LMFDB - Embedded newform 3120.2.l.p.1249.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,2,Mod(1249,3120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3120.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-2,0,0,0,-10,0,-10,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.13266844647424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 $ x^10 + 16x^8 + 90x^6 + 208x^4 + 169x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$-2.26036i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.p.1249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.23266 + 0.123438i) q^{5} -4.96953i q^{7} -1.00000 q^{9} +3.21640 q^{11} -1.00000i q^{13} +(-0.123438 - 2.23266i) q^{15} -0.448809i q^{17} -7.23291 q^{19} +4.96953 q^{21} -6.26338i q^{23} +(4.96953 - 0.551191i) q^{25} -1.00000i q^{27} +2.21844 q^{29} -2.19148 q^{31} +3.21640i q^{33} +(0.613429 + 11.0953i) q^{35} +8.26338i q^{37} +1.00000 q^{39} +1.79807 q^{41} +6.21844i q^{43} +(2.23266 - 0.123438i) q^{45} -1.66521i q^{47} -17.6962 q^{49} +0.448809 q^{51} -1.16100i q^{53} +(-7.18113 + 0.397027i) q^{55} -7.23291i q^{57} -5.88365 q^{59} -1.76963 q^{61} +4.96953i q^{63} +(0.123438 + 2.23266i) q^{65} +2.73916i q^{67} +6.26338 q^{69} -4.11402 q^{71} +10.4369i q^{73} +(0.551191 + 4.96953i) q^{75} -15.9840i q^{77} +1.05336 q^{79} +1.00000 q^{81} -5.77601i q^{83} +(0.0554002 + 1.00204i) q^{85} +2.21844i q^{87} -11.1326 q^{89} -4.96953 q^{91} -2.19148i q^{93} +(16.1486 - 0.892818i) q^{95} -8.37418i q^{97} -3.21640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 16 * q^19 + 10 * q^21 + 10 * q^25 - 16 * q^29 - 24 * q^31 - 12 * q^35 + 10 * q^39 + 10 * q^41 + 2 * q^45 - 44 * q^49 + 10 * q^51 - 2 * q^55 + 16 * q^59 + 26 * q^61 + 14 * q^69 - 10 * q^71 + 2 * q^79 + 10 * q^81 - 4 * q^85 - 38 * q^89 - 10 * q^91 + 48 * q^95 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−2.23266	+	0.123438i	−0.998475	+	0.0552033i
$5$	−2.23266	+	0.123438i	−0.998475	+	0.0552033i
$6$		0			0
$7$		−	4.96953i		−	1.87830i	−0.343501	−	0.939152i	$-0.611613\pi$
$7$		−	4.96953i		−	1.87830i	0.343501	−	0.939152i	$-0.388387\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.21640			0.969782			0.484891	−	0.874575i	$-0.338859\pi$
$11$	3.21640			0.969782			0.484891	+	0.874575i	$0.338859\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	−0.123438	−	2.23266i	−0.0318716	−	0.576470i
$16$		0			0
$17$		−	0.448809i		−	0.108852i	−0.998518	−	0.0544261i	$-0.982667\pi$
$17$		−	0.448809i		−	0.108852i	0.998518	−	0.0544261i	$-0.0173329\pi$
$18$		0			0
$19$	−7.23291			−1.65934			−0.829672	−	0.558252i	$-0.811472\pi$
$19$	−7.23291			−1.65934			−0.829672	+	0.558252i	$0.811472\pi$
$20$		0			0
$21$	4.96953			1.08444
$22$		0			0
$23$		−	6.26338i		−	1.30601i	−0.757355	−	0.653003i	$-0.773509\pi$
$23$		−	6.26338i		−	1.30601i	0.757355	−	0.653003i	$-0.226491\pi$
$24$		0			0
$25$	4.96953	−	0.551191i	0.993905	−	0.110238i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.21844			0.411954			0.205977	−	0.978557i	$-0.433963\pi$
$29$	2.21844			0.411954			0.205977	+	0.978557i	$0.433963\pi$
$30$		0			0
$31$	−2.19148			−0.393601			−0.196800	−	0.980444i	$-0.563055\pi$
$31$	−2.19148			−0.393601			−0.196800	+	0.980444i	$0.563055\pi$
$32$		0			0
$33$			3.21640i			0.559904i
$34$		0			0
$35$	0.613429	+	11.0953i	0.103689	+	1.87544i
$36$		0			0
$37$			8.26338i			1.35849i	0.733911	+	0.679246i	$0.237693\pi$
$37$			8.26338i			1.35849i	−0.733911	+	0.679246i	$0.762307\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	1.79807			0.280811			0.140405	−	0.990094i	$-0.455159\pi$
$41$	1.79807			0.280811			0.140405	+	0.990094i	$0.455159\pi$
$42$		0			0
$43$			6.21844i			0.948303i	0.880443	+	0.474152i	$0.157245\pi$
$43$			6.21844i			0.948303i	−0.880443	+	0.474152i	$0.842755\pi$
$44$		0			0
$45$	2.23266	−	0.123438i	0.332825	−	0.0184011i
$46$		0			0
$47$		−	1.66521i		−	0.242896i	−0.992598	−	0.121448i	$-0.961246\pi$
$47$		−	1.66521i		−	0.242896i	0.992598	−	0.121448i	$-0.0387538\pi$
$48$		0			0
$49$	−17.6962			−2.52803
$50$		0			0
$51$	0.448809			0.0628459
$52$		0			0
$53$		−	1.16100i		−	0.159476i	−0.996816	−	0.0797380i	$-0.974592\pi$
$53$		−	1.16100i		−	0.159476i	0.996816	−	0.0797380i	$-0.0254084\pi$
$54$		0			0
$55$	−7.18113	+	0.397027i	−0.968303	+	0.0535351i
$56$		0			0
$57$		−	7.23291i		−	0.958022i
$58$		0			0
$59$	−5.88365			−0.765986			−0.382993	−	0.923751i	$-0.625107\pi$
$59$	−5.88365			−0.765986			−0.382993	+	0.923751i	$0.625107\pi$
$60$		0			0
$61$	−1.76963			−0.226578			−0.113289	−	0.993562i	$-0.536139\pi$
$61$	−1.76963			−0.226578			−0.113289	+	0.993562i	$0.536139\pi$
$62$		0			0
$63$			4.96953i			0.626101i
$64$		0			0
$65$	0.123438	+	2.23266i	0.0153106	+	0.276927i
$66$		0			0
$67$			2.73916i			0.334641i	0.985903	+	0.167321i	$0.0535115\pi$
$67$			2.73916i			0.334641i	−0.985903	+	0.167321i	$0.946488\pi$
$68$		0			0
$69$	6.26338			0.754023
$70$		0			0
$71$	−4.11402			−0.488244			−0.244122	−	0.969744i	$-0.578500\pi$
$71$	−4.11402			−0.488244			−0.244122	+	0.969744i	$0.578500\pi$
$72$		0			0
$73$			10.4369i			1.22154i	0.791806	+	0.610772i	$0.209141\pi$
$73$			10.4369i			1.22154i	−0.791806	+	0.610772i	$0.790859\pi$
$74$		0			0
$75$	0.551191	+	4.96953i	0.0636460	+	0.573831i
$76$		0			0
$77$		−	15.9840i		−	1.82155i
$78$		0			0
$79$	1.05336			0.118513			0.0592563	−	0.998243i	$-0.481127\pi$
$79$	1.05336			0.118513			0.0592563	+	0.998243i	$0.481127\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.77601i		−	0.634000i	−0.948426	−	0.317000i	$-0.897325\pi$
