LMFDB - Embedded newform 3360.2.t.e.2689.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(2689,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3360.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.t (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:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3360.2689
Dual form			3360.2.t.e.2689.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{13} +(-2.00000 - 1.00000i) q^{15} -6.00000 q^{19} -1.00000 q^{21} +8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -6.00000 q^{29} -10.0000 q^{31} -2.00000i q^{33} +(-2.00000 - 1.00000i) q^{35} -8.00000i q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} -2.00000i q^{53} +(2.00000 - 4.00000i) q^{55} +6.00000i q^{57} -12.0000 q^{59} +14.0000 q^{61} +1.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +8.00000i q^{67} +8.00000 q^{69} -2.00000 q^{71} +2.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -2.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{87} -6.00000 q^{89} +2.00000 q^{91} +10.0000i q^{93} +(-6.00000 + 12.0000i) q^{95} -14.0000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} - 12 q^{29} - 20 q^{31} - 4 q^{35} + 4 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{49} + 4 q^{55} - 24 q^{59} + 28 q^{61} + 8 q^{65} + 16 q^{69} - 4 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{89} + 4 q^{91} - 12 q^{95} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 + 4 * q^11 - 4 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^25 - 12 * q^29 - 20 * q^31 - 4 * q^35 + 4 * q^39 - 12 * q^41 - 2 * q^45 - 2 * q^49 + 4 * q^55 - 24 * q^59 + 28 * q^61 + 8 * q^65 + 16 * q^69 - 4 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^89 + 4 * q^91 - 12 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	−2.00000	−	1.00000i	−0.516398	−	0.258199i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−10.0000			−1.79605			−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000			−1.79605			−0.898027	+	0.439941i	$0.854999\pi$
$32$		0			0
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	2.00000			0.320256
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$	2.00000	−	4.00000i	0.269680	−	0.539360i
$56$		0			0
$57$			6.00000i			0.794719i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$			10.0000i			1.03695i
$94$		0			0
$95$	−6.00000	+	12.0000i	−0.615587	+	1.23117i
$96$		0			0
$97$		−	14.0000i		−	1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$		−	14.0000i		−	1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$	−1.00000	+	2.00000i	−0.0975900	+	0.195180i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	16.0000	+	8.00000i	1.49201	+	0.746004i
$116$		0			0
$117$		−	2.00000i		−	0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$			6.00000i			0.541002i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$			6.00000i			0.520266i
$134$		0			0
$135$	2.00000	+	1.00000i	0.172133	+	0.0860663i
$136$		0			0
$137$		−	22.0000i		−	1.87959i	−0.341743	−	0.939793i	$-0.611017\pi$
$137$		−	22.0000i		−	1.87959i	0.341743	−	0.939793i	$-0.388983\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$			4.00000i			0.334497i
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$	−20.0000			−1.62758			−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000			−1.62758			−0.813788	+	0.581161i	$0.802599\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.0000	+	20.0000i	−0.803219	+	1.60644i
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$		0			0
$159$	−2.00000			−0.158610
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	−4.00000	−	2.00000i	−0.311400	−	0.155700i
$166$		0			0
$167$			4.00000i			0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$			4.00000i			0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	6.00000			0.458831
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$			12.0000i			0.901975i
$178$		0			0
$179$	14.0000			1.04641			0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000			1.04641			0.523205	+	0.852207i	$0.324736\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		−	14.0000i		−	1.03491i
$184$		0			0
$185$	−16.0000	−	8.00000i	−1.17634	−	0.588172i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$	2.00000	−	4.00000i	0.143223	−	0.286446i
$196$		0			0
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$		0			0
$201$	8.00000			0.564276
$202$		0			0
$203$			6.00000i			0.421117i
$204$		0			0
