LMFDB - Embedded newform 5040.2.f.i.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(881,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5040.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.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:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$-1.91681i$ of defining polynomial
Character	$\chi$	$=$	5040.881
Dual form			5040.2.f.i.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +(-2.27220 - 1.35539i) q^{7} +O(q^{10})$	$q+1.00000 q^{5} +(-2.27220 - 1.35539i) q^{7} -2.71078i q^{11} +6.54441i q^{13} -1.53186 q^{17} -2.30177i q^{19} +3.83363i q^{23} +1.00000 q^{25} -3.83363i q^{29} -3.25519i q^{31} +(-2.27220 - 1.35539i) q^{35} +3.01255 q^{37} +6.54441 q^{41} +0.468142 q^{43} -9.11980 q^{47} +(3.32583 + 6.15945i) q^{49} -9.25519i q^{53} -2.71078i q^{55} +11.1961 q^{59} -4.78705i q^{61} +6.54441i q^{65} -13.5570 q^{67} -2.30177i q^{71} -11.4216i q^{73} +(-3.67417 + 6.15945i) q^{77} +12.6768 q^{79} +13.0888 q^{83} -1.53186 q^{85} +9.60812 q^{89} +(8.87024 - 14.8702i) q^{91} -2.30177i q^{95} -16.9224i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} + 4 q^{7}+O(q^{10}) $ 8 * q + 8 * q^5 + 4 * q^7	$ 8 q + 8 q^{5} + 4 q^{7} + 8 q^{25} + 4 q^{35} - 8 q^{37} + 8 q^{41} + 16 q^{43} - 40 q^{47} + 4 q^{49} - 32 q^{67} - 52 q^{77} - 8 q^{79} + 16 q^{83} + 8 q^{89} + 4 q^{91}+O(q^{100}) $ 8 * q + 8 * q^5 + 4 * q^7 + 8 * q^25 + 4 * q^35 - 8 * q^37 + 8 * q^41 + 16 * q^43 - 40 * q^47 + 4 * q^49 - 32 * q^67 - 52 * q^77 - 8 * q^79 + 16 * q^83 + 8 * q^89 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$1$	$-1$	$1$	$-1$	$1$

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
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−2.27220	−	1.35539i	−0.858813	−	0.512290i
$7$	−2.27220	−	1.35539i	−0.858813	−	0.512290i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.71078i		−	0.817332i	−0.912684	−	0.408666i	$-0.865994\pi$
$11$		−	2.71078i		−	0.817332i	0.912684	−	0.408666i	$-0.134006\pi$
$12$		0			0
$13$			6.54441i			1.81509i	0.419952	+	0.907546i	$0.362047\pi$
$13$			6.54441i			1.81509i	−0.419952	+	0.907546i	$0.637953\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.53186			−0.371530			−0.185765	−	0.982594i	$-0.559476\pi$
$17$	−1.53186			−0.371530			−0.185765	+	0.982594i	$0.559476\pi$
$18$		0			0
$19$		−	2.30177i		−	0.528062i	−0.964514	−	0.264031i	$-0.914948\pi$
$19$		−	2.30177i		−	0.528062i	0.964514	−	0.264031i	$-0.0850521\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.83363i			0.799366i	0.916653	+	0.399683i	$0.130880\pi$
$23$			3.83363i			0.799366i	−0.916653	+	0.399683i	$0.869120\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.83363i		−	0.711887i	−0.934508	−	0.355943i	$-0.884160\pi$
$29$		−	3.83363i		−	0.711887i	0.934508	−	0.355943i	$-0.115840\pi$
$30$		0			0
$31$		−	3.25519i		−	0.584650i	−0.956319	−	0.292325i	$-0.905571\pi$
$31$		−	3.25519i		−	0.584650i	0.956319	−	0.292325i	$-0.0944289\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.27220	−	1.35539i	−0.384073	−	0.229103i
$36$		0			0
$37$	3.01255			0.495260			0.247630	−	0.968855i	$-0.420348\pi$
$37$	3.01255			0.495260			0.247630	+	0.968855i	$0.420348\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.54441			1.02207			0.511033	−	0.859561i	$-0.329263\pi$
$41$	6.54441			1.02207			0.511033	+	0.859561i	$0.329263\pi$
$42$		0			0
$43$	0.468142			0.0713910			0.0356955	−	0.999363i	$-0.488635\pi$
$43$	0.468142			0.0713910			0.0356955	+	0.999363i	$0.488635\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.11980			−1.33026			−0.665130	−	0.746728i	$-0.731624\pi$
$47$	−9.11980			−1.33026			−0.665130	+	0.746728i	$0.731624\pi$
$48$		0			0
$49$	3.32583	+	6.15945i	0.475118	+	0.879922i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.25519i		−	1.27130i	−0.771978	−	0.635649i	$-0.780732\pi$
$53$		−	9.25519i		−	1.27130i	0.771978	−	0.635649i	$-0.219268\pi$
$54$		0			0
$55$		−	2.71078i		−	0.365522i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11.1961			1.45760			0.728802	−	0.684725i	$-0.240078\pi$
$59$	11.1961			1.45760			0.728802	+	0.684725i	$0.240078\pi$
$60$		0			0
$61$		−	4.78705i		−	0.612919i	−0.951884	−	0.306459i	$-0.900856\pi$
$61$		−	4.78705i		−	0.612919i	0.951884	−	0.306459i	$-0.0991444\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.54441i			0.811734i
$66$		0			0
$67$	−13.5570			−1.65625			−0.828123	−	0.560546i	$-0.810591\pi$
$67$	−13.5570			−1.65625			−0.828123	+	0.560546i	$0.810591\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.30177i		−	0.273170i	−0.990628	−	0.136585i	$-0.956387\pi$
$71$		−	2.30177i		−	0.273170i	0.990628	−	0.136585i	$-0.0436126\pi$
$72$		0			0
$73$		−	11.4216i		−	1.33679i	−0.743805	−	0.668397i	$-0.766981\pi$
$73$		−	11.4216i		−	1.33679i	0.743805	−	0.668397i	$-0.233019\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.67417	+	6.15945i	−0.418711	+	0.701935i
$78$		0			0
$79$	12.6768			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$79$	12.6768			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.0888			1.43668			0.718342	−	0.695690i	$-0.244901\pi$
