LMFDB - Embedded newform 539.2.f.c.344.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,2,Mod(148,539)] 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("539.148"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.f (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,-2,0] 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:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.37515625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 $ x^8 - x^7 - x^6 + 3x^5 - x^4 + 6x^3 - 4x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{5}]$

Embedding invariants

Embedding label			344.2
Root			$-0.373058 - 1.36412i$ of defining polynomial
Character	$\chi$	$=$	539.344
Dual form			539.2.f.c.246.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.78570 - 1.29738i) q^{2} +(0.887471 - 2.73136i) q^{4} +(-0.594713 - 1.83034i) q^{8} +(2.42705 - 1.76336i) q^{9} +(-0.0629004 - 3.31603i) q^{11} +(1.21023 + 0.879283i) q^{16} +(2.04623 - 6.29764i) q^{18} +(-4.41448 - 5.83981i) q^{22} -2.56038 q^{23} +(-1.54508 - 4.75528i) q^{25} +(-2.42225 + 7.45492i) q^{29} +7.15093 q^{32} +(-2.66241 - 8.19407i) q^{36} +(-3.68322 + 11.3358i) q^{37} +12.8185 q^{43} +(-9.11308 - 2.77108i) q^{44} +(-4.57206 + 3.32180i) q^{46} +(-8.92848 - 6.48692i) q^{50} +(-11.5776 + 8.41162i) q^{53} +(5.34649 + 16.4548i) q^{58} +(10.3489 - 7.51894i) q^{64} -12.5669 q^{67} +(12.9884 + 9.43662i) q^{71} +(-4.67094 - 3.39364i) q^{72} +(8.12975 + 25.0208i) q^{74} +(-2.31271 + 1.68028i) q^{79} +(2.78115 - 8.55951i) q^{81} +(22.8899 - 16.6305i) q^{86} +(-6.03205 + 2.08722i) q^{88} +(-2.27227 + 6.99331i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 2 q^{4} + 16 q^{8} + 6 q^{9} - 4 q^{11} - 28 q^{16} - 9 q^{18} - 4 q^{22} - 16 q^{23} + 10 q^{25} - 4 q^{29} + 100 q^{32} - 6 q^{36} - 18 q^{37} + 24 q^{43} + 9 q^{44} - 31 q^{46} + 10 q^{50}+ \cdots - 48 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 + 2 * q^4 + 16 * q^8 + 6 * q^9 - 4 * q^11 - 28 * q^16 - 9 * q^18 - 4 * q^22 - 16 * q^23 + 10 * q^25 - 4 * q^29 + 100 * q^32 - 6 * q^36 - 18 * q^37 + 24 * q^43 + 9 * q^44 - 31 * q^46 + 10 * q^50 - 30 * q^53 + 71 * q^58 - 74 * q^64 - 8 * q^67 + 48 * q^71 + 57 * q^72 + 12 * q^74 + 24 * q^79 - 18 * q^81 + 29 * q^86 - 38 * q^88 + 71 * q^92 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$1$	$e\left(\frac{4}{5}\right)$

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 $

$2$	1.78570	−	1.29738i	1.26268	−	0.917389i	0.263792	−	0.964580i	$-0.415027\pi$
$2$	1.78570	−	1.29738i	1.26268	−	0.917389i	0.998886	+	0.0471903i	$0.0150267\pi$
$3$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$3$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$4$	0.887471	−	2.73136i	0.443736	−	1.36568i
$5$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$5$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−0.594713	−	1.83034i	−0.210263	−	0.647123i
$9$	2.42705	−	1.76336i	0.809017	−	0.587785i
$10$		0			0
$11$	−0.0629004	−	3.31603i	−0.0189652	−	0.999820i
$11$	−0.0629004	−	3.31603i	−0.0189652	−	0.999820i
$12$		0			0
$13$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$13$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$14$		0			0
$15$		0			0
$16$	1.21023	+	0.879283i	0.302557	+	0.219821i
$17$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$17$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$18$	2.04623	−	6.29764i	0.482300	−	1.48437i
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$		0			0
$21$		0			0
$22$	−4.41448	−	5.83981i	−0.941171	−	1.24505i
$23$	−2.56038			−0.533877			−0.266938	−	0.963714i	$-0.586012\pi$
$23$	−2.56038			−0.533877			−0.266938	+	0.963714i	$0.586012\pi$
$24$		0			0
$25$	−1.54508	−	4.75528i	−0.309017	−	0.951057i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.42225	+	7.45492i	−0.449801	+	1.38434i	0.427331	+	0.904095i	$0.359454\pi$
$29$	−2.42225	+	7.45492i	−0.449801	+	1.38434i	−0.877132	+	0.480249i	$0.840546\pi$
$30$		0			0
$31$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$31$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$32$	7.15093			1.26412
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.66241	−	8.19407i	−0.443736	−	1.36568i
$37$	−3.68322	+	11.3358i	−0.605517	+	1.86359i	−0.112320	+	0.993672i	$0.535828\pi$
$37$	−3.68322	+	11.3358i	−0.605517	+	1.86359i	−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$41$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$42$		0			0
$43$	12.8185			1.95480			0.977399	−	0.211402i	$-0.0678028\pi$
$43$	12.8185			1.95480			0.977399	+	0.211402i	$0.0678028\pi$
$44$	−9.11308	−	2.77108i	−1.37385	−	0.417756i
$45$		0			0
$46$	−4.57206	+	3.32180i	−0.674114	+	0.489773i
$47$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$47$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$48$		0			0
$49$		0			0
