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Properties

Label 185.2.m.a
Level $185$
Weight $2$
Character orbit 185.m
Analytic conductor $1.477$
Analytic rank $0$
Dimension $28$
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] = [185,2,Mod(11,185)] 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("185.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 6 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{7} - 18 q^{9} - 4 q^{10} + 4 q^{11} + 2 q^{12} - 30 q^{13} - 20 q^{16} + 30 q^{18} - 2 q^{21} - 30 q^{22} + 30 q^{24} + 14 q^{25} + 12 q^{26} + 32 q^{27} + 2 q^{28}+ \cdots - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.32618 + 1.34302i 1.52962 − 2.64937i 2.60740 − 4.51616i 0.866025 + 0.500000i 8.21722i 0.284485 − 0.492742i 8.63511i −3.17945 − 5.50697i −2.68604
11.2 −2.20795 + 1.27476i −1.58363 + 2.74293i 2.25003 − 3.89716i −0.866025 − 0.500000i − 8.07501i 2.10035 − 3.63792i 6.37394i −3.51579 − 6.08952i 2.54952
11.3 −1.91807 + 1.10740i 0.262986 − 0.455505i 1.45267 − 2.51610i −0.866025 − 0.500000i 1.16492i −0.973099 + 1.68546i 2.00515i 1.36168 + 2.35849i 2.21480
11.4 −1.86389 + 1.07612i −0.823223 + 1.42586i 1.31605 − 2.27946i 0.866025 + 0.500000i − 3.54353i −2.34537 + 4.06229i 1.36041i 0.144607 + 0.250467i −2.15223
11.5 −1.61472 + 0.932256i −0.00733364 + 0.0127022i 0.738203 − 1.27861i 0.866025 + 0.500000i − 0.0273473i 2.28706 − 3.96130i − 0.976246i 1.49989 + 2.59789i −1.86451
11.6 −0.754635 + 0.435689i 0.996238 − 1.72553i −0.620350 + 1.07448i −0.866025 − 0.500000i 1.73620i 1.67414 − 2.89970i − 2.82387i −0.484979 − 0.840008i 0.871378
11.7 −0.596292 + 0.344270i −0.622433 + 1.07808i −0.762957 + 1.32148i −0.866025 − 0.500000i − 0.857138i −0.273058 + 0.472950i − 2.42773i 0.725155 + 1.25601i 0.688539
11.8 −0.217929 + 0.125821i −1.61547 + 2.79808i −0.968338 + 1.67721i 0.866025 + 0.500000i − 0.813045i 0.127669 − 0.221129i − 0.990637i −3.71951 − 6.44238i −0.251643
11.9 0.108589 − 0.0626936i 0.156745 − 0.271490i −0.992139 + 1.71844i 0.866025 + 0.500000i − 0.0393076i −0.790882 + 1.36985i 0.499578i 1.45086 + 2.51297i 0.125387
11.10 1.22813 − 0.709060i −1.32488 + 2.29477i 0.00553227 − 0.00958218i −0.866025 − 0.500000i 3.75769i −1.47091 + 2.54770i 2.82055i −2.01063 − 3.48252i −1.41812
11.11 1.31801 − 0.760954i 1.33553 − 2.31321i 0.158102 − 0.273841i −0.866025 − 0.500000i − 4.06511i −0.696862 + 1.20700i 2.56258i −2.06729 − 3.58065i −1.52191
11.12 1.49679 − 0.864172i 0.833596 − 1.44383i 0.493588 − 0.854919i 0.866025 + 0.500000i − 2.88148i −0.0212161 + 0.0367474i 1.75051i 0.110234 + 0.190931i 1.72834
11.13 2.05130 − 1.18432i −1.07393 + 1.86010i 1.80523 − 3.12676i 0.866025 + 0.500000i 5.08750i 0.824280 − 1.42769i − 3.81462i −0.806637 − 1.39714i 2.36864
11.14 2.29684 − 1.32608i −0.0638062 + 0.110516i 2.51697 − 4.35953i −0.866025 − 0.500000i 0.338448i −1.72659 + 2.99054i − 8.04652i 1.49186 + 2.58397i −2.65216
101.1 −2.32618 − 1.34302i 1.52962 + 2.64937i 2.60740 + 4.51616i 0.866025 − 0.500000i − 8.21722i 0.284485 + 0.492742i − 8.63511i −3.17945 + 5.50697i −2.68604
101.2 −2.20795 − 1.27476i −1.58363 − 2.74293i 2.25003 + 3.89716i −0.866025 + 0.500000i 8.07501i 2.10035 + 3.63792i − 6.37394i −3.51579 + 6.08952i 2.54952
101.3 −1.91807 − 1.10740i 0.262986 + 0.455505i 1.45267 + 2.51610i −0.866025 + 0.500000i − 1.16492i −0.973099 − 1.68546i − 2.00515i 1.36168 − 2.35849i 2.21480
101.4 −1.86389 − 1.07612i −0.823223 − 1.42586i 1.31605 + 2.27946i 0.866025 − 0.500000i 3.54353i −2.34537 − 4.06229i − 1.36041i 0.144607 − 0.250467i −2.15223
101.5 −1.61472 − 0.932256i −0.00733364 − 0.0127022i 0.738203 + 1.27861i 0.866025 − 0.500000i 0.0273473i 2.28706 + 3.96130i 0.976246i 1.49989 − 2.59789i −1.86451
101.6 −0.754635 − 0.435689i 0.996238 + 1.72553i −0.620350 − 1.07448i −0.866025 + 0.500000i − 1.73620i 1.67414 + 2.89970i 2.82387i −0.484979 + 0.840008i 0.871378
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.m.a ✓ 28
5.b even 2 1 925.2.n.d 28
5.c odd 4 1 925.2.m.c 28
5.c odd 4 1 925.2.m.d 28
37.e even 6 1 inner 185.2.m.a ✓ 28
37.g odd 12 1 6845.2.a.n 14
37.g odd 12 1 6845.2.a.o 14
185.l even 6 1 925.2.n.d 28
185.r odd 12 1 925.2.m.c 28
185.r odd 12 1 925.2.m.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.m.a ✓ 28 1.a even 1 1 trivial
185.2.m.a ✓ 28 37.e even 6 1 inner
925.2.m.c 28 5.c odd 4 1
925.2.m.c 28 185.r odd 12 1
925.2.m.d 28 5.c odd 4 1
925.2.m.d 28 185.r odd 12 1
925.2.n.d 28 5.b even 2 1
925.2.n.d 28 185.l even 6 1
6845.2.a.n 14 37.g odd 12 1
6845.2.a.o 14 37.g odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).