LMFDB - Embedded newform 350.5.b.a.251.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,5,Mod(251,350)] 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("350.251"); 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([0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	350.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,32,0,0,76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$36.1794870793$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.1308672.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 72x^{2} + 1278 $ x^4 + 72*x^2 + 1278
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.4
Root			$6.34371i$ of defining polynomial
Character	$\chi$	$=$	350.251
Dual form			350.5.b.a.251.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.82843 q^{2} +12.6874i q^{3} +8.00000 q^{4} +35.8854i q^{6} +(-6.45584 - 48.5729i) q^{7} +22.6274 q^{8} -79.9706 q^{9} +191.823 q^{11} +101.499i q^{12} +48.5729i q^{13} +(-18.2599 - 137.385i) q^{14} +64.0000 q^{16} +181.977i q^{17} -226.191 q^{18} +599.915i q^{19} +(616.264 - 81.9080i) q^{21} +542.558 q^{22} +469.529 q^{23} +287.083i q^{24} +137.385i q^{26} +13.0609i q^{27} +(-51.6468 - 388.583i) q^{28} -338.881 q^{29} +267.556i q^{31} +181.019 q^{32} +2433.74i q^{33} +514.710i q^{34} -639.765 q^{36} +668.530 q^{37} +1696.82i q^{38} -616.264 q^{39} +1323.85i q^{41} +(1743.06 - 231.671i) q^{42} -1940.23 q^{43} +1534.59 q^{44} +1328.03 q^{46} -2936.89i q^{47} +811.995i q^{48} +(-2317.64 + 627.158i) q^{49} -2308.82 q^{51} +388.583i q^{52} +1460.94 q^{53} +36.9418i q^{54} +(-146.079 - 1099.08i) q^{56} -7611.38 q^{57} -958.501 q^{58} -1730.83i q^{59} +246.343i q^{61} +756.763i q^{62} +(516.277 + 3884.40i) q^{63} +512.000 q^{64} +6883.67i q^{66} +1076.59 q^{67} +1455.82i q^{68} +5957.11i q^{69} -2276.39 q^{71} -1809.53 q^{72} +7106.94i q^{73} +1890.89 q^{74} +4799.32i q^{76} +(-1238.38 - 9317.41i) q^{77} -1743.06 q^{78} +7012.38 q^{79} -6643.32 q^{81} +3744.40i q^{82} +1448.36i q^{83} +(4930.11 - 655.264i) q^{84} -5487.81 q^{86} -4299.53i q^{87} +4340.47 q^{88} -2133.73i q^{89} +(2359.32 - 313.579i) q^{91} +3756.23 q^{92} -3394.60 q^{93} -8306.77i q^{94} +2296.67i q^{96} +5898.76i q^{97} +(-6555.29 + 1773.87i) q^{98} -15340.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4} + 76 q^{7} - 252 q^{9} + 360 q^{11} - 288 q^{14} + 256 q^{16} - 192 q^{18} + 768 q^{21} + 1152 q^{22} + 792 q^{23} + 608 q^{28} + 1224 q^{29} - 2016 q^{36} + 3896 q^{37} - 768 q^{39} + 4800 q^{42}+ \cdots - 29592 q^{99}+O(q^{100}) $ 4 * q + 32 * q^4 + 76 * q^7 - 252 * q^9 + 360 * q^11 - 288 * q^14 + 256 * q^16 - 192 * q^18 + 768 * q^21 + 1152 * q^22 + 792 * q^23 + 608 * q^28 + 1224 * q^29 - 2016 * q^36 + 3896 * q^37 - 768 * q^39 + 4800 * q^42 - 3688 * q^43 + 2880 * q^44 + 3072 * q^46 - 1532 * q^49 - 11136 * q^51 - 5832 * q^53 - 2304 * q^56 - 12864 * q^57 - 7296 * q^58 - 3060 * q^63 + 2048 * q^64 + 1048 * q^67 - 21528 * q^71 - 1536 * q^72 - 3456 * q^74 - 3528 * q^77 - 4800 * q^78 + 12776 * q^79 - 29628 * q^81 + 6144 * q^84 - 11520 * q^86 + 9216 * q^88 + 5568 * q^91 + 6336 * q^92 - 38016 * q^93 - 21888 * q^98 - 29592 * 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 $

$2$	2.82843			0.707107
$2$	2.82843			0.707107
$3$			12.6874i			1.40971i	0.709350	+	0.704857i	$0.248989\pi$
$3$			12.6874i			1.40971i	−0.709350	+	0.704857i	$0.751011\pi$
$4$	8.00000			0.500000
$5$		0			0
$5$		0			0
$6$			35.8854i			0.996818i
$7$	−6.45584	−	48.5729i	−0.131752	−	0.991283i
$7$	−6.45584	−	48.5729i	−0.131752	−	0.991283i
$8$	22.6274			0.353553
$9$	−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	$0.208696\pi$
$12$			101.499i			0.704857i
$13$			48.5729i			0.287413i	0.989620	+	0.143707i	$0.0459022\pi$
$13$			48.5729i			0.287413i	−0.989620	+	0.143707i	$0.954098\pi$
$14$	−18.2599	−	137.385i	−0.0931627	−	0.700943i
$15$		0			0
$16$	64.0000			0.250000
$17$			181.977i			0.629680i	0.949145	+	0.314840i	$0.101951\pi$
$17$			181.977i			0.629680i	−0.949145	+	0.314840i	$0.898049\pi$
$18$	−226.191			−0.698120
$19$			599.915i			1.66182i	0.556410	+	0.830908i	$0.312178\pi$
$19$			599.915i			1.66182i	−0.556410	+	0.830908i	$0.687822\pi$
$20$		0			0
$21$	616.264	−	81.9080i	1.39742	−	0.185732i
$22$	542.558			1.12099
$23$	469.529			0.887578			0.443789	−	0.896131i	$-0.353634\pi$
$23$	469.529			0.887578			0.443789	+	0.896131i	$0.353634\pi$
$24$			287.083i			0.498409i
$25$		0			0
$26$			137.385i			0.203232i
$27$			13.0609i			0.0179162i
$28$	−51.6468	−	388.583i	−0.0658760	−	0.495641i
$29$	−338.881			−0.402951			−0.201475	−	0.979494i	$-0.564574\pi$
$29$	−338.881			−0.402951			−0.201475	+	0.979494i	$0.564574\pi$
$30$		0			0
$31$			267.556i			0.278414i	0.990263	+	0.139207i	$0.0444554\pi$
$31$			267.556i			0.278414i	−0.990263	+	0.139207i	$0.955545\pi$
$32$	181.019			0.176777
$33$			2433.74i			2.23484i
$34$			514.710i			0.445251i
$35$		0			0
$36$	−639.765			−0.493645
$37$	668.530			0.488334			0.244167	−	0.969733i	$-0.421485\pi$
$37$	668.530			0.488334			0.244167	+	0.969733i	$0.421485\pi$
$38$			1696.82i			1.17508i
$39$	−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	$0.871172\pi$
$42$	1743.06	−	231.671i	0.988128	−	0.131333i
$43$	−1940.23			−1.04934			−0.524671	−	0.851305i	$-0.675812\pi$
$43$	−1940.23			−1.04934			−0.524671	+	0.851305i	$0.675812\pi$
$44$	1534.59			0.792659
$45$		0			0
$46$	1328.03			0.627613
$47$		−	2936.89i		−	1.32951i	−0.747062	−	0.664755i	$-0.768536\pi$
$47$		−	2936.89i		−	1.32951i	0.747062	−	0.664755i	$-0.231464\pi$
$48$			811.995i			0.352428i
$49$	−2317.64	+	627.158i	−0.965283	+	0.261207i
$50$		0			0
$51$	−2308.82			−0.887668
$52$			388.583i			0.143707i
$53$	1460.94			0.520091			0.260046	−	0.965596i	$-0.416262\pi$
$53$	1460.94			0.520091			0.260046	+	0.965596i	$0.416262\pi$
$54$			36.9418i			0.0126687i
$55$		0			0
$56$	−146.079	−	1099.08i	−0.0465813	−	0.350471i
$57$	−7611.38			−2.34268
$58$	−958.501			−0.284929
$59$		−	1730.83i		−	0.497223i	−0.968603	−	0.248612i	$-0.920026\pi$
$59$		−	1730.83i		−	0.497223i	0.968603	−	0.248612i	$-0.0799743\pi$
$60$		0			0