$50$	−8.92848	−	6.48692i	−1.26268	−	0.917389i
$51$		0			0
$52$		0			0
$53$	−11.5776	+	8.41162i	−1.59031	+	1.15542i	−0.686803	+	0.726844i	$0.740986\pi$
$53$	−11.5776	+	8.41162i	−1.59031	+	1.15542i	−0.903503	+	0.428581i	$0.859014\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	5.34649	+	16.4548i	0.702029	+	2.16062i
$59$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$59$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$60$		0			0
$61$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$61$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$62$		0			0
$63$		0			0
$64$	10.3489	−	7.51894i	1.29362	−	0.939868i
$65$		0			0
$66$		0			0
$67$	−12.5669			−1.53529			−0.767644	−	0.640877i	$-0.778571\pi$
$67$	−12.5669			−1.53529			−0.767644	+	0.640877i	$0.778571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.9884	+	9.43662i	1.54144	+	1.11992i	0.949425	+	0.313993i	$0.101667\pi$
$71$	12.9884	+	9.43662i	1.54144	+	1.11992i	0.592014	+	0.805928i	$0.298333\pi$
$72$	−4.67094	−	3.39364i	−0.550475	−	0.399944i
$73$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$73$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$74$	8.12975	+	25.0208i	0.945064	+	2.90861i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.31271	+	1.68028i	−0.260200	+	0.189046i	−0.710235	−	0.703964i	$-0.751411\pi$
$79$	−2.31271	+	1.68028i	−0.260200	+	0.189046i	0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	2.78115	−	8.55951i	0.309017	−	0.951057i
$82$		0			0
$83$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$83$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$84$		0			0
$85$		0			0
$86$	22.8899	−	16.6305i	2.46828	−	1.79331i
$87$		0			0
$88$	−6.03205	+	2.08722i	−0.643019	+	0.222498i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−2.27227	+	6.99331i	−0.236900	+	0.729103i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$97$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$98$		0			0
$99$	−6.00000	−	7.93725i	−0.603023	−	0.797724i
$100$	−14.3596			−1.43596
$101$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$101$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$102$		0			0
$103$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$103$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$104$		0			0
$105$		0			0
$106$	−9.76098	+	30.0412i	−0.948070	+	2.91786i
$107$	−3.46496	−	10.6641i	−0.334971	−	1.03093i	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−3.46496	−	10.6641i	−0.334971	−	1.03093i	0.631766	−	0.775159i	$-0.282330\pi$
$108$		0			0
$109$	15.6273			1.49683			0.748414	−	0.663232i	$-0.230816\pi$
$109$	15.6273			1.49683			0.748414	+	0.663232i	$0.230816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.34450	+	10.2933i	0.314624	+	0.968314i	0.975909	+	0.218179i	$0.0700116\pi$
$113$	3.34450	+	10.2933i	0.314624	+	0.968314i	−0.661285	+	0.750135i	$0.729988\pi$
$114$		0			0
$115$		0			0
$116$	18.2124	+	13.2321i	1.69098	+	1.22857i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.9921	+	0.417159i	−0.999281	+	0.0379235i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.2142	−	11.7803i	−1.43878	−	1.04533i	−0.988297	−	0.152545i	$-0.951253\pi$
$127$	−16.2142	−	11.7803i	−1.43878	−	1.04533i	−0.450479	−	0.892787i	$-0.648747\pi$
$128$	4.30558	−	13.2512i	0.380563	−	1.17125i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−22.4406	+	16.3041i	−1.93857	+	1.40846i
$135$		0			0
$136$		0			0
$137$	−13.7856	−	10.0158i	−1.17778	−	0.855708i	−0.185861	−	0.982576i	$-0.559507\pi$
$137$	−13.7856	−	10.0158i	−1.17778	−	0.855708i	−0.991920	+	0.126868i	$0.959507\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$		0			0
$142$	35.4363			2.97374
$143$		0			0
$144$	4.48778			0.373981
$145$		0			0
$146$		0			0
$147$		0			0
$148$	27.6933	+	20.1203i	2.27637	+	1.65388i
$149$	−17.7984	−	12.9313i	−1.45810	−	1.05937i	−0.983853	−	0.178979i	$-0.942720\pi$
$149$	−17.7984	−	12.9313i	−1.45810	−	1.05937i	−0.474247	−	0.880392i	$-0.657280\pi$
$150$		0			0
$151$	−0.736662	−	2.26721i	−0.0599487	−	0.184503i	0.916597	−	0.399811i	$-0.130924\pi$
$151$	−0.736662	−	2.26721i	−0.0599487	−	0.184503i	−0.976546	+	0.215308i	$0.930924\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$157$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$158$	−1.94983	+	6.00095i	−0.155120	+	0.477410i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−6.13868	−	18.8929i	−0.482300	−	1.48437i
$163$	17.2142	−	12.5068i	1.34832	−	0.979611i	0.349225	−	0.937039i	$-0.386445\pi$
$163$	17.2142	−	12.5068i	1.34832	−	0.979611i	0.999093	−	0.0425718i	$-0.0135551\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$167$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$168$		0			0
$169$	−4.01722	+	12.3637i	−0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$	11.3760	−	35.0118i	0.867414	−	2.66963i
