LMFDB - Embedded newform 240.8.f.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(49,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("240.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	240.49
Dual form			240.8.f.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} +(-50.0000 - 275.000i) q^{5} -1126.00i q^{7} -729.000 q^{9} +5518.00 q^{11} +12798.0i q^{13} +(-7425.00 + 1350.00i) q^{15} +32206.0i q^{17} -4440.00 q^{19} -30402.0 q^{21} +95452.0i q^{23} +(-73125.0 + 27500.0i) q^{25} +19683.0i q^{27} -19440.0 q^{29} +240248. q^{31} -148986. i q^{33} +(-309650. + 56300.0i) q^{35} -77834.0i q^{37} +345546. q^{39} +299522. q^{41} +416212. i q^{43} +(36450.0 + 200475. i) q^{45} -322976. i q^{47} -444333. q^{49} +869562. q^{51} +880878. i q^{53} +(-275900. - 1.51745e6i) q^{55} +119880. i q^{57} -1.84511e6 q^{59} -861718. q^{61} +820854. i q^{63} +(3.51945e6 - 639900. i) q^{65} +673864. i q^{67} +2.57720e6 q^{69} +3.42695e6 q^{71} +4.67875e6i q^{73} +(742500. + 1.97438e6i) q^{75} -6.21327e6i q^{77} -3.13776e6 q^{79} +531441. q^{81} +484132. i q^{83} +(8.85665e6 - 1.61030e6i) q^{85} +524880. i q^{87} -6.25871e6 q^{89} +1.44105e7 q^{91} -6.48670e6i q^{93} +(222000. + 1.22100e6i) q^{95} +8.65758e6i q^{97} -4.02262e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 100 q^{5} - 1458 q^{9} + 11036 q^{11} - 14850 q^{15} - 8880 q^{19} - 60804 q^{21} - 146250 q^{25} - 38880 q^{29} + 480496 q^{31} - 619300 q^{35} + 691092 q^{39} + 599044 q^{41} + 72900 q^{45} - 888666 q^{49}+ \cdots - 8045244 q^{99}+O(q^{100}) $ 2 * q - 100 * q^5 - 1458 * q^9 + 11036 * q^11 - 14850 * q^15 - 8880 * q^19 - 60804 * q^21 - 146250 * q^25 - 38880 * q^29 + 480496 * q^31 - 619300 * q^35 + 691092 * q^39 + 599044 * q^41 + 72900 * q^45 - 888666 * q^49 + 1739124 * q^51 - 551800 * q^55 - 3690220 * q^59 - 1723436 * q^61 + 7038900 * q^65 + 5154408 * q^69 + 6853896 * q^71 + 1485000 * q^75 - 6275520 * q^79 + 1062882 * q^81 + 17713300 * q^85 - 12517420 * q^89 + 28821096 * q^91 + 444000 * q^95 - 8045244 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	27.0000i		−	0.577350i
$3$		−	27.0000i		−	0.577350i
$4$		0			0
$5$	−50.0000	−	275.000i	−0.178885	−	0.983870i
$5$	−50.0000	−	275.000i	−0.178885	−	0.983870i
$6$		0			0
$7$		−	1126.00i		−	1.24078i	−0.784293	−	0.620391i	$-0.786974\pi$
$7$		−	1126.00i		−	1.24078i	0.784293	−	0.620391i	$-0.213026\pi$
$8$		0			0
$9$	−729.000			−0.333333
$10$		0			0
$11$	5518.00			1.24999			0.624996	−	0.780628i	$-0.285101\pi$
$11$	5518.00			1.24999			0.624996	+	0.780628i	$0.285101\pi$
$12$		0			0
$13$			12798.0i			1.61562i	0.589440	+	0.807812i	$0.299348\pi$
$13$			12798.0i			1.61562i	−0.589440	+	0.807812i	$0.700652\pi$
$14$		0			0
$15$	−7425.00	+	1350.00i	−0.568038	+	0.103280i
$16$		0			0
$17$			32206.0i			1.58988i	0.606685	+	0.794942i	$0.292499\pi$
$17$			32206.0i			1.58988i	−0.606685	+	0.794942i	$0.707501\pi$
$18$		0			0
$19$	−4440.00			−0.148506			−0.0742532	−	0.997239i	$-0.523657\pi$
$19$	−4440.00			−0.148506			−0.0742532	+	0.997239i	$0.523657\pi$
$20$		0			0
$21$	−30402.0			−0.716365
$22$		0			0
$23$			95452.0i			1.63583i	0.575341	+	0.817914i	$0.304870\pi$
$23$			95452.0i			1.63583i	−0.575341	+	0.817914i	$0.695130\pi$
$24$		0			0
$25$	−73125.0	+	27500.0i	−0.936000	+	0.352000i
$26$		0			0
$27$			19683.0i			0.192450i
$28$		0			0
$29$	−19440.0			−0.148014			−0.0740071	−	0.997258i	$-0.523579\pi$
$29$	−19440.0			−0.148014			−0.0740071	+	0.997258i	$0.523579\pi$
$30$		0			0
$31$	240248.			1.44842			0.724209	−	0.689581i	$-0.242205\pi$
$31$	240248.			1.44842			0.724209	+	0.689581i	$0.242205\pi$
$32$		0			0
$33$		−	148986.i		−	0.721683i
$34$		0			0
$35$	−309650.	+	56300.0i	−1.22077	+	0.221958i
$36$		0			0
$37$		−	77834.0i		−	0.252617i	−0.991991	−	0.126309i	$-0.959687\pi$
$37$		−	77834.0i		−	0.252617i	0.991991	−	0.126309i	$-0.0403130\pi$
$38$		0			0
$39$	345546.			0.932781
$40$		0			0
$41$	299522.			0.678712			0.339356	−	0.940658i	$-0.389791\pi$
$41$	299522.			0.678712			0.339356	+	0.940658i	$0.389791\pi$
$42$		0			0
$43$			416212.i			0.798316i	0.916882	+	0.399158i	$0.130698\pi$
$43$			416212.i			0.798316i	−0.916882	+	0.399158i	$0.869302\pi$
$44$		0			0
$45$	36450.0	+	200475.i	0.0596285	+	0.327957i
$46$		0			0
$47$		−	322976.i		−	0.453762i	−0.973923	−	0.226881i	$-0.927147\pi$
$47$		−	322976.i		−	0.453762i	0.973923	−	0.226881i	$-0.0728528\pi$
$48$		0			0
$49$	−444333.			−0.539538
$50$		0			0
$51$	869562.			0.917920
$52$		0			0
$53$			880878.i			0.812737i	0.913709	+	0.406369i	$0.133205\pi$
$53$			880878.i			0.812737i	−0.913709	+	0.406369i	$0.866795\pi$
$54$		0			0
$55$	−275900.	−	1.51745e6i	−0.223605	−	1.22983i
$56$		0			0
$57$			119880.i			0.0857402i
$58$		0			0
$59$	−1.84511e6			−1.16961			−0.584804	−	0.811175i	$-0.698829\pi$
$59$	−1.84511e6			−1.16961			−0.584804	+	0.811175i	$0.698829\pi$
$60$		0			0
$61$	−861718.			−0.486083			−0.243042	−	0.970016i	$-0.578145\pi$
$61$	−861718.			−0.486083			−0.243042	+	0.970016i	$0.578145\pi$
$62$		0			0
$63$			820854.i			0.413594i
$64$		0			0
$65$	3.51945e6	−	639900.i	1.58956	−	0.289012i
$66$		0			0
$67$			673864.i			0.273722i	0.990590	+	0.136861i	$0.0437014\pi$
$67$			673864.i			0.273722i	−0.990590	+	0.136861i	$0.956299\pi$
$68$		0			0
$69$	2.57720e6			0.944446
$70$		0			0
$71$	3.42695e6			1.13633			0.568163	−	0.822916i	$-0.307654\pi$
$71$	3.42695e6			1.13633			0.568163	+	0.822916i	$0.307654\pi$
$72$		0			0
$73$			4.67875e6i			1.40767i	0.710365	+	0.703833i	$0.248530\pi$
$73$			4.67875e6i			1.40767i	−0.710365	+	0.703833i	$0.751470\pi$
$74$		0			0
$75$	742500.	+	1.97438e6i	0.203227	+	0.540400i
$76$		0			0
$77$		−	6.21327e6i		−	1.55097i
$78$		0			0
$79$	−3.13776e6			−0.716020			−0.358010	−	0.933718i	$-0.616545\pi$
