LMFDB - Embedded newform 720.6.f.o.289.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.f (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:	$115.476350265$
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} + 142x^{6} + 5761x^{4} + 52020x^{2} + 129600 $ x^8 + 142x^6 + 5761x^4 + 52020*x^2 + 129600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.8
Root			$2.61388i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.6.f.o.289.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(45.4388 + 32.5625i) q^{5} +242.714i q^{7} +142.542 q^{11} +1080.72i q^{13} +1329.42i q^{17} +1551.52 q^{19} +1087.52i q^{23} +(1004.36 + 2959.20i) q^{25} +4374.67 q^{29} -2623.69 q^{31} +(-7903.37 + 11028.6i) q^{35} +2081.24i q^{37} +15998.1 q^{41} -4083.17i q^{43} +4819.20i q^{47} -42102.9 q^{49} +20686.4i q^{53} +(6476.94 + 4641.53i) q^{55} +12562.3 q^{59} +41811.0 q^{61} +(-35190.9 + 49106.5i) q^{65} -60479.5i q^{67} -27123.5 q^{71} -57295.5i q^{73} +34596.9i q^{77} -27152.5 q^{79} -74516.5i q^{83} +(-43289.1 + 60407.0i) q^{85} +80563.6 q^{89} -262305. q^{91} +(70499.4 + 50521.6i) q^{95} -88537.3i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 66 q^{5} + 684 q^{11} + 3040 q^{19} - 564 q^{25} + 8004 q^{29} - 6024 q^{31} - 4476 q^{35} + 37800 q^{41} - 34368 q^{49} - 39968 q^{55} - 86028 q^{59} - 38960 q^{61} - 84204 q^{65} + 162072 q^{71}+ \cdots + 42048 q^{95}+O(q^{100}) $ 8 * q + 66 * q^5 + 684 * q^11 + 3040 * q^19 - 564 * q^25 + 8004 * q^29 - 6024 * q^31 - 4476 * q^35 + 37800 * q^41 - 34368 * q^49 - 39968 * q^55 - 86028 * q^59 - 38960 * q^61 - 84204 * q^65 + 162072 * q^71 + 169976 * q^79 + 240588 * q^85 + 452976 * q^89 - 610456 * q^91 + 42048 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	45.4388	+	32.5625i	0.812833	+	0.582496i
$5$	45.4388	+	32.5625i	0.812833	+	0.582496i
$6$		0			0
$7$			242.714i			1.87219i	0.351752	+	0.936093i	$0.385586\pi$
$7$			242.714i			1.87219i	−0.351752	+	0.936093i	$0.614414\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	142.542			0.355190			0.177595	−	0.984104i	$-0.443168\pi$
$11$	142.542			0.355190			0.177595	+	0.984104i	$0.443168\pi$
$12$		0			0
$13$			1080.72i			1.77359i	0.462160	+	0.886797i	$0.347075\pi$
$13$			1080.72i			1.77359i	−0.462160	+	0.886797i	$0.652925\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1329.42i			1.11568i	0.829950	+	0.557838i	$0.188369\pi$
$17$			1329.42i			1.11568i	−0.829950	+	0.557838i	$0.811631\pi$
$18$		0			0
$19$	1551.52			0.985995			0.492998	−	0.870031i	$-0.335901\pi$
$19$	1551.52			0.985995			0.492998	+	0.870031i	$0.335901\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1087.52i			0.428666i	0.976761	+	0.214333i	$0.0687577\pi$
$23$			1087.52i			0.428666i	−0.976761	+	0.214333i	$0.931242\pi$
$24$		0			0
$25$	1004.36	+	2959.20i	0.321396	+	0.946945i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4374.67			0.965940			0.482970	−	0.875637i	$-0.339558\pi$
$29$	4374.67			0.965940			0.482970	+	0.875637i	$0.339558\pi$
$30$		0			0
$31$	−2623.69			−0.490352			−0.245176	−	0.969479i	$-0.578846\pi$
$31$	−2623.69			−0.490352			−0.245176	+	0.969479i	$0.578846\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7903.37	+	11028.6i	−1.09054	+	1.52178i
$36$		0			0
$37$			2081.24i			0.249930i	0.992161	+	0.124965i	$0.0398818\pi$
$37$			2081.24i			0.249930i	−0.992161	+	0.124965i	$0.960118\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	15998.1			1.48631			0.743155	−	0.669120i	$-0.233329\pi$
$41$	15998.1			1.48631			0.743155	+	0.669120i	$0.233329\pi$
$42$		0			0
$43$		−	4083.17i		−	0.336764i	−0.985722	−	0.168382i	$-0.946146\pi$
$43$		−	4083.17i		−	0.336764i	0.985722	−	0.168382i	$-0.0538543\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4819.20i			0.318222i	0.987261	+	0.159111i	$0.0508627\pi$
$47$			4819.20i			0.318222i	−0.987261	+	0.159111i	$0.949137\pi$
$48$		0			0
$49$	−42102.9			−2.50508
$50$		0			0
$51$		0			0
$52$		0			0
$53$			20686.4i			1.01157i	0.862659	+	0.505785i	$0.168797\pi$
$53$			20686.4i			1.01157i	−0.862659	+	0.505785i	$0.831203\pi$
$54$		0			0
$55$	6476.94	+	4641.53i	0.288711	+	0.206897i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12562.3			0.469827			0.234913	−	0.972016i	$-0.424519\pi$
$59$	12562.3			0.469827			0.234913	+	0.972016i	$0.424519\pi$
$60$		0			0
$61$	41811.0			1.43869			0.719343	−	0.694655i	$-0.244443\pi$
$61$	41811.0			1.43869			0.719343	+	0.694655i	$0.244443\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−35190.9	+	49106.5i	−1.03311	+	1.44164i
$66$		0			0
$67$		−	60479.5i		−	1.64597i	−0.568065	−	0.822984i	$-0.692308\pi$
$67$		−	60479.5i		−	1.64597i	0.568065	−	0.822984i	$-0.307692\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−27123.5			−0.638557			−0.319278	−	0.947661i	$-0.603441\pi$
$71$	−27123.5			−0.638557			−0.319278	+	0.947661i	$0.603441\pi$
$72$		0			0
$73$		−	57295.5i		−	1.25838i	−0.777250	−	0.629192i	$-0.783386\pi$
$73$		−	57295.5i		−	1.25838i	0.777250	−	0.629192i	$-0.216614\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			34596.9i			0.664983i
$78$		0			0
$79$	−27152.5			−0.489488			−0.244744	−	0.969588i	$-0.578704\pi$
