Properties

Label 925.2.f.f
Level $925$
Weight $2$
Character orbit 925.f
Analytic conductor $7.386$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(43,925)] 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("925.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0] 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: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 40 q^{4} + 8 q^{14} + 64 q^{16} + 12 q^{19} - 52 q^{24} - 16 q^{26} + 20 q^{31} + 32 q^{39} - 24 q^{46} - 36 q^{51} - 116 q^{54} - 28 q^{56} + 28 q^{59} - 20 q^{61} - 24 q^{64} - 76 q^{66} + 68 q^{69}+ \cdots - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 − 2.75472i −1.09682 + 1.09682i −5.58848 0 3.02143 + 3.02143i −1.32452 + 1.32452i 9.88526i 0.593972i 0
43.2 − 2.42940i 1.67335 − 1.67335i −3.90197 0 −4.06524 − 4.06524i 1.37550 − 1.37550i 4.62064i − 2.60022i 0
43.3 − 2.30263i −1.40559 + 1.40559i −3.30213 0 3.23656 + 3.23656i 1.16450 − 1.16450i 2.99832i − 0.951370i 0
43.4 − 2.29119i 0.219692 − 0.219692i −3.24954 0 −0.503355 − 0.503355i −2.77283 + 2.77283i 2.86294i 2.90347i 0
43.5 − 1.84351i −1.23056 + 1.23056i −1.39852 0 2.26855 + 2.26855i 2.61463 − 2.61463i − 1.10884i − 0.0285668i 0
43.6 − 1.72281i 1.57574 − 1.57574i −0.968057 0 −2.71470 − 2.71470i −0.0954098 + 0.0954098i − 1.77784i − 1.96593i 0
43.7 − 1.23147i −0.816861 + 0.816861i 0.483484 0 1.00594 + 1.00594i −2.25739 + 2.25739i − 3.05833i 1.66548i 0
43.8 − 0.892090i 0.959592 − 0.959592i 1.20418 0 −0.856042 − 0.856042i 2.14902 − 2.14902i − 2.85841i 1.15837i 0
43.9 − 0.749237i 2.39550 − 2.39550i 1.43864 0 −1.79479 − 1.79479i −2.74187 + 2.74187i − 2.57636i − 8.47680i 0
43.10 − 0.715481i 0.0427756 − 0.0427756i 1.48809 0 −0.0306051 − 0.0306051i −0.0275714 + 0.0275714i − 2.49566i 2.99634i 0
43.11 − 0.452300i −0.789769 + 0.789769i 1.79542 0 0.357213 + 0.357213i 0.641403 − 0.641403i − 1.71667i 1.75253i 0
43.12 − 0.0334797i −2.24135 + 2.24135i 1.99888 0 0.0750395 + 0.0750395i −0.905280 + 0.905280i − 0.133881i − 7.04726i 0
43.13 0.0334797i 2.24135 − 2.24135i 1.99888 0 0.0750395 + 0.0750395i 0.905280 − 0.905280i 0.133881i − 7.04726i 0
43.14 0.452300i 0.789769 − 0.789769i 1.79542 0 0.357213 + 0.357213i −0.641403 + 0.641403i 1.71667i 1.75253i 0
43.15 0.715481i −0.0427756 + 0.0427756i 1.48809 0 −0.0306051 − 0.0306051i 0.0275714 − 0.0275714i 2.49566i 2.99634i 0
43.16 0.749237i −2.39550 + 2.39550i 1.43864 0 −1.79479 − 1.79479i 2.74187 − 2.74187i 2.57636i − 8.47680i 0
43.17 0.892090i −0.959592 + 0.959592i 1.20418 0 −0.856042 − 0.856042i −2.14902 + 2.14902i 2.85841i 1.15837i 0
43.18 1.23147i 0.816861 − 0.816861i 0.483484 0 1.00594 + 1.00594i 2.25739 − 2.25739i 3.05833i 1.66548i 0
43.19 1.72281i −1.57574 + 1.57574i −0.968057 0 −2.71470 − 2.71470i 0.0954098 − 0.0954098i 1.77784i − 1.96593i 0
43.20 1.84351i 1.23056 − 1.23056i −1.39852 0 2.26855 + 2.26855i −2.61463 + 2.61463i 1.10884i − 0.0285668i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 43.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
185.f even 4 1 inner
185.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.f.f ✓ 48
5.b even 2 1 inner 925.2.f.f ✓ 48
5.c odd 4 2 925.2.k.f yes 48
37.d odd 4 1 925.2.k.f yes 48
185.f even 4 1 inner 925.2.f.f ✓ 48
185.j odd 4 1 925.2.k.f yes 48
185.k even 4 1 inner 925.2.f.f ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.f.f ✓ 48 1.a even 1 1 trivial
925.2.f.f ✓ 48 5.b even 2 1 inner
925.2.f.f ✓ 48 185.f even 4 1 inner
925.2.f.f ✓ 48 185.k even 4 1 inner
925.2.k.f yes 48 5.c odd 4 2
925.2.k.f yes 48 37.d odd 4 1
925.2.k.f yes 48 185.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\):

\( T_{2}^{24} + 34 T_{2}^{22} + 492 T_{2}^{20} + 3962 T_{2}^{18} + 19498 T_{2}^{16} + 60710 T_{2}^{14} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{48} + 326 T_{3}^{44} + 38383 T_{3}^{40} + 2085720 T_{3}^{36} + 59465183 T_{3}^{32} + 942180736 T_{3}^{28} + \cdots + 10000 \) Copy content Toggle raw display