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Properties

Label 9350.2.a.dc
Level $9350$
Weight $2$
Character orbit 9350.a
Self dual yes
Analytic conductor $74.660$
Analytic rank $1$
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] = [9350,2,Mod(1,9350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9350, 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("9350.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9350 = 2 \cdot 5^{2} \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9350.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,7,-3,7,0,-3,-4,7,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.6601258899\)
Analytic rank: \(1\)
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: \( x^{7} - 3x^{6} - 6x^{5} + 20x^{4} + 3x^{3} - 25x^{2} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1870)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + (\beta_{3} - 1) q^{7} + q^{8} + (\beta_{4} + \beta_{2}) q^{9} - q^{11} - \beta_1 q^{12} + ( - \beta_{6} + \beta_{4} + 1) q^{13} + (\beta_{3} - 1) q^{14}+ \cdots + ( - \beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 3 q^{6} - 4 q^{7} + 7 q^{8} - 7 q^{11} - 3 q^{12} + 9 q^{13} - 4 q^{14} + 7 q^{16} - 7 q^{17} - 3 q^{19} - 4 q^{21} - 7 q^{22} - 6 q^{23} - 3 q^{24} + 9 q^{26} - 3 q^{27}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 6x^{5} + 20x^{4} + 3x^{3} - 25x^{2} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 16\nu^{3} - 9\nu^{2} - 5\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 18\nu^{3} - 5\nu^{2} + 13\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 4\nu^{4} - 16\nu^{3} + 11\nu^{2} + 5\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 11\nu^{2} + 7\nu - 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 6\nu^{3} + 25\nu^{2} - 9\nu - 14 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + 8\beta_{4} + 2\beta_{3} + 9\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{6} - 8\beta_{5} + 12\beta_{4} + 11\beta_{3} + 14\beta_{2} + 30\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{6} - 12\beta_{5} + 61\beta_{4} + 25\beta_{3} + 73\beta_{2} + 19\beta _1 + 105 \) 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
2.91909
1.82121
1.43275
0.724668
−0.711702
−0.885051
−2.30096
1.00000 −2.91909 1.00000 0 −2.91909 2.20842 1.00000 5.52106 0
1.2 1.00000 −1.82121 1.00000 0 −1.82121 −4.00712 1.00000 0.316806 0
1.3 1.00000 −1.43275 1.00000 0 −1.43275 −2.91631 1.00000 −0.947238 0
1.4 1.00000 −0.724668 1.00000 0 −0.724668 3.02718 1.00000 −2.47486 0
1.5 1.00000 0.711702 1.00000 0 0.711702 −0.217124 1.00000 −2.49348 0
1.6 1.00000 0.885051 1.00000 0 0.885051 1.31420 1.00000 −2.21668 0
1.7 1.00000 2.30096 1.00000 0 2.30096 −3.40924 1.00000 2.29440 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
\(2\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)
\(17\) \( +1 \)

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 9350.2.a.dc 7
5.b even 2 1 9350.2.a.db 7
5.c odd 4 2 1870.2.b.d ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1870.2.b.d ✓ 14 5.c odd 4 2
9350.2.a.db 7 5.b even 2 1
9350.2.a.dc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9350))\):

\( T_{3}^{7} + 3T_{3}^{6} - 6T_{3}^{5} - 20T_{3}^{4} + 3T_{3}^{3} + 25T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{7} + 4T_{7}^{6} - 18T_{7}^{5} - 64T_{7}^{4} + 114T_{7}^{3} + 258T_{7}^{2} - 300T_{7} - 76 \) Copy content Toggle raw display
\( T_{13}^{7} - 9T_{13}^{6} + 16T_{13}^{5} + 32T_{13}^{4} - 93T_{13}^{3} + 5T_{13}^{2} + 104T_{13} - 52 \) Copy content Toggle raw display
\( T_{19}^{7} + 3T_{19}^{6} - 36T_{19}^{5} - 72T_{19}^{4} + 239T_{19}^{3} + 489T_{19}^{2} + 48T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 3 T^{6} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + 4 T^{6} + \cdots - 76 \) Copy content Toggle raw display
$11$ \( (T + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 9 T^{6} + \cdots - 52 \) Copy content Toggle raw display
$17$ \( (T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} + 3 T^{6} + \cdots - 4 \) Copy content Toggle raw display
$23$ \( T^{7} + 6 T^{6} + \cdots + 604 \) Copy content Toggle raw display
$29$ \( T^{7} + 9 T^{6} + \cdots - 3994 \) Copy content Toggle raw display
$31$ \( T^{7} + 9 T^{6} + \cdots - 41878 \) Copy content Toggle raw display
$37$ \( T^{7} + 6 T^{6} + \cdots - 168944 \) Copy content Toggle raw display
$41$ \( T^{7} - 112 T^{5} + \cdots - 1256 \) Copy content Toggle raw display
$43$ \( T^{7} - 256 T^{5} + \cdots - 935584 \) Copy content Toggle raw display
$47$ \( T^{7} + 15 T^{6} + \cdots + 10400 \) Copy content Toggle raw display
$53$ \( T^{7} - 17 T^{6} + \cdots + 24316 \) Copy content Toggle raw display
$59$ \( T^{7} + 27 T^{6} + \cdots - 16 \) Copy content Toggle raw display
$61$ \( T^{7} + 33 T^{6} + \cdots + 40538 \) Copy content Toggle raw display
$67$ \( T^{7} + 16 T^{6} + \cdots + 991328 \) Copy content Toggle raw display
$71$ \( T^{7} + T^{6} + \cdots - 69022 \) Copy content Toggle raw display
$73$ \( T^{7} - T^{6} + \cdots + 87412 \) Copy content Toggle raw display
$79$ \( T^{7} + 38 T^{6} + \cdots - 11296 \) Copy content Toggle raw display
$83$ \( T^{7} + 8 T^{6} + \cdots - 226616 \) Copy content Toggle raw display
$89$ \( T^{7} - 31 T^{6} + \cdots - 1068976 \) Copy content Toggle raw display
$97$ \( T^{7} - 3 T^{6} + \cdots - 24652 \) Copy content Toggle raw display
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