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(26,1425)] 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("1425.26"); 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, 0, 2])) 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.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.711167643002\)
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: \(\Z[a_1, a_2, a_3]\)
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: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 1151.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1425.1151
Dual form 1425.1.t.a.26.1

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

Character values

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

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

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 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(4\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −1.00000
\(13\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(29\) 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
\(37\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 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
\(43\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−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
\(63\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(68\) 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
\(73\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(74\) 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 \(-0.666667\pi\)
1.00000 \(0\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −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
\(93\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 − 1.73205i 1.00000 − 1.73205i 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(98\) 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.t.a.1151.1 2
3.2 odd 2 CM 1425.1.t.a.1151.1 2
5.2 odd 4 1425.1.o.a.524.1 4
5.3 odd 4 1425.1.o.a.524.2 4
5.4 even 2 57.1.h.a.11.1 ✓ 2
15.2 even 4 1425.1.o.a.524.1 4
15.8 even 4 1425.1.o.a.524.2 4
15.14 odd 2 57.1.h.a.11.1 ✓ 2
19.7 even 3 inner 1425.1.t.a.26.1 2
20.19 odd 2 912.1.bl.a.353.1 2
35.4 even 6 2793.1.bi.b.1892.1 2
35.9 even 6 2793.1.n.a.410.1 2
35.19 odd 6 2793.1.n.b.410.1 2
35.24 odd 6 2793.1.bi.a.1892.1 2
35.34 odd 2 2793.1.bf.a.638.1 2
40.19 odd 2 3648.1.bl.a.2177.1 2
40.29 even 2 3648.1.bl.b.2177.1 2
45.4 even 6 1539.1.n.a.1322.1 2
45.14 odd 6 1539.1.n.a.1322.1 2
45.29 odd 6 1539.1.j.a.296.1 2
45.34 even 6 1539.1.j.a.296.1 2
57.26 odd 6 inner 1425.1.t.a.26.1 2
60.59 even 2 912.1.bl.a.353.1 2
95.4 even 18 1083.1.l.a.62.1 6
95.7 odd 12 1425.1.o.a.824.2 4
95.9 even 18 1083.1.l.a.245.1 6
95.14 odd 18 1083.1.l.b.389.1 6
95.24 even 18 1083.1.l.a.389.1 6
95.29 odd 18 1083.1.l.b.245.1 6
95.34 odd 18 1083.1.l.b.62.1 6
95.44 even 18 1083.1.l.a.776.1 6
95.49 even 6 1083.1.b.b.362.1 1
95.54 even 18 1083.1.l.a.821.1 6
95.59 odd 18 1083.1.l.b.956.1 6
95.64 even 6 57.1.h.a.26.1 yes 2
95.69 odd 6 1083.1.h.a.653.1 2
95.74 even 18 1083.1.l.a.956.1 6
95.79 odd 18 1083.1.l.b.821.1 6
95.83 odd 12 1425.1.o.a.824.1 4
95.84 odd 6 1083.1.b.a.362.1 1
95.89 odd 18 1083.1.l.b.776.1 6
95.94 odd 2 1083.1.h.a.68.1 2
105.44 odd 6 2793.1.n.a.410.1 2
105.59 even 6 2793.1.bi.a.1892.1 2
105.74 odd 6 2793.1.bi.b.1892.1 2
105.89 even 6 2793.1.n.b.410.1 2
105.104 even 2 2793.1.bf.a.638.1 2
120.29 odd 2 3648.1.bl.b.2177.1 2
120.59 even 2 3648.1.bl.a.2177.1 2
285.14 even 18 1083.1.l.b.389.1 6
285.29 even 18 1083.1.l.b.245.1 6
285.44 odd 18 1083.1.l.a.776.1 6
285.59 even 18 1083.1.l.b.956.1 6
285.74 odd 18 1083.1.l.a.956.1 6
285.83 even 12 1425.1.o.a.824.1 4
285.89 even 18 1083.1.l.b.776.1 6
285.104 odd 18 1083.1.l.a.245.1 6
285.119 odd 18 1083.1.l.a.389.1 6
285.149 odd 18 1083.1.l.a.821.1 6
285.164 even 6 1083.1.h.a.653.1 2
285.179 even 6 1083.1.b.a.362.1 1
285.194 odd 18 1083.1.l.a.62.1 6
