LMFDB - Embedded newform 304.5.e.e.113.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,5,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.4244687775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 $ x^8 + 450x^6 + 68229x^4 + 4001228*x^2 + 77475204
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.6
Root			$6.38941i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.e.113.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.80363i q^{3} -33.0971 q^{5} -16.0783 q^{7} +20.1034 q^{9} +O(q^{10})$	$q+7.80363i q^{3} -33.0971 q^{5} -16.0783 q^{7} +20.1034 q^{9} +215.592 q^{11} -281.497i q^{13} -258.277i q^{15} +226.582 q^{17} +(-13.9969 + 360.729i) q^{19} -125.469i q^{21} -414.609 q^{23} +470.416 q^{25} +788.973i q^{27} +606.612i q^{29} +478.655i q^{31} +1682.40i q^{33} +532.145 q^{35} -104.604i q^{37} +2196.70 q^{39} +1891.12i q^{41} -315.912 q^{43} -665.364 q^{45} +474.327 q^{47} -2142.49 q^{49} +1768.16i q^{51} -774.381i q^{53} -7135.46 q^{55} +(-2814.99 - 109.227i) q^{57} -567.212i q^{59} +4799.51 q^{61} -323.229 q^{63} +9316.74i q^{65} -4464.01i q^{67} -3235.45i q^{69} +7636.34i q^{71} +8396.65 q^{73} +3670.95i q^{75} -3466.35 q^{77} +9706.71i q^{79} -4528.48 q^{81} +10589.4 q^{83} -7499.21 q^{85} -4733.77 q^{87} -10281.7i q^{89} +4526.01i q^{91} -3735.24 q^{93} +(463.257 - 11939.1i) q^{95} +15477.3i q^{97} +4334.13 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) $ 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9	$ 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) $ 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9 + 6 * q^11 + 510 * q^17 + 12 * q^19 + 396 * q^23 + 3458 * q^25 - 1002 * q^35 + 6588 * q^39 + 8654 * q^43 - 10334 * q^45 - 3210 * q^47 + 9222 * q^49 - 17146 * q^55 - 14076 * q^57 + 1314 * q^61 - 29938 * q^63 + 23398 * q^73 - 44622 * q^77 - 20368 * q^81 + 10440 * q^83 + 21274 * q^85 + 14316 * q^87 + 19416 * q^93 + 34686 * q^95 + 56798 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			7.80363i			0.867070i	0.901137	+	0.433535i	$0.142734\pi$
$3$			7.80363i			0.867070i	−0.901137	+	0.433535i	$0.857266\pi$
$4$		0			0
$5$	−33.0971			−1.32388			−0.661941	−	0.749556i	$-0.730267\pi$
$5$	−33.0971			−1.32388			−0.661941	+	0.749556i	$0.730267\pi$
$6$		0			0
$7$	−16.0783			−0.328129			−0.164064	−	0.986450i	$-0.552461\pi$
$7$	−16.0783			−0.328129			−0.164064	+	0.986450i	$0.552461\pi$
$8$		0			0
$9$	20.1034			0.248190
$10$		0			0
$11$	215.592			1.78175			0.890875	−	0.454248i	$-0.150092\pi$
$11$	215.592			1.78175			0.890875	+	0.454248i	$0.150092\pi$
$12$		0			0
$13$		−	281.497i		−	1.66567i	−0.553525	−	0.832833i	$-0.686718\pi$
$13$		−	281.497i		−	1.66567i	0.553525	−	0.832833i	$-0.313282\pi$
$14$		0			0
$15$		−	258.277i		−	1.14790i
$16$		0			0
$17$	226.582			0.784022			0.392011	−	0.919961i	$-0.371780\pi$
$17$	226.582			0.784022			0.392011	+	0.919961i	$0.371780\pi$
$18$		0			0
$19$	−13.9969	+	360.729i	−0.0387726	+	0.999248i
$19$	−13.9969	+	360.729i	−0.0387726	+	0.999248i
$20$		0			0
$21$		−	125.469i		−	0.284511i
$22$		0			0
$23$	−414.609			−0.783760			−0.391880	−	0.920016i	$-0.628175\pi$
$23$	−414.609			−0.783760			−0.391880	+	0.920016i	$0.628175\pi$
$24$		0			0
$25$	470.416			0.752666
$26$		0			0
$27$			788.973i			1.08227i
$28$		0			0
$29$			606.612i			0.721298i	0.932702	+	0.360649i	$0.117445\pi$
$29$			606.612i			0.721298i	−0.932702	+	0.360649i	$0.882555\pi$
$30$		0			0
$31$			478.655i			0.498080i	0.968493	+	0.249040i	$0.0801151\pi$
$31$			478.655i			0.498080i	−0.968493	+	0.249040i	$0.919885\pi$
$32$		0			0
$33$			1682.40i			1.54490i
$34$		0			0
$35$	532.145			0.434404
$36$		0			0
$37$		−	104.604i		−	0.0764094i	−0.999270	−	0.0382047i	$-0.987836\pi$
$37$		−	104.604i		−	0.0764094i	0.999270	−	0.0382047i	$-0.0121639\pi$
$38$		0			0
$39$	2196.70			1.44425
$40$		0			0
$41$			1891.12i			1.12500i	0.826799	+	0.562498i	$0.190159\pi$
$41$			1891.12i			1.12500i	−0.826799	+	0.562498i	$0.809841\pi$
$42$		0			0
$43$	−315.912			−0.170855			−0.0854277	−	0.996344i	$-0.527226\pi$
$43$	−315.912			−0.170855			−0.0854277	+	0.996344i	$0.527226\pi$
$44$		0			0
$45$	−665.364			−0.328575
$46$		0			0
$47$	474.327			0.214725			0.107362	−	0.994220i	$-0.465759\pi$
$47$	474.327			0.214725			0.107362	+	0.994220i	$0.465759\pi$
$48$		0			0
$49$	−2142.49			−0.892331
$50$		0			0
$51$			1768.16i			0.679801i
$52$		0			0
$53$		−	774.381i		−	0.275679i	−0.990455	−	0.137839i	$-0.955984\pi$
$53$		−	774.381i		−	0.275679i	0.990455	−	0.137839i	$-0.0440158\pi$
$54$		0			0
$55$	−7135.46			−2.35883
$56$		0			0
$57$	−2814.99	−	109.227i	−0.866418	−	0.0336186i
$58$		0			0
$59$		−	567.212i		−	0.162945i	−0.996676	−	0.0814725i	$-0.974038\pi$
$59$		−	567.212i		−	0.162945i	0.996676	−	0.0814725i	$-0.0259623\pi$
$60$		0			0
$61$	4799.51			1.28984			0.644922	−	0.764248i	$-0.276890\pi$
$61$	4799.51			1.28984			0.644922	+	0.764248i	$0.276890\pi$
$62$		0			0
$63$	−323.229			−0.0814384
$64$		0			0
$65$			9316.74i			2.20515i
$66$		0			0
$67$		−	4464.01i		−	0.994433i	−0.867627	−	0.497216i	$-0.834356\pi$
$67$		−	4464.01i		−	0.994433i	0.867627	−	0.497216i	$-0.165644\pi$
$68$		0			0
$69$		−	3235.45i		−	0.679575i
$70$		0			0
$71$			7636.34i			1.51485i	0.652924	+	0.757423i	$0.273542\pi$
$71$			7636.34i			1.51485i	−0.652924	+	0.757423i	$0.726458\pi$
$72$		0			0
