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Properties

Label 1425.1.o.a.524.1
Level $1425$
Weight $1$
Character 1425.524
Analytic conductor $0.711$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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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] = [1425,1,Mod(524,1425)] 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("1425.524"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, 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: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1425.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.711167643002\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: no (minimal twist has level 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of \(\Q(\sqrt[3]{19})\)
Artin image: $S_3\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 524.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.524
Dual form 1425.1.o.a.824.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{7} +(0.500000 + 0.866025i) q^{9} -1.00000i q^{12} +(-0.866025 + 0.500000i) q^{13} +(-0.500000 + 0.866025i) q^{16} -1.00000 q^{19} +(0.500000 - 0.866025i) q^{21} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} -1.00000 q^{31} +(-0.500000 + 0.866025i) q^{36} +1.00000i q^{37} +1.00000 q^{39} +(0.866025 + 0.500000i) q^{43} +(0.866025 - 0.500000i) q^{48} +(-0.866025 - 0.500000i) q^{52} +(0.866025 + 0.500000i) q^{57} +(0.500000 + 0.866025i) q^{61} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(0.866025 - 0.500000i) q^{67} +(0.866025 + 0.500000i) q^{73} +(-0.500000 - 0.866025i) q^{76} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +1.00000 q^{84} +(-0.500000 - 0.866025i) q^{91} +(0.866025 + 0.500000i) q^{93} +(1.73205 + 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{16} - 4 q^{19} + 2 q^{21} - 4 q^{31} - 2 q^{36} + 4 q^{39} + 2 q^{61} - 4 q^{64} - 2 q^{76} - 2 q^{79} - 2 q^{81} + 4 q^{84} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 1.00000i − 1.00000i
\(13\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 1.00000i
\(28\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −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 0
\(35\) 0 0
\(36\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(37\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(38\) 0 0
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(53\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(92\) 0 0
\(93\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(98\) 0 0
\(99\) 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 1425.1.o.a.524.1 4
3.2 odd 2 CM 1425.1.o.a.524.1 4
5.2 odd 4 57.1.h.a.11.1 ✓ 2
5.3 odd 4 1425.1.t.a.1151.1 2
5.4 even 2 inner 1425.1.o.a.524.2 4
15.2 even 4 57.1.h.a.11.1 ✓ 2
15.8 even 4 1425.1.t.a.1151.1 2
15.14 odd 2 inner 1425.1.o.a.524.2 4
19.7 even 3 inner 1425.1.o.a.824.2 4
