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Properties

Label 372.2.p.a
Level $372$
Weight $2$
Character orbit 372.p
Analytic conductor $2.970$
Analytic rank $0$
Dimension $120$
Inner twists $8$

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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] = [372,2,Mod(191,372)] 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("372.191"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(372, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 372.p (of order \(6\), degree \(2\), 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: \(2.97043495519\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(60\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{4} - 2 q^{9} + 4 q^{10} - 4 q^{12} - 4 q^{16} - 2 q^{18} - 14 q^{21} - 6 q^{22} - 4 q^{24} + 48 q^{25} + 6 q^{28} - 48 q^{30} + 4 q^{33} + 28 q^{34} + 2 q^{36} - 8 q^{37} + 36 q^{40} - 4 q^{42}+ \cdots + 48 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 −1.40808 − 0.131599i −1.09284 − 1.34377i 1.96536 + 0.370602i 1.29603 − 0.748262i 1.36196 + 2.03594i 2.28079 + 1.31682i −2.71861 − 0.780475i −0.611419 + 2.93703i −1.92338 + 0.883056i
191.2 −1.40808 + 0.131599i 1.71015 + 0.274540i 1.96536 − 0.370602i −1.29603 + 0.748262i −2.44416 − 0.161520i −2.28079 − 1.31682i −2.71861 + 0.780475i 2.84926 + 0.939012i 1.72644 − 1.22417i
191.3 −1.37673 − 0.323454i 0.0744353 + 1.73045i 1.79075 + 0.890617i 0.0937097 − 0.0541033i 0.457245 − 2.40643i −2.83750 − 1.63823i −2.17731 − 1.80536i −2.98892 + 0.257613i −0.146513 + 0.0441747i
191.4 −1.37673 + 0.323454i −1.53583 + 0.800762i 1.79075 − 0.890617i −0.0937097 + 0.0541033i 1.85541 − 1.59920i 2.83750 + 1.63823i −2.17731 + 1.80536i 1.71756 − 2.45967i 0.111513 − 0.104796i
191.5 −1.37366 − 0.336245i −1.53913 − 0.794402i 1.77388 + 0.923772i −3.42031 + 1.97472i 1.84713 + 1.60876i −2.26131 − 1.30557i −2.12609 − 1.86541i 1.73785 + 2.44538i 5.36233 − 1.56253i
191.6 −1.37366 + 0.336245i 1.45754 + 0.935726i 1.77388 − 0.923772i 3.42031 − 1.97472i −2.31679 − 0.795278i 2.26131 + 1.30557i −2.12609 + 1.86541i 1.24884 + 2.72771i −4.03435 + 3.86265i
191.7 −1.34879 − 0.425158i 1.23409 − 1.21532i 1.63848 + 1.14690i 1.84773 − 1.06679i −2.18124 + 1.11454i 0.0817739 + 0.0472122i −1.72236 − 2.24354i 0.0459713 − 2.99965i −2.94576 + 0.653297i
191.8 −1.34879 + 0.425158i 0.435456 − 1.67642i 1.63848 − 1.14690i −1.84773 + 1.06679i 0.125404 + 2.44628i −0.0817739 − 0.0472122i −1.72236 + 2.24354i −2.62076 − 1.46001i 2.03865 − 2.22445i
191.9 −1.23525 − 0.688592i −1.22507 + 1.22442i 1.05168 + 1.70117i −1.40690 + 0.812275i 2.35639 − 0.668891i 1.22636 + 0.708038i −0.127678 − 2.82554i 0.00158919 − 3.00000i 2.29720 − 0.0345804i
191.10 −1.23525 + 0.688592i −0.447845 + 1.67315i 1.05168 − 1.70117i 1.40690 − 0.812275i −0.598919 − 2.37514i −1.22636 − 0.708038i −0.127678 + 2.82554i −2.59887 − 1.49862i −1.17855 + 1.97214i
