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Properties

Label 2793.1.n.a.410.1
Level $2793$
Weight $1$
Character 2793.410
Analytic conductor $1.394$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
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] = [2793,1,Mod(410,2793)] 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("2793.410"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2793, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2793.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1] 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: \(1.39388858028\)
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: $C_3\times S_3$
Artin field: Galois closure of 6.0.23402547.3

Embedding invariants

Embedding label 410.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2793.410
Dual form 2793.1.n.a.1451.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +1.00000 q^{4} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{12} +(0.500000 + 0.866025i) q^{13} +1.00000 q^{16} +(-0.500000 - 0.866025i) q^{19} +1.00000 q^{25} +1.00000 q^{27} +(0.500000 - 0.866025i) q^{31} +(-0.500000 + 0.866025i) q^{36} +(0.500000 + 0.866025i) q^{37} +(0.500000 - 0.866025i) q^{39} +(0.500000 - 0.866025i) q^{43} +(-0.500000 - 0.866025i) q^{48} +(0.500000 + 0.866025i) q^{52} +(-0.500000 + 0.866025i) q^{57} +(0.500000 - 0.866025i) q^{61} +1.00000 q^{64} -1.00000 q^{67} +(0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(-0.500000 - 0.866025i) q^{76} -1.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{93} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{4} - q^{9} - q^{12} + q^{13} + 2 q^{16} - q^{19} + 2 q^{25} + 2 q^{27} + q^{31} - q^{36} + q^{37} + q^{39} + q^{43} - q^{48} + q^{52} - q^{57} + q^{61} + 2 q^{64} - 2 q^{67}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times\).

\(n\) \(932\) \(2110\) \(2206\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(4\) 1.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(37\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(40\) 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 \(-0.666667\pi\)
1.00000 \(0\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −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 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 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 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(76\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −1.00000
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 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 2793.1.n.a.410.1 2
3.2 odd 2 CM 2793.1.n.a.410.1 2
7.2 even 3 2793.1.bi.b.1892.1 2
7.3 odd 6 2793.1.bf.a.638.1 2
7.4 even 3 57.1.h.a.11.1 ✓ 2
7.5 odd 6 2793.1.bi.a.1892.1 2
7.6 odd 2 2793.1.n.b.410.1 2
19.7 even 3 2793.1.bi.b.2762.1 2
21.2 odd 6 2793.1.bi.b.1892.1 2