$61$			246.343i			0.0662034i	0.999452	+	0.0331017i	$0.0105385\pi$
$61$			246.343i			0.0662034i	−0.999452	+	0.0331017i	$0.989461\pi$
$62$			756.763i			0.196869i
$63$	516.277	+	3884.40i	0.130077	+	0.978684i
$64$	512.000			0.125000
$65$		0			0
$66$			6883.67i			1.58027i
$67$	1076.59			0.239828			0.119914	−	0.992784i	$-0.461738\pi$
$67$	1076.59			0.239828			0.119914	+	0.992784i	$0.461738\pi$
$68$			1455.82i			0.314840i
$69$			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	$0.572495\pi$
$72$	−1809.53			−0.349060
$73$			7106.94i			1.33363i	0.745221	+	0.666817i	$0.232344\pi$
$73$			7106.94i			1.33363i	−0.745221	+	0.666817i	$0.767656\pi$
$74$	1890.89			0.345305
$75$		0			0
$76$			4799.32i			0.830908i
$77$	−1238.38	−	9317.41i	−0.208869	−	1.57150i
$78$	−1743.06			−0.286499
$79$	7012.38			1.12360			0.561799	−	0.827274i	$-0.310109\pi$
$79$	7012.38			1.12360			0.561799	+	0.827274i	$0.310109\pi$
$80$		0			0
$81$	−6643.32			−1.01255
$82$			3744.40i			0.556871i
$83$			1448.36i			0.210243i	0.994459	+	0.105121i	$0.0335231\pi$
$83$			1448.36i			0.210243i	−0.994459	+	0.105121i	$0.966477\pi$
$84$	4930.11	−	655.264i	0.698712	−	0.0928662i
$85$		0			0
$86$	−5487.81			−0.741997
$87$		−	4299.53i		−	0.568045i
$88$	4340.47			0.560494
$89$		−	2133.73i		−	0.269376i	−0.990888	−	0.134688i	$-0.956997\pi$
$89$		−	2133.73i		−	0.269376i	0.990888	−	0.134688i	$-0.0430032\pi$
$90$		0			0
$91$	2359.32	−	313.579i	0.284908	−	0.0378673i
$92$	3756.23			0.443789
$93$	−3394.60			−0.392484
$94$		−	8306.77i		−	0.940105i
$95$		0			0
$96$			2296.67i			0.249204i
$97$			5898.76i			0.626928i	0.949600	+	0.313464i	$0.101489\pi$
$97$			5898.76i			0.626928i	−0.949600	+	0.313464i	$0.898511\pi$
$98$	−6555.29	+	1773.87i	−0.682558	+	0.184701i
$99$	−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	$0.851578\pi$
$102$	−6530.34			−0.627676
$103$		−	3906.46i		−	0.368222i	−0.982905	−	0.184111i	$-0.941059\pi$
$103$		−	3906.46i		−	0.368222i	0.982905	−	0.184111i	$-0.0589406\pi$
$104$			1099.08i			0.101616i
$105$		0			0
$106$	4132.15			0.367760
$107$	12141.3			1.06047			0.530233	−	0.847852i	$-0.322104\pi$
$107$	12141.3			1.06047			0.530233	+	0.847852i	$0.322104\pi$
$108$			104.487i			0.00895809i
$109$	6808.34			0.573044			0.286522	−	0.958074i	$-0.407501\pi$
$109$	6808.34			0.573044			0.286522	+	0.958074i	$0.407501\pi$
$110$		0			0
$111$			8481.92i			0.688411i
$112$	−413.174	−	3108.66i	−0.0329380	−	0.247821i
$113$	4764.20			0.373107			0.186553	−	0.982445i	$-0.440268\pi$
$113$	4764.20			0.373107			0.186553	+	0.982445i	$0.440268\pi$
$114$	−21528.2			−1.65653
$115$		0			0
$116$	−2711.05			−0.201475
$117$		−	3884.40i		−	0.283761i
$118$		−	4895.54i		−	0.351590i
$119$	8839.17	−	1174.82i	0.624191	−	0.0829615i
$120$		0			0
$121$	22155.2			1.51323
$122$			696.763i			0.0468129i
$123$	−16796.2			−1.11020
$124$			2140.45i			0.139207i
$125$		0			0
$126$	1460.25	+	10986.7i	0.0919787	+	0.692034i
$127$	27968.9			1.73408			0.867038	−	0.498242i	$-0.166021\pi$
$127$	27968.9			1.73408			0.867038	+	0.498242i	$0.166021\pi$
$128$	1448.15			0.0883883
$129$		−	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	$-0.246701\pi$
$132$			19469.9i			1.11742i
$133$	29139.6	−	3872.96i	1.64733	−	0.218947i
$134$	3045.05			0.169584
$135$		0			0
$136$			4117.68i			0.222625i
$137$	4162.00			0.221748			0.110874	−	0.993834i	$-0.464635\pi$
$137$	4162.00			0.221748			0.110874	+	0.993834i	$0.464635\pi$
$138$			16849.3i			0.884754i
$139$			26365.8i			1.36462i	0.731064	+	0.682309i	$0.239024\pi$
$139$			26365.8i			1.36462i	−0.731064	+	0.682309i	$0.760976\pi$
$140$		0			0
$141$	37261.5			1.87423
$142$	−6438.59			−0.319311
$143$			9317.41i			0.455641i
$144$	−5118.12			−0.246823
$145$		0			0
$146$			20101.5i			0.943022i
$147$	−7957.01	−	29404.9i	−0.368227	−	1.36077i
$148$	5348.24			0.244167
$149$	−6576.57			−0.296229			−0.148114	−	0.988970i	$-0.547320\pi$
$149$	−6576.57			−0.296229			−0.148114	+	0.988970i	$0.547320\pi$
$150$		0			0
$151$	−22930.4			−1.00568			−0.502839	−	0.864380i	$-0.667711\pi$
$151$	−22930.4			−1.00568			−0.502839	+	0.864380i	$0.667711\pi$
$152$			13574.5i			0.587540i
$153$		−	14552.8i		−	0.621677i
$154$	−3502.67	−	26353.6i	−0.147692	−	1.11122i
$155$		0			0
$156$	−4930.11			−0.202585
$157$		−	37292.9i		−	1.51296i	−0.654017	−	0.756480i	$-0.726918\pi$
$157$		−	37292.9i		−	1.51296i	0.654017	−	0.756480i	$-0.273082\pi$
$158$	19834.0			0.794504
$159$			18535.5i			0.733180i
$160$		0			0
$161$	−3031.21	−	22806.4i	−0.116940	−	0.879841i
$162$	−18790.2			−0.715979
$163$	−40854.0			−1.53766			−0.768828	−	0.639455i	$-0.779160\pi$
$163$	−40854.0			−1.53766			−0.768828	+	0.639455i	$0.779160\pi$
$164$			10590.8i			0.393767i
$165$		0			0
$166$			4096.58i			0.148664i
$167$			34774.9i			1.24690i	0.781862	+	0.623452i	$0.214270\pi$
$167$			34774.9i			1.24690i	−0.781862	+	0.623452i	$0.785730\pi$
$168$	13944.5	−	1853.37i	0.494064	−	0.0656663i
$169$	26201.7			0.917394
$170$		0			0
$171$		−	47975.6i		−	1.64070i
$172$	−15521.9			−0.524671
$173$		−	31600.1i		−	1.05583i	−0.849296	−	0.527917i	$-0.822973\pi$
$173$		−	31600.1i		−	1.05583i	0.849296	−	0.527917i	$-0.177027\pi$
$174$		−	12160.9i		−	0.401668i
$175$		0			0
$176$	12276.7			0.396329
$177$	21959.8			0.700942
$178$		−	6035.09i		−	0.190478i
$179$	22750.7			0.710048			0.355024	−	0.934857i	$-0.384473\pi$
$179$	22750.7			0.710048			0.355024	+	0.934857i	$0.384473\pi$
$180$		0			0
$181$		−	55434.4i		−	1.69208i	−0.533116	−	0.846042i	$-0.678979\pi$
$181$		−	55434.4i		−	1.69208i	0.533116	−	0.846042i	$-0.321021\pi$
$182$	6673.17	−	886.935i	0.201460	−	0.0267762i
$183$	−3125.45			−0.0933278
$184$	10624.2			0.313806
$185$		0			0
$186$	−9601.37			−0.277528
$187$			34907.5i			0.998242i
$188$		−	23495.1i		−	0.664755i
$189$	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	$0.745258\pi$
$192$			6495.96i			0.176214i
$193$	1248.34			0.0335134			0.0167567	−	0.999860i	$-0.494666\pi$
$193$	1248.34			0.0335134			0.0167567	+	0.999860i	$0.494666\pi$
$194$			16684.2i			0.443305i
$195$		0			0
$196$	−18541.2	+	5017.26i	−0.482641	+	0.130603i
