.469132
$87$	0	0
$88$	915.595	1.10912
$89$	−119.403	−0.142209	−0.0711047	−	0.997469i	$-0.522652\pi$
$89$	−119.403	−0.142209	−0.0711047	+	0.997469i	$0.522652\pi$
$90$	0	0
$91$	0	0
$92$	380.532	0.431231
$93$	0	0
$94$	−62.3515	−0.0684157
$95$	2274.97	2.45692
$96$	0	0
$97$	554.650	0.580579	0.290290	−	0.956939i	$-0.406248\pi$
$97$	554.650	0.580579	0.290290	+	0.956939i	$0.406248\pi$
$98$	−74.0080	−0.0762851
$99$	0	0
$100$	−818.449	−0.818449
$101$	462.565	0.455712	0.227856	−	0.973695i	$-0.426828\pi$
$101$	462.565	0.455712	0.227856	+	0.973695i	$0.426828\pi$
$102$	0	0
$103$	1122.07	1.07341	0.536704	−	0.843771i	$-0.319669\pi$
$103$	1122.07	1.07341	0.536704	+	0.843771i	$0.319669\pi$
$104$	0	0
$105$	0	0
$106$	207.851	0.190456
$107$	−602.756	−0.544585	−0.272293	−	0.962214i	$-0.587782\pi$
$107$	−602.756	−0.544585	−0.272293	+	0.962214i	$0.587782\pi$
$108$	0	0
$109$	1421.89	1.24947	0.624735	−	0.780837i	$-0.285207\pi$
$109$	1421.89	1.24947	0.624735	+	0.780837i	$0.285207\pi$
$110$	−934.440	−0.809958
$111$	0	0
$112$	−916.174	−0.772949
$113$	396.719	0.330267	0.165134	−	0.986271i	$-0.447194\pi$
$113$	396.719	0.330267	0.165134	+	0.986271i	$0.447194\pi$
$114$	0	0
$115$	−822.386	−0.666851
$116$	−276.499	−0.221313
$117$	0	0
$118$	130.295	0.101650
$119$	911.752	0.702354
$120$	0	0
$121$	3004.17	2.25708
$122$	−244.784	−0.181653
$123$	0	0
$124$	−632.729	−0.458232
$125$	−165.013	−0.118074
$126$	0	0
$127$	437.868	0.305941	0.152970	−	0.988231i	$-0.451116\pi$
$127$	437.868	0.305941	0.152970	+	0.988231i	$0.451116\pi$
$128$	1415.32	0.977324
$129$	0	0
$130$	0	0
$131$	−1657.44	−1.10543	−0.552715	−	0.833370i	$-0.686408\pi$
$131$	−1657.44	−1.10543	−0.552715	+	0.833370i	$0.686408\pi$
$132$	0	0
$133$	−3026.85	−1.97339
$134$	377.212	0.243180
$135$	0	0
$136$	−615.973	−0.388377
$137$	−1967.42	−1.22692	−0.613460	−	0.789725i	$-0.710223\pi$
$137$	−1967.42	−1.22692	−0.613460	+	0.789725i	$0.710223\pi$
$138$	0	0
$139$	2825.13	1.72392	0.861960	−	0.506977i	$-0.169237\pi$
$139$	2825.13	1.72392	0.861960	+	0.506977i	$0.169237\pi$
$140$	2279.48	1.37608
$141$	0	0
$142$	−83.9445	−0.0496089
$143$	0	0
$144$	0	0
$145$	597.555	0.342236
$146$	−57.9164	−0.0328301
$147$	0	0
$148$	−565.220	−0.313924
$149$	−797.658	−0.438569	−0.219284	−	0.975661i	$-0.570372\pi$
$149$	−797.658	−0.438569	−0.219284	+	0.975661i	$0.570372\pi$
$150$	0	0
$151$	−161.987	−0.0873003	−0.0436501	−	0.999047i	$-0.513899\pi$
$151$	−161.987	−0.0873003	−0.0436501	+	0.999047i	$0.513899\pi$
$152$	2044.91	1.09121
$153$	0	0
$154$	1243.27	0.650555
$155$	1367.42	0.708606
$156$	0	0
$157$	−342.000	−0.173851	−0.0869255	−	0.996215i	$-0.527704\pi$
$157$	−342.000	−0.173851	−0.0869255	+	0.996215i	$0.527704\pi$
$158$	263.392	0.132622
$159$	0	0
$160$	−2352.74	−1.16250
$161$	1094.18	0.535613
$162$	0	0
$163$	−556.597	−0.267460	−0.133730	−	0.991018i	$-0.542696\pi$
$163$	−556.597	−0.267460	−0.133730	+	0.991018i	$0.542696\pi$
$164$	−2541.00	−1.20987
$165$	0	0
$166$	−342.926	−0.160339
$167$	−3136.42	−1.45332	−0.726658	−	0.686999i	$-0.758928\pi$
$167$	−3136.42	−1.45332	−0.726658	+	0.686999i	$0.758928\pi$
$168$	0	0
$169$	0	0
$170$	628.650	0.283619
$171$	0	0
$172$	2919.53	1.29426
$173$	−3324.19	−1.46089	−0.730444	−	0.682972i	$-0.760687\pi$
$173$	−3324.19	−1.46089	−0.730444	+	0.682972i	$0.760687\pi$
$174$	0	0
$175$	−2353.37	−1.01656
$176$	2930.66	1.25515
$177$	0	0
$178$	109.537	0.0461243
$179$	3313.25	1.38349	0.691743	−	0.722144i	$-0.256843\pi$
$179$	3313.25	1.38349	0.691743	+	0.722144i	$0.256843\pi$
$180$	0	0
$181$	76.0118	0.0312150	0.0156075	−	0.999878i	$-0.495032\pi$
$181$	76.0118	0.0312150	0.0156075	+	0.999878i	$0.495032\pi$
$182$	0	0
$183$	0	0
$184$	−739.221	−0.296174
$185$	1221.52	0.485450
$186$	0	0
$187$	−2916.51	−1.14052
$188$	−486.540	−0.188748
$189$	0	0
$190$	−2087.00	−0.796879
$191$	4073.39	1.54314	0.771572	−	0.636142i	$-0.219471\pi$
$191$	4073.39	1.54314	0.771572	+	0.636142i	$0.219471\pi$
$192$	0	0
$193$	867.581	0.323574	0.161787	−	0.986826i	$-0.448274\pi$
$193$	867.581	0.323574	0.161787	+	0.986826i	$0.448274\pi$
$194$	−508.821	−0.188305
$195$	0	0
$196$	−577.497	−0.210458
$197$	1887.69	0.682703	0.341352	−	0.939936i	$-0.389115\pi$
