LMFDB - Embedded newform 520.2.k.a.441.6

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)

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);

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")

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			441.6
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	520.441
Dual form			520.2.k.a.441.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432 q^{3} +1.00000i q^{5} -2.90321i q^{7} +1.90321 q^{9} +5.73975i q^{11} +(3.59210 - 0.311108i) q^{13} +2.21432i q^{15} +5.80642 q^{17} -5.11753i q^{19} -6.42864i q^{21} +5.59210 q^{23} -1.00000 q^{25} -2.42864 q^{27} -8.57628 q^{29} -1.44446i q^{31} +12.7096i q^{33} +2.90321 q^{35} +0.474572i q^{37} +(7.95407 - 0.688892i) q^{39} +9.47949i q^{41} -12.6430 q^{43} +1.90321i q^{45} -8.70964i q^{47} -1.42864 q^{49} +12.8573 q^{51} -9.47949 q^{53} -5.73975 q^{55} -11.3319i q^{57} +0.688892i q^{59} -3.85236 q^{61} -5.52543i q^{63} +(0.311108 + 3.59210i) q^{65} -5.52543i q^{67} +12.3827 q^{69} -12.7304i q^{71} +11.1383i q^{73} -2.21432 q^{75} +16.6637 q^{77} -0.561993 q^{79} -11.0874 q^{81} +7.03657i q^{83} +5.80642i q^{85} -18.9906 q^{87} -1.80642i q^{89} +(-0.903212 - 10.4286i) q^{91} -3.19850i q^{93} +5.11753 q^{95} +12.6637i q^{97} +10.9240i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.21432			1.27844			0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432			1.27844			0.639219	+	0.769025i	$0.279258\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	2.90321i		−	1.09731i	−0.836049	−	0.548655i	$-0.815140\pi$
$7$		−	2.90321i		−	1.09731i	0.836049	−	0.548655i	$-0.184860\pi$
$8$		0			0
$9$	1.90321			0.634404
$10$		0			0
$11$			5.73975i			1.73060i	0.501255	+	0.865299i	$0.332872\pi$
$11$			5.73975i			1.73060i	−0.501255	+	0.865299i	$0.667128\pi$
$12$		0			0
$13$	3.59210	−	0.311108i	0.996270	−	0.0862858i
$13$	3.59210	−	0.311108i	0.996270	−	0.0862858i
$14$		0			0
$15$			2.21432i			0.571735i
$16$		0			0
$17$	5.80642			1.40826			0.704132	−	0.710069i	$-0.251336\pi$
$17$	5.80642			1.40826			0.704132	+	0.710069i	$0.251336\pi$
$18$		0			0
$19$		−	5.11753i		−	1.17404i	−0.809572	−	0.587021i	$-0.800301\pi$
$19$		−	5.11753i		−	1.17404i	0.809572	−	0.587021i	$-0.199699\pi$
$20$		0			0
$21$		−	6.42864i		−	1.40284i
$22$		0			0
$23$	5.59210			1.16603			0.583017	−	0.812460i	$-0.301872\pi$
$23$	5.59210			1.16603			0.583017	+	0.812460i	$0.301872\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−2.42864			−0.467392
$28$		0			0
$29$	−8.57628			−1.59258			−0.796288	−	0.604918i	$-0.793206\pi$
$29$	−8.57628			−1.59258			−0.796288	+	0.604918i	$0.793206\pi$
$30$		0			0
$31$		−	1.44446i		−	0.259433i	−0.991551	−	0.129716i	$-0.958593\pi$
$31$		−	1.44446i		−	0.259433i	0.991551	−	0.129716i	$-0.0414067\pi$
$32$		0			0
$33$			12.7096i			2.21246i
$34$		0			0
$35$	2.90321			0.490732
$36$		0			0
$37$			0.474572i			0.0780192i	0.999239	+	0.0390096i	$0.0124203\pi$
$37$			0.474572i			0.0780192i	−0.999239	+	0.0390096i	$0.987580\pi$
$38$		0			0
$39$	7.95407	−	0.688892i	1.27367	−	0.110311i
$40$		0			0
$41$			9.47949i			1.48045i	0.672360	+	0.740224i	$0.265281\pi$
$41$			9.47949i			1.48045i	−0.672360	+	0.740224i	$0.734719\pi$
$42$		0			0
$43$	−12.6430			−1.92803			−0.964017	−	0.265842i	$-0.914350\pi$
$43$	−12.6430			−1.92803			−0.964017	+	0.265842i	$0.914350\pi$
$44$		0			0
$45$			1.90321i			0.283714i
$46$		0			0
$47$		−	8.70964i		−	1.27043i	−0.772335	−	0.635215i	$-0.780911\pi$
$47$		−	8.70964i		−	1.27043i	0.772335	−	0.635215i	$-0.219089\pi$
$48$		0			0
$49$	−1.42864			−0.204091
$50$		0			0
$51$	12.8573			1.80038
$52$		0			0
$53$	−9.47949			−1.30211			−0.651054	−	0.759031i	$-0.725673\pi$
$53$	−9.47949			−1.30211			−0.651054	+	0.759031i	$0.725673\pi$
$54$		0			0
$55$	−5.73975			−0.773947
$56$		0			0
$57$		−	11.3319i		−	1.50094i
$58$		0			0
$59$			0.688892i			0.0896861i	0.998994	+	0.0448431i	$0.0142788\pi$
$59$			0.688892i			0.0896861i	−0.998994	+	0.0448431i	$0.985721\pi$
$60$		0			0
$61$	−3.85236			−0.493244			−0.246622	−	0.969112i	$-0.579321\pi$
$61$	−3.85236			−0.493244			−0.246622	+	0.969112i	$0.579321\pi$
$62$		0			0
$63$		−	5.52543i		−	0.696138i
$64$		0			0
$65$	0.311108	+	3.59210i	0.0385882	+	0.445546i
$66$		0			0
$67$		−	5.52543i		−	0.675038i	−0.941319	−	0.337519i	$-0.890412\pi$
$67$		−	5.52543i		−	0.675038i	0.941319	−	0.337519i	$-0.109588\pi$
$68$		0			0
$69$	12.3827			1.49070
$70$		0			0
$71$		−	12.7304i		−	1.51082i	−0.655254	−	0.755409i	$-0.727438\pi$
$71$		−	12.7304i		−	1.51082i	0.655254	−	0.755409i	$-0.272562\pi$
$72$		0			0
$73$			11.1383i			1.30364i	0.758375	+	0.651818i	$0.225994\pi$
$73$			11.1383i			1.30364i	−0.758375	+	0.651818i	$0.774006\pi$
$74$		0			0
$75$	−2.21432			−0.255688
$76$		0			0
$77$	16.6637			1.89901
$78$		0			0
$79$	−0.561993			−0.0632291			−0.0316146	−	0.999500i	$-0.510065\pi$
$79$	−0.561993			−0.0632291			−0.0316146	+	0.999500i	$0.510065\pi$
$80$		0			0
$81$	−11.0874			−1.23194
$82$		0			0
$83$			7.03657i			0.772363i	0.922423	+	0.386182i	$0.126206\pi$
$83$			7.03657i			0.772363i	−0.922423	+	0.386182i	$0.873794\pi$
$84$		0			0
$85$			5.80642i			0.629795i
$86$		0			0
$87$	−18.9906			−2.03601
$88$		0			0
$89$		−	1.80642i		−	0.191481i	−0.995406	−	0.0957403i	$-0.969478\pi$
