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Properties

Label 925.2.f.d
Level $925$
Weight $2$
Character orbit 925.f
Analytic conductor $7.386$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [925,2,Mod(43,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 185)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} - 32 q^{4} - 14 q^{6} - 6 q^{7} + 36 q^{12} + 28 q^{14} + 24 q^{16} + 16 q^{17} - 4 q^{19} - 56 q^{22} + 18 q^{24} - 40 q^{26} + 26 q^{27} + 2 q^{28} + 4 q^{29} + 6 q^{31} + 32 q^{33} + 6 q^{37}+ \cdots - 36 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 − 2.69386i 0.437197 − 0.437197i −5.25686 0 −1.17775 − 1.17775i −0.705745 + 0.705745i 8.77351i 2.61772i 0
43.2 − 2.35455i −2.15478 + 2.15478i −3.54389 0 5.07354 + 5.07354i −0.853954 + 0.853954i 3.63515i − 6.28619i 0
43.3 − 1.89107i 1.70537 − 1.70537i −1.57614 0 −3.22498 − 3.22498i −0.431915 + 0.431915i − 0.801541i − 2.81659i 0
43.4 − 1.55778i −0.374562 + 0.374562i −0.426681 0 0.583485 + 0.583485i −2.24838 + 2.24838i − 2.45089i 2.71941i 0
43.5 − 0.165127i 1.77117 − 1.77117i 1.97273 0 −0.292468 − 0.292468i 0.477215 − 0.477215i − 0.656006i − 3.27410i 0
43.6 − 0.0562904i 0.608697 − 0.608697i 1.99683 0 −0.0342638 − 0.0342638i −3.37670 + 3.37670i − 0.224983i 2.25898i 0
43.7 0.917493i −1.17965 + 1.17965i 1.15821 0 −1.08232 − 1.08232i 2.45118 − 2.45118i 2.89763i 0.216841i 0
43.8 1.40625i −1.57253 + 1.57253i 0.0224648 0 −2.21137 − 2.21137i 0.0416138 − 0.0416138i 2.84409i − 1.94570i 0
43.9 1.64338i 1.31039 − 1.31039i −0.700698 0 2.15347 + 2.15347i 2.64033 − 2.64033i 2.13525i − 0.434259i 0
43.10 1.72182i 0.229804 − 0.229804i −0.964650 0 0.395681 + 0.395681i −1.16936 + 1.16936i 1.78268i 2.89438i 0
43.11 2.38796i −0.646256 + 0.646256i −3.70235 0 −1.54323 − 1.54323i −2.20744 + 2.20744i − 4.06513i 2.16471i 0
43.12 2.64177i −2.13485 + 2.13485i −4.97897 0 −5.63980 − 5.63980i 2.38316 − 2.38316i − 7.86977i − 6.11519i 0
882.1 − 2.64177i −2.13485 − 2.13485i −4.97897 0 −5.63980 + 5.63980i 2.38316 + 2.38316i 7.86977i 6.11519i 0
882.2 − 2.38796i −0.646256 − 0.646256i −3.70235 0 −1.54323 + 1.54323i −2.20744 − 2.20744i 4.06513i − 2.16471i 0
882.3 − 1.72182i 0.229804 + 0.229804i −0.964650 0 0.395681 − 0.395681i −1.16936 − 1.16936i − 1.78268i − 2.89438i 0
882.4 − 1.64338i 1.31039 + 1.31039i −0.700698 0 2.15347 − 2.15347i 2.64033 + 2.64033i − 2.13525i 0.434259i 0
882.5 − 1.40625i −1.57253 − 1.57253i 0.0224648 0 −2.21137 + 2.21137i 0.0416138 + 0.0416138i − 2.84409i 1.94570i 0
882.6 − 0.917493i −1.17965 − 1.17965i 1.15821 0 −1.08232 + 1.08232i 2.45118 + 2.45118i − 2.89763i − 0.216841i 0
882.7 0.0562904i 0.608697 + 0.608697i 1.99683 0 −0.0342638 + 0.0342638i −3.37670 − 3.37670i 0.224983i − 2.25898i 0
882.8 0.165127i 1.77117 + 1.77117i 1.97273 0 −0.292468 + 0.292468i 0.477215 + 0.477215i 0.656006i 3.27410i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f 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.d 24
5.b even 2 1 185.2.f.d ✓ 24
5.c odd 4 1 185.2.k.d yes 24
5.c odd 4 1 925.2.k.d 24
37.d odd 4 1 925.2.k.d 24
185.f even 4 1 inner 925.2.f.d 24
185.j odd 4 1 185.2.k.d yes 24
185.k even 4 1 185.2.f.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.f.d ✓ 24 5.b even 2 1
185.2.f.d ✓ 24 185.k even 4 1
185.2.k.d yes 24 5.c odd 4 1
185.2.k.d yes 24 185.j odd 4 1
925.2.f.d 24 1.a even 1 1 trivial
925.2.f.d 24 185.f even 4 1 inner
925.2.k.d 24 5.c odd 4 1
925.2.k.d 24 37.d odd 4 1

Hecke kernels

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

\( T_{2}^{24} + 40 T_{2}^{22} + 698 T_{2}^{20} + 6984 T_{2}^{18} + 44289 T_{2}^{16} + 185636 T_{2}^{14} + \cdots + 16 \) Copy content Toggle raw display
\( T_{3}^{24} + 4 T_{3}^{23} + 8 T_{3}^{22} - 6 T_{3}^{21} + 88 T_{3}^{20} + 344 T_{3}^{19} + 690 T_{3}^{18} + \cdots + 1024 \) Copy content Toggle raw display