LMFDB - Embedded newform 920.2.e.b.369.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 $ x^14 - 2x^11 + 39x^10 - 10x^9 + 2x^8 - 26x^7 + 297x^6 - 116x^5 + 24x^4 - 20x^3 + 64x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.9
Root			$1.55369 - 1.55369i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.b.369.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.356372i q^{3} +(0.376266 + 2.20418i) q^{5} +2.46376i q^{7} +2.87300 q^{9} +O(q^{10})$	$q+0.356372i q^{3} +(0.376266 + 2.20418i) q^{5} +2.46376i q^{7} +2.87300 q^{9} +1.61574 q^{11} +2.62553i q^{13} +(-0.785509 + 0.134090i) q^{15} -2.58924i q^{17} -4.02550 q^{19} -0.878013 q^{21} -1.00000i q^{23} +(-4.71685 + 1.65872i) q^{25} +2.09297i q^{27} +7.08580 q^{29} -4.58371 q^{31} +0.575803i q^{33} +(-5.43057 + 0.927026i) q^{35} +2.96054i q^{37} -0.935665 q^{39} -5.71155 q^{41} +2.30448i q^{43} +(1.08101 + 6.33262i) q^{45} +6.88317i q^{47} +0.929907 q^{49} +0.922731 q^{51} +6.76467i q^{53} +(0.607946 + 3.56138i) q^{55} -1.43457i q^{57} -2.53432 q^{59} +9.25564 q^{61} +7.07837i q^{63} +(-5.78715 + 0.987896i) q^{65} -15.7285i q^{67} +0.356372 q^{69} -5.25113 q^{71} -6.03858i q^{73} +(-0.591120 - 1.68095i) q^{75} +3.98078i q^{77} +1.35712 q^{79} +7.87312 q^{81} +8.59500i q^{83} +(5.70716 - 0.974241i) q^{85} +2.52518i q^{87} +8.84084 q^{89} -6.46867 q^{91} -1.63350i q^{93} +(-1.51466 - 8.87293i) q^{95} +8.01314i q^{97} +4.64201 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 4 * q^9	$ 14 q + 2 q^{5} - 4 q^{9} - 14 q^{11} - 6 q^{15} + 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} - 20 q^{31} - 2 q^{35} + 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} - 14 q^{51} - 38 q^{55} + 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} - 28 q^{71} - 24 q^{75} + 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} - 14 q^{91} - 30 q^{95} + 122 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 4 * q^9 - 14 * q^11 - 6 * q^15 + 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 - 20 * q^31 - 2 * q^35 + 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 - 14 * q^51 - 38 * q^55 + 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 - 28 * q^71 - 24 * q^75 + 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 - 14 * q^91 - 30 * q^95 + 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			0.356372i			0.205751i	0.994694	+	0.102876i	$0.0328044\pi$
$3$			0.356372i			0.205751i	−0.994694	+	0.102876i	$0.967196\pi$
$4$		0			0
$5$	0.376266	+	2.20418i	0.168271	+	0.985741i
$5$	0.376266	+	2.20418i	0.168271	+	0.985741i
$6$		0			0
$7$			2.46376i			0.931212i	0.884992	+	0.465606i	$0.154164\pi$
$7$			2.46376i			0.931212i	−0.884992	+	0.465606i	$0.845836\pi$
$8$		0			0
$9$	2.87300			0.957666
$10$		0			0
$11$	1.61574			0.487163			0.243581	−	0.969880i	$-0.421678\pi$
$11$	1.61574			0.487163			0.243581	+	0.969880i	$0.421678\pi$
$12$		0			0
$13$			2.62553i			0.728191i	0.931362	+	0.364096i	$0.118622\pi$
$13$			2.62553i			0.728191i	−0.931362	+	0.364096i	$0.881378\pi$
$14$		0			0
$15$	−0.785509	+	0.134090i	−0.202817	+	0.0346220i
$16$		0			0
$17$		−	2.58924i		−	0.627982i	−0.949426	−	0.313991i	$-0.898334\pi$
$17$		−	2.58924i		−	0.627982i	0.949426	−	0.313991i	$-0.101666\pi$
$18$		0			0
$19$	−4.02550			−0.923512			−0.461756	−	0.887007i	$-0.652780\pi$
$19$	−4.02550			−0.923512			−0.461756	+	0.887007i	$0.652780\pi$
$20$		0			0
$21$	−0.878013			−0.191598
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.71685	+	1.65872i	−0.943370	+	0.331743i
$26$		0			0
$27$			2.09297i			0.402793i
$28$		0			0
$29$	7.08580			1.31580			0.657900	−	0.753106i	$-0.271445\pi$
$29$	7.08580			1.31580			0.657900	+	0.753106i	$0.271445\pi$
$30$		0			0
$31$	−4.58371			−0.823258			−0.411629	−	0.911352i	$-0.635040\pi$
$31$	−4.58371			−0.823258			−0.411629	+	0.911352i	$0.635040\pi$
$32$		0			0
$33$			0.575803i			0.100234i
$34$		0			0
$35$	−5.43057	+	0.927026i	−0.917934	+	0.156696i
$36$		0			0
$37$			2.96054i			0.486710i	0.969937	+	0.243355i	$0.0782480\pi$
$37$			2.96054i			0.486710i	−0.969937	+	0.243355i	$0.921752\pi$
$38$		0			0
$39$	−0.935665			−0.149826
$40$		0			0
$41$	−5.71155			−0.891995			−0.445997	−	0.895034i	$-0.647151\pi$
$41$	−5.71155			−0.891995			−0.445997	+	0.895034i	$0.647151\pi$
$42$		0			0
$43$			2.30448i			0.351430i	0.984441	+	0.175715i	$0.0562237\pi$
$43$			2.30448i			0.351430i	−0.984441	+	0.175715i	$0.943776\pi$
$44$		0			0
$45$	1.08101	+	6.33262i	0.161148	+	0.944011i
$46$		0			0
$47$			6.88317i			1.00401i	0.864864	+	0.502007i	$0.167405\pi$
$47$			6.88317i			1.00401i	−0.864864	+	0.502007i	$0.832595\pi$
$48$		0			0
$49$	0.929907			0.132844
$50$		0			0
$51$	0.922731			0.129208
$52$		0			0
$53$			6.76467i			0.929199i	0.885521	+	0.464600i	$0.153802\pi$
$53$			6.76467i			0.929199i	−0.885521	+	0.464600i	$0.846198\pi$
$54$		0			0
$55$	0.607946	+	3.56138i	0.0819754	+	0.480216i
$56$		0			0
$57$		−	1.43457i		−	0.190014i
$58$		0			0
$59$	−2.53432			−0.329941			−0.164970	−	0.986299i	$-0.552753\pi$
$59$	−2.53432			−0.329941			−0.164970	+	0.986299i	$0.552753\pi$
$60$		0			0
$61$	9.25564			1.18506			0.592532	−	0.805547i	$-0.298129\pi$
$61$	9.25564			1.18506			0.592532	+	0.805547i	$0.298129\pi$
$62$		0			0
$63$			7.07837i			0.891791i
$64$		0			0
$65$	−5.78715	+	0.987896i	−0.717808	+	0.122533i
$66$		0			0
$67$		−	15.7285i		−	1.92154i	−0.277347	−	0.960770i	$-0.589455\pi$
$67$		−	15.7285i		−	1.92154i	0.277347	−	0.960770i	$-0.410545\pi$
$68$		0			0
$69$	0.356372			0.0429021
$70$		0			0
$71$	−5.25113			−0.623194			−0.311597	−	0.950214i	$-0.600864\pi$
$71$	−5.25113			−0.623194			−0.311597	+	0.950214i	$0.600864\pi$