$89$		−	1.80642i		−	0.191481i	0.995406	−	0.0957403i	$-0.0305218\pi$
$90$		0			0
$91$	−0.903212	−	10.4286i	−0.0946823	−	1.09322i
$92$		0			0
$93$		−	3.19850i		−	0.331669i
$94$		0			0
$95$	5.11753			0.525048
$96$		0			0
$97$			12.6637i			1.28580i	0.765948	+	0.642902i	$0.222270\pi$
$97$			12.6637i			1.28580i	−0.765948	+	0.642902i	$0.777730\pi$
$98$		0			0
$99$			10.9240i			1.09790i
$100$		0			0
$101$	2.81579			0.280182			0.140091	−	0.990139i	$-0.455261\pi$
$101$	2.81579			0.280182			0.140091	+	0.990139i	$0.455261\pi$
$102$		0			0
$103$	9.69381			0.955160			0.477580	−	0.878588i	$-0.341514\pi$
$103$	9.69381			0.955160			0.477580	+	0.878588i	$0.341514\pi$
$104$		0			0
$105$	6.42864			0.627371
$106$		0			0
$107$	−14.0207			−1.35544			−0.677718	−	0.735322i	$-0.737031\pi$
$107$	−14.0207			−1.35544			−0.677718	+	0.735322i	$0.737031\pi$
$108$		0			0
$109$			3.80642i			0.364589i	0.983244	+	0.182295i	$0.0583525\pi$
$109$			3.80642i			0.364589i	−0.983244	+	0.182295i	$0.941648\pi$
$110$		0			0
$111$			1.05086i			0.0997428i
$112$		0			0
$113$	−0.387152			−0.0364202			−0.0182101	−	0.999834i	$-0.505797\pi$
$113$	−0.387152			−0.0364202			−0.0182101	+	0.999834i	$0.505797\pi$
$114$		0			0
$115$			5.59210i			0.521466i
$116$		0			0
$117$	6.83654	−	0.592104i	0.632038	−	0.0547400i
$118$		0			0
$119$		−	16.8573i		−	1.54530i
$120$		0			0
$121$	−21.9447			−1.99497
$122$		0			0
$123$			20.9906i			1.89266i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−3.20495			−0.284393			−0.142197	−	0.989838i	$-0.545417\pi$
$127$	−3.20495			−0.284393			−0.142197	+	0.989838i	$0.545417\pi$
$128$		0			0
$129$	−27.9956			−2.46487
$130$		0			0
$131$	0.949145			0.0829272			0.0414636	−	0.999140i	$-0.486798\pi$
$131$	0.949145			0.0829272			0.0414636	+	0.999140i	$0.486798\pi$
$132$		0			0
$133$	−14.8573			−1.28829
$134$		0			0
$135$		−	2.42864i		−	0.209024i
$136$		0			0
$137$		−	12.7971i		−	1.09333i	−0.837353	−	0.546663i	$-0.815898\pi$
$137$		−	12.7971i		−	1.09333i	0.837353	−	0.546663i	$-0.184102\pi$
$138$		0			0
$139$	6.66370			0.565208			0.282604	−	0.959237i	$-0.408802\pi$
$139$	6.66370			0.565208			0.282604	+	0.959237i	$0.408802\pi$
$140$		0			0
$141$		−	19.2859i		−	1.62417i
$142$		0			0
$143$	1.78568	+	20.6178i	0.149326	+	1.72414i
$144$		0			0
$145$		−	8.57628i		−	0.712222i
$146$		0			0
$147$	−3.16346			−0.260918
$148$		0			0
$149$		−	6.94914i		−	0.569296i	−0.958632	−	0.284648i	$-0.908123\pi$
$149$		−	6.94914i		−	0.569296i	0.958632	−	0.284648i	$-0.0918767\pi$
$150$		0			0
$151$		−	8.73038i		−	0.710468i	−0.934777	−	0.355234i	$-0.884401\pi$
$151$		−	8.73038i		−	0.710468i	0.934777	−	0.355234i	$-0.115599\pi$
$152$		0			0
$153$	11.0509			0.893409
$154$		0			0
$155$	1.44446			0.116022
$156$		0			0
$157$	−0.428639			−0.0342092			−0.0171046	−	0.999854i	$-0.505445\pi$
$157$	−0.428639			−0.0342092			−0.0171046	+	0.999854i	$0.505445\pi$
$158$		0			0
$159$	−20.9906			−1.66467
$160$		0			0
$161$		−	16.2351i		−	1.27950i
$162$		0			0
$163$		−	11.1985i		−	0.877134i	−0.898698	−	0.438567i	$-0.855486\pi$
$163$		−	11.1985i		−	0.877134i	0.898698	−	0.438567i	$-0.144514\pi$
$164$		0			0
$165$	−12.7096			−0.989444
$166$		0			0
$167$			6.38271i			0.493909i	0.969027	+	0.246954i	$0.0794298\pi$
$167$			6.38271i			0.493909i	−0.969027	+	0.246954i	$0.920570\pi$
$168$		0			0
$169$	12.8064	−	2.23506i	0.985110	−	0.171928i
$170$		0			0
$171$		−	9.73975i		−	0.744817i
$172$		0			0
$173$	−3.67307			−0.279258			−0.139629	−	0.990204i	$-0.544591\pi$
$173$	−3.67307			−0.279258			−0.139629	+	0.990204i	$0.544591\pi$
$174$		0			0
$175$			2.90321i			0.219462i
$176$		0			0
$177$			1.52543i			0.114658i
$178$		0			0
$179$	1.08250			0.0809097			0.0404548	−	0.999181i	$-0.487119\pi$
$179$	1.08250			0.0809097			0.0404548	+	0.999181i	$0.487119\pi$
$180$		0			0
$181$	1.82071			0.135333			0.0676663	−	0.997708i	$-0.478445\pi$
$181$	1.82071			0.135333			0.0676663	+	0.997708i	$0.478445\pi$
$182$		0			0
$183$	−8.53035			−0.630582
$184$		0			0
$185$	−0.474572			−0.0348913
$186$		0			0
$187$			33.3274i			2.43714i
$188$		0			0
$189$			7.05086i			0.512874i
$190$		0			0
$191$	15.2257			1.10169			0.550846	−	0.834607i	$-0.314305\pi$
$191$	15.2257			1.10169			0.550846	+	0.834607i	$0.314305\pi$
$192$		0			0
$193$			5.18421i			0.373167i	0.982439	+	0.186584i	$0.0597416\pi$
$193$			5.18421i			0.373167i	−0.982439	+	0.186584i	$0.940258\pi$
$194$		0			0
$195$	0.688892	+	7.95407i	0.0493326	+	0.569603i
$196$		0			0
$197$			15.9684i			1.13770i	0.822442	+	0.568849i	$0.192611\pi$
$197$			15.9684i			1.13770i	−0.822442	+	0.568849i	$0.807389\pi$
$198$		0			0
$199$	−18.7971			−1.33249			−0.666244	−	0.745734i	$-0.732099\pi$
$199$	−18.7971			−1.33249			−0.666244	+	0.745734i	$0.732099\pi$
$200$		0			0
$201$		−	12.2351i		−	0.862995i
$202$		0			0
$203$			24.8988i			1.74755i
$204$		0			0
$205$	−9.47949			−0.662077
$206$		0			0
$207$	10.6430			0.739737
$208$		0			0
$209$	29.3733			2.03180
$210$		0			0
$211$	−6.91750			−0.476220			−0.238110	−	0.971238i	$-0.576528\pi$
$211$	−6.91750			−0.476220			−0.238110	+	0.971238i	$0.576528\pi$
$212$		0			0
$213$		−	28.1891i		−	1.93149i
$214$		0			0
$215$		−	12.6430i		−	0.862243i
$216$		0			0
$217$	−4.19358			−0.284678
$218$		0			0
$219$			24.6637i			1.66662i
