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Properties

Label 925.2.y.a
Level $925$
Weight $2$
Character orbit 925.y
Analytic conductor $7.386$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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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] = [925,2,Mod(193,925)] 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("925.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] 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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 36 q^{4} + 8 q^{14} - 72 q^{16} + 16 q^{19} - 36 q^{21} - 4 q^{24} + 8 q^{26} + 24 q^{31} + 60 q^{34} - 24 q^{41} + 24 q^{44} + 12 q^{49} + 84 q^{51} + 28 q^{54} + 104 q^{56} + 4 q^{59} - 24 q^{61}+ \cdots + 120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 −2.41884 + 1.39652i −0.623186 − 2.32576i 2.90051 − 5.02383i 0 4.75535 + 4.75535i −1.09325 − 4.08007i 10.6164i −2.42273 + 1.39876i 0
193.2 −2.25962 + 1.30459i 0.755810 + 2.82072i 2.40393 − 4.16373i 0 −5.38774 − 5.38774i 0.120281 + 0.448895i 7.32622i −4.78714 + 2.76386i 0
193.3 −1.88869 + 1.09044i −0.177680 − 0.663112i 1.37810 − 2.38694i 0 1.05866 + 1.05866i 1.19194 + 4.44838i 1.64918i 2.18993 − 1.26436i 0
193.4 −1.53267 + 0.884889i 0.223454 + 0.833940i 0.566056 − 0.980437i 0 −1.08043 − 1.08043i −0.456178 − 1.70248i − 1.53597i 1.95255 − 1.12731i 0
193.5 −0.926614 + 0.534981i 0.561303 + 2.09481i −0.427591 + 0.740609i 0 −1.64079 − 1.64079i −0.642189 − 2.39668i − 3.05494i −1.47509 + 0.851645i 0
193.6 −0.841127 + 0.485625i −0.777162 − 2.90041i −0.528337 + 0.915107i 0 2.06220 + 2.06220i 0.0821598 + 0.306625i − 2.96879i −5.21031 + 3.00817i 0
193.7 −0.331719 + 0.191518i −0.222410 − 0.830044i −0.926642 + 1.60499i 0 0.232746 + 0.232746i −0.206604 − 0.771056i − 1.47595i 1.95857 − 1.13078i 0
193.8 0.331719 − 0.191518i 0.222410 + 0.830044i −0.926642 + 1.60499i 0 0.232746 + 0.232746i 0.206604 + 0.771056i 1.47595i 1.95857 − 1.13078i 0
193.9 0.841127 − 0.485625i 0.777162 + 2.90041i −0.528337 + 0.915107i 0 2.06220 + 2.06220i −0.0821598 − 0.306625i 2.96879i −5.21031 + 3.00817i 0
193.10 0.926614 − 0.534981i −0.561303 − 2.09481i −0.427591 + 0.740609i 0 −1.64079 − 1.64079i 0.642189 + 2.39668i 3.05494i −1.47509 + 0.851645i 0
193.11 1.53267 − 0.884889i −0.223454 − 0.833940i 0.566056 − 0.980437i 0 −1.08043 − 1.08043i 0.456178 + 1.70248i 1.53597i 1.95255 − 1.12731i 0
193.12 1.88869 − 1.09044i 0.177680 + 0.663112i 1.37810 − 2.38694i 0 1.05866 + 1.05866i −1.19194 − 4.44838i − 1.64918i 2.18993 − 1.26436i 0
193.13 2.25962 − 1.30459i −0.755810 − 2.82072i 2.40393 − 4.16373i 0 −5.38774 − 5.38774i −0.120281 − 0.448895i − 7.32622i −4.78714 + 2.76386i 0
193.14 2.41884 − 1.39652i 0.623186 + 2.32576i 2.90051 − 5.02383i 0 4.75535 + 4.75535i 1.09325 + 4.08007i − 10.6164i −2.42273 + 1.39876i 0
393.1 −2.37208 − 1.36952i −2.22164 − 0.595286i 2.75116 + 4.76515i 0 4.45464 + 4.45464i 2.91092 + 0.779980i − 9.59298i 1.98324 + 1.14502i 0
393.2 −2.18069 − 1.25902i 1.64641 + 0.441154i 2.17027 + 3.75901i 0 −3.03489 − 3.03489i −3.03103 − 0.812162i − 5.89357i −0.0820259 − 0.0473577i 0
393.3 −1.53162 − 0.884282i 0.529279 + 0.141820i 0.563910 + 0.976721i 0 −0.685247 − 0.685247i 2.53925 + 0.680389i 1.54251i −2.33805 − 1.34988i 0
393.4 −1.23383 − 0.712354i −2.41055 − 0.645905i 0.0148964 + 0.0258014i 0 2.51411 + 2.51411i −0.508749 − 0.136319i 2.80697i 2.79549 + 1.61398i 0
393.5 −1.19478 − 0.689808i 3.31268 + 0.887629i −0.0483289 − 0.0837081i 0 −3.34563 − 3.34563i 1.23915 + 0.332030i 2.89258i 7.58786 + 4.38085i 0
393.6 −0.520353 − 0.300426i −0.384911 − 0.103137i −0.819489 − 1.41940i 0 0.169305 + 0.169305i −3.80207 − 1.01876i 2.18649i −2.46056 − 1.42060i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.14
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.p even 12 1 inner
185.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.y.a yes 56
5.b even 2 1 inner 925.2.y.a yes 56
5.c odd 4 2 925.2.t.a ✓ 56
37.g odd 12 1 925.2.t.a ✓ 56
185.p even 12 1 inner 925.2.y.a yes 56
185.q odd 12 1 925.2.t.a ✓ 56
185.u even 12 1 inner 925.2.y.a yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.t.a ✓ 56 5.c odd 4 2
925.2.t.a ✓ 56 37.g odd 12 1
925.2.t.a ✓ 56 185.q odd 12 1
925.2.y.a yes 56 1.a even 1 1 trivial
925.2.y.a yes 56 5.b even 2 1 inner
925.2.y.a yes 56 185.p even 12 1 inner
925.2.y.a yes 56 185.u even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 46 T_{2}^{54} + 1186 T_{2}^{52} - 21008 T_{2}^{50} + 282297 T_{2}^{48} - 3008832 T_{2}^{46} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display