LMFDB - Embedded newform 500.3.d.a.499.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,3,Mod(499,500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("500.499");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 500 = 2^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	500.d (of order $2$, degree $1$, not minimal)

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:	$13.6240132180$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			499.2
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	500.499

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.01719 q^{3} +4.00000 q^{4} +6.03437 q^{6} -4.64990 q^{7} -8.00000 q^{8} +0.103412 q^{9} +O(q^{10})$	$q-2.00000 q^{2} -3.01719 q^{3} +4.00000 q^{4} +6.03437 q^{6} -4.64990 q^{7} -8.00000 q^{8} +0.103412 q^{9} -12.0687 q^{12} +9.29980 q^{14} +16.0000 q^{16} -0.206823 q^{18} +14.0296 q^{21} -27.7108 q^{23} +24.1375 q^{24} +26.8427 q^{27} -18.5996 q^{28} -57.8374 q^{29} -32.0000 q^{32} +0.413647 q^{36} +70.1981 q^{41} -28.0592 q^{42} +14.7940 q^{43} +55.4216 q^{46} +88.0823 q^{47} -48.2750 q^{48} -27.3784 q^{49} -53.6853 q^{54} +37.1992 q^{56} +115.675 q^{58} +110.011 q^{61} -0.480854 q^{63} +64.0000 q^{64} +116.000 q^{67} +83.6086 q^{69} -0.827294 q^{72} -81.9200 q^{81} -140.396 q^{82} +148.231 q^{83} +56.1184 q^{84} -29.5880 q^{86} +174.506 q^{87} +51.7927 q^{89} -110.843 q^{92} -176.165 q^{94} +96.5500 q^{96} +54.7569 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9} - 16 q^{12} - 8 q^{14} + 64 q^{16} - 76 q^{18} + 16 q^{21} - 44 q^{23} + 32 q^{24} + 8 q^{27} + 16 q^{28} + 22 q^{29} - 128 q^{32} + 152 q^{36} - 62 q^{41} - 32 q^{42} + 76 q^{43} + 88 q^{46} + 4 q^{47} - 64 q^{48} + 278 q^{49} - 16 q^{54} - 32 q^{56} - 44 q^{58} + 58 q^{61} + 648 q^{63} + 256 q^{64} + 464 q^{67} - 176 q^{69} - 304 q^{72} + 500 q^{81} + 124 q^{82} + 76 q^{83} + 64 q^{84} - 152 q^{86} + 468 q^{87} + 142 q^{89} - 176 q^{92} - 8 q^{94} + 128 q^{96} - 556 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9 - 16 * q^12 - 8 * q^14 + 64 * q^16 - 76 * q^18 + 16 * q^21 - 44 * q^23 + 32 * q^24 + 8 * q^27 + 16 * q^28 + 22 * q^29 - 128 * q^32 + 152 * q^36 - 62 * q^41 - 32 * q^42 + 76 * q^43 + 88 * q^46 + 4 * q^47 - 64 * q^48 + 278 * q^49 - 16 * q^54 - 32 * q^56 - 44 * q^58 + 58 * q^61 + 648 * q^63 + 256 * q^64 + 464 * q^67 - 176 * q^69 - 304 * q^72 + 500 * q^81 + 124 * q^82 + 76 * q^83 + 64 * q^84 - 152 * q^86 + 468 * q^87 + 142 * q^89 - 176 * q^92 - 8 * q^94 + 128 * q^96 - 556 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/500\mathbb{Z}\right)^\times$.