$79$	−27152.5			−0.489488			−0.244744	+	0.969588i	$0.578704\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	74516.5i		−	1.18729i	−0.804727	−	0.593645i	$-0.797688\pi$
$83$		−	74516.5i		−	1.18729i	0.804727	−	0.593645i	$-0.202312\pi$
$84$		0			0
$85$	−43289.1	+	60407.0i	−0.649877	+	0.906859i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	80563.6			1.07811			0.539056	−	0.842270i	$-0.318781\pi$
$89$	80563.6			1.07811			0.539056	+	0.842270i	$0.318781\pi$
$90$		0			0
$91$	−262305.			−3.32050
$92$		0			0
$93$		0			0
$94$		0			0
$95$	70499.4	+	50521.6i	0.801450	+	0.574338i
$96$		0			0
$97$		−	88537.3i		−	0.955426i	−0.878516	−	0.477713i	$-0.841466\pi$
$97$		−	88537.3i		−	0.955426i	0.878516	−	0.477713i	$-0.158534\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	97292.1			0.949018			0.474509	−	0.880251i	$-0.342626\pi$
$101$	97292.1			0.949018			0.474509	+	0.880251i	$0.342626\pi$
$102$		0			0
$103$		−	93587.8i		−	0.869213i	−0.900620	−	0.434607i	$-0.856887\pi$
$103$		−	93587.8i		−	0.869213i	0.900620	−	0.434607i	$-0.143113\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	218604.i		−	1.84586i	−0.384971	−	0.922929i	$-0.625788\pi$
$107$		−	218604.i		−	1.84586i	0.384971	−	0.922929i	$-0.374212\pi$
$108$		0			0
$109$	107862.			0.869562			0.434781	−	0.900536i	$-0.356826\pi$
$109$	107862.			0.869562			0.434781	+	0.900536i	$0.356826\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			125811.i			0.926875i	0.886130	+	0.463438i	$0.153384\pi$
$113$			125811.i			0.926875i	−0.886130	+	0.463438i	$0.846616\pi$
$114$		0			0
$115$	−35412.5	+	49415.7i	−0.249696	+	0.348434i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−322667.			−2.08875
$120$		0			0
$121$	−140733.			−0.873840
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−50722.0	+	167167.i	−0.290350	+	0.956921i
$126$		0			0
$127$			114751.i			0.631318i	0.948873	+	0.315659i	$0.102226\pi$
$127$			114751.i			0.631318i	−0.948873	+	0.315659i	$0.897774\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	125378.			0.638329			0.319164	−	0.947699i	$-0.396598\pi$
$131$	125378.			0.638329			0.319164	+	0.947699i	$0.396598\pi$
$132$		0			0
$133$			376576.i			1.84597i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	144988.i		−	0.659982i	−0.943984	−	0.329991i	$-0.892954\pi$
$137$		−	144988.i		−	0.659982i	0.943984	−	0.329991i	$-0.107046\pi$
$138$		0			0
$139$	414038.			1.81762			0.908810	−	0.417210i	$-0.136992\pi$
$139$	414038.			1.81762			0.908810	+	0.417210i	$0.136992\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			154048.i			0.629963i
$144$		0			0
$145$	198780.	+	142450.i	0.785148	+	0.562656i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	127390.			0.470077			0.235038	−	0.971986i	$-0.424478\pi$
$149$	127390.			0.470077			0.235038	+	0.971986i	$0.424478\pi$
$150$		0			0
$151$	156233.			0.557609			0.278805	−	0.960348i	$-0.410062\pi$
$151$	156233.			0.557609			0.278805	+	0.960348i	$0.410062\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−119217.	−	85433.9i	−0.398575	−	0.285628i
$156$		0			0
$157$			472338.i			1.52934i	0.644424	+	0.764669i	$0.277097\pi$
$157$			472338.i			1.52934i	−0.644424	+	0.764669i	$0.722903\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−263957.			−0.802543
$162$		0			0
$163$			175546.i			0.517514i	0.965942	+	0.258757i	$0.0833128\pi$
$163$			175546.i			0.517514i	−0.965942	+	0.258757i	$0.916687\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	102842.i		−	0.285350i	−0.989770	−	0.142675i	$-0.954430\pi$
$167$		−	102842.i		−	0.285350i	0.989770	−	0.142675i	$-0.0455703\pi$
$168$		0			0
$169$	−796658.			−2.14563
$170$		0			0
$171$		0			0
$172$		0			0
$173$			309138.i			0.785302i	0.919687	+	0.392651i	$0.128442\pi$
$173$			309138.i			0.785302i	−0.919687	+	0.392651i	$0.871558\pi$
$174$		0			0
$175$	−718239.	+	243773.i	−1.77286	+	0.601714i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−774268.			−1.80617			−0.903086	−	0.429461i	$-0.858704\pi$
$179$	−774268.			−1.80617			−0.903086	+	0.429461i	$0.858704\pi$
$180$		0			0
$181$	−142119.			−0.322446			−0.161223	−	0.986918i	$-0.551544\pi$
$181$	−142119.			−0.322446			−0.161223	+	0.986918i	$0.551544\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−67770.5	+	94569.1i	−0.145583	+	0.203151i
$186$		0			0
$187$			189498.i			0.396278i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−557037.			−1.10484			−0.552422	−	0.833565i	$-0.686296\pi$
$191$	−557037.			−1.10484			−0.552422	+	0.833565i	$0.686296\pi$
$192$		0			0
$193$		−	127912.i		−	0.247183i	−0.992333	−	0.123591i	$-0.960559\pi$
$193$		−	127912.i		−	0.247183i	0.992333	−	0.123591i	$-0.0394412\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	916132.i		−	1.68187i	−0.541136	−	0.840935i	$-0.682006\pi$
$197$		−	916132.i		−	1.68187i	0.541136	−	0.840935i	$-0.317994\pi$
$198$		0			0
$199$	−443136.			−0.793240			−0.396620	−	0.917983i	$-0.629817\pi$
$199$	−443136.			−0.793240			−0.396620	+	0.917983i	$0.629817\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.06179e6i			1.80842i
$204$		0			0
$205$	726935.	+	520939.i	1.20812	+	0.865770i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	221158.			0.350216
$210$		0			0
$211$	−496576.			−0.767856			−0.383928	−	0.923363i	$-0.625429\pi$
