LMFDB - Embedded newform 4080.2.m.t.2449.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4080,2,Mod(2449,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, 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("4080.2449"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-12,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$32.5789640247$
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} - 2 x^{11} + 2 x^{10} + 6 x^{9} - 14 x^{8} - 34 x^{7} + 114 x^{6} - 290 x^{5} + 529 x^{4} + \cdots + 64 $ x^12 - 2x^11 + 2x^10 + 6x^9 - 14x^8 - 34x^7 + 114x^6 - 290x^5 + 529x^4 - 456x^3 + 288x^2 - 192*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 2040)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.10
Root			$-2.16636 + 1.04179i$ of defining polynomial
Character	$\chi$	$=$	4080.2449
Dual form			4080.2.m.t.2449.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(0.634216 + 2.14424i) q^{5} +0.237858i q^{7} -1.00000 q^{9} -5.46771 q^{11} -0.941367i q^{13} +(-2.14424 + 0.634216i) q^{15} +1.00000i q^{17} -3.46771 q^{19} -0.237858 q^{21} -2.95988i q^{23} +(-4.19554 + 2.71982i) q^{25} -1.00000i q^{27} -1.17922 q^{29} +3.43560 q^{31} -5.46771i q^{33} +(-0.510025 + 0.150853i) q^{35} -0.870677i q^{37} +0.941367 q^{39} +2.71762 q^{41} +5.10854i q^{43} +(-0.634216 - 2.14424i) q^{45} -12.1717i q^{47} +6.94342 q^{49} -1.00000 q^{51} -9.27643i q^{53} +(-3.46771 - 11.7241i) q^{55} -3.46771i q^{57} -4.55440 q^{59} +0.0645705 q^{61} -0.237858i q^{63} +(2.01852 - 0.597030i) q^{65} -10.0552i q^{67} +2.95988 q^{69} -11.8771 q^{71} +5.42165i q^{73} +(-2.71982 - 4.19554i) q^{75} -1.30054i q^{77} +0.248382 q^{79} +1.00000 q^{81} -7.97651i q^{83} +(-2.14424 + 0.634216i) q^{85} -1.17922i q^{87} +5.20386 q^{89} +0.223912 q^{91} +3.43560i q^{93} +(-2.19928 - 7.43560i) q^{95} +7.64540i q^{97} +5.46771 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} - 4 q^{11} + 20 q^{19} + 4 q^{21} - 4 q^{29} - 16 q^{31} + 12 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{49} - 12 q^{51} + 20 q^{55} - 8 q^{59} + 8 q^{61} - 16 q^{65} - 8 q^{69} - 32 q^{71} + 4 q^{75}+ \cdots + 4 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 - 4 * q^11 + 20 * q^19 + 4 * q^21 - 4 * q^29 - 16 * q^31 + 12 * q^35 + 8 * q^39 - 4 * q^41 + 8 * q^49 - 12 * q^51 + 20 * q^55 - 8 * q^59 + 8 * q^61 - 16 * q^65 - 8 * q^69 - 32 * q^71 + 4 * q^75 + 24 * q^79 + 12 * q^81 + 40 * q^89 - 56 * q^91 + 20 * q^95 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$5$	0.634216	+	2.14424i	0.283630	+	0.958934i
$5$	0.634216	+	2.14424i	0.283630	+	0.958934i
$6$		0			0
$7$			0.237858i			0.0899019i	0.998989	+	0.0449510i	$0.0143132\pi$
$7$			0.237858i			0.0899019i	−0.998989	+	0.0449510i	$0.985687\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.46771			−1.64858			−0.824288	−	0.566171i	$-0.808424\pi$
$11$	−5.46771			−1.64858			−0.824288	+	0.566171i	$0.808424\pi$
$12$		0			0
$13$		−	0.941367i		−	0.261088i	−0.991443	−	0.130544i	$-0.958328\pi$
$13$		−	0.941367i		−	0.261088i	0.991443	−	0.130544i	$-0.0416724\pi$
$14$		0			0
$15$	−2.14424	+	0.634216i	−0.553641	+	0.163754i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	−3.46771			−0.795547			−0.397773	−	0.917484i	$-0.630217\pi$
$19$	−3.46771			−0.795547			−0.397773	+	0.917484i	$0.630217\pi$
$20$		0			0
$21$	−0.237858			−0.0519049
$22$		0			0
$23$		−	2.95988i		−	0.617179i	−0.951195	−	0.308589i	$-0.900143\pi$
$23$		−	2.95988i		−	0.617179i	0.951195	−	0.308589i	$-0.0998569\pi$
$24$		0			0
$25$	−4.19554	+	2.71982i	−0.839108	+	0.543965i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−1.17922			−0.218977			−0.109488	−	0.993988i	$-0.534921\pi$
$29$	−1.17922			−0.218977			−0.109488	+	0.993988i	$0.534921\pi$
$30$		0			0
$31$	3.43560			0.617052			0.308526	−	0.951216i	$-0.400164\pi$
$31$	3.43560			0.617052			0.308526	+	0.951216i	$0.400164\pi$
$32$		0			0
$33$		−	5.46771i		−	0.951806i
$34$		0			0
$35$	−0.510025	+	0.150853i	−0.0862100	+	0.0254989i
$36$		0			0
$37$		−	0.870677i		−	0.143138i	−0.997436	−	0.0715692i	$-0.977199\pi$
$37$		−	0.870677i		−	0.143138i	0.997436	−	0.0715692i	$-0.0228007\pi$
$38$		0			0
$39$	0.941367			0.150739
$40$		0			0
$41$	2.71762			0.424421			0.212211	−	0.977224i	$-0.431934\pi$
$41$	2.71762			0.424421			0.212211	+	0.977224i	$0.431934\pi$
$42$		0			0
$43$			5.10854i			0.779044i	0.921017	+	0.389522i	$0.127360\pi$
$43$			5.10854i			0.779044i	−0.921017	+	0.389522i	$0.872640\pi$
$44$		0			0
$45$	−0.634216	−	2.14424i	−0.0945433	−	0.319645i
$46$		0			0
$47$		−	12.1717i		−	1.77543i	−0.460393	−	0.887715i	$-0.652291\pi$
$47$		−	12.1717i		−	1.77543i	0.460393	−	0.887715i	$-0.347709\pi$
$48$		0			0
$49$	6.94342			0.991918
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	9.27643i		−	1.27422i	−0.770775	−	0.637108i	$-0.780131\pi$
$53$		−	9.27643i		−	1.27422i	0.770775	−	0.637108i	$-0.219869\pi$
$54$		0			0
$55$	−3.46771	−	11.7241i	−0.467586	−	1.58088i
$56$		0			0
$57$		−	3.46771i		−	0.459309i
$58$		0			0
$59$	−4.55440			−0.592932			−0.296466	−	0.955043i	$-0.595808\pi$
$59$	−4.55440			−0.592932			−0.296466	+	0.955043i	$0.595808\pi$
$60$		0			0
$61$	0.0645705			0.00826741			0.00413371	−	0.999991i	$-0.498684\pi$
$61$	0.0645705			0.00826741			0.00413371	+	0.999991i	$0.498684\pi$
$62$		0			0
$63$		−	0.237858i		−	0.0299673i
$64$		0			0
$65$	2.01852	−	0.597030i	0.250366	−	0.0740524i
$66$		0			0
$67$		−	10.0552i		−	1.22844i	−0.789136	−	0.614219i	$-0.789471\pi$
$67$		−	10.0552i		−	1.22844i	0.789136	−	0.614219i	$-0.210529\pi$
$68$		0			0
$69$	2.95988			0.356328