$205$	−6.00000	+	12.0000i	−0.419058	+	0.838116i
$206$		0			0
$207$		−	8.00000i		−	0.556038i
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$			2.00000i			0.137038i
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$			10.0000i			0.678844i
$218$		0			0
$219$	2.00000			0.135147
$220$		0			0
$221$		0			0
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$		0			0
$231$	−2.00000			−0.131590
$232$		0			0
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	−16.0000	−	8.00000i	−1.04372	−	0.521862i
$236$		0			0
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−1.00000	+	2.00000i	−0.0638877	+	0.127775i
$246$		0			0
$247$		−	12.0000i		−	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	12.0000i		−	0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$		−	12.0000i		−	0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	8.00000i		−	0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$		−	8.00000i		−	0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$		0			0
$265$	−4.00000	−	2.00000i	−0.245718	−	0.122859i
$266$		0			0
$267$			6.00000i			0.367194i
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	10.0000			0.607457			0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000			0.607457			0.303728	+	0.952759i	$0.401768\pi$
$272$		0			0
$273$		−	2.00000i		−	0.121046i
$274$		0			0
$275$	−6.00000	−	8.00000i	−0.361814	−	0.482418i
$276$		0			0
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$		0			0
$279$	10.0000			0.598684
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		0			0
$285$	12.0000	+	6.00000i	0.710819	+	0.355409i
$286$		0			0
$287$			6.00000i			0.354169i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	−14.0000			−0.820695
$292$		0			0
$293$		−	32.0000i		−	1.86946i	−0.355359	−	0.934730i	$-0.615641\pi$
$293$		−	32.0000i		−	1.86946i	0.355359	−	0.934730i	$-0.384359\pi$
$294$		0			0
$295$	−12.0000	+	24.0000i	−0.698667	+	1.39733i
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$	−16.0000			−0.925304
$300$		0			0
$301$	−4.00000			−0.230556
$302$		0			0
$303$			6.00000i			0.344691i
$304$		0			0
$305$	14.0000	−	28.0000i	0.801638	−	1.60328i
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	18.0000i		−	1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$		−	18.0000i		−	1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$		0			0
$315$	2.00000	+	1.00000i	0.112687	+	0.0563436i
$316$		0			0
$317$			10.0000i			0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$			10.0000i			0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$		0			0
$327$		−	14.0000i		−	0.774202i
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$	16.0000	+	8.00000i	0.874173	+	0.437087i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	−20.0000			−1.08306
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	8.00000	−	16.0000i	0.430706	−	0.861411i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−22.0000			−1.17763			−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000			−1.17763			−0.588817	+	0.808267i	$0.700406\pi$
$350$		0			0
$351$	−2.00000			−0.106752
$352$		0			0
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	−2.00000	+	4.00000i	−0.106149	+	0.212298i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$	4.00000	+	2.00000i	0.209370	+	0.104685i
$366$		0			0
$367$		−	24.0000i		−	1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$		−	24.0000i		−	1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	−2.00000			−0.103835
$372$		0			0
$373$			36.0000i			1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$			36.0000i			1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$	2.00000	+	11.0000i	0.103280	+	0.568038i
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	−4.00000			−0.204926
$382$		0			0
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$		0			0
$385$	−4.00000	−	2.00000i	−0.203859	−	0.101929i
$386$		0			0
$387$			4.00000i			0.203331i
$388$		0			0
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			8.00000i			0.403547i
$394$		0			0
$395$	−8.00000	+	16.0000i	−0.402524	+	0.805047i
$396$		0			0
$397$			38.0000i			1.90717i	0.301131	+	0.953583i	$0.402636\pi$
$397$			38.0000i			1.90717i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$	6.00000			0.300376
$400$		0			0
$401$	−22.0000			−1.09863			−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000			−1.09863			−0.549314	+	0.835616i	$0.685111\pi$
$402$		0			0
