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Properties

Label 9196.2.a.q
Level $9196$
Weight $2$
Character orbit 9196.a
Self dual yes
Analytic conductor $73.430$
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] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9196.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-1,0,0,0,-2,0,9,0,0,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.4304296988\)
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: \( x^{8} - x^{7} - 16x^{6} + 14x^{5} + 80x^{4} - 64x^{3} - 140x^{2} + 81x + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{6} + \beta_{3} - \beta_{2}) q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{5} + \beta_{2} + 1) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{15}+ \cdots + ( - 2 \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 2 q^{7} + 9 q^{9} + 5 q^{13} - 9 q^{15} - 3 q^{17} - 8 q^{19} - 5 q^{21} + q^{23} - q^{27} - q^{29} + 3 q^{31} - 8 q^{35} - 23 q^{37} - 9 q^{39} - 10 q^{41} - 2 q^{43} + 9 q^{45} - 8 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 16x^{6} + 14x^{5} + 80x^{4} - 64x^{3} - 140x^{2} + 81x + 72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 16\nu^{5} + 2\nu^{4} - 75\nu^{3} - 14\nu^{2} + 99\nu + 33 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + \nu^{6} - 16\nu^{5} - 15\nu^{4} + 73\nu^{3} + 56\nu^{2} - 97\nu - 63 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} + \nu^{6} - 29\nu^{5} - 17\nu^{4} + 115\nu^{3} + 61\nu^{2} - 127\nu - 72 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{7} + 4\nu^{6} - 90\nu^{5} - 64\nu^{4} + 379\nu^{3} + 231\nu^{2} - 463\nu - 261 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{7} - 4\nu^{6} + 90\nu^{5} + 67\nu^{4} - 382\nu^{3} - 258\nu^{2} + 478\nu + 297 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + 10\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{6} - 21\beta_{5} - 23\beta_{4} + \beta_{3} + 14\beta_{2} + 32\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{7} + 28\beta_{6} - 30\beta_{5} - 27\beta_{4} + 3\beta_{3} + 90\beta_{2} + 8\beta _1 + 189 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} + 105\beta_{6} - 190\beta_{5} - 222\beta_{4} + 13\beta_{3} + 155\beta_{2} + 236\beta _1 + 192 \) 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
3.07096
1.81131
1.66154
1.51360
−0.564557
−1.42458
−2.38457
−2.68371
0 −3.07096 0 2.71290 0 −0.988966 0 6.43080 0
1.2 0 −1.81131 0 −1.57147 0 −5.23259 0 0.280842 0
1.3 0 −1.66154 0 −1.44886 0 4.59395 0 −0.239269 0
1.4 0 −1.51360 0 1.33319 0 1.09646 0 −0.709002 0
1.5 0 0.564557 0 1.37233 0 0.329426 0 −2.68128 0
1.6 0 1.42458 0 −2.11046 0 3.55881 0 −0.970558 0
1.7 0 2.38457 0 3.01177 0 −3.00456 0 2.68616 0
1.8 0 2.68371 0 −3.29940 0 −2.35252 0 4.20230 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
\(2\) \( -1 \)
\(11\) \( -1 \)
\(19\) \( +1 \)

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 9196.2.a.q ✓ 8
11.b odd 2 1 9196.2.a.r yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9196.2.a.q ✓ 8 1.a even 1 1 trivial
9196.2.a.r yes 8 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):

\( T_{3}^{8} + T_{3}^{7} - 16T_{3}^{6} - 14T_{3}^{5} + 80T_{3}^{4} + 64T_{3}^{3} - 140T_{3}^{2} - 81T_{3} + 72 \) Copy content Toggle raw display
\( T_{5}^{8} - 20T_{5}^{6} + 127T_{5}^{4} + 9T_{5}^{3} - 300T_{5}^{2} - 18T_{5} + 237 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} - 37T_{7}^{6} - 63T_{7}^{5} + 343T_{7}^{4} + 553T_{7}^{3} - 577T_{7}^{2} - 537T_{7} + 216 \) Copy content Toggle raw display
\( T_{13}^{8} - 5T_{13}^{7} - 23T_{13}^{6} + 75T_{13}^{5} + 225T_{13}^{4} - 219T_{13}^{3} - 631T_{13}^{2} + 176T_{13} + 509 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 72 \) Copy content Toggle raw display
$5$ \( T^{8} - 20 T^{6} + \cdots + 237 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 216 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots + 509 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots + 891 \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - T^{7} + \cdots + 96066 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + \cdots + 3681 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + \cdots - 5346 \) Copy content Toggle raw display
$37$ \( T^{8} + 23 T^{7} + \cdots + 70227 \) Copy content Toggle raw display
$41$ \( T^{8} + 10 T^{7} + \cdots - 18063 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} + \cdots - 51746 \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{7} + \cdots + 13122 \) Copy content Toggle raw display
$53$ \( T^{8} - 6 T^{7} + \cdots + 85833 \) Copy content Toggle raw display
$59$ \( T^{8} + 11 T^{7} + \cdots - 486972 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} + \cdots - 703242 \) Copy content Toggle raw display
$67$ \( T^{8} - 9 T^{7} + \cdots + 25758 \) Copy content Toggle raw display
$71$ \( T^{8} + 3 T^{7} + \cdots + 78732 \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} + \cdots + 62428806 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + \cdots - 1163718 \) Copy content Toggle raw display
$83$ \( T^{8} + 34 T^{7} + \cdots - 31901832 \) Copy content Toggle raw display
$89$ \( T^{8} + 11 T^{7} + \cdots + 3346557 \) Copy content Toggle raw display
$97$ \( T^{8} - 6 T^{7} + \cdots - 10531849 \) Copy content Toggle raw display
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