LMFDB - Embedded newform 2600.2.k.b.2001.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2600,2,Mod(2001,2600)] 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("2600.2001"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$20.7611045255$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.1
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	2600.2001
Dual form			2600.2.k.b.2001.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21432 q^{3} -2.90321i q^{7} +1.90321 q^{9} -5.73975i q^{11} +(-3.59210 - 0.311108i) q^{13} -5.80642 q^{17} +5.11753i q^{19} +6.42864i q^{21} -5.59210 q^{23} +2.42864 q^{27} -8.57628 q^{29} +1.44446i q^{31} +12.7096i q^{33} +0.474572i q^{37} +(7.95407 + 0.688892i) q^{39} -9.47949i q^{41} +12.6430 q^{43} -8.70964i q^{47} -1.42864 q^{49} +12.8573 q^{51} +9.47949 q^{53} -11.3319i q^{57} -0.688892i q^{59} -3.85236 q^{61} -5.52543i q^{63} -5.52543i q^{67} +12.3827 q^{69} +12.7304i q^{71} +11.1383i q^{73} -16.6637 q^{77} -0.561993 q^{79} -11.0874 q^{81} +7.03657i q^{83} +18.9906 q^{87} +1.80642i q^{89} +(-0.903212 + 10.4286i) q^{91} -3.19850i q^{93} +12.6637i q^{97} -10.9240i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9} - 8 q^{13} - 8 q^{17} - 20 q^{23} - 12 q^{27} - 12 q^{29} + 8 q^{39} + 36 q^{43} + 18 q^{49} + 24 q^{51} + 4 q^{53} - 36 q^{61} + 8 q^{69} - 20 q^{77} + 24 q^{79} - 26 q^{81} + 60 q^{87} + 8 q^{91}+O(q^{100}) $ 6 * q - 2 * q^9 - 8 * q^13 - 8 * q^17 - 20 * q^23 - 12 * q^27 - 12 * q^29 + 8 * q^39 + 36 * q^43 + 18 * q^49 + 24 * q^51 + 4 * q^53 - 36 * q^61 + 8 * q^69 - 20 * q^77 + 24 * q^79 - 26 * q^81 + 60 * q^87 + 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$1$

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	$-0.720742\pi$
$3$	−2.21432			−1.27844			−0.639219	+	0.769025i	$0.720742\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		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	$-0.667128\pi$
$11$		−	5.73975i		−	1.73060i	0.501255	−	0.865299i	$-0.332872\pi$
$12$		0			0
$13$	−3.59210	−	0.311108i	−0.996270	−	0.0862858i
$13$	−3.59210	−	0.311108i	−0.996270	−	0.0862858i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.80642			−1.40826			−0.704132	−	0.710069i	$-0.748664\pi$
$17$	−5.80642			−1.40826			−0.704132	+	0.710069i	$0.748664\pi$
$18$		0			0
$19$			5.11753i			1.17404i	0.809572	+	0.587021i	$0.199699\pi$
$19$			5.11753i			1.17404i	−0.809572	+	0.587021i	$0.800301\pi$
$20$		0			0
$21$			6.42864i			1.40284i
$22$		0			0
$23$	−5.59210			−1.16603			−0.583017	−	0.812460i	$-0.698128\pi$
$23$	−5.59210			−1.16603			−0.583017	+	0.812460i	$0.698128\pi$
$24$		0			0
$25$		0			0
$26$		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	$0.0414067\pi$
$31$			1.44446i			0.259433i	−0.991551	+	0.129716i	$0.958593\pi$
$32$		0			0
$33$			12.7096i			2.21246i
$34$		0			0
$35$		0			0
$36$		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	$-0.734719\pi$
$41$		−	9.47949i		−	1.48045i	0.672360	−	0.740224i	$-0.265281\pi$
$42$		0			0
$43$	12.6430			1.92803			0.964017	−	0.265842i	$-0.0856500\pi$
$43$	12.6430			1.92803			0.964017	+	0.265842i	$0.0856500\pi$
$44$		0			0
$45$		0			0
$46$		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	$-0.274327\pi$
$53$	9.47949			1.30211			0.651054	+	0.759031i	$0.274327\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	11.3319i		−	1.50094i
$58$		0			0