$70$		0			0
$71$	−11.8771			−1.40956			−0.704779	−	0.709427i	$-0.748954\pi$
$71$	−11.8771			−1.40956			−0.704779	+	0.709427i	$0.748954\pi$
$72$		0			0
$73$			5.42165i			0.634557i	0.948332	+	0.317278i	$0.102769\pi$
$73$			5.42165i			0.634557i	−0.948332	+	0.317278i	$0.897231\pi$
$74$		0			0
$75$	−2.71982	−	4.19554i	−0.314058	−	0.484459i
$76$		0			0
$77$		−	1.30054i		−	0.148210i
$78$		0			0
$79$	0.248382			0.0279451			0.0139726	−	0.999902i	$-0.495552\pi$
$79$	0.248382			0.0279451			0.0139726	+	0.999902i	$0.495552\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.97651i		−	0.875536i	−0.899088	−	0.437768i	$-0.855769\pi$
$83$		−	7.97651i		−	0.875536i	0.899088	−	0.437768i	$-0.144231\pi$
$84$		0			0
$85$	−2.14424	+	0.634216i	−0.232576	+	0.0687904i
$86$		0			0
$87$		−	1.17922i		−	0.126426i
$88$		0			0
$89$	5.20386			0.551608			0.275804	−	0.961214i	$-0.411056\pi$
$89$	5.20386			0.551608			0.275804	+	0.961214i	$0.411056\pi$
$90$		0			0
$91$	0.223912			0.0234723
$92$		0			0
$93$			3.43560i			0.356255i
$94$		0			0
$95$	−2.19928	−	7.43560i	−0.225641	−	0.762877i
$96$		0			0
$97$			7.64540i			0.776273i	0.921602	+	0.388136i	$0.126881\pi$
$97$			7.64540i			0.776273i	−0.921602	+	0.388136i	$0.873119\pi$
$98$		0			0
$99$	5.46771			0.549525
$100$		0			0
$101$	8.36374			0.832224			0.416112	−	0.909313i	$-0.363393\pi$
$101$	8.36374			0.832224			0.416112	+	0.909313i	$0.363393\pi$
$102$		0			0
$103$			7.19702i			0.709144i	0.935029	+	0.354572i	$0.115373\pi$
$103$			7.19702i			0.709144i	−0.935029	+	0.354572i	$0.884627\pi$
$104$		0			0
$105$	−0.150853	−	0.510025i	−0.0147218	−	0.0497734i
$106$		0			0
$107$			2.96331i			0.286474i	0.989688	+	0.143237i	$0.0457511\pi$
$107$			2.96331i			0.286474i	−0.989688	+	0.143237i	$0.954249\pi$
$108$		0			0
$109$	−0.131115			−0.0125586			−0.00627928	−	0.999980i	$-0.501999\pi$
$109$	−0.131115			−0.0125586			−0.00627928	+	0.999980i	$0.501999\pi$
$110$		0			0
$111$	0.870677			0.0826410
$112$		0			0
$113$		−	16.4424i		−	1.54677i	−0.633935	−	0.773387i	$-0.718561\pi$
$113$		−	16.4424i		−	1.54677i	0.633935	−	0.773387i	$-0.281439\pi$
$114$		0			0
$115$	6.34671	−	1.87721i	0.591833	−	0.175050i
$116$		0			0
$117$			0.941367i			0.0870294i
$118$		0			0
$119$	−0.237858			−0.0218044
$120$		0			0
$121$	18.8958			1.71780
$122$		0			0
$123$			2.71762i			0.245040i
$124$		0			0
$125$	−8.49284	−	7.27129i	−0.759622	−	0.650364i
$126$		0			0
$127$		−	7.84882i		−	0.696470i	−0.937407	−	0.348235i	$-0.886781\pi$
$127$		−	7.84882i		−	0.696470i	0.937407	−	0.348235i	$-0.113219\pi$
$128$		0			0
$129$	−5.10854			−0.449781
$130$		0			0
$131$	−7.73839			−0.676107			−0.338053	−	0.941127i	$-0.609768\pi$
$131$	−7.73839			−0.676107			−0.338053	+	0.941127i	$0.609768\pi$
$132$		0			0
$133$		−	0.824822i		−	0.0715212i
$134$		0			0
$135$	2.14424	−	0.634216i	0.184547	−	0.0545846i
$136$		0			0
$137$		−	7.63928i		−	0.652668i	−0.945255	−	0.326334i	$-0.894187\pi$
$137$		−	7.63928i		−	0.652668i	0.945255	−	0.326334i	$-0.105813\pi$
$138$		0			0
$139$	−6.45970			−0.547904			−0.273952	−	0.961743i	$-0.588331\pi$
$139$	−6.45970			−0.547904			−0.273952	+	0.961743i	$0.588331\pi$
$140$		0			0
$141$	12.1717			1.02505
$142$		0			0
$143$			5.14712i			0.430424i
$144$		0			0
$145$	−0.747883	−	2.52854i	−0.0621083	−	0.209984i
$146$		0			0
$147$			6.94342i			0.572684i
$148$		0			0
$149$	−4.91285			−0.402476			−0.201238	−	0.979542i	$-0.564497\pi$
$149$	−4.91285			−0.402476			−0.201238	+	0.979542i	$0.564497\pi$
$150$		0			0
$151$	−0.0521719			−0.00424569			−0.00212285	−	0.999998i	$-0.500676\pi$
$151$	−0.0521719			−0.00424569			−0.00212285	+	0.999998i	$0.500676\pi$
$152$		0			0
$153$		−	1.00000i		−	0.0808452i
$154$		0			0
$155$	2.17891	+	7.36676i	0.175014	+	0.591712i
$156$		0			0
$157$			12.5482i			1.00145i	0.865605	+	0.500727i	$0.166934\pi$
$157$			12.5482i			1.00145i	−0.865605	+	0.500727i	$0.833066\pi$
$158$		0			0
$159$	9.27643			0.735668
$160$		0			0
$161$	0.704033			0.0554855
$162$		0			0
$163$		−	20.1316i		−	1.57683i	−0.615144	−	0.788415i	$-0.710902\pi$
$163$		−	20.1316i		−	1.57683i	0.615144	−	0.788415i	$-0.289098\pi$
$164$		0			0
$165$	11.7241	−	3.46771i	0.912719	−	0.269961i
$166$		0			0
$167$			14.8093i			1.14598i	0.819562	+	0.572990i	$0.194217\pi$
$167$			14.8093i			1.14598i	−0.819562	+	0.572990i	$0.805783\pi$
$168$		0			0
$169$	12.1138			0.931833
$170$		0			0
$171$	3.46771			0.265182
$172$		0			0
$173$		−	3.04012i		−	0.231136i	−0.993300	−	0.115568i	$-0.963131\pi$
$173$		−	3.04012i		−	0.231136i	0.993300	−	0.115568i	$-0.0368688\pi$
$174$		0			0
$175$	−0.646932	−	0.997943i	−0.0489035	−	0.0754374i
$176$		0			0
$177$		−	4.55440i		−	0.342330i
$178$		0			0
$179$	−19.6620			−1.46961			−0.734804	−	0.678280i	$-0.762726\pi$
$179$	−19.6620			−1.46961			−0.734804	+	0.678280i	$0.762726\pi$
$180$		0			0
$181$	−22.2908			−1.65686			−0.828430	−	0.560092i	$-0.810766\pi$
$181$	−22.2908			−1.65686			−0.828430	+	0.560092i	$0.810766\pi$
$182$		0			0
$183$			0.0645705i			0.00477319i
$184$		0			0
$185$	1.86694	−	0.552197i	0.137260	−	0.0405984i
$186$		0			0
$187$		−	5.46771i		−	0.399838i
$188$		0			0
$189$	0.237858			0.0173016
$190$		0			0
$191$	21.5548			1.55965			0.779826	−	0.625997i	$-0.215308\pi$
$191$	21.5548			1.55965			0.779826	+	0.625997i	$0.215308\pi$
$192$		0			0
$193$			0.185329i			0.0133403i	0.999978	+	0.00667013i	$0.00212319\pi$
$193$			0.185329i			0.0133403i	−0.999978	+	0.00667013i	$0.997877\pi$
$194$		0			0
$195$	0.597030	+	2.01852i	0.0427542	+	0.144549i
$196$		0			0
$197$			3.32896i			0.237178i	0.992943	+	0.118589i	$0.0378371\pi$