$79$	−3.13776e6			−0.716020			−0.358010	+	0.933718i	$0.616545\pi$
$80$		0			0
$81$	531441.			0.111111
$82$		0			0
$83$			484132.i			0.0929374i	0.998920	+	0.0464687i	$0.0147968\pi$
$83$			484132.i			0.0929374i	−0.998920	+	0.0464687i	$0.985203\pi$
$84$		0			0
$85$	8.85665e6	−	1.61030e6i	1.56424	−	0.284407i
$86$		0			0
$87$			524880.i			0.0854560i
$88$		0			0
$89$	−6.25871e6			−0.941065			−0.470533	−	0.882383i	$-0.655938\pi$
$89$	−6.25871e6			−0.941065			−0.470533	+	0.882383i	$0.655938\pi$
$90$		0			0
$91$	1.44105e7			2.00464
$92$		0			0
$93$		−	6.48670e6i		−	0.836244i
$94$		0			0
$95$	222000.	+	1.22100e6i	0.0265656	+	0.146111i
$96$		0			0
$97$			8.65758e6i			0.963153i	0.876404	+	0.481576i	$0.159936\pi$
$97$			8.65758e6i			0.963153i	−0.876404	+	0.481576i	$0.840064\pi$
$98$		0			0
$99$	−4.02262e6			−0.416664
$100$		0			0
$101$	−7.52329e6			−0.726579			−0.363290	−	0.931676i	$-0.618346\pi$
$101$	−7.52329e6			−0.726579			−0.363290	+	0.931676i	$0.618346\pi$
$102$		0			0
$103$		−	1.40261e7i		−	1.26476i	−0.774659	−	0.632380i	$-0.782078\pi$
$103$		−	1.40261e7i		−	1.26476i	0.774659	−	0.632380i	$-0.217922\pi$
$104$		0			0
$105$	1.52010e6	+	8.36055e6i	0.128147	+	0.704810i
$106$		0			0
$107$			1.59380e6i			0.125774i	0.998021	+	0.0628871i	$0.0200308\pi$
$107$			1.59380e6i			0.125774i	−0.998021	+	0.0628871i	$0.979969\pi$
$108$		0			0
$109$	1.66129e7			1.22872			0.614360	−	0.789026i	$-0.289414\pi$
$109$	1.66129e7			1.22872			0.614360	+	0.789026i	$0.289414\pi$
$110$		0			0
$111$	−2.10152e6			−0.145849
$112$		0			0
$113$			536598.i			0.0349844i	0.999847	+	0.0174922i	$0.00556823\pi$
$113$			536598.i			0.0349844i	−0.999847	+	0.0174922i	$0.994432\pi$
$114$		0			0
$115$	2.62493e7	−	4.77260e6i	1.60944	−	0.292626i
$116$		0			0
$117$		−	9.32974e6i		−	0.538541i
$118$		0			0
$119$	3.62640e7			1.97270
$120$		0			0
$121$	1.09612e7			0.562480
$122$		0			0
$123$		−	8.08709e6i		−	0.391854i
$124$		0			0
$125$	1.12188e7	+	1.87344e7i	0.513759	+	0.857935i
$126$		0			0
$127$			3.90459e7i			1.69146i	0.533608	+	0.845732i	$0.320836\pi$
$127$			3.90459e7i			1.69146i	−0.533608	+	0.845732i	$0.679164\pi$
$128$		0			0
$129$	1.12377e7			0.460908
$130$		0			0
$131$	−4.08046e6			−0.158584			−0.0792921	−	0.996851i	$-0.525266\pi$
$131$	−4.08046e6			−0.158584			−0.0792921	+	0.996851i	$0.525266\pi$
$132$		0			0
$133$			4.99944e6i			0.184264i
$134$		0			0
$135$	5.41282e6	−	984150.i	0.189346	−	0.0344265i
$136$		0			0
$137$		−	2.34170e7i		−	0.778052i	−0.921227	−	0.389026i	$-0.872812\pi$
$137$		−	2.34170e7i		−	0.778052i	0.921227	−	0.389026i	$-0.127188\pi$
$138$		0			0
$139$	−1.35950e6			−0.0429366			−0.0214683	−	0.999770i	$-0.506834\pi$
$139$	−1.35950e6			−0.0429366			−0.0214683	+	0.999770i	$0.506834\pi$
$140$		0			0
$141$	−8.72035e6			−0.261979
$142$		0			0
$143$			7.06194e7i			2.01952i
$144$		0			0
$145$	972000.	+	5.34600e6i	0.0264776	+	0.145627i
$146$		0			0
$147$			1.19970e7i			0.311503i
$148$		0			0
$149$	6.79776e7			1.68350			0.841752	−	0.539865i	$-0.181525\pi$
$149$	6.79776e7			1.68350			0.841752	+	0.539865i	$0.181525\pi$
$150$		0			0
$151$	−3.00991e7			−0.711434			−0.355717	−	0.934594i	$-0.615763\pi$
$151$	−3.00991e7			−0.711434			−0.355717	+	0.934594i	$0.615763\pi$
$152$		0			0
$153$		−	2.34782e7i		−	0.529961i
$154$		0			0
$155$	−1.20124e7	−	6.60682e7i	−0.259101	−	1.42505i
$156$		0			0
$157$			1.09929e7i			0.226706i	0.993555	+	0.113353i	$0.0361590\pi$
$157$			1.09929e7i			0.226706i	−0.993555	+	0.113353i	$0.963841\pi$
$158$		0			0
$159$	2.37837e7			0.469234
$160$		0			0
$161$	1.07479e8			2.02970
$162$		0			0
$163$		−	8.99208e7i		−	1.62631i	−0.582047	−	0.813155i	$-0.697748\pi$
$163$		−	8.99208e7i		−	1.62631i	0.582047	−	0.813155i	$-0.302252\pi$
$164$		0			0
$165$	−4.09711e7	+	7.44930e6i	−0.710043	+	0.129099i
$166$		0			0
$167$		−	4.57557e7i		−	0.760217i	−0.924942	−	0.380108i	$-0.875887\pi$
$167$		−	4.57557e7i		−	0.760217i	0.924942	−	0.380108i	$-0.124113\pi$
$168$		0			0
$169$	−1.01040e8			−1.61024
$170$		0			0
$171$	3.23676e6			0.0495022
$172$		0			0
$173$			9.47012e7i			1.39057i	0.718732	+	0.695287i	$0.244723\pi$
$173$			9.47012e7i			1.39057i	−0.718732	+	0.695287i	$0.755277\pi$
$174$		0			0
$175$	3.09650e7	+	8.23388e7i	0.436755	+	1.16137i
$176$		0			0
$177$			4.98180e7i			0.675273i
$178$		0			0
$179$	7.12070e7			0.927977			0.463989	−	0.885841i	$-0.346418\pi$
$179$	7.12070e7			0.927977			0.463989	+	0.885841i	$0.346418\pi$
$180$		0			0
$181$	6.10292e7			0.765003			0.382501	−	0.923955i	$-0.375063\pi$
$181$	6.10292e7			0.765003			0.382501	+	0.923955i	$0.375063\pi$
$182$		0			0
$183$			2.32664e7i			0.280640i
$184$		0			0
$185$	−2.14044e7	+	3.89170e6i	−0.248543	+	0.0451896i
$186$		0			0
$187$			1.77713e8i			1.98734i
$188$		0			0
$189$	2.21631e7			0.238788
$190$		0			0
$191$	−5.63172e7			−0.584823			−0.292411	−	0.956293i	$-0.594458\pi$
$191$	−5.63172e7			−0.584823			−0.292411	+	0.956293i	$0.594458\pi$
$192$		0			0
$193$			9.97730e7i			0.998993i	0.866316	+	0.499496i	$0.166482\pi$
$193$			9.97730e7i			0.998993i	−0.866316	+	0.499496i	$0.833518\pi$
$194$		0			0
$195$	−1.72773e7	−	9.50252e7i	−0.166861	−	0.917735i
$196$		0			0
$197$		−	1.25457e8i		−	1.16913i	−0.811346	−	0.584566i	$-0.801265\pi$
$197$		−	1.25457e8i		−	1.16913i	0.811346	−	0.584566i	$-0.198735\pi$
$198$		0			0
$199$	−2.32320e6			−0.0208978			−0.0104489	−	0.999945i	$-0.503326\pi$
$199$	−2.32320e6			−0.0208978			−0.0104489	+	0.999945i	$0.503326\pi$
$200$		0			0
$201$	1.81943e7			0.158034
$202$		0			0
$203$			2.18894e7i			0.183653i
$204$		0			0
$205$	−1.49761e7	−	8.23686e7i	−0.121412	−	0.667764i
$206$		0			0
$207$		−	6.95845e7i		−	0.545276i
$208$		0			0
$209$	−2.44999e7			−0.185632
$210$		0			0
$211$	−8.85076e7			−0.648622			−0.324311	−	0.945950i	$-0.605132\pi$