20.7 even 4 912.1.bl.a.353.1 2
35.2 odd 12 2793.1.n.a.410.1 2
35.12 even 12 2793.1.n.b.410.1 2
35.17 even 12 2793.1.bi.a.1892.1 2
35.27 even 4 2793.1.bf.a.638.1 2
35.32 odd 12 2793.1.bi.b.1892.1 2
40.27 even 4 3648.1.bl.a.2177.1 2
40.37 odd 4 3648.1.bl.b.2177.1 2
45.2 even 12 1539.1.j.a.296.1 2
45.7 odd 12 1539.1.j.a.296.1 2
45.22 odd 12 1539.1.n.a.1322.1 2
45.32 even 12 1539.1.n.a.1322.1 2
57.26 odd 6 inner 1425.1.o.a.824.2 4
60.47 odd 4 912.1.bl.a.353.1 2
95.2 even 36 1083.1.l.b.956.1 6
95.7 odd 12 57.1.h.a.26.1 yes 2
95.12 even 12 1083.1.h.a.653.1 2
95.17 odd 36 1083.1.l.a.956.1 6
95.22 even 36 1083.1.l.b.821.1 6
95.27 even 12 1083.1.b.a.362.1 1
95.32 even 36 1083.1.l.b.776.1 6
95.37 even 4 1083.1.h.a.68.1 2
95.42 odd 36 1083.1.l.a.62.1 6
95.47 odd 36 1083.1.l.a.245.1 6
95.52 even 36 1083.1.l.b.389.1 6
95.62 odd 36 1083.1.l.a.389.1 6
95.64 even 6 inner 1425.1.o.a.824.1 4
95.67 even 36 1083.1.l.b.245.1 6
95.72 even 36 1083.1.l.b.62.1 6
95.82 odd 36 1083.1.l.a.776.1 6
95.83 odd 12 1425.1.t.a.26.1 2
95.87 odd 12 1083.1.b.b.362.1 1
95.92 odd 36 1083.1.l.a.821.1 6
105.2 even 12 2793.1.n.a.410.1 2
105.17 odd 12 2793.1.bi.a.1892.1 2
105.32 even 12 2793.1.bi.b.1892.1 2
105.47 odd 12 2793.1.n.b.410.1 2
105.62 odd 4 2793.1.bf.a.638.1 2
120.77 even 4 3648.1.bl.b.2177.1 2
120.107 odd 4 3648.1.bl.a.2177.1 2
285.2 odd 36 1083.1.l.b.956.1 6
285.17 even 36 1083.1.l.a.956.1 6
285.32 odd 36 1083.1.l.b.776.1 6
285.47 even 36 1083.1.l.a.245.1 6
285.62 even 36 1083.1.l.a.389.1 6
285.83 even 12 1425.1.t.a.26.1 2
285.92 even 36 1083.1.l.a.821.1 6
285.107 odd 12 1083.1.h.a.653.1 2
285.122 odd 12 1083.1.b.a.362.1 1
285.137 even 36 1083.1.l.a.62.1 6
285.167 odd 36 1083.1.l.b.62.1 6
285.182 even 12 1083.1.b.b.362.1 1
285.197 even 12 57.1.h.a.26.1 yes 2
285.212 odd 36 1083.1.l.b.821.1 6
285.227 odd 4 1083.1.h.a.68.1 2
285.242 odd 36 1083.1.l.b.389.1 6
285.254 odd 6 inner 1425.1.o.a.824.1 4
285.257 odd 36 1083.1.l.b.245.1 6
285.272 even 36 1083.1.l.a.776.1 6
380.7 even 12 912.1.bl.a.881.1 2
665.102 odd 12 2793.1.n.a.1451.1 2
665.292 even 12 2793.1.bi.a.2762.1 2
665.387 odd 12 2793.1.bi.b.2762.1 2
665.482 even 12 2793.1.bf.a.197.1 2
665.577 even 12 2793.1.n.b.1451.1 2
760.197 odd 12 3648.1.bl.b.1793.1 2
760.387 even 12 3648.1.bl.a.1793.1 2
855.7 odd 12 1539.1.n.a.539.1 2
855.292 odd 12 1539.1.j.a.26.1 2
855.482 even 12 1539.1.j.a.26.1 2
855.767 even 12 1539.1.n.a.539.1 2
1140.767 odd 12 912.1.bl.a.881.1 2
1995.482 odd 12 2793.1.bf.a.197.1 2
1995.767 even 12 2793.1.n.a.1451.1 2
1995.1052 even 12 2793.1.bi.b.2762.1 2
1995.1622 odd 12 2793.1.bi.a.2762.1 2
1995.1907 odd 12 2793.1.n.b.1451.1 2
2280.197 even 12 3648.1.bl.b.1793.1 2
2280.1907 odd 12 3648.1.bl.a.1793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 5.2 odd 4
57.1.h.a.11.1 ✓ 2 15.2 even 4
57.1.h.a.26.1 yes 2 95.7 odd 12
57.1.h.a.26.1 yes 2 285.197 even 12