$83$		−	5.77601i		−	0.634000i	0.948426	−	0.317000i	$-0.102675\pi$
$84$		0			0
$85$	0.0554002	+	1.00204i	0.00600900	+	0.108686i
$86$		0			0
$87$			2.21844i			0.237842i
$88$		0			0
$89$	−11.1326			−1.18005			−0.590025	−	0.807385i	$-0.700882\pi$
$89$	−11.1326			−1.18005			−0.590025	+	0.807385i	$0.700882\pi$
$90$		0			0
$91$	−4.96953			−0.520948
$92$		0			0
$93$		−	2.19148i		−	0.227246i
$94$		0			0
$95$	16.1486	−	0.892818i	1.65681	−	0.0916012i
$96$		0			0
$97$		−	8.37418i		−	0.850270i	−0.905130	−	0.425135i	$-0.860227\pi$
$97$		−	8.37418i		−	0.850270i	0.905130	−	0.425135i	$-0.139773\pi$
$98$		0			0
$99$	−3.21640			−0.323261
$100$		0			0
$101$	5.50625			0.547892			0.273946	−	0.961745i	$-0.411671\pi$
$101$	5.50625			0.547892			0.273946	+	0.961745i	$0.411671\pi$
$102$		0			0
$103$		−	1.32082i		−	0.130144i	−0.997881	−	0.0650722i	$-0.979272\pi$
$103$		−	1.32082i		−	0.130144i	0.997881	−	0.0650722i	$-0.0207278\pi$
$104$		0			0
$105$	−11.0953	+	0.613429i	−1.08279	+	0.0598646i
$106$		0			0
$107$			7.49024i			0.724109i	0.932157	+	0.362055i	$0.117925\pi$
$107$			7.49024i			0.724109i	−0.932157	+	0.362055i	$0.882075\pi$
$108$		0			0
$109$	−14.0330			−1.34412			−0.672060	−	0.740497i	$-0.734590\pi$
$109$	−14.0330			−1.34412			−0.672060	+	0.740497i	$0.734590\pi$
$110$		0			0
$111$	−8.26338			−0.784326
$112$		0			0
$113$			11.2269i			1.05613i	0.849203	+	0.528067i	$0.177083\pi$
$113$			11.2269i			1.05613i	−0.849203	+	0.528067i	$0.822917\pi$
$114$		0			0
$115$	0.773141	+	13.9840i	0.0720958	+	1.30401i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	−2.23037			−0.204458
$120$		0			0
$121$	−0.654755			−0.0595232
$122$		0			0
$123$			1.79807i			0.162126i
$124$		0			0
$125$	−11.0272	+	1.84405i	−0.986304	+	0.164937i
$126$		0			0
$127$			1.50217i			0.133296i	0.997777	+	0.0666481i	$0.0212305\pi$
$127$			1.50217i			0.133296i	−0.997777	+	0.0666481i	$0.978770\pi$
$128$		0			0
$129$	−6.21844			−0.547503
$130$		0			0
$131$	−12.5477			−1.09630			−0.548148	−	0.836381i	$-0.684667\pi$
$131$	−12.5477			−1.09630			−0.548148	+	0.836381i	$0.684667\pi$
$132$		0			0
$133$			35.9441i			3.11675i
$134$		0			0
$135$	0.123438	+	2.23266i	0.0106239	+	0.192157i
$136$		0			0
$137$		−	10.6797i		−	0.912427i	−0.889870	−	0.456213i	$-0.849205\pi$
$137$		−	10.6797i		−	0.912427i	0.889870	−	0.456213i	$-0.150795\pi$
$138$		0			0
$139$	−20.6962			−1.75543			−0.877714	−	0.479185i	$-0.840932\pi$
$139$	−20.6962			−1.75543			−0.877714	+	0.479185i	$0.840932\pi$
$140$		0			0
$141$	1.66521			0.140236
$142$		0			0
$143$		−	3.21640i		−	0.268969i
$144$		0			0
$145$	−4.95302	+	0.273840i	−0.411326	+	0.0227412i
$146$		0			0
$147$		−	17.6962i		−	1.45956i
$148$		0			0
$149$	10.3457			0.847557			0.423778	−	0.905766i	$-0.360704\pi$
$149$	10.3457			0.847557			0.423778	+	0.905766i	$0.360704\pi$
$150$		0			0
$151$	−9.34779			−0.760712			−0.380356	−	0.924840i	$-0.624199\pi$
$151$	−9.34779			−0.760712			−0.380356	+	0.924840i	$0.624199\pi$
$152$		0			0
$153$			0.448809i			0.0362841i
$154$		0			0
$155$	4.89282	−	0.270512i	0.393001	−	0.0217280i
$156$		0			0
$157$			0.987506i			0.0788115i	0.999223	+	0.0394058i	$0.0125465\pi$
$157$			0.987506i			0.0788115i	−0.999223	+	0.0394058i	$0.987454\pi$
$158$		0			0
$159$	1.16100			0.0920735
$160$		0			0
$161$	−31.1260			−2.45308
$162$		0			0
$163$			11.0635i			0.866559i	0.901260	+	0.433280i	$0.142644\pi$
$163$			11.0635i			0.866559i	−0.901260	+	0.433280i	$0.857356\pi$
$164$		0			0
$165$	−0.397027	−	7.18113i	−0.0309085	−	0.559050i
$166$		0			0
$167$		−	19.6373i		−	1.51958i	−0.650170	−	0.759789i	$-0.725302\pi$
$167$		−	19.6373i		−	1.51958i	0.650170	−	0.759789i	$-0.274698\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	7.23291			0.553114
$172$		0			0
$173$		−	15.4783i		−	1.17679i	−0.808572	−	0.588397i	$-0.799759\pi$
$173$		−	15.4783i		−	1.17679i	0.808572	−	0.588397i	$-0.200241\pi$
$174$		0			0
$175$	−2.73916	−	24.6962i	−0.207061	−	1.86686i
$176$		0			0
$177$		−	5.88365i		−	0.442242i
$178$		0			0
$179$	−3.10238			−0.231883			−0.115941	−	0.993256i	$-0.536988\pi$
$179$	−3.10238			−0.231883			−0.115941	+	0.993256i	$0.536988\pi$
$180$		0			0
$181$	12.8111			0.952239			0.476119	−	0.879381i	$-0.342043\pi$
$181$	12.8111			0.952239			0.476119	+	0.879381i	$0.342043\pi$
$182$		0			0
$183$		−	1.76963i		−	0.130815i
$184$		0			0
$185$	−1.02002	−	18.4493i	−0.0749932	−	1.35642i
$186$		0			0
$187$		−	1.44355i		−	0.105563i
$188$		0			0
$189$	−4.96953			−0.361480
$190$		0			0
$191$	−15.1780			−1.09824			−0.549121	−	0.835743i	$-0.685037\pi$
$191$	−15.1780			−1.09824			−0.549121	+	0.835743i	$0.685037\pi$
$192$		0			0
$193$			0.173496i			0.0124885i	0.999981	+	0.00624427i	$0.00198763\pi$
$193$			0.173496i			0.0124885i	−0.999981	+	0.00624427i	$0.998012\pi$
$194$		0			0
$195$	−2.23266	+	0.123438i	−0.159884	+	0.00883960i
$196$		0			0
$197$		−	4.27989i		−	0.304930i	−0.988309	−	0.152465i	$-0.951279\pi$
$197$		−	4.27989i		−	0.304930i	0.988309	−	0.152465i	$-0.0487211\pi$
$198$		0			0
$199$	16.6843			1.18272			0.591358	−	0.806409i	$-0.298592\pi$
$199$	16.6843			1.18272			0.591358	+	0.806409i	$0.298592\pi$
$200$		0			0
$201$	−2.73916			−0.193205
$202$		0			0
$203$		−	11.0246i		−	0.773775i
$204$		0			0
$205$	−4.01447	+	0.221950i	−0.280383	+	0.0155017i
$206$		0			0
$207$			6.26338i			0.435335i
$208$		0			0
$209$	−23.2639			−1.60920
$210$		0			0
$211$	24.3013			1.67297			0.836485	−	0.547989i	$-0.184606\pi$
