LMFDB - Embedded newform 240.9.e.b.31.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,9,Mod(31,240)]

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([1, 0, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("240.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	240.e (of order $2$, degree $1$, 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:	$97.7708664147$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 67817 x^{10} - 5043842 x^{9} + 3585199422 x^{8} - 247197333458 x^{7} + \cdots + 19\!\cdots\!76 $ x^12 - x^11 + 67817x^10 - 5043842x^9 + 3585199422x^8 - 247197333458x^7 + 78207057514777x^6 - 7698760819999837x^5 + 1322781431164317033x^4 - 84365949624552374152x^3 + 4325893825548246639488x^2 - 107206956340388913620480x + 1994880980828089481138176
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{18}\cdot 5^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.4
Root			$20.8564 - 36.1244i$ of defining polynomial
Character	$\chi$	$=$	240.31
Dual form			240.9.e.b.31.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +279.508 q^{5} -2706.52i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +279.508 q^{5} -2706.52i q^{7} -2187.00 q^{9} -6857.00i q^{11} -39260.2 q^{13} -13071.3i q^{15} -15254.5 q^{17} -51488.7i q^{19} -126571. q^{21} +402835. i q^{23} +78125.0 q^{25} +102276. i q^{27} -998230. q^{29} +209399. i q^{31} -320670. q^{33} -756494. i q^{35} +1.81498e6 q^{37} +1.83602e6i q^{39} -300185. q^{41} -259837. i q^{43} -611285. q^{45} -1.99494e6i q^{47} -1.56042e6 q^{49} +713382. i q^{51} -1.95550e6 q^{53} -1.91659e6i q^{55} -2.40789e6 q^{57} +1.36781e7i q^{59} -3.32124e6 q^{61} +5.91915e6i q^{63} -1.09736e7 q^{65} -3.00865e7i q^{67} +1.88387e7 q^{69} +3.85083e6i q^{71} -3.52393e7 q^{73} -3.65354e6i q^{75} -1.85586e7 q^{77} -1.59780e7i q^{79} +4.78297e6 q^{81} +6.22797e7i q^{83} -4.26376e6 q^{85} +4.66826e7i q^{87} +6.90961e6 q^{89} +1.06258e8i q^{91} +9.79260e6 q^{93} -1.43915e7i q^{95} +1.48092e8 q^{97} +1.49963e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 26244 q^{9}+O(q^{10}) $ 12 * q - 26244 * q^9	$ 12 q - 26244 q^{9} - 41616 q^{13} + 44016 q^{17} + 407592 q^{21} + 937500 q^{25} - 1521456 q^{29} + 1465128 q^{33} - 665760 q^{37} - 3352200 q^{41} - 4335444 q^{49} - 8855232 q^{53} + 1550016 q^{57} + 11684328 q^{61} + 7305000 q^{65} + 47784816 q^{69} + 119457096 q^{73} - 256261344 q^{77} + 57395628 q^{81} + 75405000 q^{85} - 162442392 q^{89} + 203718240 q^{93} + 12509688 q^{97}+O(q^{100}) $ 12 * q - 26244 * q^9 - 41616 * q^13 + 44016 * q^17 + 407592 * q^21 + 937500 * q^25 - 1521456 * q^29 + 1465128 * q^33 - 665760 * q^37 - 3352200 * q^41 - 4335444 * q^49 - 8855232 * q^53 + 1550016 * q^57 + 11684328 * q^61 + 7305000 * q^65 + 47784816 * q^69 + 119457096 * q^73 - 256261344 * q^77 + 57395628 * q^81 + 75405000 * q^85 - 162442392 * q^89 + 203718240 * q^93 + 12509688 * q^97

Display 100 coefficients 10 coefficients

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
$3$	− 46.7654i	− 0.577350i
$3$	− 46.7654i	− 0.577350i
$4$	0	0
$5$	279.508	0.447214
$5$	279.508	0.447214
$6$	0	0
$7$	− 2706.52i	− 1.12725i	−0.826032	−	0.563623i	$-0.809407\pi$
$7$	− 2706.52i	− 1.12725i	0.826032	−	0.563623i	$-0.190593\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 6857.00i	− 0.468342i	−0.972195	−	0.234171i	$-0.924762\pi$
$11$	− 6857.00i	− 0.468342i	0.972195	−	0.234171i	$-0.0752376\pi$
$12$	0	0
$13$	−39260.2	−1.37461	−0.687305	−	0.726369i	$-0.741206\pi$
$13$	−39260.2	−1.37461	−0.687305	+	0.726369i	$0.741206\pi$
$14$	0	0
$15$	− 13071.3i	− 0.258199i
$16$	0	0
$17$	−15254.5	−0.182643	−0.0913213	−	0.995821i	$-0.529109\pi$
$17$	−15254.5	−0.182643	−0.0913213	+	0.995821i	$0.529109\pi$
$18$	0	0
$19$	− 51488.7i	− 0.395092i	−0.980294	−	0.197546i	$-0.936703\pi$
$19$	− 51488.7i	− 0.395092i	0.980294	−	0.197546i	$-0.0632971\pi$
$20$	0	0
$21$	−126571.	−0.650815
$22$	0	0
$23$	402835.i	1.43951i	0.694227	+	0.719756i	$0.255746\pi$
$23$	402835.i	1.43951i	−0.694227	+	0.719756i	$0.744254\pi$
$24$	0	0
$25$	78125.0	0.200000
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	−998230.	−1.41136	−0.705681	−	0.708529i	$-0.749359\pi$
$29$	−998230.	−1.41136	−0.705681	+	0.708529i	$0.749359\pi$
$30$	0	0
$31$	209399.i	0.226739i	0.993553	+	0.113370i	$0.0361644\pi$
$31$	209399.i	0.226739i	−0.993553	+	0.113370i	$0.963836\pi$
$32$	0	0
$33$	−320670.	−0.270398
$34$	0	0
$35$	− 756494.i	− 0.504119i
$36$	0	0
$37$	1.81498e6	0.968423	0.484212	−	0.874951i	$-0.339106\pi$
$37$	1.81498e6	0.968423	0.484212	+	0.874951i	$0.339106\pi$
$38$	0	0
$39$	1.83602e6i	0.793631i
$40$	0	0
$41$	−300185.	−0.106232	−0.0531158	−	0.998588i	$-0.516915\pi$
$41$	−300185.	−0.106232	−0.0531158	+	0.998588i	$0.516915\pi$
$42$	0	0
$43$	− 259837.i	− 0.0760025i	−0.999278	−	0.0380012i	$-0.987901\pi$
$43$	− 259837.i	− 0.0760025i	0.999278	−	0.0380012i	$-0.0120991\pi$
$44$	0	0
$45$	−611285.	−0.149071
$46$	0	0
$47$	− 1.99494e6i	− 0.408826i	−0.978885	−	0.204413i	$-0.934471\pi$
$47$	− 1.99494e6i	− 0.408826i	0.978885	−	0.204413i	$-0.0655285\pi$
$48$	0	0
$49$	−1.56042e6	−0.270681
$50$	0	0
$51$	713382.i	0.105449i
$52$	0	0
