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Properties

Label 350.8.a.d
Level $350$
Weight $8$
Character orbit 350.a
Self dual yes
Analytic conductor $109.335$
Analytic rank $1$
Dimension $1$
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] = [350,8,Mod(1,350)] 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("350.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-8,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.334758919\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 66 q^{3} + 64 q^{4} - 528 q^{6} + 343 q^{7} - 512 q^{8} + 2169 q^{9} + 40 q^{11} + 4224 q^{12} + 4452 q^{13} - 2744 q^{14} + 4096 q^{16} - 36502 q^{17} - 17352 q^{18} - 46222 q^{19} + 22638 q^{21}+ \cdots + 86760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0
−8.00000 66.0000 64.0000 0 −528.000 343.000 −512.000 2169.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 350.8.a.d 1
5.b even 2 1 14.8.a.b ✓ 1
5.c odd 4 2 350.8.c.b 2
15.d odd 2 1 126.8.a.c 1
20.d odd 2 1 112.8.a.d 1
35.c odd 2 1 98.8.a.c 1
35.i odd 6 2 98.8.c.a 2
35.j even 6 2 98.8.c.b 2
40.e odd 2 1 448.8.a.b 1
40.f even 2 1 448.8.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.8.a.b ✓ 1 5.b even 2 1
98.8.a.c 1 35.c odd 2 1
98.8.c.a 2 35.i odd 6 2
98.8.c.b 2 35.j even 6 2
112.8.a.d 1 20.d odd 2 1
126.8.a.c 1 15.d odd 2 1
350.8.a.d 1 1.a even 1 1 trivial
350.8.c.b 2 5.c odd 4 2
448.8.a.b 1 40.e odd 2 1
448.8.a.i 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 66 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(350))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T - 66 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 343 \) Copy content Toggle raw display
$11$ \( T - 40 \) Copy content Toggle raw display
$13$ \( T - 4452 \) Copy content Toggle raw display
$17$ \( T + 36502 \) Copy content Toggle raw display
$19$ \( T + 46222 \) Copy content Toggle raw display
$23$ \( T - 105200 \) Copy content Toggle raw display
$29$ \( T + 126334 \) Copy content Toggle raw display
$31$ \( T + 170964 \) Copy content Toggle raw display
$37$ \( T + 20954 \) Copy content Toggle raw display
$41$ \( T - 318486 \) Copy content Toggle raw display
$43$ \( T + 77744 \) Copy content Toggle raw display
$47$ \( T + 703716 \) Copy content Toggle raw display
$53$ \( T + 1603278 \) Copy content Toggle raw display
$59$ \( T + 1171894 \) Copy content Toggle raw display
$61$ \( T + 2068872 \) Copy content Toggle raw display
$67$ \( T - 994268 \) Copy content Toggle raw display
$71$ \( T - 33280 \) Copy content Toggle raw display
$73$ \( T - 2971454 \) Copy content Toggle raw display
$79$ \( T + 2376168 \) Copy content Toggle raw display
$83$ \( T - 2122358 \) Copy content Toggle raw display
$89$ \( T - 6920346 \) Copy content Toggle raw display
$97$ \( T + 4952710 \) Copy content Toggle raw display
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