LMFDB - Embedded newform 512.4.e.n.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [512,4,Mod(129,512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(512, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("512.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	512.e (of order $4$, degree $2$, 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:	no
Analytic conductor:	$30.2089779229$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			385.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	512.385
Dual form			512.4.e.n.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.47214 - 4.47214i) q^{3} +(11.0000 + 11.0000i) q^{5} +26.8328i q^{7} -13.0000i q^{9} +O(q^{10})$	\(q+(4.47214 - 4.47214i) q^{3} +(11.0000 + 11.0000i) q^{5} +26.8328i q^{7} -13.0000i q^{9} +(-31.3050 - 31.3050i) q^{11} +(-23.0000 + 23.0000i) q^{13} +98.3870 q^{15} +96.0000 q^{17} +(-76.0263 + 76.0263i) q^{19} +(120.000 + 120.000i) q^{21} +116.276i q^{23} +117.000i q^{25} +(62.6099 + 62.6099i) q^{27} +(103.000 - 103.000i) q^{29} -196.774 q^{31} -280.000 q^{33} +(-295.161 + 295.161i) q^{35} +(-75.0000 - 75.0000i) q^{37} +205.718i q^{39} +312.000i q^{41} +(-254.912 - 254.912i) q^{43} +(143.000 - 143.000i) q^{45} +89.4427 q^{47} -377.000 q^{49} +(429.325 - 429.325i) q^{51} +(325.000 + 325.000i) q^{53} -688.709i q^{55} +680.000i q^{57} +(263.856 + 263.856i) q^{59} +(-39.0000 + 39.0000i) q^{61} +348.827 q^{63} -506.000 q^{65} +(424.853 - 424.853i) q^{67} +(520.000 + 520.000i) q^{69} +366.715i q^{71} +1138.00i q^{73} +(523.240 + 523.240i) q^{75} +(840.000 - 840.000i) q^{77} +911.000 q^{81} +(523.240 - 523.240i) q^{83} +(1056.00 + 1056.00i) q^{85} -921.260i q^{87} -1410.00i q^{89} +(-617.155 - 617.155i) q^{91} +(-880.000 + 880.000i) q^{93} -1672.58 q^{95} +464.000 q^{97} +(-406.964 + 406.964i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 44 q^{5}+O(q^{10}) $ 4 * q + 44 * q^5	$ 4 q + 44 q^{5} - 92 q^{13} + 384 q^{17} + 480 q^{21} + 412 q^{29} - 1120 q^{33} - 300 q^{37} + 572 q^{45} - 1508 q^{49} + 1300 q^{53} - 156 q^{61} - 2024 q^{65} + 2080 q^{69} + 3360 q^{77} + 3644 q^{81} + 4224 q^{85} - 3520 q^{93} + 1856 q^{97}+O(q^{100}) $ 4 * q + 44 * q^5 - 92 * q^13 + 384 * q^17 + 480 * q^21 + 412 * q^29 - 1120 * q^33 - 300 * q^37 + 572 * q^45 - 1508 * q^49 + 1300 * q^53 - 156 * q^61 - 2024 * q^65 + 2080 * q^69 + 3360 * q^77 + 3644 * q^81 + 4224 * q^85 - 3520 * q^93 + 1856 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/512\mathbb{Z}\right)^\times$.

$n$	$5$	$511$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	4.47214	−	4.47214i	0.860663	−	0.860663i	−0.130752	−	0.991415i	$-0.541739\pi$
$3$	4.47214	−	4.47214i	0.860663	−	0.860663i	0.991415	+	0.130752i	$0.0417392\pi$
$4$		0			0
$5$	11.0000	+	11.0000i	0.983870	+	0.983870i	0.999872	−	0.0160020i	$-0.00509383\pi$
$5$	11.0000	+	11.0000i	0.983870	+	0.983870i	−0.0160020	+	0.999872i	$0.505094\pi$
$6$		0			0
$7$			26.8328i			1.44884i	0.689361	+	0.724418i	$0.257891\pi$
$7$			26.8328i			1.44884i	−0.689361	+	0.724418i	$0.742109\pi$
$8$		0			0
$9$		−	13.0000i		−	0.481481i
$10$		0			0
$11$	−31.3050	−	31.3050i	−0.858073	−	0.858073i	0.133038	−	0.991111i	$-0.457527\pi$
$11$	−31.3050	−	31.3050i	−0.858073	−	0.858073i	−0.991111	+	0.133038i	$0.957527\pi$
$12$		0			0
$13$	−23.0000	+	23.0000i	−0.490696	+	0.490696i	−0.908526	−	0.417829i	$-0.862791\pi$
$13$	−23.0000	+	23.0000i	−0.490696	+	0.490696i	0.417829	+	0.908526i	$0.362791\pi$
$14$		0			0
$15$	98.3870			1.69356
$16$		0			0
$17$	96.0000			1.36961			0.684806	−	0.728725i	$-0.259887\pi$
$17$	96.0000			1.36961			0.684806	+	0.728725i	$0.259887\pi$
$18$		0			0
$19$	−76.0263	+	76.0263i	−0.917981	+	0.917981i	−0.996882	−	0.0789018i	$-0.974859\pi$
$19$	−76.0263	+	76.0263i	−0.917981	+	0.917981i	0.0789018	+	0.996882i	$0.474859\pi$
$20$		0			0
$21$	120.000	+	120.000i	1.24696	+	1.24696i
$22$		0			0
$23$			116.276i			1.05414i	0.849823	+	0.527068i	$0.176709\pi$
$23$			116.276i			1.05414i	−0.849823	+	0.527068i	$0.823291\pi$
$24$		0			0
$25$			117.000i			0.936000i
$26$		0			0
$27$	62.6099	+	62.6099i	0.446270	+	0.446270i
$28$		0			0
$29$	103.000	−	103.000i	0.659539	−	0.659539i	−0.295732	−	0.955271i	$-0.595564\pi$
$29$	103.000	−	103.000i	0.659539	−	0.659539i	0.955271	+	0.295732i	$0.0955636\pi$
$30$		0			0
$31$	−196.774			−1.14005			−0.570027	−	0.821626i	$-0.693067\pi$
$31$	−196.774			−1.14005			−0.570027	+	0.821626i	$0.693067\pi$
$32$		0			0
$33$	−280.000			−1.47702
$34$		0			0
$35$	−295.161	+	295.161i	−1.42547	+	1.42547i
$36$		0			0
$37$	−75.0000	−	75.0000i	−0.333241	−	0.333241i	0.520575	−	0.853816i	$-0.325718\pi$
$37$	−75.0000	−	75.0000i	−0.333241	−	0.333241i	−0.853816	+	0.520575i	$0.825718\pi$
$38$		0			0
$39$			205.718i			0.844648i
$40$		0			0
$41$			312.000i			1.18844i	0.804301	+	0.594222i	$0.202540\pi$
$41$			312.000i			1.18844i	−0.804301	+	0.594222i	$0.797460\pi$
$42$		0			0
$43$	−254.912	−	254.912i	−0.904039	−	0.904039i	0.0917437	−	0.995783i	$-0.470756\pi$
$43$	−254.912	−	254.912i	−0.904039	−	0.904039i	−0.995783	+	0.0917437i	$0.970756\pi$
$44$		0			0
$45$	143.000	−	143.000i	0.473715	−	0.473715i
$46$		0			0
$47$	89.4427			0.277586			0.138793	−	0.990321i	$-0.455678\pi$
$47$	89.4427			0.277586			0.138793	+	0.990321i	$0.455678\pi$
$48$		0			0
$49$	−377.000			−1.09913
$50$		0			0
$51$	429.325	−	429.325i	1.17878	−	1.17878i
$52$		0			0
$53$	325.000	+	325.000i	0.842305	+	0.842305i	0.989158	−	0.146853i	$-0.0469144\pi$
$53$	325.000	+	325.000i	0.842305	+	0.842305i	−0.146853	+	0.989158i	$0.546914\pi$
$54$		0			0
$55$		−	688.709i		−	1.68846i
$56$		0			0
$57$			680.000i			1.58014i
$58$		0			0
$59$	263.856	+	263.856i	0.582223	+	0.582223i	0.935513	−	0.353291i	$-0.114937\pi$
$59$	263.856	+	263.856i	0.582223	+	0.582223i	−0.353291	+	0.935513i	$0.614937\pi$
$60$		0			0
$61$	−39.0000	+	39.0000i	−0.0818596	+	0.0818596i	−0.746851	−	0.664991i	$-0.768435\pi$
$61$	−39.0000	+	39.0000i	−0.0818596	+	0.0818596i	0.664991	+	0.746851i	$0.268435\pi$
$62$		0			0
$63$	348.827			0.697588
$64$		0			0
$65$	−506.000			−0.965563
$66$		0			0
$67$	424.853	−	424.853i	0.774687	−	0.774687i	−0.204235	−	0.978922i	$-0.565471\pi$
$67$	424.853	−	424.853i	0.774687	−	0.774687i	0.978922	+	0.204235i	$0.0654706\pi$