$173$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$173$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$174$		0			0
$175$		0			0
$176$	2.83960	−	4.06846i	0.214043	−	0.306672i
$177$		0			0
$178$		0			0
$179$	8.15763	+	25.1066i	0.609730	+	1.87656i	0.460243	+	0.887793i	$0.347762\pi$
$179$	8.15763	+	25.1066i	0.609730	+	1.87656i	0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$181$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$182$		0			0
$183$		0			0
$184$	1.52269	+	4.68637i	0.112254	+	0.345484i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.80563	−	8.63483i	0.203008	−	0.624795i	−0.796781	−	0.604268i	$-0.793466\pi$
$191$	2.80563	−	8.63483i	0.203008	−	0.624795i	0.999789	−	0.0205267i	$-0.00653431\pi$
$192$		0			0
$193$	1.71612	+	1.24683i	0.123529	+	0.0897489i	0.647834	−	0.761781i	$-0.275675\pi$
$193$	1.71612	+	1.24683i	0.123529	+	0.0897489i	−0.524305	+	0.851530i	$0.675675\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.03059			0.144674			0.0723369	−	0.997380i	$-0.476954\pi$
$197$	2.03059			0.144674			0.0723369	+	0.997380i	$0.476954\pi$
$198$	−21.0118	−	6.38922i	−1.49325	−	0.454062i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−7.78490	+	5.65606i	−0.550475	+	0.399944i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.21418	+	4.51486i	−0.431915	+	0.313805i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	23.3570	−	16.9698i	1.60796	−	1.16825i	0.738490	−	0.674264i	$-0.235539\pi$
$211$	23.3570	−	16.9698i	1.60796	−	1.16825i	0.869469	−	0.493987i	$-0.164461\pi$
$212$	12.7003	+	39.0876i	0.872263	+	2.68455i
$213$		0			0
$214$	−20.0228	−	14.5474i	−1.36873	−	0.994439i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	27.9057	−	20.2747i	1.89001	−	1.37317i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$223$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$	−12.1353	−	8.81678i	−0.809017	−	0.587785i
$226$	19.3267	+	14.0416i	1.28559	+	0.934036i
$227$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$227$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$228$		0			0
$229$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$229$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$230$		0			0
$231$		0			0
$232$	15.0856			0.990417
$233$	−17.7984	+	12.9313i	−1.16601	+	0.847156i	−0.990526	−	0.137326i	$-0.956149\pi$
$233$	−17.7984	+	12.9313i	−1.16601	+	0.847156i	−0.175484	+	0.984482i	$0.556149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.80563	−	27.1009i	−0.569589	−	1.75301i	−0.653907	−	0.756575i	$-0.726871\pi$
$239$	−8.80563	−	27.1009i	−0.569589	−	1.75301i	0.0843185	−	0.996439i	$-0.473129\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−19.0873	+	15.0059i	−1.22698	+	0.964615i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$251$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$252$		0			0
$253$	0.161049	+	8.49030i	0.0101251	+	0.533781i
$254$	−44.2372			−2.77569
$255$		0			0
$256$	−1.59757	−	4.91683i	−0.0998484	−	0.307302i
$257$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$257$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.26675	+	22.3648i	0.449801	+	1.38434i
$262$		0			0
$263$	−22.7783			−1.40457			−0.702284	−	0.711897i	$-0.747836\pi$
$263$	−22.7783			−1.40457			−0.702284	+	0.711897i	$0.747836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−11.1527	+	34.3246i	−0.681262	+	2.09671i
$269$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$269$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$	−37.6112			−2.27217
$275$	−15.6715	+	5.42265i	−0.945025	+	0.326998i
$276$		0			0
$277$	26.9284	−	19.5646i	1.61797	−	1.17552i	0.803679	−	0.595063i	$-0.202873\pi$
$277$	26.9284	−	19.5646i	1.61797	−	1.17552i	0.814289	−	0.580460i	$-0.197127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−27.0823	−	19.6764i	−1.61559	−	1.17380i	−0.840077	−	0.542467i	$-0.817490\pi$
$281$	−27.0823	−	19.6764i	−1.61559	−	1.17380i	−0.775515	−	0.631329i	$-0.782510\pi$
$282$		0			0
$283$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$283$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$284$	37.3016	−	27.1012i	2.21344	−	1.60816i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	17.3557	−	12.6096i	1.02269	−	0.743030i
$289$	−5.25329	−	16.1680i	−0.309017	−	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$293$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$294$		0			0
$295$		0			0
$296$	22.9388			1.33329
$297$		0			0
$298$	−48.5593			−2.81297
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−4.25690	−	3.09282i	−0.244957	−	0.177972i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$313$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$314$		0			0
$315$		0			0
$316$	2.53698	+	7.80803i	0.142716	+	0.439236i
$317$	−17.2208	+	12.5116i	−0.967215	+	0.702723i	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−17.2208	+	12.5116i	−0.967215	+	0.702723i	−0.0123997	+	0.999923i	$0.503947\pi$