$83$	13.0888			1.43668			0.718342	+	0.695690i	$0.244901\pi$
$84$		0			0
$85$	−1.53186			−0.166153
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.60812			1.01846			0.509230	−	0.860631i	$-0.329930\pi$
$89$	9.60812			1.01846			0.509230	+	0.860631i	$0.329930\pi$
$90$		0			0
$91$	8.87024	−	14.8702i	0.929853	−	1.55882i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	2.30177i		−	0.236156i
$96$		0			0
$97$		−	16.9224i		−	1.71821i	−0.511796	−	0.859107i	$-0.671020\pi$
$97$		−	16.9224i		−	1.71821i	0.511796	−	0.859107i	$-0.328980\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−17.7405			−1.76524			−0.882622	−	0.470084i	$-0.844224\pi$
$101$	−17.7405			−1.76524			−0.882622	+	0.470084i	$0.844224\pi$
$102$		0			0
$103$		−	4.05913i		−	0.399958i	−0.979800	−	0.199979i	$-0.935913\pi$
$103$		−	4.05913i		−	0.399958i	0.979800	−	0.199979i	$-0.0640874\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.31891i		−	0.610872i	−0.952213	−	0.305436i	$-0.901198\pi$
$107$		−	6.31891i		−	0.610872i	0.952213	−	0.305436i	$-0.0988022\pi$
$108$		0			0
$109$	−12.0251			−1.15180			−0.575898	−	0.817522i	$-0.695347\pi$
$109$	−12.0251			−1.15180			−0.575898	+	0.817522i	$0.695347\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.06372i			0.852643i	0.904572	+	0.426321i	$0.140191\pi$
$113$			9.06372i			0.852643i	−0.904572	+	0.426321i	$0.859809\pi$
$114$		0			0
$115$			3.83363i			0.357487i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.48069	+	2.07627i	0.319075	+	0.190331i
$120$		0			0
$121$	3.65166			0.331969
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−21.6332			−1.91964			−0.959819	−	0.280619i	$-0.909460\pi$
$127$	−21.6332			−1.91964			−0.959819	+	0.280619i	$0.909460\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.51931			−0.220113			−0.110056	−	0.993925i	$-0.535103\pi$
$131$	−2.51931			−0.220113			−0.110056	+	0.993925i	$0.535103\pi$
$132$		0			0
$133$	−3.11980	+	5.23009i	−0.270521	+	0.453506i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.39646i		−	0.119308i	−0.998219	−	0.0596539i	$-0.981000\pi$
$137$		−	1.39646i		−	0.119308i	0.998219	−	0.0596539i	$-0.0189997\pi$
$138$		0			0
$139$		−	6.13539i		−	0.520397i	−0.965555	−	0.260199i	$-0.916212\pi$
$139$		−	6.13539i		−	0.520397i	0.965555	−	0.260199i	$-0.0837881\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.7405			1.48353
$144$		0			0
$145$		−	3.83363i		−	0.318365i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	4.65166i		−	0.381078i	−0.981680	−	0.190539i	$-0.938976\pi$
$149$		−	4.65166i		−	0.381078i	0.981680	−	0.190539i	$-0.0610236\pi$
$150$		0			0
$151$	3.58794			0.291982			0.145991	−	0.989286i	$-0.453363\pi$
$151$	3.58794			0.291982			0.145991	+	0.989286i	$0.453363\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3.25519i		−	0.261463i
$156$		0			0
$157$		−	13.1228i		−	1.04732i	−0.851928	−	0.523658i	$-0.824567\pi$
$157$		−	13.1228i		−	1.04732i	0.851928	−	0.523658i	$-0.175433\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.19606	−	8.71078i	0.409507	−	0.686506i
$162$		0			0
$163$	−16.6207			−1.30183			−0.650916	−	0.759150i	$-0.725615\pi$
$163$	−16.6207			−1.30183			−0.650916	+	0.759150i	$0.725615\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.62068			0.667088			0.333544	−	0.942735i	$-0.391755\pi$
$167$	8.62068			0.667088			0.333544	+	0.942735i	$0.391755\pi$
$168$		0			0
$169$	−29.8293			−2.29456
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.6517			0.809830			0.404915	−	0.914354i	$-0.367301\pi$
$173$	10.6517			0.809830			0.404915	+	0.914354i	$0.367301\pi$
$174$		0			0
$175$	−2.27220	−	1.35539i	−0.171763	−	0.102458i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	23.1961i		−	1.73376i	−0.498521	−	0.866878i	$-0.666123\pi$
$179$		−	23.1961i		−	1.73376i	0.498521	−	0.866878i	$-0.333877\pi$
$180$		0			0
$181$		−	9.11980i		−	0.677869i	−0.940810	−	0.338935i	$-0.889933\pi$
$181$		−	9.11980i		−	0.677869i	0.940810	−	0.338935i	$-0.110067\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.01255			0.221487
$186$		0			0
$187$			4.15253i			0.303663i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.69823i		−	0.701739i	−0.936424	−	0.350870i	$-0.885886\pi$
$191$		−	9.69823i		−	0.701739i	0.936424	−	0.350870i	$-0.114114\pi$
$192$		0			0
$193$	−18.1525			−1.30665			−0.653324	−	0.757078i	$-0.726626\pi$
$193$	−18.1525			−1.30665			−0.653324	+	0.757078i	$0.726626\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.60354i			0.327988i	0.986461	+	0.163994i	$0.0524378\pi$
$197$			4.60354i			0.327988i	−0.986461	+	0.163994i	$0.947562\pi$
$198$		0			0
$199$		−	11.7405i		−	0.832260i	−0.909305	−	0.416130i	$-0.863386\pi$
$199$		−	11.7405i		−	0.832260i	0.909305	−	0.416130i	$-0.136614\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.19606	+	8.71078i	−0.364692	+	0.611377i
$204$		0			0
$205$	6.54441			0.457081
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.23960			−0.431602
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.468142			0.0319270
$216$		0			0
$217$	−4.41206	+	7.39646i	−0.299510	+	0.502105i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	10.0251i		−	0.674361i