$59$		−	0.688892i		−	0.0896861i	−0.998994	−	0.0448431i	$-0.985721\pi$
$59$		−	0.688892i		−	0.0896861i	0.998994	−	0.0448431i	$-0.0142788\pi$
$60$		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			0
$66$		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	$0.272562\pi$
$71$			12.7304i			1.51082i	−0.655254	+	0.755409i	$0.727438\pi$
$72$		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
$75$		0			0
$76$		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
$85$		0			0
$86$		0			0
$87$	18.9906			2.03601
$88$		0			0
$89$			1.80642i			0.191481i	0.995406	+	0.0957403i	$0.0305218\pi$
$89$			1.80642i			0.191481i	−0.995406	+	0.0957403i	$0.969478\pi$
$90$		0			0
$91$	−0.903212	+	10.4286i	−0.0946823	+	1.09322i
$92$		0			0
$93$		−	3.19850i		−	0.331669i
$94$		0			0
$95$		0			0
$96$		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	$-0.658486\pi$
$103$	−9.69381			−0.955160			−0.477580	+	0.878588i	$0.658486\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.0207			1.35544			0.677718	−	0.735322i	$-0.262969\pi$
$107$	14.0207			1.35544			0.677718	+	0.735322i	$0.262969\pi$
$108$		0			0
$109$		−	3.80642i		−	0.364589i	−0.983244	−	0.182295i	$-0.941648\pi$
$109$		−	3.80642i		−	0.364589i	0.983244	−	0.182295i	$-0.0583525\pi$
$110$		0			0
$111$		−	1.05086i		−	0.0997428i
$112$		0			0
$113$	0.387152			0.0364202			0.0182101	−	0.999834i	$-0.494203\pi$
$113$	0.387152			0.0364202			0.0182101	+	0.999834i	$0.494203\pi$
$114$		0			0
$115$		0			0
$116$		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
$125$		0			0
$126$		0			0
$127$	3.20495			0.284393			0.142197	−	0.989838i	$-0.454583\pi$
$127$	3.20495			0.284393			0.142197	+	0.989838i	$0.454583\pi$
$128$		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
$135$		0			0
$136$		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
$145$		0			0
$146$		0			0
$147$	3.16346			0.260918
$148$		0			0
$149$			6.94914i			0.569296i	0.958632	+	0.284648i	$0.0918767\pi$
$149$			6.94914i			0.569296i	−0.958632	+	0.284648i	$0.908123\pi$
$150$		0			0
$151$			8.73038i			0.710468i	0.934777	+	0.355234i	$0.115599\pi$
$151$			8.73038i			0.710468i	−0.934777	+	0.355234i	$0.884401\pi$
$152$		0			0
$153$	−11.0509			−0.893409
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.428639			0.0342092			0.0171046	−	0.999854i	$-0.494555\pi$
$157$	0.428639			0.0342092			0.0171046	+	0.999854i	$0.494555\pi$
$158$		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
$165$		0			0
$166$		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	$-0.455409\pi$
$173$	3.67307			0.279258			0.139629	+	0.990204i	$0.455409\pi$
$174$		0			0
$175$		0			0
$176$		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			0
$186$		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			0
$196$		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
$205$		0			0
$206$		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
$215$		0			0
$216$		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
$225$		0			0
$226$		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	$0.177818\pi$
$229$			16.0415i			1.06005i	−0.847982	+	0.530026i	$0.822182\pi$
$230$		0			0
$231$	36.8988			2.42776
$232$		0			0
$233$	−28.6637			−1.87782			−0.938911	−	0.344161i	$-0.888164\pi$
$233$	−28.6637			−1.87782			−0.938911	+	0.344161i	$0.888164\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.24443			0.0808345
$238$		0			0
$239$		−	21.7397i		−	1.40623i	−0.711077	−	0.703114i	$-0.751792\pi$