$197$	−64454.6			−1.66082			−0.830408	−	0.557155i	$-0.811893\pi$
$197$	−64454.6			−1.66082			−0.830408	+	0.557155i	$0.811893\pi$
$198$	−43388.7			−1.10674
$199$		−	2352.60i		−	0.0594076i	−0.999559	−	0.0297038i	$-0.990544\pi$
$199$		−	2352.60i		−	0.0594076i	0.999559	−	0.0297038i	$-0.00945641\pi$
$200$		0			0
$201$			13659.1i			0.338088i
$202$			25942.5i			0.635784i
$203$	2187.77	+	16460.4i	0.0530895	+	0.399438i
$204$	−18470.6			−0.443834
$205$		0			0
$206$		−	11049.2i		−	0.260372i
$207$	−37548.5			−0.876298
$208$			3108.66i			0.0718533i
$209$			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	$0.760772\pi$
$212$	11687.5			0.260046
$213$		−	28881.5i		−	0.636590i
$214$	34340.7			0.749863
$215$		0			0
$216$			295.534i			0.00633433i
$217$	12996.0	−	1727.30i	0.275987	−	0.0366816i
$218$	19256.9			0.405203
$219$	−90168.7			−1.88004
$220$		0			0
$221$	−8839.17			−0.180978
$222$			23990.5i			0.486780i
$223$		−	30412.4i		−	0.611563i	−0.952102	−	0.305781i	$-0.901082\pi$
$223$		−	30412.4i		−	0.611563i	0.952102	−	0.305781i	$-0.0989177\pi$
$224$	−1168.63	−	8792.63i	−0.0232907	−	0.175236i
$225$		0			0
$226$	13475.2			0.263826
$227$		−	52125.5i		−	1.01158i	−0.862658	−	0.505788i	$-0.831202\pi$
$227$		−	52125.5i		−	1.01158i	0.862658	−	0.505788i	$-0.168798\pi$
$228$	−60891.0			−1.17134
$229$		−	81280.2i		−	1.54994i	−0.632000	−	0.774968i	$-0.717766\pi$
$229$		−	81280.2i		−	1.54994i	0.632000	−	0.774968i	$-0.282234\pi$
$230$		0			0
$231$	118214.	−	15711.9i	2.21536	−	0.294445i
$232$	−7668.01			−0.142465
$233$	41718.9			0.768459			0.384229	−	0.923238i	$-0.374467\pi$
$233$	41718.9			0.768459			0.384229	+	0.923238i	$0.374467\pi$
$234$		−	10986.7i		−	0.200649i
$235$		0			0
$236$		−	13846.7i		−	0.248612i
$237$			88968.9i			1.58395i
$238$	25000.9	−	3322.89i	0.441370	−	0.0586627i
$239$	−3936.55			−0.0689160			−0.0344580	−	0.999406i	$-0.510970\pi$
$239$	−3936.55			−0.0689160			−0.0344580	+	0.999406i	$0.510970\pi$
$240$		0			0
$241$		−	70511.5i		−	1.21402i	−0.794694	−	0.607010i	$-0.792369\pi$
$241$		−	70511.5i		−	1.21402i	0.794694	−	0.607010i	$-0.207631\pi$
$242$	62664.4			1.07002
$243$		−	83228.7i		−	1.40949i
$244$			1970.74i			0.0331017i
$245$		0			0
$246$	−47506.8			−0.785028
$247$	−29139.6			−0.477628
$248$			6054.11i			0.0984343i
$249$	−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	$-0.193737\pi$
$252$	4130.22	+	31075.2i	0.0650387	+	0.489342i
$253$	90066.6			1.40709
$254$	79108.1			1.22618
$255$		0			0
$256$	4096.00			0.0625000
$257$		−	65615.5i		−	0.993436i	−0.867912	−	0.496718i	$-0.834538\pi$
$257$		−	65615.5i		−	0.993436i	0.867912	−	0.496718i	$-0.165462\pi$
$258$		−	69626.1i		−	1.04600i
$259$	−4315.92	−	32472.4i	−0.0643390	−	0.484078i
$260$		0			0
$261$	27100.5			0.397829
$262$		−	67929.0i		−	0.989583i
$263$	38706.1			0.559588			0.279794	−	0.960060i	$-0.409734\pi$
$263$	38706.1			0.559588			0.279794	+	0.960060i	$0.409734\pi$
$264$			55069.3i			0.790136i
$265$		0			0
$266$	82419.2	−	10954.4i	1.16484	−	0.154819i
$267$	27071.5			0.379743
$268$	8612.70			0.119914
$269$		−	87226.9i		−	1.20544i	−0.797952	−	0.602721i	$-0.794083\pi$
$269$		−	87226.9i		−	1.20544i	0.797952	−	0.602721i	$-0.205917\pi$
$270$		0			0
$271$			105362.i			1.43465i	0.696739	+	0.717324i	$0.254633\pi$
$271$			105362.i			1.43465i	−0.696739	+	0.717324i	$0.745367\pi$
$272$			11646.6i			0.157420i
$273$	3978.50	+	29933.7i	0.0533820	+	0.401638i
$274$	11771.9			0.156800
$275$		0			0
$276$			47656.9i			0.625615i
$277$	−36178.9			−0.471515			−0.235758	−	0.971812i	$-0.575757\pi$
$277$	−36178.9			−0.471515			−0.235758	+	0.971812i	$0.575757\pi$
$278$			74573.7i			0.964931i
$279$		−	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.			1.32528
$283$			4153.27i			0.0518581i	0.999664	+	0.0259291i	$0.00825441\pi$
$283$			4153.27i			0.0518581i	−0.999664	+	0.0259291i	$0.991746\pi$
$284$	−18211.1			−0.225787
$285$		0			0
$286$			26353.6i			0.322187i
$287$	64302.9	−	8546.54i	0.780669	−	0.103759i
$288$	−14476.2			−0.174530
$289$	50405.2			0.603503
$290$		0			0
$291$	−74840.1			−0.883788
$292$			56855.5i			0.666817i
$293$			20239.4i			0.235756i	0.993028	+	0.117878i	$0.0376092\pi$
$293$			20239.4i			0.235756i	−0.993028	+	0.117878i	$0.962391\pi$
$294$	−22505.8	−	83169.7i	−0.260376	−	0.962211i
$295$		0			0
$296$	15127.1			0.172652
$297$			2505.39i			0.0284028i
$298$	−18601.3			−0.209465
$299$			22806.4i			0.255102i
$300$		0			0
$301$	12525.8	+	94242.7i	0.138253	+	1.04019i
$302$	−64857.1			−0.711121
$303$	−116370.			−1.26752
$304$			38394.6i			0.415454i
$305$		0			0
$306$		−	41161.7i		−	0.439592i
$307$		−	63269.8i		−	0.671305i	−0.941986	−	0.335652i	$-0.891043\pi$
$307$		−	63269.8i		−	0.671305i	0.941986	−	0.335652i	$-0.108957\pi$
$308$	−9907.05	−	74539.3i	−0.104434	−	0.785749i
$309$	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	$-0.0236766\pi$
$312$	−13944.5			−0.143249
$313$		−	36763.0i		−	0.375252i	−0.982241	−	0.187626i	$-0.939921\pi$
$313$		−	36763.0i		−	0.375252i	0.982241	−	0.187626i	$-0.0600792\pi$
$314$		−	105480.i		−	1.06982i
$315$		0			0
$316$	56099.0			0.561799
$317$	125556.			1.24945			0.624726	−	0.780844i	$-0.285211\pi$
$317$	125556.			1.24945			0.624726	+	0.780844i	$0.285211\pi$
$318$			52426.4i			0.518436i
$319$	−65005.4			−0.638804
$320$		0			0
$321$			154041.i			1.49495i
$322$	−8573.55	−	64506.1i	−0.0826892	−	0.622142i
$323$	−109171.			−1.04641
$324$	−53146.6			−0.506274
$325$		0			0
$326$	−115553.			−1.08729
$327$			86380.2i			0.807828i
$328$			29955.2i			0.278435i
$329$	−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	$0.492189\pi$
$332$			11586.9i			0.105121i
$333$	−53462.7			−0.482128
$334$			98358.3i			0.881694i
$335$		0			0
$336$	39440.9	−	5242.11i	0.349356	−	0.0464331i
$337$	−2202.27			−0.0193914			−0.00969572	−	0.999953i	$-0.503086\pi$
$337$	−2202.27			−0.0193914			−0.00969572	+	0.999953i	$0.503086\pi$
$338$	74109.5			0.648695
$339$			60445.4i			0.525974i
$340$		0			0
$341$			51323.5i			0.441375i
$342$		−	135695.i		−	1.16015i
$343$	45425.2	+	108526.i	0.386108	+	0.922454i
$344$	−43902.5			−0.370998
$345$		0			0
$346$		−	89378.5i		−	0.746587i