$211$	−8.85076e7			−0.648622			−0.324311	+	0.945950i	$0.605132\pi$
$212$		0			0
$213$		−	9.25276e7i		−	0.656059i
$214$		0			0
$215$	1.14458e8	−	2.08106e7i	0.785439	−	0.142807i
$216$		0			0
$217$		−	2.70519e8i		−	1.79717i
$218$		0			0
$219$	1.26326e8			0.812716
$220$		0			0
$221$	−4.12172e8			−2.56866
$222$		0			0
$223$		−	629858.i		−	0.00380343i	−0.999998	−	0.00190172i	$-0.999395\pi$
$223$		−	629858.i		−	0.00380343i	0.999998	−	0.00190172i	$-0.000605335\pi$
$224$		0			0
$225$	5.33081e7	−	2.00475e7i	0.312000	−	0.117333i
$226$		0			0
$227$		−	1.74337e8i		−	0.989236i	−0.869111	−	0.494618i	$-0.835308\pi$
$227$		−	1.74337e8i		−	0.989236i	0.869111	−	0.494618i	$-0.164692\pi$
$228$		0			0
$229$	2.00644e8			1.10408			0.552042	−	0.833816i	$-0.313849\pi$
$229$	2.00644e8			1.10408			0.552042	+	0.833816i	$0.313849\pi$
$230$		0			0
$231$	−1.67758e8			−0.895451
$232$		0			0
$233$		−	1.66522e8i		−	0.862434i	−0.902248	−	0.431217i	$-0.858084\pi$
$233$		−	1.66522e8i		−	0.862434i	0.902248	−	0.431217i	$-0.141916\pi$
$234$		0			0
$235$	−8.88184e7	+	1.61488e7i	−0.446442	+	0.0811713i
$236$		0			0
$237$			8.47195e7i			0.413394i
$238$		0			0
$239$	−1.17669e8			−0.557531			−0.278765	−	0.960359i	$-0.589925\pi$
$239$	−1.17669e8			−0.557531			−0.278765	+	0.960359i	$0.589925\pi$
$240$		0			0
$241$	−4.04961e8			−1.86360			−0.931802	−	0.362967i	$-0.881764\pi$
$241$	−4.04961e8			−1.86360			−0.931802	+	0.362967i	$0.881764\pi$
$242$		0			0
$243$		−	1.43489e7i		−	0.0641500i
$244$		0			0
$245$	2.22166e7	+	1.22192e8i	0.0965155	+	0.530836i
$246$		0			0
$247$		−	5.68231e7i		−	0.239931i
$248$		0			0
$249$	1.30716e7			0.0536574
$250$		0			0
$251$	5.93170e7			0.236767			0.118383	−	0.992968i	$-0.462229\pi$
$251$	5.93170e7			0.236767			0.118383	+	0.992968i	$0.462229\pi$
$252$		0			0
$253$			5.26704e8i			2.04477i
$254$		0			0
$255$	−4.34781e7	−	2.39130e8i	−0.164203	−	0.903114i
$256$		0			0
$257$			2.48636e7i			0.0913688i	0.998956	+	0.0456844i	$0.0145469\pi$
$257$			2.48636e7i			0.0913688i	−0.998956	+	0.0456844i	$0.985453\pi$
$258$		0			0
$259$	−8.76411e7			−0.313443
$260$		0			0
$261$	1.41718e7			0.0493381
$262$		0			0
$263$		−	5.36503e6i		−	0.0181856i	−0.999959	−	0.00909278i	$-0.997106\pi$
$263$		−	5.36503e6i		−	0.0181856i	0.999959	−	0.00909278i	$-0.00289436\pi$
$264$		0			0
$265$	2.42241e8	−	4.40439e7i	0.799628	−	0.145387i
$266$		0			0
$267$			1.68985e8i			0.543324i
$268$		0			0
$269$	1.83159e8			0.573713			0.286856	−	0.957974i	$-0.407390\pi$
$269$	1.83159e8			0.573713			0.286856	+	0.957974i	$0.407390\pi$
$270$		0			0
$271$	−2.54405e8			−0.776486			−0.388243	−	0.921557i	$-0.626918\pi$
$271$	−2.54405e8			−0.776486			−0.388243	+	0.921557i	$0.626918\pi$
$272$		0			0
$273$		−	3.89085e8i		−	1.15738i
$274$		0			0
$275$	−4.03504e8	+	1.51745e8i	−1.16999	+	0.439997i
$276$		0			0
$277$			5.92863e8i			1.67601i	0.545666	+	0.838003i	$0.316277\pi$
$277$			5.92863e8i			1.67601i	−0.545666	+	0.838003i	$0.683723\pi$
$278$		0			0
$279$	−1.75141e8			−0.482806
$280$		0			0
$281$	−6.22030e8			−1.67240			−0.836199	−	0.548427i	$-0.815227\pi$
$281$	−6.22030e8			−1.67240			−0.836199	+	0.548427i	$0.815227\pi$
$282$		0			0
$283$			3.94755e8i			1.03532i	0.855586	+	0.517661i	$0.173197\pi$
$283$			3.94755e8i			1.03532i	−0.855586	+	0.517661i	$0.826803\pi$
$284$		0			0
$285$	3.29670e7	−	5.99400e6i	0.0843573	−	0.0153377i
$286$		0			0
$287$		−	3.37262e8i		−	0.842133i
$288$		0			0
$289$	−6.26888e8			−1.52773
$290$		0			0
$291$	2.33755e8			0.556077
$292$		0			0
$293$		−	6.89404e8i		−	1.60117i	−0.599220	−	0.800585i	$-0.704522\pi$
$293$		−	6.89404e8i		−	1.60117i	0.599220	−	0.800585i	$-0.295478\pi$
$294$		0			0
$295$	9.22555e7	+	5.07405e8i	0.209226	+	1.15074i
$296$		0			0
$297$			1.08611e8i			0.240561i
$298$		0			0
$299$	−1.22159e9			−2.64288
$300$		0			0
$301$	4.68655e8			0.990536
$302$		0			0
$303$			2.03129e8i			0.419491i
$304$		0			0
$305$	4.30859e7	+	2.36972e8i	0.0869532	+	0.478243i
$306$		0			0
$307$			2.67769e8i			0.528174i	0.964499	+	0.264087i	$0.0850705\pi$
$307$			2.67769e8i			0.528174i	−0.964499	+	0.264087i	$0.914929\pi$
$308$		0			0
$309$	−3.78706e8			−0.730209
$310$		0			0
$311$	1.94551e8			0.366752			0.183376	−	0.983043i	$-0.441297\pi$
$311$	1.94551e8			0.366752			0.183376	+	0.983043i	$0.441297\pi$
$312$		0			0
$313$			8.81802e8i			1.62542i	0.582667	+	0.812711i	$0.302009\pi$
$313$			8.81802e8i			1.62542i	−0.582667	+	0.812711i	$0.697991\pi$
$314$		0			0
$315$	2.25735e8	−	4.10427e7i	0.406922	−	0.0739859i
$316$		0			0
$317$		−	4.88296e8i		−	0.860946i	−0.902603	−	0.430473i	$-0.858347\pi$
$317$		−	4.88296e8i		−	0.860946i	0.902603	−	0.430473i	$-0.141653\pi$
$318$		0			0
$319$	−1.07270e8			−0.185017
$320$		0			0
$321$	4.30327e7			0.0726158
$322$		0			0
$323$		−	1.42995e8i		−	0.236108i
$324$		0			0
$325$	−3.51945e8	−	9.35854e8i	−0.568700	−	1.51222i
$326$		0			0
$327$		−	4.48549e8i		−	0.709402i
$328$		0			0
$329$	−3.63671e8			−0.563019
$330$		0			0
$331$	7.08199e8			1.07339			0.536695	−	0.843777i	$-0.319673\pi$
$331$	7.08199e8			1.07339			0.536695	+	0.843777i	$0.319673\pi$
$332$		0			0
$333$			5.67410e7i			0.0842058i
$334$		0			0
$335$	1.85313e8	−	3.36932e7i	0.269307	−	0.0489650i
$336$		0			0
$337$			6.69707e8i			0.953192i	0.879122	+	0.476596i	$0.158130\pi$
$337$			6.69707e8i			0.953192i	−0.879122	+	0.476596i	$0.841870\pi$
$338$		0			0
$339$	1.44881e7			0.0201983
$340$		0			0
$341$	1.32569e9			1.81051
$342$		0			0
$343$		−	4.26990e8i		−	0.571332i
$344$		0			0
$345$	−1.28860e8	−	7.08731e8i	−0.168948	−	0.929212i
$346$		0			0
$347$		−	1.01172e9i		−	1.29989i	−0.759982	−	0.649945i	$-0.774792\pi$
$347$		−	1.01172e9i		−	1.29989i	0.759982	−	0.649945i	$-0.225208\pi$
$348$		0			0
$349$	−3.46636e8			−0.436501			−0.218250	−	0.975893i	$-0.570035\pi$
