Properties

Label 925.2.bq.a
Level $925$
Weight $2$
Character orbit 925.bq
Analytic conductor $7.386$
Analytic rank $0$
Dimension $144$
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(32,925)] 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("925.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bq (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(12\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 24 q^{14} + 60 q^{16} + 24 q^{19} + 36 q^{21} + 12 q^{29} - 12 q^{31} - 60 q^{34} + 60 q^{39} + 24 q^{41} - 60 q^{44} + 132 q^{49} - 120 q^{51} - 12 q^{54} - 96 q^{56} - 24 q^{61} - 12 q^{64} - 216 q^{66}+ \cdots - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.850922 − 2.33789i 0.694124 + 1.48855i −3.20957 + 2.69315i 0 2.88943 − 2.88943i −0.0925011 + 0.132105i 4.71815 + 2.72402i 0.194377 − 0.231650i 0
32.2 −0.771426 − 2.11947i −0.886187 − 1.90043i −2.36499 + 1.98446i 0 −3.34429 + 3.34429i 2.21949 − 3.16976i 2.12379 + 1.22617i −0.897959 + 1.07015i 0
32.3 −0.595060 − 1.63491i −0.831862 − 1.78393i −0.786759 + 0.660169i 0 −2.42157 + 2.42157i −2.36047 + 3.37111i −1.46600 − 0.846397i −0.562066 + 0.669844i 0
32.4 −0.450237 − 1.23702i 0.222974 + 0.478168i 0.204595 − 0.171675i 0 0.491111 − 0.491111i −1.29540 + 1.85002i −2.58456 − 1.49220i 1.74944 − 2.08490i 0
32.5 −0.427687 − 1.17506i 1.29161 + 2.76987i 0.334240 − 0.280461i 0 2.70236 − 2.70236i 1.89689 − 2.70903i −2.63839 − 1.52327i −4.07558 + 4.85708i 0
32.6 −0.0987344 − 0.271270i −0.656375 − 1.40760i 1.46825 − 1.23201i 0 −0.317034 + 0.317034i 0.734500 − 1.04897i −0.979182 − 0.565331i 0.377850 − 0.450304i 0
32.7 0.0987344 + 0.271270i 0.656375 + 1.40760i 1.46825 − 1.23201i 0 −0.317034 + 0.317034i −0.734500 + 1.04897i 0.979182 + 0.565331i 0.377850 − 0.450304i 0
32.8 0.427687 + 1.17506i −1.29161 − 2.76987i 0.334240 − 0.280461i 0 2.70236 − 2.70236i −1.89689 + 2.70903i 2.63839 + 1.52327i −4.07558 + 4.85708i 0
32.9 0.450237 + 1.23702i −0.222974 − 0.478168i 0.204595 − 0.171675i 0 0.491111 − 0.491111i 1.29540 − 1.85002i 2.58456 + 1.49220i 1.74944 − 2.08490i 0
32.10 0.595060 + 1.63491i 0.831862 + 1.78393i −0.786759 + 0.660169i 0 −2.42157 + 2.42157i 2.36047 − 3.37111i 1.46600 + 0.846397i −0.562066 + 0.669844i 0
32.11 0.771426 + 2.11947i 0.886187 + 1.90043i −2.36499 + 1.98446i 0 −3.34429 + 3.34429i −2.21949 + 3.16976i −2.12379 − 1.22617i −0.897959 + 1.07015i 0
32.12 0.850922 + 2.33789i −0.694124 − 1.48855i −3.20957 + 2.69315i 0 2.88943 − 2.88943i 0.0925011 − 0.132105i −4.71815 − 2.72402i 0.194377 − 0.231650i 0
57.1 −2.72658 + 0.480770i 0.779590 − 0.545875i 5.32373 − 1.93768i 0 −1.86318 + 1.86318i −0.277870 − 3.17607i −8.78859 + 5.07410i −0.716279 + 1.96796i 0
57.2 −2.30917 + 0.407168i −2.31475 + 1.62080i 3.28708 − 1.19640i 0 4.68520 − 4.68520i 0.0195235 + 0.223154i −3.04198 + 1.75629i 1.70499 − 4.68443i 0
57.3 −1.84316 + 0.324998i 1.98541 − 1.39020i 1.41222 − 0.514005i 0 −3.20760 + 3.20760i 0.184994 + 2.11450i 0.805805 − 0.465232i 0.983133 − 2.70114i 0
57.4 −1.46445 + 0.258221i −0.214667 + 0.150311i 0.198540 − 0.0722628i 0 0.275555 − 0.275555i −0.165575 − 1.89253i 2.30353 − 1.32995i −1.00257 + 2.75454i 0
57.5 −0.288673 + 0.0509009i 0.188313 − 0.131858i −1.79864 + 0.654653i 0 −0.0476493 + 0.0476493i 0.409344 + 4.67883i 0.993609 − 0.573660i −1.00799 + 2.76942i 0
57.6 −0.0770985 + 0.0135945i −2.33324 + 1.63376i −1.87363 + 0.681944i 0 0.157679 − 0.157679i 0.0851926 + 0.973756i 0.270781 − 0.156336i 1.74881 − 4.80482i 0
57.7 0.0770985 − 0.0135945i 2.33324 − 1.63376i −1.87363 + 0.681944i 0 0.157679 − 0.157679i −0.0851926 − 0.973756i −0.270781 + 0.156336i 1.74881 − 4.80482i 0
57.8 0.288673 − 0.0509009i −0.188313 + 0.131858i −1.79864 + 0.654653i 0 −0.0476493 + 0.0476493i −0.409344 − 4.67883i −0.993609 + 0.573660i −1.00799 + 2.76942i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 32.12
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.z even 36 1 inner
185.bc even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.bq.a yes 144
5.b even 2 1 inner 925.2.bq.a yes 144
5.c odd 4 2 925.2.bn.a ✓ 144
37.i odd 36 1 925.2.bn.a ✓ 144
185.z even 36 1 inner 925.2.bq.a yes 144
185.ba odd 36 1 925.2.bn.a ✓ 144
185.bc even 36 1 inner 925.2.bq.a yes 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.bn.a ✓ 144 5.c odd 4 2
925.2.bn.a ✓ 144 37.i odd 36 1
925.2.bn.a ✓ 144 185.ba odd 36 1
925.2.bq.a yes 144 1.a even 1 1 trivial
925.2.bq.a yes 144 5.b even 2 1 inner
925.2.bq.a yes 144 185.z even 36 1 inner
925.2.bq.a yes 144 185.bc even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - 15 T_{2}^{140} - 898 T_{2}^{138} - 576 T_{2}^{136} + 17874 T_{2}^{134} + 522721 T_{2}^{132} + \cdots + 1073283121 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display