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Properties

Label 7704.2.a.bp
Level $7704$
Weight $2$
Character orbit 7704.a
Self dual yes
Analytic conductor $61.517$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7704,2,Mod(1,7704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7704 = 2^{3} \cdot 3^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7704.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.5167497172\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 8x^{5} + 19x^{4} + 27x^{3} - 28x^{2} - 38x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2568)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{6} - 1) q^{11} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{3} - 1) q^{17} + (\beta_{4} + 2) q^{19} + (\beta_{6} + \beta_{4} + \beta_{2} + \beta_1) q^{23} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{25}+ \cdots + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{5} + 7 q^{7} - 5 q^{11} + 5 q^{13} - 9 q^{17} + 14 q^{19} - 4 q^{23} + 7 q^{25} + 4 q^{29} + 7 q^{31} - 8 q^{35} + 7 q^{37} - 3 q^{41} + 14 q^{43} + 2 q^{47} + 14 q^{49} - 2 q^{53} + 7 q^{55} - 13 q^{59} + 11 q^{61} + 2 q^{65} + 35 q^{67} - 4 q^{71} + 10 q^{73} + 2 q^{77} + 19 q^{79} - 3 q^{83} + 6 q^{85} + q^{89} + 15 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 8x^{5} + 19x^{4} + 27x^{3} - 28x^{2} - 38x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 10\nu + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 15\nu^{3} + 11\nu^{2} - 16\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 5\nu^{5} - 25\nu^{3} + 11\nu^{2} + 32\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 8\nu^{4} + 21\nu^{3} + 21\nu^{2} - 32\nu - 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 4\nu^{5} - 4\nu^{4} + 22\nu^{3} + 7\nu^{2} - 30\nu - 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} + 5\beta_{5} - 2\beta_{4} - 3\beta_{3} + 3\beta_{2} + \beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{6} + 7\beta_{5} - 3\beta_{4} - 4\beta_{3} + 5\beta_{2} + 6\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{6} + 26\beta_{5} - 13\beta_{4} - 11\beta_{3} + 15\beta_{2} + 17\beta _1 + 69 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -45\beta_{6} + 81\beta_{5} - 42\beta_{4} - 31\beta_{3} + 51\beta_{2} + 70\beta _1 + 230 \) 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.

comment: embeddings in the coefficient field
 
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
1.85213
2.26374
−0.281063
−0.805738
3.22174
−1.46191
−1.78889
0 0 0 −3.27388 0 3.50678 0 0 0
1.2 0 0 0 −2.40296 0 −0.401727 0 0 0
1.3 0 0 0 −2.35888 0 2.94623 0 0 0
1.4 0 0 0 −0.739310 0 −2.66347 0 0 0
1.5 0 0 0 0.936110 0 4.11394 0 0 0
1.6 0 0 0 2.06102 0 −3.22715 0 0 0
1.7 0 0 0 3.77790 0 2.72539 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(107\) \( +1 \)

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 7704.2.a.bp 7
3.b odd 2 1 2568.2.a.p ✓ 7
12.b even 2 1 5136.2.a.bh 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2568.2.a.p ✓ 7 3.b odd 2 1
5136.2.a.bh 7 12.b even 2 1
7704.2.a.bp 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7704))\):

\( T_{5}^{7} + 2T_{5}^{6} - 19T_{5}^{5} - 41T_{5}^{4} + 78T_{5}^{3} + 163T_{5}^{2} - 68T_{5} - 100 \) Copy content Toggle raw display
\( T_{7}^{7} - 7T_{7}^{6} - 7T_{7}^{5} + 128T_{7}^{4} - 110T_{7}^{3} - 612T_{7}^{2} + 776T_{7} + 400 \) Copy content Toggle raw display
\( T_{11}^{7} + 5T_{11}^{6} - 27T_{11}^{5} - 140T_{11}^{4} + 180T_{11}^{3} + 1136T_{11}^{2} - 2048 \) Copy content Toggle raw display
\( T_{13}^{7} - 5T_{13}^{6} - 22T_{13}^{5} + 77T_{13}^{4} + 167T_{13}^{3} - 58T_{13}^{2} - 76T_{13} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 2 T^{6} + \cdots - 100 \) Copy content Toggle raw display
$7$ \( T^{7} - 7 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{7} + 5 T^{6} + \cdots - 2048 \) Copy content Toggle raw display
$13$ \( T^{7} - 5 T^{6} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{7} + 9 T^{6} + \cdots + 1280 \) Copy content Toggle raw display
$19$ \( T^{7} - 14 T^{6} + \cdots - 160 \) Copy content Toggle raw display
$23$ \( T^{7} + 4 T^{6} + \cdots - 928 \) Copy content Toggle raw display
$29$ \( T^{7} - 4 T^{6} + \cdots - 3136 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} + \cdots + 2000 \) Copy content Toggle raw display
$37$ \( T^{7} - 7 T^{6} + \cdots - 3568 \) Copy content Toggle raw display
$41$ \( T^{7} + 3 T^{6} + \cdots + 512 \) Copy content Toggle raw display
$43$ \( T^{7} - 14 T^{6} + \cdots - 23648 \) Copy content Toggle raw display
$47$ \( T^{7} - 2 T^{6} + \cdots - 1960 \) Copy content Toggle raw display
$53$ \( T^{7} + 2 T^{6} + \cdots - 45824 \) Copy content Toggle raw display
$59$ \( T^{7} + 13 T^{6} + \cdots - 908032 \) Copy content Toggle raw display
$61$ \( T^{7} - 11 T^{6} + \cdots + 454840 \) Copy content Toggle raw display
$67$ \( T^{7} - 35 T^{6} + \cdots + 770048 \) Copy content Toggle raw display
$71$ \( T^{7} + 4 T^{6} + \cdots - 4864 \) Copy content Toggle raw display
$73$ \( T^{7} - 10 T^{6} + \cdots - 659968 \) Copy content Toggle raw display
$79$ \( T^{7} - 19 T^{6} + \cdots + 162304 \) Copy content Toggle raw display
$83$ \( T^{7} + 3 T^{6} + \cdots + 12800 \) Copy content Toggle raw display
$89$ \( T^{7} - T^{6} + \cdots + 21248 \) Copy content Toggle raw display
$97$ \( T^{7} - 8 T^{6} + \cdots + 228736 \) Copy content Toggle raw display
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