$53$	−1.95550e6	−0.247831	−0.123915	−	0.992293i	$-0.539545\pi$
$53$	−1.95550e6	−0.247831	−0.123915	+	0.992293i	$0.539545\pi$
$54$	0	0
$55$	− 1.91659e6i	− 0.209449i
$56$	0	0
$57$	−2.40789e6	−0.228106
$58$	0	0
$59$	1.36781e7i	1.12880i	0.825502	+	0.564399i	$0.190892\pi$
$59$	1.36781e7i	1.12880i	−0.825502	+	0.564399i	$0.809108\pi$
$60$	0	0
$61$	−3.32124e6	−0.239873	−0.119936	−	0.992782i	$-0.538269\pi$
$61$	−3.32124e6	−0.239873	−0.119936	+	0.992782i	$0.538269\pi$
$62$	0	0
$63$	5.91915e6i	0.375748i
$64$	0	0
$65$	−1.09736e7	−0.614744
$66$	0	0
$67$	− 3.00865e7i	− 1.49304i	−0.665362	−	0.746521i	$-0.731723\pi$
$67$	− 3.00865e7i	− 1.49304i	0.665362	−	0.746521i	$-0.268277\pi$
$68$	0	0
$69$	1.88387e7	0.831103
$70$	0	0
$71$	3.85083e6i	0.151538i	0.997125	+	0.0757690i	$0.0241411\pi$
$71$	3.85083e6i	0.151538i	−0.997125	+	0.0757690i	$0.975859\pi$
$72$	0	0
$73$	−3.52393e7	−1.24090	−0.620448	−	0.784248i	$-0.713049\pi$
$73$	−3.52393e7	−1.24090	−0.620448	+	0.784248i	$0.713049\pi$
$74$	0	0
$75$	− 3.65354e6i	− 0.115470i
$76$	0	0
$77$	−1.85586e7	−0.527937
$78$	0	0
$79$	− 1.59780e7i	− 0.410218i	−0.978739	−	0.205109i	$-0.934245\pi$
$79$	− 1.59780e7i	− 0.410218i	0.978739	−	0.205109i	$-0.0657549\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	6.22797e7i	1.31230i	0.754629	+	0.656151i	$0.227817\pi$
$83$	6.22797e7i	1.31230i	−0.754629	+	0.656151i	$0.772183\pi$
$84$	0	0
$85$	−4.26376e6	−0.0816803
$86$	0	0
$87$	4.66826e7i	0.814851i
$88$	0	0
$89$	6.90961e6	0.110127	0.0550635	−	0.998483i	$-0.482464\pi$
$89$	6.90961e6	0.110127	0.0550635	+	0.998483i	$0.482464\pi$
$90$	0	0
$91$	1.06258e8i	1.54952i
$92$	0	0
$93$	9.79260e6	0.130908
$94$	0	0
$95$	− 1.43915e7i	− 0.176690i
$96$	0	0
$97$	1.48092e8	1.67280	0.836401	−	0.548118i	$-0.184655\pi$
$97$	1.48092e8	1.67280	0.836401	+	0.548118i	$0.184655\pi$
$98$	0	0
$99$	1.49963e7i	0.156114i
$100$	0	0
$101$	1.42191e8	1.36643	0.683213	−	0.730219i	$-0.260582\pi$
$101$	1.42191e8	1.36643	0.683213	+	0.730219i	$0.260582\pi$
$102$	0	0
$103$	2.00478e7i	0.178122i	0.996026	+	0.0890612i	$0.0283867\pi$
$103$	2.00478e7i	0.178122i	−0.996026	+	0.0890612i	$0.971613\pi$
$104$	0	0
$105$	−3.53777e7	−0.291053
$106$	0	0
$107$	1.46058e8i	1.11427i	0.830422	+	0.557135i	$0.188099\pi$
$107$	1.46058e8i	1.11427i	−0.830422	+	0.557135i	$0.811901\pi$
$108$	0	0
$109$	4.57174e7	0.323874	0.161937	−	0.986801i	$-0.448226\pi$
$109$	4.57174e7	0.323874	0.161937	+	0.986801i	$0.448226\pi$
$110$	0	0
$111$	− 8.48783e7i	− 0.559119i
$112$	0	0
$113$	−1.99243e8	−1.22199	−0.610996	−	0.791634i	$-0.709231\pi$
$113$	−1.99243e8	−1.22199	−0.610996	+	0.791634i	$0.709231\pi$
$114$	0	0
$115$	1.12596e8i	0.643769i
$116$	0	0
$117$	8.58621e7	0.458203
$118$	0	0
$119$	4.12865e7i	0.205883i
$120$	0	0
$121$	1.67340e8	0.780655
$122$	0	0
$123$	1.40383e7i	0.0613328i
$124$	0	0
$125$	2.18366e7	0.0894427
$126$	0	0
$127$	4.19834e8i	1.61385i	0.590655	+	0.806925i	$0.298870\pi$
$127$	4.19834e8i	1.61385i	−0.590655	+	0.806925i	$0.701130\pi$
$128$	0	0
$129$	−1.21514e7	−0.0438801
$130$	0	0
$131$	7.21042e7i	0.244836i	0.992479	+	0.122418i	$0.0390649\pi$
$131$	7.21042e7i	0.244836i	−0.992479	+	0.122418i	$0.960935\pi$
$132$	0	0
$133$	−1.39355e8	−0.445365
$134$	0	0
$135$	2.85870e7i	0.0860663i
$136$	0	0
$137$	−3.08705e8	−0.876318	−0.438159	−	0.898898i	$-0.644369\pi$
$137$	−3.08705e8	−0.876318	−0.438159	+	0.898898i	$0.644369\pi$
$138$	0	0
$139$	2.67302e8i	0.716050i	0.933712	+	0.358025i	$0.116550\pi$
$139$	2.67302e8i	0.716050i	−0.933712	+	0.358025i	$0.883450\pi$
$140$	0	0
$141$	−9.32941e7	−0.236036
$142$	0	0
$143$	2.69207e8i	0.643788i
$144$	0	0
$145$	−2.79014e8	−0.631181
$146$	0	0
$147$	7.29738e7i	0.156278i
$148$	0	0
$149$	−9.65077e7	−0.195802	−0.0979009	−	0.995196i	$-0.531213\pi$
$149$	−9.65077e7	−0.195802	−0.0979009	+	0.995196i	$0.531213\pi$
$150$	0	0
$151$	7.95652e8i	1.53044i	0.643771	+	0.765218i	$0.277369\pi$
$151$	7.95652e8i	1.53044i	−0.643771	+	0.765218i	$0.722631\pi$
$152$	0	0
$153$	3.33616e7	0.0608809
$154$	0	0
$155$	5.85287e7i	0.101401i
$156$	0	0
$157$	−8.88036e8	−1.46161	−0.730806	−	0.682585i	$-0.760856\pi$
$157$	−8.88036e8	−1.46161	−0.730806	+	0.682585i	$0.760856\pi$
$158$	0	0
$159$	9.14498e7i	0.143085i
$160$	0	0
$161$	1.09028e9	1.62268
$162$	0	0
$163$	3.96867e8i	0.562205i	0.959678	+	0.281102i	$0.0907001\pi$
$163$	3.96867e8i	0.562205i	−0.959678	+	0.281102i	$0.909300\pi$
$164$	0	0
$165$	−8.96300e7	−0.120925
$166$	0	0
$167$	8.43839e7i	0.108491i	0.998528	+	0.0542455i	$0.0172754\pi$
$167$	8.43839e7i	0.108491i	−0.998528	+	0.0542455i	$0.982725\pi$
$168$	0	0
$169$	7.25635e8	0.889552
$170$	0	0
$171$	1.12606e8i	0.131697i
$172$	0	0
$173$	2.56179e8	0.285996	0.142998	−	0.989723i	$-0.454326\pi$
$173$	2.56179e8	0.285996	0.142998	+	0.989723i	$0.454326\pi$
$174$	0	0
$175$	− 2.11447e8i	− 0.225449i
$176$	0	0
$177$	6.39659e8	0.651712
$178$	0	0
$179$	8.41367e8i	0.819546i	0.912187	+	0.409773i	$0.134392\pi$
$179$	8.41367e8i	0.819546i	−0.912187	+	0.409773i	$0.865608\pi$
$180$	0	0
$181$	−1.63396e9	−1.52240	−0.761198	−	0.648520i	$-0.775388\pi$
$181$	−1.63396e9	−1.52240	−0.761198	+	0.648520i	$0.775388\pi$
$182$	0	0
$183$	1.55319e8i	0.138491i
$184$	0	0
$185$	5.07303e8	0.433092
$186$	0	0