$197$	1887.69	0.682703	0.341352	+	0.939936i	$0.389115\pi$
$198$	0	0
$199$	2786.22	0.992511	0.496256	−	0.868176i	$-0.334708\pi$
$199$	2786.22	0.992511	0.496256	+	0.868176i	$0.334708\pi$
$200$	1589.92	0.562120
$201$	0	0
$202$	−424.345	−0.147806
$203$	−795.045	−0.274883
$204$	0	0
$205$	5491.47	1.87093
$206$	−1029.36	−0.348150
$207$	0	0
$208$	0	0
$209$	9682.28	3.20448
$210$	0	0
$211$	708.789	0.231256	0.115628	−	0.993293i	$-0.463112\pi$
$211$	708.789	0.231256	0.115628	+	0.993293i	$0.463112\pi$
$212$	1621.90	0.525436
$213$	0	0
$214$	552.953	0.176631
$215$	−6309.54	−2.00143
$216$	0	0
$217$	−1819.35	−0.569150
$218$	−1304.40	−0.405254
$219$	0	0
$220$	−7291.59	−2.23454
$221$	0	0
$222$	0	0
$223$	5396.64	1.62056	0.810281	−	0.586041i	$-0.199314\pi$
$223$	5396.64	1.62056	0.810281	+	0.586041i	$0.199314\pi$
$224$	3130.32	0.933720
$225$	0	0
$226$	−363.940	−0.107119
$227$	−3181.79	−0.930321	−0.465160	−	0.885226i	$-0.654003\pi$
$227$	−3181.79	−0.930321	−0.465160	+	0.885226i	$0.654003\pi$
$228$	0	0
$229$	2034.00	0.586945	0.293473	−	0.955967i	$-0.405189\pi$
$229$	2034.00	0.586945	0.293473	+	0.955967i	$0.405189\pi$
$230$	754.435	0.216287
$231$	0	0
$232$	537.126	0.152000
$233$	2794.22	0.785645	0.392823	−	0.919614i	$-0.371499\pi$
$233$	2794.22	0.785645	0.392823	+	0.919614i	$0.371499\pi$
$234$	0	0
$235$	1051.48	0.291878
$236$	1016.72	0.280435
$237$	0	0
$238$	−836.417	−0.227802
$239$	5493.81	1.48688	0.743442	−	0.668800i	$-0.233192\pi$
$239$	5493.81	1.48688	0.743442	+	0.668800i	$0.233192\pi$
$240$	0	0
$241$	4061.27	1.08552	0.542758	−	0.839889i	$-0.317380\pi$
$241$	4061.27	1.08552	0.542758	+	0.839889i	$0.317380\pi$
$242$	−2755.95	−0.732062
$243$	0	0
$244$	−1910.09	−0.501152
$245$	1248.06	0.325451
$246$	0	0
$247$	0	0
$248$	1229.14	0.314719
$249$	0	0
$250$	151.379	0.0382961
$251$	3570.88	0.897977	0.448988	−	0.893538i	$-0.351784\pi$
$251$	3570.88	0.897977	0.448988	+	0.893538i	$0.351784\pi$
$252$	0	0
$253$	−3500.07	−0.869753
$254$	−401.688	−0.0992290
$255$	0	0
$256$	434.179	0.106001
$257$	7518.26	1.82481	0.912405	−	0.409288i	$-0.134223\pi$
$257$	7518.26	1.82481	0.912405	+	0.409288i	$0.134223\pi$
$258$	0	0
$259$	−1625.23	−0.389912
$260$	0	0
$261$	0	0
$262$	1520.49	0.358536
$263$	4660.18	1.09262	0.546310	−	0.837583i	$-0.316032\pi$
$263$	4660.18	1.09262	0.546310	+	0.837583i	$0.316032\pi$
$264$	0	0
$265$	−3505.16	−0.812530
$266$	2776.75	0.640050
$267$	0	0
$268$	2943.45	0.670895
$269$	−5347.44	−1.21204	−0.606021	−	0.795449i	$-0.707235\pi$
$269$	−5347.44	−1.21204	−0.606021	+	0.795449i	$0.707235\pi$
$270$	0	0
$271$	2973.08	0.666427	0.333214	−	0.942851i	$-0.391867\pi$
$271$	2973.08	0.666427	0.333214	+	0.942851i	$0.391867\pi$
$272$	−1971.62	−0.439511
$273$	0	0
$274$	1804.86	0.397940
$275$	7527.96	1.65074
$276$	0	0
$277$	−764.153	−0.165753	−0.0828764	−	0.996560i	$-0.526411\pi$
$277$	−764.153	−0.165753	−0.0828764	+	0.996560i	$0.526411\pi$
$278$	−2591.70	−0.559137
$279$	0	0
$280$	−4428.11	−0.945107
$281$	7040.34	1.49463	0.747316	−	0.664469i	$-0.231342\pi$
$281$	7040.34	1.49463	0.747316	+	0.664469i	$0.231342\pi$
$282$	0	0
$283$	−9035.17	−1.89783	−0.948913	−	0.315537i	$-0.897815\pi$
$283$	−9035.17	−1.89783	−0.948913	+	0.315537i	$0.897815\pi$
$284$	−655.033	−0.136863
$285$	0	0
$286$	0	0
$287$	−7306.39	−1.50273
$288$	0	0
$289$	−2950.90	−0.600630
$290$	−548.181	−0.111001
$291$	0	0
$292$	−451.931	−0.0905729
$293$	−1785.23	−0.355954	−0.177977	−	0.984035i	$-0.556955\pi$
$293$	−1785.23	−0.355954	−0.177977	+	0.984035i	$0.556955\pi$
$294$	0	0
$295$	−2197.27	−0.433662
$296$	1098.00	0.215607
$297$	0	0
$298$	731.751	0.142246
$299$	0	0
$300$	0	0
$301$	8394.83	1.60754
$302$	148.603	0.0283150
$303$	0	0
$304$	6545.40	1.23488
$305$	4127.99	0.774977
$306$	0	0
$307$	5323.13	0.989600	0.494800	−	0.869007i	$-0.335241\pi$
$307$	5323.13	0.989600	0.494800	+	0.869007i	$0.335241\pi$
$308$	9701.45	1.79478
$309$	0	0
$310$	−1254.44	−0.229830
$311$	−6265.64	−1.14242	−0.571209	−	0.820805i	$-0.693525\pi$
$311$	−6265.64	−1.14242	−0.571209	+	0.820805i	$0.693525\pi$
$312$	0	0
$313$	7193.77	1.29909	0.649547	−	0.760322i	$-0.274959\pi$
$313$	7193.77	1.29909	0.649547	+	0.760322i	$0.274959\pi$
$314$	313.742	0.0563869
$315$	0	0
$316$	2055.29	0.365884
