LMFDB - Embedded newform 4140.2.f.b.829.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
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} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 $ x^12 + 24x^10 + 188x^8 + 530x^6 + 508x^4 + 80*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.5
Root			$-0.116918i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.b.829.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.52160 - 1.63852i) q^{5} +1.80495i q^{7} +O(q^{10})$	$q+(-1.52160 - 1.63852i) q^{5} +1.80495i q^{7} +2.90652 q^{11} +2.25256i q^{13} -2.14477i q^{17} -0.339824 q^{19} -1.00000i q^{23} +(-0.369473 + 4.98633i) q^{25} -5.60395 q^{29} +5.92083 q^{31} +(2.95744 - 2.74641i) q^{35} -8.98088i q^{37} -1.89222 q^{41} +9.47322i q^{43} +7.83384i q^{47} +3.74216 q^{49} +6.47764i q^{53} +(-4.42256 - 4.76239i) q^{55} +5.17914 q^{59} -9.12565 q^{61} +(3.69085 - 3.42749i) q^{65} -9.25423i q^{67} +4.60255 q^{71} +11.3300i q^{73} +5.24613i q^{77} -7.94972 q^{79} +5.37849i q^{83} +(-3.51425 + 3.26348i) q^{85} +12.9258 q^{89} -4.06575 q^{91} +(0.517076 + 0.556807i) q^{95} -2.43210i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) $ 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.52160	−	1.63852i	−0.680480	−	0.732767i
$5$	−1.52160	−	1.63852i	−0.680480	−	0.732767i
$6$		0			0
$7$			1.80495i			0.682207i	0.940026	+	0.341103i	$0.110801\pi$
$7$			1.80495i			0.682207i	−0.940026	+	0.341103i	$0.889199\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.90652			0.876350			0.438175	−	0.898890i	$-0.355625\pi$
$11$	2.90652			0.876350			0.438175	+	0.898890i	$0.355625\pi$
$12$		0			0
$13$			2.25256i			0.624747i	0.949959	+	0.312373i	$0.101124\pi$
$13$			2.25256i			0.624747i	−0.949959	+	0.312373i	$0.898876\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.14477i		−	0.520184i	−0.965584	−	0.260092i	$-0.916247\pi$
$17$		−	2.14477i		−	0.520184i	0.965584	−	0.260092i	$-0.0837529\pi$
$18$		0			0
$19$	−0.339824			−0.0779610			−0.0389805	−	0.999240i	$-0.512411\pi$
$19$	−0.339824			−0.0779610			−0.0389805	+	0.999240i	$0.512411\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−0.369473	+	4.98633i	−0.0738946	+	0.997266i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.60395			−1.04063			−0.520313	−	0.853975i	$-0.674185\pi$
$29$	−5.60395			−1.04063			−0.520313	+	0.853975i	$0.674185\pi$
$30$		0			0
$31$	5.92083			1.06341			0.531706	−	0.846929i	$-0.321551\pi$
$31$	5.92083			1.06341			0.531706	+	0.846929i	$0.321551\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.95744	−	2.74641i	0.499898	−	0.464228i
$36$		0			0
$37$		−	8.98088i		−	1.47645i	−0.674556	−	0.738224i	$-0.735665\pi$
$37$		−	8.98088i		−	1.47645i	0.674556	−	0.738224i	$-0.264335\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.89222			−0.295515			−0.147757	−	0.989024i	$-0.547205\pi$
$41$	−1.89222			−0.295515			−0.147757	+	0.989024i	$0.547205\pi$
$42$		0			0
$43$			9.47322i			1.44465i	0.691552	+	0.722327i	$0.256927\pi$
$43$			9.47322i			1.44465i	−0.691552	+	0.722327i	$0.743073\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.83384i			1.14268i	0.820712	+	0.571342i	$0.193577\pi$
$47$			7.83384i			1.14268i	−0.820712	+	0.571342i	$0.806423\pi$
$48$		0			0
$49$	3.74216			0.534594
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.47764i			0.889772i	0.895587	+	0.444886i	$0.146756\pi$
$53$			6.47764i			0.889772i	−0.895587	+	0.444886i	$0.853244\pi$
$54$		0			0
$55$	−4.42256	−	4.76239i	−0.596338	−	0.642160i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.17914			0.674267			0.337134	−	0.941457i	$-0.390543\pi$
$59$	5.17914			0.674267			0.337134	+	0.941457i	$0.390543\pi$
$60$		0			0
$61$	−9.12565			−1.16842			−0.584210	−	0.811602i	$-0.698596\pi$
$61$	−9.12565			−1.16842			−0.584210	+	0.811602i	$0.698596\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.69085	−	3.42749i	0.457794	−	0.425127i
$66$		0			0
$67$		−	9.25423i		−	1.13058i	−0.824891	−	0.565292i	$-0.808764\pi$
$67$		−	9.25423i		−	1.13058i	0.824891	−	0.565292i	$-0.191236\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.60255			0.546222			0.273111	−	0.961982i	$-0.411947\pi$
$71$	4.60255			0.546222			0.273111	+	0.961982i	$0.411947\pi$
$72$		0			0
$73$			11.3300i			1.32608i	0.748585	+	0.663038i	$0.230733\pi$
$73$			11.3300i			1.32608i	−0.748585	+	0.663038i	$0.769267\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			5.24613i			0.597852i
$78$		0			0
$79$	−7.94972			−0.894414			−0.447207	−	0.894431i	$-0.647581\pi$
$79$	−7.94972			−0.894414			−0.447207	+	0.894431i	$0.647581\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.37849i			0.590366i	0.955441	+	0.295183i	$0.0953806\pi$
$83$			5.37849i			0.590366i	−0.955441	+	0.295183i	$0.904619\pi$
$84$		0			0
$85$	−3.51425	+	3.26348i	−0.381174	+	0.353975i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.9258			1.37014			0.685069	−	0.728479i	$-0.259772\pi$
$89$	12.9258			1.37014			0.685069	+	0.728479i	$0.259772\pi$
$90$		0			0
