LMFDB - Embedded newform 520.2.k.a.441.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [520,2,Mod(441,520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("520.441"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$4.15222090511$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.2
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	520.441
Dual form			520.2.k.a.441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67513 q^{3} +1.00000i q^{5} -0.806063i q^{7} -0.193937 q^{9} -3.83146i q^{11} +(3.28726 + 1.48119i) q^{13} -1.67513i q^{15} +1.61213 q^{17} +0.869067i q^{19} +1.35026i q^{21} +5.28726 q^{23} -1.00000 q^{25} +5.35026 q^{27} +8.46898 q^{29} -10.4060i q^{31} +6.41819i q^{33} +0.806063 q^{35} +6.15633i q^{37} +(-5.50659 - 2.48119i) q^{39} -9.66291i q^{41} -0.974607 q^{43} -0.193937i q^{45} -2.41819i q^{47} +6.35026 q^{49} -2.70052 q^{51} +9.66291 q^{53} +3.83146 q^{55} -1.45580i q^{57} +2.48119i q^{59} -13.1187 q^{61} +0.156325i q^{63} +(-1.48119 + 3.28726i) q^{65} +0.156325i q^{67} -8.85685 q^{69} +1.64481i q^{71} -2.93207i q^{73} +1.67513 q^{75} -3.08840 q^{77} -3.53690 q^{79} -8.38058 q^{81} +15.6932i q^{83} +1.61213i q^{85} -14.1866 q^{87} +2.38787i q^{89} +(1.19394 - 2.64974i) q^{91} +17.4314i q^{93} -0.869067 q^{95} -7.08840i q^{97} +0.743059i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77}+ \cdots + 4 q^{95}+O(q^{100}) $ 6 * q - 2 * q^9 + 8 * q^13 + 8 * q^17 + 20 * q^23 - 6 * q^25 + 12 * q^27 - 12 * q^29 + 4 * q^35 + 8 * q^39 - 36 * q^43 + 18 * q^49 + 24 * q^51 - 4 * q^53 - 8 * q^55 - 36 * q^61 + 2 * q^65 + 8 * q^69 + 20 * q^77 + 24 * q^79 - 26 * q^81 - 60 * q^87 + 8 * q^91 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	−1.67513			−0.967137			−0.483569	−	0.875306i	$-0.660660\pi$
$3$	−1.67513			−0.967137			−0.483569	+	0.875306i	$0.660660\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	0.806063i		−	0.304663i	−0.988329	−	0.152332i	$-0.951322\pi$
$7$		−	0.806063i		−	0.304663i	0.988329	−	0.152332i	$-0.0486782\pi$
$8$		0			0
$9$	−0.193937			−0.0646455
$10$		0			0
$11$		−	3.83146i		−	1.15523i	−0.816310	−	0.577614i	$-0.803984\pi$
$11$		−	3.83146i		−	1.15523i	0.816310	−	0.577614i	$-0.196016\pi$
$12$		0			0
$13$	3.28726	+	1.48119i	0.911721	+	0.410809i
$13$	3.28726	+	1.48119i	0.911721	+	0.410809i
$14$		0			0
$15$		−	1.67513i		−	0.432517i
$16$		0			0
$17$	1.61213			0.390998			0.195499	−	0.980704i	$-0.437367\pi$
$17$	1.61213			0.390998			0.195499	+	0.980704i	$0.437367\pi$
$18$		0			0
$19$			0.869067i			0.199378i	0.995019	+	0.0996889i	$0.0317847\pi$
$19$			0.869067i			0.199378i	−0.995019	+	0.0996889i	$0.968215\pi$
$20$		0			0
$21$			1.35026i			0.294651i
$22$		0			0
$23$	5.28726			1.10247			0.551235	−	0.834350i	$-0.314157\pi$
$23$	5.28726			1.10247			0.551235	+	0.834350i	$0.314157\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.35026			1.02966
$28$		0			0
$29$	8.46898			1.57265			0.786325	−	0.617814i	$-0.211981\pi$
$29$	8.46898			1.57265			0.786325	+	0.617814i	$0.211981\pi$
$30$		0			0
$31$		−	10.4060i		−	1.86897i	−0.356006	−	0.934484i	$-0.615862\pi$
$31$		−	10.4060i		−	1.86897i	0.356006	−	0.934484i	$-0.384138\pi$
$32$		0			0
$33$			6.41819i			1.11726i
$34$		0			0
$35$	0.806063			0.136250
$36$		0			0
$37$			6.15633i			1.01209i	0.862506	+	0.506047i	$0.168894\pi$
$37$			6.15633i			1.01209i	−0.862506	+	0.506047i	$0.831106\pi$
$38$		0			0
$39$	−5.50659	−	2.48119i	−0.881760	−	0.397309i
$40$		0			0
$41$		−	9.66291i		−	1.50909i	−0.656246	−	0.754547i	$-0.727857\pi$
$41$		−	9.66291i		−	1.50909i	0.656246	−	0.754547i	$-0.272143\pi$
$42$		0			0
$43$	−0.974607			−0.148626			−0.0743131	−	0.997235i	$-0.523676\pi$
$43$	−0.974607			−0.148626			−0.0743131	+	0.997235i	$0.523676\pi$
$44$		0			0
$45$		−	0.193937i		−	0.0289104i
$46$		0			0
$47$		−	2.41819i		−	0.352729i	−0.984325	−	0.176365i	$-0.943566\pi$
$47$		−	2.41819i		−	0.352729i	0.984325	−	0.176365i	$-0.0564338\pi$
$48$		0			0
$49$	6.35026			0.907180
$50$		0			0
$51$	−2.70052			−0.378149
$52$		0			0
$53$	9.66291			1.32730			0.663652	−	0.748042i	$-0.269006\pi$
$53$	9.66291			1.32730			0.663652	+	0.748042i	$0.269006\pi$
$54$		0			0
$55$	3.83146			0.516633
$56$		0			0
$57$		−	1.45580i		−	0.192826i
$58$		0			0
$59$			2.48119i			0.323024i	0.986871	+	0.161512i	$0.0516370\pi$
$59$			2.48119i			0.323024i	−0.986871	+	0.161512i	$0.948363\pi$
$60$		0			0
$61$	−13.1187			−1.67968			−0.839840	−	0.542835i	$-0.817351\pi$
$61$	−13.1187			−1.67968			−0.839840	+	0.542835i	$0.817351\pi$
$62$		0			0
$63$			0.156325i			0.0196951i
$64$		0			0
$65$	−1.48119	+	3.28726i	−0.183720	+	0.407734i
$66$		0			0
$67$			0.156325i			0.0190982i	0.999954	+	0.00954908i	$0.00303961\pi$
$67$			0.156325i			0.0190982i	−0.999954	+	0.00954908i	$0.996960\pi$
$68$		0			0
$69$	−8.85685			−1.06624
$70$		0			0
$71$			1.64481i			0.195203i	0.995226	+	0.0976017i	$0.0311171\pi$
$71$			1.64481i			0.195203i	−0.995226	+	0.0976017i	$0.968883\pi$
$72$		0			0
$73$		−	2.93207i		−	0.343173i	−0.985169	−	0.171587i	$-0.945111\pi$
$73$		−	2.93207i		−	0.343173i	0.985169	−	0.171587i	$-0.0548893\pi$
$74$		0			0
$75$	1.67513			0.193427
$76$		0			0
$77$	−3.08840			−0.351955
$78$		0			0
$79$	−3.53690			−0.397933			−0.198966	−	0.980006i	$-0.563758\pi$
$79$	−3.53690			−0.397933			−0.198966	+	0.980006i	$0.563758\pi$
$80$		0			0
$81$	−8.38058			−0.931175
