LMFDB - Embedded newform 3528.1.fb.a.2845.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3528,1,Mod(397,3528)] 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("3528.397"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 21, 0, 29])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3528.fb (of order $42$, degree $12$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.76070136457$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{42})$
Coefficient field:	$\Q(\zeta_{84})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 $ x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Embedding invariants

Embedding label			2845.2
Root			$-0.680173 - 0.733052i$ of defining polynomial
Character	$\chi$	$=$	3528.2845
Dual form			3528.1.fb.a.1405.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.930874 + 0.365341i) q^{2} +(0.733052 + 0.680173i) q^{4} +(0.246289 + 0.167917i) q^{5} +(0.826239 - 0.563320i) q^{7} +(0.433884 + 0.900969i) q^{8} +(0.167917 + 0.246289i) q^{10} +(0.218511 + 1.44973i) q^{11} +(0.974928 - 0.222521i) q^{14} +(0.0747301 + 0.997204i) q^{16} +(0.0663300 + 0.290611i) q^{20} +(-0.326239 + 1.42935i) q^{22} +(-0.332879 - 0.848162i) q^{25} +(0.988831 + 0.149042i) q^{28} +(-0.712362 + 0.162592i) q^{29} +(-1.72721 - 0.997204i) q^{31} +(-0.294755 + 0.955573i) q^{32} +0.298085 q^{35} +(-0.0444272 + 0.294755i) q^{40} +(-0.825886 + 1.21135i) q^{44} +(0.365341 - 0.930874i) q^{49} -0.911146i q^{50} +(0.680173 - 0.733052i) q^{53} +(-0.189617 + 0.393744i) q^{55} +(0.866025 + 0.500000i) q^{56} +(-0.722521 - 0.108903i) q^{58} +(1.43109 - 0.975699i) q^{59} +(-1.24349 - 1.55929i) q^{62} +(-0.623490 + 0.781831i) q^{64} +(0.277479 + 0.108903i) q^{70} +(-0.807782 + 0.317031i) q^{73} +(0.997204 + 1.07473i) q^{77} +(0.826239 + 1.43109i) q^{79} +(-0.149042 + 0.258149i) q^{80} +(-1.16078 + 1.45557i) q^{83} +(-1.21135 + 0.825886i) q^{88} -0.589510i q^{97} +(0.680173 - 0.733052i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{4} + 2 q^{7} + 6 q^{10} + 2 q^{16} + 10 q^{22} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 22 q^{40} + 2 q^{49} - 14 q^{55} - 16 q^{58} + 4 q^{64} + 8 q^{70} + 2 q^{79} - 2 q^{88}+O(q^{100}) $ 24 * q - 2 * q^4 + 2 * q^7 + 6 * q^10 + 2 * q^16 + 10 * q^22 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 22 * q^40 + 2 * q^49 - 14 * q^55 - 16 * q^58 + 4 * q^64 + 8 * q^70 + 2 * q^79 - 2 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{1}{42}\right)$	$-1$	$1$

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 $

$2$	0.930874	+	0.365341i	0.930874	+	0.365341i
$2$	0.930874	+	0.365341i	0.930874	+	0.365341i
$3$		0			0
$3$		0			0
$4$	0.733052	+	0.680173i	0.733052	+	0.680173i
$5$	0.246289	+	0.167917i	0.246289	+	0.167917i	0.680173	−	0.733052i	$-0.261905\pi$
$5$	0.246289	+	0.167917i	0.246289	+	0.167917i	−0.433884	+	0.900969i	$0.642857\pi$
$6$		0			0
$7$	0.826239	−	0.563320i	0.826239	−	0.563320i
$7$	0.826239	−	0.563320i	0.826239	−	0.563320i
$8$	0.433884	+	0.900969i	0.433884	+	0.900969i
$9$		0			0
$10$	0.167917	+	0.246289i	0.167917	+	0.246289i
$11$	0.218511	+	1.44973i	0.218511	+	1.44973i	0.781831	+	0.623490i	$0.214286\pi$
$11$	0.218511	+	1.44973i	0.218511	+	1.44973i	−0.563320	+	0.826239i	$0.690476\pi$
$12$		0			0
$13$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$13$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$14$	0.974928	−	0.222521i	0.974928	−	0.222521i
$15$		0			0
$16$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$17$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$17$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	0.0663300	+	0.290611i	0.0663300	+	0.290611i
$21$		0			0
$22$	−0.326239	+	1.42935i	−0.326239	+	1.42935i
$23$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$23$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$24$		0			0
$25$	−0.332879	−	0.848162i	−0.332879	−	0.848162i
$26$		0			0
$27$		0			0
$28$	0.988831	+	0.149042i	0.988831	+	0.149042i
$29$	−0.712362	+	0.162592i	−0.712362	+	0.162592i	−0.563320	−	0.826239i	$-0.690476\pi$
$29$	−0.712362	+	0.162592i	−0.712362	+	0.162592i	−0.149042	+	0.988831i	$0.547619\pi$
$30$		0			0
$31$	−1.72721	−	0.997204i	−1.72721	−	0.997204i	−0.900969	−	0.433884i	$-0.857143\pi$
$31$	−1.72721	−	0.997204i	−1.72721	−	0.997204i	−0.826239	−	0.563320i	$-0.809524\pi$
$32$	−0.294755	+	0.955573i	−0.294755	+	0.955573i
$33$		0			0
$34$		0			0
$35$	0.298085			0.298085
$36$		0			0
$37$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$37$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$38$		0			0
$39$		0			0
$40$	−0.0444272	+	0.294755i	−0.0444272	+	0.294755i
$41$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$41$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$42$		0			0
$43$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$43$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$44$	−0.825886	+	1.21135i	−0.825886	+	1.21135i
$45$		0			0
$46$		0			0
$47$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$47$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$48$		0			0
$49$	0.365341	−	0.930874i	0.365341	−	0.930874i
$50$		−	0.911146i		−	0.911146i
$51$		0			0
$52$		0			0
$53$	0.680173	−	0.733052i	0.680173	−	0.733052i	−0.294755	−	0.955573i	$-0.595238\pi$
$53$	0.680173	−	0.733052i	0.680173	−	0.733052i	0.974928	+	0.222521i	$0.0714286\pi$
$54$		0			0