$403$		−	20.0000i		−	0.996271i
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$	−34.0000			−1.68119			−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000			−1.68119			−0.840596	+	0.541663i	$0.817795\pi$
$410$		0			0
$411$	−22.0000			−1.08518
$412$		0			0
$413$			12.0000i			0.590481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	2.00000i		−	0.0979404i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	14.0000i		−	0.677507i
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	−18.0000			−0.867029			−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000			−0.867029			−0.433515	+	0.901146i	$0.642727\pi$
$432$		0			0
$433$			22.0000i			1.05725i	0.848855	+	0.528626i	$0.177293\pi$
$433$			22.0000i			1.05725i	−0.848855	+	0.528626i	$0.822707\pi$
$434$		0			0
$435$	12.0000	+	6.00000i	0.575356	+	0.287678i
$436$		0			0
$437$		−	48.0000i		−	2.29615i
$438$		0			0
$439$	26.0000			1.24091			0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000			1.24091			0.620456	+	0.784241i	$0.286947\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$	−6.00000	+	12.0000i	−0.284427	+	0.568855i
$446$		0			0
$447$		−	22.0000i		−	1.04056i
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$		0			0
$453$			20.0000i			0.939682i
$454$		0			0
$455$	2.00000	−	4.00000i	0.0937614	−	0.187523i
$456$		0			0
$457$			4.00000i			0.187112i	0.995614	+	0.0935561i	$0.0298234\pi$
$457$			4.00000i			0.187112i	−0.995614	+	0.0935561i	$0.970177\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$		−	12.0000i		−	0.557687i	−0.960337	−	0.278844i	$-0.910049\pi$
$463$		−	12.0000i		−	0.557687i	0.960337	−	0.278844i	$-0.0899511\pi$
$464$		0			0
$465$	20.0000	+	10.0000i	0.927478	+	0.463739i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$	18.0000			0.829396
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	18.0000	+	24.0000i	0.825897	+	1.10120i
$476$		0			0
$477$			2.00000i			0.0915737i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$		0			0
$483$		−	8.00000i		−	0.364013i
$484$		0			0
$485$	−28.0000	−	14.0000i	−1.27141	−	0.635707i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	18.0000			0.812329			0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000			0.812329			0.406164	+	0.913800i	$0.366866\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−2.00000	+	4.00000i	−0.0898933	+	0.179787i
$496$		0			0
$497$			2.00000i			0.0897123i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	4.00000			0.178707
$502$		0			0
$503$		−	20.0000i		−	0.891756i	−0.895094	−	0.445878i	$-0.852892\pi$
$503$		−	20.0000i		−	0.891756i	0.895094	−	0.445878i	$-0.147108\pi$
$504$		0			0
$505$	−6.00000	+	12.0000i	−0.266996	+	0.533993i
$506$		0			0
$507$		−	9.00000i		−	0.399704i
$508$		0			0
$509$	34.0000			1.50702			0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000			1.50702			0.753512	+	0.657434i	$0.228358\pi$
$510$		0			0
$511$	2.00000			0.0884748
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$	32.0000	+	16.0000i	1.41009	+	0.705044i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$			28.0000i			1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$			28.0000i			1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$		0			0
$525$	3.00000	+	4.00000i	0.130931	+	0.174574i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−24.0000	−	12.0000i	−1.03761	−	0.518805i
$536$		0			0
$537$		−	14.0000i		−	0.604145i
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$			6.00000i			0.257485i
$544$		0			0
$545$	14.0000	−	28.0000i	0.599694	−	1.19939i
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$	−14.0000			−0.597505
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$			8.00000i			0.340195i
$554$		0			0
$555$	−8.00000	+	16.0000i	−0.339581	+	0.679162i
$556$		0			0
$557$		−	38.0000i		−	1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$		−	38.0000i		−	1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$	−12.0000	−	6.00000i	−0.504844	−	0.252422i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$		−	18.0000i		−	0.751961i
$574$		0			0
$575$	32.0000	−	24.0000i	1.33449	−	1.00087i
$576$		0			0
$577$		−	14.0000i		−	0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$		−	14.0000i		−	0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$		0			0
$579$	−16.0000			−0.664937
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	−4.00000	−	2.00000i	−0.165380	−	0.0826898i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	60.0000			2.47226