$n$	$251$	$377$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−2.00000	−1.00000
$2$	−2.00000	−1.00000
$3$	−3.01719	−1.00573	−0.502864	−	0.864365i	$-0.667720\pi$
$3$	−3.01719	−1.00573	−0.502864	+	0.864365i	$0.667720\pi$
$4$	4.00000	1.00000
$5$	0	0
$5$	0	0
$6$	6.03437	1.00573
$7$	−4.64990	−0.664271	−0.332136	−	0.943232i	$-0.607769\pi$
$7$	−4.64990	−0.664271	−0.332136	+	0.943232i	$0.607769\pi$
$8$	−8.00000	−1.00000
$9$	0.103412	0.0114902
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−12.0687	−1.00573
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	9.29980	0.664271
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−0.206823	−0.0114902
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	14.0296	0.668077
$22$	0	0
$23$	−27.7108	−1.20482	−0.602408	−	0.798188i	$-0.705792\pi$
$23$	−27.7108	−1.20482	−0.602408	+	0.798188i	$0.705792\pi$
$24$	24.1375	1.00573
$25$	0	0
$26$	0	0
$27$	26.8427	0.994173
$28$	−18.5996	−0.664271
$29$	−57.8374	−1.99439	−0.997197	−	0.0748216i	$-0.976161\pi$
$29$	−57.8374	−1.99439	−0.997197	+	0.0748216i	$0.976161\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−32.0000	−1.00000
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0.413647	0.0114902
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	70.1981	1.71215	0.856074	−	0.516853i	$-0.172896\pi$
$41$	70.1981	1.71215	0.856074	+	0.516853i	$0.172896\pi$
$42$	−28.0592	−0.668077
$43$	14.7940	0.344046	0.172023	−	0.985093i	$-0.444970\pi$
$43$	14.7940	0.344046	0.172023	+	0.985093i	$0.444970\pi$
$44$	0	0
$45$	0	0
$46$	55.4216	1.20482
$47$	88.0823	1.87409	0.937045	−	0.349208i	$-0.113549\pi$
$47$	88.0823	1.87409	0.937045	+	0.349208i	$0.113549\pi$
$48$	−48.2750	−1.00573
$49$	−27.3784	−0.558744
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−53.6853	−0.994173
$55$	0	0
$56$	37.1992	0.664271
$57$	0	0
$58$	115.675	1.99439
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	110.011	1.80345	0.901727	−	0.432306i	$-0.142300\pi$
$61$	110.011	1.80345	0.901727	+	0.432306i	$0.142300\pi$
$62$	0	0
$63$	−0.480854	−0.00763261
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	116.000	1.73134	0.865672	−	0.500612i	$-0.166892\pi$
$67$	116.000	1.73134	0.865672	+	0.500612i	$0.166892\pi$
$68$	0	0
$69$	83.6086	1.21172
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.827294	−0.0114902
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−81.9200	−1.01136
$82$	−140.396	−1.71215
$83$	148.231	1.78591	0.892957	−	0.450141i	$-0.148626\pi$
$83$	148.231	1.78591	0.892957	+	0.450141i	$0.148626\pi$
$84$	56.1184	0.668077
$85$	0	0
$86$	−29.5880	−0.344046
$87$	174.506	2.00582
$88$	0	0
$89$	51.7927	0.581940	0.290970	−	0.956732i	$-0.406022\pi$
$89$	51.7927	0.581940	0.290970	+	0.956732i	$0.406022\pi$
$90$	0	0
$91$	0	0
$92$	−110.843	−1.20482
$93$	0	0
$94$	−176.165	−1.87409
$95$	0	0
$96$	96.5500	1.00573
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	54.7569	0.558744
$99$	0	0
$100$	0	0
$101$	−193.332	−1.91417	−0.957087	−	0.289799i	$-0.906411\pi$
$101$	−193.332	−1.91417	−0.957087	+	0.289799i	$0.906411\pi$
$102$	0	0
$103$	44.0000	0.427184	0.213592	−	0.976923i	$-0.431484\pi$
$103$	44.0000	0.427184	0.213592	+	0.976923i	$0.431484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−204.195	−1.90836	−0.954182	−	0.299226i	$-0.903272\pi$
$107$	−204.195	−1.90836	−0.954182	+	0.299226i	$0.903272\pi$
$108$	107.371	0.994173
$109$	215.899	1.98072	0.990362	−	0.138506i	$-0.0442299\pi$
$109$	215.899	1.98072	0.990362	+	0.138506i	$0.0442299\pi$
$110$	0	0
$111$	0	0
$112$	−74.3984	−0.664271
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−231.350	−1.99439
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	−220.021	−1.80345
$123$	−211.801	−1.72196