$55$	−0.189617	+	0.393744i	−0.189617	+	0.393744i
$56$	0.866025	+	0.500000i	0.866025	+	0.500000i
$57$		0			0
$58$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$59$	1.43109	−	0.975699i	1.43109	−	0.975699i	0.433884	−	0.900969i	$-0.357143\pi$
$59$	1.43109	−	0.975699i	1.43109	−	0.975699i	0.997204	−	0.0747301i	$-0.0238095\pi$
$60$		0			0
$61$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$61$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$62$	−1.24349	−	1.55929i	−1.24349	−	1.55929i
$63$		0			0
$64$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	0.277479	+	0.108903i	0.277479	+	0.108903i
$71$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$71$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$72$		0			0
$73$	−0.807782	+	0.317031i	−0.807782	+	0.317031i	−0.733052	−	0.680173i	$-0.761905\pi$
$73$	−0.807782	+	0.317031i	−0.807782	+	0.317031i	−0.0747301	+	0.997204i	$0.523810\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.997204	+	1.07473i	0.997204	+	1.07473i
$78$		0			0
$79$	0.826239	+	1.43109i	0.826239	+	1.43109i	0.900969	+	0.433884i	$0.142857\pi$
$79$	0.826239	+	1.43109i	0.826239	+	1.43109i	−0.0747301	+	0.997204i	$0.523810\pi$
$80$	−0.149042	+	0.258149i	−0.149042	+	0.258149i
$81$		0			0
$82$		0			0
$83$	−1.16078	+	1.45557i	−1.16078	+	1.45557i	−0.294755	+	0.955573i	$0.595238\pi$
$83$	−1.16078	+	1.45557i	−1.16078	+	1.45557i	−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$89$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$89$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	0.589510i		−	0.589510i	−0.955573	−	0.294755i	$-0.904762\pi$
$97$		−	0.589510i		−	0.589510i	0.955573	−	0.294755i	$-0.0952381\pi$
$98$	0.680173	−	0.733052i	0.680173	−	0.733052i
$99$		0			0
$100$	0.332879	−	0.848162i	0.332879	−	0.848162i
$101$	0.145713	−	1.94440i	0.145713	−	1.94440i	−0.149042	−	0.988831i	$-0.547619\pi$
$101$	0.145713	−	1.94440i	0.145713	−	1.94440i	0.294755	−	0.955573i	$-0.404762\pi$
$102$		0			0
$103$	1.09839	−	1.61105i	1.09839	−	1.61105i	0.365341	−	0.930874i	$-0.380952\pi$
$103$	1.09839	−	1.61105i	1.09839	−	1.61105i	0.733052	−	0.680173i	$-0.238095\pi$
$104$		0			0
$105$		0			0
$106$	0.900969	−	0.433884i	0.900969	−	0.433884i
$107$	−0.246289	+	1.63402i	−0.246289	+	1.63402i	0.433884	+	0.900969i	$0.357143\pi$
$107$	−0.246289	+	1.63402i	−0.246289	+	1.63402i	−0.680173	+	0.733052i	$0.738095\pi$
$108$		0			0
$109$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$109$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$110$	−0.320360	+	0.297251i	−0.320360	+	0.297251i
$111$		0			0
$112$	0.623490	+	0.781831i	0.623490	+	0.781831i
$113$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$113$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$114$		0			0
$115$		0			0
$116$	−0.632789	−	0.365341i	−0.632789	−	0.365341i
$117$		0			0
$118$	1.68862	−	0.385418i	1.68862	−	0.385418i
$119$		0			0
$120$		0			0
$121$	−1.09839	+	0.338809i	−1.09839	+	0.338809i
$122$		0			0
$123$		0			0
$124$	−0.587862	−	1.90580i	−0.587862	−	1.90580i
$125$	0.126766	−	0.555400i	0.126766	−	0.555400i
$126$		0			0
$127$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	0.955573	−	0.294755i	$-0.0952381\pi$
$127$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	−0.988831	+	0.149042i	$0.952381\pi$
$128$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	0.866025	+	0.500000i	$0.166667\pi$
$131$	0.0841939	+	1.12349i	0.0841939	+	1.12349i	−0.781831	+	0.623490i	$0.785714\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$137$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$138$		0			0
$139$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$139$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$140$	0.218511	+	0.202749i	0.218511	+	0.202749i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−0.202749	−	0.0795731i	−0.202749	−	0.0795731i
$146$	−0.867767			−0.867767
$147$		0			0
$148$		0			0
$149$	−1.67738	−	0.658322i	−1.67738	−	0.658322i	−0.680173	−	0.733052i	$-0.738095\pi$
$149$	−1.67738	−	0.658322i	−1.67738	−	0.658322i	−0.997204	+	0.0747301i	$0.976190\pi$
$150$		0			0
$151$	−0.733052	−	0.680173i	−0.733052	−	0.680173i	0.222521	−	0.974928i	$-0.428571\pi$
$151$	−0.733052	−	0.680173i	−0.733052	−	0.680173i	−0.955573	+	0.294755i	$0.904762\pi$
$152$		0			0
$153$		0			0
$154$	0.535628	+	1.36476i	0.535628	+	1.36476i
$155$	−0.257945	−	0.535628i	−0.257945	−	0.535628i
$156$		0			0
$157$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$157$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$158$	0.246289	+	1.63402i	0.246289	+	1.63402i
$159$		0			0
$160$	−0.233052	+	0.185853i	−0.233052	+	0.185853i
$161$		0			0
$162$		0			0
$163$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$163$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$164$		0			0
$165$		0			0
$166$	−1.61232	+	0.930874i	−1.61232	+	0.930874i
$167$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$167$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$168$		0			0
$169$	0.222521	−	0.974928i	0.222521	−	0.974928i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.49419	−	0.460898i	1.49419	−	0.460898i	0.563320	−	0.826239i	$-0.309524\pi$
$173$	1.49419	−	0.460898i	1.49419	−	0.460898i	0.930874	+	0.365341i	$0.119048\pi$
$174$		0			0
$175$	−0.752824	−	0.513267i	−0.752824	−	0.513267i
$176$	−1.42935	+	0.326239i	−1.42935	+	0.326239i