$590$		0			0
$591$	22.0000			0.904959
$592$		0			0
$593$		−	8.00000i		−	0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$		−	8.00000i		−	0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			14.0000i			0.572982i
$598$		0			0
$599$	46.0000			1.87951			0.939755	−	0.341850i	$-0.111053\pi$
$599$	46.0000			1.87951			0.939755	+	0.341850i	$0.111053\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		−	8.00000i		−	0.325785i
$604$		0			0
$605$	−7.00000	+	14.0000i	−0.284590	+	0.569181i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$	6.00000			0.243132
$610$		0			0
$611$	16.0000			0.647291
$612$		0			0
$613$		−	32.0000i		−	1.29247i	−0.763139	−	0.646234i	$-0.776343\pi$
$613$		−	32.0000i		−	1.29247i	0.763139	−	0.646234i	$-0.223657\pi$
$614$		0			0
$615$	12.0000	+	6.00000i	0.483887	+	0.241943i
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$			6.00000i			0.240385i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			12.0000i			0.479234i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		−	8.00000i		−	0.317971i
$634$		0			0
$635$	−8.00000	−	4.00000i	−0.317470	−	0.158735i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$		0			0
$639$	2.00000			0.0791188
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$		−	28.0000i		−	1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$		−	28.0000i		−	1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$		0			0
$645$	−4.00000	+	8.00000i	−0.157500	+	0.315000i
$646$		0			0
$647$		−	28.0000i		−	1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$		−	28.0000i		−	1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$		0			0
$649$	−24.0000			−0.942082
$650$		0			0
$651$	10.0000			0.391931
$652$		0			0
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		0			0
$655$	−8.00000	+	16.0000i	−0.312586	+	0.625172i
$656$		0			0
$657$		−	2.00000i		−	0.0780274i
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000	+	6.00000i	0.465340	+	0.232670i
$666$		0			0
$667$		−	48.0000i		−	1.85857i
$668$		0			0
$669$	24.0000			0.927894
$670$		0			0
$671$	28.0000			1.08093
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$		0			0
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$		0			0
$677$			32.0000i			1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$677$			32.0000i			1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$678$		0			0
$679$	−14.0000			−0.537271
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	−44.0000	−	22.0000i	−1.68115	−	0.840577i
$686$		0			0
$687$		−	2.00000i		−	0.0763048i
$688$		0			0
$689$	4.00000			0.152388
$690$		0			0
$691$	2.00000			0.0760836			0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000			0.0760836			0.0380418	+	0.999276i	$0.487888\pi$
$692$		0			0
$693$			2.00000i			0.0759737i
$694$		0			0
$695$	2.00000	−	4.00000i	0.0758643	−	0.151729i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−14.0000			−0.529529
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$			48.0000i			1.81035i
$704$		0			0
$705$	−8.00000	+	16.0000i	−0.301297	+	0.602595i
$706$		0			0
$707$			6.00000i			0.225653i
$708$		0			0
$709$	14.0000			0.525781			0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000			0.525781			0.262891	+	0.964826i	$0.415324\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$		−	80.0000i		−	2.99602i
$714$		0			0
$715$	8.00000	+	4.00000i	0.299183	+	0.149592i
$716$		0			0
$717$			6.00000i			0.224074i
$718$		0			0
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		−	14.0000i		−	0.520666i
$724$		0			0
$725$	18.0000	+	24.0000i	0.668503	+	0.891338i
$726$		0			0
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$			2.00000i			0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$			2.00000i			0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$		0			0
$735$	2.00000	+	1.00000i	0.0737711	+	0.0368856i
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$	−12.0000			−0.440831
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	22.0000	−	44.0000i	0.806018	−	1.61204i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$			28.0000i			1.02038i
$754$		0			0
$755$	−20.0000	+	40.0000i	−0.727875	+	1.45575i
$756$		0			0
$757$			20.0000i			0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$			20.0000i			0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$		0			0
$759$	16.0000			0.580763
$760$		0			0