$124$	0	0
$125$	0	0
$126$	0.961708	0.00763261
$127$	162.246	1.27753	0.638765	−	0.769402i	$-0.279446\pi$
$127$	162.246	1.27753	0.638765	+	0.769402i	$0.279446\pi$
$128$	−128.000	−1.00000
$129$	−44.6362	−0.346017
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	−232.000	−1.73134
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−167.217	−1.21172
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−265.761	−1.88483
$142$	0	0
$143$	0	0
$144$	1.65459	0.0114902
$145$	0	0
$146$	0	0
$147$	82.6059	0.561945
$148$	0	0
$149$	−287.994	−1.93285	−0.966424	−	0.256951i	$-0.917282\pi$
$149$	−287.994	−1.93285	−0.966424	+	0.256951i	$0.917282\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	128.852	0.800325
$162$	163.840	1.01136
$163$	32.9265	0.202003	0.101002	−	0.994886i	$-0.467795\pi$
$163$	32.9265	0.202003	0.101002	+	0.994886i	$0.467795\pi$
$164$	280.792	1.71215
$165$	0	0
$166$	−296.462	−1.78591
$167$	141.516	0.847400	0.423700	−	0.905803i	$-0.360731\pi$
$167$	141.516	0.847400	0.423700	+	0.905803i	$0.360731\pi$
$168$	−112.237	−0.668077
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	59.1760	0.344046
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−349.013	−2.00582
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−103.585	−0.581940
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−59.5890	−0.329221	−0.164611	−	0.986359i	$-0.552637\pi$
$181$	−59.5890	−0.329221	−0.164611	+	0.986359i	$0.552637\pi$
$182$	0	0
$183$	−331.923	−1.81379
$184$	221.686	1.20482
$185$	0	0
$186$	0	0
$187$	0	0
$188$	352.329	1.87409
$189$	−124.816	−0.660400
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−193.100	−1.00573
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−109.514	−0.558744
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−349.994	−1.74126
$202$	386.663	1.91417
$203$	268.938	1.32482
$204$	0	0
$205$	0	0
$206$	−88.0000	−0.427184
$207$	−2.86562	−0.0138436
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	408.390	1.90836
$215$	0	0
$216$	−214.741	−0.994173
$217$	0	0
$218$	−431.798	−1.98072
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−224.050	−1.00471	−0.502354	−	0.864662i	$-0.667532\pi$
$223$	−224.050	−1.00471	−0.502354	+	0.864662i	$0.667532\pi$
$224$	148.797	0.664271
$225$	0	0
$226$	0	0
$227$	−453.615	−1.99831	−0.999153	−	0.0411555i	$-0.986896\pi$
$227$	−453.615	−1.99831	−0.999153	+	0.0411555i	$0.986896\pi$
$228$	0	0
$229$	−8.84461	−0.0386228	−0.0193114	−	0.999814i	$-0.506147\pi$
$229$	−8.84461	−0.0386228	−0.0193114	+	0.999814i	$0.506147\pi$
$230$	0	0
$231$	0	0
$232$	462.699	1.99439
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−23.5161	−0.0975770	−0.0487885	−	0.998809i	$-0.515536\pi$
$241$	−23.5161	−0.0975770	−0.0487885	+	0.998809i	$0.515536\pi$
$242$	−242.000	−1.00000
$243$	5.58396	0.0229793
$244$	440.043	1.80345
$245$	0	0
$246$	423.601	1.72196
$247$	0	0
$248$	0	0
$249$	−447.240	−1.79615
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.92342	−0.00763261
$253$	0	0
$254$	−324.493	−1.27753
$255$	0	0
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	89.2725	0.346017
$259$	0	0
$260$	0	0
$261$	−5.98107	−0.0229160
$262$	0	0
$263$	508.833	1.93473	0.967363	−	0.253394i	$-0.0815470\pi$
$263$	508.833	1.93473	0.967363	+	0.253394i	$0.0815470\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−156.268	−0.585274
$268$	464.000	1.73134
$269$	38.0000	0.141264	0.0706320	−	0.997502i	$-0.477498\pi$
$269$	38.0000	0.141264	0.0706320	+	0.997502i	$0.477498\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	334.434	1.21172
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	558.976	1.98924	0.994620	−	0.103595i	$-0.0330347\pi$
$281$	558.976	1.98924	0.994620	+	0.103595i	$0.0330347\pi$