$222$		0			0
$223$			26.4513i			1.77131i	0.464347	+	0.885654i	$0.346289\pi$
$223$			26.4513i			1.77131i	−0.464347	+	0.885654i	$0.653711\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.0637			0.999814			0.499907	−	0.866079i	$-0.333368\pi$
$227$	15.0637			0.999814			0.499907	+	0.866079i	$0.333368\pi$
$228$		0			0
$229$			11.3655i			0.751052i	0.926812	+	0.375526i	$0.122538\pi$
$229$			11.3655i			0.751052i	−0.926812	+	0.375526i	$0.877462\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.9957i		−	1.37547i	−0.725961	−	0.687736i	$-0.758605\pi$
$233$		−	20.9957i		−	1.37547i	0.725961	−	0.687736i	$-0.241395\pi$
$234$		0			0
$235$	−9.11980			−0.594910
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.69823i			0.627326i	0.949534	+	0.313663i	$0.101556\pi$
$239$			9.69823i			0.627326i	−0.949534	+	0.313663i	$0.898444\pi$
$240$		0			0
$241$		−	24.7019i		−	1.59119i	−0.605831	−	0.795593i	$-0.707159\pi$
$241$		−	24.7019i		−	1.59119i	0.605831	−	0.795593i	$-0.292841\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.32583	+	6.15945i	0.212479	+	0.393513i
$246$		0			0
$247$	15.0637			0.958481
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.60812			0.227743			0.113871	−	0.993495i	$-0.463675\pi$
$251$	3.60812			0.227743			0.113871	+	0.993495i	$0.463675\pi$
$252$		0			0
$253$	10.3921			0.653348
$254$		0			0
$255$		0			0
$256$		0			0
$257$	28.7983			1.79639			0.898195	−	0.439598i	$-0.144879\pi$
$257$	28.7983			1.79639			0.898195	+	0.439598i	$0.144879\pi$
$258$		0			0
$259$	−6.84513	−	4.08319i	−0.425336	−	0.253717i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			12.7699i			0.787426i	0.919233	+	0.393713i	$0.128810\pi$
$263$			12.7699i			0.787426i	−0.919233	+	0.393713i	$0.871190\pi$
$264$		0			0
$265$		−	9.25519i		−	0.568542i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.43716			0.514423			0.257211	−	0.966355i	$-0.417196\pi$
$269$	8.43716			0.514423			0.257211	+	0.966355i	$0.417196\pi$
$270$		0			0
$271$			2.16637i			0.131598i	0.997833	+	0.0657989i	$0.0209596\pi$
$271$			2.16637i			0.131598i	−0.997833	+	0.0657989i	$0.979040\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	2.71078i		−	0.163466i
$276$		0			0
$277$	5.92373			0.355923			0.177961	−	0.984037i	$-0.443050\pi$
$277$	5.92373			0.355923			0.177961	+	0.984037i	$0.443050\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.3566i		−	1.63196i	−0.578083	−	0.815978i	$-0.696199\pi$
$281$		−	27.3566i		−	1.63196i	0.578083	−	0.815978i	$-0.303801\pi$
$282$		0			0
$283$		−	18.0594i		−	1.07352i	−0.843735	−	0.536759i	$-0.819648\pi$
$283$		−	18.0594i		−	1.07352i	0.843735	−	0.536759i	$-0.180352\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.8702	−	8.87024i	−0.877762	−	0.523594i
$288$		0			0
$289$	−14.6534			−0.861965
$290$		0			0
$291$		0			0
$292$		0			0
$293$	29.1139			1.70085			0.850427	−	0.526094i	$-0.176344\pi$
$293$	29.1139			1.70085			0.850427	+	0.526094i	$0.176344\pi$
$294$		0			0
$295$	11.1961			0.651860
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−25.0888			−1.45092
$300$		0			0
$301$	−1.06372	−	0.634516i	−0.0613115	−	0.0365729i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4.78705i		−	0.274106i
$306$		0			0
$307$			7.39646i			0.422138i	0.977471	+	0.211069i	$0.0676945\pi$
$307$			7.39646i			0.422138i	−0.977471	+	0.211069i	$0.932305\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.97490			0.111986			0.0559931	−	0.998431i	$-0.482168\pi$
$311$	1.97490			0.111986			0.0559931	+	0.998431i	$0.482168\pi$
$312$		0			0
$313$		−	0.961388i		−	0.0543409i	−0.999631	−	0.0271704i	$-0.991350\pi$
$313$		−	0.961388i		−	0.0543409i	0.999631	−	0.0271704i	$-0.00864968\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.1620i			0.963916i	0.876194	+	0.481958i	$0.160074\pi$
$317$			17.1620i			0.963916i	−0.876194	+	0.481958i	$0.839926\pi$
$318$		0			0
$319$	−10.3921			−0.581848
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.52598i			0.196191i
$324$		0			0
$325$			6.54441i			0.363019i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	20.7220	+	12.3609i	1.14244	+	0.681478i
$330$		0			0
$331$	−9.74047			−0.535385			−0.267692	−	0.963504i	$-0.586261\pi$
$331$	−9.74047			−0.535385			−0.267692	+	0.963504i	$0.586261\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.5570			−0.740696
$336$		0			0
$337$	3.15078			0.171634			0.0858169	−	0.996311i	$-0.472650\pi$
$337$	3.15078			0.171634			0.0858169	+	0.996311i	$0.472650\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.82412			−0.477853
$342$		0			0
$343$	0.791511	−	18.5033i	0.0427376	−	0.999086i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			30.0594i			1.61367i	0.590775	+	0.806836i	$0.298822\pi$
$347$			30.0594i			1.61367i	−0.590775	+	0.806836i	$0.701178\pi$
$348$		0			0
$349$		−	27.5180i		−	1.47301i	−0.676434	−	0.736503i	$-0.736476\pi$
$349$		−	27.5180i		−	1.47301i	0.676434	−	0.736503i	$-0.263524\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.5956			−0.883293			−0.441647	−	0.897189i	$-0.645605\pi$
$353$	−16.5956			−0.883293			−0.441647	+	0.897189i	$0.645605\pi$
