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Properties

Label 296.2.bj.a
Level $296$
Weight $2$
Character orbit 296.bj
Analytic conductor $2.364$
Analytic rank $0$
Dimension $432$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [296,2,Mod(19,296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([18, 18, 35])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("296.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 296.bj (of order \(36\), degree \(12\), 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: \(2.36357189983\)
Analytic rank: \(0\)
Dimension: \(432\)
Relative dimension: \(36\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 12 q^{2} - 24 q^{3} - 18 q^{4} - 12 q^{6} - 6 q^{8} - 24 q^{9} - 6 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{14} + 18 q^{16} - 24 q^{17} - 42 q^{18} - 24 q^{19} - 12 q^{20} - 24 q^{22} - 84 q^{24}+ \cdots - 132 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41357 + 0.0427573i −1.57411 − 1.87595i 1.99634 − 0.120881i −0.992822 + 0.462961i 2.30532 + 2.58448i 1.70316 − 4.67940i −2.81680 + 0.256231i −0.520428 + 2.95149i 1.38363 − 0.696876i
19.2 −1.40393 − 0.170250i 0.454193 + 0.541286i 1.94203 + 0.478039i 2.89649 − 1.35066i −0.545500 − 0.837254i 0.668851 − 1.83765i −2.64508 − 1.00176i 0.434245 − 2.46273i −4.29642 + 1.40310i
19.3 −1.39952 + 0.203356i −1.57411 − 1.87595i 1.91729 − 0.569199i 0.992822 − 0.462961i 2.58448 + 2.30532i −1.70316 + 4.67940i −2.56753 + 1.18650i −0.520428 + 2.95149i −1.29533 + 0.849817i
19.4 −1.35304 + 0.411453i 0.454193 + 0.541286i 1.66141 − 1.11342i −2.89649 + 1.35066i −0.837254 − 0.545500i −0.668851 + 1.83765i −1.78983 + 2.19009i 0.434245 − 2.46273i 3.36333 − 3.01926i
19.5 −1.30146 − 0.553347i 1.55564 + 1.85393i 1.38761 + 1.44032i −0.370035 + 0.172550i −0.998734 − 3.27363i −0.963572 + 2.64739i −1.00893 − 2.64236i −0.496126 + 2.81367i 0.577067 − 0.0198100i
19.6 −1.26092 − 0.640372i −0.0222193 − 0.0264799i 1.17985 + 1.61492i −1.94098 + 0.905095i 0.0110598 + 0.0476177i 0.153626 − 0.422084i −0.453549 − 2.79183i 0.520737 − 2.95325i 3.02702 + 0.101696i
19.7 −1.18560 + 0.770937i 1.55564 + 1.85393i 0.811312 − 1.82805i 0.370035 − 0.172550i −3.27363 − 0.998734i 0.963572 − 2.64739i 0.447418 + 2.79282i −0.496126 + 2.81367i −0.305689 + 0.489849i
19.8 −1.13057 + 0.849600i −0.0222193 − 0.0264799i 0.556360 − 1.92106i 1.94098 − 0.905095i 0.0476177 + 0.0110598i −0.153626 + 0.422084i 1.00313 + 2.64457i 0.520737 − 2.95325i −1.42544 + 2.67233i
19.9 −0.909329 − 1.08311i −1.09340 − 1.30306i −0.346241 + 1.96980i 3.31963 − 1.54797i −0.417097 + 2.36919i −0.0461237 + 0.126724i 2.44835 − 1.41618i 0.0184929 − 0.104878i −4.69526 − 2.18790i
19.10 −0.766172 − 1.18869i 1.64673 + 1.96249i −0.825961 + 1.82148i 1.88463 − 0.878817i 1.07112 − 3.46105i 0.782841 − 2.15084i 2.79800 − 0.413756i −0.618722 + 3.50895i −2.48859 − 1.56691i
19.11 −0.741052 − 1.20451i −1.99237 − 2.37441i −0.901683 + 1.78521i −1.81016 + 0.844089i −1.38355 + 4.15938i −0.0212805 + 0.0584676i 2.81849 − 0.236847i −1.14735 + 6.50696i 2.35813 + 1.55483i
19.12 −0.707435 + 1.22456i −1.09340 − 1.30306i −0.999072 − 1.73259i −3.31963 + 1.54797i 2.36919 − 0.417097i 0.0461237 − 0.126724i 2.82843 + 0.00227264i 0.0184929 − 0.104878i 0.452848 − 5.16016i
19.13 −0.625887 − 1.26817i 0.394884 + 0.470604i −1.21653 + 1.58747i −2.94726 + 1.37433i 0.349655 − 0.795327i 1.11776 − 3.07102i 2.77460 + 0.549195i 0.455409 − 2.58276i 3.58755 + 2.87746i
19.14 −0.548118 + 1.30367i 1.64673 + 1.96249i −1.39913 − 1.42914i −1.88463 + 0.878817i −3.46105 + 1.07112i −0.782841 + 2.15084i 2.63002 − 1.04068i −0.618722 + 3.50895i −0.112691 − 2.93864i
19.15 −0.520633 + 1.31489i −1.99237 − 2.37441i −1.45788 − 1.36915i 1.81016 − 0.844089i 4.15938 − 1.38355i 0.0212805 − 0.0584676i 2.55931 − 1.20413i −1.14735 + 6.50696i 0.167459 + 2.81962i
19.16 −0.396163 + 1.35759i 0.394884 + 0.470604i −1.68611 − 1.07565i 2.94726 − 1.37433i −0.795327 + 0.349655i −1.11776 + 3.07102i 2.12827 − 1.86291i 0.455409 − 2.58276i 0.698185 + 4.54564i
19.17 −0.203547 − 1.39949i −0.111309 − 0.132653i −1.91714 + 0.569723i −1.52536 + 0.711286i −0.162990 + 0.182777i −1.41720 + 3.89371i 1.18755 + 2.56705i 0.515737 − 2.92489i 1.30592 + 1.98994i
19.18 0.0425643 + 1.41357i −0.111309 − 0.132653i −1.99638 + 0.120336i 1.52536 − 0.711286i 0.182777 − 0.162990i 1.41720 − 3.89371i −0.255078 − 2.81690i 0.515737 − 2.92489i 1.07038 + 2.12593i
19.19 0.0503325 − 1.41332i −0.664239 − 0.791609i −1.99493 − 0.142272i 1.97716 − 0.921964i −1.15223 + 0.898937i 0.455696 − 1.25201i −0.301485 + 2.81231i 0.335513 − 1.90279i −1.20351 − 2.84076i
19.20 0.0962239 − 1.41094i 2.07362 + 2.47124i −1.98148 − 0.271532i −2.60899 + 1.21659i 3.68629 − 2.68795i −0.605843 + 1.66454i −0.573780 + 2.76962i −1.28620 + 7.29438i 1.46549 + 3.79819i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
37.i odd 36 1 inner
296.bj even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.2.bj.a ✓ 432
8.d odd 2 1 inner 296.2.bj.a ✓ 432
37.i odd 36 1 inner 296.2.bj.a ✓ 432
296.bj even 36 1 inner 296.2.bj.a ✓ 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.bj.a ✓ 432 1.a even 1 1 trivial
296.2.bj.a ✓ 432 8.d odd 2 1 inner
296.2.bj.a ✓ 432 37.i odd 36 1 inner
296.2.bj.a ✓ 432 296.bj even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(296, [\chi])\).