$68$		0			0
$69$	520.000	+	520.000i	0.907256	+	0.907256i
$70$		0			0
$71$			366.715i			0.612973i	0.951875	+	0.306486i	$0.0991534\pi$
$71$			366.715i			0.612973i	−0.951875	+	0.306486i	$0.900847\pi$
$72$		0			0
$73$			1138.00i			1.82456i	0.409568	+	0.912280i	$0.365679\pi$
$73$			1138.00i			1.82456i	−0.409568	+	0.912280i	$0.634321\pi$
$74$		0			0
$75$	523.240	+	523.240i	0.805581	+	0.805581i
$76$		0			0
$77$	840.000	−	840.000i	1.24321	−	1.24321i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	911.000			1.24966
$82$		0			0
$83$	523.240	−	523.240i	0.691964	−	0.691964i	−0.270700	−	0.962664i	$-0.587255\pi$
$83$	523.240	−	523.240i	0.691964	−	0.691964i	0.962664	+	0.270700i	$0.0872550\pi$
$84$		0			0
$85$	1056.00	+	1056.00i	1.34752	+	1.34752i
$86$		0			0
$87$		−	921.260i		−	1.13528i
$88$		0			0
$89$		−	1410.00i		−	1.67932i	−0.543110	−	0.839661i	$-0.682754\pi$
$89$		−	1410.00i		−	1.67932i	0.543110	−	0.839661i	$-0.317246\pi$
$90$		0			0
$91$	−617.155	−	617.155i	−0.710938	−	0.710938i
$92$		0			0
$93$	−880.000	+	880.000i	−0.981202	+	0.981202i
$94$		0			0
$95$	−1672.58			−1.80635
$96$		0			0
$97$	464.000			0.485691			0.242846	−	0.970065i	$-0.421919\pi$
$97$	464.000			0.485691			0.242846	+	0.970065i	$0.421919\pi$
$98$		0			0
$99$	−406.964	+	406.964i	−0.413146	+	0.413146i
$100$		0			0
$101$	949.000	+	949.000i	0.934941	+	0.934941i	0.998009	−	0.0630683i	$-0.0200886\pi$
$101$	949.000	+	949.000i	0.934941	+	0.934941i	−0.0630683	+	0.998009i	$0.520089\pi$
$102$		0			0
$103$		−	1744.13i		−	1.66849i	−0.551393	−	0.834245i	$-0.685904\pi$
$103$		−	1744.13i		−	1.66849i	0.551393	−	0.834245i	$-0.314096\pi$
$104$		0			0
$105$			2640.00i			2.45369i
$106$		0			0
$107$	−174.413	−	174.413i	−0.157581	−	0.157581i	0.623913	−	0.781494i	$-0.285542\pi$
$107$	−174.413	−	174.413i	−0.157581	−	0.157581i	−0.781494	+	0.623913i	$0.785542\pi$
$108$		0			0
$109$	583.000	−	583.000i	0.512305	−	0.512305i	−0.402927	−	0.915232i	$-0.632007\pi$
$109$	583.000	−	583.000i	0.512305	−	0.512305i	0.915232	+	0.402927i	$0.132007\pi$
$110$		0			0
$111$	−670.820			−0.573617
$112$		0			0
$113$	−1118.00			−0.930731			−0.465365	−	0.885119i	$-0.654077\pi$
$113$	−1118.00			−0.930731			−0.465365	+	0.885119i	$0.654077\pi$
$114$		0			0
$115$	−1279.03	+	1279.03i	−1.03713	+	1.03713i
$116$		0			0
$117$	299.000	+	299.000i	0.236261	+	0.236261i
$118$		0			0
$119$			2575.95i			1.98434i
$120$		0			0
$121$			629.000i			0.472577i
$122$		0			0
$123$	1395.31	+	1395.31i	1.02285	+	1.02285i
$124$		0			0
$125$	88.0000	−	88.0000i	0.0629677	−	0.0629677i
$126$		0			0
$127$	2092.96			1.46236			0.731182	−	0.682183i	$-0.238969\pi$
$127$	2092.96			1.46236			0.731182	+	0.682183i	$0.238969\pi$
$128$		0			0
$129$	−2280.00			−1.55615
$130$		0			0
$131$	693.181	−	693.181i	0.462317	−	0.462317i	−0.437097	−	0.899414i	$-0.643993\pi$
$131$	693.181	−	693.181i	0.462317	−	0.462317i	0.899414	+	0.437097i	$0.143993\pi$
$132$		0			0
$133$	−2040.00	−	2040.00i	−1.33000	−	1.33000i
$134$		0			0
$135$			1377.42i			0.878143i
$136$		0			0
$137$		−	1144.00i		−	0.713420i	−0.934215	−	0.356710i	$-0.883898\pi$
$137$		−	1144.00i		−	0.713420i	0.934215	−	0.356710i	$-0.116102\pi$
$138$		0			0
$139$	−809.457	−	809.457i	−0.493937	−	0.493937i	0.415607	−	0.909544i	$-0.363569\pi$
$139$	−809.457	−	809.457i	−0.493937	−	0.493937i	−0.909544	+	0.415607i	$0.863569\pi$
$140$		0			0
$141$	400.000	−	400.000i	0.238908	−	0.238908i
$142$		0			0
$143$	1440.03			0.842106
$144$		0			0
$145$	2266.00			1.29780
$146$		0			0
$147$	−1686.00	+	1686.00i	−0.945976	+	0.945976i
$148$		0			0
$149$	507.000	+	507.000i	0.278759	+	0.278759i	0.832613	−	0.553855i	$-0.186844\pi$
$149$	507.000	+	507.000i	0.278759	+	0.278759i	−0.553855	+	0.832613i	$0.686844\pi$
$150$		0			0
$151$			2298.68i			1.23883i	0.785063	+	0.619416i	$0.212631\pi$
$151$			2298.68i			1.23883i	−0.785063	+	0.619416i	$0.787369\pi$
$152$		0			0
$153$		−	1248.00i		−	0.659443i
$154$		0			0
$155$	−2164.51	−	2164.51i	−1.12166	−	1.12166i
$156$		0			0
$157$	−519.000	+	519.000i	−0.263826	+	0.263826i	−0.826606	−	0.562780i	$-0.809732\pi$
$157$	−519.000	+	519.000i	−0.263826	+	0.263826i	0.562780	+	0.826606i	$0.309732\pi$
$158$		0			0
$159$	2906.89			1.44988
$160$		0			0
$161$	−3120.00			−1.52727
$162$		0			0
$163$	988.342	−	988.342i	0.474926	−	0.474926i	−0.428579	−	0.903505i	$-0.640985\pi$
$163$	988.342	−	988.342i	0.474926	−	0.474926i	0.903505	+	0.428579i	$0.140985\pi$
$164$		0			0
$165$	−3080.00	−	3080.00i	−1.45320	−	1.45320i
$166$		0			0
$167$		−	1976.68i		−	0.915931i	−0.888970	−	0.457965i	$-0.848578\pi$
$167$		−	1976.68i		−	0.915931i	0.888970	−	0.457965i	$-0.151422\pi$
$168$		0			0
$169$			1139.00i			0.518434i
$170$		0			0
$171$	988.342	+	988.342i	0.441991	+	0.441991i
$172$		0			0
$173$	1625.00	−	1625.00i	0.714141	−	0.714141i	−0.253258	−	0.967399i	$-0.581502\pi$
$173$	1625.00	−	1625.00i	0.714141	−	0.714141i	0.967399	+	0.253258i	$0.0815020\pi$
$174$		0			0
$175$	−3139.44			−1.35611
$176$		0			0
$177$	2360.00			1.00219
$178$		0			0
$179$	326.466	−	326.466i	0.136320	−	0.136320i	−0.635654	−	0.771974i	$-0.719270\pi$
$179$	326.466	−	326.466i	0.136320	−	0.136320i	0.771974	+	0.635654i	$0.219270\pi$
$180$		0			0
$181$	299.000	+	299.000i	0.122787	+	0.122787i	0.765830	−	0.643043i	$-0.222328\pi$
$181$	299.000	+	299.000i	0.122787	+	0.122787i	−0.643043	+	0.765830i	$0.722328\pi$
$182$		0			0
$183$			348.827i			0.140907i
$184$		0			0
$185$		−	1650.00i		−	0.655732i
$186$		0			0
$187$	−3005.28	−	3005.28i	−1.17523	−	1.17523i
$188$		0			0
$189$	−1680.00	+	1680.00i	−0.646572	+	0.646572i
$190$		0			0
$191$	−1735.19			−0.657350			−0.328675	−	0.944443i	$-0.606602\pi$
$191$	−1735.19			−0.657350			−0.328675	+	0.944443i	$0.606602\pi$
$192$		0			0
$193$	1872.00			0.698184			0.349092	−	0.937088i	$-0.386490\pi$
$193$	1872.00			0.698184			0.349092	+	0.937088i	$0.386490\pi$
$194$		0			0
$195$	−2262.90	+	2262.90i	−0.831024	+	0.831024i
$196$		0			0
$197$	−1029.00	−	1029.00i	−0.372148	−	0.372148i	0.496111	−	0.868259i	$-0.334761\pi$
$197$	−1029.00	−	1029.00i	−0.372148	−	0.372148i	−0.868259	+	0.496111i	$0.834761\pi$
$198$		0			0
$199$			4606.30i			1.64086i	0.571744	+	0.820432i	$0.306267\pi$
$199$			4606.30i			1.64086i	−0.571744	+	0.820432i	$0.693733\pi$