$239$		−	21.7397i		−	1.40623i	0.711077	−	0.703114i	$-0.248208\pi$
$240$		0			0
$241$			12.5303i			0.807151i	0.914946	+	0.403575i	$0.132233\pi$
$241$			12.5303i			0.807151i	−0.914946	+	0.403575i	$0.867767\pi$
$242$		0			0
$243$	17.2652			1.10756
$244$		0			0
$245$		0			0
$246$		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
$255$		0			0
$256$		0			0
$257$	−21.6128			−1.34817			−0.674086	−	0.738653i	$-0.735462\pi$
$257$	−21.6128			−1.34817			−0.674086	+	0.738653i	$0.735462\pi$
$258$		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	$-0.462168\pi$
$263$	3.84590			0.237149			0.118574	+	0.992945i	$0.462168\pi$
$264$		0			0
$265$		0			0
$266$		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	$0.0384265\pi$
$271$			3.96497i			0.240855i	−0.992722	+	0.120427i	$0.961574\pi$
$272$		0			0
$273$	2.00000	−	23.0923i	0.121046	−	1.39761i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.99063			−0.420026			−0.210013	−	0.977699i	$-0.567351\pi$
$277$	−6.99063			−0.420026			−0.210013	+	0.977699i	$0.567351\pi$
$278$		0			0
$279$			2.74912i			0.164585i
$280$		0			0
$281$			18.0415i			1.07626i	0.842860	+	0.538132i	$0.180870\pi$
$281$			18.0415i			1.07626i	−0.842860	+	0.538132i	$0.819130\pi$
$282$		0			0
$283$	−26.3160			−1.56433			−0.782163	−	0.623074i	$-0.785883\pi$
$283$	−26.3160			−1.56433			−0.782163	+	0.623074i	$0.785883\pi$
$284$		0			0
$285$		0			0
$286$		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			0
$296$		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
$305$		0			0
$306$		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	$-0.670941\pi$
$313$	−18.1017			−1.02317			−0.511585	+	0.859233i	$0.670941\pi$
$314$		0			0
$315$		0			0
$316$		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
$325$		0			0
$326$		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	$-0.846424\pi$
$331$		−	16.8825i		−	0.927944i	0.885850	−	0.463972i	$-0.153576\pi$
$332$		0			0
$333$			0.903212i			0.0494957i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.78415			0.369556			0.184778	−	0.982780i	$-0.440843\pi$
$337$	6.78415			0.369556			0.184778	+	0.982780i	$0.440843\pi$
$338$		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
$345$		0			0
$346$		0			0
$347$	16.0207			0.860039			0.430019	−	0.902820i	$-0.358507\pi$
$347$	16.0207			0.860039			0.430019	+	0.902820i	$0.358507\pi$
$348$		0			0
$349$			14.7052i			0.787151i	0.919292	+	0.393575i	$0.128762\pi$
$349$			14.7052i			0.787151i	−0.919292	+	0.393575i	$0.871238\pi$
$350$		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
$355$		0			0
$356$		0			0
$357$		−	37.3274i		−	1.97558i
$358$		0			0
$359$		−	7.01582i		−	0.370281i	−0.982712	−	0.185140i	$-0.940726\pi$
$359$		−	7.01582i		−	0.370281i	0.982712	−	0.185140i	$-0.0592740\pi$
$360$		0			0
$361$	−7.18913			−0.378375
$362$		0			0
$363$	48.5926			2.55045
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.43954			0.127343			0.0636714	−	0.997971i	$-0.479719\pi$
$367$	2.43954			0.127343			0.0636714	+	0.997971i	$0.479719\pi$
$368$		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	$-0.360161\pi$
$373$	16.4286			0.850643			0.425321	+	0.905042i	$0.360161\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	30.8069	+	2.66815i	1.58664	+	0.137417i
$378$		0			0
$379$			0.0251894i			0.00129389i	1.00000		0.000646945i	$0.000205929\pi$