$282$	531.521	1.88483
$283$	−316.000	−1.11661	−0.558304	−	0.829637i	$-0.688548\pi$
$283$	−316.000	−1.11661	−0.558304	+	0.829637i	$0.688548\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−326.414	−1.13733
$288$	−3.30918	−0.0114902
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−165.212	−0.561945
$295$	0	0
$296$	0	0
$297$	0	0
$298$	575.989	1.93285
$299$	0	0
$300$	0	0
$301$	−68.7906	−0.228540
$302$	0	0
$303$	583.318	1.92514
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−568.920	−1.85316	−0.926579	−	0.376099i	$-0.877265\pi$
$307$	−568.920	−1.85316	−0.926579	+	0.376099i	$0.877265\pi$
$308$	0	0
$309$	−132.756	−0.429632
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	616.094	1.91930
$322$	−257.705	−0.800325
$323$	0	0
$324$	−327.680	−1.01136
$325$	0	0
$326$	−65.8530	−0.202003
$327$	−651.407	−1.99207
$328$	−561.585	−1.71215
$329$	−409.574	−1.24490
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	592.924	1.78591
$333$	0	0
$334$	−283.032	−0.847400
$335$	0	0
$336$	224.474	0.668077
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−338.000	−1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	355.152	1.03543
$344$	−118.352	−0.344046
$345$	0	0
$346$	0	0
$347$	−496.030	−1.42948	−0.714741	−	0.699389i	$-0.753455\pi$
$347$	−496.030	−1.42948	−0.714741	+	0.699389i	$0.753455\pi$
$348$	698.025	2.00582
$349$	427.869	1.22598	0.612992	−	0.790089i	$-0.289966\pi$
$349$	427.869	1.22598	0.612992	+	0.790089i	$0.289966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	207.171	0.581940
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	119.178	0.329221
$363$	−365.080	−1.00573
$364$	0	0
$365$	0	0
$366$	663.846	1.81379
$367$	656.702	1.78938	0.894689	−	0.446689i	$-0.147397\pi$
$367$	656.702	1.78938	0.894689	+	0.446689i	$0.147397\pi$
$368$	−443.372	−1.20482
$369$	7.25931	0.0196729
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−704.658	−1.87409
$377$	0	0
$378$	249.631	0.660400
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−489.527	−1.28485
$382$	0	0
$383$	−713.710	−1.86347	−0.931736	−	0.363137i	$-0.881706\pi$
$383$	−713.710	−1.86347	−0.931736	+	0.363137i	$0.881706\pi$
$384$	386.200	1.00573
$385$	0	0
$386$	0	0
$387$	1.52987	0.00395316
$388$	0	0
$389$	652.125	1.67641	0.838207	−	0.545352i	$-0.183604\pi$
$389$	652.125	1.67641	0.838207	+	0.545352i	$0.183604\pi$
$390$	0	0
$391$	0	0
$392$	219.028	0.558744
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	765.237	1.90832	0.954160	−	0.299296i	$-0.0967519\pi$
$401$	765.237	1.90832	0.954160	+	0.299296i	$0.0967519\pi$
$402$	699.987	1.74126
$403$	0	0
$404$	−773.327	−1.91417
$405$	0	0
$406$	−537.876	−1.32482
$407$	0	0
$408$	0	0
$409$	743.463	1.81776	0.908879	−	0.417059i	$-0.136939\pi$
$409$	743.463	1.81776	0.908879	+	0.417059i	$0.136939\pi$
$410$	0	0
$411$	0	0
$412$	176.000	0.427184
$413$	0	0
$414$	5.73124	0.0138436
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	65.8191	0.156340	0.0781699	−	0.996940i	$-0.475092\pi$
$421$	65.8191	0.156340	0.0781699	+	0.996940i	$0.475092\pi$
$422$	0	0
$423$	9.10874	0.0215337
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−511.539	−1.19798
$428$	−816.780	−1.90836
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	429.483	0.994173
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	863.595	1.98072
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−2.83125	−0.00642007
$442$	0	0
$443$	415.284	0.937437	0.468718	−	0.883348i	$-0.344716\pi$
$443$	415.284	0.937437	0.468718	+	0.883348i	$0.344716\pi$
$444$	0	0
$445$	0	0
$446$	448.099	1.00471
$447$	868.933	1.94392
$448$	−297.594	−0.664271
$449$	398.000	0.886414	0.443207	−	0.896419i	$-0.353841\pi$
$449$	398.000	0.886414	0.443207	+	0.896419i	$0.353841\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	907.231	1.99831
