LMFDB - Embedded newform 480.2.d.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(49,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.839056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 8x^{2} + 1 $ x^6 + 6x^4 + 8x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-0.373087i$ of defining polynomial
Character	$\chi$	$=$	480.49
Dual form			480.2.d.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(-1.86081 + 1.23992i) q^{5} +0.746175i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +(-1.86081 + 1.23992i) q^{5} +0.746175i q^{7} +1.00000 q^{9} -5.36068i q^{11} -2.92520 q^{13} +(1.86081 - 1.23992i) q^{15} -2.13466i q^{17} +1.73367i q^{19} -0.746175i q^{21} -7.49534i q^{23} +(1.92520 - 4.61450i) q^{25} -1.00000 q^{27} -6.74916i q^{29} -2.64681 q^{31} +5.36068i q^{33} +(-0.925197 - 1.38849i) q^{35} -1.07480 q^{37} +2.92520 q^{39} -11.2936 q^{41} -7.44322 q^{43} +(-1.86081 + 1.23992i) q^{45} +1.73367i q^{47} +6.44322 q^{49} +2.13466i q^{51} +7.72161 q^{53} +(6.64681 + 9.97518i) q^{55} -1.73367i q^{57} +6.85302i q^{59} -6.45203i q^{61} +0.746175i q^{63} +(5.44322 - 3.62701i) q^{65} +7.44322 q^{67} +7.49534i q^{69} -13.2936 q^{71} -0.690358i q^{73} +(-1.92520 + 4.61450i) q^{75} +4.00000 q^{77} +2.64681 q^{79} +1.00000 q^{81} +5.85039 q^{83} +(2.64681 + 3.97219i) q^{85} +6.74916i q^{87} -7.59283 q^{89} -2.18271i q^{91} +2.64681 q^{93} +(-2.14961 - 3.22601i) q^{95} +14.1887i q^{97} -5.36068i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} - 8 q^{13} + 2 q^{25} - 6 q^{27} + 16 q^{31} + 4 q^{35} - 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} + 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} - 2 q^{75} + 24 q^{77} - 16 q^{79} + 6 q^{81} + 16 q^{83} - 16 q^{85} - 20 q^{89} - 16 q^{93} - 32 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 8 * q^13 + 2 * q^25 - 6 * q^27 + 16 * q^31 + 4 * q^35 - 16 * q^37 + 8 * q^39 - 4 * q^41 - 6 * q^49 + 24 * q^53 + 8 * q^55 - 12 * q^65 - 16 * q^71 - 2 * q^75 + 24 * q^77 - 16 * q^79 + 6 * q^81 + 16 * q^83 - 16 * q^85 - 20 * q^89 - 16 * q^93 - 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−1.86081	+	1.23992i	−0.832178	+	0.554509i
$5$	−1.86081	+	1.23992i	−0.832178	+	0.554509i
$6$		0			0
$7$			0.746175i			0.282028i	0.990008	+	0.141014i	$0.0450362\pi$
$7$			0.746175i			0.282028i	−0.990008	+	0.141014i	$0.954964\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.36068i		−	1.61630i	−0.588974	−	0.808152i	$-0.700468\pi$
$11$		−	5.36068i		−	1.61630i	0.588974	−	0.808152i	$-0.299532\pi$
$12$		0			0
$13$	−2.92520			−0.811304			−0.405652	−	0.914028i	$-0.632955\pi$
$13$	−2.92520			−0.811304			−0.405652	+	0.914028i	$0.632955\pi$
$14$		0			0
$15$	1.86081	−	1.23992i	0.480458	−	0.320146i
$16$		0			0
$17$		−	2.13466i		−	0.517731i	−0.965913	−	0.258866i	$-0.916651\pi$
$17$		−	2.13466i		−	0.517731i	0.965913	−	0.258866i	$-0.0833487\pi$
$18$		0			0
$19$			1.73367i			0.397730i	0.980027	+	0.198865i	$0.0637255\pi$
$19$			1.73367i			0.397730i	−0.980027	+	0.198865i	$0.936274\pi$
$20$		0			0
$21$		−	0.746175i		−	0.162829i
$22$		0			0
$23$		−	7.49534i		−	1.56289i	−0.623977	−	0.781443i	$-0.714484\pi$
$23$		−	7.49534i		−	1.56289i	0.623977	−	0.781443i	$-0.285516\pi$
$24$		0			0
$25$	1.92520	−	4.61450i	0.385039	−	0.922900i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	6.74916i		−	1.25329i	−0.779306	−	0.626644i	$-0.784428\pi$
$29$		−	6.74916i		−	1.25329i	0.779306	−	0.626644i	$-0.215572\pi$
$30$		0			0
$31$	−2.64681			−0.475381			−0.237690	−	0.971341i	$-0.576390\pi$
$31$	−2.64681			−0.475381			−0.237690	+	0.971341i	$0.576390\pi$
$32$		0			0
$33$			5.36068i			0.933174i
$34$		0			0
$35$	−0.925197	−	1.38849i	−0.156387	−	0.234697i
$36$		0			0
$37$	−1.07480			−0.176697			−0.0883483	−	0.996090i	$-0.528159\pi$
$37$	−1.07480			−0.176697			−0.0883483	+	0.996090i	$0.528159\pi$
$38$		0			0
$39$	2.92520			0.468406
$40$		0			0
$41$	−11.2936			−1.76377			−0.881883	−	0.471468i	$-0.843724\pi$
$41$	−11.2936			−1.76377			−0.881883	+	0.471468i	$0.843724\pi$
$42$		0			0
$43$	−7.44322			−1.13508			−0.567540	−	0.823346i	$-0.692105\pi$
$43$	−7.44322			−1.13508			−0.567540	+	0.823346i	$0.692105\pi$
$44$		0			0
$45$	−1.86081	+	1.23992i	−0.277393	+	0.184836i
$46$		0			0
$47$			1.73367i			0.252881i	0.991974	+	0.126441i	$0.0403553\pi$
$47$			1.73367i			0.252881i	−0.991974	+	0.126441i	$0.959645\pi$
$48$		0			0
$49$	6.44322			0.920460
$50$		0			0
$51$			2.13466i			0.298912i
$52$		0			0
$53$	7.72161			1.06064			0.530322	−	0.847796i	$-0.322071\pi$
$53$	7.72161			1.06064			0.530322	+	0.847796i	$0.322071\pi$
$54$		0			0
$55$	6.64681	+	9.97518i	0.896255	+	1.34505i
$56$		0			0
$57$		−	1.73367i		−	0.229630i
$58$		0			0
$59$			6.85302i			0.892188i	0.894986	+	0.446094i	$0.147185\pi$
$59$			6.85302i			0.892188i	−0.894986	+	0.446094i	$0.852815\pi$
$60$		0			0
$61$		−	6.45203i		−	0.826098i	−0.910709	−	0.413049i	$-0.864464\pi$
$61$		−	6.45203i		−	0.826098i	0.910709	−	0.413049i	$-0.135536\pi$
$62$		0			0
$63$			0.746175i			0.0940092i
$64$		0			0
$65$	5.44322	−	3.62701i	0.675149	−	0.449875i
$66$		0			0
$67$	7.44322			0.909334			0.454667	−	0.890661i	$-0.349758\pi$
$67$	7.44322			0.909334			0.454667	+	0.890661i	$0.349758\pi$
$68$		0			0
$69$			7.49534i			0.902332i
$70$		0			0
$71$	−13.2936			−1.57766			−0.788831	−	0.614610i	$-0.789313\pi$
$71$	−13.2936			−1.57766			−0.788831	+	0.614610i	$0.789313\pi$
$72$		0			0
$73$		−	0.690358i		−	0.0808003i	−0.999184	−	0.0404002i	$-0.987137\pi$
$73$		−	0.690358i		−	0.0808003i	0.999184	−	0.0404002i	$-0.0128633\pi$
$74$		0			0
$75$	−1.92520	+	4.61450i	−0.222303	+	0.532837i
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	2.64681			0.297789			0.148895	−	0.988853i	$-0.452428\pi$
