LMFDB - Embedded newform 1521.4.a.w.1.2

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:	$4$
Coefficient field:	4.4.1362828.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 23x^{2} + 48 $ x^4 - 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.52356$ 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)

$f(q)$	$=$	$q-1.52356 q^{2} -5.67878 q^{4} +9.65841 q^{5} -22.3639 q^{7} +20.8404 q^{8} +O(q^{10})$	$q-1.52356 q^{2} -5.67878 q^{4} +9.65841 q^{5} -22.3639 q^{7} +20.8404 q^{8} -14.7151 q^{10} -50.3050 q^{11} +34.0727 q^{14} +13.6788 q^{16} -86.1454 q^{17} -116.880 q^{19} -54.8480 q^{20} +76.6424 q^{22} -72.0000 q^{23} -31.7151 q^{25} +127.000 q^{28} -14.1454 q^{29} +196.215 q^{31} -187.563 q^{32} +131.247 q^{34} -216.000 q^{35} +154.424 q^{37} +178.073 q^{38} +201.285 q^{40} -265.726 q^{41} +211.855 q^{43} +285.671 q^{44} +109.696 q^{46} +67.5535 q^{47} +157.145 q^{49} +48.3197 q^{50} -686.581 q^{53} -485.866 q^{55} -466.073 q^{56} +21.5512 q^{58} -91.9304 q^{59} +329.006 q^{61} -298.945 q^{62} +176.333 q^{64} -768.370 q^{67} +489.201 q^{68} +329.088 q^{70} +264.969 q^{71} -771.306 q^{73} -235.273 q^{74} +663.734 q^{76} +1125.02 q^{77} +1226.86 q^{79} +132.115 q^{80} +404.849 q^{82} -514.019 q^{83} -832.027 q^{85} -322.772 q^{86} -1048.38 q^{88} +527.889 q^{89} +408.872 q^{92} -102.921 q^{94} -1128.87 q^{95} +74.2755 q^{97} -239.420 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{4}+O(q^{10}) $ 4 * q + 14 * q^4	$ 4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100}) $ 4 * q + 14 * q^4 + 88 * q^10 - 84 * q^14 + 18 * q^16 + 96 * q^17 + 380 * q^22 - 288 * q^23 + 20 * q^25 + 384 * q^29 - 864 * q^35 + 492 * q^38 + 952 * q^40 + 1288 * q^43 + 188 * q^49 - 984 * q^53 - 328 * q^55 - 1644 * q^56 + 288 * q^61 + 1668 * q^62 - 1314 * q^64 + 4380 * q^68 - 3144 * q^74 + 1416 * q^77 + 4320 * q^79 + 3088 * q^82 - 1036 * q^88 - 1008 * q^92 - 1660 * q^94 - 1872 * 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 $

$2$	−1.52356	−0.538658	−0.269329	−	0.963048i	$-0.586802\pi$
$2$	−1.52356	−0.538658	−0.269329	+	0.963048i	$0.586802\pi$
$3$	0	0
$3$	0	0
$4$	−5.67878	−0.709847
$5$	9.65841	0.863874	0.431937	−	0.901904i	$-0.357830\pi$
$5$	9.65841	0.863874	0.431937	+	0.901904i	$0.357830\pi$
$6$	0	0
$7$	−22.3639	−1.20754	−0.603769	−	0.797159i	$-0.706335\pi$
$7$	−22.3639	−1.20754	−0.603769	+	0.797159i	$0.706335\pi$
$8$	20.8404	0.921023
$9$	0	0
$10$	−14.7151	−0.465333
$11$	−50.3050	−1.37887	−0.689433	−	0.724349i	$-0.742140\pi$
$11$	−50.3050	−1.37887	−0.689433	+	0.724349i	$0.742140\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	34.0727	0.650450
$15$	0	0
$16$	13.6788	0.213731
$17$	−86.1454	−1.22902	−0.614509	−	0.788910i	$-0.710646\pi$
$17$	−86.1454	−1.22902	−0.614509	+	0.788910i	$0.710646\pi$
$18$	0	0
$19$	−116.880	−1.41127	−0.705633	−	0.708578i	$-0.749337\pi$
$19$	−116.880	−1.41127	−0.705633	+	0.708578i	$0.749337\pi$
$20$	−54.8480	−0.613219
$21$	0	0
$22$	76.6424	0.742737
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	−31.7151	−0.253721
$26$	0	0
$27$	0	0
$28$	127.000	0.857168
$29$	−14.1454	−0.0905768	−0.0452884	−	0.998974i	$-0.514421\pi$
$29$	−14.1454	−0.0905768	−0.0452884	+	0.998974i	$0.514421\pi$
$30$	0	0
$31$	196.215	1.13682	0.568408	−	0.822747i	$-0.307559\pi$
$31$	196.215	1.13682	0.568408	+	0.822747i	$0.307559\pi$
$32$	−187.563	−1.03615
$33$	0	0
$34$	131.247	0.662021
$35$	−216.000	−1.04316
$36$	0	0
$37$	154.424	0.686138	0.343069	−	0.939310i	$-0.388533\pi$
$37$	154.424	0.686138	0.343069	+	0.939310i	$0.388533\pi$
$38$	178.073	0.760190
$39$	0	0
$40$	201.285	0.795648
$41$	−265.726	−1.01218	−0.506091	−	0.862480i	$-0.668910\pi$
$41$	−265.726	−1.01218	−0.506091	+	0.862480i	$0.668910\pi$
$42$	0	0
$43$	211.855	0.751338	0.375669	−	0.926754i	$-0.377413\pi$
$43$	211.855	0.751338	0.375669	+	0.926754i	$0.377413\pi$
$44$	285.671	0.978785
$45$	0	0
$46$	109.696	0.351604
$47$	67.5535	0.209653	0.104827	−	0.994491i	$-0.466571\pi$
$47$	67.5535	0.209653	0.104827	+	0.994491i	$0.466571\pi$
$48$	0	0
$49$	157.145	0.458150
$50$	48.3197	0.136669
$51$	0	0
$52$	0	0
$53$	−686.581	−1.77942	−0.889710	−	0.456527i	$-0.849093\pi$
$53$	−686.581	−1.77942	−0.889710	+	0.456527i	$0.849093\pi$
$54$	0	0
$55$	−485.866	−1.19117
$56$	−466.073	−1.11217
$57$	0	0
$58$	21.5512	0.0487899
$59$	−91.9304	−0.202853	−0.101426	−	0.994843i	$-0.532341\pi$
$59$	−91.9304	−0.202853	−0.101426	+	0.994843i	$0.532341\pi$
$60$	0	0
$61$	329.006	0.690572	0.345286	−	0.938498i	$-0.387782\pi$
$61$	329.006	0.690572	0.345286	+	0.938498i	$0.387782\pi$
$62$	−298.945	−0.612355
$63$	0	0
$64$	176.333	0.344400
$65$	0	0
$66$	0	0
$67$	−768.370	−1.40106	−0.700532	−	0.713621i	$-0.747054\pi$
$67$	−768.370	−1.40106	−0.700532	+	0.713621i	$0.747054\pi$
$68$	489.201	0.872416
$69$	0	0
$70$	329.088	0.561908
$71$	264.969	0.442902	0.221451	−	0.975171i	$-0.428921\pi$
$71$	264.969	0.442902	0.221451	+	0.975171i	$0.428921\pi$
$72$	0	0