$187$	1.04600e8i	0.0855393i
$188$	0	0
$189$	2.76811e8	0.216938
$190$	0	0
$191$	1.85915e9i	1.39695i	0.715635	+	0.698475i	$0.246138\pi$
$191$	1.85915e9i	1.39695i	−0.715635	+	0.698475i	$0.753862\pi$
$192$	0	0
$193$	5.55576e8	0.400419	0.200209	−	0.979753i	$-0.435838\pi$
$193$	5.55576e8	0.400419	0.200209	+	0.979753i	$0.435838\pi$
$194$	0	0
$195$	5.13183e8i	0.354923i
$196$	0	0
$197$	−1.63390e9	−1.08483	−0.542414	−	0.840112i	$-0.682489\pi$
$197$	−1.63390e9	−1.08483	−0.542414	+	0.840112i	$0.682489\pi$
$198$	0	0
$199$	2.78358e9i	1.77497i	0.460833	+	0.887487i	$0.347551\pi$
$199$	2.78358e9i	1.77497i	−0.460833	+	0.887487i	$0.652449\pi$
$200$	0	0
$201$	−1.40700e9	−0.862008
$202$	0	0
$203$	2.70172e9i	1.59095i
$204$	0	0
$205$	−8.39042e7	−0.0475082
$206$	0	0
$207$	− 8.80999e8i	− 0.479837i
$208$	0	0
$209$	−3.53058e8	−0.185038
$210$	0	0
$211$	− 2.38013e9i	− 1.20080i	−0.799700	−	0.600400i	$-0.795008\pi$
$211$	− 2.38013e9i	− 1.20080i	0.799700	−	0.600400i	$-0.204992\pi$
$212$	0	0
$213$	1.80086e8	0.0874905
$214$	0	0
$215$	− 7.26268e7i	− 0.0339893i
$216$	0	0
$217$	5.66740e8	0.255591
$218$	0	0
$219$	1.64798e9i	0.716431i
$220$	0	0
$221$	5.98895e8	0.251062
$222$	0	0
$223$	− 2.89222e9i	− 1.16953i	−0.811202	−	0.584766i	$-0.801186\pi$
$223$	− 2.89222e9i	− 1.16953i	0.811202	−	0.584766i	$-0.198814\pi$
$224$	0	0
$225$	−1.70859e8	−0.0666667
$226$	0	0
$227$	− 1.45903e8i	− 0.0549491i	−0.999623	−	0.0274745i	$-0.991253\pi$
$227$	− 1.45903e8i	− 0.0549491i	0.999623	−	0.0274745i	$-0.00874652\pi$
$228$	0	0
$229$	1.66630e8	0.0605915	0.0302957	−	0.999541i	$-0.490355\pi$
$229$	1.66630e8	0.0605915	0.0302957	+	0.999541i	$0.490355\pi$
$230$	0	0
$231$	8.67899e8i	0.304804i
$232$	0	0
$233$	5.02696e9	1.70562	0.852809	−	0.522224i	$-0.174897\pi$
$233$	5.02696e9	1.70562	0.852809	+	0.522224i	$0.174897\pi$
$234$	0	0
$235$	− 5.57602e8i	− 0.182832i
$236$	0	0
$237$	−7.47219e8	−0.236840
$238$	0	0
$239$	− 5.73951e9i	− 1.75907i	−0.475835	−	0.879534i	$-0.657854\pi$
$239$	− 5.73951e9i	− 1.75907i	0.475835	−	0.879534i	$-0.342146\pi$
$240$	0	0
$241$	−2.40081e9	−0.711687	−0.355844	−	0.934546i	$-0.615806\pi$
$241$	−2.40081e9	−0.711687	−0.355844	+	0.934546i	$0.615806\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	−4.36152e8	−0.121052
$246$	0	0
$247$	2.02146e9i	0.543097i
$248$	0	0
$249$	2.91253e9	0.757658
$250$	0	0
$251$	− 1.01780e9i	− 0.256428i	−0.991746	−	0.128214i	$-0.959075\pi$
$251$	− 1.01780e9i	− 0.256428i	0.991746	−	0.128214i	$-0.0409245\pi$
$252$	0	0
$253$	2.76224e9	0.674184
$254$	0	0
$255$	1.99396e8i	0.0471581i
$256$	0	0
$257$	−4.00949e9	−0.919087	−0.459544	−	0.888155i	$-0.651987\pi$
$257$	−4.00949e9	−0.919087	−0.459544	+	0.888155i	$0.651987\pi$
$258$	0	0
$259$	− 4.91227e9i	− 1.09165i
$260$	0	0
$261$	2.18313e9	0.470454
$262$	0	0
$263$	1.31385e8i	0.0274615i	0.999906	+	0.0137307i	$0.00437076\pi$
$263$	1.31385e8i	0.0274615i	−0.999906	+	0.0137307i	$0.995629\pi$
$264$	0	0
$265$	−5.46579e8	−0.110833
$266$	0	0
$267$	− 3.23130e8i	− 0.0635818i
$268$	0	0
$269$	9.32683e9	1.78125	0.890625	−	0.454739i	$-0.150268\pi$
$269$	9.32683e9	1.78125	0.890625	+	0.454739i	$0.150268\pi$
$270$	0	0
$271$	4.69280e9i	0.870071i	0.900413	+	0.435035i	$0.143264\pi$
$271$	4.69280e9i	0.870071i	−0.900413	+	0.435035i	$0.856736\pi$
$272$	0	0
$273$	4.96921e9	0.894617
$274$	0	0
$275$	− 5.35703e8i	− 0.0936685i
$276$	0	0
$277$	−7.64519e9	−1.29858	−0.649290	−	0.760541i	$-0.724934\pi$
$277$	−7.64519e9	−1.29858	−0.649290	+	0.760541i	$0.724934\pi$
$278$	0	0
$279$	− 4.57955e8i	− 0.0755798i
$280$	0	0
$281$	−1.30067e9	−0.208613	−0.104307	−	0.994545i	$-0.533262\pi$
$281$	−1.30067e9	−0.208613	−0.104307	+	0.994545i	$0.533262\pi$
$282$	0	0
$283$	− 3.33025e9i	− 0.519195i	−0.965717	−	0.259598i	$-0.916410\pi$
$283$	− 3.33025e9i	− 0.519195i	0.965717	−	0.259598i	$-0.0835899\pi$
$284$	0	0
$285$	−6.73026e8	−0.102012
$286$	0	0
$287$	8.12455e8i	0.119749i
$288$	0	0
$289$	−6.74306e9	−0.966642
$290$	0	0
$291$	− 6.92558e9i	− 0.965793i
$292$	0	0
$293$	−4.66702e9	−0.633241	−0.316620	−	0.948552i	$-0.602548\pi$
$293$	−4.66702e9	−0.633241	−0.316620	+	0.948552i	$0.602548\pi$
$294$	0	0
$295$	3.82313e9i	0.504814i
$296$	0	0
$297$	7.01306e8	0.0901325
$298$	0	0
$299$	− 1.58154e10i	− 1.97877i
$300$	0	0
$301$	−7.03254e8	−0.0856734
$302$	0	0
$303$	− 6.64961e9i	− 0.788907i
$304$	0	0
$305$	−9.28315e8	−0.107274
$306$	0	0
$307$	− 1.26133e10i	− 1.41996i	−0.704221	−	0.709981i	$-0.748704\pi$
$307$	− 1.26133e10i	− 1.41996i	0.704221	−	0.709981i	$-0.251296\pi$
$308$	0	0
$309$	9.37544e8	0.102839
$310$	0	0
$311$	− 6.35390e8i	− 0.0679202i	−0.999423	−	0.0339601i	$-0.989188\pi$
$311$	− 6.35390e8i	− 0.0679202i	0.999423	−	0.0339601i	$-0.0108119\pi$
$312$	0	0
$313$	−1.34527e10	−1.40163	−0.700813	−	0.713345i	$-0.747179\pi$
$313$	−1.34527e10	−1.40163	−0.700813	+	0.713345i	$0.747179\pi$
$314$	0	0
$315$	1.65445e9i	0.168040i
$316$	0	0
$317$	3.04591e9	0.301634	0.150817	−	0.988562i	$-0.451810\pi$
$317$	3.04591e9	0.301634	0.150817	+	0.988562i	$0.451810\pi$
$318$	0	0
$319$	6.84486e9i	0.661001i
$320$	0	0
$321$	6.83046e9	0.643324
$322$	0	0
$323$	7.85435e8i	0.0721606i
$324$	0	0
$325$	−3.06721e9	−0.274922
$326$	0	0
$327$	− 2.13799e9i	− 0.186989i