$220$		0			0
$221$	20.8573	−	1.80642i	1.40301	−	0.121513i
$222$		0			0
$223$		−	5.56691i		−	0.372788i	−0.982475	−	0.186394i	$-0.940320\pi$
$223$		−	5.56691i		−	0.372788i	0.982475	−	0.186394i	$-0.0596801\pi$
$224$		0			0
$225$	−1.90321			−0.126881
$226$		0			0
$227$		−	9.43356i		−	0.626127i	−0.949732	−	0.313064i	$-0.898645\pi$
$227$		−	9.43356i		−	0.626127i	0.949732	−	0.313064i	$-0.101355\pi$
$228$		0			0
$229$		−	16.0415i		−	1.06005i	−0.847982	−	0.530026i	$-0.822182\pi$
$229$		−	16.0415i		−	1.06005i	0.847982	−	0.530026i	$-0.177818\pi$
$230$		0			0
$231$	36.8988			2.42776
$232$		0			0
$233$	28.6637			1.87782			0.938911	−	0.344161i	$-0.111836\pi$
$233$	28.6637			1.87782			0.938911	+	0.344161i	$0.111836\pi$
$234$		0			0
$235$	8.70964			0.568154
$236$		0			0
$237$	−1.24443			−0.0808345
$238$		0			0
$239$			21.7397i			1.40623i	0.711077	+	0.703114i	$0.248208\pi$
$239$			21.7397i			1.40623i	−0.711077	+	0.703114i	$0.751792\pi$
$240$		0			0
$241$		−	12.5303i		−	0.807151i	−0.914946	−	0.403575i	$-0.867767\pi$
$241$		−	12.5303i		−	0.807151i	0.914946	−	0.403575i	$-0.132233\pi$
$242$		0			0
$243$	−17.2652			−1.10756
$244$		0			0
$245$		−	1.42864i		−	0.0912724i
$246$		0			0
$247$	−1.59210	−	18.3827i	−0.101303	−	1.16966i
$248$		0			0
$249$			15.5812i			0.987419i
$250$		0			0
$251$	17.8479			1.12655			0.563275	−	0.826269i	$-0.309541\pi$
$251$	17.8479			1.12655			0.563275	+	0.826269i	$0.309541\pi$
$252$		0			0
$253$			32.0973i			2.01794i
$254$		0			0
$255$			12.8573i			0.805154i
$256$		0			0
$257$	21.6128			1.34817			0.674086	−	0.738653i	$-0.264538\pi$
$257$	21.6128			1.34817			0.674086	+	0.738653i	$0.264538\pi$
$258$		0			0
$259$	1.37778			0.0856114
$260$		0			0
$261$	−16.3225			−1.01034
$262$		0			0
$263$	−3.84590			−0.237149			−0.118574	−	0.992945i	$-0.537832\pi$
$263$	−3.84590			−0.237149			−0.118574	+	0.992945i	$0.537832\pi$
$264$		0			0
$265$		−	9.47949i		−	0.582321i
$266$		0			0
$267$		−	4.00000i		−	0.244796i
$268$		0			0
$269$	−5.57136			−0.339692			−0.169846	−	0.985471i	$-0.554327\pi$
$269$	−5.57136			−0.339692			−0.169846	+	0.985471i	$0.554327\pi$
$270$		0			0
$271$		−	3.96497i		−	0.240855i	−0.992722	−	0.120427i	$-0.961574\pi$
$271$		−	3.96497i		−	0.240855i	0.992722	−	0.120427i	$-0.0384265\pi$
$272$		0			0
$273$	−2.00000	−	23.0923i	−0.121046	−	1.39761i
$274$		0			0
$275$		−	5.73975i		−	0.346120i
$276$		0			0
$277$	6.99063			0.420026			0.210013	−	0.977699i	$-0.432649\pi$
$277$	6.99063			0.420026			0.210013	+	0.977699i	$0.432649\pi$
$278$		0			0
$279$		−	2.74912i		−	0.164585i
$280$		0			0
$281$		−	18.0415i		−	1.07626i	−0.842860	−	0.538132i	$-0.819130\pi$
$281$		−	18.0415i		−	1.07626i	0.842860	−	0.538132i	$-0.180870\pi$
$282$		0			0
$283$	26.3160			1.56433			0.782163	−	0.623074i	$-0.214117\pi$
$283$	26.3160			1.56433			0.782163	+	0.623074i	$0.214117\pi$
$284$		0			0
$285$	11.3319			0.671241
$286$		0			0
$287$	27.5210			1.62451
$288$		0			0
$289$	16.7146			0.983209
$290$		0			0
$291$			28.0415i			1.64382i
$292$		0			0
$293$			20.1160i			1.17519i	0.809155	+	0.587595i	$0.199925\pi$
$293$			20.1160i			1.17519i	−0.809155	+	0.587595i	$0.800075\pi$
$294$		0			0
$295$	−0.688892			−0.0401089
$296$		0			0
$297$		−	13.9398i		−	0.808868i
$298$		0			0
$299$	20.0874	−	1.73975i	1.16169	−	0.100612i
$300$		0			0
$301$			36.7052i			2.11565i
$302$		0			0
$303$	6.23506			0.358195
$304$		0			0
$305$		−	3.85236i		−	0.220585i
$306$		0			0
$307$		−	3.06515i		−	0.174937i	−0.996167	−	0.0874685i	$-0.972122\pi$
$307$		−	3.06515i		−	0.174937i	0.996167	−	0.0874685i	$-0.0278777\pi$
$308$		0			0
$309$	21.4652			1.22111
$310$		0			0
$311$	−28.2449			−1.60162			−0.800811	−	0.598917i	$-0.795598\pi$
$311$	−28.2449			−1.60162			−0.800811	+	0.598917i	$0.795598\pi$
$312$		0			0
$313$	18.1017			1.02317			0.511585	−	0.859233i	$-0.329059\pi$
$313$	18.1017			1.02317			0.511585	+	0.859233i	$0.329059\pi$
$314$		0			0
$315$	5.52543			0.311323
$316$		0			0
$317$			16.2908i			0.914985i	0.889214	+	0.457492i	$0.151252\pi$
$317$			16.2908i			0.914985i	−0.889214	+	0.457492i	$0.848748\pi$
$318$		0			0
$319$		−	49.2257i		−	2.75611i
$320$		0			0
$321$	−31.0464			−1.73284
$322$		0			0
$323$		−	29.7146i		−	1.65336i
$324$		0			0
$325$	−3.59210	+	0.311108i	−0.199254	+	0.0172572i
$326$		0			0
$327$			8.42864i			0.466105i
$328$		0			0
$329$	−25.2859			−1.39406
$330$		0			0
$331$			16.8825i			0.927944i	0.885850	+	0.463972i	$0.153576\pi$
$331$			16.8825i			0.927944i	−0.885850	+	0.463972i	$0.846424\pi$
$332$		0			0
$333$			0.903212i			0.0494957i
$334$		0			0
$335$	5.52543			0.301886
$336$		0			0
$337$	−6.78415			−0.369556			−0.184778	−	0.982780i	$-0.559157\pi$
$337$	−6.78415			−0.369556			−0.184778	+	0.982780i	$0.559157\pi$
$338$		0			0
$339$	−0.857279			−0.0465610
$340$		0			0
$341$	8.29084			0.448974
$342$		0			0
$343$		−	16.1748i		−	0.873359i
$344$		0			0
$345$			12.3827i			0.666663i
$346$		0			0
$347$	−16.0207			−0.860039			−0.430019	−	0.902820i	$-0.641493\pi$
$347$	−16.0207			−0.860039			−0.430019	+	0.902820i	$0.641493\pi$
$348$		0			0
$349$		−	14.7052i		−	0.787151i	−0.919292	−	0.393575i	$-0.871238\pi$
$349$		−	14.7052i		−	0.787151i	0.919292	−	0.393575i	$-0.128762\pi$
$350$		0			0
$351$	−8.72393	+	0.755569i	−0.465649	+	0.0403293i
