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Properties

Label 925.2.bb.b
Level $925$
Weight $2$
Character orbit 925.bb
Analytic conductor $7.386$
Analytic rank $0$
Dimension $72$
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(151,925)] 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("925.151"); 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([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1 −1.68876 + 2.01259i −0.343451 + 0.288189i −0.851300 − 4.82796i 0 − 1.17791i −1.98541 − 0.722629i 6.60383 + 3.81272i −0.486039 + 2.75647i 0
151.2 −1.29158 + 1.53924i −0.583938 + 0.489982i −0.353796 − 2.00647i 0 − 1.53167i 1.63114 + 0.593686i 0.0651292 + 0.0376024i −0.420044 + 2.38219i 0
151.3 −0.829311 + 0.988334i 1.25448 − 1.05263i 0.0582485 + 0.330344i 0 2.11280i −2.68169 − 0.976055i −2.60945 − 1.50657i −0.0552654 + 0.313426i 0
151.4 −0.551402 + 0.657135i −1.62051 + 1.35977i 0.219514 + 1.24492i 0 − 1.81467i 0.115935 + 0.0421968i −2.42493 − 1.40003i 0.256133 − 1.45260i 0
151.5 −0.535006 + 0.637595i 0.837162 − 0.702462i 0.227000 + 1.28738i 0 0.909592i 0.707269 + 0.257425i −2.38390 − 1.37634i −0.313558 + 1.77827i 0
151.6 −0.0948413 + 0.113027i −2.27065 + 1.90530i 0.343516 + 1.94818i 0 − 0.437347i 1.01836 + 0.370651i −0.508336 − 0.293488i 1.00473 − 5.69811i 0
151.7 0.301512 − 0.359328i 2.39156 − 2.00676i 0.309089 + 1.75293i 0 − 1.46442i −1.06448 − 0.387441i 1.53553 + 0.886536i 1.17154 − 6.64413i 0
151.8 0.590034 − 0.703175i 0.605440 − 0.508024i 0.200981 + 1.13982i 0 − 0.725482i −3.19522 − 1.16297i 2.50998 + 1.44914i −0.412476 + 2.33927i 0
151.9 0.761031 − 0.906961i −1.36011 + 1.14127i 0.103886 + 0.589165i 0 2.10211i −2.61191 − 0.950658i 2.66408 + 1.53811i 0.0264639 − 0.150084i 0
151.10 1.28058 − 1.52614i 2.12261 − 1.78108i −0.341910 − 1.93907i 0 − 5.52022i 2.45400 + 0.893181i 0.0535129 + 0.0308957i 0.812283 − 4.60669i 0
151.11 1.42759 − 1.70134i −1.89399 + 1.58924i −0.509236 − 2.88802i 0 5.49111i 0.556964 + 0.202718i −1.79371 − 1.03560i 0.540547 − 3.06559i 0
151.12 1.50953 − 1.79899i −0.670695 + 0.562780i −0.610385 − 3.46166i 0 2.05611i 4.17567 + 1.51982i −3.08132 − 1.77900i −0.387834 + 2.19951i 0
176.1 −2.40518 + 0.424099i 0.541060 − 3.06850i 3.72567 − 1.35603i 0 7.60978i 3.04240 + 2.55288i −4.15566 + 2.39927i −6.30389 − 2.29443i 0
176.2 −2.17919 + 0.384249i −0.156421 + 0.887108i 2.72182 − 0.990661i 0 − 1.99328i 2.30968 + 1.93805i −1.71800 + 0.991889i 2.05658 + 0.748535i 0
176.3 −1.80790 + 0.318781i −0.168356 + 0.954792i 1.28749 − 0.468607i 0 − 1.77983i 1.19566 + 1.00328i 1.00141 − 0.578166i 1.93579 + 0.704571i 0
176.4 −1.67806 + 0.295887i −0.576068 + 3.26704i 0.848938 − 0.308988i 0 − 5.65273i −3.03262 − 2.54467i 1.61818 − 0.934254i −7.52263 − 2.73801i 0
176.5 −1.24400 + 0.219350i 0.234364 − 1.32915i −0.379974 + 0.138299i 0 1.70486i −2.83337 − 2.37748i 2.63025 − 1.51858i 1.10737 + 0.403051i 0
176.6 −0.772863 + 0.136277i 0.381820 − 2.16541i −1.30064 + 0.473394i 0 1.72560i 0.668410 + 0.560863i 2.29999 − 1.32790i −1.72413 − 0.627533i 0
176.7 −0.105693 + 0.0186364i −0.492384 + 2.79245i −1.86856 + 0.680101i 0 − 0.304317i 3.01176 + 2.52717i 0.370707 − 0.214028i −4.73626 − 1.72386i 0
176.8 0.540933 − 0.0953810i −0.128393 + 0.728150i −1.59587 + 0.580851i 0 0.406126i −1.81619 − 1.52397i −1.75924 + 1.01569i 2.30536 + 0.839082i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.12
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 6 T_{2}^{71} + 21 T_{2}^{70} + 48 T_{2}^{69} + 72 T_{2}^{68} + 126 T_{2}^{67} + \cdots + 11449 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display