$328$	0	0
$329$	−5.39933e9	−0.460847
$330$	0	0
$331$	− 1.55298e9i	− 0.129376i	−0.997906	−	0.0646881i	$-0.979395\pi$
$331$	− 1.55298e9i	− 0.129376i	0.997906	−	0.0646881i	$-0.0206053\pi$
$332$	0	0
$333$	−3.96936e9	−0.322808
$334$	0	0
$335$	− 8.40942e9i	− 0.667708i
$336$	0	0
$337$	−1.10622e10	−0.857670	−0.428835	−	0.903383i	$-0.641076\pi$
$337$	−1.10622e10	−0.857670	−0.428835	+	0.903383i	$0.641076\pi$
$338$	0	0
$339$	9.31765e9i	0.705517i
$340$	0	0
$341$	1.43585e9	0.106192
$342$	0	0
$343$	− 1.13792e10i	− 0.822121i
$344$	0	0
$345$	5.26558e9	0.371680
$346$	0	0
$347$	− 1.51344e10i	− 1.04387i	−0.852985	−	0.521936i	$-0.825210\pi$
$347$	− 1.51344e10i	− 1.04387i	0.852985	−	0.521936i	$-0.174790\pi$
$348$	0	0
$349$	4.33978e9	0.292527	0.146264	−	0.989246i	$-0.453275\pi$
$349$	4.33978e9	0.292527	0.146264	+	0.989246i	$0.453275\pi$
$350$	0	0
$351$	− 4.01537e9i	− 0.264544i
$352$	0	0
$353$	−2.45286e9	−0.157970	−0.0789849	−	0.996876i	$-0.525168\pi$
$353$	−2.45286e9	−0.157970	−0.0789849	+	0.996876i	$0.525168\pi$
$354$	0	0
$355$	1.07634e9i	0.0677698i
$356$	0	0
$357$	1.93078e9	0.118867
$358$	0	0
$359$	− 8.66426e9i	− 0.521619i	−0.965390	−	0.260810i	$-0.916010\pi$
$359$	− 8.66426e9i	− 0.521619i	0.965390	−	0.260810i	$-0.0839895\pi$
$360$	0	0
$361$	1.43325e10	0.843903
$362$	0	0
$363$	− 7.82574e9i	− 0.450712i
$364$	0	0
$365$	−9.84967e9	−0.554945
$366$	0	0
$367$	− 2.37131e10i	− 1.30714i	−0.756864	−	0.653572i	$-0.773270\pi$
$367$	− 2.37131e10i	− 1.30714i	0.756864	−	0.653572i	$-0.226730\pi$
$368$	0	0
$369$	6.56504e8	0.0354105
$370$	0	0
$371$	5.29260e9i	0.279366i
$372$	0	0
$373$	−8.27572e9	−0.427534	−0.213767	−	0.976885i	$-0.568573\pi$
$373$	−8.27572e9	−0.427534	−0.213767	+	0.976885i	$0.568573\pi$
$374$	0	0
$375$	− 1.02120e9i	− 0.0516398i
$376$	0	0
$377$	3.91907e10	1.94007
$378$	0	0
$379$	− 1.23938e10i	− 0.600685i	−0.953831	−	0.300342i	$-0.902899\pi$
$379$	− 1.23938e10i	− 0.600685i	0.953831	−	0.300342i	$-0.0971009\pi$
$380$	0	0
$381$	1.96337e10	0.931756
$382$	0	0
$383$	2.52043e10i	1.17133i	0.810553	+	0.585666i	$0.199167\pi$
$383$	2.52043e10i	1.17133i	−0.810553	+	0.585666i	$0.800833\pi$
$384$	0	0
$385$	−5.18728e9	−0.236100
$386$	0	0
$387$	5.68264e8i	0.0253342i
$388$	0	0
$389$	3.41024e10	1.48932	0.744658	−	0.667446i	$-0.232613\pi$
$389$	3.41024e10	1.48932	0.744658	+	0.667446i	$0.232613\pi$
$390$	0	0
$391$	− 6.14504e9i	− 0.262916i
$392$	0	0
$393$	3.37198e9	0.141356
$394$	0	0
$395$	− 4.46600e9i	− 0.183455i
$396$	0	0
$397$	−4.52778e10	−1.82274	−0.911368	−	0.411593i	$-0.864972\pi$
$397$	−4.52778e10	−1.82274	−0.911368	+	0.411593i	$0.864972\pi$
$398$	0	0
$399$	6.51699e9i	0.257132i
$400$	0	0
$401$	2.50353e9	0.0968222	0.0484111	−	0.998827i	$-0.484584\pi$
$401$	2.50353e9	0.0968222	0.0484111	+	0.998827i	$0.484584\pi$
$402$	0	0
$403$	− 8.22103e9i	− 0.311678i
$404$	0	0
$405$	1.33688e9	0.0496904
$406$	0	0
$407$	− 1.24453e10i	− 0.453554i
$408$	0	0
$409$	−2.09888e10	−0.750056	−0.375028	−	0.927013i	$-0.622367\pi$
$409$	−2.09888e10	−0.750056	−0.375028	+	0.927013i	$0.622367\pi$
$410$	0	0
$411$	1.44367e10i	0.505942i
$412$	0	0
$413$	3.70199e10	1.27243
$414$	0	0
$415$	1.74077e10i	0.586880i
$416$	0	0
$417$	1.25005e10	0.413411
$418$	0	0
$419$	− 2.35864e9i	− 0.0765255i	−0.999268	−	0.0382627i	$-0.987818\pi$
$419$	− 2.35864e9i	− 0.0765255i	0.999268	−	0.0382627i	$-0.0121824\pi$
$420$	0	0
$421$	−4.27740e10	−1.36161	−0.680803	−	0.732467i	$-0.738369\pi$
$421$	−4.27740e10	−1.36161	−0.680803	+	0.732467i	$0.738369\pi$
$422$	0	0
$423$	4.36293e9i	0.136275i
$424$	0	0
$425$	−1.19176e9	−0.0365285
$426$	0	0
$427$	8.98899e9i	0.270395i
$428$	0	0
$429$	1.25896e10	0.371691
$430$	0	0
$431$	6.60013e10i	1.91268i	0.292249	+	0.956342i	$0.405596\pi$
$431$	6.60013e10i	1.91268i	−0.292249	+	0.956342i	$0.594404\pi$
$432$	0	0
$433$	−5.60404e10	−1.59422	−0.797112	−	0.603831i	$-0.793640\pi$
$433$	−5.60404e10	−1.59422	−0.797112	+	0.603831i	$0.793640\pi$
$434$	0	0
$435$	1.30482e10i	0.364412i
$436$	0	0
$437$	2.07414e10	0.568739
$438$	0	0
$439$	3.01675e10i	0.812235i	0.913821	+	0.406118i	$0.133118\pi$
$439$	3.01675e10i	0.812235i	−0.913821	+	0.406118i	$0.866882\pi$
$440$	0	0
$441$	3.41265e9	0.0902271
$442$	0	0
$443$	− 3.20257e10i	− 0.831540i	−0.909470	−	0.415770i	$-0.863512\pi$
$443$	− 3.20257e10i	− 0.831540i	0.909470	−	0.415770i	$-0.136488\pi$
$444$	0	0
$445$	1.93129e9	0.0492503
$446$	0	0
$447$	4.51322e9i	0.113046i
$448$	0	0
$449$	−1.89561e10	−0.466406	−0.233203	−	0.972428i	$-0.574921\pi$
$449$	−1.89561e10	−0.466406	−0.233203	+	0.972428i	$0.574921\pi$
$450$	0	0
$451$	2.05837e9i	0.0497527i
$452$	0	0
$453$	3.72090e10	0.883598
$454$	0	0
$455$	2.97001e10i	0.692967i
$456$	0	0
$457$	3.40082e10	0.779686	0.389843	−	0.920881i	$-0.372529\pi$
$457$	3.40082e10	0.779686	0.389843	+	0.920881i	$0.372529\pi$
$458$	0	0
$459$	− 1.56017e9i	− 0.0351496i
$460$	0	0
$461$	8.76590e9	0.194085	0.0970427	−	0.995280i	$-0.469062\pi$
$461$	8.76590e9	0.194085	0.0970427	+	0.995280i	$0.469062\pi$
$462$	0	0
$463$	2.51903e10i	0.548162i	0.961707	+	0.274081i	$0.0883737\pi$
$463$	2.51903e10i	0.548162i	−0.961707	+	0.274081i	$0.911626\pi$
$464$	0	0
$465$	2.73711e9	0.0585438
$466$	0	0