$318$		0			0
$319$	24.8731	+	7.56333i	1.39263	+	0.423465i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−20.9109	−	15.1926i	−1.16171	−	0.844035i
$325$		0			0
$326$	14.5131	−	44.6668i	0.803808	−	2.47387i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.1571			0.888076			0.444038	−	0.896008i	$-0.353545\pi$
$331$	16.1571			0.888076			0.444038	+	0.896008i	$0.353545\pi$
$332$		0			0
$333$	11.0496	+	34.0073i	0.605517	+	1.86359i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.35579	+	10.3281i	−0.182801	+	0.562605i	−0.999904	−	0.0138879i	$-0.995579\pi$
$337$	−3.35579	+	10.3281i	−0.182801	+	0.562605i	0.817102	+	0.576493i	$0.195579\pi$
$338$	8.86698	+	27.2898i	0.482300	+	1.48437i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−7.62332	−	23.4622i	−0.411022	−	1.26499i
$345$		0			0
$346$		0			0
$347$	−14.9958	−	10.8951i	−0.805016	−	0.584878i	0.107366	−	0.994220i	$-0.465758\pi$
$347$	−14.9958	−	10.8951i	−0.805016	−	0.584878i	−0.912381	+	0.409342i	$0.865758\pi$
$348$		0			0
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$		0			0
$351$		0			0
$352$	−0.449797	−	23.7127i	−0.0239742	−	1.26389i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	47.1400	+	34.2492i	2.49143	+	1.81013i
$359$	11.6499	−	35.8546i	0.614856	−	1.89233i	0.211112	−	0.977462i	$-0.432292\pi$
$359$	11.6499	−	35.8546i	0.614856	−	1.89233i	0.403745	−	0.914872i	$-0.367708\pi$
$360$		0			0
$361$	15.3713	−	11.1679i	0.809017	−	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$367$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$368$	−3.09865	−	2.25130i	−0.161528	−	0.117357i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−25.4997	−	18.5267i	−1.30983	−	0.951650i	−1.00000		0.000859657i	$-0.999726\pi$
$379$	−25.4997	−	18.5267i	−1.30983	−	0.951650i	−0.309834	−	0.950791i	$-0.600274\pi$
$380$		0			0
$381$		0			0
$382$	−6.19270	−	19.0592i	−0.316846	−	0.975152i
$383$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$383$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$384$		0			0
$385$		0			0
$386$	4.68208			0.238312
$387$	31.1111	−	22.6035i	1.58147	−	1.14900i
$388$		0			0
$389$	−0.518410	+	1.59550i	−0.0262844	+	0.0808951i	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−0.518410	+	1.59550i	−0.0262844	+	0.0808951i	0.937054	+	0.349185i	$0.113542\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	3.62602	−	2.63446i	0.182676	−	0.132722i
$395$		0			0
$396$	−27.0043	+	9.34405i	−1.35702	+	0.469556i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	2.31133	−	7.11355i	0.115567	−	0.355677i
$401$	24.7856	+	18.0078i	1.23773	+	0.899265i	0.997445	−	0.0714367i	$-0.0227584\pi$
$401$	24.7856	+	18.0078i	1.23773	+	0.899265i	0.240287	+	0.970702i	$0.422758\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	37.8214	+	11.5006i	1.87474	+	0.570065i
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−5.23912	+	16.1244i	−0.257489	+	0.792469i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	6.84802	+	21.0760i	0.333752	+	1.02718i	0.967333	+	0.253507i	$0.0815842\pi$
$421$	6.84802	+	21.0760i	0.333752	+	1.02718i	−0.633581	+	0.773676i	$0.718416\pi$
$422$	19.6921	−	60.6059i	0.958594	−	2.95025i
$423$		0			0
$424$	22.2815	+	16.1884i	1.08208	+	0.786180i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−32.2024			−1.55656
$429$		0			0
$430$		0			0
$431$	12.3570	−	8.97786i	0.595214	−	0.432448i	−0.248963	−	0.968513i	$-0.580090\pi$
$431$	12.3570	−	8.97786i	0.595214	−	0.432448i	0.844177	+	0.536065i	$0.180090\pi$
$432$		0			0
$433$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$433$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$434$		0			0
$435$		0			0
$436$	13.8688	−	42.6838i	0.664196	−	2.04418i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.72788	−	5.31786i	−0.0820940	−	0.252659i	0.901582	−	0.432608i	$-0.142407\pi$
$443$	−1.72788	−	5.31786i	−0.0820940	−	0.252659i	−0.983676	+	0.179949i	$0.942407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.0711	+	24.0276i	−1.56072	+	1.13393i	−0.625310	+	0.780376i	$0.715028\pi$
$449$	−33.0711	+	24.0276i	−1.56072	+	1.13393i	−0.935413	+	0.353556i	$0.884972\pi$
$450$	−33.1086			−1.56076
$451$		0			0
$452$	31.0829			1.46201
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	34.0711	+	24.7541i	1.59378	+	1.15795i	0.898279	+	0.439425i	$0.144818\pi$
$457$	34.0711	+	24.7541i	1.59378	+	1.15795i	0.695501	+	0.718525i	$0.255182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	23.0299			1.07029			0.535145	−	0.844760i	$-0.320257\pi$
$463$	23.0299			1.07029			0.535145	+	0.844760i	$0.320257\pi$
$464$	−9.48646	+	6.89232i	−0.440398	+	0.319968i
$465$		0			0
$466$	−15.0057	+	46.1827i	−0.695124	+	2.13937i