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	17.6892	0.0386228
$459$	0	0
$460$	0	0
$461$	−460.385	−0.998666	−0.499333	−	0.866410i	$-0.666422\pi$
$461$	−460.385	−0.998666	−0.499333	+	0.866410i	$0.666422\pi$
$462$	0	0
$463$	−925.642	−1.99923	−0.999613	−	0.0278164i	$-0.991145\pi$
$463$	−925.642	−1.99923	−0.999613	+	0.0278164i	$0.991145\pi$
$464$	−925.399	−1.99439
$465$	0	0
$466$	0	0
$467$	644.450	1.37998	0.689989	−	0.723820i	$-0.257615\pi$
$467$	644.450	1.37998	0.689989	+	0.723820i	$0.257615\pi$
$468$	0	0
$469$	−539.388	−1.15008
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	47.0321	0.0975770
$483$	−388.771	−0.804910
$484$	484.000	1.00000
$485$	0	0
$486$	−11.1679	−0.0229793
$487$	−953.429	−1.95776	−0.978880	−	0.204435i	$-0.934464\pi$
$487$	−953.429	−1.95776	−0.978880	+	0.204435i	$0.934464\pi$
$488$	−880.086	−1.80345
$489$	−99.3454	−0.203160
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−847.203	−1.72196
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	894.481	1.79615
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−426.980	−0.852255
$502$	0	0
$503$	985.525	1.95929	0.979647	−	0.200730i	$-0.0643313\pi$
$503$	985.525	1.95929	0.979647	+	0.200730i	$0.0643313\pi$
$504$	3.84683	0.00763261
$505$	0	0
$506$	0	0
$507$	−509.904	−1.00573
$508$	648.985	1.27753
$509$	−982.000	−1.92927	−0.964637	−	0.263584i	$-0.915095\pi$
$509$	−982.000	−1.92927	−0.964637	+	0.263584i	$0.915095\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−178.545	−0.346017
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−491.436	−0.943256	−0.471628	−	0.881798i	$-0.656333\pi$
$521$	−491.436	−0.943256	−0.471628	+	0.881798i	$0.656333\pi$
$522$	11.9621	0.0229160
$523$	−931.823	−1.78169	−0.890844	−	0.454309i	$-0.849886\pi$
$523$	−931.823	−1.78169	−0.890844	+	0.454309i	$0.849886\pi$
$524$	0	0
$525$	0	0
$526$	−1017.67	−1.93473
$527$	0	0
$528$	0	0
$529$	238.887	0.451583
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	312.536	0.585274
$535$	0	0
$536$	−928.000	−1.73134
$537$	0	0
$538$	−76.0000	−0.141264
$539$	0	0
$540$	0	0
$541$	−857.878	−1.58573	−0.792863	−	0.609400i	$-0.791410\pi$
$541$	−857.878	−1.58573	−0.792863	+	0.609400i	$0.791410\pi$
$542$	0	0
$543$	179.791	0.331107
$544$	0	0
$545$	0	0
$546$	0	0
$547$	790.229	1.44466	0.722330	−	0.691549i	$-0.243071\pi$
$547$	790.229	1.44466	0.722330	+	0.691549i	$0.243071\pi$
$548$	0	0
$549$	11.3764	0.0207220
$550$	0	0
$551$	0	0
$552$	−668.869	−1.21172
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−1117.95	−1.98924
$563$	1124.00	1.99645	0.998224	−	0.0595755i	$-0.0189747\pi$
$563$	1124.00	1.99645	0.998224	+	0.0595755i	$0.0189747\pi$
$564$	−1063.04	−1.88483
$565$	0	0
$566$	632.000	1.11661
$567$	380.920	0.671816
$568$	0	0
$569$	−1023.00	−1.79788	−0.898941	−	0.438069i	$-0.855663\pi$
$569$	−1023.00	−1.79788	−0.898941	+	0.438069i	$0.855663\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	652.828	1.13733
$575$	0	0
$576$	6.61835	0.0114902
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−578.000	−1.00000
$579$	0	0
$580$	0	0
$581$	−689.259	−1.18633
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1076.00	1.83305	0.916525	−	0.399978i	$-0.130982\pi$
$587$	1076.00	1.83305	0.916525	+	0.399978i	$0.130982\pi$
$588$	330.423	0.561945
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	−1151.98	−1.93285
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	942.651	1.56847	0.784235	−	0.620463i	$-0.213055\pi$
$601$	942.651	1.56847	0.784235	+	0.620463i	$0.213055\pi$
$602$	137.581	0.228540
$603$	11.9958	0.0198935
$604$	0	0
$605$	0	0
$606$	−1166.64	−1.92514
$607$	−964.000	−1.58814	−0.794069	−	0.607827i	$-0.792041\pi$