$177$		0			0
$178$		0			0
$179$	−0.367554	+	1.19158i	−0.367554	+	1.19158i	0.563320	+	0.826239i	$0.309524\pi$
$179$	−0.367554	+	1.19158i	−0.367554	+	1.19158i	−0.930874	+	0.365341i	$0.880952\pi$
$180$		0			0
$181$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$181$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$191$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$192$		0			0
$193$	0.123490	−	1.64786i	0.123490	−	1.64786i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	0.123490	−	1.64786i	0.123490	−	1.64786i	0.623490	−	0.781831i	$-0.285714\pi$
$194$	0.215372	−	0.548760i	0.215372	−	0.548760i
$195$		0			0
$196$	0.900969	−	0.433884i	0.900969	−	0.433884i
$197$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$197$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$198$		0			0
$199$	0.865341	+	0.0648483i	0.865341	+	0.0648483i	0.500000	−	0.866025i	$-0.333333\pi$
$199$	0.865341	+	0.0648483i	0.865341	+	0.0648483i	0.365341	+	0.930874i	$0.380952\pi$
$200$	0.619736	−	0.667917i	0.619736	−	0.667917i
$201$		0			0
$202$	0.846011	−	1.75676i	0.846011	−	1.75676i
$203$	−0.496990	+	0.535628i	−0.496990	+	0.535628i
$204$		0			0
$205$		0			0
$206$	1.61105	−	1.09839i	1.61105	−	1.09839i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$211$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$212$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$213$		0			0
$214$	−0.826239	+	1.43109i	−0.826239	+	1.43109i
$215$		0			0
$216$		0			0
$217$	−1.98883	+	0.149042i	−1.98883	+	0.149042i
$218$		0			0
$219$		0			0
$220$	−0.406813	+	0.159662i	−0.406813	+	0.159662i
$221$		0			0
$222$		0			0
$223$	−0.574730	−	0.131178i	−0.574730	−	0.131178i	−0.0747301	−	0.997204i	$-0.523810\pi$
$223$	−0.574730	−	0.131178i	−0.574730	−	0.131178i	−0.500000	+	0.866025i	$0.666667\pi$
$224$	0.294755	+	0.955573i	0.294755	+	0.955573i
$225$		0			0
$226$		0			0
$227$	−0.680173	+	1.17809i	−0.680173	+	1.17809i	0.294755	+	0.955573i	$0.404762\pi$
$227$	−0.680173	+	1.17809i	−0.680173	+	1.17809i	−0.974928	+	0.222521i	$0.928571\pi$
$228$		0			0
$229$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$229$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$230$		0			0
$231$		0			0
$232$	−0.455573	−	0.571270i	−0.455573	−	0.571270i
$233$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$233$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$234$		0			0
$235$		0			0
$236$	1.71271	+	0.258149i	1.71271	+	0.258149i
$237$		0			0
$238$		0			0
$239$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$239$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$240$		0			0
$241$	−1.26631	+	1.36476i	−1.26631	+	1.36476i	−0.365341	+	0.930874i	$0.619048\pi$
$241$	−1.26631	+	1.36476i	−1.26631	+	1.36476i	−0.900969	+	0.433884i	$0.857143\pi$
$242$	−1.14625	−	0.0858993i	−1.14625	−	0.0858993i
$243$		0			0
$244$		0			0
$245$	0.246289	−	0.167917i	0.246289	−	0.167917i
$246$		0			0
$247$		0			0
$248$	0.149042	−	1.98883i	0.149042	−	1.98883i
$249$		0			0
$250$	0.320914	−	0.470694i	0.320914	−	0.470694i
$251$	−1.79690	−	0.865341i	−1.79690	−	0.865341i	−0.930874	−	0.365341i	$-0.880952\pi$
$251$	−1.79690	−	0.865341i	−1.79690	−	0.865341i	−0.866025	−	0.500000i	$-0.833333\pi$
$252$		0			0
$253$		0			0
$254$	0.0222759	−	0.147791i	0.0222759	−	0.147791i
$255$		0			0
$256$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$257$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$257$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.332083	+	1.07659i	−0.332083	+	1.07659i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	0.290611	−	0.0663300i	0.290611	−	0.0663300i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.680173	−	1.73305i	−0.680173	−	1.73305i	−0.680173	−	0.733052i	$-0.738095\pi$
$269$	−0.680173	−	1.73305i	−0.680173	−	1.73305i		−	1.00000i	$-0.5\pi$
$270$		0			0
$271$	0.173761	+	0.563320i	0.173761	+	0.563320i	1.00000			$0$
$271$	0.173761	+	0.563320i	0.173761	+	0.563320i	−0.826239	+	0.563320i	$0.809524\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.15687	−	0.667917i	1.15687	−	0.667917i
$276$		0			0
$277$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$277$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$278$		0			0
$279$		0			0
$280$	0.129334	+	0.268565i	0.129334	+	0.268565i
$281$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$281$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$282$		0			0
$283$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$283$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.826239	+	0.563320i	0.826239	+	0.563320i
$290$	−0.159662	−	0.148145i	−0.159662	−	0.148145i
$291$		0			0
$292$	−0.807782	−	0.317031i	−0.807782	−	0.317031i
$293$	0.589510			0.589510			0.294755	−	0.955573i	$-0.404762\pi$
$293$	0.589510			0.589510			0.294755	+	0.955573i	$0.404762\pi$
$294$		0			0
$295$	0.516298			0.516298
$296$		0			0
$297$		0			0
$298$	−1.32091	−	1.22563i	−1.32091	−	1.22563i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$307$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$308$			1.46610i			1.46610i
$309$		0			0
$310$	−0.0444272	−	0.592840i	−0.0444272	−	0.592840i
$311$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$311$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$312$		0			0
$313$	0.975699	−	0.563320i	0.975699	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$313$	0.975699	−	0.563320i	0.975699	−	0.563320i	0.900969	+	0.433884i	$0.142857\pi$