$197$			3.32896i			0.237178i	−0.992943	+	0.118589i	$0.962163\pi$
$198$		0			0
$199$	13.7768			0.976610			0.488305	−	0.872673i	$-0.337615\pi$
$199$	13.7768			0.976610			0.488305	+	0.872673i	$0.337615\pi$
$200$		0			0
$201$	10.0552			0.709239
$202$		0			0
$203$		−	0.280488i		−	0.0196864i
$204$		0			0
$205$	1.72356	+	5.82724i	0.120379	+	0.406992i
$206$		0			0
$207$			2.95988i			0.205726i
$208$		0			0
$209$	18.9604			1.31152
$210$		0			0
$211$	−11.2327			−0.773292			−0.386646	−	0.922228i	$-0.626366\pi$
$211$	−11.2327			−0.773292			−0.386646	+	0.922228i	$0.626366\pi$
$212$		0			0
$213$		−	11.8771i		−	0.813808i
$214$		0			0
$215$	−10.9539	+	3.23991i	−0.747052	+	0.220960i
$216$		0			0
$217$			0.817186i			0.0554742i
$218$		0			0
$219$	−5.42165			−0.366361
$220$		0			0
$221$	0.941367			0.0633232
$222$		0			0
$223$			15.5450i			1.04097i	0.853870	+	0.520486i	$0.174249\pi$
$223$			15.5450i			1.04097i	−0.853870	+	0.520486i	$0.825751\pi$
$224$		0			0
$225$	4.19554	−	2.71982i	0.279703	−	0.181322i
$226$		0			0
$227$			11.8480i			0.786379i	0.919457	+	0.393190i	$0.128628\pi$
$227$			11.8480i			0.786379i	−0.919457	+	0.393190i	$0.871372\pi$
$228$		0			0
$229$	−1.84802			−0.122121			−0.0610604	−	0.998134i	$-0.519448\pi$
$229$	−1.84802			−0.122121			−0.0610604	+	0.998134i	$0.519448\pi$
$230$		0			0
$231$	1.30054			0.0855692
$232$		0			0
$233$		−	21.5297i		−	1.41045i	−0.708981	−	0.705227i	$-0.750845\pi$
$233$		−	21.5297i		−	1.41045i	0.708981	−	0.705227i	$-0.249155\pi$
$234$		0			0
$235$	26.0991	−	7.71951i	1.70252	−	0.503565i
$236$		0			0
$237$			0.248382i			0.0161341i
$238$		0			0
$239$	23.6721			1.53122			0.765610	−	0.643305i	$-0.222437\pi$
$239$	23.6721			1.53122			0.765610	+	0.643305i	$0.222437\pi$
$240$		0			0
$241$	−1.95306			−0.125808			−0.0629038	−	0.998020i	$-0.520036\pi$
$241$	−1.95306			−0.125808			−0.0629038	+	0.998020i	$0.520036\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	4.40363	+	14.8884i	0.281338	+	0.951183i
$246$		0			0
$247$			3.26438i			0.207708i
$248$		0			0
$249$	7.97651			0.505491
$250$		0			0
$251$	−6.76133			−0.426771			−0.213386	−	0.976968i	$-0.568449\pi$
$251$	−6.76133			−0.426771			−0.213386	+	0.976968i	$0.568449\pi$
$252$		0			0
$253$			16.1838i			1.01747i
$254$		0			0
$255$	−0.634216	−	2.14424i	−0.0397161	−	0.134278i
$256$		0			0
$257$			0.502790i			0.0313632i	0.999877	+	0.0156816i	$0.00499181\pi$
$257$			0.502790i			0.0313632i	−0.999877	+	0.0156816i	$0.995008\pi$
$258$		0			0
$259$	0.207098			0.0128684
$260$		0			0
$261$	1.17922			0.0729922
$262$		0			0
$263$		−	17.5859i		−	1.08440i	−0.840251	−	0.542198i	$-0.817592\pi$
$263$		−	17.5859i		−	1.08440i	0.840251	−	0.542198i	$-0.182408\pi$
$264$		0			0
$265$	19.8909	−	5.88326i	1.22189	−	0.361406i
$266$		0			0
$267$			5.20386i			0.318471i
$268$		0			0
$269$	−13.6430			−0.831831			−0.415915	−	0.909403i	$-0.636539\pi$
$269$	−13.6430			−0.831831			−0.415915	+	0.909403i	$0.636539\pi$
$270$		0			0
$271$	−26.5514			−1.61288			−0.806442	−	0.591314i	$-0.798609\pi$
$271$	−26.5514			−1.61288			−0.806442	+	0.591314i	$0.798609\pi$
$272$		0			0
$273$			0.223912i			0.0135518i
$274$		0			0
$275$	22.9400	−	14.8712i	1.38333	−	0.896767i
$276$		0			0
$277$		−	16.3523i		−	0.982514i	−0.871015	−	0.491257i	$-0.836538\pi$
$277$		−	16.3523i		−	0.982514i	0.871015	−	0.491257i	$-0.163462\pi$
$278$		0			0
$279$	−3.43560			−0.205684
$280$		0			0
$281$	−30.0849			−1.79472			−0.897359	−	0.441302i	$-0.854517\pi$
$281$	−30.0849			−1.79472			−0.897359	+	0.441302i	$0.854517\pi$
$282$		0			0
$283$		−	6.11059i		−	0.363237i	−0.983369	−	0.181619i	$-0.941866\pi$
$283$		−	6.11059i		−	0.363237i	0.983369	−	0.181619i	$-0.0581336\pi$
$284$		0			0
$285$	7.43560	−	2.19928i	0.440447	−	0.130274i
$286$		0			0
$287$			0.646408i			0.0381563i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−7.64540			−0.448181
$292$		0			0
$293$		−	23.7212i		−	1.38581i	−0.721031	−	0.692903i	$-0.756331\pi$
$293$		−	23.7212i		−	1.38581i	0.721031	−	0.692903i	$-0.243669\pi$
$294$		0			0
$295$	−2.88847	−	9.76573i	−0.168173	−	0.568583i
$296$		0			0
$297$			5.46771i			0.317269i
$298$		0			0
$299$	−2.78634			−0.161138
$300$		0			0
$301$	−1.21511			−0.0700376
$302$		0			0
$303$			8.36374i			0.480484i
$304$		0			0
$305$	0.0409517	+	0.138455i	0.00234489	+	0.00792790i
$306$		0			0
$307$		−	13.9842i		−	0.798123i	−0.916924	−	0.399062i	$-0.869336\pi$
$307$		−	13.9842i		−	0.798123i	0.916924	−	0.399062i	$-0.130664\pi$
$308$		0			0
$309$	−7.19702			−0.409424
$310$		0			0
$311$	−17.3286			−0.982613			−0.491306	−	0.870987i	$-0.663480\pi$
$311$	−17.3286			−0.982613			−0.491306	+	0.870987i	$0.663480\pi$
$312$		0			0
$313$			12.1159i			0.684830i	0.939549	+	0.342415i	$0.111245\pi$
$313$			12.1159i			0.684830i	−0.939549	+	0.342415i	$0.888755\pi$
$314$		0			0
$315$	0.510025	−	0.150853i	0.0287367	−	0.00849963i
$316$		0			0
$317$			16.3901i			0.920560i	0.887774	+	0.460280i	$0.152251\pi$
$317$			16.3901i			0.920560i	−0.887774	+	0.460280i	$0.847749\pi$
$318$		0			0
$319$	6.44766			0.360999
$320$		0			0
$321$	−2.96331			−0.165396
$322$		0			0
$323$		−	3.46771i		−	0.192948i
$324$		0			0
$325$	2.56035	+	3.94954i	0.142023	+	0.219081i
$326$		0			0
$327$		−	0.131115i		−	0.00725069i
$328$		0			0
$329$	2.89515			0.159615
$330$		0			0
$331$	22.7926			1.25279			0.626397	−	0.779504i	$-0.284529\pi$
$331$	22.7926			1.25279			0.626397	+	0.779504i	$0.284529\pi$
$332$		0			0
$333$			0.870677i			0.0477128i
$334$		0			0
$335$	21.5608	−	6.37717i	1.17799	−	0.348422i
$336$		0			0
$337$		−	20.5355i		−	1.11864i	−0.828952	−	0.559320i	$-0.811062\pi$
