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Properties

Label 300.8.a.n
Level $300$
Weight $8$
Character orbit 300.a
Self dual yes
Analytic conductor $93.716$
Analytic rank $0$
Dimension $3$
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] = [300,8,Mod(1,300)] 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("300.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,81,0,0,0,351,0,2187,0,3138] 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: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 572x - 3990 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 27 q^{3} + (\beta_1 + 117) q^{7} + 729 q^{9} + (\beta_{2} + \beta_1 + 1046) q^{11} + ( - \beta_{2} + 6 \beta_1 + 1195) q^{13} + (\beta_{2} + 7 \beta_1 + 1742) q^{17} + (\beta_{2} - 13 \beta_1 + 7065) q^{19}+ \cdots + (729 \beta_{2} + 729 \beta_1 + 762534) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 81 q^{3} + 351 q^{7} + 2187 q^{9} + 3138 q^{11} + 3585 q^{13} + 5226 q^{17} + 21195 q^{19} + 9477 q^{21} + 40434 q^{23} + 59049 q^{27} + 26598 q^{29} - 143463 q^{31} + 84726 q^{33} + 536658 q^{37}+ \cdots + 2287602 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 572x - 3990 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 60\nu - 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 40\nu^{2} - 440\nu - 15120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 20 ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 22\beta _1 + 45800 ) / 120 \) 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
−18.3302
−7.97258
27.3028
0 27.0000 0 0 0 −1002.81 0 729.000 0
1.2 0 27.0000 0 0 0 −381.355 0 729.000 0
1.3 0 27.0000 0 0 0 1735.17 0 729.000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)

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 300.8.a.n yes 3
5.b even 2 1 300.8.a.m ✓ 3
5.c odd 4 2 300.8.d.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.8.a.m ✓ 3 5.b even 2 1
300.8.a.n yes 3 1.a even 1 1 trivial
300.8.d.h 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(300))\):

\( T_{7}^{3} - 351T_{7}^{2} - 2019333T_{7} - 663574813 \) Copy content Toggle raw display
\( T_{11}^{3} - 3138T_{11}^{2} - 65608452T_{11} + 287673763464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 27)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 351 T^{2} + \cdots - 663574813 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 287673763464 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 712072782125 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 49150196712 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 1414307134375 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 41285443261752 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 773565070226904 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 15\!\cdots\!39 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 747888011444056 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 23\!\cdots\!37 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 34\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 67\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 86\!\cdots\!43 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 52\!\cdots\!03 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 54\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 56\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 35\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 46\!\cdots\!41 \) Copy content Toggle raw display
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