Properties

Label 925.2.bb.c
Level $925$
Weight $2$
Character orbit 925.bb
Analytic conductor $7.386$
Analytic rank $0$
Dimension $78$
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(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 −1.79172 + 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
151.2 −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
151.3 −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
151.4 −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
151.5 −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
151.6 −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
151.7 −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 0.599501 − 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
151.9 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
151.10 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
151.11 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
151.12 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
151.13 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
176.1 −2.76476 + 0.487501i 0.0865123 − 0.490635i 5.52683 − 2.01160i 0 1.39866i 2.82487 + 2.37035i −9.43712 + 5.44852i 2.58584 + 0.941168i 0
176.2 −2.28319 + 0.402589i −0.377725 + 2.14218i 3.17152 − 1.15434i 0 − 5.04309i −2.51308 − 2.10873i −2.76085 + 1.59398i −1.62720 − 0.592251i 0
176.3 −2.12999 + 0.375574i 0.547085 − 3.10267i 2.51640 − 0.915894i 0 6.81412i −3.65886 − 3.07015i −1.26975 + 0.733089i −6.50821 − 2.36879i 0
176.4 −1.75463 + 0.309389i 0.178573 − 1.01274i 1.10363 − 0.401687i 0 1.83223i 0.766686 + 0.643326i 1.27382 − 0.735438i 1.82533 + 0.664365i 0
176.5 −0.696718 + 0.122850i −0.0964514 + 0.547003i −1.40906 + 0.512856i 0 − 0.392956i −0.0119186 − 0.0100009i 2.14408 − 1.23789i 2.52917 + 0.920542i 0
176.6 −0.595611 + 0.105022i −0.376028 + 2.13256i −1.53566 + 0.558935i 0 − 1.30967i −1.62721 − 1.36539i 1.90350 − 1.09899i −1.58734 − 0.577744i 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 80-90
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.c ✓ 78
5.b even 2 1 925.2.bb.d yes 78
5.c odd 4 2 925.2.ba.d 156
37.h even 18 1 inner 925.2.bb.c ✓ 78
185.v even 18 1 925.2.bb.d yes 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 1.a even 1 1 trivial
925.2.bb.c ✓ 78 37.h even 18 1 inner
925.2.bb.d yes 78 5.b even 2 1
925.2.bb.d yes 78 185.v even 18 1

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