$337$		−	20.5355i		−	1.11864i	0.828952	−	0.559320i	$-0.188938\pi$
$338$		0			0
$339$	16.4424			0.893030
$340$		0			0
$341$	−18.7849			−1.01726
$342$		0			0
$343$			3.31656i			0.179077i
$344$		0			0
$345$	1.87721	+	6.34671i	0.101065	+	0.341695i
$346$		0			0
$347$		−	10.5140i		−	0.564421i	−0.959352	−	0.282211i	$-0.908932\pi$
$347$		−	10.5140i		−	0.564421i	0.959352	−	0.282211i	$-0.0910677\pi$
$348$		0			0
$349$	−16.4847			−0.882405			−0.441203	−	0.897407i	$-0.645448\pi$
$349$	−16.4847			−0.882405			−0.441203	+	0.897407i	$0.645448\pi$
$350$		0			0
$351$	−0.941367			−0.0502464
$352$		0			0
$353$		−	22.1798i		−	1.18051i	−0.807216	−	0.590257i	$-0.799027\pi$
$353$		−	22.1798i		−	1.18051i	0.807216	−	0.590257i	$-0.200973\pi$
$354$		0			0
$355$	−7.53267	−	25.4674i	−0.399793	−	1.35167i
$356$		0			0
$357$		−	0.237858i		−	0.0125888i
$358$		0			0
$359$	16.6102			0.876654			0.438327	−	0.898816i	$-0.355571\pi$
$359$	16.6102			0.876654			0.438327	+	0.898816i	$0.355571\pi$
$360$		0			0
$361$	−6.97501			−0.367106
$362$		0			0
$363$			18.8958i			0.991773i
$364$		0			0
$365$	−11.6253	+	3.43850i	−0.608498	+	0.179979i
$366$		0			0
$367$		−	23.9411i		−	1.24971i	−0.780740	−	0.624857i	$-0.785157\pi$
$367$		−	23.9411i		−	1.24971i	0.780740	−	0.624857i	$-0.214843\pi$
$368$		0			0
$369$	−2.71762			−0.141474
$370$		0			0
$371$	2.20647			0.114554
$372$		0			0
$373$		−	10.4789i		−	0.542579i	−0.962498	−	0.271290i	$-0.912550\pi$
$373$		−	10.4789i		−	0.542579i	0.962498	−	0.271290i	$-0.0874501\pi$
$374$		0			0
$375$	7.27129	−	8.49284i	0.375488	−	0.438568i
$376$		0			0
$377$			1.11008i			0.0571722i
$378$		0			0
$379$	31.9399			1.64064			0.820320	−	0.571904i	$-0.193795\pi$
$379$	31.9399			1.64064			0.820320	+	0.571904i	$0.193795\pi$
$380$		0			0
$381$	7.84882			0.402107
$382$		0			0
$383$			18.7728i			0.959246i	0.877475	+	0.479623i	$0.159227\pi$
$383$			18.7728i			0.959246i	−0.877475	+	0.479623i	$0.840773\pi$
$384$		0			0
$385$	2.78867	−	0.824822i	0.142124	−	0.0420368i
$386$		0			0
$387$		−	5.10854i		−	0.259681i
$388$		0			0
$389$	16.7608			0.849809			0.424904	−	0.905238i	$-0.360308\pi$
$389$	16.7608			0.849809			0.424904	+	0.905238i	$0.360308\pi$
$390$		0			0
$391$	2.95988			0.149688
$392$		0			0
$393$		−	7.73839i		−	0.390350i
$394$		0			0
$395$	0.157528	+	0.532590i	0.00792607	+	0.0267975i
$396$		0			0
$397$			25.2347i			1.26649i	0.773950	+	0.633246i	$0.218278\pi$
$397$			25.2347i			1.26649i	−0.773950	+	0.633246i	$0.781722\pi$
$398$		0			0
$399$	0.824822			0.0412928
$400$		0			0
$401$	−34.4003			−1.71787			−0.858935	−	0.512085i	$-0.828873\pi$
$401$	−34.4003			−1.71787			−0.858935	+	0.512085i	$0.828873\pi$
$402$		0			0
$403$		−	3.23416i		−	0.161105i
$404$		0			0
$405$	0.634216	+	2.14424i	0.0315144	+	0.106548i
$406$		0			0
$407$			4.76061i			0.235975i
$408$		0			0
$409$	−16.2056			−0.801317			−0.400658	−	0.916228i	$-0.631219\pi$
$409$	−16.2056			−0.801317			−0.400658	+	0.916228i	$0.631219\pi$
$410$		0			0
$411$	7.63928			0.376818
$412$		0			0
$413$		−	1.08330i		−	0.0533058i
$414$		0			0
$415$	17.1036	−	5.05883i	0.839581	−	0.248328i
$416$		0			0
$417$		−	6.45970i		−	0.316333i
$418$		0			0
$419$	−31.0412			−1.51646			−0.758232	−	0.651985i	$-0.773936\pi$
$419$	−31.0412			−1.51646			−0.758232	+	0.651985i	$0.773936\pi$
$420$		0			0
$421$	−20.4277			−0.995583			−0.497792	−	0.867297i	$-0.665856\pi$
$421$	−20.4277			−0.995583			−0.497792	+	0.867297i	$0.665856\pi$
$422$		0			0
$423$			12.1717i			0.591810i
$424$		0			0
$425$	−2.71982	−	4.19554i	−0.131931	−	0.203514i
$426$		0			0
$427$			0.0153586i			0.000743256i
$428$		0			0
$429$	−5.14712			−0.248505
$430$		0			0
$431$	2.38091			0.114685			0.0573423	−	0.998355i	$-0.481737\pi$
$431$	2.38091			0.114685			0.0573423	+	0.998355i	$0.481737\pi$
$432$		0			0
$433$		−	29.2344i		−	1.40492i	−0.711726	−	0.702458i	$-0.752086\pi$
$433$		−	29.2344i		−	1.40492i	0.711726	−	0.702458i	$-0.247914\pi$
$434$		0			0
$435$	2.52854	−	0.747883i	0.121234	−	0.0358583i
$436$		0			0
$437$			10.2640i			0.490994i
$438$		0			0
$439$	30.8338			1.47162			0.735808	−	0.677190i	$-0.236803\pi$
$439$	30.8338			1.47162			0.735808	+	0.677190i	$0.236803\pi$
$440$		0			0
$441$	−6.94342			−0.330639
$442$		0			0
$443$		−	3.07751i		−	0.146217i	−0.997324	−	0.0731084i	$-0.976708\pi$
$443$		−	3.07751i		−	0.146217i	0.997324	−	0.0731084i	$-0.0232919\pi$
$444$		0			0
$445$	3.30037	+	11.1583i	0.156453	+	0.528956i
$446$		0			0
$447$		−	4.91285i		−	0.232370i
$448$		0			0
$449$	−16.7147			−0.788815			−0.394408	−	0.918936i	$-0.629050\pi$
$449$	−16.7147			−0.788815			−0.394408	+	0.918936i	$0.629050\pi$
$450$		0			0
$451$	−14.8592			−0.699691
$452$		0			0
$453$		−	0.0521719i		−	0.00245125i
$454$		0			0
$455$	0.142008	+	0.480121i	0.00665746	+	0.0225084i
$456$		0			0
$457$			1.86106i			0.0870566i	0.999052	+	0.0435283i	$0.0138599\pi$
$457$			1.86106i			0.0870566i	−0.999052	+	0.0435283i	$0.986140\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	−9.99434			−0.465483			−0.232741	−	0.972539i	$-0.574770\pi$
$461$	−9.99434			−0.465483			−0.232741	+	0.972539i	$0.574770\pi$
$462$		0			0
$463$			2.47098i			0.114836i	0.998350	+	0.0574181i	$0.0182868\pi$
$463$			2.47098i			0.114836i	−0.998350	+	0.0574181i	$0.981713\pi$
$464$		0			0
$465$	−7.36676	+	2.17891i	−0.341625	+	0.101045i
$466$		0			0
$467$		−	24.0755i		−	1.11408i	−0.830485	−	0.557040i	$-0.811937\pi$
$467$		−	24.0755i		−	1.11408i	0.830485	−	0.557040i	$-0.188063\pi$
$468$		0			0
$469$	2.39171			0.110439
$470$		0			0
$471$	−12.5482			−0.578190
