LMFDB - Embedded newform 350.5.d.a.349.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,5,Mod(349,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.349"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	350.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.1794870793$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6850489614336.26
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 60x^{6} + 1800x^{4} + 38340x^{2} + 408321 $ x^8 - 60x^6 + 1800x^4 + 38340*x^2 + 408321
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.8
Root			$-1.31383 + 3.17185i$ of defining polynomial
Character	$\chi$	$=$	350.349
Dual form			350.5.d.a.349.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.82843i q^{2} +12.6874 q^{3} -8.00000 q^{4} +35.8854i q^{6} +(48.5729 - 6.45584i) q^{7} -22.6274i q^{8} +79.9706 q^{9} +191.823 q^{11} -101.499 q^{12} +48.5729 q^{13} +(18.2599 + 137.385i) q^{14} +64.0000 q^{16} -181.977 q^{17} +226.191i q^{18} -599.915i q^{19} +(616.264 - 81.9080i) q^{21} +542.558i q^{22} -469.529i q^{23} -287.083i q^{24} +137.385i q^{26} -13.0609 q^{27} +(-388.583 + 51.6468i) q^{28} +338.881 q^{29} +267.556i q^{31} +181.019i q^{32} +2433.74 q^{33} -514.710i q^{34} -639.765 q^{36} +668.530i q^{37} +1696.82 q^{38} +616.264 q^{39} +1323.85i q^{41} +(231.671 + 1743.06i) q^{42} +1940.23i q^{43} -1534.59 q^{44} +1328.03 q^{46} +2936.89 q^{47} +811.995 q^{48} +(2317.64 - 627.158i) q^{49} -2308.82 q^{51} -388.583 q^{52} -1460.94i q^{53} -36.9418i q^{54} +(-146.079 - 1099.08i) q^{56} -7611.38i q^{57} +958.501i q^{58} +1730.83i q^{59} +246.343i q^{61} -756.763 q^{62} +(3884.40 - 516.277i) q^{63} -512.000 q^{64} +6883.67i q^{66} +1076.59i q^{67} +1455.82 q^{68} -5957.11i q^{69} -2276.39 q^{71} -1809.53i q^{72} +7106.94 q^{73} -1890.89 q^{74} +4799.32i q^{76} +(9317.41 - 1238.38i) q^{77} +1743.06i q^{78} -7012.38 q^{79} -6643.32 q^{81} -3744.40 q^{82} +1448.36 q^{83} +(-4930.11 + 655.264i) q^{84} -5487.81 q^{86} +4299.53 q^{87} -4340.47i q^{88} +2133.73i q^{89} +(2359.32 - 313.579i) q^{91} +3756.23i q^{92} +3394.60i q^{93} +8306.77i q^{94} +2296.67i q^{96} -5898.76 q^{97} +(1773.87 + 6555.29i) q^{98} +15340.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4} + 504 q^{9} + 720 q^{11} + 576 q^{14} + 512 q^{16} + 1536 q^{21} - 2448 q^{29} - 4032 q^{36} + 1536 q^{39} - 5760 q^{44} + 6144 q^{46} + 3064 q^{49} - 22272 q^{51} - 4608 q^{56} - 4096 q^{64}+ \cdots + 59184 q^{99}+O(q^{100}) $ 8 * q - 64 * q^4 + 504 * q^9 + 720 * q^11 + 576 * q^14 + 512 * q^16 + 1536 * q^21 - 2448 * q^29 - 4032 * q^36 + 1536 * q^39 - 5760 * q^44 + 6144 * q^46 + 3064 * q^49 - 22272 * q^51 - 4608 * q^56 - 4096 * q^64 - 43056 * q^71 + 6912 * q^74 - 25552 * q^79 - 59256 * q^81 - 12288 * q^84 - 23040 * q^86 + 11136 * q^91 + 59184 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$			2.82843i			0.707107i
$2$			2.82843i			0.707107i
$3$	12.6874			1.40971			0.704857	−	0.709350i	$-0.251011\pi$
$3$	12.6874			1.40971			0.704857	+	0.709350i	$0.251011\pi$
$4$	−8.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			35.8854i			0.996818i
$7$	48.5729	−	6.45584i	0.991283	−	0.131752i
$7$	48.5729	−	6.45584i	0.991283	−	0.131752i
$8$		−	22.6274i		−	0.353553i
$9$	79.9706			0.987291
$10$		0			0
$11$	191.823			1.58532			0.792659	−	0.609666i	$-0.208696\pi$
$11$	191.823			1.58532			0.792659	+	0.609666i	$0.208696\pi$
$12$	−101.499			−0.704857
$13$	48.5729			0.287413			0.143707	−	0.989620i	$-0.454098\pi$
$13$	48.5729			0.287413			0.143707	+	0.989620i	$0.454098\pi$
$14$	18.2599	+	137.385i	0.0931627	+	0.700943i
$15$		0			0
$16$	64.0000			0.250000
$17$	−181.977			−0.629680			−0.314840	−	0.949145i	$-0.601951\pi$
$17$	−181.977			−0.629680			−0.314840	+	0.949145i	$0.601951\pi$
$18$			226.191i			0.698120i
$19$		−	599.915i		−	1.66182i	−0.556410	−	0.830908i	$-0.687822\pi$
$19$		−	599.915i		−	1.66182i	0.556410	−	0.830908i	$-0.312178\pi$
$20$		0			0
$21$	616.264	−	81.9080i	1.39742	−	0.185732i
$22$			542.558i			1.12099i
$23$		−	469.529i		−	0.887578i	−0.896131	−	0.443789i	$-0.853634\pi$
$23$		−	469.529i		−	0.887578i	0.896131	−	0.443789i	$-0.146366\pi$
$24$		−	287.083i		−	0.498409i
$25$		0			0
$26$			137.385i			0.203232i
$27$	−13.0609			−0.0179162
$28$	−388.583	+	51.6468i	−0.495641	+	0.0658760i
$29$	338.881			0.402951			0.201475	−	0.979494i	$-0.435426\pi$
$29$	338.881			0.402951			0.201475	+	0.979494i	$0.435426\pi$
$30$		0			0
$31$			267.556i			0.278414i	0.990263	+	0.139207i	$0.0444554\pi$
$31$			267.556i			0.278414i	−0.990263	+	0.139207i	$0.955545\pi$
$32$			181.019i			0.176777i
$33$	2433.74			2.23484
$34$		−	514.710i		−	0.445251i
$35$		0			0
$36$	−639.765			−0.493645
$37$			668.530i			0.488334i	0.969733	+	0.244167i	$0.0785146\pi$
$37$			668.530i			0.488334i	−0.969733	+	0.244167i	$0.921485\pi$
$38$	1696.82			1.17508
$39$	616.264			0.405170
$40$		0			0
$41$			1323.85i			0.787534i	0.919210	+	0.393767i	$0.128828\pi$
$41$			1323.85i			0.787534i	−0.919210	+	0.393767i	$0.871172\pi$
$42$	231.671	+	1743.06i	0.131333	+	0.988128i
$43$			1940.23i			1.04934i	0.851305	+	0.524671i	$0.175812\pi$
$43$			1940.23i			1.04934i	−0.851305	+	0.524671i	$0.824188\pi$
$44$	−1534.59			−0.792659
$45$		0			0
$46$	1328.03			0.627613
$47$	2936.89			1.32951			0.664755	−	0.747062i	$-0.268536\pi$
$47$	2936.89			1.32951			0.664755	+	0.747062i	$0.268536\pi$
$48$	811.995			0.352428
$49$	2317.64	−	627.158i	0.965283	−	0.261207i
$50$		0			0
$51$	−2308.82			−0.887668
$52$	−388.583			−0.143707
$53$		−	1460.94i		−	0.520091i	−0.965596	−	0.260046i	$-0.916262\pi$
$53$		−	1460.94i		−	0.520091i	0.965596	−	0.260046i	$-0.0837376\pi$
$54$		−	36.9418i		−	0.0126687i
$55$		0			0
$56$	−146.079	−	1099.08i	−0.0465813	−	0.350471i
$57$		−	7611.38i		−	2.34268i
$58$			958.501i			0.284929i
$59$			1730.83i			0.497223i	0.968603	+	0.248612i	$0.0799743\pi$
$59$			1730.83i			0.497223i	−0.968603	+	0.248612i	$0.920026\pi$
$60$		0			0
$61$			246.343i			0.0662034i	0.999452	+	0.0331017i	$0.0105385\pi$
$61$			246.343i			0.0662034i	−0.999452	+	0.0331017i	$0.989461\pi$
$62$	−756.763			−0.196869
$63$	3884.40	−	516.277i	0.978684	−	0.130077i
$64$	−512.000			−0.125000
$65$		0			0
$66$			6883.67i			1.58027i
$67$			1076.59i			0.239828i	0.992784	+	0.119914i	$0.0382619\pi$
$67$			1076.59i			0.239828i	−0.992784	+	0.119914i	$0.961738\pi$
$68$	1455.82			0.314840
$69$		−	5957.11i		−	1.25123i
$70$		0			0
$71$	−2276.39			−0.451574			−0.225787	−	0.974177i	$-0.572495\pi$