$467$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$467$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.806287	−	42.5064i	−0.0370731	−	1.95445i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−13.2668	+	40.8309i	−0.607443	+	1.86952i
$478$	−50.8845	−	36.9698i	−2.32740	−	1.69096i
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−8.61575	+	30.3935i	−0.391625	+	1.38152i
$485$		0			0
$486$		0			0
$487$	12.7279	+	39.1724i	0.576755	+	1.77507i	0.630123	+	0.776495i	$0.283004\pi$
$487$	12.7279	+	39.1724i	0.576755	+	1.77507i	−0.0533681	+	0.998575i	$0.516996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.5967	−	41.8465i	0.613613	−	1.88851i	0.193249	−	0.981150i	$-0.438097\pi$
$491$	13.5967	−	41.8465i	0.613613	−	1.88851i	0.420363	−	0.907356i	$-0.361903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.8056	+	42.4894i	−0.618025	+	1.90208i	−0.303812	+	0.952732i	$0.598259\pi$
$499$	−13.8056	+	42.4894i	−0.618025	+	1.90208i	−0.314213	+	0.949352i	$0.601741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$503$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$504$		0			0
$505$		0			0
$506$	11.3028	+	14.9522i	0.502469	+	0.664704i
$507$		0			0
$508$	−46.5658	+	33.8320i	−2.06602	+	1.50105i
$509$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$509$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$510$		0			0
$511$		0			0
$512$	13.3125	+	9.67211i	0.588336	+	0.427451i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$521$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$522$	41.9919	+	30.5089i	1.83794	+	1.33534i
$523$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$523$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$524$		0			0
$525$		0			0
$526$	−40.6751	+	29.5522i	−1.77352	+	1.28854i
$527$		0			0
$528$		0			0
$529$	−16.4444			−0.714976
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	7.47368	+	23.0016i	0.322814	+	0.993519i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−15.9284	+	11.5726i	−0.684814	+	0.497546i	−0.874951	−	0.484211i	$-0.839107\pi$
$541$	−15.9284	+	11.5726i	−0.684814	+	0.497546i	0.190138	+	0.981757i	$0.439107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.5967	+	41.8465i	0.581355	+	1.78923i	0.613440	+	0.789741i	$0.289785\pi$
$547$	13.5967	+	41.8465i	0.581355	+	1.78923i	−0.0320849	+	0.999485i	$0.510215\pi$
$548$	−39.5910	+	28.7646i	−1.69124	+	1.22876i
$549$		0			0
$550$	−20.9492	+	30.0151i	−0.893277	+	1.27985i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	22.7031	−	69.8729i	0.964561	−	2.96861i
$555$		0			0
$556$		0			0
$557$	−13.4223	+	41.3094i	−0.568719	+	1.75034i	0.0879152	+	0.996128i	$0.471980\pi$
$557$	−13.4223	+	41.3094i	−0.568719	+	1.75034i	−0.656634	+	0.754209i	$0.728020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−73.8886			−3.11680
$563$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$563$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	9.54785	−	29.3853i	0.400619	−	1.23298i
$569$	6.79837	+	20.9232i	0.285003	+	0.877148i	0.986398	+	0.164375i	$0.0525608\pi$
$569$	6.79837	+	20.9232i	0.285003	+	0.877148i	−0.701395	+	0.712773i	$0.747439\pi$
$570$		0			0
$571$	46.5287			1.94717			0.973583	−	0.228332i	$-0.0733271\pi$
$571$	46.5287			1.94717			0.973583	+	0.228332i	$0.0733271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.95601	+	12.1753i	0.164977	+	0.507747i
$576$	11.8588	−	36.4977i	0.494118	−	1.52074i
$577$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$577$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$578$	−30.3568	−	22.0555i	−1.26268	−	0.917389i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	28.6214	+	37.8626i	1.18538	+	1.56811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$587$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−14.4249	+	10.4803i	−0.592859	+	0.430737i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−51.1155	+	37.1376i	−2.09377	+	1.52121i
$597$		0			0
$598$		0			0
$599$	−36.4997	−	26.5186i	−1.49134	−	1.08352i	−0.973676	−	0.227937i	$-0.926802\pi$
$599$	−36.4997	−	26.5186i	−1.49134	−	1.08352i	−0.517663	−	0.855584i	$-0.673198\pi$
$600$		0			0
$601$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$601$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$602$		0			0
$603$	−30.5004	+	22.1599i	−1.24207	+	0.902419i
$604$	−6.84633			−0.278573
$605$		0			0
$606$		0			0
$607$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$607$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	3.73325	+	11.4898i	0.150784	+	0.464067i	0.997709	−	0.0676456i	$-0.0215487\pi$
$613$	3.73325	+	11.4898i	0.150784	+	0.464067i	−0.846925	+	0.531712i	$0.821549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.84770			−0.154902			−0.0774512	−	0.996996i	$-0.524678\pi$
$617$	−3.84770			−0.154902			−0.0774512	+	0.996996i	$0.524678\pi$
$618$		0			0