$72$		0			0
$73$		−	6.03858i		−	0.706762i	−0.935479	−	0.353381i	$-0.885032\pi$
$73$		−	6.03858i		−	0.706762i	0.935479	−	0.353381i	$-0.114968\pi$
$74$		0			0
$75$	−0.591120	−	1.68095i	−0.0682566	−	0.194100i
$76$		0			0
$77$			3.98078i			0.453652i
$78$		0			0
$79$	1.35712			0.152688			0.0763439	−	0.997082i	$-0.475675\pi$
$79$	1.35712			0.152688			0.0763439	+	0.997082i	$0.475675\pi$
$80$		0			0
$81$	7.87312			0.874791
$82$		0			0
$83$			8.59500i			0.943423i	0.881753	+	0.471712i	$0.156364\pi$
$83$			8.59500i			0.943423i	−0.881753	+	0.471712i	$0.843636\pi$
$84$		0			0
$85$	5.70716	−	0.974241i	0.619028	−	0.105671i
$86$		0			0
$87$			2.52518i			0.270727i
$88$		0			0
$89$	8.84084			0.937127			0.468564	−	0.883430i	$-0.344772\pi$
$89$	8.84084			0.937127			0.468564	+	0.883430i	$0.344772\pi$
$90$		0			0
$91$	−6.46867			−0.678100
$92$		0			0
$93$		−	1.63350i		−	0.169386i
$94$		0			0
$95$	−1.51466	−	8.87293i	−0.155400	−	0.910344i
$96$		0			0
$97$			8.01314i			0.813611i	0.913515	+	0.406806i	$0.133357\pi$
$97$			8.01314i			0.813611i	−0.913515	+	0.406806i	$0.866643\pi$
$98$		0			0
$99$	4.64201			0.466539
$100$		0			0
$101$	−6.33185			−0.630043			−0.315021	−	0.949085i	$-0.602012\pi$
$101$	−6.33185			−0.630043			−0.315021	+	0.949085i	$0.602012\pi$
$102$		0			0
$103$		−	13.1901i		−	1.29966i	−0.760078	−	0.649831i	$-0.774839\pi$
$103$		−	13.1901i		−	1.29966i	0.760078	−	0.649831i	$-0.225161\pi$
$104$		0			0
$105$	−0.330366	−	1.93530i	−0.0322404	−	0.188866i
$106$		0			0
$107$			9.54923i			0.923159i	0.887099	+	0.461580i	$0.152717\pi$
$107$			9.54923i			0.923159i	−0.887099	+	0.461580i	$0.847283\pi$
$108$		0			0
$109$	17.5622			1.68215			0.841077	−	0.540915i	$-0.181922\pi$
$109$	17.5622			1.68215			0.841077	+	0.540915i	$0.181922\pi$
$110$		0			0
$111$	−1.05505			−0.100141
$112$		0			0
$113$			1.54267i			0.145122i	0.997364	+	0.0725611i	$0.0231172\pi$
$113$			1.54267i			0.145122i	−0.997364	+	0.0725611i	$0.976883\pi$
$114$		0			0
$115$	2.20418	−	0.376266i	0.205541	−	0.0350869i
$116$		0			0
$117$			7.54315i			0.697364i
$118$		0			0
$119$	6.37925			0.584785
$120$		0			0
$121$	−8.38940			−0.762673
$122$		0			0
$123$		−	2.03544i		−	0.183529i
$124$		0			0
$125$	−5.43090	−	9.77268i	−0.485755	−	0.874095i
$126$		0			0
$127$		−	12.4256i		−	1.10260i	−0.834308	−	0.551299i	$-0.814132\pi$
$127$		−	12.4256i		−	1.10260i	0.834308	−	0.551299i	$-0.185868\pi$
$128$		0			0
$129$	−0.821251			−0.0723071
$130$		0			0
$131$	−11.3191			−0.988957			−0.494479	−	0.869190i	$-0.664641\pi$
$131$	−11.3191			−0.988957			−0.494479	+	0.869190i	$0.664641\pi$
$132$		0			0
$133$		−	9.91784i		−	0.859986i
$134$		0			0
$135$	−4.61329	+	0.787513i	−0.397049	+	0.0677783i
$136$		0			0
$137$		−	4.94093i		−	0.422132i	−0.977472	−	0.211066i	$-0.932306\pi$
$137$		−	4.94093i		−	0.422132i	0.977472	−	0.211066i	$-0.0676936\pi$
$138$		0			0
$139$	2.93101			0.248605			0.124303	−	0.992244i	$-0.460331\pi$
$139$	2.93101			0.248605			0.124303	+	0.992244i	$0.460331\pi$
$140$		0			0
$141$	−2.45297			−0.206577
$142$		0			0
$143$			4.24216i			0.354748i
$144$		0			0
$145$	2.66614	+	15.6184i	0.221411	+	1.29704i
$146$		0			0
$147$			0.331393i			0.0273328i
$148$		0			0
$149$	5.09023			0.417008			0.208504	−	0.978022i	$-0.433141\pi$
$149$	5.09023			0.417008			0.208504	+	0.978022i	$0.433141\pi$
$150$		0			0
$151$	−14.3645			−1.16897			−0.584485	−	0.811405i	$-0.698703\pi$
$151$	−14.3645			−1.16897			−0.584485	+	0.811405i	$0.698703\pi$
$152$		0			0
$153$		−	7.43888i		−	0.601398i
$154$		0			0
$155$	−1.72469	−	10.1033i	−0.138530	−	0.811519i
$156$		0			0
$157$			7.34146i			0.585912i	0.956126	+	0.292956i	$0.0946390\pi$
$157$			7.34146i			0.585912i	−0.956126	+	0.292956i	$0.905361\pi$
$158$		0			0
$159$	−2.41074			−0.191184
$160$		0			0
$161$	2.46376			0.194171
$162$		0			0
$163$		−	2.12762i		−	0.166648i	−0.996523	−	0.0833240i	$-0.973446\pi$
$163$		−	2.12762i		−	0.166648i	0.996523	−	0.0833240i	$-0.0265536\pi$
$164$		0			0
$165$	−1.26917	+	0.216655i	−0.0988051	+	0.0168665i
$166$		0			0
$167$		−	2.16080i		−	0.167208i	−0.996499	−	0.0836039i	$-0.973357\pi$
$167$		−	2.16080i		−	0.167208i	0.996499	−	0.0836039i	$-0.0266431\pi$
$168$		0			0
$169$	6.10659			0.469738
$170$		0			0
$171$	−11.5652			−0.884417
$172$		0			0
$173$		−	18.6210i		−	1.41573i	−0.706350	−	0.707863i	$-0.749659\pi$
$173$		−	18.6210i		−	1.41573i	0.706350	−	0.707863i	$-0.250341\pi$
$174$		0			0
$175$	−4.08667	−	11.6212i	−0.308923	−	0.878477i
$176$		0			0
$177$		−	0.903162i		−	0.0678858i
$178$		0			0
$179$	13.4670			1.00657			0.503287	−	0.864119i	$-0.332124\pi$
$179$	13.4670			1.00657			0.503287	+	0.864119i	$0.332124\pi$
$180$		0			0
$181$	12.2927			0.913706			0.456853	−	0.889542i	$-0.348977\pi$
$181$	12.2927			0.913706			0.456853	+	0.889542i	$0.348977\pi$
$182$		0			0
$183$			3.29845i			0.243828i
$184$		0			0
$185$	−6.52558	+	1.11395i	−0.479770	+	0.0818992i
$186$		0			0
$187$		−	4.18352i		−	0.305930i
$188$		0			0
$189$	−5.15657			−0.375085
$190$		0			0
$191$	−2.07099			−0.149852			−0.0749258	−	0.997189i	$-0.523872\pi$
$191$	−2.07099			−0.149852			−0.0749258	+	0.997189i	$0.523872\pi$
$192$		0			0
$193$		−	25.7996i		−	1.85710i	−0.371209	−	0.928549i	$-0.621057\pi$
$193$		−	25.7996i		−	1.85710i	0.371209	−	0.928549i	$-0.378943\pi$
$194$		0			0
$195$	−0.352058	−	2.06238i	−0.0252114	−	0.147690i
$196$		0			0
$197$		−	4.21443i		−	0.300266i	−0.988666	−	0.150133i	$-0.952030\pi$
$197$		−	4.21443i		−	0.300266i	0.988666	−	0.150133i	$-0.0479702\pi$
$198$		0			0
$199$	22.0428			1.56257			0.781285	−	0.624175i	$-0.214565\pi$
$199$	22.0428			1.56257			0.781285	+	0.624175i	$0.214565\pi$