$347$	−222201.			−1.84538			−0.922691	−	0.385541i	$-0.874015\pi$
$347$	−222201.			−1.84538			−0.922691	+	0.385541i	$0.874015\pi$
$348$		−	34396.2i		−	0.284022i
$349$		−	102679.i		−	0.843006i	−0.906827	−	0.421503i	$-0.861503\pi$
$349$		−	102679.i		−	0.843006i	0.906827	−	0.421503i	$-0.138497\pi$
$350$		0			0
$351$	−634.405			−0.00514935
$352$	34723.7			0.280247
$353$		−	62595.6i		−	0.502336i	−0.967943	−	0.251168i	$-0.919185\pi$
$353$		−	62595.6i		−	0.502336i	0.967943	−	0.251168i	$-0.0808147\pi$
$354$	62111.8			0.495641
$355$		0			0
$356$		−	17069.8i		−	0.134688i
$357$	14905.4	+	112146.i	0.116952	+	0.879930i
$358$	64348.6			0.502080
$359$	95505.9			0.741040			0.370520	−	0.928825i	$-0.379180\pi$
$359$	95505.9			0.741040			0.370520	+	0.928825i	$0.379180\pi$
$360$		0			0
$361$	−229577.			−1.76163
$362$		−	156792.i		−	1.19648i
$363$			281092.i			2.13322i
$364$	18874.6	−	2508.63i	0.142454	−	0.0189336i
$365$		0			0
$366$	−8840.12			−0.0659927
$367$			82330.9i			0.611267i	0.952149	+	0.305633i	$0.0988682\pi$
$367$			82330.9i			0.611267i	−0.952149	+	0.305633i	$0.901132\pi$
$368$	30049.9			0.221895
$369$		−	105869.i		−	0.777525i
$370$		0			0
$371$	−9431.58	−	70961.9i	−0.0685230	−	0.515558i
$372$	−27156.8			−0.196242
$373$	−130223.			−0.935991			−0.467995	−	0.883731i	$-0.655024\pi$
$373$	−130223.			−0.935991			−0.467995	+	0.883731i	$0.655024\pi$
$374$			98733.4i			0.705864i
$375$		0			0
$376$		−	66454.2i		−	0.470053i
$377$		−	16460.4i		−	0.115813i
$378$	1794.37	−	238.491i	0.0125582	−	0.00166912i
$379$	192349.			1.33909			0.669546	−	0.742770i	$-0.266489\pi$
$379$	192349.			1.33909			0.669546	+	0.742770i	$0.266489\pi$
$380$		0			0
$381$			354853.i			2.44455i
$382$	−143734.			−0.984992
$383$			101933.i			0.694891i	0.937700	+	0.347446i	$0.112951\pi$
$383$			101933.i			0.694891i	−0.937700	+	0.347446i	$0.887049\pi$
$384$			18373.3i			0.124602i
$385$		0			0
$386$	3530.84			0.0236975
$387$	155162.			1.03601
$388$			47190.1i			0.313464i
$389$	191074.			1.26271			0.631353	−	0.775495i	$-0.282500\pi$
$389$	191074.			1.26271			0.631353	+	0.775495i	$0.282500\pi$
$390$		0			0
$391$			85443.7i			0.558890i
$392$	−52442.3	+	14191.0i	−0.341279	+	0.0923506i
$393$	304708.			1.97287
$394$	−182305.			−1.17437
$395$		0			0
$396$	−122722.			−0.782585
$397$		−	201143.i		−	1.27622i	−0.769947	−	0.638108i	$-0.779717\pi$
$397$		−	201143.i		−	1.27622i	0.769947	−	0.638108i	$-0.220283\pi$
$398$		−	6654.16i		−	0.0420076i
$399$	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	$0.539672\pi$
$402$			38633.8i			0.239065i
$403$	−12996.0			−0.0800200
$404$			73376.6i			0.449567i
$405$		0			0
$406$	6187.93	+	46557.1i	0.0375399	+	0.282445i
$407$	128240.			0.774165
$408$	−52242.7			−0.313838
$409$		−	80655.7i		−	0.482157i	−0.970506	−	0.241078i	$-0.922499\pi$
$409$		−	80655.7i		−	0.482157i	0.970506	−	0.241078i	$-0.0775011\pi$
$410$		0			0
$411$			52805.0i			0.312602i
$412$		−	31251.7i		−	0.184111i
$413$	−84071.6	+	11174.0i	−0.492889	+	0.0655101i
$414$	−106203.			−0.619636
$415$		0			0
$416$			8792.63i			0.0508080i
$417$	−334514.			−1.92372
$418$			325489.i			1.86288i
$419$			252034.i			1.43559i	0.696254	+	0.717795i	$0.254849\pi$
$419$			252034.i			1.43559i	−0.696254	+	0.717795i	$0.745151\pi$
$420$		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.			−1.03326
$423$			234864.i			1.31261i
$424$	33057.2			0.183880
$425$		0			0
$426$		−	81689.1i		−	0.450137i
$427$	11965.6	−	1590.35i	0.0656263	−	0.00872242i
$428$	97130.2			0.530233
$429$	−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	$0.611516\pi$
$432$			835.898i			0.00447905i
$433$			233539.i			1.24562i	0.782375	+	0.622808i	$0.214008\pi$
$433$			233539.i			1.24562i	−0.782375	+	0.622808i	$0.785992\pi$
$434$	36758.2	−	4885.55i	0.195153	−	0.0259378i
$435$		0			0
$436$	54466.7			0.286522
$437$			281678.i			1.47499i
$438$	−255036.			−1.32939
$439$		−	304238.i		−	1.57864i	−0.613980	−	0.789322i	$-0.710432\pi$
$439$		−	304238.i		−	1.57864i	0.613980	−	0.789322i	$-0.289568\pi$
$440$		0			0
$441$	185343.	−	50154.1i	0.953015	−	0.257887i
$442$	−25000.9			−0.127971
$443$	87061.0			0.443625			0.221813	−	0.975089i	$-0.428803\pi$
$443$	87061.0			0.443625			0.221813	+	0.975089i	$0.428803\pi$
$444$			67855.3i			0.344206i
$445$		0			0
$446$		−	86019.3i		−	0.432440i
$447$		−	83439.7i		−	0.417597i
$448$	−3305.39	−	24869.3i	−0.0164690	−	0.123910i
$449$	−91141.4			−0.452088			−0.226044	−	0.974117i	$-0.572579\pi$
$449$	−91141.4			−0.452088			−0.226044	+	0.974117i	$0.572579\pi$
$450$		0			0
$451$			253944.i			1.24849i
$452$	38113.6			0.186553
$453$		−	290928.i		−	1.41772i
$454$		−	147433.i		−	0.715292i
$455$		0			0
$456$	−172226.			−0.828263
$457$	411928.			1.97237			0.986187	−	0.165634i	$-0.0529670\pi$
$457$	411928.			1.97237			0.986187	+	0.165634i	$0.0529670\pi$
$458$		−	229895.i		−	1.09597i
$459$	−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	$-0.120748\pi$
$462$	334359.	−	44439.9i	1.56650	−	0.208204i
$463$	−245557.			−1.14549			−0.572745	−	0.819734i	$-0.694121\pi$
$463$	−245557.			−1.14549			−0.572745	+	0.819734i	$0.694121\pi$
$464$	−21688.4			−0.100738
$465$		0			0
$466$	117999.			0.543383
$467$			33069.2i			0.151632i	0.997122	+	0.0758158i	$0.0241561\pi$
$467$			33069.2i			0.151632i	−0.997122	+	0.0758158i	$0.975844\pi$
$468$		−	31075.2i		−	0.141880i
$469$	−6950.28	−	52292.9i	−0.0315978	−	0.237737i
$470$		0			0
$471$	473151.			2.13284
$472$		−	39164.3i		−	0.175795i
$473$	−372182.			−1.66354
$474$			251642.i			1.12002i
$475$		0			0
$476$	70713.3	−	9398.55i	0.312095	−	0.0414808i
$477$	−116832.			−0.513482
$478$	−11134.2			−0.0487309
$479$			384277.i			1.67484i	0.546559	+	0.837421i	$0.315937\pi$
$479$			384277.i			1.67484i	−0.546559	+	0.837421i	$0.684063\pi$
$480$		0			0
$481$			32472.4i			0.140354i
$482$		−	199437.i		−	0.858442i
$483$	289354.	−	38458.2i	1.24032	−	0.164852i
$484$	177242.			0.756615
$485$		0			0
$486$		−	235406.i		−	0.996657i
$487$	173526.			0.731657			0.365829	−	0.930682i	$-0.380786\pi$
$487$	173526.			0.731657			0.365829	+	0.930682i	$0.380786\pi$
$488$			5574.10i			0.0234064i