$211$	24.3013			1.67297			0.836485	+	0.547989i	$0.184606\pi$
$212$		0			0
$213$		−	4.11402i		−	0.281888i
$214$		0			0
$215$	−0.767593	−	13.8837i	−0.0523494	−	0.946857i
$216$		0			0
$217$			10.8906i			0.739302i
$218$		0			0
$219$	−10.4369			−0.705259
$220$		0			0
$221$	−0.448809			−0.0301902
$222$		0			0
$223$			0.821018i			0.0549794i	0.999622	+	0.0274897i	$0.00875135\pi$
$223$			0.821018i			0.0549794i	−0.999622	+	0.0274897i	$0.991249\pi$
$224$		0			0
$225$	−4.96953	+	0.551191i	−0.331302	+	0.0367461i
$226$		0			0
$227$		−	6.24591i		−	0.414555i	−0.978282	−	0.207278i	$-0.933540\pi$
$227$		−	6.24591i		−	0.414555i	0.978282	−	0.207278i	$-0.0664604\pi$
$228$		0			0
$229$	−20.5766			−1.35974			−0.679871	−	0.733332i	$-0.737964\pi$
$229$	−20.5766			−1.35974			−0.679871	+	0.733332i	$0.737964\pi$
$230$		0			0
$231$	15.9840			1.05167
$232$		0			0
$233$		−	5.19402i		−	0.340271i	−0.985421	−	0.170136i	$-0.945579\pi$
$233$		−	5.19402i		−	0.340271i	0.985421	−	0.170136i	$-0.0544206\pi$
$234$		0			0
$235$	0.205551	+	3.71785i	0.0134087	+	0.242526i
$236$		0			0
$237$			1.05336i			0.0684233i
$238$		0			0
$239$	−12.3229			−0.797100			−0.398550	−	0.917147i	$-0.630486\pi$
$239$	−12.3229			−0.797100			−0.398550	+	0.917147i	$0.630486\pi$
$240$		0			0
$241$	−2.41597			−0.155626			−0.0778131	−	0.996968i	$-0.524794\pi$
$241$	−2.41597			−0.155626			−0.0778131	+	0.996968i	$0.524794\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	39.5095	−	2.18439i	2.52417	−	0.139555i
$246$		0			0
$247$			7.23291i			0.460219i
$248$		0			0
$249$	5.77601			0.366040
$250$		0			0
$251$	−21.6428			−1.36608			−0.683042	−	0.730380i	$-0.739343\pi$
$251$	−21.6428			−1.36608			−0.683042	+	0.730380i	$0.739343\pi$
$252$		0			0
$253$		−	20.1456i		−	1.26654i
$254$		0			0
$255$	−1.00204	+	0.0554002i	−0.0627500	+	0.00346930i
$256$		0			0
$257$			16.9027i			1.05436i	0.849753	+	0.527181i	$0.176751\pi$
$257$			16.9027i			1.05436i	−0.849753	+	0.527181i	$0.823249\pi$
$258$		0			0
$259$	41.0651			2.55166
$260$		0			0
$261$	−2.21844			−0.137318
$262$		0			0
$263$			20.1864i			1.24475i	0.782720	+	0.622374i	$0.213832\pi$
$263$			20.1864i			1.24475i	−0.782720	+	0.622374i	$0.786168\pi$
$264$		0			0
$265$	0.143312	+	2.59212i	0.00880359	+	0.159233i
$266$		0			0
$267$		−	11.1326i		−	0.681302i
$268$		0			0
$269$	29.6277			1.80643			0.903215	−	0.429188i	$-0.141200\pi$
$269$	29.6277			1.80643			0.903215	+	0.429188i	$0.141200\pi$
$270$		0			0
$271$	−22.2703			−1.35282			−0.676411	−	0.736524i	$-0.736466\pi$
$271$	−22.2703			−1.35282			−0.676411	+	0.736524i	$0.736466\pi$
$272$		0			0
$273$		−	4.96953i		−	0.300769i
$274$		0			0
$275$	15.9840	−	1.77285i	0.963871	−	0.106907i
$276$		0			0
$277$		−	1.88512i		−	0.113266i	−0.998395	−	0.0566331i	$-0.981963\pi$
$277$		−	1.88512i		−	0.113266i	0.998395	−	0.0566331i	$-0.0180365\pi$
$278$		0			0
$279$	2.19148			0.131200
$280$		0			0
$281$	−8.84827			−0.527843			−0.263922	−	0.964544i	$-0.585016\pi$
$281$	−8.84827			−0.527843			−0.263922	+	0.964544i	$0.585016\pi$
$282$		0			0
$283$		−	24.0219i		−	1.42795i	−0.700169	−	0.713977i	$-0.746892\pi$
$283$		−	24.0219i		−	1.42795i	0.700169	−	0.713977i	$-0.253108\pi$
$284$		0			0
$285$	0.892818	+	16.1486i	0.0528859	+	0.956561i
$286$		0			0
$287$		−	8.93554i		−	0.527448i
$288$		0			0
$289$	16.7986			0.988151
$290$		0			0
$291$	8.37418			0.490903
$292$		0			0
$293$			22.6019i			1.32042i	0.751082	+	0.660208i	$0.229532\pi$
$293$			22.6019i			1.32042i	−0.751082	+	0.660208i	$0.770468\pi$
$294$		0			0
$295$	13.1362	−	0.726268i	0.764818	−	0.0422849i
$296$		0			0
$297$		−	3.21640i		−	0.186635i
$298$		0			0
$299$	−6.26338			−0.362221
$300$		0			0
$301$	30.9027			1.78120
$302$		0			0
$303$			5.50625i			0.316326i
$304$		0			0
$305$	3.95098	−	0.218440i	0.226233	−	0.0125078i
$306$		0			0
$307$			2.90858i			0.166001i	0.996549	+	0.0830007i	$0.0264504\pi$
$307$			2.90858i			0.166001i	−0.996549	+	0.0830007i	$0.973550\pi$
$308$		0			0
$309$	1.32082			0.0751389
$310$		0			0
$311$	−7.13290			−0.404469			−0.202235	−	0.979337i	$-0.564820\pi$
$311$	−7.13290			−0.404469			−0.202235	+	0.979337i	$0.564820\pi$
$312$		0			0
$313$		−	26.8418i		−	1.51719i	−0.651565	−	0.758593i	$-0.725887\pi$
$313$		−	26.8418i		−	1.51719i	0.651565	−	0.758593i	$-0.274113\pi$
$314$		0			0
$315$	−0.613429	−	11.0953i	−0.0345628	−	0.625147i
$316$		0			0
$317$		−	8.19000i		−	0.459996i	−0.973191	−	0.229998i	$-0.926128\pi$
$317$		−	8.19000i		−	0.459996i	0.973191	−	0.229998i	$-0.0738720\pi$
$318$		0			0
$319$	7.13540			0.399505
$320$		0			0
$321$	−7.49024			−0.418065
$322$		0			0
$323$			3.24620i			0.180623i
$324$		0			0
$325$	−0.551191	−	4.96953i	−0.0305746	−	0.275660i
$326$		0			0
$327$		−	14.0330i		−	0.776027i
$328$		0			0
$329$	−8.27531			−0.456233
$330$		0			0
$331$	4.75599			0.261413			0.130707	−	0.991421i	$-0.458275\pi$
$331$	4.75599			0.261413			0.130707	+	0.991421i	$0.458275\pi$
$332$		0			0
$333$		−	8.26338i		−	0.452831i
$334$		0			0
$335$	−0.338117	−	6.11560i	−0.0184733	−	0.334131i
$336$		0			0
$337$			4.54361i			0.247506i	0.992313	+	0.123753i	$0.0394930\pi$
$337$			4.54361i			0.247506i	−0.992313	+	0.123753i	$0.960507\pi$
$338$		0			0
$339$	−11.2269			−0.609759
$340$		0			0
$341$	−7.04867			−0.381707
$342$		0			0
$343$			53.1550i			2.87010i
$344$		0			0
$345$	−13.9840	+	0.773141i	−0.752873	+	0.0416245i
$346$		0			0
$347$			33.2648i			1.78575i	0.450307	+	0.892874i	$0.351314\pi$