$761$	26.0000			0.942499			0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000			0.942499			0.471250	+	0.882000i	$0.343803\pi$
$762$		0			0
$763$		−	14.0000i		−	0.506834i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	24.0000i		−	0.866590i
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$		−	48.0000i		−	1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$		−	48.0000i		−	1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$		0			0
$775$	30.0000	+	40.0000i	1.07763	+	1.43684i
$776$		0			0
$777$			8.00000i			0.286998i
$778$		0			0
$779$	36.0000			1.28983
$780$		0			0
$781$	−4.00000			−0.143131
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$		0			0
$785$	36.0000	+	18.0000i	1.28490	+	0.642448i
$786$		0			0
$787$		−	20.0000i		−	0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$		−	20.0000i		−	0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$		0			0
$789$	−8.00000			−0.284808
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$			28.0000i			0.994309i
$794$		0			0
$795$	−2.00000	+	4.00000i	−0.0709327	+	0.141865i
$796$		0			0
$797$		−	48.0000i		−	1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$		−	48.0000i		−	1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$			4.00000i			0.141157i
$804$		0			0
$805$	8.00000	−	16.0000i	0.281963	−	0.563926i
$806$		0			0
$807$		−	14.0000i		−	0.492823i
$808$		0			0
$809$	−38.0000			−1.33601			−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000			−1.33601			−0.668004	+	0.744157i	$0.732851\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		−	10.0000i		−	0.350715i
$814$		0			0
$815$	8.00000	+	4.00000i	0.280228	+	0.140114i
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$	−2.00000			−0.0698857
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$		0			0
$825$	−8.00000	+	6.00000i	−0.278524	+	0.208893i
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	8.00000			0.277517
$832$		0			0
$833$		0			0
$834$		0			0
$835$	8.00000	+	4.00000i	0.276851	+	0.138426i
$836$		0			0
$837$		−	10.0000i		−	0.345651i
$838$		0			0
$839$	−28.0000			−0.966667			−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000			−0.966667			−0.483334	+	0.875436i	$0.660574\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			10.0000i			0.344418i
$844$		0			0
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$	64.0000			2.19389
$852$		0			0
$853$		−	46.0000i		−	1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$		−	46.0000i		−	1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$		0			0
$855$	6.00000	−	12.0000i	0.205196	−	0.410391i
$856$		0			0
$857$			12.0000i			0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$			12.0000i			0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$		0			0
$859$	14.0000			0.477674			0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000			0.477674			0.238837	+	0.971060i	$0.423234\pi$
$860$		0			0
$861$	6.00000			0.204479
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	17.0000i		−	0.577350i
$868$		0			0
$869$	−16.0000			−0.542763
$870$		0			0
$871$	−16.0000			−0.542139
$872$		0			0
$873$			14.0000i			0.473828i
$874$		0			0
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$		0			0
$877$			16.0000i			0.540282i	0.962821	+	0.270141i	$0.0870703\pi$
$877$			16.0000i			0.540282i	−0.962821	+	0.270141i	$0.912930\pi$
$878$		0			0
$879$	−32.0000			−1.07933
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$			24.0000i			0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$			24.0000i			0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$		0			0
$885$	24.0000	+	12.0000i	0.806751	+	0.403376i
$886$		0			0
$887$		−	48.0000i		−	1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$		−	48.0000i		−	1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$		0			0
$889$	−4.00000			−0.134156
$890$		0			0
$891$	2.00000			0.0670025
$892$		0			0
$893$			48.0000i			1.60626i
$894$		0			0
$895$	14.0000	−	28.0000i	0.467968	−	0.935937i
$896$		0			0
$897$			16.0000i			0.534224i
$898$		0			0
$899$	60.0000			2.00111
$900$		0			0
$901$		0			0
$902$		0			0
$903$			4.00000i			0.133112i
$904$		0			0
$905$	−6.00000	+	12.0000i	−0.199447	+	0.398893i
$906$		0			0
$907$		−	40.0000i		−	1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$		−	40.0000i		−	1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	−46.0000			−1.52405			−0.762024	−	0.647549i	$-0.775794\pi$