$211$	−496576.			−0.767856			−0.383928	+	0.923363i	$0.625429\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	132958.	−	185534.i	0.196164	−	0.273733i
$216$		0			0
$217$		−	636805.i		−	0.918031i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.43672e6			−1.97876
$222$		0			0
$223$			94509.5i			0.127266i	0.997973	+	0.0636332i	$0.0202688\pi$
$223$			94509.5i			0.127266i	−0.997973	+	0.0636332i	$0.979731\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	1.16373e6i		−	1.49895i	−0.662030	−	0.749477i	$-0.730305\pi$
$227$		−	1.16373e6i		−	1.49895i	0.662030	−	0.749477i	$-0.269695\pi$
$228$		0			0
$229$	−820259.			−1.03362			−0.516811	−	0.856099i	$-0.672881\pi$
$229$	−820259.			−1.03362			−0.516811	+	0.856099i	$0.672881\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.22012e6i			1.47236i	0.676788	+	0.736178i	$0.263371\pi$
$233$			1.22012e6i			1.47236i	−0.676788	+	0.736178i	$0.736629\pi$
$234$		0			0
$235$	−156925.	+	218978.i	−0.185363	+	0.258661i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.02798e6			1.16410			0.582048	−	0.813154i	$-0.302252\pi$
$239$	1.02798e6			1.16410			0.582048	+	0.813154i	$0.302252\pi$
$240$		0			0
$241$	−755090.			−0.837445			−0.418723	−	0.908114i	$-0.637522\pi$
$241$	−755090.			−0.837445			−0.418723	+	0.908114i	$0.637522\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.91311e6	−	1.37098e6i	−2.03622	−	1.45920i
$246$		0			0
$247$			1.67676e6i			1.74875i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.87333e6			1.87686			0.938428	−	0.345476i	$-0.112283\pi$
$251$	1.87333e6			1.87686			0.938428	+	0.345476i	$0.112283\pi$
$252$		0			0
$253$			155018.i			0.152258i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	55931.3i		−	0.0528229i	−0.999651	−	0.0264114i	$-0.991592\pi$
$257$		−	55931.3i		−	0.0528229i	0.999651	−	0.0264114i	$-0.00840800\pi$
$258$		0			0
$259$	−505146.			−0.467915
$260$		0			0
$261$		0			0
$262$		0			0
$263$			613008.i			0.546483i	0.961945	+	0.273242i	$0.0880959\pi$
$263$			613008.i			0.546483i	−0.961945	+	0.273242i	$0.911904\pi$
$264$		0			0
$265$	−673603.	+	939967.i	−0.589236	+	0.822239i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	452657.			0.381407			0.190704	−	0.981648i	$-0.438923\pi$
$269$	452657.			0.381407			0.190704	+	0.981648i	$0.438923\pi$
$270$		0			0
$271$	550844.			0.455623			0.227811	−	0.973705i	$-0.426843\pi$
$271$	550844.			0.455623			0.227811	+	0.973705i	$0.426843\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	143164.	+	421811.i	0.114157	+	0.336346i
$276$		0			0
$277$		−	992154.i		−	0.776926i	−0.921464	−	0.388463i	$-0.873006\pi$
$277$		−	992154.i		−	0.776926i	0.921464	−	0.388463i	$-0.126994\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	731905.			0.552954			0.276477	−	0.961021i	$-0.410833\pi$
$281$	731905.			0.552954			0.276477	+	0.961021i	$0.410833\pi$
$282$		0			0
$283$			459405.i			0.340981i	0.985359	+	0.170490i	$0.0545352\pi$
$283$			459405.i			0.340981i	−0.985359	+	0.170490i	$0.945465\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3.88296e6i			2.78265i
$288$		0			0
$289$	−347487.			−0.244734
$290$		0			0
$291$		0			0
$292$		0			0
$293$			364973.i			0.248366i	0.992259	+	0.124183i	$0.0396309\pi$
$293$			364973.i			0.248366i	−0.992259	+	0.124183i	$0.960369\pi$
$294$		0			0
$295$	570813.	+	409059.i	0.381891	+	0.273672i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.17531e6			−0.760279
$300$		0			0
$301$	991041.			0.630486
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.89984e6	+	1.36147e6i	1.16941	+	0.838030i
$306$		0			0
$307$		−	307960.i		−	0.186487i	−0.995643	−	0.0932434i	$-0.970277\pi$
$307$		−	307960.i		−	0.186487i	0.995643	−	0.0932434i	$-0.0297235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.30555e6			−0.765405			−0.382703	−	0.923872i	$-0.625007\pi$
$311$	−1.30555e6			−0.765405			−0.382703	+	0.923872i	$0.625007\pi$
$312$		0			0
$313$		−	2.66546e6i		−	1.53784i	−0.639344	−	0.768921i	$-0.720794\pi$
$313$		−	2.66546e6i		−	1.53784i	0.639344	−	0.768921i	$-0.279206\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.23744e6i		−	1.25056i	−0.780402	−	0.625278i	$-0.784986\pi$
$317$		−	2.23744e6i		−	1.25056i	0.780402	−	0.625278i	$-0.215014\pi$
$318$		0			0
$319$	623574.			0.343093
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.06262e6i			1.10005i
$324$		0			0
$325$	−3.19806e6	+	1.08543e6i	−1.67949	+	0.570027i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.16968e6			−0.595771
$330$		0			0
$331$	−2.09233e6			−1.04969			−0.524844	−	0.851199i	$-0.675876\pi$
$331$	−2.09233e6			−1.04969			−0.524844	+	0.851199i	$0.675876\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.96937e6	−	2.74811e6i	0.958770	−	1.33790i
$336$		0			0
$337$			1.01282e6i			0.485802i	0.970051	+	0.242901i	$0.0780990\pi$
$337$			1.01282e6i			0.485802i	−0.970051	+	0.242901i	$0.921901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−373986.			−0.174168
$342$		0			0
$343$		−	6.13967e6i		−	2.81780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.64429e6i			0.733084i	0.930402	+	0.366542i	$0.119458\pi$
$347$			1.64429e6i			0.733084i	−0.930402	+	0.366542i	$0.880542\pi$
$348$		0			0
$349$	−898528.			−0.394882			−0.197441	−	0.980315i	$-0.563263\pi$