$489$		−	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.			−0.555099
$493$		−	61668.8i		−	0.253730i
$494$	−82419.2			−0.337734
$495$		0			0
$496$			17123.6i			0.0696036i
$497$	14696.0	+	110571.i	0.0594958	+	0.447638i
$498$	−51975.1			−0.209574
$499$	−428368.			−1.72035			−0.860174	−	0.510000i	$-0.829645\pi$
$499$	−428368.			−1.72035			−0.860174	+	0.510000i	$0.829645\pi$
$500$		0			0
$501$	−441204.			−1.75778
$502$		−	203766.i		−	0.808584i
$503$		−	36793.4i		−	0.145423i	−0.997353	−	0.0727116i	$-0.976835\pi$
$503$		−	36793.4i		−	0.145423i	0.997353	−	0.0727116i	$-0.0231653\pi$
$504$	11682.0	+	87893.9i	0.0459893	+	0.346017i
$505$		0			0
$506$	254747.			0.994965
$507$			332432.i			1.29326i
$508$	223751.			0.867038
$509$		−	137334.i		−	0.530080i	−0.964237	−	0.265040i	$-0.914615\pi$
$509$		−	137334.i		−	0.530080i	0.964237	−	0.265040i	$-0.0853852\pi$
$510$		0			0
$511$	345204.	−	45881.3i	1.32201	−	0.175709i
$512$	11585.2			0.0441942
$513$	−7835.43			−0.0297734
$514$		−	185589.i		−	0.702466i
$515$		0			0
$516$		−	196932.i		−	0.739636i
$517$		−	563363.i		−	2.10769i
$518$	−12207.3	−	91845.8i	−0.0454945	−	0.342294i
$519$	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	$0.939373\pi$
$522$	76651.9			0.281308
$523$			314194.i			1.14867i	0.818621	+	0.574334i	$0.194739\pi$
$523$			314194.i			1.14867i	−0.818621	+	0.574334i	$0.805261\pi$
$524$		−	192132.i		−	0.699741i
$525$		0			0
$526$	109478.			0.395689
$527$	−48689.2			−0.175312
$528$			155760.i			0.558711i
$529$	−59383.5			−0.212204
$530$		0			0
$531$			138416.i			0.490904i
$532$	233117.	−	30983.7i	0.823664	−	0.109474i
$533$	−64302.9			−0.226348
$534$	76569.7			0.268519
$535$		0			0
$536$	24360.4			0.0847919
$537$			288647.i			1.00096i
$538$		−	246715.i		−	0.852376i
$539$	−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.i			1.01445i
$543$	703319.			2.38535
$544$			32941.4i			0.111313i
$545$		0			0
$546$	11252.9	+	84665.3i	0.0377468	+	0.284001i
$547$	−361462.			−1.20806			−0.604029	−	0.796963i	$-0.706439\pi$
$547$	−361462.			−1.20806			−0.604029	+	0.796963i	$0.706439\pi$
$548$	33296.0			0.110874
$549$		−	19700.2i		−	0.0653620i
$550$		0			0
$551$		−	203300.i		−	0.669629i
$552$			134794.i			0.442377i
$553$	−45270.8	−	340611.i	−0.148036	−	1.11380i
$554$	−102329.			−0.333412
$555$		0			0
$556$			210926.i			0.682309i
$557$	−112424.			−0.362367			−0.181183	−	0.983449i	$-0.557993\pi$
$557$	−112424.			−0.362367			−0.181183	+	0.983449i	$0.557993\pi$
$558$		−	60518.8i		−	0.194367i
$559$		−	94242.7i		−	0.301595i
$560$		0			0
$561$	−442886.			−1.40724
$562$	280416.			0.887832
$563$		−	441530.i		−	1.39298i	−0.717569	−	0.696488i	$-0.754745\pi$
$563$		−	441530.i		−	1.39298i	0.717569	−	0.696488i	$-0.245255\pi$
$564$	298092.			0.937113
$565$		0			0
$566$			11747.2i			0.0366692i
$567$	42888.3	+	322685.i	0.133405	+	1.00372i
$568$	−51508.8			−0.159656
$569$	397273.			1.22706			0.613529	−	0.789673i	$-0.289750\pi$
$569$	397273.			1.22706			0.613529	+	0.789673i	$0.289750\pi$
$570$		0			0
$571$	−67235.8			−0.206219			−0.103109	−	0.994670i	$-0.532879\pi$
$571$	−67235.8			−0.206219			−0.103109	+	0.994670i	$0.532879\pi$
$572$			74539.3i			0.227821i
$573$		−	644744.i		−	1.96371i
$574$	181876.	−	24173.3i	0.552016	−	0.0733688i
$575$		0			0
$576$	−40944.9			−0.123411
$577$			64756.1i			0.194504i	0.995260	+	0.0972521i	$0.0310053\pi$
$577$			64756.1i			0.194504i	−0.995260	+	0.0972521i	$0.968995\pi$
$578$	142567.			0.426741
$579$			15838.2i			0.0472443i
$580$		0			0
$581$	70351.0	−	9350.39i	0.208410	−	0.0276999i
$582$	−211680.			−0.624933
$583$	280242.			0.824510
$584$			160812.i			0.471511i
$585$		0			0
$586$			57245.8i			0.166705i
$587$			338967.i			0.983741i	0.870668	+	0.491870i	$0.163687\pi$
$587$			338967.i			0.983741i	−0.870668	+	0.491870i	$0.836313\pi$
$588$	−63656.1	−	235239.i	−0.184113	−	0.680386i
$589$	−160511.			−0.462673
$590$		0			0
$591$		−	817763.i		−	2.34128i
$592$	42785.9			0.122084
$593$		−	255668.i		−	0.727054i	−0.931584	−	0.363527i	$-0.881572\pi$
$593$		−	255668.i		−	0.727054i	0.931584	−	0.363527i	$-0.118428\pi$
$594$			7086.30i			0.0200838i
$595$		0			0
$596$	−52612.6			−0.148114
$597$	29848.4			0.0837477
$598$			64506.1i			0.180384i
$599$	129521.			0.360982			0.180491	−	0.983577i	$-0.442231\pi$
$599$	129521.			0.360982			0.180491	+	0.983577i	$0.442231\pi$
$600$		0			0
$601$		−	377277.i		−	1.04451i	−0.852790	−	0.522254i	$-0.825091\pi$
$601$		−	377277.i		−	1.04451i	0.852790	−	0.522254i	$-0.174909\pi$
$602$	35428.4	+	266559.i	0.0977595	+	0.735529i
$603$	−86095.3			−0.236780
$604$	−183444.			−0.502839
$605$		0			0
$606$	−329144.			−0.896274
$607$		−	421091.i		−	1.14287i	−0.820646	−	0.571437i	$-0.806386\pi$
$607$		−	421091.i		−	1.14287i	0.820646	−	0.571437i	$-0.193614\pi$
$608$			108596.i			0.293770i
$609$	−208840.	+	27757.1i	−0.563093	+	0.0748410i
$610$		0			0
$611$	142653.			0.382119
$612$		−	116423.i		−	0.310839i
$613$	−412173.			−1.09688			−0.548440	−	0.836190i	$-0.684778\pi$
$613$	−412173.			−1.09688			−0.548440	+	0.836190i	$0.684778\pi$
$614$		−	178954.i		−	0.474684i
$615$		0			0
$616$	−28021.4	−	210829.i	−0.0738462	−	0.555608i
$617$	262676.			0.690002			0.345001	−	0.938602i	$-0.387879\pi$
$617$	262676.			0.690002			0.345001	+	0.938602i	$0.387879\pi$
$618$	140185.			0.367050
$619$		−	373975.i		−	0.976026i	−0.872836	−	0.488013i	$-0.837722\pi$
$619$		−	373975.i		−	0.976026i	0.872836	−	0.488013i	$-0.162278\pi$
$620$		0			0
$621$			6132.47i			0.0159020i
$622$		−	40659.7i		−	0.105095i
$623$	−103641.	+	13775.0i	−0.267028	+	0.0354908i
$624$	−39440.9			−0.101293
$625$		0			0
$626$		−	103982.i		−	0.265343i
$627$	−1.46004e6			−3.71389
$628$		−	298344.i		−	0.756480i
$629$			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.			0.397252
$633$		−	825397.i		−	2.05995i
$634$	355126.			0.883495
$635$		0			0
$636$			148284.i			0.366590i
$637$	−30462.8	−	112575.i	−0.0750743	−	0.277435i
$638$	−183863.			−0.451703
$639$	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.i			1.05709i
$643$			323445.i			0.782310i	0.920325	+	0.391155i	$0.127924\pi$