$314$		0			0
$315$		0			0
$316$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$317$	0.563320	+	1.82624i	0.563320	+	1.82624i	0.563320	+	0.826239i	$0.309524\pi$
$317$	0.563320	+	1.82624i	0.563320	+	1.82624i			1.00000i	$0.5\pi$
$318$		0			0
$319$	−0.391374	−	0.997204i	−0.391374	−	0.997204i
$320$	−0.284841	+	0.0878620i	−0.284841	+	0.0878620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$331$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$332$	−1.84095	+	0.277479i	−1.84095	+	0.277479i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$337$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$338$	0.563320	−	0.826239i	0.563320	−	0.826239i
$339$		0			0
$340$		0			0
$341$	1.06826	−	2.72188i	1.06826	−	2.72188i
$342$		0			0
$343$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$344$		0			0
$345$		0			0
$346$	1.55929	+	0.116853i	1.55929	+	0.116853i
$347$	−0.848162	+	0.914101i	−0.848162	+	0.914101i	−0.997204	−	0.0747301i	$-0.976190\pi$
$347$	−0.848162	+	0.914101i	−0.848162	+	0.914101i	0.149042	+	0.988831i	$0.452381\pi$
$348$		0			0
$349$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$349$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$350$	−0.513267	−	0.752824i	−0.513267	−	0.752824i
$351$		0			0
$352$	−1.44973	−	0.218511i	−1.44973	−	0.218511i
$353$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$353$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.777479	+	0.974928i	−0.777479	+	0.974928i
$359$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$359$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.252183	−	0.0575591i	−0.252183	−	0.0575591i
$366$		0			0
$367$	−1.61232	+	0.632789i	−1.61232	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$367$	−1.61232	+	0.632789i	−1.61232	+	0.632789i	−0.623490	+	0.781831i	$0.714286\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.149042	−	0.988831i	0.149042	−	0.988831i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$379$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$383$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$384$		0			0
$385$	0.0651349	+	0.432142i	0.0651349	+	0.432142i
$386$	0.716983	−	1.48883i	0.716983	−	1.48883i
$387$		0			0
$388$	0.400969	−	0.432142i	0.400969	−	0.432142i
$389$	−1.24349	−	0.0931869i	−1.24349	−	0.0931869i	−0.563320	−	0.826239i	$-0.690476\pi$
$389$	−1.24349	−	0.0931869i	−1.24349	−	0.0931869i	−0.680173	+	0.733052i	$0.738095\pi$
$390$		0			0
$391$		0			0
$392$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$393$		0			0
$394$	0.162592	−	0.414278i	0.162592	−	0.414278i
$395$	−0.0368104	+	0.491201i	−0.0368104	+	0.491201i
$396$		0			0
$397$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$397$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$398$	0.781831	+	0.376510i	0.781831	+	0.376510i
$399$		0			0
$400$	0.820914	−	0.395331i	0.820914	−	0.395331i
$401$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$401$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$402$		0			0
$403$		0			0
$404$	1.42935	−	1.32624i	1.42935	−	1.32624i
$405$		0			0
$406$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$407$		0			0
$408$		0			0
$409$	0.510531	−	1.65510i	0.510531	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$409$	0.510531	−	1.65510i	0.510531	−	1.65510i	0.733052	−	0.680173i	$-0.238095\pi$
$410$		0			0
$411$		0			0
$412$	1.90097	−	0.433884i	1.90097	−	0.433884i
$413$	0.632789	−	1.61232i	0.632789	−	1.61232i
$414$		0			0
$415$	−0.530303	+	0.163577i	−0.530303	+	0.163577i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.347948	−	1.52446i	0.347948	−	1.52446i	−0.433884	−	0.900969i	$-0.642857\pi$
$419$	0.347948	−	1.52446i	0.347948	−	1.52446i	0.781831	−	0.623490i	$-0.214286\pi$
$420$		0			0
$421$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$421$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$422$		0			0
$423$		0			0
$424$	0.955573	+	0.294755i	0.955573	+	0.294755i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.29196	+	1.03030i	−1.29196	+	1.03030i
$429$		0			0
$430$		0			0
$431$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$431$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$432$		0			0
$433$	0.678448	+	1.40881i	0.678448	+	1.40881i	0.900969	+	0.433884i	$0.142857\pi$
$433$	0.678448	+	1.40881i	0.678448	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$434$	−1.90580	−	0.587862i	−1.90580	−	0.587862i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.26631	+	0.496990i	1.26631	+	0.496990i	0.900969	−	0.433884i	$-0.142857\pi$
$439$	1.26631	+	0.496990i	1.26631	+	0.496990i	0.365341	+	0.930874i	$0.380952\pi$
$440$	−0.437023			−0.437023
$441$		0			0
$442$		0			0
$443$	1.77904	+	0.698220i	1.77904	+	0.698220i	0.997204	+	0.0747301i	$0.0238095\pi$
$443$	1.77904	+	0.698220i	1.77904	+	0.698220i	0.781831	+	0.623490i	$0.214286\pi$
$444$		0			0
$445$		0			0
$446$	−0.487076	−	0.332083i	−0.487076	−	0.332083i
$447$		0			0
$448$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$449$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$449$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−1.06356	+	0.848162i	−1.06356	+	0.848162i
$455$		0			0
$456$		0			0
$457$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	0.955573	+	0.294755i	$0.0952381\pi$
