LMFDB - Embedded newform 1521.4.a.bg.1.4

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.22799$ 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)

$f(q)$	$=$	$q-2.22799 q^{2} -3.03607 q^{4} +8.20685 q^{5} +8.35495 q^{7} +24.5882 q^{8} +O(q^{10})$	$q-2.22799 q^{2} -3.03607 q^{4} +8.20685 q^{5} +8.35495 q^{7} +24.5882 q^{8} -18.2848 q^{10} +9.69898 q^{11} -18.6147 q^{14} -30.4937 q^{16} -44.6219 q^{17} +87.7418 q^{19} -24.9166 q^{20} -21.6092 q^{22} -107.053 q^{23} -57.6475 q^{25} -25.3662 q^{28} +14.0430 q^{29} +171.090 q^{31} -128.766 q^{32} +99.4170 q^{34} +68.5678 q^{35} -413.954 q^{37} -195.488 q^{38} +201.792 q^{40} -258.282 q^{41} +61.0718 q^{43} -29.4468 q^{44} +238.512 q^{46} +68.7115 q^{47} -273.195 q^{49} +128.438 q^{50} -328.701 q^{53} +79.5981 q^{55} +205.433 q^{56} -31.2876 q^{58} -147.144 q^{59} -97.8083 q^{61} -381.186 q^{62} +530.839 q^{64} +677.597 q^{67} +135.475 q^{68} -152.768 q^{70} -786.767 q^{71} +997.675 q^{73} +922.285 q^{74} -266.390 q^{76} +81.0345 q^{77} +383.897 q^{79} -250.258 q^{80} +575.448 q^{82} -519.718 q^{83} -366.205 q^{85} -136.067 q^{86} +238.481 q^{88} -683.222 q^{89} +325.019 q^{92} -153.088 q^{94} +720.085 q^{95} -347.294 q^{97} +608.675 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8	$ 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8} - 147 q^{10} - 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} + 161 q^{19} - 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} + 344 q^{28} - 280 q^{29} + 706 q^{31} - 680 q^{32} + 216 q^{34} - 20 q^{35} + 298 q^{37} + 739 q^{38} + 13 q^{40} - 1201 q^{41} - 533 q^{43} - 355 q^{44} - 840 q^{46} - 956 q^{47} + 403 q^{49} + 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} - 2877 q^{58} - 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} - 931 q^{67} + 1536 q^{68} - 4854 q^{70} - 2046 q^{71} - 45 q^{73} + 1990 q^{74} - 3608 q^{76} + 718 q^{77} + 412 q^{79} + 787 q^{80} + 2757 q^{82} - 3709 q^{83} - 2106 q^{85} - 125 q^{86} - 636 q^{88} - 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} - 1087 q^{97} + 282 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8 - 147 * q^10 - 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 + 161 * q^19 - 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 + 344 * q^28 - 280 * q^29 + 706 * q^31 - 680 * q^32 + 216 * q^34 - 20 * q^35 + 298 * q^37 + 739 * q^38 + 13 * q^40 - 1201 * q^41 - 533 * q^43 - 355 * q^44 - 840 * q^46 - 956 * q^47 + 403 * q^49 + 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 - 2877 * q^58 - 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 - 931 * q^67 + 1536 * q^68 - 4854 * q^70 - 2046 * q^71 - 45 * q^73 + 1990 * q^74 - 3608 * q^76 + 718 * q^77 + 412 * q^79 + 787 * q^80 + 2757 * q^82 - 3709 * q^83 - 2106 * q^85 - 125 * q^86 - 636 * q^88 - 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 - 1087 * q^97 + 282 * q^98

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 $

$2$	−2.22799	−0.787713	−0.393856	−	0.919172i	$-0.628859\pi$
$2$	−2.22799	−0.787713	−0.393856	+	0.919172i	$0.628859\pi$
$3$	0	0
$3$	0	0
$4$	−3.03607	−0.379509
$5$	8.20685	0.734043	0.367022	−	0.930212i	$-0.380377\pi$
$5$	8.20685	0.734043	0.367022	+	0.930212i	$0.380377\pi$
$6$	0	0
$7$	8.35495	0.451125	0.225562	−	0.974229i	$-0.427578\pi$
$7$	8.35495	0.451125	0.225562	+	0.974229i	$0.427578\pi$
$8$	24.5882	1.08666
$9$	0	0
$10$	−18.2848	−0.578215
$11$	9.69898	0.265850	0.132925	−	0.991126i	$-0.457563\pi$
$11$	9.69898	0.265850	0.132925	+	0.991126i	$0.457563\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−18.6147	−0.355357
$15$	0	0
$16$	−30.4937	−0.476465
$17$	−44.6219	−0.636612	−0.318306	−	0.947988i	$-0.603114\pi$
$17$	−44.6219	−0.636612	−0.318306	+	0.947988i	$0.603114\pi$
$18$	0	0
$19$	87.7418	1.05944	0.529720	−	0.848173i	$-0.322297\pi$
$19$	87.7418	1.05944	0.529720	+	0.848173i	$0.322297\pi$
$20$	−24.9166	−0.278576
$21$	0	0
$22$	−21.6092	−0.209414
$23$	−107.053	−0.970522	−0.485261	−	0.874369i	$-0.661275\pi$
$23$	−107.053	−0.970522	−0.485261	+	0.874369i	$0.661275\pi$
$24$	0	0
$25$	−57.6475	−0.461180
$26$	0	0
$27$	0	0
$28$	−25.3662	−0.171206
$29$	14.0430	0.0899214	0.0449607	−	0.998989i	$-0.485684\pi$
$29$	14.0430	0.0899214	0.0449607	+	0.998989i	$0.485684\pi$
$30$	0	0
$31$	171.090	0.991247	0.495624	−	0.868537i	$-0.334940\pi$
$31$	171.090	0.991247	0.495624	+	0.868537i	$0.334940\pi$
$32$	−128.766	−0.711339
$33$	0	0
$34$	99.4170	0.501467
$35$	68.5678	0.331145
$36$	0	0
$37$	−413.954	−1.83929	−0.919643	−	0.392754i	$-0.871522\pi$
$37$	−413.954	−1.83929	−0.919643	+	0.392754i	$0.871522\pi$
$38$	−195.488	−0.834534
$39$	0	0
$40$	201.792	0.797653
$41$	−258.282	−0.983825	−0.491912	−	0.870645i	$-0.663702\pi$
$41$	−258.282	−0.983825	−0.491912	+	0.870645i	$0.663702\pi$
$42$	0	0
$43$	61.0718	0.216590	0.108295	−	0.994119i	$-0.465461\pi$
$43$	61.0718	0.216590	0.108295	+	0.994119i	$0.465461\pi$
$44$	−29.4468	−0.100892
$45$	0	0
$46$	238.512	0.764492
$47$	68.7115	0.213247	0.106623	−	0.994299i	$-0.465996\pi$
$47$	68.7115	0.213247	0.106623	+	0.994299i	$0.465996\pi$
$48$	0	0
$49$	−273.195	−0.796486
$50$	128.438	0.363278
$51$	0	0
$52$	0	0
$53$	−328.701	−0.851896	−0.425948	−	0.904748i	$-0.640059\pi$