$200$		0			0
$201$		−	3800.00i		−	1.33349i
$202$		0			0
$203$	2763.78	+	2763.78i	0.955563	+	0.955563i
$204$		0			0
$205$	−3432.00	+	3432.00i	−1.16927	+	1.16927i
$206$		0			0
$207$	1511.58			0.507547
$208$		0			0
$209$	4760.00			1.57539
$210$		0			0
$211$	809.457	−	809.457i	0.264101	−	0.264101i	−0.562617	−	0.826718i	$-0.690205\pi$
$211$	809.457	−	809.457i	0.264101	−	0.264101i	0.826718	+	0.562617i	$0.190205\pi$
$212$		0			0
$213$	1640.00	+	1640.00i	0.527563	+	0.527563i
$214$		0			0
$215$		−	5608.06i		−	1.77891i
$216$		0			0
$217$		−	5280.00i		−	1.65175i
$218$		0			0
$219$	5089.29	+	5089.29i	1.57033	+	1.57033i
$220$		0			0
$221$	−2208.00	+	2208.00i	−0.672064	+	0.672064i
$222$		0			0
$223$	−5134.01			−1.54170			−0.770850	−	0.637017i	$-0.780168\pi$
$223$	−5134.01			−1.54170			−0.770850	+	0.637017i	$0.780168\pi$
$224$		0			0
$225$	1521.00			0.450667
$226$		0			0
$227$	755.791	−	755.791i	0.220985	−	0.220985i	−0.587928	−	0.808913i	$-0.700056\pi$
$227$	755.791	−	755.791i	0.220985	−	0.220985i	0.808913	+	0.587928i	$0.200056\pi$
$228$		0			0
$229$	−1557.00	−	1557.00i	−0.449299	−	0.449299i	0.445822	−	0.895121i	$-0.352911\pi$
$229$	−1557.00	−	1557.00i	−0.449299	−	0.449299i	−0.895121	+	0.445822i	$0.852911\pi$
$230$		0			0
$231$		−	7513.19i		−	2.13996i
$232$		0			0
$233$		−	4558.00i		−	1.28156i	−0.767723	−	0.640782i	$-0.778610\pi$
$233$		−	4558.00i		−	1.28156i	0.767723	−	0.640782i	$-0.221390\pi$
$234$		0			0
$235$	983.870	+	983.870i	0.273109	+	0.273109i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3041.05			−0.823052			−0.411526	−	0.911398i	$-0.635004\pi$
$239$	−3041.05			−0.823052			−0.411526	+	0.911398i	$0.635004\pi$
$240$		0			0
$241$	1632.00			0.436209			0.218104	−	0.975925i	$-0.430013\pi$
$241$	1632.00			0.436209			0.218104	+	0.975925i	$0.430013\pi$
$242$		0			0
$243$	2383.65	−	2383.65i	0.629264	−	0.629264i
$244$		0			0
$245$	−4147.00	−	4147.00i	−1.08140	−	1.08140i
$246$		0			0
$247$		−	3497.21i		−	0.900899i
$248$		0			0
$249$		−	4680.00i		−	1.19110i
$250$		0			0
$251$	−3009.75	−	3009.75i	−0.756867	−	0.756867i	0.218884	−	0.975751i	$-0.429758\pi$
$251$	−3009.75	−	3009.75i	−0.756867	−	0.756867i	−0.975751	+	0.218884i	$0.929758\pi$
$252$		0			0
$253$	3640.00	−	3640.00i	0.904525	−	0.904525i
$254$		0			0
$255$	9445.15			2.31952
$256$		0			0
$257$	−814.000			−0.197572			−0.0987858	−	0.995109i	$-0.531496\pi$
$257$	−814.000			−0.197572			−0.0987858	+	0.995109i	$0.531496\pi$
$258$		0			0
$259$	2012.46	−	2012.46i	0.482812	−	0.482812i
$260$		0			0
$261$	−1339.00	−	1339.00i	−0.317556	−	0.317556i
$262$		0			0
$263$			2441.79i			0.572498i	0.958155	+	0.286249i	$0.0924085\pi$
$263$			2441.79i			0.572498i	−0.958155	+	0.286249i	$0.907592\pi$
$264$		0			0
$265$			7150.00i			1.65744i
$266$		0			0
$267$	−6305.71	−	6305.71i	−1.44533	−	1.44533i
$268$		0			0
$269$	247.000	−	247.000i	0.0559846	−	0.0559846i	−0.678560	−	0.734545i	$-0.737396\pi$
$269$	247.000	−	247.000i	0.0559846	−	0.0559846i	0.734545	+	0.678560i	$0.237396\pi$
$270$		0			0
$271$	89.4427			0.0200489			0.0100245	−	0.999950i	$-0.496809\pi$
$271$	89.4427			0.0200489			0.0100245	+	0.999950i	$0.496809\pi$
$272$		0			0
$273$	−5520.00			−1.22376
$274$		0			0
$275$	3662.68	−	3662.68i	0.803156	−	0.803156i
$276$		0			0
$277$	−2795.00	−	2795.00i	−0.606265	−	0.606265i	0.335703	−	0.941968i	$-0.391026\pi$
$277$	−2795.00	−	2795.00i	−0.606265	−	0.606265i	−0.941968	+	0.335703i	$0.891026\pi$
$278$		0			0
$279$			2558.06i			0.548915i
$280$		0			0
$281$			930.000i			0.197435i	0.995116	+	0.0987173i	$0.0314740\pi$
$281$			930.000i			0.197435i	−0.995116	+	0.0987173i	$0.968526\pi$
$282$		0			0
$283$	5871.91	+	5871.91i	1.23339	+	1.23339i	0.962654	+	0.270735i	$0.0872668\pi$
$283$	5871.91	+	5871.91i	1.23339	+	1.23339i	0.270735	+	0.962654i	$0.412733\pi$
$284$		0			0
$285$	−7480.00	+	7480.00i	−1.55466	+	1.55466i
$286$		0			0
$287$	−8371.84			−1.72186
$288$		0			0
$289$	4303.00			0.875840
$290$		0			0
$291$	2075.07	−	2075.07i	0.418017	−	0.418017i
$292$		0			0
$293$	283.000	+	283.000i	0.0564267	+	0.0564267i	0.734757	−	0.678330i	$-0.237296\pi$
$293$	283.000	+	283.000i	0.0564267	+	0.0564267i	−0.678330	+	0.734757i	$0.737296\pi$
$294$		0			0
$295$			5804.83i			1.14566i
$296$		0			0
$297$		−	3920.00i		−	0.765864i
$298$		0			0
$299$	−2674.34	−	2674.34i	−0.517261	−	0.517261i
$300$		0			0
$301$	6840.00	−	6840.00i	1.30980	−	1.30980i
$302$		0			0
$303$	8488.11			1.60934
$304$		0			0
$305$	−858.000			−0.161078
$306$		0			0
$307$	84.9706	−	84.9706i	0.0157965	−	0.0157965i	−0.699164	−	0.714961i	$-0.746444\pi$
$307$	84.9706	−	84.9706i	0.0157965	−	0.0157965i	0.714961	+	0.699164i	$0.246444\pi$
$308$		0			0
$309$	−7800.00	−	7800.00i	−1.43601	−	1.43601i
$310$		0			0
$311$			2745.89i			0.500660i	0.968161	+	0.250330i	$0.0805391\pi$
$311$			2745.89i			0.500660i	−0.968161	+	0.250330i	$0.919461\pi$
$312$		0			0
$313$		−	4408.00i		−	0.796022i	−0.917381	−	0.398011i	$-0.869701\pi$
$313$		−	4408.00i		−	0.796022i	0.917381	−	0.398011i	$-0.130299\pi$
$314$		0			0
$315$	3837.09	+	3837.09i	0.686335	+	0.686335i
$316$		0			0
$317$	−585.000	+	585.000i	−0.103649	+	0.103649i	−0.757030	−	0.653380i	$-0.773350\pi$
$317$	−585.000	+	585.000i	−0.103649	+	0.103649i	0.653380	+	0.757030i	$0.273350\pi$
$318$		0			0
$319$	−6448.82			−1.13186
$320$		0			0
$321$	−1560.00			−0.271248
$322$		0			0
$323$	−7298.53	+	7298.53i	−1.25728	+	1.25728i
$324$		0			0
$325$	−2691.00	−	2691.00i	−0.459292	−	0.459292i
$326$		0			0
$327$		−	5214.51i		−	0.881844i
$328$		0			0
$329$			2400.00i			0.402177i
$330$		0			0
$331$	58.1378	+	58.1378i	0.00965420	+	0.00965420i	0.711917	−	0.702263i	$-0.247827\pi$
$331$	58.1378	+	58.1378i	0.00965420	+	0.00965420i	−0.702263	+	0.711917i	$0.747827\pi$
$332$		0			0
$333$	−975.000	+	975.000i	−0.160449	+	0.160449i
$334$		0			0
$335$	9346.76			1.52438
$336$		0			0
$337$	−9394.00			−1.51847			−0.759234	−	0.650818i	$-0.774426\pi$
$337$	−9394.00			−1.51847			−0.759234	+	0.650818i	$0.774426\pi$
$338$		0			0
$339$	−4999.85	+	4999.85i	−0.801046	+	0.801046i
$340$		0			0
$341$	6160.00	+	6160.00i	0.978248	+	0.978248i
$342$		0			0
$343$		−	912.316i		−	0.143616i
$344$		0			0
$345$			11440.0i			1.78524i
$346$		0			0