$349$	−3.46636e8			−0.436501			−0.218250	+	0.975893i	$0.570035\pi$
$350$		0			0
$351$	−2.51903e8			−0.310927
$352$		0			0
$353$		−	2.93766e6i		−	0.00355460i	−0.999998	−	0.00177730i	$-0.999434\pi$
$353$		−	2.93766e6i		−	0.00355460i	0.999998	−	0.00177730i	$-0.000565732\pi$
$354$		0			0
$355$	−1.71347e8	−	9.42411e8i	−0.203272	−	1.11800i
$356$		0			0
$357$		−	9.79127e8i		−	1.13894i
$358$		0			0
$359$	1.64977e9			1.88188			0.940942	−	0.338568i	$-0.109942\pi$
$359$	1.64977e9			1.88188			0.940942	+	0.338568i	$0.109942\pi$
$360$		0			0
$361$	−8.74158e8			−0.977946
$362$		0			0
$363$		−	2.95951e8i		−	0.324748i
$364$		0			0
$365$	1.28666e9	−	2.33937e8i	1.38496	−	0.251811i
$366$		0			0
$367$			4.33054e8i			0.457311i	0.973507	+	0.228655i	$0.0734329\pi$
$367$			4.33054e8i			0.457311i	−0.973507	+	0.228655i	$0.926567\pi$
$368$		0			0
$369$	−2.18352e8			−0.226237
$370$		0			0
$371$	9.91869e8			1.00843
$372$		0			0
$373$			7.13249e8i			0.711640i	0.934555	+	0.355820i	$0.115798\pi$
$373$			7.13249e8i			0.711640i	−0.934555	+	0.355820i	$0.884202\pi$
$374$		0			0
$375$	5.05828e8	−	3.02906e8i	0.495329	−	0.296619i
$376$		0			0
$377$		−	2.48793e8i		−	0.239135i
$378$		0			0
$379$	1.92795e8			0.181911			0.0909555	−	0.995855i	$-0.471008\pi$
$379$	1.92795e8			0.181911			0.0909555	+	0.995855i	$0.471008\pi$
$380$		0			0
$381$	1.05424e9			0.976567
$382$		0			0
$383$			1.30544e9i			1.18730i	0.804724	+	0.593649i	$0.202313\pi$
$383$			1.30544e9i			1.18730i	−0.804724	+	0.593649i	$0.797687\pi$
$384$		0			0
$385$	−1.70865e9	+	3.10663e8i	−1.52595	+	0.277445i
$386$		0			0
$387$		−	3.03419e8i		−	0.266105i
$388$		0			0
$389$	−2.53075e7			−0.0217985			−0.0108992	−	0.999941i	$-0.503469\pi$
$389$	−2.53075e7			−0.0217985			−0.0108992	+	0.999941i	$0.503469\pi$
$390$		0			0
$391$	−3.07413e9			−2.60078
$392$		0			0
$393$			1.10172e8i			0.0915586i
$394$		0			0
$395$	1.56888e8	+	8.62884e8i	0.128085	+	0.704470i
$396$		0			0
$397$			1.34997e9i			1.08282i	0.840759	+	0.541410i	$0.182109\pi$
$397$			1.34997e9i			1.08282i	−0.840759	+	0.541410i	$0.817891\pi$
$398$		0			0
$399$	1.34985e8			0.106385
$400$		0			0
$401$	−9.20762e7			−0.0713086			−0.0356543	−	0.999364i	$-0.511352\pi$
$401$	−9.20762e7			−0.0713086			−0.0356543	+	0.999364i	$0.511352\pi$
$402$		0			0
$403$			3.07469e9i			2.34010i
$404$		0			0
$405$	−2.65720e7	−	1.46146e8i	−0.0198762	−	0.109319i
$406$		0			0
$407$		−	4.29488e8i		−	0.315770i
$408$		0			0
$409$	2.68396e9			1.93975			0.969874	−	0.243609i	$-0.0783314\pi$
$409$	2.68396e9			1.93975			0.969874	+	0.243609i	$0.0783314\pi$
$410$		0			0
$411$	−6.32258e8			−0.449209
$412$		0			0
$413$			2.07759e9i			1.45123i
$414$		0			0
$415$	1.33136e8	−	2.42066e7i	0.0914383	−	0.0166251i
$416$		0			0
$417$			3.67065e7i			0.0247894i
$418$		0			0
$419$	8.45201e8			0.561320			0.280660	−	0.959807i	$-0.409447\pi$
$419$	8.45201e8			0.561320			0.280660	+	0.959807i	$0.409447\pi$
$420$		0			0
$421$	2.66227e9			1.73886			0.869430	−	0.494056i	$-0.164486\pi$
$421$	2.66227e9			1.73886			0.869430	+	0.494056i	$0.164486\pi$
$422$		0			0
$423$			2.35450e8i			0.151254i
$424$		0			0
$425$	−8.85665e8	−	2.35506e9i	−0.559639	−	1.48813i
$426$		0			0
$427$			9.70294e8i			0.603123i
$428$		0			0
$429$	1.90672e9			1.16597
$430$		0			0
$431$	9.83169e8			0.591504			0.295752	−	0.955265i	$-0.404430\pi$
$431$	9.83169e8			0.591504			0.295752	+	0.955265i	$0.404430\pi$
$432$		0			0
$433$		−	8.30277e8i		−	0.491491i	−0.969334	−	0.245745i	$-0.920967\pi$
$433$		−	8.30277e8i		−	0.491491i	0.969334	−	0.245745i	$-0.0790327\pi$
$434$		0			0
$435$	1.44342e8	−	2.62440e7i	0.0840776	−	0.0152868i
$436$		0			0
$437$		−	4.23807e8i		−	0.242931i
$438$		0			0
$439$	2.79662e9			1.57764			0.788820	−	0.614625i	$-0.210692\pi$
$439$	2.79662e9			1.57764			0.788820	+	0.614625i	$0.210692\pi$
$440$		0			0
$441$	3.23919e8			0.179846
$442$		0			0
$443$			3.97081e8i			0.217003i	0.994096	+	0.108502i	$0.0346052\pi$
$443$			3.97081e8i			0.217003i	−0.994096	+	0.108502i	$0.965395\pi$
$444$		0			0
$445$	3.12936e8	+	1.72115e9i	0.168343	+	0.925886i
$446$		0			0
$447$		−	1.83540e9i		−	0.971971i
$448$		0			0
$449$	−1.74461e8			−0.0909569			−0.0454785	−	0.998965i	$-0.514481\pi$
$449$	−1.74461e8			−0.0909569			−0.0454785	+	0.998965i	$0.514481\pi$
$450$		0			0
$451$	1.65276e9			0.848384
$452$		0			0
$453$			8.12676e8i			0.410746i
$454$		0			0
$455$	−7.20527e8	−	3.96290e9i	−0.358600	−	1.97230i
$456$		0			0
$457$			3.60027e8i			0.176453i	0.996100	+	0.0882264i	$0.0281199\pi$
$457$			3.60027e8i			0.176453i	−0.996100	+	0.0882264i	$0.971880\pi$
$458$		0			0
$459$	−6.33911e8			−0.305973
$460$		0			0
$461$	−1.33487e8			−0.0634579			−0.0317290	−	0.999497i	$-0.510101\pi$
$461$	−1.33487e8			−0.0634579			−0.0317290	+	0.999497i	$0.510101\pi$
$462$		0			0
$463$			2.93605e9i			1.37477i	0.726294	+	0.687384i	$0.241241\pi$
$463$			2.93605e9i			1.37477i	−0.726294	+	0.687384i	$0.758759\pi$
$464$		0			0
$465$	−1.78384e9	+	3.24335e8i	−0.822756	+	0.149592i
$466$		0			0
$467$		−	1.72777e9i		−	0.785011i	−0.919750	−	0.392506i	$-0.871608\pi$
$467$		−	1.72777e9i		−	0.785011i	0.919750	−	0.392506i	$-0.128392\pi$
$468$		0			0
$469$	7.58771e8			0.339630
$470$		0			0
$471$	2.96807e8			0.130889
$472$		0			0
$473$			2.29666e9i			0.997889i
$474$		0			0
$475$	3.24675e8	−	1.22100e8i	0.139002	−	0.0522743i
$476$		0			0
$477$		−	6.42160e8i		−	0.270912i
$478$		0			0
$479$	−1.69230e9			−0.703562			−0.351781	−	0.936082i	$-0.614424\pi$
$479$	−1.69230e9			−0.703562			−0.351781	+	0.936082i	$0.614424\pi$
$480$		0			0
$481$	9.96120e8			0.408135
$482$		0			0
$483$		−	2.90193e9i		−	1.17185i
$484$		0			0
$485$	2.38083e9	−	4.32879e8i	0.947617	−	0.172294i
$486$		0			0
$487$			1.46319e9i			0.574048i	0.957923	+	0.287024i	$0.0926660\pi$