$71$	−2276.39			−0.451574			−0.225787	+	0.974177i	$0.572495\pi$
$72$		−	1809.53i		−	0.349060i
$73$	7106.94			1.33363			0.666817	−	0.745221i	$-0.267656\pi$
$73$	7106.94			1.33363			0.666817	+	0.745221i	$0.267656\pi$
$74$	−1890.89			−0.345305
$75$		0			0
$76$			4799.32i			0.830908i
$77$	9317.41	−	1238.38i	1.57150	−	0.208869i
$78$			1743.06i			0.286499i
$79$	−7012.38			−1.12360			−0.561799	−	0.827274i	$-0.689891\pi$
$79$	−7012.38			−1.12360			−0.561799	+	0.827274i	$0.689891\pi$
$80$		0			0
$81$	−6643.32			−1.01255
$82$	−3744.40			−0.556871
$83$	1448.36			0.210243			0.105121	−	0.994459i	$-0.466477\pi$
$83$	1448.36			0.210243			0.105121	+	0.994459i	$0.466477\pi$
$84$	−4930.11	+	655.264i	−0.698712	+	0.0928662i
$85$		0			0
$86$	−5487.81			−0.741997
$87$	4299.53			0.568045
$88$		−	4340.47i		−	0.560494i
$89$			2133.73i			0.269376i	0.990888	+	0.134688i	$0.0430032\pi$
$89$			2133.73i			0.269376i	−0.990888	+	0.134688i	$0.956997\pi$
$90$		0			0
$91$	2359.32	−	313.579i	0.284908	−	0.0378673i
$92$			3756.23i			0.443789i
$93$			3394.60i			0.392484i
$94$			8306.77i			0.940105i
$95$		0			0
$96$			2296.67i			0.249204i
$97$	−5898.76			−0.626928			−0.313464	−	0.949600i	$-0.601489\pi$
$97$	−5898.76			−0.626928			−0.313464	+	0.949600i	$0.601489\pi$
$98$	1773.87	+	6555.29i	0.184701	+	0.682558i
$99$	15340.2			1.56517
$100$		0			0
$101$			9172.07i			0.899135i	0.893246	+	0.449567i	$0.148422\pi$
$101$			9172.07i			0.899135i	−0.893246	+	0.449567i	$0.851578\pi$
$102$		−	6530.34i		−	0.627676i
$103$	−3906.46			−0.368222			−0.184111	−	0.982905i	$-0.558941\pi$
$103$	−3906.46			−0.368222			−0.184111	+	0.982905i	$0.558941\pi$
$104$		−	1099.08i		−	0.101616i
$105$		0			0
$106$	4132.15			0.367760
$107$			12141.3i			1.06047i	0.847852	+	0.530233i	$0.177896\pi$
$107$			12141.3i			1.06047i	−0.847852	+	0.530233i	$0.822104\pi$
$108$	104.487			0.00895809
$109$	−6808.34			−0.573044			−0.286522	−	0.958074i	$-0.592499\pi$
$109$	−6808.34			−0.573044			−0.286522	+	0.958074i	$0.592499\pi$
$110$		0			0
$111$			8481.92i			0.688411i
$112$	3108.66	−	413.174i	0.247821	−	0.0329380i
$113$		−	4764.20i		−	0.373107i	−0.982445	−	0.186553i	$-0.940268\pi$
$113$		−	4764.20i		−	0.373107i	0.982445	−	0.186553i	$-0.0597317\pi$
$114$	21528.2			1.65653
$115$		0			0
$116$	−2711.05			−0.201475
$117$	3884.40			0.283761
$118$	−4895.54			−0.351590
$119$	−8839.17	+	1174.82i	−0.624191	+	0.0829615i
$120$		0			0
$121$	22155.2			1.51323
$122$	−696.763			−0.0468129
$123$			16796.2i			1.11020i
$124$		−	2140.45i		−	0.139207i
$125$		0			0
$126$	1460.25	+	10986.7i	0.0919787	+	0.692034i
$127$			27968.9i			1.73408i	0.498242	+	0.867038i	$0.333979\pi$
$127$			27968.9i			1.73408i	−0.498242	+	0.867038i	$0.666021\pi$
$128$		−	1448.15i		−	0.0883883i
$129$			24616.6i			1.47927i
$130$		0			0
$131$		−	24016.5i		−	1.39948i	−0.714397	−	0.699741i	$-0.753299\pi$
$131$		−	24016.5i		−	1.39948i	0.714397	−	0.699741i	$-0.246701\pi$
$132$	−19469.9			−1.11742
$133$	−3872.96	−	29139.6i	−0.218947	−	1.64733i
$134$	−3045.05			−0.169584
$135$		0			0
$136$			4117.68i			0.222625i
$137$			4162.00i			0.221748i	0.993834	+	0.110874i	$0.0353651\pi$
$137$			4162.00i			0.221748i	−0.993834	+	0.110874i	$0.964635\pi$
$138$	16849.3			0.884754
$139$		−	26365.8i		−	1.36462i	−0.731064	−	0.682309i	$-0.760976\pi$
$139$		−	26365.8i		−	1.36462i	0.731064	−	0.682309i	$-0.239024\pi$
$140$		0			0
$141$	37261.5			1.87423
$142$		−	6438.59i		−	0.319311i
$143$	9317.41			0.455641
$144$	5118.12			0.246823
$145$		0			0
$146$			20101.5i			0.943022i
$147$	29404.9	−	7957.01i	1.36077	−	0.368227i
$148$		−	5348.24i		−	0.244167i
$149$	6576.57			0.296229			0.148114	−	0.988970i	$-0.452680\pi$
$149$	6576.57			0.296229			0.148114	+	0.988970i	$0.452680\pi$
$150$		0			0
$151$	−22930.4			−1.00568			−0.502839	−	0.864380i	$-0.667711\pi$
$151$	−22930.4			−1.00568			−0.502839	+	0.864380i	$0.667711\pi$
$152$	−13574.5			−0.587540
$153$	−14552.8			−0.621677
$154$	3502.67	+	26353.6i	0.147692	+	1.11122i
$155$		0			0
$156$	−4930.11			−0.202585
$157$	37292.9			1.51296			0.756480	−	0.654017i	$-0.226918\pi$
$157$	37292.9			1.51296			0.756480	+	0.654017i	$0.226918\pi$
$158$		−	19834.0i		−	0.794504i
$159$		−	18535.5i		−	0.733180i
$160$		0			0
$161$	−3031.21	−	22806.4i	−0.116940	−	0.879841i
$162$		−	18790.2i		−	0.715979i
$163$			40854.0i			1.53766i	0.639455	+	0.768828i	$0.279160\pi$
$163$			40854.0i			1.53766i	−0.639455	+	0.768828i	$0.720840\pi$
$164$		−	10590.8i		−	0.393767i
$165$		0			0
$166$			4096.58i			0.148664i
$167$	−34774.9			−1.24690			−0.623452	−	0.781862i	$-0.714270\pi$
$167$	−34774.9			−1.24690			−0.623452	+	0.781862i	$0.714270\pi$
$168$	−1853.37	−	13944.5i	−0.0656663	−	0.494064i
$169$	−26201.7			−0.917394
$170$		0			0
$171$		−	47975.6i		−	1.64070i
$172$		−	15521.9i		−	0.524671i
$173$	−31600.1			−1.05583			−0.527917	−	0.849296i	$-0.677027\pi$
$173$	−31600.1			−1.05583			−0.527917	+	0.849296i	$0.677027\pi$
$174$			12160.9i			0.401668i
$175$		0			0
$176$	12276.7			0.396329
$177$			21959.8i			0.700942i
$178$	−6035.09			−0.190478
$179$	−22750.7			−0.710048			−0.355024	−	0.934857i	$-0.615527\pi$
$179$	−22750.7			−0.710048			−0.355024	+	0.934857i	$0.615527\pi$
$180$		0			0
$181$		−	55434.4i		−	1.69208i	−0.533116	−	0.846042i	$-0.678979\pi$
$181$		−	55434.4i		−	1.69208i	0.533116	−	0.846042i	$-0.321021\pi$
$182$	886.935	+	6673.17i	0.0267762	+	0.201460i
$183$			3125.45i			0.0933278i
$184$	−10624.2			−0.313806
$185$		0			0
$186$	−9601.37			−0.277528
$187$	−34907.5			−0.998242
$188$	−23495.1			−0.664755
$189$	−634.405	+	84.3191i	−0.0177600	+	0.00236049i
$190$		0			0
$191$	−50817.6			−1.39299			−0.696494	−	0.717562i	$-0.745258\pi$
$191$	−50817.6			−1.39299			−0.696494	+	0.717562i	$0.745258\pi$
$192$	−6495.96			−0.176214
$193$		−	1248.34i		−	0.0335134i	−0.999860	−	0.0167567i	$-0.994666\pi$
$193$		−	1248.34i		−	0.0335134i	0.999860	−	0.0167567i	$-0.00533407\pi$
$194$		−	16684.2i		−	0.443305i
$195$		0			0
$196$	−18541.2	+	5017.26i	−0.482641	+	0.130603i
$197$		−	64454.6i		−	1.66082i	−0.557155	−	0.830408i	$-0.688107\pi$
$197$		−	64454.6i		−	1.66082i	0.557155	−	0.830408i	$-0.311893\pi$
$198$			43388.7i			1.10674i
$199$			2352.60i			0.0594076i	0.999559	+	0.0297038i	$0.00945641\pi$
$199$			2352.60i			0.0594076i	−0.999559	+	0.0297038i	$0.990544\pi$
$200$		0			0
$201$			13659.1i			0.338088i
$202$	−25942.5			−0.635784
$203$	16460.4	−	2187.77i	0.399438	−	0.0530895i