$347$	−6623.23	−	6623.23i	−1.02465	−	1.02465i	−0.999688	−	0.0249623i	$-0.992053\pi$
$347$	−6623.23	−	6623.23i	−1.02465	−	1.02465i	−0.0249623	−	0.999688i	$-0.507947\pi$
$348$		0			0
$349$	7127.00	−	7127.00i	1.09312	−	1.09312i	0.0979285	−	0.995193i	$-0.468778\pi$
$349$	7127.00	−	7127.00i	1.09312	−	1.09312i	0.995193	−	0.0979285i	$-0.0312217\pi$
$350$		0			0
$351$	−2880.06			−0.437966
$352$		0			0
$353$	−2418.00			−0.364581			−0.182291	−	0.983245i	$-0.558351\pi$
$353$	−2418.00			−0.364581			−0.182291	+	0.983245i	$0.558351\pi$
$354$		0			0
$355$	−4033.87	+	4033.87i	−0.603086	+	0.603086i
$356$		0			0
$357$	11520.0	+	11520.0i	1.70785	+	1.70785i
$358$		0			0
$359$			6162.60i			0.905988i	0.891514	+	0.452994i	$0.149644\pi$
$359$			6162.60i			0.905988i	−0.891514	+	0.452994i	$0.850356\pi$
$360$		0			0
$361$		−	4701.00i		−	0.685377i
$362$		0			0
$363$	2812.97	+	2812.97i	0.406730	+	0.406730i
$364$		0			0
$365$	−12518.0	+	12518.0i	−1.79513	+	1.79513i
$366$		0			0
$367$	−1055.42			−0.150116			−0.0750582	−	0.997179i	$-0.523914\pi$
$367$	−1055.42			−0.150116			−0.0750582	+	0.997179i	$0.523914\pi$
$368$		0			0
$369$	4056.00			0.572214
$370$		0			0
$371$	−8720.67	+	8720.67i	−1.22036	+	1.22036i
$372$		0			0
$373$	2613.00	+	2613.00i	0.362724	+	0.362724i	0.864815	−	0.502091i	$-0.167436\pi$
$373$	2613.00	+	2613.00i	0.362724	+	0.362724i	−0.502091	+	0.864815i	$0.667436\pi$
$374$		0			0
$375$		−	787.096i		−	0.108388i
$376$		0			0
$377$			4738.00i			0.647266i
$378$		0			0
$379$	639.515	+	639.515i	0.0866747	+	0.0866747i	0.749115	−	0.662440i	$-0.230479\pi$
$379$	639.515	+	639.515i	0.0866747	+	0.0866747i	−0.662440	+	0.749115i	$0.730479\pi$
$380$		0			0
$381$	9360.00	−	9360.00i	1.25860	−	1.25860i
$382$		0			0
$383$	6296.77			0.840078			0.420039	−	0.907506i	$-0.362016\pi$
$383$	6296.77			0.840078			0.420039	+	0.907506i	$0.362016\pi$
$384$		0			0
$385$	18480.0			2.44631
$386$		0			0
$387$	−3313.85	+	3313.85i	−0.435278	+	0.435278i
$388$		0			0
$389$	6037.00	+	6037.00i	0.786859	+	0.786859i	0.980978	−	0.194119i	$-0.0621849\pi$
$389$	6037.00	+	6037.00i	0.786859	+	0.786859i	−0.194119	+	0.980978i	$0.562185\pi$
$390$		0			0
$391$			11162.5i			1.44376i
$392$		0			0
$393$		−	6200.00i		−	0.795798i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−585.000	+	585.000i	−0.0739554	+	0.0739554i	−0.743117	−	0.669162i	$-0.766653\pi$
$397$	−585.000	+	585.000i	−0.0739554	+	0.0739554i	0.669162	+	0.743117i	$0.266653\pi$
$398$		0			0
$399$	−18246.3			−2.28937
$400$		0			0
$401$	−2080.00			−0.259028			−0.129514	−	0.991578i	$-0.541342\pi$
$401$	−2080.00			−0.259028			−0.129514	+	0.991578i	$0.541342\pi$
$402$		0			0
$403$	4525.80	−	4525.80i	0.559420	−	0.559420i
$404$		0			0
$405$	10021.0	+	10021.0i	1.22950	+	1.22950i
$406$		0			0
$407$			4695.74i			0.571890i
$408$		0			0
$409$			13000.0i			1.57166i	0.618443	+	0.785830i	$0.287764\pi$
$409$			13000.0i			1.57166i	−0.618443	+	0.785830i	$0.712236\pi$
$410$		0			0
$411$	−5116.12	−	5116.12i	−0.614014	−	0.614014i
$412$		0			0
$413$	−7080.00	+	7080.00i	−0.843545	+	0.843545i
$414$		0			0
$415$	11511.3			1.36161
$416$		0			0
$417$	−7240.00			−0.850226
$418$		0			0
$419$	8599.92	−	8599.92i	1.00271	−	1.00271i	0.00270882	−	0.999996i	$-0.499138\pi$
$419$	8599.92	−	8599.92i	1.00271	−	1.00271i	0.999996	−	0.00270882i	$-0.000862246\pi$
$420$		0			0
$421$	−1029.00	−	1029.00i	−0.119122	−	0.119122i	0.645033	−	0.764155i	$-0.276844\pi$
$421$	−1029.00	−	1029.00i	−0.119122	−	0.119122i	−0.764155	+	0.645033i	$0.776844\pi$
$422$		0			0
$423$		−	1162.76i		−	0.133653i
$424$		0			0
$425$			11232.0i			1.28196i
$426$		0			0
$427$	−1046.48	−	1046.48i	−0.118601	−	0.118601i
$428$		0			0
$429$	6440.00	−	6440.00i	0.724770	−	0.724770i
$430$		0			0
$431$	−5330.79			−0.595765			−0.297883	−	0.954602i	$-0.596280\pi$
$431$	−5330.79			−0.595765			−0.297883	+	0.954602i	$0.596280\pi$
$432$		0			0
$433$	7808.00			0.866579			0.433289	−	0.901255i	$-0.357353\pi$
$433$	7808.00			0.866579			0.433289	+	0.901255i	$0.357353\pi$
$434$		0			0
$435$	10133.9	−	10133.9i	1.11697	−	1.11697i
$436$		0			0
$437$	−8840.00	−	8840.00i	−0.967676	−	0.967676i
$438$		0			0
$439$		−	4695.74i		−	0.510514i	−0.966873	−	0.255257i	$-0.917840\pi$
$439$		−	4695.74i		−	0.510514i	0.966873	−	0.255257i	$-0.0821601\pi$
$440$		0			0
$441$			4901.00i			0.529209i
$442$		0			0
$443$	3027.64	+	3027.64i	0.324712	+	0.324712i	0.850571	−	0.525860i	$-0.176256\pi$
$443$	3027.64	+	3027.64i	0.324712	+	0.324712i	−0.525860	+	0.850571i	$0.676256\pi$
$444$		0			0
$445$	15510.0	−	15510.0i	1.65223	−	1.65223i
$446$		0			0
$447$	4534.75			0.479835
$448$		0			0
$449$	−12720.0			−1.33696			−0.668479	−	0.743731i	$-0.733054\pi$
$449$	−12720.0			−1.33696			−0.668479	+	0.743731i	$0.733054\pi$
$450$		0			0
$451$	9767.14	−	9767.14i	1.01977	−	1.01977i
$452$		0			0
$453$	10280.0	+	10280.0i	1.06622	+	1.06622i
$454$		0			0
$455$		−	13577.4i		−	1.39894i
$456$		0			0
$457$			56.0000i			0.00573210i	0.999996	+	0.00286605i	$0.000912293\pi$
$457$			56.0000i			0.00573210i	−0.999996	+	0.00286605i	$0.999088\pi$
$458$		0			0
$459$	6010.55	+	6010.55i	0.611217	+	0.611217i
$460$		0			0
$461$	−12729.0	+	12729.0i	−1.28601	+	1.28601i	−0.348814	+	0.937192i	$0.613415\pi$
$461$	−12729.0	+	12729.0i	−1.28601	+	1.28601i	−0.937192	+	0.348814i	$0.886585\pi$
$462$		0			0
$463$	13953.1			1.40055			0.700274	−	0.713874i	$-0.253061\pi$
$463$	13953.1			1.40055			0.700274	+	0.713874i	$0.253061\pi$
$464$		0			0
$465$	−19360.0			−1.93075
$466$		0			0
$467$	−1220.89	+	1220.89i	−0.120977	+	0.120977i	−0.765003	−	0.644026i	$-0.777263\pi$
$467$	−1220.89	+	1220.89i	−0.120977	+	0.120977i	0.644026	+	0.765003i	$0.277263\pi$
$468$		0			0
$469$	11400.0	+	11400.0i	1.12239	+	1.12239i
$470$		0			0
$471$			4642.08i			0.454131i
$472$		0			0
$473$			15960.0i			1.55146i
$474$		0			0
$475$	−8895.08	−	8895.08i	−0.859230	−	0.859230i
$476$		0			0
$477$	4225.00	−	4225.00i	0.405554	−	0.405554i
$478$		0			0
$479$	14507.6			1.38386			0.691931	−	0.721964i	$-0.256760\pi$
$479$	14507.6			1.38386			0.691931	+	0.721964i	$0.256760\pi$
$480$		0			0
$481$	3450.00			0.327040
$482$		0			0
$483$	−13953.1	+	13953.1i	−1.31446	+	1.31446i
$484$		0			0
$485$	5104.00	+	5104.00i	0.477857	+	0.477857i
$486$		0			0