$457$	0.0546039	+	0.728639i	0.0546039	+	0.728639i	−0.900969	+	0.433884i	$0.857143\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.193096	+	0.846011i	0.193096	+	0.846011i	0.974928	+	0.222521i	$0.0714286\pi$
$461$	0.193096	+	0.846011i	0.193096	+	0.846011i	−0.781831	+	0.623490i	$0.785714\pi$
$462$		0			0
$463$	0.277479	−	1.21572i	0.277479	−	1.21572i	−0.623490	−	0.781831i	$-0.714286\pi$
$463$	0.277479	−	1.21572i	0.277479	−	1.21572i	0.900969	−	0.433884i	$-0.142857\pi$
$464$	−0.215372	−	0.698220i	−0.215372	−	0.698220i
$465$		0			0
$466$		0			0
$467$	−1.86323	+	0.574730i	−1.86323	+	0.574730i	−0.866025	+	0.500000i	$0.833333\pi$
$467$	−1.86323	+	0.574730i	−1.86323	+	0.574730i	−0.997204	+	0.0747301i	$0.976190\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.50000	+	0.866025i	1.50000	+	0.866025i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$479$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$480$		0			0
$481$		0			0
$482$	−1.67738	+	0.807782i	−1.67738	+	0.807782i
$483$		0			0
$484$	−1.03563	−	0.498732i	−1.03563	−	0.498732i
$485$	0.0989888	−	0.145190i	0.0989888	−	0.145190i
$486$		0			0
$487$	−0.0111692	+	0.149042i	−0.0111692	+	0.149042i	0.988831	+	0.149042i	$0.0476190\pi$
$487$	−0.0111692	+	0.149042i	−0.0111692	+	0.149042i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	0.290611	−	0.0663300i	0.290611	−	0.0663300i
$491$		−	1.97766i		−	1.97766i	−0.149042	−	0.988831i	$-0.547619\pi$
$491$		−	1.97766i		−	1.97766i	0.149042	−	0.988831i	$-0.452381\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.865341	−	1.79690i	0.865341	−	1.79690i
$497$		0			0
$498$		0			0
$499$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$499$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$500$	0.470694	−	0.320914i	0.470694	−	0.320914i
$501$		0			0
$502$	−1.35654	−	1.46200i	−1.35654	−	1.46200i
$503$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$503$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$504$		0			0
$505$	0.362386	−	0.454418i	0.362386	−	0.454418i
$506$		0			0
$507$		0			0
$508$	0.0747301	−	0.129436i	0.0747301	−	0.129436i
$509$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$509$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$510$		0			0
$511$	−0.488831	+	0.716983i	−0.488831	+	0.716983i
$512$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$513$		0			0
$514$		0			0
$515$	0.541044	−	0.212344i	0.541044	−	0.212344i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$523$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$524$	−0.702449	+	0.880843i	−0.702449	+	0.880843i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$530$	0.294755	+	0.0444272i	0.294755	+	0.0444272i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.335038	+	0.361085i	−0.335038	+	0.361085i
$536$		0			0
$537$		0			0
$538$		−	1.86175i		−	1.86175i
$539$	1.42935	+	0.326239i	1.42935	+	0.326239i
$540$		0			0
$541$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$541$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$542$	−0.0440542	+	0.587862i	−0.0440542	+	0.587862i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$547$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$548$		0			0
$549$		0			0
$550$	1.32091	−	0.199096i	1.32091	−	0.199096i
$551$		0			0
$552$		0			0
$553$	1.48883	+	0.716983i	1.48883	+	0.716983i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.71271	−	0.988831i	−1.71271	−	0.988831i	−0.930874	−	0.365341i	$-0.880952\pi$
$557$	−1.71271	−	0.988831i	−1.71271	−	0.988831i	−0.781831	−	0.623490i	$-0.785714\pi$
$558$		0			0
$559$		0			0
$560$	0.0222759	+	0.297251i	0.0222759	+	0.297251i
$561$		0			0
$562$		0			0
$563$	0.411608	+	1.04876i	0.411608	+	1.04876i	0.974928	+	0.222521i	$0.0714286\pi$
$563$	0.411608	+	1.04876i	0.411608	+	1.04876i	−0.563320	+	0.826239i	$0.690476\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$571$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.297251	+	1.97213i	0.297251	+	1.97213i	0.222521	+	0.974928i	$0.428571\pi$
$577$	0.297251	+	1.97213i	0.297251	+	1.97213i	0.0747301	+	0.997204i	$0.476190\pi$
$578$	0.563320	+	0.826239i	0.563320	+	0.826239i
$579$		0			0
$580$	−0.0945021	−	0.196236i	−0.0945021	−	0.196236i
$581$	−0.139129	+	1.85654i	−0.139129	+	1.85654i
$582$		0			0
$583$	1.21135	+	0.825886i	1.21135	+	0.825886i
$584$	−0.636119	−	0.590232i	−0.636119	−	0.590232i
$585$		0			0
$586$	0.548760	+	0.215372i	0.548760	+	0.215372i
$587$	−1.36035			−1.36035			−0.680173	−	0.733052i	$-0.738095\pi$
$587$	−1.36035			−1.36035			−0.680173	+	0.733052i	$0.738095\pi$
$588$		0			0
$589$		0			0
$590$	0.480608	+	0.188625i	0.480608	+	0.188625i
$591$		0			0
$592$		0			0
$593$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$593$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$594$		0			0
$595$		0			0
$596$	−0.781831	−	1.62349i	−0.781831	−	1.62349i
$597$		0			0
$598$		0			0
$599$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$599$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$600$		0			0
$601$	−1.55929	+	1.24349i	−1.55929	+	1.24349i	−0.733052	+	0.680173i	$0.761905\pi$
$601$	−1.55929	+	1.24349i	−1.55929	+	1.24349i	−0.826239	+	0.563320i	$0.809524\pi$
$602$		0			0
$603$		0			0
$604$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$605$	−0.327414	−	0.100994i	−0.327414	−	0.100994i
$606$		0			0