$379$			0.0251894i			0.00129389i	−1.00000		0.000646945i	$0.999794\pi$
$380$		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
$385$		0			0
$386$		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			0
$396$		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	$-0.649807\pi$
$401$		−	35.6958i		−	1.78256i	0.453449	−	0.891282i	$-0.350193\pi$
$402$		0			0
$403$	0.449383	−	5.18865i	0.0223854	−	0.258465i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.72393			0.135020
$408$		0			0
$409$			25.8666i			1.27902i	0.768781	+	0.639512i	$0.220863\pi$
$409$			25.8666i			1.27902i	−0.768781	+	0.639512i	$0.779137\pi$
$410$		0			0
$411$			28.3368i			1.39775i
$412$		0			0
$413$	−2.00000			−0.0984136
$414$		0			0
$415$		0			0
$416$		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	$-0.886054\pi$
$421$		−	14.3783i		−	0.700754i	0.936609	−	0.350377i	$-0.113946\pi$
$422$		0			0
$423$		−	16.5763i		−	0.805966i
$424$		0			0
$425$		0			0
$426$		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	$-0.837658\pi$
$431$		−	20.2701i		−	0.976376i	0.872738	−	0.488188i	$-0.162342\pi$
$432$		0			0
$433$	−9.34614			−0.449147			−0.224573	−	0.974457i	$-0.572099\pi$
$433$	−9.34614			−0.449147			−0.224573	+	0.974457i	$0.572099\pi$
$434$		0			0
$435$		0			0
$436$		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	$-0.670355\pi$
$443$	−21.4686			−1.02000			−0.510002	+	0.860173i	$0.670355\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	15.3876i		−	0.727810i
$448$		0			0
$449$		−	5.83807i		−	0.275515i	−0.990466	−	0.137758i	$-0.956010\pi$
$449$		−	5.83807i		−	0.275515i	0.990466	−	0.137758i	$-0.0439895\pi$
$450$		0			0
$451$	−54.4099			−2.56206
$452$		0			0
$453$		−	19.3319i		−	0.908290i
$454$		0			0
$455$		0			0
$456$		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	$0.0216429\pi$
$461$			2.91750i			0.135882i	−0.997689	+	0.0679408i	$0.978357\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$	−12.5698			−0.581662			−0.290831	−	0.956774i	$-0.593932\pi$
$467$	−12.5698			−0.581662			−0.290831	+	0.956774i	$0.593932\pi$
$468$		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
$475$		0			0
$476$		0			0
$477$	18.0415			0.826063
$478$		0			0
$479$		−	39.0988i		−	1.78647i	−0.449590	−	0.893235i	$-0.648430\pi$
$479$		−	39.0988i		−	1.78647i	0.449590	−	0.893235i	$-0.351570\pi$
$480$		0			0
$481$	0.147643	−	1.70471i	0.00673195	−	0.0777282i
$482$		0			0
$483$		−	35.9496i		−	1.63576i
$484$		0			0
$485$		0			0
$486$		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
$495$		0			0
$496$		0			0
$497$	36.9590			1.65784
$498$		0			0
$499$			36.8134i			1.64799i	0.566596	+	0.823996i	$0.308260\pi$
$499$			36.8134i			1.64799i	−0.566596	+	0.823996i	$0.691740\pi$
$500$		0			0
$501$		−	14.1334i		−	0.631432i
$502$		0			0
$503$	−28.9382			−1.29029			−0.645146	−	0.764059i	$-0.723204\pi$
$503$	−28.9382			−1.29029			−0.645146	+	0.764059i	$0.723204\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−28.3575	−	4.94914i	−1.25940	−	0.219799i
$508$		0			0
$509$			37.8765i			1.67885i	0.543479	+	0.839423i	$0.317107\pi$
$509$			37.8765i			1.67885i	−0.543479	+	0.839423i	$0.682893\pi$
$510$		0			0
$511$	32.3368			1.43049
$512$		0			0
$513$			12.4286i			0.548738i
$514$		0			0
$515$		0			0
$516$		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	$-0.441060\pi$
$523$	8.42080			0.368216			0.184108	+	0.982906i	$0.441060\pi$
$524$		0			0
$525$		0			0