$73$	−771.306	−1.23664	−0.618319	−	0.785927i	$-0.712186\pi$
$73$	−771.306	−1.23664	−0.618319	+	0.785927i	$0.712186\pi$
$74$	−235.273	−0.369594
$75$	0	0
$76$	663.734	1.00178
$77$	1125.02	1.66503
$78$	0	0
$79$	1226.86	1.74725	0.873624	−	0.486602i	$-0.161764\pi$
$79$	1226.86	1.74725	0.873624	+	0.486602i	$0.161764\pi$
$80$	132.115	0.184637
$81$	0	0
$82$	404.849	0.545220
$83$	−514.019	−0.679771	−0.339885	−	0.940467i	$-0.610388\pi$
$83$	−514.019	−0.679771	−0.339885	+	0.940467i	$0.610388\pi$
$84$	0	0
$85$	−832.027	−1.06172
$86$	−322.772	−0.404714
$87$	0	0
$88$	−1048.38	−1.26997
$89$	527.889	0.628720	0.314360	−	0.949304i	$-0.398210\pi$
$89$	527.889	0.628720	0.314360	+	0.949304i	$0.398210\pi$
$90$	0	0
$91$	0	0
$92$	408.872	0.463346
$93$	0	0
$94$	−102.921	−0.112931
$95$	−1128.87	−1.21916
$96$	0	0
$97$	74.2755	0.0777478	0.0388739	−	0.999244i	$-0.487623\pi$
$97$	74.2755	0.0777478	0.0388739	+	0.999244i	$0.487623\pi$
$98$	−239.420	−0.246786
$99$	0	0
$100$	180.103	0.180103
$101$	−609.419	−0.600390	−0.300195	−	0.953878i	$-0.597052\pi$
$101$	−609.419	−0.600390	−0.300195	+	0.953878i	$0.597052\pi$
$102$	0	0
$103$	−32.4361	−0.0310293	−0.0155147	−	0.999880i	$-0.504939\pi$
$103$	−32.4361	−0.0310293	−0.0155147	+	0.999880i	$0.504939\pi$
$104$	0	0
$105$	0	0
$106$	1046.04	0.958498
$107$	1725.45	1.55893	0.779467	−	0.626444i	$-0.215490\pi$
$107$	1725.45	1.55893	0.779467	+	0.626444i	$0.215490\pi$
$108$	0	0
$109$	−1273.43	−1.11902	−0.559508	−	0.828825i	$-0.689010\pi$
$109$	−1273.43	−1.11902	−0.559508	+	0.828825i	$0.689010\pi$
$110$	740.244	0.641632
$111$	0	0
$112$	−305.911	−0.258088
$113$	−1114.73	−0.928006	−0.464003	−	0.885834i	$-0.653587\pi$
$113$	−1114.73	−0.928006	−0.464003	+	0.885834i	$0.653587\pi$
$114$	0	0
$115$	−695.406	−0.563886
$116$	80.3284	0.0642957
$117$	0	0
$118$	140.061	0.109268
$119$	1926.55	1.48409
$120$	0	0
$121$	1199.59	0.901272
$122$	−501.259	−0.371982
$123$	0	0
$124$	−1114.26	−0.806966
$125$	−1513.62	−1.08306
$126$	0	0
$127$	1174.01	0.820289	0.410144	−	0.912021i	$-0.365478\pi$
$127$	1174.01	0.820289	0.410144	+	0.912021i	$0.365478\pi$
$128$	1231.85	0.850637
$129$	0	0
$130$	0	0
$131$	1445.16	0.963851	0.481925	−	0.876212i	$-0.339938\pi$
$131$	1445.16	0.963851	0.481925	+	0.876212i	$0.339938\pi$
$132$	0	0
$133$	2613.89	1.70416
$134$	1170.65	0.754695
$135$	0	0
$136$	−1795.30	−1.13195
$137$	508.793	0.317293	0.158646	−	0.987335i	$-0.449287\pi$
$137$	508.793	0.317293	0.158646	+	0.987335i	$0.449287\pi$
$138$	0	0
$139$	−757.018	−0.461938	−0.230969	−	0.972961i	$-0.574190\pi$
$139$	−757.018	−0.461938	−0.230969	+	0.972961i	$0.574190\pi$
$140$	1226.62	0.740486
$141$	0	0
$142$	−403.695	−0.238573
$143$	0	0
$144$	0	0
$145$	−136.622	−0.0782470
$146$	1175.13	0.666125
$147$	0	0
$148$	−876.939	−0.487054
$149$	−3247.79	−1.78570	−0.892851	−	0.450352i	$-0.851298\pi$
$149$	−3247.79	−1.78570	−0.892851	+	0.450352i	$0.851298\pi$
$150$	0	0
$151$	−795.296	−0.428611	−0.214305	−	0.976767i	$-0.568749\pi$
$151$	−795.296	−0.428611	−0.214305	+	0.976767i	$0.568749\pi$
$152$	−2435.82	−1.29981
$153$	0	0
$154$	−1714.03	−0.896884
$155$	1895.13	0.982067
$156$	0	0
$157$	−65.2732	−0.0331807	−0.0165903	−	0.999862i	$-0.505281\pi$
$157$	−65.2732	−0.0331807	−0.0165903	+	0.999862i	$0.505281\pi$
$158$	−1869.19	−0.941169
$159$	0	0
$160$	−1811.56	−0.895104
$161$	1610.20	0.788210
$162$	0	0
$163$	1855.39	0.891568	0.445784	−	0.895140i	$-0.352925\pi$
$163$	1855.39	0.891568	0.445784	+	0.895140i	$0.352925\pi$
$164$	1509.00	0.718495
$165$	0	0
$166$	783.137	0.366164
$167$	−3532.54	−1.63686	−0.818432	−	0.574604i	$-0.805156\pi$
$167$	−3532.54	−1.63686	−0.818432	+	0.574604i	$0.805156\pi$
$168$	0	0
$169$	0	0
$170$	1267.64	0.571903
$171$	0	0
$172$	−1203.08	−0.533335
$173$	3178.36	1.39680	0.698400	−	0.715708i	$-0.253896\pi$
$173$	3178.36	1.39680	0.698400	+	0.715708i	$0.253896\pi$
$174$	0	0
$175$	709.275	0.306378
$176$	−688.111	−0.294706
$177$	0	0
$178$	−804.267	−0.338665
$179$	−2741.16	−1.14460	−0.572302	−	0.820043i	$-0.693950\pi$
$179$	−2741.16	−1.14460	−0.572302	+	0.820043i	$0.693950\pi$
$180$	0	0
$181$	3871.09	1.58970	0.794850	−	0.606806i	$-0.207550\pi$
$181$	3871.09	1.58970	0.794850	+	0.606806i	$0.207550\pi$
$182$	0	0
$183$	0	0
$184$	−1500.51	−0.601189
$185$	1491.49	0.592737
$186$	0	0
$187$	4333.54	1.69465
$188$	−383.621	−0.148822
$189$	0	0
$190$	1719.90	0.656708
$191$	928.291	0.351669	0.175834	−	0.984420i	$-0.443738\pi$
$191$	928.291	0.351669	0.175834	+	0.984420i	$0.443738\pi$
$192$	0	0
$193$	2261.51	0.843456	0.421728	−	0.906722i	$-0.361424\pi$
$193$	2261.51	0.843456	0.421728	+	0.906722i	$0.361424\pi$
$194$	−113.163	−0.0418795
$195$	0	0
$196$	−892.394	−0.325216
$197$	−2265.59	−0.819374	−0.409687	−	0.912226i	$-0.634362\pi$
$197$	−2265.59	−0.819374	−0.409687	+	0.912226i	$0.634362\pi$
$198$	0	0
$199$	−260.895	−0.0929366	−0.0464683	−	0.998920i	$-0.514797\pi$
$199$	−260.895	−0.0929366	−0.0464683	+	0.998920i	$0.514797\pi$
$200$	−660.955	−0.233683
$201$	0	0
$202$	928.483	0.323405