$79$	2.64681			0.297789			0.148895	+	0.988853i	$0.452428\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	5.85039			0.642164			0.321082	−	0.947051i	$-0.395953\pi$
$83$	5.85039			0.642164			0.321082	+	0.947051i	$0.395953\pi$
$84$		0			0
$85$	2.64681	+	3.97219i	0.287087	+	0.430844i
$86$		0			0
$87$			6.74916i			0.723586i
$88$		0			0
$89$	−7.59283			−0.804838			−0.402419	−	0.915456i	$-0.631831\pi$
$89$	−7.59283			−0.804838			−0.402419	+	0.915456i	$0.631831\pi$
$90$		0			0
$91$		−	2.18271i		−	0.228810i
$92$		0			0
$93$	2.64681			0.274461
$94$		0			0
$95$	−2.14961	−	3.22601i	−0.220545	−	0.330982i
$96$		0			0
$97$			14.1887i			1.44064i	0.693641	+	0.720321i	$0.256006\pi$
$97$			14.1887i			1.44064i	−0.693641	+	0.720321i	$0.743994\pi$
$98$		0			0
$99$		−	5.36068i		−	0.538768i
$100$		0			0
$101$		−	7.43952i		−	0.740260i	−0.928980	−	0.370130i	$-0.879313\pi$
$101$		−	7.43952i		−	0.740260i	0.928980	−	0.370130i	$-0.120687\pi$
$102$		0			0
$103$		−	7.19820i		−	0.709260i	−0.935007	−	0.354630i	$-0.884607\pi$
$103$		−	7.19820i		−	0.709260i	0.935007	−	0.354630i	$-0.115393\pi$
$104$		0			0
$105$	0.925197	+	1.38849i	0.0902900	+	0.135502i
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$			19.9504i			1.91090i	0.295158	+	0.955449i	$0.404628\pi$
$109$			19.9504i			1.91090i	−0.295158	+	0.955449i	$0.595372\pi$
$110$		0			0
$111$	1.07480			0.102016
$112$		0			0
$113$			12.0540i			1.13395i	0.823736	+	0.566973i	$0.191886\pi$
$113$			12.0540i			1.13395i	−0.823736	+	0.566973i	$0.808114\pi$
$114$		0			0
$115$	9.29362	+	13.9474i	0.866634	+	1.30060i
$116$		0			0
$117$	−2.92520			−0.270435
$118$		0			0
$119$	1.59283			0.146014
$120$		0			0
$121$	−17.7368			−1.61244
$122$		0			0
$123$	11.2936			1.01831
$124$		0			0
$125$	2.13919	+	10.9738i	0.191335	+	0.981525i
$126$		0			0
$127$		−	4.21351i		−	0.373888i	−0.982371	−	0.186944i	$-0.940142\pi$
$127$		−	4.21351i		−	0.373888i	0.982371	−	0.186944i	$-0.0598583\pi$
$128$		0			0
$129$	7.44322			0.655339
$130$		0			0
$131$		−	10.3204i		−	0.901694i	−0.892601	−	0.450847i	$-0.851122\pi$
$131$		−	10.3204i		−	0.901694i	0.892601	−	0.450847i	$-0.148878\pi$
$132$		0			0
$133$	−1.29362			−0.112171
$134$		0			0
$135$	1.86081	−	1.23992i	0.160153	−	0.106715i
$136$		0			0
$137$		−	15.0387i		−	1.28484i	−0.766351	−	0.642422i	$-0.777930\pi$
$137$		−	15.0387i		−	1.28484i	0.766351	−	0.642422i	$-0.222070\pi$
$138$		0			0
$139$		−	9.47032i		−	0.803262i	−0.915802	−	0.401631i	$-0.868443\pi$
$139$		−	9.47032i		−	0.803262i	0.915802	−	0.401631i	$-0.131557\pi$
$140$		0			0
$141$		−	1.73367i		−	0.146001i
$142$		0			0
$143$			15.6810i			1.31131i
$144$		0			0
$145$	8.36842	+	12.5589i	0.694959	+	1.04296i
$146$		0			0
$147$	−6.44322			−0.531428
$148$		0			0
$149$		−	1.78948i		−	0.146600i	−0.997310	−	0.0733000i	$-0.976647\pi$
$149$		−	1.78948i		−	0.146600i	0.997310	−	0.0733000i	$-0.0233531\pi$
$150$		0			0
$151$	−10.6468			−0.866425			−0.433212	−	0.901292i	$-0.642620\pi$
$151$	−10.6468			−0.866425			−0.433212	+	0.901292i	$0.642620\pi$
$152$		0			0
$153$		−	2.13466i		−	0.172577i
$154$		0			0
$155$	4.92520	−	3.28183i	0.395601	−	0.263603i
$156$		0			0
$157$	6.92520			0.552691			0.276345	−	0.961058i	$-0.410877\pi$
$157$	6.92520			0.552691			0.276345	+	0.961058i	$0.410877\pi$
$158$		0			0
$159$	−7.72161			−0.612364
$160$		0			0
$161$	5.59283			0.440777
$162$		0			0
$163$	−7.70079			−0.603172			−0.301586	−	0.953439i	$-0.597516\pi$
$163$	−7.70079			−0.603172			−0.301586	+	0.953439i	$0.597516\pi$
$164$		0			0
$165$	−6.64681	−	9.97518i	−0.517453	−	0.776566i
$166$		0			0
$167$		−	3.22601i		−	0.249637i	−0.992180	−	0.124818i	$-0.960165\pi$
$167$		−	3.22601i		−	0.249637i	0.992180	−	0.124818i	$-0.0398348\pi$
$168$		0			0
$169$	−4.44322			−0.341786
$170$		0			0
$171$			1.73367i			0.132577i
$172$		0			0
$173$	−6.42799			−0.488711			−0.244356	−	0.969686i	$-0.578576\pi$
$173$	−6.42799			−0.488711			−0.244356	+	0.969686i	$0.578576\pi$
$174$		0			0
$175$	3.44322	+	1.43653i	0.260283	+	0.108592i
$176$		0			0
$177$		−	6.85302i		−	0.515105i
$178$		0			0
$179$			8.13765i			0.608236i	0.952634	+	0.304118i	$0.0983618\pi$
$179$			8.13765i			0.608236i	−0.952634	+	0.304118i	$0.901638\pi$
$180$		0			0
$181$		−	1.49235i		−	0.110925i	−0.998461	−	0.0554627i	$-0.982337\pi$
$181$		−	1.49235i		−	0.110925i	0.998461	−	0.0554627i	$-0.0176634\pi$
$182$		0			0
$183$			6.45203i			0.476948i
$184$		0			0
$185$	2.00000	−	1.33267i	0.147043	−	0.0979798i
$186$		0			0
$187$	−11.4432			−0.836811
$188$		0			0
$189$		−	0.746175i		−	0.0542762i
$190$		0			0
$191$	−6.88645			−0.498286			−0.249143	−	0.968467i	$-0.580149\pi$
$191$	−6.88645			−0.498286			−0.249143	+	0.968467i	$0.580149\pi$
$192$		0			0
$193$		−	16.4830i		−	1.18647i	−0.805028	−	0.593237i	$-0.797850\pi$
$193$		−	16.4830i		−	1.18647i	0.805028	−	0.593237i	$-0.202150\pi$
$194$		0			0
$195$	−5.44322	+	3.62701i	−0.389797	+	0.259736i
$196$		0			0
$197$	−13.5720			−0.966965			−0.483483	−	0.875354i	$-0.660628\pi$
$197$	−13.5720			−0.966965			−0.483483	+	0.875354i	$0.660628\pi$
$198$		0			0
$199$	9.05398			0.641820			0.320910	−	0.947110i	$-0.396011\pi$
$199$	9.05398			0.641820			0.320910	+	0.947110i	$0.396011\pi$
$200$		0			0
$201$	−7.44322			−0.525005
$202$		0			0
$203$	5.03605			0.353462
$204$		0			0
$205$	21.0152	−	14.0032i	1.46777	−	0.978025i
$206$		0			0
$207$		−	7.49534i		−	0.520962i
$208$		0			0
$209$	9.29362			0.642853
$210$		0			0
$211$		−	2.53566i		−	0.174562i	−0.996184	−	0.0872809i	$-0.972182\pi$
$211$		−	2.53566i		−	0.174562i	0.996184	−	0.0872809i	$-0.0278178\pi$
$212$		0			0
$213$	13.2936			0.910864
$214$		0			0
$215$	13.8504	−	9.22900i	0.944589	−	0.629413i
$216$		0			0
$217$		−	1.97498i		−	0.134070i
$218$		0			0
$219$			0.690358i			0.0466501i
$220$		0			0
$221$			6.24430i			0.420037i
$222$		0			0