$352$		0			0
$353$			1.81087i			0.0963829i	0.998838	+	0.0481914i	$0.0153458\pi$
$353$			1.81087i			0.0963829i	−0.998838	+	0.0481914i	$0.984654\pi$
$354$		0			0
$355$	12.7304			0.675658
$356$		0			0
$357$		−	37.3274i		−	1.97558i
$358$		0			0
$359$			7.01582i			0.370281i	0.982712	+	0.185140i	$0.0592740\pi$
$359$			7.01582i			0.370281i	−0.982712	+	0.185140i	$0.940726\pi$
$360$		0			0
$361$	−7.18913			−0.378375
$362$		0			0
$363$	−48.5926			−2.55045
$364$		0			0
$365$	−11.1383			−0.583004
$366$		0			0
$367$	−2.43954			−0.127343			−0.0636714	−	0.997971i	$-0.520281\pi$
$367$	−2.43954			−0.127343			−0.0636714	+	0.997971i	$0.520281\pi$
$368$		0			0
$369$			18.0415i			0.939202i
$370$		0			0
$371$			27.5210i			1.42882i
$372$		0			0
$373$	−16.4286			−0.850643			−0.425321	−	0.905042i	$-0.639839\pi$
$373$	−16.4286			−0.850643			−0.425321	+	0.905042i	$0.639839\pi$
$374$		0			0
$375$		−	2.21432i		−	0.114347i
$376$		0			0
$377$	−30.8069	+	2.66815i	−1.58664	+	0.137417i
$378$		0			0
$379$		−	0.0251894i		−	0.00129389i	−1.00000		0.000646945i	$-0.999794\pi$
$379$		−	0.0251894i		−	0.00129389i	1.00000		0.000646945i	$-0.000205929\pi$
$380$		0			0
$381$	−7.09679			−0.363579
$382$		0			0
$383$		−	3.89384i		−	0.198966i	−0.995039	−	0.0994831i	$-0.968281\pi$
$383$		−	3.89384i		−	0.198966i	0.995039	−	0.0994831i	$-0.0317189\pi$
$384$		0			0
$385$			16.6637i			0.849261i
$386$		0			0
$387$	−24.0622			−1.22315
$388$		0			0
$389$	24.1017			1.22201			0.611003	−	0.791629i	$-0.290766\pi$
$389$	24.1017			1.22201			0.611003	+	0.791629i	$0.290766\pi$
$390$		0			0
$391$	32.4701			1.64208
$392$		0			0
$393$	2.10171			0.106017
$394$		0			0
$395$		−	0.561993i		−	0.0282769i
$396$		0			0
$397$			33.2114i			1.66683i	0.552646	+	0.833416i	$0.313618\pi$
$397$			33.2114i			1.66683i	−0.552646	+	0.833416i	$0.686382\pi$
$398$		0			0
$399$	−32.8988			−1.64700
$400$		0			0
$401$			35.6958i			1.78256i	0.453449	+	0.891282i	$0.350193\pi$
$401$			35.6958i			1.78256i	−0.453449	+	0.891282i	$0.649807\pi$
$402$		0			0
$403$	−0.449383	−	5.18865i	−0.0223854	−	0.258465i
$404$		0			0
$405$		−	11.0874i		−	0.550938i
$406$		0			0
$407$	−2.72393			−0.135020
$408$		0			0
$409$		−	25.8666i		−	1.27902i	−0.768781	−	0.639512i	$-0.779137\pi$
$409$		−	25.8666i		−	1.27902i	0.768781	−	0.639512i	$-0.220863\pi$
$410$		0			0
$411$		−	28.3368i		−	1.39775i
$412$		0			0
$413$	2.00000			0.0984136
$414$		0			0
$415$	−7.03657			−0.345411
$416$		0			0
$417$	14.7556			0.722583
$418$		0			0
$419$	12.6035			0.615720			0.307860	−	0.951432i	$-0.400387\pi$
$419$	12.6035			0.615720			0.307860	+	0.951432i	$0.400387\pi$
$420$		0			0
$421$			14.3783i			0.700754i	0.936609	+	0.350377i	$0.113946\pi$
$421$			14.3783i			0.700754i	−0.936609	+	0.350377i	$0.886054\pi$
$422$		0			0
$423$		−	16.5763i		−	0.805966i
$424$		0			0
$425$	−5.80642			−0.281653
$426$		0			0
$427$			11.1842i			0.541242i
$428$		0			0
$429$	3.95407	+	45.6543i	0.190904	+	2.20421i
$430$		0			0
$431$			20.2701i			0.976376i	0.872738	+	0.488188i	$0.162342\pi$
$431$			20.2701i			0.976376i	−0.872738	+	0.488188i	$0.837658\pi$
$432$		0			0
$433$	9.34614			0.449147			0.224573	−	0.974457i	$-0.427901\pi$
$433$	9.34614			0.449147			0.224573	+	0.974457i	$0.427901\pi$
$434$		0			0
$435$		−	18.9906i		−	0.910531i
$436$		0			0
$437$		−	28.6178i		−	1.36897i
$438$		0			0
$439$	−6.50177			−0.310313			−0.155156	−	0.987890i	$-0.549588\pi$
$439$	−6.50177			−0.310313			−0.155156	+	0.987890i	$0.549588\pi$
$440$		0			0
$441$	−2.71900			−0.129476
$442$		0			0
$443$	21.4686			1.02000			0.510002	−	0.860173i	$-0.329645\pi$
$443$	21.4686			1.02000			0.510002	+	0.860173i	$0.329645\pi$
$444$		0			0
$445$	1.80642			0.0856327
$446$		0			0
$447$		−	15.3876i		−	0.727810i
$448$		0			0
$449$			5.83807i			0.275515i	0.990466	+	0.137758i	$0.0439895\pi$
$449$			5.83807i			0.275515i	−0.990466	+	0.137758i	$0.956010\pi$
$450$		0			0
$451$	−54.4099			−2.56206
$452$		0			0
$453$		−	19.3319i		−	0.908290i
$454$		0			0
$455$	10.4286	−	0.903212i	0.488902	−	0.0423432i
$456$		0			0
$457$			12.0731i			0.564757i	0.959303	+	0.282379i	$0.0911234\pi$
$457$			12.0731i			0.564757i	−0.959303	+	0.282379i	$0.908877\pi$
$458$		0			0
$459$	−14.1017			−0.658211
$460$		0			0
$461$		−	2.91750i		−	0.135882i	−0.997689	−	0.0679408i	$-0.978357\pi$
$461$		−	2.91750i		−	0.135882i	0.997689	−	0.0679408i	$-0.0216429\pi$
$462$		0			0
$463$			13.9826i			0.649829i	0.945743	+	0.324914i	$0.105335\pi$
$463$			13.9826i			0.649829i	−0.945743	+	0.324914i	$0.894665\pi$
$464$		0			0
$465$	3.19850			0.148327
$466$		0			0
$467$	12.5698			0.581662			0.290831	−	0.956774i	$-0.406068\pi$
$467$	12.5698			0.581662			0.290831	+	0.956774i	$0.406068\pi$
$468$		0			0
$469$	−16.0415			−0.740727
$470$		0			0
$471$	−0.949145			−0.0437343
$472$		0			0
$473$		−	72.5674i		−	3.33665i
$474$		0			0
$475$			5.11753i			0.234808i
$476$		0			0
$477$	−18.0415			−0.826063
$478$		0			0
$479$			39.0988i			1.78647i	0.449590	+	0.893235i	$0.351570\pi$
$479$			39.0988i			1.78647i	−0.449590	+	0.893235i	$0.648430\pi$
$480$		0			0
$481$	0.147643	+	1.70471i	0.00673195	+	0.0777282i
$482$		0			0
$483$		−	35.9496i		−	1.63576i
$484$		0			0
$485$	−12.6637			−0.575029
$486$		0			0
$487$			17.9541i			0.813576i	0.913522	+	0.406788i	$0.133351\pi$