$347$			33.2648i			1.78575i	−0.450307	+	0.892874i	$0.648686\pi$
$348$		0			0
$349$	−19.6460			−1.05163			−0.525813	−	0.850600i	$-0.676239\pi$
$349$	−19.6460			−1.05163			−0.525813	+	0.850600i	$0.676239\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$		−	12.1609i		−	0.647262i	−0.946183	−	0.323631i	$-0.895096\pi$
$353$		−	12.1609i		−	0.647262i	0.946183	−	0.323631i	$-0.104904\pi$
$354$		0			0
$355$	9.18520	−	0.507827i	0.487500	−	0.0269527i
$356$		0			0
$357$		−	2.23037i		−	0.118044i
$358$		0			0
$359$	−23.5378			−1.24228			−0.621139	−	0.783701i	$-0.713330\pi$
$359$	−23.5378			−1.24228			−0.621139	+	0.783701i	$0.713330\pi$
$360$		0			0
$361$	33.3150			1.75342
$362$		0			0
$363$		−	0.654755i		−	0.0343657i
$364$		0			0
$365$	−1.28831	−	23.3020i	−0.0674332	−	1.21968i
$366$		0			0
$367$			12.5944i			0.657421i	0.944431	+	0.328710i	$0.106614\pi$
$367$			12.5944i			0.657421i	−0.944431	+	0.328710i	$0.893386\pi$
$368$		0			0
$369$	−1.79807			−0.0936036
$370$		0			0
$371$	−5.76963			−0.299544
$372$		0			0
$373$		−	10.6585i		−	0.551875i	−0.961176	−	0.275938i	$-0.911012\pi$
$373$		−	10.6585i		−	0.551875i	0.961176	−	0.275938i	$-0.0889883\pi$
$374$		0			0
$375$	−1.84405	−	11.0272i	−0.0952263	−	0.569443i
$376$		0			0
$377$		−	2.21844i		−	0.114255i
$378$		0			0
$379$	−22.5065			−1.15608			−0.578040	−	0.816009i	$-0.696182\pi$
$379$	−22.5065			−1.15608			−0.578040	+	0.816009i	$0.696182\pi$
$380$		0			0
$381$	−1.50217			−0.0769586
$382$		0			0
$383$		−	6.22399i		−	0.318031i	−0.987276	−	0.159015i	$-0.949168\pi$
$383$		−	6.22399i		−	0.318031i	0.987276	−	0.159015i	$-0.0508320\pi$
$384$		0			0
$385$	1.97304	+	35.6868i	0.100555	+	1.81877i
$386$		0			0
$387$		−	6.21844i		−	0.316101i
$388$		0			0
$389$	6.49059			0.329086			0.164543	−	0.986370i	$-0.447385\pi$
$389$	6.49059			0.329086			0.164543	+	0.986370i	$0.447385\pi$
$390$		0			0
$391$	−2.81106			−0.142162
$392$		0			0
$393$		−	12.5477i		−	0.632947i
$394$		0			0
$395$	−2.35180	+	0.130025i	−0.118332	+	0.00654228i
$396$		0			0
$397$			13.0832i			0.656628i	0.944569	+	0.328314i	$0.106480\pi$
$397$			13.0832i			0.656628i	−0.944569	+	0.328314i	$0.893520\pi$
$398$		0			0
$399$	−35.9441			−1.79946
$400$		0			0
$401$	24.2326			1.21012			0.605060	−	0.796180i	$-0.293149\pi$
$401$	24.2326			1.21012			0.605060	+	0.796180i	$0.293149\pi$
$402$		0			0
$403$			2.19148i			0.109165i
$404$		0			0
$405$	−2.23266	+	0.123438i	−0.110942	+	0.00613369i
$406$		0			0
$407$			26.5784i			1.31744i
$408$		0			0
$409$	−2.52576			−0.124891			−0.0624454	−	0.998048i	$-0.519890\pi$
$409$	−2.52576			−0.124891			−0.0624454	+	0.998048i	$0.519890\pi$
$410$		0			0
$411$	10.6797			0.526790
$412$		0			0
$413$			29.2390i			1.43876i
$414$		0			0
$415$	0.712981	+	12.8959i	0.0349988	+	0.633033i
$416$		0			0
$417$		−	20.6962i		−	1.01350i
$418$		0			0
$419$	29.1917			1.42611			0.713054	−	0.701109i	$-0.247312\pi$
$419$	29.1917			1.42611			0.713054	+	0.701109i	$0.247312\pi$
$420$		0			0
$421$	−22.3719			−1.09034			−0.545169	−	0.838326i	$-0.683534\pi$
$421$	−22.3719			−1.09034			−0.545169	+	0.838326i	$0.683534\pi$
$422$		0			0
$423$			1.66521i			0.0809654i
$424$		0			0
$425$	−0.247380	−	2.23037i	−0.0119997	−	0.108189i
$426$		0			0
$427$			8.79423i			0.425582i
$428$		0			0
$429$	3.21640			0.155289
$430$		0			0
$431$	26.6111			1.28181			0.640906	−	0.767619i	$-0.278559\pi$
$431$	26.6111			1.28181			0.640906	+	0.767619i	$0.278559\pi$
$432$		0			0
$433$			10.4369i			0.501564i	0.968044	+	0.250782i	$0.0806878\pi$
$433$			10.4369i			0.501564i	−0.968044	+	0.250782i	$0.919312\pi$
$434$		0			0
$435$	−0.273840	−	4.95302i	−0.0131296	−	0.237479i
$436$		0			0
$437$			45.3025i			2.16711i
$438$		0			0
$439$	−28.0916			−1.34074			−0.670370	−	0.742027i	$-0.733865\pi$
$439$	−28.0916			−1.34074			−0.670370	+	0.742027i	$0.733865\pi$
$440$		0			0
$441$	17.6962			0.842676
$442$		0			0
$443$		−	28.9146i		−	1.37378i	−0.726764	−	0.686888i	$-0.758976\pi$
$443$		−	28.9146i		−	1.37378i	0.726764	−	0.686888i	$-0.241024\pi$
$444$		0			0
$445$	24.8552	−	1.37418i	1.17825	−	0.0651426i
$446$		0			0
$447$			10.3457i			0.489337i
$448$		0			0
$449$	−27.9652			−1.31976			−0.659879	−	0.751372i	$-0.729392\pi$
$449$	−27.9652			−1.31976			−0.659879	+	0.751372i	$0.729392\pi$
$450$		0			0
$451$	5.78331			0.272325
$452$		0			0
$453$		−	9.34779i		−	0.439197i
$454$		0			0
$455$	11.0953	−	0.613429i	0.520154	−	0.0287580i
$456$		0			0
$457$			28.7251i			1.34370i	0.740685	+	0.671852i	$0.234501\pi$
$457$			28.7251i			1.34370i	−0.740685	+	0.671852i	$0.765499\pi$
$458$		0			0
$459$	−0.448809			−0.0209486
$460$		0			0
$461$	5.81017			0.270607			0.135303	−	0.990804i	$-0.456799\pi$
$461$	5.81017			0.270607			0.135303	+	0.990804i	$0.456799\pi$
$462$		0			0
$463$			27.1743i			1.26290i	0.775418	+	0.631448i	$0.217539\pi$
$463$			27.1743i			1.26290i	−0.775418	+	0.631448i	$0.782461\pi$
$464$		0			0
$465$	0.270512	+	4.89282i	0.0125447	+	0.226899i
$466$		0			0
$467$			23.1290i			1.07028i	0.844763	+	0.535141i	$0.179742\pi$
$467$			23.1290i			1.07028i	−0.844763	+	0.535141i	$0.820258\pi$
$468$		0			0
$469$	13.6123			0.628558
$470$		0			0
$471$	−0.987506			−0.0455019
$472$		0			0
$473$			20.0010i			0.919647i
$474$		0			0
$475$	−35.9441	+	3.98671i	−1.64923	+	0.182923i
$476$		0			0
$477$			1.16100i			0.0531586i
$478$		0			0
$479$	23.3243			1.06571			0.532856	−	0.846206i	$-0.321119\pi$
$479$	23.3243			1.06571			0.532856	+	0.846206i	$0.321119\pi$
$480$		0			0
$481$	8.26338			0.376778
$482$		0			0