$911$	−46.0000			−1.52405			−0.762024	+	0.647549i	$0.775794\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−28.0000	−	14.0000i	−0.925651	−	0.462826i
$916$		0			0
$917$			8.00000i			0.264183i
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	−28.0000			−0.922631
$922$		0			0
$923$		−	4.00000i		−	0.131662i
$924$		0			0
$925$	−32.0000	+	24.0000i	−1.05215	+	0.789115i
$926$		0			0
$927$		−	16.0000i		−	0.525509i
$928$		0			0
$929$	−34.0000			−1.11550			−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000			−1.11550			−0.557752	+	0.830008i	$0.688336\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	34.0000i		−	1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$		−	34.0000i		−	1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$		0			0
$939$	−18.0000			−0.587408
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$		−	48.0000i		−	1.56310i
$944$		0			0
$945$	1.00000	−	2.00000i	0.0325300	−	0.0650600i
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	−4.00000			−0.129845
$950$		0			0
$951$	10.0000			0.324272
$952$		0			0
$953$			34.0000i			1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$			34.0000i			1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$		0			0
$955$	18.0000	−	36.0000i	0.582466	−	1.16493i
$956$		0			0
$957$			12.0000i			0.387905i
$958$		0			0
$959$	−22.0000			−0.710417
$960$		0			0
$961$	69.0000			2.22581
$962$		0			0
$963$			12.0000i			0.386695i
$964$		0			0
$965$	−32.0000	−	16.0000i	−1.03012	−	0.515058i
$966$		0			0
$967$		−	40.0000i		−	1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$		−	40.0000i		−	1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−52.0000			−1.66876			−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000			−1.66876			−0.834380	+	0.551190i	$0.814174\pi$
$972$		0			0
$973$		−	2.00000i		−	0.0641171i
$974$		0			0
$975$	−6.00000	−	8.00000i	−0.192154	−	0.256205i
$976$		0			0
$977$			14.0000i			0.447900i	0.974601	+	0.223950i	$0.0718952\pi$
$977$			14.0000i			0.447900i	−0.974601	+	0.223950i	$0.928105\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$			48.0000i			1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$			48.0000i			1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$		0			0
$985$	44.0000	+	22.0000i	1.40196	+	0.700978i
$986$		0			0
$987$			8.00000i			0.254643i
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$	−14.0000	+	28.0000i	−0.443830	+	0.887660i
$996$		0			0
$997$		−	26.0000i		−	0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$		−	26.0000i		−	0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.t.e.2689.1	yes	2
4.3	odd	2		3360.2.t.d.2689.2	yes	2
5.4	even	2	inner	3360.2.t.e.2689.2	yes	2
20.19	odd	2		3360.2.t.d.2689.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.t.d.2689.1	✓	2	20.19	odd	2
3360.2.t.d.2689.2	yes	2	4.3	odd	2
3360.2.t.e.2689.1	yes	2	1.1	even	1	trivial
3360.2.t.e.2689.2	yes	2	5.4	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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{13} +(-2.00000 - 1.00000i) q^{15} -6.00000 q^{19} -1.00000 q^{21} +8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -6.00000 q^{29} -10.0000 q^{31} -2.00000i q^{33} +(-2.00000 - 1.00000i) q^{35} -8.00000i q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} -2.00000i q^{53} +(2.00000 - 4.00000i) q^{55} +6.00000i q^{57} -12.0000 q^{59} +14.0000 q^{61} +1.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +8.00000i q^{67} +8.00000 q^{69} -2.00000 q^{71} +2.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -2.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{87} -6.00000 q^{89} +2.00000 q^{91} +10.0000i q^{93} +(-6.00000 + 12.0000i) q^{95} -14.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} - 12 q^{29} - 20 q^{31} - 4 q^{35} + 4 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{49} + 4 q^{55} - 24 q^{59} + 28 q^{61} + 8 q^{65} + 16 q^{69} - 4 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{89} + 4 q^{91} - 12 q^{95} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 + 4 * q^11 - 4 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^25 - 12 * q^29 - 20 * q^31 - 4 * q^35 + 4 * q^39 - 12 * q^41 - 2 * q^45 - 2 * q^49 + 4 * q^55 - 24 * q^59 + 28 * q^61 + 8 * q^65 + 16 * q^69 - 4 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^89 + 4 * q^91 - 12 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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