$317$	9576.87	1.69682	0.848408	−	0.529343i	$-0.177561\pi$
$317$	9576.87	1.69682	0.848408	+	0.529343i	$0.177561\pi$
$318$	0	0
$319$	2543.19	0.446368
$320$	−3350.41	−0.585293
$321$	0	0
$322$	−1003.77	−0.173721
$323$	−6513.81	−1.12210
$324$	0	0
$325$	0	0
$326$	510.608	0.0867483
$327$	0	0
$328$	4936.14	0.830953
$329$	−1399.00	−0.234435
$330$	0	0
$331$	5773.13	0.958670	0.479335	−	0.877632i	$-0.340878\pi$
$331$	5773.13	0.958670	0.479335	+	0.877632i	$0.340878\pi$
$332$	−2675.91	−0.442348
$333$	0	0
$334$	2877.27	0.471370
$335$	−6361.23	−1.03747
$336$	0	0
$337$	1238.09	0.200127	0.100063	−	0.994981i	$-0.468095\pi$
$337$	1238.09	0.200127	0.100063	+	0.994981i	$0.468095\pi$
$338$	0	0
$339$	0	0
$340$	4905.47	0.782460
$341$	5819.74	0.924213
$342$	0	0
$343$	5399.55	0.849995
$344$	−5671.48	−0.888912
$345$	0	0
$346$	3049.53	0.473826
$347$	−5449.97	−0.843140	−0.421570	−	0.906796i	$-0.638521\pi$
$347$	−5449.97	−0.843140	−0.421570	+	0.906796i	$0.638521\pi$
$348$	0	0
$349$	−1374.03	−0.210746	−0.105373	−	0.994433i	$-0.533604\pi$
$349$	−1374.03	−0.210746	−0.105373	+	0.994433i	$0.533604\pi$
$350$	2158.92	0.329711
$351$	0	0
$352$	−10013.3	−1.51622
$353$	5970.44	0.900211	0.450106	−	0.892975i	$-0.351386\pi$
$353$	5970.44	0.900211	0.450106	+	0.892975i	$0.351386\pi$
$354$	0	0
$355$	1415.62	0.211644
$356$	854.734	0.127249
$357$	0	0
$358$	−3039.49	−0.448720
$359$	7813.71	1.14872	0.574362	−	0.818602i	$-0.305250\pi$
$359$	7813.71	1.14872	0.574362	+	0.818602i	$0.305250\pi$
$360$	0	0
$361$	14765.6	2.15274
$362$	−69.7313	−0.0101243
$363$	0	0
$364$	0	0
$365$	976.690	0.140061
$366$	0	0
$367$	1688.39	0.240145	0.120073	−	0.992765i	$-0.461687\pi$
$367$	1688.39	0.240145	0.120073	+	0.992765i	$0.461687\pi$
$368$	−2366.11	−0.335169
$369$	0	0
$370$	−1120.59	−0.157451
$371$	4663.61	0.652622
$372$	0	0
$373$	1870.61	0.259670	0.129835	−	0.991536i	$-0.458555\pi$
$373$	1870.61	0.259670	0.129835	+	0.991536i	$0.458555\pi$
$374$	2675.53	0.369916
$375$	0	0
$376$	945.151	0.129634
$377$	0	0
$378$	0	0
$379$	−11667.0	−1.58125	−0.790627	−	0.612298i	$-0.790245\pi$
$379$	−11667.0	−1.58125	−0.790627	+	0.612298i	$0.790245\pi$
$380$	−16285.2	−2.19846
$381$	0	0
$382$	−3736.82	−0.500504
$383$	−6676.60	−0.890753	−0.445376	−	0.895343i	$-0.646930\pi$
$383$	−6676.60	−0.890753	−0.445376	+	0.895343i	$0.646930\pi$
$384$	0	0
$385$	−20966.3	−2.77543
$386$	−795.896	−0.104948
$387$	0	0
$388$	−3970.42	−0.519504
$389$	−14285.3	−1.86194	−0.930969	−	0.365099i	$-0.881035\pi$
$389$	−14285.3	−1.86194	−0.930969	+	0.365099i	$0.881035\pi$
$390$	0	0
$391$	2354.69	0.304558
$392$	1121.85	0.144545
$393$	0	0
$394$	−1731.72	−0.221428
$395$	−4441.79	−0.565800
$396$	0	0
$397$	−3569.28	−0.451227	−0.225614	−	0.974217i	$-0.572439\pi$
$397$	−3569.28	−0.451227	−0.225614	+	0.974217i	$0.572439\pi$
$398$	−2556.00	−0.321912
$399$	0	0
$400$	5089.04	0.636130
$401$	499.348	0.0621852	0.0310926	−	0.999517i	$-0.490101\pi$
$401$	499.348	0.0621852	0.0310926	+	0.999517i	$0.490101\pi$
$402$	0	0
$403$	0	0
$404$	−3311.23	−0.407772
$405$	0	0
$406$	729.354	0.0891557
$407$	5198.80	0.633158
$408$	0	0
$409$	−10457.5	−1.26428	−0.632140	−	0.774855i	$-0.717823\pi$
$409$	−10457.5	−1.26428	−0.632140	+	0.774855i	$0.717823\pi$
$410$	−5037.73	−0.606819
$411$	0	0
$412$	−8032.27	−0.960489
$413$	2923.47	0.348316
$414$	0	0
$415$	5783.03	0.684043
$416$	0	0
$417$	0	0
$418$	−8882.27	−1.03934
$419$	1705.42	0.198843	0.0994215	−	0.995045i	$-0.468301\pi$
$419$	1705.42	0.198843	0.0994215	+	0.995045i	$0.468301\pi$
$420$	0	0
$421$	8765.57	1.01475	0.507373	−	0.861727i	$-0.330617\pi$
$421$	8765.57	1.01475	0.507373	+	0.861727i	$0.330617\pi$
$422$	−650.225	−0.0750058
$423$	0	0
$424$	−3150.70	−0.360876
$425$	−5064.48	−0.578032
$426$	0	0
$427$	−5492.28	−0.622459
$428$	4314.79	0.487297
$429$	0	0
$430$	5788.21	0.649145
$431$	−8080.37	−0.903057	−0.451529	−	0.892257i	$-0.649121\pi$
$431$	−8080.37	−0.903057	−0.451529	+	0.892257i	$0.649121\pi$
$432$	0	0
$433$	−4124.48	−0.457760	−0.228880	−	0.973455i	$-0.573506\pi$
$433$	−4124.48	−0.457760	−0.228880	+	0.973455i	$0.573506\pi$
$434$	1669.02	0.184598
$435$	0	0
$436$	−10178.5	−1.11803
$437$	−7817.14	−0.855709
$438$	0	0
$439$	−6115.52	−0.664870	−0.332435	−	0.943126i	$-0.607870\pi$
