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Properties

Label 161.3.d.a
Level $161$
Weight $3$
Character orbit 161.d
Analytic conductor $4.387$
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] = [161,3,Mod(22,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.22");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 161.d (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: \(4.38693225620\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 4 q^{3} + 34 q^{4} + 30 q^{6} + 20 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 4 q^{3} + 34 q^{4} + 30 q^{6} + 20 q^{8} + 52 q^{9} - 62 q^{12} - 4 q^{13} - 6 q^{16} - 104 q^{18} + 44 q^{23} + 48 q^{24} - 208 q^{25} - 14 q^{26} + 128 q^{27} + 100 q^{29} - 16 q^{31} + 198 q^{32} + 28 q^{35} - 268 q^{36} + 12 q^{39} - 36 q^{41} - 14 q^{46} - 96 q^{47} - 158 q^{48} - 168 q^{49} - 178 q^{50} + 146 q^{52} + 554 q^{54} - 156 q^{55} + 170 q^{58} - 8 q^{59} - 326 q^{62} + 64 q^{64} + 8 q^{69} + 112 q^{70} - 36 q^{71} - 456 q^{72} + 188 q^{73} + 40 q^{75} - 112 q^{77} - 402 q^{78} + 488 q^{81} - 598 q^{82} + 424 q^{85} - 40 q^{87} - 2 q^{92} + 236 q^{93} + 306 q^{94} - 168 q^{95} - 258 q^{96} + 14 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −3.47619 −1.40762 8.08387 − 0.0658372i 4.89315 2.64575i −14.1963 −7.01861 0.228863i
22.2 −3.47619 −1.40762 8.08387 0.0658372i 4.89315 − 2.64575i −14.1963 −7.01861 − 0.228863i
22.3 −2.92729 −5.43259 4.56904 − 7.31745i 15.9028 − 2.64575i −1.66576 20.5131 21.4203i
22.4 −2.92729 −5.43259 4.56904 7.31745i 15.9028 2.64575i −1.66576 20.5131 − 21.4203i
22.5 −2.43346 1.21827 1.92174 − 8.65587i −2.96460 2.64575i 5.05738 −7.51583 21.0637i
22.6 −2.43346 1.21827 1.92174 8.65587i −2.96460 − 2.64575i 5.05738 −7.51583 − 21.0637i
22.7 −2.05724 4.52370 0.232234 − 1.14818i −9.30634 − 2.64575i 7.75120 11.4639 2.36207i
22.8 −2.05724 4.52370 0.232234 1.14818i −9.30634 2.64575i 7.75120 11.4639 − 2.36207i
22.9 −1.51406 −1.78414 −1.70762 − 3.34926i 2.70130 − 2.64575i 8.64168 −5.81684 5.07098i
22.10 −1.51406 −1.78414 −1.70762 3.34926i 2.70130 2.64575i 8.64168 −5.81684 − 5.07098i
22.11 −0.216529 −4.48588 −3.95312 − 5.66881i 0.971321 2.64575i 1.72208 11.1231 1.22746i
22.12 −0.216529 −4.48588 −3.95312 5.66881i 0.971321 − 2.64575i 1.72208 11.1231 − 1.22746i
22.13 −0.100994 2.00093 −3.98980 − 5.53327i −0.202081 − 2.64575i 0.806921 −4.99629 0.558827i
22.14 −0.100994 2.00093 −3.98980 5.53327i −0.202081 2.64575i 0.806921 −4.99629 − 0.558827i
22.15 1.09177 5.62429 −2.80804 − 8.70874i 6.14043 2.64575i −7.43281 22.6326 − 9.50794i
22.16 1.09177 5.62429 −2.80804 8.70874i 6.14043 − 2.64575i −7.43281 22.6326 9.50794i
22.17 1.63397 −2.37487 −1.33014 − 3.10230i −3.88047 2.64575i −8.70929 −3.35998 − 5.06906i
22.18 1.63397 −2.37487 −1.33014 3.10230i −3.88047 − 2.64575i −8.70929 −3.35998 5.06906i
22.19 2.37565 −2.27397 1.64372 − 8.23547i −5.40215 − 2.64575i −5.59771 −3.82908 − 19.5646i
22.20 2.37565 −2.27397 1.64372 8.23547i −5.40215 2.64575i −5.59771 −3.82908 19.5646i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.3.d.a ✓ 24
23.b odd 2 1 inner 161.3.d.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.3.d.a ✓ 24 1.a even 1 1 trivial
161.3.d.a ✓ 24 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(161, [\chi])\).