$223$		−	12.1579i		−	0.814152i	−0.913394	−	0.407076i	$-0.866548\pi$
$223$		−	12.1579i		−	0.814152i	0.913394	−	0.407076i	$-0.133452\pi$
$224$		0			0
$225$	1.92520	−	4.61450i	0.128346	−	0.307633i
$226$		0			0
$227$	−20.7368			−1.37635			−0.688176	−	0.725544i	$-0.741588\pi$
$227$	−20.7368			−1.37635			−0.688176	+	0.725544i	$0.741588\pi$
$228$		0			0
$229$		−	19.9504i		−	1.31836i	−0.751987	−	0.659178i	$-0.770904\pi$
$229$		−	19.9504i		−	1.31836i	0.751987	−	0.659178i	$-0.229096\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$		−	13.3386i		−	0.873844i	−0.899499	−	0.436922i	$-0.856069\pi$
$233$		−	13.3386i		−	0.873844i	0.899499	−	0.436922i	$-0.143931\pi$
$234$		0			0
$235$	−2.14961	−	3.22601i	−0.140225	−	0.210442i
$236$		0			0
$237$	−2.64681			−0.171929
$238$		0			0
$239$	22.8864			1.48040			0.740201	−	0.672386i	$-0.234731\pi$
$239$	22.8864			1.48040			0.740201	+	0.672386i	$0.234731\pi$
$240$		0			0
$241$	3.59283			0.231435			0.115717	−	0.993282i	$-0.463083\pi$
$241$	3.59283			0.231435			0.115717	+	0.993282i	$0.463083\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−11.9896	+	7.98908i	−0.765987	+	0.510404i
$246$		0			0
$247$		−	5.07131i		−	0.322680i
$248$		0			0
$249$	−5.85039			−0.370754
$250$		0			0
$251$			8.82801i			0.557219i	0.960404	+	0.278609i	$0.0898735\pi$
$251$			8.82801i			0.557219i	−0.960404	+	0.278609i	$0.910127\pi$
$252$		0			0
$253$	−40.1801			−2.52610
$254$		0			0
$255$	−2.64681	−	3.97219i	−0.165750	−	0.248748i
$256$		0			0
$257$			22.2927i			1.39058i	0.718728	+	0.695291i	$0.244725\pi$
$257$			22.2927i			1.39058i	−0.718728	+	0.695291i	$0.755275\pi$
$258$		0			0
$259$		−	0.801991i		−	0.0498333i
$260$		0			0
$261$		−	6.74916i		−	0.417763i
$262$		0			0
$263$			21.2014i			1.30733i	0.756783	+	0.653667i	$0.226770\pi$
$263$			21.2014i			1.30733i	−0.756783	+	0.653667i	$0.773230\pi$
$264$		0			0
$265$	−14.3684	+	9.57418i	−0.882645	+	0.588137i
$266$		0			0
$267$	7.59283			0.464674
$268$		0			0
$269$		−	14.6935i		−	0.895881i	−0.894063	−	0.447940i	$-0.852158\pi$
$269$		−	14.6935i		−	0.895881i	0.894063	−	0.447940i	$-0.147842\pi$
$270$		0			0
$271$	20.2396			1.22947			0.614735	−	0.788734i	$-0.289263\pi$
$271$	20.2396			1.22947			0.614735	+	0.788734i	$0.289263\pi$
$272$		0			0
$273$			2.18271i			0.132103i
$274$		0			0
$275$	−24.7368	−	10.3204i	−1.49169	−	0.622341i
$276$		0			0
$277$	0.518027			0.0311252			0.0155626	−	0.999879i	$-0.495046\pi$
$277$	0.518027			0.0311252			0.0155626	+	0.999879i	$0.495046\pi$
$278$		0			0
$279$	−2.64681			−0.158460
$280$		0			0
$281$	13.7008			0.817320			0.408660	−	0.912687i	$-0.365996\pi$
$281$	13.7008			0.817320			0.408660	+	0.912687i	$0.365996\pi$
$282$		0			0
$283$	−18.0305			−1.07180			−0.535900	−	0.844282i	$-0.680027\pi$
$283$	−18.0305			−1.07180			−0.535900	+	0.844282i	$0.680027\pi$
$284$		0			0
$285$	2.14961	+	3.22601i	0.127332	+	0.191093i
$286$		0			0
$287$		−	8.42701i		−	0.497431i
$288$		0			0
$289$	12.4432			0.731954
$290$		0			0
$291$		−	14.1887i		−	0.831755i
$292$		0			0
$293$	15.9792			0.933513			0.466757	−	0.884386i	$-0.345422\pi$
$293$	15.9792			0.933513			0.466757	+	0.884386i	$0.345422\pi$
$294$		0			0
$295$	−8.49720	−	12.7521i	−0.494726	−	0.742459i
$296$		0			0
$297$			5.36068i			0.311058i
$298$		0			0
$299$			21.9253i			1.26797i
$300$		0			0
$301$		−	5.55394i		−	0.320124i
$302$		0			0
$303$			7.43952i			0.427389i
$304$		0			0
$305$	8.00000	+	12.0060i	0.458079	+	0.687460i
$306$		0			0
$307$	22.5872			1.28912			0.644561	−	0.764553i	$-0.277040\pi$
$307$	22.5872			1.28912			0.644561	+	0.764553i	$0.277040\pi$
$308$		0			0
$309$			7.19820i			0.409492i
$310$		0			0
$311$	18.5872			1.05399			0.526993	−	0.849870i	$-0.323320\pi$
$311$	18.5872			1.05399			0.526993	+	0.849870i	$0.323320\pi$
$312$		0			0
$313$		−	29.3871i		−	1.66106i	−0.556977	−	0.830528i	$-0.688039\pi$
$313$		−	29.3871i		−	1.66106i	0.556977	−	0.830528i	$-0.311961\pi$
$314$		0			0
$315$	−0.925197	−	1.38849i	−0.0521289	−	0.0782323i
$316$		0			0
$317$	−5.57201			−0.312955			−0.156478	−	0.987682i	$-0.550014\pi$
$317$	−5.57201			−0.312955			−0.156478	+	0.987682i	$0.550014\pi$
$318$		0			0
$319$	−36.1801			−2.02569
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	3.70079			0.205917
$324$		0			0
$325$	−5.63158	+	13.4983i	−0.312384	+	0.748752i
$326$		0			0
$327$		−	19.9504i		−	1.10326i
$328$		0			0
$329$	−1.29362			−0.0713194
$330$		0			0
$331$		−	13.7396i		−	0.755199i	−0.925969	−	0.377599i	$-0.876750\pi$
$331$		−	13.7396i		−	0.755199i	0.925969	−	0.377599i	$-0.123250\pi$
$332$		0			0
$333$	−1.07480			−0.0588989
$334$		0			0
$335$	−13.8504	+	9.22900i	−0.756728	+	0.504234i
$336$		0			0
$337$		−	20.7523i		−	1.13045i	−0.824936	−	0.565226i	$-0.808789\pi$
$337$		−	20.7523i		−	1.13045i	0.824936	−	0.565226i	$-0.191211\pi$
$338$		0			0
$339$		−	12.0540i		−	0.654685i
$340$		0			0
$341$			14.1887i			0.768360i
$342$		0			0
$343$			10.0310i			0.541623i
$344$		0			0
$345$	−9.29362	−	13.9474i	−0.500352	−	0.750901i
$346$		0			0
$347$	4.73684			0.254287			0.127143	−	0.991884i	$-0.459419\pi$
$347$	4.73684			0.254287			0.127143	+	0.991884i	$0.459419\pi$
$348$		0			0
$349$		−	0.482632i		−	0.0258347i	−0.999917	−	0.0129174i	$-0.995888\pi$
$349$		−	0.482632i		−	0.0258347i	0.999917	−	0.0129174i	$-0.00411184\pi$
$350$		0			0
$351$	2.92520			0.156135
$352$		0			0
$353$		−	2.13466i		−	0.113617i	−0.998385	−	0.0568083i	$-0.981908\pi$
$353$		−	2.13466i		−	0.113617i	0.998385	−	0.0568083i	$-0.0180924\pi$
$354$		0			0
$355$	24.7368	−	16.4830i	1.31290	−	0.874828i
$356$		0			0
$357$	−1.59283			−0.0843015
$358$		0			0
$359$	−9.59283			−0.506290			−0.253145	−	0.967428i	$-0.581465\pi$
$359$	−9.59283			−0.506290			−0.253145	+	0.967428i	$0.581465\pi$
$360$		0			0
$361$	15.9944			0.841811