$53$	−328.701	−0.851896	−0.425948	+	0.904748i	$0.640059\pi$
$54$	0	0
$55$	79.5981	0.195146
$56$	205.433	0.490218
$57$	0	0
$58$	−31.2876	−0.0708322
$59$	−147.144	−0.324687	−0.162344	−	0.986734i	$-0.551905\pi$
$59$	−147.144	−0.324687	−0.162344	+	0.986734i	$0.551905\pi$
$60$	0	0
$61$	−97.8083	−0.205296	−0.102648	−	0.994718i	$-0.532732\pi$
$61$	−97.8083	−0.205296	−0.102648	+	0.994718i	$0.532732\pi$
$62$	−381.186	−0.780818
$63$	0	0
$64$	530.839	1.03680
$65$	0	0
$66$	0	0
$67$	677.597	1.23555	0.617774	−	0.786356i	$-0.288035\pi$
$67$	677.597	1.23555	0.617774	+	0.786356i	$0.288035\pi$
$68$	135.475	0.241600
$69$	0	0
$70$	−152.768	−0.260847
$71$	−786.767	−1.31510	−0.657549	−	0.753411i	$-0.728407\pi$
$71$	−786.767	−1.31510	−0.657549	+	0.753411i	$0.728407\pi$
$72$	0	0
$73$	997.675	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$73$	997.675	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$74$	922.285	1.44883
$75$	0	0
$76$	−266.390	−0.402067
$77$	81.0345	0.119932
$78$	0	0
$79$	383.897	0.546731	0.273366	−	0.961910i	$-0.411863\pi$
$79$	383.897	0.546731	0.273366	+	0.961910i	$0.411863\pi$
$80$	−250.258	−0.349746
$81$	0	0
$82$	575.448	0.774971
$83$	−519.718	−0.687307	−0.343654	−	0.939096i	$-0.611665\pi$
$83$	−519.718	−0.687307	−0.343654	+	0.939096i	$0.611665\pi$
$84$	0	0
$85$	−366.205	−0.467301
$86$	−136.067	−0.170610
$87$	0	0
$88$	238.481	0.288888
$89$	−683.222	−0.813723	−0.406862	−	0.913490i	$-0.633377\pi$
$89$	−683.222	−0.813723	−0.406862	+	0.913490i	$0.633377\pi$
$90$	0	0
$91$	0	0
$92$	325.019	0.368321
$93$	0	0
$94$	−153.088	−0.167977
$95$	720.085	0.777675
$96$	0	0
$97$	−347.294	−0.363530	−0.181765	−	0.983342i	$-0.558181\pi$
$97$	−347.294	−0.363530	−0.181765	+	0.983342i	$0.558181\pi$
$98$	608.675	0.627402
$99$	0	0
$100$	175.022	0.175022
$101$	−554.794	−0.546575	−0.273288	−	0.961932i	$-0.588111\pi$
$101$	−554.794	−0.546575	−0.273288	+	0.961932i	$0.588111\pi$
$102$	0	0
$103$	1137.13	1.08781	0.543905	−	0.839147i	$-0.316945\pi$
$103$	1137.13	1.08781	0.543905	+	0.839147i	$0.316945\pi$
$104$	0	0
$105$	0	0
$106$	732.341	0.671050
$107$	1556.14	1.40596	0.702980	−	0.711210i	$-0.251853\pi$
$107$	1556.14	1.40596	0.702980	+	0.711210i	$0.251853\pi$
$108$	0	0
$109$	−71.6448	−0.0629572	−0.0314786	−	0.999504i	$-0.510022\pi$
$109$	−71.6448	−0.0629572	−0.0314786	+	0.999504i	$0.510022\pi$
$110$	−177.344	−0.153719
$111$	0	0
$112$	−254.774	−0.214945
$113$	980.872	0.816573	0.408286	−	0.912854i	$-0.366126\pi$
$113$	980.872	0.816573	0.408286	+	0.912854i	$0.366126\pi$
$114$	0	0
$115$	−878.565	−0.712405
$116$	−42.6355	−0.0341259
$117$	0	0
$118$	327.836	0.255760
$119$	−372.814	−0.287191
$120$	0	0
$121$	−1236.93	−0.929324
$122$	217.916	0.161714
$123$	0	0
$124$	−519.441	−0.376187
$125$	−1498.96	−1.07257
$126$	0	0
$127$	−2177.31	−1.52130	−0.760649	−	0.649164i	$-0.775119\pi$
$127$	−2177.31	−1.52130	−0.760649	+	0.649164i	$0.775119\pi$
$128$	−152.575	−0.105358
$129$	0	0
$130$	0	0
$131$	1919.81	1.28041	0.640207	−	0.768202i	$-0.278848\pi$
$131$	1919.81	1.28041	0.640207	+	0.768202i	$0.278848\pi$
$132$	0	0
$133$	733.079	0.477940
$134$	−1509.68	−0.973257
$135$	0	0
$136$	−1097.17	−0.691778
$137$	744.013	0.463980	0.231990	−	0.972718i	$-0.425476\pi$
$137$	744.013	0.463980	0.231990	+	0.972718i	$0.425476\pi$
$138$	0	0
$139$	2820.00	1.72078	0.860392	−	0.509633i	$-0.170219\pi$
$139$	2820.00	1.72078	0.860392	+	0.509633i	$0.170219\pi$
$140$	−208.177	−0.125672
$141$	0	0
$142$	1752.91	1.03592
$143$	0	0
$144$	0	0
$145$	115.249	0.0660062
$146$	−2222.81	−1.26001
$147$	0	0
$148$	1256.79	0.698025
$149$	−2894.60	−1.59151	−0.795755	−	0.605619i	$-0.792926\pi$
$149$	−2894.60	−1.59151	−0.795755	+	0.605619i	$0.792926\pi$
$150$	0	0
$151$	494.004	0.266235	0.133118	−	0.991100i	$-0.457501\pi$
$151$	494.004	0.266235	0.133118	+	0.991100i	$0.457501\pi$
$152$	2157.42	1.15125
$153$	0	0
$154$	−180.544	−0.0944717
$155$	1404.11	0.727618
$156$	0	0
$157$	50.7450	0.0257955	0.0128977	−	0.999917i	$-0.495894\pi$
$157$	50.7450	0.0257955	0.0128977	+	0.999917i	$0.495894\pi$
$158$	−855.318	−0.430667
$159$	0	0
$160$	−1056.77	−0.522154
$161$	−894.419	−0.437826
$162$	0	0
$163$	−751.801	−0.361261	−0.180631	−	0.983551i	$-0.557814\pi$
$163$	−751.801	−0.361261	−0.180631	+	0.983551i	$0.557814\pi$
$164$	784.161	0.373370
$165$	0	0
$166$	1157.93	0.541401
$167$	−2974.32	−1.37820	−0.689102	−	0.724664i	$-0.741995\pi$
$167$	−2974.32	−1.37820	−0.689102	+	0.724664i	$0.741995\pi$
$168$	0	0
$169$	0	0
$170$	815.901	0.368099
$171$	0	0
$172$	−185.418	−0.0821976
$173$	−1633.74	−0.717982	−0.358991	−	0.933341i	$-0.616879\pi$
$173$	−1633.74	−0.717982	−0.358991	+	0.933341i	$0.616879\pi$
$174$	0	0
$175$	−481.642	−0.208050
$176$	−295.758	−0.126668
$177$	0	0
$178$	1522.21	0.640980
$179$	−3392.65	−1.41664	−0.708320	−	0.705892i	$-0.750546\pi$
$179$	−3392.65	−1.41664	−0.708320	+	0.705892i	$0.750546\pi$
$180$	0	0
$181$	−3801.07	−1.56095	−0.780473	−	0.625190i	$-0.785021\pi$
$181$	−3801.07	−1.56095	−0.780473	+	0.625190i	$0.785021\pi$
$182$	0	0
$183$	0	0
$184$	−2632.23	−1.05462
$185$	−3397.26	−1.35012
$186$	0	0
$187$	−432.787	−0.169243
$188$	−208.613	−0.0809290
$189$	0	0
$190$	−1604.34	−0.612584
$191$	−1265.51	−0.479419	−0.239710	−	0.970845i	$-0.577052\pi$