$204$	18470.6			0.443834
$205$		0			0
$206$		−	11049.2i		−	0.260372i
$207$		−	37548.5i		−	0.876298i
$208$	3108.66			0.0718533
$209$		−	115078.i		−	2.63450i
$210$		0			0
$211$	−65056.4			−1.46125			−0.730626	−	0.682778i	$-0.760772\pi$
$211$	−65056.4			−1.46125			−0.730626	+	0.682778i	$0.760772\pi$
$212$			11687.5i			0.260046i
$213$	−28881.5			−0.636590
$214$	−34340.7			−0.749863
$215$		0			0
$216$			295.534i			0.00633433i
$217$	1727.30	+	12996.0i	0.0366816	+	0.275987i
$218$		−	19256.9i		−	0.405203i
$219$	90168.7			1.88004
$220$		0			0
$221$	−8839.17			−0.180978
$222$	−23990.5			−0.486780
$223$	−30412.4			−0.611563			−0.305781	−	0.952102i	$-0.598918\pi$
$223$	−30412.4			−0.611563			−0.305781	+	0.952102i	$0.598918\pi$
$224$	1168.63	+	8792.63i	0.0232907	+	0.175236i
$225$		0			0
$226$	13475.2			0.263826
$227$	52125.5			1.01158			0.505788	−	0.862658i	$-0.331202\pi$
$227$	52125.5			1.01158			0.505788	+	0.862658i	$0.331202\pi$
$228$			60891.0i			1.17134i
$229$			81280.2i			1.54994i	0.632000	+	0.774968i	$0.282234\pi$
$229$			81280.2i			1.54994i	−0.632000	+	0.774968i	$0.717766\pi$
$230$		0			0
$231$	118214.	−	15711.9i	2.21536	−	0.294445i
$232$		−	7668.01i		−	0.142465i
$233$		−	41718.9i		−	0.768459i	−0.923238	−	0.384229i	$-0.874467\pi$
$233$		−	41718.9i		−	0.768459i	0.923238	−	0.384229i	$-0.125533\pi$
$234$			10986.7i			0.200649i
$235$		0			0
$236$		−	13846.7i		−	0.248612i
$237$	−88968.9			−1.58395
$238$	−3322.89	−	25000.9i	−0.0586627	−	0.441370i
$239$	3936.55			0.0689160			0.0344580	−	0.999406i	$-0.489030\pi$
$239$	3936.55			0.0689160			0.0344580	+	0.999406i	$0.489030\pi$
$240$		0			0
$241$		−	70511.5i		−	1.21402i	−0.794694	−	0.607010i	$-0.792369\pi$
$241$		−	70511.5i		−	1.21402i	0.794694	−	0.607010i	$-0.207631\pi$
$242$			62664.4i			1.07002i
$243$	−83228.7			−1.40949
$244$		−	1970.74i		−	0.0331017i
$245$		0			0
$246$	−47506.8			−0.785028
$247$		−	29139.6i		−	0.477628i
$248$	6054.11			0.0984343
$249$	18376.0			0.296382
$250$		0			0
$251$		−	72042.3i		−	1.14351i	−0.820424	−	0.571755i	$-0.806263\pi$
$251$		−	72042.3i		−	1.14351i	0.820424	−	0.571755i	$-0.193737\pi$
$252$	−31075.2	+	4130.22i	−0.489342	+	0.0650387i
$253$		−	90066.6i		−	1.40709i
$254$	−79108.1			−1.22618
$255$		0			0
$256$	4096.00			0.0625000
$257$	65615.5			0.993436			0.496718	−	0.867912i	$-0.334538\pi$
$257$	65615.5			0.993436			0.496718	+	0.867912i	$0.334538\pi$
$258$	−69626.1			−1.04600
$259$	4315.92	+	32472.4i	0.0643390	+	0.484078i
$260$		0			0
$261$	27100.5			0.397829
$262$	67929.0			0.989583
$263$		−	38706.1i		−	0.559588i	−0.960060	−	0.279794i	$-0.909734\pi$
$263$		−	38706.1i		−	0.559588i	0.960060	−	0.279794i	$-0.0902662\pi$
$264$		−	55069.3i		−	0.790136i
$265$		0			0
$266$	82419.2	−	10954.4i	1.16484	−	0.154819i
$267$			27071.5i			0.379743i
$268$		−	8612.70i		−	0.119914i
$269$			87226.9i			1.20544i	0.797952	+	0.602721i	$0.205917\pi$
$269$			87226.9i			1.20544i	−0.797952	+	0.602721i	$0.794083\pi$
$270$		0			0
$271$			105362.i			1.43465i	0.696739	+	0.717324i	$0.254633\pi$
$271$			105362.i			1.43465i	−0.696739	+	0.717324i	$0.745367\pi$
$272$	−11646.6			−0.157420
$273$	29933.7	−	3978.50i	0.401638	−	0.0533820i
$274$	−11771.9			−0.156800
$275$		0			0
$276$			47656.9i			0.625615i
$277$		−	36178.9i		−	0.471515i	−0.971812	−	0.235758i	$-0.924243\pi$
$277$		−	36178.9i		−	0.471515i	0.971812	−	0.235758i	$-0.0757571\pi$
$278$	74573.7			0.964931
$279$			21396.6i			0.274876i
$280$		0			0
$281$	99142.2			1.25558			0.627792	−	0.778381i	$-0.283959\pi$
$281$	99142.2			1.25558			0.627792	+	0.778381i	$0.283959\pi$
$282$			105391.i			1.32528i
$283$	4153.27			0.0518581			0.0259291	−	0.999664i	$-0.491746\pi$
$283$	4153.27			0.0518581			0.0259291	+	0.999664i	$0.491746\pi$
$284$	18211.1			0.225787
$285$		0			0
$286$			26353.6i			0.322187i
$287$	8546.54	+	64302.9i	0.103759	+	0.780669i
$288$			14476.2i			0.174530i
$289$	−50405.2			−0.603503
$290$		0			0
$291$	−74840.1			−0.883788
$292$	−56855.5			−0.666817
$293$	20239.4			0.235756			0.117878	−	0.993028i	$-0.462391\pi$
$293$	20239.4			0.235756			0.117878	+	0.993028i	$0.462391\pi$
$294$	22505.8	+	83169.7i	0.260376	+	0.962211i
$295$		0			0
$296$	15127.1			0.172652
$297$	−2505.39			−0.0284028
$298$			18601.3i			0.209465i
$299$		−	22806.4i		−	0.255102i
$300$		0			0
$301$	12525.8	+	94242.7i	0.138253	+	1.04019i
$302$		−	64857.1i		−	0.711121i
$303$			116370.i			1.26752i
$304$		−	38394.6i		−	0.415454i
$305$		0			0
$306$		−	41161.7i		−	0.439592i
$307$	63269.8			0.671305			0.335652	−	0.941986i	$-0.391043\pi$
$307$	63269.8			0.671305			0.335652	+	0.941986i	$0.391043\pi$
$308$	−74539.3	+	9907.05i	−0.785749	+	0.104434i
$309$	−49563.0			−0.519087
$310$		0			0
$311$		−	14375.4i		−	0.148627i	−0.997235	−	0.0743137i	$-0.976323\pi$
$311$		−	14375.4i		−	0.148627i	0.997235	−	0.0743137i	$-0.0236766\pi$
$312$		−	13944.5i		−	0.143249i
$313$	−36763.0			−0.375252			−0.187626	−	0.982241i	$-0.560079\pi$
$313$	−36763.0			−0.375252			−0.187626	+	0.982241i	$0.560079\pi$
$314$			105480.i			1.06982i
$315$		0			0
$316$	56099.0			0.561799
$317$			125556.i			1.24945i	0.780844	+	0.624726i	$0.214789\pi$
$317$			125556.i			1.24945i	−0.780844	+	0.624726i	$0.785211\pi$
$318$	52426.4			0.518436
$319$	65005.4			0.638804
$320$		0			0
$321$			154041.i			1.49495i
$322$	64506.1	−	8573.55i	0.622142	−	0.0826892i
$323$			109171.i			1.04641i
$324$	53146.6			0.506274
$325$		0			0
$326$	−115553.			−1.08729
$327$	−86380.2			−0.807828
$328$	29955.2			0.278435
$329$	142653.	−	18960.1i	1.31792	−	0.175165i
$330$		0			0
$331$	5376.54			0.0490735			0.0245367	−	0.999699i	$-0.492189\pi$
$331$	5376.54			0.0490735			0.0245367	+	0.999699i	$0.492189\pi$
$332$	−11586.9			−0.105121
$333$			53462.7i			0.482128i
$334$		−	98358.3i		−	0.881694i
$335$		0			0
$336$	39440.9	−	5242.11i	0.349356	−	0.0464331i
$337$		−	2202.27i		−	0.0193914i	−0.999953	−	0.00969572i	$-0.996914\pi$
$337$		−	2202.27i		−	0.0193914i	0.999953	−	0.00969572i	$-0.00308629\pi$
$338$		−	74109.5i		−	0.648695i
$339$		−	60445.4i		−	0.525974i
$340$		0			0
$341$			51323.5i			0.441375i
$342$	135695.			1.16015
$343$	108526.	−	45425.2i	0.922454	−	0.386108i
$344$	43902.5			0.370998
$345$		0			0
$346$		−	89378.5i		−	0.746587i
$347$		−	222201.i		−	1.84538i	−0.385541	−	0.922691i	$-0.625985\pi$
$347$		−	222201.i		−	1.84538i	0.385541	−	0.922691i	$-0.374015\pi$
$348$	−34396.2			−0.284022