$472$		0			0
$473$		−	27.9320i		−	1.28431i
$474$		0			0
$475$	14.5489	−	9.43155i	0.667550	−	0.432749i
$476$		0			0
$477$			9.27643i			0.424738i
$478$		0			0
$479$	−4.48365			−0.204863			−0.102432	−	0.994740i	$-0.532662\pi$
$479$	−4.48365			−0.204863			−0.102432	+	0.994740i	$0.532662\pi$
$480$		0			0
$481$	−0.819627			−0.0373718
$482$		0			0
$483$			0.704033i			0.0320346i
$484$		0			0
$485$	−16.3936	+	4.84883i	−0.744394	+	0.220174i
$486$		0			0
$487$		−	3.11573i		−	0.141187i	−0.997505	−	0.0705937i	$-0.977511\pi$
$487$		−	3.11573i		−	0.141187i	0.997505	−	0.0705937i	$-0.0224894\pi$
$488$		0			0
$489$	20.1316			0.910383
$490$		0			0
$491$	−37.6749			−1.70025			−0.850123	−	0.526584i	$-0.823472\pi$
$491$	−37.6749			−1.70025			−0.850123	+	0.526584i	$0.823472\pi$
$492$		0			0
$493$		−	1.17922i		−	0.0531096i
$494$		0			0
$495$	3.46771	+	11.7241i	0.155862	+	0.526958i
$496$		0			0
$497$		−	2.82507i		−	0.126722i
$498$		0			0
$499$	19.6519			0.879742			0.439871	−	0.898061i	$-0.355024\pi$
$499$	19.6519			0.879742			0.439871	+	0.898061i	$0.355024\pi$
$500$		0			0
$501$	−14.8093			−0.661632
$502$		0			0
$503$			29.1386i			1.29923i	0.760264	+	0.649614i	$0.225069\pi$
$503$			29.1386i			1.29923i	−0.760264	+	0.649614i	$0.774931\pi$
$504$		0			0
$505$	5.30442	+	17.9339i	0.236044	+	0.798047i
$506$		0			0
$507$			12.1138i			0.537994i
$508$		0			0
$509$	10.4267			0.462155			0.231078	−	0.972935i	$-0.425775\pi$
$509$	10.4267			0.462155			0.231078	+	0.972935i	$0.425775\pi$
$510$		0			0
$511$	−1.28958			−0.0570479
$512$		0			0
$513$			3.46771i			0.153103i
$514$		0			0
$515$	−15.4321	+	4.56447i	−0.680022	+	0.201134i
$516$		0			0
$517$			66.5515i			2.92693i
$518$		0			0
$519$	3.04012			0.133446
$520$		0			0
$521$	−35.5147			−1.55593			−0.777964	−	0.628309i	$-0.783747\pi$
$521$	−35.5147			−1.55593			−0.777964	+	0.628309i	$0.783747\pi$
$522$		0			0
$523$			27.3395i			1.19548i	0.801692	+	0.597738i	$0.203934\pi$
$523$			27.3395i			1.19548i	−0.801692	+	0.597738i	$0.796066\pi$
$524$		0			0
$525$	0.997943	−	0.646932i	0.0435538	−	0.0282344i
$526$		0			0
$527$			3.43560i			0.149657i
$528$		0			0
$529$	14.2391			0.619091
$530$		0			0
$531$	4.55440			0.197644
$532$		0			0
$533$		−	2.55828i		−	0.110811i
$534$		0			0
$535$	−6.35405	+	1.87938i	−0.274709	+	0.0812525i
$536$		0			0
$537$		−	19.6620i		−	0.848478i
$538$		0			0
$539$	−37.9646			−1.63525
$540$		0			0
$541$	17.8793			0.768690			0.384345	−	0.923189i	$-0.374427\pi$
$541$	17.8793			0.768690			0.384345	+	0.923189i	$0.374427\pi$
$542$		0			0
$543$		−	22.2908i		−	0.956589i
$544$		0			0
$545$	−0.0831554	−	0.281143i	−0.00356199	−	0.0120428i
$546$		0			0
$547$			7.18432i			0.307179i	0.988135	+	0.153590i	$0.0490834\pi$
$547$			7.18432i			0.307179i	−0.988135	+	0.153590i	$0.950917\pi$
$548$		0			0
$549$	−0.0645705			−0.00275580
$550$		0			0
$551$	4.08921			0.174206
$552$		0			0
$553$			0.0590796i			0.00251232i
$554$		0			0
$555$	0.552197	+	1.86694i	0.0234395	+	0.0792473i
$556$		0			0
$557$		−	34.8845i		−	1.47810i	−0.673650	−	0.739051i	$-0.735274\pi$
$557$		−	34.8845i		−	1.47810i	0.673650	−	0.739051i	$-0.264726\pi$
$558$		0			0
$559$	4.80901			0.203399
$560$		0			0
$561$	5.46771			0.230847
$562$		0			0
$563$			9.06376i			0.381992i	0.981591	+	0.190996i	$0.0611717\pi$
$563$			9.06376i			0.381992i	−0.981591	+	0.190996i	$0.938828\pi$
$564$		0			0
$565$	35.2565	−	10.4280i	1.48325	−	0.438711i
$566$		0			0
$567$			0.237858i			0.00998910i
$568$		0			0
$569$	0.974497			0.0408531			0.0204265	−	0.999791i	$-0.493498\pi$
$569$	0.974497			0.0408531			0.0204265	+	0.999791i	$0.493498\pi$
$570$		0			0
$571$	8.29549			0.347155			0.173578	−	0.984820i	$-0.444467\pi$
$571$	8.29549			0.347155			0.173578	+	0.984820i	$0.444467\pi$
$572$		0			0
$573$			21.5548i			0.900465i
$574$		0			0
$575$	8.05036	+	12.4183i	0.335723	+	0.517879i
$576$		0			0
$577$			0.745840i			0.0310497i	0.999879	+	0.0155249i	$0.00494192\pi$
$577$			0.745840i			0.0310497i	−0.999879	+	0.0155249i	$0.995058\pi$
$578$		0			0
$579$	−0.185329			−0.00770201
$580$		0			0
$581$	1.89728			0.0787124
$582$		0			0
$583$			50.7208i			2.10064i
$584$		0			0
$585$	−2.01852	+	0.597030i	−0.0834554	+	0.0246841i
$586$		0			0
$587$			28.9061i			1.19308i	0.802583	+	0.596541i	$0.203459\pi$
$587$			28.9061i			1.19308i	−0.802583	+	0.596541i	$0.796541\pi$
$588$		0			0
$589$	−11.9137			−0.490894
$590$		0			0
$591$	−3.32896			−0.136935
$592$		0			0
$593$		−	42.7973i		−	1.75748i	−0.477305	−	0.878738i	$-0.658386\pi$
$593$		−	42.7973i		−	1.75748i	0.477305	−	0.878738i	$-0.341614\pi$
$594$		0			0
$595$	−0.150853	−	0.510025i	−0.00618439	−	0.0209090i
$596$		0			0
$597$			13.7768i			0.563846i
$598$		0			0
$599$	−23.4692			−0.958926			−0.479463	−	0.877562i	$-0.659169\pi$
$599$	−23.4692			−0.958926			−0.479463	+	0.877562i	$0.659169\pi$
$600$		0			0
$601$	−28.5231			−1.16348			−0.581742	−	0.813374i	$-0.697629\pi$
$601$	−28.5231			−1.16348			−0.581742	+	0.813374i	$0.697629\pi$
$602$		0			0
$603$			10.0552i			0.409479i
$604$		0			0
$605$	11.9840	+	40.5172i	0.487220	+	1.64726i
$606$		0			0
$607$			6.76033i			0.274393i	0.990544	+	0.137197i	$0.0438092\pi$
$607$			6.76033i			0.274393i	−0.990544	+	0.137197i	$0.956191\pi$
$608$		0			0
$609$	0.280488			0.0113660
$610$		0			0
$611$	−11.4581			−0.463544
$612$		0			0
$613$			8.74154i			0.353067i	0.984295	+	0.176534i	$0.0564885\pi$
$613$			8.74154i			0.353067i	−0.984295	+	0.176534i	$0.943512\pi$
$614$		0			0
$615$	−5.82724	+	1.72356i	−0.234977	+	0.0695006i
$616$		0			0
$617$			3.01764i			0.121486i	0.998153	+	0.0607429i	$0.0193470\pi$