$191$	−1265.51	−0.479419	−0.239710	+	0.970845i	$0.577052\pi$
$192$	0	0
$193$	−2212.32	−0.825109	−0.412555	−	0.910933i	$-0.635363\pi$
$193$	−2212.32	−0.825109	−0.412555	+	0.910933i	$0.635363\pi$
$194$	773.767	0.286357
$195$	0	0
$196$	829.438	0.302273
$197$	4357.25	1.57585	0.787923	−	0.615774i	$-0.211157\pi$
$197$	4357.25	1.57585	0.787923	+	0.615774i	$0.211157\pi$
$198$	0	0
$199$	1090.63	0.388507	0.194254	−	0.980951i	$-0.437772\pi$
$199$	1090.63	0.388507	0.194254	+	0.980951i	$0.437772\pi$
$200$	−1417.45	−0.501145
$201$	0	0
$202$	1236.08	0.430544
$203$	117.329	0.0405658
$204$	0	0
$205$	−2119.68	−0.722170
$206$	−2533.50	−0.856881
$207$	0	0
$208$	0	0
$209$	851.007	0.281652
$210$	0	0
$211$	−727.448	−0.237344	−0.118672	−	0.992934i	$-0.537864\pi$
$211$	−727.448	−0.237344	−0.118672	+	0.992934i	$0.537864\pi$
$212$	997.958	0.323302
$213$	0	0
$214$	−3467.06	−1.10749
$215$	501.207	0.158986
$216$	0	0
$217$	1429.45	0.447176
$218$	159.624	0.0495922
$219$	0	0
$220$	−241.665	−0.0740595
$221$	0	0
$222$	0	0
$223$	−1015.45	−0.304931	−0.152465	−	0.988309i	$-0.548721\pi$
$223$	−1015.45	−0.304931	−0.152465	+	0.988309i	$0.548721\pi$
$224$	−1075.83	−0.320903
$225$	0	0
$226$	−2185.37	−0.643225
$227$	−5292.59	−1.54749	−0.773747	−	0.633494i	$-0.781620\pi$
$227$	−5292.59	−1.54749	−0.773747	+	0.633494i	$0.781620\pi$
$228$	0	0
$229$	−3010.03	−0.868597	−0.434298	−	0.900769i	$-0.643004\pi$
$229$	−3010.03	−0.868597	−0.434298	+	0.900769i	$0.643004\pi$
$230$	1957.43	0.561171
$231$	0	0
$232$	345.292	0.0977136
$233$	2373.96	0.667482	0.333741	−	0.942665i	$-0.391689\pi$
$233$	2373.96	0.667482	0.333741	+	0.942665i	$0.391689\pi$
$234$	0	0
$235$	563.905	0.156532
$236$	446.740	0.123222
$237$	0	0
$238$	830.624	0.226224
$239$	783.439	0.212035	0.106018	−	0.994364i	$-0.466190\pi$
$239$	783.439	0.212035	0.106018	+	0.994364i	$0.466190\pi$
$240$	0	0
$241$	3517.40	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$241$	3517.40	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$242$	2755.86	0.732040
$243$	0	0
$244$	296.953	0.0779117
$245$	−2242.07	−0.584656
$246$	0	0
$247$	0	0
$248$	4206.80	1.07715
$249$	0	0
$250$	3339.67	0.844877
$251$	−1416.96	−0.356326	−0.178163	−	0.984001i	$-0.557015\pi$
$251$	−1416.96	−0.356326	−0.178163	+	0.984001i	$0.557015\pi$
$252$	0	0
$253$	−1038.30	−0.258013
$254$	4851.01	1.19835
$255$	0	0
$256$	−3906.78	−0.953804
$257$	1482.63	0.359861	0.179930	−	0.983679i	$-0.442413\pi$
$257$	1482.63	0.359861	0.179930	+	0.983679i	$0.442413\pi$
$258$	0	0
$259$	−3458.56	−0.829748
$260$	0	0
$261$	0	0
$262$	−4277.31	−1.00860
$263$	−7229.64	−1.69505	−0.847526	−	0.530753i	$-0.821909\pi$
$263$	−7229.64	−1.69505	−0.847526	+	0.530753i	$0.821909\pi$
$264$	0	0
$265$	−2697.60	−0.625329
$266$	−1633.29	−0.376479
$267$	0	0
$268$	−2057.23	−0.468901
$269$	494.904	0.112174	0.0560870	−	0.998426i	$-0.482138\pi$
$269$	494.904	0.112174	0.0560870	+	0.998426i	$0.482138\pi$
$270$	0	0
$271$	4205.58	0.942696	0.471348	−	0.881947i	$-0.343768\pi$
$271$	4205.58	0.942696	0.471348	+	0.881947i	$0.343768\pi$
$272$	1360.69	0.303323
$273$	0	0
$274$	−1657.65	−0.365483
$275$	−559.123	−0.122605
$276$	0	0
$277$	4208.55	0.912878	0.456439	−	0.889755i	$-0.349125\pi$
$277$	4208.55	0.912878	0.456439	+	0.889755i	$0.349125\pi$
$278$	−6282.92	−1.35548
$279$	0	0
$280$	1685.96	0.359841
$281$	−4740.83	−1.00646	−0.503228	−	0.864153i	$-0.667855\pi$
$281$	−4740.83	−1.00646	−0.503228	+	0.864153i	$0.667855\pi$
$282$	0	0
$283$	−3742.82	−0.786176	−0.393088	−	0.919501i	$-0.628593\pi$
$283$	−3742.82	−0.786176	−0.393088	+	0.919501i	$0.628593\pi$
$284$	2388.68	0.499091
$285$	0	0
$286$	0	0
$287$	−2157.93	−0.443828
$288$	0	0
$289$	−2921.89	−0.594726
$290$	−256.773	−0.0519939
$291$	0	0
$292$	−3029.01	−0.607053
$293$	5254.97	1.04778	0.523889	−	0.851787i	$-0.324481\pi$
$293$	5254.97	1.04778	0.523889	+	0.851787i	$0.324481\pi$
$294$	0	0
$295$	−1207.59	−0.238334
$296$	−10178.4	−1.99867
$297$	0	0
$298$	6449.14	1.25365
$299$	0	0
$300$	0	0
$301$	510.251	0.0977090
$302$	−1100.64	−0.209717
$303$	0	0
$304$	−2675.58	−0.504786
$305$	−802.698	−0.150696
$306$	0	0
$307$	252.464	0.0469344	0.0234672	−	0.999725i	$-0.492529\pi$
$307$	252.464	0.0469344	0.0234672	+	0.999725i	$0.492529\pi$
$308$	−246.026	−0.0455151
$309$	0	0
$310$	−3128.34	−0.573154
$311$	−2561.20	−0.466986	−0.233493	−	0.972359i	$-0.575016\pi$
$311$	−2561.20	−0.466986	−0.233493	+	0.972359i	$0.575016\pi$
$312$	0	0
$313$	−695.893	−0.125668	−0.0628342	−	0.998024i	$-0.520014\pi$
$313$	−695.893	−0.125668	−0.0628342	+	0.998024i	$0.520014\pi$
$314$	−113.059	−0.0203194
$315$	0	0
$316$	−1165.54	−0.207489
$317$	−5747.37	−1.01831	−0.509155	−	0.860675i	$-0.670042\pi$
$317$	−5747.37	−1.01831	−0.509155	+	0.860675i	$0.670042\pi$
$318$	0	0
$319$	136.203	0.0239056
$320$	4356.52	0.761053
$321$	0	0
$322$	1992.75	0.344881
$323$	−3915.21	−0.674452
$324$	0	0
$325$	0	0
$326$	1675.00	0.284570
$327$	0	0
$328$	−6350.69	−1.06908
$329$	574.081	0.0962010
$330$	0	0
$331$	−4244.76	−0.704874	−0.352437	−	0.935836i	$-0.614647\pi$
$331$	−4244.76	−0.704874	−0.352437	+	0.935836i	$0.614647\pi$
$332$	1577.90	0.260839
$333$	0	0
$334$	6626.76	1.08563
$335$	5560.94	0.906946
$336$	0	0
$337$	−7122.49	−1.15130	−0.575648	−	0.817698i	$-0.695250\pi$