$73$	8396.65			1.57565			0.787826	−	0.615898i	$-0.211207\pi$
$73$	8396.65			1.57565			0.787826	+	0.615898i	$0.211207\pi$
$74$		0			0
$75$			3670.95i			0.652614i
$76$		0			0
$77$	−3466.35			−0.584644
$78$		0			0
$79$			9706.71i			1.55531i	0.628689	+	0.777657i	$0.283592\pi$
$79$			9706.71i			1.55531i	−0.628689	+	0.777657i	$0.716408\pi$
$80$		0			0
$81$	−4528.48			−0.690212
$82$		0			0
$83$	10589.4			1.53715			0.768575	−	0.639760i	$-0.220966\pi$
$83$	10589.4			1.53715			0.768575	+	0.639760i	$0.220966\pi$
$84$		0			0
$85$	−7499.21			−1.03795
$86$		0			0
$87$	−4733.77			−0.625416
$88$		0			0
$89$		−	10281.7i		−	1.29804i	−0.760773	−	0.649018i	$-0.775180\pi$
$89$		−	10281.7i		−	1.29804i	0.760773	−	0.649018i	$-0.224820\pi$
$90$		0			0
$91$			4526.01i			0.546553i
$92$		0			0
$93$	−3735.24			−0.431870
$94$		0			0
$95$	463.257	−	11939.1i	0.0513304	−	1.32289i
$96$		0			0
$97$			15477.3i			1.64494i	0.568806	+	0.822472i	$0.307406\pi$
$97$			15477.3i			1.64494i	−0.568806	+	0.822472i	$0.692594\pi$
$98$		0			0
$99$	4334.13			0.442213
$100$		0			0
$101$	10464.5			1.02583			0.512914	−	0.858440i	$-0.328566\pi$
$101$	10464.5			1.02583			0.512914	+	0.858440i	$0.328566\pi$
$102$		0			0
$103$			3224.53i			0.303943i	0.988385	+	0.151972i	$0.0485622\pi$
$103$			3224.53i			0.303943i	−0.988385	+	0.151972i	$0.951438\pi$
$104$		0			0
$105$			4152.66i			0.376659i
$106$		0			0
$107$			7740.93i			0.676123i	0.941124	+	0.338061i	$0.109771\pi$
$107$			7740.93i			0.676123i	−0.941124	+	0.338061i	$0.890229\pi$
$108$		0			0
$109$			12613.1i			1.06162i	0.847492	+	0.530809i	$0.178112\pi$
$109$			12613.1i			1.06162i	−0.847492	+	0.530809i	$0.821888\pi$
$110$		0			0
$111$	816.294			0.0662523
$112$		0			0
$113$		−	900.876i		−	0.0705518i	−0.999378	−	0.0352759i	$-0.988769\pi$
$113$		−	900.876i		−	0.0705518i	0.999378	−	0.0352759i	$-0.0112310\pi$
$114$		0			0
$115$	13722.3			1.03761
$116$		0			0
$117$		−	5659.06i		−	0.413402i
$118$		0			0
$119$	−3643.06			−0.257260
$120$		0			0
$121$	31838.8			2.17464
$122$		0			0
$123$	−14757.6			−0.975449
$124$		0			0
$125$	5116.28			0.327442
$126$		0			0
$127$			16688.1i			1.03467i	0.855784	+	0.517334i	$0.173075\pi$
$127$			16688.1i			1.03467i	−0.855784	+	0.517334i	$0.826925\pi$
$128$		0			0
$129$		−	2465.26i		−	0.148144i
$130$		0			0
$131$	−9365.24			−0.545728			−0.272864	−	0.962053i	$-0.587971\pi$
$131$	−9365.24			−0.545728			−0.272864	+	0.962053i	$0.587971\pi$
$132$		0			0
$133$	225.047	−	5799.91i	0.0127224	−	0.327882i
$134$		0			0
$135$		−	26112.7i		−	1.43280i
$136$		0			0
$137$	−36021.1			−1.91918			−0.959591	−	0.281399i	$-0.909201\pi$
$137$	−36021.1			−1.91918			−0.959591	+	0.281399i	$0.909201\pi$
$138$		0			0
$139$	10371.8			0.536813			0.268406	−	0.963306i	$-0.413503\pi$
$139$	10371.8			0.536813			0.268406	+	0.963306i	$0.413503\pi$
$140$		0			0
$141$			3701.47i			0.186181i
$142$		0			0
$143$		−	60688.6i		−	2.96780i
$144$		0			0
$145$		−	20077.1i		−	0.954914i
$146$		0			0
$147$		−	16719.2i		−	0.773714i
$148$		0			0
$149$	1353.29			0.0609562			0.0304781	−	0.999535i	$-0.490297\pi$
$149$	1353.29			0.0609562			0.0304781	+	0.999535i	$0.490297\pi$
$150$		0			0
$151$			29132.7i			1.27769i	0.769334	+	0.638847i	$0.220588\pi$
$151$			29132.7i			1.27769i	−0.769334	+	0.638847i	$0.779412\pi$
$152$		0			0
$153$	4555.07			0.194586
$154$		0			0
$155$		−	15842.1i		−	0.659399i
$156$		0			0
$157$	−11934.0			−0.484156			−0.242078	−	0.970257i	$-0.577829\pi$
$157$	−11934.0			−0.484156			−0.242078	+	0.970257i	$0.577829\pi$
$158$		0			0
$159$	6042.98			0.239033
$160$		0			0
$161$	6666.22			0.257174
$162$		0			0
$163$	−13818.7			−0.520108			−0.260054	−	0.965594i	$-0.583740\pi$
$163$	−13818.7			−0.520108			−0.260054	+	0.965594i	$0.583740\pi$
$164$		0			0
$165$		−	55682.5i		−	2.04527i
$166$		0			0
$167$			10267.5i			0.368157i	0.982912	+	0.184079i	$0.0589301\pi$
$167$			10267.5i			0.368157i	−0.982912	+	0.184079i	$0.941070\pi$
$168$		0			0
$169$	−50679.8			−1.77444
$170$		0			0
$171$	−281.386	+	7251.87i	−0.00962298	+	0.248003i
$172$		0			0
$173$		−	29001.7i		−	0.969015i	−0.874787	−	0.484508i	$-0.838999\pi$
$173$		−	29001.7i		−	0.969015i	0.874787	−	0.484508i	$-0.161001\pi$
$174$		0			0
$175$	−7563.50			−0.246971
$176$		0			0
$177$	4426.31			0.141285
$178$		0			0
$179$		−	25479.5i		−	0.795217i	−0.917555	−	0.397608i	$-0.869840\pi$
$179$		−	25479.5i		−	0.795217i	0.917555	−	0.397608i	$-0.130160\pi$
$180$		0			0
$181$			31158.1i			0.951073i	0.879696	+	0.475536i	$0.157746\pi$
$181$			31158.1i			0.951073i	−0.879696	+	0.475536i	$0.842254\pi$
$182$		0			0
$183$			37453.6i			1.11839i
$184$		0			0
$185$			3462.10i			0.101157i
$186$		0			0
$187$	48849.3			1.39693
$188$		0			0
$189$		−	12685.4i		−	0.355123i
$190$		0			0
$191$	−17888.8			−0.490359			−0.245179	−	0.969478i	$-0.578847\pi$
$191$	−17888.8			−0.490359			−0.245179	+	0.969478i	$0.578847\pi$
$192$		0			0
$193$			784.195i			0.0210528i	0.999945	+	0.0105264i	$0.00335071\pi$
$193$			784.195i			0.0210528i	−0.999945	+	0.0105264i	$0.996649\pi$
$194$		0			0
$195$	−72704.4			−1.91202
$196$		0			0
$197$	42272.4			1.08924			0.544621	−	0.838682i	$-0.316673\pi$
$197$	42272.4			1.08924			0.544621	+	0.838682i	$0.316673\pi$
$198$		0			0
$199$	15361.7			0.387911			0.193955	−	0.981010i	$-0.437868\pi$
$199$	15361.7			0.387911			0.193955	+	0.981010i	$0.437868\pi$