$91$	−4.06575			−0.426206
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.517076	+	0.556807i	0.0530508	+	0.0571272i
$96$		0			0
$97$		−	2.43210i		−	0.246942i	−0.992348	−	0.123471i	$-0.960597\pi$
$97$		−	2.43210i		−	0.246942i	0.992348	−	0.123471i	$-0.0394026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.89635			−0.288198			−0.144099	−	0.989563i	$-0.546028\pi$
$101$	−2.89635			−0.288198			−0.144099	+	0.989563i	$0.546028\pi$
$102$		0			0
$103$		−	10.3524i		−	1.02005i	−0.860159	−	0.510027i	$-0.829636\pi$
$103$		−	10.3524i		−	1.02005i	0.860159	−	0.510027i	$-0.170364\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.2395i			1.47326i	0.676296	+	0.736630i	$0.263584\pi$
$107$			15.2395i			1.47326i	−0.676296	+	0.736630i	$0.736416\pi$
$108$		0			0
$109$	7.00555			0.671010			0.335505	−	0.942038i	$-0.391093\pi$
$109$	7.00555			0.671010			0.335505	+	0.942038i	$0.391093\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			6.48460i			0.610020i	0.952349	+	0.305010i	$0.0986597\pi$
$113$			6.48460i			0.610020i	−0.952349	+	0.305010i	$0.901340\pi$
$114$		0			0
$115$	−1.63852	+	1.52160i	−0.152792	+	0.141890i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.87121			0.354873
$120$		0			0
$121$	−2.55212			−0.232011
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.73237	−	6.98181i	0.781047	−	0.624472i
$126$		0			0
$127$		−	6.28324i		−	0.557547i	−0.960357	−	0.278774i	$-0.910072\pi$
$127$		−	6.28324i		−	0.557547i	0.960357	−	0.278774i	$-0.0899279\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.90426			0.690598			0.345299	−	0.938493i	$-0.387777\pi$
$131$	7.90426			0.690598			0.345299	+	0.938493i	$0.387777\pi$
$132$		0			0
$133$		−	0.613365i		−	0.0531855i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.28955i			0.708224i	0.935203	+	0.354112i	$0.115217\pi$
$137$			8.28955i			0.708224i	−0.935203	+	0.354112i	$0.884783\pi$
$138$		0			0
$139$	8.97515			0.761262			0.380631	−	0.924727i	$-0.375707\pi$
$139$	8.97515			0.761262			0.380631	+	0.924727i	$0.375707\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			6.54711i			0.547497i
$144$		0			0
$145$	8.52696	+	9.18216i	0.708125	+	0.762537i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.39792			0.442215			0.221107	−	0.975249i	$-0.429033\pi$
$149$	5.39792			0.442215			0.221107	+	0.975249i	$0.429033\pi$
$150$		0			0
$151$	22.6083			1.83984			0.919919	−	0.392108i	$-0.128254\pi$
$151$	22.6083			1.83984			0.919919	+	0.392108i	$0.128254\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.00913	−	9.70138i	−0.723631	−	0.779234i
$156$		0			0
$157$			8.00033i			0.638496i	0.947671	+	0.319248i	$0.103430\pi$
$157$			8.00033i			0.638496i	−0.947671	+	0.319248i	$0.896570\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.80495			0.142250
$162$		0			0
$163$			18.6504i			1.46081i	0.683015	+	0.730404i	$0.260668\pi$
$163$			18.6504i			1.46081i	−0.683015	+	0.730404i	$0.739332\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.13547i			0.242630i	0.992614	+	0.121315i	$0.0387111\pi$
$167$			3.13547i			0.242630i	−0.992614	+	0.121315i	$0.961289\pi$
$168$		0			0
$169$	7.92599			0.609692
$170$		0			0
$171$		0			0
$172$		0			0
$173$			7.56301i			0.575005i	0.957780	+	0.287503i	$0.0928250\pi$
$173$			7.56301i			0.575005i	−0.957780	+	0.287503i	$0.907175\pi$
$174$		0			0
$175$	−9.00007	−	0.666880i	−0.680342	−	0.0504114i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.73608			0.727709			0.363855	−	0.931456i	$-0.381461\pi$
$179$	9.73608			0.727709			0.363855	+	0.931456i	$0.381461\pi$
$180$		0			0
$181$	0.260581			0.0193688			0.00968440	−	0.999953i	$-0.496917\pi$
$181$	0.260581			0.0193688			0.00968440	+	0.999953i	$0.496917\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.7153	+	13.6653i	−1.08189	+	1.00469i
$186$		0			0
$187$		−	6.23384i		−	0.455863i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.1730			1.31495			0.657476	−	0.753475i	$-0.271624\pi$
$191$	18.1730			1.31495			0.657476	+	0.753475i	$0.271624\pi$
$192$		0			0
$193$			2.15073i			0.154813i	0.997000	+	0.0774063i	$0.0246639\pi$
$193$			2.15073i			0.154813i	−0.997000	+	0.0774063i	$0.975336\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			27.0886i			1.92998i	0.262279	+	0.964992i	$0.415526\pi$
$197$			27.0886i			1.92998i	−0.262279	+	0.964992i	$0.584474\pi$
$198$		0			0
$199$	14.2178			1.00787			0.503937	−	0.863741i	$-0.331884\pi$
$199$	14.2178			1.00787			0.503937	+	0.863741i	$0.331884\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	10.1148i		−	0.709922i
$204$		0			0
$205$	2.87920	+	3.10043i	0.201092	+	0.216543i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.987706			−0.0683211
$210$		0			0
$211$	1.89855			0.130701			0.0653506	−	0.997862i	$-0.479183\pi$
$211$	1.89855			0.130701			0.0653506	+	0.997862i	$0.479183\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.5220	−	14.4144i	1.05859	−	0.983057i
$216$		0			0
$217$			10.6868i			0.725467i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.83122			0.324983