$203$	316.346	0.109375
$204$	0	0
$205$	−2566.49	−0.874399
$206$	49.4181	0.0167142
$207$	0	0
$208$	0	0
$209$	5879.63	1.94595
$210$	0	0
$211$	5851.22	1.90907	0.954537	−	0.298092i	$-0.0963502\pi$
$211$	5851.22	1.90907	0.954537	+	0.298092i	$0.0963502\pi$
$212$	3898.94	1.26312
$213$	0	0
$214$	−2628.82	−0.839732
$215$	2046.18	0.649062
$216$	0	0
$217$	−4388.15	−1.37275
$218$	1940.15	0.602767
$219$	0	0
$220$	2759.13	0.845547
$221$	0	0
$222$	0	0
$223$	3463.60	1.04009	0.520045	−	0.854139i	$-0.325915\pi$
$223$	3463.60	1.04009	0.520045	+	0.854139i	$0.325915\pi$
$224$	4194.65	1.25119
$225$	0	0
$226$	1698.35	0.499878
$227$	5329.15	1.55819	0.779093	−	0.626908i	$-0.215680\pi$
$227$	5329.15	1.55819	0.779093	+	0.626908i	$0.215680\pi$
$228$	0	0
$229$	−4773.95	−1.37761	−0.688803	−	0.724949i	$-0.741863\pi$
$229$	−4773.95	−1.37761	−0.688803	+	0.724949i	$0.741863\pi$
$230$	1059.49	0.303742
$231$	0	0
$232$	−294.795	−0.0834233
$233$	4813.78	1.35348	0.676741	−	0.736221i	$-0.263392\pi$
$233$	4813.78	1.35348	0.676741	+	0.736221i	$0.263392\pi$
$234$	0	0
$235$	652.459	0.181114
$236$	522.052	0.143995
$237$	0	0
$238$	−2935.20	−0.799416
$239$	1683.19	0.455549	0.227775	−	0.973714i	$-0.426855\pi$
$239$	1683.19	0.455549	0.227775	+	0.973714i	$0.426855\pi$
$240$	0	0
$241$	664.861	0.177707	0.0888537	−	0.996045i	$-0.471680\pi$
$241$	664.861	0.177707	0.0888537	+	0.996045i	$0.471680\pi$
$242$	−1827.65	−0.485477
$243$	0	0
$244$	−1868.35	−0.490201
$245$	1517.77	0.395784
$246$	0	0
$247$	0	0
$248$	4089.20	1.04703
$249$	0	0
$250$	2306.08	0.583398
$251$	−2142.04	−0.538662	−0.269331	−	0.963048i	$-0.586802\pi$
$251$	−2142.04	−0.538662	−0.269331	+	0.963048i	$0.586802\pi$
$252$	0	0
$253$	3621.96	0.900042
$254$	−1788.67	−0.441855
$255$	0	0
$256$	−3287.46	−0.802603
$257$	−3152.62	−0.765194	−0.382597	−	0.923915i	$-0.624970\pi$
$257$	−3152.62	−0.765194	−0.382597	+	0.923915i	$0.624970\pi$
$258$	0	0
$259$	−3453.52	−0.828539
$260$	0	0
$261$	0	0
$262$	−2201.79	−0.519186
$263$	2167.71	0.508238	0.254119	−	0.967173i	$-0.418214\pi$
$263$	2167.71	0.508238	0.254119	+	0.967173i	$0.418214\pi$
$264$	0	0
$265$	−6631.28	−1.53719
$266$	−3982.40	−0.917958
$267$	0	0
$268$	4363.40	0.994542
$269$	3248.98	0.736409	0.368204	−	0.929745i	$-0.379973\pi$
$269$	3248.98	0.736409	0.368204	+	0.929745i	$0.379973\pi$
$270$	0	0
$271$	4897.70	1.09784	0.548919	−	0.835876i	$-0.315040\pi$
$271$	4897.70	1.09784	0.548919	+	0.835876i	$0.315040\pi$
$272$	−1178.36	−0.262679
$273$	0	0
$274$	−775.175	−0.170912
$275$	1595.43	0.349847
$276$	0	0
$277$	2900.33	0.629111	0.314555	−	0.949239i	$-0.398145\pi$
$277$	2900.33	0.629111	0.314555	+	0.949239i	$0.398145\pi$
$278$	1153.36	0.248827
$279$	0	0
$280$	−4501.52	−0.960776
$281$	4396.53	0.933364	0.466682	−	0.884425i	$-0.345449\pi$
$281$	4396.53	0.933364	0.466682	+	0.884425i	$0.345449\pi$
$282$	0	0
$283$	−559.151	−0.117449	−0.0587245	−	0.998274i	$-0.518703\pi$
$283$	−559.151	−0.117449	−0.0587245	+	0.998274i	$0.518703\pi$
$284$	−1504.70	−0.314393
$285$	0	0
$286$	0	0
$287$	5942.69	1.22225
$288$	0	0
$289$	2508.02	0.510487
$290$	208.151	0.0421484
$291$	0	0
$292$	4380.08	0.877825
$293$	−6918.65	−1.37950	−0.689748	−	0.724050i	$-0.742278\pi$
$293$	−6918.65	−1.37950	−0.689748	+	0.724050i	$0.742278\pi$
$294$	0	0
$295$	−887.901	−0.175239
$296$	3218.25	0.631949
$297$	0	0
$298$	4948.19	0.961883
$299$	0	0
$300$	0	0
$301$	−4737.90	−0.907270
$302$	1211.68	0.230875
$303$	0	0
$304$	−1598.77	−0.301631
$305$	3177.67	0.596567
$306$	0	0
$307$	8980.94	1.66961	0.834803	−	0.550548i	$-0.185581\pi$
$307$	8980.94	1.66961	0.834803	+	0.550548i	$0.185581\pi$
$308$	−6388.73	−1.18192
$309$	0	0
$310$	−2887.33	−0.528998
$311$	7943.13	1.44827	0.724137	−	0.689656i	$-0.242238\pi$
$311$	7943.13	1.44827	0.724137	+	0.689656i	$0.242238\pi$
$312$	0	0
$313$	−5059.57	−0.913686	−0.456843	−	0.889547i	$-0.651020\pi$
$313$	−5059.57	−0.913686	−0.456843	+	0.889547i	$0.651020\pi$
$314$	99.4473	0.0178730
$315$	0	0
$316$	−6967.07	−1.24028
$317$	−8702.12	−1.54183	−0.770914	−	0.636939i	$-0.780200\pi$
$317$	−8702.12	−1.54183	−0.770914	+	0.636939i	$0.780200\pi$
$318$	0	0
$319$	711.582	0.124893
$320$	1703.10	0.297519
$321$	0	0
$322$	−2453.23	−0.424576
$323$	10068.6	1.73447
$324$	0	0
$325$	0	0
$326$	−2826.79	−0.480250
$327$	0	0
$328$	−5537.84	−0.932244
$329$	−1510.76	−0.253164
$330$	0	0
$331$	1737.65	0.288549	0.144275	−	0.989538i	$-0.453915\pi$
$331$	1737.65	0.288549	0.144275	+	0.989538i	$0.453915\pi$
$332$	2919.00	0.482533
$333$	0	0
$334$	5382.02	0.881710
$335$	−7421.23	−1.21034
$336$	0	0
$337$	2917.47	0.471586	0.235793	−	0.971803i	$-0.424231\pi$
$337$	2917.47	0.471586	0.235793	+	0.971803i	$0.424231\pi$
$338$	0	0
$339$	0	0
$340$	4724.90	0.753658
$341$	−9870.61	−1.56752
$342$	0	0
$343$	4156.44	0.654305
$344$	4415.13	0.692000
$345$	0	0
$346$	−4842.41	−0.752397
$347$	−5081.23	−0.786095	−0.393047	−	0.919518i	$-0.628579\pi$
$347$	−5081.23	−0.786095	−0.393047	+	0.919518i	$0.628579\pi$
$348$	0	0