$487$		−	16394.9i		−	1.52551i	−0.646690	−	0.762753i	$-0.723847\pi$
$487$		−	16394.9i		−	1.52551i	0.646690	−	0.762753i	$-0.276153\pi$
$488$		0			0
$489$		−	8840.00i		−	0.817502i
$490$		0			0
$491$	2929.25	+	2929.25i	0.269237	+	0.269237i	0.828793	−	0.559556i	$-0.189028\pi$
$491$	2929.25	+	2929.25i	0.269237	+	0.269237i	−0.559556	+	0.828793i	$0.689028\pi$
$492$		0			0
$493$	9888.00	−	9888.00i	0.903313	−	0.903313i
$494$		0			0
$495$	−8953.22			−0.812964
$496$		0			0
$497$	−9840.00			−0.888097
$498$		0			0
$499$	−2499.92	+	2499.92i	−0.224272	+	0.224272i	−0.810295	−	0.586022i	$-0.800693\pi$
$499$	−2499.92	+	2499.92i	−0.224272	+	0.224272i	0.586022	+	0.810295i	$0.300693\pi$
$500$		0			0
$501$	−8840.00	−	8840.00i	−0.788308	−	0.788308i
$502$		0			0
$503$		−	3962.31i		−	0.351234i	−0.984459	−	0.175617i	$-0.943808\pi$
$503$		−	3962.31i		−	0.351234i	0.984459	−	0.175617i	$-0.0561921\pi$
$504$		0			0
$505$			20878.0i			1.83972i
$506$		0			0
$507$	5093.76	+	5093.76i	0.446197	+	0.446197i
$508$		0			0
$509$	10313.0	−	10313.0i	0.898066	−	0.898066i	−0.0971988	−	0.995265i	$-0.530988\pi$
$509$	10313.0	−	10313.0i	0.898066	−	0.898066i	0.995265	+	0.0971988i	$0.0309883\pi$
$510$		0			0
$511$	−30535.7			−2.64349
$512$		0			0
$513$	−9520.00			−0.819334
$514$		0			0
$515$	19185.5	−	19185.5i	1.64158	−	1.64158i
$516$		0			0
$517$	−2800.00	−	2800.00i	−0.238189	−	0.238189i
$518$		0			0
$519$		−	14534.4i		−	1.22927i
$520$		0			0
$521$			4200.00i			0.353177i	0.984285	+	0.176589i	$0.0565062\pi$
$521$			4200.00i			0.353177i	−0.984285	+	0.176589i	$0.943494\pi$
$522$		0			0
$523$	11453.1	+	11453.1i	0.957573	+	0.957573i	0.999136	−	0.0415626i	$-0.0132336\pi$
$523$	11453.1	+	11453.1i	0.957573	+	0.957573i	−0.0415626	+	0.999136i	$0.513234\pi$
$524$		0			0
$525$	−14040.0	+	14040.0i	−1.16715	+	1.16715i
$526$		0			0
$527$	−18890.3			−1.56143
$528$		0			0
$529$	−1353.00			−0.111202
$530$		0			0
$531$	3430.13	−	3430.13i	0.280329	−	0.280329i
$532$		0			0
$533$	−7176.00	−	7176.00i	−0.583165	−	0.583165i
$534$		0			0
$535$		−	3837.09i		−	0.310078i
$536$		0			0
$537$		−	2920.00i		−	0.234650i
$538$		0			0
$539$	11802.0	+	11802.0i	0.943129	+	0.943129i
$540$		0			0
$541$	−6471.00	+	6471.00i	−0.514251	+	0.514251i	−0.915826	−	0.401575i	$-0.868463\pi$
$541$	−6471.00	+	6471.00i	−0.514251	+	0.514251i	0.401575	+	0.915826i	$0.368463\pi$
$542$		0			0
$543$	2674.34			0.211357
$544$		0			0
$545$	12826.0			1.00808
$546$		0			0
$547$	353.299	−	353.299i	0.0276160	−	0.0276160i	−0.693164	−	0.720780i	$-0.743784\pi$
$547$	353.299	−	353.299i	0.0276160	−	0.0276160i	0.720780	+	0.693164i	$0.243784\pi$
$548$		0			0
$549$	507.000	+	507.000i	0.0394139	+	0.0394139i
$550$		0			0
$551$			15661.4i			1.21089i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−7379.02	−	7379.02i	−0.564364	−	0.564364i
$556$		0			0
$557$	6809.00	−	6809.00i	0.517965	−	0.517965i	−0.398990	−	0.916955i	$-0.630639\pi$
$557$	6809.00	−	6809.00i	0.517965	−	0.517965i	0.916955	+	0.398990i	$0.130639\pi$
$558$		0			0
$559$	11725.9			0.887217
$560$		0			0
$561$	−26880.0			−2.02295
$562$		0			0
$563$	−11954.0	+	11954.0i	−0.894852	+	0.894852i	−0.994975	−	0.100123i	$-0.968076\pi$
$563$	−11954.0	+	11954.0i	−0.894852	+	0.894852i	0.100123	+	0.994975i	$0.468076\pi$
$564$		0			0
$565$	−12298.0	−	12298.0i	−0.915718	−	0.915718i
$566$		0			0
$567$			24444.7i			1.81055i
$568$		0			0
$569$		−	11240.0i		−	0.828129i	−0.910248	−	0.414064i	$-0.864109\pi$
$569$		−	11240.0i		−	0.828129i	0.910248	−	0.414064i	$-0.135891\pi$
$570$		0			0
$571$	−17580.0	−	17580.0i	−1.28844	−	1.28844i	−0.935735	−	0.352704i	$-0.885262\pi$
$571$	−17580.0	−	17580.0i	−1.28844	−	1.28844i	−0.352704	−	0.935735i	$-0.614738\pi$
$572$		0			0
$573$	−7760.00	+	7760.00i	−0.565757	+	0.565757i
$574$		0			0
$575$	−13604.2			−0.986671
$576$		0			0
$577$	26654.0			1.92309			0.961543	−	0.274655i	$-0.0885635\pi$
$577$	26654.0			1.92309			0.961543	+	0.274655i	$0.0885635\pi$
$578$		0			0
$579$	8371.84	−	8371.84i	0.600901	−	0.600901i
$580$		0			0
$581$	14040.0	+	14040.0i	1.00254	+	1.00254i
$582$		0			0
$583$		−	20348.2i		−	1.44552i
$584$		0			0
$585$			6578.00i			0.464901i
$586$		0			0
$587$	988.342	+	988.342i	0.0694944	+	0.0694944i	0.741000	−	0.671505i	$-0.234352\pi$
$587$	988.342	+	988.342i	0.0694944	+	0.0694944i	−0.671505	+	0.741000i	$0.734352\pi$
$588$		0			0
$589$	14960.0	−	14960.0i	1.04655	−	1.04655i
$590$		0			0
$591$	−9203.66			−0.640589
$592$		0			0
$593$	11538.0			0.799003			0.399502	−	0.916733i	$-0.369183\pi$
$593$	11538.0			0.799003			0.399502	+	0.916733i	$0.369183\pi$
$594$		0			0
$595$	−28335.5	+	28335.5i	−1.95234	+	1.95234i
$596$		0			0
$597$	20600.0	+	20600.0i	1.41223	+	1.41223i
$598$		0			0
$599$		−	18362.6i		−	1.25255i	−0.779604	−	0.626273i	$-0.784580\pi$
$599$		−	18362.6i		−	1.25255i	0.779604	−	0.626273i	$-0.215420\pi$
$600$		0			0
$601$		−	14210.0i		−	0.964456i	−0.876046	−	0.482228i	$-0.839828\pi$
$601$		−	14210.0i		−	0.964456i	0.876046	−	0.482228i	$-0.160172\pi$
$602$		0			0
$603$	−5523.09	−	5523.09i	−0.372998	−	0.372998i
$604$		0			0
$605$	−6919.00	+	6919.00i	−0.464954	+	0.464954i
$606$		0			0
$607$	21752.5			1.45454			0.727270	−	0.686352i	$-0.240789\pi$
$607$	21752.5			1.45454			0.727270	+	0.686352i	$0.240789\pi$
$608$		0			0
$609$	24720.0			1.64484
$610$		0			0
$611$	−2057.18	+	2057.18i	−0.136211	+	0.136211i
$612$		0			0
$613$	12315.0	+	12315.0i	0.811416	+	0.811416i	0.984846	−	0.173430i	$-0.0554850\pi$
$613$	12315.0	+	12315.0i	0.811416	+	0.811416i	−0.173430	+	0.984846i	$0.555485\pi$
$614$		0			0
$615$			30696.7i			2.01270i
$616$		0			0
$617$		−	1586.00i		−	0.103485i	−0.998660	−	0.0517423i	$-0.983523\pi$
$617$		−	1586.00i		−	0.103485i	0.998660	−	0.0517423i	$-0.0164774\pi$
$618$		0			0
$619$	−12732.2	−	12732.2i	−0.826735	−	0.826735i	0.160328	−	0.987064i	$-0.448745\pi$
$619$	−12732.2	−	12732.2i	−0.826735	−	0.826735i	−0.987064	+	0.160328i	$0.948745\pi$
$620$		0			0
$621$	−7280.00	+	7280.00i	−0.470429	+	0.470429i
$622$		0			0
$623$	37834.3			2.43306
$624$		0			0
$625$	16561.0			1.05990
$626$		0			0
$627$	21287.4	−	21287.4i	1.35588	−	1.35588i
$628$		0			0
$629$	−7200.00	−	7200.00i	−0.456411	−	0.456411i
$630$		0			0
$631$			2119.79i			0.133736i	0.997762	+	0.0668681i	$0.0213007\pi$