$439$	−6115.52	−0.664870	−0.332435	+	0.943126i	$0.607870\pi$
$440$	14164.6	1.53471
$441$	0	0
$442$	0	0
$443$	11058.8	1.18605	0.593025	−	0.805184i	$-0.297933\pi$
$443$	11058.8	1.18605	0.593025	+	0.805184i	$0.297933\pi$
$444$	0	0
$445$	−1847.21	−0.196777
$446$	−4950.73	−0.525614
$447$	0	0
$448$	4457.72	0.470106
$449$	242.012	0.0254371	0.0127185	−	0.999919i	$-0.495951\pi$
$449$	242.012	0.0254371	0.0127185	+	0.999919i	$0.495951\pi$
$450$	0	0
$451$	23371.7	2.44020
$452$	−2839.88	−0.295524
$453$	0	0
$454$	2918.89	0.301741
$455$	0	0
$456$	0	0
$457$	−11052.2	−1.13129	−0.565644	−	0.824650i	$-0.691372\pi$
$457$	−11052.2	−1.13129	−0.565644	+	0.824650i	$0.691372\pi$
$458$	−1865.94	−0.190370
$459$	0	0
$460$	5886.99	0.596700
$461$	−273.763	−0.0276582	−0.0138291	−	0.999904i	$-0.504402\pi$
$461$	−273.763	−0.0276582	−0.0138291	+	0.999904i	$0.504402\pi$
$462$	0	0
$463$	11579.2	1.16227	0.581134	−	0.813808i	$-0.302609\pi$
$463$	11579.2	1.16227	0.581134	+	0.813808i	$0.302609\pi$
$464$	1719.25	0.172013
$465$	0	0
$466$	−2563.34	−0.254817
$467$	−902.915	−0.0894688	−0.0447344	−	0.998999i	$-0.514244\pi$
$467$	−902.915	−0.0894688	−0.0447344	+	0.998999i	$0.514244\pi$
$468$	0	0
$469$	8463.59	0.833289
$470$	−964.604	−0.0946678
$471$	0	0
$472$	−1975.07	−0.192606
$473$	−26853.4	−2.61040
$474$	0	0
$475$	16813.1	1.62408
$476$	−6526.71	−0.628469
$477$	0	0
$478$	−5039.88	−0.482257
$479$	12068.8	1.15122	0.575611	−	0.817723i	$-0.304764\pi$
$479$	12068.8	1.15122	0.575611	+	0.817723i	$0.304764\pi$
$480$	0	0
$481$	0	0
$482$	−3725.70	−0.352077
$483$	0	0
$484$	−21505.1	−2.01964
$485$	8580.66	0.803356
$486$	0	0
$487$	11492.5	1.06936	0.534678	−	0.845056i	$-0.320433\pi$
$487$	11492.5	1.06936	0.534678	+	0.845056i	$0.320433\pi$
$488$	3710.54	0.344197
$489$	0	0
$490$	−1144.93	−0.105557
$491$	−15704.2	−1.44342	−0.721710	−	0.692196i	$-0.756643\pi$
$491$	−15704.2	−1.44342	−0.721710	+	0.692196i	$0.756643\pi$
$492$	0	0
$493$	−1710.95	−0.156303
$494$	0	0
$495$	0	0
$496$	3934.25	0.356156
$497$	−1883.48	−0.169991
$498$	0	0
$499$	9019.80	0.809181	0.404591	−	0.914498i	$-0.367414\pi$
$499$	9019.80	0.809181	0.404591	+	0.914498i	$0.367414\pi$
$500$	1181.23	0.105653
$501$	0	0
$502$	−3275.83	−0.291250
$503$	6033.84	0.534862	0.267431	−	0.963577i	$-0.413825\pi$
$503$	6033.84	0.534862	0.267431	+	0.963577i	$0.413825\pi$
$504$	0	0
$505$	7156.06	0.630576
$506$	3210.87	0.282096
$507$	0	0
$508$	−3134.44	−0.273757
$509$	−22451.5	−1.95510	−0.977551	−	0.210699i	$-0.932426\pi$
$509$	−22451.5	−1.95510	−0.977551	+	0.210699i	$0.932426\pi$
$510$	0	0
$511$	−1299.48	−0.112497
$512$	−11720.8	−1.01170
$513$	0	0
$514$	−6897.06	−0.591860
$515$	17358.9	1.48529
$516$	0	0
$517$	4475.11	0.380687
$518$	1490.95	0.126464
$519$	0	0
$520$	0	0
$521$	−15674.7	−1.31808	−0.659040	−	0.752108i	$-0.729037\pi$
$521$	−15674.7	−1.31808	−0.659040	+	0.752108i	$0.729037\pi$
$522$	0	0
$523$	13510.8	1.12961	0.564804	−	0.825225i	$-0.308952\pi$
$523$	13510.8	1.12961	0.564804	+	0.825225i	$0.308952\pi$
$524$	11864.7	0.989142
$525$	0	0
$526$	−4275.13	−0.354381
$527$	−3915.27	−0.323627
$528$	0	0
$529$	−9341.16	−0.767746
$530$	3215.55	0.263537
$531$	0	0
$532$	21667.4	1.76580
$533$	0	0
$534$	0	0
$535$	−9324.89	−0.753551
$536$	−5717.94	−0.460778
$537$	0	0
$538$	4905.60	0.393114
$539$	5311.73	0.424475
$540$	0	0
$541$	12103.6	0.961875	0.480937	−	0.876755i	$-0.340296\pi$
$541$	12103.6	0.961875	0.480937	+	0.876755i	$0.340296\pi$
$542$	−2727.43	−0.216149
$543$	0	0
$544$	6736.49	0.530928
$545$	21997.2	1.72891
$546$	0	0
$547$	−15228.6	−1.19036	−0.595181	−	0.803592i	$-0.702920\pi$
$547$	−15228.6	−1.19036	−0.595181	+	0.803592i	$0.702920\pi$
$548$	14083.6	1.09785
$549$	0	0
$550$	−6905.95	−0.535401
$551$	5680.03	0.439160
$552$	0	0
$553$	5909.79	0.454448
$554$	701.014	0.0537603
$555$	0	0
$556$	−20223.5	−1.54257
$557$	−23596.9	−1.79503	−0.897516	−	0.440982i	$-0.854630\pi$
$557$	−23596.9	−1.79503	−0.897516	+	0.440982i	$0.854630\pi$
$558$	0	0
$559$	0	0
$560$	−14173.6	−1.06954
$561$	0	0
$562$	−6458.63	−0.484770
$563$	−7941.62	−0.594493	−0.297246	−	0.954801i	$-0.596068\pi$
$563$	−7941.62	−0.594493	−0.297246	+	0.954801i	$0.596068\pi$
$564$	0	0
$565$	6137.40	0.456996