$354$		0			0
$355$		−	2.30177i		−	0.122165i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			15.0076i			0.792073i	0.918235	+	0.396036i	$0.129615\pi$
$359$			15.0076i			0.792073i	−0.918235	+	0.396036i	$0.870385\pi$
$360$		0			0
$361$	13.7019			0.721151
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	11.4216i		−	0.597832i
$366$		0			0
$367$		−	14.2598i		−	0.744354i	−0.928162	−	0.372177i	$-0.878611\pi$
$367$		−	14.2598i		−	0.744354i	0.928162	−	0.372177i	$-0.121389\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.5444	+	21.0297i	−0.651273	+	1.09181i
$372$		0			0
$373$	8.98745			0.465352			0.232676	−	0.972554i	$-0.425252\pi$
$373$	8.98745			0.465352			0.232676	+	0.972554i	$0.425252\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	25.0888			1.29214
$378$		0			0
$379$	34.5447			1.77444			0.887220	−	0.461346i	$-0.152633\pi$
$379$	34.5447			1.77444			0.887220	+	0.461346i	$0.152633\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.88020			0.147171			0.0735857	−	0.997289i	$-0.476556\pi$
$383$	2.88020			0.147171			0.0735857	+	0.997289i	$0.476556\pi$
$384$		0			0
$385$	−3.67417	+	6.15945i	−0.187253	+	0.313915i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			35.1620i			1.78279i	0.453231	+	0.891393i	$0.350271\pi$
$389$			35.1620i			1.78279i	−0.453231	+	0.891393i	$0.649729\pi$
$390$		0			0
$391$		−	5.87257i		−	0.296989i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.6768			0.637837
$396$		0			0
$397$			16.1185i			0.808965i	0.914546	+	0.404482i	$0.132548\pi$
$397$			16.1185i			0.808965i	−0.914546	+	0.404482i	$0.867452\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	16.6256i		−	0.830243i	−0.909766	−	0.415121i	$-0.863739\pi$
$401$		−	16.6256i		−	0.830243i	0.909766	−	0.415121i	$-0.136261\pi$
$402$		0			0
$403$	21.3033			1.06119
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.16637i		−	0.404792i
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−25.4397	−	15.1751i	−1.25181	−	0.746715i
$414$		0			0
$415$	13.0888			0.642505
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−36.2849			−1.77263			−0.886316	−	0.463080i	$-0.846744\pi$
$419$	−36.2849			−1.77263			−0.886316	+	0.463080i	$0.846744\pi$
$420$		0			0
$421$	19.2414			0.937766			0.468883	−	0.883260i	$-0.344657\pi$
$421$	19.2414			0.937766			0.468883	+	0.883260i	$0.344657\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.53186			−0.0743060
$426$		0			0
$427$	−6.48833	+	10.8772i	−0.313992	+	0.526383i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	17.2974i		−	0.833188i	−0.909093	−	0.416594i	$-0.863224\pi$
$431$		−	17.2974i		−	0.833188i	0.909093	−	0.416594i	$-0.136776\pi$
$432$		0			0
$433$		−	31.6473i		−	1.52087i	−0.649412	−	0.760437i	$-0.724985\pi$
$433$		−	31.6473i		−	1.52087i	0.649412	−	0.760437i	$-0.275015\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.82412			0.422115
$438$		0			0
$439$		−	13.0095i		−	0.620910i	−0.950588	−	0.310455i	$-0.899519\pi$
$439$		−	13.0095i		−	0.620910i	0.950588	−	0.310455i	$-0.100481\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	3.78551i		−	0.179855i	−0.995948	−	0.0899275i	$-0.971336\pi$
$443$		−	3.78551i		−	0.179855i	0.995948	−	0.0899275i	$-0.0286635\pi$
$444$		0			0
$445$	9.60812			0.455469
$446$		0			0
$447$		0			0
$448$		0			0
$449$			24.9307i			1.17655i	0.808661	+	0.588275i	$0.200193\pi$
$449$			24.9307i			1.17655i	−0.808661	+	0.588275i	$0.799807\pi$
$450$		0			0
$451$		−	17.7405i		−	0.835366i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.87024	−	14.8702i	0.415843	−	0.697127i
$456$		0			0
$457$	−31.3535			−1.46666			−0.733328	−	0.679875i	$-0.762034\pi$
$457$	−31.3535			−1.46666			−0.733328	+	0.679875i	$0.762034\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.52598			−0.443669			−0.221835	−	0.975084i	$-0.571205\pi$
$461$	−9.52598			−0.443669			−0.221835	+	0.975084i	$0.571205\pi$
$462$		0			0
$463$	−34.0003			−1.58013			−0.790063	−	0.613026i	$-0.789952\pi$
$463$	−34.0003			−1.58013			−0.790063	+	0.613026i	$0.789952\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	39.2665			1.81703			0.908517	−	0.417847i	$-0.137215\pi$
$467$	39.2665			1.81703			0.908517	+	0.417847i	$0.137215\pi$
$468$		0			0
$469$	30.8042	+	18.3750i	1.42241	+	0.848478i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1.26903i		−	0.0583502i
$474$		0			0
$475$		−	2.30177i		−	0.105612i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	34.0251			1.55465			0.777323	−	0.629101i	$-0.216577\pi$
$479$	34.0251			1.55465			0.777323	+	0.629101i	$0.216577\pi$
$480$		0			0
$481$			19.7154i			0.898944i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	16.9224i		−	0.768409i
$486$		0			0
$487$	12.8091			0.580436			0.290218	−	0.956961i	$-0.406272\pi$
$487$	12.8091			0.580436			0.290218	+	0.956961i	$0.406272\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11.6471i		−	0.525625i	−0.964847	−	0.262812i	$-0.915350\pi$
$491$		−	11.6471i		−	0.525625i	0.964847	−	0.262812i	$-0.0846500\pi$
$492$		0			0
$493$			5.87257i			0.264487i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.11980	+	5.23009i	−0.139942	+	0.234602i