$337$	−7122.49	−1.15130	−0.575648	+	0.817698i	$0.695250\pi$
$338$	0	0
$339$	0	0
$340$	1111.82	0.177345
$341$	1659.40	0.263523
$342$	0	0
$343$	−5148.28	−0.810440
$344$	1501.65	0.235359
$345$	0	0
$346$	3639.95	0.565563
$347$	3367.90	0.521033	0.260517	−	0.965469i	$-0.416107\pi$
$347$	3367.90	0.521033	0.260517	+	0.965469i	$0.416107\pi$
$348$	0	0
$349$	6079.71	0.932491	0.466246	−	0.884655i	$-0.345606\pi$
$349$	6079.71	0.932491	0.466246	+	0.884655i	$0.345606\pi$
$350$	1073.09	0.163884
$351$	0	0
$352$	−1248.90	−0.189110
$353$	−3230.99	−0.487162	−0.243581	−	0.969881i	$-0.578322\pi$
$353$	−3230.99	−0.487162	−0.243581	+	0.969881i	$0.578322\pi$
$354$	0	0
$355$	−6456.88	−0.965340
$356$	2074.31	0.308815
$357$	0	0
$358$	7558.78	1.11590
$359$	−5345.81	−0.785908	−0.392954	−	0.919558i	$-0.628547\pi$
$359$	−5345.81	−0.785908	−0.392954	+	0.919558i	$0.628547\pi$
$360$	0	0
$361$	839.632	0.122413
$362$	8468.73	1.22958
$363$	0	0
$364$	0	0
$365$	8187.78	1.17416
$366$	0	0
$367$	−12171.3	−1.73117	−0.865583	−	0.500765i	$-0.833052\pi$
$367$	−12171.3	−1.73117	−0.865583	+	0.500765i	$0.833052\pi$
$368$	3264.43	0.462419
$369$	0	0
$370$	7569.05	1.06350
$371$	−2746.28	−0.384312
$372$	0	0
$373$	−4463.95	−0.619664	−0.309832	−	0.950791i	$-0.600273\pi$
$373$	−4463.95	−0.619664	−0.309832	+	0.950791i	$0.600273\pi$
$374$	964.244	0.133315
$375$	0	0
$376$	1689.49	0.231726
$377$	0	0
$378$	0	0
$379$	5369.06	0.727679	0.363839	−	0.931462i	$-0.381466\pi$
$379$	5369.06	0.727679	0.363839	+	0.931462i	$0.381466\pi$
$380$	−2186.23	−0.295134
$381$	0	0
$382$	2819.54	0.377645
$383$	−195.084	−0.0260269	−0.0130135	−	0.999915i	$-0.504142\pi$
$383$	−195.084	−0.0260269	−0.0130135	+	0.999915i	$0.504142\pi$
$384$	0	0
$385$	665.038	0.0880351
$386$	4929.02	0.649949
$387$	0	0
$388$	1054.41	0.137963
$389$	−9120.52	−1.18876	−0.594381	−	0.804183i	$-0.702603\pi$
$389$	−9120.52	−1.18876	−0.594381	+	0.804183i	$0.702603\pi$
$390$	0	0
$391$	4776.89	0.617845
$392$	−6717.38	−0.865507
$393$	0	0
$394$	−9707.91	−1.24131
$395$	3150.59	0.401325
$396$	0	0
$397$	−6613.46	−0.836071	−0.418036	−	0.908431i	$-0.637281\pi$
$397$	−6613.46	−0.836071	−0.418036	+	0.908431i	$0.637281\pi$
$398$	−2429.92	−0.306032
$399$	0	0
$400$	1757.89	0.219736
$401$	−8589.26	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$401$	−8589.26	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$402$	0	0
$403$	0	0
$404$	1684.39	0.207430
$405$	0	0
$406$	−261.407	−0.0319542
$407$	−4014.93	−0.488975
$408$	0	0
$409$	7515.59	0.908612	0.454306	−	0.890846i	$-0.349887\pi$
$409$	7515.59	0.908612	0.454306	+	0.890846i	$0.349887\pi$
$410$	4722.62	0.568863
$411$	0	0
$412$	−3452.39	−0.412833
$413$	−1229.38	−0.146474
$414$	0	0
$415$	−4265.25	−0.504513
$416$	0	0
$417$	0	0
$418$	−1896.03	−0.221861
$419$	4557.33	0.531361	0.265680	−	0.964061i	$-0.414403\pi$
$419$	4557.33	0.531361	0.265680	+	0.964061i	$0.414403\pi$
$420$	0	0
$421$	2225.19	0.257599	0.128800	−	0.991671i	$-0.458888\pi$
$421$	2225.19	0.257599	0.128800	+	0.991671i	$0.458888\pi$
$422$	1620.75	0.186959
$423$	0	0
$424$	−8082.17	−0.925719
$425$	2572.34	0.293593
$426$	0	0
$427$	−817.183	−0.0926142
$428$	−4724.54	−0.533574
$429$	0	0
$430$	−1116.68	−0.125235
$431$	−396.028	−0.0442599	−0.0221299	−	0.999755i	$-0.507045\pi$
$431$	−396.028	−0.0442599	−0.0221299	+	0.999755i	$0.507045\pi$
$432$	0	0
$433$	5188.06	0.575802	0.287901	−	0.957660i	$-0.407043\pi$
$433$	5188.06	0.575802	0.287901	+	0.957660i	$0.407043\pi$
$434$	−3184.79	−0.352246
$435$	0	0
$436$	217.519	0.0238928
$437$	−9392.99	−1.02821
$438$	0	0
$439$	11329.9	1.23176	0.615882	−	0.787838i	$-0.288800\pi$
$439$	11329.9	1.23176	0.615882	+	0.787838i	$0.288800\pi$
$440$	1957.18	0.212056
$441$	0	0
$442$	0	0
$443$	−15625.2	−1.67579	−0.837897	−	0.545828i	$-0.816215\pi$
$443$	−15625.2	−1.67579	−0.837897	+	0.545828i	$0.816215\pi$
$444$	0	0
$445$	−5607.10	−0.597308
$446$	2262.41	0.240198
$447$	0	0
$448$	4435.14	0.467724
$449$	−13686.6	−1.43855	−0.719276	−	0.694725i	$-0.755526\pi$
$449$	−13686.6	−1.43855	−0.719276	+	0.694725i	$0.755526\pi$
$450$	0	0
$451$	−2505.07	−0.261550
$452$	−2978.00	−0.309896
$453$	0	0
$454$	11791.8	1.21898
$455$	0	0
$456$	0	0
$457$	12673.1	1.29720	0.648602	−	0.761127i	$-0.275354\pi$
$457$	12673.1	1.29720	0.648602	+	0.761127i	$0.275354\pi$
$458$	6706.32	0.684205
$459$	0	0
$460$	2667.38	0.270364
$461$	−9154.85	−0.924911	−0.462456	−	0.886642i	$-0.653032\pi$
$461$	−9154.85	−0.924911	−0.462456	+	0.886642i	$0.653032\pi$
$462$	0	0
$463$	6910.59	0.693655	0.346827	−	0.937929i	$-0.387259\pi$
$463$	6910.59	0.693655	0.346827	+	0.937929i	$0.387259\pi$
$464$	−428.223	−0.0428443
$465$	0	0
$466$	−5289.16	−0.525784
$467$	−2920.19	−0.289359	−0.144679	−	0.989479i	$-0.546215\pi$
$467$	−2920.19	−0.289359	−0.144679	+	0.989479i	$0.546215\pi$
$468$	0	0
$469$	5661.29	0.557386
$470$	−1256.37	−0.123303
$471$	0	0
$472$	−3618.02	−0.352823
$473$	592.334	0.0575804
$474$	0	0
$475$	−5058.10	−0.488593
$476$	1131.89	0.108992
$477$	0	0
$478$	−1745.49	−0.167023
$479$	17088.6	1.63006	0.815030	−	0.579419i	$-0.196720\pi$
$479$	17088.6	1.63006	0.815030	+	0.579419i	$0.196720\pi$
$480$	0	0
$481$	0	0
$482$	−7836.73	−0.740567
$483$	0	0
$484$	3755.40	0.352686
$485$	−2850.19	−0.266846
$486$	0	0
$487$	−13813.0	−1.28527	−0.642634	−	0.766174i	$-0.722158\pi$