$200$		0			0
$201$	5.60519			0.395359
$202$		0			0
$203$			17.4577i			1.22529i
$204$		0			0
$205$	−2.14906	−	12.5893i	−0.150097	−	0.879276i
$206$		0			0
$207$		−	2.87300i		−	0.199687i
$208$		0			0
$209$	−6.50414			−0.449901
$210$		0			0
$211$	28.9217			1.99105			0.995524	−	0.0945049i	$-0.0301268\pi$
$211$	28.9217			1.99105			0.995524	+	0.0945049i	$0.0301268\pi$
$212$		0			0
$213$		−	1.87135i		−	0.128223i
$214$		0			0
$215$	−5.07949	+	0.867096i	−0.346418	+	0.0591354i
$216$		0			0
$217$		−	11.2931i		−	0.766628i
$218$		0			0
$219$	2.15198			0.145417
$220$		0			0
$221$	6.79812			0.457291
$222$		0			0
$223$		−	6.63731i		−	0.444467i	−0.974993	−	0.222234i	$-0.928665\pi$
$223$		−	6.63731i		−	0.444467i	0.974993	−	0.222234i	$-0.0713348\pi$
$224$		0			0
$225$	−13.5515	+	4.76549i	−0.903433	+	0.317699i
$226$		0			0
$227$			0.370367i			0.0245821i	0.999924	+	0.0122911i	$0.00391246\pi$
$227$			0.370367i			0.0245821i	−0.999924	+	0.0122911i	$0.996088\pi$
$228$		0			0
$229$	11.5625			0.764073			0.382036	−	0.924147i	$-0.375223\pi$
$229$	11.5625			0.764073			0.382036	+	0.924147i	$0.375223\pi$
$230$		0			0
$231$	−1.41864			−0.0933395
$232$		0			0
$233$		−	16.8534i		−	1.10410i	−0.833810	−	0.552051i	$-0.813845\pi$
$233$		−	16.8534i		−	1.10410i	0.833810	−	0.552051i	$-0.186155\pi$
$234$		0			0
$235$	−15.1718	+	2.58990i	−0.989697	+	0.168946i
$236$		0			0
$237$			0.483639i			0.0314157i
$238$		0			0
$239$	17.7742			1.14972			0.574859	−	0.818252i	$-0.305057\pi$
$239$	17.7742			1.14972			0.574859	+	0.818252i	$0.305057\pi$
$240$		0			0
$241$	−26.6318			−1.71551			−0.857754	−	0.514061i	$-0.828141\pi$
$241$	−26.6318			−1.71551			−0.857754	+	0.514061i	$0.828141\pi$
$242$		0			0
$243$			9.08467i			0.582782i
$244$		0			0
$245$	0.349892	+	2.04969i	0.0223538	+	0.130950i
$246$		0			0
$247$		−	10.5691i		−	0.672493i
$248$		0			0
$249$	−3.06301			−0.194111
$250$		0			0
$251$	16.8599			1.06419			0.532094	−	0.846685i	$-0.321405\pi$
$251$	16.8599			1.06419			0.532094	+	0.846685i	$0.321405\pi$
$252$		0			0
$253$		−	1.61574i		−	0.101580i
$254$		0			0
$255$	0.347192	+	2.03387i	0.0217420	+	0.127366i
$256$		0			0
$257$		−	8.33024i		−	0.519626i	−0.965659	−	0.259813i	$-0.916339\pi$
$257$		−	8.33024i		−	0.519626i	0.965659	−	0.259813i	$-0.0836610\pi$
$258$		0			0
$259$	−7.29405			−0.453230
$260$		0			0
$261$	20.3575			1.26010
$262$		0			0
$263$			15.0389i			0.927337i	0.886009	+	0.463669i	$0.153467\pi$
$263$			15.0389i			0.927337i	−0.886009	+	0.463669i	$0.846533\pi$
$264$		0			0
$265$	−14.9106	+	2.54531i	−0.915950	+	0.156357i
$266$		0			0
$267$			3.15063i			0.192815i
$268$		0			0
$269$	7.01922			0.427969			0.213985	−	0.976837i	$-0.431356\pi$
$269$	7.01922			0.427969			0.213985	+	0.976837i	$0.431356\pi$
$270$		0			0
$271$	4.40129			0.267359			0.133680	−	0.991025i	$-0.457321\pi$
$271$	4.40129			0.267359			0.133680	+	0.991025i	$0.457321\pi$
$272$		0			0
$273$		−	2.30525i		−	0.139520i
$274$		0			0
$275$	−7.62118	+	2.68005i	−0.459575	+	0.161613i
$276$		0			0
$277$		−	8.41544i		−	0.505634i	−0.967514	−	0.252817i	$-0.918643\pi$
$277$		−	8.41544i		−	0.505634i	0.967514	−	0.252817i	$-0.0813571\pi$
$278$		0			0
$279$	−13.1690			−0.788406
$280$		0			0
$281$	−15.0068			−0.895233			−0.447617	−	0.894226i	$-0.647727\pi$
$281$	−15.0068			−0.895233			−0.447617	+	0.894226i	$0.647727\pi$
$282$		0			0
$283$		−	7.09122i		−	0.421529i	−0.977537	−	0.210765i	$-0.932405\pi$
$283$		−	7.09122i		−	0.421529i	0.977537	−	0.210765i	$-0.0675953\pi$
$284$		0			0
$285$	3.16206	−	0.539781i	0.187304	−	0.0319738i
$286$		0			0
$287$		−	14.0719i		−	0.830637i
$288$		0			0
$289$	10.2958			0.605638
$290$		0			0
$291$	−2.85566			−0.167402
$292$		0			0
$293$			28.0567i			1.63909i	0.573015	+	0.819545i	$0.305774\pi$
$293$			28.0567i			1.63909i	−0.573015	+	0.819545i	$0.694226\pi$
$294$		0			0
$295$	−0.953579	−	5.58612i	−0.0555195	−	0.325236i
$296$		0			0
$297$			3.38169i			0.196225i
$298$		0			0
$299$	2.62553			0.151838
$300$		0			0
$301$	−5.67767			−0.327256
$302$		0			0
$303$		−	2.25649i		−	0.129632i
$304$		0			0
$305$	3.48258	+	20.4011i	0.199412	+	1.16817i
$306$		0			0
$307$		−	29.2707i		−	1.67057i	−0.549820	−	0.835283i	$-0.685304\pi$
$307$		−	29.2707i		−	1.67057i	0.549820	−	0.835283i	$-0.314696\pi$
$308$		0			0
$309$	4.70059			0.267407
$310$		0			0
$311$	−15.8111			−0.896563			−0.448281	−	0.893893i	$-0.647964\pi$
$311$	−15.8111			−0.896563			−0.448281	+	0.893893i	$0.647964\pi$
$312$		0			0
$313$			9.98602i			0.564443i	0.959349	+	0.282222i	$0.0910713\pi$
$313$			9.98602i			0.564443i	−0.959349	+	0.282222i	$0.908929\pi$
$314$		0			0
$315$	−15.6020	+	2.66335i	−0.879074	+	0.150063i
$316$		0			0
$317$			13.7683i			0.773302i	0.922226	+	0.386651i	$0.126368\pi$
$317$			13.7683i			0.773302i	−0.922226	+	0.386651i	$0.873632\pi$
$318$		0			0
$319$	11.4488			0.641008
$320$		0			0
$321$	−3.40308			−0.189941
$322$		0			0
$323$			10.4230i			0.579950i
$324$		0			0
$325$	−4.35501	−	12.3842i	−0.241572	−	0.686953i
$326$		0			0
$327$			6.25868i			0.346106i
$328$		0			0
$329$	−16.9584			−0.934950
$330$		0			0
$331$	18.6765			1.02656			0.513278	−	0.858223i	$-0.328431\pi$
$331$	18.6765			1.02656			0.513278	+	0.858223i	$0.328431\pi$
$332$		0			0
$333$			8.50564i			0.466106i
$334$		0			0
$335$	34.6685	−	5.91809i	1.89414	−	0.323340i
$336$		0			0
$337$			13.8915i			0.756716i	0.925659	+	0.378358i	$0.123511\pi$
$337$			13.8915i			0.756716i	−0.925659	+	0.378358i	$0.876489\pi$
$338$		0			0
$339$	−0.549764			−0.0298591
$340$		0			0
$341$	−7.40606			−0.401060
$342$		0			0
$343$			19.5374i			1.05492i
$344$		0			0