$362$		0			0
$363$	17.7368			0.930943
$364$		0			0
$365$	0.855989	+	1.28462i	0.0448045	+	0.0672402i
$366$		0			0
$367$			34.0832i			1.77913i	0.456809	+	0.889565i	$0.348992\pi$
$367$			34.0832i			1.77913i	−0.456809	+	0.889565i	$0.651008\pi$
$368$		0			0
$369$	−11.2936			−0.587922
$370$		0			0
$371$			5.76167i			0.299131i
$372$		0			0
$373$	4.33796			0.224611			0.112306	−	0.993674i	$-0.464176\pi$
$373$	4.33796			0.224611			0.112306	+	0.993674i	$0.464176\pi$
$374$		0			0
$375$	−2.13919	−	10.9738i	−0.110468	−	0.566684i
$376$		0			0
$377$			19.7426i			1.01680i
$378$		0			0
$379$		−	6.90107i		−	0.354484i	−0.984167	−	0.177242i	$-0.943282\pi$
$379$		−	6.90107i		−	0.354484i	0.984167	−	0.177242i	$-0.0567176\pi$
$380$		0			0
$381$			4.21351i			0.215864i
$382$		0			0
$383$		−	22.3744i		−	1.14328i	−0.820506	−	0.571639i	$-0.806308\pi$
$383$		−	22.3744i		−	1.14328i	0.820506	−	0.571639i	$-0.193692\pi$
$384$		0			0
$385$	−7.44322	+	4.95968i	−0.379342	+	0.252769i
$386$		0			0
$387$	−7.44322			−0.378360
$388$		0			0
$389$			11.0185i			0.558659i	0.960195	+	0.279330i	$0.0901122\pi$
$389$			11.0185i			0.558659i	−0.960195	+	0.279330i	$0.909888\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$			10.3204i			0.520593i
$394$		0			0
$395$	−4.92520	+	3.28183i	−0.247814	+	0.165127i
$396$		0			0
$397$	25.2549			1.26751			0.633753	−	0.773536i	$-0.281514\pi$
$397$	25.2549			1.26751			0.633753	+	0.773536i	$0.281514\pi$
$398$		0			0
$399$	1.29362			0.0647619
$400$		0			0
$401$	7.29362			0.364226			0.182113	−	0.983278i	$-0.441706\pi$
$401$	7.29362			0.364226			0.182113	+	0.983278i	$0.441706\pi$
$402$		0			0
$403$	7.74244			0.385678
$404$		0			0
$405$	−1.86081	+	1.23992i	−0.0924642	+	0.0616121i
$406$		0			0
$407$			5.76167i			0.285595i
$408$		0			0
$409$	−15.8504			−0.783752			−0.391876	−	0.920018i	$-0.628174\pi$
$409$	−15.8504			−0.783752			−0.391876	+	0.920018i	$0.628174\pi$
$410$		0			0
$411$			15.0387i			0.741805i
$412$		0			0
$413$	−5.11355			−0.251622
$414$		0			0
$415$	−10.8864	+	7.25402i	−0.534395	+	0.356086i
$416$		0			0
$417$			9.47032i			0.463763i
$418$		0			0
$419$		−	8.02602i		−	0.392097i	−0.980594	−	0.196048i	$-0.937189\pi$
$419$		−	8.02602i		−	0.392097i	0.980594	−	0.196048i	$-0.0628109\pi$
$420$		0			0
$421$			22.9351i			1.11779i	0.829240	+	0.558893i	$0.188774\pi$
$421$			22.9351i			1.11779i	−0.829240	+	0.558893i	$0.811226\pi$
$422$		0			0
$423$			1.73367i			0.0842937i
$424$		0			0
$425$	−9.85039	−	4.10964i	−0.477814	−	0.199347i
$426$		0			0
$427$	4.81434			0.232982
$428$		0			0
$429$		−	15.6810i		−	0.757087i
$430$		0			0
$431$	35.0665			1.68909			0.844547	−	0.535481i	$-0.179870\pi$
$431$	35.0665			1.68909			0.844547	+	0.535481i	$0.179870\pi$
$432$		0			0
$433$			17.0773i			0.820682i	0.911932	+	0.410341i	$0.134590\pi$
$433$			17.0773i			0.820682i	−0.911932	+	0.410341i	$0.865410\pi$
$434$		0			0
$435$	−8.36842	−	12.5589i	−0.401235	−	0.602152i
$436$		0			0
$437$	12.9944			0.621607
$438$		0			0
$439$	8.53885			0.407537			0.203769	−	0.979019i	$-0.434681\pi$
$439$	8.53885			0.407537			0.203769	+	0.979019i	$0.434681\pi$
$440$		0			0
$441$	6.44322			0.306820
$442$		0			0
$443$	20.7368			0.985237			0.492619	−	0.870245i	$-0.336040\pi$
$443$	20.7368			0.985237			0.492619	+	0.870245i	$0.336040\pi$
$444$		0			0
$445$	14.1288	−	9.41450i	0.669769	−	0.446290i
$446$		0			0
$447$			1.78948i			0.0846396i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$			60.5414i			2.85078i
$452$		0			0
$453$	10.6468			0.500231
$454$		0			0
$455$	2.70638	+	4.06160i	0.126877	+	0.190411i
$456$		0			0
$457$		−	1.28462i		−	0.0600921i	−0.999549	−	0.0300461i	$-0.990435\pi$
$457$		−	1.28462i		−	0.0600921i	0.999549	−	0.0300461i	$-0.00956540\pi$
$458$		0			0
$459$			2.13466i			0.0996374i
$460$		0			0
$461$			15.7033i			0.731374i	0.930738	+	0.365687i	$0.119166\pi$
$461$			15.7033i			0.731374i	−0.930738	+	0.365687i	$0.880834\pi$
$462$		0			0
$463$		−	18.7215i		−	0.870064i	−0.900415	−	0.435032i	$-0.856737\pi$
$463$		−	18.7215i		−	0.870064i	0.900415	−	0.435032i	$-0.143263\pi$
$464$		0			0
$465$	−4.92520	+	3.28183i	−0.228401	+	0.152191i
$466$		0			0
$467$	2.14961			0.0994719			0.0497360	−	0.998762i	$-0.484162\pi$
$467$	2.14961			0.0994719			0.0497360	+	0.998762i	$0.484162\pi$
$468$		0			0
$469$			5.55394i			0.256457i
$470$		0			0
$471$	−6.92520			−0.319096
$472$		0			0
$473$			39.9007i			1.83464i
$474$		0			0
$475$	8.00000	+	3.33765i	0.367065	+	0.153142i
$476$		0			0
$477$	7.72161			0.353548
$478$		0			0
$479$	−12.1801			−0.556521			−0.278261	−	0.960506i	$-0.589758\pi$
$479$	−12.1801			−0.556521			−0.278261	+	0.960506i	$0.589758\pi$
$480$		0			0
$481$	3.14401			0.143355
$482$		0			0
$483$	−5.59283			−0.254483
$484$		0			0
$485$	−17.5928	−	26.4024i	−0.798849	−	1.19887i
$486$		0			0
$487$			25.7678i			1.16765i	0.811879	+	0.583826i	$0.198445\pi$
$487$			25.7678i			1.16765i	−0.811879	+	0.583826i	$0.801555\pi$
$488$		0			0
$489$	7.70079			0.348242
$490$		0			0
$491$			16.7724i			0.756927i	0.925616	+	0.378464i	$0.123547\pi$
$491$			16.7724i			0.756927i	−0.925616	+	0.378464i	$0.876453\pi$
$492$		0			0
$493$	−14.4072			−0.648866
$494$		0			0
$495$	6.64681	+	9.97518i	0.298752	+	0.448351i
$496$		0			0
$497$		−	9.91936i		−	0.444944i
$498$		0			0
$499$			17.6224i			0.788888i	0.918920	+	0.394444i	$0.129063\pi$
$499$			17.6224i			0.788888i	−0.918920	+	0.394444i	$0.870937\pi$
$500$		0			0
$501$			3.22601i			0.144128i
$502$		0			0
$503$			27.1263i			1.20950i	0.796414	+	0.604752i	$0.206728\pi$
$503$			27.1263i			1.20950i	−0.796414	+	0.604752i	$0.793272\pi$
$504$		0			0
$505$	9.22441	+	13.8435i	0.410481	+	0.616028i
$506$		0			0
$507$	4.44322			0.197330
$508$		0			0
$509$			15.9782i			0.708220i	0.935204	+	0.354110i	$0.115216\pi$
