Properties

Label 495.1.bd.a.428.1
Level $495$
Weight $1$
Character 495.428
Analytic conductor $0.247$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,1,Mod(32,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.32"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 3, 6])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 495.bd (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.247037181253\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 428.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 495.428
Dual form 495.1.bd.a.362.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} +(-0.866025 - 0.500000i) q^{4} -1.00000i q^{5} +(0.500000 + 0.866025i) q^{9} +(0.866025 - 0.500000i) q^{11} +(-0.500000 - 0.866025i) q^{12} +(0.500000 - 0.866025i) q^{15} +(0.500000 + 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{20} +(-0.366025 - 1.36603i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +(-0.866025 + 1.50000i) q^{31} +1.00000 q^{33} -1.00000i q^{36} +(-1.36603 + 1.36603i) q^{37} -1.00000 q^{44} +(0.866025 - 0.500000i) q^{45} +(-0.133975 + 0.500000i) q^{47} +1.00000i q^{48} +(0.866025 + 0.500000i) q^{49} +(-1.36603 + 1.36603i) q^{53} +(-0.500000 - 0.866025i) q^{55} +(0.500000 - 0.866025i) q^{59} +(-0.866025 + 0.500000i) q^{60} -1.00000i q^{64} +(-0.133975 - 0.500000i) q^{67} +(0.366025 - 1.36603i) q^{69} -1.73205i q^{71} +(-0.866025 - 0.500000i) q^{75} +(0.866025 - 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.366025 + 1.36603i) q^{92} +(-1.50000 + 0.866025i) q^{93} +(0.500000 + 0.133975i) q^{97} +(0.866025 + 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 2 q^{20} + 2 q^{23} - 4 q^{25} + 4 q^{33} - 2 q^{37} - 4 q^{44} - 4 q^{47} - 2 q^{53} - 2 q^{55} + 2 q^{59} - 4 q^{67} - 2 q^{69} - 2 q^{81} + 2 q^{92} - 6 q^{93}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(4\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(5\) − 1.00000i − 1.00000i
\(6\) 0 0
\(7\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(12\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(13\) 0 0 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(14\) 0 0
\(15\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.366025 − 1.36603i −0.366025 − 1.36603i −0.866025 − 0.500000i \(-0.833333\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 1.00000 1.00000
\(34\) 0 0
\(35\) 0 0
\(36\) − 1.00000i − 1.00000i
\(37\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(44\) −1.00000 −1.00000
\(45\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(46\) 0 0
\(47\) −0.133975 + 0.500000i −0.133975 + 0.500000i 0.866025 + 0.500000i \(0.166667\pi\)
−1.00000 \(\pi\)
\(48\) 1.00000i 1.00000i
\(49\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(60\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) − 1.00000i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.133975 − 0.500000i −0.133975 − 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0.366025 − 1.36603i 0.366025 − 1.36603i
\(70\) 0 0
\(71\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(93\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.500000 + 0.133975i 0.500000 + 0.133975i 0.500000 − 0.866025i \(-0.333333\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0.866025 + 0.500000i 0.866025 + 0.500000i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 495.1.bd.a.428.1 yes 4
3.2 odd 2 1485.1.be.a.98.1 4
5.2 odd 4 495.1.bd.b.32.1 yes 4
5.3 odd 4 2475.1.cr.a.32.1 4
5.4 even 2 2475.1.cr.b.1418.1 4
9.2 odd 6 495.1.bd.b.263.1 yes 4
9.7 even 3 1485.1.be.b.1088.1 4
11.10 odd 2 CM 495.1.bd.a.428.1 yes 4
15.2 even 4 1485.1.be.b.692.1 4
33.32 even 2 1485.1.be.a.98.1 4
45.2 even 12 inner 495.1.bd.a.362.1 ✓ 4
45.7 odd 12 1485.1.be.a.197.1 4
45.29 odd 6 2475.1.cr.a.2243.1 4
45.38 even 12 2475.1.cr.b.857.1 4
55.32 even 4 495.1.bd.b.32.1 yes 4
55.43 even 4 2475.1.cr.a.32.1 4
55.54 odd 2 2475.1.cr.b.1418.1 4
99.43 odd 6 1485.1.be.b.1088.1 4
99.65 even 6 495.1.bd.b.263.1 yes 4
165.32 odd 4 1485.1.be.b.692.1 4
495.142 even 12 1485.1.be.a.197.1 4
495.164 even 6 2475.1.cr.a.2243.1 4
495.263 odd 12 2475.1.cr.b.857.1 4
495.362 odd 12 inner 495.1.bd.a.362.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.1.bd.a.362.1 ✓ 4 45.2 even 12 inner
495.1.bd.a.362.1 ✓ 4 495.362 odd 12 inner
495.1.bd.a.428.1 yes 4 1.1 even 1 trivial
495.1.bd.a.428.1 yes 4 11.10 odd 2 CM
495.1.bd.b.32.1 yes 4 5.2 odd 4
495.1.bd.b.32.1 yes 4 55.32 even 4
495.1.bd.b.263.1 yes 4 9.2 odd 6
495.1.bd.b.263.1 yes 4 99.65 even 6
1485.1.be.a.98.1 4 3.2 odd 2
1485.1.be.a.98.1 4 33.32 even 2
1485.1.be.a.197.1 4 45.7 odd 12
1485.1.be.a.197.1 4 495.142 even 12
1485.1.be.b.692.1 4 15.2 even 4
1485.1.be.b.692.1 4 165.32 odd 4
1485.1.be.b.1088.1 4 9.7 even 3
1485.1.be.b.1088.1 4 99.43 odd 6
2475.1.cr.a.32.1 4 5.3 odd 4
2475.1.cr.a.32.1 4 55.43 even 4
2475.1.cr.a.2243.1 4 45.29 odd 6
2475.1.cr.a.2243.1 4 495.164 even 6
2475.1.cr.b.857.1 4 45.38 even 12
2475.1.cr.b.857.1 4 495.263 odd 12
2475.1.cr.b.1418.1 4 5.4 even 2
2475.1.cr.b.1418.1 4 55.54 odd 2