$566$	8288.63	0.615542
$567$	0	0
$568$	1272.47	0.0939991
$569$	−2274.65	−0.167590	−0.0837948	−	0.996483i	$-0.526704\pi$
$569$	−2274.65	−0.167590	−0.0837948	+	0.996483i	$0.526704\pi$
$570$	0	0
$571$	−4499.84	−0.329794	−0.164897	−	0.986311i	$-0.552729\pi$
$571$	−4499.84	−0.329794	−0.164897	+	0.986311i	$0.552729\pi$
$572$	0	0
$573$	0	0
$574$	6702.69	0.487395
$575$	−6077.82	−0.440804
$576$	0	0
$577$	−25253.3	−1.82202	−0.911011	−	0.412381i	$-0.864697\pi$
$577$	−25253.3	−1.82202	−0.911011	+	0.412381i	$0.864697\pi$
$578$	2707.08	0.194809
$579$	0	0
$580$	−4277.55	−0.306234
$581$	−7694.31	−0.549421
$582$	0	0
$583$	−14918.0	−1.05976
$584$	877.921	0.0622066
$585$	0	0
$586$	1637.73	0.115450
$587$	11285.2	0.793508	0.396754	−	0.917925i	$-0.370137\pi$
$587$	11285.2	0.793508	0.396754	+	0.917925i	$0.370137\pi$
$588$	0	0
$589$	12997.9	0.909289
$590$	2015.72	0.140654
$591$	0	0
$592$	3514.49	0.243994
$593$	−12824.5	−0.888090	−0.444045	−	0.896005i	$-0.646457\pi$
$593$	−12824.5	−0.888090	−0.444045	+	0.896005i	$0.646457\pi$
$594$	0	0
$595$	14105.2	0.971859
$596$	5709.98	0.392433
$597$	0	0
$598$	0	0
$599$	26180.3	1.78581	0.892905	−	0.450245i	$-0.148664\pi$
$599$	26180.3	1.78581	0.892905	+	0.450245i	$0.148664\pi$
$600$	0	0
$601$	16012.6	1.08680	0.543399	−	0.839474i	$-0.317137\pi$
$601$	16012.6	1.08680	0.543399	+	0.839474i	$0.317137\pi$
$602$	−7701.20	−0.521391
$603$	0	0
$604$	1159.57	0.0781166
$605$	46475.8	3.12316
$606$	0	0
$607$	−9531.48	−0.637349	−0.318674	−	0.947864i	$-0.603238\pi$
$607$	−9531.48	−0.637349	−0.318674	+	0.947864i	$0.603238\pi$
$608$	−22363.9	−1.49174
$609$	0	0
$610$	−3786.91	−0.251356
$611$	0	0
$612$	0	0
$613$	10187.1	0.671211	0.335605	−	0.942003i	$-0.391059\pi$
$613$	10187.1	0.671211	0.335605	+	0.942003i	$0.391059\pi$
$614$	−4883.30	−0.320967
$615$	0	0
$616$	−18846.0	−1.23267
$617$	−8012.62	−0.522813	−0.261407	−	0.965229i	$-0.584186\pi$
$617$	−8012.62	−0.522813	−0.261407	+	0.965229i	$0.584186\pi$
$618$	0	0
$619$	1886.59	0.122501	0.0612506	−	0.998122i	$-0.480491\pi$
$619$	1886.59	0.122501	0.0612506	+	0.998122i	$0.480491\pi$
$620$	−9788.58	−0.634063
$621$	0	0
$622$	5747.94	0.370533
$623$	2457.70	0.158051
$624$	0	0
$625$	−16844.5	−1.07805
$626$	−6599.38	−0.421349
$627$	0	0
$628$	2448.18	0.155562
$629$	−3497.53	−0.221710
$630$	0	0
$631$	14956.1	0.943569	0.471784	−	0.881714i	$-0.343610\pi$
$631$	14956.1	0.943569	0.471784	+	0.881714i	$0.343610\pi$
$632$	−3992.61	−0.251293
$633$	0	0
$634$	−8785.57	−0.550346
$635$	6773.99	0.423335
$636$	0	0
$637$	0	0
$638$	−2333.06	−0.144775
$639$	0	0
$640$	21895.5	1.35234
$641$	−23691.7	−1.45985	−0.729927	−	0.683525i	$-0.760446\pi$
$641$	−23691.7	−1.45985	−0.729927	+	0.683525i	$0.760446\pi$
$642$	0	0
$643$	13651.3	0.837254	0.418627	−	0.908158i	$-0.362511\pi$
$643$	13651.3	0.837254	0.418627	+	0.908158i	$0.362511\pi$
$644$	−7832.62	−0.479268
$645$	0	0
$646$	5975.60	0.363943
$647$	18132.3	1.10178	0.550892	−	0.834577i	$-0.314288\pi$
$647$	18132.3	1.10178	0.550892	+	0.834577i	$0.314288\pi$
$648$	0	0
$649$	−9351.59	−0.565612
$650$	0	0
$651$	0	0
$652$	3984.36	0.239324
$653$	−6363.47	−0.381351	−0.190675	−	0.981653i	$-0.561068\pi$
$653$	−6363.47	−0.381351	−0.190675	+	0.981653i	$0.561068\pi$
$654$	0	0
$655$	−25641.3	−1.52960
$656$	15799.7	0.940358
$657$	0	0
$658$	1283.40	0.0760369
$659$	1051.43	0.0621518	0.0310759	−	0.999517i	$-0.490107\pi$
$659$	1051.43	0.0621518	0.0310759	+	0.999517i	$0.490107\pi$
$660$	0	0
$661$	8119.75	0.477794	0.238897	−	0.971045i	$-0.423214\pi$
$661$	8119.75	0.477794	0.238897	+	0.971045i	$0.423214\pi$
$662$	−5296.12	−0.310936
$663$	0	0
$664$	5198.21	0.303810
$665$	−46826.5	−2.73061
$666$	0	0
$667$	−2053.29	−0.119196
$668$	22451.9	1.30043
$669$	0	0
$670$	5835.62	0.336492
$671$	17568.7	1.01078
$672$	0	0
$673$	−190.264	−0.0108977	−0.00544885	−	0.999985i	$-0.501734\pi$
$673$	−190.264	−0.0108977	−0.00544885	+	0.999985i	$0.501734\pi$
$674$	−1135.79	−0.0649093
$675$	0	0
$676$	0	0
$677$	−4861.93	−0.276010	−0.138005	−	0.990431i	$-0.544069\pi$
$677$	−4861.93	−0.276010	−0.138005	+	0.990431i	$0.544069\pi$
$678$	0	0
$679$	−11416.5	−0.645253
$680$	−9529.35	−0.537403
$681$	0	0
$682$	−5338.88	−0.299760