$643$			323445.i			0.782310i	−0.920325	+	0.391155i	$0.872076\pi$
$644$	−24249.6	−	182451.i	−0.0584701	−	0.439921i
$645$		0			0
$646$	−308782.			−0.739925
$647$		−	592372.i		−	1.41510i	−0.706665	−	0.707548i	$-0.749801\pi$
$647$		−	592372.i		−	1.41510i	0.706665	−	0.707548i	$-0.250199\pi$
$648$	−150321.			−0.357990
$649$		−	332015.i		−	0.788257i
$650$		0			0
$651$	21915.0	+	164885.i	0.0517106	+	0.389063i
$652$	−326832.			−0.768828
$653$	329810.			0.773459			0.386730	−	0.922193i	$-0.373605\pi$
$653$	329810.			0.773459			0.386730	+	0.922193i	$0.373605\pi$
$654$			244320.i			0.571221i
$655$		0			0
$656$			84726.1i			0.196884i
$657$		−	568346.i		−	1.31669i
$658$	−403483.	+	53627.2i	−0.931910	+	0.123861i
$659$	−526737.			−1.21290			−0.606448	−	0.795123i	$-0.707406\pi$
$659$	−526737.			−1.21290			−0.606448	+	0.795123i	$0.707406\pi$
$660$		0			0
$661$		−	144047.i		−	0.329686i	−0.986320	−	0.164843i	$-0.947288\pi$
$661$		−	144047.i		−	0.329686i	0.986320	−	0.164843i	$-0.0527117\pi$
$662$	15207.2			0.0347002
$663$		−	112146.i		−	0.255128i
$664$			32772.7i			0.0743320i
$665$		0			0
$666$	−151215.			−0.340916
$667$	−159115.			−0.357650
$668$			278199.i			0.623452i
$669$	385855.			0.862128
$670$		0			0
$671$			47254.3i			0.104953i
$672$	111556.	−	14826.9i	0.247032	−	0.0328332i
$673$	−620117.			−1.36913			−0.684563	−	0.728953i	$-0.740007\pi$
$673$	−620117.			−1.36913			−0.684563	+	0.728953i	$0.740007\pi$
$674$	−6228.95			−0.0137118
$675$		0			0
$676$	209613.			0.458697
$677$			626172.i			1.36621i	0.730322	+	0.683103i	$0.239370\pi$
$677$			626172.i			1.36621i	−0.730322	+	0.683103i	$0.760630\pi$
$678$			170965.i			0.371920i
$679$	286520.	−	38081.5i	0.621463	−	0.0825989i
$680$		0			0
$681$	661338.			1.42603
$682$			145165.i			0.312099i
$683$	−185558.			−0.397775			−0.198888	−	0.980022i	$-0.563733\pi$
$683$	−185558.			−0.397775			−0.198888	+	0.980022i	$0.563733\pi$
$684$		−	383805.i		−	0.820348i
$685$		0			0
$686$	128482.	+	306957.i	0.273019	+	0.652273i
$687$	1.03124e6			2.18497
$688$	−124175.			−0.262336
$689$			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.i		−	0.527917i
$693$	99034.1	+	745118.i	0.206214	+	1.55153i
$694$	−628478.			−1.30488
$695$		0			0
$696$		−	97287.2i		−	0.200834i
$697$	−240910.			−0.495894
$698$		−	290420.i		−	0.596095i
$699$			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	$0.323483\pi$
$702$	−1794.37			−0.00364114
$703$			401061.i			0.811522i
$704$	98213.6			0.198165
$705$		0			0
$706$		−	177047.i		−	0.355205i
$707$	445514.	−	59213.5i	0.891297	−	0.118463i
$708$	175679.			0.350471
$709$	6035.96			0.0120075			0.00600377	−	0.999982i	$-0.498089\pi$
$709$	6035.96			0.0120075			0.00600377	+	0.999982i	$0.498089\pi$
$710$		0			0
$711$	−560784.			−1.10932
$712$		−	48280.7i		−	0.0952388i
$713$			125625.i			0.247115i
$714$	42158.9	+	317197.i	0.0826975	+	0.622204i
$715$		0			0
$716$	182005.			0.355024
$717$		−	49944.6i		−	0.0971517i
$718$	270132.			0.523994
$719$			748658.i			1.44819i	0.689700	+	0.724095i	$0.257742\pi$
$719$			748658.i			1.44819i	−0.689700	+	0.724095i	$0.742258\pi$
$720$		0			0
$721$	−189748.	+	25219.5i	−0.365012	+	0.0485139i
$722$	−649343.			−1.24566
$723$	894609.			1.71142
$724$		−	443475.i		−	0.846042i
$725$		0			0
$726$			795049.i			1.50841i
$727$			370335.i			0.700691i	0.936621	+	0.350345i	$0.113936\pi$
$727$			370335.i			0.700691i	−0.936621	+	0.350345i	$0.886064\pi$
$728$	53385.4	−	7095.48i	0.100730	−	0.0133881i
$729$	517848.			0.974422
$730$		0			0
$731$		−	353079.i		−	0.660750i
$732$	−25003.6			−0.0466639
$733$		−	144545.i		−	0.269027i	−0.990912	−	0.134513i	$-0.957053\pi$
$733$		−	144545.i		−	0.269027i	0.990912	−	0.134513i	$-0.0429471\pi$
$734$			232867.i			0.432231i
$735$		0			0
$736$	84993.8			0.156903
$737$	206515.			0.380203
$738$		−	299442.i		−	0.549793i
$739$	4650.26			0.00851506			0.00425753	−	0.999991i	$-0.498645\pi$
$739$	4650.26			0.00851506			0.00425753	+	0.999991i	$0.498645\pi$
$740$		0			0
$741$		−	369706.i		−	0.673318i
$742$	−26676.5	−	200710.i	−0.0484531	−	0.364554i
$743$	−500204.			−0.906087			−0.453043	−	0.891489i	$-0.649662\pi$
$743$	−500204.			−0.906087			−0.453043	+	0.891489i	$0.649662\pi$
$744$	−76811.0			−0.138764
$745$		0			0
$746$	−368327.			−0.661845
$747$		−	115826.i		−	0.207571i
$748$			279260.i			0.499121i
$749$	−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.i		−	0.332377i
$753$	914031.			1.61202
$754$		−	46557.1i		−	0.0818924i
$755$		0			0
$756$	5075.24	−	674.553i	0.00888000	−	0.00118025i
$757$	654028.			1.14131			0.570656	−	0.821189i	$-0.306689\pi$
$757$	654028.			1.14131			0.570656	+	0.821189i	$0.306689\pi$
$758$	544044.			0.946881
$759$			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	$0.812438\pi$
$762$			1.00368e6i			1.72856i
$763$	−43953.6	−	330700.i	−0.0754997	−	0.568049i
$764$	−406541.			−0.696494
$765$		0			0
$766$			288310.i			0.491362i
$767$	84071.6			0.142909
$768$			51967.7i			0.0881071i
$769$		−	103697.i		−	0.175354i	−0.996149	−	0.0876768i	$-0.972056\pi$
$769$		−	103697.i		−	0.175354i	0.996149	−	0.0876768i	$-0.0279443\pi$
$770$		0			0
$771$	832491.			1.40046
$772$	9986.72			0.0167567
$773$			796380.i			1.33279i	0.745599	+	0.666394i	$0.232163\pi$
$773$			796380.i			1.33279i	−0.745599	+	0.666394i	$0.767837\pi$
$774$	438863.			0.732567
$775$		0			0
$776$			133474.i			0.221652i
$777$	411991.	−	54757.9i	0.682410	−	0.0906995i
$778$	540439.			0.892868
$779$	−794195.			−1.30874
$780$		0			0
$781$	−436664.			−0.715889
$782$			241671.i			0.395195i
$783$		−	4426.10i		−	0.00721934i
$784$	−148329.	+	40138.1i	−0.241321	+	0.0653017i
$785$		0			0
$786$	861843.			1.39503
$787$		−	1.16895e6i		−	1.88733i	−0.330905	−	0.943664i	$-0.607354\pi$
$787$		−	1.16895e6i		−	1.88733i	0.330905	−	0.943664i	$-0.392646\pi$
$788$	−515637.			−0.830408
$789$			491081.i			0.788859i
$790$		0			0
$791$	−30756.9	−	231411.i	−0.0491575	−	0.369854i
$792$	−347110.			−0.553371
$793$	−11965.6			−0.0190277
$794$		−	568919.i		−	0.902421i
$795$		0			0
$796$		−	18820.8i		−	0.0297038i
$797$			808048.i			1.27210i	0.771649	+	0.636049i	$0.219432\pi$