$82$		0			0
$83$			15.6932i			1.72256i	0.508134	+	0.861278i	$0.330335\pi$
$83$			15.6932i			1.72256i	−0.508134	+	0.861278i	$0.669665\pi$
$84$		0			0
$85$			1.61213i			0.174860i
$86$		0			0
$87$	−14.1866			−1.52097
$88$		0			0
$89$			2.38787i			0.253114i	0.991959	+	0.126557i	$0.0403927\pi$
$89$			2.38787i			0.253114i	−0.991959	+	0.126557i	$0.959607\pi$
$90$		0			0
$91$	1.19394	−	2.64974i	0.125159	−	0.277768i
$92$		0			0
$93$			17.4314i			1.80755i
$94$		0			0
$95$	−0.869067			−0.0891644
$96$		0			0
$97$		−	7.08840i		−	0.719718i	−0.933007	−	0.359859i	$-0.882825\pi$
$97$		−	7.08840i		−	0.719718i	0.933007	−	0.359859i	$-0.117175\pi$
$98$		0			0
$99$			0.743059i			0.0746803i
$100$		0			0
$101$	3.42548			0.340848			0.170424	−	0.985371i	$-0.445486\pi$
$101$	3.42548			0.340848			0.170424	+	0.985371i	$0.445486\pi$
$102$		0			0
$103$	−13.3380			−1.31424			−0.657118	−	0.753788i	$-0.728225\pi$
$103$	−13.3380			−1.31424			−0.657118	+	0.753788i	$0.728225\pi$
$104$		0			0
$105$	−1.35026			−0.131772
$106$		0			0
$107$	−5.93700			−0.573951			−0.286976	−	0.957938i	$-0.592650\pi$
$107$	−5.93700			−0.573951			−0.286976	+	0.957938i	$0.592650\pi$
$108$		0			0
$109$		−	0.387873i		−	0.0371515i	−0.999827	−	0.0185758i	$-0.994087\pi$
$109$		−	0.387873i		−	0.0371515i	0.999827	−	0.0185758i	$-0.00591319\pi$
$110$		0			0
$111$		−	10.3127i		−	0.978833i
$112$		0			0
$113$	−8.77575			−0.825553			−0.412776	−	0.910832i	$-0.635441\pi$
$113$	−8.77575			−0.825553			−0.412776	+	0.910832i	$0.635441\pi$
$114$		0			0
$115$			5.28726i			0.493039i
$116$		0			0
$117$	−0.637519	−	0.287258i	−0.0589387	−	0.0265570i
$118$		0			0
$119$		−	1.29948i		−	0.119123i
$120$		0			0
$121$	−3.68006			−0.334550
$122$		0			0
$123$			16.1866i			1.45950i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	5.48849			0.487025			0.243512	−	0.969898i	$-0.421700\pi$
$127$	5.48849			0.487025			0.243512	+	0.969898i	$0.421700\pi$
$128$		0			0
$129$	1.63259			0.143742
$130$		0			0
$131$	12.3127			1.07576			0.537881	−	0.843021i	$-0.319225\pi$
$131$	12.3127			1.07576			0.537881	+	0.843021i	$0.319225\pi$
$132$		0			0
$133$	0.700523			0.0607431
$134$		0			0
$135$			5.35026i			0.460477i
$136$		0			0
$137$		−	3.79877i		−	0.324551i	−0.986746	−	0.162275i	$-0.948117\pi$
$137$		−	3.79877i		−	0.324551i	0.986746	−	0.162275i	$-0.0518833\pi$
$138$		0			0
$139$	−13.0884			−1.11014			−0.555072	−	0.831803i	$-0.687309\pi$
$139$	−13.0884			−1.11014			−0.555072	+	0.831803i	$0.687309\pi$
$140$		0			0
$141$			4.05079i			0.341138i
$142$		0			0
$143$	5.67513	−	12.5950i	0.474578	−	1.05325i
$144$		0			0
$145$			8.46898i			0.703310i
$146$		0			0
$147$	−10.6375			−0.877368
$148$		0			0
$149$		−	18.3127i		−	1.50023i	−0.661307	−	0.750115i	$-0.729998\pi$
$149$		−	18.3127i		−	1.50023i	0.661307	−	0.750115i	$-0.270002\pi$
$150$		0			0
$151$			5.64481i			0.459368i	0.973265	+	0.229684i	$0.0737693\pi$
$151$			5.64481i			0.459368i	−0.973265	+	0.229684i	$0.926231\pi$
$152$		0			0
$153$	−0.312650			−0.0252763
$154$		0			0
$155$	10.4060			0.835828
$156$		0			0
$157$	7.35026			0.586615			0.293307	−	0.956018i	$-0.405244\pi$
$157$	7.35026			0.586615			0.293307	+	0.956018i	$0.405244\pi$
$158$		0			0
$159$	−16.1866			−1.28368
$160$		0			0
$161$		−	4.26187i		−	0.335882i
$162$		0			0
$163$			9.43136i			0.738721i	0.929286	+	0.369361i	$0.120423\pi$
$163$			9.43136i			0.738721i	−0.929286	+	0.369361i	$0.879577\pi$
$164$		0			0
$165$	−6.41819			−0.499655
$166$		0			0
$167$		−	14.8568i		−	1.14966i	−0.818274	−	0.574829i	$-0.805069\pi$
$167$		−	14.8568i		−	1.14966i	0.818274	−	0.574829i	$-0.194931\pi$
$168$		0			0
$169$	8.61213	+	9.73813i	0.662471	+	0.749087i
$170$		0			0
$171$		−	0.168544i		−	0.0128889i
$172$		0			0
$173$	11.2750			0.857225			0.428613	−	0.903488i	$-0.359003\pi$
$173$	11.2750			0.857225			0.428613	+	0.903488i	$0.359003\pi$
$174$		0			0
$175$			0.806063i			0.0609327i
$176$		0			0
$177$		−	4.15633i		−	0.312409i
$178$		0			0
$179$	23.1998			1.73404			0.867018	−	0.498277i	$-0.166034\pi$
$179$	23.1998			1.73404			0.867018	+	0.498277i	$0.166034\pi$
$180$		0			0
$181$	−22.3938			−1.66451			−0.832257	−	0.554390i	$-0.812952\pi$
$181$	−22.3938			−1.66451			−0.832257	+	0.554390i	$0.812952\pi$
$182$		0			0
$183$	21.9756			1.62448
$184$		0			0
$185$	−6.15633			−0.452622
$186$		0			0
$187$		−	6.17679i		−	0.451692i
$188$		0			0
$189$		−	4.31265i		−	0.313699i
$190$		0			0
$191$	−1.55149			−0.112262			−0.0561310	−	0.998423i	$-0.517876\pi$
$191$	−1.55149			−0.112262			−0.0561310	+	0.998423i	$0.517876\pi$
$192$		0			0
$193$			4.57452i			0.329281i	0.986354	+	0.164640i	$0.0526464\pi$
$193$			4.57452i			0.329281i	−0.986354	+	0.164640i	$0.947354\pi$
$194$		0			0
$195$	2.48119	−	5.50659i	0.177682	−	0.394335i
$196$		0			0
$197$		−	17.5125i		−	1.24771i	−0.781539	−	0.623856i	$-0.785565\pi$
$197$		−	17.5125i		−	1.24771i	0.781539	−	0.623856i	$-0.214435\pi$
$198$		0			0
$199$	−9.79877			−0.694616			−0.347308	−	0.937751i	$-0.612904\pi$
$199$	−9.79877			−0.694616			−0.347308	+	0.937751i	$0.612904\pi$
$200$		0			0
$201$		−	0.261865i		−	0.0184705i
$202$		0			0
$203$		−	6.82653i		−	0.479129i
$204$		0			0
$205$	9.66291			0.674887
$206$		0			0
$207$	−1.02539			−0.0712697
$208$		0			0
$209$	3.32979			0.230327
$210$		0			0
$211$	15.1998			1.04640			0.523199	−	0.852210i	$-0.324738\pi$