$483$		−	31.1260i		−	1.41628i
$484$		0			0
$485$	1.03369	+	18.6967i	0.0469376	+	0.848973i
$486$		0			0
$487$		−	7.23466i		−	0.327834i	−0.986474	−	0.163917i	$-0.947587\pi$
$487$		−	7.23466i		−	0.327834i	0.986474	−	0.163917i	$-0.0524129\pi$
$488$		0			0
$489$	−11.0635			−0.500308
$490$		0			0
$491$	42.9453			1.93809			0.969047	−	0.246874i	$-0.0794035\pi$
$491$	42.9453			1.93809			0.969047	+	0.246874i	$0.0794035\pi$
$492$		0			0
$493$		−	0.995656i		−	0.0448421i
$494$		0			0
$495$	7.18113	−	0.397027i	0.322768	−	0.0178450i
$496$		0			0
$497$			20.4447i			0.917072i
$498$		0			0
$499$	−18.0487			−0.807969			−0.403985	−	0.914766i	$-0.632375\pi$
$499$	−18.0487			−0.807969			−0.403985	+	0.914766i	$0.632375\pi$
$500$		0			0
$501$	19.6373			0.877329
$502$		0			0
$503$		−	10.3252i		−	0.460376i	−0.973146	−	0.230188i	$-0.926066\pi$
$503$		−	10.3252i		−	0.460376i	0.973146	−	0.230188i	$-0.0739342\pi$
$504$		0			0
$505$	−12.2936	+	0.679681i	−0.547057	+	0.0302454i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	8.57786			0.380207			0.190104	−	0.981764i	$-0.439118\pi$
$509$	8.57786			0.380207			0.190104	+	0.981764i	$0.439118\pi$
$510$		0			0
$511$	51.8663			2.29443
$512$		0			0
$513$			7.23291i			0.319341i
$514$		0			0
$515$	0.163040	+	2.94894i	0.00718440	+	0.129946i
$516$		0			0
$517$		−	5.35599i		−	0.235556i
$518$		0			0
$519$	15.4783			0.679423
$520$		0			0
$521$	8.44706			0.370072			0.185036	−	0.982732i	$-0.440760\pi$
$521$	8.44706			0.370072			0.185036	+	0.982732i	$0.440760\pi$
$522$		0			0
$523$			31.5650i			1.38024i	0.723694	+	0.690121i	$0.242443\pi$
$523$			31.5650i			1.38024i	−0.723694	+	0.690121i	$0.757557\pi$
$524$		0			0
$525$	24.6962	−	2.73916i	1.07783	−	0.119547i
$526$		0			0
$527$			0.983555i			0.0428443i
$528$		0			0
$529$	−16.2300			−0.705651
$530$		0			0
$531$	5.88365			0.255329
$532$		0			0
$533$		−	1.79807i		−	0.0778829i
$534$		0			0
$535$	−0.924582	−	16.7232i	−0.0399732	−	0.723005i
$536$		0			0
$537$		−	3.10238i		−	0.133878i
$538$		0			0
$539$	−56.9181			−2.45163
$540$		0			0
$541$	24.7478			1.06399			0.531995	−	0.846747i	$-0.321442\pi$
$541$	24.7478			1.06399			0.531995	+	0.846747i	$0.321442\pi$
$542$		0			0
$543$			12.8111i			0.549775i
$544$		0			0
$545$	31.3309	−	1.73221i	1.34207	−	0.0741997i
$546$		0			0
$547$			25.9207i			1.10829i	0.832420	+	0.554145i	$0.186955\pi$
$547$			25.9207i			1.10829i	−0.832420	+	0.554145i	$0.813045\pi$
$548$		0			0
$549$	1.76963			0.0755260
$550$		0			0
$551$	−16.0458			−0.683573
$552$		0			0
$553$		−	5.23471i		−	0.222603i
$554$		0			0
$555$	18.4493	−	1.02002i	0.783130	−	0.0432973i
$556$		0			0
$557$		−	35.3538i		−	1.49799i	−0.662577	−	0.748993i	$-0.730537\pi$
$557$		−	35.3538i		−	1.49799i	0.662577	−	0.748993i	$-0.269463\pi$
$558$		0			0
$559$	6.21844			0.263012
$560$		0			0
$561$	1.44355			0.0609468
$562$		0			0
$563$		−	22.3919i		−	0.943708i	−0.881677	−	0.471854i	$-0.843585\pi$
$563$		−	22.3919i		−	0.943708i	0.881677	−	0.471854i	$-0.156415\pi$
$564$		0			0
$565$	−1.38582	−	25.0657i	−0.0583020	−	1.05452i
$566$		0			0
$567$		−	4.96953i		−	0.208700i
$568$		0			0
$569$	−41.1034			−1.72314			−0.861572	−	0.507636i	$-0.830520\pi$
$569$	−41.1034			−1.72314			−0.861572	+	0.507636i	$0.830520\pi$
$570$		0			0
$571$	−28.3848			−1.18787			−0.593933	−	0.804514i	$-0.702426\pi$
$571$	−28.3848			−1.18787			−0.593933	+	0.804514i	$0.702426\pi$
$572$		0			0
$573$		−	15.1780i		−	0.634071i
$574$		0			0
$575$	−3.45232	−	31.1260i	−0.143972	−	1.29805i
$576$		0			0
$577$		−	5.44592i		−	0.226716i	−0.993554	−	0.113358i	$-0.963839\pi$
$577$		−	5.44592i		−	0.226716i	0.993554	−	0.113358i	$-0.0361608\pi$
$578$		0			0
$579$	−0.173496			−0.00721026
$580$		0			0
$581$	−28.7040			−1.19084
$582$		0			0
$583$		−	3.73425i		−	0.154657i
$584$		0			0
$585$	−0.123438	−	2.23266i	−0.00510354	−	0.0923091i
$586$		0			0
$587$		−	18.7717i		−	0.774790i	−0.921914	−	0.387395i	$-0.873375\pi$
$587$		−	18.7717i		−	0.774790i	0.921914	−	0.387395i	$-0.126625\pi$
$588$		0			0
$589$	15.8508			0.653119
$590$		0			0
$591$	4.27989			0.176051
$592$		0			0
$593$		−	39.9300i		−	1.63973i	−0.572559	−	0.819863i	$-0.694049\pi$
$593$		−	39.9300i		−	1.63973i	0.572559	−	0.819863i	$-0.305951\pi$
$594$		0			0
$595$	4.97965	−	0.275313i	0.204146	−	0.0112867i
$596$		0			0
$597$			16.6843i			0.682841i
$598$		0			0
$599$	36.7310			1.50079			0.750393	−	0.660992i	$-0.229864\pi$
$599$	36.7310			1.50079			0.750393	+	0.660992i	$0.229864\pi$
$600$		0			0
$601$	−25.7376			−1.04986			−0.524930	−	0.851146i	$-0.675908\pi$
$601$	−25.7376			−1.04986			−0.524930	+	0.851146i	$0.675908\pi$
$602$		0			0
$603$		−	2.73916i		−	0.111547i
$604$		0			0
$605$	1.46184	−	0.0808218i	0.0594324	−	0.00328587i
$606$		0			0
$607$			3.95975i			0.160721i	0.996766	+	0.0803606i	$0.0256072\pi$
$607$			3.95975i			0.160721i	−0.996766	+	0.0803606i	$0.974393\pi$
$608$		0			0
$609$	11.0246			0.446739
$610$		0			0
$611$	−1.66521			−0.0673673
$612$		0			0
$613$			14.0707i			0.568311i	0.958778	+	0.284156i	$0.0917132\pi$
$613$			14.0707i			0.568311i	−0.958778	+	0.284156i	$0.908287\pi$
$614$		0			0
$615$	−0.221950	−	4.01447i	−0.00894990	−	0.161879i
$616$		0			0
$617$			31.6852i			1.27560i	0.770204	+	0.637798i	$0.220155\pi$
$617$			31.6852i			1.27560i	−0.770204	+	0.637798i	$0.779845\pi$
$618$		0			0
$619$	12.4566			0.500673			0.250337	−	0.968159i	$-0.419459\pi$
$619$	12.4566			0.500673			0.250337	+	0.968159i	$0.419459\pi$
$620$		0			0
$621$	−6.26338			−0.251341
