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Properties

Label 9300.2.a.be
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: \( x^{7} - x^{6} - 21x^{5} + 31x^{4} + 113x^{3} - 187x^{2} - 161x + 281 \) 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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{4} - 1) q^{7} + q^{9} + ( - \beta_{5} + \beta_{3}) q^{11} + (\beta_{6} - \beta_{4} - 1) q^{13} + ( - \beta_{4} + \beta_{2} - 2) q^{17} + ( - 2 \beta_{6} - \beta_{3} - \beta_{2}) q^{19}+ \cdots + ( - \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} - 4 q^{7} + 7 q^{9} + 4 q^{11} - 10 q^{13} - 14 q^{17} - 6 q^{19} - 4 q^{21} - 8 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} - 12 q^{37} - 10 q^{39} + 10 q^{41} - 8 q^{43} - 16 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 21x^{5} + 31x^{4} + 113x^{3} - 187x^{2} - 161x + 281 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 15\nu^{4} - 14\nu^{3} + 71\nu^{2} + 18\nu - 115 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 6\nu^{5} - 3\nu^{4} - 62\nu^{3} - 49\nu^{2} + 134\nu + 153 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} - 6\nu^{5} + 45\nu^{4} + 50\nu^{3} - 181\nu^{2} - 94\nu + 201 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 2\nu^{5} + 57\nu^{4} + 2\nu^{3} - 293\nu^{2} + 38\nu + 413 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{6} - 14\nu^{5} + 197\nu^{4} + 98\nu^{3} - 997\nu^{2} - 126\nu + 1433 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{5} + 5\beta_{4} + 4\beta_{3} - \beta_{2} + 13\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 17\beta_{5} - 22\beta_{4} - 19\beta_{3} + 15\beta_{2} - 39\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{6} - 69\beta_{5} + 96\beta_{4} + 77\beta_{3} - 29\beta_{2} + 184\beta _1 - 202 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 21\beta_{6} + 266\beta_{5} - 381\beta_{4} - 312\beta_{3} + 206\beta_{2} - 647\beta _1 + 1007 \) 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
1.64087
2.69460
1.78206
−1.52495
−3.94538
2.61743
−2.26463
0 1.00000 0 0 0 −3.92312 0 1.00000 0
1.2 0 1.00000 0 0 0 −3.07303 0 1.00000 0
1.3 0 1.00000 0 0 0 −2.05452 0 1.00000 0
1.4 0 1.00000 0 0 0 −0.846429 0 1.00000 0
1.5 0 1.00000 0 0 0 0.0132572 0 1.00000 0
1.6 0 1.00000 0 0 0 1.73400 0 1.00000 0
1.7 0 1.00000 0 0 0 4.14985 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.be 7
5.b even 2 1 9300.2.a.bd 7
5.c odd 4 2 1860.2.g.b ✓ 14
15.e even 4 2 5580.2.g.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.g.b ✓ 14 5.c odd 4 2
5580.2.g.e 14 15.e even 4 2
9300.2.a.bd 7 5.b even 2 1
9300.2.a.be 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9300))\):

\( T_{7}^{7} + 4T_{7}^{6} - 17T_{7}^{5} - 82T_{7}^{4} - 10T_{7}^{3} + 216T_{7}^{2} + 148T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{7} - 4T_{11}^{6} - 24T_{11}^{5} + 60T_{11}^{4} + 96T_{11}^{3} - 104T_{11}^{2} - 96T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{7} + 10T_{13}^{6} + 6T_{13}^{5} - 192T_{13}^{4} - 604T_{13}^{3} - 482T_{13}^{2} + 160T_{13} + 224 \) Copy content Toggle raw display
\( T_{17}^{7} + 14T_{17}^{6} + 26T_{17}^{5} - 424T_{17}^{4} - 2716T_{17}^{3} - 6556T_{17}^{2} - 7008T_{17} - 2688 \) 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
$7$ \( T^{7} + 4 T^{6} + \cdots - 2 \) Copy content Toggle raw display
$11$ \( T^{7} - 4 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{7} + 10 T^{6} + \cdots + 224 \) Copy content Toggle raw display
$17$ \( T^{7} + 14 T^{6} + \cdots - 2688 \) Copy content Toggle raw display
$19$ \( T^{7} + 6 T^{6} + \cdots - 1008 \) Copy content Toggle raw display
$23$ \( T^{7} + 8 T^{6} + \cdots - 5184 \) Copy content Toggle raw display
$29$ \( T^{7} + 2 T^{6} + \cdots - 8632 \) Copy content Toggle raw display
$31$ \( (T - 1)^{7} \) Copy content Toggle raw display
$37$ \( T^{7} + 12 T^{6} + \cdots - 5832 \) Copy content Toggle raw display
$41$ \( T^{7} - 10 T^{6} + \cdots + 62184 \) Copy content Toggle raw display
$43$ \( T^{7} + 8 T^{6} + \cdots - 368864 \) Copy content Toggle raw display
$47$ \( T^{7} + 16 T^{6} + \cdots - 16944 \) Copy content Toggle raw display
$53$ \( T^{7} + 12 T^{6} + \cdots + 107264 \) Copy content Toggle raw display
$59$ \( T^{7} + 2 T^{6} + \cdots - 562 \) Copy content Toggle raw display
$61$ \( T^{7} - 2 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{7} + 18 T^{6} + \cdots - 3848 \) Copy content Toggle raw display
$71$ \( T^{7} - 6 T^{6} + \cdots + 1213926 \) Copy content Toggle raw display
$73$ \( T^{7} + 22 T^{6} + \cdots + 12498984 \) Copy content Toggle raw display
$79$ \( T^{7} + 24 T^{6} + \cdots - 1587088 \) Copy content Toggle raw display
$83$ \( T^{7} + 12 T^{6} + \cdots - 1836144 \) Copy content Toggle raw display
$89$ \( T^{7} + 12 T^{6} + \cdots - 8865336 \) Copy content Toggle raw display
$97$ \( T^{7} + 36 T^{6} + \cdots - 584856 \) Copy content Toggle raw display
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