$345$	0.134090	+	0.785509i	0.00721919	+	0.0422904i
$346$		0			0
$347$		−	0.706854i		−	0.0379459i	−0.999820	−	0.0189730i	$-0.993960\pi$
$347$		−	0.706854i		−	0.0379459i	0.999820	−	0.0189730i	$-0.00603964\pi$
$348$		0			0
$349$	−6.93817			−0.371392			−0.185696	−	0.982607i	$-0.559454\pi$
$349$	−6.93817			−0.371392			−0.185696	+	0.982607i	$0.559454\pi$
$350$		0			0
$351$	−5.49516			−0.293310
$352$		0			0
$353$			0.413170i			0.0219908i	0.999940	+	0.0109954i	$0.00350002\pi$
$353$			0.413170i			0.0219908i	−0.999940	+	0.0109954i	$0.996500\pi$
$354$		0			0
$355$	−1.97582	−	11.5744i	−0.104866	−	0.614308i
$356$		0			0
$357$			2.27338i			0.120320i
$358$		0			0
$359$	32.6209			1.72167			0.860834	−	0.508886i	$-0.169943\pi$
$359$	32.6209			1.72167			0.860834	+	0.508886i	$0.169943\pi$
$360$		0			0
$361$	−2.79537			−0.147125
$362$		0			0
$363$		−	2.98974i		−	0.156921i
$364$		0			0
$365$	13.3101	−	2.27211i	0.696685	−	0.118928i
$366$		0			0
$367$		−	13.5293i		−	0.706223i	−0.935581	−	0.353112i	$-0.885124\pi$
$367$		−	13.5293i		−	0.706223i	0.935581	−	0.353112i	$-0.114876\pi$
$368$		0			0
$369$	−16.4093			−0.854234
$370$		0			0
$371$	−16.6665			−0.865282
$372$		0			0
$373$		−	20.7690i		−	1.07538i	−0.843143	−	0.537689i	$-0.819297\pi$
$373$		−	20.7690i		−	1.07538i	0.843143	−	0.537689i	$-0.180703\pi$
$374$		0			0
$375$	3.48271	−	1.93542i	0.179846	−	0.0999447i
$376$		0			0
$377$			18.6040i			0.958153i
$378$		0			0
$379$	25.0528			1.28688			0.643438	−	0.765498i	$-0.277507\pi$
$379$	25.0528			1.28688			0.643438	+	0.765498i	$0.277507\pi$
$380$		0			0
$381$	4.42815			0.226861
$382$		0			0
$383$			4.51677i			0.230796i	0.993319	+	0.115398i	$0.0368144\pi$
$383$			4.51677i			0.230796i	−0.993319	+	0.115398i	$0.963186\pi$
$384$		0			0
$385$	−8.77437	+	1.49783i	−0.447183	+	0.0763365i
$386$		0			0
$387$			6.62076i			0.336552i
$388$		0			0
$389$	−0.455188			−0.0230789			−0.0115395	−	0.999933i	$-0.503673\pi$
$389$	−0.455188			−0.0230789			−0.0115395	+	0.999933i	$0.503673\pi$
$390$		0			0
$391$	−2.58924			−0.130943
$392$		0			0
$393$		−	4.03382i		−	0.203479i
$394$		0			0
$395$	0.510637	+	2.99134i	0.0256929	+	0.150511i
$396$		0			0
$397$		−	15.0798i		−	0.756832i	−0.925636	−	0.378416i	$-0.876469\pi$
$397$		−	15.0798i		−	0.756832i	0.925636	−	0.378416i	$-0.123531\pi$
$398$		0			0
$399$	3.53444			0.176943
$400$		0			0
$401$	−38.0845			−1.90185			−0.950924	−	0.309424i	$-0.899864\pi$
$401$	−38.0845			−1.90185			−0.950924	+	0.309424i	$0.899864\pi$
$402$		0			0
$403$		−	12.0347i		−	0.599489i
$404$		0			0
$405$	2.96238	+	17.3538i	0.147202	+	0.862317i
$406$		0			0
$407$			4.78345i			0.237107i
$408$		0			0
$409$	−16.3672			−0.809305			−0.404653	−	0.914470i	$-0.632608\pi$
$409$	−16.3672			−0.809305			−0.404653	+	0.914470i	$0.632608\pi$
$410$		0			0
$411$	1.76081			0.0868543
$412$		0			0
$413$		−	6.24396i		−	0.307245i
$414$		0			0
$415$	−18.9449	+	3.23400i	−0.929971	+	0.158751i
$416$		0			0
$417$			1.04453i			0.0511509i
$418$		0			0
$419$	5.32621			0.260202			0.130101	−	0.991501i	$-0.458470\pi$
$419$	5.32621			0.260202			0.130101	+	0.991501i	$0.458470\pi$
$420$		0			0
$421$	−11.3000			−0.550730			−0.275365	−	0.961340i	$-0.588799\pi$
$421$	−11.3000			−0.550730			−0.275365	+	0.961340i	$0.588799\pi$
$422$		0			0
$423$			19.7753i			0.961510i
$424$		0			0
$425$	4.29481	+	12.2130i	0.208329	+	0.592420i
$426$		0			0
$427$			22.8036i			1.10355i
$428$		0			0
$429$	−1.51179			−0.0729898
$430$		0			0
$431$	33.6364			1.62021			0.810105	−	0.586285i	$-0.199410\pi$
$431$	33.6364			1.62021			0.810105	+	0.586285i	$0.199410\pi$
$432$		0			0
$433$			18.3932i			0.883920i	0.897035	+	0.441960i	$0.145717\pi$
$433$			18.3932i			0.883920i	−0.897035	+	0.441960i	$0.854283\pi$
$434$		0			0
$435$	−5.56595	+	0.950137i	−0.266867	+	0.0455556i
$436$		0			0
$437$			4.02550i			0.192566i
$438$		0			0
$439$	−28.7628			−1.37277			−0.686387	−	0.727236i	$-0.740804\pi$
$439$	−28.7628			−1.37277			−0.686387	+	0.727236i	$0.740804\pi$
$440$		0			0
$441$	2.67162			0.127220
$442$		0			0
$443$			22.2359i			1.05646i	0.849102	+	0.528229i	$0.177144\pi$
$443$			22.2359i			1.05646i	−0.849102	+	0.528229i	$0.822856\pi$
$444$		0			0
$445$	3.32650	+	19.4868i	0.157691	+	0.923764i
$446$		0			0
$447$			1.81401i			0.0857999i
$448$		0			0
$449$	−28.4538			−1.34282			−0.671409	−	0.741087i	$-0.734311\pi$
$449$	−28.4538			−1.34282			−0.671409	+	0.741087i	$0.734311\pi$
$450$		0			0
$451$	−9.22836			−0.434547
$452$		0			0
$453$		−	5.11912i		−	0.240517i
$454$		0			0
$455$	−2.43394	−	14.2581i	−0.114105	−	0.668431i
$456$		0			0
$457$		−	7.58512i		−	0.354817i	−0.984137	−	0.177408i	$-0.943229\pi$
$457$		−	7.58512i		−	0.354817i	0.984137	−	0.177408i	$-0.0567714\pi$
$458$		0			0
$459$	5.41920			0.252947
$460$		0			0
$461$	6.02951			0.280822			0.140411	−	0.990093i	$-0.455158\pi$
$461$	6.02951			0.280822			0.140411	+	0.990093i	$0.455158\pi$
$462$		0			0
$463$			24.2763i			1.12821i	0.825701	+	0.564107i	$0.190780\pi$
$463$			24.2763i			1.12821i	−0.825701	+	0.564107i	$0.809220\pi$
$464$		0			0
$465$	3.60054	−	0.614631i	0.166971	−	0.0285028i
$466$		0			0
$467$		−	5.99614i		−	0.277468i	−0.990330	−	0.138734i	$-0.955697\pi$
$467$		−	5.99614i		−	0.277468i	0.990330	−	0.138734i	$-0.0443034\pi$
$468$		0			0
$469$	38.7511			1.78936
$470$		0			0
$471$	−2.61629			−0.120552
$472$		0			0
$473$			3.72343i			0.171203i
$474$		0			0
$475$	18.9877	−	6.67716i	0.871214	−	0.306369i
$476$		0			0
$477$			19.4349i			0.889863i
$478$		0			0
$479$	17.5065			0.799892			0.399946	−	0.916539i	$-0.369029\pi$
$479$	17.5065			0.799892			0.399946	+	0.916539i	$0.369029\pi$
$480$		0			0