$467$	1.72069e10i	0.361772i	0.983504	+	0.180886i	$0.0578965\pi$
$467$	1.72069e10i	0.361772i	−0.983504	+	0.180886i	$0.942103\pi$
$468$	0	0
$469$	−8.14295e10	−1.68302
$470$	0	0
$471$	4.15293e10i	0.843862i
$472$	0	0
$473$	−1.78171e9	−0.0355952
$474$	0	0
$475$	− 4.02256e9i	− 0.0790183i
$476$	0	0
$477$	4.27668e9	0.0826102
$478$	0	0
$479$	− 4.79698e10i	− 0.911226i	−0.890178	−	0.455613i	$-0.849420\pi$
$479$	− 4.79698e10i	− 0.911226i	0.890178	−	0.455613i	$-0.150580\pi$
$480$	0	0
$481$	−7.12566e10	−1.33120
$482$	0	0
$483$	− 5.09872e10i	− 0.936856i
$484$	0	0
$485$	4.13930e10	0.748100
$486$	0	0
$487$	5.58966e10i	0.993733i	0.867827	+	0.496867i	$0.165516\pi$
$487$	5.58966e10i	0.993733i	−0.867827	+	0.496867i	$0.834484\pi$
$488$	0	0
$489$	1.85596e10	0.324589
$490$	0	0
$491$	6.46741e10i	1.11277i	0.830926	+	0.556384i	$0.187811\pi$
$491$	6.46741e10i	1.11277i	−0.830926	+	0.556384i	$0.812189\pi$
$492$	0	0
$493$	1.52275e10	0.257775
$494$	0	0
$495$	4.19158e9i	0.0698164i
$496$	0	0
$497$	1.04223e10	0.170820
$498$	0	0
$499$	− 1.24563e10i	− 0.200904i	−0.994942	−	0.100452i	$-0.967971\pi$
$499$	− 1.24563e10i	− 0.200904i	0.994942	−	0.100452i	$-0.0320288\pi$
$500$	0	0
$501$	3.94625e9	0.0626373
$502$	0	0
$503$	1.08937e11i	1.70178i	0.525346	+	0.850889i	$0.323936\pi$
$503$	1.08937e11i	1.70178i	−0.525346	+	0.850889i	$0.676064\pi$
$504$	0	0
$505$	3.97436e10	0.611085
$506$	0	0
$507$	− 3.39346e10i	− 0.513583i
$508$	0	0
$509$	4.15193e10	0.618555	0.309278	−	0.950972i	$-0.399913\pi$
$509$	4.15193e10	0.618555	0.309278	+	0.950972i	$0.399913\pi$
$510$	0	0
$511$	9.53756e10i	1.39879i
$512$	0	0
$513$	5.26606e9	0.0760354
$514$	0	0
$515$	5.60354e9i	0.0796588i
$516$	0	0
$517$	−1.36793e10	−0.191470
$518$	0	0
$519$	− 1.19803e10i	− 0.165120i
$520$	0	0
$521$	−1.13757e10	−0.154393	−0.0771965	−	0.997016i	$-0.524597\pi$
$521$	−1.13757e10	−0.154393	−0.0771965	+	0.997016i	$0.524597\pi$
$522$	0	0
$523$	7.93483e10i	1.06055i	0.847826	+	0.530274i	$0.177911\pi$
$523$	7.93483e10i	1.06055i	−0.847826	+	0.530274i	$0.822089\pi$
$524$	0	0
$525$	−9.88837e9	−0.130163
$526$	0	0
$527$	− 3.19427e9i	− 0.0414123i
$528$	0	0
$529$	−8.39647e10	−1.07220
$530$	0	0
$531$	− 2.99139e10i	− 0.376266i
$532$	0	0
$533$	1.17853e10	0.146027
$534$	0	0
$535$	4.08245e10i	0.498317i
$536$	0	0
$537$	3.93469e10	0.473165
$538$	0	0
$539$	1.06998e10i	0.126772i
$540$	0	0
$541$	−7.26875e10	−0.848537	−0.424268	−	0.905536i	$-0.639469\pi$
$541$	−7.26875e10	−0.848537	−0.424268	+	0.905536i	$0.639469\pi$
$542$	0	0
$543$	7.64128e10i	0.878955i
$544$	0	0
$545$	1.27784e10	0.144841
$546$	0	0
$547$	− 9.99239e10i	− 1.11614i	−0.829793	−	0.558072i	$-0.811541\pi$
$547$	− 9.99239e10i	− 1.11614i	0.829793	−	0.558072i	$-0.188459\pi$
$548$	0	0
$549$	7.26355e9	0.0799576
$550$	0	0
$551$	5.13976e10i	0.557618i
$552$	0	0
$553$	−4.32448e10	−0.462416
$554$	0	0
$555$	− 2.37242e10i	− 0.250046i
$556$	0	0
$557$	−1.22643e11	−1.27415	−0.637076	−	0.770801i	$-0.719856\pi$
$557$	−1.22643e11	−1.27415	−0.637076	+	0.770801i	$0.719856\pi$
$558$	0	0
$559$	1.02013e10i	0.104474i
$560$	0	0
$561$	4.89166e9	0.0493861
$562$	0	0
$563$	− 1.63342e11i	− 1.62579i	−0.582411	−	0.812895i	$-0.697891\pi$
$563$	− 1.63342e11i	− 1.62579i	0.582411	−	0.812895i	$-0.302109\pi$
$564$	0	0
$565$	−5.56900e10	−0.546491
$566$	0	0
$567$	− 1.29452e10i	− 0.125249i
$568$	0	0
$569$	−1.24740e11	−1.19003	−0.595014	−	0.803715i	$-0.702854\pi$
$569$	−1.24740e11	−1.19003	−0.595014	+	0.803715i	$0.702854\pi$
$570$	0	0
$571$	1.83334e11i	1.72464i	0.506367	+	0.862318i	$0.330988\pi$
$571$	1.83334e11i	1.72464i	−0.506367	+	0.862318i	$0.669012\pi$
$572$	0	0
$573$	8.69438e10	0.806529
$574$	0	0
$575$	3.14714e10i	0.287902i
$576$	0	0
$577$	−6.15605e10	−0.555391	−0.277695	−	0.960669i	$-0.589571\pi$
$577$	−6.15605e10	−0.555391	−0.277695	+	0.960669i	$0.589571\pi$
$578$	0	0
$579$	− 2.59817e10i	− 0.231182i
$580$	0	0
$581$	1.68561e11	1.47929
$582$	0	0
$583$	1.34089e10i	0.116070i
$584$	0	0
$585$	2.39992e10	0.204915
$586$	0	0
$587$	− 2.17925e11i	− 1.83550i	−0.397160	−	0.917749i	$-0.630004\pi$
$587$	− 2.17925e11i	− 1.83550i	0.397160	−	0.917749i	$-0.369996\pi$
$588$	0	0
$589$	1.07817e10	0.0895828
$590$	0	0
$591$	7.64099e10i	0.626325i
$592$	0	0
$593$	8.17212e10	0.660870	0.330435	−	0.943829i	$-0.392805\pi$
$593$	8.17212e10	0.660870	0.330435	+	0.943829i	$0.392805\pi$
$594$	0	0
$595$	1.15399e10i	0.0920737i
$596$	0	0
$597$	1.30175e11	1.02478
$598$	0	0
$599$	2.37785e11i	1.84704i	0.383546	+	0.923522i	$0.374703\pi$
$599$	2.37785e11i	1.84704i	−0.383546	+	0.923522i	$0.625297\pi$
$600$	0	0
$601$	−8.15453e10	−0.625030	−0.312515	−	0.949913i	$-0.601171\pi$
$601$	−8.15453e10	−0.625030	−0.312515	+	0.949913i	$0.601171\pi$
$602$	0	0
$603$	6.57991e10i	0.497680i
$604$	0	0
$605$	4.67731e10	0.349120
$606$	0	0
$607$	− 2.08864e11i	− 1.53854i	−0.638922	−	0.769272i	$-0.720619\pi$
$607$	− 2.08864e11i	− 1.53854i	0.638922	−	0.769272i	$-0.279381\pi$
$608$	0	0
$609$	1.26347e11	0.918536
$610$	0	0
$611$	7.83218e10i	0.561976i
$612$	0	0
$613$	−6.97158e10	−0.493730	−0.246865	−	0.969050i	$-0.579400\pi$
$613$	−6.97158e10	−0.493730	−0.246865	+	0.969050i	$0.579400\pi$
$614$	0	0
$615$	3.92381e9i	0.0274289i
$616$	0	0