$222$		0			0
$223$			20.5574i			1.37663i	0.725414	+	0.688313i	$0.241648\pi$
$223$			20.5574i			1.37663i	−0.725414	+	0.688313i	$0.758352\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	7.67464i		−	0.509384i	−0.967022	−	0.254692i	$-0.918026\pi$
$227$		−	7.67464i		−	0.509384i	0.967022	−	0.254692i	$-0.0819740\pi$
$228$		0			0
$229$	−8.50379			−0.561946			−0.280973	−	0.959716i	$-0.590657\pi$
$229$	−8.50379			−0.561946			−0.280973	+	0.959716i	$0.590657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.37058i		−	0.155302i	−0.996981	−	0.0776511i	$-0.975258\pi$
$233$		−	2.37058i		−	0.155302i	0.996981	−	0.0776511i	$-0.0247420\pi$
$234$		0			0
$235$	12.8359	−	11.9200i	0.837320	−	0.777573i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.298079			−0.0192811			−0.00964057	−	0.999954i	$-0.503069\pi$
$239$	−0.298079			−0.0192811			−0.00964057	+	0.999954i	$0.503069\pi$
$240$		0			0
$241$	29.7779			1.91817			0.959083	−	0.283126i	$-0.0913714\pi$
$241$	29.7779			1.91817			0.959083	+	0.283126i	$0.0913714\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.69406	−	6.13159i	−0.363780	−	0.391733i
$246$		0			0
$247$		−	0.765472i		−	0.0487058i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.1766			−0.642341			−0.321171	−	0.947021i	$-0.604076\pi$
$251$	−10.1766			−0.642341			−0.321171	+	0.947021i	$0.604076\pi$
$252$		0			0
$253$		−	2.90652i		−	0.182732i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	28.2919i		−	1.76480i	−0.470501	−	0.882400i	$-0.655927\pi$
$257$		−	28.2919i		−	1.76480i	0.470501	−	0.882400i	$-0.344073\pi$
$258$		0			0
$259$	16.2100			1.00724
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.9182i		−	1.53652i	−0.640138	−	0.768260i	$-0.721123\pi$
$263$		−	24.9182i		−	1.53652i	0.640138	−	0.768260i	$-0.278877\pi$
$264$		0			0
$265$	10.6137	−	9.85637i	0.651996	−	0.605472i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.8948			1.57883			0.789416	−	0.613859i	$-0.210384\pi$
$269$	25.8948			1.57883			0.789416	+	0.613859i	$0.210384\pi$
$270$		0			0
$271$	3.49603			0.212369			0.106184	−	0.994346i	$-0.466137\pi$
$271$	3.49603			0.212369			0.106184	+	0.994346i	$0.466137\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.07388	+	14.4929i	−0.0647576	+	0.873954i
$276$		0			0
$277$		−	1.38317i		−	0.0831064i	−0.999136	−	0.0415532i	$-0.986769\pi$
$277$		−	1.38317i		−	0.0831064i	0.999136	−	0.0415532i	$-0.0132306\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.3144			−0.913582			−0.456791	−	0.889574i	$-0.651001\pi$
$281$	−15.3144			−0.913582			−0.456791	+	0.889574i	$0.651001\pi$
$282$		0			0
$283$			11.8266i			0.703019i	0.936184	+	0.351510i	$0.114332\pi$
$283$			11.8266i			0.703019i	−0.936184	+	0.351510i	$0.885668\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3.41536i		−	0.201602i
$288$		0			0
$289$	12.3999			0.729409
$290$		0			0
$291$		0			0
$292$		0			0
$293$			31.4274i			1.83601i	0.396569	+	0.918005i	$0.370201\pi$
$293$			31.4274i			1.83601i	−0.396569	+	0.918005i	$0.629799\pi$
$294$		0			0
$295$	−7.88058	−	8.48611i	−0.458825	−	0.494081i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.25256			0.130269
$300$		0			0
$301$	−17.0987			−0.985552
$302$		0			0
$303$		0			0
$304$		0			0
$305$	13.8856	+	14.9525i	0.795086	+	0.856180i
$306$		0			0
$307$		−	1.43504i		−	0.0819018i	−0.999161	−	0.0409509i	$-0.986961\pi$
$307$		−	1.43504i		−	0.0819018i	0.999161	−	0.0409509i	$-0.0130387\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.9698			−1.18909			−0.594544	−	0.804063i	$-0.702667\pi$
$311$	−20.9698			−1.18909			−0.594544	+	0.804063i	$0.702667\pi$
$312$		0			0
$313$			0.535929i			0.0302925i	0.999885	+	0.0151463i	$0.00482139\pi$
$313$			0.535929i			0.0302925i	−0.999885	+	0.0151463i	$0.995179\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.61790i		−	0.371698i	−0.982578	−	0.185849i	$-0.940496\pi$
$317$		−	6.61790i		−	0.371698i	0.982578	−	0.185849i	$-0.0595036\pi$
$318$		0			0
$319$	−16.2880			−0.911953
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.728845i			0.0405540i
$324$		0			0
$325$	−11.2320	−	0.832259i	−0.623038	−	0.0461654i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−14.1397			−0.779546
$330$		0			0
$331$	−1.81617			−0.0998255			−0.0499127	−	0.998754i	$-0.515894\pi$
$331$	−1.81617			−0.0998255			−0.0499127	+	0.998754i	$0.515894\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.1632	+	14.0812i	−0.828454	+	0.769339i
$336$		0			0
$337$			16.3308i			0.889593i	0.895632	+	0.444796i	$0.146724\pi$
$337$			16.3308i			0.889593i	−0.895632	+	0.444796i	$0.853276\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17.2090			0.931922
$342$		0			0
$343$			19.3891i			1.04691i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.1771i			0.653700i	0.945076	+	0.326850i	$0.105987\pi$
$347$			12.1771i			0.653700i	−0.945076	+	0.326850i	$0.894013\pi$
$348$		0			0
$349$	−12.0364			−0.644293			−0.322147	−	0.946690i	$-0.604404\pi$