$683$	14539.6	0.814556	0.407278	−	0.913304i	$-0.366478\pi$
$683$	14539.6	0.814556	0.407278	+	0.913304i	$0.366478\pi$
$684$	0	0
$685$	−30436.8	−1.69771
$686$	−4953.40	−0.275688
$687$	0	0
$688$	−18153.4	−1.00595
$689$	0	0
$690$	0	0
$691$	−22106.5	−1.21703	−0.608517	−	0.793541i	$-0.708235\pi$
$691$	−22106.5	−1.21703	−0.608517	+	0.793541i	$0.708235\pi$
$692$	23796.0	1.30721
$693$	0	0
$694$	4999.66	0.273465
$695$	43706.0	2.38541
$696$	0	0
$697$	−15723.4	−0.854474
$698$	1260.50	0.0683534
$699$	0	0
$700$	16846.4	0.909620
$701$	−229.971	−0.0123907	−0.00619535	−	0.999981i	$-0.501972\pi$
$701$	−229.971	−0.0123907	−0.00619535	+	0.999981i	$0.501972\pi$
$702$	0	0
$703$	11611.1	0.622934
$704$	−14259.4	−0.763380
$705$	0	0
$706$	−5477.13	−0.291975
$707$	−9521.12	−0.506476
$708$	0	0
$709$	−4802.22	−0.254374	−0.127187	−	0.991879i	$-0.540595\pi$
$709$	−4802.22	−0.254374	−0.127187	+	0.991879i	$0.540595\pi$
$710$	−1298.66	−0.0686446
$711$	0	0
$712$	−1660.40	−0.0873965
$713$	−4698.66	−0.246797
$714$	0	0
$715$	0	0
$716$	−23717.6	−1.23795
$717$	0	0
$718$	−7168.09	−0.372578
$719$	11016.9	0.571435	0.285718	−	0.958314i	$-0.407768\pi$
$719$	11016.9	0.571435	0.285718	+	0.958314i	$0.407768\pi$
$720$	0	0
$721$	−23096.0	−1.19298
$722$	−13545.6	−0.698221
$723$	0	0
$724$	−544.125	−0.0279313
$725$	4416.21	0.226226
$726$	0	0
$727$	−13498.2	−0.688612	−0.344306	−	0.938857i	$-0.611886\pi$
$727$	−13498.2	−0.688612	−0.344306	+	0.938857i	$0.611886\pi$
$728$	0	0
$729$	0	0
$730$	−895.990	−0.0454275
$731$	18065.8	0.914073
$732$	0	0
$733$	−17014.7	−0.857368	−0.428684	−	0.903455i	$-0.641023\pi$
$733$	−17014.7	−0.857368	−0.428684	+	0.903455i	$0.641023\pi$
$734$	−1548.88	−0.0778888
$735$	0	0
$736$	8084.38	0.404883
$737$	−27073.4	−1.35313
$738$	0	0
$739$	13392.6	0.666651	0.333326	−	0.942812i	$-0.391829\pi$
$739$	13392.6	0.666651	0.333326	+	0.942812i	$0.391829\pi$
$740$	−8744.19	−0.434382
$741$	0	0
$742$	−4278.27	−0.211672
$743$	−11796.5	−0.582467	−0.291233	−	0.956652i	$-0.594066\pi$
$743$	−11796.5	−0.582467	−0.291233	+	0.956652i	$0.594066\pi$
$744$	0	0
$745$	−12340.1	−0.606854
$746$	−1716.05	−0.0842214
$747$	0	0
$748$	20877.6	1.02054
$749$	12406.7	0.605250
$750$	0	0
$751$	−12450.5	−0.604961	−0.302480	−	0.953156i	$-0.597815\pi$
$751$	−12450.5	−0.604961	−0.302480	+	0.953156i	$0.597815\pi$
$752$	3025.26	0.146702
$753$	0	0
$754$	0	0
$755$	−2506.01	−0.120799
$756$	0	0
$757$	−7029.84	−0.337522	−0.168761	−	0.985657i	$-0.553977\pi$
$757$	−7029.84	−0.337522	−0.168761	+	0.985657i	$0.553977\pi$
$758$	10703.0	0.512865
$759$	0	0
$760$	31635.6	1.50993
$761$	−17740.6	−0.845067	−0.422533	−	0.906347i	$-0.638859\pi$
$761$	−17740.6	−0.845067	−0.422533	+	0.906347i	$0.638859\pi$
$762$	0	0
$763$	−29267.2	−1.38866
$764$	−29159.1	−1.38081
$765$	0	0
$766$	6124.94	0.288907
$767$	0	0
$768$	0	0
$769$	31976.5	1.49948	0.749741	−	0.661732i	$-0.230178\pi$
$769$	31976.5	1.49948	0.749741	+	0.661732i	$0.230178\pi$
$770$	19233.9	0.900184
$771$	0	0
$772$	−6210.51	−0.289535
$773$	2135.11	0.0993463	0.0496732	−	0.998766i	$-0.484182\pi$
$773$	2135.11	0.0993463	0.0496732	+	0.998766i	$0.484182\pi$
$774$	0	0
$775$	10105.9	0.468405
$776$	7712.93	0.356802
$777$	0	0
$778$	13105.0	0.603902
$779$	52198.9	2.40080
$780$	0	0
$781$	6024.89	0.276040
$782$	−2160.14	−0.0987804
$783$	0	0
$784$	3590.83	0.163576
$785$	−5290.89	−0.240560
$786$	0	0
$787$	−11704.4	−0.530134	−0.265067	−	0.964230i	$-0.585394\pi$
$787$	−11704.4	−0.530134	−0.265067	+	0.964230i	$0.585394\pi$
$788$	−13512.9	−0.610885
$789$	0	0
$790$	4074.78	0.183512
$791$	−8165.80	−0.367057
$792$	0	0
$793$	0	0
$794$	3274.37	0.146351
$795$	0	0
$796$	−19944.9	−0.888102
$797$	149.733	0.00665474	0.00332737	−	0.999994i	$-0.498941\pi$
$797$	149.733	0.00665474	0.00332737	+	0.999994i	$0.498941\pi$
$798$	0	0
$799$	−3010.66	−0.133304
$800$	−17387.9	−0.768443
$801$	0	0
$802$	−458.089	−0.0201692
$803$	4156.79	0.182677
$804$	0	0
$805$	16927.4	0.741135
$806$	0	0
$807$	0	0
$808$	6432.40	0.280063
$809$	−23520.8	−1.02219	−0.511093	−	0.859526i	$-0.670759\pi$
$809$	−23520.8	−1.02219	−0.511093	+	0.859526i	$0.670759\pi$
$810$	0	0
$811$	29604.8	1.28183	0.640915	−	0.767612i	$-0.278555\pi$
$811$	29604.8	1.28183	0.640915	+	0.767612i	$0.278555\pi$