$200$		0			0
$201$	34835.5			0.862243
$202$		0			0
$203$		−	9753.30i		−	0.236679i
$204$		0			0
$205$		−	62590.4i		−	1.48936i
$206$		0			0
$207$	−8335.05			−0.194521
$208$		0			0
$209$	−3017.62	+	77770.1i	−0.0690831	+	1.78041i
$210$		0			0
$211$		−	36036.6i		−	0.809429i	−0.914443	−	0.404714i	$-0.867371\pi$
$211$		−	36036.6i		−	0.809429i	0.914443	−	0.404714i	$-0.132629\pi$
$212$		0			0
$213$	−59591.2			−1.31348
$214$		0			0
$215$	10455.8			0.226193
$216$		0			0
$217$		−	7695.96i		−	0.163434i
$218$		0			0
$219$			65524.3i			1.36620i
$220$		0			0
$221$		−	63782.3i		−	1.30592i
$222$		0			0
$223$		−	44099.4i		−	0.886795i	−0.896325	−	0.443398i	$-0.853773\pi$
$223$		−	44099.4i		−	0.886795i	0.896325	−	0.443398i	$-0.146227\pi$
$224$		0			0
$225$	9456.96			0.186804
$226$		0			0
$227$		−	34866.5i		−	0.676638i	−0.941032	−	0.338319i	$-0.890142\pi$
$227$		−	34866.5i		−	0.676638i	0.941032	−	0.338319i	$-0.109858\pi$
$228$		0			0
$229$	26085.5			0.497425			0.248712	−	0.968577i	$-0.419993\pi$
$229$	26085.5			0.497425			0.248712	+	0.968577i	$0.419993\pi$
$230$		0			0
$231$		−	27050.1i		−	0.506927i
$232$		0			0
$233$	75466.8			1.39009			0.695047	−	0.718964i	$-0.255383\pi$
$233$	75466.8			1.39009			0.695047	+	0.718964i	$0.255383\pi$
$234$		0			0
$235$	−15698.8			−0.284271
$236$		0			0
$237$	−75747.5			−1.34856
$238$		0			0
$239$	42891.8			0.750895			0.375447	−	0.926844i	$-0.377489\pi$
$239$	42891.8			0.750895			0.375447	+	0.926844i	$0.377489\pi$
$240$		0			0
$241$			15629.3i			0.269095i	0.990907	+	0.134547i	$0.0429580\pi$
$241$			15629.3i			0.269095i	−0.990907	+	0.134547i	$0.957042\pi$
$242$		0			0
$243$			28568.3i			0.483806i
$244$		0			0
$245$	70910.1			1.18134
$246$		0			0
$247$	101544.	+	3940.10i	1.66441	+	0.0645822i
$248$		0			0
$249$			82635.9i			1.33282i
$250$		0			0
$251$	−76998.8			−1.22218			−0.611092	−	0.791560i	$-0.709269\pi$
$251$	−76998.8			−1.22218			−0.611092	+	0.791560i	$0.709269\pi$
$252$		0			0
$253$	−89386.3			−1.39646
$254$		0			0
$255$		−	58521.0i		−	0.899977i
$256$		0			0
$257$			8903.45i			0.134801i	0.997726	+	0.0674003i	$0.0214705\pi$
$257$			8903.45i			0.134801i	−0.997726	+	0.0674003i	$0.978530\pi$
$258$		0			0
$259$			1681.86i			0.0250721i
$260$		0			0
$261$			12195.0i			0.179019i
$262$		0			0
$263$	79232.7			1.14549			0.572747	−	0.819732i	$-0.305878\pi$
$263$	79232.7			1.14549			0.572747	+	0.819732i	$0.305878\pi$
$264$		0			0
$265$			25629.8i			0.364966i
$266$		0			0
$267$	80234.9			1.12549
$268$		0			0
$269$		−	132817.i		−	1.83548i	−0.397177	−	0.917742i	$-0.630010\pi$
$269$		−	132817.i		−	1.83548i	0.397177	−	0.917742i	$-0.369990\pi$
$270$		0			0
$271$	−73016.0			−0.994213			−0.497106	−	0.867690i	$-0.665604\pi$
$271$	−73016.0			−0.994213			−0.497106	+	0.867690i	$0.665604\pi$
$272$		0			0
$273$	−35319.3			−0.473900
$274$		0			0
$275$	101418.			1.34106
$276$		0			0
$277$	66280.9			0.863831			0.431916	−	0.901914i	$-0.357838\pi$
$277$	66280.9			0.863831			0.431916	+	0.901914i	$0.357838\pi$
$278$		0			0
$279$			9622.59i			0.123619i
$280$		0			0
$281$			112860.i			1.42932i	0.699473	+	0.714659i	$0.253418\pi$
$281$			112860.i			1.42932i	−0.699473	+	0.714659i	$0.746582\pi$
$282$		0			0
$283$	−80766.6			−1.00846			−0.504230	−	0.863569i	$-0.668224\pi$
$283$	−80766.6			−1.00846			−0.504230	+	0.863569i	$0.668224\pi$
$284$		0			0
$285$	93168.0	+	3615.08i	1.14704	+	0.0445070i
$286$		0			0
$287$		−	30406.0i		−	0.369144i
$288$		0			0
$289$	−32181.5			−0.385310
$290$		0			0
$291$	−120779.			−1.42628
$292$		0			0
$293$		−	133666.i		−	1.55699i	−0.627653	−	0.778493i	$-0.715984\pi$
$293$		−	133666.i		−	1.55699i	0.627653	−	0.778493i	$-0.284016\pi$
$294$		0			0
$295$			18773.0i			0.215720i
$296$		0			0
$297$			170096.i			1.92833i
$298$		0			0
$299$			116711.i			1.30548i
$300$		0			0
$301$	5079.33			0.0560626
$302$		0			0
$303$			81660.8i			0.889464i
$304$		0			0
$305$	−158850.			−1.70760
$306$		0			0
$307$		−	100857.i		−	1.07011i	−0.844816	−	0.535057i	$-0.820290\pi$
$307$		−	100857.i		−	1.07011i	0.844816	−	0.535057i	$-0.179710\pi$
$308$		0			0
$309$	−25163.1			−0.263540
$310$		0			0
$311$	−121531.			−1.25651			−0.628254	−	0.778009i	$-0.716230\pi$
$311$	−121531.			−1.25651			−0.628254	+	0.778009i	$0.716230\pi$
$312$		0			0
$313$	189932.			1.93869			0.969346	−	0.245700i	$-0.0790180\pi$
$313$	189932.			1.93869			0.969346	+	0.245700i	$0.0790180\pi$
$314$		0			0
$315$	10697.9			0.107815
$316$		0			0
$317$			67336.1i			0.670085i	0.942203	+	0.335042i	$0.108751\pi$
$317$			67336.1i			0.670085i	−0.942203	+	0.335042i	$0.891249\pi$
$318$		0			0
$319$			130781.i			1.28517i
$320$		0			0
$321$	−60407.3			−0.586246
$322$		0			0
$323$	−3171.45	+	81734.7i	−0.0303986	+	0.783432i
$324$		0			0
$325$		−	132421.i		−	1.25369i
$326$		0			0
$327$	−98427.7			−0.920496
$328$		0			0
$329$	−7626.38			−0.0704574
$330$		0			0
$331$			154310.i			1.40844i	0.709984	+	0.704218i	$0.248702\pi$
$331$			154310.i			1.40844i	−0.709984	+	0.704218i	$0.751298\pi$
$332$		0			0
$333$		−	2102.90i		−	0.0189640i
$334$		0			0
$335$			147746.i			1.31651i
$336$		0			0
$337$			127543.i			1.12304i	0.827463	+	0.561521i	$0.189783\pi$
$337$			127543.i			1.12304i	−0.827463	+	0.561521i	$0.810217\pi$
$338$		0			0
$339$	7030.10			0.0611733
$340$		0			0
$341$			103194.i			0.887454i
