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Properties

Label 368.3.r.b
Level $368$
Weight $3$
Character orbit 368.r
Analytic conductor $10.027$
Analytic rank $0$
Dimension $160$
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] = [368,3,Mod(31,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.r (of order \(22\), degree \(10\), 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: \(10.0272737285\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{5} + 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{5} + 68 q^{9} - 4 q^{13} - 114 q^{17} - 22 q^{21} + 20 q^{25} + 38 q^{29} + 246 q^{33} - 48 q^{37} - 44 q^{41} + 372 q^{45} + 360 q^{49} - 180 q^{53} + 630 q^{57} + 420 q^{61} + 554 q^{65} - 110 q^{69} - 480 q^{73} - 1070 q^{77} - 88 q^{81} - 4 q^{85} + 640 q^{89} + 76 q^{93} - 1622 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −4.90567 + 2.24035i 0 6.16408 + 7.11373i 0 2.33708 − 3.63656i 0 13.1527 − 15.1791i 0
31.2 0 −4.39152 + 2.00554i 0 −2.94882 − 3.40312i 0 −5.72591 + 8.90970i 0 9.36949 − 10.8130i 0
31.3 0 −3.82019 + 1.74462i 0 2.11571 + 2.44166i 0 5.26092 − 8.18615i 0 5.65641 − 6.52785i 0
31.4 0 −3.44741 + 1.57438i 0 0.639641 + 0.738185i 0 −5.23392 + 8.14414i 0 3.51222 − 4.05332i 0
31.5 0 −2.53264 + 1.15662i 0 −6.33903 − 7.31563i 0 3.01183 − 4.68650i 0 −0.817246 + 0.943152i 0
31.6 0 −2.08013 + 0.949965i 0 −1.22553 − 1.41434i 0 4.20550 − 6.54388i 0 −2.46923 + 2.84964i 0
31.7 0 −1.46513 + 0.669103i 0 −1.63487 − 1.88674i 0 0.136183 − 0.211905i 0 −4.19483 + 4.84110i 0
31.8 0 −0.391376 + 0.178735i 0 4.33208 + 4.99949i 0 1.59336 − 2.47931i 0 −5.77252 + 6.66184i 0
31.9 0 0.391376 − 0.178735i 0 4.33208 + 4.99949i 0 −1.59336 + 2.47931i 0 −5.77252 + 6.66184i 0
31.10 0 1.46513 − 0.669103i 0 −1.63487 − 1.88674i 0 −0.136183 + 0.211905i 0 −4.19483 + 4.84110i 0
31.11 0 2.08013 − 0.949965i 0 −1.22553 − 1.41434i 0 −4.20550 + 6.54388i 0 −2.46923 + 2.84964i 0
31.12 0 2.53264 − 1.15662i 0 −6.33903 − 7.31563i 0 −3.01183 + 4.68650i 0 −0.817246 + 0.943152i 0
31.13 0 3.44741 − 1.57438i 0 0.639641 + 0.738185i 0 5.23392 − 8.14414i 0 3.51222 − 4.05332i 0
31.14 0 3.82019 − 1.74462i 0 2.11571 + 2.44166i 0 −5.26092 + 8.18615i 0 5.65641 − 6.52785i 0
31.15 0 4.39152 − 2.00554i 0 −2.94882 − 3.40312i 0 5.72591 − 8.90970i 0 9.36949 − 10.8130i 0
31.16 0 4.90567 − 2.24035i 0 6.16408 + 7.11373i 0 −2.33708 + 3.63656i 0 13.1527 − 15.1791i 0
95.1 0 −4.90567 − 2.24035i 0 6.16408 − 7.11373i 0 2.33708 + 3.63656i 0 13.1527 + 15.1791i 0
95.2 0 −4.39152 − 2.00554i 0 −2.94882 + 3.40312i 0 −5.72591 − 8.90970i 0 9.36949 + 10.8130i 0
95.3 0 −3.82019 − 1.74462i 0 2.11571 − 2.44166i 0 5.26092 + 8.18615i 0 5.65641 + 6.52785i 0
95.4 0 −3.44741 − 1.57438i 0 0.639641 − 0.738185i 0 −5.23392 − 8.14414i 0 3.51222 + 4.05332i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.c even 11 1 inner
92.g odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.r.b ✓ 160
4.b odd 2 1 inner 368.3.r.b ✓ 160
23.c even 11 1 inner 368.3.r.b ✓ 160
92.g odd 22 1 inner 368.3.r.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.3.r.b ✓ 160 1.a even 1 1 trivial
368.3.r.b ✓ 160 4.b odd 2 1 inner
368.3.r.b ✓ 160 23.c even 11 1 inner
368.3.r.b ✓ 160 92.g odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} - 106 T_{3}^{158} + 6747 T_{3}^{156} - 338655 T_{3}^{154} + 14687667 T_{3}^{152} + \cdots + 30\!\cdots\!81 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display