$349$	−898528.			−0.394882			−0.197441	+	0.980315i	$0.563263\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	853045.i		−	0.364364i	−0.983265	−	0.182182i	$-0.941684\pi$
$353$		−	853045.i		−	0.364364i	0.983265	−	0.182182i	$-0.0583160\pi$
$354$		0			0
$355$	−1.23246e6	−	883209.i	−0.519040	−	0.371957i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.05799e6			0.842767			0.421384	−	0.906882i	$-0.361545\pi$
$359$	2.05799e6			0.842767			0.421384	+	0.906882i	$0.361545\pi$
$360$		0			0
$361$	−68869.5			−0.0278137
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.86569e6	−	2.60344e6i	0.733004	−	1.02286i
$366$		0			0
$367$			4.38094e6i			1.69786i	0.528505	+	0.848930i	$0.322753\pi$
$367$			4.38094e6i			1.69786i	−0.528505	+	0.848930i	$0.677247\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.02088e6			−1.89385
$372$		0			0
$373$		−	456550.i		−	0.169909i	−0.996385	−	0.0849545i	$-0.972926\pi$
$373$		−	456550.i		−	0.169909i	0.996385	−	0.0849545i	$-0.0270745\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			4.72778e6i			1.71318i
$378$		0			0
$379$	935559.			0.334559			0.167280	−	0.985909i	$-0.446502\pi$
$379$	935559.			0.334559			0.167280	+	0.985909i	$0.446502\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	2.78295e6i		−	0.969413i	−0.874677	−	0.484707i	$-0.838926\pi$
$383$		−	2.78295e6i		−	0.969413i	0.874677	−	0.484707i	$-0.161074\pi$
$384$		0			0
$385$	−1.12656e6	+	1.57204e6i	−0.387350	+	0.540520i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.18050e6			−0.730604			−0.365302	−	0.930889i	$-0.619034\pi$
$389$	−2.18050e6			−0.730604			−0.365302	+	0.930889i	$0.619034\pi$
$390$		0			0
$391$	−1.44577e6			−0.478252
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.23378e6	−	884154.i	−0.397872	−	0.285125i
$396$		0			0
$397$		−	1.05032e6i		−	0.334462i	−0.985918	−	0.167231i	$-0.946517\pi$
$397$		−	1.05032e6i		−	0.334462i	0.985918	−	0.167231i	$-0.0534826\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.63701e6			0.818937			0.409469	−	0.912324i	$-0.365714\pi$
$401$	2.63701e6			0.818937			0.409469	+	0.912324i	$0.365714\pi$
$402$		0			0
$403$		−	2.83547e6i		−	0.869685i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			296664.i			0.0887727i
$408$		0			0
$409$	2.72798e6			0.806368			0.403184	−	0.915119i	$-0.367904\pi$
$409$	2.72798e6			0.806368			0.403184	+	0.915119i	$0.367904\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			3.04903e6i			0.879603i
$414$		0			0
$415$	2.42644e6	−	3.38594e6i	0.691592	−	0.965069i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.94093e6			1.65318			0.826589	−	0.562806i	$-0.190278\pi$
$419$	5.94093e6			1.65318			0.826589	+	0.562806i	$0.190278\pi$
$420$		0			0
$421$	474301.			0.130422			0.0652108	−	0.997872i	$-0.479228\pi$
$421$	474301.			0.130422			0.0652108	+	0.997872i	$0.479228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.93401e6	+	1.33522e6i	−1.05648	+	0.358574i
$426$		0			0
$427$			1.01481e7i			2.69349i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.17898e6			0.824319			0.412159	−	0.911112i	$-0.364775\pi$
$431$	3.17898e6			0.824319			0.412159	+	0.911112i	$0.364775\pi$
$432$		0			0
$433$		−	2.09103e6i		−	0.535970i	−0.963423	−	0.267985i	$-0.913642\pi$
$433$		−	2.09103e6i		−	0.535970i	0.963423	−	0.267985i	$-0.0863578\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.68732e6i			0.422662i
$438$		0			0
$439$	−758627.			−0.187874			−0.0939371	−	0.995578i	$-0.529945\pi$
$439$	−758627.			−0.187874			−0.0939371	+	0.995578i	$0.529945\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			940168.i			0.227613i	0.993503	+	0.113806i	$0.0363043\pi$
$443$			940168.i			0.227613i	−0.993503	+	0.113806i	$0.963696\pi$
$444$		0			0
$445$	3.66071e6	+	2.62335e6i	0.876325	+	0.627996i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.83158e6			0.896937			0.448469	−	0.893799i	$-0.351970\pi$
$449$	3.83158e6			0.896937			0.448469	+	0.893799i	$0.351970\pi$
$450$		0			0
$451$	2.28040e6			0.527923
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.19188e7	−	8.54131e6i	−2.69901	−	1.93418i
$456$		0			0
$457$		−	2.01368e6i		−	0.451024i	−0.974240	−	0.225512i	$-0.927594\pi$
$457$		−	2.01368e6i		−	0.451024i	0.974240	−	0.225512i	$-0.0724055\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	152998.			0.0335300			0.0167650	−	0.999859i	$-0.494663\pi$
$461$	152998.			0.0335300			0.0167650	+	0.999859i	$0.494663\pi$
$462$		0			0
$463$		−	5.40324e6i		−	1.17139i	−0.810531	−	0.585696i	$-0.800821\pi$
$463$		−	5.40324e6i		−	1.17139i	0.810531	−	0.585696i	$-0.199179\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.41706e6i		−	0.300674i	−0.988635	−	0.150337i	$-0.951964\pi$
$467$		−	1.41706e6i		−	0.300674i	0.988635	−	0.150337i	$-0.0480359\pi$
$468$		0			0
$469$	1.46792e7			3.08156
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	582023.i		−	0.119615i
$474$		0			0
$475$	1.55830e6	+	4.59128e6i	0.316895	+	0.933683i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.46163e6			0.490213			0.245106	−	0.969496i	$-0.421177\pi$
$479$	2.46163e6			0.490213			0.245106	+	0.969496i	$0.421177\pi$
$480$		0			0
$481$	−2.24924e6			−0.443274
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.88300e6	−	4.02303e6i	0.556532	−	0.776602i