$481$	−7.77299			−0.354418
$482$		0			0
$483$			0.878013i			0.0399510i
$484$		0			0
$485$	−17.6624	+	3.01507i	−0.802010	+	0.136907i
$486$		0			0
$487$			31.6269i			1.43315i	0.697509	+	0.716576i	$0.254292\pi$
$487$			31.6269i			1.43315i	−0.697509	+	0.716576i	$0.745708\pi$
$488$		0			0
$489$	0.758224			0.0342881
$490$		0			0
$491$	24.7642			1.11759			0.558797	−	0.829305i	$-0.311263\pi$
$491$	24.7642			1.11759			0.558797	+	0.829305i	$0.311263\pi$
$492$		0			0
$493$		−	18.3468i		−	0.826299i
$494$		0			0
$495$	1.74663	+	10.2318i	0.0785051	+	0.459887i
$496$		0			0
$497$		−	12.9375i		−	0.580326i
$498$		0			0
$499$	−37.7906			−1.69174			−0.845871	−	0.533388i	$-0.820919\pi$
$499$	−37.7906			−1.69174			−0.845871	+	0.533388i	$0.820919\pi$
$500$		0			0
$501$	0.770049			0.0344032
$502$		0			0
$503$			1.40481i			0.0626376i	0.999509	+	0.0313188i	$0.00997071\pi$
$503$			1.40481i			0.0626376i	−0.999509	+	0.0313188i	$0.990029\pi$
$504$		0			0
$505$	−2.38246	−	13.9566i	−0.106018	−	0.621059i
$506$		0			0
$507$			2.17622i			0.0966492i
$508$		0			0
$509$	−30.4044			−1.34765			−0.673825	−	0.738891i	$-0.735350\pi$
$509$	−30.4044			−1.34765			−0.673825	+	0.738891i	$0.735350\pi$
$510$		0			0
$511$	14.8776			0.658146
$512$		0			0
$513$		−	8.42525i		−	0.371984i
$514$		0			0
$515$	29.0735	−	4.96299i	1.28113	−	0.218696i
$516$		0			0
$517$			11.1214i			0.489118i
$518$		0			0
$519$	6.63599			0.291288
$520$		0			0
$521$	−13.3500			−0.584873			−0.292437	−	0.956285i	$-0.594466\pi$
$521$	−13.3500			−0.584873			−0.292437	+	0.956285i	$0.594466\pi$
$522$		0			0
$523$			8.71026i			0.380873i	0.981699	+	0.190437i	$0.0609904\pi$
$523$			8.71026i			0.380873i	−0.981699	+	0.190437i	$0.939010\pi$
$524$		0			0
$525$	4.14145	−	1.45637i	0.180748	−	0.0635614i
$526$		0			0
$527$			11.8683i			0.516991i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−7.28111			−0.315973
$532$		0			0
$533$		−	14.9959i		−	0.649543i
$534$		0			0
$535$	−21.0483	+	3.59305i	−0.909996	+	0.155341i
$536$		0			0
$537$			4.79927i			0.207104i
$538$		0			0
$539$	1.50248			0.0647166
$540$		0			0
$541$	21.2922			0.915423			0.457711	−	0.889101i	$-0.348669\pi$
$541$	21.2922			0.915423			0.457711	+	0.889101i	$0.348669\pi$
$542$		0			0
$543$			4.38075i			0.187996i
$544$		0			0
$545$	6.60806	+	38.7103i	0.283058	+	1.65817i
$546$		0			0
$547$		−	7.27222i		−	0.310937i	−0.987841	−	0.155469i	$-0.950311\pi$
$547$		−	7.27222i		−	0.310937i	0.987841	−	0.155469i	$-0.0496888\pi$
$548$		0			0
$549$	26.5914			1.13490
$550$		0			0
$551$	−28.5238			−1.21516
$552$		0			0
$553$			3.34361i			0.142185i
$554$		0			0
$555$	−0.396980	−	2.32553i	−0.0168509	−	0.0987133i
$556$		0			0
$557$			27.0665i			1.14684i	0.819260	+	0.573422i	$0.194384\pi$
$557$			27.0665i			1.14684i	−0.819260	+	0.573422i	$0.805616\pi$
$558$		0			0
$559$	−6.05048			−0.255908
$560$		0			0
$561$	1.49089			0.0629454
$562$		0			0
$563$			12.7921i			0.539124i	0.962983	+	0.269562i	$0.0868790\pi$
$563$			12.7921i			0.539124i	−0.962983	+	0.269562i	$0.913121\pi$
$564$		0			0
$565$	−3.40033	+	0.580454i	−0.143053	+	0.0244199i
$566$		0			0
$567$			19.3974i			0.814616i
$568$		0			0
$569$	13.3013			0.557621			0.278810	−	0.960346i	$-0.410060\pi$
$569$	13.3013			0.557621			0.278810	+	0.960346i	$0.410060\pi$
$570$		0			0
$571$	31.8239			1.33179			0.665894	−	0.746046i	$-0.268050\pi$
$571$	31.8239			1.33179			0.665894	+	0.746046i	$0.268050\pi$
$572$		0			0
$573$		−	0.738043i		−	0.0308322i
$574$		0			0
$575$	1.65872	+	4.71685i	0.0691733	+	0.196706i
$576$		0			0
$577$		−	26.0091i		−	1.08277i	−0.840774	−	0.541386i	$-0.817900\pi$
$577$		−	26.0091i		−	1.08277i	0.840774	−	0.541386i	$-0.182100\pi$
$578$		0			0
$579$	9.19426			0.382101
$580$		0			0
$581$	−21.1760			−0.878527
$582$		0			0
$583$			10.9299i			0.452671i
$584$		0			0
$585$	−16.6265	+	2.83823i	−0.687420	+	0.117346i
$586$		0			0
$587$			39.4259i			1.62728i	0.581367	+	0.813641i	$0.302518\pi$
$587$			39.4259i			1.62728i	−0.581367	+	0.813641i	$0.697482\pi$
$588$		0			0
$589$	18.4517			0.760289
$590$		0			0
$591$	1.50191			0.0617801
$592$		0			0
$593$			20.8918i			0.857922i	0.903323	+	0.428961i	$0.141120\pi$
$593$			20.8918i			0.857922i	−0.903323	+	0.428961i	$0.858880\pi$
$594$		0			0
$595$	2.40029	+	14.0610i	0.0984024	+	0.576446i
$596$		0			0
$597$			7.85542i			0.321501i
$598$		0			0
$599$	−12.7986			−0.522938			−0.261469	−	0.965212i	$-0.584207\pi$
$599$	−12.7986			−0.522938			−0.261469	+	0.965212i	$0.584207\pi$
$600$		0			0
$601$	−18.7566			−0.765099			−0.382550	−	0.923935i	$-0.624954\pi$
$601$	−18.7566			−0.765099			−0.382550	+	0.923935i	$0.624954\pi$
$602$		0			0
$603$		−	45.1879i		−	1.84019i
$604$		0			0
$605$	−3.15664	−	18.4918i	−0.128336	−	0.751797i
$606$		0			0
$607$			1.51153i			0.0613512i	0.999529	+	0.0306756i	$0.00976588\pi$
$607$			1.51153i			0.0613512i	−0.999529	+	0.0306756i	$0.990234\pi$
$608$		0			0
$609$	−6.22142			−0.252105
$610$		0			0
$611$	−18.0720			−0.731114
$612$		0			0
$613$		−	39.5673i		−	1.59811i	−0.601259	−	0.799054i	$-0.705334\pi$
$613$		−	39.5673i		−	1.59811i	0.601259	−	0.799054i	$-0.294666\pi$
$614$		0			0
$615$	4.48648	−	0.765865i	0.180912	−	0.0308826i
$616$		0			0
$617$		−	41.0023i		−	1.65069i	−0.564629	−	0.825345i	$-0.690981\pi$
$617$		−	41.0023i		−	1.65069i	0.564629	−	0.825345i	$-0.309019\pi$
$618$		0			0
$619$	−10.9089			−0.438464			−0.219232	−	0.975673i	$-0.570355\pi$
$619$	−10.9089			−0.438464			−0.219232	+	0.975673i	$0.570355\pi$
$620$		0			0
$621$	2.09297			0.0839880
$622$		0			0
$623$			21.7817i			0.872664i
$624$		0			0
$625$	19.4973	−	15.6478i	0.779893	−	0.625913i
$626$		0			0