$342$		0			0
$343$	73051.6			0.620929
$344$		0			0
$345$			107084.i			0.899677i
$346$		0			0
$347$	71084.5			0.590359			0.295179	−	0.955442i	$-0.404621\pi$
$347$	71084.5			0.590359			0.295179	+	0.955442i	$0.404621\pi$
$348$		0			0
$349$	83601.6			0.686378			0.343189	−	0.939266i	$-0.388493\pi$
$349$	83601.6			0.686378			0.343189	+	0.939266i	$0.388493\pi$
$350$		0			0
$351$	222094.			1.80270
$352$		0			0
$353$	−40368.7			−0.323963			−0.161981	−	0.986794i	$-0.551788\pi$
$353$	−40368.7			−0.323963			−0.161981	+	0.986794i	$0.551788\pi$
$354$		0			0
$355$		−	252740.i		−	2.00548i
$356$		0			0
$357$		−	28429.1i		−	0.223063i
$358$		0			0
$359$	−143079.			−1.11016			−0.555081	−	0.831796i	$-0.687313\pi$
$359$	−143079.			−1.11016			−0.555081	+	0.831796i	$0.687313\pi$
$360$		0			0
$361$	−129929.	−	10098.2i	−0.996993	−	0.0774869i
$362$		0			0
$363$			248458.i			1.88556i
$364$		0			0
$365$	−277904.			−2.08598
$366$		0			0
$367$	30831.7			0.228910			0.114455	−	0.993428i	$-0.463488\pi$
$367$	30831.7			0.228910			0.114455	+	0.993428i	$0.463488\pi$
$368$		0			0
$369$			38017.9i			0.279213i
$370$		0			0
$371$			12450.8i			0.0904582i
$372$		0			0
$373$			199882.i			1.43666i	0.695701	+	0.718332i	$0.255094\pi$
$373$			199882.i			1.43666i	−0.695701	+	0.718332i	$0.744906\pi$
$374$		0			0
$375$			39925.5i			0.283915i
$376$		0			0
$377$	170760.			1.20144
$378$		0			0
$379$			49499.6i			0.344606i	0.985044	+	0.172303i	$0.0551209\pi$
$379$			49499.6i			0.344606i	−0.985044	+	0.172303i	$0.944879\pi$
$380$		0			0
$381$	−130228.			−0.897129
$382$		0			0
$383$		−	49593.2i		−	0.338084i	−0.985609	−	0.169042i	$-0.945933\pi$
$383$		−	49593.2i		−	0.338084i	0.985609	−	0.169042i	$-0.0540674\pi$
$384$		0			0
$385$	114726.			0.774000
$386$		0			0
$387$	−6350.90			−0.0424046
$388$		0			0
$389$	−15826.9			−0.104592			−0.0522958	−	0.998632i	$-0.516654\pi$
$389$	−15826.9			−0.104592			−0.0522958	+	0.998632i	$0.516654\pi$
$390$		0			0
$391$	−93943.1			−0.614485
$392$		0			0
$393$		−	73082.8i		−	0.473184i
$394$		0			0
$395$		−	321264.i		−	2.05905i
$396$		0			0
$397$	−2861.99			−0.0181588			−0.00907939	−	0.999959i	$-0.502890\pi$
$397$	−2861.99			−0.0181588			−0.00907939	+	0.999959i	$0.502890\pi$
$398$		0			0
$399$	45260.3	+	1756.18i	0.284297	+	0.0110312i
$400$		0			0
$401$		−	226034.i		−	1.40567i	−0.711351	−	0.702837i	$-0.751916\pi$
$401$		−	226034.i		−	1.40567i	0.711351	−	0.702837i	$-0.248084\pi$
$402$		0			0
$403$	134740.			0.829634
$404$		0			0
$405$	149879.			0.913759
$406$		0			0
$407$		−	22551.9i		−	0.136142i
$408$		0			0
$409$		−	300321.i		−	1.79531i	−0.440700	−	0.897655i	$-0.645270\pi$
$409$		−	300321.i		−	1.79531i	0.440700	−	0.897655i	$-0.354730\pi$
$410$		0			0
$411$		−	281095.i		−	1.66406i
$412$		0			0
$413$			9119.81i			0.0534670i
$414$		0			0
$415$	−350479.			−2.03501
$416$		0			0
$417$			80937.4i			0.465454i
$418$		0			0
$419$	88915.8			0.506467			0.253233	−	0.967405i	$-0.418506\pi$
$419$	88915.8			0.506467			0.253233	+	0.967405i	$0.418506\pi$
$420$		0			0
$421$		−	52919.8i		−	0.298575i	−0.988794	−	0.149288i	$-0.952302\pi$
$421$		−	52919.8i		−	0.298575i	0.988794	−	0.149288i	$-0.0476980\pi$
$422$		0			0
$423$	9535.59			0.0532926
$424$		0			0
$425$	106588.			0.590106
$426$		0			0
$427$	−77168.1			−0.423235
$428$		0			0
$429$	473591.			2.57329
$430$		0			0
$431$		−	18064.2i		−	0.0972442i	−0.998817	−	0.0486221i	$-0.984517\pi$
$431$		−	18064.2i		−	0.0972442i	0.998817	−	0.0486221i	$-0.0154830\pi$
$432$		0			0
$433$			188974.i			1.00792i	0.863726	+	0.503961i	$0.168125\pi$
$433$			188974.i			1.00792i	−0.863726	+	0.503961i	$0.831875\pi$
$434$		0			0
$435$	156674.			0.827977
$436$		0			0
$437$	5803.25	−	149561.i	0.0303884	−	0.783171i
$438$		0			0
$439$		−	249410.i		−	1.29415i	−0.762427	−	0.647074i	$-0.775992\pi$
$439$		−	249410.i		−	1.29415i	0.762427	−	0.647074i	$-0.224008\pi$
$440$		0			0
$441$	−43071.3			−0.221468
$442$		0			0
$443$	−43050.1			−0.219365			−0.109682	−	0.993967i	$-0.534983\pi$
$443$	−43050.1			−0.219365			−0.109682	+	0.993967i	$0.534983\pi$
$444$		0			0
$445$			340296.i			1.71845i
$446$		0			0
$447$			10560.6i			0.0528533i
$448$		0			0
$449$			19223.8i			0.0953556i	0.998863	+	0.0476778i	$0.0151821\pi$
$449$			19223.8i			0.0953556i	−0.998863	+	0.0476778i	$0.984818\pi$
$450$		0			0
$451$			407709.i			2.00446i
$452$		0			0
$453$	−227341.			−1.10785
$454$		0			0
$455$		−	149798.i		−	0.723572i
$456$		0			0
$457$	38484.7			0.184271			0.0921353	−	0.995747i	$-0.470631\pi$
$457$	38484.7			0.184271			0.0921353	+	0.995747i	$0.470631\pi$
$458$		0			0
$459$			178767.i			0.848521i
$460$		0			0
$461$	−203940.			−0.959623			−0.479812	−	0.877371i	$-0.659295\pi$
$461$	−203940.			−0.959623			−0.479812	+	0.877371i	$0.659295\pi$
$462$		0			0
$463$	203200.			0.947900			0.473950	−	0.880552i	$-0.342828\pi$
$463$	203200.			0.947900			0.473950	+	0.880552i	$0.342828\pi$
$464$		0			0
$465$	123626.			0.571745
$466$		0			0
$467$	121674.			0.557912			0.278956	−	0.960304i	$-0.410012\pi$
$467$	121674.			0.557912			0.278956	+	0.960304i	$0.410012\pi$
$468$		0			0
$469$			71773.8i			0.326302i
$470$		0			0
$471$		−	93128.2i		−	0.419797i
$472$		0			0
$473$	−68108.0			−0.304422
$474$		0			0
$475$	−6584.37	+	169693.i	−0.0291828	+	0.752100i
$476$		0			0
$477$		−	15567.7i		−	0.0684207i