$211$	15.1998			1.04640			0.523199	+	0.852210i	$0.324738\pi$
$212$		0			0
$213$		−	2.75528i		−	0.188789i
$214$		0			0
$215$		−	0.974607i		−	0.0664677i
$216$		0			0
$217$	−8.38787			−0.569406
$218$		0			0
$219$			4.91160i			0.331895i
$220$		0			0
$221$	5.29948	+	2.38787i	0.356481	+	0.160626i
$222$		0			0
$223$			16.2823i			1.09035i	0.838324	+	0.545173i	$0.183536\pi$
$223$			16.2823i			1.09035i	−0.838324	+	0.545173i	$0.816464\pi$
$224$		0			0
$225$	0.193937			0.0129291
$226$		0			0
$227$			23.1695i			1.53781i	0.639361	+	0.768907i	$0.279199\pi$
$227$			23.1695i			1.53781i	−0.639361	+	0.768907i	$0.720801\pi$
$228$		0			0
$229$			0.126008i			0.00832684i	0.999991	+	0.00416342i	$0.00132526\pi$
$229$			0.126008i			0.00832684i	−0.999991	+	0.00416342i	$0.998675\pi$
$230$		0			0
$231$	5.17347			0.340389
$232$		0			0
$233$	8.91160			0.583819			0.291909	−	0.956446i	$-0.405709\pi$
$233$	8.91160			0.583819			0.291909	+	0.956446i	$0.405709\pi$
$234$		0			0
$235$	2.41819			0.157745
$236$		0			0
$237$	5.92478			0.384856
$238$		0			0
$239$			12.1685i			0.787118i	0.919299	+	0.393559i	$0.128756\pi$
$239$			12.1685i			0.787118i	−0.919299	+	0.393559i	$0.871244\pi$
$240$		0			0
$241$			17.9756i			1.15791i	0.815360	+	0.578954i	$0.196539\pi$
$241$			17.9756i			1.15791i	−0.815360	+	0.578954i	$0.803461\pi$
$242$		0			0
$243$	−2.01222			−0.129084
$244$		0			0
$245$			6.35026i			0.405703i
$246$		0			0
$247$	−1.28726	+	2.85685i	−0.0819062	+	0.181777i
$248$		0			0
$249$		−	26.2882i		−	1.66595i
$250$		0			0
$251$	−2.51388			−0.158675			−0.0793374	−	0.996848i	$-0.525280\pi$
$251$	−2.51388			−0.158675			−0.0793374	+	0.996848i	$0.525280\pi$
$252$		0			0
$253$		−	20.2579i		−	1.27360i
$254$		0			0
$255$		−	2.70052i		−	0.169113i
$256$		0			0
$257$	13.2243			0.824906			0.412453	−	0.910979i	$-0.364672\pi$
$257$	13.2243			0.824906			0.412453	+	0.910979i	$0.364672\pi$
$258$		0			0
$259$	4.96239			0.308348
$260$		0			0
$261$	−1.64244			−0.101665
$262$		0			0
$263$	−1.17584			−0.0725053			−0.0362526	−	0.999343i	$-0.511542\pi$
$263$	−1.17584			−0.0725053			−0.0362526	+	0.999343i	$0.511542\pi$
$264$		0			0
$265$			9.66291i			0.593588i
$266$		0			0
$267$		−	4.00000i		−	0.244796i
$268$		0			0
$269$	−13.3503			−0.813980			−0.406990	−	0.913433i	$-0.633422\pi$
$269$	−13.3503			−0.813980			−0.406990	+	0.913433i	$0.633422\pi$
$270$		0			0
$271$		−	32.0689i		−	1.94805i	−0.226449	−	0.974023i	$-0.572712\pi$
$271$		−	32.0689i		−	1.94805i	0.226449	−	0.974023i	$-0.427288\pi$
$272$		0			0
$273$	−2.00000	+	4.43866i	−0.121046	+	0.268640i
$274$		0			0
$275$			3.83146i			0.231045i
$276$		0			0
$277$	2.18664			0.131383			0.0656913	−	0.997840i	$-0.479075\pi$
$277$	2.18664			0.131383			0.0656913	+	0.997840i	$0.479075\pi$
$278$		0			0
$279$			2.01810i			0.120820i
$280$		0			0
$281$		−	1.87399i		−	0.111793i	−0.998437	−	0.0558965i	$-0.982198\pi$
$281$		−	1.87399i		−	0.111793i	0.998437	−	0.0558965i	$-0.0178017\pi$
$282$		0			0
$283$	−0.300432			−0.0178588			−0.00892940	−	0.999960i	$-0.502842\pi$
$283$	−0.300432			−0.0178588			−0.00892940	+	0.999960i	$0.502842\pi$
$284$		0			0
$285$	1.45580			0.0862342
$286$		0			0
$287$	−7.78892			−0.459765
$288$		0			0
$289$	−14.4010			−0.847120
$290$		0			0
$291$			11.8740i			0.696066i
$292$		0			0
$293$		−	22.6312i		−	1.32213i	−0.750330	−	0.661064i	$-0.770105\pi$
$293$		−	22.6312i		−	1.32213i	0.750330	−	0.661064i	$-0.229895\pi$
$294$		0			0
$295$	−2.48119			−0.144461
$296$		0			0
$297$		−	20.4993i		−	1.18949i
$298$		0			0
$299$	17.3806	+	7.83146i	1.00514	+	0.452905i
$300$		0			0
$301$			0.785595i			0.0452810i
$302$		0			0
$303$	−5.73813			−0.329647
$304$		0			0
$305$		−	13.1187i		−	0.751175i
$306$		0			0
$307$			28.3185i			1.61622i	0.589029	+	0.808112i	$0.299510\pi$
$307$			28.3185i			1.61622i	−0.589029	+	0.808112i	$0.700490\pi$
$308$		0			0
$309$	22.3430			1.27105
$310$		0			0
$311$	33.3766			1.89261			0.946307	−	0.323270i	$-0.104782\pi$
$311$	33.3766			1.89261			0.946307	+	0.323270i	$0.104782\pi$
$312$		0			0
$313$	−4.62530			−0.261437			−0.130719	−	0.991419i	$-0.541728\pi$
$313$	−4.62530			−0.261437			−0.130719	+	0.991419i	$0.541728\pi$
$314$		0			0
$315$	−0.156325			−0.00880793
$316$		0			0
$317$		−	31.8700i		−	1.79000i	−0.446067	−	0.894999i	$-0.647176\pi$
$317$		−	31.8700i		−	1.79000i	0.446067	−	0.894999i	$-0.352824\pi$
$318$		0			0
$319$		−	32.4485i		−	1.81677i
$320$		0			0
$321$	9.94525			0.555089
$322$		0			0
$323$			1.40105i			0.0779563i
$324$		0			0
$325$	−3.28726	−	1.48119i	−0.182344	−	0.0821619i
$326$		0			0
$327$			0.649738i			0.0359306i
$328$		0			0
$329$	−1.94921			−0.107464
$330$		0			0
$331$			22.8691i			1.25700i	0.777811	+	0.628499i	$0.216330\pi$
$331$			22.8691i			1.25700i	−0.777811	+	0.628499i	$0.783670\pi$
$332$		0			0
$333$		−	1.19394i		−	0.0654273i
$334$		0			0
$335$	−0.156325			−0.00854096
$336$		0			0
$337$	26.0870			1.42105			0.710524	−	0.703673i	$-0.248458\pi$
$337$	26.0870			1.42105			0.710524	+	0.703673i	$0.248458\pi$
$338$		0			0
$339$	14.7005			0.798423
$340$		0			0
$341$	−39.8700			−2.15908
$342$		0			0
$343$		−	10.7612i		−	0.581048i
$344$		0			0
$345$		−	8.85685i		−	0.476837i
$346$		0			0
$347$	−7.93700			−0.426080			−0.213040	−	0.977043i	$-0.568337\pi$
$347$	−7.93700			−0.426080			−0.213040	+	0.977043i	$0.568337\pi$
$348$		0			0