$619$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$619$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.2254	+	14.6946i	−0.809017	+	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	15.5241	−	47.7782i	0.618003	−	1.90202i	0.299528	−	0.954087i	$-0.403171\pi$
$631$	15.5241	−	47.7782i	0.618003	−	1.90202i	0.318475	−	0.947931i	$-0.396829\pi$
$632$	4.45088	+	3.23376i	0.177047	+	0.128632i
$633$		0			0
$634$	−14.5187	+	44.6839i	−0.576611	+	1.77463i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	54.2283	−	18.7641i	2.14692	−	0.742879i
$639$	48.1636			1.90532
$640$		0			0
$641$	−7.65550	−	23.5612i	−0.302374	−	0.930611i	−0.980644	−	0.195799i	$-0.937270\pi$
$641$	−7.65550	−	23.5612i	−0.302374	−	0.930611i	0.678270	−	0.734813i	$-0.262730\pi$
$642$		0			0
$643$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$643$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$647$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$648$	−17.3208			−0.680425
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−18.8835	−	58.1175i	−0.739536	−	2.27606i
$653$	7.88484	−	24.2670i	0.308558	−	0.949643i	−0.669768	−	0.742571i	$-0.733606\pi$
$653$	7.88484	−	24.2670i	0.308558	−	0.949643i	0.978326	−	0.207072i	$-0.0663936\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.0000			1.71400			0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000			1.71400			0.856998	+	0.515319i	$0.172327\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	28.8517	−	20.9620i	1.12135	−	0.814712i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	63.8519	+	46.3911i	2.47421	+	1.79762i
$667$	6.20189	−	19.0874i	0.240138	−	0.739069i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	39.7653	−	28.8912i	1.53284	−	1.11367i	0.578208	−	0.815890i	$-0.303752\pi$
$673$	39.7653	−	28.8912i	1.53284	−	1.11367i	0.954633	−	0.297784i	$-0.0962476\pi$
$674$	7.40704	+	22.7965i	0.285308	+	0.878089i
$675$		0			0
$676$	30.2046	+	21.9449i	1.16171	+	0.844035i
$677$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$677$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.0364			−0.422295			−0.211147	−	0.977454i	$-0.567720\pi$
$683$	−11.0364			−0.422295			−0.211147	+	0.977454i	$0.567720\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	15.5133	+	11.2711i	0.591438	+	0.429705i
$689$		0			0
$690$		0			0
$691$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$691$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$692$		0			0
$693$		0			0
$694$	−40.9130			−1.55304
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.7423	+	48.4499i	0.594579	+	1.82993i	0.556810	+	0.830640i	$0.312025\pi$
$701$	15.7423	+	48.4499i	0.594579	+	1.82993i	0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$		0			0
$704$	−25.5840	−	33.8444i	−0.964233	−	1.27556i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−29.0896	−	21.1349i	−1.09248	−	0.793736i	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−29.0896	−	21.1349i	−1.09248	−	0.793736i	−0.979817	+	0.199896i	$0.935939\pi$
$710$		0			0
$711$	−2.65013	+	8.15626i	−0.0993876	+	0.305884i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	75.8148			2.83333
$717$		0			0
$718$	−25.7141	−	79.1398i	−0.959641	−	2.95347i
$719$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$719$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$720$		0			0
$721$		0			0
$722$	12.9594	−	39.8850i	0.482300	−	1.48437i
$723$		0			0
$724$		0			0
$725$	39.1928			1.45559
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−8.34346	−	25.6785i	−0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$733$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$734$		0			0
$735$		0			0
$736$	−18.3091			−0.674883
$737$	0.790461	+	41.6721i	0.0291170	+	1.53501i
$738$		0			0
$739$	−26.4857	+	19.2430i	−0.974291	+	0.707864i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−26.4857	+	19.2430i	−0.974291	+	0.707864i	−0.0178655	+	0.999840i	$0.505687\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−43.7942	−	31.8183i	−1.60665	−	1.16730i	−0.872923	−	0.487858i	$-0.837778\pi$
$743$	−43.7942	−	31.8183i	−1.60665	−	1.16730i	−0.733729	−	0.679442i	$-0.762222\pi$
$744$		0			0
$745$		0			0
$746$	39.2853	−	28.5425i	1.43834	−	1.04501i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−12.3593	−	38.0379i	−0.450996	−	1.38802i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−12.3593	−	38.0379i	−0.450996	−	1.38802i	0.424777	−	0.905298i	$-0.360353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−40.3760	+	29.3349i	−1.46749	+	1.06619i	−0.486158	+	0.873871i	$0.661602\pi$
$757$	−40.3760	+	29.3349i	−1.46749	+	1.06619i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	−69.5710			−2.52693
$759$		0			0
$760$		0			0
$761$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$761$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$762$		0			0
$763$		0			0
$764$	−21.0949	−	15.3263i	−0.763186	−	0.554487i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	4.92855	−	3.58080i	0.177382	−	0.128876i