$526$		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
$535$		0			0
$536$		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	$-0.839288\pi$
$541$		−	22.5018i		−	0.967427i	0.875227	−	0.483713i	$-0.160712\pi$
$542$		0			0
$543$	−4.03164			−0.173014
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−10.4494			−0.446783			−0.223392	−	0.974729i	$-0.571713\pi$
$547$	−10.4494			−0.446783			−0.223392	+	0.974729i	$0.571713\pi$
$548$		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
$555$		0			0
$556$		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	$-0.179152\pi$
$563$	40.1354			1.69150			0.845752	+	0.533576i	$0.179152\pi$
$564$		0			0
$565$		0			0
$566$		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
$575$		0			0
$576$		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			0
$586$		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
$595$		0			0
$596$		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
$605$		0			0
$606$		0			0
$607$	17.3254			0.703216			0.351608	−	0.936147i	$-0.385635\pi$
$607$	17.3254			0.703216			0.351608	+	0.936147i	$0.385635\pi$
$608$		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
$615$		0			0
$616$		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	$0.0803992\pi$
$619$			12.4351i			0.499809i	−0.968271	+	0.249904i	$0.919601\pi$
$620$		0			0
$621$	−13.5812			−0.544995
$622$		0			0
$623$	5.24443			0.210114
$624$		0			0
$625$		0			0
$626$		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	$0.168373\pi$
$631$			25.3526i			1.00927i	−0.863333	+	0.504635i	$0.831627\pi$
$632$		0			0
$633$	15.3176			0.608818
$634$		0			0
$635$		0			0
$636$		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
$645$		0			0
$646$		0			0
$647$	31.4084			1.23479			0.617395	−	0.786653i	$-0.288188\pi$
$647$	31.4084			1.23479			0.617395	+	0.786653i	$0.288188\pi$
$648$		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	$-0.569640\pi$
$653$	−11.0923			−0.434077			−0.217038	+	0.976163i	$0.569640\pi$
$654$		0			0
$655$		0			0
$656$		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	$0.0254183\pi$
$661$			4.10171i			0.159538i	−0.996813	+	0.0797690i	$0.974582\pi$
$662$		0			0
$663$	−46.1847	−	4.00000i	−1.79366	−	0.155347i
$664$		0			0
$665$		0			0
$666$		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	$-0.421978\pi$
$673$	12.5906			0.485331			0.242665	+	0.970110i	$0.421978\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.53035			−0.174115			−0.0870577	−	0.996203i	$-0.527746\pi$
$677$	−4.53035			−0.174115			−0.0870577	+	0.996203i	$0.527746\pi$
$678$		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
$685$		0			0
$686$		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	$-0.893519\pi$
$691$		−	17.2607i		−	0.656629i	0.944568	−	0.328315i	$-0.106481\pi$
$692$		0			0
$693$	−31.7146			−1.20474
$694$		0			0
$695$		0			0
$696$		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
$705$		0			0
$706$		0			0
$707$		−	8.17484i		−	0.307447i
$708$		0			0
$709$		−	26.4385i		−	0.992918i	−0.868060	−	0.496459i	$-0.834633\pi$
$709$		−	26.4385i		−	0.992918i	0.868060	−	0.496459i	$-0.165367\pi$
$710$		0			0
$711$	−1.06959			−0.0401128
$712$		0			0
$713$		−	8.07758i		−	0.302508i
$714$		0			0
$715$		0			0
$716$		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
$725$		0			0
$726$		0			0
$727$	−5.04302			−0.187035			−0.0935176	−	0.995618i	$-0.529811\pi$