$607$	−964.000	−1.58814	−0.794069	+	0.607827i	$0.792041\pi$
$608$	0	0
$609$	−811.436	−1.33241
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1137.84	1.85316
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	265.512	0.429632
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−743.831	−1.19780
$622$	0	0
$623$	−240.831	−0.386566
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	209.768	0.327251	0.163626	−	0.986522i	$-0.447681\pi$
$641$	209.768	0.327251	0.163626	+	0.986522i	$0.447681\pi$
$642$	−1232.19	−1.91930
$643$	390.780	0.607745	0.303873	−	0.952713i	$-0.401720\pi$
$643$	390.780	0.607745	0.303873	+	0.952713i	$0.401720\pi$
$644$	515.409	0.800325
$645$	0	0
$646$	0	0
$647$	956.000	1.47759	0.738794	−	0.673931i	$-0.235395\pi$
$647$	956.000	1.47759	0.738794	+	0.673931i	$0.235395\pi$
$648$	655.360	1.01136
$649$	0	0
$650$	0	0
$651$	0	0
$652$	131.706	0.202003
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	1302.81	1.99207
$655$	0	0
$656$	1123.17	1.71215
$657$	0	0
$658$	819.147	1.24490
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1317.02	−1.99247	−0.996236	−	0.0866803i	$-0.972374\pi$
$661$	−1317.02	−1.99247	−0.996236	+	0.0866803i	$0.972374\pi$
$662$	0	0
$663$	0	0
$664$	−1185.85	−1.78591
$665$	0	0
$666$	0	0
$667$	1602.72	2.40288
$668$	566.063	0.847400
$669$	676.000	1.01046
$670$	0	0
$671$	0	0
$672$	−448.948	−0.668077
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	676.000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1368.64	2.00975
$682$	0	0
$683$	1014.84	1.48586	0.742931	−	0.669367i	$-0.233435\pi$
$683$	1014.84	1.48586	0.742931	+	0.669367i	$0.233435\pi$
$684$	0	0
$685$	0	0
$686$	−710.304	−1.03543
$687$	26.6858	0.0388440
$688$	236.704	0.344046
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	992.061	1.42948
$695$	0	0
$696$	−1396.05	−2.00582
$697$	0	0
$698$	−855.737	−1.22598
$699$	0	0
$700$	0	0
$701$	902.000	1.28673	0.643367	−	0.765558i	$-0.277537\pi$
$701$	902.000	1.28673	0.643367	+	0.765558i	$0.277537\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	898.973	1.27153
$708$	0	0
$709$	160.815	0.226820	0.113410	−	0.993548i	$-0.463823\pi$
$709$	160.815	0.226820	0.113410	+	0.993548i	$0.463823\pi$
$710$	0	0
$711$	0	0
$712$	−414.341	−0.581940
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−204.596	−0.283766
$722$	−722.000	−1.00000
$723$	70.9523	0.0981360
$724$	−238.356	−0.329221
$725$	0	0
$726$	730.159	1.00573
$727$	660.664	0.908754	0.454377	−	0.890809i	$-0.349862\pi$
$727$	660.664	0.908754	0.454377	+	0.890809i	$0.349862\pi$
$728$	0	0
$729$	720.432	0.988247
$730$	0	0
$731$	0	0
$732$	−1327.69	−1.81379
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1313.40	−1.78938
$735$	0	0
$736$	886.745	1.20482
$737$	0	0
$738$	−14.5186	−0.0196729
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	764.000	1.02826	0.514132	−	0.857711i	$-0.328114\pi$
$743$	764.000	1.02826	0.514132	+	0.857711i	$0.328114\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.3288	0.0205205
$748$	0	0
$749$	949.486	1.26767
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1409.32	1.87409
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−499.263	−0.660400
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1079.01	−1.41788	−0.708942	−	0.705266i	$-0.750827\pi$
$761$	−1079.01	−1.41788	−0.708942	+	0.705266i	$0.750827\pi$
$762$	979.055	1.28485
$763$	−1003.91	−1.31574
$764$	0	0
$765$	0	0
$766$	1427.42	1.86347
$767$	0	0
$768$	−772.400	−1.00573
$769$	299.846	0.389917	0.194958	−	0.980812i	$-0.437543\pi$
$769$	299.846	0.389917	0.194958	+	0.980812i	$0.437543\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−3.05975	−0.00395316
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−1304.25	−1.67641