$773$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$773$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$774$	26.2295	−	80.7261i	0.942799	−	2.90164i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.14426	+	3.52166i	0.0410236	+	0.126258i
$779$		0			0
$780$		0			0
$781$	30.4751	−	43.6635i	1.09049	−	1.56240i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$787$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$788$	1.80209	−	5.54628i	0.0641969	−	0.197578i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−10.9596	+	15.7024i	−0.389432	+	0.557961i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$797$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$798$		0			0
$799$		0			0
$800$	−11.0488	−	34.0047i	−0.390634	−	1.20225i
$801$		0			0
$802$	67.6225			2.38783
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23.0711	−	16.7622i	−0.811138	−	0.589326i	0.103022	−	0.994679i	$-0.467149\pi$
$809$	−23.0711	−	16.7622i	−0.811138	−	0.589326i	−0.914160	+	0.405353i	$0.867149\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$	82.4583	−	28.5323i	2.89016	−	1.00006i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.79837	−	20.9232i	0.237265	−	0.730226i	−0.759548	−	0.650451i	$-0.774580\pi$
$821$	6.79837	−	20.9232i	0.237265	−	0.730226i	0.996813	−	0.0797750i	$-0.0254202\pi$
$822$		0			0
$823$	−1.70206	+	1.23662i	−0.0593301	+	0.0431058i	−0.617055	−	0.786920i	$-0.711674\pi$
$823$	−1.70206	+	1.23662i	−0.0593301	+	0.0431058i	0.557725	+	0.830026i	$0.311674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.5967	+	25.8626i	−1.23782	+	0.899329i	−0.997451	−	0.0713526i	$-0.977268\pi$
$827$	−35.5967	+	25.8626i	−1.23782	+	0.899329i	−0.240369	+	0.970682i	$0.577268\pi$
$828$	6.81680	+	20.9799i	0.236900	+	0.729103i
$829$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$829$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$839$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$840$		0			0
$841$	−26.2471	−	19.0696i	−0.905071	−	0.657573i
$842$	39.5722	+	28.7509i	1.36375	+	0.990821i
$843$		0			0
$844$	−25.6220	−	78.8564i	−0.881946	−	2.71435i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−21.4077			−0.735145
$849$		0			0
$850$		0			0
$851$	9.43044	−	29.0239i	0.323271	−	0.994927i
$852$		0			0
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$		0			0
$855$		0			0
$856$	−17.4582	+	12.6841i	−0.596709	+	0.433534i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	10.4180	−	32.0635i	0.354840	−	1.09209i
$863$	−6.47214	−	4.70228i	−0.220314	−	0.160068i	0.472154	−	0.881516i	$-0.343477\pi$
$863$	−6.47214	−	4.70228i	−0.220314	−	0.160068i	−0.692468	+	0.721449i	$0.743477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.71733	+	7.56332i	0.193947	+	0.256568i
$870$		0			0
$871$		0			0
$872$	−9.29379	−	28.6033i	−0.314727	−	0.968631i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−18.2668	−	56.2193i	−0.616824	−	1.89839i	−0.367885	−	0.929871i	$-0.619918\pi$
$877$	−18.2668	−	56.2193i	−0.616824	−	1.89839i	−0.248939	−	0.968519i	$-0.580082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−3.70820	−	11.4127i	−0.124791	−	0.384067i	0.869072	−	0.494686i	$-0.164717\pi$
$883$	−3.70820	−	11.4127i	−0.124791	−	0.384067i	−0.993863	+	0.110619i	$0.964717\pi$
$884$		0			0
$885$		0			0
$886$	−9.98478	−	7.25436i	−0.335445	−	0.243715i
$887$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$887$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−28.5585	−	8.68399i	−0.956746	−	0.290924i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−27.8820	+	85.8120i	−0.930434	+	2.86358i
$899$		0			0
$900$	−34.8514	+	25.3211i	−1.16171	+	0.844035i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	16.8512	−	12.2431i	0.560464	−	0.407201i
$905$		0			0
$906$		0			0
$907$	41.7868	+	30.3599i	1.38751	+	1.00808i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$	41.7868	+	30.3599i	1.38751	+	1.00808i	0.391373	+	0.920232i	$0.372000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.9443	+	9.40456i	−0.428863	+	0.311587i	−0.781194	−	0.624288i	$-0.785389\pi$
$911$	−12.9443	+	9.40456i	−0.428863	+	0.311587i	0.352331	+	0.935875i	$0.385389\pi$
$912$		0			0
$913$		0			0
$914$	92.9563			3.07472
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−13.8026	−	10.0282i	−0.455306	−	0.330799i	0.336381	−	0.941726i	$-0.390797\pi$
$919$	−13.8026	−	10.0282i	−0.455306	−	0.330799i	−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	59.5957			1.95949
$926$	41.1244	−	29.8786i	1.35143	−	0.981872i
$927$		0			0
$928$	−17.3214	+	53.3097i	−0.568601	+	1.74998i
$929$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$929$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$930$		0			0
$931$		0			0