$622$		0			0
$623$			55.3236i			2.21649i
$624$		0			0
$625$	24.3924	−	5.47831i	0.975695	−	0.219133i
$626$		0			0
$627$		−	23.2639i		−	0.929073i
$628$		0			0
$629$	3.70868			0.147875
$630$		0			0
$631$	14.8500			0.591167			0.295584	−	0.955317i	$-0.404486\pi$
$631$	14.8500			0.591167			0.295584	+	0.955317i	$0.404486\pi$
$632$		0			0
$633$			24.3013i			0.965890i
$634$		0			0
$635$	−0.185425	−	3.35384i	−0.00735838	−	0.133093i
$636$		0			0
$637$			17.6962i			0.701149i
$638$		0			0
$639$	4.11402			0.162748
$640$		0			0
$641$	13.2878			0.524837			0.262418	−	0.964954i	$-0.415480\pi$
$641$	13.2878			0.524837			0.262418	+	0.964954i	$0.415480\pi$
$642$		0			0
$643$		−	18.7409i		−	0.739069i	−0.929217	−	0.369535i	$-0.879517\pi$
$643$		−	18.7409i		−	0.739069i	0.929217	−	0.369535i	$-0.120483\pi$
$644$		0			0
$645$	13.8837	−	0.767593i	0.546668	−	0.0302240i
$646$		0			0
$647$			30.9106i			1.21522i	0.794236	+	0.607610i	$0.207871\pi$
$647$			30.9106i			1.21522i	−0.794236	+	0.607610i	$0.792129\pi$
$648$		0			0
$649$	−18.9242			−0.742840
$650$		0			0
$651$	−10.8906			−0.426836
$652$		0			0
$653$		−	1.99040i		−	0.0778903i	−0.999241	−	0.0389452i	$-0.987600\pi$
$653$		−	1.99040i		−	0.0778903i	0.999241	−	0.0389452i	$-0.0123998\pi$
$654$		0			0
$655$	28.0147	−	1.54886i	1.09462	−	0.0605191i
$656$		0			0
$657$		−	10.4369i		−	0.407181i
$658$		0			0
$659$	38.5954			1.50346			0.751731	−	0.659470i	$-0.229219\pi$
$659$	38.5954			1.50346			0.751731	+	0.659470i	$0.229219\pi$
$660$		0			0
$661$	−10.3150			−0.401206			−0.200603	−	0.979673i	$-0.564290\pi$
$661$	−10.3150			−0.401206			−0.200603	+	0.979673i	$0.564290\pi$
$662$		0			0
$663$		−	0.448809i		−	0.0174303i
$664$		0			0
$665$	−4.43688	−	80.2510i	−0.172055	−	3.11200i
$666$		0			0
$667$		−	13.8949i		−	0.538014i
$668$		0			0
$669$	−0.821018			−0.0317424
$670$		0			0
$671$	−5.69185			−0.219731
$672$		0			0
$673$			19.1024i			0.736343i	0.929758	+	0.368171i	$0.120016\pi$
$673$			19.1024i			0.736343i	−0.929758	+	0.368171i	$0.879984\pi$
$674$		0			0
$675$	−0.551191	−	4.96953i	−0.0212153	−	0.191277i
$676$		0			0
$677$			18.2482i			0.701336i	0.936500	+	0.350668i	$0.114045\pi$
$677$			18.2482i			0.701336i	−0.936500	+	0.350668i	$0.885955\pi$
$678$		0			0
$679$	−41.6157			−1.59706
$680$		0			0
$681$	6.24591			0.239344
$682$		0			0
$683$		−	45.2264i		−	1.73054i	−0.501306	−	0.865270i	$-0.667147\pi$
$683$		−	45.2264i		−	1.73054i	0.501306	−	0.865270i	$-0.332853\pi$
$684$		0			0
$685$	1.31828	+	23.8441i	0.0503689	+	0.911035i
$686$		0			0
$687$		−	20.5766i		−	0.785047i
$688$		0			0
$689$	−1.16100			−0.0442307
$690$		0			0
$691$	−12.4293			−0.472831			−0.236416	−	0.971652i	$-0.575973\pi$
$691$	−12.4293			−0.472831			−0.236416	+	0.971652i	$0.575973\pi$
$692$		0			0
$693$			15.9840i			0.607182i
$694$		0			0
$695$	46.2075	−	2.55470i	1.75275	−	0.0969053i
$696$		0			0
$697$		−	0.806989i		−	0.0305669i
$698$		0			0
$699$	5.19402			0.196456
$700$		0			0
$701$	−30.9611			−1.16939			−0.584693	−	0.811254i	$-0.698785\pi$
$701$	−30.9611			−1.16939			−0.584693	+	0.811254i	$0.698785\pi$
$702$		0			0
$703$		−	59.7683i		−	2.25420i
$704$		0			0
$705$	−3.71785	+	0.205551i	−0.140022	+	0.00774149i
$706$		0			0
$707$		−	27.3634i		−	1.02911i
$708$		0			0
$709$	−2.20069			−0.0826486			−0.0413243	−	0.999146i	$-0.513158\pi$
$709$	−2.20069			−0.0826486			−0.0413243	+	0.999146i	$0.513158\pi$
$710$		0			0
$711$	−1.05336			−0.0395042
$712$		0			0
$713$			13.7261i			0.514045i
$714$		0			0
$715$	0.397027	+	7.18113i	0.0148480	+	0.268559i
$716$		0			0
$717$		−	12.3229i		−	0.460206i
$718$		0			0
$719$	−44.5913			−1.66297			−0.831487	−	0.555543i	$-0.812510\pi$
$719$	−44.5913			−1.66297			−0.831487	+	0.555543i	$0.812510\pi$
$720$		0			0
$721$	−6.56386			−0.244451
$722$		0			0
$723$		−	2.41597i		−	0.0898508i
$724$		0			0
$725$	11.0246	−	1.22278i	0.409443	−	0.0454130i
$726$		0			0
$727$		−	34.7808i		−	1.28995i	−0.764204	−	0.644974i	$-0.776868\pi$
$727$		−	34.7808i		−	1.28995i	0.764204	−	0.644974i	$-0.223132\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.79089			0.103225
$732$		0			0
$733$		−	31.0158i		−	1.14560i	−0.819697	−	0.572798i	$-0.805858\pi$
$733$		−	31.0158i		−	1.14560i	0.819697	−	0.572798i	$-0.194142\pi$
$734$		0			0
$735$	2.18439	+	39.5095i	0.0805723	+	1.45733i
$736$		0			0
$737$			8.81023i			0.324529i
$738$		0			0
$739$	37.9029			1.39428			0.697141	−	0.716934i	$-0.254455\pi$
$739$	37.9029			1.39428			0.697141	+	0.716934i	$0.254455\pi$
$740$		0			0
$741$	−7.23291			−0.265708
$742$		0			0
$743$		−	14.6939i		−	0.539066i	−0.962991	−	0.269533i	$-0.913131\pi$
$743$		−	14.6939i		−	0.539066i	0.962991	−	0.269533i	$-0.0868694\pi$
$744$		0			0
$745$	−23.0985	+	1.27706i	−0.846264	+	0.0467879i
$746$		0			0
$747$			5.77601i			0.211333i
$748$		0			0
$749$	37.2230			1.36010
$750$		0			0
$751$	32.5622			1.18821			0.594106	−	0.804387i	$-0.297506\pi$
$751$	32.5622			1.18821			0.594106	+	0.804387i	$0.297506\pi$
$752$		0			0
$753$		−	21.6428i		−	0.788708i
$754$		0			0
$755$	20.8704	−	1.15387i	0.759552	−	0.0419938i
$756$		0			0
$757$			15.2365i			0.553779i	0.960902	+	0.276889i	$0.0893035\pi$
$757$			15.2365i			0.553779i	−0.960902	+	0.276889i	$0.910696\pi$
$758$		0			0
$759$	20.1456			0.731238
$760$		0			0
$761$	51.5327			1.86806			0.934030	−	0.357194i	$-0.116266\pi$
$761$	51.5327			1.86806			0.934030	+	0.357194i	$0.116266\pi$
$762$		0			0
$763$			69.7374i			2.52466i
$764$		0			0