$349$	−12.0364			−0.644293			−0.322147	+	0.946690i	$0.604404\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			30.0447i			1.59912i	0.600588	+	0.799559i	$0.294933\pi$
$353$			30.0447i			1.59912i	−0.600588	+	0.799559i	$0.705067\pi$
$354$		0			0
$355$	−7.00324	−	7.54135i	−0.371693	−	0.400254i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−29.6859			−1.56676			−0.783380	−	0.621543i	$-0.786506\pi$
$359$	−29.6859			−1.56676			−0.783380	+	0.621543i	$0.786506\pi$
$360$		0			0
$361$	−18.8845			−0.993922
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.5644	−	17.2397i	0.971705	−	0.902368i
$366$		0			0
$367$			12.4413i			0.649428i	0.945812	+	0.324714i	$0.105268\pi$
$367$			12.4413i			0.649428i	−0.945812	+	0.324714i	$0.894732\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.6918			−0.607008
$372$		0			0
$373$		−	28.5580i		−	1.47867i	−0.673335	−	0.739337i	$-0.735139\pi$
$373$		−	28.5580i		−	1.47867i	0.673335	−	0.739337i	$-0.264861\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.6232i		−	0.650128i
$378$		0			0
$379$	−14.8436			−0.762465			−0.381233	−	0.924479i	$-0.624500\pi$
$379$	−14.8436			−0.762465			−0.381233	+	0.924479i	$0.624500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			7.00776i			0.358080i	0.983842	+	0.179040i	$0.0572991\pi$
$383$			7.00776i			0.358080i	−0.983842	+	0.179040i	$0.942701\pi$
$384$		0			0
$385$	8.59587	−	7.98250i	0.438086	−	0.406826i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	29.2715			1.48413			0.742063	−	0.670330i	$-0.233847\pi$
$389$	29.2715			1.48413			0.742063	+	0.670330i	$0.233847\pi$
$390$		0			0
$391$	−2.14477			−0.108466
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0963	+	13.0258i	0.608630	+	0.655397i
$396$		0			0
$397$		−	32.9288i		−	1.65265i	−0.563196	−	0.826324i	$-0.690428\pi$
$397$		−	32.9288i		−	1.65265i	0.563196	−	0.826324i	$-0.309572\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.3403			1.21550			0.607749	−	0.794129i	$-0.292073\pi$
$401$	24.3403			1.21550			0.607749	+	0.794129i	$0.292073\pi$
$402$		0			0
$403$			13.3370i			0.664363i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	26.1031i		−	1.29389i
$408$		0			0
$409$	5.84661			0.289096			0.144548	−	0.989498i	$-0.453827\pi$
$409$	5.84661			0.289096			0.144548	+	0.989498i	$0.453827\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			9.34809i			0.459990i
$414$		0			0
$415$	8.81274	−	8.18390i	0.432600	−	0.401732i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.0557			−1.37061			−0.685304	−	0.728257i	$-0.740331\pi$
$419$	−28.0557			−1.37061			−0.685304	+	0.728257i	$0.740331\pi$
$420$		0			0
$421$	−30.0719			−1.46561			−0.732807	−	0.680437i	$-0.761790\pi$
$421$	−30.0719			−1.46561			−0.732807	+	0.680437i	$0.761790\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.6945	+	0.792436i	0.518762	+	0.0384388i
$426$		0			0
$427$		−	16.4713i		−	0.797104i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.92111			0.477883			0.238941	−	0.971034i	$-0.423200\pi$
$431$	9.92111			0.477883			0.238941	+	0.971034i	$0.423200\pi$
$432$		0			0
$433$			9.82152i			0.471992i	0.971754	+	0.235996i	$0.0758353\pi$
$433$			9.82152i			0.471992i	−0.971754	+	0.235996i	$0.924165\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			0.339824i			0.0162560i
$438$		0			0
$439$	−24.4026			−1.16467			−0.582336	−	0.812948i	$-0.697861\pi$
$439$	−24.4026			−1.16467			−0.582336	+	0.812948i	$0.697861\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			8.35646i			0.397027i	0.980098	+	0.198514i	$0.0636114\pi$
$443$			8.35646i			0.397027i	−0.980098	+	0.198514i	$0.936389\pi$
$444$		0			0
$445$	−19.6680	−	21.1792i	−0.932351	−	1.00399i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.4109			0.727286			0.363643	−	0.931538i	$-0.381533\pi$
$449$	15.4109			0.727286			0.363643	+	0.931538i	$0.381533\pi$
$450$		0			0
$451$	−5.49978			−0.258974
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.18644	+	6.66180i	0.290025	+	0.312310i
$456$		0			0
$457$			22.9314i			1.07269i	0.844000	+	0.536343i	$0.180195\pi$
$457$			22.9314i			1.07269i	−0.844000	+	0.536343i	$0.819805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.71207			0.452336			0.226168	−	0.974088i	$-0.427380\pi$
$461$	9.71207			0.452336			0.226168	+	0.974088i	$0.427380\pi$
$462$		0			0
$463$		−	13.6091i		−	0.632469i	−0.948681	−	0.316234i	$-0.897581\pi$
$463$		−	13.6091i		−	0.632469i	0.948681	−	0.316234i	$-0.102419\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	34.9381i		−	1.61674i	−0.588674	−	0.808371i	$-0.700350\pi$
$467$		−	34.9381i		−	1.61674i	0.588674	−	0.808371i	$-0.299650\pi$
$468$		0			0
$469$	16.7034			0.771292
$470$		0			0
$471$		0			0
$472$		0			0
$473$			27.5342i			1.26602i
$474$		0			0
$475$	0.125556	−	1.69447i	0.00576090	−	0.0777478i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.87962			−0.314338			−0.157169	−	0.987572i	$-0.550237\pi$
$479$	−6.87962			−0.314338			−0.157169	+	0.987572i	$0.550237\pi$
$480$		0			0
$481$	20.2299			0.922406
