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Properties

Label 296.3.bi.b
Level $296$
Weight $3$
Character orbit 296.bi
Analytic conductor $8.065$
Analytic rank $0$
Dimension $120$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [296,3,Mod(17,296)] 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("296.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 0, 7])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 296.bi (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.06541582506\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{5} - 36 q^{11} + 24 q^{13} + 72 q^{15} + 12 q^{17} - 12 q^{19} + 60 q^{21} - 126 q^{23} - 24 q^{25} + 180 q^{29} - 36 q^{31} - 240 q^{33} + 192 q^{35} - 84 q^{37} + 78 q^{39} - 132 q^{41} + 84 q^{43}+ \cdots + 468 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −4.90109 − 0.864195i 0 −0.570413 − 6.51985i 0 7.27901 + 6.10781i 0 14.8167 + 5.39282i 0
17.2 0 −4.29622 − 0.757539i 0 0.0541265 + 0.618668i 0 −0.149149 − 0.125151i 0 9.42637 + 3.43092i 0
17.3 0 −3.88870 − 0.685683i 0 0.199604 + 2.28148i 0 −6.03287 − 5.06218i 0 6.19460 + 2.25465i 0
17.4 0 −1.31288 − 0.231496i 0 0.775872 + 8.86825i 0 3.37539 + 2.83229i 0 −6.78717 − 2.47033i 0
17.5 0 −0.667551 − 0.117707i 0 −0.830559 − 9.49334i 0 1.37285 + 1.15196i 0 −8.02546 − 2.92103i 0
17.6 0 0.721302 + 0.127185i 0 −0.0299958 − 0.342854i 0 −4.68217 − 3.92881i 0 −7.95313 − 2.89470i 0
17.7 0 0.817411 + 0.144132i 0 −0.0594943 − 0.680023i 0 5.74116 + 4.81740i 0 −7.80985 − 2.84255i 0
17.8 0 4.12282 + 0.726965i 0 0.839727 + 9.59813i 0 −6.41922 − 5.38637i 0 8.01196 + 2.91611i 0
17.9 0 4.60509 + 0.812002i 0 0.170250 + 1.94597i 0 10.3434 + 8.67915i 0 12.0903 + 4.40051i 0
17.10 0 4.79981 + 0.846336i 0 −0.506310 − 5.78715i 0 −4.54663 − 3.81508i 0 13.8647 + 5.04633i 0
57.1 0 −5.69786 − 1.00469i 0 0.0747078 − 0.00653609i 0 4.74911 + 3.98498i 0 22.9990 + 8.37094i 0
57.2 0 −3.72007 − 0.655949i 0 −7.81957 + 0.684124i 0 −7.30764 − 6.13184i 0 4.95144 + 1.80218i 0
57.3 0 −3.34573 − 0.589943i 0 6.58072 − 0.575738i 0 −3.26564 − 2.74020i 0 2.38865 + 0.869399i 0
57.4 0 −1.88602 − 0.332556i 0 0.216779 − 0.0189657i 0 4.91304 + 4.12253i 0 −5.01076 − 1.82377i 0
57.5 0 −0.394399 − 0.0695432i 0 5.70541 − 0.499159i 0 −8.40119 − 7.04943i 0 −8.30652 − 3.02333i 0
57.6 0 0.569964 + 0.100500i 0 −6.84961 + 0.599263i 0 3.94979 + 3.31427i 0 −8.14247 − 2.96362i 0
57.7 0 1.85806 + 0.327625i 0 −3.80635 + 0.333012i 0 −3.38884 − 2.84357i 0 −5.11220 − 1.86069i 0
57.8 0 2.74608 + 0.484208i 0 7.15226 − 0.625742i 0 6.90361 + 5.79282i 0 −1.15073 − 0.418832i 0
57.9 0 4.24104 + 0.747810i 0 3.28844 − 0.287701i 0 −6.95793 − 5.83840i 0 8.96999 + 3.26481i 0
57.10 0 5.62894 + 0.992534i 0 −4.05351 + 0.354636i 0 5.58810 + 4.68897i 0 22.2426 + 8.09564i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.3.bi.b ✓ 120
37.i odd 36 1 inner 296.3.bi.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.3.bi.b ✓ 120 1.a even 1 1 trivial
296.3.bi.b ✓ 120 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 126 T_{3}^{116} - 1080 T_{3}^{115} - 74754 T_{3}^{114} + 39282 T_{3}^{113} + \cdots + 69\!\cdots\!36 \) acting on \(S_{3}^{\mathrm{new}}(296, [\chi])\). Copy content Toggle raw display