$509$			15.9782i			0.708220i	−0.935204	+	0.354110i	$0.884784\pi$
$510$		0			0
$511$	0.515128			0.0227879
$512$		0			0
$513$		−	1.73367i		−	0.0765432i
$514$		0			0
$515$	8.92520	+	13.3945i	0.393291	+	0.590230i
$516$		0			0
$517$	9.29362			0.408733
$518$		0			0
$519$	6.42799			0.282158
$520$		0			0
$521$	0.886447			0.0388359			0.0194180	−	0.999811i	$-0.493819\pi$
$521$	0.886447			0.0388359			0.0194180	+	0.999811i	$0.493819\pi$
$522$		0			0
$523$	−41.7729			−1.82660			−0.913301	−	0.407286i	$-0.866475\pi$
$523$	−41.7729			−1.82660			−0.913301	+	0.407286i	$0.866475\pi$
$524$		0			0
$525$	−3.44322	−	1.43653i	−0.150275	−	0.0626954i
$526$		0			0
$527$			5.65004i			0.246120i
$528$		0			0
$529$	−33.1801			−1.44261
$530$		0			0
$531$			6.85302i			0.297396i
$532$		0			0
$533$	33.0361			1.43095
$534$		0			0
$535$	−7.44322	+	4.95968i	−0.321799	+	0.214426i
$536$		0			0
$537$		−	8.13765i		−	0.351165i
$538$		0			0
$539$		−	34.5400i		−	1.48774i
$540$		0			0
$541$		−	4.47705i		−	0.192483i	−0.995358	−	0.0962417i	$-0.969318\pi$
$541$		−	4.47705i		−	0.192483i	0.995358	−	0.0962417i	$-0.0306822\pi$
$542$		0			0
$543$			1.49235i			0.0640428i
$544$		0			0
$545$	−24.7368	−	37.1237i	−1.05961	−	1.59021i
$546$		0			0
$547$	14.3297			0.612692			0.306346	−	0.951920i	$-0.400893\pi$
$547$	14.3297			0.612692			0.306346	+	0.951920i	$0.400893\pi$
$548$		0			0
$549$		−	6.45203i		−	0.275366i
$550$		0			0
$551$	11.7008			0.498470
$552$		0			0
$553$			1.97498i			0.0839848i
$554$		0			0
$555$	−2.00000	+	1.33267i	−0.0848953	+	0.0565687i
$556$		0			0
$557$	2.68556			0.113791			0.0568954	−	0.998380i	$-0.481880\pi$
$557$	2.68556			0.113791			0.0568954	+	0.998380i	$0.481880\pi$
$558$		0			0
$559$	21.7729			0.920895
$560$		0			0
$561$	11.4432			0.483133
$562$		0			0
$563$	−20.7368			−0.873954			−0.436977	−	0.899473i	$-0.643951\pi$
$563$	−20.7368			−0.873954			−0.436977	+	0.899473i	$0.643951\pi$
$564$		0			0
$565$	−14.9460	−	22.4302i	−0.628784	−	0.943645i
$566$		0			0
$567$			0.746175i			0.0313364i
$568$		0			0
$569$	−4.40717			−0.184758			−0.0923791	−	0.995724i	$-0.529447\pi$
$569$	−4.40717			−0.184758			−0.0923791	+	0.995724i	$0.529447\pi$
$570$		0			0
$571$		−	23.6590i		−	0.990098i	−0.868865	−	0.495049i	$-0.835150\pi$
$571$		−	23.6590i		−	0.990098i	0.868865	−	0.495049i	$-0.164850\pi$
$572$		0			0
$573$	6.88645			0.287685
$574$		0			0
$575$	−34.5872	−	14.4300i	−1.44239	−	0.601772i
$576$		0			0
$577$			6.56366i			0.273249i	0.990623	+	0.136624i	$0.0436253\pi$
$577$			6.56366i			0.273249i	−0.990623	+	0.136624i	$0.956375\pi$
$578$		0			0
$579$			16.4830i			0.685011i
$580$		0			0
$581$			4.36542i			0.181108i
$582$		0			0
$583$		−	41.3931i		−	1.71433i
$584$		0			0
$585$	5.44322	−	3.62701i	0.225050	−	0.149958i
$586$		0			0
$587$	16.2992			0.672741			0.336370	−	0.941730i	$-0.390801\pi$
$587$	16.2992			0.672741			0.336370	+	0.941730i	$0.390801\pi$
$588$		0			0
$589$		−	4.58868i		−	0.189073i
$590$		0			0
$591$	13.5720			0.558278
$592$		0			0
$593$		−	16.3233i		−	0.670319i	−0.942161	−	0.335160i	$-0.891210\pi$
$593$		−	16.3233i		−	0.670319i	0.942161	−	0.335160i	$-0.108790\pi$
$594$		0			0
$595$	−2.96395	+	1.97498i	−0.121510	+	0.0809663i
$596$		0			0
$597$	−9.05398			−0.370555
$598$		0			0
$599$	−25.5928			−1.04569			−0.522847	−	0.852426i	$-0.675130\pi$
$599$	−25.5928			−1.04569			−0.522847	+	0.852426i	$0.675130\pi$
$600$		0			0
$601$	29.9225			1.22056			0.610282	−	0.792184i	$-0.291056\pi$
$601$	29.9225			1.22056			0.610282	+	0.792184i	$0.291056\pi$
$602$		0			0
$603$	7.44322			0.303111
$604$		0			0
$605$	33.0048	−	21.9923i	1.34184	−	0.894113i
$606$		0			0
$607$		−	20.6965i		−	0.840046i	−0.907513	−	0.420023i	$-0.862022\pi$
$607$		−	20.6965i		−	0.840046i	0.907513	−	0.420023i	$-0.137978\pi$
$608$		0			0
$609$	−5.03605			−0.204071
$610$		0			0
$611$		−	5.07131i		−	0.205163i
$612$		0			0
$613$	22.6676			0.915537			0.457769	−	0.889071i	$-0.348649\pi$
$613$	22.6676			0.915537			0.457769	+	0.889071i	$0.348649\pi$
$614$		0			0
$615$	−21.0152	+	14.0032i	−0.847416	+	0.564663i
$616$		0			0
$617$		−	22.1966i		−	0.893603i	−0.894633	−	0.446802i	$-0.852563\pi$
$617$		−	22.1966i		−	0.893603i	0.894633	−	0.446802i	$-0.147437\pi$
$618$		0			0
$619$		−	16.8204i		−	0.676070i	−0.941133	−	0.338035i	$-0.890238\pi$
$619$		−	16.8204i		−	0.676070i	0.941133	−	0.338035i	$-0.109762\pi$
$620$		0			0
$621$			7.49534i			0.300777i
$622$		0			0
$623$		−	5.66558i		−	0.226987i
$624$		0			0
$625$	−17.5872	−	17.7676i	−0.703489	−	0.710706i
$626$		0			0
$627$	−9.29362			−0.371151
$628$		0			0
$629$			2.29434i			0.0914813i
$630$		0			0
$631$	44.1205			1.75641			0.878204	−	0.478285i	$-0.158742\pi$
$631$	44.1205			1.75641			0.878204	+	0.478285i	$0.158742\pi$
$632$		0			0
$633$			2.53566i			0.100783i
$634$		0			0
$635$	5.22441	+	7.84052i	0.207324	+	0.311141i
$636$		0			0
$637$	−18.8477			−0.746773
$638$		0			0
$639$	−13.2936			−0.525887
$640$		0			0
$641$	−1.18566			−0.0468307			−0.0234154	−	0.999726i	$-0.507454\pi$
$641$	−1.18566			−0.0468307			−0.0234154	+	0.999726i	$0.507454\pi$
$642$		0			0
$643$	22.5872			0.890754			0.445377	−	0.895343i	$-0.353070\pi$
$643$	22.5872			0.890754			0.445377	+	0.895343i	$0.353070\pi$
$644$		0			0
$645$	−13.8504	+	9.22900i	−0.545359	+	0.363392i
$646$		0			0
$647$		−	19.7090i		−	0.774842i	−0.921903	−	0.387421i	$-0.873366\pi$
$647$		−	19.7090i		−	0.774842i	0.921903	−	0.387421i	$-0.126634\pi$
$648$		0			0
$649$	36.7368			1.44205
$650$		0			0
$651$			1.97498i			0.0774056i
$652$		0			0
$653$	−44.4585			−1.73979			−0.869897	−	0.493234i	$-0.835815\pi$
$653$	−44.4585			−1.73979			−0.869897	+	0.493234i	$0.835815\pi$
$654$		0			0
$655$	12.7964	+	19.2042i	0.499997	+	0.750369i
$656$		0			0
$657$		−	0.690358i		−	0.0269334i
$658$		0			0