$349$			21.2144i			1.13558i	0.823173	+	0.567791i	$0.192202\pi$
$349$			21.2144i			1.13558i	−0.823173	+	0.567791i	$0.807798\pi$
$350$		0			0
$351$	17.5877	+	7.92478i	0.938761	+	0.422993i
$352$		0			0
$353$			27.2447i			1.45009i	0.688701	+	0.725045i	$0.258181\pi$
$353$			27.2447i			1.45009i	−0.688701	+	0.725045i	$0.741819\pi$
$354$		0			0
$355$	−1.64481			−0.0872976
$356$		0			0
$357$			2.17679i			0.115208i
$358$		0			0
$359$			23.7562i			1.25381i	0.779097	+	0.626903i	$0.215678\pi$
$359$			23.7562i			1.25381i	−0.779097	+	0.626903i	$0.784322\pi$
$360$		0			0
$361$	18.2447			0.960249
$362$		0			0
$363$	6.16457			0.323556
$364$		0			0
$365$	2.93207			0.153472
$366$		0			0
$367$	−36.2252			−1.89094			−0.945470	−	0.325708i	$-0.894397\pi$
$367$	−36.2252			−1.89094			−0.945470	+	0.325708i	$0.894397\pi$
$368$		0			0
$369$			1.87399i			0.0975561i
$370$		0			0
$371$		−	7.78892i		−	0.404381i
$372$		0			0
$373$	−8.64974			−0.447866			−0.223933	−	0.974604i	$-0.571890\pi$
$373$	−8.64974			−0.447866			−0.223933	+	0.974604i	$0.571890\pi$
$374$		0			0
$375$			1.67513i			0.0865034i
$376$		0			0
$377$	27.8397	+	12.5442i	1.43382	+	0.646059i
$378$		0			0
$379$		−	21.5696i		−	1.10796i	−0.832531	−	0.553978i	$-0.813109\pi$
$379$		−	21.5696i		−	1.10796i	0.832531	−	0.553978i	$-0.186891\pi$
$380$		0			0
$381$	−9.19394			−0.471020
$382$		0			0
$383$			3.00729i			0.153666i	0.997044	+	0.0768328i	$0.0244808\pi$
$383$			3.00729i			0.153666i	−0.997044	+	0.0768328i	$0.975519\pi$
$384$		0			0
$385$		−	3.08840i		−	0.157399i
$386$		0			0
$387$	0.189012			0.00960802
$388$		0			0
$389$	1.37470			0.0697000			0.0348500	−	0.999393i	$-0.488905\pi$
$389$	1.37470			0.0697000			0.0348500	+	0.999393i	$0.488905\pi$
$390$		0			0
$391$	8.52373			0.431064
$392$		0			0
$393$	−20.6253			−1.04041
$394$		0			0
$395$		−	3.53690i		−	0.177961i
$396$		0			0
$397$			36.4544i			1.82959i	0.403915	+	0.914797i	$0.367649\pi$
$397$			36.4544i			1.82959i	−0.403915	+	0.914797i	$0.632351\pi$
$398$		0			0
$399$	−1.17347			−0.0587469
$400$		0			0
$401$		−	5.02776i		−	0.251074i	−0.992089	−	0.125537i	$-0.959935\pi$
$401$		−	5.02776i		−	0.251074i	0.992089	−	0.125537i	$-0.0400655\pi$
$402$		0			0
$403$	15.4133	−	34.2071i	0.767789	−	1.70398i
$404$		0			0
$405$		−	8.38058i		−	0.416434i
$406$		0			0
$407$	23.5877			1.16920
$408$		0			0
$409$		−	15.1128i		−	0.747282i	−0.927573	−	0.373641i	$-0.878109\pi$
$409$		−	15.1128i		−	0.747282i	0.927573	−	0.373641i	$-0.121891\pi$
$410$		0			0
$411$			6.36344i			0.313885i
$412$		0			0
$413$	2.00000			0.0984136
$414$		0			0
$415$	−15.6932			−0.770350
$416$		0			0
$417$	21.9248			1.07366
$418$		0			0
$419$	−0.589104			−0.0287796			−0.0143898	−	0.999896i	$-0.504581\pi$
$419$	−0.589104			−0.0287796			−0.0143898	+	0.999896i	$0.504581\pi$
$420$		0			0
$421$		−	36.4894i		−	1.77839i	−0.457532	−	0.889193i	$-0.651266\pi$
$421$		−	36.4894i		−	1.77839i	0.457532	−	0.889193i	$-0.348734\pi$
$422$		0			0
$423$			0.468976i			0.0228024i
$424$		0			0
$425$	−1.61213			−0.0781996
$426$		0			0
$427$			10.5745i			0.511737i
$428$		0			0
$429$	−9.50659	+	21.0982i	−0.458982	+	1.01863i
$430$		0			0
$431$		−	19.8070i		−	0.954071i	−0.878884	−	0.477035i	$-0.841711\pi$
$431$		−	19.8070i		−	0.954071i	0.878884	−	0.477035i	$-0.158289\pi$
$432$		0			0
$433$	−20.5501			−0.987574			−0.493787	−	0.869583i	$-0.664388\pi$
$433$	−20.5501			−0.987574			−0.493787	+	0.869583i	$0.664388\pi$
$434$		0			0
$435$		−	14.1866i		−	0.680197i
$436$		0			0
$437$			4.59498i			0.219808i
$438$		0			0
$439$	−16.0362			−0.765366			−0.382683	−	0.923880i	$-0.625000\pi$
$439$	−16.0362			−0.765366			−0.382683	+	0.923880i	$0.625000\pi$
$440$		0			0
$441$	−1.23155			−0.0586451
$442$		0			0
$443$	−39.2384			−1.86427			−0.932136	−	0.362109i	$-0.882057\pi$
$443$	−39.2384			−1.86427			−0.932136	+	0.362109i	$0.882057\pi$
$444$		0			0
$445$	−2.38787			−0.113196
$446$		0			0
$447$			30.6761i			1.45093i
$448$		0			0
$449$			35.1246i			1.65763i	0.559521	+	0.828816i	$0.310985\pi$
$449$			35.1246i			1.65763i	−0.559521	+	0.828816i	$0.689015\pi$
$450$		0			0
$451$	−37.0230			−1.74335
$452$		0			0
$453$		−	9.45580i		−	0.444272i
$454$		0			0
$455$	2.64974	+	1.19394i	0.124222	+	0.0559726i
$456$		0			0
$457$			29.3865i			1.37464i	0.726354	+	0.687320i	$0.241213\pi$
$457$			29.3865i			1.37464i	−0.726354	+	0.687320i	$0.758787\pi$
$458$		0			0
$459$	8.62530			0.402595
$460$		0			0
$461$			19.1998i			0.894225i	0.894478	+	0.447112i	$0.147548\pi$
$461$			19.1998i			0.894225i	−0.894478	+	0.447112i	$0.852452\pi$
$462$		0			0
$463$		−	39.5183i		−	1.83657i	−0.395916	−	0.918287i	$-0.629573\pi$
$463$		−	39.5183i		−	1.83657i	0.395916	−	0.918287i	$-0.370427\pi$
$464$		0			0
$465$	−17.4314			−0.808360
$466$		0			0
$467$	−16.4119			−0.759450			−0.379725	−	0.925099i	$-0.623981\pi$
$467$	−16.4119			−0.759450			−0.379725	+	0.925099i	$0.623981\pi$
$468$		0			0
$469$	0.126008			0.00581851
$470$		0			0
$471$	−12.3127			−0.567337
$472$		0			0
$473$			3.73417i			0.171697i
$474$		0			0
$475$		−	0.869067i		−	0.0398755i
$476$		0			0
$477$	−1.87399			−0.0858042
$478$		0			0
$479$			23.5042i			1.07394i	0.843603	+	0.536968i	$0.180430\pi$
$479$			23.5042i			1.07394i	−0.843603	+	0.536968i	$0.819570\pi$
$480$		0			0
$481$	−9.11871	+	20.2374i	−0.415778	+	0.922747i
$482$		0			0
$483$			7.13918i			0.324844i
$484$		0			0
$485$	7.08840			0.321868
