Properties

Label 161.6.c.b
Level $161$
Weight $6$
Character orbit 161.c
Analytic conductor $25.822$
Analytic rank $0$
Dimension $76$
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] = [161,6,Mod(160,161)]
 
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([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.160");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 161.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: \(25.8217949899\)
Analytic rank: \(0\)
Dimension: \(76\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 16 q^{2} + 1056 q^{4} + 1236 q^{8} - 5724 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 16 q^{2} + 1056 q^{4} + 1236 q^{8} - 5724 q^{9} + 8816 q^{16} + 6708 q^{18} - 1512 q^{23} + 43124 q^{25} - 42336 q^{29} + 41448 q^{32} + 3836 q^{35} - 59508 q^{36} - 26120 q^{39} - 73628 q^{46} + 56224 q^{49} - 209744 q^{50} + 82204 q^{58} + 2780 q^{64} - 104048 q^{70} - 153328 q^{71} + 280636 q^{72} - 130956 q^{77} - 132724 q^{78} - 98012 q^{81} - 10168 q^{85} + 553260 q^{92} + 580672 q^{93} - 154344 q^{95} - 48972 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1 −10.2721 25.0021i 73.5163 51.3975 256.825i 128.551 16.7787i −426.461 −382.105 −527.961
160.2 −10.2721 25.0021i 73.5163 −51.3975 256.825i −128.551 + 16.7787i −426.461 −382.105 527.961
160.3 −10.2721 25.0021i 73.5163 −51.3975 256.825i −128.551 16.7787i −426.461 −382.105 527.961
160.4 −10.2721 25.0021i 73.5163 51.3975 256.825i 128.551 + 16.7787i −426.461 −382.105 −527.961
160.5 −9.51992 10.0279i 58.6288 107.196 95.4647i −42.1636 + 122.594i −253.504 142.441 −1020.50
160.6 −9.51992 10.0279i 58.6288 −107.196 95.4647i 42.1636 122.594i −253.504 142.441 1020.50
160.7 −9.51992 10.0279i 58.6288 −107.196 95.4647i 42.1636 + 122.594i −253.504 142.441 1020.50
160.8 −9.51992 10.0279i 58.6288 107.196 95.4647i −42.1636 122.594i −253.504 142.441 −1020.50
160.9 −8.50063 9.26458i 40.2607 8.77494 78.7548i −115.888 58.1113i −70.2213 157.168 −74.5925
160.10 −8.50063 9.26458i 40.2607 −8.77494 78.7548i 115.888 + 58.1113i −70.2213 157.168 74.5925
160.11 −8.50063 9.26458i 40.2607 −8.77494 78.7548i 115.888 58.1113i −70.2213 157.168 74.5925
160.12 −8.50063 9.26458i 40.2607 8.77494 78.7548i −115.888 + 58.1113i −70.2213 157.168 −74.5925
160.13 −7.69648 23.6280i 27.2358 52.2503 181.853i −79.9898 + 102.023i 36.6673 −315.283 −402.144
160.14 −7.69648 23.6280i 27.2358 −52.2503 181.853i 79.9898 + 102.023i 36.6673 −315.283 402.144
160.15 −7.69648 23.6280i 27.2358 −52.2503 181.853i 79.9898 102.023i 36.6673 −315.283 402.144
160.16 −7.69648 23.6280i 27.2358 52.2503 181.853i −79.9898 102.023i 36.6673 −315.283 −402.144
160.17 −5.36240 14.2998i −3.24467 33.8739 76.6815i −9.98159 + 129.257i 188.996 38.5144 −181.646
160.18 −5.36240 14.2998i −3.24467 −33.8739 76.6815i 9.98159 129.257i 188.996 38.5144 181.646
160.19 −5.36240 14.2998i −3.24467 −33.8739 76.6815i 9.98159 + 129.257i 188.996 38.5144 181.646
160.20 −5.36240 14.2998i −3.24467 33.8739 76.6815i −9.98159 129.257i 188.996 38.5144 −181.646
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.76
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.6.c.b 76
7.b odd 2 1 inner 161.6.c.b 76
23.b odd 2 1 inner 161.6.c.b 76
161.c even 2 1 inner 161.6.c.b 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.6.c.b 76 1.a even 1 1 trivial
161.6.c.b 76 7.b odd 2 1 inner
161.6.c.b 76 23.b odd 2 1 inner
161.6.c.b 76 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{19} - 4 T_{2}^{18} - 428 T_{2}^{17} + 1545 T_{2}^{16} + 75709 T_{2}^{15} - 242562 T_{2}^{14} - 7182298 T_{2}^{13} + 19955241 T_{2}^{12} + 397008040 T_{2}^{11} - 926477020 T_{2}^{10} + \cdots - 2870938054656 \) acting on \(S_{6}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display