$659$		−	41.5863i		−	1.61997i	−0.586448	−	0.809987i	$-0.699474\pi$
$659$		−	41.5863i		−	1.61997i	0.586448	−	0.809987i	$-0.300526\pi$
$660$		0			0
$661$		−	12.0060i		−	0.466978i	−0.972359	−	0.233489i	$-0.924986\pi$
$661$		−	12.0060i		−	0.466978i	0.972359	−	0.233489i	$-0.0750143\pi$
$662$		0			0
$663$		−	6.24430i		−	0.242509i
$664$		0			0
$665$	2.40717	−	1.60398i	0.0933461	−	0.0621997i
$666$		0			0
$667$	−50.5872			−1.95875
$668$		0			0
$669$			12.1579i			0.470051i
$670$		0			0
$671$	−34.5872			−1.33523
$672$		0			0
$673$			14.5080i			0.559244i	0.960110	+	0.279622i	$0.0902091\pi$
$673$			14.5080i			0.559244i	−0.960110	+	0.279622i	$0.909791\pi$
$674$		0			0
$675$	−1.92520	+	4.61450i	−0.0741009	+	0.177612i
$676$		0			0
$677$	−43.8600			−1.68568			−0.842839	−	0.538166i	$-0.819117\pi$
$677$	−43.8600			−1.68568			−0.842839	+	0.538166i	$0.819117\pi$
$678$		0			0
$679$	−10.5872			−0.406301
$680$		0			0
$681$	20.7368			0.794637
$682$		0			0
$683$	−5.33527			−0.204148			−0.102074	−	0.994777i	$-0.532548\pi$
$683$	−5.33527			−0.204148			−0.102074	+	0.994777i	$0.532548\pi$
$684$		0			0
$685$	18.6468	+	27.9841i	0.712458	+	1.06922i
$686$		0			0
$687$			19.9504i			0.761153i
$688$		0			0
$689$	−22.5872			−0.860505
$690$		0			0
$691$		−	39.7710i		−	1.51296i	−0.654016	−	0.756480i	$-0.726917\pi$
$691$		−	39.7710i		−	1.51296i	0.654016	−	0.756480i	$-0.273083\pi$
$692$		0			0
$693$	4.00000			0.151947
$694$		0			0
$695$	11.7424	+	17.6224i	0.445416	+	0.668457i
$696$		0			0
$697$			24.1080i			0.913157i
$698$		0			0
$699$			13.3386i			0.504514i
$700$		0			0
$701$		−	27.5015i		−	1.03872i	−0.854556	−	0.519359i	$-0.826171\pi$
$701$		−	27.5015i		−	1.03872i	0.854556	−	0.519359i	$-0.173829\pi$
$702$		0			0
$703$		−	1.86335i		−	0.0702775i
$704$		0			0
$705$	2.14961	+	3.22601i	0.0809589	+	0.121499i
$706$		0			0
$707$	5.55118			0.208774
$708$		0			0
$709$		−	0.111632i		−	0.00419244i	−0.999998	−	0.00209622i	$-0.999333\pi$
$709$		−	0.111632i		−	0.00419244i	0.999998	−	0.00209622i	$-0.000667249\pi$
$710$		0			0
$711$	2.64681			0.0992631
$712$		0			0
$713$			19.8387i			0.742966i
$714$		0			0
$715$	−19.4432	−	29.1794i	−0.727135	−	1.09125i
$716$		0			0
$717$	−22.8864			−0.854710
$718$		0			0
$719$	10.7064			0.399281			0.199640	−	0.979869i	$-0.436023\pi$
$719$	10.7064			0.399281			0.199640	+	0.979869i	$0.436023\pi$
$720$		0			0
$721$	5.37112			0.200031
$722$		0			0
$723$	−3.59283			−0.133619
$724$		0			0
$725$	−31.1440	−	12.9935i	−1.15666	−	0.482565i
$726$		0			0
$727$		−	25.6562i		−	0.951536i	−0.879571	−	0.475768i	$-0.842170\pi$
$727$		−	25.6562i		−	0.951536i	0.879571	−	0.475768i	$-0.157830\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			15.8888i			0.587667i
$732$		0			0
$733$	−30.3684			−1.12168			−0.560842	−	0.827923i	$-0.689522\pi$
$733$	−30.3684			−1.12168			−0.560842	+	0.827923i	$0.689522\pi$
$734$		0			0
$735$	11.9896	−	7.98908i	0.442243	−	0.294682i
$736$		0			0
$737$		−	39.9007i		−	1.46976i
$738$		0			0
$739$			20.1917i			0.742763i	0.928480	+	0.371381i	$0.121116\pi$
$739$			20.1917i			0.742763i	−0.928480	+	0.371381i	$0.878884\pi$
$740$		0			0
$741$			5.07131i			0.186299i
$742$		0			0
$743$		−	46.3863i		−	1.70175i	−0.525369	−	0.850875i	$-0.676073\pi$
$743$		−	46.3863i		−	1.70175i	0.525369	−	0.850875i	$-0.323927\pi$
$744$		0			0
$745$	2.21881	+	3.32988i	0.0812911	+	0.121997i
$746$		0			0
$747$	5.85039			0.214055
$748$		0			0
$749$			2.98470i			0.109059i
$750$		0			0
$751$	27.1261			0.989845			0.494922	−	0.868937i	$-0.335196\pi$
$751$	27.1261			0.989845			0.494922	+	0.868937i	$0.335196\pi$
$752$		0			0
$753$		−	8.82801i		−	0.321710i
$754$		0			0
$755$	19.8116	−	13.2012i	0.721020	−	0.480441i
$756$		0			0
$757$	−45.2549			−1.64482			−0.822408	−	0.568898i	$-0.807370\pi$
$757$	−45.2549			−1.64482			−0.822408	+	0.568898i	$0.807370\pi$
$758$		0			0
$759$	40.1801			1.45844
$760$		0			0
$761$	16.8864			0.612133			0.306067	−	0.952010i	$-0.400987\pi$
$761$	16.8864			0.612133			0.306067	+	0.952010i	$0.400987\pi$
$762$		0			0
$763$	−14.8864			−0.538926
$764$		0			0
$765$	2.64681	+	3.97219i	0.0956956	+	0.143615i
$766$		0			0
$767$		−	20.0464i		−	0.723835i
$768$		0			0
$769$	−16.3297			−0.588863			−0.294431	−	0.955673i	$-0.595130\pi$
$769$	−16.3297			−0.588863			−0.294431	+	0.955673i	$0.595130\pi$
$770$		0			0
$771$		−	22.2927i		−	0.802853i
$772$		0			0
$773$	41.3144			1.48598			0.742989	−	0.669304i	$-0.233408\pi$
$773$	41.3144			1.48598			0.742989	+	0.669304i	$0.233408\pi$
$774$		0			0
$775$	−5.09563	+	12.2137i	−0.183040	+	0.438729i
$776$		0			0
$777$			0.801991i			0.0287713i
$778$		0			0
$779$		−	19.5794i		−	0.701503i
$780$		0			0
$781$			71.2628i			2.54998i
$782$		0			0
$783$			6.74916i			0.241195i
$784$		0			0
$785$	−12.8864	+	8.58669i	−0.459937	+	0.306472i
$786$		0			0
$787$	11.4849			0.409391			0.204696	−	0.978826i	$-0.434380\pi$
$787$	11.4849			0.409391			0.204696	+	0.978826i	$0.434380\pi$
$788$		0			0
$789$		−	21.2014i		−	0.754789i
$790$		0			0
$791$	−8.99440			−0.319804
$792$		0			0
$793$			18.8735i			0.670216i
$794$		0			0
$795$	14.3684	−	9.57418i	0.509595	−	0.339561i
$796$		0			0
$797$	45.4945			1.61150			0.805749	−	0.592257i	$-0.201763\pi$
$797$	45.4945			1.61150			0.805749	+	0.592257i	$0.201763\pi$
$798$		0			0
$799$	3.70079			0.130924
$800$		0			0
$801$	−7.59283			−0.268279
$802$		0			0
$803$	−3.70079			−0.130598
$804$		0			0
$805$	−10.4072	+	6.93466i	−0.366805	+	0.244415i
$806$		0			0
$807$			14.6935i			0.517237i
$808$		0			0
$809$	36.0721			1.26823			0.634114	−	0.773240i	$-0.281365\pi$
$809$	36.0721			1.26823			0.634114	+	0.773240i	$0.281365\pi$
$810$		0			0
$811$			44.5230i			1.56341i	0.623646	+	0.781707i	$0.285651\pi$
$811$			44.5230i			1.56341i	−0.623646	+	0.781707i	$0.714349\pi$
$812$		0			0