$478$		0			0
$479$	208037.			0.906711			0.453355	−	0.891330i	$-0.350227\pi$
$479$	208037.			0.906711			0.453355	+	0.891330i	$0.350227\pi$
$480$		0			0
$481$	−29445.9			−0.127272
$482$		0			0
$483$			52020.7i			0.222988i
$484$		0			0
$485$		−	512253.i		−	2.17771i
$486$		0			0
$487$			47417.7i			0.199932i	0.994991	+	0.0999660i	$0.0318734\pi$
$487$			47417.7i			0.199932i	−0.994991	+	0.0999660i	$0.968127\pi$
$488$		0			0
$489$		−	107836.i		−	0.450970i
$490$		0			0
$491$	218823.			0.907676			0.453838	−	0.891084i	$-0.350054\pi$
$491$	218823.			0.907676			0.453838	+	0.891084i	$0.350054\pi$
$492$		0			0
$493$			137447.i			0.565513i
$494$		0			0
$495$	−143447.			−0.585438
$496$		0			0
$497$		−	122780.i		−	0.497065i
$498$		0			0
$499$	−147213.			−0.591214			−0.295607	−	0.955310i	$-0.595522\pi$
$499$	−147213.			−0.591214			−0.295607	+	0.955310i	$0.595522\pi$
$500$		0			0
$501$	−80124.1			−0.319218
$502$		0			0
$503$	−189477.			−0.748895			−0.374448	−	0.927248i	$-0.622168\pi$
$503$	−189477.			−0.748895			−0.374448	+	0.927248i	$0.622168\pi$
$504$		0			0
$505$	−346343.			−1.35807
$506$		0			0
$507$		−	395486.i		−	1.53856i
$508$		0			0
$509$		−	117829.i		−	0.454797i	−0.973802	−	0.227398i	$-0.926978\pi$
$509$		−	117829.i		−	0.454797i	0.973802	−	0.227398i	$-0.0730220\pi$
$510$		0			0
$511$	−135004.			−0.517017
$512$		0			0
$513$	−284605.	−	11043.2i	−1.08145	−	0.0419624i
$514$		0			0
$515$		−	106723.i		−	0.402385i
$516$		0			0
$517$	102261.			0.382586
$518$		0			0
$519$	226318.			0.840204
$520$		0			0
$521$		−	254449.i		−	0.937402i	−0.883357	−	0.468701i	$-0.844722\pi$
$521$		−	254449.i		−	0.937402i	0.883357	−	0.468701i	$-0.155278\pi$
$522$		0			0
$523$			33047.5i			0.120819i	0.998174	+	0.0604094i	$0.0192406\pi$
$523$			33047.5i			0.120819i	−0.998174	+	0.0604094i	$0.980759\pi$
$524$		0			0
$525$		−	59022.7i		−	0.214141i
$526$		0			0
$527$			108455.i			0.390505i
$528$		0			0
$529$	−107940.			−0.385720
$530$		0			0
$531$		−	11402.9i		−	0.0404414i
$532$		0			0
$533$	532345.			1.87387
$534$		0			0
$535$		−	256202.i		−	0.895107i
$536$		0			0
$537$	198833.			0.689508
$538$		0			0
$539$	−461903.			−1.58991
$540$		0			0
$541$	−363239.			−1.24108			−0.620538	−	0.784176i	$-0.713086\pi$
$541$	−363239.			−1.24108			−0.620538	+	0.784176i	$0.713086\pi$
$542$		0			0
$543$	−243146.			−0.824647
$544$		0			0
$545$		−	417456.i		−	1.40546i
$546$		0			0
$547$		−	899.921i		−	0.00300767i	−0.999999	−	0.00150383i	$-0.999521\pi$
$547$		−	899.921i		−	0.00300767i	0.999999	−	0.00150383i	$-0.000478685\pi$
$548$		0			0
$549$	96486.5			0.320127
$550$		0			0
$551$	−218822.	−	8490.69i	−0.720756	−	0.0279666i
$552$		0			0
$553$		−	156068.i		−	0.510343i
$554$		0			0
$555$	−27016.9			−0.0877102
$556$		0			0
$557$	340311.			1.09690			0.548448	−	0.836184i	$-0.315219\pi$
$557$	340311.			1.09690			0.548448	+	0.836184i	$0.315219\pi$
$558$		0			0
$559$			88928.3i			0.284588i
$560$		0			0
$561$			381202.i			1.21124i
$562$		0			0
$563$		−	631686.i		−	1.99289i	−0.0842180	−	0.996447i	$-0.526839\pi$
$563$		−	631686.i		−	1.99289i	0.0842180	−	0.996447i	$-0.473161\pi$
$564$		0			0
$565$			29816.3i			0.0934023i
$566$		0			0
$567$	72810.3			0.226478
$568$		0			0
$569$			5519.19i			0.0170471i	0.999964	+	0.00852356i	$0.00271317\pi$
$569$			5519.19i			0.0170471i	−0.999964	+	0.00852356i	$0.997287\pi$
$570$		0			0
$571$	478673.			1.46814			0.734068	−	0.679076i	$-0.237619\pi$
$571$	478673.			1.46814			0.734068	+	0.679076i	$0.237619\pi$
$572$		0			0
$573$		−	139597.i		−	0.425175i
$574$		0			0
$575$	−195039.			−0.589909
$576$		0			0
$577$	65914.9			0.197985			0.0989925	−	0.995088i	$-0.468438\pi$
$577$	65914.9			0.197985			0.0989925	+	0.995088i	$0.468438\pi$
$578$		0			0
$579$	−6119.56			−0.0182542
$580$		0			0
$581$	−170260.			−0.504383
$582$		0			0
$583$		−	166950.i		−	0.491191i
$584$		0			0
$585$			187298.i			0.547295i
$586$		0			0
$587$	−289436.			−0.839994			−0.419997	−	0.907526i	$-0.637969\pi$
$587$	−289436.			−0.839994			−0.419997	+	0.907526i	$0.637969\pi$
$588$		0			0
$589$	−172664.	−	6699.69i	−0.497705	−	0.0193119i
$590$		0			0
$591$			329878.i			0.944449i
$592$		0			0
$593$	388285.			1.10418			0.552092	−	0.833783i	$-0.313830\pi$
$593$	388285.			1.10418			0.552092	+	0.833783i	$0.313830\pi$
$594$		0			0
$595$	120575.			0.340582
$596$		0			0
$597$			119877.i			0.336346i
$598$		0			0
$599$		−	601872.i		−	1.67745i	−0.544553	−	0.838727i	$-0.683301\pi$
$599$		−	601872.i		−	1.67745i	0.544553	−	0.838727i	$-0.316699\pi$
$600$		0			0
$601$		−	176095.i		−	0.487527i	−0.969835	−	0.243764i	$-0.921618\pi$
$601$		−	176095.i		−	0.487527i	0.969835	−	0.243764i	$-0.0783821\pi$
$602$		0			0
$603$		−	89741.7i		−	0.246808i
$604$		0			0
$605$	−1.05377e6			−2.87896
$606$		0			0
$607$			607939.i			1.64999i	0.565137	+	0.824997i	$0.308823\pi$
$607$			607939.i			1.64999i	−0.565137	+	0.824997i	$0.691177\pi$
$608$		0			0
$609$	76111.1			0.205217
$610$		0			0
$611$		−	133522.i		−	0.357660i
$612$		0			0
$613$	194933.			0.518757			0.259379	−	0.965776i	$-0.416482\pi$
$613$	194933.			0.518757			0.259379	+	0.965776i	$0.416482\pi$
$614$		0			0
$615$	488432.			1.29138
$616$		0			0
$617$	148271.			0.389480			0.194740	−	0.980855i	$-0.437614\pi$
$617$	148271.			0.389480			0.194740	+	0.980855i	$0.437614\pi$
$618$		0			0