$631$			2119.79i			0.133736i	−0.997762	+	0.0668681i	$0.978699\pi$
$632$		0			0
$633$		−	7240.00i		−	0.454604i
$634$		0			0
$635$	23022.6	+	23022.6i	1.43878	+	1.43878i
$636$		0			0
$637$	8671.00	−	8671.00i	0.539337	−	0.539337i
$638$		0			0
$639$	4767.30			0.295135
$640$		0			0
$641$	−14992.0			−0.923788			−0.461894	−	0.886935i	$-0.652830\pi$
$641$	−14992.0			−0.923788			−0.461894	+	0.886935i	$0.652830\pi$
$642$		0			0
$643$	−20504.7	+	20504.7i	−1.25759	+	1.25759i	−0.305345	+	0.952242i	$0.598772\pi$
$643$	−20504.7	+	20504.7i	−1.25759	+	1.25759i	−0.952242	+	0.305345i	$0.901228\pi$
$644$		0			0
$645$	−25080.0	−	25080.0i	−1.53105	−	1.53105i
$646$		0			0
$647$			19257.0i			1.17013i	0.810988	+	0.585063i	$0.198930\pi$
$647$			19257.0i			1.17013i	−0.810988	+	0.585063i	$0.801070\pi$
$648$		0			0
$649$		−	16520.0i		−	0.999178i
$650$		0			0
$651$	−23612.9	−	23612.9i	−1.42160	−	1.42160i
$652$		0			0
$653$	−18473.0	+	18473.0i	−1.10705	+	1.10705i	−0.113515	+	0.993536i	$0.536211\pi$
$653$	−18473.0	+	18473.0i	−1.10705	+	1.10705i	−0.993536	+	0.113515i	$0.963789\pi$
$654$		0			0
$655$	15250.0			0.909719
$656$		0			0
$657$	14794.0			0.878491
$658$		0			0
$659$	14476.3	−	14476.3i	0.855716	−	0.855716i	−0.135114	−	0.990830i	$-0.543140\pi$
$659$	14476.3	−	14476.3i	0.855716	−	0.855716i	0.990830	+	0.135114i	$0.0431401\pi$
$660$		0			0
$661$	−16341.0	−	16341.0i	−0.961560	−	0.961560i	0.0377279	−	0.999288i	$-0.487988\pi$
$661$	−16341.0	−	16341.0i	−0.961560	−	0.961560i	−0.999288	+	0.0377279i	$0.987988\pi$
$662$		0			0
$663$			19749.0i			1.15684i
$664$		0			0
$665$		−	44880.0i		−	2.61710i
$666$		0			0
$667$	11976.4	+	11976.4i	0.695243	+	0.695243i
$668$		0			0
$669$	−22960.0	+	22960.0i	−1.32688	+	1.32688i
$670$		0			0
$671$	2441.79			0.140483
$672$		0			0
$673$	−17168.0			−0.983325			−0.491663	−	0.870786i	$-0.663611\pi$
$673$	−17168.0			−0.983325			−0.491663	+	0.870786i	$0.663611\pi$
$674$		0			0
$675$	−7325.36	+	7325.36i	−0.417708	+	0.417708i
$676$		0			0
$677$	3029.00	+	3029.00i	0.171956	+	0.171956i	0.787838	−	0.615882i	$-0.211200\pi$
$677$	3029.00	+	3029.00i	0.171956	+	0.171956i	−0.615882	+	0.787838i	$0.711200\pi$
$678$		0			0
$679$			12450.4i			0.703687i
$680$		0			0
$681$		−	6760.00i		−	0.380387i
$682$		0			0
$683$	16220.4	+	16220.4i	0.908723	+	0.908723i	0.996169	−	0.0874463i	$-0.0278706\pi$
$683$	16220.4	+	16220.4i	0.908723	+	0.908723i	−0.0874463	+	0.996169i	$0.527871\pi$
$684$		0			0
$685$	12584.0	−	12584.0i	0.701912	−	0.701912i
$686$		0			0
$687$	−13926.2			−0.773390
$688$		0			0
$689$	−14950.0			−0.826632
$690$		0			0
$691$	11220.6	−	11220.6i	0.617730	−	0.617730i	−0.327219	−	0.944949i	$-0.606111\pi$
$691$	11220.6	−	11220.6i	0.617730	−	0.617730i	0.944949	+	0.327219i	$0.106111\pi$
$692$		0			0
$693$	−10920.0	−	10920.0i	−0.598581	−	0.598581i
$694$		0			0
$695$		−	17808.0i		−	0.971939i
$696$		0			0
$697$			29952.0i			1.62771i
$698$		0			0
$699$	−20384.0	−	20384.0i	−1.10300	−	1.10300i
$700$		0			0
$701$	10569.0	−	10569.0i	0.569452	−	0.569452i	−0.362523	−	0.931975i	$-0.618085\pi$
$701$	10569.0	−	10569.0i	0.569452	−	0.569452i	0.931975	+	0.362523i	$0.118085\pi$
$702$		0			0
$703$	11403.9			0.611818
$704$		0			0
$705$	8800.00			0.470109
$706$		0			0
$707$	−25464.3	+	25464.3i	−1.35458	+	1.35458i
$708$		0			0
$709$	−21307.0	−	21307.0i	−1.12863	−	1.12863i	−0.990400	−	0.138234i	$-0.955857\pi$
$709$	−21307.0	−	21307.0i	−1.12863	−	1.12863i	−0.138234	−	0.990400i	$-0.544143\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	22880.0i		−	1.20177i
$714$		0			0
$715$	15840.3	+	15840.3i	0.828523	+	0.828523i
$716$		0			0
$717$	−13600.0	+	13600.0i	−0.708370	+	0.708370i
$718$		0			0
$719$	−21788.2			−1.13013			−0.565066	−	0.825046i	$-0.691149\pi$
$719$	−21788.2			−1.13013			−0.565066	+	0.825046i	$0.691149\pi$
$720$		0			0
$721$	46800.0			2.41737
$722$		0			0
$723$	7298.53	−	7298.53i	0.375429	−	0.375429i
$724$		0			0
$725$	12051.0	+	12051.0i	0.617328	+	0.617328i
$726$		0			0
$727$			15160.5i			0.773416i	0.922202	+	0.386708i	$0.126388\pi$
$727$			15160.5i			0.773416i	−0.922202	+	0.386708i	$0.873612\pi$
$728$		0			0
$729$			3277.00i			0.166489i
$730$		0			0
$731$	−24471.5	−	24471.5i	−1.23818	−	1.23818i
$732$		0			0
$733$	−12105.0	+	12105.0i	−0.609971	+	0.609971i	−0.942938	−	0.332968i	$-0.891950\pi$
$733$	−12105.0	+	12105.0i	−0.609971	+	0.609971i	0.332968	+	0.942938i	$0.391950\pi$
$734$		0			0
$735$	−37091.9			−1.86144
$736$		0			0
$737$	−26600.0			−1.32948
$738$		0			0
$739$	−6453.29	+	6453.29i	−0.321229	+	0.321229i	−0.849238	−	0.528010i	$-0.822939\pi$
$739$	−6453.29	+	6453.29i	−0.321229	+	0.321229i	0.528010	+	0.849238i	$0.322939\pi$
$740$		0			0
$741$	−15640.0	−	15640.0i	−0.775371	−	0.775371i
$742$		0			0
$743$			7074.92i			0.349332i	0.984628	+	0.174666i	$0.0558846\pi$
$743$			7074.92i			0.349332i	−0.984628	+	0.174666i	$0.944115\pi$
$744$		0			0
$745$			11154.0i			0.548525i
$746$		0			0
$747$	−6802.12	−	6802.12i	−0.333168	−	0.333168i
$748$		0			0
$749$	4680.00	−	4680.00i	0.228309	−	0.228309i
$750$		0			0
$751$	−6511.43			−0.316385			−0.158193	−	0.987408i	$-0.550567\pi$
$751$	−6511.43			−0.316385			−0.158193	+	0.987408i	$0.550567\pi$
$752$		0			0
$753$	−26920.0			−1.30281
$754$		0			0
$755$	−25285.5	+	25285.5i	−1.21885	+	1.21885i
$756$		0			0
$757$	−27365.0	−	27365.0i	−1.31387	−	1.31387i	−0.918540	−	0.395327i	$-0.870631\pi$
$757$	−27365.0	−	27365.0i	−1.31387	−	1.31387i	−0.395327	−	0.918540i	$-0.629369\pi$
$758$		0			0
$759$		−	32557.1i		−	1.55698i
$760$		0			0
$761$		−	29400.0i		−	1.40046i	−0.713918	−	0.700229i	$-0.753081\pi$
$761$		−	29400.0i		−	1.40046i	0.713918	−	0.700229i	$-0.246919\pi$
$762$		0			0
$763$	15643.5	+	15643.5i	0.742246	+	0.742246i
$764$		0			0
$765$	13728.0	−	13728.0i	0.648806	−	0.648806i
$766$		0			0
$767$	−12137.4			−0.571389
$768$		0			0
$769$	8944.00			0.419413			0.209707	−	0.977764i	$-0.432749\pi$
$769$	8944.00			0.419413			0.209707	+	0.977764i	$0.432749\pi$
$770$		0			0
$771$	−3640.32	+	3640.32i	−0.170043	+	0.170043i
$772$		0			0
$773$	−17275.0	−	17275.0i	−0.803802	−	0.803802i	0.179886	−	0.983687i	$-0.442427\pi$
$773$	−17275.0	−	17275.0i	−0.803802	−	0.803802i	−0.983687	+	0.179886i	$0.942427\pi$
$774$		0			0