21.5 even 6 2793.1.bi.a.1892.1 2
21.11 odd 6 57.1.h.a.11.1 ✓ 2
21.17 even 6 2793.1.bf.a.638.1 2
21.20 even 2 2793.1.n.b.410.1 2
28.11 odd 6 912.1.bl.a.353.1 2
35.4 even 6 1425.1.t.a.1151.1 2
35.18 odd 12 1425.1.o.a.524.1 4
35.32 odd 12 1425.1.o.a.524.2 4
56.11 odd 6 3648.1.bl.a.2177.1 2
56.53 even 6 3648.1.bl.b.2177.1 2
57.26 odd 6 2793.1.bi.b.2762.1 2
63.4 even 3 1539.1.n.a.1322.1 2
63.11 odd 6 1539.1.j.a.296.1 2
63.25 even 3 1539.1.j.a.296.1 2
63.32 odd 6 1539.1.n.a.1322.1 2
84.11 even 6 912.1.bl.a.353.1 2
105.32 even 12 1425.1.o.a.524.2 4
105.53 even 12 1425.1.o.a.524.1 4
105.74 odd 6 1425.1.t.a.1151.1 2
133.4 even 9 1083.1.l.a.62.1 6
133.11 even 3 1083.1.b.b.362.1 1
133.18 odd 6 1083.1.h.a.68.1 2
133.25 even 9 1083.1.l.a.776.1 6
133.26 odd 6 2793.1.n.b.1451.1 2
133.32 odd 18 1083.1.l.b.776.1 6
133.45 odd 6 2793.1.bf.a.197.1 2
133.46 odd 6 1083.1.b.a.362.1 1
133.53 odd 18 1083.1.l.b.62.1 6
133.60 odd 18 1083.1.l.b.821.1 6
133.67 odd 18 1083.1.l.b.245.1 6
133.74 even 9 1083.1.l.a.956.1 6
133.81 even 9 1083.1.l.a.389.1 6
133.83 odd 6 2793.1.bi.a.2762.1 2
133.88 odd 6 1083.1.h.a.653.1 2
133.102 even 3 57.1.h.a.26.1 yes 2
133.109 odd 18 1083.1.l.b.389.1 6
133.116 odd 18 1083.1.l.b.956.1 6
133.121 even 3 inner 2793.1.n.a.1451.1 2
133.123 even 9 1083.1.l.a.245.1 6
133.130 even 9 1083.1.l.a.821.1 6
168.11 even 6 3648.1.bl.a.2177.1 2
168.53 odd 6 3648.1.bl.b.2177.1 2
399.11 odd 6 1083.1.b.b.362.1 1
399.26 even 6 2793.1.n.b.1451.1 2
399.32 even 18 1083.1.l.b.776.1 6
399.53 even 18 1083.1.l.b.62.1 6
399.74 odd 18 1083.1.l.a.956.1 6
399.83 even 6 2793.1.bi.a.2762.1 2
399.116 even 18 1083.1.l.b.956.1 6
399.137 odd 18 1083.1.l.a.62.1 6
399.158 odd 18 1083.1.l.a.776.1 6
399.179 even 6 1083.1.b.a.362.1 1
399.200 even 18 1083.1.l.b.245.1 6
399.221 even 6 1083.1.h.a.653.1 2
399.242 even 18 1083.1.l.b.389.1 6
399.254 odd 6 inner 2793.1.n.a.1451.1 2
399.263 odd 18 1083.1.l.a.821.1 6
399.284 even 6 1083.1.h.a.68.1 2
399.311 even 6 2793.1.bf.a.197.1 2
399.326 even 18 1083.1.l.b.821.1 6
399.347 odd 18 1083.1.l.a.389.1 6
399.368 odd 6 57.1.h.a.26.1 yes 2
399.389 odd 18 1083.1.l.a.245.1 6
532.235 odd 6 912.1.bl.a.881.1 2
665.102 odd 12 1425.1.o.a.824.1 4
665.368 odd 12 1425.1.o.a.824.2 4
665.634 even 6 1425.1.t.a.26.1 2
1064.235 odd 6 3648.1.bl.a.1793.1 2
1064.501 even 6 3648.1.bl.b.1793.1 2
1197.634 even 3 1539.1.j.a.26.1 2
1197.767 odd 6 1539.1.n.a.539.1 2
1197.1033 even 3 1539.1.n.a.539.1 2
1197.1166 odd 6 1539.1.j.a.26.1 2
1596.767 even 6 912.1.bl.a.881.1 2
1995.368 even 12 1425.1.o.a.824.2 4
1995.767 even 12 1425.1.o.a.824.1 4
1995.1964 odd 6 1425.1.t.a.26.1 2
3192.1565 odd 6 3648.1.bl.b.1793.1 2
3192.2363 even 6 3648.1.bl.a.1793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 7.4 even 3
57.1.h.a.11.1 ✓ 2 21.11 odd 6
57.1.h.a.26.1 yes 2 133.102 even 3
57.1.h.a.26.1 yes 2 399.368 odd 6