$627$		−	2.31789i		−	0.0925677i
$628$		0			0
$629$	7.66555			0.305645
$630$		0			0
$631$	−24.7516			−0.985347			−0.492674	−	0.870214i	$-0.663980\pi$
$631$	−24.7516			−0.985347			−0.492674	+	0.870214i	$0.663980\pi$
$632$		0			0
$633$			10.3069i			0.409661i
$634$		0			0
$635$	27.3884	−	4.67534i	1.08688	−	0.185535i
$636$		0			0
$637$			2.44150i			0.0967357i
$638$		0			0
$639$	−15.0865			−0.596812
$640$		0			0
$641$	33.5110			1.32360			0.661802	−	0.749679i	$-0.269792\pi$
$641$	33.5110			1.32360			0.661802	+	0.749679i	$0.269792\pi$
$642$		0			0
$643$			9.15767i			0.361143i	0.983562	+	0.180572i	$0.0577947\pi$
$643$			9.15767i			0.361143i	−0.983562	+	0.180572i	$0.942205\pi$
$644$		0			0
$645$	−0.309008	−	1.81019i	−0.0121672	−	0.0712761i
$646$		0			0
$647$		−	30.8011i		−	1.21092i	−0.795877	−	0.605458i	$-0.792990\pi$
$647$		−	30.8011i		−	1.21092i	0.795877	−	0.605458i	$-0.207010\pi$
$648$		0			0
$649$	−4.09480			−0.160735
$650$		0			0
$651$	4.02455			0.157735
$652$		0			0
$653$			25.4779i			0.997026i	0.866882	+	0.498513i	$0.166120\pi$
$653$			25.4779i			0.997026i	−0.866882	+	0.498513i	$0.833880\pi$
$654$		0			0
$655$	−4.25900	−	24.9494i	−0.166413	−	0.974855i
$656$		0			0
$657$		−	17.3488i		−	0.676843i
$658$		0			0
$659$	−2.34417			−0.0913159			−0.0456580	−	0.998957i	$-0.514538\pi$
$659$	−2.34417			−0.0913159			−0.0456580	+	0.998957i	$0.514538\pi$
$660$		0			0
$661$	43.4164			1.68870			0.844351	−	0.535791i	$-0.179987\pi$
$661$	43.4164			1.68870			0.844351	+	0.535791i	$0.179987\pi$
$662$		0			0
$663$			2.42266i			0.0940883i
$664$		0			0
$665$	21.8607	−	3.73174i	0.847723	−	0.144711i
$666$		0			0
$667$		−	7.08580i		−	0.274363i
$668$		0			0
$669$	2.36535			0.0914498
$670$		0			0
$671$	14.9547			0.577319
$672$		0			0
$673$		−	48.4902i		−	1.86916i	−0.355754	−	0.934580i	$-0.615776\pi$
$673$		−	48.4902i		−	1.86916i	0.355754	−	0.934580i	$-0.384224\pi$
$674$		0			0
$675$	−3.47165	−	9.87223i	−0.133624	−	0.379982i
$676$		0			0
$677$			46.9213i			1.80333i	0.432435	+	0.901665i	$0.357655\pi$
$677$			46.9213i			1.80333i	−0.432435	+	0.901665i	$0.642345\pi$
$678$		0			0
$679$	−19.7424			−0.757645
$680$		0			0
$681$	−0.131988			−0.00505780
$682$		0			0
$683$		−	15.3660i		−	0.587965i	−0.955811	−	0.293983i	$-0.905019\pi$
$683$		−	15.3660i		−	0.587965i	0.955811	−	0.293983i	$-0.0949808\pi$
$684$		0			0
$685$	10.8907	−	1.85910i	0.416113	−	0.0710327i
$686$		0			0
$687$			4.12056i			0.157209i
$688$		0			0
$689$	−17.7609			−0.676635
$690$		0			0
$691$	−40.0235			−1.52257			−0.761283	−	0.648419i	$-0.775430\pi$
$691$	−40.0235			−1.52257			−0.761283	+	0.648419i	$0.775430\pi$
$692$		0			0
$693$			11.4368i			0.434447i
$694$		0			0
$695$	1.10284	+	6.46049i	0.0418331	+	0.245060i
$696$		0			0
$697$			14.7886i			0.560157i
$698$		0			0
$699$	6.00607			0.227171
$700$		0			0
$701$	−28.2943			−1.06866			−0.534331	−	0.845276i	$-0.679436\pi$
$701$	−28.2943			−1.06866			−0.534331	+	0.845276i	$0.679436\pi$
$702$		0			0
$703$		−	11.9177i		−	0.449483i
$704$		0			0
$705$	−0.922967	−	5.40679i	−0.0347610	−	0.203631i
$706$		0			0
$707$		−	15.6001i		−	0.586703i
$708$		0			0
$709$	−26.8823			−1.00959			−0.504793	−	0.863240i	$-0.668431\pi$
$709$	−26.8823			−1.00959			−0.504793	+	0.863240i	$0.668431\pi$
$710$		0			0
$711$	3.89900			0.146224
$712$		0			0
$713$			4.58371i			0.171661i
$714$		0			0
$715$	−9.35051	+	1.59618i	−0.349689	+	0.0596937i
$716$		0			0
$717$			6.33423i			0.236556i
$718$		0			0
$719$	−11.2624			−0.420015			−0.210008	−	0.977700i	$-0.567349\pi$
$719$	−11.2624			−0.420015			−0.210008	+	0.977700i	$0.567349\pi$
$720$		0			0
$721$	32.4973			1.21026
$722$		0			0
$723$		−	9.49084i		−	0.352968i
$724$		0			0
$725$	−33.4226	+	11.7533i	−1.24129	+	0.436508i
$726$		0			0
$727$			2.07629i			0.0770054i	0.999258	+	0.0385027i	$0.0122588\pi$
$727$			2.07629i			0.0770054i	−0.999258	+	0.0385027i	$0.987741\pi$
$728$		0			0
$729$	20.3818			0.754883
$730$		0			0
$731$	5.96684			0.220692
$732$		0			0
$733$		−	16.6294i		−	0.614221i	−0.951674	−	0.307111i	$-0.900638\pi$
$733$		−	16.6294i		−	0.614221i	0.951674	−	0.307111i	$-0.0993621\pi$
$734$		0			0
$735$	−0.730450	+	0.124692i	−0.0269431	+	0.00459932i
$736$		0			0
$737$		−	25.4131i		−	0.936103i
$738$		0			0
$739$	−50.8052			−1.86890			−0.934450	−	0.356095i	$-0.884108\pi$
$739$	−50.8052			−1.86890			−0.934450	+	0.356095i	$0.884108\pi$
$740$		0			0
$741$	3.76652			0.138366
$742$		0			0
$743$		−	31.3667i		−	1.15073i	−0.817896	−	0.575366i	$-0.804859\pi$
$743$		−	31.3667i		−	1.15073i	0.817896	−	0.575366i	$-0.195141\pi$
$744$		0			0
$745$	1.91528	+	11.2198i	0.0701703	+	0.411062i
$746$		0			0
$747$			24.6934i			0.903485i
$748$		0			0
$749$	−23.5270			−0.859657
$750$		0			0
$751$	41.2551			1.50542			0.752711	−	0.658351i	$-0.228746\pi$
$751$	41.2551			1.50542			0.752711	+	0.658351i	$0.228746\pi$
$752$		0			0
$753$			6.00839i			0.218958i
$754$		0			0
$755$	−5.40488	−	31.6621i	−0.196704	−	1.15230i
$756$		0			0
$757$			11.3081i			0.411000i	0.978657	+	0.205500i	$0.0658820\pi$
$757$			11.3081i			0.411000i	−0.978657	+	0.205500i	$0.934118\pi$
$758$		0			0
$759$	0.575803			0.0209003
$760$		0			0
$761$	−4.32880			−0.156919			−0.0784594	−	0.996917i	$-0.525000\pi$
$761$	−4.32880			−0.156919			−0.0784594	+	0.996917i	$0.525000\pi$
$762$		0			0
$763$			43.2690i			1.56644i
$764$		0			0
$765$	16.3967	−	2.79899i	0.592822	−	0.101198i
$766$		0			0
$767$		−	6.65395i		−	0.240260i
$768$		0			0
$769$	−40.6517			−1.46594			−0.732968	−	0.680263i	$-0.761866\pi$
$769$	−40.6517			−1.46594			−0.732968	+	0.680263i	$0.761866\pi$
$770$		0			0
$771$	2.96866			0.106914