$498$		0			0
$499$	−35.5511			−1.59149			−0.795743	−	0.605635i	$-0.792919\pi$
$499$	−35.5511			−1.59149			−0.795743	+	0.605635i	$0.792919\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.23372			0.367123			0.183562	−	0.983008i	$-0.441237\pi$
$503$	8.23372			0.367123			0.183562	+	0.983008i	$0.441237\pi$
$504$		0			0
$505$	−17.7405			−0.789441
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−31.0888			−1.37799			−0.688994	−	0.724767i	$-0.741947\pi$
$509$	−31.0888			−1.37799			−0.688994	+	0.724767i	$0.741947\pi$
$510$		0			0
$511$	−15.4807	+	25.9521i	−0.684826	+	1.14805i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	4.05913i		−	0.178867i
$516$		0			0
$517$			24.7218i			1.08726i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.39363			−0.0610561			−0.0305281	−	0.999534i	$-0.509719\pi$
$521$	−1.39363			−0.0610561			−0.0305281	+	0.999534i	$0.509719\pi$
$522$		0			0
$523$			20.4853i			0.895759i	0.894094	+	0.447879i	$0.147821\pi$
$523$			20.4853i			0.895759i	−0.894094	+	0.447879i	$0.852179\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.98649i			0.217215i
$528$		0			0
$529$	8.30331			0.361013
$530$		0			0
$531$		0			0
$532$		0			0
$533$			42.8293i			1.85514i
$534$		0			0
$535$		−	6.31891i		−	0.273190i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.6969	−	9.01560i	0.719188	−	0.388329i
$540$		0			0
$541$	−0.0870615			−0.00374306			−0.00187153	−	0.999998i	$-0.500596\pi$
$541$	−0.0870615			−0.00374306			−0.00187153	+	0.999998i	$0.500596\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0251			−0.515099
$546$		0			0
$547$	5.40443			0.231077			0.115538	−	0.993303i	$-0.463141\pi$
$547$	5.40443			0.231077			0.115538	+	0.993303i	$0.463141\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.82412			−0.375920
$552$		0			0
$553$	−28.8042	−	17.1820i	−1.22488	−	0.730652i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	0.127431i		−	0.00539940i	−0.999996	−	0.00269970i	$-0.999141\pi$
$557$		−	0.127431i		−	0.00539940i	0.999996	−	0.00269970i	$-0.000859343\pi$
$558$		0			0
$559$			3.06372i			0.129581i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.23960			−0.262968			−0.131484	−	0.991318i	$-0.541974\pi$
$563$	−6.23960			−0.262968			−0.131484	+	0.991318i	$0.541974\pi$
$564$		0			0
$565$			9.06372i			0.381313i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	11.6391i		−	0.487937i	−0.969783	−	0.243968i	$-0.921551\pi$
$569$		−	11.6391i		−	0.487937i	0.969783	−	0.243968i	$-0.0784493\pi$
$570$		0			0
$571$	35.9181			1.50313			0.751563	−	0.659661i	$-0.229300\pi$
$571$	35.9181			1.50313			0.751563	+	0.659661i	$0.229300\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.83363i			0.159873i
$576$		0			0
$577$			7.15687i			0.297944i	0.988841	+	0.148972i	$0.0475965\pi$
$577$			7.15687i			0.297944i	−0.988841	+	0.148972i	$0.952404\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−29.7405	−	17.7405i	−1.23384	−	0.735999i
$582$		0			0
$583$	−25.0888			−1.03907
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.15253			−0.171393			−0.0856967	−	0.996321i	$-0.527312\pi$
$587$	−4.15253			−0.171393			−0.0856967	+	0.996321i	$0.527312\pi$
$588$		0			0
$589$	−7.49270			−0.308731
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−29.7966			−1.22360			−0.611799	−	0.791013i	$-0.709554\pi$
$593$	−29.7966			−1.22360			−0.611799	+	0.791013i	$0.709554\pi$
$594$		0			0
$595$	3.48069	+	2.07627i	0.142695	+	0.0851186i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	35.4249i		−	1.44742i	−0.690104	−	0.723710i	$-0.742435\pi$
$599$		−	35.4249i		−	1.44742i	0.690104	−	0.723710i	$-0.257565\pi$
$600$		0			0
$601$		−	27.4948i		−	1.12154i	−0.827973	−	0.560768i	$-0.810506\pi$
$601$		−	27.4948i		−	1.12154i	0.827973	−	0.560768i	$-0.189494\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.65166			0.148461
$606$		0			0
$607$		−	4.76499i		−	0.193405i	−0.995313	−	0.0967025i	$-0.969170\pi$
$607$		−	4.76499i		−	0.193405i	0.995313	−	0.0967025i	$-0.0308296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	59.6837i		−	2.41454i
$612$		0			0
$613$	10.7981			0.436130			0.218065	−	0.975934i	$-0.430026\pi$
$613$	10.7981			0.436130			0.218065	+	0.975934i	$0.430026\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	34.9025i		−	1.40512i	−0.711623	−	0.702561i	$-0.752040\pi$
$617$		−	34.9025i		−	1.40512i	0.711623	−	0.702561i	$-0.247960\pi$
$618$		0			0
$619$		−	1.26107i		−	0.0506866i	−0.999679	−	0.0253433i	$-0.991932\pi$
$619$		−	1.26107i		−	0.0506866i	0.999679	−	0.0253433i	$-0.00806789\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−21.8316	−	13.0228i	−0.874666	−	0.521746i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.61480			−0.184004
$630$		0			0
$631$	−16.2145			−0.645489			−0.322744	−	0.946486i	$-0.604605\pi$
$631$	−16.2145			−0.645489			−0.322744	+	0.946486i	$0.604605\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21.6332			−0.858488
$636$		0			0
$637$	−40.3100	+	21.7656i	−1.59714	+	0.862384i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	19.6893i		−	0.777681i	−0.921305	−	0.388840i	$-0.872876\pi$