$487$			1.46319e9i			0.574048i	−0.957923	+	0.287024i	$0.907334\pi$
$488$		0			0
$489$	−2.42786e9			−0.938951
$490$		0			0
$491$	2.86378e9			1.09183			0.545915	−	0.837840i	$-0.316182\pi$
$491$	2.86378e9			1.09183			0.545915	+	0.837840i	$0.316182\pi$
$492$		0			0
$493$		−	6.26085e8i		−	0.235325i
$494$		0			0
$495$	2.01131e8	+	1.10622e9i	0.0745351	+	0.409943i
$496$		0			0
$497$		−	3.85874e9i		−	1.40993i
$498$		0			0
$499$	−1.17178e9			−0.422176			−0.211088	−	0.977467i	$-0.567701\pi$
$499$	−1.17178e9			−0.422176			−0.211088	+	0.977467i	$0.567701\pi$
$500$		0			0
$501$	−1.23540e9			−0.438911
$502$		0			0
$503$		−	3.67913e9i		−	1.28901i	−0.764599	−	0.644506i	$-0.777063\pi$
$503$		−	3.67913e9i		−	1.28901i	0.764599	−	0.644506i	$-0.222937\pi$
$504$		0			0
$505$	3.76164e8	+	2.06890e9i	0.129974	+	0.714859i
$506$		0			0
$507$			2.72809e9i			0.929674i
$508$		0			0
$509$	4.56520e9			1.53443			0.767216	−	0.641389i	$-0.221641\pi$
$509$	4.56520e9			1.53443			0.767216	+	0.641389i	$0.221641\pi$
$510$		0			0
$511$	5.26827e9			1.74661
$512$		0			0
$513$		−	8.73925e7i		−	0.0285801i
$514$		0			0
$515$	−3.85719e9	+	7.01307e8i	−1.24436	+	0.226247i
$516$		0			0
$517$		−	1.78218e9i		−	0.567198i
$518$		0			0
$519$	2.55693e9			0.802848
$520$		0			0
$521$	−5.34662e9			−1.65633			−0.828166	−	0.560484i	$-0.810615\pi$
$521$	−5.34662e9			−1.65633			−0.828166	+	0.560484i	$0.810615\pi$
$522$		0			0
$523$			5.14610e9i			1.57298i	0.617605	+	0.786488i	$0.288103\pi$
$523$			5.14610e9i			1.57298i	−0.617605	+	0.786488i	$0.711897\pi$
$524$		0			0
$525$	2.22315e9	−	8.36055e8i	0.670518	−	0.252161i
$526$		0			0
$527$			7.73743e9i			2.30282i
$528$		0			0
$529$	−5.70626e9			−1.67593
$530$		0			0
$531$	1.34509e9			0.389869
$532$		0			0
$533$			3.83328e9i			1.09654i
$534$		0			0
$535$	4.38296e8	−	7.96902e7i	0.123745	−	0.0224992i
$536$		0			0
$537$		−	1.92259e9i		−	0.535768i
$538$		0			0
$539$	−2.45183e9			−0.674419
$540$		0			0
$541$	−2.32073e8			−0.0630137			−0.0315069	−	0.999504i	$-0.510031\pi$
$541$	−2.32073e8			−0.0630137			−0.0315069	+	0.999504i	$0.510031\pi$
$542$		0			0
$543$		−	1.64779e9i		−	0.441675i
$544$		0			0
$545$	−8.30646e8	−	4.56855e9i	−0.219800	−	1.20890i
$546$		0			0
$547$		−	7.09990e9i		−	1.85480i	−0.374075	−	0.927399i	$-0.622040\pi$
$547$		−	7.09990e9i		−	1.85480i	0.374075	−	0.927399i	$-0.377960\pi$
$548$		0			0
$549$	6.28192e8			0.162028
$550$		0			0
$551$	8.63136e7			0.0219811
$552$		0			0
$553$			3.53312e9i			0.888424i
$554$		0			0
$555$	1.05076e8	+	5.77917e8i	0.0260902	+	0.143496i
$556$		0			0
$557$			2.92398e9i			0.716937i	0.933542	+	0.358469i	$0.116701\pi$
$557$			2.92398e9i			0.716937i	−0.933542	+	0.358469i	$0.883299\pi$
$558$		0			0
$559$	−5.32668e9			−1.28978
$560$		0			0
$561$	4.79824e9			1.14739
$562$		0			0
$563$			3.17221e9i			0.749174i	0.927192	+	0.374587i	$0.122215\pi$
$563$			3.17221e9i			0.749174i	−0.927192	+	0.374587i	$0.877785\pi$
$564$		0			0
$565$	1.47564e8	−	2.68299e7i	0.0344201	−	0.00625820i
$566$		0			0
$567$		−	5.98403e8i		−	0.137865i
$568$		0			0
$569$	−2.75561e9			−0.627082			−0.313541	−	0.949575i	$-0.601515\pi$
$569$	−2.75561e9			−0.627082			−0.313541	+	0.949575i	$0.601515\pi$
$570$		0			0
$571$	−4.36061e9			−0.980213			−0.490107	−	0.871662i	$-0.663042\pi$
$571$	−4.36061e9			−0.980213			−0.490107	+	0.871662i	$0.663042\pi$
$572$		0			0
$573$			1.52056e9i			0.337648i
$574$		0			0
$575$	−2.62493e9	−	6.97993e9i	−0.575811	−	1.53113i
$576$		0			0
$577$		−	6.07328e9i		−	1.31616i	−0.752949	−	0.658079i	$-0.771369\pi$
$577$		−	6.07328e9i		−	1.31616i	0.752949	−	0.658079i	$-0.228631\pi$
$578$		0			0
$579$	2.69387e9			0.576769
$580$		0			0
$581$	5.45133e8			0.115315
$582$		0			0
$583$			4.86068e9i			1.01592i
$584$		0			0
$585$	−2.56568e9	+	4.66487e8i	−0.529855	+	0.0963372i
$586$		0			0
$587$		−	4.35319e9i		−	0.888329i	−0.895945	−	0.444165i	$-0.853501\pi$
$587$		−	4.35319e9i		−	0.888329i	0.895945	−	0.444165i	$-0.146499\pi$
$588$		0			0
$589$	−1.06670e9			−0.215099
$590$		0			0
$591$	−3.38734e9			−0.674999
$592$		0			0
$593$			2.22483e9i			0.438133i	0.975710	+	0.219067i	$0.0703012\pi$
$593$			2.22483e9i			0.438133i	−0.975710	+	0.219067i	$0.929699\pi$
$594$		0			0
$595$	−1.81320e9	−	9.97259e9i	−0.352887	−	1.94088i
$596$		0			0
$597$			6.27264e7i			0.0120654i
$598$		0			0
$599$	−3.63376e9			−0.690815			−0.345408	−	0.938453i	$-0.612259\pi$
$599$	−3.63376e9			−0.690815			−0.345408	+	0.938453i	$0.612259\pi$
$600$		0			0
$601$	−6.00206e9			−1.12782			−0.563909	−	0.825837i	$-0.690703\pi$
$601$	−6.00206e9			−1.12782			−0.563909	+	0.825837i	$0.690703\pi$
$602$		0			0
$603$		−	4.91247e8i		−	0.0912408i
$604$		0			0
$605$	−5.48058e8	−	3.01432e9i	−0.100620	−	0.553408i
$606$		0			0
$607$		−	2.80807e9i		−	0.509621i	−0.966991	−	0.254811i	$-0.917987\pi$
$607$		−	2.80807e9i		−	0.509621i	0.966991	−	0.254811i	$-0.0820131\pi$
$608$		0			0
$609$	5.91015e8			0.106032
$610$		0			0
$611$	4.13345e9			0.733108
$612$		0			0
$613$			8.34916e9i			1.46397i	0.681323	+	0.731983i	$0.261405\pi$
$613$			8.34916e9i			1.46397i	−0.681323	+	0.731983i	$0.738595\pi$
$614$		0			0
$615$	−2.22395e9	+	4.04355e8i	−0.385534	+	0.0700970i
$616$		0			0
$617$			2.16207e9i			0.370571i	0.982685	+	0.185286i	$0.0593211\pi$
$617$			2.16207e9i			0.370571i	−0.982685	+	0.185286i	$0.940679\pi$
$618$		0			0
$619$	−2.82281e9			−0.478370			−0.239185	−	0.970974i	$-0.576880\pi$
$619$	−2.82281e9			−0.478370			−0.239185	+	0.970974i	$0.576880\pi$
$620$		0			0
$621$	−1.87878e9			−0.314815
$622$		0			0
$623$			7.04731e9i			1.16766i
$624$		0			0
$625$	4.59102e9	−	4.02188e9i	0.752192	−	0.658944i
$626$		0			0
$627$			6.61498e8i			0.107175i
$628$		0			0
$629$	2.50672e9			0.401633
$630$		0			0