$812$	5691.27	0.245966
$813$	0	0
$814$	−4769.25	−0.205359
$815$	−8610.79	−0.370089
$816$	0	0
$817$	−59975.0	−2.56825
$818$	9593.44	0.410057
$819$	0	0
$820$	−39310.3	−1.67412
$821$	−43034.7	−1.82938	−0.914691	−	0.404154i	$-0.867566\pi$
$821$	−43034.7	−1.82938	−0.914691	+	0.404154i	$0.867566\pi$
$822$	0	0
$823$	5584.95	0.236548	0.118274	−	0.992981i	$-0.462264\pi$
$823$	5584.95	0.236548	0.118274	+	0.992981i	$0.462264\pi$
$824$	15603.5	0.659675
$825$	0	0
$826$	−2681.91	−0.112973
$827$	4788.13	0.201330	0.100665	−	0.994920i	$-0.467903\pi$
$827$	4788.13	0.201330	0.100665	+	0.994920i	$0.467903\pi$
$828$	0	0
$829$	−32392.2	−1.35709	−0.678546	−	0.734558i	$-0.737390\pi$
$829$	−32392.2	−1.35709	−0.678546	+	0.734558i	$0.737390\pi$
$830$	−5305.20	−0.221863
$831$	0	0
$832$	0	0
$833$	−3573.50	−0.148637
$834$	0	0
$835$	−48521.8	−2.01098
$836$	−69309.9	−2.86738
$837$	0	0
$838$	−1564.51	−0.0644929
$839$	−13598.9	−0.559577	−0.279788	−	0.960062i	$-0.590264\pi$
$839$	−13598.9	−0.559577	−0.279788	+	0.960062i	$0.590264\pi$
$840$	0	0
$841$	−22897.1	−0.938827
$842$	−8041.31	−0.329123
$843$	0	0
$844$	−5073.82	−0.206929
$845$	0	0
$846$	0	0
$847$	−61835.9	−2.50851
$848$	−10084.8	−0.408390
$849$	0	0
$850$	4646.02	0.187479
$851$	−4197.34	−0.169075
$852$	0	0
$853$	41037.0	1.64722	0.823611	−	0.567155i	$-0.191956\pi$
$853$	41037.0	1.64722	0.823611	+	0.567155i	$0.191956\pi$
$854$	5038.47	0.201889
$855$	0	0
$856$	−8381.89	−0.334681
$857$	39959.3	1.59275	0.796374	−	0.604804i	$-0.206749\pi$
$857$	39959.3	1.59275	0.796374	+	0.604804i	$0.206749\pi$
$858$	0	0
$859$	−32570.5	−1.29371	−0.646853	−	0.762615i	$-0.723915\pi$
$859$	−32570.5	−1.29371	−0.646853	+	0.762615i	$0.723915\pi$
$860$	45166.4	1.79088
$861$	0	0
$862$	7412.72	0.292898
$863$	16951.8	0.668652	0.334326	−	0.942457i	$-0.391491\pi$
$863$	16951.8	0.668652	0.334326	+	0.942457i	$0.391491\pi$
$864$	0	0
$865$	−51426.6	−2.02145
$866$	3783.69	0.148470
$867$	0	0
$868$	13023.7	0.509277
$869$	−18904.3	−0.737955
$870$	0	0
$871$	0	0
$872$	19772.7	0.767876
$873$	0	0
$874$	7171.24	0.277541
$875$	3396.52	0.131227
$876$	0	0
$877$	−45939.2	−1.76882	−0.884411	−	0.466709i	$-0.845439\pi$
$877$	−45939.2	−1.76882	−0.884411	+	0.466709i	$0.845439\pi$
$878$	5610.22	0.215644
$879$	0	0
$880$	45338.5	1.73677
$881$	−37960.5	−1.45167	−0.725836	−	0.687868i	$-0.758547\pi$
$881$	−37960.5	−1.45167	−0.725836	+	0.687868i	$0.758547\pi$
$882$	0	0
$883$	43172.9	1.64539	0.822697	−	0.568480i	$-0.192468\pi$
$883$	43172.9	1.64539	0.822697	+	0.568480i	$0.192468\pi$
$884$	0	0
$885$	0	0
$886$	−10145.1	−0.384684
$887$	−28706.3	−1.08665	−0.543327	−	0.839521i	$-0.682836\pi$
$887$	−28706.3	−1.08665	−0.543327	+	0.839521i	$0.682836\pi$
$888$	0	0
$889$	−9012.78	−0.340021
$890$	1694.58	0.0638229
$891$	0	0
$892$	−38631.4	−1.45008
$893$	9994.83	0.374540
$894$	0	0
$895$	51257.3	1.91435
$896$	−29131.9	−1.08619
$897$	0	0
$898$	−222.015	−0.00825027
$899$	3414.10	0.126659
$900$	0	0
$901$	10036.2	0.371091
$902$	−21440.6	−0.791456
$903$	0	0
$904$	5516.75	0.202969
$905$	1175.93	0.0431927
$906$	0	0
$907$	−22356.1	−0.818435	−0.409218	−	0.912437i	$-0.634198\pi$
$907$	−22356.1	−0.818435	−0.409218	+	0.912437i	$0.634198\pi$
$908$	22776.6	0.832454
$909$	0	0
$910$	0	0
$911$	−6953.80	−0.252897	−0.126449	−	0.991973i	$-0.540358\pi$
$911$	−6953.80	−0.252897	−0.126449	+	0.991973i	$0.540358\pi$
$912$	0	0
$913$	24612.6	0.892176
$914$	10139.0	0.366923
$915$	0	0
$916$	−14560.2	−0.525200
$917$	34115.7	1.22857
$918$	0	0
$919$	40625.2	1.45822	0.729108	−	0.684399i	$-0.239935\pi$
$919$	40625.2	1.45822	0.729108	+	0.684399i	$0.239935\pi$
$920$	−11436.0	−0.409821
$921$	0	0
$922$	251.143	0.00897068
$923$	0	0
$924$	0	0
$925$	9027.64	0.320894
$926$	−10622.4	−0.376971
$927$	0	0
$928$	−5874.20	−0.207791
$929$	−42813.8	−1.51203	−0.756014	−	0.654555i	$-0.772856\pi$
$929$	−42813.8	−1.51203	−0.756014	+	0.654555i	$0.772856\pi$
$930$	0	0
$931$	11863.3	0.417621
$932$	−20002.2	−0.702998
$933$	0	0
$934$	828.311	0.0290184
$935$	−45119.7	−1.57815
$936$	0	0
$937$	43484.1	1.51608	0.758038	−	0.652210i	$-0.226158\pi$
$937$	43484.1	1.51608	0.758038	+	0.652210i	$0.226158\pi$
$938$	−7764.28	−0.270269
$939$	0	0
$940$	−7526.97	−0.261173