$617$	2.12217e11	1.46433	0.732167	−	0.681125i	$-0.238509\pi$
$617$	2.12217e11	1.46433	0.732167	+	0.681125i	$0.238509\pi$
$618$	0	0
$619$	− 1.46426e11i	− 0.997368i	−0.866784	−	0.498684i	$-0.833817\pi$
$619$	− 1.46426e11i	− 0.997368i	0.866784	−	0.498684i	$-0.166183\pi$
$620$	0	0
$621$	−4.12002e10	−0.277034
$622$	0	0
$623$	− 1.87010e10i	− 0.124140i
$624$	0	0
$625$	6.10352e9	0.0400000
$626$	0	0
$627$	1.65109e10i	0.106832i
$628$	0	0
$629$	−2.76866e10	−0.176875
$630$	0	0
$631$	2.20149e11i	1.38867i	0.719650	+	0.694337i	$0.244302\pi$
$631$	2.20149e11i	1.38867i	−0.719650	+	0.694337i	$0.755698\pi$
$632$	0	0
$633$	−1.11308e11	−0.693282
$634$	0	0
$635$	1.17347e11i	0.721735i
$636$	0	0
$637$	6.12626e10	0.372081
$638$	0	0
$639$	− 8.42177e9i	− 0.0505126i
$640$	0	0
$641$	2.61222e11	1.54731	0.773657	−	0.633605i	$-0.218425\pi$
$641$	2.61222e11	1.54731	0.773657	+	0.633605i	$0.218425\pi$
$642$	0	0
$643$	− 2.28629e11i	− 1.33748i	−0.743496	−	0.668740i	$-0.766834\pi$
$643$	− 2.28629e11i	− 1.33748i	0.743496	−	0.668740i	$-0.233166\pi$
$644$	0	0
$645$	−3.39642e9	−0.0196238
$646$	0	0
$647$	8.21741e9i	0.0468941i	0.999725	+	0.0234470i	$0.00746411\pi$
$647$	8.21741e9i	0.0468941i	−0.999725	+	0.0234470i	$0.992536\pi$
$648$	0	0
$649$	9.37904e10	0.528664
$650$	0	0
$651$	− 2.65038e10i	− 0.147565i
$652$	0	0
$653$	−2.63276e11	−1.44796	−0.723982	−	0.689819i	$-0.757690\pi$
$653$	−2.63276e11	−1.44796	−0.723982	+	0.689819i	$0.757690\pi$
$654$	0	0
$655$	2.01537e10i	0.109494i
$656$	0	0
$657$	7.70682e10	0.413632
$658$	0	0
$659$	− 3.48892e11i	− 1.84990i	−0.380084	−	0.924952i	$-0.624105\pi$
$659$	− 3.48892e11i	− 1.84990i	0.380084	−	0.924952i	$-0.375895\pi$
$660$	0	0
$661$	1.13622e11	0.595194	0.297597	−	0.954692i	$-0.403815\pi$
$661$	1.13622e11	0.595194	0.297597	+	0.954692i	$0.403815\pi$
$662$	0	0
$663$	− 2.80076e10i	− 0.144951i
$664$	0	0
$665$	−3.89509e10	−0.199173
$666$	0	0
$667$	− 4.02121e11i	− 2.03167i
$668$	0	0
$669$	−1.35256e11	−0.675230
$670$	0	0
$671$	2.27737e10i	0.112343i
$672$	0	0
$673$	−5.03704e10	−0.245536	−0.122768	−	0.992435i	$-0.539177\pi$
$673$	−5.03704e10	−0.245536	−0.122768	+	0.992435i	$0.539177\pi$
$674$	0	0
$675$	7.99030e9i	0.0384900i
$676$	0	0
$677$	7.42494e10	0.353458	0.176729	−	0.984260i	$-0.443448\pi$
$677$	7.42494e10	0.353458	0.176729	+	0.984260i	$0.443448\pi$
$678$	0	0
$679$	− 4.00813e11i	− 1.88566i
$680$	0	0
$681$	−6.82320e9	−0.0317249
$682$	0	0
$683$	− 1.67148e11i	− 0.768101i	−0.923312	−	0.384051i	$-0.874529\pi$
$683$	− 1.67148e11i	− 0.768101i	0.923312	−	0.384051i	$-0.125471\pi$
$684$	0	0
$685$	−8.62857e10	−0.391901
$686$	0	0
$687$	− 7.79252e9i	− 0.0349825i
$688$	0	0
$689$	7.67735e10	0.340670
$690$	0	0
$691$	1.90747e10i	0.0836653i	0.999125	+	0.0418326i	$0.0133196\pi$
$691$	1.90747e10i	0.0836653i	−0.999125	+	0.0418326i	$0.986680\pi$
$692$	0	0
$693$	4.05876e10	0.175979
$694$	0	0
$695$	7.47132e10i	0.320227i
$696$	0	0
$697$	4.57917e9	0.0194024
$698$	0	0
$699$	− 2.35088e11i	− 0.984739i
$700$	0	0
$701$	−4.34596e11	−1.79976	−0.899878	−	0.436141i	$-0.856345\pi$
$701$	−4.34596e11	−1.79976	−0.899878	+	0.436141i	$0.856345\pi$
$702$	0	0
$703$	− 9.34511e10i	− 0.382616i
$704$	0	0
$705$	−2.60765e10	−0.105558
$706$	0	0
$707$	− 3.84842e11i	− 1.54030i
$708$	0	0
$709$	−1.49310e11	−0.590886	−0.295443	−	0.955360i	$-0.595467\pi$
$709$	−1.49310e11	−0.590886	−0.295443	+	0.955360i	$0.595467\pi$
$710$	0	0
$711$	3.49440e10i	0.136739i
$712$	0	0
$713$	−8.43530e10	−0.326394
$714$	0	0
$715$	7.52458e10i	0.287911i
$716$	0	0
$717$	−2.68410e11	−1.01560
$718$	0	0
$719$	1.93505e11i	0.724063i	0.932166	+	0.362032i	$0.117917\pi$
$719$	1.93505e11i	0.724063i	−0.932166	+	0.362032i	$0.882083\pi$
$720$	0	0
$721$	5.42598e10	0.200788
$722$	0	0
$723$	1.12275e11i	0.410893i
$724$	0	0
$725$	−7.79867e10	−0.282273
$726$	0	0
$727$	2.51281e11i	0.899543i	0.893144	+	0.449772i	$0.148495\pi$
$727$	2.51281e11i	0.899543i	−0.893144	+	0.449772i	$0.851505\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	3.96369e9i	0.0138813i
$732$	0	0
$733$	−2.87966e10	−0.0997527	−0.0498764	−	0.998755i	$-0.515883\pi$
$733$	−2.87966e10	−0.0997527	−0.0498764	+	0.998755i	$0.515883\pi$
$734$	0	0
$735$	2.03968e10i	0.0698896i
$736$	0	0
$737$	−2.06303e11	−0.699254
$738$	0	0
$739$	− 2.60744e11i	− 0.874252i	−0.899400	−	0.437126i	$-0.855996\pi$
$739$	− 2.60744e11i	− 0.874252i	0.899400	−	0.437126i	$-0.144004\pi$
$740$	0	0
$741$	9.45343e10	0.313557
$742$	0	0
$743$	− 5.51507e11i	− 1.80965i	−0.425780	−	0.904827i	$-0.640000\pi$
$743$	− 5.51507e11i	− 1.80965i	0.425780	−	0.904827i	$-0.360000\pi$
$744$	0	0
$745$	−2.69747e10	−0.0875653
$746$	0	0
$747$	− 1.36206e11i	− 0.437434i
$748$	0	0
$749$	3.95309e11	1.25606
$750$	0	0
$751$	4.15518e11i	1.30626i	0.757245	+	0.653131i	$0.226545\pi$
$751$	4.15518e11i	1.30626i	−0.757245	+	0.653131i	$0.773455\pi$
$752$	0	0
$753$	−4.75976e10	−0.148049
$754$	0	0
$755$	2.22392e11i	0.684432i
$756$	0	0
$757$	−1.27137e11	−0.387159	−0.193580	−	0.981085i	$-0.562010\pi$
$757$	−1.27137e11	−0.387159	−0.193580	+	0.981085i	$0.562010\pi$
$758$	0	0
$759$	− 1.29177e11i	− 0.389241i
$760$	0	0
$761$	2.83958e11	0.846674	0.423337	−	0.905972i	$-0.360859\pi$
$761$	2.83958e11	0.846674	0.423337	+	0.905972i	$0.360859\pi$