$486$		0			0
$487$		−	2.88009e6i		−	0.550280i	−0.961404	−	0.275140i	$-0.911276\pi$
$487$		−	2.88009e6i		−	0.550280i	0.961404	−	0.275140i	$-0.0887242\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.55154e6			−0.477637			−0.238819	−	0.971064i	$-0.576760\pi$
$491$	−2.55154e6			−0.477637			−0.238819	+	0.971064i	$0.576760\pi$
$492$		0			0
$493$			5.81575e6i			1.07768i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.58324e6i		−	1.19550i
$498$		0			0
$499$	−8.58040e6			−1.54261			−0.771305	−	0.636466i	$-0.780396\pi$
$499$	−8.58040e6			−1.54261			−0.771305	+	0.636466i	$0.780396\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.22134e6i		−	1.27262i	−0.771434	−	0.636309i	$-0.780460\pi$
$503$		−	7.22134e6i		−	1.27262i	0.771434	−	0.636309i	$-0.219540\pi$
$504$		0			0
$505$	4.42083e6	+	3.16808e6i	0.771393	+	0.552799i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−73769.6			−0.0126207			−0.00631035	−	0.999980i	$-0.502009\pi$
$509$	−73769.6			−0.0126207			−0.00631035	+	0.999980i	$0.502009\pi$
$510$		0			0
$511$	1.39064e7			2.35593
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.04746e6	−	4.25252e6i	0.506313	−	0.706526i
$516$		0			0
$517$			686938.i			0.113029i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.33407e6			−1.02232			−0.511162	−	0.859484i	$-0.670785\pi$
$521$	−6.33407e6			−1.02232			−0.511162	+	0.859484i	$0.670785\pi$
$522$		0			0
$523$			411025.i			0.0657074i	0.999460	+	0.0328537i	$0.0104595\pi$
$523$			411025.i			0.0657074i	−0.999460	+	0.0328537i	$0.989540\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.48797e6i		−	0.547074i
$528$		0			0
$529$	5.25364e6			0.816246
$530$		0			0
$531$		0			0
$532$		0			0
$533$			1.72895e7i			2.63611i
$534$		0			0
$535$	7.11829e6	−	9.93309e6i	1.07520	−	1.50037i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.00144e6			−0.889782
$540$		0			0
$541$	239548.			0.0351884			0.0175942	−	0.999845i	$-0.494399\pi$
$541$	239548.			0.0351884			0.0175942	+	0.999845i	$0.494399\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.90110e6	+	3.51225e6i	0.706809	+	0.506517i
$546$		0			0
$547$		−	1.55307e6i		−	0.221933i	−0.993824	−	0.110967i	$-0.964605\pi$
$547$		−	1.55307e6i		−	0.221933i	0.993824	−	0.110967i	$-0.0353946\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.78741e6			0.952412
$552$		0			0
$553$		−	6.59028e6i		−	0.916413i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1.19125e7i		−	1.62691i	−0.581627	−	0.813455i	$-0.697584\pi$
$557$		−	1.19125e7i		−	1.62691i	0.581627	−	0.813455i	$-0.302416\pi$
$558$		0			0
$559$	4.41275e6			0.597283
$560$		0			0
$561$		0			0
$562$		0			0
$563$			1.67271e6i			0.222408i	0.993798	+	0.111204i	$0.0354707\pi$
$563$			1.67271e6i			0.222408i	−0.993798	+	0.111204i	$0.964529\pi$
$564$		0			0
$565$	−4.09671e6	+	5.71668e6i	−0.539901	+	0.753395i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.26524e7			1.63829			0.819146	−	0.573585i	$-0.194448\pi$
$569$	1.26524e7			1.63829			0.819146	+	0.573585i	$0.194448\pi$
$570$		0			0
$571$	5.80638e6			0.745273			0.372637	−	0.927977i	$-0.378454\pi$
$571$	5.80638e6			0.745273			0.372637	+	0.927977i	$0.378454\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.21820e6	+	1.09227e6i	−0.405923	+	0.137772i
$576$		0			0
$577$		−	1.15669e7i		−	1.44636i	−0.690658	−	0.723182i	$-0.742679\pi$
$577$		−	1.15669e7i		−	1.44636i	0.690658	−	0.723182i	$-0.257321\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.80862e7			2.22283
$582$		0			0
$583$			2.94869e6i			0.359300i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			829454.i			0.0993566i	0.998765	+	0.0496783i	$0.0158196\pi$
$587$			829454.i			0.0993566i	−0.998765	+	0.0496783i	$0.984180\pi$
$588$		0			0
$589$	−4.07072e6			−0.483485
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.42831e7i		−	1.66796i	−0.551798	−	0.833978i	$-0.686058\pi$
$593$		−	1.42831e7i		−	1.66796i	0.551798	−	0.833978i	$-0.313942\pi$
$594$		0			0
$595$	−1.46616e7	−	1.05069e7i	−1.69781	−	1.21669i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.51569e7			1.72601			0.863007	−	0.505192i	$-0.168579\pi$
$599$	1.51569e7			1.72601			0.863007	+	0.505192i	$0.168579\pi$
$600$		0			0
$601$	−4.20502e6			−0.474877			−0.237439	−	0.971403i	$-0.576308\pi$
$601$	−4.20502e6			−0.474877			−0.237439	+	0.971403i	$0.576308\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.39472e6	−	4.58261e6i	−0.710286	−	0.509008i
$606$		0			0
$607$			242069.i			0.0266666i	0.999911	+	0.0133333i	$0.00424425\pi$
$607$			242069.i			0.0266666i	−0.999911	+	0.0133333i	$0.995756\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.20819e6			−0.564396
$612$		0			0
$613$			1.13279e7i			1.21759i	0.793329	+	0.608793i	$0.208346\pi$
$613$			1.13279e7i			1.21759i	−0.793329	+	0.608793i	$0.791654\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.54069e7i		−	1.62931i	−0.579947	−	0.814654i	$-0.696927\pi$
$617$		−	1.54069e7i		−	1.62931i	0.579947	−	0.814654i	$-0.303073\pi$
$618$		0			0
$619$	1.49269e7			1.56583			0.782914	−	0.622129i	$-0.213732\pi$
$619$	1.49269e7			1.56583			0.782914	+	0.622129i	$0.213732\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.95539e7i			2.01843i
$624$		0			0