912.1.bl.a.353.1 2 28.11 odd 6
912.1.bl.a.353.1 2 84.11 even 6
912.1.bl.a.881.1 2 532.235 odd 6
912.1.bl.a.881.1 2 1596.767 even 6
1083.1.b.a.362.1 1 133.46 odd 6
1083.1.b.a.362.1 1 399.179 even 6
1083.1.b.b.362.1 1 133.11 even 3
1083.1.b.b.362.1 1 399.11 odd 6
1083.1.h.a.68.1 2 133.18 odd 6
1083.1.h.a.68.1 2 399.284 even 6
1083.1.h.a.653.1 2 133.88 odd 6
1083.1.h.a.653.1 2 399.221 even 6
1083.1.l.a.62.1 6 133.4 even 9
1083.1.l.a.62.1 6 399.137 odd 18
1083.1.l.a.245.1 6 133.123 even 9
1083.1.l.a.245.1 6 399.389 odd 18
1083.1.l.a.389.1 6 133.81 even 9
1083.1.l.a.389.1 6 399.347 odd 18
1083.1.l.a.776.1 6 133.25 even 9
1083.1.l.a.776.1 6 399.158 odd 18
1083.1.l.a.821.1 6 133.130 even 9
1083.1.l.a.821.1 6 399.263 odd 18
1083.1.l.a.956.1 6 133.74 even 9
1083.1.l.a.956.1 6 399.74 odd 18
1083.1.l.b.62.1 6 133.53 odd 18
1083.1.l.b.62.1 6 399.53 even 18
1083.1.l.b.245.1 6 133.67 odd 18
1083.1.l.b.245.1 6 399.200 even 18
1083.1.l.b.389.1 6 133.109 odd 18
1083.1.l.b.389.1 6 399.242 even 18
1083.1.l.b.776.1 6 133.32 odd 18
1083.1.l.b.776.1 6 399.32 even 18
1083.1.l.b.821.1 6 133.60 odd 18
1083.1.l.b.821.1 6 399.326 even 18
1083.1.l.b.956.1 6 133.116 odd 18
1083.1.l.b.956.1 6 399.116 even 18
1425.1.o.a.524.1 4 35.18 odd 12
1425.1.o.a.524.1 4 105.53 even 12
1425.1.o.a.524.2 4 35.32 odd 12
1425.1.o.a.524.2 4 105.32 even 12
1425.1.o.a.824.1 4 665.102 odd 12
1425.1.o.a.824.1 4 1995.767 even 12
1425.1.o.a.824.2 4 665.368 odd 12
1425.1.o.a.824.2 4 1995.368 even 12
1425.1.t.a.26.1 2 665.634 even 6
1425.1.t.a.26.1 2 1995.1964 odd 6
1425.1.t.a.1151.1 2 35.4 even 6
1425.1.t.a.1151.1 2 105.74 odd 6
1539.1.j.a.26.1 2 1197.634 even 3
1539.1.j.a.26.1 2 1197.1166 odd 6
1539.1.j.a.296.1 2 63.11 odd 6
1539.1.j.a.296.1 2 63.25 even 3
1539.1.n.a.539.1 2 1197.767 odd 6
1539.1.n.a.539.1 2 1197.1033 even 3
1539.1.n.a.1322.1 2 63.4 even 3
1539.1.n.a.1322.1 2 63.32 odd 6
2793.1.n.a.410.1 2 1.1 even 1 trivial
2793.1.n.a.410.1 2 3.2 odd 2 CM
2793.1.n.a.1451.1 2 133.121 even 3 inner
2793.1.n.a.1451.1 2 399.254 odd 6 inner
2793.1.n.b.410.1 2 7.6 odd 2
2793.1.n.b.410.1 2 21.20 even 2
2793.1.n.b.1451.1 2 133.26 odd 6
2793.1.n.b.1451.1 2 399.26 even 6
2793.1.bf.a.197.1 2 133.45 odd 6
2793.1.bf.a.197.1 2 399.311 even 6
2793.1.bf.a.638.1 2 7.3 odd 6
2793.1.bf.a.638.1 2 21.17 even 6
2793.1.bi.a.1892.1 2 7.5 odd 6
2793.1.bi.a.1892.1 2 21.5 even 6
2793.1.bi.a.2762.1 2 133.83 odd 6
2793.1.bi.a.2762.1 2 399.83 even 6
2793.1.bi.b.1892.1 2 7.2 even 3
2793.1.bi.b.1892.1 2 21.2 odd 6
2793.1.bi.b.2762.1 2 19.7 even 3
2793.1.bi.b.2762.1 2 57.26 odd 6
3648.1.bl.a.1793.1 2 1064.235 odd 6
3648.1.bl.a.1793.1 2 3192.2363 even 6
3648.1.bl.a.2177.1 2 56.11 odd 6
3648.1.bl.a.2177.1 2 168.11 even 6
3648.1.bl.b.1793.1 2 1064.501 even 6
3648.1.bl.b.1793.1 2 3192.1565 odd 6
3648.1.bl.b.2177.1 2 56.53 even 6
3648.1.bl.b.2177.1 2 168.53 odd 6