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Properties

Label 231.3.h.e
Level $231$
Weight $3$
Character orbit 231.h
Analytic conductor $6.294$
Analytic rank $0$
Dimension $48$
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] = [231,3,Mod(230,231)] 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("231.230"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 231.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,56,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.29429410672\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 56 q^{4} - 108 q^{9} - 60 q^{15} - 24 q^{16} - 16 q^{22} - 8 q^{25} - 504 q^{36} - 40 q^{37} + 20 q^{42} - 576 q^{49} + 976 q^{58} + 528 q^{60} + 296 q^{64} - 104 q^{67} + 704 q^{70} + 712 q^{78}+ \cdots + 260 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
230.1 −3.61990 −0.855150 − 2.87554i 9.10367 1.63873 3.09556 + 10.4092i −4.95924 − 4.94024i −18.4748 −7.53744 + 4.91803i −5.93203
230.2 −3.61990 −0.855150 + 2.87554i 9.10367 1.63873 3.09556 − 10.4092i −4.95924 + 4.94024i −18.4748 −7.53744 − 4.91803i −5.93203
230.3 −3.61990 0.855150 − 2.87554i 9.10367 −1.63873 −3.09556 + 10.4092i 4.95924 + 4.94024i −18.4748 −7.53744 − 4.91803i 5.93203
230.4 −3.61990 0.855150 + 2.87554i 9.10367 −1.63873 −3.09556 − 10.4092i 4.95924 − 4.94024i −18.4748 −7.53744 + 4.91803i 5.93203
230.5 −2.79736 −2.26534 − 1.96678i 3.82525 −5.34447 6.33698 + 5.50181i 0.424866 + 6.98709i 0.488843 1.26352 + 8.91086i 14.9504
230.6 −2.79736 −2.26534 + 1.96678i 3.82525 −5.34447 6.33698 − 5.50181i 0.424866 − 6.98709i 0.488843 1.26352 − 8.91086i 14.9504
230.7 −2.79736 2.26534 − 1.96678i 3.82525 5.34447 −6.33698 + 5.50181i −0.424866 − 6.98709i 0.488843 1.26352 − 8.91086i −14.9504
230.8 −2.79736 2.26534 + 1.96678i 3.82525 5.34447 −6.33698 − 5.50181i −0.424866 + 6.98709i 0.488843 1.26352 + 8.91086i −14.9504
230.9 −2.18332 −0.798986 − 2.89165i 0.766888 9.25203 1.74444 + 6.31339i −3.46525 + 6.08211i 7.05892 −7.72324 + 4.62077i −20.2001
230.10 −2.18332 −0.798986 + 2.89165i 0.766888 9.25203 1.74444 − 6.31339i −3.46525 − 6.08211i 7.05892 −7.72324 − 4.62077i −20.2001
230.11 −2.18332 0.798986 − 2.89165i 0.766888 −9.25203 −1.74444 + 6.31339i 3.46525 − 6.08211i 7.05892 −7.72324 − 4.62077i 20.2001
230.12 −2.18332 0.798986 + 2.89165i 0.766888 −9.25203 −1.74444 − 6.31339i 3.46525 + 6.08211i 7.05892 −7.72324 + 4.62077i 20.2001
230.13 −2.01477 −2.47008 − 1.70256i 0.0592996 0.175820 4.97664 + 3.43027i 5.46195 − 4.37802i 7.93961 3.20255 + 8.41092i −0.354238
230.14 −2.01477 −2.47008 + 1.70256i 0.0592996 0.175820 4.97664 − 3.43027i 5.46195 + 4.37802i 7.93961 3.20255 − 8.41092i −0.354238
230.15 −2.01477 2.47008 − 1.70256i 0.0592996 −0.175820 −4.97664 + 3.43027i −5.46195 + 4.37802i 7.93961 3.20255 − 8.41092i 0.354238
230.16 −2.01477 2.47008 + 1.70256i 0.0592996 −0.175820 −4.97664 − 3.43027i −5.46195 − 4.37802i 7.93961 3.20255 + 8.41092i 0.354238
230.17 −0.909212 −0.785304 − 2.89539i −3.17333 −3.00590 0.714008 + 2.63253i −6.62219 − 2.26860i 6.52208 −7.76660 + 4.54752i 2.73300
230.18 −0.909212 −0.785304 + 2.89539i −3.17333 −3.00590 0.714008 − 2.63253i −6.62219 + 2.26860i 6.52208 −7.76660 − 4.54752i 2.73300
230.19 −0.909212 0.785304 − 2.89539i −3.17333 3.00590 −0.714008 + 2.63253i 6.62219 + 2.26860i 6.52208 −7.76660 − 4.54752i −2.73300
230.20 −0.909212 0.785304 + 2.89539i −3.17333 3.00590 −0.714008 − 2.63253i 6.62219 − 2.26860i 6.52208 −7.76660 + 4.54752i −2.73300
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 230.48
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
7.b odd 2 1 inner
11.b odd 2 1 inner
21.c even 2 1 inner
33.d even 2 1 inner
77.b even 2 1 inner
231.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.3.h.e ✓ 48
3.b odd 2 1 inner 231.3.h.e ✓ 48
7.b odd 2 1 inner 231.3.h.e ✓ 48
11.b odd 2 1 inner 231.3.h.e ✓ 48
21.c even 2 1 inner 231.3.h.e ✓ 48
33.d even 2 1 inner 231.3.h.e ✓ 48
77.b even 2 1 inner 231.3.h.e ✓ 48
231.h odd 2 1 inner 231.3.h.e ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.3.h.e ✓ 48 1.a even 1 1 trivial
231.3.h.e ✓ 48 3.b odd 2 1 inner
231.3.h.e ✓ 48 7.b odd 2 1 inner
231.3.h.e ✓ 48 11.b odd 2 1 inner
231.3.h.e ✓ 48 21.c even 2 1 inner
231.3.h.e ✓ 48 33.d even 2 1 inner
231.3.h.e ✓ 48 77.b even 2 1 inner
231.3.h.e ✓ 48 231.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(231, [\chi])\):

\( T_{2}^{12} - 31T_{2}^{10} + 344T_{2}^{8} - 1702T_{2}^{6} + 3721T_{2}^{4} - 2923T_{2}^{2} + 686 \) Copy content Toggle raw display
\( T_{5}^{12} - 149T_{5}^{10} + 6718T_{5}^{8} - 119528T_{5}^{6} + 788520T_{5}^{4} - 1393792T_{5}^{2} + 42336 \) Copy content Toggle raw display