$486$		0			0
$487$			4.49341i			0.203616i	0.994804	+	0.101808i	$0.0324627\pi$
$487$			4.49341i			0.203616i	−0.994804	+	0.101808i	$0.967537\pi$
$488$		0			0
$489$		−	15.7988i		−	0.714445i
$490$		0			0
$491$	−14.7005			−0.663425			−0.331713	−	0.943380i	$-0.607626\pi$
$491$	−14.7005			−0.663425			−0.331713	+	0.943380i	$0.607626\pi$
$492$		0			0
$493$	13.6531			0.614903
$494$		0			0
$495$	−0.743059			−0.0333980
$496$		0			0
$497$	1.32582			0.0594713
$498$		0			0
$499$			9.89683i			0.443043i	0.975155	+	0.221521i	$0.0711023\pi$
$499$			9.89683i			0.443043i	−0.975155	+	0.221521i	$0.928898\pi$
$500$		0			0
$501$			24.8872i			1.11188i
$502$		0			0
$503$	−1.26282			−0.0563064			−0.0281532	−	0.999604i	$-0.508963\pi$
$503$	−1.26282			−0.0563064			−0.0281532	+	0.999604i	$0.508963\pi$
$504$		0			0
$505$			3.42548i			0.152432i
$506$		0			0
$507$	−14.4264	−	16.3127i	−0.640701	−	0.724470i
$508$		0			0
$509$			22.5256i			0.998431i	0.866478	+	0.499216i	$0.166378\pi$
$509$			22.5256i			0.998431i	−0.866478	+	0.499216i	$0.833622\pi$
$510$		0			0
$511$	−2.36344			−0.104552
$512$		0			0
$513$			4.64974i			0.205291i
$514$		0			0
$515$		−	13.3380i		−	0.587744i
$516$		0			0
$517$	−9.26519			−0.407483
$518$		0			0
$519$	−18.8872			−0.829055
$520$		0			0
$521$	−18.8568			−0.826134			−0.413067	−	0.910701i	$-0.635542\pi$
$521$	−18.8568			−0.826134			−0.413067	+	0.910701i	$0.635542\pi$
$522$		0			0
$523$	−32.5985			−1.42543			−0.712716	−	0.701452i	$-0.752535\pi$
$523$	−32.5985			−1.42543			−0.712716	+	0.701452i	$0.752535\pi$
$524$		0			0
$525$		−	1.35026i		−	0.0589303i
$526$		0			0
$527$		−	16.7757i		−	0.730763i
$528$		0			0
$529$	4.95509			0.215439
$530$		0			0
$531$		−	0.481194i		−	0.0208821i
$532$		0			0
$533$	14.3127	−	31.7645i	0.619950	−	1.37587i
$534$		0			0
$535$		−	5.93700i		−	0.256679i
$536$		0			0
$537$	−38.8627			−1.67705
$538$		0			0
$539$		−	24.3307i		−	1.04800i
$540$		0			0
$541$			32.0362i			1.37734i	0.725073	+	0.688672i	$0.241806\pi$
$541$			32.0362i			1.37734i	−0.725073	+	0.688672i	$0.758194\pi$
$542$		0			0
$543$	37.5125			1.60981
$544$		0			0
$545$	0.387873			0.0166147
$546$		0			0
$547$	−5.41327			−0.231455			−0.115727	−	0.993281i	$-0.536920\pi$
$547$	−5.41327			−0.231455			−0.115727	+	0.993281i	$0.536920\pi$
$548$		0			0
$549$	2.54420			0.108584
$550$		0			0
$551$			7.36011i			0.313551i
$552$		0			0
$553$			2.85097i			0.121236i
$554$		0			0
$555$	10.3127			0.437748
$556$		0			0
$557$			24.7064i			1.04684i	0.852074	+	0.523422i	$0.175345\pi$
$557$			24.7064i			1.04684i	−0.852074	+	0.523422i	$0.824655\pi$
$558$		0			0
$559$	−3.20379	−	1.44358i	−0.135506	−	0.0610571i
$560$		0			0
$561$			10.3469i			0.436848i
$562$		0			0
$563$	−33.1974			−1.39911			−0.699553	−	0.714581i	$-0.746617\pi$
$563$	−33.1974			−1.39911			−0.699553	+	0.714581i	$0.746617\pi$
$564$		0			0
$565$		−	8.77575i		−	0.369198i
$566$		0			0
$567$			6.75528i			0.283695i
$568$		0			0
$569$	−34.3693			−1.44084			−0.720418	−	0.693540i	$-0.756050\pi$
$569$	−34.3693			−1.44084			−0.720418	+	0.693540i	$0.756050\pi$
$570$		0			0
$571$	24.3127			1.01745			0.508726	−	0.860928i	$-0.330116\pi$
$571$	24.3127			1.01745			0.508726	+	0.860928i	$0.330116\pi$
$572$		0			0
$573$	2.59895			0.108573
$574$		0			0
$575$	−5.28726			−0.220494
$576$		0			0
$577$			26.6312i			1.10867i	0.832293	+	0.554335i	$0.187027\pi$
$577$			26.6312i			1.10867i	−0.832293	+	0.554335i	$0.812973\pi$
$578$		0			0
$579$		−	7.66291i		−	0.318460i
$580$		0			0
$581$	12.6497			0.524800
$582$		0			0
$583$		−	37.0230i		−	1.53334i
$584$		0			0
$585$	0.287258	−	0.637519i	0.0118766	−	0.0263582i
$586$		0			0
$587$		−	18.8813i		−	0.779314i	−0.920960	−	0.389657i	$-0.872594\pi$
$587$		−	18.8813i		−	0.779314i	0.920960	−	0.389657i	$-0.127406\pi$
$588$		0			0
$589$	9.04349			0.372631
$590$		0			0
$591$			29.3357i			1.20671i
$592$		0			0
$593$		−	36.4894i		−	1.49844i	−0.662320	−	0.749221i	$-0.730428\pi$
$593$		−	36.4894i		−	1.49844i	0.662320	−	0.749221i	$-0.269572\pi$
$594$		0			0
$595$	1.29948			0.0532733
$596$		0			0
$597$	16.4142			0.671789
$598$		0			0
$599$	−23.3503			−0.954066			−0.477033	−	0.878885i	$-0.658288\pi$
$599$	−23.3503			−0.954066			−0.477033	+	0.878885i	$0.658288\pi$
$600$		0			0
$601$	18.3733			0.749462			0.374731	−	0.927134i	$-0.377735\pi$
$601$	18.3733			0.749462			0.374731	+	0.927134i	$0.377735\pi$
$602$		0			0
$603$		−	0.0303172i		−	0.00123461i
$604$		0			0
$605$		−	3.68006i		−	0.149616i
$606$		0			0
$607$	4.48707			0.182125			0.0910624	−	0.995845i	$-0.470974\pi$
$607$	4.48707			0.182125			0.0910624	+	0.995845i	$0.470974\pi$
$608$		0			0
$609$			11.4353i			0.463383i
$610$		0			0
$611$	3.58181	−	7.94921i	0.144905	−	0.321591i
$612$		0			0
$613$			38.3536i			1.54909i	0.632521	+	0.774543i	$0.282020\pi$
$613$			38.3536i			1.54909i	−0.632521	+	0.774543i	$0.717980\pi$
$614$		0			0
$615$	−16.1866			−0.652708
$616$		0			0
$617$		−	17.1998i		−	0.692439i	−0.938154	−	0.346219i	$-0.887465\pi$
$617$		−	17.1998i		−	0.692439i	0.938154	−	0.346219i	$-0.112535\pi$
$618$		0			0
$619$		−	16.5926i		−	0.666913i	−0.942765	−	0.333457i	$-0.891785\pi$
$619$		−	16.5926i		−	0.666913i	0.942765	−	0.333457i	$-0.108215\pi$
$620$		0			0
$621$	28.2882			1.13517
$622$		0			0
$623$	1.92478			0.0771146
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−5.57784			−0.222757