$772$		0			0
$773$		−	37.9716i		−	1.36574i	−0.730538	−	0.682872i	$-0.760731\pi$
$773$		−	37.9716i		−	1.36574i	0.730538	−	0.682872i	$-0.239269\pi$
$774$		0			0
$775$	21.6206	−	7.60307i	0.776636	−	0.273110i
$776$		0			0
$777$		−	2.59940i		−	0.0932528i
$778$		0			0
$779$	22.9918			0.823768
$780$		0			0
$781$	−8.48443			−0.303597
$782$		0			0
$783$			14.8304i			0.529994i
$784$		0			0
$785$	−16.1819	+	2.76234i	−0.577557	+	0.0985920i
$786$		0			0
$787$		−	55.1654i		−	1.96643i	−0.182443	−	0.983216i	$-0.558401\pi$
$787$		−	55.1654i		−	1.96643i	0.182443	−	0.983216i	$-0.441599\pi$
$788$		0			0
$789$	−5.35944			−0.190801
$790$		0			0
$791$	−3.80076			−0.135140
$792$		0			0
$793$			24.3010i			0.862952i
$794$		0			0
$795$	−0.907078	−	5.31371i	−0.0321707	−	0.188458i
$796$		0			0
$797$			33.0601i			1.17105i	0.810655	+	0.585524i	$0.199111\pi$
$797$			33.0601i			1.17105i	−0.810655	+	0.585524i	$0.800889\pi$
$798$		0			0
$799$	17.8222			0.630503
$800$		0			0
$801$	25.3997			0.897455
$802$		0			0
$803$		−	9.75675i		−	0.344308i
$804$		0			0
$805$	0.927026	+	5.43057i	0.0326734	+	0.191402i
$806$		0			0
$807$			2.50145i			0.0880553i
$808$		0			0
$809$	26.4318			0.929291			0.464646	−	0.885497i	$-0.346182\pi$
$809$	26.4318			0.929291			0.464646	+	0.885497i	$0.346182\pi$
$810$		0			0
$811$	5.49057			0.192800			0.0964000	−	0.995343i	$-0.469267\pi$
$811$	5.49057			0.192800			0.0964000	+	0.995343i	$0.469267\pi$
$812$		0			0
$813$			1.56850i			0.0550096i
$814$		0			0
$815$	4.68966	−	0.800550i	0.164272	−	0.0280420i
$816$		0			0
$817$		−	9.27667i		−	0.324550i
$818$		0			0
$819$	−18.5845			−0.649394
$820$		0			0
$821$	−42.2732			−1.47534			−0.737672	−	0.675160i	$-0.764075\pi$
$821$	−42.2732			−1.47534			−0.737672	+	0.675160i	$0.764075\pi$
$822$		0			0
$823$		−	52.2599i		−	1.82167i	−0.412774	−	0.910833i	$-0.635440\pi$
$823$		−	52.2599i		−	1.82167i	0.412774	−	0.910833i	$-0.364560\pi$
$824$		0			0
$825$	−0.955093	−	2.71597i	−0.0332521	−	0.0945581i
$826$		0			0
$827$		−	46.7667i		−	1.62624i	−0.582097	−	0.813119i	$-0.697768\pi$
$827$		−	46.7667i		−	1.62624i	0.582097	−	0.813119i	$-0.302232\pi$
$828$		0			0
$829$	−15.2175			−0.528525			−0.264262	−	0.964451i	$-0.585129\pi$
$829$	−15.2175			−0.528525			−0.264262	+	0.964451i	$0.585129\pi$
$830$		0			0
$831$	2.99902			0.104035
$832$		0			0
$833$		−	2.40775i		−	0.0834236i
$834$		0			0
$835$	4.76280	−	0.813035i	0.164824	−	0.0281362i
$836$		0			0
$837$		−	9.59356i		−	0.331602i
$838$		0			0
$839$	−54.7388			−1.88979			−0.944897	−	0.327369i	$-0.893838\pi$
$839$	−54.7388			−1.88979			−0.944897	+	0.327369i	$0.893838\pi$
$840$		0			0
$841$	21.2085			0.731328
$842$		0			0
$843$		−	5.34802i		−	0.184195i
$844$		0			0
$845$	2.29770	+	13.4600i	0.0790433	+	0.463040i
$846$		0			0
$847$		−	20.6694i		−	0.710210i
$848$		0			0
$849$	2.52711			0.0867302
$850$		0			0
$851$	2.96054			0.101486
$852$		0			0
$853$		−	16.8683i		−	0.577559i	−0.957396	−	0.288779i	$-0.906751\pi$
$853$		−	16.8683i		−	0.577559i	0.957396	−	0.288779i	$-0.0932494\pi$
$854$		0			0
$855$	−4.35160	−	25.4919i	−0.148822	−	0.871806i
$856$		0			0
$857$		−	39.9513i		−	1.36471i	−0.731021	−	0.682355i	$-0.760956\pi$
$857$		−	39.9513i		−	1.36471i	0.731021	−	0.682355i	$-0.239044\pi$
$858$		0			0
$859$	−21.5728			−0.736053			−0.368027	−	0.929815i	$-0.619966\pi$
$859$	−21.5728			−0.736053			−0.368027	+	0.929815i	$0.619966\pi$
$860$		0			0
$861$	5.01482			0.170905
$862$		0			0
$863$			2.71384i			0.0923803i	0.998933	+	0.0461902i	$0.0147080\pi$
$863$			2.71384i			0.0923803i	−0.998933	+	0.0461902i	$0.985292\pi$
$864$		0			0
$865$	41.0440	−	7.00643i	1.39554	−	0.238226i
$866$		0			0
$867$			3.66915i			0.124611i
$868$		0			0
$869$	2.19275			0.0743838
$870$		0			0
$871$	41.2956			1.39925
$872$		0			0
$873$			23.0217i			0.779168i
$874$		0			0
$875$	24.0775	−	13.3804i	0.813968	−	0.452341i
$876$		0			0
$877$			40.8294i			1.37871i	0.724423	+	0.689355i	$0.242106\pi$
$877$			40.8294i			1.37871i	−0.724423	+	0.689355i	$0.757894\pi$
$878$		0			0
$879$	−9.99862			−0.337245
$880$		0			0
$881$	36.3676			1.22525			0.612627	−	0.790372i	$-0.290113\pi$
$881$	36.3676			1.22525			0.612627	+	0.790372i	$0.290113\pi$
$882$		0			0
$883$			18.2495i			0.614146i	0.951686	+	0.307073i	$0.0993495\pi$
$883$			18.2495i			0.614146i	−0.951686	+	0.307073i	$0.900650\pi$
$884$		0			0
$885$	1.99073	−	0.339829i	0.0669178	−	0.0114232i
$886$		0			0
$887$		−	3.33546i		−	0.111994i	−0.998431	−	0.0559968i	$-0.982166\pi$
$887$		−	3.33546i		−	0.111994i	0.998431	−	0.0559968i	$-0.0178337\pi$
$888$		0			0
$889$	30.6138			1.02675
$890$		0			0
$891$	12.7209			0.426166
$892$		0			0
$893$		−	27.7082i		−	0.927219i
$894$		0			0
$895$	5.06718	+	29.6838i	0.169377	+	0.992220i
$896$		0			0
$897$			0.935665i			0.0312409i
$898$		0			0
$899$	−32.4792			−1.08324
$900$		0			0
$901$	17.5153			0.583521
$902$		0			0
$903$		−	2.02336i		−	0.0673333i
$904$		0			0
$905$	4.62530	+	27.0953i	0.153750	+	0.900677i
$906$		0			0
$907$		−	5.50057i		−	0.182643i	−0.995821	−	0.0913217i	$-0.970891\pi$
$907$		−	5.50057i		−	0.182643i	0.995821	−	0.0913217i	$-0.0291092\pi$
$908$		0			0
$909$	−18.1914			−0.603371
$910$		0			0
$911$	11.9531			0.396024			0.198012	−	0.980200i	$-0.436551\pi$
$911$	11.9531			0.396024			0.198012	+	0.980200i	$0.436551\pi$
$912$		0			0
$913$			13.8872i			0.459601i
$914$		0			0
$915$	−7.27039	+	1.24109i	−0.240352	+	0.0410293i
$916$		0			0
$917$		−	27.8876i		−	0.920929i
$918$		0			0
$919$	−22.3756			−0.738102			−0.369051	−	0.929409i	$-0.620317\pi$
$919$	−22.3756			−0.738102			−0.369051	+	0.929409i	$0.620317\pi$
$920$		0			0
$921$	10.4312			0.343721