$349$			102679.i			0.843006i	0.906827	+	0.421503i	$0.138497\pi$
$349$			102679.i			0.843006i	−0.906827	+	0.421503i	$0.861503\pi$
$350$		0			0
$351$	−634.405			−0.00514935
$352$			34723.7i			0.280247i
$353$	−62595.6			−0.502336			−0.251168	−	0.967943i	$-0.580815\pi$
$353$	−62595.6			−0.502336			−0.251168	+	0.967943i	$0.580815\pi$
$354$	−62111.8			−0.495641
$355$		0			0
$356$		−	17069.8i		−	0.134688i
$357$	−112146.	+	14905.4i	−0.879930	+	0.116952i
$358$		−	64348.6i		−	0.502080i
$359$	−95505.9			−0.741040			−0.370520	−	0.928825i	$-0.620820\pi$
$359$	−95505.9			−0.741040			−0.370520	+	0.928825i	$0.620820\pi$
$360$		0			0
$361$	−229577.			−1.76163
$362$	156792.			1.19648
$363$	281092.			2.13322
$364$	−18874.6	+	2508.63i	−0.142454	+	0.0189336i
$365$		0			0
$366$	−8840.12			−0.0659927
$367$	−82330.9			−0.611267			−0.305633	−	0.952149i	$-0.598868\pi$
$367$	−82330.9			−0.611267			−0.305633	+	0.952149i	$0.598868\pi$
$368$		−	30049.9i		−	0.221895i
$369$			105869.i			0.777525i
$370$		0			0
$371$	−9431.58	−	70961.9i	−0.0685230	−	0.515558i
$372$		−	27156.8i		−	0.196242i
$373$			130223.i			0.935991i	0.883731	+	0.467995i	$0.155024\pi$
$373$			130223.i			0.935991i	−0.883731	+	0.467995i	$0.844976\pi$
$374$		−	98733.4i		−	0.705864i
$375$		0			0
$376$		−	66454.2i		−	0.470053i
$377$	16460.4			0.115813
$378$	−238.491	−	1794.37i	−0.00166912	−	0.0125582i
$379$	−192349.			−1.33909			−0.669546	−	0.742770i	$-0.733511\pi$
$379$	−192349.			−1.33909			−0.669546	+	0.742770i	$0.733511\pi$
$380$		0			0
$381$			354853.i			2.44455i
$382$		−	143734.i		−	0.984992i
$383$	101933.			0.694891			0.347446	−	0.937700i	$-0.387049\pi$
$383$	101933.			0.694891			0.347446	+	0.937700i	$0.387049\pi$
$384$		−	18373.3i		−	0.124602i
$385$		0			0
$386$	3530.84			0.0236975
$387$			155162.i			1.03601i
$388$	47190.1			0.313464
$389$	−191074.			−1.26271			−0.631353	−	0.775495i	$-0.717500\pi$
$389$	−191074.			−1.26271			−0.631353	+	0.775495i	$0.717500\pi$
$390$		0			0
$391$			85443.7i			0.558890i
$392$	−14191.0	−	52442.3i	−0.0923506	−	0.341279i
$393$		−	304708.i		−	1.97287i
$394$	182305.			1.17437
$395$		0			0
$396$	−122722.			−0.782585
$397$	201143.			1.27622			0.638108	−	0.769947i	$-0.279717\pi$
$397$	201143.			1.27622			0.638108	+	0.769947i	$0.279717\pi$
$398$	−6654.16			−0.0420076
$399$	−49137.9	−	369706.i	−0.308653	−	2.32226i
$400$		0			0
$401$	−39978.4			−0.248621			−0.124310	−	0.992243i	$-0.539672\pi$
$401$	−39978.4			−0.248621			−0.124310	+	0.992243i	$0.539672\pi$
$402$	−38633.8			−0.239065
$403$			12996.0i			0.0800200i
$404$		−	73376.6i		−	0.449567i
$405$		0			0
$406$	6187.93	+	46557.1i	0.0375399	+	0.282445i
$407$			128240.i			0.774165i
$408$			52242.7i			0.313838i
$409$			80655.7i			0.482157i	0.970506	+	0.241078i	$0.0775011\pi$
$409$			80655.7i			0.482157i	−0.970506	+	0.241078i	$0.922499\pi$
$410$		0			0
$411$			52805.0i			0.312602i
$412$	31251.7			0.184111
$413$	11174.0	+	84071.6i	0.0655101	+	0.492889i
$414$	106203.			0.619636
$415$		0			0
$416$			8792.63i			0.0508080i
$417$		−	334514.i		−	1.92372i
$418$	325489.			1.86288
$419$		−	252034.i		−	1.43559i	−0.696254	−	0.717795i	$-0.745151\pi$
$419$		−	252034.i		−	1.43559i	0.696254	−	0.717795i	$-0.254849\pi$
$420$		0			0
$421$	−84439.3			−0.476410			−0.238205	−	0.971215i	$-0.576559\pi$
$421$	−84439.3			−0.476410			−0.238205	+	0.971215i	$0.576559\pi$
$422$		−	184007.i		−	1.03326i
$423$	234864.			1.31261
$424$	−33057.2			−0.183880
$425$		0			0
$426$		−	81689.1i		−	0.450137i
$427$	1590.35	+	11965.6i	0.00872242	+	0.0656263i
$428$		−	97130.2i		−	0.530233i
$429$	118214.			0.642323
$430$		0			0
$431$	−127512.			−0.686431			−0.343215	−	0.939257i	$-0.611516\pi$
$431$	−127512.			−0.686431			−0.343215	+	0.939257i	$0.611516\pi$
$432$	−835.898			−0.00447905
$433$	233539.			1.24562			0.622808	−	0.782375i	$-0.285992\pi$
$433$	233539.			1.24562			0.622808	+	0.782375i	$0.285992\pi$
$434$	−36758.2	+	4885.55i	−0.195153	+	0.0259378i
$435$		0			0
$436$	54466.7			0.286522
$437$	−281678.			−1.47499
$438$			255036.i			1.32939i
$439$			304238.i			1.57864i	0.613980	+	0.789322i	$0.289568\pi$
$439$			304238.i			1.57864i	−0.613980	+	0.789322i	$0.710432\pi$
$440$		0			0
$441$	185343.	−	50154.1i	0.953015	−	0.257887i
$442$		−	25000.9i		−	0.127971i
$443$		−	87061.0i		−	0.443625i	−0.975089	−	0.221813i	$-0.928803\pi$
$443$		−	87061.0i		−	0.443625i	0.975089	−	0.221813i	$-0.0711974\pi$
$444$		−	67855.3i		−	0.344206i
$445$		0			0
$446$		−	86019.3i		−	0.432440i
$447$	83439.7			0.417597
$448$	−24869.3	+	3305.39i	−0.123910	+	0.0164690i
$449$	91141.4			0.452088			0.226044	−	0.974117i	$-0.427421\pi$
$449$	91141.4			0.452088			0.226044	+	0.974117i	$0.427421\pi$
$450$		0			0
$451$			253944.i			1.24849i
$452$			38113.6i			0.186553i
$453$	−290928.			−1.41772
$454$			147433.i			0.715292i
$455$		0			0
$456$	−172226.			−0.828263
$457$			411928.i			1.97237i	0.165634	+	0.986187i	$0.447033\pi$
$457$			411928.i			1.97237i	−0.165634	+	0.986187i	$0.552967\pi$
$458$	−229895.			−1.09597
$459$	2376.79			0.0112815
$460$		0			0
$461$		−	157397.i		−	0.740617i	−0.928909	−	0.370309i	$-0.879252\pi$
$461$		−	157397.i		−	0.740617i	0.928909	−	0.370309i	$-0.120748\pi$
$462$	44439.9	+	334359.i	0.208204	+	1.56650i
$463$			245557.i			1.14549i	0.819734	+	0.572745i	$0.194121\pi$
$463$			245557.i			1.14549i	−0.819734	+	0.572745i	$0.805879\pi$
$464$	21688.4			0.100738
$465$		0			0
$466$	117999.			0.543383
$467$	−33069.2			−0.151632			−0.0758158	−	0.997122i	$-0.524156\pi$
$467$	−33069.2			−0.151632			−0.0758158	+	0.997122i	$0.524156\pi$
$468$	−31075.2			−0.141880
$469$	6950.28	+	52292.9i	0.0315978	+	0.237737i
$470$		0			0
$471$	473151.			2.13284
$472$	39164.3			0.175795
$473$			372182.i			1.66354i
$474$		−	251642.i		−	1.12002i
$475$		0			0
$476$	70713.3	−	9398.55i	0.312095	−	0.0414808i
$477$		−	116832.i		−	0.513482i
$478$			11134.2i			0.0487309i
$479$		−	384277.i		−	1.67484i	−0.546559	−	0.837421i	$-0.684063\pi$
$479$		−	384277.i		−	1.67484i	0.546559	−	0.837421i	$-0.315937\pi$
$480$		0			0
$481$			32472.4i			0.140354i
$482$	199437.			0.858442
$483$	−38458.2	−	289354.i	−0.164852	−	1.24032i
$484$	−177242.			−0.756615
$485$		0			0
$486$		−	235406.i		−	0.996657i
$487$			173526.i			0.731657i	0.930682	+	0.365829i	$0.119214\pi$
$487$			173526.i			0.731657i	−0.930682	+	0.365829i	$0.880786\pi$
$488$	5574.10			0.0234064
$489$			518332.i			2.16765i
$490$		0			0
$491$	−20006.3			−0.0829858			−0.0414929	−	0.999139i	$-0.513211\pi$