$619$	572353.			1.49377			0.746884	−	0.664955i	$-0.231549\pi$
$619$	572353.			1.49377			0.746884	+	0.664955i	$0.231549\pi$
$620$		0			0
$621$		−	327115.i		−	0.848238i
$622$		0			0
$623$			165313.i			0.425923i
$624$		0			0
$625$	−463344.			−1.18616
$626$		0			0
$627$	−606889.	−	23548.4i	−1.54374	−	0.0598999i
$628$		0			0
$629$		−	23701.5i		−	0.0599066i
$630$		0			0
$631$	−236439.			−0.593829			−0.296914	−	0.954904i	$-0.595958\pi$
$631$	−236439.			−0.593829			−0.296914	+	0.954904i	$0.595958\pi$
$632$		0			0
$633$	281216.			0.701831
$634$		0			0
$635$		−	552329.i		−	1.36978i
$636$		0			0
$637$			603105.i			1.48633i
$638$		0			0
$639$			153516.i			0.375970i
$640$		0			0
$641$			247053.i			0.601275i	0.953738	+	0.300638i	$0.0971995\pi$
$641$			247053.i			0.601275i	−0.953738	+	0.300638i	$0.902801\pi$
$642$		0			0
$643$	−543389.			−1.31428			−0.657142	−	0.753767i	$-0.728235\pi$
$643$	−543389.			−1.31428			−0.657142	+	0.753767i	$0.728235\pi$
$644$		0			0
$645$			81592.8i			0.196125i
$646$		0			0
$647$	263444.			0.629332			0.314666	−	0.949202i	$-0.398107\pi$
$647$	263444.			0.629332			0.314666	+	0.949202i	$0.398107\pi$
$648$		0			0
$649$		−	122286.i		−	0.290327i
$650$		0			0
$651$	60056.4			0.141709
$652$		0			0
$653$	−261207.			−0.612574			−0.306287	−	0.951939i	$-0.599087\pi$
$653$	−261207.			−0.612574			−0.306287	+	0.951939i	$0.599087\pi$
$654$		0			0
$655$	309962.			0.722480
$656$		0			0
$657$	168801.			0.391061
$658$		0			0
$659$			532996.i			1.22731i	0.789576	+	0.613653i	$0.210301\pi$
$659$			532996.i			1.22731i	−0.789576	+	0.613653i	$0.789699\pi$
$660$		0			0
$661$			495346.i			1.13372i	0.823814	+	0.566860i	$0.191842\pi$
$661$			495346.i			1.13372i	−0.823814	+	0.566860i	$0.808158\pi$
$662$		0			0
$663$	497734.			1.13232
$664$		0			0
$665$	−7448.39	+	191960.i	−0.0168430	+	0.434078i
$666$		0			0
$667$		−	251507.i		−	0.565325i
$668$		0			0
$669$	344136.			0.768913
$670$		0			0
$671$	1.03474e6			2.29818
$672$		0			0
$673$		−	425648.i		−	0.939768i	−0.882728	−	0.469884i	$-0.844296\pi$
$673$		−	425648.i		−	0.939768i	0.882728	−	0.469884i	$-0.155704\pi$
$674$		0			0
$675$			371146.i			0.814586i
$676$		0			0
$677$			423249.i			0.923461i	0.887020	+	0.461730i	$0.152771\pi$
$677$			423249.i			0.923461i	−0.887020	+	0.461730i	$0.847229\pi$
$678$		0			0
$679$		−	248849.i		−	0.539754i
$680$		0			0
$681$	272085.			0.586692
$682$		0			0
$683$			310900.i			0.666467i	0.942844	+	0.333233i	$0.108140\pi$
$683$			310900.i			0.666467i	−0.942844	+	0.333233i	$0.891860\pi$
$684$		0			0
$685$	1.19219e6			2.54077
$686$		0			0
$687$			203561.i			0.431302i
$688$		0			0
$689$	−217986.			−0.459188
$690$		0			0
$691$	−314072.			−0.657769			−0.328885	−	0.944370i	$-0.606673\pi$
$691$	−314072.			−0.657769			−0.328885	+	0.944370i	$0.606673\pi$
$692$		0			0
$693$	−69685.5			−0.145103
$694$		0			0
$695$	−343275.			−0.710677
$696$		0			0
$697$			428494.i			0.882021i
$698$		0			0
$699$			588915.i			1.20531i
$700$		0			0
$701$	607472.			1.23620			0.618102	−	0.786098i	$-0.287902\pi$
$701$	607472.			1.23620			0.618102	+	0.786098i	$0.287902\pi$
$702$		0			0
$703$	37733.8	+	1464.14i	0.0763519	+	0.00296259i
$704$		0			0
$705$		−	122508.i		−	0.246482i
$706$		0			0
$707$	−168251.			−0.336604
$708$		0			0
$709$	−293984.			−0.584833			−0.292416	−	0.956291i	$-0.594459\pi$
$709$	−293984.			−0.584833			−0.292416	+	0.956291i	$0.594459\pi$
$710$		0			0
$711$			195138.i			0.386013i
$712$		0			0
$713$		−	198455.i		−	0.390375i
$714$		0			0
$715$			2.00861e6i			3.92902i
$716$		0			0
$717$			334712.i			0.651078i
$718$		0			0
$719$	−911462.			−1.76312			−0.881558	−	0.472075i	$-0.843505\pi$
$719$	−911462.			−1.76312			−0.881558	+	0.472075i	$0.843505\pi$
$720$		0			0
$721$		−	51845.1i		−	0.0997326i
$722$		0			0
$723$	−121965.			−0.233324
$724$		0			0
$725$			285360.i			0.542896i
$726$		0			0
$727$	174790.			0.330710			0.165355	−	0.986234i	$-0.447123\pi$
$727$	174790.			0.330710			0.165355	+	0.986234i	$0.447123\pi$
$728$		0			0
$729$	−589743.			−1.10971
$730$		0			0
$731$	−71580.0			−0.133954
$732$		0			0
$733$	−401715.			−0.747671			−0.373835	−	0.927495i	$-0.621958\pi$
$733$	−401715.			−0.747671			−0.373835	+	0.927495i	$0.621958\pi$
$734$		0			0
$735$			553356.i			1.02431i
$736$		0			0
$737$		−	962404.i		−	1.77183i
$738$		0			0
$739$	302531.			0.553963			0.276982	−	0.960875i	$-0.410666\pi$
$739$	302531.			0.553963			0.276982	+	0.960875i	$0.410666\pi$
$740$		0			0
$741$	−30747.0	+	792413.i	−0.0559973	+	1.44316i
$742$		0			0
$743$		−	909052.i		−	1.64669i	−0.567543	−	0.823344i	$-0.692106\pi$
$743$		−	909052.i		−	1.64669i	0.567543	−	0.823344i	$-0.307894\pi$
$744$		0			0
$745$	−44789.9			−0.0806989
$746$		0			0
$747$	212883.			0.381505
$748$		0			0
$749$		−	124461.i		−	0.221855i
$750$		0			0
$751$			370317.i			0.656589i	0.944575	+	0.328295i	$0.106474\pi$
$751$			370317.i			0.656589i	−0.944575	+	0.328295i	$0.893526\pi$
$752$		0			0
$753$		−	600870.i		−	1.05972i
$754$		0			0
$755$		−	964208.i		−	1.69152i
$756$		0			0
$757$	154581.			0.269751			0.134876	−	0.990863i	$-0.456937\pi$
$757$	154581.			0.269751			0.134876	+	0.990863i	$0.456937\pi$
$758$		0			0
$759$		−	697538.i		−	1.21083i
$760$		0			0
$761$	−722058.			−1.24682			−0.623409	−	0.781896i	$-0.714253\pi$
$761$	−722058.			−1.24682			−0.623409	+	0.781896i	$0.714253\pi$
$762$		0			0