$797$			808048.i			1.27210i	−0.771649	+	0.636049i	$0.780568\pi$
$798$	138983.	+	1.04569e6i	0.218251	+	1.64209i
$799$	534447.			0.837165
$800$		0			0
$801$			170635.i			0.265952i
$802$	−113076.			−0.175801
$803$			1.36328e6i			2.11423i
$804$			109273.i			0.169044i
$805$		0			0
$806$	−36758.2			−0.0565827
$807$	1.10668e6			1.69933
$808$			207540.i			0.317892i
$809$	−692708.			−1.05841			−0.529204	−	0.848495i	$-0.677509\pi$
$809$	−692708.			−1.05841			−0.529204	+	0.848495i	$0.677509\pi$
$810$		0			0
$811$			382267.i			0.581200i	0.956845	+	0.290600i	$0.0938548\pi$
$811$			382267.i			0.581200i	−0.956845	+	0.290600i	$0.906145\pi$
$812$	17502.1	+	131683.i	0.0265448	+	0.199719i
$813$	−1.33677e6			−2.02244
$814$	362717.			0.547417
$815$		0			0
$816$	−147765.			−0.221917
$817$		−	1.16398e6i		−	1.74381i
$818$		−	228129.i		−	0.340936i
$819$	−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.i			0.221043i
$823$	−640526.			−0.945665			−0.472832	−	0.881152i	$-0.656768\pi$
$823$	−640526.			−0.945665			−0.472832	+	0.881152i	$0.656768\pi$
$824$		−	88393.2i		−	0.130186i
$825$		0			0
$826$	−237790.	+	31604.8i	−0.348525	+	0.0463227i
$827$	158299.			0.231455			0.115727	−	0.993281i	$-0.463080\pi$
$827$	158299.			0.231455			0.115727	+	0.993281i	$0.463080\pi$
$828$	−300388.			−0.438149
$829$		−	680501.i		−	0.990192i	−0.868838	−	0.495096i	$-0.835133\pi$
$829$		−	680501.i		−	0.990192i	0.868838	−	0.495096i	$-0.164867\pi$
$830$		0			0
$831$		−	459017.i		−	0.664701i
$832$			24869.3i			0.0359267i
$833$	−114129.	−	421759.i	−0.164477	−	0.607819i
$834$	−946148.			−1.36028
$835$		0			0
$836$			920622.i			1.31725i
$837$	−3494.53			−0.00498812
$838$			712859.i			1.01512i
$839$			622397.i			0.884186i	0.896969	+	0.442093i	$0.145764\pi$
$839$			622397.i			0.884186i	−0.896969	+	0.442093i	$0.854236\pi$
$840$		0			0
$841$	−592440.			−0.837631
$842$	−238831.			−0.336873
$843$			1.25786e6i			1.77001i
$844$	−520451.			−0.730626
$845$		0			0
$846$			664297.i			0.928157i
$847$	−143031.	−	1.07614e6i	−0.199371	−	1.50004i
$848$	93500.0			0.130023
$849$	−52694.2			−0.0731051
$850$		0			0
$851$	313894.			0.433435
$852$		−	231052.i		−	0.318295i
$853$		−	826596.i		−	1.13604i	−0.823013	−	0.568022i	$-0.807709\pi$
$853$		−	826596.i		−	1.13604i	0.823013	−	0.568022i	$-0.192291\pi$
$854$	33843.7	−	4498.19i	0.0464048	−	0.00616768i
$855$		0			0
$856$	274726.			0.374931
$857$		−	539076.i		−	0.733987i	−0.930223	−	0.366994i	$-0.880387\pi$
$857$		−	539076.i		−	0.733987i	0.930223	−	0.366994i	$-0.119613\pi$
$858$	−334359.			−0.454191
$859$			499964.i			0.677567i	0.940864	+	0.338783i	$0.110015\pi$
$859$			499964.i			0.677567i	−0.940864	+	0.338783i	$0.889985\pi$
$860$		0			0
$861$	108433.	+	815838.i	0.146271	+	1.10052i
$862$	−360659.			−0.485380
$863$	−1.16944e6			−1.57021			−0.785103	−	0.619365i	$-0.787390\pi$
$863$	−1.16944e6			−1.57021			−0.785103	+	0.619365i	$0.787390\pi$
$864$			2364.28i			0.00316716i
$865$		0			0
$866$			660548.i			0.880783i
$867$			639512.i			0.850766i
$868$	103968.	−	13818.4i	0.137994	−	0.0183408i
$869$	1.34514e6			1.78126
$870$		0			0
$871$			52292.9i			0.0689297i
$872$	154055.			0.202602
$873$		−	471727.i		−	0.618960i
$874$			796705.i			1.04298i
$875$		0			0
$876$	−721350.			−0.940021
$877$	−187496.			−0.243777			−0.121888	−	0.992544i	$-0.538895\pi$
$877$	−187496.			−0.243777			−0.121888	+	0.992544i	$0.538895\pi$
$878$		−	860515.i		−	1.11627i
$879$	−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	$-0.0369254\pi$
$882$	524230.	−	141857.i	0.673883	−	0.182354i
$883$	1.04658e6			1.34231			0.671155	−	0.741317i	$-0.265799\pi$
$883$	1.04658e6			1.34231			0.671155	+	0.741317i	$0.265799\pi$
$884$	−70713.3			−0.0904892
$885$		0			0
$886$	246246.			0.313691
$887$			1.30029e6i			1.65270i	0.563158	+	0.826349i	$0.309586\pi$
$887$			1.30029e6i			1.65270i	−0.563158	+	0.826349i	$0.690414\pi$
$888$			191924.i			0.243390i
$889$	−180563.	−	1.35853e6i	−0.228468	−	1.71896i
$890$		0			0
$891$	−1.27434e6			−1.60521
$892$		−	243299.i		−	0.305781i
$893$	1.76188e6			2.20940
$894$		−	236003.i		−	0.295286i
$895$		0			0
$896$	−9349.06	−	70341.0i	−0.0116453	−	0.0876178i
$897$	−289354.			−0.359620
$898$	−257787.			−0.319675
$899$		−	90669.8i		−	0.112187i
$900$		0			0
$901$			265858.i			0.327491i
$902$			718263.i			0.882817i
$903$	−1.19570e6	+	158921.i	−1.46638	+	0.194897i
$904$	107802.			0.131913
$905$		0			0
$906$		−	822869.i		−	1.00248i
$907$	−1.18388e6			−1.43911			−0.719556	−	0.694435i	$-0.755654\pi$
$907$	−1.18388e6			−1.43911			−0.719556	+	0.694435i	$0.755654\pi$
$908$		−	417004.i		−	0.505788i
$909$		−	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.			−0.585671
$913$			277830.i			0.333301i
$914$	1.16511e6			1.39468
$915$		0			0
$916$		−	650242.i		−	0.774968i
$917$	−1.16655e6	+	155047.i	−1.38728	+	0.184384i
$918$	−6722.58			−0.00797720
$919$	698775.			0.827382			0.413691	−	0.910417i	$-0.364239\pi$
$919$	698775.			0.827382			0.413691	+	0.910417i	$0.364239\pi$
$920$		0			0
$921$	802730.			0.946347
$922$		−	445185.i		−	0.523696i
$923$		−	110571.i		−	0.129789i
$924$	945711.	−	125695.i	1.10768	−	0.147222i
$925$		0			0
$926$	−694541.			−0.809984
$927$			312402.i			0.363542i
$928$	−61344.1			−0.0712323
$929$			1.39460e6i			1.61591i	0.589244	+	0.807955i	$0.299426\pi$
$929$			1.39460e6i			1.61591i	−0.589244	+	0.807955i	$0.700574\pi$
$930$		0			0
$931$	−376241.	−	1.39039e6i	−0.434077	−	1.60412i
$932$	333751.			0.384229
$933$	182387.			0.209522
$934$			93533.8i			0.107220i
$935$		0			0
$936$		−	87893.9i		−	0.100325i
$937$		−	509380.i		−	0.580180i	−0.956999	−	0.290090i	$-0.906315\pi$
$937$		−	509380.i		−	0.580180i	0.956999	−	0.290090i	$-0.0936852\pi$
$938$	−19658.4	−	147907.i	−0.0223430	−	0.168106i
$939$	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	$0.630117\pi$
$942$	1.33827e6			1.50814
$943$			621584.i			0.698998i
$944$		−	110773.i		−	0.124306i
$945$		0			0
$946$	−1.05269e6			−1.17630
$947$	−1.22462e6			−1.36553			−0.682763	−	0.730640i	$-0.739222\pi$
$947$	−1.22462e6			−1.36553			−0.682763	+	0.730640i	$0.739222\pi$
