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Properties

Label 7704.2.a.bs
Level $7704$
Weight $2$
Character orbit 7704.a
Self dual yes
Analytic conductor $61.517$
Analytic rank $1$
Dimension $10$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 23x^{8} + 21x^{7} + 182x^{6} - 137x^{5} - 580x^{4} + 281x^{3} + 650x^{2} - 70x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 856)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + \beta_{6} q^{7} + ( - \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{11}+ \cdots + (2 \beta_{9} - \beta_{8} - \beta_{6} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{5} - 4 q^{7} - 6 q^{11} + 4 q^{13} - 17 q^{17} + 4 q^{19} + 4 q^{23} + 31 q^{25} - 21 q^{29} + 6 q^{31} - 9 q^{35} + 5 q^{37} - 36 q^{41} - q^{43} - q^{47} + 36 q^{49} - 8 q^{53} + q^{55} - 23 q^{59} + 3 q^{61} - 18 q^{65} - 28 q^{67} - 9 q^{71} + 34 q^{73} + 6 q^{77} - 9 q^{79} - 2 q^{83} - 8 q^{85} - 34 q^{89} - 32 q^{91} + 12 q^{95} + 26 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 23x^{8} + 21x^{7} + 182x^{6} - 137x^{5} - 580x^{4} + 281x^{3} + 650x^{2} - 70x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{9} - 768 \nu^{8} - 425 \nu^{7} + 15564 \nu^{6} + 8940 \nu^{5} - 98207 \nu^{4} + \cdots - 11566 ) / 4802 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51 \nu^{9} - 22 \nu^{8} + 1007 \nu^{7} + 478 \nu^{6} - 5800 \nu^{5} - 3225 \nu^{4} + 6805 \nu^{3} + \cdots + 7120 ) / 1372 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 180 \nu^{9} + 461 \nu^{8} - 3312 \nu^{7} - 9455 \nu^{6} + 15830 \nu^{5} + 62792 \nu^{4} + \cdots + 12076 ) / 4802 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 505 \nu^{9} - 374 \nu^{8} - 11693 \nu^{7} + 6754 \nu^{6} + 89364 \nu^{5} - 36989 \nu^{4} + \cdots - 29880 ) / 9604 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 115 \nu^{9} - 58 \nu^{8} - 2459 \nu^{7} + 886 \nu^{6} + 16764 \nu^{5} - 3139 \nu^{4} - 38561 \nu^{3} + \cdots - 936 ) / 1372 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 411 \nu^{9} + 428 \nu^{8} + 9003 \nu^{7} - 9424 \nu^{6} - 66958 \nu^{5} + 63271 \nu^{4} + \cdots + 24128 ) / 4802 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 244 \nu^{9} - 38 \nu^{8} + 5450 \nu^{7} + 545 \nu^{6} - 41200 \nu^{5} - 2577 \nu^{4} + 120298 \nu^{3} + \cdots + 1291 ) / 2401 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + \beta_{3} + 10\beta_{2} - \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{9} - 14 \beta_{8} + 2 \beta_{7} + 11 \beta_{6} - 2 \beta_{5} + 15 \beta_{4} - 15 \beta_{3} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{9} + 2\beta_{8} - 6\beta_{7} + 30\beta_{5} + 20\beta_{4} + 19\beta_{3} + 90\beta_{2} - 21\beta _1 + 304 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 141 \beta_{9} - 159 \beta_{8} + 39 \beta_{7} + 96 \beta_{6} - 38 \beta_{5} + 169 \beta_{4} - 176 \beta_{3} + \cdots - 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 72 \beta_{9} + 51 \beta_{8} - 122 \beta_{7} - 7 \beta_{6} + 352 \beta_{5} + 145 \beta_{4} + \cdots + 2631 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1442 \beta_{9} - 1684 \beta_{8} + 539 \beta_{7} + 781 \beta_{6} - 520 \beta_{5} + 1736 \beta_{4} + \cdots - 1229 \) 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