191.11 −1.09739 − 0.892042i 1.20202 + 1.24706i 0.408523 + 1.95783i −0.0699686 + 0.0403964i −0.206654 − 2.44076i 0.885120 + 0.511024i 1.29816 − 2.51292i −0.110303 + 2.99797i 0.112818 + 0.0180844i
191.12 −1.09739 + 0.892042i −1.68099 − 0.417450i 0.408523 − 1.95783i 0.0699686 − 0.0403964i 2.21708 − 1.04141i −0.885120 − 0.511024i 1.29816 + 2.51292i 2.65147 + 1.40346i −0.0407475 + 0.106745i
191.13 −0.979289 − 1.02029i −0.580862 − 1.63175i −0.0819876 + 1.99832i 1.50717 − 0.870163i −1.09603 + 2.19060i −2.62156 − 1.51356i 2.11916 − 1.87328i −2.32520 + 1.89564i −2.36377 − 0.685608i
191.14 −0.979289 + 1.02029i 1.70357 − 0.312832i −0.0819876 − 1.99832i −1.50717 + 0.870163i −1.34910 + 2.04449i 2.62156 + 1.51356i 2.11916 + 1.87328i 2.80427 − 1.06586i 0.588131 − 2.38989i
191.15 −0.920658 − 1.07349i 0.212154 − 1.71901i −0.304778 + 1.97664i −2.47217 + 1.42731i −2.04067 + 1.35487i 4.10556 + 2.37034i 2.40251 − 1.49263i −2.90998 − 0.729388i 3.80823 + 1.33980i
191.16 −0.920658 + 1.07349i 1.38263 − 1.04323i −0.304778 − 1.97664i 2.47217 − 1.42731i −0.153021 + 2.44471i −4.10556 − 2.37034i 2.40251 + 1.49263i 0.823322 − 2.88481i −0.743816 + 3.96792i
191.17 −0.873830 − 1.11194i 1.72250 − 0.181663i −0.472843 + 1.94330i −3.08078 + 1.77869i −1.70717 − 1.75658i −2.77350 − 1.60128i 2.57403 − 1.17234i 2.93400 − 0.625827i 4.66988 + 1.87138i
191.18 −0.873830 + 1.11194i −0.703924 − 1.58256i −0.472843 − 1.94330i 3.08078 − 1.77869i 2.37483 + 0.600161i 2.77350 + 1.60128i 2.57403 + 1.17234i −2.00898 + 2.22800i −0.714273 + 4.97993i
191.19 −0.751547 − 1.19799i −0.138319 + 1.72652i −0.870354 + 1.80069i 3.47137 − 2.00420i 2.17230 − 1.13186i 1.31109 + 0.756957i 2.81132 − 0.310628i −2.96174 − 0.477621i −5.00990 − 2.65241i
191.20 −0.751547 + 1.19799i −1.42605 + 0.983047i −0.870354 − 1.80069i −3.47137 + 2.00420i −0.105936 − 2.44720i −1.31109 − 0.756957i 2.81132 + 0.310628i 1.06724 − 2.80375i 0.207893 − 5.66491i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
31.c even 3 1 inner
93.h odd 6 1 inner
124.i odd 6 1 inner
372.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 372.2.p.a ✓ 120
3.b odd 2 1 inner 372.2.p.a ✓ 120
4.b odd 2 1 inner 372.2.p.a ✓ 120
12.b even 2 1 inner 372.2.p.a ✓ 120
31.c even 3 1 inner 372.2.p.a ✓ 120
93.h odd 6 1 inner 372.2.p.a ✓ 120
124.i odd 6 1 inner 372.2.p.a ✓ 120
372.p even 6 1 inner 372.2.p.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
372.2.p.a ✓ 120 1.a even 1 1 trivial
372.2.p.a ✓ 120 3.b odd 2 1 inner
372.2.p.a ✓ 120 4.b odd 2 1 inner
372.2.p.a ✓ 120 12.b even 2 1 inner
372.2.p.a ✓ 120 31.c even 3 1 inner
372.2.p.a ✓ 120 93.h odd 6 1 inner
372.2.p.a ✓ 120 124.i odd 6 1 inner
372.2.p.a ✓ 120 372.p even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(372, [\chi])\).