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Properties

Label 228.3.g.a
Level $228$
Weight $3$
Character orbit 228.g
Analytic conductor $6.213$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [228,3,Mod(115,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.115");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.g (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: \(6.21255002741\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} - 108 q^{9} + 8 q^{10} + 24 q^{12} - 24 q^{13} - 12 q^{14} + 4 q^{16} - 40 q^{17} - 12 q^{18} - 80 q^{20} + 12 q^{22} + 36 q^{24} + 284 q^{25} - 112 q^{26} - 48 q^{28} + 104 q^{29} + 24 q^{30} + 44 q^{32} + 48 q^{33} + 140 q^{34} - 36 q^{36} - 184 q^{37} + 180 q^{40} - 200 q^{41} + 48 q^{42} + 96 q^{44} - 24 q^{45} - 28 q^{46} - 144 q^{48} - 332 q^{49} + 176 q^{50} + 276 q^{52} + 264 q^{53} - 192 q^{56} - 184 q^{58} - 180 q^{60} + 40 q^{61} - 240 q^{62} - 372 q^{64} + 176 q^{65} - 120 q^{66} - 104 q^{68} - 60 q^{70} + 60 q^{72} + 424 q^{73} - 104 q^{74} - 400 q^{77} - 180 q^{78} + 704 q^{80} + 324 q^{81} + 528 q^{82} + 312 q^{84} - 128 q^{85} + 668 q^{86} - 496 q^{88} - 520 q^{89} - 24 q^{90} - 456 q^{92} - 32 q^{94} + 300 q^{96} - 440 q^{97} - 472 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −1.98096 − 0.275326i − 1.73205i 3.84839 + 1.09082i −1.55324 −0.476878 + 3.43112i 5.27603i −7.32317 − 3.22043i −3.00000 3.07690 + 0.427647i
115.2 −1.98096 + 0.275326i 1.73205i 3.84839 − 1.09082i −1.55324 −0.476878 − 3.43112i − 5.27603i −7.32317 + 3.22043i −3.00000 3.07690 − 0.427647i
115.3 −1.90744 − 0.601394i 1.73205i 3.27665 + 2.29424i 8.03115 1.04164 − 3.30378i − 8.62572i −4.87027 − 6.34669i −3.00000 −15.3189 − 4.82988i
115.4 −1.90744 + 0.601394i − 1.73205i 3.27665 − 2.29424i 8.03115 1.04164 + 3.30378i 8.62572i −4.87027 + 6.34669i −3.00000 −15.3189 + 4.82988i
115.5 −1.90181 − 0.618963i 1.73205i 3.23377 + 2.35430i −8.79379 1.07208 − 3.29403i 0.210673i −4.69279 − 6.47902i −3.00000 16.7241 + 5.44303i
115.6 −1.90181 + 0.618963i − 1.73205i 3.23377 − 2.35430i −8.79379 1.07208 + 3.29403i − 0.210673i −4.69279 + 6.47902i −3.00000 16.7241 − 5.44303i
115.7 −1.56678 − 1.24306i 1.73205i 0.909627 + 3.89520i 2.29903 2.15303 − 2.71375i 9.46637i 3.41676 − 7.23366i −3.00000 −3.60208 − 2.85782i
115.8 −1.56678 + 1.24306i − 1.73205i 0.909627 − 3.89520i 2.29903 2.15303 + 2.71375i − 9.46637i 3.41676 + 7.23366i −3.00000 −3.60208 + 2.85782i
115.9 −1.56188 − 1.24921i − 1.73205i 0.878937 + 3.90224i −5.44888 −2.16370 + 2.70526i − 11.4186i 3.50193 − 7.19281i −3.00000 8.51050 + 6.80681i
115.10 −1.56188 + 1.24921i 1.73205i 0.878937 − 3.90224i −5.44888 −2.16370 − 2.70526i 11.4186i 3.50193 + 7.19281i −3.00000 8.51050 − 6.80681i
115.11 −1.38888 − 1.43910i − 1.73205i −0.142008 + 3.99748i 1.77514 −2.49259 + 2.40562i 3.79275i 5.95000 − 5.34767i −3.00000 −2.46546 − 2.55460i
115.12 −1.38888 + 1.43910i 1.73205i −0.142008 − 3.99748i 1.77514 −2.49259 − 2.40562i − 3.79275i 5.95000 + 5.34767i −3.00000 −2.46546 + 2.55460i
115.13 −0.459070 − 1.94660i − 1.73205i −3.57851 + 1.78725i −0.507724 −3.37161 + 0.795132i 7.82103i 5.12185 + 6.14546i −3.00000 0.233081 + 0.988336i
115.14 −0.459070 + 1.94660i 1.73205i −3.57851 − 1.78725i −0.507724 −3.37161 − 0.795132i − 7.82103i 5.12185 − 6.14546i −3.00000 0.233081 − 0.988336i
115.15 −0.156655 − 1.99386i − 1.73205i −3.95092 + 0.624694i 5.18748 −3.45346 + 0.271334i − 9.10140i 1.86448 + 7.77970i −3.00000 −0.812643 − 10.3431i
115.16 −0.156655 + 1.99386i 1.73205i −3.95092 − 0.624694i 5.18748 −3.45346 − 0.271334i 9.10140i 1.86448 − 7.77970i −3.00000 −0.812643 + 10.3431i
115.17 0.157997 − 1.99375i 1.73205i −3.95007 − 0.630012i 9.57494 3.45328 + 0.273658i 3.50228i −1.88018 + 7.77592i −3.00000 1.51281 − 19.0900i
115.18 0.157997 + 1.99375i − 1.73205i −3.95007 + 0.630012i 9.57494 3.45328 − 0.273658i − 3.50228i −1.88018 − 7.77592i −3.00000 1.51281 + 19.0900i
115.19 0.234708 − 1.98618i 1.73205i −3.88982 − 0.932347i −0.816108 3.44017 + 0.406527i − 3.29246i −2.76478 + 7.50706i −3.00000 −0.191547 + 1.62094i
115.20 0.234708 + 1.98618i − 1.73205i −3.88982 + 0.932347i −0.816108 3.44017 − 0.406527i 3.29246i −2.76478 − 7.50706i −3.00000 −0.191547 − 1.62094i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.3.g.a ✓ 36
3.b odd 2 1 684.3.g.c 36
4.b odd 2 1 inner 228.3.g.a ✓ 36
12.b even 2 1 684.3.g.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.g.a ✓ 36 1.a even 1 1 trivial
228.3.g.a ✓ 36 4.b odd 2 1 inner
684.3.g.c 36 3.b odd 2 1
684.3.g.c 36 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(228, [\chi])\).