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.98503	+	3.70068i	−0.180951	+	0.168039i
$486$		0			0
$487$		−	31.8903i		−	1.44509i	−0.691325	−	0.722544i	$-0.742973\pi$
$487$		−	31.8903i		−	1.44509i	0.691325	−	0.722544i	$-0.257027\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−22.1719			−1.00061			−0.500303	−	0.865851i	$-0.666778\pi$
$491$	−22.1719			−1.00061			−0.500303	+	0.865851i	$0.666778\pi$
$492$		0			0
$493$			12.0192i			0.541317i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			8.30737i			0.372636i
$498$		0			0
$499$	−11.5721			−0.518037			−0.259019	−	0.965872i	$-0.583399\pi$
$499$	−11.5721			−0.518037			−0.259019	+	0.965872i	$0.583399\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	39.2532i		−	1.75021i	−0.483931	−	0.875106i	$-0.660792\pi$
$503$		−	39.2532i		−	1.75021i	0.483931	−	0.875106i	$-0.339208\pi$
$504$		0			0
$505$	4.40709	+	4.74572i	0.196113	+	0.211182i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.61391			−0.160184			−0.0800919	−	0.996787i	$-0.525521\pi$
$509$	−3.61391			−0.160184			−0.0800919	+	0.996787i	$0.525521\pi$
$510$		0			0
$511$	−20.4501			−0.904658
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.9626	+	15.7522i	−0.747461	+	0.694126i
$516$		0			0
$517$			22.7692i			1.00139i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.64494			−0.334931			−0.167465	−	0.985878i	$-0.553558\pi$
$521$	−7.64494			−0.334931			−0.167465	+	0.985878i	$0.553558\pi$
$522$		0			0
$523$		−	5.62852i		−	0.246118i	−0.992399	−	0.123059i	$-0.960730\pi$
$523$		−	5.62852i		−	0.246118i	0.992399	−	0.123059i	$-0.0392704\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.6988i		−	0.553170i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4.26233i		−	0.184622i
$534$		0			0
$535$	24.9702	−	23.1884i	1.07956	−	1.00252i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.8767			0.468492
$540$		0			0
$541$	0.537263			0.0230987			0.0115494	−	0.999933i	$-0.496324\pi$
$541$	0.537263			0.0230987			0.0115494	+	0.999933i	$0.496324\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10.6596	−	11.4787i	−0.456609	−	0.491694i
$546$		0			0
$547$			41.7590i			1.78548i	0.450568	+	0.892742i	$0.351222\pi$
$547$			41.7590i			1.78548i	−0.450568	+	0.892742i	$0.648778\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.90435			0.0811282
$552$		0			0
$553$		−	14.3488i		−	0.610175i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			22.5405i			0.955071i	0.878613	+	0.477535i	$0.158470\pi$
$557$			22.5405i			0.955071i	−0.878613	+	0.477535i	$0.841530\pi$
$558$		0			0
$559$	−21.3390			−0.902542
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	31.5366i		−	1.32911i	−0.747240	−	0.664554i	$-0.768622\pi$
$563$		−	31.5366i		−	1.32911i	0.747240	−	0.664554i	$-0.231378\pi$
$564$		0			0
$565$	10.6251	−	9.86696i	0.447002	−	0.415106i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.49650			0.104659			0.0523294	−	0.998630i	$-0.483335\pi$
$569$	2.49650			0.104659			0.0523294	+	0.998630i	$0.483335\pi$
$570$		0			0
$571$	−43.6658			−1.82736			−0.913678	−	0.406440i	$-0.866770\pi$
$571$	−43.6658			−1.82736			−0.913678	+	0.406440i	$0.866770\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.98633	+	0.369473i	0.207944	+	0.0154081i
$576$		0			0
$577$			33.7402i			1.40462i	0.711869	+	0.702312i	$0.247849\pi$
$577$			33.7402i			1.40462i	−0.711869	+	0.702312i	$0.752151\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.70790			−0.402751
$582$		0			0
$583$			18.8274i			0.779752i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			13.8050i			0.569795i	0.958558	+	0.284898i	$0.0919596\pi$
$587$			13.8050i			0.569795i	−0.958558	+	0.284898i	$0.908040\pi$
$588$		0			0
$589$	−2.01204			−0.0829047
$590$		0			0
$591$		0			0
$592$		0			0
$593$			43.3336i			1.77950i	0.456450	+	0.889749i	$0.349121\pi$
$593$			43.3336i			1.77950i	−0.456450	+	0.889749i	$0.650879\pi$
$594$		0			0
$595$	−5.89042	−	6.34304i	−0.241484	−	0.260039i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−27.2077			−1.11168			−0.555838	−	0.831290i	$-0.687603\pi$
$599$	−27.2077			−1.11168			−0.555838	+	0.831290i	$0.687603\pi$
$600$		0			0
$601$	39.6299			1.61654			0.808269	−	0.588813i	$-0.200405\pi$
$601$	39.6299			1.61654			0.808269	+	0.588813i	$0.200405\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.88330	+	4.18168i	0.157878	+	0.170010i
$606$		0			0
$607$		−	1.10233i		−	0.0447421i	−0.999750	−	0.0223711i	$-0.992878\pi$
$607$		−	1.10233i		−	0.0447421i	0.999750	−	0.0223711i	$-0.00712152\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17.6462			−0.713887
$612$		0			0
$613$			13.6705i			0.552145i	0.961137	+	0.276073i	$0.0890331\pi$
$613$			13.6705i			0.552145i	−0.961137	+	0.276073i	$0.910967\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.9011i		−	0.801187i	−0.916256	−	0.400594i	$-0.868804\pi$
$617$		−	19.9011i		−	0.801187i	0.916256	−	0.400594i	$-0.131196\pi$
$618$		0			0
$619$	32.1024			1.29030			0.645151	−	0.764055i	$-0.276794\pi$