$727$	−5.04302			−0.187035			−0.0935176	+	0.995618i	$0.529811\pi$
$728$		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
$735$		0			0
$736$		0			0
$737$	−31.7146			−1.16822
$738$		0			0
$739$		−	17.1363i		−	0.630368i	−0.949031	−	0.315184i	$-0.897934\pi$
$739$		−	17.1363i		−	0.630368i	0.949031	−	0.315184i	$-0.102066\pi$
$740$		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
$745$		0			0
$746$		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
$755$		0			0
$756$		0			0
$757$	16.1432			0.586734			0.293367	−	0.956000i	$-0.405224\pi$
$757$	16.1432			0.586734			0.293367	+	0.956000i	$0.405224\pi$
$758$		0			0
$759$		−	71.0736i		−	2.57981i
$760$		0			0
$761$		−	26.1936i		−	0.949516i	−0.880116	−	0.474758i	$-0.842536\pi$
$761$		−	26.1936i		−	0.949516i	0.880116	−	0.474758i	$-0.157464\pi$
$762$		0			0
$763$	−11.0509			−0.400068
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.214320	+	2.47457i	−0.00773864	+	0.0893516i
$768$		0			0
$769$			20.8287i			0.751102i	0.926802	+	0.375551i	$0.122547\pi$
$769$			20.8287i			0.751102i	−0.926802	+	0.375551i	$0.877453\pi$
$770$		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
$775$		0			0
$776$		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			0
$786$		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
$795$		0			0
$796$		0			0
$797$	−24.9273			−0.882972			−0.441486	−	0.897268i	$-0.645548\pi$
$797$	−24.9273			−0.882972			−0.441486	+	0.897268i	$0.645548\pi$
$798$		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
$805$		0			0
$806$		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	$-0.989298\pi$
$811$		−	1.91459i		−	0.0672303i	0.999435	−	0.0336151i	$-0.0107020\pi$
$812$		0			0
$813$		−	8.77970i		−	0.307918i
$814$		0			0
$815$		0			0
$816$		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	$0.361576\pi$
$821$			51.9724i			1.81385i	−0.421294	+	0.906924i	$0.638424\pi$
$822$		0			0
$823$	−28.8909			−1.00707			−0.503537	−	0.863974i	$-0.667968\pi$
$823$	−28.8909			−1.00707			−0.503537	+	0.863974i	$0.667968\pi$
$824$		0			0
$825$		0			0
$826$		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
$835$		0			0
$836$		0			0
$837$			3.50807i			0.121257i
$838$		0			0
$839$			20.1871i			0.696937i	0.937321	+	0.348468i	$0.113298\pi$
$839$			20.1871i			0.696937i	−0.937321	+	0.348468i	$0.886702\pi$
$840$		0			0
$841$	44.5526			1.53630
$842$		0			0
$843$		−	39.9496i		−	1.37594i
$844$		0			0
$845$		0			0
$846$		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
$855$		0			0
$856$		0			0
$857$	−8.51744			−0.290950			−0.145475	−	0.989362i	$-0.546471\pi$
$857$	−8.51744			−0.290950			−0.145475	+	0.989362i	$0.546471\pi$
$858$		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
$865$		0			0
$866$		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
$875$		0			0
$876$		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	$-0.714343\pi$
$883$	−37.0627			−1.24726			−0.623630	+	0.781720i	$0.714343\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.3892			−0.483141			−0.241570	−	0.970383i	$-0.577662\pi$
$887$	−14.3892			−0.483141			−0.241570	+	0.970383i	$0.577662\pi$
$888$		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
$895$		0			0
$896$		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
$905$		0			0
$906$		0			0
$907$	−3.47166			−0.115275			−0.0576373	−	0.998338i	$-0.518357\pi$
$907$	−3.47166			−0.115275			−0.0576373	+	0.998338i	$0.518357\pi$
