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Properties

Label 925.2.y.b
Level $925$
Weight $2$
Character orbit 925.y
Analytic conductor $7.386$
Analytic rank $0$
Dimension $68$
Inner twists $2$

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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 = [68] 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: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 185)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 6 q^{2} + 4 q^{3} + 30 q^{4} - 8 q^{6} + 2 q^{7} + 10 q^{12} + 6 q^{13} - 26 q^{16} + 10 q^{17} + 8 q^{18} - 4 q^{19} - 12 q^{21} + 14 q^{22} - 24 q^{26} - 68 q^{27} - 14 q^{28} - 14 q^{29} - 24 q^{31}+ \cdots - 32 q^{99}+O(q^{100}) \) 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 −2.10002 + 1.21245i −0.586925 − 2.19044i 1.94006 − 3.36028i 0 3.88834 + 3.88834i 0.132002 + 0.492639i 4.55908i −1.85545 + 1.07124i 0
193.2 −2.08716 + 1.20502i 0.740695 + 2.76431i 1.90415 − 3.29809i 0 −4.87700 − 4.87700i −0.856324 − 3.19584i 4.35808i −4.49471 + 2.59502i 0
193.3 −1.80193 + 1.04034i 0.294240 + 1.09812i 1.16463 − 2.01721i 0 −1.67262 − 1.67262i 0.0206204 + 0.0769563i 0.685107i 1.47879 − 0.853779i 0
193.4 −1.11517 + 0.643846i 0.352022 + 1.31376i −0.170924 + 0.296050i 0 −1.23843 − 1.23843i 0.459253 + 1.71395i − 3.01558i 0.996020 − 0.575052i 0
193.5 −0.971183 + 0.560713i −0.168506 − 0.628874i −0.371203 + 0.642942i 0 0.516268 + 0.516268i 0.377247 + 1.40791i − 3.07540i 2.23099 − 1.28806i 0
193.6 −0.877658 + 0.506716i −0.458707 − 1.71192i −0.486478 + 0.842604i 0 1.27004 + 1.27004i −1.12037 − 4.18129i − 3.01289i −0.122169 + 0.0705344i 0
193.7 −0.407096 + 0.235037i 0.838017 + 3.12752i −0.889515 + 1.54069i 0 −1.07624 − 1.07624i 0.968227 + 3.61347i − 1.77643i −6.48105 + 3.74183i 0
193.8 0.138202 − 0.0797909i −0.607844 − 2.26850i −0.987267 + 1.71000i 0 −0.265011 − 0.265011i 0.345018 + 1.28763i 0.634263i −2.17856 + 1.25779i 0
193.9 0.212566 − 0.122725i 0.491666 + 1.83492i −0.969877 + 1.67988i 0 0.329702 + 0.329702i −1.00868 − 3.76446i 0.967013i −0.527123 + 0.304334i 0
193.10 0.614089 − 0.354544i 0.0751630 + 0.280512i −0.748597 + 1.29661i 0 0.145611 + 0.145611i 0.977086 + 3.64653i 2.47982i 2.52504 − 1.45783i 0
193.11 0.713738 − 0.412077i −0.475172 − 1.77336i −0.660386 + 1.14382i 0 −1.06991 − 1.06991i −0.320840 − 1.19739i 2.73682i −0.320958 + 0.185305i 0
193.12 1.25283 − 0.723320i 0.212077 + 0.791482i 0.0463849 − 0.0803411i 0 0.838191 + 0.838191i −0.425785 − 1.58905i 2.75908i 2.01661 − 1.16429i 0
193.13 1.54509 − 0.892058i 0.471934 + 1.76128i 0.591534 − 1.02457i 0 2.30034 + 2.30034i 0.134169 + 0.500724i 1.45750i −0.281310 + 0.162414i 0
193.14 1.72086 − 0.993536i −0.630160 − 2.35179i 0.974229 − 1.68741i 0 −3.42100 − 3.42100i −0.704548 − 2.62941i 0.102418i −2.53574 + 1.46401i 0
193.15 1.90916 − 1.10225i 0.817378 + 3.05050i 1.42993 − 2.47671i 0 4.92293 + 4.92293i −0.0394498 − 0.147229i − 1.89556i −6.03934 + 3.48681i 0
193.16 2.10315 − 1.21425i −0.180195 − 0.672498i 1.94883 − 3.37547i 0 −1.19556 − 1.19556i 1.19887 + 4.47423i − 4.60849i 2.17829 − 1.25764i 0
193.17 2.38259 − 1.37559i −0.185681 − 0.692973i 2.78450 − 4.82289i 0 −1.39565 − 1.39565i −0.502511 − 1.87540i − 9.81895i 2.15234 − 1.24266i 0
393.1 −2.23824 − 1.29225i 2.39703 + 0.642284i 2.33980 + 4.05266i 0 −4.53514 − 4.53514i −1.07329 − 0.287588i − 6.92543i 2.73517 + 1.57915i 0
393.2 −2.07760 − 1.19950i −0.543256 − 0.145565i 1.87760 + 3.25211i 0 0.954062 + 0.954062i −1.62066 − 0.434254i − 4.21075i −2.32414 − 1.34184i 0
393.3 −2.01885 − 1.16559i −0.204992 − 0.0549274i 1.71718 + 2.97424i 0 0.349826 + 0.349826i 4.70941 + 1.26188i − 3.34372i −2.55907 − 1.47748i 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.b 68
5.b even 2 1 185.2.u.a yes 68
5.c odd 4 1 185.2.p.a ✓ 68
5.c odd 4 1 925.2.t.b 68
37.g odd 12 1 925.2.t.b 68
185.p even 12 1 185.2.u.a yes 68
185.q odd 12 1 185.2.p.a ✓ 68
185.u even 12 1 inner 925.2.y.b 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.p.a ✓ 68 5.c odd 4 1
185.2.p.a ✓ 68 185.q odd 12 1
185.2.u.a yes 68 5.b even 2 1
185.2.u.a yes 68 185.p even 12 1
925.2.t.b 68 5.c odd 4 1
925.2.t.b 68 37.g odd 12 1
925.2.y.b 68 1.a even 1 1 trivial
925.2.y.b 68 185.u even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{68} - 6 T_{2}^{67} - 31 T_{2}^{66} + 258 T_{2}^{65} + 492 T_{2}^{64} - 6174 T_{2}^{63} + \cdots + 11881 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display