$617$			3.01764i			0.121486i	−0.998153	+	0.0607429i	$0.980653\pi$
$618$		0			0
$619$	25.1179			1.00958			0.504788	−	0.863244i	$-0.331571\pi$
$619$	25.1179			1.00958			0.504788	+	0.863244i	$0.331571\pi$
$620$		0			0
$621$	−2.95988			−0.118776
$622$		0			0
$623$			1.23778i			0.0495906i
$624$		0			0
$625$	10.2051	−	22.8223i	0.408205	−	0.912890i
$626$		0			0
$627$			18.9604i			0.757206i
$628$		0			0
$629$	0.870677			0.0347162
$630$		0			0
$631$	−0.585256			−0.0232987			−0.0116493	−	0.999932i	$-0.503708\pi$
$631$	−0.585256			−0.0232987			−0.0116493	+	0.999932i	$0.503708\pi$
$632$		0			0
$633$		−	11.2327i		−	0.446460i
$634$		0			0
$635$	16.8298	−	4.97785i	0.667869	−	0.197540i
$636$		0			0
$637$		−	6.53631i		−	0.258978i
$638$		0			0
$639$	11.8771			0.469852
$640$		0			0
$641$	7.15321			0.282535			0.141267	−	0.989971i	$-0.454882\pi$
$641$	7.15321			0.282535			0.141267	+	0.989971i	$0.454882\pi$
$642$		0			0
$643$			20.4232i			0.805411i	0.915330	+	0.402706i	$0.131930\pi$
$643$			20.4232i			0.805411i	−0.915330	+	0.402706i	$0.868070\pi$
$644$		0			0
$645$	−3.23991	−	10.9539i	−0.127572	−	0.431311i
$646$		0			0
$647$		−	19.1766i		−	0.753910i	−0.926232	−	0.376955i	$-0.876971\pi$
$647$		−	19.1766i		−	0.753910i	0.926232	−	0.376955i	$-0.123029\pi$
$648$		0			0
$649$	24.9021			0.977494
$650$		0			0
$651$	−0.817186			−0.0320280
$652$		0			0
$653$			37.0109i			1.44835i	0.689617	+	0.724174i	$0.257779\pi$
$653$			37.0109i			1.44835i	−0.689617	+	0.724174i	$0.742221\pi$
$654$		0			0
$655$	−4.90781	−	16.5930i	−0.191764	−	0.648341i
$656$		0			0
$657$		−	5.42165i		−	0.211519i
$658$		0			0
$659$	−19.4076			−0.756013			−0.378007	−	0.925803i	$-0.623390\pi$
$659$	−19.4076			−0.756013			−0.378007	+	0.925803i	$0.623390\pi$
$660$		0			0
$661$	−13.1741			−0.512413			−0.256206	−	0.966622i	$-0.582473\pi$
$661$	−13.1741			−0.512413			−0.256206	+	0.966622i	$0.582473\pi$
$662$		0			0
$663$			0.941367i			0.0365597i
$664$		0			0
$665$	1.76862	−	0.523116i	0.0685841	−	0.0202855i
$666$		0			0
$667$			3.49037i			0.135148i
$668$		0			0
$669$	−15.5450			−0.601006
$670$		0			0
$671$	−0.353053			−0.0136295
$672$		0			0
$673$		−	34.7871i		−	1.34094i	−0.741935	−	0.670471i	$-0.766092\pi$
$673$		−	34.7871i		−	1.34094i	0.741935	−	0.670471i	$-0.233908\pi$
$674$		0			0
$675$	2.71982	+	4.19554i	0.104686	+	0.161486i
$676$		0			0
$677$			0.132559i			0.00509465i	0.999997	+	0.00254733i	$0.000810840\pi$
$677$			0.132559i			0.00509465i	−0.999997	+	0.00254733i	$0.999189\pi$
$678$		0			0
$679$	−1.81852			−0.0697884
$680$		0			0
$681$	−11.8480			−0.454016
$682$		0			0
$683$			28.5717i			1.09327i	0.837372	+	0.546633i	$0.184091\pi$
$683$			28.5717i			1.09327i	−0.837372	+	0.546633i	$0.815909\pi$
$684$		0			0
$685$	16.3805	−	4.84495i	0.625865	−	0.185116i
$686$		0			0
$687$		−	1.84802i		−	0.0705065i
$688$		0			0
$689$	−8.73252			−0.332683
$690$		0			0
$691$	29.6645			1.12849			0.564246	−	0.825607i	$-0.309167\pi$
$691$	29.6645			1.12849			0.564246	+	0.825607i	$0.309167\pi$
$692$		0			0
$693$			1.30054i			0.0494034i
$694$		0			0
$695$	−4.09684	−	13.8512i	−0.155402	−	0.525404i
$696$		0			0
$697$			2.71762i			0.102937i
$698$		0			0
$699$	21.5297			0.814326
$700$		0			0
$701$	−51.7973			−1.95636			−0.978179	−	0.207763i	$-0.933382\pi$
$701$	−51.7973			−1.95636			−0.978179	+	0.207763i	$0.933382\pi$
$702$		0			0
$703$			3.01925i			0.113873i
$704$		0			0
$705$	7.71951	+	26.0991i	0.290734	+	0.982951i
$706$		0			0
$707$			1.98938i			0.0748185i
$708$		0			0
$709$	−30.9913			−1.16390			−0.581952	−	0.813223i	$-0.697711\pi$
$709$	−30.9913			−1.16390			−0.581952	+	0.813223i	$0.697711\pi$
$710$		0			0
$711$	−0.248382			−0.00931504
$712$		0			0
$713$		−	10.1690i		−	0.380831i
$714$		0			0
$715$	−11.0367	+	3.26438i	−0.412748	+	0.122081i
$716$		0			0
$717$			23.6721i			0.884050i
$718$		0			0
$719$	−53.3625			−1.99009			−0.995043	−	0.0994473i	$-0.968293\pi$
$719$	−53.3625			−1.99009			−0.995043	+	0.0994473i	$0.968293\pi$
$720$		0			0
$721$	−1.71187			−0.0637534
$722$		0			0
$723$		−	1.95306i		−	0.0726350i
$724$		0			0
$725$	4.94749	−	3.20728i	0.183745	−	0.119116i
$726$		0			0
$727$			25.8597i			0.959085i	0.877519	+	0.479542i	$0.159197\pi$
$727$			25.8597i			0.959085i	−0.877519	+	0.479542i	$0.840803\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−5.10854			−0.188946
$732$		0			0
$733$		−	2.96042i		−	0.109346i	−0.998504	−	0.0546728i	$-0.982588\pi$
$733$		−	2.96042i		−	0.109346i	0.998504	−	0.0546728i	$-0.0174116\pi$
$734$		0			0
$735$	−14.8884	+	4.40363i	−0.549166	+	0.162430i
$736$		0			0
$737$			54.9789i			2.02517i
$738$		0			0
$739$	30.7761			1.13212			0.566059	−	0.824365i	$-0.308467\pi$
$739$	30.7761			1.13212			0.566059	+	0.824365i	$0.308467\pi$
$740$		0			0
$741$	−3.26438			−0.119920
$742$		0			0
$743$		−	14.6599i		−	0.537818i	−0.963166	−	0.268909i	$-0.913337\pi$
$743$		−	14.6599i		−	0.537818i	0.963166	−	0.268909i	$-0.0866631\pi$
$744$		0			0
$745$	−3.11581	−	10.5343i	−0.114154	−	0.385948i
$746$		0			0
$747$			7.97651i			0.291845i
$748$		0			0
$749$	−0.704847			−0.0257545
$750$		0			0
$751$	−31.3676			−1.14462			−0.572310	−	0.820038i	$-0.693952\pi$
$751$	−31.3676			−1.14462			−0.572310	+	0.820038i	$0.693952\pi$
$752$		0			0
$753$		−	6.76133i		−	0.246397i
$754$		0			0
$755$	−0.0330883	−	0.111869i	−0.00120421	−	0.00407134i
$756$		0			0
$757$		−	25.4449i		−	0.924812i	−0.886668	−	0.462406i	$-0.846986\pi$
$757$		−	25.4449i		−	0.924812i	0.886668	−	0.462406i	$-0.153014\pi$
$758$		0			0
$759$	−16.1838			−0.587434
$760$		0			0