$628$		0			0
$629$			9.92478i			0.395727i
$630$		0			0
$631$		−	7.39280i		−	0.294303i	−0.989114	−	0.147151i	$-0.952990\pi$
$631$		−	7.39280i		−	0.294303i	0.989114	−	0.147151i	$-0.0470104\pi$
$632$		0			0
$633$	−25.4617			−1.01201
$634$		0			0
$635$			5.48849i			0.217804i
$636$		0			0
$637$	20.8749	+	9.40597i	0.827096	+	0.372678i
$638$		0			0
$639$		−	0.318990i		−	0.0126190i
$640$		0			0
$641$	25.9511			1.02501			0.512504	−	0.858685i	$-0.328718\pi$
$641$	25.9511			1.02501			0.512504	+	0.858685i	$0.328718\pi$
$642$		0			0
$643$			12.6439i			0.498625i	0.968423	+	0.249313i	$0.0802046\pi$
$643$			12.6439i			0.498625i	−0.968423	+	0.249313i	$0.919795\pi$
$644$		0			0
$645$			1.63259i			0.0642834i
$646$		0			0
$647$	22.7391			0.893966			0.446983	−	0.894542i	$-0.352498\pi$
$647$	22.7391			0.893966			0.446983	+	0.894542i	$0.352498\pi$
$648$		0			0
$649$	9.50659			0.373166
$650$		0			0
$651$	14.0508			0.550694
$652$		0			0
$653$	−16.4387			−0.643294			−0.321647	−	0.946860i	$-0.604237\pi$
$653$	−16.4387			−0.643294			−0.321647	+	0.946860i	$0.604237\pi$
$654$		0			0
$655$			12.3127i			0.481095i
$656$		0			0
$657$			0.568636i			0.0221846i
$658$		0			0
$659$	37.4372			1.45835			0.729174	−	0.684328i	$-0.239904\pi$
$659$	37.4372			1.45835			0.729174	+	0.684328i	$0.239904\pi$
$660$		0			0
$661$			18.6253i			0.724440i	0.932093	+	0.362220i	$0.117981\pi$
$661$			18.6253i			0.724440i	−0.932093	+	0.362220i	$0.882019\pi$
$662$		0			0
$663$	−8.87732	−	4.00000i	−0.344766	−	0.155347i
$664$		0			0
$665$			0.700523i			0.0271651i
$666$		0			0
$667$	44.7777			1.73380
$668$		0			0
$669$		−	27.2750i		−	1.05451i
$670$		0			0
$671$			50.2638i			1.94041i
$672$		0			0
$673$	24.4749			0.943436			0.471718	−	0.881749i	$-0.343634\pi$
$673$	24.4749			0.943436			0.471718	+	0.881749i	$0.343634\pi$
$674$		0			0
$675$	−5.35026			−0.205932
$676$		0			0
$677$	−25.9756			−0.998322			−0.499161	−	0.866509i	$-0.666358\pi$
$677$	−25.9756			−0.998322			−0.499161	+	0.866509i	$0.666358\pi$
$678$		0			0
$679$	−5.71370			−0.219272
$680$		0			0
$681$		−	38.8119i		−	1.48728i
$682$		0			0
$683$		−	2.08110i		−	0.0796312i	−0.999207	−	0.0398156i	$-0.987323\pi$
$683$		−	2.08110i		−	0.0796312i	0.999207	−	0.0398156i	$-0.0126770\pi$
$684$		0			0
$685$	3.79877			0.145144
$686$		0			0
$687$		−	0.211080i		−	0.00805320i
$688$		0			0
$689$	31.7645	+	14.3127i	1.21013	+	0.545269i
$690$		0			0
$691$		−	27.6204i		−	1.05073i	−0.850877	−	0.525364i	$-0.823929\pi$
$691$		−	27.6204i		−	1.05073i	0.850877	−	0.525364i	$-0.176071\pi$
$692$		0			0
$693$	0.598953			0.0227523
$694$		0			0
$695$		−	13.0884i		−	0.496471i
$696$		0			0
$697$		−	15.5778i		−	0.590053i
$698$		0			0
$699$	−14.9281			−0.564633
$700$		0			0
$701$	17.8251			0.673245			0.336623	−	0.941640i	$-0.390715\pi$
$701$	17.8251			0.673245			0.336623	+	0.941640i	$0.390715\pi$
$702$		0			0
$703$	−5.35026			−0.201789
$704$		0			0
$705$	−4.05079			−0.152561
$706$		0			0
$707$		−	2.76116i		−	0.103844i
$708$		0			0
$709$		−	30.9887i		−	1.16381i	−0.813258	−	0.581903i	$-0.802308\pi$
$709$		−	30.9887i		−	1.16381i	0.813258	−	0.581903i	$-0.197692\pi$
$710$		0			0
$711$	0.685935			0.0257246
$712$		0			0
$713$		−	55.0191i		−	2.06048i
$714$		0			0
$715$	12.5950	+	5.67513i	0.471026	+	0.212238i
$716$		0			0
$717$		−	20.3839i		−	0.761251i
$718$		0			0
$719$	24.3127			0.906709			0.453354	−	0.891330i	$-0.350227\pi$
$719$	24.3127			0.906709			0.453354	+	0.891330i	$0.350227\pi$
$720$		0			0
$721$			10.7513i			0.400400i
$722$		0			0
$723$		−	30.1114i		−	1.11986i
$724$		0			0
$725$	−8.46898			−0.314530
$726$		0			0
$727$	25.6361			0.950791			0.475395	−	0.879772i	$-0.342305\pi$
$727$	25.6361			0.950791			0.475395	+	0.879772i	$0.342305\pi$
$728$		0			0
$729$	28.5125			1.05602
$730$		0			0
$731$	−1.57119			−0.0581126
$732$		0			0
$733$		−	30.1866i		−	1.11497i	−0.830187	−	0.557485i	$-0.811767\pi$
$733$		−	30.1866i		−	1.11497i	0.830187	−	0.557485i	$-0.188233\pi$
$734$		0			0
$735$		−	10.6375i		−	0.392371i
$736$		0			0
$737$	0.598953			0.0220627
$738$		0			0
$739$			20.7576i			0.763582i	0.924249	+	0.381791i	$0.124693\pi$
$739$			20.7576i			0.763582i	−0.924249	+	0.381791i	$0.875307\pi$
$740$		0			0
$741$	2.15633	−	4.78560i	0.0792146	−	0.175803i
$742$		0			0
$743$			14.2315i			0.522105i	0.965325	+	0.261052i	$0.0840695\pi$
$743$			14.2315i			0.522105i	−0.965325	+	0.261052i	$0.915930\pi$
$744$		0			0
$745$	18.3127			0.670924
$746$		0			0
$747$		−	3.04349i		−	0.111356i
$748$		0			0
$749$			4.78560i			0.174862i
$750$		0			0
$751$	4.81194			0.175590			0.0877951	−	0.996139i	$-0.472018\pi$
$751$	4.81194			0.175590			0.0877951	+	0.996139i	$0.472018\pi$
$752$		0			0
$753$	4.21108			0.153460
$754$		0			0
$755$	−5.64481			−0.205436
$756$		0			0
$757$	22.7513			0.826910			0.413455	−	0.910524i	$-0.364322\pi$
$757$	22.7513			0.826910			0.413455	+	0.910524i	$0.364322\pi$
$758$		0			0
$759$			33.9346i			1.23175i
$760$		0			0
$761$			30.3879i			1.10156i	0.834651	+	0.550780i	$0.185670\pi$
$761$			30.3879i			1.10156i	−0.834651	+	0.550780i	$0.814330\pi$
$762$		0			0
$763$	−0.312650			−0.0113187
$764$		0			0
$765$		−	0.312650i		−	0.0113039i
$766$		0			0
$767$	−3.67513	+	8.15633i	−0.132701	+	0.294508i
$768$		0			0
$769$		−	45.3112i		−	1.63396i	−0.576662	−	0.816982i	$-0.695645\pi$
$769$		−	45.3112i		−	1.63396i	0.576662	−	0.816982i	$-0.304355\pi$