$491$	−20006.3			−0.0829858			−0.0414929	+	0.999139i	$0.513211\pi$
$492$		−	134369.i		−	0.555099i
$493$	−61668.8			−0.253730
$494$	82419.2			0.337734
$495$		0			0
$496$			17123.6i			0.0696036i
$497$	−110571.	+	14696.0i	−0.447638	+	0.0594958i
$498$			51975.1i			0.209574i
$499$	428368.			1.72035			0.860174	−	0.510000i	$-0.170355\pi$
$499$	428368.			1.72035			0.860174	+	0.510000i	$0.170355\pi$
$500$		0			0
$501$	−441204.			−1.75778
$502$	203766.			0.808584
$503$	−36793.4			−0.145423			−0.0727116	−	0.997353i	$-0.523165\pi$
$503$	−36793.4			−0.145423			−0.0727116	+	0.997353i	$0.523165\pi$
$504$	−11682.0	−	87893.9i	−0.0459893	−	0.346017i
$505$		0			0
$506$	254747.			0.994965
$507$	−332432.			−1.29326
$508$		−	223751.i		−	0.867038i
$509$			137334.i			0.530080i	0.964237	+	0.265040i	$0.0853852\pi$
$509$			137334.i			0.530080i	−0.964237	+	0.265040i	$0.914615\pi$
$510$		0			0
$511$	345204.	−	45881.3i	1.32201	−	0.175709i
$512$			11585.2i			0.0441942i
$513$			7835.43i			0.0297734i
$514$			185589.i			0.702466i
$515$		0			0
$516$		−	196932.i		−	0.739636i
$517$	563363.			2.10769
$518$	−91845.8	+	12207.3i	−0.342294	+	0.0454945i
$519$	−400923.			−1.48842
$520$		0			0
$521$			102775.i			0.378629i	0.981916	+	0.189315i	$0.0606266\pi$
$521$			102775.i			0.378629i	−0.981916	+	0.189315i	$0.939373\pi$
$522$			76651.9i			0.281308i
$523$	314194.			1.14867			0.574334	−	0.818621i	$-0.305261\pi$
$523$	314194.			1.14867			0.574334	+	0.818621i	$0.305261\pi$
$524$			192132.i			0.699741i
$525$		0			0
$526$	109478.			0.395689
$527$		−	48689.2i		−	0.175312i
$528$	155760.			0.558711
$529$	59383.5			0.212204
$530$		0			0
$531$			138416.i			0.490904i
$532$	30983.7	+	233117.i	0.109474	+	0.823664i
$533$			64302.9i			0.226348i
$534$	−76569.7			−0.268519
$535$		0			0
$536$	24360.4			0.0847919
$537$	−288647.			−1.00096
$538$	−246715.			−0.852376
$539$	444578.	−	120303.i	1.53028	−	0.414096i
$540$		0			0
$541$	298384.			1.01948			0.509742	−	0.860327i	$-0.329741\pi$
$541$	298384.			1.01948			0.509742	+	0.860327i	$0.329741\pi$
$542$	−298009.			−1.01445
$543$		−	703319.i		−	2.38535i
$544$		−	32941.4i		−	0.111313i
$545$		0			0
$546$	11252.9	+	84665.3i	0.0377468	+	0.284001i
$547$		−	361462.i		−	1.20806i	−0.796963	−	0.604029i	$-0.793561\pi$
$547$		−	361462.i		−	1.20806i	0.796963	−	0.604029i	$-0.206439\pi$
$548$		−	33296.0i		−	0.110874i
$549$			19700.2i			0.0653620i
$550$		0			0
$551$		−	203300.i		−	0.669629i
$552$	−134794.			−0.442377
$553$	−340611.	+	45270.8i	−1.11380	+	0.148036i
$554$	102329.			0.333412
$555$		0			0
$556$			210926.i			0.682309i
$557$		−	112424.i		−	0.362367i	−0.983449	−	0.181183i	$-0.942007\pi$
$557$		−	112424.i		−	0.362367i	0.983449	−	0.181183i	$-0.0579928\pi$
$558$	−60518.8			−0.194367
$559$			94242.7i			0.301595i
$560$		0			0
$561$	−442886.			−1.40724
$562$			280416.i			0.887832i
$563$	−441530.			−1.39298			−0.696488	−	0.717569i	$-0.745255\pi$
$563$	−441530.			−1.39298			−0.696488	+	0.717569i	$0.745255\pi$
$564$	−298092.			−0.937113
$565$		0			0
$566$			11747.2i			0.0366692i
$567$	−322685.	+	42888.3i	−1.00372	+	0.133405i
$568$			51508.8i			0.159656i
$569$	−397273.			−1.22706			−0.613529	−	0.789673i	$-0.710250\pi$
$569$	−397273.			−1.22706			−0.613529	+	0.789673i	$0.710250\pi$
$570$		0			0
$571$	−67235.8			−0.206219			−0.103109	−	0.994670i	$-0.532879\pi$
$571$	−67235.8			−0.206219			−0.103109	+	0.994670i	$0.532879\pi$
$572$	−74539.3			−0.227821
$573$	−644744.			−1.96371
$574$	−181876.	+	24173.3i	−0.552016	+	0.0733688i
$575$		0			0
$576$	−40944.9			−0.123411
$577$	−64756.1			−0.194504			−0.0972521	−	0.995260i	$-0.531005\pi$
$577$	−64756.1			−0.194504			−0.0972521	+	0.995260i	$0.531005\pi$
$578$		−	142567.i		−	0.426741i
$579$		−	15838.2i		−	0.0472443i
$580$		0			0
$581$	70351.0	−	9350.39i	0.208410	−	0.0276999i
$582$		−	211680.i		−	0.624933i
$583$		−	280242.i		−	0.824510i
$584$		−	160812.i		−	0.471511i
$585$		0			0
$586$			57245.8i			0.166705i
$587$	−338967.			−0.983741			−0.491870	−	0.870668i	$-0.663687\pi$
$587$	−338967.			−0.983741			−0.491870	+	0.870668i	$0.663687\pi$
$588$	−235239.	+	63656.1i	−0.680386	+	0.184113i
$589$	160511.			0.462673
$590$		0			0
$591$		−	817763.i		−	2.34128i
$592$			42785.9i			0.122084i
$593$	−255668.			−0.727054			−0.363527	−	0.931584i	$-0.618428\pi$
$593$	−255668.			−0.727054			−0.363527	+	0.931584i	$0.618428\pi$
$594$		−	7086.30i		−	0.0200838i
$595$		0			0
$596$	−52612.6			−0.148114
$597$			29848.4i			0.0837477i
$598$	64506.1			0.180384
$599$	−129521.			−0.360982			−0.180491	−	0.983577i	$-0.557769\pi$
$599$	−129521.			−0.360982			−0.180491	+	0.983577i	$0.557769\pi$
$600$		0			0
$601$		−	377277.i		−	1.04451i	−0.852790	−	0.522254i	$-0.825091\pi$
$601$		−	377277.i		−	1.04451i	0.852790	−	0.522254i	$-0.174909\pi$
$602$	−266559.	+	35428.4i	−0.735529	+	0.0977595i
$603$			86095.3i			0.236780i
$604$	183444.			0.502839
$605$		0			0
$606$	−329144.			−0.896274
$607$	421091.			1.14287			0.571437	−	0.820646i	$-0.306386\pi$
$607$	421091.			1.14287			0.571437	+	0.820646i	$0.306386\pi$
$608$	108596.			0.293770
$609$	208840.	−	27757.1i	0.563093	−	0.0748410i
$610$		0			0
$611$	142653.			0.382119
$612$	116423.			0.310839
$613$			412173.i			1.09688i	0.836190	+	0.548440i	$0.184778\pi$
$613$			412173.i			1.09688i	−0.836190	+	0.548440i	$0.815222\pi$
$614$			178954.i			0.474684i
$615$		0			0
$616$	−28021.4	−	210829.i	−0.0738462	−	0.555608i
$617$			262676.i			0.690002i	0.938602	+	0.345001i	$0.112121\pi$
$617$			262676.i			0.690002i	−0.938602	+	0.345001i	$0.887879\pi$
$618$		−	140185.i		−	0.367050i
$619$			373975.i			0.976026i	0.872836	+	0.488013i	$0.162278\pi$
$619$			373975.i			0.976026i	−0.872836	+	0.488013i	$0.837722\pi$
$620$		0			0
$621$			6132.47i			0.0159020i
$622$	40659.7			0.105095
$623$	13775.0	+	103641.i	0.0354908	+	0.267028i
$624$	39440.9			0.101293
$625$		0			0
$626$		−	103982.i		−	0.265343i
$627$		−	1.46004e6i		−	3.71389i
$628$	−298344.			−0.756480
$629$		−	121657.i		−	0.307494i
$630$		0			0
$631$	−408746.			−1.02659			−0.513293	−	0.858214i	$-0.671574\pi$
$631$	−408746.			−1.02659			−0.513293	+	0.858214i	$0.671574\pi$
$632$			158672.i			0.397252i
$633$	−825397.			−2.05995
$634$	−355126.			−0.883495
$635$		0			0
$636$			148284.i			0.366590i
$637$	112575.	−	30462.8i	0.277435	−	0.0750743i
$638$			183863.i			0.451703i
$639$	−182044.			−0.445835
$640$		0			0
$641$	−528074.			−1.28522			−0.642612	−	0.766192i	$-0.722149\pi$