$487$	−13813.0	−1.28527	−0.642634	+	0.766174i	$0.722158\pi$
$488$	−2404.93	−0.223086
$489$	0	0
$490$	4995.31	0.460541
$491$	11848.4	1.08903	0.544513	−	0.838752i	$-0.316714\pi$
$491$	11848.4	1.08903	0.544513	+	0.838752i	$0.316714\pi$
$492$	0	0
$493$	−626.625	−0.0572450
$494$	0	0
$495$	0	0
$496$	−5217.17	−0.472294
$497$	−6573.40	−0.593274
$498$	0	0
$499$	−5224.46	−0.468696	−0.234348	−	0.972153i	$-0.575295\pi$
$499$	−5224.46	−0.468696	−0.234348	+	0.972153i	$0.575295\pi$
$500$	4550.95	0.407049
$501$	0	0
$502$	3156.97	0.280682
$503$	−7884.39	−0.698902	−0.349451	−	0.936955i	$-0.613632\pi$
$503$	−7884.39	−0.698902	−0.349451	+	0.936955i	$0.613632\pi$
$504$	0	0
$505$	−4553.12	−0.401210
$506$	2313.32	0.203241
$507$	0	0
$508$	6610.45	0.577345
$509$	4026.56	0.350637	0.175318	−	0.984512i	$-0.443905\pi$
$509$	4026.56	0.350637	0.175318	+	0.984512i	$0.443905\pi$
$510$	0	0
$511$	8335.53	0.721609
$512$	9924.86	0.856681
$513$	0	0
$514$	−3303.29	−0.283467
$515$	9332.23	0.798499
$516$	0	0
$517$	666.432	0.0566918
$518$	7705.64	0.653603
$519$	0	0
$520$	0	0
$521$	−6196.12	−0.521030	−0.260515	−	0.965470i	$-0.583892\pi$
$521$	−6196.12	−0.521030	−0.260515	+	0.965470i	$0.583892\pi$
$522$	0	0
$523$	7899.39	0.660452	0.330226	−	0.943902i	$-0.392875\pi$
$523$	7899.39	0.660452	0.330226	+	0.943902i	$0.392875\pi$
$524$	−5828.67	−0.485928
$525$	0	0
$526$	16107.6	1.33521
$527$	−7634.36	−0.631040
$528$	0	0
$529$	−706.752	−0.0580876
$530$	6010.22	0.492580
$531$	0	0
$532$	−2225.68	−0.181382
$533$	0	0
$534$	0	0
$535$	12771.0	1.03203
$536$	16660.9	1.34262
$537$	0	0
$538$	−1102.64	−0.0883609
$539$	−2649.71	−0.211746
$540$	0	0
$541$	−6146.22	−0.488441	−0.244220	−	0.969720i	$-0.578532\pi$
$541$	−6146.22	−0.488441	−0.244220	+	0.969720i	$0.578532\pi$
$542$	−9369.98	−0.742574
$543$	0	0
$544$	5745.79	0.452847
$545$	−587.979	−0.0462133
$546$	0	0
$547$	5555.49	0.434252	0.217126	−	0.976144i	$-0.430332\pi$
$547$	5555.49	0.434252	0.217126	+	0.976144i	$0.430332\pi$
$548$	−2258.87	−0.176084
$549$	0	0
$550$	1245.72	0.0965775
$551$	1232.16	0.0952663
$552$	0	0
$553$	3207.44	0.246644
$554$	−9376.59	−0.719085
$555$	0	0
$556$	−8561.70	−0.653052
$557$	−7580.71	−0.576669	−0.288335	−	0.957530i	$-0.593102\pi$
$557$	−7580.71	−0.576669	−0.288335	+	0.957530i	$0.593102\pi$
$558$	0	0
$559$	0	0
$560$	−2090.89	−0.157779
$561$	0	0
$562$	10562.5	0.792799
$563$	13594.6	1.01766	0.508831	−	0.860866i	$-0.330078\pi$
$563$	13594.6	1.01766	0.508831	+	0.860866i	$0.330078\pi$
$564$	0	0
$565$	8049.88	0.599400
$566$	8338.97	0.619281
$567$	0	0
$568$	−19345.2	−1.42906
$569$	−5650.14	−0.416285	−0.208143	−	0.978099i	$-0.566742\pi$
$569$	−5650.14	−0.416285	−0.208143	+	0.978099i	$0.566742\pi$
$570$	0	0
$571$	6297.53	0.461547	0.230773	−	0.973008i	$-0.425874\pi$
$571$	6297.53	0.461547	0.230773	+	0.973008i	$0.425874\pi$
$572$	0	0
$573$	0	0
$574$	4807.84	0.349609
$575$	6171.32	0.447586
$576$	0	0
$577$	17838.9	1.28707	0.643537	−	0.765415i	$-0.277466\pi$
$577$	17838.9	1.28707	0.643537	+	0.765415i	$0.277466\pi$
$578$	6509.93	0.468473
$579$	0	0
$580$	−349.903	−0.0250499
$581$	−4342.22	−0.310061
$582$	0	0
$583$	−3188.06	−0.226477
$584$	24531.1	1.73819
$585$	0	0
$586$	−11708.0	−0.825348
$587$	455.091	0.0319993	0.0159997	−	0.999872i	$-0.494907\pi$
$587$	455.091	0.0319993	0.0159997	+	0.999872i	$0.494907\pi$
$588$	0	0
$589$	15011.7	1.05017
$590$	2690.50	0.187739
$591$	0	0
$592$	12623.0	0.876355
$593$	16240.6	1.12466	0.562330	−	0.826913i	$-0.309905\pi$
$593$	16240.6	1.12466	0.562330	+	0.826913i	$0.309905\pi$
$594$	0	0
$595$	−3059.63	−0.210811
$596$	8788.21	0.603992
$597$	0	0
$598$	0	0
$599$	6704.05	0.457296	0.228648	−	0.973509i	$-0.426570\pi$
$599$	6704.05	0.457296	0.228648	+	0.973509i	$0.426570\pi$
$600$	0	0
$601$	26413.2	1.79271	0.896354	−	0.443339i	$-0.146206\pi$
$601$	26413.2	1.79271	0.896354	+	0.443339i	$0.146206\pi$
$602$	−1136.83	−0.0769666
$603$	0	0
$604$	−1499.83	−0.101039
$605$	−10151.3	−0.682164
$606$	0	0
$607$	−3326.37	−0.222427	−0.111214	−	0.993797i	$-0.535474\pi$
$607$	−3326.37	−0.222427	−0.111214	+	0.993797i	$0.535474\pi$
$608$	−11298.2	−0.753621
$609$	0	0
$610$	1788.40	0.118705
$611$	0	0
$612$	0	0
$613$	−26258.7	−1.73014	−0.865072	−	0.501647i	$-0.832728\pi$
$613$	−26258.7	−1.73014	−0.865072	+	0.501647i	$0.832728\pi$
$614$	−562.486	−0.0369708
$615$	0	0
$616$	1992.50	0.130325
$617$	27352.9	1.78474	0.892370	−	0.451304i	$-0.149041\pi$
$617$	27352.9	1.78474	0.892370	+	0.451304i	$0.149041\pi$
$618$	0	0
$619$	13056.5	0.847795	0.423897	−	0.905710i	$-0.360662\pi$
$619$	13056.5	0.847795	0.423897	+	0.905710i	$0.360662\pi$
$620$	−4262.98	−0.276137
$621$	0	0
$622$	5706.33	0.367851
$623$	−5708.28	−0.367091
$624$	0	0
$625$	−5095.82	−0.326132
$626$	1550.44	0.0989905
$627$	0	0
$628$	−154.065	−0.00978961
$629$	18471.4	1.17091
$630$	0	0
$631$	−6552.91	−0.413419	−0.206709	−	0.978402i	$-0.566275\pi$
$631$	−6552.91	−0.413419	−0.206709	+	0.978402i	$0.566275\pi$
$632$	9439.35	0.594109
$633$	0	0
$634$	12805.1	0.802137
$635$	−17868.8	−1.11670
$636$	0	0
$637$	0	0
$638$	−303.458	−0.0188308
$639$	0	0
$640$	−1252.16	−0.0773373
$641$	5764.77	0.355218	0.177609	−	0.984101i	$-0.443164\pi$
$641$	5764.77	0.355218	0.177609	+	0.984101i	$0.443164\pi$