$607$	−1.17809	+	0.680173i	−1.17809	+	0.680173i	−0.955573	−	0.294755i	$-0.904762\pi$
$607$	−1.17809	+	0.680173i	−1.17809	+	0.680173i	−0.222521	+	0.974928i	$0.571429\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$613$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$614$		0			0
$615$		0			0
$616$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$617$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$617$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$	0.175233	−	0.568090i	0.175233	−	0.568090i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.543435	+	0.504234i	−0.543435	+	0.504234i
$626$	1.11406	−	0.167917i	1.11406	−	0.167917i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.72188	+	0.829215i	1.72188	+	0.829215i	0.988831	+	0.149042i	$0.0476190\pi$
$631$	1.72188	+	0.829215i	1.72188	+	0.829215i	0.733052	+	0.680173i	$0.238095\pi$
$632$	−0.930874	+	1.36534i	−0.930874	+	1.36534i
$633$		0			0
$634$	−0.142820	+	1.90580i	−0.142820	+	1.90580i
$635$	0.0162766	−	0.0414721i	0.0162766	−	0.0414721i
$636$		0			0
$637$		0			0
$638$		−	1.07126i		−	1.07126i
$639$		0			0
$640$	−0.297251	−	0.0222759i	−0.297251	−	0.0222759i
$641$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$641$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$642$		0			0
$643$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$643$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$647$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$648$		0			0
$649$	1.72721	+	1.86149i	1.72721	+	1.86149i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.97213	+	0.147791i	−1.97213	+	0.147791i	−0.997204	−	0.0747301i	$-0.976190\pi$
$653$	−1.97213	+	0.147791i	−1.97213	+	0.147791i	−0.974928	+	0.222521i	$0.928571\pi$
$654$		0			0
$655$	−0.167917	+	0.290841i	−0.167917	+	0.290841i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.433884	+	0.0990311i	0.433884	+	0.0990311i	0.433884	−	0.900969i	$-0.357143\pi$
$659$	0.433884	+	0.0990311i	0.433884	+	0.0990311i			1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$661$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$662$		0			0
$663$		0			0
$664$	−1.81507	−	0.414278i	−1.81507	−	0.414278i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.914101	+	1.14625i	0.914101	+	1.14625i	0.988831	+	0.149042i	$0.0476190\pi$
$673$	0.914101	+	1.14625i	0.914101	+	1.14625i	−0.0747301	+	0.997204i	$0.523810\pi$
$674$	0.680173	+	0.733052i	0.680173	+	0.733052i
$675$		0			0
$676$	0.826239	−	0.563320i	0.826239	−	0.563320i
$677$	1.34515	+	0.202749i	1.34515	+	0.202749i	0.781831	−	0.623490i	$-0.214286\pi$
$677$	1.34515	+	0.202749i	1.34515	+	0.202749i	0.563320	+	0.826239i	$0.309524\pi$
$678$		0			0
$679$	−0.332083	−	0.487076i	−0.332083	−	0.487076i
$680$		0			0
$681$		0			0
$682$	1.98883	−	2.14345i	1.98883	−	2.14345i
$683$	−1.97213	−	0.147791i	−1.97213	−	0.147791i	−0.974928	−	0.222521i	$-0.928571\pi$
$683$	−1.97213	−	0.147791i	−1.97213	−	0.147791i	−0.997204	+	0.0747301i	$0.976190\pi$
$684$		0			0
$685$		0			0
$686$	0.149042	−	0.988831i	0.149042	−	0.988831i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$691$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$692$	1.40881	+	0.678448i	1.40881	+	0.678448i
$693$		0			0
$694$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.202749	−	0.888301i	−0.202749	−	0.888301i
$701$	0.116853	+	0.0931869i	0.116853	+	0.0931869i	0.680173	−	0.733052i	$-0.261905\pi$
$701$	0.116853	+	0.0931869i	0.116853	+	0.0931869i	−0.563320	+	0.826239i	$0.690476\pi$
$702$		0			0
$703$		0			0
$704$	−1.26968	−	0.733052i	−1.26968	−	0.733052i
$705$		0			0
$706$		0			0
$707$	−0.974928	−	1.68862i	−0.974928	−	1.68862i
$708$		0			0
$709$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$709$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$717$		0			0
$718$		0			0
$719$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$719$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$720$		0			0
$721$		−	1.94986i		−	1.94986i
$722$	0.781831	−	0.623490i	0.781831	−	0.623490i
$723$		0			0
$724$		0			0
$725$	0.375035	+	0.550075i	0.375035	+	0.550075i
$726$		0			0
$727$	0.807782	+	1.67738i	0.807782	+	1.67738i	0.733052	+	0.680173i	$0.238095\pi$
$727$	0.807782	+	1.67738i	0.807782	+	1.67738i	0.0747301	+	0.997204i	$0.476190\pi$
$728$		0			0
$729$		0			0
$730$	−0.213722	−	0.145713i	−0.213722	−	0.145713i
$731$		0			0
$732$		0			0
$733$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$733$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$734$	−1.73205			−1.73205
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$739$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$740$		0			0
$741$		0			0
$742$	0.500000	−	0.866025i	0.500000	−	0.866025i
$743$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$743$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$744$		0			0
$745$	−0.302576	−	0.443797i	−0.302576	−	0.443797i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.716983	+	1.48883i	0.716983	+	1.48883i
$750$		0			0
$751$	0.109562	+	1.46200i	0.109562	+	1.46200i	0.733052	+	0.680173i	$0.238095\pi$
$751$	0.109562	+	1.46200i	0.109562	+	1.46200i	−0.623490	+	0.781831i	$0.714286\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.0663300	−	0.290611i	−0.0663300	−	0.290611i
$756$		0			0