$641$		−	19.6893i		−	0.777681i	0.921305	−	0.388840i	$-0.127124\pi$
$642$		0			0
$643$		−	17.1508i		−	0.676361i	−0.941081	−	0.338180i	$-0.890189\pi$
$643$		−	17.1508i		−	0.676361i	0.941081	−	0.338180i	$-0.109811\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.0578			1.14238			0.571191	−	0.820817i	$-0.306482\pi$
$647$	29.0578			1.14238			0.571191	+	0.820817i	$0.306482\pi$
$648$		0			0
$649$		−	30.3501i		−	1.19135i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	9.11183i		−	0.356574i	−0.983979	−	0.178287i	$-0.942945\pi$
$653$		−	9.11183i		−	0.356574i	0.983979	−	0.178287i	$-0.0570555\pi$
$654$		0			0
$655$	−2.51931			−0.0984374
$656$		0			0
$657$		0			0
$658$		0			0
$659$			37.5539i			1.46289i	0.681899	+	0.731446i	$0.261154\pi$
$659$			37.5539i			1.46289i	−0.681899	+	0.731446i	$0.738846\pi$
$660$		0			0
$661$			9.50275i			0.369614i	0.982775	+	0.184807i	$0.0591660\pi$
$661$			9.50275i			0.369614i	−0.982775	+	0.184807i	$0.940834\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.11980	+	5.23009i	−0.120981	+	0.202814i
$666$		0			0
$667$	14.6967			0.569058
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.9767			−0.500958
$672$		0			0
$673$	49.6335			1.91323			0.956615	−	0.291355i	$-0.0941060\pi$
$673$	49.6335			1.91323			0.956615	+	0.291355i	$0.0941060\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.5312			−1.44244			−0.721220	−	0.692706i	$-0.756418\pi$
$677$	−37.5312			−1.44244			−0.721220	+	0.692706i	$0.756418\pi$
$678$		0			0
$679$	−22.9365	+	38.4513i	−0.880224	+	1.47562i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			9.01560i			0.344972i	0.985012	+	0.172486i	$0.0551800\pi$
$683$			9.01560i			0.344972i	−0.985012	+	0.172486i	$0.944820\pi$
$684$		0			0
$685$		−	1.39646i		−	0.0533561i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	60.5698			2.30752
$690$		0			0
$691$			8.15841i			0.310361i	0.987886	+	0.155180i	$0.0495958\pi$
$691$			8.15841i			0.310361i	−0.987886	+	0.155180i	$0.950404\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	6.13539i		−	0.232729i
$696$		0			0
$697$	−10.0251			−0.379728
$698$		0			0
$699$		0			0
$700$		0			0
$701$			34.3440i			1.29716i	0.761148	+	0.648578i	$0.224636\pi$
$701$			34.3440i			1.29716i	−0.761148	+	0.648578i	$0.775364\pi$
$702$		0			0
$703$		−	6.93420i		−	0.261528i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	40.3100	+	24.0453i	1.51601	+	0.904316i
$708$		0			0
$709$	−5.06372			−0.190172			−0.0950859	−	0.995469i	$-0.530313\pi$
$709$	−5.06372			−0.190172			−0.0950859	+	0.995469i	$0.530313\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.4792			0.467349
$714$		0			0
$715$	17.7405			0.663456
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.1274			0.676039			0.338020	−	0.941139i	$-0.390243\pi$
$719$	18.1274			0.676039			0.338020	+	0.941139i	$0.390243\pi$
$720$		0			0
$721$	−5.50171	+	9.22317i	−0.204894	+	0.343489i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	3.83363i		−	0.142377i
$726$		0			0
$727$			11.1168i			0.412298i	0.978521	+	0.206149i	$0.0660931\pi$
$727$			11.1168i			0.412298i	−0.978521	+	0.206149i	$0.933907\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.717127			−0.0265239
$732$		0			0
$733$			20.8342i			0.769529i	0.923015	+	0.384765i	$0.125717\pi$
$733$			20.8342i			0.769529i	−0.923015	+	0.384765i	$0.874283\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			36.7500i			1.35370i
$738$		0			0
$739$	−30.9818			−1.13968			−0.569842	−	0.821754i	$-0.692996\pi$
$739$	−30.9818			−1.13968			−0.569842	+	0.821754i	$0.692996\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	45.8449i		−	1.68189i	−0.541124	−	0.840943i	$-0.682001\pi$
$743$		−	45.8449i		−	1.68189i	0.541124	−	0.840943i	$-0.317999\pi$
$744$		0			0
$745$		−	4.65166i		−	0.170423i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.56459	+	14.3579i	−0.312943	+	0.524624i
$750$		0			0
$751$	35.2665			1.28689			0.643446	−	0.765492i	$-0.277504\pi$
$751$	35.2665			1.28689			0.643446	+	0.765492i	$0.277504\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.58794			0.130579
$756$		0			0
$757$	−12.3661			−0.449452			−0.224726	−	0.974422i	$-0.572149\pi$
$757$	−12.3661			−0.449452			−0.224726	+	0.974422i	$0.572149\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.50580			0.0545851			0.0272926	−	0.999627i	$-0.491311\pi$
$761$	1.50580			0.0545851			0.0272926	+	0.999627i	$0.491311\pi$
$762$		0			0
$763$	27.3235	+	16.2987i	0.989177	+	0.590053i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			73.2716i			2.64569i
$768$		0			0
$769$		−	3.76558i		−	0.135790i	−0.997692	−	0.0678951i	$-0.978372\pi$
$769$		−	3.76558i		−	0.135790i	0.997692	−	0.0678951i	$-0.0216283\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.1911			0.546388			0.273194	−	0.961959i	$-0.411920\pi$
$773$	15.1911			0.546388			0.273194	+	0.961959i	$0.411920\pi$
$774$		0			0
$775$		−	3.25519i		−	0.116930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	15.0637i		−	0.539714i
$780$		0			0
$781$	−6.23960			−0.223270
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	13.1228i		−	0.468374i
