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Properties

Label 6975.2.a.ce
Level $6975$
Weight $2$
Character orbit 6975.a
Self dual yes
Analytic conductor $55.696$
Analytic rank $1$
Dimension $8$
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] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.3057647616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 4x^{4} - 20x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1395)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{4} + \beta_{7} q^{7} + ( - \beta_{5} - \beta_{2}) q^{8} - \beta_{3} q^{11} + ( - \beta_{7} - \beta_{6}) q^{13} + \beta_{4} q^{14} + (2 \beta_1 - 1) q^{16} + \beta_{5} q^{17}+ \cdots + (\beta_{5} - 5 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{16} - 16 q^{19} + 8 q^{31} - 8 q^{34} + 8 q^{46} - 32 q^{49} - 32 q^{61} - 32 q^{64} - 40 q^{79} - 24 q^{91} - 56 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 4x^{4} - 20x^{3} - 4x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} + 6\nu^{6} - 66\nu^{5} + 3\nu^{4} + 254\nu^{3} - 62\nu^{2} - 142\nu + 14 ) / 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 16\nu^{6} - \nu^{5} - 92\nu^{4} + 69\nu^{3} + 143\nu^{2} - 87\nu - 46 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} - 13\nu^{6} - 32\nu^{5} + 81\nu^{4} + 108\nu^{3} - 149\nu^{2} - 134\nu + 53 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{7} + 58\nu^{6} + 37\nu^{5} - 246\nu^{4} + 22\nu^{3} + 109\nu^{2} + 19\nu + 27 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -44\nu^{7} + 193\nu^{6} + 102\nu^{5} - 916\nu^{4} + 162\nu^{3} + 789\nu^{2} - 76\nu - 108 ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46\nu^{7} - 187\nu^{6} - 168\nu^{5} + 919\nu^{4} + 92\nu^{3} - 851\nu^{2} - 16\nu + 97 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -61\nu^{7} + 267\nu^{6} + 138\nu^{5} - 1254\nu^{4} + 253\nu^{3} + 1066\nu^{2} - 194\nu - 177 ) / 25 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} + 8\beta_{6} + 5\beta_{5} - 6\beta_{4} + 3\beta_{3} - 6\beta_{2} - 9\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{7} + 12\beta_{6} + 4\beta_{5} - 14\beta_{4} + 4\beta_{3} - 16\beta_{2} - 13\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 58\beta_{7} + 87\beta_{6} + 49\beta_{5} - 100\beta_{4} + 40\beta_{3} - 110\beta_{2} - 99\beta _1 + 161 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 113\beta_{7} + 148\beta_{6} + 73\beta_{5} - 193\beta_{4} + 67\beta_{3} - 219\beta_{2} - 165\beta _1 + 297 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 760\beta_{7} + 1064\beta_{6} + 603\beta_{5} - 1400\beta_{4} + 525\beta_{3} - 1568\beta_{2} - 1203\beta _1 + 2045 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.534943
3.61132
−0.276907
−1.86936
1.13731
−0.455144
−0.879270
2.19711
−1.93185 0 1.73205 0 0 −1.12603 0.517638 0 0
1.2 −1.93185 0 1.73205 0 0 1.12603 0.517638 0 0
1.3 −0.517638 0 −1.73205 0 0 −2.17533 1.93185 0 0
1.4 −0.517638 0 −1.73205 0 0 2.17533 1.93185 0 0
1.5 0.517638 0 −1.73205 0 0 −2.17533 −1.93185 0 0
1.6 0.517638 0 −1.73205 0 0 2.17533 −1.93185 0 0
1.7 1.93185 0 1.73205 0 0 −1.12603 −0.517638 0 0
1.8 1.93185 0 1.73205 0 0 1.12603 −0.517638 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(31\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6975.2.a.ce 8
3.b odd 2 1 inner 6975.2.a.ce 8
5.b even 2 1 inner 6975.2.a.ce 8
5.c odd 4 2 1395.2.c.d ✓ 8
15.d odd 2 1 inner 6975.2.a.ce 8
15.e even 4 2 1395.2.c.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1395.2.c.d ✓ 8 5.c odd 4 2
1395.2.c.d ✓ 8 15.e even 4 2
6975.2.a.ce 8 1.a even 1 1 trivial
6975.2.a.ce 8 3.b odd 2 1 inner
6975.2.a.ce 8 5.b even 2 1 inner
6975.2.a.ce 8 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):

\( T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{2} + 6 \) Copy content Toggle raw display
\( T_{11}^{4} - 12T_{11}^{2} + 24 \) Copy content Toggle raw display
\( T_{13}^{4} - 18T_{13}^{2} + 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 2 \) Copy content Toggle raw display
\( T_{29}^{4} - 54T_{29}^{2} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{2} + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 12 T^{2} + 24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 18 T^{2} + 6)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$19$ \( (T + 2)^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 54 T^{2} + 54)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 54 T^{2} + 726)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 36 T^{2} + 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{2} + 24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 76 T^{2} + 676)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{2} + 6)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 354 T^{2} + 30246)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 114 T^{2} + 3174)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 54 T^{2} + 726)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 162 T^{2} + 4374)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 108 T^{2} + 216)^{2} \) Copy content Toggle raw display
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