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Properties

Label 222.3.b.a
Level $222$
Weight $3$
Character orbit 222.b
Analytic conductor $6.049$
Analytic rank $0$
Dimension $24$
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] = [222,3,Mod(149,222)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(222, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("222.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 222.b (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: \(6.04906186880\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 48 q^{4} - 8 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 48 q^{4} - 8 q^{7} - 20 q^{9} + 16 q^{10} - 8 q^{12} + 8 q^{13} + 44 q^{15} + 96 q^{16} - 64 q^{19} + 12 q^{21} + 16 q^{22} - 72 q^{25} - 104 q^{27} + 16 q^{28} + 48 q^{30} + 40 q^{31} - 20 q^{33} - 96 q^{34} + 40 q^{36} + 76 q^{39} - 32 q^{40} - 80 q^{42} + 48 q^{43} + 28 q^{45} - 16 q^{46} + 16 q^{48} + 144 q^{49} + 100 q^{51} - 16 q^{52} - 120 q^{54} + 248 q^{55} - 212 q^{57} + 32 q^{58} - 88 q^{60} - 160 q^{61} + 244 q^{63} - 192 q^{64} + 192 q^{66} - 176 q^{67} + 8 q^{69} + 112 q^{70} - 456 q^{73} - 260 q^{75} + 128 q^{76} + 304 q^{79} + 140 q^{81} + 144 q^{82} - 24 q^{84} + 48 q^{85} + 144 q^{87} - 32 q^{88} + 128 q^{90} + 264 q^{91} - 168 q^{93} - 192 q^{94} + 256 q^{97} - 548 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 − 1.41421i −2.94231 − 0.585511i −2.00000 2.90689i −0.828037 + 4.16105i 8.77745 2.82843i 8.31435 + 3.44551i 4.11096
149.2 − 1.41421i −2.91502 − 0.708971i −2.00000 − 4.46234i −1.00264 + 4.12247i −8.66229 2.82843i 7.99472 + 4.13333i −6.31070
149.3 − 1.41421i −2.05640 + 2.18431i −2.00000 − 1.35739i 3.08908 + 2.90819i 0.240843 2.82843i −0.542403 − 8.98364i −1.91964
149.4 − 1.41421i −1.38453 − 2.66141i −2.00000 5.90576i −3.76380 + 1.95801i 2.25362 2.82843i −5.16618 + 7.36957i 8.35201
149.5 − 1.41421i −0.393473 + 2.97408i −2.00000 5.41303i 4.20599 + 0.556455i −5.49247 2.82843i −8.69036 − 2.34045i 7.65518
149.6 − 1.41421i 0.359117 − 2.97843i −2.00000 − 5.74163i −4.21213 − 0.507869i 2.95072 2.82843i −8.74207 − 2.13921i −8.11990
149.7 − 1.41421i 0.514762 + 2.95551i −2.00000 − 9.83335i 4.17972 − 0.727983i −5.67060 2.82843i −8.47004 + 3.04276i −13.9065
149.8 − 1.41421i 1.46779 − 2.61641i −2.00000 1.00949i −3.70016 − 2.07577i −11.0273 2.82843i −4.69117 − 7.68068i 1.42763
149.9 − 1.41421i 1.52524 + 2.58334i −2.00000 0.964535i 3.65339 − 2.15702i 7.66494 2.82843i −4.34726 + 7.88044i 1.36406
149.10 − 1.41421i 2.22945 − 2.00737i −2.00000 8.72019i −2.83885 − 3.15292i 4.81421 2.82843i 0.940911 − 8.95068i 12.3322
149.11 − 1.41421i 2.64656 + 1.41270i −2.00000 5.04654i 1.99786 − 3.74280i −10.6067 2.82843i 5.00855 + 7.47759i 7.13688
149.12 − 1.41421i 2.94881 − 0.551840i −2.00000 − 2.91487i −0.780420 − 4.17025i 10.7575 2.82843i 8.39094 − 3.25454i −4.12224
149.13 1.41421i −2.94231 + 0.585511i −2.00000 − 2.90689i −0.828037 − 4.16105i 8.77745 − 2.82843i 8.31435 − 3.44551i 4.11096
149.14 1.41421i −2.91502 + 0.708971i −2.00000 4.46234i −1.00264 − 4.12247i −8.66229 − 2.82843i 7.99472 − 4.13333i −6.31070
149.15 1.41421i −2.05640 − 2.18431i −2.00000 1.35739i 3.08908 − 2.90819i 0.240843 − 2.82843i −0.542403 + 8.98364i −1.91964
149.16 1.41421i −1.38453 + 2.66141i −2.00000 − 5.90576i −3.76380 − 1.95801i 2.25362 − 2.82843i −5.16618 − 7.36957i 8.35201
149.17 1.41421i −0.393473 − 2.97408i −2.00000 − 5.41303i 4.20599 − 0.556455i −5.49247 − 2.82843i −8.69036 + 2.34045i 7.65518
149.18 1.41421i 0.359117 + 2.97843i −2.00000 5.74163i −4.21213 + 0.507869i 2.95072 − 2.82843i −8.74207 + 2.13921i −8.11990
149.19 1.41421i 0.514762 − 2.95551i −2.00000 9.83335i 4.17972 + 0.727983i −5.67060 − 2.82843i −8.47004 − 3.04276i −13.9065
149.20 1.41421i 1.46779 + 2.61641i −2.00000 − 1.00949i −3.70016 + 2.07577i −11.0273 − 2.82843i −4.69117 + 7.68068i 1.42763
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 222.3.b.a ✓ 24
3.b odd 2 1 inner 222.3.b.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.3.b.a ✓ 24 1.a even 1 1 trivial
222.3.b.a ✓ 24 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(222, [\chi])\).