$641$	−528074.			−1.28522			−0.642612	+	0.766192i	$0.722149\pi$
$642$	−435695.			−1.05709
$643$	323445.			0.782310			0.391155	−	0.920325i	$-0.372076\pi$
$643$	323445.			0.782310			0.391155	+	0.920325i	$0.372076\pi$
$644$	24249.6	+	182451.i	0.0584701	+	0.439921i
$645$		0			0
$646$	−308782.			−0.739925
$647$	592372.			1.41510			0.707548	−	0.706665i	$-0.249801\pi$
$647$	592372.			1.41510			0.707548	+	0.706665i	$0.249801\pi$
$648$			150321.i			0.357990i
$649$			332015.i			0.788257i
$650$		0			0
$651$	21915.0	+	164885.i	0.0517106	+	0.389063i
$652$		−	326832.i		−	0.768828i
$653$		−	329810.i		−	0.773459i	−0.922193	−	0.386730i	$-0.873605\pi$
$653$		−	329810.i		−	0.773459i	0.922193	−	0.386730i	$-0.126395\pi$
$654$		−	244320.i		−	0.571221i
$655$		0			0
$656$			84726.1i			0.196884i
$657$	568346.			1.31669
$658$	53627.2	+	403483.i	0.123861	+	0.931910i
$659$	526737.			1.21290			0.606448	−	0.795123i	$-0.292594\pi$
$659$	526737.			1.21290			0.606448	+	0.795123i	$0.292594\pi$
$660$		0			0
$661$		−	144047.i		−	0.329686i	−0.986320	−	0.164843i	$-0.947288\pi$
$661$		−	144047.i		−	0.329686i	0.986320	−	0.164843i	$-0.0527117\pi$
$662$			15207.2i			0.0347002i
$663$	−112146.			−0.255128
$664$		−	32772.7i		−	0.0743320i
$665$		0			0
$666$	−151215.			−0.340916
$667$		−	159115.i		−	0.357650i
$668$	278199.			0.623452
$669$	−385855.			−0.862128
$670$		0			0
$671$			47254.3i			0.104953i
$672$	14826.9	+	111556.i	0.0328332	+	0.247032i
$673$			620117.i			1.36913i	0.728953	+	0.684563i	$0.240007\pi$
$673$			620117.i			1.36913i	−0.728953	+	0.684563i	$0.759993\pi$
$674$	6228.95			0.0137118
$675$		0			0
$676$	209613.			0.458697
$677$	−626172.			−1.36621			−0.683103	−	0.730322i	$-0.739370\pi$
$677$	−626172.			−1.36621			−0.683103	+	0.730322i	$0.739370\pi$
$678$	170965.			0.371920
$679$	−286520.	+	38081.5i	−0.621463	+	0.0825989i
$680$		0			0
$681$	661338.			1.42603
$682$	−145165.			−0.312099
$683$			185558.i			0.397775i	0.980022	+	0.198888i	$0.0637328\pi$
$683$			185558.i			0.397775i	−0.980022	+	0.198888i	$0.936267\pi$
$684$			383805.i			0.820348i
$685$		0			0
$686$	128482.	+	306957.i	0.273019	+	0.652273i
$687$			1.03124e6i			2.18497i
$688$			124175.i			0.262336i
$689$		−	70961.9i		−	0.149481i
$690$		0			0
$691$		−	548881.i		−	1.14953i	−0.818317	−	0.574767i	$-0.805093\pi$
$691$		−	548881.i		−	1.14953i	0.818317	−	0.574767i	$-0.194907\pi$
$692$	252800.			0.527917
$693$	745118.	−	99034.1i	1.55153	−	0.206214i
$694$	628478.			1.30488
$695$		0			0
$696$		−	97287.2i		−	0.200834i
$697$		−	240910.i		−	0.495894i
$698$	−290420.			−0.596095
$699$		−	529305.i		−	1.08331i
$700$		0			0
$701$	517501.			1.05311			0.526556	−	0.850140i	$-0.323483\pi$
$701$	517501.			1.05311			0.526556	+	0.850140i	$0.323483\pi$
$702$		−	1794.37i		−	0.00364114i
$703$	401061.			0.811522
$704$	−98213.6			−0.198165
$705$		0			0
$706$		−	177047.i		−	0.355205i
$707$	59213.5	+	445514.i	0.118463	+	0.891297i
$708$		−	175679.i		−	0.350471i
$709$	−6035.96			−0.0120075			−0.00600377	−	0.999982i	$-0.501911\pi$
$709$	−6035.96			−0.0120075			−0.00600377	+	0.999982i	$0.501911\pi$
$710$		0			0
$711$	−560784.			−1.10932
$712$	48280.7			0.0952388
$713$	125625.			0.247115
$714$	−42158.9	−	317197.i	−0.0826975	−	0.622204i
$715$		0			0
$716$	182005.			0.355024
$717$	49944.6			0.0971517
$718$		−	270132.i		−	0.523994i
$719$		−	748658.i		−	1.44819i	−0.689700	−	0.724095i	$-0.742258\pi$
$719$		−	748658.i		−	1.44819i	0.689700	−	0.724095i	$-0.257742\pi$
$720$		0			0
$721$	−189748.	+	25219.5i	−0.365012	+	0.0485139i
$722$		−	649343.i		−	1.24566i
$723$		−	894609.i		−	1.71142i
$724$			443475.i			0.846042i
$725$		0			0
$726$			795049.i			1.50841i
$727$	−370335.			−0.700691			−0.350345	−	0.936621i	$-0.613936\pi$
$727$	−370335.			−0.700691			−0.350345	+	0.936621i	$0.613936\pi$
$728$	−7095.48	−	53385.4i	−0.0133881	−	0.100730i
$729$	−517848.			−0.974422
$730$		0			0
$731$		−	353079.i		−	0.660750i
$732$		−	25003.6i		−	0.0466639i
$733$	−144545.			−0.269027			−0.134513	−	0.990912i	$-0.542947\pi$
$733$	−144545.			−0.269027			−0.134513	+	0.990912i	$0.542947\pi$
$734$		−	232867.i		−	0.432231i
$735$		0			0
$736$	84993.8			0.156903
$737$			206515.i			0.380203i
$738$	−299442.			−0.549793
$739$	−4650.26			−0.00851506			−0.00425753	−	0.999991i	$-0.501355\pi$
$739$	−4650.26			−0.00851506			−0.00425753	+	0.999991i	$0.501355\pi$
$740$		0			0
$741$		−	369706.i		−	0.673318i
$742$	200710.	−	26676.5i	0.364554	−	0.0484531i
$743$			500204.i			0.906087i	0.891489	+	0.453043i	$0.149662\pi$
$743$			500204.i			0.906087i	−0.891489	+	0.453043i	$0.850338\pi$
$744$	76811.0			0.138764
$745$		0			0
$746$	−368327.			−0.661845
$747$	115826.			0.207571
$748$	279260.			0.499121
$749$	78382.2	+	589737.i	0.139718	+	1.05122i
$750$		0			0
$751$	944576.			1.67478			0.837389	−	0.546608i	$-0.184081\pi$
$751$	944576.			1.67478			0.837389	+	0.546608i	$0.184081\pi$
$752$	187961.			0.332377
$753$		−	914031.i		−	1.61202i
$754$			46557.1i			0.0818924i
$755$		0			0
$756$	5075.24	−	674.553i	0.00888000	−	0.00118025i
$757$			654028.i			1.14131i	0.821189	+	0.570656i	$0.193311\pi$
$757$			654028.i			1.14131i	−0.821189	+	0.570656i	$0.806689\pi$
$758$		−	544044.i		−	0.946881i
$759$		−	1.14271e6i		−	1.98360i
$760$		0			0
$761$			643673.i			1.11147i	0.831361	+	0.555733i	$0.187562\pi$
$761$			643673.i			1.11147i	−0.831361	+	0.555733i	$0.812438\pi$
$762$	−1.00368e6			−1.72856
$763$	−330700.	+	43953.6i	−0.568049	+	0.0754997i
$764$	406541.			0.696494
$765$		0			0
$766$			288310.i			0.491362i
$767$			84071.6i			0.142909i
$768$	51967.7			0.0881071
$769$			103697.i			0.175354i	0.996149	+	0.0876768i	$0.0279443\pi$
$769$			103697.i			0.175354i	−0.996149	+	0.0876768i	$0.972056\pi$
$770$		0			0
$771$	832491.			1.40046
$772$			9986.72i			0.0167567i
$773$	796380.			1.33279			0.666394	−	0.745599i	$-0.267837\pi$
$773$	796380.			1.33279			0.666394	+	0.745599i	$0.267837\pi$
$774$	−438863.			−0.732567
$775$		0			0
$776$			133474.i			0.221652i
$777$	54757.9	+	411991.i	0.0906995	+	0.682410i
$778$		−	540439.i		−	0.892868i
$779$	794195.			1.30874
$780$		0			0
$781$	−436664.			−0.715889
$782$	−241671.			−0.395195
$783$	−4426.10			−0.00721934
$784$	148329.	−	40138.1i	0.241321	−	0.0653017i
$785$		0			0
$786$	861843.			1.39503
$787$	1.16895e6			1.88733			0.943664	−	0.330905i	$-0.107354\pi$
$787$	1.16895e6			1.88733			0.943664	+	0.330905i	$0.107354\pi$
$788$			515637.i			0.830408i