$762$	0	0
$763$	− 1.23735e11i	− 0.365085i
$764$	0	0
$765$	9.32485e9	0.0272268
$766$	0	0
$767$	− 5.37004e11i	− 1.55166i
$768$	0	0
$769$	2.58924e11	0.740401	0.370200	−	0.928952i	$-0.379289\pi$
$769$	2.58924e11	0.740401	0.370200	+	0.928952i	$0.379289\pi$
$770$	0	0
$771$	1.87505e11i	0.530635i
$772$	0	0
$773$	2.66185e11	0.745531	0.372766	−	0.927926i	$-0.378410\pi$
$773$	2.66185e11	0.745531	0.372766	+	0.927926i	$0.378410\pi$
$774$	0	0
$775$	1.63593e10i	0.0453479i
$776$	0	0
$777$	−2.29724e11	−0.630265
$778$	0	0
$779$	1.54561e10i	0.0419712i
$780$	0	0
$781$	2.64052e10	0.0709716
$782$	0	0
$783$	− 1.02095e11i	− 0.271617i
$784$	0	0
$785$	−2.48214e11	−0.653653
$786$	0	0
$787$	− 2.47585e11i	− 0.645395i	−0.946502	−	0.322698i	$-0.895410\pi$
$787$	− 2.47585e11i	− 0.645395i	0.946502	−	0.322698i	$-0.104590\pi$
$788$	0	0
$789$	6.14428e9	0.0158549
$790$	0	0
$791$	5.39253e11i	1.37748i
$792$	0	0
$793$	1.30393e11	0.329731
$794$	0	0
$795$	2.55610e10i	0.0639896i
$796$	0	0
$797$	6.25586e9	0.0155043	0.00775217	−	0.999970i	$-0.497532\pi$
$797$	6.25586e9	0.0155043	0.00775217	+	0.999970i	$0.497532\pi$
$798$	0	0
$799$	3.04318e10i	0.0746690i
$800$	0	0
$801$	−1.51113e10	−0.0367090
$802$	0	0
$803$	2.41636e11i	0.581164i
$804$	0	0
$805$	3.04742e11	0.725686
$806$	0	0
$807$	− 4.36172e11i	− 1.02840i
$808$	0	0
$809$	−1.90012e10	−0.0443596	−0.0221798	−	0.999754i	$-0.507061\pi$
$809$	−1.90012e10	−0.0443596	−0.0221798	+	0.999754i	$0.507061\pi$
$810$	0	0
$811$	− 7.50225e9i	− 0.0173423i	−0.999962	−	0.00867117i	$-0.997240\pi$
$811$	− 7.50225e9i	− 0.0173423i	0.999962	−	0.00867117i	$-0.00276016\pi$
$812$	0	0
$813$	2.19460e11	0.502336
$814$	0	0
$815$	1.10928e11i	0.251426i
$816$	0	0
$817$	−1.33787e10	−0.0300280
$818$	0	0
$819$	− 2.32387e11i	− 0.516507i
$820$	0	0
$821$	−5.11743e11	−1.12636	−0.563182	−	0.826333i	$-0.690423\pi$
$821$	−5.11743e11	−1.12636	−0.563182	+	0.826333i	$0.690423\pi$
$822$	0	0
$823$	5.47555e11i	1.19352i	0.802421	+	0.596758i	$0.203545\pi$
$823$	5.47555e11i	1.19352i	−0.802421	+	0.596758i	$0.796455\pi$
$824$	0	0
$825$	−2.50524e10	−0.0540795
$826$	0	0
$827$	− 8.27481e11i	− 1.76903i	−0.466508	−	0.884517i	$-0.654488\pi$
$827$	− 8.27481e11i	− 1.76903i	0.466508	−	0.884517i	$-0.345512\pi$
$828$	0	0
$829$	−3.11028e11	−0.658539	−0.329269	−	0.944236i	$-0.606802\pi$
$829$	−3.11028e11	−0.658539	−0.329269	+	0.944236i	$0.606802\pi$
$830$	0	0
$831$	3.57530e11i	0.749736i
$832$	0	0
$833$	2.38035e10	0.0494380
$834$	0	0
$835$	2.35860e10i	0.0485187i
$836$	0	0
$837$	−2.14164e10	−0.0436360
$838$	0	0
$839$	− 4.82063e11i	− 0.972873i	−0.873716	−	0.486437i	$-0.838296\pi$
$839$	− 4.82063e11i	− 0.972873i	0.873716	−	0.486437i	$-0.161704\pi$
$840$	0	0
$841$	4.96217e11	0.991945
$842$	0	0
$843$	6.08264e10i	0.120443i
$844$	0	0
$845$	2.02821e11	0.397820
$846$	0	0
$847$	− 4.52909e11i	− 0.879990i
$848$	0	0
$849$	−1.55740e11	−0.299758
$850$	0	0
$851$	7.31137e11i	1.39406i
$852$	0	0
$853$	9.86599e11	1.86356	0.931782	−	0.363017i	$-0.118253\pi$
$853$	9.86599e11	1.86356	0.931782	+	0.363017i	$0.118253\pi$
$854$	0	0
$855$	3.14743e10i	0.0588968i
$856$	0	0
$857$	−2.43003e11	−0.450494	−0.225247	−	0.974302i	$-0.572319\pi$
$857$	−2.43003e11	−0.450494	−0.225247	+	0.974302i	$0.572319\pi$
$858$	0	0
$859$	4.24291e11i	0.779276i	0.920968	+	0.389638i	$0.127400\pi$
$859$	4.24291e11i	0.779276i	−0.920968	+	0.389638i	$0.872600\pi$
$860$	0	0
$861$	3.79948e10	0.0691371
$862$	0	0
$863$	2.81695e11i	0.507850i	0.967224	+	0.253925i	$0.0817217\pi$
$863$	2.81695e11i	0.507850i	−0.967224	+	0.253925i	$0.918278\pi$
$864$	0	0
$865$	7.16043e10	0.127901
$866$	0	0
$867$	3.15342e11i	0.558091i
$868$	0	0
$869$	−1.09561e11	−0.192123
$870$	0	0
$871$	1.18120e12i	2.05235i
$872$	0	0
$873$	−3.23877e11	−0.557601
$874$	0	0
$875$	− 5.91011e10i	− 0.100824i
$876$	0	0
$877$	8.14031e10	0.137608	0.0688038	−	0.997630i	$-0.478082\pi$
$877$	8.14031e10	0.137608	0.0688038	+	0.997630i	$0.478082\pi$
$878$	0	0
$879$	2.18255e11i	0.365602i
$880$	0	0
$881$	−5.73301e11	−0.951653	−0.475827	−	0.879539i	$-0.657851\pi$
$881$	−5.73301e11	−0.951653	−0.475827	+	0.879539i	$0.657851\pi$
$882$	0	0
$883$	2.70032e8i	0 0.000444193i	1.00000		0.000222097i	$7.06955e-5\pi$
$883$	2.70032e8i	0 0.000444193i	−1.00000		0.000222097i	$0.999929\pi$
$884$	0	0
$885$	1.78790e11	0.291455
$886$	0	0
$887$	3.25567e11i	0.525953i	0.964802	+	0.262976i	$0.0847041\pi$
$887$	3.25567e11i	0.525953i	−0.964802	+	0.262976i	$0.915296\pi$
$888$	0	0
$889$	1.13629e12	1.81920
$890$	0	0
$891$	− 3.27968e10i	− 0.0520380i
$892$	0	0
$893$	−1.02717e11	−0.161524
$894$	0	0
$895$	2.35169e11i	0.366512i
$896$	0	0
$897$	−7.39612e11	−1.14244
$898$	0	0
$899$	− 2.09028e11i	− 0.320011i
$900$	0	0
$901$	2.98302e10	0.0452644
$902$	0	0
$903$	3.28879e10i	0.0494636i
$904$	0	0
$905$	−4.56706e11	−0.680836
$906$	0	0
$907$	− 9.10684e11i	− 1.34567i	−0.739792	−	0.672835i	$-0.765076\pi$
$907$	− 9.10684e11i	− 1.34567i	0.739792	−	0.672835i	$-0.234924\pi$
$908$	0	0
$909$	−3.10972e11	−0.455476
$910$	0	0
$911$	8.41502e11i	1.22175i	0.791728	+	0.610874i	$0.209182\pi$
$911$	8.41502e11i	1.22175i	−0.791728	+	0.610874i	$0.790818\pi$
$912$	0	0
$913$	4.27052e11	0.614607
$914$	0	0