$941$	7108.58	0.246263	0.123131	−	0.992390i	$-0.460706\pi$
$941$	7108.58	0.246263	0.123131	+	0.992390i	$0.460706\pi$
$942$	0	0
$943$	−18869.5	−0.651618
$944$	−6321.85	−0.217965
$945$	0	0
$946$	24634.6	0.846660
$947$	−1938.99	−0.0665351	−0.0332676	−	0.999446i	$-0.510591\pi$
$947$	−1938.99	−0.0665351	−0.0332676	+	0.999446i	$0.510591\pi$
$948$	0	0
$949$	0	0
$950$	−15423.9	−0.526756
$951$	0	0
$952$	12678.8	0.431640
$953$	47806.7	1.62498	0.812492	−	0.582972i	$-0.198110\pi$
$953$	47806.7	1.62498	0.812492	+	0.582972i	$0.198110\pi$
$954$	0	0
$955$	63017.1	2.13527
$956$	−39327.1	−1.33047
$957$	0	0
$958$	−11071.6	−0.373388
$959$	40496.1	1.36359
$960$	0	0
$961$	−21978.3	−0.737750
$962$	0	0
$963$	0	0
$964$	−29072.3	−0.971323
$965$	13421.8	0.447735
$966$	0	0
$967$	−2832.71	−0.0942025	−0.0471013	−	0.998890i	$-0.514998\pi$
$967$	−2832.71	−0.0942025	−0.0471013	+	0.998890i	$0.514998\pi$
$968$	41775.8	1.38711
$969$	0	0
$970$	−7871.68	−0.260561
$971$	41276.3	1.36418	0.682090	−	0.731268i	$-0.261071\pi$
$971$	41276.3	1.36418	0.682090	+	0.731268i	$0.261071\pi$
$972$	0	0
$973$	−58150.7	−1.91596
$974$	−10542.9	−0.346835
$975$	0	0
$976$	11876.8	0.389515
$977$	−4278.89	−0.140117	−0.0700583	−	0.997543i	$-0.522319\pi$
$977$	−4278.89	−0.140117	−0.0700583	+	0.997543i	$0.522319\pi$
$978$	0	0
$979$	−7861.70	−0.256651
$980$	−8934.12	−0.291214
$981$	0	0
$982$	14406.6	0.468159
$983$	−22652.9	−0.735011	−0.367505	−	0.930021i	$-0.619788\pi$
$983$	−22652.9	−0.735011	−0.367505	+	0.930021i	$0.619788\pi$
$984$	0	0
$985$	29203.4	0.944667
$986$	1569.58	0.0506953
$987$	0	0
$988$	0	0
$989$	21680.5	0.697068
$990$	0	0
$991$	22060.2	0.707131	0.353565	−	0.935410i	$-0.384969\pi$
$991$	22060.2	0.707131	0.353565	+	0.935410i	$0.384969\pi$
$992$	−13442.3	−0.430235
$993$	0	0
$994$	1727.86	0.0551351
$995$	43103.9	1.37335
$996$	0	0
$997$	35635.5	1.13198	0.565992	−	0.824411i	$-0.308493\pi$
$997$	35635.5	1.13198	0.565992	+	0.824411i	$0.308493\pi$
$998$	−8274.52	−0.262450
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bk.1.5		10
3.2	odd	2		507.4.a.r.1.6		10
13.6	odd	12		117.4.q.e.10.3		10
13.11	odd	12		117.4.q.e.82.3		10
13.12	even	2	inner	1521.4.a.bk.1.6		10
39.5	even	4		507.4.b.i.337.5		10
39.8	even	4		507.4.b.i.337.6		10
39.11	even	12		39.4.j.c.4.3	✓	10
39.32	even	12		39.4.j.c.10.3	yes	10
39.38	odd	2		507.4.a.r.1.5		10
156.11	odd	12		624.4.bv.h.433.2		10
156.71	odd	12		624.4.bv.h.49.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.3	✓	10	39.11	even	12
39.4.j.c.10.3	yes	10	39.32	even	12
117.4.q.e.10.3		10	13.6	odd	12
117.4.q.e.82.3		10	13.11	odd	12
507.4.a.r.1.5		10	39.38	odd	2
507.4.a.r.1.6		10	3.2	odd	2
507.4.b.i.337.5		10	39.5	even	4
507.4.b.i.337.6		10	39.8	even	4
624.4.bv.h.49.4		10	156.71	odd	12
624.4.bv.h.433.2		10	156.11	odd	12
1521.4.a.bk.1.5		10	1.1	even	1	trivial
1521.4.a.bk.1.6		10	13.12	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-0.917374 q^{2} -7.15843 q^{4} +15.4704 q^{5} -20.5833 q^{7} +13.9059 q^{8} +O(q^{10})\)	\(q-0.917374 q^{2} -7.15843 q^{4} +15.4704 q^{5} -20.5833 q^{7} +13.9059 q^{8} -14.1922 q^{10} +65.8420 q^{11} +18.8826 q^{14} +44.5105 q^{16} -44.2956 q^{17} +147.053 q^{19} -110.744 q^{20} -60.4017 q^{22} -53.1586 q^{23} +114.334 q^{25} +147.344 q^{28} +38.6257 q^{29} +88.3894 q^{31} -152.080 q^{32} +40.6357 q^{34} -318.433 q^{35} +78.9587 q^{37} -134.903 q^{38} +215.131 q^{40} +354.966 q^{41} -407.846 q^{43} -471.325 q^{44} +48.7663 q^{46} +67.9674 q^{47} +80.6738 q^{49} -104.887 q^{50} -226.572 q^{53} +1018.60 q^{55} -286.231 q^{56} -35.4342 q^{58} -142.031 q^{59} +266.831 q^{61} -81.0862 q^{62} -216.569 q^{64} -411.187 q^{67} +317.087 q^{68} +292.122 q^{70} +91.5052 q^{71} +63.1328 q^{73} -72.4347 q^{74} -1052.67 q^{76} -1355.25 q^{77} -287.115 q^{79} +688.595 q^{80} -325.637 q^{82} +373.812 q^{83} -685.272 q^{85} +374.147 q^{86} +915.595 q^{88} -119.403 q^{89} +380.532 q^{92} -62.3515 q^{94} +2274.97 q^{95} +554.650 q^{97} -74.0080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 60 q^{4}+O(q^{10}) \) 10 * q + 60 * q^4	\( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

Introduction

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