$349$	−4266.14	−0.654330	−0.327165	−	0.944967i	$-0.606093\pi$
$349$	−4266.14	−0.654330	−0.327165	+	0.944967i	$0.606093\pi$
$350$	−1080.62	−0.165033
$351$	0	0
$352$	9435.38	1.42871
$353$	3264.49	0.492213	0.246106	−	0.969243i	$-0.420849\pi$
$353$	3264.49	0.492213	0.246106	+	0.969243i	$0.420849\pi$
$354$	0	0
$355$	2559.18	0.382612
$356$	−2997.76	−0.446295
$357$	0	0
$358$	4176.31	0.616550
$359$	−4416.98	−0.649357	−0.324679	−	0.945824i	$-0.605256\pi$
$359$	−4416.98	−0.649357	−0.324679	+	0.945824i	$0.605256\pi$
$360$	0	0
$361$	6801.87	0.991670
$362$	−5897.82	−0.856305
$363$	0	0
$364$	0	0
$365$	−7449.59	−1.06830
$366$	0	0
$367$	−1740.22	−0.247516	−0.123758	−	0.992312i	$-0.539495\pi$
$367$	−1740.22	−0.247516	−0.123758	+	0.992312i	$0.539495\pi$
$368$	−984.872	−0.139511
$369$	0	0
$370$	−2272.37	−0.319283
$371$	15354.7	2.14872
$372$	0	0
$373$	1176.28	0.163285	0.0816427	−	0.996662i	$-0.473983\pi$
$373$	1176.28	0.163285	0.0816427	+	0.996662i	$0.473983\pi$
$374$	−6602.39	−0.912838
$375$	0	0
$376$	1407.84	0.193095
$377$	0	0
$378$	0	0
$379$	7135.54	0.967092	0.483546	−	0.875319i	$-0.339349\pi$
$379$	7135.54	0.967092	0.483546	+	0.875319i	$0.339349\pi$
$380$	6410.62	0.865415
$381$	0	0
$382$	−1414.30	−0.189429
$383$	1942.87	0.259207	0.129603	−	0.991566i	$-0.458630\pi$
$383$	1942.87	0.259207	0.129603	+	0.991566i	$0.458630\pi$
$384$	0	0
$385$	10865.9	1.43838
$386$	−3445.53	−0.454334
$387$	0	0
$388$	−421.794	−0.0551891
$389$	7545.92	0.983531	0.491766	−	0.870728i	$-0.336352\pi$
$389$	7545.92	0.983531	0.491766	+	0.870728i	$0.336352\pi$
$390$	0	0
$391$	6202.47	0.802231
$392$	3274.97	0.421967
$393$	0	0
$394$	3451.75	0.441362
$395$	11849.5	1.50940
$396$	0	0
$397$	415.922	0.0525806	0.0262903	−	0.999654i	$-0.491631\pi$
$397$	415.922	0.0525806	0.0262903	+	0.999654i	$0.491631\pi$
$398$	397.489	0.0500611
$399$	0	0
$400$	−433.824	−0.0542280
$401$	958.178	0.119324	0.0596622	−	0.998219i	$-0.480998\pi$
$401$	958.178	0.119324	0.0596622	+	0.998219i	$0.480998\pi$
$402$	0	0
$403$	0	0
$404$	3460.75	0.426185
$405$	0	0
$406$	−481.970	−0.0589157
$407$	−7768.29	−0.946093
$408$	0	0
$409$	4284.83	0.518022	0.259011	−	0.965874i	$-0.416603\pi$
$409$	4284.83	0.518022	0.259011	+	0.965874i	$0.416603\pi$
$410$	3910.20	0.471002
$411$	0	0
$412$	184.197	0.0220261
$413$	2055.92	0.244953
$414$	0	0
$415$	−4964.61	−0.587236
$416$	0	0
$417$	0	0
$418$	−8957.95	−1.04820
$419$	9949.01	1.16000	0.580001	−	0.814616i	$-0.303052\pi$
$419$	9949.01	1.16000	0.580001	+	0.814616i	$0.303052\pi$
$420$	0	0
$421$	−377.250	−0.0436724	−0.0218362	−	0.999762i	$-0.506951\pi$
$421$	−377.250	−0.0436724	−0.0218362	+	0.999762i	$0.506951\pi$
$422$	−8914.66	−1.02834
$423$	0	0
$424$	−14308.6	−1.63889
$425$	2732.11	0.311828
$426$	0	0
$427$	−7357.86	−0.833892
$428$	−9798.47	−1.10661
$429$	0	0
$430$	−3117.47	−0.349622
$431$	−2437.13	−0.272373	−0.136186	−	0.990683i	$-0.543485\pi$
$431$	−2437.13	−0.272373	−0.136186	+	0.990683i	$0.543485\pi$
$432$	0	0
$433$	11215.1	1.24471	0.622357	−	0.782733i	$-0.286175\pi$
$433$	11215.1	1.24471	0.622357	+	0.782733i	$0.286175\pi$
$434$	6685.58	0.739443
$435$	0	0
$436$	7231.55	0.794331
$437$	8415.34	0.921191
$438$	0	0
$439$	−1835.19	−0.199519	−0.0997596	−	0.995012i	$-0.531807\pi$
$439$	−1835.19	−0.199519	−0.0997596	+	0.995012i	$0.531807\pi$
$440$	−10125.6	−1.09709
$441$	0	0
$442$	0	0
$443$	−11610.1	−1.24518	−0.622588	−	0.782550i	$-0.713919\pi$
$443$	−11610.1	−1.24518	−0.622588	+	0.782550i	$0.713919\pi$
$444$	0	0
$445$	5098.56	0.543135
$446$	−5276.99	−0.560253
$447$	0	0
$448$	−3943.50	−0.415877
$449$	−14087.0	−1.48064	−0.740319	−	0.672255i	$-0.765326\pi$
$449$	−14087.0	−1.48064	−0.740319	+	0.672255i	$0.765326\pi$
$450$	0	0
$451$	13367.4	1.39566
$452$	6330.29	0.658743
$453$	0	0
$454$	−8119.26	−0.839330
$455$	0	0
$456$	0	0
$457$	−2375.01	−0.243103	−0.121552	−	0.992585i	$-0.538787\pi$
$457$	−2375.01	−0.243103	−0.121552	+	0.992585i	$0.538787\pi$
$458$	7273.38	0.742058
$459$	0	0
$460$	3949.05	0.400273
$461$	−6372.06	−0.643766	−0.321883	−	0.946779i	$-0.604316\pi$
$461$	−6372.06	−0.643766	−0.321883	+	0.946779i	$0.604316\pi$
$462$	0	0
$463$	63.4732	0.00637117	0.00318558	−	0.999995i	$-0.498986\pi$
$463$	63.4732	0.00637117	0.00318558	+	0.999995i	$0.498986\pi$
$464$	−193.491	−0.0193591
$465$	0	0
$466$	−7334.06	−0.729064
$467$	7855.78	0.778420	0.389210	−	0.921149i	$-0.372748\pi$
$467$	7855.78	0.778420	0.389210	+	0.921149i	$0.372748\pi$
$468$	0	0
$469$	17183.8	1.69184
$470$	−994.058	−0.0975584
$471$	0	0
$472$	−1915.86	−0.186832
$473$	−10657.3	−1.03599
$474$	0	0
$475$	3706.85	0.358068
$476$	−10940.4	−1.05348
$477$	0	0
$478$	−2564.43	−0.245385
$479$	−13033.3	−1.24323	−0.621613	−	0.783324i	$-0.713523\pi$
$479$	−13033.3	−1.24323	−0.621613	+	0.783324i	$0.713523\pi$
$480$	0	0
$481$	0	0
$482$	−1012.95	−0.0957235
$483$	0	0
$484$	−6812.23	−0.639766
$485$	717.384	0.0671643
$486$	0	0
$487$	69.8976	0.00650383	0.00325191	−	0.999995i	$-0.498965\pi$