$813$	−20.2396			−0.709835
$814$		0			0
$815$	14.3297	−	9.54836i	0.501946	−	0.334464i
$816$		0			0
$817$		−	12.9041i		−	0.451456i
$818$		0			0
$819$		−	2.18271i		−	0.0762700i
$820$		0			0
$821$		−	34.1613i		−	1.19224i	−0.802897	−	0.596118i	$-0.796709\pi$
$821$		−	34.1613i		−	1.19224i	0.802897	−	0.596118i	$-0.203291\pi$
$822$		0			0
$823$			2.12689i			0.0741388i	0.999313	+	0.0370694i	$0.0118023\pi$
$823$			2.12689i			0.0741388i	−0.999313	+	0.0370694i	$0.988198\pi$
$824$		0			0
$825$	24.7368	+	10.3204i	0.861226	+	0.359309i
$826$		0			0
$827$	−38.5872			−1.34181			−0.670905	−	0.741543i	$-0.734094\pi$
$827$	−38.5872			−1.34181			−0.670905	+	0.741543i	$0.734094\pi$
$828$		0			0
$829$		−	34.2351i		−	1.18904i	−0.804083	−	0.594518i	$-0.797343\pi$
$829$		−	34.2351i		−	1.18904i	0.804083	−	0.594518i	$-0.202657\pi$
$830$		0			0
$831$	−0.518027			−0.0179701
$832$		0			0
$833$		−	13.7541i		−	0.476551i
$834$		0			0
$835$	4.00000	+	6.00299i	0.138426	+	0.207742i
$836$		0			0
$837$	2.64681			0.0914871
$838$		0			0
$839$	−41.5928			−1.43594			−0.717972	−	0.696072i	$-0.754929\pi$
$839$	−41.5928			−1.43594			−0.717972	+	0.696072i	$0.754929\pi$
$840$		0			0
$841$	−16.5512			−0.570730
$842$		0			0
$843$	−13.7008			−0.471880
$844$		0			0
$845$	8.26798	−	5.50924i	0.284427	−	0.189524i
$846$		0			0
$847$		−	13.2348i		−	0.454752i
$848$		0			0
$849$	18.0305			0.618804
$850$		0			0
$851$			8.05601i			0.276156i
$852$		0			0
$853$	23.1828			0.793763			0.396881	−	0.917870i	$-0.370092\pi$
$853$	23.1828			0.793763			0.396881	+	0.917870i	$0.370092\pi$
$854$		0			0
$855$	−2.14961	−	3.22601i	−0.0735150	−	0.110327i
$856$		0			0
$857$		−	9.38868i		−	0.320711i	−0.987059	−	0.160356i	$-0.948736\pi$
$857$		−	9.38868i		−	0.320711i	0.987059	−	0.160356i	$-0.0512641\pi$
$858$		0			0
$859$			8.98769i			0.306656i	0.988175	+	0.153328i	$0.0489991\pi$
$859$			8.98769i			0.306656i	−0.988175	+	0.153328i	$0.951001\pi$
$860$		0			0
$861$			8.42701i			0.287192i
$862$		0			0
$863$			12.2473i			0.416903i	0.978033	+	0.208451i	$0.0668423\pi$
$863$			12.2473i			0.416903i	−0.978033	+	0.208451i	$0.933158\pi$
$864$		0			0
$865$	11.9612	−	7.97020i	0.406695	−	0.270995i
$866$		0			0
$867$	−12.4432			−0.422594
$868$		0			0
$869$		−	14.1887i		−	0.481318i
$870$		0			0
$871$	−21.7729			−0.737746
$872$		0			0
$873$			14.1887i			0.480214i
$874$		0			0
$875$	−8.18836	+	1.59621i	−0.276817	+	0.0539618i
$876$		0			0
$877$	26.1109			0.881701			0.440850	−	0.897581i	$-0.354677\pi$
$877$	26.1109			0.881701			0.440850	+	0.897581i	$0.354677\pi$
$878$		0			0
$879$	−15.9792			−0.538964
$880$		0			0
$881$	−38.4793			−1.29640			−0.648200	−	0.761470i	$-0.724478\pi$
$881$	−38.4793			−1.29640			−0.648200	+	0.761470i	$0.724478\pi$
$882$		0			0
$883$	6.58723			0.221678			0.110839	−	0.993838i	$-0.464646\pi$
$883$	6.58723			0.221678			0.110839	+	0.993838i	$0.464646\pi$
$884$		0			0
$885$	8.49720	+	12.7521i	0.285630	+	0.428659i
$886$		0			0
$887$			50.9595i			1.71105i	0.517760	+	0.855526i	$0.326766\pi$
$887$			50.9595i			1.71105i	−0.517760	+	0.855526i	$0.673234\pi$
$888$		0			0
$889$	3.14401			0.105447
$890$		0			0
$891$		−	5.36068i		−	0.179589i
$892$		0			0
$893$	−3.00560			−0.100578
$894$		0			0
$895$	−10.0900	−	15.1426i	−0.337273	−	0.506161i
$896$		0			0
$897$		−	21.9253i		−	0.732066i
$898$		0			0
$899$			17.8637i			0.595789i
$900$		0			0
$901$		−	16.4830i		−	0.549129i
$902$		0			0
$903$			5.55394i			0.184824i
$904$		0			0
$905$	1.85039	+	2.77697i	0.0615092	+	0.0923097i
$906$		0			0
$907$	24.5568			0.815394			0.407697	−	0.913117i	$-0.366332\pi$
$907$	24.5568			0.815394			0.407697	+	0.913117i	$0.366332\pi$
$908$		0			0
$909$		−	7.43952i		−	0.246753i
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		−	31.3621i		−	1.03793i
$914$		0			0
$915$	−8.00000	−	12.0060i	−0.264472	−	0.396905i
$916$		0			0
$917$	7.70079			0.254302
$918$		0			0
$919$	28.7548			0.948532			0.474266	−	0.880382i	$-0.342713\pi$
$919$	28.7548			0.948532			0.474266	+	0.880382i	$0.342713\pi$
$920$		0			0
$921$	−22.5872			−0.744275
$922$		0			0
$923$	38.8864			1.27996
$924$		0			0
$925$	−2.06921	+	4.95968i	−0.0680351	+	0.163073i
$926$		0			0
$927$		−	7.19820i		−	0.236420i
$928$		0			0
$929$	−28.2880			−0.928100			−0.464050	−	0.885809i	$-0.653604\pi$
$929$	−28.2880			−0.928100			−0.464050	+	0.885809i	$0.653604\pi$
$930$		0			0
$931$			11.1704i			0.366095i
$932$		0			0
$933$	−18.5872			−0.608519
$934$		0			0
$935$	21.2936	−	14.1887i	0.696376	−	0.464020i
$936$		0			0
$937$			33.9313i			1.10849i	0.832354	+	0.554244i	$0.186992\pi$
$937$			33.9313i			1.10849i	−0.832354	+	0.554244i	$0.813008\pi$
$938$		0			0
$939$			29.3871i			0.959011i
$940$		0			0
$941$		−	38.8016i		−	1.26490i	−0.774603	−	0.632448i	$-0.782050\pi$
$941$		−	38.8016i		−	1.26490i	0.774603	−	0.632448i	$-0.217950\pi$
$942$		0			0
$943$			84.6495i			2.75657i
$944$		0			0
$945$	0.925197	+	1.38849i	0.0300967	+	0.0451675i
$946$		0			0
$947$	−17.7729			−0.577541			−0.288771	−	0.957398i	$-0.593247\pi$
$947$	−17.7729			−0.577541			−0.288771	+	0.957398i	$0.593247\pi$
$948$		0			0
$949$			2.01943i			0.0655536i
$950$		0			0
$951$	5.57201			0.180685
$952$		0			0
$953$		−	46.3047i		−	1.49996i	−0.661463	−	0.749978i	$-0.730064\pi$
$953$		−	46.3047i		−	1.49996i	0.661463	−	0.749978i	$-0.269936\pi$
$954$		0			0
$955$	12.8143	−	8.53864i	0.414662	−	0.276304i
$956$		0			0
$957$	36.1801			1.16954
$958$		0			0
$959$	11.2215			0.362361
$960$		0			0
$961$	−23.9944			−0.774013
$962$		0			0
$963$	4.00000			0.128898
$964$		0			0
$965$	20.4376	+	30.6717i	0.657911	+	0.987357i
$966$		0			0
$967$			9.28482i			0.298580i	0.988793	+	0.149290i	$0.0476987\pi$
$967$			9.28482i			0.298580i	−0.988793	+	0.149290i	$0.952301\pi$
$968$		0			0
$969$	−3.70079			−0.118886