$763$		−	202797.i		−	0.348347i
$764$		0			0
$765$	−150760.			−0.257610
$766$		0			0
$767$	−159669.			−0.271412
$768$		0			0
$769$	496131.			0.838965			0.419482	−	0.907763i	$-0.362212\pi$
$769$	496131.			0.838965			0.419482	+	0.907763i	$0.362212\pi$
$770$		0			0
$771$	−69479.2			−0.116882
$772$		0			0
$773$			317854.i			0.531948i	0.963980	+	0.265974i	$0.0856935\pi$
$773$			317854.i			0.531948i	−0.963980	+	0.265974i	$0.914306\pi$
$774$		0			0
$775$			225167.i			0.374888i
$776$		0			0
$777$	−13124.6			−0.0217393
$778$		0			0
$779$	−682180.	−	26469.8i	−1.12415	−	0.0436190i
$780$		0			0
$781$			1.64633e6i			2.69908i
$782$		0			0
$783$	−478601.			−0.780638
$784$		0			0
$785$	394979.			0.640966
$786$		0			0
$787$		−	206097.i		−	0.332752i	−0.986062	−	0.166376i	$-0.946793\pi$
$787$		−	206097.i		−	0.332752i	0.986062	−	0.166376i	$-0.0532066\pi$
$788$		0			0
$789$			618302.i			0.993223i
$790$		0			0
$791$			14484.6i			0.0231501i
$792$		0			0
$793$		−	1.35105e6i		−	2.14845i
$794$		0			0
$795$	−200005.			−0.316451
$796$		0			0
$797$		−	406943.i		−	0.640644i	−0.947309	−	0.320322i	$-0.896209\pi$
$797$		−	406943.i		−	0.640644i	0.947309	−	0.320322i	$-0.103791\pi$
$798$		0			0
$799$	107474.			0.168349
$800$		0			0
$801$		−	206698.i		−	0.322160i
$802$		0			0
$803$	1.81025e6			2.80742
$804$		0			0
$805$	−220632.			−0.340469
$806$		0			0
$807$	1.03646e6			1.59149
$808$		0			0
$809$	−301170.			−0.460166			−0.230083	−	0.973171i	$-0.573900\pi$
$809$	−301170.			−0.460166			−0.230083	+	0.973171i	$0.573900\pi$
$810$		0			0
$811$			454953.i			0.691711i	0.938288	+	0.345855i	$0.112411\pi$
$811$			454953.i			0.691711i	−0.938288	+	0.345855i	$0.887589\pi$
$812$		0			0
$813$		−	569789.i		−	0.862052i
$814$		0			0
$815$	457360.			0.688562
$816$		0			0
$817$	4421.79	−	113958.i	0.00662451	−	0.170727i
$818$		0			0
$819$			90988.1i			0.135649i
$820$		0			0
$821$	−833823.			−1.23705			−0.618526	−	0.785764i	$-0.712270\pi$
$821$	−833823.			−1.23705			−0.618526	+	0.785764i	$0.712270\pi$
$822$		0			0
$823$	−903550.			−1.33399			−0.666995	−	0.745063i	$-0.732420\pi$
$823$	−903550.			−1.33399			−0.666995	+	0.745063i	$0.732420\pi$
$824$		0			0
$825$			791427.i			1.16279i
$826$		0			0
$827$		−	273012.i		−	0.399182i	−0.979879	−	0.199591i	$-0.936039\pi$
$827$		−	273012.i		−	0.399182i	0.979879	−	0.199591i	$-0.0639613\pi$
$828$		0			0
$829$			29206.8i			0.0424986i	0.999774	+	0.0212493i	$0.00676438\pi$
$829$			29206.8i			0.0424986i	−0.999774	+	0.0212493i	$0.993236\pi$
$830$		0			0
$831$			517232.i			0.749002i
$832$		0			0
$833$	−485450.			−0.699607
$834$		0			0
$835$		−	339826.i		−	0.487397i
$836$		0			0
$837$	−377646.			−0.539056
$838$		0			0
$839$			563481.i			0.800488i	0.916409	+	0.400244i	$0.131075\pi$
$839$			563481.i			0.800488i	−0.916409	+	0.400244i	$0.868925\pi$
$840$		0			0
$841$	339303.			0.479729
$842$		0			0
$843$	−880720.			−1.23932
$844$		0			0
$845$	1.67735e6			2.34915
$846$		0			0
$847$	−511915.			−0.713561
$848$		0			0
$849$		−	630272.i		−	0.874405i
$850$		0			0
$851$			43369.9i			0.0598866i
$852$		0			0
$853$	−852435.			−1.17156			−0.585778	−	0.810472i	$-0.699211\pi$
$853$	−852435.			−1.17156			−0.585778	+	0.810472i	$0.699211\pi$
$854$		0			0
$855$	9313.04	−	240016.i	0.0127397	−	0.328328i
$856$		0			0
$857$			103865.i			0.141419i	0.997497	+	0.0707097i	$0.0225264\pi$
$857$			103865.i			0.141419i	−0.997497	+	0.0707097i	$0.977474\pi$
$858$		0			0
$859$	74101.6			0.100425			0.0502125	−	0.998739i	$-0.484010\pi$
$859$	74101.6			0.100425			0.0502125	+	0.998739i	$0.484010\pi$
$860$		0			0
$861$	237277.			0.320073
$862$		0			0
$863$		−	434232.i		−	0.583042i	−0.956564	−	0.291521i	$-0.905839\pi$
$863$		−	434232.i		−	0.583042i	0.956564	−	0.291521i	$-0.0941613\pi$
$864$		0			0
$865$			959870.i			1.28286i
$866$		0			0
$867$		−	251132.i		−	0.334091i
$868$		0			0
$869$			2.09269e6i			2.77118i
$870$		0			0
$871$	−1.25661e6			−1.65639
$872$		0			0
$873$			311146.i			0.408259i
$874$		0			0
$875$	−82261.1			−0.107443
$876$		0			0
$877$			493156.i			0.641188i	0.947217	+	0.320594i	$0.103883\pi$
$877$			493156.i			0.641188i	−0.947217	+	0.320594i	$0.896117\pi$
$878$		0			0
$879$	1.04308e6			1.35002
$880$		0			0
$881$	−699253.			−0.900913			−0.450456	−	0.892798i	$-0.648739\pi$
$881$	−699253.			−0.900913			−0.450456	+	0.892798i	$0.648739\pi$
$882$		0			0
$883$	−657183.			−0.842878			−0.421439	−	0.906857i	$-0.638475\pi$
$883$	−657183.			−0.842878			−0.421439	+	0.906857i	$0.638475\pi$
$884$		0			0
$885$	−146498.			−0.187044
$886$		0			0
$887$			123664.i			0.157179i	0.996907	+	0.0785895i	$0.0250416\pi$
$887$			123664.i			0.157179i	−0.996907	+	0.0785895i	$0.974958\pi$
$888$		0			0
$889$		−	268317.i		−	0.339504i
$890$		0			0
$891$	−976303.			−1.22978
$892$		0			0
$893$	−6639.12	+	171103.i	−0.00832545	+	0.214563i
$894$		0			0
$895$			843298.i			1.05277i
$896$		0			0
$897$	−910772.			−1.13194
$898$		0			0
$899$	−290358.			−0.359264
$900$		0			0
$901$		−	175461.i		−	0.216138i
$902$		0			0
$903$			39637.2i			0.0486102i
$904$		0			0
$905$		−	1.03124e6i		−	1.25911i
$906$		0			0
$907$		−	30174.9i		−	0.0366801i	−0.999832	−	0.0183401i	$-0.994162\pi$
$907$		−	30174.9i		−	0.0366801i	0.999832	−	0.0183401i	$-0.00583815\pi$
$908$		0			0
$909$	210371.			0.254600
$910$		0			0