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1552.51	−1.98277
$784$	−438.055	−0.558744
$785$	0	0
$786$	0	0
$787$	1528.74	1.94249	0.971244	−	0.238086i	$-0.0765200\pi$
$787$	1528.74	1.94249	0.971244	+	0.238086i	$0.0765200\pi$
$788$	0	0
$789$	−1535.24	−1.94581
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5.35597	0.00668661
$802$	−1530.47	−1.90832
$803$	0	0
$804$	−1399.97	−1.74126
$805$	0	0
$806$	0	0
$807$	−114.653	−0.142073
$808$	1546.65	1.91417
$809$	−482.314	−0.596186	−0.298093	−	0.954537i	$-0.596351\pi$
$809$	−482.314	−0.596186	−0.298093	+	0.954537i	$0.596351\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1075.75	1.32482
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1486.93	−1.81776
$819$	0	0
$820$	0	0
$821$	−1418.80	−1.72813	−0.864067	−	0.503377i	$-0.832091\pi$
$821$	−1418.80	−1.72813	−0.864067	+	0.503377i	$0.832091\pi$
$822$	0	0
$823$	−1396.00	−1.69623	−0.848117	−	0.529810i	$-0.822263\pi$
$823$	−1396.00	−1.69623	−0.848117	+	0.529810i	$0.822263\pi$
$824$	−352.000	−0.427184
$825$	0	0
$826$	0	0
$827$	596.000	0.720677	0.360339	−	0.932822i	$-0.382661\pi$
$827$	596.000	0.720677	0.360339	+	0.932822i	$0.382661\pi$
$828$	−11.4625	−0.0138436
$829$	1171.27	1.41288	0.706438	−	0.707775i	$-0.250301\pi$
$829$	1171.27	1.41288	0.706438	+	0.707775i	$0.250301\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2504.17	2.97761
$842$	−131.638	−0.156340
$843$	−1686.54	−2.00063
$844$	0	0
$845$	0	0
$846$	−18.2175	−0.0215337
$847$	−562.638	−0.664271
$848$	0	0
$849$	953.431	1.12300
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	1023.08	1.19798
$855$	0	0
$856$	1633.56	1.90836
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	984.852	1.14385
$862$	0	0
$863$	1646.88	1.90832	0.954158	−	0.299304i	$-0.0967544\pi$
$863$	1646.88	1.90832	0.954158	+	0.299304i	$0.0967544\pi$
$864$	−858.965	−0.994173
$865$	0	0
$866$	0	0
$867$	−871.967	−1.00573
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1727.19	−1.98072
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	898.919	1.02034	0.510170	−	0.860074i	$-0.329583\pi$
$881$	898.919	1.02034	0.510170	+	0.860074i	$0.329583\pi$
$882$	5.66251	0.00642007
$883$	1749.93	1.98180	0.990900	−	0.134603i	$-0.0429760\pi$
$883$	1749.93	1.98180	0.990900	+	0.134603i	$0.0429760\pi$
$884$	0	0
$885$	0	0
$886$	−830.569	−0.937437
$887$	−95.4499	−0.107610	−0.0538049	−	0.998551i	$-0.517135\pi$
$887$	−95.4499	−0.107610	−0.0538049	+	0.998551i	$0.517135\pi$
$888$	0	0
$889$	−754.429	−0.848627
$890$	0	0
$891$	0	0
$892$	−896.199	−1.00471
$893$	0	0
$894$	−1737.87	−1.94392
$895$	0	0
$896$	595.187	0.664271
$897$	0	0
$898$	−796.000	−0.886414
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	207.554	0.229849
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1303.16	−1.43678	−0.718391	−	0.695640i	$-0.755121\pi$
$907$	−1303.16	−1.43678	−0.718391	+	0.695640i	$0.755121\pi$
$908$	−1814.46	−1.99831
$909$	−19.9928	−0.0219942
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−35.3785	−0.0386228
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1716.54	1.86378
$922$	920.770	0.998666
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1851.28	1.99923
$927$	4.55012	0.00490843
$928$	1850.80	1.99439
$929$	1495.62	1.60992	0.804960	−	0.593329i	$-0.202187\pi$
$929$	1495.62	1.60992	0.804960	+	0.593329i	$0.202187\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−1288.90	−1.37998
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	1078.78	1.15008
$939$	0	0
$940$	0	0
$941$	−118.000	−0.125399	−0.0626993	−	0.998032i	$-0.519971\pi$
$941$	−118.000	−0.125399	−0.0626993	+	0.998032i	$0.519971\pi$
$942$	0	0
$943$	−1945.24	−2.06283
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1106.14	−1.16804	−0.584021	−	0.811738i	$-0.698522\pi$
