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Properties

Label 925.2.bq.b
Level $925$
Weight $2$
Character orbit 925.bq
Analytic conductor $7.386$
Analytic rank $0$
Dimension $204$
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(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 = [204] 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: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 185)
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.

\(\operatorname{Tr}(f)(q) = \) \( 204 q + 12 q^{2} + 18 q^{3} - 24 q^{6} + 12 q^{7} + 18 q^{8} - 36 q^{11} + 12 q^{12} + 12 q^{13} + 24 q^{14} - 24 q^{16} - 12 q^{17} - 66 q^{18} - 24 q^{21} + 24 q^{22} + 18 q^{23} - 36 q^{24} - 12 q^{26}+ \cdots - 192 q^{99}+O(q^{100}) \) 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 −0.832436 − 2.28710i −0.328351 − 0.704150i −3.00578 + 2.52215i 0 −1.33713 + 1.33713i 0.721419 − 1.03029i 4.05492 + 2.34111i 1.54035 − 1.83572i 0
32.2 −0.821903 − 2.25816i −0.361505 − 0.775249i −2.89168 + 2.42640i 0 −1.45352 + 1.45352i −1.04388 + 1.49081i 3.69362 + 2.13251i 1.45804 − 1.73762i 0
32.3 −0.660073 − 1.81354i 0.883904 + 1.89554i −1.32113 + 1.10856i 0 2.85418 − 2.85418i −0.950354 + 1.35725i −0.460283 − 0.265744i −0.883415 + 1.05281i 0
32.4 −0.458203 − 1.25890i −0.797932 − 1.71117i 0.157204 − 0.131910i 0 −1.78858 + 1.78858i 2.01560 − 2.87857i −2.55851 − 1.47716i −0.363050 + 0.432666i 0
32.5 −0.450855 − 1.23871i 0.673579 + 1.44450i 0.200946 − 0.168613i 0 1.48563 − 1.48563i −1.15432 + 1.64853i −2.58267 − 1.49111i 0.295505 − 0.352169i 0
32.6 −0.376760 − 1.03514i −1.25887 − 2.69966i 0.602522 − 0.505576i 0 −2.32023 + 2.32023i −2.37053 + 3.38546i −2.65833 − 1.53479i −3.77503 + 4.49891i 0
32.7 −0.107303 − 0.294811i −0.322978 − 0.692629i 1.45669 − 1.22231i 0 −0.169538 + 0.169538i 1.89134 − 2.70111i −1.06006 − 0.612023i 1.55294 − 1.85073i 0
32.8 0.0170756 + 0.0469149i 0.925781 + 1.98534i 1.53018 − 1.28397i 0 −0.0773339 + 0.0773339i 1.38729 − 1.98126i 0.172840 + 0.0997893i −1.15616 + 1.37786i 0
32.9 0.121408 + 0.333566i −0.0309295 − 0.0663285i 1.43556 − 1.20458i 0 0.0183698 − 0.0183698i 0.315022 − 0.449897i 1.19093 + 0.687582i 1.92492 − 2.29403i 0
32.10 0.135305 + 0.371747i 1.40717 + 3.01769i 1.41220 − 1.18498i 0 −0.931421 + 0.931421i −1.00196 + 1.43095i 1.31680 + 0.760254i −5.19796 + 6.19469i 0
32.11 0.396661 + 1.08982i −1.15993 − 2.48749i 0.501728 − 0.421000i 0 2.25080 − 2.25080i −0.770113 + 1.09983i 2.66659 + 1.53956i −2.91378 + 3.47251i 0
32.12 0.488034 + 1.34086i 0.0854494 + 0.183247i −0.0276467 + 0.0231983i 0 −0.204007 + 0.204007i −2.20476 + 3.14872i 2.42689 + 1.40117i 1.90209 − 2.26682i 0
32.13 0.534665 + 1.46898i −0.0387270 − 0.0830502i −0.339950 + 0.285252i 0 0.101293 − 0.101293i −0.202819 + 0.289656i 2.10685 + 1.21639i 1.92297 − 2.29170i 0
32.14 0.696044 + 1.91237i 0.612122 + 1.31270i −1.64058 + 1.37661i 0 −2.08430 + 2.08430i 1.92897 − 2.75485i −0.249600 − 0.144107i 0.579876 − 0.691069i 0
32.15 0.748641 + 2.05687i −1.09354 − 2.34511i −2.13818 + 1.79415i 0 4.00493 − 4.00493i 1.77798 − 2.53922i −1.49981 − 0.865918i −2.37536 + 2.83084i 0
32.16 0.861852 + 2.36792i 1.22003 + 2.61637i −3.33217 + 2.79602i 0 −5.14386 + 5.14386i 0.753014 − 1.07542i −5.12801 − 2.96066i −3.42855 + 4.08598i 0
32.17 0.926609 + 2.54584i −0.616531 − 1.32215i −4.09060 + 3.43242i 0 2.79471 − 2.79471i −1.37362 + 1.96173i −7.83625 − 4.52426i 0.560381 − 0.667836i 0
57.1 −2.40601 + 0.424244i −0.363472 + 0.254506i 3.72951 − 1.35743i 0 0.766544 − 0.766544i 0.242719 + 2.77429i −4.16574 + 2.40509i −0.958722 + 2.63407i 0
57.2 −2.25339 + 0.397334i 2.48480 − 1.73988i 3.04053 − 1.10666i 0 −4.90793 + 4.90793i −0.232457 − 2.65700i −2.44859 + 1.41370i 2.12101 − 5.82742i 0
57.3 −1.89240 + 0.333682i −1.08638 + 0.760692i 1.59046 − 0.578880i 0 1.80204 − 1.80204i −0.282561 − 3.22968i 0.511672 − 0.295414i −0.424490 + 1.16628i 0
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 32.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.b 204
5.b even 2 1 185.2.bc.a yes 204
5.c odd 4 1 185.2.z.a ✓ 204
5.c odd 4 1 925.2.bn.b 204
37.i odd 36 1 925.2.bn.b 204
185.z even 36 1 185.2.bc.a yes 204
185.ba odd 36 1 185.2.z.a ✓ 204
185.bc even 36 1 inner 925.2.bq.b 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.z.a ✓ 204 5.c odd 4 1
185.2.z.a ✓ 204 185.ba odd 36 1
185.2.bc.a yes 204 5.b even 2 1
185.2.bc.a yes 204 185.z even 36 1
925.2.bn.b 204 5.c odd 4 1
925.2.bn.b 204 37.i odd 36 1
925.2.bq.b 204 1.a even 1 1 trivial
925.2.bq.b 204 185.bc even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{204} - 12 T_{2}^{203} + 72 T_{2}^{202} - 294 T_{2}^{201} + 942 T_{2}^{200} + \cdots + 11\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display