$775$		−	23022.6i		−	1.06709i
$776$		0			0
$777$		−	18000.0i		−	0.831076i
$778$		0			0
$779$	−23720.2	−	23720.2i	−1.09097	−	1.09097i
$780$		0			0
$781$	11480.0	−	11480.0i	0.525975	−	0.525975i
$782$		0			0
$783$	12897.6			0.588664
$784$		0			0
$785$	−11418.0			−0.519141
$786$		0			0
$787$	24010.9	−	24010.9i	1.08754	−	1.08754i	0.0917617	−	0.995781i	$-0.470750\pi$
$787$	24010.9	−	24010.9i	1.08754	−	1.08754i	0.995781	−	0.0917617i	$-0.0292498\pi$
$788$		0			0
$789$	10920.0	+	10920.0i	0.492728	+	0.492728i
$790$		0			0
$791$		−	29999.1i		−	1.34848i
$792$		0			0
$793$		−	1794.00i		−	0.0803365i
$794$		0			0
$795$	31975.8	+	31975.8i	1.42650	+	1.42650i
$796$		0			0
$797$	−1175.00	+	1175.00i	−0.0522216	+	0.0522216i	−0.732735	−	0.680514i	$-0.761757\pi$
$797$	−1175.00	+	1175.00i	−0.0522216	+	0.0522216i	0.680514	+	0.732735i	$0.261757\pi$
$798$		0			0
$799$	8586.50			0.380186
$800$		0			0
$801$	−18330.0			−0.808563
$802$		0			0
$803$	35625.0	−	35625.0i	1.56560	−	1.56560i
$804$		0			0
$805$	−34320.0	−	34320.0i	−1.50263	−	1.50263i
$806$		0			0
$807$		−	2209.24i		−	0.0963677i
$808$		0			0
$809$			17304.0i			0.752010i	0.926618	+	0.376005i	$0.122702\pi$
$809$			17304.0i			0.752010i	−0.926618	+	0.376005i	$0.877298\pi$
$810$		0			0
$811$	20522.6	+	20522.6i	0.888591	+	0.888591i	0.994388	−	0.105797i	$-0.0337394\pi$
$811$	20522.6	+	20522.6i	0.888591	+	0.888591i	−0.105797	+	0.994388i	$0.533739\pi$
$812$		0			0
$813$	400.000	−	400.000i	0.0172554	−	0.0172554i
$814$		0			0
$815$	21743.5			0.934531
$816$		0			0
$817$	38760.0			1.65978
$818$		0			0
$819$	−8023.01	+	8023.01i	−0.342304	+	0.342304i
$820$		0			0
$821$	10229.0	+	10229.0i	0.434829	+	0.434829i	0.890267	−	0.455438i	$-0.150517\pi$
$821$	10229.0	+	10229.0i	0.434829	+	0.434829i	−0.455438	+	0.890267i	$0.650517\pi$
$822$		0			0
$823$		−	34140.3i		−	1.44600i	−0.690850	−	0.722999i	$-0.742763\pi$
$823$		−	34140.3i		−	1.44600i	0.690850	−	0.722999i	$-0.257237\pi$
$824$		0			0
$825$		−	32760.0i		−	1.38249i
$826$		0			0
$827$	9735.84	+	9735.84i	0.409369	+	0.409369i	0.881519	−	0.472149i	$-0.156522\pi$
$827$	9735.84	+	9735.84i	0.409369	+	0.409369i	−0.472149	+	0.881519i	$0.656522\pi$
$828$		0			0
$829$	−2647.00	+	2647.00i	−0.110898	+	0.110898i	−0.760378	−	0.649481i	$-0.774986\pi$
$829$	−2647.00	+	2647.00i	−0.110898	+	0.110898i	0.649481	+	0.760378i	$0.274986\pi$
$830$		0			0
$831$	−24999.2			−1.04358
$832$		0			0
$833$	−36192.0			−1.50538
$834$		0			0
$835$	21743.5	−	21743.5i	0.901157	−	0.901157i
$836$		0			0
$837$	−12320.0	−	12320.0i	−0.508771	−	0.508771i
$838$		0			0
$839$			8255.56i			0.339706i	0.985469	+	0.169853i	$0.0543294\pi$
$839$			8255.56i			0.339706i	−0.985469	+	0.169853i	$0.945671\pi$
$840$		0			0
$841$			3171.00i			0.130018i
$842$		0			0
$843$	4159.09	+	4159.09i	0.169925	+	0.169925i
$844$		0			0
$845$	−12529.0	+	12529.0i	−0.510072	+	0.510072i
$846$		0			0
$847$	−16877.8			−0.684687
$848$		0			0
$849$	52520.0			2.12306
$850$		0			0
$851$	8720.67	−	8720.67i	0.351281	−	0.351281i
$852$		0			0
$853$	28005.0	+	28005.0i	1.12412	+	1.12412i	0.991116	+	0.133003i	$0.0424619\pi$
$853$	28005.0	+	28005.0i	1.12412	+	1.12412i	0.133003	+	0.991116i	$0.457538\pi$
$854$		0			0
$855$			21743.5i			0.869723i
$856$		0			0
$857$		−	30056.0i		−	1.19801i	−0.800746	−	0.599004i	$-0.795563\pi$
$857$		−	30056.0i		−	1.19801i	0.800746	−	0.599004i	$-0.204437\pi$
$858$		0			0
$859$	6220.74	+	6220.74i	0.247088	+	0.247088i	0.819775	−	0.572686i	$-0.194099\pi$
$859$	6220.74	+	6220.74i	0.247088	+	0.247088i	−0.572686	+	0.819775i	$0.694099\pi$
$860$		0			0
$861$	−37440.0	+	37440.0i	−1.48194	+	1.48194i
$862$		0			0
$863$	27458.9			1.08310			0.541548	−	0.840670i	$-0.317838\pi$
$863$	27458.9			1.08310			0.541548	+	0.840670i	$0.317838\pi$
$864$		0			0
$865$	35750.0			1.40524
$866$		0			0
$867$	19243.6	−	19243.6i	0.753803	−	0.753803i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			19543.2i			0.760272i
$872$		0			0
$873$		−	6032.00i		−	0.233851i
$874$		0			0
$875$	2361.29	+	2361.29i	0.0912298	+	0.0912298i
$876$		0			0
$877$	13479.0	−	13479.0i	0.518989	−	0.518989i	−0.398276	−	0.917265i	$-0.630391\pi$
$877$	13479.0	−	13479.0i	0.518989	−	0.518989i	0.917265	+	0.398276i	$0.130391\pi$
$878$		0			0
$879$	2531.23			0.0971288
$880$		0			0
$881$	30350.0			1.16063			0.580316	−	0.814391i	$-0.302929\pi$
$881$	30350.0			1.16063			0.580316	+	0.814391i	$0.302929\pi$
$882$		0			0
$883$	1140.39	−	1140.39i	0.0434624	−	0.0434624i	−0.685042	−	0.728504i	$-0.740216\pi$
$883$	1140.39	−	1140.39i	0.0434624	−	0.0434624i	0.728504	+	0.685042i	$0.240216\pi$
$884$		0			0
$885$	25960.0	+	25960.0i	0.986029	+	0.986029i
$886$		0			0
$887$			43603.3i			1.65057i	0.564716	+	0.825285i	$0.308986\pi$
$887$			43603.3i			1.65057i	−0.564716	+	0.825285i	$0.691014\pi$
$888$		0			0
$889$			56160.0i			2.11872i
$890$		0			0
$891$	−28518.8	−	28518.8i	−1.07230	−	1.07230i
$892$		0			0
$893$	−6800.00	+	6800.00i	−0.254819	+	0.254819i
$894$		0			0
$895$	7182.25			0.268242
$896$		0			0
$897$	−23920.0			−0.890374
$898$		0			0
$899$	−20267.7	+	20267.7i	−0.751909	+	0.751909i
$900$		0			0
$901$	31200.0	+	31200.0i	1.15363	+	1.15363i
$902$		0			0
$903$		−	61178.8i		−	2.25460i
$904$		0			0
$905$			6578.00i			0.241613i
$906$		0			0
$907$	−11721.5	−	11721.5i	−0.429112	−	0.429112i	0.459214	−	0.888326i	$-0.348131\pi$
$907$	−11721.5	−	11721.5i	−0.429112	−	0.429112i	−0.888326	+	0.459214i	$0.848131\pi$
$908$		0			0
$909$	12337.0	−	12337.0i	0.450157	−	0.450157i
$910$		0			0
$911$	41734.0			1.51779			0.758896	−	0.651212i	$-0.225739\pi$
$911$	41734.0			1.51779			0.758896	+	0.651212i	$0.225739\pi$
$912$		0			0
$913$	−32760.0			−1.18751
$914$		0			0
$915$	−3837.09	+	3837.09i	−0.138634	+	0.138634i
$916$		0			0
$917$	18600.0	+	18600.0i	0.669821	+	0.669821i
$918$		0			0
$919$			14999.5i			0.538400i	0.963084	+	0.269200i	$0.0867592\pi$
$919$			14999.5i			0.538400i	−0.963084	+	0.269200i	$0.913241\pi$
$920$		0			0
$921$		−	760.000i		−	0.0271909i
$922$		0			0
$923$	−8434.45	−	8434.45i	−0.300784	−	0.300784i
$924$		0			0
$925$	8775.00	−	8775.00i	0.311914	−	0.311914i
$926$		0			0
$927$	−22673.7			−0.803347
$928$		0			0
$929$	11440.0			0.404020			0.202010	−	0.979383i	$-0.435253\pi$
$929$	11440.0			0.404020			0.202010	+	0.979383i	$0.435253\pi$