$789$		−	491081.i		−	0.788859i
$790$		0			0
$791$	−30756.9	−	231411.i	−0.0491575	−	0.369854i
$792$		−	347110.i		−	0.553371i
$793$			11965.6i			0.0190277i
$794$			568919.i			0.902421i
$795$		0			0
$796$		−	18820.8i		−	0.0297038i
$797$	−808048.			−1.27210			−0.636049	−	0.771649i	$-0.719432\pi$
$797$	−808048.			−1.27210			−0.636049	+	0.771649i	$0.719432\pi$
$798$	1.04569e6	−	138983.i	1.64209	−	0.218251i
$799$	−534447.			−0.837165
$800$		0			0
$801$			170635.i			0.265952i
$802$		−	113076.i		−	0.175801i
$803$	1.36328e6			2.11423
$804$		−	109273.i		−	0.169044i
$805$		0			0
$806$	−36758.2			−0.0565827
$807$			1.10668e6i			1.69933i
$808$	207540.			0.317892
$809$	692708.			1.05841			0.529204	−	0.848495i	$-0.322491\pi$
$809$	692708.			1.05841			0.529204	+	0.848495i	$0.322491\pi$
$810$		0			0
$811$			382267.i			0.581200i	0.956845	+	0.290600i	$0.0938548\pi$
$811$			382267.i			0.581200i	−0.956845	+	0.290600i	$0.906145\pi$
$812$	−131683.	+	17502.1i	−0.199719	+	0.0265448i
$813$			1.33677e6i			2.02244i
$814$	−362717.			−0.547417
$815$		0			0
$816$	−147765.			−0.221917
$817$	1.16398e6			1.74381
$818$	−228129.			−0.340936
$819$	188676.	−	25077.1i	0.281287	−	0.0373860i
$820$		0			0
$821$	116392.			0.172678			0.0863392	−	0.996266i	$-0.472483\pi$
$821$	116392.			0.172678			0.0863392	+	0.996266i	$0.472483\pi$
$822$	−149355.			−0.221043
$823$			640526.i			0.945665i	0.881152	+	0.472832i	$0.156768\pi$
$823$			640526.i			0.945665i	−0.881152	+	0.472832i	$0.843232\pi$
$824$			88393.2i			0.130186i
$825$		0			0
$826$	−237790.	+	31604.8i	−0.348525	+	0.0463227i
$827$			158299.i			0.231455i	0.993281	+	0.115727i	$0.0369199\pi$
$827$			158299.i			0.231455i	−0.993281	+	0.115727i	$0.963080\pi$
$828$			300388.i			0.438149i
$829$			680501.i			0.990192i	0.868838	+	0.495096i	$0.164867\pi$
$829$			680501.i			0.990192i	−0.868838	+	0.495096i	$0.835133\pi$
$830$		0			0
$831$		−	459017.i		−	0.664701i
$832$	−24869.3			−0.0359267
$833$	−421759.	+	114129.i	−0.607819	+	0.164477i
$834$	946148.			1.36028
$835$		0			0
$836$			920622.i			1.31725i
$837$		−	3494.53i		−	0.00498812i
$838$	712859.			1.01512
$839$		−	622397.i		−	0.884186i	−0.896969	−	0.442093i	$-0.854236\pi$
$839$		−	622397.i		−	0.884186i	0.896969	−	0.442093i	$-0.145764\pi$
$840$		0			0
$841$	−592440.			−0.837631
$842$		−	238831.i		−	0.336873i
$843$	1.25786e6			1.77001
$844$	520451.			0.730626
$845$		0			0
$846$			664297.i			0.928157i
$847$	1.07614e6	−	143031.i	1.50004	−	0.199371i
$848$		−	93500.0i		−	0.130023i
$849$	52694.2			0.0731051
$850$		0			0
$851$	313894.			0.433435
$852$	231052.			0.318295
$853$	−826596.			−1.13604			−0.568022	−	0.823013i	$-0.692291\pi$
$853$	−826596.			−1.13604			−0.568022	+	0.823013i	$0.692291\pi$
$854$	−33843.7	+	4498.19i	−0.0464048	+	0.00616768i
$855$		0			0
$856$	274726.			0.374931
$857$	539076.			0.733987			0.366994	−	0.930223i	$-0.380387\pi$
$857$	539076.			0.733987			0.366994	+	0.930223i	$0.380387\pi$
$858$			334359.i			0.454191i
$859$		−	499964.i		−	0.677567i	−0.940864	−	0.338783i	$-0.889985\pi$
$859$		−	499964.i		−	0.677567i	0.940864	−	0.338783i	$-0.110015\pi$
$860$		0			0
$861$	108433.	+	815838.i	0.146271	+	1.10052i
$862$		−	360659.i		−	0.485380i
$863$			1.16944e6i			1.57021i	0.619365	+	0.785103i	$0.287390\pi$
$863$			1.16944e6i			1.57021i	−0.619365	+	0.785103i	$0.712610\pi$
$864$		−	2364.28i		−	0.00316716i
$865$		0			0
$866$			660548.i			0.880783i
$867$	−639512.			−0.850766
$868$	−13818.4	−	103968.i	−0.0183408	−	0.137994i
$869$	−1.34514e6			−1.78126
$870$		0			0
$871$			52292.9i			0.0689297i
$872$			154055.i			0.202602i
$873$	−471727.			−0.618960
$874$		−	796705.i		−	1.04298i
$875$		0			0
$876$	−721350.			−0.940021
$877$		−	187496.i		−	0.243777i	−0.992544	−	0.121888i	$-0.961105\pi$
$877$		−	187496.i		−	0.243777i	0.992544	−	0.121888i	$-0.0388950\pi$
$878$	−860515.			−1.11627
$879$	256786.			0.332349
$880$		0			0
$881$		−	179673.i		−	0.231489i	−0.993279	−	0.115745i	$-0.963075\pi$
$881$		−	179673.i		−	0.231489i	0.993279	−	0.115745i	$-0.0369254\pi$
$882$	141857.	+	524230.i	0.182354	+	0.673883i
$883$		−	1.04658e6i		−	1.34231i	−0.741317	−	0.671155i	$-0.765799\pi$
$883$		−	1.04658e6i		−	1.34231i	0.741317	−	0.671155i	$-0.234201\pi$
$884$	70713.3			0.0904892
$885$		0			0
$886$	246246.			0.313691
$887$	−1.30029e6			−1.65270			−0.826349	−	0.563158i	$-0.809586\pi$
$887$	−1.30029e6			−1.65270			−0.826349	+	0.563158i	$0.809586\pi$
$888$	191924.			0.243390
$889$	180563.	+	1.35853e6i	0.228468	+	1.71896i
$890$		0			0
$891$	−1.27434e6			−1.60521
$892$	243299.			0.305781
$893$		−	1.76188e6i		−	2.20940i
$894$			236003.i			0.295286i
$895$		0			0
$896$	−9349.06	−	70341.0i	−0.0116453	−	0.0876178i
$897$		−	289354.i		−	0.359620i
$898$			257787.i			0.319675i
$899$			90669.8i			0.112187i
$900$		0			0
$901$			265858.i			0.327491i
$902$	−718263.			−0.882817
$903$	158921.	+	1.19570e6i	0.194897	+	1.46638i
$904$	−107802.			−0.131913
$905$		0			0
$906$		−	822869.i		−	1.00248i
$907$		−	1.18388e6i		−	1.43911i	−0.694435	−	0.719556i	$-0.744346\pi$
$907$		−	1.18388e6i		−	1.43911i	0.694435	−	0.719556i	$-0.255654\pi$
$908$	−417004.			−0.505788
$909$			733496.i			0.887708i
$910$		0			0
$911$	−1.30731e6			−1.57522			−0.787612	−	0.616172i	$-0.788683\pi$
$911$	−1.30731e6			−1.57522			−0.787612	+	0.616172i	$0.788683\pi$
$912$		−	487128.i		−	0.585671i
$913$	277830.			0.333301
$914$	−1.16511e6			−1.39468
$915$		0			0
$916$		−	650242.i		−	0.774968i
$917$	−155047.	−	1.16655e6i	−0.184384	−	1.38728i
$918$			6722.58i			0.00797720i
$919$	−698775.			−0.827382			−0.413691	−	0.910417i	$-0.635761\pi$
$919$	−698775.			−0.827382			−0.413691	+	0.910417i	$0.635761\pi$
$920$		0			0
$921$	802730.			0.946347
$922$	445185.			0.523696
$923$	−110571.			−0.129789
$924$	−945711.	+	125695.i	−1.10768	+	0.147222i
$925$		0			0
$926$	−694541.			−0.809984
$927$	−312402.			−0.363542
$928$			61344.1i			0.0712323i
$929$		−	1.39460e6i		−	1.61591i	−0.589244	−	0.807955i	$-0.700574\pi$
$929$		−	1.39460e6i		−	1.61591i	0.589244	−	0.807955i	$-0.299426\pi$
$930$		0			0
$931$	−376241.	−	1.39039e6i	−0.434077	−	1.60412i
$932$			333751.i			0.384229i
$933$		−	182387.i		−	0.209522i
$934$		−	93533.8i		−	0.107220i
$935$		0			0
$936$		−	87893.9i		−	0.100325i
$937$	509380.			0.580180			0.290090	−	0.956999i	$-0.406315\pi$
$937$	509380.			0.580180			0.290090	+	0.956999i	$0.406315\pi$
$938$	−147907.	+	19658.4i	−0.168106	+	0.0223430i