$625$	−7.74813e6	+	5.94423e6i	−0.793409	+	0.608689i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.76683e6			−0.278841
$630$		0			0
$631$	−1.64446e7			−1.64418			−0.822091	−	0.569357i	$-0.807192\pi$
$631$	−1.64446e7			−1.64418			−0.822091	+	0.569357i	$0.807192\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.73659e6	+	5.21416e6i	−0.367740	+	0.513156i
$636$		0			0
$637$		−	4.55014e7i		−	4.44300i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.47616e6			−0.334160			−0.167080	−	0.985943i	$-0.553434\pi$
$641$	−3.47616e6			−0.334160			−0.167080	+	0.985943i	$0.553434\pi$
$642$		0			0
$643$		−	1.62205e7i		−	1.54716i	−0.633697	−	0.773581i	$-0.718463\pi$
$643$		−	1.62205e7i		−	1.54716i	0.633697	−	0.773581i	$-0.281537\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	7.87206e6i		−	0.739312i	−0.929169	−	0.369656i	$-0.879476\pi$
$647$		−	7.87206e6i		−	0.739312i	0.929169	−	0.369656i	$-0.120524\pi$
$648$		0			0
$649$	1.79065e6			0.166878
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.37332e7i			1.26034i	0.776457	+	0.630171i	$0.217015\pi$
$653$			1.37332e7i			1.26034i	−0.776457	+	0.630171i	$0.782985\pi$
$654$		0			0
$655$	5.69704e6	+	4.08264e6i	0.518855	+	0.371824i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.18405e6			−0.375305			−0.187652	−	0.982236i	$-0.560088\pi$
$659$	−4.18405e6			−0.375305			−0.187652	+	0.982236i	$0.560088\pi$
$660$		0			0
$661$	1.95785e6			0.174291			0.0871455	−	0.996196i	$-0.472226\pi$
$661$	1.95785e6			0.174291			0.0871455	+	0.996196i	$0.472226\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.22623e7	+	1.71112e7i	−1.07527	+	1.50046i
$666$		0			0
$667$			4.75755e6i			0.414066i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.95983e6			0.511008
$672$		0			0
$673$		−	8.85319e6i		−	0.753463i	−0.926322	−	0.376732i	$-0.877048\pi$
$673$		−	8.85319e6i		−	0.753463i	0.926322	−	0.376732i	$-0.122952\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			7.29523e6i			0.611740i	0.952073	+	0.305870i	$0.0989474\pi$
$677$			7.29523e6i			0.611740i	−0.952073	+	0.305870i	$0.901053\pi$
$678$		0			0
$679$	2.14892e7			1.78874
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	8.10456e6i		−	0.664779i	−0.943142	−	0.332390i	$-0.892145\pi$
$683$		−	8.10456e6i		−	0.664779i	0.943142	−	0.332390i	$-0.107855\pi$
$684$		0			0
$685$	4.72119e6	−	6.58810e6i	0.384437	−	0.536455i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.23562e7			−1.79411
$690$		0			0
$691$	−2.30304e7			−1.83487			−0.917437	−	0.397882i	$-0.869745\pi$
$691$	−2.30304e7			−1.83487			−0.917437	+	0.397882i	$0.869745\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.88134e7	+	1.34821e7i	1.47742	+	1.05876i
$696$		0			0
$697$			2.12681e7i			1.65824i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.60749e7			−1.23553			−0.617766	−	0.786362i	$-0.711962\pi$
$701$	−1.60749e7			−1.23553			−0.617766	+	0.786362i	$0.711962\pi$
$702$		0			0
$703$			3.22910e6i			0.246430i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			2.36141e7i			1.77674i
$708$		0			0
$709$	−2.10037e7			−1.56921			−0.784605	−	0.619996i	$-0.787134\pi$
$709$	−2.10037e7			−1.56921			−0.784605	+	0.619996i	$0.787134\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.85332e6i		−	0.210197i
$714$		0			0
$715$	−5.01618e6	+	6.99974e6i	−0.366951	+	0.512055i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.42531e7			1.02822			0.514112	−	0.857723i	$-0.328122\pi$
$719$	1.42531e7			1.02822			0.514112	+	0.857723i	$0.328122\pi$
$720$		0			0
$721$	2.27150e7			1.62733
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.39376e6	+	1.29455e7i	0.310450	+	0.914692i
$726$		0			0
$727$			4.59136e6i			0.322185i	0.986939	+	0.161092i	$0.0515017\pi$
$727$			4.59136e6i			0.322185i	−0.986939	+	0.161092i	$0.948498\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.42822e6			0.375720
$732$		0			0
$733$		−	4.59016e6i		−	0.315550i	−0.987475	−	0.157775i	$-0.949568\pi$
$733$		−	4.59016e6i		−	0.315550i	0.987475	−	0.157775i	$-0.0504321\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8.62087e6i		−	0.584632i
$738$		0			0
$739$	307856.			0.0207365			0.0103683	−	0.999946i	$-0.496700\pi$
$739$	307856.			0.0207365			0.0103683	+	0.999946i	$0.496700\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	6.44444e6i		−	0.428266i	−0.976805	−	0.214133i	$-0.931307\pi$
$743$		−	6.44444e6i		−	0.428266i	0.976805	−	0.214133i	$-0.0686926\pi$
$744$		0			0
$745$	5.78844e6	+	4.14813e6i	0.382094	+	0.273818i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.30581e7			3.45579
$750$		0			0
$751$	5.56856e6			0.360282			0.180141	−	0.983641i	$-0.442345\pi$
$751$	5.56856e6			0.360282			0.180141	+	0.983641i	$0.442345\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.09903e6	+	5.08734e6i	0.453244	+	0.324805i
$756$		0			0
$757$		−	7.47540e6i		−	0.474127i	−0.971494	−	0.237064i	$-0.923815\pi$
$757$		−	7.47540e6i		−	0.474127i	0.971494	−	0.237064i	$-0.0761850\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.31205e7			−0.821278			−0.410639	−	0.911798i	$-0.634694\pi$
$761$	−1.31205e7			−0.821278			−0.410639	+	0.911798i	$0.634694\pi$
$762$		0			0
$763$			2.61795e7i			1.62798i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.35763e7i			0.833281i