$908$		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
$915$		0			0
$916$		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			0
$926$		0			0
$927$	−18.4494			−0.605957
$928$		0			0
$929$			56.8256i			1.86439i	0.361958	+	0.932194i	$0.382108\pi$
$929$			56.8256i			1.86439i	−0.361958	+	0.932194i	$0.617892\pi$
$930$		0			0
$931$		−	7.31111i		−	0.239612i
$932$		0			0
$933$	62.5433			2.04757
$934$		0			0
$935$		0			0
$936$		0			0
$937$	51.8894			1.69515			0.847576	−	0.530673i	$-0.178061\pi$
$937$	51.8894			1.69515			0.847576	+	0.530673i	$0.178061\pi$
$938$		0			0
$939$	40.0830			1.30806
$940$		0			0
$941$		−	50.4929i		−	1.64602i	−0.568026	−	0.823010i	$-0.692293\pi$
$941$		−	50.4929i		−	1.64602i	0.568026	−	0.823010i	$-0.307707\pi$
$942$		0			0
$943$			53.0103i			1.72625i
$944$		0			0
$945$		0			0
$946$		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	$-0.592577\pi$
$953$	−17.7047			−0.573512			−0.286756	+	0.958004i	$0.592577\pi$
$954$		0			0
$955$		0			0
$956$		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
$965$		0			0
$966$		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
$975$		0			0
$976$		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
$985$		0			0
$986$		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
$995$		0			0
$996$		0			0
$997$	−53.0005			−1.67854			−0.839271	−	0.543713i	$-0.817018\pi$
$997$	−53.0005			−1.67854			−0.839271	+	0.543713i	$0.817018\pi$
$998$		0			0
$999$			1.15257i			0.0364656i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.a.441.5	✓	6	5.4	even	2
520.2.k.a.441.6	yes	6	65.64	even	2
1040.2.k.b.961.1		6	20.19	odd	2
1040.2.k.b.961.2		6	260.259	odd	2
2600.2.f.c.649.1		6	5.3	odd	4
2600.2.f.c.649.6		6	65.12	odd	4
2600.2.f.d.649.1		6	65.38	odd	4
2600.2.f.d.649.6		6	5.2	odd	4
2600.2.k.b.2001.1		6	1.1	even	1	trivial
2600.2.k.b.2001.2		6	13.12	even	2	inner
4680.2.g.j.2521.1		6	195.194	odd	2
4680.2.g.j.2521.6		6	15.14	odd	2
6760.2.a.u.1.3		3	65.44	odd	4
6760.2.a.v.1.3		3	65.34	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:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.21432 q^{3} -2.90321i q^{7} +1.90321 q^{9} -5.73975i q^{11} +(-3.59210 - 0.311108i) q^{13} -5.80642 q^{17} +5.11753i q^{19} +6.42864i q^{21} -5.59210 q^{23} +2.42864 q^{27} -8.57628 q^{29} +1.44446i q^{31} +12.7096i q^{33} +0.474572i q^{37} +(7.95407 + 0.688892i) q^{39} -9.47949i q^{41} +12.6430 q^{43} -8.70964i q^{47} -1.42864 q^{49} +12.8573 q^{51} +9.47949 q^{53} -11.3319i q^{57} -0.688892i q^{59} -3.85236 q^{61} -5.52543i q^{63} -5.52543i q^{67} +12.3827 q^{69} +12.7304i q^{71} +11.1383i q^{73} -16.6637 q^{77} -0.561993 q^{79} -11.0874 q^{81} +7.03657i q^{83} +18.9906 q^{87} +1.80642i q^{89} +(-0.903212 + 10.4286i) q^{91} -3.19850i q^{93} +12.6637i q^{97} -10.9240i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} - 8 q^{13} - 8 q^{17} - 20 q^{23} - 12 q^{27} - 12 q^{29} + 8 q^{39} + 36 q^{43} + 18 q^{49} + 24 q^{51} + 4 q^{53} - 36 q^{61} + 8 q^{69} - 20 q^{77} + 24 q^{79} - 26 q^{81} + 60 q^{87} + 8 q^{91}+O(q^{100}) \) 6 * q - 2 * q^9 - 8 * q^13 - 8 * q^17 - 20 * q^23 - 12 * q^27 - 12 * q^29 + 8 * q^39 + 36 * q^43 + 18 * q^49 + 24 * q^51 + 4 * q^53 - 36 * q^61 + 8 * q^69 - 20 * q^77 + 24 * q^79 - 26 * q^81 + 60 * q^87 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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