$948$			711752.i			0.791976i
$949$	−345204.			−0.383304
$950$		0			0
$951$			1.59298e6i			1.76137i
$952$	200007.	−	26583.1i	0.220685	−	0.0293313i
$953$	1.14847e6			1.26454			0.632269	−	0.774749i	$-0.282124\pi$
$953$	1.14847e6			1.26454			0.632269	+	0.774749i	$0.282124\pi$
$954$	−330451.			−0.363086
$955$		0			0
$956$	−31492.4			−0.0344580
$957$		−	824750.i		−	0.900531i
$958$			1.08690e6i			1.18429i
$959$	−26869.2	−	202160.i	−0.0292158	−	0.219815i
$960$		0			0
$961$	851935.			0.922485
$962$			91845.8i			0.0992451i
$963$	−970945.			−1.04699
$964$		−	564092.i		−	0.607010i
$965$		0			0
$966$	818416.	−	108776.i	0.877041	−	0.116568i
$967$	733668.			0.784597			0.392298	−	0.919838i	$-0.371680\pi$
$967$	733668.			0.784597			0.392298	+	0.919838i	$0.371680\pi$
$968$	501315.			0.535008
$969$		−	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.i		−	0.704743i
$973$	1.28066e6	−	170213.i	1.35272	−	0.179791i
$974$	490807.			0.517360
$975$		0			0
$976$			15765.9i			0.0165508i
$977$	−837039.			−0.876913			−0.438457	−	0.898752i	$-0.644475\pi$
$977$	−837039.			−0.876913			−0.438457	+	0.898752i	$0.644475\pi$
$978$		−	1.46606e6i		−	1.53276i
$979$		−	409299.i		−	0.427046i
$980$		0			0
$981$	−544467.			−0.565761
$982$	−56586.3			−0.0586798
$983$		−	644129.i		−	0.666600i	−0.942821	−	0.333300i	$-0.891838\pi$
$983$		−	644129.i		−	0.666600i	0.942821	−	0.333300i	$-0.108162\pi$
$984$	−380054.			−0.392514
$985$		0			0
$986$		−	174426.i		−	0.179414i
$987$	−240554.	−	1.80990e6i	−0.246933	−	1.85789i
$988$	−233117.			−0.238814
$989$	−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	$0.485001\pi$
$992$			48432.8i			0.0492172i
$993$			68214.4i			0.0691795i
$994$	41566.6	+	312741.i	0.0420699	+	0.316528i
$995$		0			0
$996$	−147008.			−0.148191
$997$		−	366383.i		−	0.368591i	−0.982871	−	0.184295i	$-0.941000\pi$
$997$		−	366383.i		−	0.368591i	0.982871	−	0.184295i	$-0.0590003\pi$
$998$	−1.21161e6			−1.21647
$999$			8731.60i			0.00874909i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.5.b.a.251.4		4
5.2	odd	4		350.5.d.a.349.8		8
5.3	odd	4		350.5.d.a.349.1		8
5.4	even	2		14.5.b.a.13.1	✓	4
7.6	odd	2	inner	350.5.b.a.251.3		4
15.14	odd	2		126.5.c.a.55.4		4
20.19	odd	2		112.5.c.c.97.4		4
35.4	even	6		98.5.d.d.19.3		8
35.9	even	6		98.5.d.d.31.4		8
35.13	even	4		350.5.d.a.349.4		8
35.19	odd	6		98.5.d.d.31.3		8
35.24	odd	6		98.5.d.d.19.4		8
35.27	even	4		350.5.d.a.349.5		8
35.34	odd	2		14.5.b.a.13.2	yes	4
40.19	odd	2		448.5.c.f.321.1		4
40.29	even	2		448.5.c.e.321.4		4
60.59	even	2		1008.5.f.h.433.3		4
105.104	even	2		126.5.c.a.55.3		4
140.139	even	2		112.5.c.c.97.1		4
280.69	odd	2		448.5.c.e.321.1		4
280.139	even	2		448.5.c.f.321.4		4
420.419	odd	2		1008.5.f.h.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.5.b.a.13.1	✓	4	5.4	even	2
14.5.b.a.13.2	yes	4	35.34	odd	2
98.5.d.d.19.3		8	35.4	even	6
98.5.d.d.19.4		8	35.24	odd	6
98.5.d.d.31.3		8	35.19	odd	6
98.5.d.d.31.4		8	35.9	even	6
112.5.c.c.97.1		4	140.139	even	2
112.5.c.c.97.4		4	20.19	odd	2
126.5.c.a.55.3		4	105.104	even	2
126.5.c.a.55.4		4	15.14	odd	2
350.5.b.a.251.3		4	7.6	odd	2	inner
350.5.b.a.251.4		4	1.1	even	1	trivial
350.5.d.a.349.1		8	5.3	odd	4
350.5.d.a.349.4		8	35.13	even	4
350.5.d.a.349.5		8	35.27	even	4
350.5.d.a.349.8		8	5.2	odd	4
448.5.c.e.321.1		4	280.69	odd	2
448.5.c.e.321.4		4	40.29	even	2
448.5.c.f.321.1		4	40.19	odd	2
448.5.c.f.321.4		4	280.139	even	2
1008.5.f.h.433.2		4	420.419	odd	2
1008.5.f.h.433.3		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	350.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +12.6874i q^{3} +8.00000 q^{4} +35.8854i q^{6} +(-6.45584 - 48.5729i) q^{7} +22.6274 q^{8} -79.9706 q^{9} +191.823 q^{11} +101.499i q^{12} +48.5729i q^{13} +(-18.2599 - 137.385i) q^{14} +64.0000 q^{16} +181.977i q^{17} -226.191 q^{18} +599.915i q^{19} +(616.264 - 81.9080i) q^{21} +542.558 q^{22} +469.529 q^{23} +287.083i q^{24} +137.385i q^{26} +13.0609i q^{27} +(-51.6468 - 388.583i) q^{28} -338.881 q^{29} +267.556i q^{31} +181.019 q^{32} +2433.74i q^{33} +514.710i q^{34} -639.765 q^{36} +668.530 q^{37} +1696.82i q^{38} -616.264 q^{39} +1323.85i q^{41} +(1743.06 - 231.671i) q^{42} -1940.23 q^{43} +1534.59 q^{44} +1328.03 q^{46} -2936.89i q^{47} +811.995i q^{48} +(-2317.64 + 627.158i) q^{49} -2308.82 q^{51} +388.583i q^{52} +1460.94 q^{53} +36.9418i q^{54} +(-146.079 - 1099.08i) q^{56} -7611.38 q^{57} -958.501 q^{58} -1730.83i q^{59} +246.343i q^{61} +756.763i q^{62} +(516.277 + 3884.40i) q^{63} +512.000 q^{64} +6883.67i q^{66} +1076.59 q^{67} +1455.82i q^{68} +5957.11i q^{69} -2276.39 q^{71} -1809.53 q^{72} +7106.94i q^{73} +1890.89 q^{74} +4799.32i q^{76} +(-1238.38 - 9317.41i) q^{77} -1743.06 q^{78} +7012.38 q^{79} -6643.32 q^{81} +3744.40i q^{82} +1448.36i q^{83} +(4930.11 - 655.264i) q^{84} -5487.81 q^{86} -4299.53i q^{87} +4340.47 q^{88} -2133.73i q^{89} +(2359.32 - 313.579i) q^{91} +3756.23 q^{92} -3394.60 q^{93} -8306.77i q^{94} +2296.67i q^{96} +5898.76i q^{97} +(-6555.29 + 1773.87i) q^{98} -15340.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4} + 76 q^{7} - 252 q^{9} + 360 q^{11} - 288 q^{14} + 256 q^{16} - 192 q^{18} + 768 q^{21} + 1152 q^{22} + 792 q^{23} + 608 q^{28} + 1224 q^{29} - 2016 q^{36} + 3896 q^{37} - 768 q^{39} + 4800 q^{42}+ \cdots - 29592 q^{99}+O(q^{100}) \) 4 * q + 32 * q^4 + 76 * q^7 - 252 * q^9 + 360 * q^11 - 288 * q^14 + 256 * q^16 - 192 * q^18 + 768 * q^21 + 1152 * q^22 + 792 * q^23 + 608 * q^28 + 1224 * q^29 - 2016 * q^36 + 3896 * q^37 - 768 * q^39 + 4800 * q^42 - 3688 * q^43 + 2880 * q^44 + 3072 * q^46 - 1532 * q^49 - 11136 * q^51 - 5832 * q^53 - 2304 * q^56 - 12864 * q^57 - 7296 * q^58 - 3060 * q^63 + 2048 * q^64 + 1048 * q^67 - 21528 * q^71 - 1536 * q^72 - 3456 * q^74 - 3528 * q^77 - 4800 * q^78 + 12776 * q^79 - 29628 * q^81 + 6144 * q^84 - 11520 * q^86 + 9216 * q^88 + 5568 * q^91 + 6336 * q^92 - 38016 * q^93 - 21888 * q^98 - 29592 * q^99

Introduction

L-functions

Modular forms

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Database

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