$487$			17.9541i			0.813576i	−0.913522	+	0.406788i	$0.866649\pi$
$488$		0			0
$489$		−	24.7971i		−	1.12136i
$490$		0			0
$491$	0.857279			0.0386885			0.0193442	−	0.999813i	$-0.493842\pi$
$491$	0.857279			0.0386885			0.0193442	+	0.999813i	$0.493842\pi$
$492$		0			0
$493$	−49.7975			−2.24277
$494$		0			0
$495$	−10.9240			−0.490995
$496$		0			0
$497$	−36.9590			−1.65784
$498$		0			0
$499$		−	36.8134i		−	1.64799i	−0.566596	−	0.823996i	$-0.691740\pi$
$499$		−	36.8134i		−	1.64799i	0.566596	−	0.823996i	$-0.308260\pi$
$500$		0			0
$501$			14.1334i			0.631432i
$502$		0			0
$503$	28.9382			1.29029			0.645146	−	0.764059i	$-0.276796\pi$
$503$	28.9382			1.29029			0.645146	+	0.764059i	$0.276796\pi$
$504$		0			0
$505$			2.81579i			0.125301i
$506$		0			0
$507$	28.3575	−	4.94914i	1.25940	−	0.219799i
$508$		0			0
$509$		−	37.8765i		−	1.67885i	−0.543479	−	0.839423i	$-0.682893\pi$
$509$		−	37.8765i		−	1.67885i	0.543479	−	0.839423i	$-0.317107\pi$
$510$		0			0
$511$	32.3368			1.43049
$512$		0			0
$513$			12.4286i			0.548738i
$514$		0			0
$515$			9.69381i			0.427160i
$516$		0			0
$517$	49.9911			2.19861
$518$		0			0
$519$	−8.13335			−0.357015
$520$		0			0
$521$	2.38271			0.104388			0.0521941	−	0.998637i	$-0.483379\pi$
$521$	2.38271			0.104388			0.0521941	+	0.998637i	$0.483379\pi$
$522$		0			0
$523$	−8.42080			−0.368216			−0.184108	−	0.982906i	$-0.558940\pi$
$523$	−8.42080			−0.368216			−0.184108	+	0.982906i	$0.558940\pi$
$524$		0			0
$525$			6.42864i			0.280569i
$526$		0			0
$527$		−	8.38715i		−	0.365350i
$528$		0			0
$529$	8.27163			0.359636
$530$		0			0
$531$			1.31111i			0.0568972i
$532$		0			0
$533$	2.94914	+	34.0513i	0.127742	+	1.47493i
$534$		0			0
$535$		−	14.0207i		−	0.606170i
$536$		0			0
$537$	2.39700			0.103438
$538$		0			0
$539$		−	8.20003i		−	0.353200i
$540$		0			0
$541$			22.5018i			0.967427i	0.875227	+	0.483713i	$0.160712\pi$
$541$			22.5018i			0.967427i	−0.875227	+	0.483713i	$0.839288\pi$
$542$		0			0
$543$	4.03164			0.173014
$544$		0			0
$545$	−3.80642			−0.163049
$546$		0			0
$547$	10.4494			0.446783			0.223392	−	0.974729i	$-0.428287\pi$
$547$	10.4494			0.446783			0.223392	+	0.974729i	$0.428287\pi$
$548$		0			0
$549$	−7.33185			−0.312916
$550$		0			0
$551$			43.8894i			1.86975i
$552$		0			0
$553$			1.63158i			0.0693820i
$554$		0			0
$555$	−1.05086			−0.0446063
$556$		0			0
$557$		−	10.8716i		−	0.460643i	−0.973115	−	0.230322i	$-0.926022\pi$
$557$		−	10.8716i		−	0.460643i	0.973115	−	0.230322i	$-0.0739778\pi$
$558$		0			0
$559$	−45.4148	+	3.93332i	−1.92084	+	0.166362i
$560$		0			0
$561$			73.7975i			3.11573i
$562$		0			0
$563$	−40.1354			−1.69150			−0.845752	−	0.533576i	$-0.820848\pi$
$563$	−40.1354			−1.69150			−0.845752	+	0.533576i	$0.820848\pi$
$564$		0			0
$565$		−	0.387152i		−	0.0162876i
$566$		0			0
$567$			32.1891i			1.35182i
$568$		0			0
$569$	20.3511			0.853161			0.426580	−	0.904450i	$-0.359718\pi$
$569$	20.3511			0.853161			0.426580	+	0.904450i	$0.359718\pi$
$570$		0			0
$571$	12.9491			0.541905			0.270952	−	0.962593i	$-0.412661\pi$
$571$	12.9491			0.541905			0.270952	+	0.962593i	$0.412661\pi$
$572$		0			0
$573$	33.7146			1.40845
$574$		0			0
$575$	−5.59210			−0.233207
$576$		0			0
$577$		−	16.1160i		−	0.670918i	−0.942055	−	0.335459i	$-0.891109\pi$
$577$		−	16.1160i		−	0.670918i	0.942055	−	0.335459i	$-0.108891\pi$
$578$		0			0
$579$			11.4795i			0.477072i
$580$		0			0
$581$	20.4286			0.847523
$582$		0			0
$583$		−	54.4099i		−	2.25343i
$584$		0			0
$585$	0.592104	+	6.83654i	0.0244805	+	0.282656i
$586$		0			0
$587$		−	28.1476i		−	1.16178i	−0.813983	−	0.580889i	$-0.802705\pi$
$587$		−	28.1476i		−	1.16178i	0.813983	−	0.580889i	$-0.197295\pi$
$588$		0			0
$589$	−7.39207			−0.304585
$590$		0			0
$591$			35.3590i			1.45448i
$592$		0			0
$593$			14.3783i			0.590444i	0.955429	+	0.295222i	$0.0953937\pi$
$593$			14.3783i			0.590444i	−0.955429	+	0.295222i	$0.904606\pi$
$594$		0			0
$595$	16.8573			0.691081
$596$		0			0
$597$	−41.6227			−1.70350
$598$		0			0
$599$	−15.5714			−0.636229			−0.318114	−	0.948052i	$-0.603050\pi$
$599$	−15.5714			−0.636229			−0.318114	+	0.948052i	$0.603050\pi$
$600$		0			0
$601$	27.9813			1.14138			0.570690	−	0.821166i	$-0.306676\pi$
$601$	27.9813			1.14138			0.570690	+	0.821166i	$0.306676\pi$
$602$		0			0
$603$		−	10.5161i		−	0.428247i
$604$		0			0
$605$		−	21.9447i		−	0.892179i
$606$		0			0
$607$	−17.3254			−0.703216			−0.351608	−	0.936147i	$-0.614365\pi$
$607$	−17.3254			−0.703216			−0.351608	+	0.936147i	$0.614365\pi$
$608$		0			0
$609$			55.1338i			2.23414i
$610$		0			0
$611$	−2.70964	−	31.2859i	−0.109620	−	1.26569i
$612$		0			0
$613$		−	40.6548i		−	1.64203i	−0.570905	−	0.821016i	$-0.693408\pi$
$613$		−	40.6548i		−	1.64203i	0.570905	−	0.821016i	$-0.306592\pi$
$614$		0			0
$615$	−20.9906			−0.846424
$616$		0			0
$617$			4.91750i			0.197971i	0.995089	+	0.0989856i	$0.0315598\pi$
$617$			4.91750i			0.197971i	−0.995089	+	0.0989856i	$0.968440\pi$
$618$		0			0
$619$		−	12.4351i		−	0.499809i	−0.968271	−	0.249904i	$-0.919601\pi$
$619$		−	12.4351i		−	0.499809i	0.968271	−	0.249904i	$-0.0803992\pi$
$620$		0			0
$621$	−13.5812			−0.544995
$622$		0			0
$623$	−5.24443			−0.210114
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	65.0420			2.59753