285.197 even 12 1425.1.o.a.824.2 4
285.224 even 18 1083.1.l.b.62.1 6
285.239 odd 6 1083.1.b.b.362.1 1
285.254 odd 6 57.1.h.a.26.1 yes 2
285.269 even 18 1083.1.l.b.821.1 6
285.284 even 2 1083.1.h.a.68.1 2
380.159 odd 6 912.1.bl.a.881.1 2
665.159 odd 6 2793.1.bi.a.2762.1 2
665.254 even 6 2793.1.bi.b.2762.1 2
665.349 odd 6 2793.1.bf.a.197.1 2
665.444 odd 6 2793.1.n.b.1451.1 2
665.634 even 6 2793.1.n.a.1451.1 2
760.349 even 6 3648.1.bl.b.1793.1 2
760.539 odd 6 3648.1.bl.a.1793.1 2
855.254 odd 6 1539.1.n.a.539.1 2
855.349 even 6 1539.1.n.a.539.1 2
855.634 even 6 1539.1.j.a.26.1 2
855.824 odd 6 1539.1.j.a.26.1 2
1140.539 even 6 912.1.bl.a.881.1 2
1995.254 odd 6 2793.1.bi.b.2762.1 2
1995.824 even 6 2793.1.bi.a.2762.1 2
1995.1109 even 6 2793.1.n.b.1451.1 2
1995.1679 even 6 2793.1.bf.a.197.1 2
1995.1964 odd 6 2793.1.n.a.1451.1 2
2280.539 even 6 3648.1.bl.a.1793.1 2
2280.1109 odd 6 3648.1.bl.b.1793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 5.4 even 2
57.1.h.a.11.1 ✓ 2 15.14 odd 2
57.1.h.a.26.1 yes 2 95.64 even 6
57.1.h.a.26.1 yes 2 285.254 odd 6
912.1.bl.a.353.1 2 20.19 odd 2
912.1.bl.a.353.1 2 60.59 even 2
912.1.bl.a.881.1 2 380.159 odd 6
912.1.bl.a.881.1 2 1140.539 even 6
1083.1.b.a.362.1 1 95.84 odd 6
1083.1.b.a.362.1 1 285.179 even 6
1083.1.b.b.362.1 1 95.49 even 6
1083.1.b.b.362.1 1 285.239 odd 6
1083.1.h.a.68.1 2 95.94 odd 2
1083.1.h.a.68.1 2 285.284 even 2
1083.1.h.a.653.1 2 95.69 odd 6
1083.1.h.a.653.1 2 285.164 even 6
1083.1.l.a.62.1 6 95.4 even 18
1083.1.l.a.62.1 6 285.194 odd 18
1083.1.l.a.245.1 6 95.9 even 18
1083.1.l.a.245.1 6 285.104 odd 18
1083.1.l.a.389.1 6 95.24 even 18
1083.1.l.a.389.1 6 285.119 odd 18
1083.1.l.a.776.1 6 95.44 even 18
1083.1.l.a.776.1 6 285.44 odd 18
1083.1.l.a.821.1 6 95.54 even 18
1083.1.l.a.821.1 6 285.149 odd 18
1083.1.l.a.956.1 6 95.74 even 18
1083.1.l.a.956.1 6 285.74 odd 18
1083.1.l.b.62.1 6 95.34 odd 18
1083.1.l.b.62.1 6 285.224 even 18
1083.1.l.b.245.1 6 95.29 odd 18
1083.1.l.b.245.1 6 285.29 even 18
1083.1.l.b.389.1 6 95.14 odd 18
1083.1.l.b.389.1 6 285.14 even 18
1083.1.l.b.776.1 6 95.89 odd 18
1083.1.l.b.776.1 6 285.89 even 18
1083.1.l.b.821.1 6 95.79 odd 18
1083.1.l.b.821.1 6 285.269 even 18
1083.1.l.b.956.1 6 95.59 odd 18
1083.1.l.b.956.1 6 285.59 even 18
1425.1.o.a.524.1 4 5.2 odd 4
1425.1.o.a.524.1 4 15.2 even 4
1425.1.o.a.524.2 4 5.3 odd 4
1425.1.o.a.524.2 4 15.8 even 4
1425.1.o.a.824.1 4 95.83 odd 12
1425.1.o.a.824.1 4 285.83 even 12
1425.1.o.a.824.2 4 95.7 odd 12
1425.1.o.a.824.2 4 285.197 even 12
1425.1.t.a.26.1 2 19.7 even 3 inner
1425.1.t.a.26.1 2 57.26 odd 6 inner
1425.1.t.a.1151.1 2 1.1 even 1 trivial
1425.1.t.a.1151.1 2 3.2 odd 2 CM
1539.1.j.a.26.1 2 855.634 even 6
1539.1.j.a.26.1 2 855.824 odd 6
1539.1.j.a.296.1 2 45.29 odd 6
1539.1.j.a.296.1 2 45.34 even 6
1539.1.n.a.539.1 2 855.254 odd 6
1539.1.n.a.539.1 2 855.349 even 6
1539.1.n.a.1322.1 2 45.4 even 6
1539.1.n.a.1322.1 2 45.14 odd 6
2793.1.n.a.410.1 2 35.9 even 6
2793.1.n.a.410.1 2 105.44 odd 6
2793.1.n.a.1451.1 2 665.634 even 6
2793.1.n.a.1451.1 2 1995.1964 odd 6
2793.1.n.b.410.1 2 35.19 odd 6
2793.1.n.b.410.1 2 105.89 even 6
2793.1.n.b.1451.1 2 665.444 odd 6
2793.1.n.b.1451.1 2 1995.1109 even 6
2793.1.bf.a.197.1 2 665.349 odd 6
2793.1.bf.a.197.1 2 1995.1679 even 6
2793.1.bf.a.638.1 2 35.34 odd 2
2793.1.bf.a.638.1 2 105.104 even 2
2793.1.bi.a.1892.1 2 35.24 odd 6
2793.1.bi.a.1892.1 2 105.59 even 6
2793.1.bi.a.2762.1 2 665.159 odd 6
2793.1.bi.a.2762.1 2 1995.824 even 6
2793.1.bi.b.1892.1 2 35.4 even 6
2793.1.bi.b.1892.1 2 105.74 odd 6
2793.1.bi.b.2762.1 2 665.254 even 6
2793.1.bi.b.2762.1 2 1995.254 odd 6
3648.1.bl.a.1793.1 2 760.539 odd 6
3648.1.bl.a.1793.1 2 2280.539 even 6
3648.1.bl.a.2177.1 2 40.19 odd 2
3648.1.bl.a.2177.1 2 120.59 even 2
3648.1.bl.b.1793.1 2 760.349 even 6
3648.1.bl.b.1793.1 2 2280.1109 odd 6
3648.1.bl.b.2177.1 2 40.29 even 2
3648.1.bl.b.2177.1 2 120.29 odd 2