$970$		0			0
$971$		−	20.9301i		−	0.671678i	−0.941919	−	0.335839i	$-0.890980\pi$
$971$		−	20.9301i		−	0.671678i	0.941919	−	0.335839i	$-0.109020\pi$
$972$		0			0
$973$	7.06651			0.226542
$974$		0			0
$975$	5.63158	−	13.4983i	0.180355	−	0.432292i
$976$		0			0
$977$			30.8314i			0.986383i	0.869921	+	0.493192i	$0.164170\pi$
$977$			30.8314i			0.986383i	−0.869921	+	0.493192i	$0.835830\pi$
$978$		0			0
$979$			40.7027i			1.30086i
$980$		0			0
$981$			19.9504i			0.636966i
$982$		0			0
$983$		−	31.3285i		−	0.999223i	−0.866250	−	0.499612i	$-0.833476\pi$
$983$		−	31.3285i		−	0.999223i	0.866250	−	0.499612i	$-0.166524\pi$
$984$		0			0
$985$	25.2549	−	16.8282i	0.804687	−	0.536191i
$986$		0			0
$987$	1.29362			0.0411763
$988$		0			0
$989$			55.7895i			1.77400i
$990$		0			0
$991$	−31.3420			−0.995611			−0.497806	−	0.867289i	$-0.665861\pi$
$991$	−31.3420			−0.995611			−0.497806	+	0.867289i	$0.665861\pi$
$992$		0			0
$993$			13.7396i			0.436014i
$994$		0			0
$995$	−16.8477	+	11.2262i	−0.534108	+	0.355895i
$996$		0			0
$997$	20.9557			0.663672			0.331836	−	0.943337i	$-0.392332\pi$
$997$	20.9557			0.663672			0.331836	+	0.943337i	$0.392332\pi$
$998$		0			0
$999$	1.07480			0.0340053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.d.a.49.2		6
3.2	odd	2		1440.2.d.e.1009.5		6
4.3	odd	2		120.2.d.a.109.5	✓	6
5.2	odd	4		2400.2.k.f.1201.9		12
5.3	odd	4		2400.2.k.f.1201.4		12
5.4	even	2		480.2.d.b.49.6		6
8.3	odd	2		120.2.d.b.109.1	yes	6
8.5	even	2		480.2.d.b.49.5		6
12.11	even	2		360.2.d.f.109.2		6
15.2	even	4		7200.2.k.u.3601.6		12
15.8	even	4		7200.2.k.u.3601.8		12
15.14	odd	2		1440.2.d.f.1009.1		6
16.3	odd	4		3840.2.f.l.769.11		12
16.5	even	4		3840.2.f.m.769.8		12
16.11	odd	4		3840.2.f.l.769.2		12
16.13	even	4		3840.2.f.m.769.5		12
20.3	even	4		600.2.k.f.301.3		12
20.7	even	4		600.2.k.f.301.10		12
20.19	odd	2		120.2.d.b.109.2	yes	6
24.5	odd	2		1440.2.d.f.1009.2		6
24.11	even	2		360.2.d.e.109.6		6
40.3	even	4		600.2.k.f.301.4		12
40.13	odd	4		2400.2.k.f.1201.10		12
40.19	odd	2		120.2.d.a.109.6	yes	6
40.27	even	4		600.2.k.f.301.9		12
40.29	even	2	inner	480.2.d.a.49.1		6
40.37	odd	4		2400.2.k.f.1201.3		12
60.23	odd	4		1800.2.k.u.901.10		12
60.47	odd	4		1800.2.k.u.901.3		12
60.59	even	2		360.2.d.e.109.5		6
80.19	odd	4		3840.2.f.l.769.5		12
80.29	even	4		3840.2.f.m.769.11		12
80.59	odd	4		3840.2.f.l.769.8		12
80.69	even	4		3840.2.f.m.769.2		12
120.29	odd	2		1440.2.d.e.1009.6		6
120.53	even	4		7200.2.k.u.3601.7		12
120.59	even	2		360.2.d.f.109.1		6
120.77	even	4		7200.2.k.u.3601.5		12
120.83	odd	4		1800.2.k.u.901.9		12
120.107	odd	4		1800.2.k.u.901.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.5	✓	6	4.3	odd	2
120.2.d.a.109.6	yes	6	40.19	odd	2
120.2.d.b.109.1	yes	6	8.3	odd	2
120.2.d.b.109.2	yes	6	20.19	odd	2
360.2.d.e.109.5		6	60.59	even	2
360.2.d.e.109.6		6	24.11	even	2
360.2.d.f.109.1		6	120.59	even	2
360.2.d.f.109.2		6	12.11	even	2
480.2.d.a.49.1		6	40.29	even	2	inner
480.2.d.a.49.2		6	1.1	even	1	trivial
480.2.d.b.49.5		6	8.5	even	2
480.2.d.b.49.6		6	5.4	even	2
600.2.k.f.301.3		12	20.3	even	4
600.2.k.f.301.4		12	40.3	even	4
600.2.k.f.301.9		12	40.27	even	4
600.2.k.f.301.10		12	20.7	even	4
1440.2.d.e.1009.5		6	3.2	odd	2
1440.2.d.e.1009.6		6	120.29	odd	2
1440.2.d.f.1009.1		6	15.14	odd	2
1440.2.d.f.1009.2		6	24.5	odd	2
1800.2.k.u.901.3		12	60.47	odd	4
1800.2.k.u.901.4		12	120.107	odd	4
1800.2.k.u.901.9		12	120.83	odd	4
1800.2.k.u.901.10		12	60.23	odd	4
2400.2.k.f.1201.3		12	40.37	odd	4
2400.2.k.f.1201.4		12	5.3	odd	4
2400.2.k.f.1201.9		12	5.2	odd	4
2400.2.k.f.1201.10		12	40.13	odd	4
3840.2.f.l.769.2		12	16.11	odd	4
3840.2.f.l.769.5		12	80.19	odd	4
3840.2.f.l.769.8		12	80.59	odd	4
3840.2.f.l.769.11		12	16.3	odd	4
3840.2.f.m.769.2		12	80.69	even	4
3840.2.f.m.769.5		12	16.13	even	4
3840.2.f.m.769.8		12	16.5	even	4
3840.2.f.m.769.11		12	80.29	even	4
7200.2.k.u.3601.5		12	120.77	even	4
7200.2.k.u.3601.6		12	15.2	even	4
7200.2.k.u.3601.7		12	120.53	even	4
7200.2.k.u.3601.8		12	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(-1.86081 + 1.23992i) q^{5} +0.746175i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +(-1.86081 + 1.23992i) q^{5} +0.746175i q^{7} +1.00000 q^{9} -5.36068i q^{11} -2.92520 q^{13} +(1.86081 - 1.23992i) q^{15} -2.13466i q^{17} +1.73367i q^{19} -0.746175i q^{21} -7.49534i q^{23} +(1.92520 - 4.61450i) q^{25} -1.00000 q^{27} -6.74916i q^{29} -2.64681 q^{31} +5.36068i q^{33} +(-0.925197 - 1.38849i) q^{35} -1.07480 q^{37} +2.92520 q^{39} -11.2936 q^{41} -7.44322 q^{43} +(-1.86081 + 1.23992i) q^{45} +1.73367i q^{47} +6.44322 q^{49} +2.13466i q^{51} +7.72161 q^{53} +(6.64681 + 9.97518i) q^{55} -1.73367i q^{57} +6.85302i q^{59} -6.45203i q^{61} +0.746175i q^{63} +(5.44322 - 3.62701i) q^{65} +7.44322 q^{67} +7.49534i q^{69} -13.2936 q^{71} -0.690358i q^{73} +(-1.92520 + 4.61450i) q^{75} +4.00000 q^{77} +2.64681 q^{79} +1.00000 q^{81} +5.85039 q^{83} +(2.64681 + 3.97219i) q^{85} +6.74916i q^{87} -7.59283 q^{89} -2.18271i q^{91} +2.64681 q^{93} +(-2.14961 - 3.22601i) q^{95} +14.1887i q^{97} -5.36068i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} - 8 q^{13} + 2 q^{25} - 6 q^{27} + 16 q^{31} + 4 q^{35} - 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} + 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} - 2 q^{75} + 24 q^{77} - 16 q^{79} + 6 q^{81} + 16 q^{83} - 16 q^{85} - 20 q^{89} - 16 q^{93} - 32 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 8 * q^13 + 2 * q^25 - 6 * q^27 + 16 * q^31 + 4 * q^35 - 16 * q^37 + 8 * q^39 - 4 * q^41 - 6 * q^49 + 24 * q^53 + 8 * q^55 - 12 * q^65 - 16 * q^71 - 2 * q^75 + 24 * q^77 - 16 * q^79 + 6 * q^81 + 16 * q^83 - 16 * q^85 - 20 * q^89 - 16 * q^93 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more