$922$		0			0
$923$		−	13.7870i		−	0.453804i
$924$		0			0
$925$	−4.91070	−	13.9644i	−0.161463	−	0.459148i
$926$		0			0
$927$		−	37.8953i		−	1.24464i
$928$		0			0
$929$	42.9751			1.40997			0.704983	−	0.709224i	$-0.250955\pi$
$929$	42.9751			1.40997			0.704983	+	0.709224i	$0.250955\pi$
$930$		0			0
$931$	−3.74334			−0.122683
$932$		0			0
$933$		−	5.63461i		−	0.184469i
$934$		0			0
$935$	9.22126	−	1.57412i	0.301567	−	0.0514791i
$936$		0			0
$937$		−	3.76556i		−	0.123015i	−0.998107	−	0.0615077i	$-0.980409\pi$
$937$		−	3.76556i		−	0.123015i	0.998107	−	0.0615077i	$-0.0195909\pi$
$938$		0			0
$939$	−3.55874			−0.116135
$940$		0			0
$941$	−15.9193			−0.518953			−0.259477	−	0.965749i	$-0.583550\pi$
$941$	−15.9193			−0.518953			−0.259477	+	0.965749i	$0.583550\pi$
$942$		0			0
$943$			5.71155i			0.185994i
$944$		0			0
$945$	−1.94024	−	11.3660i	−0.0631160	−	0.369737i
$946$		0			0
$947$			51.8553i			1.68507i	0.538642	+	0.842535i	$0.318938\pi$
$947$			51.8553i			1.68507i	−0.538642	+	0.842535i	$0.681062\pi$
$948$		0			0
$949$	15.8545			0.514658
$950$		0			0
$951$	−4.90662			−0.159108
$952$		0			0
$953$		−	44.6848i		−	1.44748i	−0.690072	−	0.723741i	$-0.742421\pi$
$953$		−	44.6848i		−	1.44748i	0.690072	−	0.723741i	$-0.257579\pi$
$954$		0			0
$955$	−0.779242	−	4.56484i	−0.0252157	−	0.147715i
$956$		0			0
$957$			4.08002i			0.131888i
$958$		0			0
$959$	12.1733			0.393095
$960$		0			0
$961$	−9.98964			−0.322247
$962$		0			0
$963$			27.4349i			0.884079i
$964$		0			0
$965$	56.8671	−	9.70751i	1.83062	−	0.312496i
$966$		0			0
$967$		−	31.5594i		−	1.01488i	−0.861687	−	0.507440i	$-0.830592\pi$
$967$		−	31.5594i		−	1.01488i	0.861687	−	0.507440i	$-0.169408\pi$
$968$		0			0
$969$	−3.71445			−0.119325
$970$		0			0
$971$	−20.6291			−0.662018			−0.331009	−	0.943628i	$-0.607389\pi$
$971$	−20.6291			−0.662018			−0.331009	+	0.943628i	$0.607389\pi$
$972$		0			0
$973$			7.22130i			0.231504i
$974$		0			0
$975$	4.41339	−	1.55200i	0.141342	−	0.0497039i
$976$		0			0
$977$		−	1.76933i		−	0.0566058i	−0.999599	−	0.0283029i	$-0.990990\pi$
$977$		−	1.76933i		−	0.0566058i	0.999599	−	0.0283029i	$-0.00901029\pi$
$978$		0			0
$979$	14.2845			0.456533
$980$		0			0
$981$	50.4562			1.61094
$982$		0			0
$983$		−	43.4073i		−	1.38448i	−0.721668	−	0.692239i	$-0.756624\pi$
$983$		−	43.4073i		−	1.38448i	0.721668	−	0.692239i	$-0.243376\pi$
$984$		0			0
$985$	9.28938	−	1.58575i	0.295984	−	0.0505261i
$986$		0			0
$987$		−	6.04351i		−	0.192367i
$988$		0			0
$989$	2.30448			0.0732781
$990$		0			0
$991$	16.2222			0.515317			0.257658	−	0.966236i	$-0.417049\pi$
$991$	16.2222			0.515317			0.257658	+	0.966236i	$0.417049\pi$
$992$		0			0
$993$			6.65579i			0.211215i
$994$		0			0
$995$	8.29393	+	48.5863i	0.262935	+	1.54029i
$996$		0			0
$997$			2.61347i			0.0827695i	0.999143	+	0.0413847i	$0.0131769\pi$
$997$			2.61347i			0.0827695i	−0.999143	+	0.0413847i	$0.986823\pi$
$998$		0			0
$999$	−6.19633			−0.196043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.b.369.9	yes	14
4.3	odd	2		1840.2.e.g.369.6		14
5.2	odd	4		4600.2.a.bi.1.4		7
5.3	odd	4		4600.2.a.bh.1.4		7
5.4	even	2	inner	920.2.e.b.369.6	✓	14
20.3	even	4		9200.2.a.dc.1.4		7
20.7	even	4		9200.2.a.cz.1.4		7
20.19	odd	2		1840.2.e.g.369.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.6	✓	14	5.4	even	2	inner
920.2.e.b.369.9	yes	14	1.1	even	1	trivial
1840.2.e.g.369.6		14	4.3	odd	2
1840.2.e.g.369.9		14	20.19	odd	2
4600.2.a.bh.1.4		7	5.3	odd	4
4600.2.a.bi.1.4		7	5.2	odd	4
9200.2.a.cz.1.4		7	20.7	even	4
9200.2.a.dc.1.4		7	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.356372i q^{3} +(0.376266 + 2.20418i) q^{5} +2.46376i q^{7} +2.87300 q^{9} +O(q^{10})\)	\(q+0.356372i q^{3} +(0.376266 + 2.20418i) q^{5} +2.46376i q^{7} +2.87300 q^{9} +1.61574 q^{11} +2.62553i q^{13} +(-0.785509 + 0.134090i) q^{15} -2.58924i q^{17} -4.02550 q^{19} -0.878013 q^{21} -1.00000i q^{23} +(-4.71685 + 1.65872i) q^{25} +2.09297i q^{27} +7.08580 q^{29} -4.58371 q^{31} +0.575803i q^{33} +(-5.43057 + 0.927026i) q^{35} +2.96054i q^{37} -0.935665 q^{39} -5.71155 q^{41} +2.30448i q^{43} +(1.08101 + 6.33262i) q^{45} +6.88317i q^{47} +0.929907 q^{49} +0.922731 q^{51} +6.76467i q^{53} +(0.607946 + 3.56138i) q^{55} -1.43457i q^{57} -2.53432 q^{59} +9.25564 q^{61} +7.07837i q^{63} +(-5.78715 + 0.987896i) q^{65} -15.7285i q^{67} +0.356372 q^{69} -5.25113 q^{71} -6.03858i q^{73} +(-0.591120 - 1.68095i) q^{75} +3.98078i q^{77} +1.35712 q^{79} +7.87312 q^{81} +8.59500i q^{83} +(5.70716 - 0.974241i) q^{85} +2.52518i q^{87} +8.84084 q^{89} -6.46867 q^{91} -1.63350i q^{93} +(-1.51466 - 8.87293i) q^{95} +8.01314i q^{97} +4.64201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 4 * q^9	\( 14 q + 2 q^{5} - 4 q^{9} - 14 q^{11} - 6 q^{15} + 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} - 20 q^{31} - 2 q^{35} + 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} - 14 q^{51} - 38 q^{55} + 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} - 28 q^{71} - 24 q^{75} + 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} - 14 q^{91} - 30 q^{95} + 122 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 4 * q^9 - 14 * q^11 - 6 * q^15 + 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 - 20 * q^31 - 2 * q^35 + 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 - 14 * q^51 - 38 * q^55 + 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 - 28 * q^71 - 24 * q^75 + 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 - 14 * q^91 - 30 * q^95 + 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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