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Properties

Label 7105.2.a.t
Level $7105$
Weight $2$
Character orbit 7105.a
Self dual yes
Analytic conductor $56.734$
Analytic rank $1$
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] = [7105,2,Mod(1,7105)] 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("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7105.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1,-4,5,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.7337106361\)
Analytic rank: \(1\)
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: \( x^{8} - x^{7} - 10x^{6} + 11x^{5} + 25x^{4} - 25x^{3} - 16x^{2} + 9x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1015)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{6} + (\beta_{3} - \beta_{2} - 1) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{9}+ \cdots + (5 \beta_{7} - \beta_{6} - 6 \beta_{5} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} + 5 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{8} + 14 q^{9} - q^{10} + q^{11} + 7 q^{12} + q^{13} + 4 q^{15} + 3 q^{16} - 22 q^{17} + 13 q^{18} - 8 q^{19} - 5 q^{20} - 3 q^{22} + 9 q^{23}+ \cdots - 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 10x^{6} + 11x^{5} + 25x^{4} - 25x^{3} - 16x^{2} + 9x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 9\nu^{4} + 2\nu^{3} + 19\nu^{2} - 4\nu - 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 11\nu^{4} + 16\nu^{3} - 23\nu^{2} + 2\nu + 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{7} + \nu^{6} + 10\nu^{5} - 10\nu^{4} - 24\nu^{3} + 18\nu^{2} + 11\nu - 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 12\nu^{4} + 17\nu^{3} - 29\nu^{2} - \nu + 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{5} - \beta_{3} + 7\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + \beta_{6} + 2\beta_{5} + 9\beta_{3} - 10\beta_{2} + 20\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{7} - 9\beta_{5} + \beta_{4} - 11\beta_{3} + 46\beta_{2} - 13\beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{7} + 9\beta_{6} + 21\beta_{5} + \beta_{4} + 65\beta_{3} - 82\beta_{2} + 112\beta _1 - 106 \) 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
−2.63979
−1.61305
−0.574934
−0.387528
0.742598
1.33564
1.90142
2.23565
−2.63979 0.519171 4.96847 −1.00000 −1.37050 0 −7.83611 −2.73046 2.63979
1.2 −1.61305 −2.82043 0.601933 −1.00000 4.54950 0 2.25515 4.95484 1.61305
1.3 −0.574934 0.252826 −1.66945 −1.00000 −0.145358 0 2.10969 −2.93608 0.574934
1.4 −0.387528 −2.91250 −1.84982 −1.00000 1.12867 0 1.49191 5.48263 0.387528
1.5 0.742598 −1.24299 −1.44855 −1.00000 −0.923043 0 −2.56088 −1.45497 −0.742598
1.6 1.33564 3.35302 −0.216072 −1.00000 4.47842 0 −2.95987 8.24276 −1.33564
1.7 1.90142 −2.54699 1.61538 −1.00000 −4.84289 0 −0.731315 3.48717 −1.90142
1.8 2.23565 1.39789 2.99811 −1.00000 3.12519 0 2.23142 −1.04589 −2.23565
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)
\(29\) \( -1 \)

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 7105.2.a.t 8
7.b odd 2 1 1015.2.a.l ✓ 8
21.c even 2 1 9135.2.a.bh 8
35.c odd 2 1 5075.2.a.ba 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1015.2.a.l ✓ 8 7.b odd 2 1
5075.2.a.ba 8 35.c odd 2 1
7105.2.a.t 8 1.a even 1 1 trivial
9135.2.a.bh 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7105))\):

\( T_{2}^{8} - T_{2}^{7} - 10T_{2}^{6} + 11T_{2}^{5} + 25T_{2}^{4} - 25T_{2}^{3} - 16T_{2}^{2} + 9T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{8} + 4T_{3}^{7} - 11T_{3}^{6} - 57T_{3}^{5} - 5T_{3}^{4} + 140T_{3}^{3} + 32T_{3}^{2} - 80T_{3} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} + 22T_{17}^{7} + 175T_{17}^{6} + 535T_{17}^{5} - 170T_{17}^{4} - 4704T_{17}^{3} - 10272T_{17}^{2} - 8576T_{17} - 2432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} - 10 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - T^{7} + \cdots - 320 \) Copy content Toggle raw display
$13$ \( T^{8} - T^{7} + \cdots - 17920 \) Copy content Toggle raw display
$17$ \( T^{8} + 22 T^{7} + \cdots - 2432 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + \cdots - 4868 \) Copy content Toggle raw display
$23$ \( T^{8} - 9 T^{7} + \cdots + 46016 \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 3632 \) Copy content Toggle raw display
$37$ \( T^{8} - 9 T^{7} + \cdots + 123320 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots - 2869580 \) Copy content Toggle raw display
$43$ \( T^{8} - 7 T^{7} + \cdots - 71768 \) Copy content Toggle raw display
$47$ \( T^{8} + 21 T^{7} + \cdots + 245968 \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} + \cdots - 98384 \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 3768320 \) Copy content Toggle raw display
$61$ \( T^{8} + 26 T^{7} + \cdots + 431936 \) Copy content Toggle raw display
$67$ \( T^{8} + 8 T^{7} + \cdots - 7071104 \) Copy content Toggle raw display
$71$ \( T^{8} + 12 T^{7} + \cdots + 173296 \) Copy content Toggle raw display
$73$ \( T^{8} - T^{7} + \cdots + 1022336 \) Copy content Toggle raw display
$79$ \( T^{8} + 14 T^{7} + \cdots + 17779648 \) Copy content Toggle raw display
$83$ \( T^{8} - 10 T^{7} + \cdots - 75488 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} + \cdots + 333596 \) Copy content Toggle raw display
$97$ \( T^{8} - 11 T^{7} + \cdots - 172480 \) Copy content Toggle raw display
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