$770$		0			0
$771$	−22.1524			−0.797798
$772$		0			0
$773$			14.6194i			0.525824i	0.964820	+	0.262912i	$0.0846829\pi$
$773$			14.6194i			0.525824i	−0.964820	+	0.262912i	$0.915317\pi$
$774$		0			0
$775$			10.4060i			0.373794i
$776$		0			0
$777$	−8.31265			−0.298215
$778$		0			0
$779$	8.39772			0.300880
$780$		0			0
$781$	6.30203			0.225504
$782$		0			0
$783$	45.3112			1.61929
$784$		0			0
$785$			7.35026i			0.262342i
$786$		0			0
$787$			7.06793i			0.251944i	0.992034	+	0.125972i	$0.0402050\pi$
$787$			7.06793i			0.251944i	−0.992034	+	0.125972i	$0.959795\pi$
$788$		0			0
$789$	1.96968			0.0701226
$790$		0			0
$791$			7.07381i			0.251516i
$792$		0			0
$793$	−43.1246	−	19.4314i	−1.53140	−	0.690028i
$794$		0			0
$795$		−	16.1866i		−	0.574081i
$796$		0			0
$797$	−46.8383			−1.65910			−0.829549	−	0.558434i	$-0.811402\pi$
$797$	−46.8383			−1.65910			−0.829549	+	0.558434i	$0.811402\pi$
$798$		0			0
$799$		−	3.89843i		−	0.137917i
$800$		0			0
$801$		−	0.463096i		−	0.0163627i
$802$		0			0
$803$	−11.2341			−0.396443
$804$		0			0
$805$	4.26187			0.150211
$806$		0			0
$807$	22.3634			0.787230
$808$		0			0
$809$	−31.4412			−1.10541			−0.552707	−	0.833376i	$-0.686405\pi$
$809$	−31.4412			−1.10541			−0.552707	+	0.833376i	$0.686405\pi$
$810$		0			0
$811$		−	13.0703i		−	0.458960i	−0.973313	−	0.229480i	$-0.926297\pi$
$811$		−	13.0703i		−	0.458960i	0.973313	−	0.229480i	$-0.0737026\pi$
$812$		0			0
$813$			53.7196i			1.88403i
$814$		0			0
$815$	−9.43136			−0.330366
$816$		0			0
$817$		−	0.847000i		−	0.0296328i
$818$		0			0
$819$	−0.231548	+	0.513881i	−0.00809094	+	0.0179565i
$820$		0			0
$821$			16.8919i			0.589532i	0.955570	+	0.294766i	$0.0952416\pi$
$821$			16.8919i			0.589532i	−0.955570	+	0.294766i	$0.904758\pi$
$822$		0			0
$823$	29.1222			1.01514			0.507568	−	0.861611i	$-0.330544\pi$
$823$	29.1222			1.01514			0.507568	+	0.861611i	$0.330544\pi$
$824$		0			0
$825$		−	6.41819i		−	0.223453i
$826$		0			0
$827$		−	32.0665i		−	1.11506i	−0.830156	−	0.557531i	$-0.811749\pi$
$827$		−	32.0665i		−	1.11506i	0.830156	−	0.557531i	$-0.188251\pi$
$828$		0			0
$829$	−19.3561			−0.672267			−0.336133	−	0.941814i	$-0.609119\pi$
$829$	−19.3561			−0.672267			−0.336133	+	0.941814i	$0.609119\pi$
$830$		0			0
$831$	−3.66291			−0.127065
$832$		0			0
$833$	10.2374			0.354706
$834$		0			0
$835$	14.8568			0.514142
$836$		0			0
$837$		−	55.6747i		−	1.92440i
$838$		0			0
$839$		−	12.4450i		−	0.429649i	−0.976653	−	0.214825i	$-0.931082\pi$
$839$		−	12.4450i		−	0.429649i	0.976653	−	0.214825i	$-0.0689180\pi$
$840$		0			0
$841$	42.7235			1.47323
$842$		0			0
$843$			3.13918i			0.108119i
$844$		0			0
$845$	−9.73813	+	8.61213i	−0.335002	+	0.296266i
$846$		0			0
$847$			2.96636i			0.101925i
$848$		0			0
$849$	0.503262			0.0172719
$850$		0			0
$851$			32.5501i			1.11580i
$852$		0			0
$853$		−	31.3054i		−	1.07187i	−0.844258	−	0.535937i	$-0.819958\pi$
$853$		−	31.3054i		−	1.07187i	0.844258	−	0.535937i	$-0.180042\pi$
$854$		0			0
$855$	0.168544			0.00576408
$856$		0			0
$857$	−45.8613			−1.56659			−0.783296	−	0.621649i	$-0.786463\pi$
$857$	−45.8613			−1.56659			−0.783296	+	0.621649i	$0.786463\pi$
$858$		0			0
$859$	34.2981			1.17023			0.585117	−	0.810949i	$-0.301048\pi$
$859$	34.2981			1.17023			0.585117	+	0.810949i	$0.301048\pi$
$860$		0			0
$861$	13.0475			0.444656
$862$		0			0
$863$			40.5052i			1.37881i	0.724375	+	0.689406i	$0.242128\pi$
$863$			40.5052i			1.37881i	−0.724375	+	0.689406i	$0.757872\pi$
$864$		0			0
$865$			11.2750i			0.383363i
$866$		0			0
$867$	24.1236			0.819282
$868$		0			0
$869$			13.5515i			0.459703i
$870$		0			0
$871$	−0.231548	+	0.513881i	−0.00784570	+	0.0174122i
$872$		0			0
$873$			1.37470i			0.0465265i
$874$		0			0
$875$	−0.806063			−0.0272499
$876$		0			0
$877$			31.1002i			1.05018i	0.851047	+	0.525089i	$0.175968\pi$
$877$			31.1002i			1.05018i	−0.851047	+	0.525089i	$0.824032\pi$
$878$		0			0
$879$			37.9102i			1.27868i
$880$		0			0
$881$	15.8192			0.532964			0.266482	−	0.963840i	$-0.414139\pi$
$881$	15.8192			0.532964			0.266482	+	0.963840i	$0.414139\pi$
$882$		0			0
$883$	−41.6408			−1.40133			−0.700663	−	0.713492i	$-0.747112\pi$
$883$	−41.6408			−1.40133			−0.700663	+	0.713492i	$0.747112\pi$
$884$		0			0
$885$	4.15633			0.139713
$886$		0			0
$887$	5.08603			0.170772			0.0853860	−	0.996348i	$-0.472788\pi$
$887$	5.08603			0.170772			0.0853860	+	0.996348i	$0.472788\pi$
$888$		0			0
$889$		−	4.42407i		−	0.148379i
$890$		0			0
$891$			32.1098i			1.07572i
$892$		0			0
$893$	2.10157			0.0703264
$894$		0			0
$895$			23.1998i			0.775484i
$896$		0			0
$897$	−29.1147	−	13.1187i	−0.972113	−	0.438021i
$898$		0			0
$899$		−	88.1279i		−	2.93923i
$900$		0			0
$901$	15.5778			0.518973
$902$		0			0
$903$		−	1.31598i		−	0.0437929i
$904$		0			0
$905$		−	22.3938i		−	0.744394i
$906$		0			0
$907$	16.2858			0.540763			0.270381	−	0.962753i	$-0.412850\pi$
$907$	16.2858			0.540763			0.270381	+	0.962753i	$0.412850\pi$
$908$		0			0
$909$	−0.664327			−0.0220343
$910$		0			0
$911$	−26.5745			−0.880453			−0.440226	−	0.897887i	$-0.645102\pi$
$911$	−26.5745			−0.880453			−0.440226	+	0.897887i	$0.645102\pi$
$912$		0			0
$913$	60.1279			1.98994
$914$		0			0
$915$			21.9756i			0.726490i
$916$		0			0
$917$		−	9.92478i		−	0.327745i
$918$		0			0
$919$	−46.1232			−1.52146			−0.760732	−	0.649067i	$-0.775160\pi$
