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Properties

Label 875.2.a.i
Level $875$
Weight $2$
Character orbit 875.a
Self dual yes
Analytic conductor $6.987$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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] = [875,2,Mod(1,875)] 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("875.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,8] 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: yes
Analytic conductor: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 10x^{6} + 30x^{5} + 29x^{4} - 79x^{3} - 43x^{2} + 62x + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{5} + \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots + 1) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{6} + q^{7} + (\beta_{6} + \beta_{4} + \beta_{2} + \cdots - 3) q^{8}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \cdots - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 8 q^{3} + 13 q^{4} + 2 q^{6} + 8 q^{7} - 12 q^{8} + 18 q^{9} - 5 q^{11} + 20 q^{12} + 6 q^{13} - q^{14} + 35 q^{16} - 13 q^{17} - 3 q^{18} + 13 q^{19} + 8 q^{21} - 22 q^{22} + 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 10x^{6} + 30x^{5} + 29x^{4} - 79x^{3} - 43x^{2} + 62x + 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 50\nu^{4} - 53\nu^{3} - 80\nu^{2} + 81\nu + 35 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 12\nu^{6} + 16\nu^{5} - 102\nu^{4} + 33\nu^{3} + 174\nu^{2} - 79\nu - 73 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{7} - 30\nu^{6} - 32\nu^{5} + 254\nu^{4} - 119\nu^{3} - 424\nu^{2} + 235\nu + 169 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 38\nu^{6} + 44\nu^{5} - 322\nu^{4} + 129\nu^{3} + 536\nu^{2} - 269\nu - 203 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -3\nu^{7} + 13\nu^{6} + 14\nu^{5} - 111\nu^{4} + 48\nu^{3} + 190\nu^{2} - 95\nu - 76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -19\nu^{7} + 78\nu^{6} + 96\nu^{5} - 658\nu^{4} + 247\nu^{3} + 1084\nu^{2} - 527\nu - 405 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 2\beta_{5} + 9\beta_{4} + 8\beta_{3} - 9\beta_{2} + 9\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{7} - 9\beta_{6} + 21\beta_{5} - 3\beta_{4} + 11\beta_{3} - 3\beta_{2} + 43\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{7} - 11\beta_{6} + 30\beta_{5} + 74\beta_{4} + 61\beta_{3} - 71\beta_{2} + 77\beta _1 + 104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -71\beta_{7} - 69\beta_{6} + 186\beta_{5} + 21\beta_{4} + 99\beta_{3} - 52\beta_{2} + 329\beta _1 + 188 \) 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
−0.453202
−2.52916
1.85173
2.88263
−1.22981
−1.24025
2.44932
1.26874
−2.78847 −1.35133 5.77559 0 3.76815 1.00000 −10.5281 −1.17391 0
1.2 −2.66234 3.18114 5.08806 0 −8.46928 1.00000 −8.22148 7.11965 0
1.3 −1.30219 2.37813 −0.304314 0 −3.09676 1.00000 3.00064 2.65549 0
1.4 −0.499926 −0.163530 −1.75007 0 0.0817529 1.00000 1.87476 −2.97326 0
1.5 0.582954 −2.60791 −1.66016 0 −1.52029 1.00000 −2.13371 3.80121 0
1.6 1.13692 2.38455 −0.707406 0 2.71105 1.00000 −3.07811 2.68607 0
1.7 1.88967 3.34505 1.57086 0 6.32104 1.00000 −0.810940 8.18935 0
1.8 2.64338 0.833911 4.98745 0 2.20434 1.00000 7.89697 −2.30459 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
\(5\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 875.2.a.i ✓ 8
3.b odd 2 1 7875.2.a.bb 8
5.b even 2 1 875.2.a.j yes 8
5.c odd 4 2 875.2.b.e 16
7.b odd 2 1 6125.2.a.v 8
15.d odd 2 1 7875.2.a.w 8
35.c odd 2 1 6125.2.a.w 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
875.2.a.i ✓ 8 1.a even 1 1 trivial
875.2.a.j yes 8 5.b even 2 1
875.2.b.e 16 5.c odd 4 2
6125.2.a.v 8 7.b odd 2 1
6125.2.a.w 8 35.c odd 2 1
7875.2.a.w 8 15.d odd 2 1
7875.2.a.bb 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(875))\):

\( T_{2}^{8} + T_{2}^{7} - 14T_{2}^{6} - 9T_{2}^{5} + 59T_{2}^{4} + 14T_{2}^{3} - 72T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{8} - 8T_{3}^{7} + 11T_{3}^{6} + 57T_{3}^{5} - 161T_{3}^{4} + 2T_{3}^{3} + 276T_{3}^{2} - 133T_{3} - 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots - 29 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots - 1229 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + \cdots + 25524 \) Copy content Toggle raw display
$17$ \( T^{8} + 13 T^{7} + \cdots - 2521 \) Copy content Toggle raw display
$19$ \( T^{8} - 13 T^{7} + \cdots + 1280 \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{7} + \cdots + 5696 \) Copy content Toggle raw display
$29$ \( T^{8} - 22 T^{7} + \cdots - 11125 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + \cdots - 46400 \) Copy content Toggle raw display
$37$ \( T^{8} + 7 T^{7} + \cdots + 30656 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots - 446464 \) Copy content Toggle raw display
$43$ \( T^{8} - T^{7} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{8} - 2 T^{7} + \cdots - 150336 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 20416 \) Copy content Toggle raw display
$59$ \( T^{8} + 11 T^{7} + \cdots + 83520 \) Copy content Toggle raw display
$61$ \( T^{8} - 45 T^{7} + \cdots + 6976 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + \cdots + 2816 \) Copy content Toggle raw display
$71$ \( T^{8} + 15 T^{7} + \cdots + 32458861 \) Copy content Toggle raw display
$73$ \( T^{8} - 32 T^{7} + \cdots - 1421 \) Copy content Toggle raw display
$79$ \( T^{8} - 12 T^{7} + \cdots + 918900 \) Copy content Toggle raw display
$83$ \( T^{8} - 15 T^{7} + \cdots + 173601 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + \cdots - 68384320 \) Copy content Toggle raw display
$97$ \( T^{8} + 20 T^{7} + \cdots + 3215936 \) Copy content Toggle raw display
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