$628$		0			0
$629$			2.75557i			0.109872i
$630$		0			0
$631$		−	25.3526i		−	1.00927i	−0.863333	−	0.504635i	$-0.831627\pi$
$631$		−	25.3526i		−	1.00927i	0.863333	−	0.504635i	$-0.168373\pi$
$632$		0			0
$633$	−15.3176			−0.608818
$634$		0			0
$635$		−	3.20495i		−	0.127185i
$636$		0			0
$637$	−5.13182	+	0.444461i	−0.203330	+	0.0176102i
$638$		0			0
$639$		−	24.2286i		−	0.958469i
$640$		0			0
$641$	−35.0607			−1.38481			−0.692407	−	0.721507i	$-0.743450\pi$
$641$	−35.0607			−1.38481			−0.692407	+	0.721507i	$0.743450\pi$
$642$		0			0
$643$			40.4429i			1.59491i	0.603376	+	0.797457i	$0.293822\pi$
$643$			40.4429i			1.59491i	−0.603376	+	0.797457i	$0.706178\pi$
$644$		0			0
$645$		−	27.9956i		−	1.10232i
$646$		0			0
$647$	−31.4084			−1.23479			−0.617395	−	0.786653i	$-0.711812\pi$
$647$	−31.4084			−1.23479			−0.617395	+	0.786653i	$0.711812\pi$
$648$		0			0
$649$	−3.95407			−0.155211
$650$		0			0
$651$	−9.28592			−0.363944
$652$		0			0
$653$	11.0923			0.434077			0.217038	−	0.976163i	$-0.430360\pi$
$653$	11.0923			0.434077			0.217038	+	0.976163i	$0.430360\pi$
$654$		0			0
$655$			0.949145i			0.0370862i
$656$		0			0
$657$			21.1985i			0.827032i
$658$		0			0
$659$	−3.21279			−0.125152			−0.0625762	−	0.998040i	$-0.519932\pi$
$659$	−3.21279			−0.125152			−0.0625762	+	0.998040i	$0.519932\pi$
$660$		0			0
$661$		−	4.10171i		−	0.159538i	−0.996813	−	0.0797690i	$-0.974582\pi$
$661$		−	4.10171i		−	0.159538i	0.996813	−	0.0797690i	$-0.0254183\pi$
$662$		0			0
$663$	46.1847	−	4.00000i	1.79366	−	0.155347i
$664$		0			0
$665$		−	14.8573i		−	0.576141i
$666$		0			0
$667$	−47.9595			−1.85700
$668$		0			0
$669$		−	12.3269i		−	0.476587i
$670$		0			0
$671$		−	22.1116i		−	0.853607i
$672$		0			0
$673$	−12.5906			−0.485331			−0.242665	−	0.970110i	$-0.578022\pi$
$673$	−12.5906			−0.485331			−0.242665	+	0.970110i	$0.578022\pi$
$674$		0			0
$675$	2.42864			0.0934784
$676$		0			0
$677$	4.53035			0.174115			0.0870577	−	0.996203i	$-0.472254\pi$
$677$	4.53035			0.174115			0.0870577	+	0.996203i	$0.472254\pi$
$678$		0			0
$679$	36.7654			1.41093
$680$		0			0
$681$		−	20.8889i		−	0.800465i
$682$		0			0
$683$			10.7699i			0.412097i	0.978542	+	0.206049i	$0.0660605\pi$
$683$			10.7699i			0.412097i	−0.978542	+	0.206049i	$0.933940\pi$
$684$		0			0
$685$	12.7971			0.488950
$686$		0			0
$687$		−	35.5210i		−	1.35521i
$688$		0			0
$689$	−34.0513	+	2.94914i	−1.29725	+	0.112353i
$690$		0			0
$691$			17.2607i			0.656629i	0.944568	+	0.328315i	$0.106481\pi$
$691$			17.2607i			0.656629i	−0.944568	+	0.328315i	$0.893519\pi$
$692$		0			0
$693$	31.7146			1.20474
$694$		0			0
$695$			6.66370i			0.252769i
$696$		0			0
$697$			55.0420i			2.08486i
$698$		0			0
$699$	63.4706			2.40068
$700$		0			0
$701$	−27.0192			−1.02050			−0.510251	−	0.860026i	$-0.670447\pi$
$701$	−27.0192			−1.02050			−0.510251	+	0.860026i	$0.670447\pi$
$702$		0			0
$703$	2.42864			0.0915979
$704$		0			0
$705$	19.2859			0.726350
$706$		0			0
$707$		−	8.17484i		−	0.307447i
$708$		0			0
$709$			26.4385i			0.992918i	0.868060	+	0.496459i	$0.165367\pi$
$709$			26.4385i			0.992918i	−0.868060	+	0.496459i	$0.834633\pi$
$710$		0			0
$711$	−1.06959			−0.0401128
$712$		0			0
$713$		−	8.07758i		−	0.302508i
$714$		0			0
$715$	−20.6178	+	1.78568i	−0.771061	+	0.0667807i
$716$		0			0
$717$			48.1388i			1.79777i
$718$		0			0
$719$	12.9491			0.482922			0.241461	−	0.970411i	$-0.422373\pi$
$719$	12.9491			0.482922			0.241461	+	0.970411i	$0.422373\pi$
$720$		0			0
$721$		−	28.1432i		−	1.04811i
$722$		0			0
$723$		−	27.7462i		−	1.03189i
$724$		0			0
$725$	8.57628			0.318515
$726$		0			0
$727$	5.04302			0.187035			0.0935176	−	0.995618i	$-0.470189\pi$
$727$	5.04302			0.187035			0.0935176	+	0.995618i	$0.470189\pi$
$728$		0			0
$729$	−4.96836			−0.184013
$730$		0			0
$731$	−73.4104			−2.71518
$732$		0			0
$733$		−	34.9906i		−	1.29241i	−0.763165	−	0.646204i	$-0.776355\pi$
$733$		−	34.9906i		−	1.29241i	0.763165	−	0.646204i	$-0.223645\pi$
$734$		0			0
$735$		−	3.16346i		−	0.116686i
$736$		0			0
$737$	31.7146			1.16822
$738$		0			0
$739$			17.1363i			0.630368i	0.949031	+	0.315184i	$0.102066\pi$
$739$			17.1363i			0.630368i	−0.949031	+	0.315184i	$0.897934\pi$
$740$		0			0
$741$	−3.52543	−	40.7052i	−0.129510	−	1.49534i
$742$		0			0
$743$			15.7190i			0.576674i	0.957529	+	0.288337i	$0.0931024\pi$
$743$			15.7190i			0.576674i	−0.957529	+	0.288337i	$0.906898\pi$
$744$		0			0
$745$	6.94914			0.254597
$746$		0			0
$747$			13.3921i			0.489990i
$748$		0			0
$749$			40.7052i			1.48734i
$750$		0			0
$751$	−13.1111			−0.478430			−0.239215	−	0.970967i	$-0.576890\pi$
$751$	−13.1111			−0.478430			−0.239215	+	0.970967i	$0.576890\pi$
$752$		0			0
$753$	39.5210			1.44022
$754$		0			0
$755$	8.73038			0.317731
$756$		0			0
$757$	−16.1432			−0.586734			−0.293367	−	0.956000i	$-0.594776\pi$
$757$	−16.1432			−0.586734			−0.293367	+	0.956000i	$0.594776\pi$
$758$		0			0
$759$			71.0736i			2.57981i
$760$		0			0
$761$			26.1936i			0.949516i	0.880116	+	0.474758i	$0.157464\pi$
$761$			26.1936i			0.949516i	−0.880116	+	0.474758i	$0.842536\pi$
$762$		0			0
$763$	11.0509			0.400068
$764$		0			0
$765$			11.0509i			0.399545i
$766$		0			0
$767$	0.214320	+	2.47457i	0.00773864	+	0.0893516i
$768$		0			0
