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Properties

Label 230.3.c.a
Level $230$
Weight $3$
Character orbit 230.c
Analytic conductor $6.267$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [230,3,Mod(229,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.229");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{4} + 8 q^{6} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{4} + 8 q^{6} - 96 q^{9} + 96 q^{16} - 16 q^{24} - 48 q^{25} - 32 q^{26} + 100 q^{29} - 124 q^{31} - 28 q^{35} + 192 q^{36} + 192 q^{39} - 116 q^{41} + 148 q^{46} - 76 q^{49} - 144 q^{50} - 16 q^{54} - 224 q^{55} + 84 q^{59} - 192 q^{64} - 340 q^{69} + 328 q^{70} + 196 q^{71} - 496 q^{75} + 1360 q^{81} + 316 q^{85} - 376 q^{94} - 368 q^{95} + 32 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 − 1.41421i − 5.45221i −2.00000 −3.90521 + 3.12239i −7.71059 0.770899 2.82843i −20.7266 4.41573 + 5.52280i
229.2 − 1.41421i − 5.45221i −2.00000 3.90521 − 3.12239i −7.71059 −0.770899 2.82843i −20.7266 −4.41573 − 5.52280i
229.3 − 1.41421i − 2.29017i −2.00000 −2.28201 − 4.44887i −3.23879 −9.92340 2.82843i 3.75511 −6.29165 + 3.22725i
229.4 − 1.41421i − 2.29017i −2.00000 2.28201 + 4.44887i −3.23879 9.92340 2.82843i 3.75511 6.29165 − 3.22725i
229.5 − 1.41421i − 0.894518i −2.00000 −3.14390 + 3.88792i −1.26504 −4.24317 2.82843i 8.19984 5.49835 + 4.44614i
229.6 − 1.41421i − 0.894518i −2.00000 3.14390 − 3.88792i −1.26504 4.24317 2.82843i 8.19984 −5.49835 − 4.44614i
229.7 − 1.41421i 1.47600i −2.00000 −4.75172 + 1.55601i 2.08738 0.788814 2.82843i 6.82142 2.20053 + 6.71994i
229.8 − 1.41421i 1.47600i −2.00000 4.75172 − 1.55601i 2.08738 −0.788814 2.82843i 6.82142 −2.20053 − 6.71994i
229.9 − 1.41421i 3.00625i −2.00000 −3.95888 − 3.05405i 4.25149 7.53698 2.82843i −0.0375672 −4.31908 + 5.59871i
229.10 − 1.41421i 3.00625i −2.00000 3.95888 + 3.05405i 4.25149 −7.53698 2.82843i −0.0375672 4.31908 − 5.59871i
229.11 − 1.41421i 5.56886i −2.00000 −0.637273 + 4.95922i 7.87556 10.0249 2.82843i −22.0122 7.01340 + 0.901240i
229.12 − 1.41421i 5.56886i −2.00000 0.637273 − 4.95922i 7.87556 −10.0249 2.82843i −22.0122 −7.01340 − 0.901240i
229.13 1.41421i − 5.56886i −2.00000 −0.637273 − 4.95922i 7.87556 10.0249 − 2.82843i −22.0122 7.01340 − 0.901240i
229.14 1.41421i − 5.56886i −2.00000 0.637273 + 4.95922i 7.87556 −10.0249 − 2.82843i −22.0122 −7.01340 + 0.901240i
229.15 1.41421i − 3.00625i −2.00000 −3.95888 + 3.05405i 4.25149 7.53698 − 2.82843i −0.0375672 −4.31908 − 5.59871i
229.16 1.41421i − 3.00625i −2.00000 3.95888 − 3.05405i 4.25149 −7.53698 − 2.82843i −0.0375672 4.31908 + 5.59871i
229.17 1.41421i − 1.47600i −2.00000 −4.75172 − 1.55601i 2.08738 0.788814 − 2.82843i 6.82142 2.20053 − 6.71994i
229.18 1.41421i − 1.47600i −2.00000 4.75172 + 1.55601i 2.08738 −0.788814 − 2.82843i 6.82142 −2.20053 + 6.71994i
229.19 1.41421i 0.894518i −2.00000 −3.14390 − 3.88792i −1.26504 −4.24317 − 2.82843i 8.19984 5.49835 − 4.44614i
229.20 1.41421i 0.894518i −2.00000 3.14390 + 3.88792i −1.26504 4.24317 − 2.82843i 8.19984 −5.49835 + 4.44614i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.c.a ✓ 24
5.b even 2 1 inner 230.3.c.a ✓ 24
5.c odd 4 2 1150.3.d.e 24
23.b odd 2 1 inner 230.3.c.a ✓ 24
115.c odd 2 1 inner 230.3.c.a ✓ 24
115.e even 4 2 1150.3.d.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.c.a ✓ 24 1.a even 1 1 trivial
230.3.c.a ✓ 24 5.b even 2 1 inner
230.3.c.a ✓ 24 23.b odd 2 1 inner
230.3.c.a ✓ 24 115.c odd 2 1 inner
1150.3.d.e 24 5.c odd 4 2
1150.3.d.e 24 115.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(230, [\chi])\).