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Properties

Label 114.3.g.a
Level $114$
Weight $3$
Character orbit 114.g
Analytic conductor $3.106$
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] = [114,3,Mod(11,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 114.g (of order \(6\), degree \(2\), 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: \(3.10627501371\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} + 24 q^{4} + 4 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} + 24 q^{4} + 4 q^{6} + 8 q^{7} - 2 q^{9} - 8 q^{12} + 28 q^{13} - 2 q^{15} - 48 q^{16} + 80 q^{18} - 96 q^{19} + 60 q^{21} - 16 q^{22} - 8 q^{24} - 24 q^{25} - 56 q^{27} + 8 q^{28} + 88 q^{30} + 88 q^{31} + 8 q^{33} - 88 q^{34} + 4 q^{36} - 208 q^{37} - 156 q^{39} + 8 q^{42} + 232 q^{43} - 460 q^{45} - 160 q^{46} - 8 q^{48} + 168 q^{49} - 38 q^{51} - 56 q^{52} + 40 q^{54} + 192 q^{55} - 150 q^{57} + 96 q^{58} + 4 q^{60} - 12 q^{61} + 232 q^{63} - 192 q^{64} - 24 q^{66} + 352 q^{67} + 380 q^{69} + 72 q^{70} + 80 q^{72} - 244 q^{73} + 144 q^{75} + 292 q^{78} + 8 q^{79} - 230 q^{81} - 184 q^{82} + 240 q^{84} + 564 q^{85} + 404 q^{87} - 64 q^{88} + 56 q^{90} - 388 q^{91} - 104 q^{93} - 160 q^{94} - 32 q^{96} - 156 q^{97} + 208 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.22474 − 0.707107i −2.74673 − 1.20644i 1.00000 + 1.73205i −1.16161 − 0.670656i 2.51096 + 3.41981i −3.67117 − 2.82843i 6.08902 + 6.62751i 0.948451 + 1.64277i
11.2 −1.22474 − 0.707107i −2.14225 + 2.10019i 1.00000 + 1.73205i −2.24295 − 1.29497i 4.10876 − 1.05740i 0.804666 − 2.82843i 0.178429 − 8.99823i 1.83136 + 3.17201i
11.3 −1.22474 − 0.707107i −0.912292 − 2.85792i 1.00000 + 1.73205i 4.75916 + 2.74770i −0.903532 + 4.14531i 10.1756 − 2.82843i −7.33545 + 5.21452i −3.88584 − 6.73047i
11.4 −1.22474 − 0.707107i 0.328743 + 2.98193i 1.00000 + 1.73205i 6.41109 + 3.70145i 1.70592 − 3.88456i −10.7270 − 2.82843i −8.78386 + 1.96058i −5.23463 − 9.06665i
11.5 −1.22474 − 0.707107i 1.35945 − 2.67430i 1.00000 + 1.73205i −5.15075 − 2.97378i −3.55600 + 2.31406i −3.94636 − 2.82843i −5.30379 − 7.27116i 4.20557 + 7.28425i
11.6 −1.22474 − 0.707107i 2.38833 + 1.81546i 1.00000 + 1.73205i −2.61495 − 1.50974i −1.64136 − 3.91228i 9.36430 − 2.82843i 2.40820 + 8.67183i 2.13510 + 3.69810i
11.7 1.22474 + 0.707107i −2.76640 − 1.16062i 1.00000 + 1.73205i 2.61495 + 1.50974i −2.56745 − 3.37760i 9.36430 2.82843i 6.30592 + 6.42148i 2.13510 + 3.69810i
11.8 1.22474 + 0.707107i −2.74680 + 1.20627i 1.00000 + 1.73205i −6.41109 − 3.70145i −4.21709 − 0.464913i −10.7270 2.82843i 6.08984 − 6.62675i −5.23463 − 9.06665i
11.9 1.22474 + 0.707107i −0.747693 + 2.90533i 1.00000 + 1.73205i 2.24295 + 1.29497i −2.97011 + 3.02959i 0.804666 2.82843i −7.88191 − 4.34459i 1.83136 + 3.17201i
11.10 1.22474 + 0.707107i 1.63629 − 2.51447i 1.00000 + 1.73205i 5.15075 + 2.97378i 3.78204 − 1.92255i −3.94636 2.82843i −3.64511 − 8.22880i 4.20557 + 7.28425i
11.11 1.22474 + 0.707107i 2.41817 + 1.77552i 1.00000 + 1.73205i 1.16161 + 0.670656i 1.70616 + 3.88446i −3.67117 2.82843i 2.69508 + 8.58700i 0.948451 + 1.64277i
11.12 1.22474 + 0.707107i 2.93118 − 0.638894i 1.00000 + 1.73205i −4.75916 − 2.74770i 4.04171 + 1.29018i 10.1756 2.82843i 8.18363 − 3.74543i −3.88584 − 6.73047i
83.1 −1.22474 + 0.707107i −2.74673 + 1.20644i 1.00000 − 1.73205i −1.16161 + 0.670656i 2.51096 − 3.41981i −3.67117 2.82843i 6.08902 − 6.62751i 0.948451 − 1.64277i
83.2 −1.22474 + 0.707107i −2.14225 − 2.10019i 1.00000 − 1.73205i −2.24295 + 1.29497i 4.10876 + 1.05740i 0.804666 2.82843i 0.178429 + 8.99823i 1.83136 − 3.17201i
83.3 −1.22474 + 0.707107i −0.912292 + 2.85792i 1.00000 − 1.73205i 4.75916 − 2.74770i −0.903532 − 4.14531i 10.1756 2.82843i −7.33545 − 5.21452i −3.88584 + 6.73047i
83.4 −1.22474 + 0.707107i 0.328743 − 2.98193i 1.00000 − 1.73205i 6.41109 − 3.70145i 1.70592 + 3.88456i −10.7270 2.82843i −8.78386 − 1.96058i −5.23463 + 9.06665i
83.5 −1.22474 + 0.707107i 1.35945 + 2.67430i 1.00000 − 1.73205i −5.15075 + 2.97378i −3.55600 − 2.31406i −3.94636 2.82843i −5.30379 + 7.27116i 4.20557 − 7.28425i
83.6 −1.22474 + 0.707107i 2.38833 − 1.81546i 1.00000 − 1.73205i −2.61495 + 1.50974i −1.64136 + 3.91228i 9.36430 2.82843i 2.40820 − 8.67183i 2.13510 − 3.69810i
83.7 1.22474 − 0.707107i −2.76640 + 1.16062i 1.00000 − 1.73205i 2.61495 − 1.50974i −2.56745 + 3.37760i 9.36430 − 2.82843i 6.30592 − 6.42148i 2.13510 − 3.69810i
83.8 1.22474 − 0.707107i −2.74680 − 1.20627i 1.00000 − 1.73205i −6.41109 + 3.70145i −4.21709 + 0.464913i −10.7270 − 2.82843i 6.08984 + 6.62675i −5.23463 + 9.06665i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.3.g.a ✓ 24
3.b odd 2 1 inner 114.3.g.a ✓ 24
19.c even 3 1 inner 114.3.g.a ✓ 24
57.h odd 6 1 inner 114.3.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.g.a ✓ 24 1.a even 1 1 trivial
114.3.g.a ✓ 24 3.b odd 2 1 inner
114.3.g.a ✓ 24 19.c even 3 1 inner
114.3.g.a ✓ 24 57.h odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(114, [\chi])\).