$765$	−0.0554002	−	1.00204i	−0.00200300	−	0.0362287i
$766$		0			0
$767$			5.88365i			0.212446i
$768$		0			0
$769$	−28.7148			−1.03548			−0.517741	−	0.855538i	$-0.673227\pi$
$769$	−28.7148			−1.03548			−0.517741	+	0.855538i	$0.673227\pi$
$770$		0			0
$771$	−16.9027			−0.608736
$772$		0			0
$773$		−	34.5120i		−	1.24131i	−0.784084	−	0.620655i	$-0.786867\pi$
$773$		−	34.5120i		−	1.24131i	0.784084	−	0.620655i	$-0.213133\pi$
$774$		0			0
$775$	−10.8906	+	1.20792i	−0.391202	+	0.0433898i
$776$		0			0
$777$			41.0651i			1.47320i
$778$		0			0
$779$	−13.0053			−0.465962
$780$		0			0
$781$	−13.2323			−0.473491
$782$		0			0
$783$		−	2.21844i		−	0.0792806i
$784$		0			0
$785$	−0.121896	−	2.20476i	−0.00435065	−	0.0786914i
$786$		0			0
$787$			34.2046i			1.21926i	0.792685	+	0.609631i	$0.208682\pi$
$787$			34.2046i			1.21926i	−0.792685	+	0.609631i	$0.791318\pi$
$788$		0			0
$789$	−20.1864			−0.718656
$790$		0			0
$791$	55.7922			1.98374
$792$		0			0
$793$			1.76963i			0.0628414i
$794$		0			0
$795$	−2.59212	+	0.143312i	−0.0919331	+	0.00508276i
$796$		0			0
$797$		−	42.9365i		−	1.52089i	−0.649402	−	0.760445i	$-0.724981\pi$
$797$		−	42.9365i		−	1.52089i	0.649402	−	0.760445i	$-0.275019\pi$
$798$		0			0
$799$	−0.747362			−0.0264398
$800$		0			0
$801$	11.1326			0.393350
$802$		0			0
$803$			33.5692i			1.18463i
$804$		0			0
$805$	69.4938	−	3.84214i	2.44934	−	0.135418i
$806$		0			0
$807$			29.6277i			1.04294i
$808$		0			0
$809$	8.42320			0.296144			0.148072	−	0.988977i	$-0.452693\pi$
$809$	8.42320			0.296144			0.148072	+	0.988977i	$0.452693\pi$
$810$		0			0
$811$	−39.8090			−1.39788			−0.698941	−	0.715180i	$-0.746345\pi$
$811$	−39.8090			−1.39788			−0.698941	+	0.715180i	$0.746345\pi$
$812$		0			0
$813$		−	22.2703i		−	0.781052i
$814$		0			0
$815$	−1.36566	−	24.7010i	−0.0478369	−	0.865238i
$816$		0			0
$817$		−	44.9774i		−	1.57356i
$818$		0			0
$819$	4.96953			0.173649
$820$		0			0
$821$	−29.3494			−1.02430			−0.512151	−	0.858895i	$-0.671151\pi$
$821$	−29.3494			−1.02430			−0.512151	+	0.858895i	$0.671151\pi$
$822$		0			0
$823$			4.65419i			0.162235i	0.996705	+	0.0811174i	$0.0258489\pi$
$823$			4.65419i			0.162235i	−0.996705	+	0.0811174i	$0.974151\pi$
$824$		0			0
$825$	1.77285	+	15.9840i	0.0617228	+	0.556491i
$826$		0			0
$827$		−	53.8430i		−	1.87231i	−0.351593	−	0.936153i	$-0.614360\pi$
$827$		−	53.8430i		−	1.87231i	0.351593	−	0.936153i	$-0.385640\pi$
$828$		0			0
$829$	4.58070			0.159094			0.0795471	−	0.996831i	$-0.474653\pi$
$829$	4.58070			0.159094			0.0795471	+	0.996831i	$0.474653\pi$
$830$		0			0
$831$	1.88512			0.0653942
$832$		0			0
$833$			7.94221i			0.275181i
$834$		0			0
$835$	2.42399	+	43.8433i	0.0838857	+	1.51726i
$836$		0			0
$837$			2.19148i			0.0757485i
$838$		0			0
$839$	21.2754			0.734509			0.367254	−	0.930121i	$-0.380298\pi$
$839$	21.2754			0.734509			0.367254	+	0.930121i	$0.380298\pi$
$840$		0			0
$841$	−24.0785			−0.830294
$842$		0			0
$843$		−	8.84827i		−	0.304751i
$844$		0			0
$845$	2.23266	−	0.123438i	0.0768058	−	0.00424640i
$846$		0			0
$847$			3.25382i			0.111803i
$848$		0			0
$849$	24.0219			0.824430
$850$		0			0
$851$	51.7567			1.77420
$852$		0			0
$853$		−	15.5689i		−	0.533070i	−0.963825	−	0.266535i	$-0.914121\pi$
$853$		−	15.5689i		−	0.533070i	0.963825	−	0.266535i	$-0.0858789\pi$
$854$		0			0
$855$	−16.1486	+	0.892818i	−0.552271	+	0.0305337i
$856$		0			0
$857$			18.5524i			0.633737i	0.948469	+	0.316869i	$0.102631\pi$
$857$			18.5524i			0.633737i	−0.948469	+	0.316869i	$0.897369\pi$
$858$		0			0
$859$	−26.2980			−0.897275			−0.448638	−	0.893714i	$-0.648091\pi$
$859$	−26.2980			−0.897275			−0.448638	+	0.893714i	$0.648091\pi$
$860$		0			0
$861$	8.93554			0.304522
$862$		0			0
$863$			0.537842i			0.0183083i	0.999958	+	0.00915417i	$0.00291390\pi$
$863$			0.537842i			0.0183083i	−0.999958	+	0.00915417i	$0.997086\pi$
$864$		0			0
$865$	1.91062	+	34.5578i	0.0649629	+	1.17500i
$866$		0			0
$867$			16.7986i			0.570509i
$868$		0			0
$869$	3.38804			0.114931
$870$		0			0
$871$	2.73916			0.0928128
$872$		0			0
$873$			8.37418i			0.283423i
$874$		0			0
$875$	9.16406	+	54.8000i	0.309802	+	1.85258i
$876$		0			0
$877$		−	49.9065i		−	1.68522i	−0.538523	−	0.842611i	$-0.681018\pi$
$877$		−	49.9065i		−	1.68522i	0.538523	−	0.842611i	$-0.318982\pi$
$878$		0			0
$879$	−22.6019			−0.762343
$880$		0			0
$881$	41.0327			1.38243			0.691213	−	0.722651i	$-0.257076\pi$
$881$	41.0327			1.38243			0.691213	+	0.722651i	$0.257076\pi$
$882$		0			0
$883$		−	6.59753i		−	0.222025i	−0.993819	−	0.111012i	$-0.964591\pi$
$883$		−	6.59753i		−	0.222025i	0.993819	−	0.111012i	$-0.0354093\pi$
$884$		0			0
$885$	0.726268	+	13.1362i	0.0244132	+	0.441568i
$886$		0			0
$887$			9.02521i			0.303037i	0.988454	+	0.151519i	$0.0484163\pi$
$887$			9.02521i			0.303037i	−0.988454	+	0.151519i	$0.951584\pi$
$888$		0			0
$889$	7.46508			0.250371
$890$		0			0
$891$	3.21640			0.107754
$892$		0			0
$893$			12.0443i			0.403048i
$894$		0			0
$895$	6.92656	−	0.382952i	0.231529	−	0.0128007i
$896$		0			0
$897$		−	6.26338i		−	0.209128i
$898$		0			0
$899$	−4.86166			−0.162145
$900$		0			0
$901$	−0.521069			−0.0173593
$902$		0			0
$903$			30.9027i			1.02838i
$904$		0			0
$905$	−28.6027	+	1.58138i	−0.950787	+	0.0525667i
$906$		0			0
$907$			15.5850i			0.517493i	0.965945	+	0.258746i	$0.0833094\pi$
$907$			15.5850i			0.517493i	−0.965945	+	0.258746i	$0.916691\pi$
$908$		0			0
$909$	−5.50625			−0.182631
$910$		0			0
$911$	−2.86303			−0.0948564			−0.0474282	−	0.998875i	$-0.515103\pi$