$768$		0			0
$769$	1.56508e7			0.954378			0.477189	−	0.878801i	$-0.341656\pi$
$769$	1.56508e7			0.954378			0.477189	+	0.878801i	$0.341656\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2.07890e7i		−	1.25137i	−0.780078	−	0.625683i	$-0.784821\pi$
$773$		−	2.07890e7i		−	1.25137i	0.780078	−	0.625683i	$-0.215179\pi$
$774$		0			0
$775$	−2.63514e6	−	7.76403e6i	−0.157597	−	0.464336i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.48215e7			1.46549
$780$		0			0
$781$	−3.86624e6			−0.226809
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.53805e7	+	2.14624e7i	−0.890833	+	1.24310i
$786$		0			0
$787$			3.31541e7i			1.90809i	0.299656	+	0.954047i	$0.403128\pi$
$787$			3.31541e7i			1.90809i	−0.299656	+	0.954047i	$0.596872\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.05360e7			−1.73528
$792$		0			0
$793$			4.51859e7i			2.55165i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	2.00986e6i		−	0.112078i	−0.998429	−	0.0560389i	$-0.982153\pi$
$797$		−	2.00986e6i		−	0.112078i	0.998429	−	0.0560389i	$-0.0178471\pi$
$798$		0			0
$799$	−6.40671e6			−0.355033
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	8.16702e6i		−	0.446966i
$804$		0			0
$805$	−1.19939e7	−	8.59510e6i	−0.652333	−	0.467478i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.55183e7			1.90801			0.954006	−	0.299787i	$-0.0969156\pi$
$809$	3.55183e7			1.90801			0.954006	+	0.299787i	$0.0969156\pi$
$810$		0			0
$811$	−8.99587e6			−0.480276			−0.240138	−	0.970739i	$-0.577193\pi$
$811$	−8.99587e6			−0.480276			−0.240138	+	0.970739i	$0.577193\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.71622e6	+	7.97659e6i	−0.301450	+	0.420652i
$816$		0			0
$817$		−	6.33514e6i		−	0.332048i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.39427e7			−1.75747			−0.878736	−	0.477308i	$-0.841613\pi$
$821$	−3.39427e7			−1.75747			−0.878736	+	0.477308i	$0.841613\pi$
$822$		0			0
$823$		−	1.15878e7i		−	0.596348i	−0.954511	−	0.298174i	$-0.903622\pi$
$823$		−	1.15878e7i		−	0.596348i	0.954511	−	0.298174i	$-0.0963776\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			743207.i			0.0377873i	0.999821	+	0.0188937i	$0.00601440\pi$
$827$			743207.i			0.0377873i	−0.999821	+	0.0188937i	$0.993986\pi$
$828$		0			0
$829$	−2.94662e7			−1.48915			−0.744574	−	0.667540i	$-0.767347\pi$
$829$	−2.94662e7			−1.48915			−0.744574	+	0.667540i	$0.767347\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	5.59723e7i		−	2.79486i
$834$		0			0
$835$	3.34878e6	−	4.67299e6i	0.166215	−	0.231942i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.18433e7			−1.56175			−0.780877	−	0.624685i	$-0.785227\pi$
$839$	−3.18433e7			−1.56175			−0.780877	+	0.624685i	$0.785227\pi$
$840$		0			0
$841$	−1.37342e6			−0.0669596
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.61992e7	−	2.59412e7i	−1.74404	−	1.24982i
$846$		0			0
$847$		−	3.41578e7i		−	1.63599i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.26340e6			−0.107136
$852$		0			0
$853$		−	9.10862e6i		−	0.428627i	−0.976765	−	0.214314i	$-0.931249\pi$
$853$		−	9.10862e6i		−	0.428627i	0.976765	−	0.214314i	$-0.0687515\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			4.41175e6i			0.205191i	0.994723	+	0.102596i	$0.0327148\pi$
$857$			4.41175e6i			0.205191i	−0.994723	+	0.102596i	$0.967285\pi$
$858$		0			0
$859$	−9.64496e6			−0.445982			−0.222991	−	0.974821i	$-0.571582\pi$
$859$	−9.64496e6			−0.445982			−0.222991	+	0.974821i	$0.571582\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.95068e7i		−	0.891579i	−0.895138	−	0.445790i	$-0.852923\pi$
$863$		−	1.95068e7i		−	0.891579i	0.895138	−	0.445790i	$-0.147077\pi$
$864$		0			0
$865$	−1.00663e7	+	1.40468e7i	−0.457436	+	0.638320i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.87037e6			−0.173861
$870$		0			0
$871$	6.53613e7			2.91928
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.05738e7	−	1.23109e7i	−1.79153	−	0.543589i
$876$		0			0
$877$		−	5.18933e6i		−	0.227831i	−0.993490	−	0.113915i	$-0.963661\pi$
$877$		−	5.18933e6i		−	0.227831i	0.993490	−	0.113915i	$-0.0363393\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.43979e7			0.624972			0.312486	−	0.949922i	$-0.398838\pi$
$881$	1.43979e7			0.624972			0.312486	+	0.949922i	$0.398838\pi$
$882$		0			0
$883$			2.09749e7i			0.905311i	0.891685	+	0.452656i	$0.149523\pi$
$883$			2.09749e7i			0.905311i	−0.891685	+	0.452656i	$0.850477\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	3.54403e7i		−	1.51248i	−0.654296	−	0.756238i	$-0.727035\pi$
$887$		−	3.54403e7i		−	1.51248i	0.654296	−	0.756238i	$-0.272965\pi$
$888$		0			0
$889$	−2.78517e7			−1.18195
$890$		0			0
$891$		0			0
$892$		0			0
$893$			7.47710e6i			0.313765i
$894$		0			0
$895$	−3.51818e7	−	2.52121e7i	−1.46812	−	1.05209i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.14778e7			−0.473651
$900$		0			0
$901$	−2.75009e7			−1.12859
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.45773e6	−	4.62777e6i	−0.262095	−	0.187824i
$906$		0			0
$907$			2.52614e7i			1.01962i	0.860286	+	0.509812i	$0.170285\pi$
$907$			2.52614e7i			1.01962i	−0.860286	+	0.509812i	$0.829715\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.45292e7			0.580022			0.290011	−	0.957023i	$-0.406341\pi$
