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Properties

Label 352.6.a.g
Level $352$
Weight $6$
Character orbit 352.a
Self dual yes
Analytic conductor $56.455$
Analytic rank $0$
Dimension $7$
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] = [352,6,Mod(1,352)] 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("352.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 352.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,-9,0,61] 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: \(56.4551045742\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 1343x^{5} + 3732x^{4} + 459776x^{3} - 2085920x^{2} - 16548352x + 80811520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{2} - \beta_1 + 9) q^{5} + (\beta_{5} - \beta_1) q^{7} + (\beta_{3} - \beta_{2} + 143) q^{9} - 121 q^{11} + (\beta_{6} - 2 \beta_{5} + \beta_{3} + \cdots - 3) q^{13}+ \cdots + ( - 121 \beta_{3} + 121 \beta_{2} - 17303) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{3} + 61 q^{5} + 1000 q^{9} - 847 q^{11} - 18 q^{13} + 1331 q^{15} + 2432 q^{17} - 2362 q^{19} + 1344 q^{21} + 1835 q^{23} + 770 q^{25} - 579 q^{27} - 1168 q^{29} + 3983 q^{31} + 1089 q^{33} - 1872 q^{35}+ \cdots - 121000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 1343x^{5} + 3732x^{4} + 459776x^{3} - 2085920x^{2} - 16548352x + 80811520 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10541 \nu^{6} - 48282 \nu^{5} + 13985443 \nu^{4} + 48914120 \nu^{3} - 4657794600 \nu^{2} + \cdots + 155438668864 ) / 157144752 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10541 \nu^{6} - 48282 \nu^{5} + 13985443 \nu^{4} + 48914120 \nu^{3} - 4500649848 \nu^{2} + \cdots + 94937939344 ) / 157144752 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1567 \nu^{6} + 9110 \nu^{5} - 2178153 \nu^{4} - 5820968 \nu^{3} + 753755176 \nu^{2} + \cdots - 20260926176 ) / 17460528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11609 \nu^{6} + 46134 \nu^{5} - 15664975 \nu^{4} - 44811380 \nu^{3} + 5330310096 \nu^{2} + \cdots - 178669891936 ) / 78572376 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 127051 \nu^{6} + 361638 \nu^{5} - 165998669 \nu^{4} - 396978304 \nu^{3} + 54298809096 \nu^{2} + \cdots - 1755910165088 ) / 157144752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 2\beta _1 + 385 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 7\beta_{5} + 7\beta_{4} + 5\beta_{3} + \beta_{2} + 659\beta _1 - 631 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 23\beta_{6} - 287\beta_{5} + 59\beta_{4} + 795\beta_{3} - 1071\beta_{2} - 1061\beta _1 + 251915 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 429\beta_{6} - 9465\beta_{5} + 6807\beta_{4} + 5279\beta_{3} - 11849\beta_{2} + 460491\beta _1 - 299565 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 33191 \beta_{6} - 369911 \beta_{5} + 79583 \beta_{4} + 611927 \beta_{3} - 935087 \beta_{2} + \cdots + 177300959 \) 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
28.5711
20.4597
5.85632
5.46174
−6.41645
−25.1581
−26.7744
0 −29.5711 0 37.4069 0 −105.127 0 631.452 0
1.2 0 −21.4597 0 −60.5699 0 247.110 0 217.518 0
1.3 0 −6.85632 0 88.9891 0 −0.0104569 0 −195.991 0
1.4 0 −6.46174 0 −22.1205 0 −246.771 0 −201.246 0
1.5 0 5.41645 0 −62.6369 0 21.9774 0 −213.662 0
1.6 0 24.1581 0 71.9871 0 191.883 0 340.611 0
1.7 0 25.7744 0 7.94416 0 −109.063 0 421.318 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
\(2\) \( -1 \)
\(11\) \( +1 \)

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 352.6.a.g ✓ 7
4.b odd 2 1 352.6.a.h yes 7
8.b even 2 1 704.6.a.bf 7
8.d odd 2 1 704.6.a.be 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.6.a.g ✓ 7 1.a even 1 1 trivial
352.6.a.h yes 7 4.b odd 2 1
704.6.a.be 7 8.d odd 2 1
704.6.a.bf 7 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 9T_{3}^{6} - 1310T_{3}^{5} - 10382T_{3}^{4} + 431493T_{3}^{3} + 3429477T_{3}^{2} - 11018808T_{3} - 94819248 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(352))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 9 T^{6} + \cdots - 94819248 \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots + 159761993132 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 30831280128 \) Copy content Toggle raw display
$11$ \( (T + 121)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 61\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 93\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 92\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 81\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 42\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 18\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 25\!\cdots\!64 \) Copy content Toggle raw display
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