$487$	69.8976	0.00650383	0.00325191	+	0.999995i	$0.498965\pi$
$488$	6856.60	0.636033
$489$	0	0
$490$	−2312.41	−0.213192
$491$	2625.66	0.241333	0.120667	−	0.992693i	$-0.461497\pi$
$491$	2625.66	0.241333	0.120667	+	0.992693i	$0.461497\pi$
$492$	0	0
$493$	1218.56	0.111321
$494$	0	0
$495$	0	0
$496$	2683.99	0.242973
$497$	−5925.75	−0.534821
$498$	0	0
$499$	−7631.34	−0.684621	−0.342310	−	0.939587i	$-0.611209\pi$
$499$	−7631.34	−0.684621	−0.342310	+	0.939587i	$0.611209\pi$
$500$	8595.51	0.768806
$501$	0	0
$502$	3263.51	0.290154
$503$	4320.14	0.382953	0.191477	−	0.981497i	$-0.438672\pi$
$503$	4320.14	0.382953	0.191477	+	0.981497i	$0.438672\pi$
$504$	0	0
$505$	−5886.01	−0.518662
$506$	−5518.26	−0.484815
$507$	0	0
$508$	−6666.95	−0.582280
$509$	−12450.7	−1.08422	−0.542109	−	0.840308i	$-0.682374\pi$
$509$	−12450.7	−1.08422	−0.542109	+	0.840308i	$0.682374\pi$
$510$	0	0
$511$	17249.4	1.49329
$512$	−4846.21	−0.418309
$513$	0	0
$514$	4803.18	0.412178
$515$	−313.281	−0.0268054
$516$	0	0
$517$	−3398.28	−0.289083
$518$	5261.63	0.446299
$519$	0	0
$520$	0	0
$521$	−14373.1	−1.20863	−0.604314	−	0.796746i	$-0.706553\pi$
$521$	−14373.1	−1.20863	−0.604314	+	0.796746i	$0.706553\pi$
$522$	0	0
$523$	−16946.7	−1.41688	−0.708439	−	0.705772i	$-0.750600\pi$
$523$	−16946.7	−1.41688	−0.708439	+	0.705772i	$0.750600\pi$
$524$	−8206.76	−0.684187
$525$	0	0
$526$	−3302.62	−0.273767
$527$	−16903.0	−1.39717
$528$	0	0
$529$	−6983.00	−0.573929
$530$	10103.1	0.828022
$531$	0	0
$532$	−14843.7	−1.20969
$533$	0	0
$534$	0	0
$535$	16665.1	1.34672
$536$	−16013.1	−1.29041
$537$	0	0
$538$	−4950.00	−0.396673
$539$	−7905.20	−0.631727
$540$	0	0
$541$	−815.667	−0.0648212	−0.0324106	−	0.999475i	$-0.510318\pi$
$541$	−815.667	−0.0648212	−0.0324106	+	0.999475i	$0.510318\pi$
$542$	−7461.92	−0.591359
$543$	0	0
$544$	16157.7	1.27345
$545$	−12299.3	−0.966689
$546$	0	0
$547$	−17971.4	−1.40476	−0.702378	−	0.711804i	$-0.747878\pi$
$547$	−17971.4	−1.40476	−0.702378	+	0.711804i	$0.747878\pi$
$548$	−2889.32	−0.225230
$549$	0	0
$550$	−2430.72	−0.188448
$551$	1653.31	0.127828
$552$	0	0
$553$	−27437.4	−2.10987
$554$	−4418.81	−0.338876
$555$	0	0
$556$	4298.94	0.327906
$557$	6760.83	0.514301	0.257151	−	0.966371i	$-0.417216\pi$
$557$	6760.83	0.514301	0.257151	+	0.966371i	$0.417216\pi$
$558$	0	0
$559$	0	0
$560$	−2954.62	−0.222956
$561$	0	0
$562$	−6698.36	−0.502764
$563$	12962.7	0.970359	0.485179	−	0.874415i	$-0.338754\pi$
$563$	12962.7	0.970359	0.485179	+	0.874415i	$0.338754\pi$
$564$	0	0
$565$	−10766.5	−0.801681
$566$	851.898	0.0632649
$567$	0	0
$568$	5522.05	0.407923
$569$	164.757	0.0121388	0.00606938	−	0.999982i	$-0.498068\pi$
$569$	164.757	0.0121388	0.00606938	+	0.999982i	$0.498068\pi$
$570$	0	0
$571$	5216.55	0.382322	0.191161	−	0.981559i	$-0.438775\pi$
$571$	5216.55	0.382322	0.191161	+	0.981559i	$0.438775\pi$
$572$	0	0
$573$	0	0
$574$	−9054.01	−0.658375
$575$	2283.49	0.165614
$576$	0	0
$577$	−12753.3	−0.920151	−0.460076	−	0.887880i	$-0.652178\pi$
$577$	−12753.3	−0.920151	−0.460076	+	0.887880i	$0.652178\pi$
$578$	−3821.11	−0.274978
$579$	0	0
$580$	775.844	0.0555434
$581$	11495.5	0.820849
$582$	0	0
$583$	34538.5	2.45358
$584$	−16074.3	−1.13897
$585$	0	0
$586$	10540.9	0.743076
$587$	1575.38	0.110771	0.0553857	−	0.998465i	$-0.482361\pi$
$587$	1575.38	0.110771	0.0553857	+	0.998465i	$0.482361\pi$
$588$	0	0
$589$	−22933.6	−1.60435
$590$	1352.77	0.0943941
$591$	0	0
$592$	2112.33	0.146649
$593$	3845.95	0.266331	0.133165	−	0.991094i	$-0.457486\pi$
$593$	3845.95	0.266331	0.133165	+	0.991094i	$0.457486\pi$
$594$	0	0
$595$	18607.4	1.28207
$596$	18443.5	1.26758
$597$	0	0
$598$	0	0
$599$	−6107.20	−0.416583	−0.208292	−	0.978067i	$-0.566790\pi$
$599$	−6107.20	−0.416583	−0.208292	+	0.978067i	$0.566790\pi$
$600$	0	0
$601$	9638.90	0.654208	0.327104	−	0.944988i	$-0.393927\pi$
$601$	9638.90	0.654208	0.327104	+	0.944988i	$0.393927\pi$
$602$	7218.46	0.488708
$603$	0	0
$604$	4516.31	0.304248
$605$	11586.2	0.778586
$606$	0	0
$607$	11821.7	0.790489	0.395244	−	0.918576i	$-0.370660\pi$
$607$	11821.7	0.790489	0.395244	+	0.918576i	$0.370660\pi$
$608$	21922.4	1.46228
$609$	0	0
$610$	−4841.36	−0.321346
$611$	0	0
$612$	0	0
$613$	−8107.86	−0.534214	−0.267107	−	0.963667i	$-0.586068\pi$
$613$	−8107.86	−0.534214	−0.267107	+	0.963667i	$0.586068\pi$
$614$	−13683.0	−0.899347
$615$	0	0
$616$	23445.8	1.53354
$617$	27647.6	1.80397	0.901985	−	0.431768i	$-0.142110\pi$
$617$	27647.6	1.80397	0.901985	+	0.431768i	$0.142110\pi$
$618$	0	0
$619$	−29181.9	−1.89486	−0.947432	−	0.319956i	$-0.896332\pi$
$619$	−29181.9	−1.89486	−0.947432	+	0.319956i	$0.896332\pi$
$620$	−10762.0	−0.697118
$621$	0	0
$622$	−12101.8	−0.780125
$623$	−11805.7	−0.759204
$624$	0	0
$625$	−10654.8	−0.681905
$626$	7708.53	0.492165
$627$	0	0
$628$	370.672	0.0235532
$629$	−13302.9	−0.843277
$630$	0	0
$631$	2209.34	0.139386	0.0696928	−	0.997569i	$-0.477798\pi$
$631$	2209.34	0.139386	0.0696928	+	0.997569i	$0.477798\pi$
$632$	25568.2	1.60926
$633$	0	0