$947$	−1106.14	−1.16804	−0.584021	+	0.811738i	$0.698522\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	−21.1162	−0.0219275
$964$	−94.0643	−0.0975770
$965$	0	0
$966$	777.543	0.804910
$967$	−1900.05	−1.96489	−0.982444	−	0.186559i	$-0.940266\pi$
$967$	−1900.05	−1.96489	−0.982444	+	0.186559i	$0.940266\pi$
$968$	−968.000	−1.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	22.3358	0.0229793
$973$	0	0
$974$	1906.86	1.95776
$975$	0	0
$976$	1760.17	1.80345
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	198.691	0.203160
$979$	0	0
$980$	0	0
$981$	22.3265	0.0227589
$982$	0	0
$983$	284.000	0.288911	0.144456	−	0.989511i	$-0.453857\pi$
$983$	284.000	0.288911	0.144456	+	0.989511i	$0.453857\pi$
$984$	1694.41	1.72196
$985$	0	0
$986$	0	0
$987$	1235.76	1.25204
$988$	0	0
$989$	−409.953	−0.414513
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−1788.96	−1.79615
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	500.3.d.a.499.2		4
4.3	odd	2		500.3.d.b.499.3		4
5.2	odd	4		500.3.b.a.251.3	✓	8
5.3	odd	4		500.3.b.a.251.6	yes	8
5.4	even	2		500.3.d.b.499.3		4
20.3	even	4		500.3.b.a.251.3	✓	8
20.7	even	4		500.3.b.a.251.6	yes	8
20.19	odd	2	CM	500.3.d.a.499.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.3.b.a.251.3	✓	8	5.2	odd	4
500.3.b.a.251.3	✓	8	20.3	even	4
500.3.b.a.251.6	yes	8	5.3	odd	4
500.3.b.a.251.6	yes	8	20.7	even	4
500.3.d.a.499.2		4	1.1	even	1	trivial
500.3.d.a.499.2		4	20.19	odd	2	CM
500.3.d.b.499.3		4	4.3	odd	2
500.3.d.b.499.3		4	5.4	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:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.01719 q^{3} +4.00000 q^{4} +6.03437 q^{6} -4.64990 q^{7} -8.00000 q^{8} +0.103412 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.01719 q^{3} +4.00000 q^{4} +6.03437 q^{6} -4.64990 q^{7} -8.00000 q^{8} +0.103412 q^{9} -12.0687 q^{12} +9.29980 q^{14} +16.0000 q^{16} -0.206823 q^{18} +14.0296 q^{21} -27.7108 q^{23} +24.1375 q^{24} +26.8427 q^{27} -18.5996 q^{28} -57.8374 q^{29} -32.0000 q^{32} +0.413647 q^{36} +70.1981 q^{41} -28.0592 q^{42} +14.7940 q^{43} +55.4216 q^{46} +88.0823 q^{47} -48.2750 q^{48} -27.3784 q^{49} -53.6853 q^{54} +37.1992 q^{56} +115.675 q^{58} +110.011 q^{61} -0.480854 q^{63} +64.0000 q^{64} +116.000 q^{67} +83.6086 q^{69} -0.827294 q^{72} -81.9200 q^{81} -140.396 q^{82} +148.231 q^{83} +56.1184 q^{84} -29.5880 q^{86} +174.506 q^{87} +51.7927 q^{89} -110.843 q^{92} -176.165 q^{94} +96.5500 q^{96} +54.7569 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} + 4 q^{7} - 32 q^{8} + 38 q^{9} - 16 q^{12} - 8 q^{14} + 64 q^{16} - 76 q^{18} + 16 q^{21} - 44 q^{23} + 32 q^{24} + 8 q^{27} + 16 q^{28} + 22 q^{29} - 128 q^{32} + 152 q^{36} - 62 q^{41} - 32 q^{42} + 76 q^{43} + 88 q^{46} + 4 q^{47} - 64 q^{48} + 278 q^{49} - 16 q^{54} - 32 q^{56} - 44 q^{58} + 58 q^{61} + 648 q^{63} + 256 q^{64} + 464 q^{67} - 176 q^{69} - 304 q^{72} + 500 q^{81} + 124 q^{82} + 76 q^{83} + 64 q^{84} - 152 q^{86} + 468 q^{87} + 142 q^{89} - 176 q^{92} - 8 q^{94} + 128 q^{96} - 556 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 + 4 * q^7 - 32 * q^8 + 38 * q^9 - 16 * q^12 - 8 * q^14 + 64 * q^16 - 76 * q^18 + 16 * q^21 - 44 * q^23 + 32 * q^24 + 8 * q^27 + 16 * q^28 + 22 * q^29 - 128 * q^32 + 152 * q^36 - 62 * q^41 - 32 * q^42 + 76 * q^43 + 88 * q^46 + 4 * q^47 - 64 * q^48 + 278 * q^49 - 16 * q^54 - 32 * q^56 - 44 * q^58 + 58 * q^61 + 648 * q^63 + 256 * q^64 + 464 * q^67 - 176 * q^69 - 304 * q^72 + 500 * q^81 + 124 * q^82 + 76 * q^83 + 64 * q^84 - 152 * q^86 + 468 * q^87 + 142 * q^89 - 176 * q^92 - 8 * q^94 + 128 * q^96 - 556 * q^98

Introduction

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