$769$		−	20.8287i		−	0.751102i	−0.926802	−	0.375551i	$-0.877453\pi$
$769$		−	20.8287i		−	0.751102i	0.926802	−	0.375551i	$-0.122547\pi$
$770$		0			0
$771$	47.8578			1.72355
$772$		0			0
$773$			11.9126i			0.428466i	0.976783	+	0.214233i	$0.0687251\pi$
$773$			11.9126i			0.428466i	−0.976783	+	0.214233i	$0.931275\pi$
$774$		0			0
$775$			1.44446i			0.0518866i
$776$		0			0
$777$	3.05086			0.109449
$778$		0			0
$779$	48.5116			1.73811
$780$		0			0
$781$	73.0692			2.61462
$782$		0			0
$783$	20.8287			0.744357
$784$		0			0
$785$		−	0.428639i		−	0.0152988i
$786$		0			0
$787$			21.1383i			0.753498i	0.926315	+	0.376749i	$0.122958\pi$
$787$			21.1383i			0.753498i	−0.926315	+	0.376749i	$0.877042\pi$
$788$		0			0
$789$	−8.51606			−0.303180
$790$		0			0
$791$			1.12399i			0.0399643i
$792$		0			0
$793$	−13.8381	+	1.19850i	−0.491404	+	0.0425599i
$794$		0			0
$795$		−	20.9906i		−	0.744461i
$796$		0			0
$797$	24.9273			0.882972			0.441486	−	0.897268i	$-0.354452\pi$
$797$	24.9273			0.882972			0.441486	+	0.897268i	$0.354452\pi$
$798$		0			0
$799$		−	50.5718i		−	1.78910i
$800$		0			0
$801$		−	3.43801i		−	0.121476i
$802$		0			0
$803$	−63.9309			−2.25607
$804$		0			0
$805$	16.2351			0.572211
$806$		0			0
$807$	−12.3368			−0.434275
$808$		0			0
$809$	−55.1195			−1.93790			−0.968950	−	0.247257i	$-0.920471\pi$
$809$	−55.1195			−1.93790			−0.968950	+	0.247257i	$0.920471\pi$
$810$		0			0
$811$			1.91459i			0.0672303i	0.999435	+	0.0336151i	$0.0107020\pi$
$811$			1.91459i			0.0672303i	−0.999435	+	0.0336151i	$0.989298\pi$
$812$		0			0
$813$		−	8.77970i		−	0.307918i
$814$		0			0
$815$	11.1985			0.392266
$816$		0			0
$817$			64.7007i			2.26359i
$818$		0			0
$819$	−1.71900	−	19.8479i	−0.0600669	−	0.693542i
$820$		0			0
$821$		−	51.9724i		−	1.81385i	−0.421294	−	0.906924i	$-0.638424\pi$
$821$		−	51.9724i		−	1.81385i	0.421294	−	0.906924i	$-0.361576\pi$
$822$		0			0
$823$	28.8909			1.00707			0.503537	−	0.863974i	$-0.332032\pi$
$823$	28.8909			1.00707			0.503537	+	0.863974i	$0.332032\pi$
$824$		0			0
$825$		−	12.7096i		−	0.442493i
$826$		0			0
$827$		−	33.0178i		−	1.14814i	−0.818805	−	0.574071i	$-0.805363\pi$
$827$		−	33.0178i		−	1.14814i	0.818805	−	0.574071i	$-0.194637\pi$
$828$		0			0
$829$	8.44293			0.293235			0.146618	−	0.989193i	$-0.453161\pi$
$829$	8.44293			0.293235			0.146618	+	0.989193i	$0.453161\pi$
$830$		0			0
$831$	15.4795			0.536978
$832$		0			0
$833$	−8.29529			−0.287415
$834$		0			0
$835$	−6.38271			−0.220883
$836$		0			0
$837$			3.50807i			0.121257i
$838$		0			0
$839$		−	20.1871i		−	0.696937i	−0.937321	−	0.348468i	$-0.886702\pi$
$839$		−	20.1871i		−	0.696937i	0.937321	−	0.348468i	$-0.113298\pi$
$840$		0			0
$841$	44.5526			1.53630
$842$		0			0
$843$		−	39.9496i		−	1.37594i
$844$		0			0
$845$	2.23506	+	12.8064i	0.0768885	+	0.440554i
$846$		0			0
$847$			63.7101i			2.18911i
$848$		0			0
$849$	58.2721			1.99989
$850$		0			0
$851$			2.65386i			0.0909731i
$852$		0			0
$853$		−	26.8430i		−	0.919086i	−0.888155	−	0.459543i	$-0.848013\pi$
$853$		−	26.8430i		−	0.919086i	0.888155	−	0.459543i	$-0.151987\pi$
$854$		0			0
$855$	9.73975			0.333092
$856$		0			0
$857$	8.51744			0.290950			0.145475	−	0.989362i	$-0.453529\pi$
$857$	8.51744			0.290950			0.145475	+	0.989362i	$0.453529\pi$
$858$		0			0
$859$	36.7368			1.25344			0.626722	−	0.779243i	$-0.284396\pi$
$859$	36.7368			1.25344			0.626722	+	0.779243i	$0.284396\pi$
$860$		0			0
$861$	60.9403			2.07684
$862$		0			0
$863$			13.9255i			0.474029i	0.971506	+	0.237015i	$0.0761689\pi$
$863$			13.9255i			0.474029i	−0.971506	+	0.237015i	$0.923831\pi$
$864$		0			0
$865$		−	3.67307i		−	0.124888i
$866$		0			0
$867$	37.0114			1.25697
$868$		0			0
$869$		−	3.22570i		−	0.109424i
$870$		0			0
$871$	−1.71900	−	19.8479i	−0.0582462	−	0.672521i
$872$		0			0
$873$			24.1017i			0.815719i
$874$		0			0
$875$	−2.90321			−0.0981465
$876$		0			0
$877$		−	28.6923i		−	0.968870i	−0.874827	−	0.484435i	$-0.839025\pi$
$877$		−	28.6923i		−	0.968870i	0.874827	−	0.484435i	$-0.160975\pi$
$878$		0			0
$879$			44.5433i			1.50241i
$880$		0			0
$881$	−9.00492			−0.303384			−0.151692	−	0.988428i	$-0.548472\pi$
$881$	−9.00492			−0.303384			−0.151692	+	0.988428i	$0.548472\pi$
$882$		0			0
$883$	37.0627			1.24726			0.623630	−	0.781720i	$-0.285657\pi$
$883$	37.0627			1.24726			0.623630	+	0.781720i	$0.285657\pi$
$884$		0			0
$885$	−1.52543			−0.0512767
$886$		0			0
$887$	14.3892			0.483141			0.241570	−	0.970383i	$-0.422338\pi$
$887$	14.3892			0.483141			0.241570	+	0.970383i	$0.422338\pi$
$888$		0			0
$889$			9.30465i			0.312068i
$890$		0			0
$891$		−	63.6390i		−	2.13199i
$892$		0			0
$893$	−44.5718			−1.49154
$894$		0			0
$895$			1.08250i			0.0361839i
$896$		0			0
$897$	44.4800	−	3.85236i	1.48514	−	0.128626i
$898$		0			0
$899$			12.3881i			0.413166i
$900$		0			0
$901$	−55.0420			−1.83371
$902$		0			0
$903$			81.2770i			2.70473i
$904$		0			0
$905$			1.82071i			0.0605226i
$906$		0			0
$907$	3.47166			0.115275			0.0576373	−	0.998338i	$-0.481643\pi$
$907$	3.47166			0.115275			0.0576373	+	0.998338i	$0.481643\pi$
$908$		0			0
$909$	5.35905			0.177748
$910$		0			0
$911$	−27.1842			−0.900653			−0.450326	−	0.892864i	$-0.648692\pi$
$911$	−27.1842			−0.900653			−0.450326	+	0.892864i	$0.648692\pi$
$912$		0			0