$761$	−3.67791			−0.133324			−0.0666620	−	0.997776i	$-0.521235\pi$
$761$	−3.67791			−0.133324			−0.0666620	+	0.997776i	$0.521235\pi$
$762$		0			0
$763$		−	0.0311868i		−	0.00112904i
$764$		0			0
$765$	2.14424	−	0.634216i	0.0775252	−	0.0229301i
$766$		0			0
$767$			4.28736i			0.154808i
$768$		0			0
$769$	−13.8136			−0.498132			−0.249066	−	0.968486i	$-0.580124\pi$
$769$	−13.8136			−0.498132			−0.249066	+	0.968486i	$0.580124\pi$
$770$		0			0
$771$	−0.502790			−0.0181075
$772$		0			0
$773$			45.0463i			1.62020i	0.586290	+	0.810102i	$0.300588\pi$
$773$			45.0463i			1.62020i	−0.586290	+	0.810102i	$0.699412\pi$
$774$		0			0
$775$	−14.4142	+	9.34423i	−0.517773	+	0.335655i
$776$		0			0
$777$			0.207098i			0.00742959i
$778$		0			0
$779$	−9.42392			−0.337647
$780$		0			0
$781$	64.9407			2.32376
$782$		0			0
$783$			1.17922i			0.0421421i
$784$		0			0
$785$	−26.9063	+	7.95826i	−0.960328	+	0.284042i
$786$		0			0
$787$		−	2.28723i		−	0.0815309i	−0.999169	−	0.0407655i	$-0.987020\pi$
$787$		−	2.28723i		−	0.0815309i	0.999169	−	0.0407655i	$-0.0129796\pi$
$788$		0			0
$789$	17.5859			0.626076
$790$		0			0
$791$	3.91096			0.139058
$792$		0			0
$793$		−	0.0607846i		−	0.00215852i
$794$		0			0
$795$	5.88326	+	19.8909i	0.208658	+	0.705457i
$796$		0			0
$797$		−	4.83940i		−	0.171420i	−0.996320	−	0.0857101i	$-0.972684\pi$
$797$		−	4.83940i		−	0.171420i	0.996320	−	0.0857101i	$-0.0273159\pi$
$798$		0			0
$799$	12.1717			0.430605
$800$		0			0
$801$	−5.20386			−0.183869
$802$		0			0
$803$		−	29.6440i		−	1.04611i
$804$		0			0
$805$	0.446509	+	1.50962i	0.0157374	+	0.0532070i
$806$		0			0
$807$		−	13.6430i		−	0.480258i
$808$		0			0
$809$	44.8465			1.57672			0.788360	−	0.615214i	$-0.210930\pi$
$809$	44.8465			1.57672			0.788360	+	0.615214i	$0.210930\pi$
$810$		0			0
$811$	11.0525			0.388105			0.194053	−	0.980991i	$-0.437837\pi$
$811$	11.0525			0.388105			0.194053	+	0.980991i	$0.437837\pi$
$812$		0			0
$813$		−	26.5514i		−	0.931199i
$814$		0			0
$815$	43.1670	−	12.7678i	1.51207	−	0.447236i
$816$		0			0
$817$		−	17.7149i		−	0.619766i
$818$		0			0
$819$	−0.223912			−0.00782411
$820$		0			0
$821$	−47.2122			−1.64772			−0.823859	−	0.566795i	$-0.808183\pi$
$821$	−47.2122			−1.64772			−0.823859	+	0.566795i	$0.808183\pi$
$822$		0			0
$823$		−	29.5297i		−	1.02934i	−0.857388	−	0.514670i	$-0.827914\pi$
$823$		−	29.5297i		−	1.02934i	0.857388	−	0.514670i	$-0.172086\pi$
$824$		0			0
$825$	14.8712	+	22.9400i	0.517749	+	0.798668i
$826$		0			0
$827$			43.4607i			1.51128i	0.654989	+	0.755639i	$0.272673\pi$
$827$			43.4607i			1.51128i	−0.654989	+	0.755639i	$0.727327\pi$
$828$		0			0
$829$	27.3352			0.949390			0.474695	−	0.880150i	$-0.342558\pi$
$829$	27.3352			0.949390			0.474695	+	0.880150i	$0.342558\pi$
$830$		0			0
$831$	16.3523			0.567255
$832$		0			0
$833$			6.94342i			0.240575i
$834$		0			0
$835$	−31.7548	+	9.39231i	−1.09892	+	0.325034i
$836$		0			0
$837$		−	3.43560i		−	0.118752i
$838$		0			0
$839$	−3.99497			−0.137922			−0.0689608	−	0.997619i	$-0.521968\pi$
$839$	−3.99497			−0.137922			−0.0689608	+	0.997619i	$0.521968\pi$
$840$		0			0
$841$	−27.6094			−0.952049
$842$		0			0
$843$		−	30.0849i		−	1.03618i
$844$		0			0
$845$	7.68278	+	25.9750i	0.264296	+	0.893566i
$846$		0			0
$847$			4.49452i			0.154434i
$848$		0			0
$849$	6.11059			0.209715
$850$		0			0
$851$	−2.57710			−0.0883420
$852$		0			0
$853$			50.1057i			1.71559i	0.513995	+	0.857793i	$0.328165\pi$
$853$			50.1057i			1.71559i	−0.513995	+	0.857793i	$0.671835\pi$
$854$		0			0
$855$	2.19928	+	7.43560i	0.0752136	+	0.254292i
$856$		0			0
$857$			39.5138i			1.34977i	0.737925	+	0.674883i	$0.235806\pi$
$857$			39.5138i			1.34977i	−0.737925	+	0.674883i	$0.764194\pi$
$858$		0			0
$859$	−25.3104			−0.863578			−0.431789	−	0.901975i	$-0.642118\pi$
$859$	−25.3104			−0.863578			−0.431789	+	0.901975i	$0.642118\pi$
$860$		0			0
$861$	−0.646408			−0.0220295
$862$		0			0
$863$			20.6077i			0.701495i	0.936470	+	0.350747i	$0.114072\pi$
$863$			20.6077i			0.701495i	−0.936470	+	0.350747i	$0.885928\pi$
$864$		0			0
$865$	6.51874	−	1.92809i	0.221644	−	0.0655570i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	−1.35808			−0.0460696
$870$		0			0
$871$	−9.46563			−0.320731
$872$		0			0
$873$		−	7.64540i		−	0.258758i
$874$		0			0
$875$	1.72954	−	2.02009i	0.0584690	−	0.0682915i
$876$		0			0
$877$			13.8568i			0.467910i	0.972247	+	0.233955i	$0.0751669\pi$
$877$			13.8568i			0.467910i	−0.972247	+	0.233955i	$0.924833\pi$
$878$		0			0
$879$	23.7212			0.800096
$880$		0			0
$881$	37.8447			1.27502			0.637510	−	0.770442i	$-0.279965\pi$
$881$	37.8447			1.27502			0.637510	+	0.770442i	$0.279965\pi$
$882$		0			0
$883$			16.5480i			0.556884i	0.960453	+	0.278442i	$0.0898180\pi$
$883$			16.5480i			0.556884i	−0.960453	+	0.278442i	$0.910182\pi$
$884$		0			0
$885$	9.76573	−	2.88847i	0.328271	−	0.0970950i
$886$		0			0
$887$		−	41.7154i		−	1.40067i	−0.713816	−	0.700333i	$-0.753035\pi$
$887$		−	41.7154i		−	1.40067i	0.713816	−	0.700333i	$-0.246965\pi$
$888$		0			0
$889$	1.86691			0.0626140
$890$		0			0
$891$	−5.46771			−0.183175
$892$		0			0
$893$			42.2080i			1.41244i
$894$		0			0
$895$	−12.4700	−	42.1601i	−0.416825	−	1.40926i
$896$		0			0
$897$		−	2.78634i		−	0.0930331i
$898$		0			0
$899$	−4.05135			−0.135120
$900$		0			0
$901$	9.27643			0.309043
$902$		0			0
$903$		−	1.21511i		−	0.0404362i
$904$		0			0
$905$	−14.1372	−	47.7968i	−0.469935	−	1.58882i
$906$		0			0
$907$		−	13.8959i		−	0.461404i	−0.973024	−	0.230702i	$-0.925898\pi$
$907$		−	13.8959i		−	0.461404i	0.973024	−	0.230702i	$-0.0741023\pi$