$757$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$757$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$761$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.880843	−	0.702449i	−0.880843	−	0.702449i	0.0747301	−	0.997204i	$-0.476190\pi$
$769$	−0.880843	−	0.702449i	−0.880843	−	0.702449i	−0.955573	+	0.294755i	$0.904762\pi$
$770$	−0.0972467	+	0.426066i	−0.0972467	+	0.426066i
$771$		0			0
$772$	1.21135	−	1.12397i	1.21135	−	1.12397i
$773$	−1.92808	+	0.290611i	−1.92808	+	0.290611i	−0.997204	−	0.0747301i	$-0.976190\pi$
$773$	−1.92808	+	0.290611i	−1.92808	+	0.290611i	−0.930874	+	0.365341i	$0.880952\pi$
$774$		0			0
$775$	−0.270839	+	1.79690i	−0.270839	+	1.79690i
$776$	0.531130	−	0.255779i	0.531130	−	0.255779i
$777$		0			0
$778$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.955573	+	0.294755i	0.955573	+	0.294755i
$785$		0			0
$786$		0			0
$787$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$787$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$788$	0.302705	−	0.326239i	0.302705	−	0.326239i
$789$		0			0
$790$	−0.213722	+	0.443797i	−0.213722	+	0.443797i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.590232	+	0.636119i	0.590232	+	0.636119i
$797$	−0.367554	−	0.460898i	−0.367554	−	0.460898i	0.563320	−	0.826239i	$-0.309524\pi$
$797$	−0.367554	−	0.460898i	−0.367554	−	0.460898i	−0.930874	+	0.365341i	$0.880952\pi$
$798$		0			0
$799$		0			0
$800$	0.908598	−	0.0680900i	0.908598	−	0.0680900i
$801$		0			0
$802$		0			0
$803$	−0.636119	−	1.10179i	−0.636119	−	1.10179i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.81507	−	0.712362i	1.81507	−	0.712362i
$809$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$809$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$810$		0			0
$811$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$811$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$812$	−0.728639	+	0.0546039i	−0.728639	+	0.0546039i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.07992	−	1.35417i	1.07992	−	1.35417i
$819$		0			0
$820$		0			0
$821$	1.12397	+	1.21135i	1.12397	+	1.21135i	0.974928	+	0.222521i	$0.0714286\pi$
$821$	1.12397	+	1.21135i	1.12397	+	1.21135i	0.149042	+	0.988831i	$0.452381\pi$
$822$		0			0
$823$	1.48883	−	1.01507i	1.48883	−	1.01507i	0.500000	−	0.866025i	$-0.333333\pi$
$823$	1.48883	−	1.01507i	1.48883	−	1.01507i	0.988831	−	0.149042i	$-0.0476190\pi$
$824$	1.92808	+	0.290611i	1.92808	+	0.290611i
$825$		0			0
$826$	1.17809	−	1.26968i	1.17809	−	1.26968i
$827$	0.858075	−	1.78181i	0.858075	−	1.78181i	0.294755	−	0.955573i	$-0.404762\pi$
$827$	0.858075	−	1.78181i	0.858075	−	1.78181i	0.563320	−	0.826239i	$-0.309524\pi$
$828$		0			0
$829$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$829$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$830$	−0.553406	−	0.0414721i	−0.553406	−	0.0414721i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	0.880843	−	1.29196i	0.880843	−	1.29196i
$839$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$839$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$840$		0			0
$841$	−0.419945	+	0.202235i	−0.419945	+	0.202235i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.218511	−	0.202749i	0.218511	−	0.202749i
$846$		0			0
$847$	−0.716677	+	0.898684i	−0.716677	+	0.898684i
$848$	0.781831	+	0.623490i	0.781831	+	0.623490i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$853$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$854$		0			0
$855$		0			0
$856$	−1.57906	+	0.487076i	−1.57906	+	0.487076i
$857$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$857$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$858$		0			0
$859$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$859$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$	0.445396	+	0.137386i	0.445396	+	0.137386i
$866$	0.116853	+	1.55929i	0.116853	+	1.55929i
$867$		0			0
$868$	−1.55929	−	1.24349i	−1.55929	−	1.24349i
$869$	−1.89415	+	1.51053i	−1.89415	+	1.51053i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.208129	−	0.530303i	−0.208129	−	0.530303i
$876$		0			0
$877$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$877$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$878$	0.997204	+	0.925270i	0.997204	+	0.925270i
$879$		0			0
$880$	−0.406813	−	0.159662i	−0.406813	−	0.159662i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	1.40097	+	1.29991i	1.40097	+	1.29991i
$887$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$887$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$888$		0			0
$889$	−0.109562	−	0.101659i	−0.109562	−	0.101659i
$890$		0			0
$891$		0			0
$892$	−0.332083	−	0.487076i	−0.332083	−	0.487076i
$893$		0			0
$894$		0			0
$895$	−0.290611	+	0.231754i	−0.290611	+	0.231754i
$896$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$897$		0			0
$898$		0			0
$899$	1.39254	+	0.429540i	1.39254	+	0.429540i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$907$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$908$	−1.29991	+	0.400969i	−1.29991	+	0.400969i
$909$		0			0
$910$		0			0
$911$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$911$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$912$		0			0
$913$	−2.36383	−	1.36476i	−2.36383	−	1.36476i
$914$	−0.215372	+	0.698220i	−0.215372	+	0.698220i
$915$		0			0
$916$		0			0
$917$	0.702449	+	0.880843i	0.702449	+	0.880843i
$918$		0			0
$919$	0.326239	−	0.302705i	0.326239	−	0.302705i	−0.500000	−	0.866025i	$-0.666667\pi$