$939$	−466428.			−0.528997
$940$		0			0
$941$			1.62505e6i			1.83522i	0.397486	+	0.917608i	$0.369883\pi$
$941$			1.62505e6i			1.83522i	−0.397486	+	0.917608i	$0.630117\pi$
$942$			1.33827e6i			1.50814i
$943$	621584.			0.698998
$944$			110773.i			0.124306i
$945$		0			0
$946$	−1.05269e6			−1.17630
$947$		−	1.22462e6i		−	1.36553i	−0.730640	−	0.682763i	$-0.760778\pi$
$947$		−	1.22462e6i		−	1.36553i	0.730640	−	0.682763i	$-0.239222\pi$
$948$	711752.			0.791976
$949$	345204.			0.383304
$950$		0			0
$951$			1.59298e6i			1.76137i
$952$	26583.1	+	200007.i	0.0293313	+	0.220685i
$953$		−	1.14847e6i		−	1.26454i	−0.774749	−	0.632269i	$-0.782124\pi$
$953$		−	1.14847e6i		−	1.26454i	0.774749	−	0.632269i	$-0.217876\pi$
$954$	330451.			0.363086
$955$		0			0
$956$	−31492.4			−0.0344580
$957$	824750.			0.900531
$958$	1.08690e6			1.18429
$959$	26869.2	+	202160.i	0.0292158	+	0.219815i
$960$		0			0
$961$	851935.			0.922485
$962$	−91845.8			−0.0992451
$963$			970945.i			1.04699i
$964$			564092.i			0.607010i
$965$		0			0
$966$	818416.	−	108776.i	0.877041	−	0.116568i
$967$			733668.i			0.784597i	0.919838	+	0.392298i	$0.128320\pi$
$967$			733668.i			0.784597i	−0.919838	+	0.392298i	$0.871680\pi$
$968$		−	501315.i		−	0.535008i
$969$			1.38510e6i			1.47514i
$970$		0			0
$971$			786749.i			0.834445i	0.908804	+	0.417223i	$0.136996\pi$
$971$			786749.i			0.834445i	−0.908804	+	0.417223i	$0.863004\pi$
$972$	665830.			0.704743
$973$	−170213.	−	1.28066e6i	−0.179791	−	1.35272i
$974$	−490807.			−0.517360
$975$		0			0
$976$			15765.9i			0.0165508i
$977$		−	837039.i		−	0.876913i	−0.898752	−	0.438457i	$-0.855525\pi$
$977$		−	837039.i		−	0.876913i	0.898752	−	0.438457i	$-0.144475\pi$
$978$	−1.46606e6			−1.53276
$979$			409299.i			0.427046i
$980$		0			0
$981$	−544467.			−0.565761
$982$		−	56586.3i		−	0.0586798i
$983$	−644129.			−0.666600			−0.333300	−	0.942821i	$-0.608162\pi$
$983$	−644129.			−0.666600			−0.333300	+	0.942821i	$0.608162\pi$
$984$	380054.			0.392514
$985$		0			0
$986$		−	174426.i		−	0.179414i
$987$	1.80990e6	−	240554.i	1.85789	−	0.246933i
$988$			233117.i			0.238814i
$989$	910996.			0.931374
$990$		0			0
$991$	92517.5			0.0942056			0.0471028	−	0.998890i	$-0.485001\pi$
$991$	92517.5			0.0942056			0.0471028	+	0.998890i	$0.485001\pi$
$992$	−48432.8			−0.0492172
$993$	68214.4			0.0691795
$994$	−41566.6	−	312741.i	−0.0420699	−	0.316528i
$995$		0			0
$996$	−147008.			−0.148191
$997$	366383.			0.368591			0.184295	−	0.982871i	$-0.441000\pi$
$997$	366383.			0.368591			0.184295	+	0.982871i	$0.441000\pi$
$998$			1.21161e6i			1.21647i
$999$		−	8731.60i		−	0.00874909i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.5.d.a.349.8		8
5.2	odd	4		14.5.b.a.13.1	✓	4
5.3	odd	4		350.5.b.a.251.4		4
5.4	even	2	inner	350.5.d.a.349.1		8
7.6	odd	2	inner	350.5.d.a.349.5		8
15.2	even	4		126.5.c.a.55.4		4
20.7	even	4		112.5.c.c.97.4		4
35.2	odd	12		98.5.d.d.31.4		8
35.12	even	12		98.5.d.d.31.3		8
35.13	even	4		350.5.b.a.251.3		4
35.17	even	12		98.5.d.d.19.4		8
35.27	even	4		14.5.b.a.13.2	yes	4
35.32	odd	12		98.5.d.d.19.3		8
35.34	odd	2	inner	350.5.d.a.349.4		8
40.27	even	4		448.5.c.f.321.1		4
40.37	odd	4		448.5.c.e.321.4		4
60.47	odd	4		1008.5.f.h.433.3		4
105.62	odd	4		126.5.c.a.55.3		4
140.27	odd	4		112.5.c.c.97.1		4
280.27	odd	4		448.5.c.f.321.4		4
280.237	even	4		448.5.c.e.321.1		4
420.167	even	4		1008.5.f.h.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.5.b.a.13.1	✓	4	5.2	odd	4
14.5.b.a.13.2	yes	4	35.27	even	4
98.5.d.d.19.3		8	35.32	odd	12
98.5.d.d.19.4		8	35.17	even	12
98.5.d.d.31.3		8	35.12	even	12
98.5.d.d.31.4		8	35.2	odd	12
112.5.c.c.97.1		4	140.27	odd	4
112.5.c.c.97.4		4	20.7	even	4
126.5.c.a.55.3		4	105.62	odd	4
126.5.c.a.55.4		4	15.2	even	4
350.5.b.a.251.3		4	35.13	even	4
350.5.b.a.251.4		4	5.3	odd	4
350.5.d.a.349.1		8	5.4	even	2	inner
350.5.d.a.349.4		8	35.34	odd	2	inner
350.5.d.a.349.5		8	7.6	odd	2	inner
350.5.d.a.349.8		8	1.1	even	1	trivial
448.5.c.e.321.1		4	280.237	even	4
448.5.c.e.321.4		4	40.37	odd	4
448.5.c.f.321.1		4	40.27	even	4
448.5.c.f.321.4		4	280.27	odd	4
1008.5.f.h.433.2		4	420.167	even	4
1008.5.f.h.433.3		4	60.47	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	350.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +12.6874 q^{3} -8.00000 q^{4} +35.8854i q^{6} +(48.5729 - 6.45584i) q^{7} -22.6274i q^{8} +79.9706 q^{9} +191.823 q^{11} -101.499 q^{12} +48.5729 q^{13} +(18.2599 + 137.385i) q^{14} +64.0000 q^{16} -181.977 q^{17} +226.191i q^{18} -599.915i q^{19} +(616.264 - 81.9080i) q^{21} +542.558i q^{22} -469.529i q^{23} -287.083i q^{24} +137.385i q^{26} -13.0609 q^{27} +(-388.583 + 51.6468i) q^{28} +338.881 q^{29} +267.556i q^{31} +181.019i q^{32} +2433.74 q^{33} -514.710i q^{34} -639.765 q^{36} +668.530i q^{37} +1696.82 q^{38} +616.264 q^{39} +1323.85i q^{41} +(231.671 + 1743.06i) q^{42} +1940.23i q^{43} -1534.59 q^{44} +1328.03 q^{46} +2936.89 q^{47} +811.995 q^{48} +(2317.64 - 627.158i) q^{49} -2308.82 q^{51} -388.583 q^{52} -1460.94i q^{53} -36.9418i q^{54} +(-146.079 - 1099.08i) q^{56} -7611.38i q^{57} +958.501i q^{58} +1730.83i q^{59} +246.343i q^{61} -756.763 q^{62} +(3884.40 - 516.277i) q^{63} -512.000 q^{64} +6883.67i q^{66} +1076.59i q^{67} +1455.82 q^{68} -5957.11i q^{69} -2276.39 q^{71} -1809.53i q^{72} +7106.94 q^{73} -1890.89 q^{74} +4799.32i q^{76} +(9317.41 - 1238.38i) q^{77} +1743.06i q^{78} -7012.38 q^{79} -6643.32 q^{81} -3744.40 q^{82} +1448.36 q^{83} +(-4930.11 + 655.264i) q^{84} -5487.81 q^{86} +4299.53 q^{87} -4340.47i q^{88} +2133.73i q^{89} +(2359.32 - 313.579i) q^{91} +3756.23i q^{92} +3394.60i q^{93} +8306.77i q^{94} +2296.67i q^{96} -5898.76 q^{97} +(1773.87 + 6555.29i) q^{98} +15340.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4} + 504 q^{9} + 720 q^{11} + 576 q^{14} + 512 q^{16} + 1536 q^{21} - 2448 q^{29} - 4032 q^{36} + 1536 q^{39} - 5760 q^{44} + 6144 q^{46} + 3064 q^{49} - 22272 q^{51} - 4608 q^{56} - 4096 q^{64}+ \cdots + 59184 q^{99}+O(q^{100}) \) 8 * q - 64 * q^4 + 504 * q^9 + 720 * q^11 + 576 * q^14 + 512 * q^16 + 1536 * q^21 - 2448 * q^29 - 4032 * q^36 + 1536 * q^39 - 5760 * q^44 + 6144 * q^46 + 3064 * q^49 - 22272 * q^51 - 4608 * q^56 - 4096 * q^64 - 43056 * q^71 + 6912 * q^74 - 25552 * q^79 - 59256 * q^81 - 12288 * q^84 - 23040 * q^86 + 11136 * q^91 + 59184 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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