$642$	0	0
$643$	−12279.5	−0.753120	−0.376560	−	0.926392i	$-0.622893\pi$
$643$	−12279.5	−0.753120	−0.376560	+	0.926392i	$0.622893\pi$
$644$	2715.52	0.166159
$645$	0	0
$646$	8723.04	0.531274
$647$	29024.3	1.76362	0.881810	−	0.471605i	$-0.156325\pi$
$647$	29024.3	1.76362	0.881810	+	0.471605i	$0.156325\pi$
$648$	0	0
$649$	−1427.15	−0.0863182
$650$	0	0
$651$	0	0
$652$	2282.52	0.137102
$653$	17167.1	1.02879	0.514396	−	0.857552i	$-0.328016\pi$
$653$	17167.1	1.02879	0.514396	+	0.857552i	$0.328016\pi$
$654$	0	0
$655$	15755.6	0.939880
$656$	7875.97	0.468758
$657$	0	0
$658$	−1279.05	−0.0757787
$659$	−18313.0	−1.08251	−0.541254	−	0.840859i	$-0.682050\pi$
$659$	−18313.0	−1.08251	−0.541254	+	0.840859i	$0.682050\pi$
$660$	0	0
$661$	−6885.30	−0.405155	−0.202577	−	0.979266i	$-0.564932\pi$
$661$	−6885.30	−0.405155	−0.202577	+	0.979266i	$0.564932\pi$
$662$	9457.28	0.555238
$663$	0	0
$664$	−12779.0	−0.746867
$665$	6016.27	0.350828
$666$	0	0
$667$	−1503.34	−0.0872706
$668$	9030.25	0.523040
$669$	0	0
$670$	−12389.7	−0.714413
$671$	−948.641	−0.0545781
$672$	0	0
$673$	6238.26	0.357307	0.178653	−	0.983912i	$-0.442826\pi$
$673$	6238.26	0.357307	0.178653	+	0.983912i	$0.442826\pi$
$674$	15868.8	0.906890
$675$	0	0
$676$	0	0
$677$	−25482.4	−1.44663	−0.723316	−	0.690517i	$-0.757383\pi$
$677$	−25482.4	−1.44663	−0.723316	+	0.690517i	$0.757383\pi$
$678$	0	0
$679$	−2901.62	−0.163997
$680$	−9004.34	−0.507795
$681$	0	0
$682$	−3697.12	−0.207581
$683$	−27426.5	−1.53652	−0.768261	−	0.640136i	$-0.778878\pi$
$683$	−27426.5	−1.53652	−0.768261	+	0.640136i	$0.778878\pi$
$684$	0	0
$685$	6106.00	0.340582
$686$	11470.3	0.638394
$687$	0	0
$688$	−1862.31	−0.103197
$689$	0	0
$690$	0	0
$691$	−13886.3	−0.764486	−0.382243	−	0.924062i	$-0.624848\pi$
$691$	−13886.3	−0.764486	−0.382243	+	0.924062i	$0.624848\pi$
$692$	4960.14	0.272480
$693$	0	0
$694$	−7503.65	−0.410425
$695$	23143.3	1.26313
$696$	0	0
$697$	11525.0	0.626314
$698$	−13545.5	−0.734535
$699$	0	0
$700$	1462.30	0.0789567
$701$	−15744.4	−0.848301	−0.424151	−	0.905592i	$-0.639427\pi$
$701$	−15744.4	−0.848301	−0.424151	+	0.905592i	$0.639427\pi$
$702$	0	0
$703$	−36321.1	−1.94861
$704$	5148.60	0.275632
$705$	0	0
$706$	7198.61	0.383744
$707$	−4635.28	−0.246574
$708$	0	0
$709$	−28689.1	−1.51967	−0.759833	−	0.650118i	$-0.774719\pi$
$709$	−28689.1	−1.51967	−0.759833	+	0.650118i	$0.774719\pi$
$710$	14385.9	0.760410
$711$	0	0
$712$	−16799.2	−0.884237
$713$	−18315.6	−0.962027
$714$	0	0
$715$	0	0
$716$	10300.3	0.537627
$717$	0	0
$718$	11910.4	0.619070
$719$	12619.1	0.654538	0.327269	−	0.944931i	$-0.393872\pi$
$719$	12619.1	0.654538	0.327269	+	0.944931i	$0.393872\pi$
$720$	0	0
$721$	9500.63	0.490738
$722$	−1870.69	−0.0964265
$723$	0	0
$724$	11540.3	0.592392
$725$	−809.544	−0.0414700
$726$	0	0
$727$	−12644.5	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$727$	−12644.5	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$728$	0	0
$729$	0	0
$730$	−18242.3	−0.924900
$731$	−2725.14	−0.137884
$732$	0	0
$733$	19109.0	0.962903	0.481451	−	0.876473i	$-0.340110\pi$
$733$	19109.0	0.962903	0.481451	+	0.876473i	$0.340110\pi$
$734$	27117.6	1.36366
$735$	0	0
$736$	13784.7	0.690370
$737$	6572.01	0.328471
$738$	0	0
$739$	−14093.6	−0.701543	−0.350771	−	0.936461i	$-0.614081\pi$
$739$	−14093.6	−0.701543	−0.350771	+	0.936461i	$0.614081\pi$
$740$	10314.3	0.512381
$741$	0	0
$742$	6118.67	0.302727
$743$	17587.4	0.868397	0.434198	−	0.900817i	$-0.357032\pi$
$743$	17587.4	0.868397	0.434198	+	0.900817i	$0.357032\pi$
$744$	0	0
$745$	−23755.6	−1.16824
$746$	9945.63	0.488117
$747$	0	0
$748$	1313.97	0.0642293
$749$	13001.5	0.634263
$750$	0	0
$751$	−16587.0	−0.805948	−0.402974	−	0.915211i	$-0.632023\pi$
$751$	−16587.0	−0.805948	−0.402974	+	0.915211i	$0.632023\pi$
$752$	−2095.27	−0.101605
$753$	0	0
$754$	0	0
$755$	4054.22	0.195428
$756$	0	0
$757$	33819.5	1.62376	0.811882	−	0.583822i	$-0.198443\pi$
$757$	33819.5	1.62376	0.811882	+	0.583822i	$0.198443\pi$
$758$	−11962.2	−0.573202
$759$	0	0
$760$	17705.6	0.845066
$761$	28131.2	1.34002	0.670011	−	0.742351i	$-0.266290\pi$
$761$	28131.2	1.34002	0.670011	+	0.742351i	$0.266290\pi$
$762$	0	0
$763$	−598.589	−0.0284015
$764$	3842.18	0.181944
$765$	0	0
$766$	434.645	0.0205018
$767$	0	0
$768$	0	0
$769$	23735.0	1.11301	0.556506	−	0.830844i	$-0.312142\pi$
$769$	23735.0	1.11301	0.556506	+	0.830844i	$0.312142\pi$
$770$	−1481.70	−0.0693463
$771$	0	0
$772$	6716.75	0.313136
$773$	−9536.25	−0.443719	−0.221860	−	0.975079i	$-0.571213\pi$
$773$	−9536.25	−0.443719	−0.221860	+	0.975079i	$0.571213\pi$
$774$	0	0
$775$	−9862.92	−0.457144
$776$	−8539.34	−0.395032
$777$	0	0
$778$	20320.4	0.936403
$779$	−22662.1	−1.04230
$780$	0	0
$781$	−7630.84	−0.349619
$782$	−10642.8	−0.486685
$783$	0	0
$784$	8330.73	0.379498
$785$	416.457	0.0189350
$786$	0	0
$787$	−17696.2	−0.801527	−0.400763	−	0.916182i	$-0.631255\pi$
$787$	−17696.2	−0.801527	−0.400763	+	0.916182i	$0.631255\pi$
$788$	−13228.9	−0.598047
$789$	0	0
$790$	−7019.47	−0.316128
$791$	8195.14	0.368376
$792$	0	0
$793$	0	0
$794$	14734.7	0.658584
$795$	0	0
$796$	−3311.24	−0.147442
$797$	32566.7	1.44739	0.723696	−	0.690118i	$-0.242442\pi$
$797$	32566.7	1.44739	0.723696	+	0.690118i	$0.242442\pi$
$798$	0	0
$799$	−3066.04	−0.135755
