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Properties

Label 9300.2.a.bc
Level $9300$
Weight $2$
Character orbit 9300.a
Self dual yes
Analytic conductor $74.261$
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] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,-7,0,0,0,-4,0,7,0,4] 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: \(74.2608738798\)
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: \( x^{7} - 3x^{6} - 15x^{5} + 49x^{4} + 13x^{3} - 69x^{2} - 35x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1860)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{2} - 1) q^{7} + q^{9} + ( - \beta_{4} + 1) q^{11} + (\beta_{6} - \beta_{5} - \beta_{4} + \cdots - 1) q^{13} + ( - \beta_{6} + \beta_{4} + \beta_{2} + \cdots - 1) q^{17} + ( - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{19}+ \cdots + ( - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{3} - 4 q^{7} + 7 q^{9} + 4 q^{11} - 10 q^{13} - 6 q^{17} + 10 q^{19} + 4 q^{21} - 7 q^{27} + 10 q^{29} - 7 q^{31} - 4 q^{33} - 16 q^{37} + 10 q^{39} + 2 q^{41} + 8 q^{43} - 12 q^{47} + 9 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 15x^{5} + 49x^{4} + 13x^{3} - 69x^{2} - 35x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - \nu^{5} + 28\nu^{4} + 12\nu^{3} - 203\nu^{2} + 39\nu + 106 ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 13\nu^{5} + 61\nu^{4} - 224\nu^{3} - 64\nu^{2} + 445\nu + 135 ) / 34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 14\nu^{5} + 33\nu^{4} - 219\nu^{3} + 156\nu^{2} + 219\nu - 39 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{6} - 38\nu^{5} - 177\nu^{4} + 592\nu^{3} - 115\nu^{2} - 422\nu - 1 ) / 34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{6} + 55\nu^{5} + 160\nu^{4} - 864\nu^{3} + 387\nu^{2} + 949\nu + 120 ) / 34 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + 2\beta_{4} + \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} - 2\beta_{3} + \beta_{2} + 10\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{6} + \beta_{5} + 28\beta_{4} - \beta_{3} + 18\beta_{2} + 7\beta _1 + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21\beta_{6} + 3\beta_{5} - 20\beta_{4} - 33\beta_{3} + 18\beta_{2} + 120\beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -170\beta_{6} + 25\beta_{5} + 386\beta_{4} - 19\beta_{3} + 261\beta_{2} + 32\beta _1 + 748 \) 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
3.53062
−0.0304029
−0.575791
−3.80500
−0.792075
2.63632
2.03634
0 −1.00000 0 0 0 −4.13530 0 1.00000 0
1.2 0 −1.00000 0 0 0 −4.07725 0 1.00000 0
1.3 0 −1.00000 0 0 0 −1.50165 0 1.00000 0
1.4 0 −1.00000 0 0 0 −0.690103 0 1.00000 0
1.5 0 −1.00000 0 0 0 0.386084 0 1.00000 0
1.6 0 −1.00000 0 0 0 1.72739 0 1.00000 0
1.7 0 −1.00000 0 0 0 4.29083 0 1.00000 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
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)
\(31\) \( +1 \)

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 9300.2.a.bc 7
5.b even 2 1 9300.2.a.bf 7
5.c odd 4 2 1860.2.g.a ✓ 14
15.e even 4 2 5580.2.g.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.g.a ✓ 14 5.c odd 4 2
5580.2.g.d 14 15.e even 4 2
9300.2.a.bc 7 1.a even 1 1 trivial
9300.2.a.bf 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9300))\):

\( T_{7}^{7} + 4T_{7}^{6} - 21T_{7}^{5} - 86T_{7}^{4} + 46T_{7}^{3} + 228T_{7}^{2} + 40T_{7} - 50 \) Copy content Toggle raw display
\( T_{11}^{7} - 4T_{11}^{6} - 28T_{11}^{5} + 108T_{11}^{4} + 128T_{11}^{3} - 456T_{11}^{2} - 192T_{11} + 384 \) Copy content Toggle raw display
\( T_{13}^{7} + 10T_{13}^{6} - 2T_{13}^{5} - 336T_{13}^{4} - 1304T_{13}^{3} - 1358T_{13}^{2} + 1040T_{13} + 1800 \) Copy content Toggle raw display
\( T_{17}^{7} + 6T_{17}^{6} - 30T_{17}^{5} - 184T_{17}^{4} + 268T_{17}^{3} + 1548T_{17}^{2} - 864T_{17} - 3456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + 4 T^{6} + \cdots - 50 \) Copy content Toggle raw display
$11$ \( T^{7} - 4 T^{6} + \cdots + 384 \) Copy content Toggle raw display
$13$ \( T^{7} + 10 T^{6} + \cdots + 1800 \) Copy content Toggle raw display
$17$ \( T^{7} + 6 T^{6} + \cdots - 3456 \) Copy content Toggle raw display
$19$ \( T^{7} - 10 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( T^{7} - 54 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{7} - 10 T^{6} + \cdots - 18624 \) Copy content Toggle raw display
$31$ \( (T + 1)^{7} \) Copy content Toggle raw display
$37$ \( T^{7} + 16 T^{6} + \cdots - 1280 \) Copy content Toggle raw display
$41$ \( T^{7} - 2 T^{6} + \cdots + 9736 \) Copy content Toggle raw display
$43$ \( T^{7} - 8 T^{6} + \cdots - 106240 \) Copy content Toggle raw display
$47$ \( T^{7} + 12 T^{6} + \cdots - 4720 \) Copy content Toggle raw display
$53$ \( T^{7} + 16 T^{6} + \cdots + 14592 \) Copy content Toggle raw display
$59$ \( T^{7} - 14 T^{6} + \cdots - 90 \) Copy content Toggle raw display
$61$ \( T^{7} - 10 T^{6} + \cdots + 344576 \) Copy content Toggle raw display
$67$ \( T^{7} + 22 T^{6} + \cdots - 1413640 \) Copy content Toggle raw display
$71$ \( T^{7} + 2 T^{6} + \cdots + 4014 \) Copy content Toggle raw display
$73$ \( T^{7} + 14 T^{6} + \cdots + 7424 \) Copy content Toggle raw display
$79$ \( T^{7} - 24 T^{6} + \cdots - 116208 \) Copy content Toggle raw display
$83$ \( T^{7} - 12 T^{6} + \cdots - 192208 \) Copy content Toggle raw display
$89$ \( T^{7} - 12 T^{6} + \cdots - 193096 \) Copy content Toggle raw display
$97$ \( T^{7} + 24 T^{6} + \cdots + 10338248 \) Copy content Toggle raw display
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