$786$		0			0
$787$		−	23.8198i		−	0.849084i	−0.905408	−	0.424542i	$-0.860435\pi$
$787$		−	23.8198i		−	0.849084i	0.905408	−	0.424542i	$-0.139565\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.2849	−	20.5946i	0.436800	−	0.732260i
$792$		0			0
$793$	31.3284			1.11250
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.6517			−0.377301			−0.188650	−	0.982044i	$-0.560411\pi$
$797$	−10.6517			−0.377301			−0.188650	+	0.982044i	$0.560411\pi$
$798$		0			0
$799$	13.9702			0.494231
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−30.9614			−1.09260
$804$		0			0
$805$	5.19606	−	8.71078i	0.183137	−	0.307015i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			32.8462i			1.15481i	0.816458	+	0.577405i	$0.195935\pi$
$809$			32.8462i			1.15481i	−0.816458	+	0.577405i	$0.804065\pi$
$810$		0			0
$811$			26.0734i			0.915562i	0.889065	+	0.457781i	$0.151356\pi$
$811$			26.0734i			0.915562i	−0.889065	+	0.457781i	$0.848644\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16.6207			−0.582197
$816$		0			0
$817$		−	1.07756i		−	0.0376989i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			14.1352i			0.493321i	0.969102	+	0.246661i	$0.0793333\pi$
$821$			14.1352i			0.493321i	−0.969102	+	0.246661i	$0.920667\pi$
$822$		0			0
$823$	16.9114			0.589496			0.294748	−	0.955575i	$-0.404764\pi$
$823$	16.9114			0.589496			0.294748	+	0.955575i	$0.404764\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			34.2959i			1.19259i	0.802767	+	0.596293i	$0.203360\pi$
$827$			34.2959i			1.19259i	−0.802767	+	0.596293i	$0.796640\pi$
$828$		0			0
$829$			53.4408i			1.85608i	0.372486	+	0.928038i	$0.378505\pi$
$829$			53.4408i			1.85608i	−0.372486	+	0.928038i	$0.621495\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.09469	−	9.43541i	−0.176521	−	0.326917i
$834$		0			0
$835$	8.62068			0.298331
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.2898			−0.493339			−0.246669	−	0.969100i	$-0.579336\pi$
$839$	−14.2898			−0.493339			−0.246669	+	0.969100i	$0.579336\pi$
$840$		0			0
$841$	14.3033			0.493218
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−29.8293			−1.02616
$846$		0			0
$847$	−8.29731	−	4.94942i	−0.285099	−	0.170064i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			11.5490i			0.395895i
$852$		0			0
$853$			47.0118i			1.60966i	0.593509	+	0.804828i	$0.297742\pi$
$853$			47.0118i			1.60966i	−0.593509	+	0.804828i	$0.702258\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.65929			0.261636			0.130818	−	0.991406i	$-0.458240\pi$
$857$	7.65929			0.261636			0.130818	+	0.991406i	$0.458240\pi$
$858$		0			0
$859$			47.8559i			1.63282i	0.577470	+	0.816412i	$0.304040\pi$
$859$			47.8559i			1.63282i	−0.577470	+	0.816412i	$0.695960\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			41.4922i			1.41241i	0.708007	+	0.706206i	$0.249595\pi$
$863$			41.4922i			1.41241i	−0.708007	+	0.706206i	$0.750405\pi$
$864$		0			0
$865$	10.6517			0.362167
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	34.3639i		−	1.16572i
$870$		0			0
$871$		−	88.7223i		−	3.00624i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.27220	−	1.35539i	−0.0768145	−	0.0458206i
$876$		0			0
$877$	17.2925			0.583927			0.291963	−	0.956429i	$-0.405691\pi$
$877$	17.2925			0.583927			0.291963	+	0.956429i	$0.405691\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.7454			1.06953			0.534765	−	0.845001i	$-0.320400\pi$
$881$	31.7454			1.06953			0.534765	+	0.845001i	$0.320400\pi$
$882$		0			0
$883$	2.07601			0.0698634			0.0349317	−	0.999390i	$-0.488879\pi$
$883$	2.07601			0.0698634			0.0349317	+	0.999390i	$0.488879\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.18994			−0.174261			−0.0871305	−	0.996197i	$-0.527770\pi$
$887$	−5.18994			−0.174261			−0.0871305	+	0.996197i	$0.527770\pi$
$888$		0			0
$889$	49.1551	+	29.3215i	1.64861	+	0.983411i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			20.9917i			0.702459i
$894$		0			0
$895$		−	23.1961i		−	0.775359i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.4792			−0.416204
$900$		0			0
$901$			14.1776i			0.472326i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	9.11980i		−	0.303152i
$906$		0			0
$907$	−32.9473			−1.09400			−0.546999	−	0.837133i	$-0.684230\pi$
$907$	−32.9473			−1.09400			−0.546999	+	0.837133i	$0.684230\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.02356i		−	0.100175i	−0.998745	−	0.0500875i	$-0.984050\pi$
$911$		−	3.02356i		−	0.100175i	0.998745	−	0.0500875i	$-0.0159500\pi$
$912$		0			0
$913$		−	35.4809i		−	1.17425i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.72438	+	3.41465i	0.189036	+	0.112762i
$918$		0			0
$919$	−13.9129			−0.458945			−0.229473	−	0.973315i	$-0.573700\pi$
$919$	−13.9129			−0.458945			−0.229473	+	0.973315i	$0.573700\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	15.0637			0.495828
$924$		0			0
$925$	3.01255			0.0990521
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.15228			0.267468			0.133734	−	0.991017i	$-0.457303\pi$
$929$	8.15228			0.267468			0.133734	+	0.991017i	$0.457303\pi$
$930$		0			0