$800$	7423.05	0.328056
$801$	0	0
$802$	19136.8	0.842572
$803$	9676.44	0.425248
$804$	0	0
$805$	−7340.36	−0.321384
$806$	0	0
$807$	0	0
$808$	−13641.4	−0.593940
$809$	−11622.3	−0.505089	−0.252544	−	0.967585i	$-0.581267\pi$
$809$	−11622.3	−0.505089	−0.252544	+	0.967585i	$0.581267\pi$
$810$	0	0
$811$	−6494.39	−0.281195	−0.140597	−	0.990067i	$-0.544902\pi$
$811$	−6494.39	−0.281195	−0.140597	+	0.990067i	$0.544902\pi$
$812$	−356.218	−0.0153951
$813$	0	0
$814$	8945.22	0.385172
$815$	−6169.92	−0.265181
$816$	0	0
$817$	5358.55	0.229464
$818$	−16744.7	−0.715725
$819$	0	0
$820$	6435.49	0.274070
$821$	34897.8	1.48349	0.741743	−	0.670685i	$-0.234000\pi$
$821$	34897.8	1.48349	0.741743	+	0.670685i	$0.234000\pi$
$822$	0	0
$823$	1502.61	0.0636426	0.0318213	−	0.999494i	$-0.489869\pi$
$823$	1502.61	0.0636426	0.0318213	+	0.999494i	$0.489869\pi$
$824$	27959.9	1.18208
$825$	0	0
$826$	2739.05	0.115380
$827$	27887.8	1.17262	0.586308	−	0.810088i	$-0.300581\pi$
$827$	27887.8	1.17262	0.586308	+	0.810088i	$0.300581\pi$
$828$	0	0
$829$	30843.7	1.29221	0.646107	−	0.763247i	$-0.276396\pi$
$829$	30843.7	1.29221	0.646107	+	0.763247i	$0.276396\pi$
$830$	9502.93	0.397412
$831$	0	0
$832$	0	0
$833$	12190.5	0.507053
$834$	0	0
$835$	−24409.8	−1.01166
$836$	−2583.71	−0.106890
$837$	0	0
$838$	−10153.7	−0.418560
$839$	−36598.8	−1.50599	−0.752997	−	0.658023i	$-0.771393\pi$
$839$	−36598.8	−1.50599	−0.752997	+	0.658023i	$0.771393\pi$
$840$	0	0
$841$	−24191.8	−0.991914
$842$	−4957.71	−0.202914
$843$	0	0
$844$	2208.58	0.0900741
$845$	0	0
$846$	0	0
$847$	−10334.5	−0.419241
$848$	10023.3	0.405898
$849$	0	0
$850$	−5731.15	−0.231267
$851$	44314.8	1.78507
$852$	0	0
$853$	−21578.4	−0.866155	−0.433077	−	0.901357i	$-0.642572\pi$
$853$	−21578.4	−0.866155	−0.433077	+	0.901357i	$0.642572\pi$
$854$	1820.67	0.0729534
$855$	0	0
$856$	38262.7	1.52779
$857$	31199.6	1.24359	0.621795	−	0.783180i	$-0.286404\pi$
$857$	31199.6	1.24359	0.621795	+	0.783180i	$0.286404\pi$
$858$	0	0
$859$	8035.71	0.319179	0.159590	−	0.987183i	$-0.448983\pi$
$859$	8035.71	0.319179	0.159590	+	0.987183i	$0.448983\pi$
$860$	−1521.70	−0.0603366
$861$	0	0
$862$	882.346	0.0348641
$863$	−8741.47	−0.344801	−0.172400	−	0.985027i	$-0.555152\pi$
$863$	−8741.47	−0.344801	−0.172400	+	0.985027i	$0.555152\pi$
$864$	0	0
$865$	−13407.9	−0.527030
$866$	−11558.9	−0.453566
$867$	0	0
$868$	−4339.90	−0.169707
$869$	3723.41	0.145349
$870$	0	0
$871$	0	0
$872$	−1761.62	−0.0684128
$873$	0	0
$874$	20927.5	0.809934
$875$	−12523.7	−0.483863
$876$	0	0
$877$	1744.13	0.0671552	0.0335776	−	0.999436i	$-0.489310\pi$
$877$	1744.13	0.0671552	0.0335776	+	0.999436i	$0.489310\pi$
$878$	−25242.8	−0.970276
$879$	0	0
$880$	−2427.24	−0.0929800
$881$	5688.81	0.217549	0.108775	−	0.994066i	$-0.465307\pi$
$881$	5688.81	0.217549	0.108775	+	0.994066i	$0.465307\pi$
$882$	0	0
$883$	−3940.14	−0.150165	−0.0750827	−	0.997177i	$-0.523922\pi$
$883$	−3940.14	−0.150165	−0.0750827	+	0.997177i	$0.523922\pi$
$884$	0	0
$885$	0	0
$886$	34812.8	1.32004
$887$	36880.4	1.39608	0.698039	−	0.716060i	$-0.254056\pi$
$887$	36880.4	1.39608	0.698039	+	0.716060i	$0.254056\pi$
$888$	0	0
$889$	−18191.3	−0.686295
$890$	12492.6	0.470507
$891$	0	0
$892$	3082.97	0.115724
$893$	6028.88	0.225922
$894$	0	0
$895$	−27843.0	−1.03987
$896$	−1274.75	−0.0475296
$897$	0	0
$898$	30493.5	1.13317
$899$	2402.62	0.0891343
$900$	0	0
$901$	14667.2	0.542327
$902$	5581.26	0.206026
$903$	0	0
$904$	24117.9	0.887334
$905$	−31194.8	−1.14580
$906$	0	0
$907$	17912.0	0.655741	0.327871	−	0.944723i	$-0.393669\pi$
$907$	17912.0	0.655741	0.327871	+	0.944723i	$0.393669\pi$
$908$	16068.7	0.587288
$909$	0	0
$910$	0	0
$911$	−51246.0	−1.86373	−0.931864	−	0.362807i	$-0.881819\pi$
$911$	−51246.0	−1.86373	−0.931864	+	0.362807i	$0.881819\pi$
$912$	0	0
$913$	−5040.74	−0.182721
$914$	−28235.5	−1.02182
$915$	0	0
$916$	9138.67	0.329640
$917$	16039.9	0.577627
$918$	0	0
$919$	25496.5	0.915182	0.457591	−	0.889163i	$-0.348712\pi$
$919$	25496.5	0.915182	0.457591	+	0.889163i	$0.348712\pi$
$920$	−21602.4	−0.774140
$921$	0	0
$922$	20396.9	0.728564
$923$	0	0
$924$	0	0
$925$	23863.4	0.848243
$926$	−15396.7	−0.546401
$927$	0	0
$928$	−1808.26	−0.0639646
$929$	−45048.6	−1.59095	−0.795477	−	0.605983i	$-0.792780\pi$
$929$	−45048.6	−1.59095	−0.795477	+	0.605983i	$0.792780\pi$
$930$	0	0
$931$	−23970.6	−0.843830
$932$	−7207.51	−0.253315
$933$	0	0
$934$	6506.16	0.227931
$935$	−3551.82	−0.124232
$936$	0	0
$937$	−2280.50	−0.0795099	−0.0397550	−	0.999209i	$-0.512658\pi$
$937$	−2280.50	−0.0795099	−0.0397550	+	0.999209i	$0.512658\pi$
$938$	−12613.3	−0.439060
$939$	0	0
$940$	−1712.06	−0.0594054
$941$	−31174.8	−1.07999	−0.539994	−	0.841669i	$-0.681573\pi$
$941$	−31174.8	−1.07999	−0.539994	+	0.841669i	$0.681573\pi$
$942$	0	0
$943$	27649.7	0.954823
$944$	4486.98	0.154702
$945$	0	0
$946$	−1319.71	−0.0453568
$947$	28394.0	0.974318	0.487159	−	0.873313i	$-0.338033\pi$
$947$	28394.0	0.974318	0.487159	+	0.873313i	$0.338033\pi$
$948$	0	0
$949$	0	0
$950$	11269.4	0.384871
$951$	0	0
$952$	−9166.83	−0.312078
$953$	33916.8	1.15286	0.576429	−	0.817147i	$-0.304446\pi$
$953$	33916.8	1.15286	0.576429	+	0.817147i	$0.304446\pi$
$954$	0	0
$955$	−10385.9	−0.351915
$956$	−2378.57	−0.0804693
$957$	0	0
