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Properties

Label 231.4.g.a
Level $231$
Weight $4$
Character orbit 231.g
Analytic conductor $13.629$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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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] = [231,4,Mod(197,231)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("231.197"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 231.g (of order \(2\), degree \(1\), 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: \(13.6294412113\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 288 q^{4} + 4 q^{9} + 16 q^{12} + 20 q^{15} + 1248 q^{16} + 384 q^{22} - 1824 q^{25} - 900 q^{27} + 264 q^{31} - 964 q^{33} - 1008 q^{34} + 48 q^{36} - 24 q^{37} + 1524 q^{45} + 192 q^{48} - 3528 q^{49}+ \cdots - 1260 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −5.54421 −4.13734 − 3.14363i 22.7383 − 1.50076i 22.9383 + 17.4289i 7.00000i −81.7123 7.23519 + 26.0125i 8.32054i
197.2 −5.54421 −4.13734 + 3.14363i 22.7383 1.50076i 22.9383 − 17.4289i − 7.00000i −81.7123 7.23519 − 26.0125i − 8.32054i
197.3 −5.11246 4.16723 − 3.10390i 18.1372 − 2.14203i −21.3048 + 15.8685i 7.00000i −51.8260 7.73167 − 25.8693i 10.9511i
197.4 −5.11246 4.16723 + 3.10390i 18.1372 2.14203i −21.3048 − 15.8685i − 7.00000i −51.8260 7.73167 + 25.8693i − 10.9511i
197.5 −5.08922 2.66331 − 4.46170i 17.9001 19.3058i −13.5542 + 22.7066i − 7.00000i −50.3840 −12.8135 − 23.7658i − 98.2512i
197.6 −5.08922 2.66331 + 4.46170i 17.9001 − 19.3058i −13.5542 − 22.7066i 7.00000i −50.3840 −12.8135 + 23.7658i 98.2512i
197.7 −4.99045 0.361427 − 5.18357i 16.9046 − 15.1552i −1.80369 + 25.8683i − 7.00000i −44.4381 −26.7387 − 3.74697i 75.6313i
197.8 −4.99045 0.361427 + 5.18357i 16.9046 15.1552i −1.80369 − 25.8683i 7.00000i −44.4381 −26.7387 + 3.74697i − 75.6313i
197.9 −4.29970 5.14904 − 0.698150i 10.4874 − 10.0912i −22.1393 + 3.00183i − 7.00000i −10.6952 26.0252 − 7.18960i 43.3891i
197.10 −4.29970 5.14904 + 0.698150i 10.4874 10.0912i −22.1393 − 3.00183i 7.00000i −10.6952 26.0252 + 7.18960i − 43.3891i
197.11 −4.20854 −4.96824 − 1.52205i 9.71180 − 14.9587i 20.9090 + 6.40560i − 7.00000i −7.20418 22.3667 + 15.1238i 62.9541i
197.12 −4.20854 −4.96824 + 1.52205i 9.71180 14.9587i 20.9090 − 6.40560i 7.00000i −7.20418 22.3667 − 15.1238i − 62.9541i
197.13 −4.03592 −4.73038 − 2.15024i 8.28864 6.50649i 19.0914 + 8.67819i − 7.00000i −1.16491 17.7529 + 20.3429i − 26.2597i
197.14 −4.03592 −4.73038 + 2.15024i 8.28864 − 6.50649i 19.0914 − 8.67819i 7.00000i −1.16491 17.7529 − 20.3429i 26.2597i
197.15 −3.86705 0.0225447 − 5.19610i 6.95404 − 11.8293i −0.0871816 + 20.0936i 7.00000i 4.04476 −26.9990 − 0.234290i 45.7445i
197.16 −3.86705 0.0225447 + 5.19610i 6.95404 11.8293i −0.0871816 − 20.0936i − 7.00000i 4.04476 −26.9990 + 0.234290i − 45.7445i
197.17 −3.16170 2.47782 − 4.56732i 1.99637 4.01053i −7.83413 + 14.4405i 7.00000i 18.9817 −14.7208 − 22.6340i − 12.6801i
197.18 −3.16170 2.47782 + 4.56732i 1.99637 − 4.01053i −7.83413 − 14.4405i − 7.00000i 18.9817 −14.7208 + 22.6340i 12.6801i
197.19 −2.91575 −0.805139 − 5.13340i 0.501627 6.15350i 2.34759 + 14.9677i − 7.00000i 21.8634 −25.7035 + 8.26620i − 17.9421i
197.20 −2.91575 −0.805139 + 5.13340i 0.501627 − 6.15350i 2.34759 − 14.9677i 7.00000i 21.8634 −25.7035 − 8.26620i 17.9421i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.72
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.4.g.a ✓ 72
3.b odd 2 1 inner 231.4.g.a ✓ 72
11.b odd 2 1 inner 231.4.g.a ✓ 72
33.d even 2 1 inner 231.4.g.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.g.a ✓ 72 1.a even 1 1 trivial
231.4.g.a ✓ 72 3.b odd 2 1 inner
231.4.g.a ✓ 72 11.b odd 2 1 inner
231.4.g.a ✓ 72 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(231, [\chi])\).