$631$	−7.87109e9			−1.24719			−0.623595	−	0.781748i	$-0.714328\pi$
$631$	−7.87109e9			−1.24719			−0.623595	+	0.781748i	$0.714328\pi$
$632$		0			0
$633$			2.38970e9i			0.374482i
$634$		0			0
$635$	1.07376e10	−	1.95230e9i	1.66418	−	0.302578i
$636$		0			0
$637$		−	5.68657e9i		−	0.871691i
$638$		0			0
$639$	−2.49825e9			−0.378776
$640$		0			0
$641$	−3.77527e9			−0.566168			−0.283084	−	0.959095i	$-0.591357\pi$
$641$	−3.77527e9			−0.566168			−0.283084	+	0.959095i	$0.591357\pi$
$642$		0			0
$643$		−	2.51923e9i		−	0.373705i	−0.982388	−	0.186852i	$-0.940171\pi$
$643$		−	2.51923e9i		−	0.373705i	0.982388	−	0.186852i	$-0.0598286\pi$
$644$		0			0
$645$	−5.61886e8	−	3.09037e9i	−0.0824498	−	0.453474i
$646$		0			0
$647$			3.83911e9i			0.557269i	0.960397	+	0.278635i	$0.0898819\pi$
$647$			3.83911e9i			0.557269i	−0.960397	+	0.278635i	$0.910118\pi$
$648$		0			0
$649$	−1.01813e10			−1.46200
$650$		0			0
$651$	−7.30402e9			−1.03760
$652$		0			0
$653$		−	1.00785e10i		−	1.41644i	−0.705989	−	0.708222i	$-0.749497\pi$
$653$		−	1.00785e10i		−	1.41644i	0.705989	−	0.708222i	$-0.250503\pi$
$654$		0			0
$655$	2.04023e8	+	1.12213e9i	0.0283684	+	0.156026i
$656$		0			0
$657$		−	3.41081e9i		−	0.469222i
$658$		0			0
$659$	−8.94469e8			−0.121749			−0.0608746	−	0.998145i	$-0.519389\pi$
$659$	−8.94469e8			−0.121749			−0.0608746	+	0.998145i	$0.519389\pi$
$660$		0			0
$661$	1.19420e10			1.60832			0.804159	−	0.594414i	$-0.202616\pi$
$661$	1.19420e10			1.60832			0.804159	+	0.594414i	$0.202616\pi$
$662$		0			0
$663$			1.11287e10i			1.48301i
$664$		0			0
$665$	1.37485e9	−	2.49972e8i	0.181292	−	0.0329622i
$666$		0			0
$667$		−	1.85559e9i		−	0.242126i
$668$		0			0
$669$	−1.70062e7			−0.00219591
$670$		0			0
$671$	−4.75496e9			−0.607600
$672$		0			0
$673$		−	1.66611e9i		−	0.210693i	−0.994436	−	0.105347i	$-0.966405\pi$
$673$		−	1.66611e9i		−	0.210693i	0.994436	−	0.105347i	$-0.0335952\pi$
$674$		0			0
$675$	−5.41282e8	−	1.43932e9i	−0.0677424	−	0.180133i
$676$		0			0
$677$		−	3.07840e9i		−	0.381298i	−0.981658	−	0.190649i	$-0.938941\pi$
$677$		−	3.07840e9i		−	0.381298i	0.981658	−	0.190649i	$-0.0610592\pi$
$678$		0			0
$679$	9.74843e9			1.19506
$680$		0			0
$681$	−4.70711e9			−0.571135
$682$		0			0
$683$			3.66304e9i			0.439915i	0.975509	+	0.219958i	$0.0705919\pi$
$683$			3.66304e9i			0.439915i	−0.975509	+	0.219958i	$0.929408\pi$
$684$		0			0
$685$	−6.43967e9	+	1.17085e9i	−0.765502	+	0.139182i
$686$		0			0
$687$		−	5.41739e9i		−	0.637443i
$688$		0			0
$689$	−1.12735e10			−1.31308
$690$		0			0
$691$	−1.03135e10			−1.18914			−0.594569	−	0.804045i	$-0.702677\pi$
$691$	−1.03135e10			−1.18914			−0.594569	+	0.804045i	$0.702677\pi$
$692$		0			0
$693$			4.52947e9i			0.516989i
$694$		0			0
$695$	6.79750e7	+	3.73862e8i	0.00768073	+	0.0422440i
$696$		0			0
$697$			9.64641e9i			1.07907i
$698$		0			0
$699$	−4.49609e9			−0.497926
$700$		0			0
$701$	−5.82173e9			−0.638321			−0.319161	−	0.947701i	$-0.603401\pi$
$701$	−5.82173e9			−0.638321			−0.319161	+	0.947701i	$0.603401\pi$
$702$		0			0
$703$			3.45583e8i			0.0375153i
$704$		0			0
$705$	4.36018e8	+	2.39810e9i	0.0468643	+	0.257754i
$706$		0			0
$707$			8.47122e9i			0.901526i
$708$		0			0
$709$	8.52806e9			0.898645			0.449323	−	0.893370i	$-0.351665\pi$
$709$	8.52806e9			0.898645			0.449323	+	0.893370i	$0.351665\pi$
$710$		0			0
$711$	2.28743e9			0.238673
$712$		0			0
$713$			2.29322e10i			2.36936i
$714$		0			0
$715$	1.94203e10	−	3.53097e9i	1.98694	−	0.361262i
$716$		0			0
$717$			3.17706e9i			0.321891i
$718$		0			0
$719$	1.73546e9			0.174126			0.0870631	−	0.996203i	$-0.472252\pi$
$719$	1.73546e9			0.174126			0.0870631	+	0.996203i	$0.472252\pi$
$720$		0			0
$721$	−1.57934e10			−1.56929
$722$		0			0
$723$			1.09339e10i			1.07595i
$724$		0			0
$725$	1.42155e9	−	5.34600e8i	0.138541	−	0.0521010i
$726$		0			0
$727$		−	1.16762e10i		−	1.12702i	−0.826110	−	0.563509i	$-0.809451\pi$
$727$		−	1.16762e10i		−	1.12702i	0.826110	−	0.563509i	$-0.190549\pi$
$728$		0			0
$729$	−3.87420e8			−0.0370370
$730$		0			0
$731$	−1.34045e10			−1.26923
$732$		0			0
$733$			4.92260e9i			0.461669i	0.972993	+	0.230834i	$0.0741456\pi$
$733$			4.92260e9i			0.461669i	−0.972993	+	0.230834i	$0.925854\pi$
$734$		0			0
$735$	3.29917e9	−	5.99850e8i	0.306478	−	0.0557233i
$736$		0			0
$737$			3.71838e9i			0.342151i
$738$		0			0
$739$	−4.46577e9			−0.407043			−0.203522	−	0.979070i	$-0.565239\pi$
$739$	−4.46577e9			−0.407043			−0.203522	+	0.979070i	$0.565239\pi$
$740$		0			0
$741$	−1.53422e9			−0.138524
$742$		0			0
$743$		−	8.36112e8i		−	0.0747831i	−0.999301	−	0.0373916i	$-0.988095\pi$
$743$		−	8.36112e8i		−	0.0747831i	0.999301	−	0.0373916i	$-0.0119049\pi$
$744$		0			0
$745$	−3.39888e9	−	1.86938e10i	−0.301154	−	1.65635i
$746$		0			0
$747$		−	3.52932e8i		−	0.0309791i
$748$		0			0
$749$	1.79462e9			0.156058
$750$		0			0
$751$	1.37161e10			1.18166			0.590829	−	0.806797i	$-0.298801\pi$
$751$	1.37161e10			1.18166			0.590829	+	0.806797i	$0.298801\pi$
$752$		0			0
$753$		−	1.60156e9i		−	0.136697i
$754$		0			0
$755$	1.50496e9	+	8.27726e9i	0.127265	+	0.699958i
$756$		0			0
$757$		−	1.41449e10i		−	1.18513i	−0.805524	−	0.592563i	$-0.798116\pi$
$757$		−	1.41449e10i		−	1.18513i	0.805524	−	0.592563i	$-0.201884\pi$
$758$		0			0
$759$	1.42210e10			1.18055
$760$		0			0
$761$	2.16828e10			1.78348			0.891741	−	0.452547i	$-0.149484\pi$
$761$	2.16828e10			1.78348			0.891741	+	0.452547i	$0.149484\pi$
$762$		0			0
$763$		−	1.87061e10i		−	1.52457i
$764$		0			0
$765$	−6.45650e9	+	1.17391e9i	−0.521413	+	0.0948024i
$766$		0			0
$767$		−	2.36137e10i		−	1.88965i
$768$		0			0
$769$	1.45804e10			1.15618			0.578092	−	0.815972i	$-0.303798\pi$
$769$	1.45804e10			1.15618			0.578092	+	0.815972i	$0.303798\pi$
$770$		0			0
$771$	6.71317e8			0.0527518
$772$		0			0