$958$	−38073.2	−1.28402
$959$	6216.19	0.209313
$960$	0	0
$961$	−519.228	−0.0174290
$962$	0	0
$963$	0	0
$964$	−10679.1	−0.356794
$965$	−18156.2	−0.605666
$966$	0	0
$967$	−10792.9	−0.358920	−0.179460	−	0.983765i	$-0.557435\pi$
$967$	−10792.9	−0.358920	−0.179460	+	0.983765i	$0.557435\pi$
$968$	−30413.9	−1.00986
$969$	0	0
$970$	6350.19	0.210198
$971$	−7431.13	−0.245599	−0.122799	−	0.992432i	$-0.539187\pi$
$971$	−7431.13	−0.245599	−0.122799	+	0.992432i	$0.539187\pi$
$972$	0	0
$973$	23560.9	0.776288
$974$	30775.1	1.01242
$975$	0	0
$976$	2982.54	0.0978164
$977$	−12422.6	−0.406790	−0.203395	−	0.979097i	$-0.565198\pi$
$977$	−12422.6	−0.406790	−0.203395	+	0.979097i	$0.565198\pi$
$978$	0	0
$979$	−6626.56	−0.216329
$980$	6807.08	0.221882
$981$	0	0
$982$	−26398.2	−0.857840
$983$	38791.6	1.25866	0.629328	−	0.777140i	$-0.283330\pi$
$983$	38791.6	1.25866	0.629328	+	0.777140i	$0.283330\pi$
$984$	0	0
$985$	35759.3	1.15674
$986$	1396.11	0.0450926
$987$	0	0
$988$	0	0
$989$	−6537.89	−0.210205
$990$	0	0
$991$	−1066.51	−0.0341863	−0.0170932	−	0.999854i	$-0.505441\pi$
$991$	−1066.51	−0.0341863	−0.0170932	+	0.999854i	$0.505441\pi$
$992$	−22030.6	−0.705113
$993$	0	0
$994$	14645.4	0.467329
$995$	8950.66	0.285181
$996$	0	0
$997$	−30651.6	−0.973665	−0.486833	−	0.873495i	$-0.661848\pi$
$997$	−30651.6	−0.973665	−0.486833	+	0.873495i	$0.661848\pi$
$998$	11640.0	0.369198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bg.1.4		9
3.2	odd	2		169.4.a.l.1.6	yes	9
13.12	even	2		1521.4.a.bh.1.6		9
39.2	even	12		169.4.e.h.147.13		36
39.5	even	4		169.4.b.g.168.6		18
39.8	even	4		169.4.b.g.168.13		18
39.11	even	12		169.4.e.h.147.6		36
39.17	odd	6		169.4.c.l.146.6		18
39.20	even	12		169.4.e.h.23.13		36
39.23	odd	6		169.4.c.l.22.6		18
39.29	odd	6		169.4.c.k.22.4		18
39.32	even	12		169.4.e.h.23.6		36
39.35	odd	6		169.4.c.k.146.4		18
39.38	odd	2		169.4.a.k.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.4	✓	9	39.38	odd	2
169.4.a.l.1.6	yes	9	3.2	odd	2
169.4.b.g.168.6		18	39.5	even	4
169.4.b.g.168.13		18	39.8	even	4
169.4.c.k.22.4		18	39.29	odd	6
169.4.c.k.146.4		18	39.35	odd	6
169.4.c.l.22.6		18	39.23	odd	6
169.4.c.l.146.6		18	39.17	odd	6
169.4.e.h.23.6		36	39.32	even	12
169.4.e.h.23.13		36	39.20	even	12
169.4.e.h.147.6		36	39.11	even	12
169.4.e.h.147.13		36	39.2	even	12
1521.4.a.bg.1.4		9	1.1	even	1	trivial
1521.4.a.bh.1.6		9	13.12	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22799 q^{2} -3.03607 q^{4} +8.20685 q^{5} +8.35495 q^{7} +24.5882 q^{8} +O(q^{10})\)	\(q-2.22799 q^{2} -3.03607 q^{4} +8.20685 q^{5} +8.35495 q^{7} +24.5882 q^{8} -18.2848 q^{10} +9.69898 q^{11} -18.6147 q^{14} -30.4937 q^{16} -44.6219 q^{17} +87.7418 q^{19} -24.9166 q^{20} -21.6092 q^{22} -107.053 q^{23} -57.6475 q^{25} -25.3662 q^{28} +14.0430 q^{29} +171.090 q^{31} -128.766 q^{32} +99.4170 q^{34} +68.5678 q^{35} -413.954 q^{37} -195.488 q^{38} +201.792 q^{40} -258.282 q^{41} +61.0718 q^{43} -29.4468 q^{44} +238.512 q^{46} +68.7115 q^{47} -273.195 q^{49} +128.438 q^{50} -328.701 q^{53} +79.5981 q^{55} +205.433 q^{56} -31.2876 q^{58} -147.144 q^{59} -97.8083 q^{61} -381.186 q^{62} +530.839 q^{64} +677.597 q^{67} +135.475 q^{68} -152.768 q^{70} -786.767 q^{71} +997.675 q^{73} +922.285 q^{74} -266.390 q^{76} +81.0345 q^{77} +383.897 q^{79} -250.258 q^{80} +575.448 q^{82} -519.718 q^{83} -366.205 q^{85} -136.067 q^{86} +238.481 q^{88} -683.222 q^{89} +325.019 q^{92} -153.088 q^{94} +720.085 q^{95} -347.294 q^{97} +608.675 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8	\( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8} - 147 q^{10} - 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} + 161 q^{19} - 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} + 344 q^{28} - 280 q^{29} + 706 q^{31} - 680 q^{32} + 216 q^{34} - 20 q^{35} + 298 q^{37} + 739 q^{38} + 13 q^{40} - 1201 q^{41} - 533 q^{43} - 355 q^{44} - 840 q^{46} - 956 q^{47} + 403 q^{49} + 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} - 2877 q^{58} - 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} - 931 q^{67} + 1536 q^{68} - 4854 q^{70} - 2046 q^{71} - 45 q^{73} + 1990 q^{74} - 3608 q^{76} + 718 q^{77} + 412 q^{79} + 787 q^{80} + 2757 q^{82} - 3709 q^{83} - 2106 q^{85} - 125 q^{86} - 636 q^{88} - 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} - 1087 q^{97} + 282 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8 - 147 * q^10 - 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 + 161 * q^19 - 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 + 344 * q^28 - 280 * q^29 + 706 * q^31 - 680 * q^32 + 216 * q^34 - 20 * q^35 + 298 * q^37 + 739 * q^38 + 13 * q^40 - 1201 * q^41 - 533 * q^43 - 355 * q^44 - 840 * q^46 - 956 * q^47 + 403 * q^49 + 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 - 2877 * q^58 - 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 - 931 * q^67 + 1536 * q^68 - 4854 * q^70 - 2046 * q^71 - 45 * q^73 + 1990 * q^74 - 3608 * q^76 + 718 * q^77 + 412 * q^79 + 787 * q^80 + 2757 * q^82 - 3709 * q^83 - 2106 * q^85 - 125 * q^86 - 636 * q^88 - 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 - 1087 * q^97 + 282 * q^98

Introduction

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