912.1.bl.a.353.1 2 20.7 even 4
912.1.bl.a.353.1 2 60.47 odd 4
912.1.bl.a.881.1 2 380.7 even 12
912.1.bl.a.881.1 2 1140.767 odd 12
1083.1.b.a.362.1 1 95.27 even 12
1083.1.b.a.362.1 1 285.122 odd 12
1083.1.b.b.362.1 1 95.87 odd 12
1083.1.b.b.362.1 1 285.182 even 12
1083.1.h.a.68.1 2 95.37 even 4
1083.1.h.a.68.1 2 285.227 odd 4
1083.1.h.a.653.1 2 95.12 even 12
1083.1.h.a.653.1 2 285.107 odd 12
1083.1.l.a.62.1 6 95.42 odd 36
1083.1.l.a.62.1 6 285.137 even 36
1083.1.l.a.245.1 6 95.47 odd 36
1083.1.l.a.245.1 6 285.47 even 36
1083.1.l.a.389.1 6 95.62 odd 36
1083.1.l.a.389.1 6 285.62 even 36
1083.1.l.a.776.1 6 95.82 odd 36
1083.1.l.a.776.1 6 285.272 even 36
1083.1.l.a.821.1 6 95.92 odd 36
1083.1.l.a.821.1 6 285.92 even 36
1083.1.l.a.956.1 6 95.17 odd 36
1083.1.l.a.956.1 6 285.17 even 36
1083.1.l.b.62.1 6 95.72 even 36
1083.1.l.b.62.1 6 285.167 odd 36
1083.1.l.b.245.1 6 95.67 even 36
1083.1.l.b.245.1 6 285.257 odd 36
1083.1.l.b.389.1 6 95.52 even 36
1083.1.l.b.389.1 6 285.242 odd 36
1083.1.l.b.776.1 6 95.32 even 36
1083.1.l.b.776.1 6 285.32 odd 36
1083.1.l.b.821.1 6 95.22 even 36
1083.1.l.b.821.1 6 285.212 odd 36
1083.1.l.b.956.1 6 95.2 even 36
1083.1.l.b.956.1 6 285.2 odd 36
1425.1.o.a.524.1 4 1.1 even 1 trivial
1425.1.o.a.524.1 4 3.2 odd 2 CM
1425.1.o.a.524.2 4 5.4 even 2 inner
1425.1.o.a.524.2 4 15.14 odd 2 inner
1425.1.o.a.824.1 4 95.64 even 6 inner
1425.1.o.a.824.1 4 285.254 odd 6 inner
1425.1.o.a.824.2 4 19.7 even 3 inner
1425.1.o.a.824.2 4 57.26 odd 6 inner
1425.1.t.a.26.1 2 95.83 odd 12
1425.1.t.a.26.1 2 285.83 even 12
1425.1.t.a.1151.1 2 5.3 odd 4
1425.1.t.a.1151.1 2 15.8 even 4
1539.1.j.a.26.1 2 855.292 odd 12
1539.1.j.a.26.1 2 855.482 even 12
1539.1.j.a.296.1 2 45.2 even 12
1539.1.j.a.296.1 2 45.7 odd 12
1539.1.n.a.539.1 2 855.7 odd 12
1539.1.n.a.539.1 2 855.767 even 12
1539.1.n.a.1322.1 2 45.22 odd 12
1539.1.n.a.1322.1 2 45.32 even 12
2793.1.n.a.410.1 2 35.2 odd 12
2793.1.n.a.410.1 2 105.2 even 12
2793.1.n.a.1451.1 2 665.102 odd 12
2793.1.n.a.1451.1 2 1995.767 even 12
2793.1.n.b.410.1 2 35.12 even 12
2793.1.n.b.410.1 2 105.47 odd 12
2793.1.n.b.1451.1 2 665.577 even 12
2793.1.n.b.1451.1 2 1995.1907 odd 12
2793.1.bf.a.197.1 2 665.482 even 12
2793.1.bf.a.197.1 2 1995.482 odd 12
2793.1.bf.a.638.1 2 35.27 even 4
2793.1.bf.a.638.1 2 105.62 odd 4
2793.1.bi.a.1892.1 2 35.17 even 12
2793.1.bi.a.1892.1 2 105.17 odd 12
2793.1.bi.a.2762.1 2 665.292 even 12
2793.1.bi.a.2762.1 2 1995.1622 odd 12
2793.1.bi.b.1892.1 2 35.32 odd 12
2793.1.bi.b.1892.1 2 105.32 even 12
2793.1.bi.b.2762.1 2 665.387 odd 12
2793.1.bi.b.2762.1 2 1995.1052 even 12
3648.1.bl.a.1793.1 2 760.387 even 12
3648.1.bl.a.1793.1 2 2280.1907 odd 12
3648.1.bl.a.2177.1 2 40.27 even 4
3648.1.bl.a.2177.1 2 120.107 odd 4
3648.1.bl.b.1793.1 2 760.197 odd 12
3648.1.bl.b.1793.1 2 2280.197 even 12
3648.1.bl.b.2177.1 2 40.37 odd 4
3648.1.bl.b.2177.1 2 120.77 even 4