$634$	13258.2	0.830518
$635$	11339.1	0.708627
$636$	0	0
$637$	0	0
$638$	−1084.13	−0.0672748
$639$	0	0
$640$	11897.8	0.734844
$641$	18256.0	1.12491	0.562455	−	0.826828i	$-0.309857\pi$
$641$	18256.0	1.12491	0.562455	+	0.826828i	$0.309857\pi$
$642$	0	0
$643$	−1281.61	−0.0786033	−0.0393016	−	0.999227i	$-0.512513\pi$
$643$	−1281.61	−0.0786033	−0.0393016	+	0.999227i	$0.512513\pi$
$644$	−9143.99	−0.559509
$645$	0	0
$646$	−15340.1	−0.934287
$647$	−16393.2	−0.996107	−0.498054	−	0.867146i	$-0.665952\pi$
$647$	−16393.2	−0.996107	−0.498054	+	0.867146i	$0.665952\pi$
$648$	0	0
$649$	4624.56	0.279707
$650$	0	0
$651$	0	0
$652$	−10536.4	−0.632878
$653$	−16759.2	−1.00434	−0.502172	−	0.864768i	$-0.667466\pi$
$653$	−16759.2	−1.00434	−0.502172	+	0.864768i	$0.667466\pi$
$654$	0	0
$655$	13958.0	0.832646
$656$	−3634.81	−0.216335
$657$	0	0
$658$	2301.73	0.136369
$659$	29659.3	1.75320	0.876601	−	0.481217i	$-0.159805\pi$
$659$	29659.3	1.75320	0.876601	+	0.481217i	$0.159805\pi$
$660$	0	0
$661$	10386.0	0.611147	0.305573	−	0.952169i	$-0.401152\pi$
$661$	10386.0	0.611147	0.305573	+	0.952169i	$0.401152\pi$
$662$	−2647.40	−0.155429
$663$	0	0
$664$	−10712.4	−0.626084
$665$	25246.0	1.47218
$666$	0	0
$667$	1018.47	0.0591232
$668$	20060.5	1.16192
$669$	0	0
$670$	11306.7	0.651962
$671$	−16550.6	−0.952206
$672$	0	0
$673$	−19449.6	−1.11400	−0.557002	−	0.830511i	$-0.688049\pi$
$673$	−19449.6	−1.11400	−0.557002	+	0.830511i	$0.688049\pi$
$674$	−4444.92	−0.254024
$675$	0	0
$676$	0	0
$677$	6629.48	0.376354	0.188177	−	0.982135i	$-0.439742\pi$
$677$	6629.48	0.376354	0.188177	+	0.982135i	$0.439742\pi$
$678$	0	0
$679$	−1661.09	−0.0938835
$680$	−17339.8	−0.977867
$681$	0	0
$682$	15038.4	0.844356
$683$	−2526.40	−0.141537	−0.0707687	−	0.997493i	$-0.522545\pi$
$683$	−2526.40	−0.141537	−0.0707687	+	0.997493i	$0.522545\pi$
$684$	0	0
$685$	4914.13	0.274101
$686$	−6332.57	−0.352447
$687$	0	0
$688$	2897.91	0.160584
$689$	0	0
$690$	0	0
$691$	−3808.76	−0.209685	−0.104842	−	0.994489i	$-0.533434\pi$
$691$	−3808.76	−0.209685	−0.104842	+	0.994489i	$0.533434\pi$
$692$	−18049.2	−0.991515
$693$	0	0
$694$	7741.54	0.423436
$695$	−7311.59	−0.399056
$696$	0	0
$697$	22891.1	1.24399
$698$	6499.69	0.352460
$699$	0	0
$700$	−4027.81	−0.217482
$701$	−33617.8	−1.81131	−0.905655	−	0.424015i	$-0.860620\pi$
$701$	−33617.8	−1.81131	−0.905655	+	0.424015i	$0.860620\pi$
$702$	0	0
$703$	−18049.0	−0.968323
$704$	−8870.43	−0.474882
$705$	0	0
$706$	−4973.63	−0.265134
$707$	13629.0	0.724994
$708$	0	0
$709$	−26606.5	−1.40935	−0.704675	−	0.709530i	$-0.748907\pi$
$709$	−26606.5	−1.40935	−0.704675	+	0.709530i	$0.748907\pi$
$710$	−3899.05	−0.206097
$711$	0	0
$712$	11001.4	0.579066
$713$	−14127.5	−0.742046
$714$	0	0
$715$	0	0
$716$	15566.5	0.812494
$717$	0	0
$718$	6729.51	0.349781
$719$	16539.3	0.857877	0.428939	−	0.903334i	$-0.358888\pi$
$719$	16539.3	0.857877	0.428939	+	0.903334i	$0.358888\pi$
$720$	0	0
$721$	725.398	0.0374691
$722$	−10363.0	−0.534171
$723$	0	0
$724$	−21983.1	−1.12844
$725$	448.622	0.0229812
$726$	0	0
$727$	−12757.5	−0.650823	−0.325411	−	0.945573i	$-0.605503\pi$
$727$	−12757.5	−0.650823	−0.325411	+	0.945573i	$0.605503\pi$
$728$	0	0
$729$	0	0
$730$	11349.9	0.575448
$731$	−18250.3	−0.923408
$732$	0	0
$733$	20523.4	1.03417	0.517087	−	0.855933i	$-0.327016\pi$
$733$	20523.4	1.03417	0.517087	+	0.855933i	$0.327016\pi$
$734$	2651.31	0.133327
$735$	0	0
$736$	13504.6	0.676338
$737$	38652.9	1.93188
$738$	0	0
$739$	−7462.66	−0.371473	−0.185736	−	0.982600i	$-0.559467\pi$
$739$	−7462.66	−0.371473	−0.185736	+	0.982600i	$0.559467\pi$
$740$	−8469.84	−0.420753
$741$	0	0
$742$	−23393.7	−1.15742
$743$	−18847.8	−0.930632	−0.465316	−	0.885145i	$-0.654059\pi$
$743$	−18847.8	−0.930632	−0.465316	+	0.885145i	$0.654059\pi$
$744$	0	0
$745$	−31368.5	−1.54262
$746$	−1792.13	−0.0879549
$747$	0	0
$748$	−24609.2	−1.20294
$749$	−38587.9	−1.88247
$750$	0	0
$751$	−20832.6	−1.01224	−0.506119	−	0.862464i	$-0.668920\pi$
$751$	−20832.6	−1.01224	−0.506119	+	0.862464i	$0.668920\pi$
$752$	924.050	0.0448093
$753$	0	0
$754$	0	0
$755$	−7681.29	−0.370266
$756$	0	0
$757$	−20860.9	−1.00159	−0.500795	−	0.865566i	$-0.666959\pi$
$757$	−20860.9	−1.00159	−0.500795	+	0.865566i	$0.666959\pi$
$758$	−10871.4	−0.520932
$759$	0	0
$760$	−23526.1	−1.12287
$761$	4464.86	0.212682	0.106341	−	0.994330i	$-0.466086\pi$
$761$	4464.86	0.212682	0.106341	+	0.994330i	$0.466086\pi$
$762$	0	0
$763$	28479.0	1.35126
$764$	−5271.56	−0.249631
$765$	0	0
$766$	−2960.08	−0.139624
$767$	0	0
$768$	0	0
$769$	23797.7	1.11595	0.557977	−	0.829857i	$-0.311578\pi$
$769$	23797.7	1.11595	0.557977	+	0.829857i	$0.311578\pi$
$770$	−16554.8	−0.774795
$771$	0	0
$772$	−12842.6	−0.598725
$773$	−29418.7	−1.36885	−0.684423	−	0.729085i	$-0.739946\pi$
$773$	−29418.7	−1.36885	−0.684423	+	0.729085i	$0.739946\pi$
$774$	0	0
$775$	−6222.99	−0.288434
$776$	1547.93	0.0716075
$777$	0	0
$778$	−11496.6	−0.529787
$779$	31058.0	1.42846
$780$	0	0
$781$	−13329.3	−0.610703