$773$		−	1.83373e9i		−	0.142793i	−0.997448	−	0.0713966i	$-0.977254\pi$
$773$		−	1.83373e9i		−	0.142793i	0.997448	−	0.0713966i	$-0.0227456\pi$
$774$		0			0
$775$	−1.75681e10	+	6.60682e9i	−1.35572	+	0.509843i
$776$		0			0
$777$			2.36631e9i			0.180966i
$778$		0			0
$779$	−1.32988e9			−0.100793
$780$		0			0
$781$	1.89099e10			1.42040
$782$		0			0
$783$		−	3.82638e8i		−	0.0284853i
$784$		0			0
$785$	3.02304e9	−	5.49643e8i	0.223049	−	0.0405543i
$786$		0			0
$787$			1.48611e10i			1.08678i	0.839482	+	0.543388i	$0.182859\pi$
$787$			1.48611e10i			1.08678i	−0.839482	+	0.543388i	$0.817141\pi$
$788$		0			0
$789$	−1.44856e8			−0.0104994
$790$		0			0
$791$	6.04209e8			0.0434080
$792$		0			0
$793$		−	1.10283e10i		−	0.785328i
$794$		0			0
$795$	−1.18919e9	−	6.54052e9i	−0.0839392	−	0.461665i
$796$		0			0
$797$			1.46595e10i			1.02569i	0.858483	+	0.512843i	$0.171408\pi$
$797$			1.46595e10i			1.02569i	−0.858483	+	0.512843i	$0.828592\pi$
$798$		0			0
$799$	1.04018e10			0.721429
$800$		0			0
$801$	4.56260e9			0.313688
$802$		0			0
$803$			2.58173e10i			1.75957i
$804$		0			0
$805$	−5.37395e9	−	2.95567e10i	−0.363085	−	1.99697i
$806$		0			0
$807$		−	4.94528e9i		−	0.331233i
$808$		0			0
$809$	9.95193e9			0.660827			0.330413	−	0.943836i	$-0.392812\pi$
$809$	9.95193e9			0.660827			0.330413	+	0.943836i	$0.392812\pi$
$810$		0			0
$811$	1.78463e10			1.17483			0.587415	−	0.809286i	$-0.300146\pi$
$811$	1.78463e10			1.17483			0.587415	+	0.809286i	$0.300146\pi$
$812$		0			0
$813$			6.86894e9i			0.448304i
$814$		0			0
$815$	−2.47282e10	+	4.49604e9i	−1.60008	+	0.290923i
$816$		0			0
$817$		−	1.84798e9i		−	0.118555i
$818$		0			0
$819$	−1.05053e10			−0.668212
$820$		0			0
$821$	−3.93673e9			−0.248276			−0.124138	−	0.992265i	$-0.539617\pi$
$821$	−3.93673e9			−0.248276			−0.124138	+	0.992265i	$0.539617\pi$
$822$		0			0
$823$			8.31828e9i			0.520156i	0.965588	+	0.260078i	$0.0837483\pi$
$823$			8.31828e9i			0.520156i	−0.965588	+	0.260078i	$0.916252\pi$
$824$		0			0
$825$	4.09712e9	+	1.08946e10i	0.254033	+	0.675496i
$826$		0			0
$827$		−	1.30781e10i		−	0.804033i	−0.915632	−	0.402017i	$-0.868309\pi$
$827$		−	1.30781e10i		−	0.804033i	0.915632	−	0.402017i	$-0.131691\pi$
$828$		0			0
$829$	1.78307e10			1.08700			0.543499	−	0.839410i	$-0.317099\pi$
$829$	1.78307e10			1.08700			0.543499	+	0.839410i	$0.317099\pi$
$830$		0			0
$831$	1.60073e10			0.967642
$832$		0			0
$833$		−	1.43102e10i		−	0.857804i
$834$		0			0
$835$	−1.25828e10	+	2.28778e9i	−0.747954	+	0.135992i
$836$		0			0
$837$			4.72880e9i			0.278748i
$838$		0			0
$839$	2.05093e10			1.19890			0.599451	−	0.800411i	$-0.295386\pi$
$839$	2.05093e10			1.19890			0.599451	+	0.800411i	$0.295386\pi$
$840$		0			0
$841$	−1.68720e10			−0.978092
$842$		0			0
$843$			1.67948e10i			0.965559i
$844$		0			0
$845$	5.05201e9	+	2.77861e10i	0.288049	+	1.58427i
$846$		0			0
$847$		−	1.23423e10i		−	0.697915i
$848$		0			0
$849$	1.06584e10			0.597744
$850$		0			0
$851$	7.42941e9			0.413239
$852$		0			0
$853$			2.30130e10i			1.26955i	0.772695	+	0.634777i	$0.218908\pi$
$853$			2.30130e10i			1.26955i	−0.772695	+	0.634777i	$0.781092\pi$
$854$		0			0
$855$	−1.61838e8	−	8.90109e8i	−0.00885521	−	0.0487037i
$856$		0			0
$857$		−	1.45901e10i		−	0.791819i	−0.918289	−	0.395910i	$-0.870429\pi$
$857$		−	1.45901e10i		−	0.791819i	0.918289	−	0.395910i	$-0.129571\pi$
$858$		0			0
$859$	3.04784e10			1.64065			0.820324	−	0.571898i	$-0.193793\pi$
$859$	3.04784e10			1.64065			0.820324	+	0.571898i	$0.193793\pi$
$860$		0			0
$861$	−9.10607e9			−0.486205
$862$		0			0
$863$		−	1.15324e10i		−	0.610775i	−0.952228	−	0.305387i	$-0.901214\pi$
$863$		−	1.15324e10i		−	0.610775i	0.952228	−	0.305387i	$-0.0987860\pi$
$864$		0			0
$865$	2.60428e10	−	4.73506e9i	1.36814	−	0.248753i
$866$		0			0
$867$			1.69260e10i			0.882037i
$868$		0			0
$869$	−1.73142e10			−0.895019
$870$		0			0
$871$	−8.62411e9			−0.442233
$872$		0			0
$873$		−	6.31137e9i		−	0.321051i
$874$		0			0
$875$	2.10949e10	−	1.26323e10i	1.06451	−	0.637463i
$876$		0			0
$877$		−	1.15209e10i		−	0.576748i	−0.957518	−	0.288374i	$-0.906885\pi$
$877$		−	1.15209e10i		−	0.576748i	0.957518	−	0.288374i	$-0.0931146\pi$
$878$		0			0
$879$	−1.86139e10			−0.924436
$880$		0			0
$881$	−7.26258e8			−0.0357829			−0.0178914	−	0.999840i	$-0.505695\pi$
$881$	−7.26258e8			−0.0357829			−0.0178914	+	0.999840i	$0.505695\pi$
$882$		0			0
$883$		−	1.81507e10i		−	0.887219i	−0.896220	−	0.443609i	$-0.853698\pi$
$883$		−	1.81507e10i		−	0.887219i	0.896220	−	0.443609i	$-0.146302\pi$
$884$		0			0
$885$	1.36999e10	−	2.49090e9i	0.664381	−	0.120797i
$886$		0			0
$887$			8.91198e9i			0.428787i	0.976747	+	0.214393i	$0.0687775\pi$
$887$			8.91198e9i			0.428787i	−0.976747	+	0.214393i	$0.931223\pi$
$888$		0			0
$889$	4.39657e10			2.09874
$890$		0			0
$891$	2.93249e9			0.138888
$892$		0			0
$893$			1.43401e9i			0.0673865i
$894$		0			0
$895$	−3.56035e9	−	1.95819e10i	−0.166002	−	0.913009i
$896$		0			0
$897$			3.29831e10i			1.52587i
$898$		0			0
$899$	−4.67042e9			−0.214386
$900$		0			0
$901$	−2.83696e10			−1.29216
$902$		0			0
$903$		−	1.26537e10i		−	0.571886i
$904$		0			0
$905$	−3.05146e9	−	1.67830e10i	−0.136848	−	0.752663i
$906$		0			0
$907$		−	1.76727e10i		−	0.786462i	−0.919440	−	0.393231i	$-0.871357\pi$
$907$		−	1.76727e10i		−	0.786462i	0.919440	−	0.393231i	$-0.128643\pi$
$908$		0			0
$909$	5.48448e9			0.242193
$910$		0			0
$911$	2.84979e10			1.24882			0.624409	−	0.781098i	$-0.285340\pi$
$911$	2.84979e10			1.24882			0.624409	+	0.781098i	$0.285340\pi$
$912$		0			0
$913$			2.67144e9i			0.116171i
$914$		0			0
$915$	6.39826e9	−	1.16332e9i	0.276114	−	0.0502025i
$916$		0			0
$917$			4.59460e9i			0.196768i
$918$		0			0
$919$	−9.46354e9			−0.402207			−0.201103	−	0.979570i	$-0.564453\pi$