$919$	−46.1232			−1.52146			−0.760732	+	0.649067i	$0.775160\pi$
$920$		0			0
$921$		−	47.4372i		−	1.56311i
$922$		0			0
$923$	−2.43629	+	5.40693i	−0.0801914	+	0.177971i
$924$		0			0
$925$		−	6.15633i		−	0.202419i
$926$		0			0
$927$	2.58673			0.0849595
$928$		0			0
$929$		−	7.78701i		−	0.255484i	−0.991807	−	0.127742i	$-0.959227\pi$
$929$		−	7.78701i		−	0.255484i	0.991807	−	0.127742i	$-0.0407729\pi$
$930$		0			0
$931$			5.51881i			0.180872i
$932$		0			0
$933$	−55.9102			−1.83042
$934$		0			0
$935$	6.17679			0.202003
$936$		0			0
$937$	−15.3601			−0.501793			−0.250896	−	0.968014i	$-0.580725\pi$
$937$	−15.3601			−0.501793			−0.250896	+	0.968014i	$0.580725\pi$
$938$		0			0
$939$	7.74798			0.252846
$940$		0			0
$941$			0.771007i			0.0251341i	0.999921	+	0.0125671i	$0.00400032\pi$
$941$			0.771007i			0.0251341i	−0.999921	+	0.0125671i	$0.996000\pi$
$942$		0			0
$943$		−	51.0903i		−	1.66373i
$944$		0			0
$945$	4.31265			0.140291
$946$		0			0
$947$		−	15.7948i		−	0.513262i	−0.966509	−	0.256631i	$-0.917387\pi$
$947$		−	15.7948i		−	0.513262i	0.966509	−	0.256631i	$-0.0826125\pi$
$948$		0			0
$949$	4.34297	−	9.63847i	0.140979	−	0.312878i
$950$		0			0
$951$			53.3865i			1.73117i
$952$		0			0
$953$	36.2374			1.17385			0.586923	−	0.809643i	$-0.300339\pi$
$953$	36.2374			1.17385			0.586923	+	0.809643i	$0.300339\pi$
$954$		0			0
$955$		−	1.55149i		−	0.0502051i
$956$		0			0
$957$			54.3555i			1.75706i
$958$		0			0
$959$	−3.06205			−0.0988787
$960$		0			0
$961$	−77.2842			−2.49304
$962$		0			0
$963$	1.15140			0.0371034
$964$		0			0
$965$	−4.57452			−0.147259
$966$		0			0
$967$		−	20.8832i		−	0.671558i	−0.941941	−	0.335779i	$-0.891000\pi$
$967$		−	20.8832i		−	0.671558i	0.941941	−	0.335779i	$-0.109000\pi$
$968$		0			0
$969$		−	2.34694i		−	0.0753945i
$970$		0			0
$971$	−45.0757			−1.44655			−0.723274	−	0.690561i	$-0.757364\pi$
$971$	−45.0757			−1.44655			−0.723274	+	0.690561i	$0.757364\pi$
$972$		0			0
$973$			10.5501i			0.338220i
$974$		0			0
$975$	5.50659	+	2.48119i	0.176352	+	0.0794618i
$976$		0			0
$977$			19.8094i			0.633758i	0.948466	+	0.316879i	$0.102635\pi$
$977$			19.8094i			0.633758i	−0.948466	+	0.316879i	$0.897365\pi$
$978$		0			0
$979$	9.14903			0.292404
$980$		0			0
$981$			0.0752228i			0.00240168i
$982$		0			0
$983$			57.6542i			1.83888i	0.393226	+	0.919442i	$0.371359\pi$
$983$			57.6542i			1.83888i	−0.393226	+	0.919442i	$0.628641\pi$
$984$		0			0
$985$	17.5125			0.557994
$986$		0			0
$987$	3.26519			0.103932
$988$		0			0
$989$	−5.15300			−0.163856
$990$		0			0
$991$	50.4650			1.60307			0.801537	−	0.597945i	$-0.204016\pi$
$991$	50.4650			1.60307			0.801537	+	0.597945i	$0.204016\pi$
$992$		0			0
$993$		−	38.3087i		−	1.21569i
$994$		0			0
$995$		−	9.79877i		−	0.310642i
$996$		0			0
$997$	−1.45183			−0.0459800			−0.0229900	−	0.999736i	$-0.507319\pi$
$997$	−1.45183			−0.0459800			−0.0229900	+	0.999736i	$0.507319\pi$
$998$		0			0
$999$			32.9380i			1.04211i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.2.k.a.441.2	yes	6
3.2	odd	2		4680.2.g.j.2521.2		6
4.3	odd	2		1040.2.k.b.961.6		6
5.2	odd	4		2600.2.f.d.649.5		6
5.3	odd	4		2600.2.f.c.649.2		6
5.4	even	2		2600.2.k.b.2001.6		6
13.5	odd	4		6760.2.a.v.1.1		3
13.8	odd	4		6760.2.a.u.1.1		3
13.12	even	2	inner	520.2.k.a.441.1	✓	6
39.38	odd	2		4680.2.g.j.2521.5		6
52.51	odd	2		1040.2.k.b.961.5		6
65.12	odd	4		2600.2.f.c.649.5		6
65.38	odd	4		2600.2.f.d.649.2		6
65.64	even	2		2600.2.k.b.2001.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.a.441.1	✓	6	13.12	even	2	inner
520.2.k.a.441.2	yes	6	1.1	even	1	trivial
1040.2.k.b.961.5		6	52.51	odd	2
1040.2.k.b.961.6		6	4.3	odd	2
2600.2.f.c.649.2		6	5.3	odd	4
2600.2.f.c.649.5		6	65.12	odd	4
2600.2.f.d.649.2		6	65.38	odd	4
2600.2.f.d.649.5		6	5.2	odd	4
2600.2.k.b.2001.5		6	65.64	even	2
2600.2.k.b.2001.6		6	5.4	even	2
4680.2.g.j.2521.2		6	3.2	odd	2
4680.2.g.j.2521.5		6	39.38	odd	2
6760.2.a.u.1.1		3	13.8	odd	4
6760.2.a.v.1.1		3	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.67513 q^{3} +1.00000i q^{5} -0.806063i q^{7} -0.193937 q^{9} -3.83146i q^{11} +(3.28726 + 1.48119i) q^{13} -1.67513i q^{15} +1.61213 q^{17} +0.869067i q^{19} +1.35026i q^{21} +5.28726 q^{23} -1.00000 q^{25} +5.35026 q^{27} +8.46898 q^{29} -10.4060i q^{31} +6.41819i q^{33} +0.806063 q^{35} +6.15633i q^{37} +(-5.50659 - 2.48119i) q^{39} -9.66291i q^{41} -0.974607 q^{43} -0.193937i q^{45} -2.41819i q^{47} +6.35026 q^{49} -2.70052 q^{51} +9.66291 q^{53} +3.83146 q^{55} -1.45580i q^{57} +2.48119i q^{59} -13.1187 q^{61} +0.156325i q^{63} +(-1.48119 + 3.28726i) q^{65} +0.156325i q^{67} -8.85685 q^{69} +1.64481i q^{71} -2.93207i q^{73} +1.67513 q^{75} -3.08840 q^{77} -3.53690 q^{79} -8.38058 q^{81} +15.6932i q^{83} +1.61213i q^{85} -14.1866 q^{87} +2.38787i q^{89} +(1.19394 - 2.64974i) q^{91} +17.4314i q^{93} -0.869067 q^{95} -7.08840i q^{97} +0.743059i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77}+ \cdots + 4 q^{95}+O(q^{100}) \) 6 * q - 2 * q^9 + 8 * q^13 + 8 * q^17 + 20 * q^23 - 6 * q^25 + 12 * q^27 - 12 * q^29 + 4 * q^35 + 8 * q^39 - 36 * q^43 + 18 * q^49 + 24 * q^51 - 4 * q^53 - 8 * q^55 - 36 * q^61 + 2 * q^65 + 8 * q^69 + 20 * q^77 + 24 * q^79 - 26 * q^81 - 60 * q^87 + 8 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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