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Properties

Label 400.8.a.u
Level $400$
Weight $8$
Character orbit 400.a
Self dual yes
Analytic conductor $124.954$
Analytic rank $0$
Dimension $2$
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] = [400,8,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-40,0,0,0,-280] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2521}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 630 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 100)
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 \(\beta = \sqrt{2521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 20) q^{3} + ( - 6 \beta - 140) q^{7} + (40 \beta + 734) q^{9} + ( - 45 \beta - 1140) q^{11} + ( - 156 \beta - 2680) q^{13} + (432 \beta + 14565) q^{17} + ( - 885 \beta - 10484) q^{19} + (260 \beta + 17926) q^{21}+ \cdots + ( - 78630 \beta - 5374560) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 40 q^{3} - 280 q^{7} + 1468 q^{9} - 2280 q^{11} - 5360 q^{13} + 29130 q^{17} - 20968 q^{19} + 35852 q^{21} - 11040 q^{23} - 143560 q^{27} - 230952 q^{29} - 214672 q^{31} + 272490 q^{33} + 734620 q^{37}+ \cdots - 10749120 q^{99}+O(q^{100}) \) 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
25.6048
−24.6048
0 −70.2096 0 0 0 −441.257 0 2742.38 0
1.2 0 30.2096 0 0 0 161.257 0 −1274.38 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 400.8.a.u 2
4.b odd 2 1 100.8.a.d yes 2
5.b even 2 1 400.8.a.be 2
5.c odd 4 2 400.8.c.r 4
20.d odd 2 1 100.8.a.b ✓ 2
20.e even 4 2 100.8.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.8.a.b ✓ 2 20.d odd 2 1
100.8.a.d yes 2 4.b odd 2 1
100.8.c.c 4 20.e even 4 2
400.8.a.u 2 1.a even 1 1 trivial
400.8.a.be 2 5.b even 2 1
400.8.c.r 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 40T_{3} - 2121 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 40T - 2121 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 280T - 71156 \) Copy content Toggle raw display
$11$ \( T^{2} + 2280 T - 3805425 \) Copy content Toggle raw display
$13$ \( T^{2} + 5360 T - 54168656 \) Copy content Toggle raw display
$17$ \( T^{2} - 29130 T - 258339879 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 1864595969 \) Copy content Toggle raw display
$23$ \( T^{2} + 11040 T - 329740164 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 3451378176 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 11464294396 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 119361420724 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 122417065719 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 52831567600 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 922416343056 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 1294466823324 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 3875694814656 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 1043925150476 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2330589677359 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 564685588944 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 2579323674551 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 5717491570196 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 14724853048809 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 36039722681439 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 30418309126564 \) Copy content Toggle raw display
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