$930$		0			0
$931$	28661.9	−	28661.9i	1.00898	−	1.00898i
$932$		0			0
$933$	12280.0	+	12280.0i	0.430899	+	0.430899i
$934$		0			0
$935$		−	66116.1i		−	2.31254i
$936$		0			0
$937$			9906.00i			0.345373i	0.984977	+	0.172687i	$0.0552448\pi$
$937$			9906.00i			0.345373i	−0.984977	+	0.172687i	$0.944755\pi$
$938$		0			0
$939$	−19713.2	−	19713.2i	−0.685107	−	0.685107i
$940$		0			0
$941$	1159.00	−	1159.00i	0.0401512	−	0.0401512i	−0.686746	−	0.726897i	$-0.740962\pi$
$941$	1159.00	−	1159.00i	0.0401512	−	0.0401512i	0.726897	+	0.686746i	$0.240962\pi$
$942$		0			0
$943$	−36278.0			−1.25278
$944$		0			0
$945$	−36960.0			−1.27228
$946$		0			0
$947$	16077.3	−	16077.3i	0.551682	−	0.551682i	−0.375244	−	0.926926i	$-0.622441\pi$
$947$	16077.3	−	16077.3i	0.551682	−	0.551682i	0.926926	+	0.375244i	$0.122441\pi$
$948$		0			0
$949$	−26174.0	−	26174.0i	−0.895305	−	0.895305i
$950$		0			0
$951$			5232.40i			0.178414i
$952$		0			0
$953$		−	49512.0i		−	1.68295i	−0.540296	−	0.841475i	$-0.681688\pi$
$953$		−	49512.0i		−	1.68295i	0.540296	−	0.841475i	$-0.318312\pi$
$954$		0			0
$955$	−19087.1	−	19087.1i	−0.646747	−	0.646747i
$956$		0			0
$957$	−28840.0	+	28840.0i	−0.974153	+	0.974153i
$958$		0			0
$959$	30696.7			1.03363
$960$		0			0
$961$	8929.00			0.299721
$962$		0			0
$963$	−2267.37	+	2267.37i	−0.0758723	+	0.0758723i
$964$		0			0
$965$	20592.0	+	20592.0i	0.686922	+	0.686922i
$966$		0			0
$967$			52225.6i			1.73678i	0.495886	+	0.868388i	$0.334844\pi$
$967$			52225.6i			1.73678i	−0.495886	+	0.868388i	$0.665156\pi$
$968$		0			0
$969$			65280.0i			2.16419i
$970$		0			0
$971$	24538.6	+	24538.6i	0.811000	+	0.811000i	0.984784	−	0.173783i	$-0.0555993\pi$
$971$	24538.6	+	24538.6i	0.811000	+	0.811000i	−0.173783	+	0.984784i	$0.555599\pi$
$972$		0			0
$973$	21720.0	−	21720.0i	0.715633	−	0.715633i
$974$		0			0
$975$	−24069.0			−0.790591
$976$		0			0
$977$	−3584.00			−0.117362			−0.0586808	−	0.998277i	$-0.518689\pi$
$977$	−3584.00			−0.117362			−0.0586808	+	0.998277i	$0.518689\pi$
$978$		0			0
$979$	−44140.0	+	44140.0i	−1.44098	+	1.44098i
$980$		0			0
$981$	−7579.00	−	7579.00i	−0.246665	−	0.246665i
$982$		0			0
$983$			32584.0i			1.05724i	0.848858	+	0.528620i	$0.177290\pi$
$983$			32584.0i			1.05724i	−0.848858	+	0.528620i	$0.822710\pi$
$984$		0			0
$985$		−	22638.0i		−	0.732291i
$986$		0			0
$987$	10733.1	+	10733.1i	0.346139	+	0.346139i
$988$		0			0
$989$	29640.0	−	29640.0i	0.952980	−	0.952980i
$990$		0			0
$991$	−19641.6			−0.629603			−0.314802	−	0.949158i	$-0.601938\pi$
$991$	−19641.6			−0.629603			−0.314802	+	0.949158i	$0.601938\pi$
$992$		0			0
$993$	520.000			0.0166180
$994$		0			0
$995$	−50669.3	+	50669.3i	−1.61440	+	1.61440i
$996$		0			0
$997$	3301.00	+	3301.00i	0.104858	+	0.104858i	0.757590	−	0.652731i	$-0.226377\pi$
$997$	3301.00	+	3301.00i	0.104858	+	0.104858i	−0.652731	+	0.757590i	$0.726377\pi$
$998$		0			0
$999$		−	9391.49i		−	0.297431i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	512.4.e.n.385.2	yes	4
4.3	odd	2	inner	512.4.e.n.385.1	yes	4
8.3	odd	2		512.4.e.k.385.2	yes	4
8.5	even	2		512.4.e.k.385.1	yes	4
16.3	odd	4	inner	512.4.e.n.129.1	yes	4
16.5	even	4		512.4.e.k.129.1	✓	4
16.11	odd	4		512.4.e.k.129.2	yes	4
16.13	even	4	inner	512.4.e.n.129.2	yes	4
32.3	odd	8		1024.4.a.g.1.1		4
32.5	even	8		1024.4.b.f.513.4		4
32.11	odd	8		1024.4.b.f.513.3		4
32.13	even	8		1024.4.a.g.1.2		4
32.19	odd	8		1024.4.a.g.1.4		4
32.21	even	8		1024.4.b.f.513.1		4
32.27	odd	8		1024.4.b.f.513.2		4
32.29	even	8		1024.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.4.e.k.129.1	✓	4	16.5	even	4
512.4.e.k.129.2	yes	4	16.11	odd	4
512.4.e.k.385.1	yes	4	8.5	even	2
512.4.e.k.385.2	yes	4	8.3	odd	2
512.4.e.n.129.1	yes	4	16.3	odd	4	inner
512.4.e.n.129.2	yes	4	16.13	even	4	inner
512.4.e.n.385.1	yes	4	4.3	odd	2	inner
512.4.e.n.385.2	yes	4	1.1	even	1	trivial
1024.4.a.g.1.1		4	32.3	odd	8
1024.4.a.g.1.2		4	32.13	even	8
1024.4.a.g.1.3		4	32.29	even	8
1024.4.a.g.1.4		4	32.19	odd	8
1024.4.b.f.513.1		4	32.21	even	8
1024.4.b.f.513.2		4	32.27	odd	8
1024.4.b.f.513.3		4	32.11	odd	8
1024.4.b.f.513.4		4	32.5	even	8

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:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	512.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.47214 - 4.47214i) q^{3} +(11.0000 + 11.0000i) q^{5} +26.8328i q^{7} -13.0000i q^{9} +O(q^{10})\)	\(q+(4.47214 - 4.47214i) q^{3} +(11.0000 + 11.0000i) q^{5} +26.8328i q^{7} -13.0000i q^{9} +(-31.3050 - 31.3050i) q^{11} +(-23.0000 + 23.0000i) q^{13} +98.3870 q^{15} +96.0000 q^{17} +(-76.0263 + 76.0263i) q^{19} +(120.000 + 120.000i) q^{21} +116.276i q^{23} +117.000i q^{25} +(62.6099 + 62.6099i) q^{27} +(103.000 - 103.000i) q^{29} -196.774 q^{31} -280.000 q^{33} +(-295.161 + 295.161i) q^{35} +(-75.0000 - 75.0000i) q^{37} +205.718i q^{39} +312.000i q^{41} +(-254.912 - 254.912i) q^{43} +(143.000 - 143.000i) q^{45} +89.4427 q^{47} -377.000 q^{49} +(429.325 - 429.325i) q^{51} +(325.000 + 325.000i) q^{53} -688.709i q^{55} +680.000i q^{57} +(263.856 + 263.856i) q^{59} +(-39.0000 + 39.0000i) q^{61} +348.827 q^{63} -506.000 q^{65} +(424.853 - 424.853i) q^{67} +(520.000 + 520.000i) q^{69} +366.715i q^{71} +1138.00i q^{73} +(523.240 + 523.240i) q^{75} +(840.000 - 840.000i) q^{77} +911.000 q^{81} +(523.240 - 523.240i) q^{83} +(1056.00 + 1056.00i) q^{85} -921.260i q^{87} -1410.00i q^{89} +(-617.155 - 617.155i) q^{91} +(-880.000 + 880.000i) q^{93} -1672.58 q^{95} +464.000 q^{97} +(-406.964 + 406.964i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 44 q^{5}+O(q^{10}) \) 4 * q + 44 * q^5	\( 4 q + 44 q^{5} - 92 q^{13} + 384 q^{17} + 480 q^{21} + 412 q^{29} - 1120 q^{33} - 300 q^{37} + 572 q^{45} - 1508 q^{49} + 1300 q^{53} - 156 q^{61} - 2024 q^{65} + 2080 q^{69} + 3360 q^{77} + 3644 q^{81} + 4224 q^{85} - 3520 q^{93} + 1856 q^{97}+O(q^{100}) \) 4 * q + 44 * q^5 - 92 * q^13 + 384 * q^17 + 480 * q^21 + 412 * q^29 - 1120 * q^33 - 300 * q^37 + 572 * q^45 - 1508 * q^49 + 1300 * q^53 - 156 * q^61 - 2024 * q^65 + 2080 * q^69 + 3360 * q^77 + 3644 * q^81 + 4224 * q^85 - 3520 * q^93 + 1856 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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