$782$	−9449.80	−0.432128
$783$	0	0
$784$	2149.56	0.0979208
$785$	−630.435	−0.0286640
$786$	0	0
$787$	23896.0	1.08234	0.541170	−	0.840913i	$-0.317982\pi$
$787$	23896.0	1.08234	0.541170	+	0.840913i	$0.317982\pi$
$788$	12865.8	0.581630
$789$	0	0
$790$	−18053.4	−0.813052
$791$	24929.7	1.12060
$792$	0	0
$793$	0	0
$794$	−633.680	−0.0283230
$795$	0	0
$796$	1481.57	0.0659708
$797$	1034.67	0.0459847	0.0229923	−	0.999736i	$-0.492681\pi$
$797$	1034.67	0.0459847	0.0229923	+	0.999736i	$0.492681\pi$
$798$	0	0
$799$	−5819.42	−0.257667
$800$	5948.59	0.262893
$801$	0	0
$802$	−1459.84	−0.0642751
$803$	38800.6	1.70516
$804$	0	0
$805$	15552.0	0.680914
$806$	0	0
$807$	0	0
$808$	−12700.5	−0.552973
$809$	−29557.9	−1.28455	−0.642275	−	0.766474i	$-0.722009\pi$
$809$	−29557.9	−1.28455	−0.642275	+	0.766474i	$0.722009\pi$
$810$	0	0
$811$	−30268.9	−1.31058	−0.655292	−	0.755376i	$-0.727454\pi$
$811$	−30268.9	−1.31058	−0.655292	+	0.755376i	$0.727454\pi$
$812$	−1796.46	−0.0776396
$813$	0	0
$814$	11835.4	0.509621
$815$	17920.2	0.770203
$816$	0	0
$817$	−24761.5	−1.06034
$818$	−6528.17	−0.279037
$819$	0	0
$820$	14574.6	0.620690
$821$	39292.7	1.67031	0.835156	−	0.550013i	$-0.185377\pi$
$821$	39292.7	1.67031	0.835156	+	0.550013i	$0.185377\pi$
$822$	0	0
$823$	−25988.8	−1.10074	−0.550372	−	0.834920i	$-0.685514\pi$
$823$	−25988.8	−1.10074	−0.550372	+	0.834920i	$0.685514\pi$
$824$	−675.980	−0.0285787
$825$	0	0
$826$	−3132.31	−0.131946
$827$	19585.1	0.823508	0.411754	−	0.911295i	$-0.364916\pi$
$827$	19585.1	0.823508	0.411754	+	0.911295i	$0.364916\pi$
$828$	0	0
$829$	666.472	0.0279222	0.0139611	−	0.999903i	$-0.495556\pi$
$829$	666.472	0.0279222	0.0139611	+	0.999903i	$0.495556\pi$
$830$	7563.86	0.316320
$831$	0	0
$832$	0	0
$833$	−13537.3	−0.563075
$834$	0	0
$835$	−34118.7	−1.41404
$836$	−33389.1	−1.38133
$837$	0	0
$838$	−15157.9	−0.624845
$839$	36379.7	1.49698	0.748490	−	0.663146i	$-0.230779\pi$
$839$	36379.7	1.49698	0.748490	+	0.663146i	$0.230779\pi$
$840$	0	0
$841$	−24188.9	−0.991796
$842$	574.762	0.0235245
$843$	0	0
$844$	−33227.8	−1.35515
$845$	0	0
$846$	0	0
$847$	−26827.6	−1.08832
$848$	−9391.60	−0.380317
$849$	0	0
$850$	−4162.52	−0.167969
$851$	−11118.5	−0.447871
$852$	0	0
$853$	−39951.4	−1.60364	−0.801822	−	0.597563i	$-0.796136\pi$
$853$	−39951.4	−1.60364	−0.801822	+	0.597563i	$0.796136\pi$
$854$	11210.1	0.449183
$855$	0	0
$856$	35959.1	1.43581
$857$	−17226.4	−0.686629	−0.343314	−	0.939221i	$-0.611550\pi$
$857$	−17226.4	−0.686629	−0.343314	+	0.939221i	$0.611550\pi$
$858$	0	0
$859$	−33392.3	−1.32634	−0.663172	−	0.748467i	$-0.730790\pi$
$859$	−33392.3	−1.32634	−0.663172	+	0.748467i	$0.730790\pi$
$860$	−11619.8	−0.460735
$861$	0	0
$862$	3713.11	0.146716
$863$	−17381.9	−0.685614	−0.342807	−	0.939406i	$-0.611378\pi$
$863$	−17381.9	−0.685614	−0.342807	+	0.939406i	$0.611378\pi$
$864$	0	0
$865$	30697.9	1.20666
$866$	−17086.8	−0.670476
$867$	0	0
$868$	24919.3	0.974443
$869$	−61717.2	−2.40922
$870$	0	0
$871$	0	0
$872$	−26538.8	−1.03064
$873$	0	0
$874$	−12821.2	−0.496207
$875$	33850.5	1.30783
$876$	0	0
$877$	−14335.6	−0.551970	−0.275985	−	0.961162i	$-0.589004\pi$
$877$	−14335.6	−0.551970	−0.275985	+	0.961162i	$0.589004\pi$
$878$	2796.02	0.107473
$879$	0	0
$880$	−6646.06	−0.254589
$881$	−5436.53	−0.207901	−0.103951	−	0.994582i	$-0.533148\pi$
$881$	−5436.53	−0.207901	−0.103951	+	0.994582i	$0.533148\pi$
$882$	0	0
$883$	21185.9	0.807430	0.403715	−	0.914885i	$-0.367719\pi$
$883$	21185.9	0.807430	0.403715	+	0.914885i	$0.367719\pi$
$884$	0	0
$885$	0	0
$886$	17688.6	0.670724
$887$	12661.4	0.479287	0.239644	−	0.970861i	$-0.422969\pi$
$887$	12661.4	0.479287	0.239644	+	0.970861i	$0.422969\pi$
$888$	0	0
$889$	−26255.5	−0.990531
$890$	−7767.94	−0.292564
$891$	0	0
$892$	−19669.0	−0.738305
$893$	−7895.63	−0.295876
$894$	0	0
$895$	−26475.3	−0.988794
$896$	−27549.1	−1.02718
$897$	0	0
$898$	21462.3	0.797558
$899$	−2775.54	−0.102969
$900$	0	0
$901$	59145.8	2.18694
$902$	−20365.9	−0.751786
$903$	0	0
$904$	−23231.3	−0.854715
$905$	37388.6	1.37330
$906$	0	0
$907$	3545.27	0.129789	0.0648946	−	0.997892i	$-0.479329\pi$
$907$	3545.27	0.129789	0.0648946	+	0.997892i	$0.479329\pi$
$908$	−30263.1	−1.10607
$909$	0	0
$910$	0	0
$911$	3913.88	0.142341	0.0711706	−	0.997464i	$-0.477327\pi$
$911$	3913.88	0.142341	0.0711706	+	0.997464i	$0.477327\pi$
$912$	0	0
$913$	25857.7	0.937313
$914$	3618.46	0.130950
$915$	0	0
$916$	27110.2	0.977890
$917$	−32319.5	−1.16389
$918$	0	0
$919$	6917.79	0.248310	0.124155	−	0.992263i	$-0.460378\pi$
$919$	6917.79	0.248310	0.124155	+	0.992263i	$0.460378\pi$
$920$	−14492.5	−0.519352
$921$	0	0
$922$	9708.18	0.346770
$923$	0	0
$924$	0	0
$925$	−4897.57	−0.174088
$926$	−96.7050	−0.00343188
$927$	0	0
$928$	2653.15	0.0938512
$929$	−3753.03	−0.132543	−0.0662717	−	0.997802i	$-0.521110\pi$
$929$	−3753.03	−0.132543	−0.0662717	+	0.997802i	$0.521110\pi$
$930$	0	0
$931$	−18367.1	−0.646571
$932$	−27336.4	−0.960765
$933$	0	0
$934$	−11968.7	−0.419302
