Properties

Label 925.2.ba.b
Level $925$
Weight $2$
Character orbit 925.ba
Analytic conductor $7.386$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) 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.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{4} - 6 q^{7} - 18 q^{8} - 42 q^{12} - 18 q^{13} + 18 q^{14} - 18 q^{16} + 12 q^{17} + 24 q^{18} + 18 q^{19} + 24 q^{22} - 24 q^{23} + 30 q^{24} - 48 q^{28} - 18 q^{29} + 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −1.79899 + 1.50953i −0.562780 + 0.670695i 0.610385 − 3.46166i 0 − 2.05611i 1.51982 + 4.17567i 1.77900 + 3.08132i 0.387834 + 2.19951i 0
99.2 −1.53924 + 1.29158i 0.489982 − 0.583938i 0.353796 − 2.00647i 0 1.53167i −0.593686 − 1.63114i 0.0376024 + 0.0651292i 0.420044 + 2.38219i 0
99.3 −0.988334 + 0.829311i −1.05263 + 1.25448i −0.0582485 + 0.330344i 0 − 2.11280i 0.976055 + 2.68169i −1.50657 − 2.60945i 0.0552654 + 0.313426i 0
99.4 −0.906961 + 0.761031i −1.14127 + 1.36011i −0.103886 + 0.589165i 0 − 2.10211i −0.950658 − 2.61191i −1.53811 − 2.66408i −0.0264639 − 0.150084i 0
99.5 −0.359328 + 0.301512i 2.00676 − 2.39156i −0.309089 + 1.75293i 0 1.46442i −0.387441 − 1.06448i −0.886536 − 1.53553i −1.17154 − 6.64413i 0
99.6 −0.113027 + 0.0948413i 1.90530 − 2.27065i −0.343516 + 1.94818i 0 0.437347i −0.370651 − 1.01836i −0.293488 − 0.508336i −1.00473 − 5.69811i 0
99.7 0.637595 − 0.535006i 0.702462 − 0.837162i −0.227000 + 1.28738i 0 − 0.909592i 0.257425 + 0.707269i 1.37634 + 2.38390i 0.313558 + 1.77827i 0
99.8 0.657135 − 0.551402i −1.35977 + 1.62051i −0.219514 + 1.24492i 0 1.81467i 0.0421968 + 0.115935i 1.40003 + 2.42493i −0.256133 − 1.45260i 0
99.9 0.703175 − 0.590034i −0.508024 + 0.605440i −0.200981 + 1.13982i 0 0.725482i 1.16297 + 3.19522i 1.44914 + 2.50998i 0.412476 + 2.33927i 0
99.10 1.52614 − 1.28058i −1.78108 + 2.12261i 0.341910 − 1.93907i 0 5.52022i −0.893181 − 2.45400i 0.0308957 + 0.0535129i −0.812283 − 4.60669i 0
99.11 1.70134 − 1.42759i 1.58924 − 1.89399i 0.509236 − 2.88802i 0 − 5.49111i −0.202718 − 0.556964i −1.03560 − 1.79371i −0.540547 − 3.06559i 0
99.12 2.01259 − 1.68876i −0.288189 + 0.343451i 0.851300 − 4.82796i 0 1.17791i −0.722629 − 1.98541i −3.81272 − 6.60383i 0.486039 + 2.75647i 0
299.1 −1.79899 − 1.50953i −0.562780 − 0.670695i 0.610385 + 3.46166i 0 2.05611i 1.51982 − 4.17567i 1.77900 − 3.08132i 0.387834 − 2.19951i 0
299.2 −1.53924 − 1.29158i 0.489982 + 0.583938i 0.353796 + 2.00647i 0 − 1.53167i −0.593686 + 1.63114i 0.0376024 − 0.0651292i 0.420044 − 2.38219i 0
299.3 −0.988334 − 0.829311i −1.05263 − 1.25448i −0.0582485 − 0.330344i 0 2.11280i 0.976055 − 2.68169i −1.50657 + 2.60945i 0.0552654 − 0.313426i 0
299.4 −0.906961 − 0.761031i −1.14127 − 1.36011i −0.103886 − 0.589165i 0 2.10211i −0.950658 + 2.61191i −1.53811 + 2.66408i −0.0264639 + 0.150084i 0
299.5 −0.359328 − 0.301512i 2.00676 + 2.39156i −0.309089 − 1.75293i 0 − 1.46442i −0.387441 + 1.06448i −0.886536 + 1.53553i −1.17154 + 6.64413i 0
299.6 −0.113027 − 0.0948413i 1.90530 + 2.27065i −0.343516 − 1.94818i 0 − 0.437347i −0.370651 + 1.01836i −0.293488 + 0.508336i −1.00473 + 5.69811i 0
299.7 0.637595 + 0.535006i 0.702462 + 0.837162i −0.227000 − 1.28738i 0 0.909592i 0.257425 − 0.707269i 1.37634 − 2.38390i 0.313558 − 1.77827i 0
299.8 0.657135 + 0.551402i −1.35977 − 1.62051i −0.219514 − 1.24492i 0 − 1.81467i 0.0421968 − 0.115935i 1.40003 − 2.42493i −0.256133 + 1.45260i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 99.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.ba.b 72
5.b even 2 1 925.2.ba.c 72
5.c odd 4 1 185.2.w.a ✓ 72
5.c odd 4 1 925.2.bb.b 72
37.h even 18 1 925.2.ba.c 72
185.v even 18 1 inner 925.2.ba.b 72
185.y odd 36 1 185.2.w.a ✓ 72
185.y odd 36 1 925.2.bb.b 72
185.z even 36 1 6845.2.a.u 36
185.bc even 36 1 6845.2.a.t 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.w.a ✓ 72 5.c odd 4 1
185.2.w.a ✓ 72 185.y odd 36 1
925.2.ba.b 72 1.a even 1 1 trivial
925.2.ba.b 72 185.v even 18 1 inner
925.2.ba.c 72 5.b even 2 1
925.2.ba.c 72 37.h even 18 1
925.2.bb.b 72 5.c odd 4 1
925.2.bb.b 72 185.y odd 36 1
6845.2.a.t 36 185.bc even 36 1
6845.2.a.u 36 185.z even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 3 T_{2}^{70} + 14 T_{2}^{69} - 42 T_{2}^{67} + 557 T_{2}^{66} - 12 T_{2}^{65} + \cdots + 11449 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display