$619$	32.1024			1.29030			0.645151	+	0.764055i	$0.276794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			23.3305i			0.934717i
$624$		0			0
$625$	−24.7270	−	3.68463i	−0.989079	−	0.147385i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.2620			−0.768024
$630$		0			0
$631$	32.3065			1.28610			0.643051	−	0.765823i	$-0.277668\pi$
$631$	32.3065			1.28610			0.643051	+	0.765823i	$0.277668\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.2952	+	9.56057i	−0.408552	+	0.379400i
$636$		0			0
$637$			8.42942i			0.333986i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.6696			0.737407			0.368703	−	0.929547i	$-0.379802\pi$
$641$	18.6696			0.737407			0.368703	+	0.929547i	$0.379802\pi$
$642$		0			0
$643$			19.9459i			0.786590i	0.919412	+	0.393295i	$0.128665\pi$
$643$			19.9459i			0.786590i	−0.919412	+	0.393295i	$0.871335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.6920i		−	1.01006i	−0.863102	−	0.505029i	$-0.831482\pi$
$647$		−	25.6920i		−	1.01006i	0.863102	−	0.505029i	$-0.168518\pi$
$648$		0			0
$649$	15.0533			0.590894
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	14.7648i		−	0.577791i	−0.957361	−	0.288896i	$-0.906712\pi$
$653$		−	14.7648i		−	0.577791i	0.957361	−	0.288896i	$-0.0932880\pi$
$654$		0			0
$655$	−12.0271	−	12.9513i	−0.469938	−	0.506047i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.53869			0.371574			0.185787	−	0.982590i	$-0.440516\pi$
$659$	9.53869			0.371574			0.185787	+	0.982590i	$0.440516\pi$
$660$		0			0
$661$	13.2912			0.516968			0.258484	−	0.966016i	$-0.416777\pi$
$661$	13.2912			0.516968			0.258484	+	0.966016i	$0.416777\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.00501	+	0.933295i	−0.0389726	+	0.0361916i
$666$		0			0
$667$			5.60395i			0.216986i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−26.5239			−1.02395
$672$		0			0
$673$		−	11.1517i		−	0.429868i	−0.976629	−	0.214934i	$-0.931046\pi$
$673$		−	11.1517i		−	0.429868i	0.976629	−	0.214934i	$-0.0689536\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	27.0771i		−	1.04066i	−0.853966	−	0.520328i	$-0.825810\pi$
$677$		−	27.0771i		−	1.04066i	0.853966	−	0.520328i	$-0.174190\pi$
$678$		0			0
$679$	4.38981			0.168466
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	23.8752i		−	0.913561i	−0.889579	−	0.456780i	$-0.849003\pi$
$683$		−	23.8752i		−	0.913561i	0.889579	−	0.456780i	$-0.150997\pi$
$684$		0			0
$685$	13.5826	−	12.6134i	0.518963	−	0.481932i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.5912			−0.555882
$690$		0			0
$691$	37.5350			1.42790			0.713950	−	0.700197i	$-0.246904\pi$
$691$	37.5350			1.42790			0.713950	+	0.700197i	$0.246904\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.6566	−	14.7059i	−0.518023	−	0.557828i
$696$		0			0
$697$			4.05838i			0.153722i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−45.8587			−1.73206			−0.866030	−	0.499992i	$-0.833336\pi$
$701$	−45.8587			−1.73206			−0.866030	+	0.499992i	$0.833336\pi$
$702$		0			0
$703$			3.05192i			0.115105i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	5.22777i		−	0.196611i
$708$		0			0
$709$	−10.2917			−0.386515			−0.193257	−	0.981148i	$-0.561905\pi$
$709$	−10.2917			−0.386515			−0.193257	+	0.981148i	$0.561905\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	5.92083i		−	0.221737i
$714$		0			0
$715$	10.7275	−	9.96207i	0.401187	−	0.372560i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.2862			−0.570080			−0.285040	−	0.958516i	$-0.592007\pi$
$719$	−15.2862			−0.570080			−0.285040	+	0.958516i	$0.592007\pi$
$720$		0			0
$721$	18.6856			0.695887
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.07051	−	27.9431i	0.0768967	−	1.03778i
$726$		0			0
$727$		−	18.5486i		−	0.687930i	−0.938982	−	0.343965i	$-0.888230\pi$
$727$		−	18.5486i		−	0.687930i	0.938982	−	0.343965i	$-0.111770\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	20.3179			0.751485
$732$		0			0
$733$			29.6045i			1.09347i	0.837307	+	0.546734i	$0.184129\pi$
$733$			29.6045i			1.09347i	−0.837307	+	0.546734i	$0.815871\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	26.8976i		−	0.990787i
$738$		0			0
$739$	−44.2455			−1.62760			−0.813799	−	0.581147i	$-0.802604\pi$
$739$	−44.2455			−1.62760			−0.813799	+	0.581147i	$0.802604\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.33740i			0.0857509i	0.999080	+	0.0428755i	$0.0136519\pi$
$743$			2.33740i			0.0857509i	−0.999080	+	0.0428755i	$0.986348\pi$
$744$		0			0
$745$	−8.21347	−	8.84458i	−0.300918	−	0.324040i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27.5066			−1.00507
$750$		0			0
$751$	−34.4606			−1.25748			−0.628742	−	0.777614i	$-0.716430\pi$
$751$	−34.4606			−1.25748			−0.628742	+	0.777614i	$0.716430\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−34.4008	−	37.0441i	−1.25197	−	1.34817i
$756$		0			0
$757$		−	26.5337i		−	0.964384i	−0.876066	−	0.482192i	$-0.839841\pi$
$757$		−	26.5337i		−	0.964384i	0.876066	−	0.482192i	$-0.160159\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	48.2585			1.74937			0.874685	−	0.484691i	$-0.161068\pi$