$913$	−40.3881			−1.33665
$914$		0			0
$915$		−	8.53035i		−	0.282005i
$916$		0			0
$917$		−	2.75557i		−	0.0909969i
$918$		0			0
$919$	−3.71762			−0.122633			−0.0613165	−	0.998118i	$-0.519530\pi$
$919$	−3.71762			−0.122633			−0.0613165	+	0.998118i	$0.519530\pi$
$920$		0			0
$921$		−	6.78721i		−	0.223646i
$922$		0			0
$923$	−3.96052	−	45.7288i	−0.130362	−	1.50518i
$924$		0			0
$925$		−	0.474572i		−	0.0156038i
$926$		0			0
$927$	18.4494			0.605957
$928$		0			0
$929$		−	56.8256i		−	1.86439i	−0.361958	−	0.932194i	$-0.617892\pi$
$929$		−	56.8256i		−	1.86439i	0.361958	−	0.932194i	$-0.382108\pi$
$930$		0			0
$931$			7.31111i			0.239612i
$932$		0			0
$933$	−62.5433			−2.04757
$934$		0			0
$935$	−33.3274			−1.08992
$936$		0			0
$937$	−51.8894			−1.69515			−0.847576	−	0.530673i	$-0.821939\pi$
$937$	−51.8894			−1.69515			−0.847576	+	0.530673i	$0.821939\pi$
$938$		0			0
$939$	40.0830			1.30806
$940$		0			0
$941$			50.4929i			1.64602i	0.568026	+	0.823010i	$0.307707\pi$
$941$			50.4929i			1.64602i	−0.568026	+	0.823010i	$0.692293\pi$
$942$		0			0
$943$			53.0103i			1.72625i
$944$		0			0
$945$	−7.05086			−0.229364
$946$		0			0
$947$			39.5353i			1.28472i	0.766401	+	0.642362i	$0.222045\pi$
$947$			39.5353i			1.28472i	−0.766401	+	0.642362i	$0.777955\pi$
$948$		0			0
$949$	3.46520	+	40.0098i	0.112485	+	1.29877i
$950$		0			0
$951$			36.0731i			1.16975i
$952$		0			0
$953$	17.7047			0.573512			0.286756	−	0.958004i	$-0.407423\pi$
$953$	17.7047			0.573512			0.286756	+	0.958004i	$0.407423\pi$
$954$		0			0
$955$			15.2257i			0.492692i
$956$		0			0
$957$		−	109.001i		−	3.52352i
$958$		0			0
$959$	−37.1526			−1.19972
$960$		0			0
$961$	28.9135			0.932695
$962$		0			0
$963$	−26.6844			−0.859894
$964$		0			0
$965$	−5.18421			−0.166886
$966$		0			0
$967$			54.1990i			1.74292i	0.490465	+	0.871461i	$0.336827\pi$
$967$			54.1990i			1.74292i	−0.490465	+	0.871461i	$0.663173\pi$
$968$		0			0
$969$		−	65.7975i		−	2.11372i
$970$		0			0
$971$	45.2226			1.45126			0.725632	−	0.688083i	$-0.241548\pi$
$971$	45.2226			1.45126			0.725632	+	0.688083i	$0.241548\pi$
$972$		0			0
$973$		−	19.3461i		−	0.620209i
$974$		0			0
$975$	−7.95407	+	0.688892i	−0.254734	+	0.0220622i
$976$		0			0
$977$		−	49.3230i		−	1.57798i	−0.614405	−	0.788991i	$-0.710604\pi$
$977$		−	49.3230i		−	1.57798i	0.614405	−	0.788991i	$-0.289396\pi$
$978$		0			0
$979$	10.3684			0.331376
$980$		0			0
$981$			7.24443i			0.231297i
$982$		0			0
$983$			32.2939i			1.03002i	0.857186	+	0.515008i	$0.172211\pi$
$983$			32.2939i			1.03002i	−0.857186	+	0.515008i	$0.827789\pi$
$984$		0			0
$985$	−15.9684			−0.508794
$986$		0			0
$987$	−55.9911			−1.78222
$988$		0			0
$989$	−70.7007			−2.24815
$990$		0			0
$991$	−30.9086			−0.981844			−0.490922	−	0.871203i	$-0.663340\pi$
$991$	−30.9086			−0.981844			−0.490922	+	0.871203i	$0.663340\pi$
$992$		0			0
$993$			37.3832i			1.18632i
$994$		0			0
$995$		−	18.7971i		−	0.595907i
$996$		0			0
$997$	53.0005			1.67854			0.839271	−	0.543713i	$-0.182982\pi$
$997$	53.0005			1.67854			0.839271	+	0.543713i	$0.182982\pi$
$998$		0			0
$999$		−	1.15257i		−	0.0364656i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.2.k.a.441.6	yes	6
3.2	odd	2		4680.2.g.j.2521.1		6
4.3	odd	2		1040.2.k.b.961.2		6
5.2	odd	4		2600.2.f.d.649.1		6
5.3	odd	4		2600.2.f.c.649.6		6
5.4	even	2		2600.2.k.b.2001.2		6
13.5	odd	4		6760.2.a.v.1.3		3
13.8	odd	4		6760.2.a.u.1.3		3
13.12	even	2	inner	520.2.k.a.441.5	✓	6
39.38	odd	2		4680.2.g.j.2521.6		6
52.51	odd	2		1040.2.k.b.961.1		6
65.12	odd	4		2600.2.f.c.649.1		6
65.38	odd	4		2600.2.f.d.649.6		6
65.64	even	2		2600.2.k.b.2001.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.a.441.5	✓	6	13.12	even	2	inner
520.2.k.a.441.6	yes	6	1.1	even	1	trivial
1040.2.k.b.961.1		6	52.51	odd	2
1040.2.k.b.961.2		6	4.3	odd	2
2600.2.f.c.649.1		6	65.12	odd	4
2600.2.f.c.649.6		6	5.3	odd	4
2600.2.f.d.649.1		6	5.2	odd	4
2600.2.f.d.649.6		6	65.38	odd	4
2600.2.k.b.2001.1		6	65.64	even	2
2600.2.k.b.2001.2		6	5.4	even	2
4680.2.g.j.2521.1		6	3.2	odd	2
4680.2.g.j.2521.6		6	39.38	odd	2
6760.2.a.u.1.3		3	13.8	odd	4
6760.2.a.v.1.3		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)

\(f(q)\)	\(=\)	\(q+2.21432 q^{3} +1.00000i q^{5} -2.90321i q^{7} +1.90321 q^{9} +5.73975i q^{11} +(3.59210 - 0.311108i) q^{13} +2.21432i q^{15} +5.80642 q^{17} -5.11753i q^{19} -6.42864i q^{21} +5.59210 q^{23} -1.00000 q^{25} -2.42864 q^{27} -8.57628 q^{29} -1.44446i q^{31} +12.7096i q^{33} +2.90321 q^{35} +0.474572i q^{37} +(7.95407 - 0.688892i) q^{39} +9.47949i q^{41} -12.6430 q^{43} +1.90321i q^{45} -8.70964i q^{47} -1.42864 q^{49} +12.8573 q^{51} -9.47949 q^{53} -5.73975 q^{55} -11.3319i q^{57} +0.688892i q^{59} -3.85236 q^{61} -5.52543i q^{63} +(0.311108 + 3.59210i) q^{65} -5.52543i q^{67} +12.3827 q^{69} -12.7304i q^{71} +11.1383i q^{73} -2.21432 q^{75} +16.6637 q^{77} -0.561993 q^{79} -11.0874 q^{81} +7.03657i q^{83} +5.80642i q^{85} -18.9906 q^{87} -1.80642i q^{89} +(-0.903212 - 10.4286i) q^{91} -3.19850i q^{93} +5.11753 q^{95} +12.6637i q^{97} +10.9240i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77}+ \cdots + 4 q^{95}+O(q^{100}) \) 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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