$919$	−9.46354e9			−0.402207			−0.201103	+	0.979570i	$0.564453\pi$
$920$		0			0
$921$	7.22977e9			0.304941
$922$		0			0
$923$			4.38581e10i			1.83588i
$924$		0			0
$925$	2.14043e9	+	5.69161e9i	0.0889213	+	0.236450i
$926$		0			0
$927$			1.02251e10i			0.421586i
$928$		0			0
$929$	−1.56995e10			−0.642438			−0.321219	−	0.947005i	$-0.604093\pi$
$929$	−1.56995e10			−0.642438			−0.321219	+	0.947005i	$0.604093\pi$
$930$		0			0
$931$	1.97284e9			0.0801249
$932$		0			0
$933$		−	5.25289e9i		−	0.211745i
$934$		0			0
$935$	4.88710e10	−	8.88564e9i	1.95529	−	0.355507i
$936$		0			0
$937$		−	8.78711e9i		−	0.348946i	−0.984662	−	0.174473i	$-0.944178\pi$
$937$		−	8.78711e9i		−	0.348946i	0.984662	−	0.174473i	$-0.0558221\pi$
$938$		0			0
$939$	2.38087e10			0.938438
$940$		0			0
$941$	−1.58721e10			−0.620971			−0.310485	−	0.950578i	$-0.600492\pi$
$941$	−1.58721e10			−0.620971			−0.310485	+	0.950578i	$0.600492\pi$
$942$		0			0
$943$			2.85900e10i			1.11026i
$944$		0			0
$945$	−1.10815e9	−	6.09484e9i	−0.0427158	−	0.234937i
$946$		0			0
$947$		−	5.08496e9i		−	0.194564i	−0.995257	−	0.0972820i	$-0.968985\pi$
$947$		−	5.08496e9i		−	0.194564i	0.995257	−	0.0972820i	$-0.0310149\pi$
$948$		0			0
$949$	−5.98786e10			−2.27426
$950$		0			0
$951$	−1.31840e10			−0.497068
$952$		0			0
$953$			4.59917e10i			1.72129i	0.509205	+	0.860645i	$0.329940\pi$
$953$			4.59917e10i			1.72129i	−0.509205	+	0.860645i	$0.670060\pi$
$954$		0			0
$955$	2.81586e9	+	1.54872e10i	0.104616	+	0.575390i
$956$		0			0
$957$			2.89629e9i			0.106819i
$958$		0			0
$959$	−2.63675e10			−0.965393
$960$		0			0
$961$	3.02065e10			1.09791
$962$		0			0
$963$		−	1.16188e9i		−	0.0419247i
$964$		0			0
$965$	2.74376e10	−	4.98865e9i	0.982879	−	0.178705i
$966$		0			0
$967$			3.82819e10i			1.36145i	0.732541	+	0.680723i	$0.238334\pi$
$967$			3.82819e10i			1.36145i	−0.732541	+	0.680723i	$0.761666\pi$
$968$		0			0
$969$	−3.86086e9			−0.136317
$970$		0			0
$971$	−2.85329e10			−1.00018			−0.500090	−	0.865974i	$-0.666700\pi$
$971$	−2.85329e10			−1.00018			−0.500090	+	0.865974i	$0.666700\pi$
$972$		0			0
$973$			1.53080e9i			0.0532749i
$974$		0			0
$975$	−2.52681e10	+	9.50252e9i	−0.873083	+	0.328339i
$976$		0			0
$977$		−	8.40644e9i		−	0.288391i	−0.989549	−	0.144195i	$-0.953941\pi$
$977$		−	8.40644e9i		−	0.288391i	0.989549	−	0.144195i	$-0.0460594\pi$
$978$		0			0
$979$	−3.45356e10			−1.17632
$980$		0			0
$981$	−1.21108e10			−0.409573
$982$		0			0
$983$			5.00100e10i			1.67927i	0.543154	+	0.839633i	$0.317230\pi$
$983$			5.00100e10i			1.67927i	−0.543154	+	0.839633i	$0.682770\pi$
$984$		0			0
$985$	−3.45007e10	+	6.27286e9i	−1.15027	+	0.209141i
$986$		0			0
$987$			9.81912e9i			0.325059i
$988$		0			0
$989$	−3.97283e10			−1.30591
$990$		0			0
$991$	1.17819e10			0.384555			0.192277	−	0.981341i	$-0.438413\pi$
$991$	1.17819e10			0.384555			0.192277	+	0.981341i	$0.438413\pi$
$992$		0			0
$993$		−	1.91214e10i		−	0.619722i
$994$		0			0
$995$	1.16160e8	+	6.38880e8i	0.00373831	+	0.0205607i
$996$		0			0
$997$		−	2.36572e10i		−	0.756014i	−0.925803	−	0.378007i	$-0.876609\pi$
$997$		−	2.36572e10i		−	0.756014i	0.925803	−	0.378007i	$-0.123391\pi$
$998$		0			0
$999$	1.53201e9			0.0486162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.f.a.49.1		2
4.3	odd	2		30.8.c.a.19.1	✓	2
5.4	even	2	inner	240.8.f.a.49.2		2
12.11	even	2		90.8.c.a.19.2		2
20.3	even	4		150.8.a.d.1.1		1
20.7	even	4		150.8.a.m.1.1		1
20.19	odd	2		30.8.c.a.19.2	yes	2
60.23	odd	4		450.8.a.y.1.1		1
60.47	odd	4		450.8.a.b.1.1		1
60.59	even	2		90.8.c.a.19.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.c.a.19.1	✓	2	4.3	odd	2
30.8.c.a.19.2	yes	2	20.19	odd	2
90.8.c.a.19.1		2	60.59	even	2
90.8.c.a.19.2		2	12.11	even	2
150.8.a.d.1.1		1	20.3	even	4
150.8.a.m.1.1		1	20.7	even	4
240.8.f.a.49.1		2	1.1	even	1	trivial
240.8.f.a.49.2		2	5.4	even	2	inner
450.8.a.b.1.1		1	60.47	odd	4
450.8.a.y.1.1		1	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} +(-50.0000 - 275.000i) q^{5} -1126.00i q^{7} -729.000 q^{9} +5518.00 q^{11} +12798.0i q^{13} +(-7425.00 + 1350.00i) q^{15} +32206.0i q^{17} -4440.00 q^{19} -30402.0 q^{21} +95452.0i q^{23} +(-73125.0 + 27500.0i) q^{25} +19683.0i q^{27} -19440.0 q^{29} +240248. q^{31} -148986. i q^{33} +(-309650. + 56300.0i) q^{35} -77834.0i q^{37} +345546. q^{39} +299522. q^{41} +416212. i q^{43} +(36450.0 + 200475. i) q^{45} -322976. i q^{47} -444333. q^{49} +869562. q^{51} +880878. i q^{53} +(-275900. - 1.51745e6i) q^{55} +119880. i q^{57} -1.84511e6 q^{59} -861718. q^{61} +820854. i q^{63} +(3.51945e6 - 639900. i) q^{65} +673864. i q^{67} +2.57720e6 q^{69} +3.42695e6 q^{71} +4.67875e6i q^{73} +(742500. + 1.97438e6i) q^{75} -6.21327e6i q^{77} -3.13776e6 q^{79} +531441. q^{81} +484132. i q^{83} +(8.85665e6 - 1.61030e6i) q^{85} +524880. i q^{87} -6.25871e6 q^{89} +1.44105e7 q^{91} -6.48670e6i q^{93} +(222000. + 1.22100e6i) q^{95} +8.65758e6i q^{97} -4.02262e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 100 q^{5} - 1458 q^{9} + 11036 q^{11} - 14850 q^{15} - 8880 q^{19} - 60804 q^{21} - 146250 q^{25} - 38880 q^{29} + 480496 q^{31} - 619300 q^{35} + 691092 q^{39} + 599044 q^{41} + 72900 q^{45} - 888666 q^{49}+ \cdots - 8045244 q^{99}+O(q^{100}) \) 2 * q - 100 * q^5 - 1458 * q^9 + 11036 * q^11 - 14850 * q^15 - 8880 * q^19 - 60804 * q^21 - 146250 * q^25 - 38880 * q^29 + 480496 * q^31 - 619300 * q^35 + 691092 * q^39 + 599044 * q^41 + 72900 * q^45 - 888666 * q^49 + 1739124 * q^51 - 551800 * q^55 - 3690220 * q^59 - 1723436 * q^61 + 7038900 * q^65 + 5154408 * q^69 + 6853896 * q^71 + 1485000 * q^75 - 6275520 * q^79 + 1062882 * q^81 + 17713300 * q^85 - 12517420 * q^89 + 28821096 * q^91 + 444000 * q^95 - 8045244 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more