$932$	19.5244	+	60.0898i	0.639542	+	1.96831i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$937$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$941$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−56.5869	−	74.8575i	−1.83980	−	2.43383i
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.7590	−	36.1906i	0.380913	−	1.17233i	−0.558489	−	0.829512i	$-0.688619\pi$
$953$	11.7590	−	36.1906i	0.380913	−	1.17233i	0.939402	−	0.342817i	$-0.111381\pi$
$954$	29.2829	+	90.1236i	0.948070	+	2.91786i
$955$		0			0
$956$	−81.8370			−2.64680
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.57953	+	29.4828i	−0.309017	+	0.951057i
$962$		0			0
$963$	−27.2142	−	19.7723i	−0.876965	−	0.637152i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	32.9320			1.05902			0.529511	−	0.848303i	$-0.322376\pi$
$967$	32.9320			1.05902			0.529511	+	0.848303i	$0.322376\pi$
$968$	7.30068	+	19.8712i	0.234653	+	0.638683i
$969$		0			0
$970$		0			0
$971$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$971$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$972$		0			0
$973$		0			0
$974$	73.5498	+	53.4370i	2.35669	+	1.71223i
$975$		0			0
$976$		0			0
$977$	−44.0711	+	32.0196i	−1.40996	+	1.02440i	−0.416632	+	0.909075i	$0.636790\pi$
$977$	−44.0711	+	32.0196i	−1.40996	+	1.02440i	−0.993328	+	0.115321i	$0.963210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	37.9284	−	27.5566i	1.21096	−	0.879813i
$982$	−30.0113	−	92.3653i	−0.957699	−	2.94750i
$983$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$983$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.8202			−1.04362
$990$		0			0
$991$	−24.0000			−0.762385			−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000			−0.762385			−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$997$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$998$	30.4724	+	93.7843i	0.964586	+	2.96869i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.f.c.344.2	yes	8
7.2	even	3		539.2.q.d.410.1		16
7.3	odd	6		539.2.q.d.520.2		16
7.4	even	3		539.2.q.d.520.2		16
7.5	odd	6		539.2.q.d.410.1		16
7.6	odd	2	CM	539.2.f.c.344.2	yes	8
11.2	odd	10		5929.2.a.bg.1.3		4
11.4	even	5	inner	539.2.f.c.246.2	✓	8
11.9	even	5		5929.2.a.bc.1.2		4
77.4	even	15		539.2.q.d.422.1		16
77.13	even	10		5929.2.a.bg.1.3		4
77.20	odd	10		5929.2.a.bc.1.2		4
77.26	odd	30		539.2.q.d.312.2		16
77.37	even	15		539.2.q.d.312.2		16
77.48	odd	10	inner	539.2.f.c.246.2	✓	8
77.59	odd	30		539.2.q.d.422.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.246.2	✓	8	11.4	even	5	inner
539.2.f.c.246.2	✓	8	77.48	odd	10	inner
539.2.f.c.344.2	yes	8	1.1	even	1	trivial
539.2.f.c.344.2	yes	8	7.6	odd	2	CM
539.2.q.d.312.2		16	77.26	odd	30
539.2.q.d.312.2		16	77.37	even	15
539.2.q.d.410.1		16	7.2	even	3
539.2.q.d.410.1		16	7.5	odd	6
539.2.q.d.422.1		16	77.4	even	15
539.2.q.d.422.1		16	77.59	odd	30
539.2.q.d.520.2		16	7.3	odd	6
539.2.q.d.520.2		16	7.4	even	3
5929.2.a.bc.1.2		4	11.9	even	5
5929.2.a.bc.1.2		4	77.20	odd	10
5929.2.a.bg.1.3		4	11.2	odd	10
5929.2.a.bg.1.3		4	77.13	even	10

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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.f (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.78570 - 1.29738i) q^{2} +(0.887471 - 2.73136i) q^{4} +(-0.594713 - 1.83034i) q^{8} +(2.42705 - 1.76336i) q^{9} +(-0.0629004 - 3.31603i) q^{11} +(1.21023 + 0.879283i) q^{16} +(2.04623 - 6.29764i) q^{18} +(-4.41448 - 5.83981i) q^{22} -2.56038 q^{23} +(-1.54508 - 4.75528i) q^{25} +(-2.42225 + 7.45492i) q^{29} +7.15093 q^{32} +(-2.66241 - 8.19407i) q^{36} +(-3.68322 + 11.3358i) q^{37} +12.8185 q^{43} +(-9.11308 - 2.77108i) q^{44} +(-4.57206 + 3.32180i) q^{46} +(-8.92848 - 6.48692i) q^{50} +(-11.5776 + 8.41162i) q^{53} +(5.34649 + 16.4548i) q^{58} +(10.3489 - 7.51894i) q^{64} -12.5669 q^{67} +(12.9884 + 9.43662i) q^{71} +(-4.67094 - 3.39364i) q^{72} +(8.12975 + 25.0208i) q^{74} +(-2.31271 + 1.68028i) q^{79} +(2.78115 - 8.55951i) q^{81} +(22.8899 - 16.6305i) q^{86} +(-6.03205 + 2.08722i) q^{88} +(-2.27227 + 6.99331i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 2 q^{4} + 16 q^{8} + 6 q^{9} - 4 q^{11} - 28 q^{16} - 9 q^{18} - 4 q^{22} - 16 q^{23} + 10 q^{25} - 4 q^{29} + 100 q^{32} - 6 q^{36} - 18 q^{37} + 24 q^{43} + 9 q^{44} - 31 q^{46} + 10 q^{50}+ \cdots - 48 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 2 * q^4 + 16 * q^8 + 6 * q^9 - 4 * q^11 - 28 * q^16 - 9 * q^18 - 4 * q^22 - 16 * q^23 + 10 * q^25 - 4 * q^29 + 100 * q^32 - 6 * q^36 - 18 * q^37 + 24 * q^43 + 9 * q^44 - 31 * q^46 + 10 * q^50 - 30 * q^53 + 71 * q^58 - 74 * q^64 - 8 * q^67 + 48 * q^71 + 57 * q^72 + 12 * q^74 + 24 * q^79 - 18 * q^81 + 29 * q^86 - 38 * q^88 + 71 * q^92 - 48 * q^99

Introduction

L-functions

Modular forms

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Database

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