$935$	41855.1	1.46397
$936$	0	0
$937$	48189.0	1.68011	0.840056	−	0.542499i	$-0.182522\pi$
$937$	48189.0	1.68011	0.840056	+	0.542499i	$0.182522\pi$
$938$	−26180.4	−0.911323
$939$	0	0
$940$	−3705.17	−0.128563
$941$	26656.4	0.923456	0.461728	−	0.887022i	$-0.347230\pi$
$941$	26656.4	0.923456	0.461728	+	0.887022i	$0.347230\pi$
$942$	0	0
$943$	19132.3	0.660693
$944$	−1257.50	−0.0433559
$945$	0	0
$946$	16237.1	0.558047
$947$	31258.9	1.07263	0.536314	−	0.844018i	$-0.319816\pi$
$947$	31258.9	1.07263	0.536314	+	0.844018i	$0.319816\pi$
$948$	0	0
$949$	0	0
$950$	−5647.60	−0.192876
$951$	0	0
$952$	40150.0	1.36688
$953$	−10602.4	−0.360382	−0.180191	−	0.983632i	$-0.557672\pi$
$953$	−10602.4	−0.360382	−0.180191	+	0.983632i	$0.557672\pi$
$954$	0	0
$955$	8965.81	0.303798
$956$	−9558.44	−0.323370
$957$	0	0
$958$	19856.9	0.669674
$959$	−11378.6	−0.383144
$960$	0	0
$961$	8709.45	0.292352
$962$	0	0
$963$	0	0
$964$	−3775.60	−0.126145
$965$	21842.6	0.728640
$966$	0	0
$967$	4815.93	0.160155	0.0800774	−	0.996789i	$-0.474483\pi$
$967$	4815.93	0.160155	0.0800774	+	0.996789i	$0.474483\pi$
$968$	25000.0	0.830092
$969$	0	0
$970$	−1092.97	−0.0361786
$971$	−56729.5	−1.87491	−0.937454	−	0.348109i	$-0.886824\pi$
$971$	−56729.5	−1.87491	−0.937454	+	0.348109i	$0.886824\pi$
$972$	0	0
$973$	16929.9	0.557808
$974$	−106.493	−0.00350334
$975$	0	0
$976$	4500.40	0.147597
$977$	−55246.2	−1.80909	−0.904546	−	0.426375i	$-0.859790\pi$
$977$	−55246.2	−1.80909	−0.904546	+	0.426375i	$0.859790\pi$
$978$	0	0
$979$	−26555.4	−0.866921
$980$	−8619.11	−0.280946
$981$	0	0
$982$	−4000.34	−0.129996
$983$	−22820.6	−0.740451	−0.370226	−	0.928942i	$-0.620720\pi$
$983$	−22820.6	−0.740451	−0.370226	+	0.928942i	$0.620720\pi$
$984$	0	0
$985$	−21882.0	−0.707836
$986$	−1856.54	−0.0599637
$987$	0	0
$988$	0	0
$989$	−15253.5	−0.490429
$990$	0	0
$991$	−46490.4	−1.49023	−0.745115	−	0.666936i	$-0.767605\pi$
$991$	−46490.4	−1.49023	−0.745115	+	0.666936i	$0.767605\pi$
$992$	−36802.8	−1.17791
$993$	0	0
$994$	9028.21	0.288086
$995$	−2519.84	−0.0802856
$996$	0	0
$997$	24943.4	0.792344	0.396172	−	0.918176i	$-0.370338\pi$
$997$	24943.4	0.792344	0.396172	+	0.918176i	$0.370338\pi$
$998$	11626.8	0.368777
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.w.1.2		4
3.2	odd	2		507.4.a.l.1.3		4
13.5	odd	4		117.4.b.e.64.3		4
13.8	odd	4		117.4.b.e.64.2		4
13.12	even	2	inner	1521.4.a.w.1.3		4
39.5	even	4		39.4.b.b.25.2	✓	4
39.8	even	4		39.4.b.b.25.3	yes	4
39.38	odd	2		507.4.a.l.1.2		4
156.47	odd	4		624.4.c.c.337.2		4
156.83	odd	4		624.4.c.c.337.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.2	✓	4	39.5	even	4
39.4.b.b.25.3	yes	4	39.8	even	4
117.4.b.e.64.2		4	13.8	odd	4
117.4.b.e.64.3		4	13.5	odd	4
507.4.a.l.1.2		4	39.38	odd	2
507.4.a.l.1.3		4	3.2	odd	2
624.4.c.c.337.2		4	156.47	odd	4
624.4.c.c.337.3		4	156.83	odd	4
1521.4.a.w.1.2		4	1.1	even	1	trivial
1521.4.a.w.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.52356 q^{2} -5.67878 q^{4} +9.65841 q^{5} -22.3639 q^{7} +20.8404 q^{8} +O(q^{10})\)	\(q-1.52356 q^{2} -5.67878 q^{4} +9.65841 q^{5} -22.3639 q^{7} +20.8404 q^{8} -14.7151 q^{10} -50.3050 q^{11} +34.0727 q^{14} +13.6788 q^{16} -86.1454 q^{17} -116.880 q^{19} -54.8480 q^{20} +76.6424 q^{22} -72.0000 q^{23} -31.7151 q^{25} +127.000 q^{28} -14.1454 q^{29} +196.215 q^{31} -187.563 q^{32} +131.247 q^{34} -216.000 q^{35} +154.424 q^{37} +178.073 q^{38} +201.285 q^{40} -265.726 q^{41} +211.855 q^{43} +285.671 q^{44} +109.696 q^{46} +67.5535 q^{47} +157.145 q^{49} +48.3197 q^{50} -686.581 q^{53} -485.866 q^{55} -466.073 q^{56} +21.5512 q^{58} -91.9304 q^{59} +329.006 q^{61} -298.945 q^{62} +176.333 q^{64} -768.370 q^{67} +489.201 q^{68} +329.088 q^{70} +264.969 q^{71} -771.306 q^{73} -235.273 q^{74} +663.734 q^{76} +1125.02 q^{77} +1226.86 q^{79} +132.115 q^{80} +404.849 q^{82} -514.019 q^{83} -832.027 q^{85} -322.772 q^{86} -1048.38 q^{88} +527.889 q^{89} +408.872 q^{92} -102.921 q^{94} -1128.87 q^{95} +74.2755 q^{97} -239.420 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{4}+O(q^{10}) \) 4 * q + 14 * q^4	\( 4 q + 14 q^{4} + 88 q^{10} - 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} + 20 q^{25} + 384 q^{29} - 864 q^{35} + 492 q^{38} + 952 q^{40} + 1288 q^{43} + 188 q^{49} - 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} - 1314 q^{64} + 4380 q^{68} - 3144 q^{74} + 1416 q^{77} + 4320 q^{79} + 3088 q^{82} - 1036 q^{88} - 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100}) \) 4 * q + 14 * q^4 + 88 * q^10 - 84 * q^14 + 18 * q^16 + 96 * q^17 + 380 * q^22 - 288 * q^23 + 20 * q^25 + 384 * q^29 - 864 * q^35 + 492 * q^38 + 952 * q^40 + 1288 * q^43 + 188 * q^49 - 984 * q^53 - 328 * q^55 - 1644 * q^56 + 288 * q^61 + 1668 * q^62 - 1314 * q^64 + 4380 * q^68 - 3144 * q^74 + 1416 * q^77 + 4320 * q^79 + 3088 * q^82 - 1036 * q^88 - 1008 * q^92 - 1660 * q^94 - 1872 * q^95

Introduction

L-functions

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