$919$	0.326239	−	0.302705i	0.326239	−	0.302705i	0.826239	+	0.563320i	$0.190476\pi$
$920$		0			0
$921$		0			0
$922$	−0.129334	+	0.858075i	−0.129334	+	0.858075i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	0.702449	−	1.03030i	0.702449	−	1.03030i
$927$		0			0
$928$	0.0546039	−	0.728639i	0.0546039	−	0.728639i
$929$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$929$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−1.94440	−	0.145713i	−1.94440	−	0.145713i
$935$		0			0
$936$		0			0
$937$	−0.865341	+	1.79690i	−0.865341	+	1.79690i	−0.365341	+	0.930874i	$0.619048\pi$
$937$	−0.865341	+	1.79690i	−0.865341	+	1.79690i	−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.487076	−	0.332083i	0.487076	−	0.332083i	−0.294755	−	0.955573i	$-0.595238\pi$
$941$	0.487076	−	0.332083i	0.487076	−	0.332083i	0.781831	+	0.623490i	$0.214286\pi$
$942$		0			0
$943$		0			0
$944$	1.07992	+	1.35417i	1.07992	+	1.35417i
$945$		0			0
$946$		0			0
$947$	1.79690	−	0.134659i	1.79690	−	0.134659i	0.866025	−	0.500000i	$-0.166667\pi$
$947$	1.79690	−	0.134659i	1.79690	−	0.134659i	0.930874	+	0.365341i	$0.119048\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$953$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.48883	+	2.57873i	1.48883	+	2.57873i
$962$		0			0
$963$		0			0
$964$	−1.85654	+	0.139129i	−1.85654	+	0.139129i
$965$	0.307117	−	0.385113i	0.307117	−	0.385113i
$966$		0			0
$967$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.955573	+	0.294755i	$0.0952381\pi$
$967$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.0747301	+	0.997204i	$0.476190\pi$
$968$	−0.781831	−	0.842614i	−0.781831	−	0.842614i
$969$		0			0
$970$	0.145190	−	0.0989888i	0.145190	−	0.0989888i
$971$	1.97213	+	0.297251i	1.97213	+	0.297251i	0.997204	+	0.0747301i	$0.0238095\pi$
$971$	1.97213	+	0.297251i	1.97213	+	0.297251i	0.974928	+	0.222521i	$0.0714286\pi$
$972$		0			0
$973$		0			0
$974$	−0.0648483	+	0.134659i	−0.0648483	+	0.134659i
$975$		0			0
$976$		0			0
$977$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$977$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$978$		0			0
$979$		0			0
$980$	0.294755	+	0.0444272i	0.294755	+	0.0444272i
$981$		0			0
$982$	0.722521	−	1.84095i	0.722521	−	1.84095i
$983$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$983$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$984$		0			0
$985$	0.0747301	−	0.109609i	0.0747301	−	0.109609i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.44973	−	0.218511i	1.44973	−	0.218511i	0.623490	−	0.781831i	$-0.285714\pi$
$991$	1.44973	−	0.218511i	1.44973	−	0.218511i	0.826239	+	0.563320i	$0.190476\pi$
$992$	1.46200	−	1.35654i	1.46200	−	1.35654i
$993$		0			0
$994$		0			0
$995$	0.202235	+	0.161277i	0.202235	+	0.161277i
$996$		0			0
$997$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$997$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.1.fb.a.2845.2	yes	24
3.2	odd	2	inner	3528.1.fb.a.2845.1	yes	24
8.5	even	2	inner	3528.1.fb.a.2845.1	yes	24
24.5	odd	2	CM	3528.1.fb.a.2845.2	yes	24
49.33	odd	42	inner	3528.1.fb.a.1405.1	✓	24
147.131	even	42	inner	3528.1.fb.a.1405.2	yes	24
392.229	odd	42	inner	3528.1.fb.a.1405.2	yes	24
1176.1013	even	42	inner	3528.1.fb.a.1405.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.1.fb.a.1405.1	✓	24	49.33	odd	42	inner
3528.1.fb.a.1405.1	✓	24	1176.1013	even	42	inner
3528.1.fb.a.1405.2	yes	24	147.131	even	42	inner
3528.1.fb.a.1405.2	yes	24	392.229	odd	42	inner
3528.1.fb.a.2845.1	yes	24	3.2	odd	2	inner
3528.1.fb.a.2845.1	yes	24	8.5	even	2	inner
3528.1.fb.a.2845.2	yes	24	1.1	even	1	trivial
3528.1.fb.a.2845.2	yes	24	24.5	odd	2	CM

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3528.fb (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.930874 + 0.365341i) q^{2} +(0.733052 + 0.680173i) q^{4} +(0.246289 + 0.167917i) q^{5} +(0.826239 - 0.563320i) q^{7} +(0.433884 + 0.900969i) q^{8} +(0.167917 + 0.246289i) q^{10} +(0.218511 + 1.44973i) q^{11} +(0.974928 - 0.222521i) q^{14} +(0.0747301 + 0.997204i) q^{16} +(0.0663300 + 0.290611i) q^{20} +(-0.326239 + 1.42935i) q^{22} +(-0.332879 - 0.848162i) q^{25} +(0.988831 + 0.149042i) q^{28} +(-0.712362 + 0.162592i) q^{29} +(-1.72721 - 0.997204i) q^{31} +(-0.294755 + 0.955573i) q^{32} +0.298085 q^{35} +(-0.0444272 + 0.294755i) q^{40} +(-0.825886 + 1.21135i) q^{44} +(0.365341 - 0.930874i) q^{49} -0.911146i q^{50} +(0.680173 - 0.733052i) q^{53} +(-0.189617 + 0.393744i) q^{55} +(0.866025 + 0.500000i) q^{56} +(-0.722521 - 0.108903i) q^{58} +(1.43109 - 0.975699i) q^{59} +(-1.24349 - 1.55929i) q^{62} +(-0.623490 + 0.781831i) q^{64} +(0.277479 + 0.108903i) q^{70} +(-0.807782 + 0.317031i) q^{73} +(0.997204 + 1.07473i) q^{77} +(0.826239 + 1.43109i) q^{79} +(-0.149042 + 0.258149i) q^{80} +(-1.16078 + 1.45557i) q^{83} +(-1.21135 + 0.825886i) q^{88} -0.589510i q^{97} +(0.680173 - 0.733052i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{4} + 2 q^{7} + 6 q^{10} + 2 q^{16} + 10 q^{22} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 22 q^{40} + 2 q^{49} - 14 q^{55} - 16 q^{58} + 4 q^{64} + 8 q^{70} + 2 q^{79} - 2 q^{88}+O(q^{100}) \) 24 * q - 2 * q^4 + 2 * q^7 + 6 * q^10 + 2 * q^16 + 10 * q^22 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 22 * q^40 + 2 * q^49 - 14 * q^55 - 16 * q^58 + 4 * q^64 + 8 * q^70 + 2 * q^79 - 2 * q^88

Introduction

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