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Properties

Label 925.2.bb.d
Level $925$
Weight $2$
Character orbit 925.bb
Analytic conductor $7.386$
Analytic rank $0$
Dimension $78$
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 = [78,3] 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: \(78\)
Relative dimension: \(13\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 78 q + 3 q^{2} + 3 q^{4} - 6 q^{7} + 18 q^{8} + 15 q^{12} - 27 q^{13} - 18 q^{14} + 9 q^{16} - 15 q^{18} + 3 q^{21} - 30 q^{22} + 15 q^{24} + 36 q^{27} + 27 q^{28} - 9 q^{29} - 18 q^{32} - 6 q^{33} - 51 q^{34}+ \cdots - 183 q^{99}+O(q^{100}) \) 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 −1.69880 + 2.02455i 1.49825 − 1.25718i −0.865587 − 4.90899i 0 5.16897i −2.38678 − 0.868717i 6.83137 + 3.94409i 0.143302 − 0.812704i 0
151.2 −1.62071 + 1.93149i −1.96713 + 1.65061i −0.756645 − 4.29115i 0 − 6.47464i −1.53693 − 0.559397i 5.14744 + 2.97188i 0.624110 − 3.53951i 0
151.3 −1.39619 + 1.66391i 0.217032 − 0.182111i −0.471965 − 2.67664i 0 0.615382i 4.31785 + 1.57157i 1.35049 + 0.779708i −0.507006 + 2.87538i 0
151.4 −0.883019 + 1.05234i −1.16902 + 0.980928i 0.0195970 + 0.111140i 0 − 2.09639i −2.28284 − 0.830886i −2.51364 − 1.45125i −0.116546 + 0.660964i 0
151.5 −0.724117 + 0.862969i 1.09398 − 0.917960i 0.126926 + 0.719834i 0 1.60878i 0.138989 + 0.0505877i −2.66430 − 1.53824i −0.166798 + 0.945961i 0
151.6 −0.599501 + 0.714457i −1.58936 + 1.33363i 0.196248 + 1.11298i 0 − 1.93505i 2.64049 + 0.961060i −2.52824 − 1.45968i 0.226551 − 1.28483i 0
151.7 0.0854553 − 0.101842i 0.586668 − 0.492273i 0.344227 + 1.95221i 0 − 0.101815i −3.05159 − 1.11069i 0.458499 + 0.264715i −0.419098 + 2.37682i 0
151.8 0.410747 − 0.489509i 1.79456 − 1.50581i 0.276390 + 1.56749i 0 − 1.49696i 3.88080 + 1.41250i 1.98762 + 1.14755i 0.432020 − 2.45011i 0
151.9 0.623566 − 0.743137i −2.53956 + 2.13094i 0.183878 + 1.04283i 0 3.21603i −4.24285 − 1.54427i 2.56988 + 1.48372i 1.38750 − 7.86890i 0
151.10 1.12720 − 1.34335i −1.57550 + 1.32200i −0.186702 − 1.05884i 0 3.60661i 3.02288 + 1.10024i 1.40451 + 0.810896i 0.213569 − 1.21121i 0
151.11 1.18664 − 1.41418i 0.148672 − 0.124750i −0.244503 − 1.38664i 0 − 0.358283i 0.0463114 + 0.0168560i 0.946410 + 0.546410i −0.514404 + 2.91733i 0
151.12 1.25731 − 1.49841i 2.41941 − 2.03012i −0.317090 − 1.79831i 0 − 6.17774i −4.20231 − 1.52951i 0.294667 + 0.170126i 1.21118 − 6.86895i 0
151.13 1.79172 − 2.13528i −1.21612 + 1.02045i −1.00190 − 5.68203i 0 4.42511i −2.98218 − 1.08542i −9.09992 − 5.25384i −0.0833059 + 0.472451i 0
176.1 −2.59510 + 0.457586i −0.544754 + 3.08945i 4.64578 − 1.69092i 0 − 8.26671i 1.39849 + 1.17348i −6.71833 + 3.87883i −6.42888 − 2.33992i 0
176.2 −2.36950 + 0.417807i 0.0811184 − 0.460045i 3.56059 − 1.29595i 0 1.12397i −0.897054 − 0.752718i −3.72795 + 2.15233i 2.61402 + 0.951424i 0
176.3 −1.64358 + 0.289808i 0.509568 − 2.88990i 0.737996 − 0.268609i 0 4.89748i 0.365344 + 0.306560i 1.75557 − 1.01358i −5.27281 − 1.91915i 0
176.4 −1.25314 + 0.220962i −0.351630 + 1.99420i −0.357850 + 0.130247i 0 − 2.57670i 0.383562 + 0.321847i 2.62364 − 1.51476i −1.03409 − 0.376379i 0
176.5 −0.853985 + 0.150581i −0.0608485 + 0.345089i −1.17277 + 0.426853i 0 − 0.303863i 3.52328 + 2.95638i 2.43921 − 1.40828i 2.70369 + 0.984064i 0
176.6 −0.589919 + 0.104019i 0.262418 − 1.48825i −1.54220 + 0.561315i 0 0.905241i −3.25364 − 2.73013i 1.88892 − 1.09057i 0.673064 + 0.244975i 0
176.7 0.346377 − 0.0610756i −0.454850 + 2.57958i −1.76314 + 0.641730i 0 0.921288i −2.14323 − 1.79838i −1.18071 + 0.681686i −3.62828 − 1.32059i 0
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.13
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.d yes 78
5.b even 2 1 925.2.bb.c ✓ 78
5.c odd 4 2 925.2.ba.d 156
37.h even 18 1 inner 925.2.bb.d yes 78
185.v even 18 1 925.2.bb.c ✓ 78
185.y odd 36 2 925.2.ba.d 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.ba.d 156 5.c odd 4 2
925.2.ba.d 156 185.y odd 36 2
925.2.bb.c ✓ 78 5.b even 2 1
925.2.bb.c ✓ 78 185.v even 18 1
925.2.bb.d yes 78 1.a even 1 1 trivial
925.2.bb.d yes 78 37.h even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{78} - 3 T_{2}^{77} + 3 T_{2}^{76} - 6 T_{2}^{75} + 18 T_{2}^{74} - 63 T_{2}^{73} + \cdots + 86607387 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display