$911$			198609.i			0.239311i	0.992815	+	0.119655i	$0.0381790\pi$
$911$			198609.i			0.239311i	−0.992815	+	0.119655i	$0.961821\pi$
$912$		0			0
$913$	2.28299e6			2.73882
$914$		0			0
$915$		−	1.23960e6i		−	1.48061i
$916$		0			0
$917$	150577.			0.179069
$918$		0			0
$919$	1.20577e6			1.42769			0.713845	−	0.700304i	$-0.246952\pi$
$919$	1.20577e6			1.42769			0.713845	+	0.700304i	$0.246952\pi$
$920$		0			0
$921$	787052.			0.927863
$922$		0			0
$923$	2.14961e6			2.52323
$924$		0			0
$925$		−	49207.6i		−	0.0575107i
$926$		0			0
$927$			64824.1i			0.0754357i
$928$		0			0
$929$	1.04643e6			1.21249			0.606246	−	0.795277i	$-0.292675\pi$
$929$	1.04643e6			1.21249			0.606246	+	0.795277i	$0.292675\pi$
$930$		0			0
$931$	29988.2	−	772856.i	0.0345980	−	0.891660i
$932$		0			0
$933$		−	948380.i		−	1.08948i
$934$		0			0
$935$	−1.61677e6			−1.84937
$936$		0			0
$937$	251416.			0.286361			0.143181	−	0.989697i	$-0.454267\pi$
$937$	251416.			0.286361			0.143181	+	0.989697i	$0.454267\pi$
$938$		0			0
$939$			1.48216e6i			1.68098i
$940$		0			0
$941$		−	1.26323e6i		−	1.42660i	−0.700858	−	0.713301i	$-0.747199\pi$
$941$		−	1.26323e6i		−	1.42660i	0.700858	−	0.713301i	$-0.252801\pi$
$942$		0			0
$943$		−	784074.i		−	0.881726i
$944$		0			0
$945$			419848.i			0.470142i
$946$		0			0
$947$	1.04458e6			1.16477			0.582386	−	0.812913i	$-0.302119\pi$
$947$	1.04458e6			1.16477			0.582386	+	0.812913i	$0.302119\pi$
$948$		0			0
$949$		−	2.36363e6i		−	2.62451i
$950$		0			0
$951$	−525466.			−0.581010
$952$		0			0
$953$		−	928965.i		−	1.02285i	−0.859327	−	0.511427i	$-0.829117\pi$
$953$		−	928965.i		−	1.02285i	0.859327	−	0.511427i	$-0.170883\pi$
$954$		0			0
$955$	592066.			0.649178
$956$		0			0
$957$	−1.02056e6			−1.11434
$958$		0			0
$959$	579159.			0.629739
$960$		0			0
$961$	694411.			0.751916
$962$		0			0
$963$			155619.i			0.167807i
$964$		0			0
$965$		−	25954.5i		−	0.0278714i
$966$		0			0
$967$	72252.5			0.0772680			0.0386340	−	0.999253i	$-0.487699\pi$
$967$	72252.5			0.0772680			0.0386340	+	0.999253i	$0.487699\pi$
$968$		0			0
$969$	−637827.	−	24748.8i	−0.679290	−	0.0263577i
$970$		0			0
$971$			217697.i			0.230895i	0.993314	+	0.115447i	$0.0368302\pi$
$971$			217697.i			0.230895i	−0.993314	+	0.115447i	$0.963170\pi$
$972$		0			0
$973$	−166761.			−0.176144
$974$		0			0
$975$	1.03336e6			1.08704
$976$		0			0
$977$		−	1.19511e6i		−	1.25204i	−0.779805	−	0.626022i	$-0.784682\pi$
$977$		−	1.19511e6i		−	1.25204i	0.779805	−	0.626022i	$-0.215318\pi$
$978$		0			0
$979$		−	2.21666e6i		−	2.31278i
$980$		0			0
$981$			253566.i			0.263483i
$982$		0			0
$983$		−	743610.i		−	0.769552i	−0.923010	−	0.384776i	$-0.874279\pi$
$983$		−	743610.i		−	0.769552i	0.923010	−	0.384776i	$-0.125721\pi$
$984$		0			0
$985$	−1.39909e6			−1.44203
$986$		0			0
$987$		−	59513.5i		−	0.0610915i
$988$		0			0
$989$	130980.			0.133910
$990$		0			0
$991$		−	213907.i		−	0.217810i	−0.994052	−	0.108905i	$-0.965266\pi$
$991$		−	213907.i		−	0.217810i	0.994052	−	0.108905i	$-0.0347345\pi$
$992$		0			0
$993$	−1.20418e6			−1.22121
$994$		0			0
$995$	−508426.			−0.513549
$996$		0			0
$997$	563076.			0.566470			0.283235	−	0.959051i	$-0.408592\pi$
$997$	563076.			0.566470			0.283235	+	0.959051i	$0.408592\pi$
$998$		0			0
$999$	82530.1			0.0826954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.e.113.6		8
4.3	odd	2		38.5.b.a.37.2	✓	8
12.11	even	2		342.5.d.a.37.8		8
19.18	odd	2	inner	304.5.e.e.113.3		8
76.75	even	2		38.5.b.a.37.7	yes	8
228.227	odd	2		342.5.d.a.37.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.b.a.37.2	✓	8	4.3	odd	2
38.5.b.a.37.7	yes	8	76.75	even	2
304.5.e.e.113.3		8	19.18	odd	2	inner
304.5.e.e.113.6		8	1.1	even	1	trivial
342.5.d.a.37.4		8	228.227	odd	2
342.5.d.a.37.8		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.80363i q^{3} -33.0971 q^{5} -16.0783 q^{7} +20.1034 q^{9} +O(q^{10})\)	\(q+7.80363i q^{3} -33.0971 q^{5} -16.0783 q^{7} +20.1034 q^{9} +215.592 q^{11} -281.497i q^{13} -258.277i q^{15} +226.582 q^{17} +(-13.9969 + 360.729i) q^{19} -125.469i q^{21} -414.609 q^{23} +470.416 q^{25} +788.973i q^{27} +606.612i q^{29} +478.655i q^{31} +1682.40i q^{33} +532.145 q^{35} -104.604i q^{37} +2196.70 q^{39} +1891.12i q^{41} -315.912 q^{43} -665.364 q^{45} +474.327 q^{47} -2142.49 q^{49} +1768.16i q^{51} -774.381i q^{53} -7135.46 q^{55} +(-2814.99 - 109.227i) q^{57} -567.212i q^{59} +4799.51 q^{61} -323.229 q^{63} +9316.74i q^{65} -4464.01i q^{67} -3235.45i q^{69} +7636.34i q^{71} +8396.65 q^{73} +3670.95i q^{75} -3466.35 q^{77} +9706.71i q^{79} -4528.48 q^{81} +10589.4 q^{83} -7499.21 q^{85} -4733.77 q^{87} -10281.7i q^{89} +4526.01i q^{91} -3735.24 q^{93} +(463.257 - 11939.1i) q^{95} +15477.3i q^{97} +4334.13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) \) 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9	\( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) \) 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9 + 6 * q^11 + 510 * q^17 + 12 * q^19 + 396 * q^23 + 3458 * q^25 - 1002 * q^35 + 6588 * q^39 + 8654 * q^43 - 10334 * q^45 - 3210 * q^47 + 9222 * q^49 - 17146 * q^55 - 14076 * q^57 + 1314 * q^61 - 29938 * q^63 + 23398 * q^73 - 44622 * q^77 - 20368 * q^81 + 10440 * q^83 + 21274 * q^85 + 14316 * q^87 + 19416 * q^93 + 34686 * q^95 + 56798 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more