$915$	4.34130e10i	0.0619349i
$916$	0	0
$917$	1.95151e11	0.275990
$918$	0	0
$919$	− 4.77344e11i	− 0.669221i	−0.942356	−	0.334611i	$-0.891395\pi$
$919$	− 4.77344e11i	− 0.669221i	0.942356	−	0.334611i	$-0.108605\pi$
$920$	0	0
$921$	−5.89867e11	−0.819815
$922$	0	0
$923$	− 1.51185e11i	− 0.208306i
$924$	0	0
$925$	1.41795e11	0.193685
$926$	0	0
$927$	− 4.38446e10i	− 0.0593741i
$928$	0	0
$929$	−6.55185e11	−0.879633	−0.439816	−	0.898088i	$-0.644956\pi$
$929$	−6.55185e11	−0.879633	−0.439816	+	0.898088i	$0.644956\pi$
$930$	0	0
$931$	8.03443e10i	0.106944i
$932$	0	0
$933$	−2.97143e10	−0.0392138
$934$	0	0
$935$	2.92366e10i	0.0382543i
$936$	0	0
$937$	−2.66622e11	−0.345890	−0.172945	−	0.984931i	$-0.555328\pi$
$937$	−2.66622e11	−0.345890	−0.172945	+	0.984931i	$0.555328\pi$
$938$	0	0
$939$	6.29121e11i	0.809229i
$940$	0	0
$941$	−2.78557e11	−0.355268	−0.177634	−	0.984097i	$-0.556844\pi$
$941$	−2.78557e11	−0.355268	−0.177634	+	0.984097i	$0.556844\pi$
$942$	0	0
$943$	− 1.20925e11i	− 0.152922i
$944$	0	0
$945$	7.73711e10	0.0970178
$946$	0	0
$947$	9.16582e11i	1.13965i	0.821766	+	0.569825i	$0.192989\pi$
$947$	9.16582e11i	1.13965i	−0.821766	+	0.569825i	$0.807011\pi$
$948$	0	0
$949$	1.38350e12	1.70575
$950$	0	0
$951$	− 1.42443e11i	− 0.174149i
$952$	0	0
$953$	3.17195e10	0.0384552	0.0192276	−	0.999815i	$-0.493879\pi$
$953$	3.17195e10	0.0384552	0.0192276	+	0.999815i	$0.493879\pi$
$954$	0	0
$955$	5.19648e11i	0.624735i
$956$	0	0
$957$	3.20103e11	0.381629
$958$	0	0
$959$	8.35515e11i	0.987825i
$960$	0	0
$961$	8.09043e11	0.948589
$962$	0	0
$963$	− 3.19429e11i	− 0.371423i
$964$	0	0
$965$	1.55288e11	0.179073
$966$	0	0
$967$	− 2.56145e10i	− 0.0292941i	−0.999893	−	0.0146471i	$-0.995338\pi$
$967$	− 2.56145e10i	− 0.0292941i	0.999893	−	0.0146471i	$-0.00466247\pi$
$968$	0	0
$969$	3.67312e10	0.0416620
$970$	0	0
$971$	1.28668e12i	1.44742i	0.690105	+	0.723710i	$0.257564\pi$
$971$	1.28668e12i	1.44742i	−0.690105	+	0.723710i	$0.742436\pi$
$972$	0	0
$973$	7.23457e11	0.807164
$974$	0	0
$975$	1.43439e11i	0.158726i
$976$	0	0
$977$	−9.08955e11	−0.997618	−0.498809	−	0.866712i	$-0.666229\pi$
$977$	−9.08955e11	−0.997618	−0.498809	+	0.866712i	$0.666229\pi$
$978$	0	0
$979$	− 4.73792e10i	− 0.0515771i
$980$	0	0
$981$	−9.99840e10	−0.107958
$982$	0	0
$983$	9.97917e11i	1.06876i	0.845244	+	0.534380i	$0.179455\pi$
$983$	9.97917e11i	1.06876i	−0.845244	+	0.534380i	$0.820545\pi$
$984$	0	0
$985$	−4.56689e11	−0.485149
$986$	0	0
$987$	2.52502e11i	0.266070i
$988$	0	0
$989$	1.04671e11	0.109407
$990$	0	0
$991$	6.71065e11i	0.695777i	0.937536	+	0.347888i	$0.113101\pi$
$991$	6.71065e11i	0.695777i	−0.937536	+	0.347888i	$0.886899\pi$
$992$	0	0
$993$	−7.26258e10	−0.0746954
$994$	0	0
$995$	7.78035e11i	0.793792i
$996$	0	0
$997$	1.70974e11	0.173041	0.0865203	−	0.996250i	$-0.472425\pi$
$997$	1.70974e11	0.173041	0.0865203	+	0.996250i	$0.472425\pi$
$998$	0	0
$999$	1.85629e11i	0.186373i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.9.e.b.31.4	✓	12
4.3	odd	2	inner	240.9.e.b.31.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.9.e.b.31.4	✓	12	1.1	even	1	trivial
240.9.e.b.31.12	yes	12	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	240.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +279.508 q^{5} -2706.52i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +279.508 q^{5} -2706.52i q^{7} -2187.00 q^{9} -6857.00i q^{11} -39260.2 q^{13} -13071.3i q^{15} -15254.5 q^{17} -51488.7i q^{19} -126571. q^{21} +402835. i q^{23} +78125.0 q^{25} +102276. i q^{27} -998230. q^{29} +209399. i q^{31} -320670. q^{33} -756494. i q^{35} +1.81498e6 q^{37} +1.83602e6i q^{39} -300185. q^{41} -259837. i q^{43} -611285. q^{45} -1.99494e6i q^{47} -1.56042e6 q^{49} +713382. i q^{51} -1.95550e6 q^{53} -1.91659e6i q^{55} -2.40789e6 q^{57} +1.36781e7i q^{59} -3.32124e6 q^{61} +5.91915e6i q^{63} -1.09736e7 q^{65} -3.00865e7i q^{67} +1.88387e7 q^{69} +3.85083e6i q^{71} -3.52393e7 q^{73} -3.65354e6i q^{75} -1.85586e7 q^{77} -1.59780e7i q^{79} +4.78297e6 q^{81} +6.22797e7i q^{83} -4.26376e6 q^{85} +4.66826e7i q^{87} +6.90961e6 q^{89} +1.06258e8i q^{91} +9.79260e6 q^{93} -1.43915e7i q^{95} +1.48092e8 q^{97} +1.49963e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 26244 q^{9}+O(q^{10}) \) 12 * q - 26244 * q^9	\( 12 q - 26244 q^{9} - 41616 q^{13} + 44016 q^{17} + 407592 q^{21} + 937500 q^{25} - 1521456 q^{29} + 1465128 q^{33} - 665760 q^{37} - 3352200 q^{41} - 4335444 q^{49} - 8855232 q^{53} + 1550016 q^{57} + 11684328 q^{61} + 7305000 q^{65} + 47784816 q^{69} + 119457096 q^{73} - 256261344 q^{77} + 57395628 q^{81} + 75405000 q^{85} - 162442392 q^{89} + 203718240 q^{93} + 12509688 q^{97}+O(q^{100}) \) 12 * q - 26244 * q^9 - 41616 * q^13 + 44016 * q^17 + 407592 * q^21 + 937500 * q^25 - 1521456 * q^29 + 1465128 * q^33 - 665760 * q^37 - 3352200 * q^41 - 4335444 * q^49 - 8855232 * q^53 + 1550016 * q^57 + 11684328 * q^61 + 7305000 * q^65 + 47784816 * q^69 + 119457096 * q^73 - 256261344 * q^77 + 57395628 * q^81 + 75405000 * q^85 - 162442392 * q^89 + 203718240 * q^93 + 12509688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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