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Properties

Label 285.1.n.b.254.1
Level $285$
Weight $1$
Character 285.254
Analytic conductor $0.142$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [285,1,Mod(239,285)] 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("285.239"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 285.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.142233528600\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.5415.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.1218375.1

Embedding invariants

Embedding label 254.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 285.254
Dual form 285.1.n.b.239.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{15} +(0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} -1.00000 q^{18} +(-0.500000 - 0.866025i) q^{19} +(-1.00000 + 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} -1.00000 q^{30} -1.00000 q^{31} +(-0.500000 + 0.866025i) q^{34} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +1.00000 q^{45} -2.00000 q^{46} +(0.500000 - 0.866025i) q^{47} +(0.500000 - 0.866025i) q^{48} +1.00000 q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{51} +(0.500000 - 0.866025i) q^{53} +(0.500000 + 0.866025i) q^{54} +(-0.500000 + 0.866025i) q^{57} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 - 0.866025i) q^{62} +1.00000 q^{64} +2.00000 q^{69} +(-0.500000 + 0.866025i) q^{72} +1.00000 q^{75} +(-1.00000 - 1.73205i) q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{83} +(0.500000 - 0.866025i) q^{85} +(0.500000 + 0.866025i) q^{90} +(0.500000 + 0.866025i) q^{93} +1.00000 q^{94} +(-0.500000 + 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{5} + q^{6} + 2 q^{8} - q^{9} + q^{10} - q^{15} + q^{16} + q^{17} - 2 q^{18} - q^{19} - 2 q^{23} - q^{24} - q^{25} + 2 q^{27} - 2 q^{30} - 2 q^{31} - q^{34} + q^{38}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(4\) 0 0
\(5\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(6\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000 1.00000
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) −1.00000 −1.00000
\(19\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) −1.00000 −1.00000
\(31\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(39\) 0 0
\(40\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(41\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 1.00000 1.00000
\(46\) −2.00000 −2.00000
\(47\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(48\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(49\) 1.00000 1.00000
\(50\) −1.00000 −1.00000
\(51\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(52\) 0 0
\(53\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(54\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 2.00000 2.00000
\(70\) 0 0
\(71\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(73\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(80\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(81\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(91\) 0 0
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(94\) 1.00000 1.00000
\(95\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(96\) 0 0
\(97\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(99\) 0 0
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 285.1.n.b.254.1 yes 2
3.2 odd 2 285.1.n.a.254.1 yes 2
5.2 odd 4 1425.1.t.b.26.1 4
5.3 odd 4 1425.1.t.b.26.2 4
5.4 even 2 285.1.n.a.254.1 yes 2
15.2 even 4 1425.1.t.b.26.2 4
15.8 even 4 1425.1.t.b.26.1 4
15.14 odd 2 CM 285.1.n.b.254.1 yes 2
19.11 even 3 inner 285.1.n.b.239.1 yes 2
57.11 odd 6 285.1.n.a.239.1 ✓ 2
95.49 even 6 285.1.n.a.239.1 ✓ 2
95.68 odd 12 1425.1.t.b.1151.1 4
95.87 odd 12 1425.1.t.b.1151.2 4
285.68 even 12 1425.1.t.b.1151.2 4
285.182 even 12 1425.1.t.b.1151.1 4
285.239 odd 6 inner 285.1.n.b.239.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.1.n.a.239.1 ✓ 2 57.11 odd 6
285.1.n.a.239.1 ✓ 2 95.49 even 6
285.1.n.a.254.1 yes 2 3.2 odd 2
285.1.n.a.254.1 yes 2 5.4 even 2
285.1.n.b.239.1 yes 2 19.11 even 3 inner
285.1.n.b.239.1 yes 2 285.239 odd 6 inner
285.1.n.b.254.1 yes 2 1.1 even 1 trivial
285.1.n.b.254.1 yes 2 15.14 odd 2 CM
1425.1.t.b.26.1 4 5.2 odd 4
1425.1.t.b.26.1 4 15.8 even 4
1425.1.t.b.26.2 4 5.3 odd 4
1425.1.t.b.26.2 4 15.2 even 4
1425.1.t.b.1151.1 4 95.68 odd 12
1425.1.t.b.1151.1 4 285.182 even 12
1425.1.t.b.1151.2 4 95.87 odd 12
1425.1.t.b.1151.2 4 285.68 even 12