$761$	48.2585			1.74937			0.874685	+	0.484691i	$0.161068\pi$
$762$		0			0
$763$			12.6447i			0.457768i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			11.6663i			0.421246i
$768$		0			0
$769$	−31.6874			−1.14268			−0.571338	−	0.820715i	$-0.693576\pi$
$769$	−31.6874			−1.14268			−0.571338	+	0.820715i	$0.693576\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	25.7906i		−	0.927623i	−0.885934	−	0.463811i	$-0.846481\pi$
$773$		−	25.7906i		−	0.927623i	0.885934	−	0.463811i	$-0.153519\pi$
$774$		0			0
$775$	−2.18759	+	29.5232i	−0.0785805	+	1.06051i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.643021			0.0230386
$780$		0			0
$781$	13.3774			0.478682
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.1087	−	12.1733i	0.467868	−	0.434483i
$786$		0			0
$787$		−	26.2714i		−	0.936475i	−0.883603	−	0.468237i	$-0.844889\pi$
$787$		−	26.2714i		−	0.936475i	0.883603	−	0.468237i	$-0.155111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.7044			−0.416159
$792$		0			0
$793$		−	20.5560i		−	0.729967i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			5.81284i			0.205901i	0.994686	+	0.102951i	$0.0328284\pi$
$797$			5.81284i			0.205901i	−0.994686	+	0.102951i	$0.967172\pi$
$798$		0			0
$799$	16.8018			0.594405
$800$		0			0
$801$		0			0
$802$		0			0
$803$			32.9309i			1.16211i
$804$		0			0
$805$	−2.74641	−	2.95744i	−0.0967982	−	0.104236i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	43.5439			1.53092			0.765462	−	0.643481i	$-0.222511\pi$
$809$	43.5439			1.53092			0.765462	+	0.643481i	$0.222511\pi$
$810$		0			0
$811$	11.0230			0.387069			0.193535	−	0.981093i	$-0.438005\pi$
$811$	11.0230			0.387069			0.193535	+	0.981093i	$0.438005\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	30.5589	−	28.3784i	1.07043	−	0.994051i
$816$		0			0
$817$		−	3.21923i		−	0.112627i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.4664			−0.644481			−0.322240	−	0.946658i	$-0.604436\pi$
$821$	−18.4664			−0.644481			−0.322240	+	0.946658i	$0.604436\pi$
$822$		0			0
$823$			11.4960i			0.400724i	0.979722	+	0.200362i	$0.0642118\pi$
$823$			11.4960i			0.400724i	−0.979722	+	0.200362i	$0.935788\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	13.9089i		−	0.483658i	−0.970319	−	0.241829i	$-0.922253\pi$
$827$		−	13.9089i		−	0.483658i	0.970319	−	0.241829i	$-0.0777474\pi$
$828$		0			0
$829$	−9.63272			−0.334558			−0.167279	−	0.985910i	$-0.553498\pi$
$829$	−9.63272			−0.334558			−0.167279	+	0.985910i	$0.553498\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	8.02608i		−	0.278087i
$834$		0			0
$835$	5.13752	−	4.77093i	0.177791	−	0.165105i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.697892			−0.0240939			−0.0120470	−	0.999927i	$-0.503835\pi$
$839$	−0.697892			−0.0240939			−0.0120470	+	0.999927i	$0.503835\pi$
$840$		0			0
$841$	2.40421			0.0829036
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0602	−	12.9869i	−0.414883	−	0.446762i
$846$		0			0
$847$		−	4.60644i		−	0.158279i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.98088			−0.307861
$852$		0			0
$853$			40.2978i			1.37977i	0.723919	+	0.689885i	$0.242339\pi$
$853$			40.2978i			1.37977i	−0.723919	+	0.689885i	$0.757661\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	48.1828i		−	1.64589i	−0.568118	−	0.822947i	$-0.692328\pi$
$857$		−	48.1828i		−	1.64589i	0.568118	−	0.822947i	$-0.307672\pi$
$858$		0			0
$859$	−55.5268			−1.89455			−0.947276	−	0.320419i	$-0.896176\pi$
$859$	−55.5268			−1.89455			−0.947276	+	0.320419i	$0.896176\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			39.8794i			1.35751i	0.734364	+	0.678756i	$0.237480\pi$
$863$			39.8794i			1.35751i	−0.734364	+	0.678756i	$0.762520\pi$
$864$		0			0
$865$	12.3921	−	11.5079i	0.421345	−	0.391279i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−23.1061			−0.783819
$870$		0			0
$871$	20.8457			0.706328
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.6018	+	15.7615i	0.426019	+	0.532836i
$876$		0			0
$877$		−	34.8562i		−	1.17701i	−0.808493	−	0.588506i	$-0.799716\pi$
$877$		−	34.8562i		−	1.17701i	0.808493	−	0.588506i	$-0.200284\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.94904			0.166737			0.0833687	−	0.996519i	$-0.473432\pi$
$881$	4.94904			0.166737			0.0833687	+	0.996519i	$0.473432\pi$
$882$		0			0
$883$			8.95960i			0.301515i	0.988571	+	0.150757i	$0.0481712\pi$
$883$			8.95960i			0.301515i	−0.988571	+	0.150757i	$0.951829\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			28.8125i			0.967429i	0.875226	+	0.483714i	$0.160713\pi$
$887$			28.8125i			0.967429i	−0.875226	+	0.483714i	$0.839287\pi$
$888$		0			0
$889$	11.3409			0.380363
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.66213i		−	0.0890847i
$894$		0			0
$895$	−14.8144	−	15.9527i	−0.495191	−	0.533241i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−33.1800			−1.10662
$900$		0			0
$901$	13.8931			0.462845
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.396499	−	0.426966i	−0.0131801	−	0.0141928i