$931$	14.1776	−	7.65529i	0.464653	−	0.250892i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			4.15253i			0.135802i
$936$		0			0
$937$			2.22576i			0.0727122i	0.999339	+	0.0363561i	$0.0115751\pi$
$937$			2.22576i			0.0727122i	−0.999339	+	0.0363561i	$0.988425\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	40.6517			1.32521			0.662603	−	0.748971i	$-0.269452\pi$
$941$	40.6517			1.32521			0.662603	+	0.748971i	$0.269452\pi$
$942$		0			0
$943$			25.0888i			0.817004i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			34.4874i			1.12069i	0.828260	+	0.560344i	$0.189331\pi$
$947$			34.4874i			1.12069i	−0.828260	+	0.560344i	$0.810669\pi$
$948$		0			0
$949$	74.7474			2.42640
$950$		0			0
$951$		0			0
$952$		0			0
$953$			23.9688i			0.776426i	0.921570	+	0.388213i	$0.126907\pi$
$953$			23.9688i			0.776426i	−0.921570	+	0.388213i	$0.873093\pi$
$954$		0			0
$955$		−	9.69823i		−	0.313827i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.89275	+	3.17305i	−0.0611202	+	0.102463i
$960$		0			0
$961$	20.4037			0.658185
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.1525			−0.584351
$966$		0			0
$967$	−6.45735			−0.207654			−0.103827	−	0.994595i	$-0.533109\pi$
$967$	−6.45735			−0.207654			−0.103827	+	0.994595i	$0.533109\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48.7471			−1.56437			−0.782185	−	0.623046i	$-0.785895\pi$
$971$	−48.7471			−1.56437			−0.782185	+	0.623046i	$0.785895\pi$
$972$		0			0
$973$	−8.31586	+	13.9409i	−0.266594	+	0.446924i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			14.4853i			0.463425i	0.972784	+	0.231713i	$0.0744329\pi$
$977$			14.4853i			0.463425i	−0.972784	+	0.231713i	$0.925567\pi$
$978$		0			0
$979$		−	26.0455i		−	0.832419i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−46.7784			−1.49200			−0.745999	−	0.665947i	$-0.768028\pi$
$983$	−46.7784			−1.49200			−0.745999	+	0.665947i	$0.768028\pi$
$984$		0			0
$985$			4.60354i			0.146681i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.79468i			0.0570676i
$990$		0			0
$991$	37.1292			1.17945			0.589724	−	0.807605i	$-0.299237\pi$
$991$	37.1292			1.17945			0.589724	+	0.807605i	$0.299237\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	11.7405i		−	0.372198i
$996$		0			0
$997$			35.5309i			1.12527i	0.826704	+	0.562637i	$0.190213\pi$
$997$			35.5309i			1.12527i	−0.826704	+	0.562637i	$0.809787\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.f.i.881.1		8
3.2	odd	2		5040.2.f.f.881.1		8
4.3	odd	2		630.2.b.b.251.4	yes	8
7.6	odd	2		5040.2.f.f.881.2		8
12.11	even	2		630.2.b.a.251.8	yes	8
20.3	even	4		3150.2.d.a.3149.5		8
20.7	even	4		3150.2.d.f.3149.4		8
20.19	odd	2		3150.2.b.f.251.5		8
21.20	even	2	inner	5040.2.f.i.881.2		8
28.27	even	2		630.2.b.a.251.4	✓	8
60.23	odd	4		3150.2.d.d.3149.5		8
60.47	odd	4		3150.2.d.c.3149.4		8
60.59	even	2		3150.2.b.e.251.1		8
84.83	odd	2		630.2.b.b.251.8	yes	8
140.27	odd	4		3150.2.d.d.3149.6		8
140.83	odd	4		3150.2.d.c.3149.3		8
140.139	even	2		3150.2.b.e.251.5		8
420.83	even	4		3150.2.d.f.3149.3		8
420.167	even	4		3150.2.d.a.3149.6		8
420.419	odd	2		3150.2.b.f.251.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.4	✓	8	28.27	even	2
630.2.b.a.251.8	yes	8	12.11	even	2
630.2.b.b.251.4	yes	8	4.3	odd	2
630.2.b.b.251.8	yes	8	84.83	odd	2
3150.2.b.e.251.1		8	60.59	even	2
3150.2.b.e.251.5		8	140.139	even	2
3150.2.b.f.251.1		8	420.419	odd	2
3150.2.b.f.251.5		8	20.19	odd	2
3150.2.d.a.3149.5		8	20.3	even	4
3150.2.d.a.3149.6		8	420.167	even	4
3150.2.d.c.3149.3		8	140.83	odd	4
3150.2.d.c.3149.4		8	60.47	odd	4
3150.2.d.d.3149.5		8	60.23	odd	4
3150.2.d.d.3149.6		8	140.27	odd	4
3150.2.d.f.3149.3		8	420.83	even	4
3150.2.d.f.3149.4		8	20.7	even	4
5040.2.f.f.881.1		8	3.2	odd	2
5040.2.f.f.881.2		8	7.6	odd	2
5040.2.f.i.881.1		8	1.1	even	1	trivial
5040.2.f.i.881.2		8	21.20	even	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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +(-2.27220 - 1.35539i) q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +(-2.27220 - 1.35539i) q^{7} -2.71078i q^{11} +6.54441i q^{13} -1.53186 q^{17} -2.30177i q^{19} +3.83363i q^{23} +1.00000 q^{25} -3.83363i q^{29} -3.25519i q^{31} +(-2.27220 - 1.35539i) q^{35} +3.01255 q^{37} +6.54441 q^{41} +0.468142 q^{43} -9.11980 q^{47} +(3.32583 + 6.15945i) q^{49} -9.25519i q^{53} -2.71078i q^{55} +11.1961 q^{59} -4.78705i q^{61} +6.54441i q^{65} -13.5570 q^{67} -2.30177i q^{71} -11.4216i q^{73} +(-3.67417 + 6.15945i) q^{77} +12.6768 q^{79} +13.0888 q^{83} -1.53186 q^{85} +9.60812 q^{89} +(8.87024 - 14.8702i) q^{91} -2.30177i q^{95} -16.9224i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 4 q^{7}+O(q^{10}) \) 8 * q + 8 * q^5 + 4 * q^7	\( 8 q + 8 q^{5} + 4 q^{7} + 8 q^{25} + 4 q^{35} - 8 q^{37} + 8 q^{41} + 16 q^{43} - 40 q^{47} + 4 q^{49} - 32 q^{67} - 52 q^{77} - 8 q^{79} + 16 q^{83} + 8 q^{89} + 4 q^{91}+O(q^{100}) \) 8 * q + 8 * q^5 + 4 * q^7 + 8 * q^25 + 4 * q^35 - 8 * q^37 + 8 * q^41 + 16 * q^43 - 40 * q^47 + 4 * q^49 - 32 * q^67 - 52 * q^77 - 8 * q^79 + 16 * q^83 + 8 * q^89 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more