$911$	1.45292e7			0.580022			0.290011	+	0.957023i	$0.406341\pi$
$912$		0			0
$913$		−	1.06217e7i		−	0.421714i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			3.04310e7i			1.19507i
$918$		0			0
$919$	2.82570e7			1.10366			0.551832	−	0.833955i	$-0.313929\pi$
$919$	2.82570e7			1.10366			0.551832	+	0.833955i	$0.313929\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2.93128e7i		−	1.13254i
$924$		0			0
$925$	−6.15881e6	+	2.09032e6i	−0.236670	+	0.0803266i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.09229e6			0.155570			0.0777852	−	0.996970i	$-0.475215\pi$
$929$	4.09229e6			0.155570			0.0777852	+	0.996970i	$0.475215\pi$
$930$		0			0
$931$	−6.53238e7			−2.47000
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.17052e6	+	8.61054e6i	−0.230830	+	0.322108i
$936$		0			0
$937$			5.13115e7i			1.90926i	0.297790	+	0.954632i	$0.403751\pi$
$937$			5.13115e7i			1.90926i	−0.297790	+	0.954632i	$0.596249\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.60558e7			0.591096			0.295548	−	0.955328i	$-0.404498\pi$
$941$	1.60558e7			0.591096			0.295548	+	0.955328i	$0.404498\pi$
$942$		0			0
$943$			1.73983e7i			0.637130i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			3.82296e7i			1.38524i	0.721302	+	0.692620i	$0.243544\pi$
$947$			3.82296e7i			1.38524i	−0.721302	+	0.692620i	$0.756456\pi$
$948$		0			0
$949$	6.19203e7			2.23186
$950$		0			0
$951$		0			0
$952$		0			0
$953$			4.76886e7i			1.70092i	0.526044	+	0.850458i	$0.323675\pi$
$953$			4.76886e7i			1.70092i	−0.526044	+	0.850458i	$0.676325\pi$
$954$		0			0
$955$	−2.53111e7	−	1.81385e7i	−0.898053	−	0.643567i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.51907e7			1.23561
$960$		0			0
$961$	−2.17454e7			−0.759555
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.16514e6	−	5.81216e6i	0.143983	−	0.200918i
$966$		0			0
$967$			2.70295e6i			0.0929548i	0.998919	+	0.0464774i	$0.0147995\pi$
$967$			2.70295e6i			0.0929548i	−0.998919	+	0.0464774i	$0.985200\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.87085e7			−0.977153			−0.488577	−	0.872521i	$-0.662484\pi$
$971$	−2.87085e7			−0.977153			−0.488577	+	0.872521i	$0.662484\pi$
$972$		0			0
$973$			1.00493e8i			3.40292i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	4.40555e6i		−	0.147660i	−0.997271	−	0.0738302i	$-0.976478\pi$
$977$		−	4.40555e6i		−	0.147660i	0.997271	−	0.0738302i	$-0.0235223\pi$
$978$		0			0
$979$	1.14837e7			0.382935
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	2.90145e7i		−	0.957705i	−0.877895	−	0.478853i	$-0.841053\pi$
$983$		−	2.90145e7i		−	0.957705i	0.877895	−	0.478853i	$-0.158947\pi$
$984$		0			0
$985$	2.98316e7	−	4.16279e7i	0.979683	−	1.36708i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.44054e6			0.144359
$990$		0			0
$991$	−1.86483e7			−0.603190			−0.301595	−	0.953436i	$-0.597519\pi$
$991$	−1.86483e7			−0.603190			−0.301595	+	0.953436i	$0.597519\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.01356e7	−	1.44296e7i	−0.644772	−	0.462059i
$996$		0			0
$997$		−	6.75822e6i		−	0.215325i	−0.994187	−	0.107663i	$-0.965663\pi$
$997$		−	6.75822e6i		−	0.215325i	0.994187	−	0.107663i	$-0.0343366\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.f.o.289.8		8
3.2	odd	2		240.6.f.f.49.5		8
4.3	odd	2		360.6.f.c.289.8		8
5.4	even	2	inner	720.6.f.o.289.7		8
12.11	even	2		120.6.f.b.49.1	✓	8
15.14	odd	2		240.6.f.f.49.1		8
20.19	odd	2		360.6.f.c.289.7		8
60.23	odd	4		600.6.a.w.1.1		4
60.47	odd	4		600.6.a.v.1.4		4
60.59	even	2		120.6.f.b.49.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.f.b.49.1	✓	8	12.11	even	2
120.6.f.b.49.5	yes	8	60.59	even	2
240.6.f.f.49.1		8	15.14	odd	2
240.6.f.f.49.5		8	3.2	odd	2
360.6.f.c.289.7		8	20.19	odd	2
360.6.f.c.289.8		8	4.3	odd	2
600.6.a.v.1.4		4	60.47	odd	4
600.6.a.w.1.1		4	60.23	odd	4
720.6.f.o.289.7		8	5.4	even	2	inner
720.6.f.o.289.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(45.4388 + 32.5625i) q^{5} +242.714i q^{7} +142.542 q^{11} +1080.72i q^{13} +1329.42i q^{17} +1551.52 q^{19} +1087.52i q^{23} +(1004.36 + 2959.20i) q^{25} +4374.67 q^{29} -2623.69 q^{31} +(-7903.37 + 11028.6i) q^{35} +2081.24i q^{37} +15998.1 q^{41} -4083.17i q^{43} +4819.20i q^{47} -42102.9 q^{49} +20686.4i q^{53} +(6476.94 + 4641.53i) q^{55} +12562.3 q^{59} +41811.0 q^{61} +(-35190.9 + 49106.5i) q^{65} -60479.5i q^{67} -27123.5 q^{71} -57295.5i q^{73} +34596.9i q^{77} -27152.5 q^{79} -74516.5i q^{83} +(-43289.1 + 60407.0i) q^{85} +80563.6 q^{89} -262305. q^{91} +(70499.4 + 50521.6i) q^{95} -88537.3i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 66 q^{5} + 684 q^{11} + 3040 q^{19} - 564 q^{25} + 8004 q^{29} - 6024 q^{31} - 4476 q^{35} + 37800 q^{41} - 34368 q^{49} - 39968 q^{55} - 86028 q^{59} - 38960 q^{61} - 84204 q^{65} + 162072 q^{71}+ \cdots + 42048 q^{95}+O(q^{100}) \) 8 * q + 66 * q^5 + 684 * q^11 + 3040 * q^19 - 564 * q^25 + 8004 * q^29 - 6024 * q^31 - 4476 * q^35 + 37800 * q^41 - 34368 * q^49 - 39968 * q^55 - 86028 * q^59 - 38960 * q^61 - 84204 * q^65 + 162072 * q^71 + 169976 * q^79 + 240588 * q^85 + 452976 * q^89 - 610456 * q^91 + 42048 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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