$906$		0			0
$907$		−	29.2784i		−	0.972174i	−0.873910	−	0.486087i	$-0.838424\pi$
$907$		−	29.2784i		−	0.972174i	0.873910	−	0.486087i	$-0.161576\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.48369			0.281077			0.140539	−	0.990075i	$-0.455117\pi$
$911$	8.48369			0.281077			0.140539	+	0.990075i	$0.455117\pi$
$912$		0			0
$913$			15.6327i			0.517367i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			14.2668i			0.471131i
$918$		0			0
$919$	−5.73942			−0.189326			−0.0946630	−	0.995509i	$-0.530177\pi$
$919$	−5.73942			−0.189326			−0.0946630	+	0.995509i	$0.530177\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			10.3675i			0.341250i
$924$		0			0
$925$	44.7816	+	3.31819i	1.47241	+	0.109102i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.72127			0.187709			0.0938544	−	0.995586i	$-0.470081\pi$
$929$	5.72127			0.187709			0.0938544	+	0.995586i	$0.470081\pi$
$930$		0			0
$931$	−1.27167			−0.0416775
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.2142	+	9.48540i	−0.334041	+	0.310206i
$936$		0			0
$937$		−	35.6492i		−	1.16461i	−0.812971	−	0.582304i	$-0.802151\pi$
$937$		−	35.6492i		−	1.16461i	0.812971	−	0.582304i	$-0.197849\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.5212			0.636372			0.318186	−	0.948028i	$-0.396926\pi$
$941$	19.5212			0.636372			0.318186	+	0.948028i	$0.396926\pi$
$942$		0			0
$943$			1.89222i			0.0616191i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	35.7799i		−	1.16269i	−0.813657	−	0.581345i	$-0.802527\pi$
$947$		−	35.7799i		−	1.16269i	0.813657	−	0.581345i	$-0.197473\pi$
$948$		0			0
$949$	−25.5215			−0.828462
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	9.97808i		−	0.323222i	−0.986855	−	0.161611i	$-0.948331\pi$
$953$		−	9.97808i		−	0.323222i	0.986855	−	0.161611i	$-0.0516689\pi$
$954$		0			0
$955$	−27.6520	−	29.7768i	−0.894799	−	0.963554i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.9622			−0.483155
$960$		0			0
$961$	4.05624			0.130847
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.52400	−	3.27254i	0.113442	−	0.105347i
$966$		0			0
$967$		−	58.6546i		−	1.88620i	−0.332505	−	0.943102i	$-0.607894\pi$
$967$		−	58.6546i		−	1.88620i	0.332505	−	0.943102i	$-0.392106\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	39.8684			1.27944			0.639719	−	0.768609i	$-0.279051\pi$
$971$	39.8684			1.27944			0.639719	+	0.768609i	$0.279051\pi$
$972$		0			0
$973$			16.1997i			0.519338i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			26.7860i			0.856962i	0.903551	+	0.428481i	$0.140951\pi$
$977$			26.7860i			0.856962i	−0.903551	+	0.428481i	$0.859049\pi$
$978$		0			0
$979$	37.5693			1.20072
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	58.5711i		−	1.86813i	−0.357104	−	0.934065i	$-0.616236\pi$
$983$		−	58.5711i		−	1.86813i	0.357104	−	0.934065i	$-0.383764\pi$
$984$		0			0
$985$	44.3851	−	41.2180i	1.41423	−	1.31331i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.47322			0.301231
$990$		0			0
$991$	−53.9818			−1.71479			−0.857394	−	0.514661i	$-0.827918\pi$
$991$	−53.9818			−1.71479			−0.857394	+	0.514661i	$0.827918\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−21.6338	−	23.2961i	−0.685838	−	0.738536i
$996$		0			0
$997$			0.592272i			0.0187575i	0.999956	+	0.00937873i	$0.00298538\pi$
$997$			0.592272i			0.0187575i	−0.999956	+	0.00937873i	$0.997015\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.b.829.5		12
3.2	odd	2		460.2.c.a.369.6	✓	12
5.4	even	2	inner	4140.2.f.b.829.6		12
12.11	even	2		1840.2.e.f.369.7		12
15.2	even	4		2300.2.a.o.1.3		6
15.8	even	4		2300.2.a.n.1.4		6
15.14	odd	2		460.2.c.a.369.7	yes	12
60.23	odd	4		9200.2.a.cy.1.3		6
60.47	odd	4		9200.2.a.cx.1.4		6
60.59	even	2		1840.2.e.f.369.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.6	✓	12	3.2	odd	2
460.2.c.a.369.7	yes	12	15.14	odd	2
1840.2.e.f.369.6		12	60.59	even	2
1840.2.e.f.369.7		12	12.11	even	2
2300.2.a.n.1.4		6	15.8	even	4
2300.2.a.o.1.3		6	15.2	even	4
4140.2.f.b.829.5		12	1.1	even	1	trivial
4140.2.f.b.829.6		12	5.4	even	2	inner
9200.2.a.cx.1.4		6	60.47	odd	4
9200.2.a.cy.1.3		6	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.52160 - 1.63852i) q^{5} +1.80495i q^{7} +O(q^{10})\)	\(q+(-1.52160 - 1.63852i) q^{5} +1.80495i q^{7} +2.90652 q^{11} +2.25256i q^{13} -2.14477i q^{17} -0.339824 q^{19} -1.00000i q^{23} +(-0.369473 + 4.98633i) q^{25} -5.60395 q^{29} +5.92083 q^{31} +(2.95744 - 2.74641i) q^{35} -8.98088i q^{37} -1.89222 q^{41} +9.47322i q^{43} +7.83384i q^{47} +3.74216 q^{49} +6.47764i q^{53} +(-4.42256 - 4.76239i) q^{55} +5.17914 q^{59} -9.12565 q^{61} +(3.69085 - 3.42749i) q^{65} -9.25423i q^{67} +4.60255 q^{71} +11.3300i q^{73} +5.24613i q^{77} -7.94972 q^{79} +5.37849i q^{83} +(-3.51425 + 3.26348i) q^{85} +12.9258 q^{89} -4.06575 q^{91} +(0.517076 + 0.556807i) q^{95} -2.43210i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) \) 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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