$911$	−2.86303			−0.0948564			−0.0474282	+	0.998875i	$0.515103\pi$
$912$		0			0
$913$		−	18.5780i		−	0.614841i
$914$		0			0
$915$	0.218440	+	3.95098i	0.00722141	+	0.130615i
$916$		0			0
$917$			62.3560i			2.05918i
$918$		0			0
$919$	−53.0874			−1.75119			−0.875596	−	0.483045i	$-0.839531\pi$
$919$	−53.0874			−1.75119			−0.875596	+	0.483045i	$0.839531\pi$
$920$		0			0
$921$	−2.90858			−0.0958409
$922$		0			0
$923$			4.11402i			0.135415i
$924$		0			0
$925$	4.55470	+	41.0651i	0.149758	+	1.35021i
$926$		0			0
$927$			1.32082i			0.0433815i
$928$		0			0
$929$	−25.0925			−0.823259			−0.411630	−	0.911351i	$-0.635040\pi$
$929$	−25.0925			−0.823259			−0.411630	+	0.911351i	$0.635040\pi$
$930$		0			0
$931$	127.995			4.19486
$932$		0			0
$933$		−	7.13290i		−	0.233521i
$934$		0			0
$935$	0.178189	+	3.22296i	0.00582742	+	0.105402i
$936$		0			0
$937$		−	54.5029i		−	1.78053i	−0.455441	−	0.890266i	$-0.650518\pi$
$937$		−	54.5029i		−	1.78053i	0.455441	−	0.890266i	$-0.349482\pi$
$938$		0			0
$939$	26.8418			0.875947
$940$		0			0
$941$	32.3538			1.05470			0.527351	−	0.849647i	$-0.323185\pi$
$941$	32.3538			1.05470			0.527351	+	0.849647i	$0.323185\pi$
$942$		0			0
$943$		−	11.2620i		−	0.366741i
$944$		0			0
$945$	11.0953	−	0.613429i	0.360929	−	0.0199549i
$946$		0			0
$947$			36.3069i			1.17981i	0.807471	+	0.589907i	$0.200836\pi$
$947$			36.3069i			1.17981i	−0.807471	+	0.589907i	$0.799164\pi$
$948$		0			0
$949$	10.4369			0.338795
$950$		0			0
$951$	8.19000			0.265579
$952$		0			0
$953$		−	13.3176i		−	0.431399i	−0.976460	−	0.215699i	$-0.930797\pi$
$953$		−	13.3176i		−	0.431399i	0.976460	−	0.215699i	$-0.0692031\pi$
$954$		0			0
$955$	33.8873	−	1.87355i	1.09657	−	0.0606266i
$956$		0			0
$957$			7.13540i			0.230655i
$958$		0			0
$959$	−53.0730			−1.71382
$960$		0			0
$961$	−26.1974			−0.845078
$962$		0			0
$963$		−	7.49024i		−	0.241370i
$964$		0			0
$965$	−0.0214161	−	0.387358i	−0.000689408	−	0.0124695i
$966$		0			0
$967$		−	46.9352i		−	1.50933i	−0.656108	−	0.754667i	$-0.727798\pi$
$967$		−	46.9352i		−	1.50933i	0.656108	−	0.754667i	$-0.272202\pi$
$968$		0			0
$969$	−3.24620			−0.104283
$970$		0			0
$971$	22.5340			0.723151			0.361575	−	0.932343i	$-0.382239\pi$
$971$	22.5340			0.723151			0.361575	+	0.932343i	$0.382239\pi$
$972$		0			0
$973$			102.850i			3.29723i
$974$		0			0
$975$	4.96953	−	0.551191i	0.159152	−	0.0176522i
$976$		0			0
$977$		−	59.9345i		−	1.91747i	−0.284296	−	0.958737i	$-0.591760\pi$
$977$		−	59.9345i		−	1.91747i	0.284296	−	0.958737i	$-0.408240\pi$
$978$		0			0
$979$	−35.8068			−1.14439
$980$		0			0
$981$	14.0330			0.448040
$982$		0			0
$983$		−	52.0879i		−	1.66135i	−0.556760	−	0.830673i	$-0.687956\pi$
$983$		−	52.0879i		−	1.66135i	0.556760	−	0.830673i	$-0.312044\pi$
$984$		0			0
$985$	0.528302	+	9.55553i	0.0168331	+	0.304465i
$986$		0			0
$987$		−	8.27531i		−	0.263406i
$988$		0			0
$989$	38.9485			1.23849
$990$		0			0
$991$	−18.6973			−0.593940			−0.296970	−	0.954887i	$-0.595976\pi$
$991$	−18.6973			−0.593940			−0.296970	+	0.954887i	$0.595976\pi$
$992$		0			0
$993$			4.75599i			0.150927i
$994$		0			0
$995$	−37.2503	+	2.05948i	−1.18091	+	0.0652898i
$996$		0			0
$997$			18.9805i			0.601118i	0.953763	+	0.300559i	$0.0971732\pi$
$997$			18.9805i			0.601118i	−0.953763	+	0.300559i	$0.902827\pi$
$998$		0			0
$999$	8.26338			0.261442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.p.1249.6		10
4.3	odd	2		195.2.c.b.79.9	yes	10
5.4	even	2	inner	3120.2.l.p.1249.1		10
12.11	even	2		585.2.c.c.469.2		10
20.3	even	4		975.2.a.s.1.5		5
20.7	even	4		975.2.a.r.1.1		5
20.19	odd	2		195.2.c.b.79.2	✓	10
60.23	odd	4		2925.2.a.bm.1.1		5
60.47	odd	4		2925.2.a.bl.1.5		5
60.59	even	2		585.2.c.c.469.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.2	✓	10	20.19	odd	2
195.2.c.b.79.9	yes	10	4.3	odd	2
585.2.c.c.469.2		10	12.11	even	2
585.2.c.c.469.9		10	60.59	even	2
975.2.a.r.1.1		5	20.7	even	4
975.2.a.s.1.5		5	20.3	even	4
2925.2.a.bl.1.5		5	60.47	odd	4
2925.2.a.bm.1.1		5	60.23	odd	4
3120.2.l.p.1249.1		10	5.4	even	2	inner
3120.2.l.p.1249.6		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.23266 + 0.123438i) q^{5} -4.96953i q^{7} -1.00000 q^{9} +3.21640 q^{11} -1.00000i q^{13} +(-0.123438 - 2.23266i) q^{15} -0.448809i q^{17} -7.23291 q^{19} +4.96953 q^{21} -6.26338i q^{23} +(4.96953 - 0.551191i) q^{25} -1.00000i q^{27} +2.21844 q^{29} -2.19148 q^{31} +3.21640i q^{33} +(0.613429 + 11.0953i) q^{35} +8.26338i q^{37} +1.00000 q^{39} +1.79807 q^{41} +6.21844i q^{43} +(2.23266 - 0.123438i) q^{45} -1.66521i q^{47} -17.6962 q^{49} +0.448809 q^{51} -1.16100i q^{53} +(-7.18113 + 0.397027i) q^{55} -7.23291i q^{57} -5.88365 q^{59} -1.76963 q^{61} +4.96953i q^{63} +(0.123438 + 2.23266i) q^{65} +2.73916i q^{67} +6.26338 q^{69} -4.11402 q^{71} +10.4369i q^{73} +(0.551191 + 4.96953i) q^{75} -15.9840i q^{77} +1.05336 q^{79} +1.00000 q^{81} -5.77601i q^{83} +(0.0554002 + 1.00204i) q^{85} +2.21844i q^{87} -11.1326 q^{89} -4.96953 q^{91} -2.19148i q^{93} +(16.1486 - 0.892818i) q^{95} -8.37418i q^{97} -3.21640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 16 * q^19 + 10 * q^21 + 10 * q^25 - 16 * q^29 - 24 * q^31 - 12 * q^35 + 10 * q^39 + 10 * q^41 + 2 * q^45 - 44 * q^49 + 10 * q^51 - 2 * q^55 + 16 * q^59 + 26 * q^61 + 14 * q^69 - 10 * q^71 + 2 * q^79 + 10 * q^81 - 4 * q^85 - 38 * q^89 - 10 * q^91 + 48 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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