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Properties

Label 725.2.e.d
Level $725$
Weight $2$
Character orbit 725.e
Analytic conductor $5.789$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(157,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.e (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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{4} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{4} + 56 q^{9} + 12 q^{14} + 80 q^{16} - 4 q^{19} - 28 q^{21} - 12 q^{29} - 40 q^{31} - 8 q^{34} - 64 q^{36} - 68 q^{39} - 28 q^{41} + 40 q^{44} - 40 q^{46} - 40 q^{56} - 24 q^{61} - 80 q^{66} - 40 q^{69} - 108 q^{76} + 72 q^{79} + 200 q^{81} - 4 q^{84} - 40 q^{89} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 − 2.65926i −1.93483 −5.07167 0 5.14522i 3.36169 + 3.36169i 8.16838i 0.743572 0
157.2 − 2.52500i −3.42491 −4.37564 0 8.64790i −1.78637 − 1.78637i 5.99850i 8.72998 0
157.3 − 2.46344i 1.09135 −4.06854 0 − 2.68848i −0.369215 − 0.369215i 5.09573i −1.80895 0
157.4 − 2.15259i 0.669815 −2.63363 0 − 1.44183i −1.74569 − 1.74569i 1.36393i −2.55135 0
157.5 − 1.63047i −0.836393 −0.658440 0 1.36372i 1.64475 + 1.64475i − 2.18738i −2.30045 0
157.6 − 1.23283i 3.00892 0.480125 0 − 3.70950i 1.53900 + 1.53900i − 3.05758i 6.05362 0
157.7 − 1.10914i −2.35179 0.769812 0 2.60846i 0.825650 + 0.825650i − 3.07210i 2.53090 0
157.8 − 0.624848i −1.25254 1.60957 0 0.782644i −2.62256 − 2.62256i − 2.25543i −1.43115 0
157.9 − 0.204638i 3.05184 1.95812 0 − 0.624521i −3.34957 − 3.34957i − 0.809982i 6.31371 0
157.10 − 0.0985170i 0.848593 1.99029 0 − 0.0836009i 0.686476 + 0.686476i − 0.393112i −2.27989 0
157.11 0.0985170i −0.848593 1.99029 0 − 0.0836009i −0.686476 − 0.686476i 0.393112i −2.27989 0
157.12 0.204638i −3.05184 1.95812 0 − 0.624521i 3.34957 + 3.34957i 0.809982i 6.31371 0
157.13 0.624848i 1.25254 1.60957 0 0.782644i 2.62256 + 2.62256i 2.25543i −1.43115 0
157.14 1.10914i 2.35179 0.769812 0 2.60846i −0.825650 − 0.825650i 3.07210i 2.53090 0
157.15 1.23283i −3.00892 0.480125 0 − 3.70950i −1.53900 − 1.53900i 3.05758i 6.05362 0
157.16 1.63047i 0.836393 −0.658440 0 1.36372i −1.64475 − 1.64475i 2.18738i −2.30045 0
157.17 2.15259i −0.669815 −2.63363 0 − 1.44183i 1.74569 + 1.74569i − 1.36393i −2.55135 0
157.18 2.46344i −1.09135 −4.06854 0 − 2.68848i 0.369215 + 0.369215i − 5.09573i −1.80895 0
157.19 2.52500i 3.42491 −4.37564 0 8.64790i 1.78637 + 1.78637i − 5.99850i 8.72998 0
157.20 2.65926i 1.93483 −5.07167 0 5.14522i −3.36169 − 3.36169i − 8.16838i 0.743572 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
145.e even 4 1 inner
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.e.d ✓ 40
5.b even 2 1 inner 725.2.e.d ✓ 40
5.c odd 4 2 725.2.j.d yes 40
29.c odd 4 1 725.2.j.d yes 40
145.e even 4 1 inner 725.2.e.d ✓ 40
145.f odd 4 1 725.2.j.d yes 40
145.j even 4 1 inner 725.2.e.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.e.d ✓ 40 1.a even 1 1 trivial
725.2.e.d ✓ 40 5.b even 2 1 inner
725.2.e.d ✓ 40 145.e even 4 1 inner
725.2.e.d ✓ 40 145.j even 4 1 inner
725.2.j.d yes 40 5.c odd 4 2
725.2.j.d yes 40 29.c odd 4 1
725.2.j.d yes 40 145.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 30 T_{2}^{18} + 370 T_{2}^{16} + 2420 T_{2}^{14} + 9043 T_{2}^{12} + 19436 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display