1.93616
−2.70781
−1.78383
2.83364
0.171421
2.98450
−1.29944
−3.13135
2.06687
−0.0701604
0 0 0 −4.33274 0 −4.40753 0 0 0
1.2 0 0 0 −3.42974 0 −1.83102 0 0 0
1.3 0 0 0 −2.99765 0 2.85417 0 0 0
1.4 0 0 0 −2.35430 0 4.08303 0 0 0
1.5 0 0 0 −2.13872 0 0.199295 0 0 0
1.6 0 0 0 0.195934 0 −3.71957 0 0 0
1.7 0 0 0 1.26945 0 4.87875 0 0 0
1.8 0 0 0 1.32141 0 −0.889576 0 0 0
1.9 0 0 0 3.51835 0 −0.769496 0 0 0
1.10 0 0 0 3.94800 0 −4.39806 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bs 10
3.b odd 2 1 856.2.a.i ✓ 10
12.b even 2 1 1712.2.a.u 10
24.f even 2 1 6848.2.a.bx 10
24.h odd 2 1 6848.2.a.by 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
856.2.a.i ✓ 10 3.b odd 2 1
1712.2.a.u 10 12.b even 2 1
6848.2.a.bx 10 24.f even 2 1
6848.2.a.by 10 24.h odd 2 1
7704.2.a.bs 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7704))\):

\( T_{5}^{10} + 5 T_{5}^{9} - 28 T_{5}^{8} - 161 T_{5}^{7} + 187 T_{5}^{6} + 1594 T_{5}^{5} + 148 T_{5}^{4} + \cdots - 1024 \) Copy content Toggle raw display
\( T_{7}^{10} + 4 T_{7}^{9} - 45 T_{7}^{8} - 194 T_{7}^{7} + 554 T_{7}^{6} + 2856 T_{7}^{5} - 616 T_{7}^{4} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{11}^{10} + 6 T_{11}^{9} - 53 T_{11}^{8} - 309 T_{11}^{7} + 1071 T_{11}^{6} + 5678 T_{11}^{5} + \cdots + 184 \) Copy content Toggle raw display
\( T_{13}^{10} - 4 T_{13}^{9} - 54 T_{13}^{8} + 187 T_{13}^{7} + 639 T_{13}^{6} - 1825 T_{13}^{5} + \cdots - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 5 T^{9} + \cdots - 1024 \) Copy content Toggle raw display
$7$ \( T^{10} + 4 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{10} + 6 T^{9} + \cdots + 184 \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{10} + 17 T^{9} + \cdots + 28672 \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + \cdots - 14272 \) Copy content Toggle raw display
$23$ \( T^{10} - 4 T^{9} + \cdots - 429184 \) Copy content Toggle raw display
$29$ \( T^{10} + 21 T^{9} + \cdots - 194672 \) Copy content Toggle raw display
$31$ \( T^{10} - 6 T^{9} + \cdots - 32178176 \) Copy content Toggle raw display
$37$ \( T^{10} - 5 T^{9} + \cdots + 39867968 \) Copy content Toggle raw display
$41$ \( T^{10} + 36 T^{9} + \cdots + 114968 \) Copy content Toggle raw display
$43$ \( T^{10} + T^{9} + \cdots - 609536 \) Copy content Toggle raw display
$47$ \( T^{10} + T^{9} + \cdots - 14680064 \) Copy content Toggle raw display
$53$ \( T^{10} + 8 T^{9} + \cdots + 740744 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 506934656 \) Copy content Toggle raw display
$61$ \( T^{10} - 3 T^{9} + \cdots - 36185716 \) Copy content Toggle raw display
$67$ \( T^{10} + 28 T^{9} + \cdots + 19871744 \) Copy content Toggle raw display
$71$ \( T^{10} + 9 T^{9} + \cdots + 3810304 \) Copy content Toggle raw display
$73$ \( T^{10} - 34 T^{9} + \cdots + 30373888 \) Copy content Toggle raw display
$79$ \( T^{10} + 9 T^{9} + \cdots + 2049208 \) Copy content Toggle raw display
$83$ \( T^{10} + 2 T^{9} + \cdots + 12148736 \) Copy content Toggle raw display
$89$ \( T^{10} + 34 T^{9} + \cdots - 15804586 \) Copy content Toggle raw display
$97$ \( T^{10} - 26 T^{9} + \cdots + 3582848 \) Copy content Toggle raw display
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