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Properties

Label 75.8.e.e
Level $75$
Weight $8$
Character orbit 75.e
Analytic conductor $23.429$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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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] = [75,8,Mod(32,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) 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("75.32"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,-828] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 828 q^{6} + 91096 q^{16} - 658848 q^{21} + 492704 q^{31} - 3405396 q^{36} - 7065936 q^{46} + 6088032 q^{51} - 21311696 q^{61} + 34621020 q^{66} + 1923688 q^{76} + 23831352 q^{81} + 45825984 q^{91}+ \cdots + 4217076 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −13.6541 + 13.6541i −34.8320 + 31.2047i − 244.867i 0 49.5271 − 901.669i 896.113 + 896.113i 1595.70 + 1595.70i 239.534 − 2173.84i 0
32.2 −13.6541 + 13.6541i 31.2047 − 34.8320i − 244.867i 0 49.5271 + 901.669i −896.113 − 896.113i 1595.70 + 1595.70i −239.534 − 2173.84i 0
32.3 −11.5625 + 11.5625i 4.54415 + 46.5441i − 139.384i 0 −590.709 − 485.625i 632.779 + 632.779i 131.626 + 131.626i −2145.70 + 423.006i 0
32.4 −11.5625 + 11.5625i 46.5441 + 4.54415i − 139.384i 0 −590.709 + 485.625i −632.779 − 632.779i 131.626 + 131.626i 2145.70 + 423.006i 0
32.5 −9.04944 + 9.04944i −41.6173 − 21.3308i − 35.7849i 0 569.645 − 183.582i 248.346 + 248.346i −834.495 − 834.495i 1277.00 + 1775.46i 0
32.6 −9.04944 + 9.04944i −21.3308 − 41.6173i − 35.7849i 0 569.645 + 183.582i −248.346 − 248.346i −834.495 − 834.495i −1277.00 + 1775.46i 0
32.7 −2.73539 + 2.73539i 1.50167 + 46.7413i 113.035i 0 −131.963 − 123.748i −186.453 − 186.453i −659.326 − 659.326i −2182.49 + 140.380i 0
32.8 −2.73539 + 2.73539i 46.7413 + 1.50167i 113.035i 0 −131.963 + 123.748i 186.453 + 186.453i −659.326 − 659.326i 2182.49 + 140.380i 0
32.9 2.73539 − 2.73539i −46.7413 − 1.50167i 113.035i 0 −131.963 + 123.748i −186.453 − 186.453i 659.326 + 659.326i 2182.49 + 140.380i 0
32.10 2.73539 − 2.73539i −1.50167 − 46.7413i 113.035i 0 −131.963 − 123.748i 186.453 + 186.453i 659.326 + 659.326i −2182.49 + 140.380i 0
32.11 9.04944 − 9.04944i 21.3308 + 41.6173i − 35.7849i 0 569.645 + 183.582i 248.346 + 248.346i 834.495 + 834.495i −1277.00 + 1775.46i 0
32.12 9.04944 − 9.04944i 41.6173 + 21.3308i − 35.7849i 0 569.645 − 183.582i −248.346 − 248.346i 834.495 + 834.495i 1277.00 + 1775.46i 0
32.13 11.5625 − 11.5625i −46.5441 − 4.54415i − 139.384i 0 −590.709 + 485.625i 632.779 + 632.779i −131.626 − 131.626i 2145.70 + 423.006i 0
32.14 11.5625 − 11.5625i −4.54415 − 46.5441i − 139.384i 0 −590.709 − 485.625i −632.779 − 632.779i −131.626 − 131.626i −2145.70 + 423.006i 0
32.15 13.6541 − 13.6541i −31.2047 + 34.8320i − 244.867i 0 49.5271 + 901.669i 896.113 + 896.113i −1595.70 − 1595.70i −239.534 − 2173.84i 0
32.16 13.6541 − 13.6541i 34.8320 − 31.2047i − 244.867i 0 49.5271 − 901.669i −896.113 − 896.113i −1595.70 − 1595.70i 239.534 − 2173.84i 0
68.1 −13.6541 − 13.6541i −34.8320 − 31.2047i 244.867i 0 49.5271 + 901.669i 896.113 − 896.113i 1595.70 − 1595.70i 239.534 + 2173.84i 0
68.2 −13.6541 − 13.6541i 31.2047 + 34.8320i 244.867i 0 49.5271 − 901.669i −896.113 + 896.113i 1595.70 − 1595.70i −239.534 + 2173.84i 0
68.3 −11.5625 − 11.5625i 4.54415 − 46.5441i 139.384i 0 −590.709 + 485.625i 632.779 − 632.779i 131.626 − 131.626i −2145.70 − 423.006i 0
68.4 −11.5625 − 11.5625i 46.5441 − 4.54415i 139.384i 0 −590.709 − 485.625i −632.779 + 632.779i 131.626 − 131.626i 2145.70 − 423.006i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.8.e.e ✓ 32
3.b odd 2 1 inner 75.8.e.e ✓ 32
5.b even 2 1 inner 75.8.e.e ✓ 32
5.c odd 4 2 inner 75.8.e.e ✓ 32
15.d odd 2 1 inner 75.8.e.e ✓ 32
15.e even 4 2 inner 75.8.e.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.8.e.e ✓ 32 1.a even 1 1 trivial
75.8.e.e ✓ 32 3.b odd 2 1 inner
75.8.e.e ✓ 32 5.b even 2 1 inner
75.8.e.e ✓ 32 5.c odd 4 2 inner
75.8.e.e ✓ 32 15.d odd 2 1 inner
75.8.e.e ✓ 32 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 237573T_{2}^{12} + 15640338276T_{2}^{8} + 270130343472000T_{2}^{4} + 59712200576640000 \) acting on \(S_{8}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display