$908$		0			0
$909$	−8.36374			−0.277408
$910$		0			0
$911$	21.3295			0.706677			0.353339	−	0.935495i	$-0.385046\pi$
$911$	21.3295			0.706677			0.353339	+	0.935495i	$0.385046\pi$
$912$		0			0
$913$			43.6132i			1.44339i
$914$		0			0
$915$	−0.138455	+	0.0409517i	−0.00457718	+	0.00135382i
$916$		0			0
$917$		−	1.84064i		−	0.0607833i
$918$		0			0
$919$	−19.1983			−0.633294			−0.316647	−	0.948543i	$-0.602557\pi$
$919$	−19.1983			−0.633294			−0.316647	+	0.948543i	$0.602557\pi$
$920$		0			0
$921$	13.9842			0.460797
$922$		0			0
$923$			11.1807i			0.368019i
$924$		0			0
$925$	2.36809	+	3.65296i	0.0778623	+	0.120109i
$926$		0			0
$927$		−	7.19702i		−	0.236381i
$928$		0			0
$929$	19.6052			0.643227			0.321613	−	0.946871i	$-0.395775\pi$
$929$	19.6052			0.643227			0.321613	+	0.946871i	$0.395775\pi$
$930$		0			0
$931$	−24.0778			−0.789117
$932$		0			0
$933$		−	17.3286i		−	0.567312i
$934$		0			0
$935$	11.7241	−	3.46771i	0.383419	−	0.113406i
$936$		0			0
$937$		−	54.7338i		−	1.78808i	−0.447991	−	0.894038i	$-0.647860\pi$
$937$		−	54.7338i		−	1.78808i	0.447991	−	0.894038i	$-0.352140\pi$
$938$		0			0
$939$	−12.1159			−0.395387
$940$		0			0
$941$	55.5419			1.81061			0.905307	−	0.424758i	$-0.139641\pi$
$941$	55.5419			1.81061			0.905307	+	0.424758i	$0.139641\pi$
$942$		0			0
$943$		−	8.04385i		−	0.261944i
$944$		0			0
$945$	0.150853	+	0.510025i	0.00490726	+	0.0165911i
$946$		0			0
$947$		−	50.1195i		−	1.62867i	−0.580399	−	0.814333i	$-0.697103\pi$
$947$		−	50.1195i		−	1.62867i	0.580399	−	0.814333i	$-0.302897\pi$
$948$		0			0
$949$	5.10377			0.165675
$950$		0			0
$951$	−16.3901			−0.531485
$952$		0			0
$953$			41.3154i			1.33834i	0.743111	+	0.669168i	$0.233349\pi$
$953$			41.3154i			1.33834i	−0.743111	+	0.669168i	$0.766651\pi$
$954$		0			0
$955$	13.6704	+	46.2187i	0.442364	+	1.49560i
$956$		0			0
$957$			6.44766i			0.208423i
$958$		0			0
$959$	1.81706			0.0586761
$960$		0			0
$961$	−19.1966			−0.619247
$962$		0			0
$963$		−	2.96331i		−	0.0954912i
$964$		0			0
$965$	−0.397390	+	0.117539i	−0.0127924	+	0.00378370i
$966$		0			0
$967$			53.3333i			1.71508i	0.514415	+	0.857541i	$0.328009\pi$
$967$			53.3333i			1.71508i	−0.514415	+	0.857541i	$0.671991\pi$
$968$		0			0
$969$	3.46771			0.111399
$970$		0			0
$971$	−38.3069			−1.22933			−0.614663	−	0.788790i	$-0.710708\pi$
$971$	−38.3069			−1.22933			−0.614663	+	0.788790i	$0.710708\pi$
$972$		0			0
$973$		−	1.53649i		−	0.0492577i
$974$		0			0
$975$	−3.94954	+	2.56035i	−0.126487	+	0.0819969i
$976$		0			0
$977$		−	46.9301i		−	1.50143i	−0.660629	−	0.750713i	$-0.729710\pi$
$977$		−	46.9301i		−	1.50143i	0.660629	−	0.750713i	$-0.270290\pi$
$978$		0			0
$979$	−28.4532			−0.909368
$980$		0			0
$981$	0.131115			0.00418619
$982$		0			0
$983$		−	12.7640i		−	0.407109i	−0.979064	−	0.203554i	$-0.934751\pi$
$983$		−	12.7640i		−	0.407109i	0.979064	−	0.203554i	$-0.0652494\pi$
$984$		0			0
$985$	−7.13808	+	2.11128i	−0.227438	+	0.0672709i
$986$		0			0
$987$			2.89515i			0.0921536i
$988$		0			0
$989$	15.1207			0.480809
$990$		0			0
$991$	−13.4619			−0.427630			−0.213815	−	0.976874i	$-0.568589\pi$
$991$	−13.4619			−0.427630			−0.213815	+	0.976874i	$0.568589\pi$
$992$		0			0
$993$			22.7926i			0.723301i
$994$		0			0
$995$	8.73745	+	29.5407i	0.276996	+	0.936504i
$996$		0			0
$997$			62.6082i			1.98282i	0.130790	+	0.991410i	$0.458249\pi$
$997$			62.6082i			1.98282i	−0.130790	+	0.991410i	$0.541751\pi$
$998$		0			0
$999$	−0.870677			−0.0275470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.m.t.2449.10		12
4.3	odd	2		2040.2.m.i.409.4	✓	12
5.4	even	2	inner	4080.2.m.t.2449.4		12
20.19	odd	2		2040.2.m.i.409.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.m.i.409.4	✓	12	4.3	odd	2
2040.2.m.i.409.10	yes	12	20.19	odd	2
4080.2.m.t.2449.4		12	5.4	even	2	inner
4080.2.m.t.2449.10		12	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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(0.634216 + 2.14424i) q^{5} +0.237858i q^{7} -1.00000 q^{9} -5.46771 q^{11} -0.941367i q^{13} +(-2.14424 + 0.634216i) q^{15} +1.00000i q^{17} -3.46771 q^{19} -0.237858 q^{21} -2.95988i q^{23} +(-4.19554 + 2.71982i) q^{25} -1.00000i q^{27} -1.17922 q^{29} +3.43560 q^{31} -5.46771i q^{33} +(-0.510025 + 0.150853i) q^{35} -0.870677i q^{37} +0.941367 q^{39} +2.71762 q^{41} +5.10854i q^{43} +(-0.634216 - 2.14424i) q^{45} -12.1717i q^{47} +6.94342 q^{49} -1.00000 q^{51} -9.27643i q^{53} +(-3.46771 - 11.7241i) q^{55} -3.46771i q^{57} -4.55440 q^{59} +0.0645705 q^{61} -0.237858i q^{63} +(2.01852 - 0.597030i) q^{65} -10.0552i q^{67} +2.95988 q^{69} -11.8771 q^{71} +5.42165i q^{73} +(-2.71982 - 4.19554i) q^{75} -1.30054i q^{77} +0.248382 q^{79} +1.00000 q^{81} -7.97651i q^{83} +(-2.14424 + 0.634216i) q^{85} -1.17922i q^{87} +5.20386 q^{89} +0.223912 q^{91} +3.43560i q^{93} +(-2.19928 - 7.43560i) q^{95} +7.64540i q^{97} +5.46771 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} - 4 q^{11} + 20 q^{19} + 4 q^{21} - 4 q^{29} - 16 q^{31} + 12 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{49} - 12 q^{51} + 20 q^{55} - 8 q^{59} + 8 q^{61} - 16 q^{65} - 8 q^{69} - 32 q^{71} + 4 q^{75}+ \cdots + 4 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 - 4 * q^11 + 20 * q^19 + 4 * q^21 - 4 * q^29 - 16 * q^31 + 12 * q^35 + 8 * q^39 - 4 * q^41 + 8 * q^49 - 12 * q^51 + 20 * q^55 - 8 * q^59 + 8 * q^61 - 16 * q^65 - 8 * q^69 - 32 * q^71 + 4 * q^75 + 24 * q^79 + 12 * q^81 + 40 * q^89 - 56 * q^91 + 20 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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