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Properties

Label 920.2.bk.a
Level $920$
Weight $2$
Character orbit 920.bk
Analytic conductor $7.346$
Analytic rank $0$
Dimension $360$
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] = [920,2,Mod(9,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 11, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.bk (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: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 360 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q + 32 q^{9} + 16 q^{11} - 20 q^{15} - 16 q^{19} - 16 q^{21} + 8 q^{25} + 12 q^{29} + 8 q^{31} + 4 q^{35} + 52 q^{39} - 8 q^{41} + 12 q^{45} + 102 q^{49} + 24 q^{51} - 8 q^{55} + 34 q^{59} - 12 q^{65} - 16 q^{69} - 8 q^{71} + 50 q^{75} - 8 q^{79} - 236 q^{81} + 78 q^{85} + 48 q^{89} - 248 q^{91} + 124 q^{95} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.56933 + 2.22634i 0 1.58809 + 1.57415i 0 −3.35844 − 1.53375i 0 1.21794 − 8.47093i 0
9.2 0 −2.43424 + 2.10928i 0 −2.19266 + 0.438460i 0 4.57544 + 2.08953i 0 1.04951 − 7.29951i 0
9.3 0 −2.26055 + 1.95878i 0 1.19584 − 1.88944i 0 −1.11664 − 0.509952i 0 0.846336 − 5.88640i 0
9.4 0 −2.10860 + 1.82711i 0 −0.810434 − 2.08403i 0 2.52446 + 1.15288i 0 0.680910 − 4.73583i 0
9.5 0 −1.99985 + 1.73288i 0 −2.07889 − 0.823530i 0 −2.57469 − 1.17582i 0 0.569588 − 3.96157i 0
9.6 0 −1.74416 + 1.51132i 0 −0.671554 + 2.13284i 0 −2.31780 − 1.05850i 0 0.331047 − 2.30248i 0
9.7 0 −1.60176 + 1.38793i 0 −1.56163 + 1.60041i 0 0.355423 + 0.162316i 0 0.212331 − 1.47680i 0
9.8 0 −1.48992 + 1.29102i 0 1.84250 − 1.26696i 0 1.57353 + 0.718607i 0 0.126176 − 0.877571i 0
9.9 0 −1.33407 + 1.15598i 0 2.03640 + 0.923616i 0 2.79867 + 1.27811i 0 0.0165126 − 0.114847i 0
9.10 0 −1.01281 + 0.877608i 0 2.21713 + 0.290373i 0 −0.477781 − 0.218195i 0 −0.171349 + 1.19176i 0
9.11 0 −0.907242 + 0.786130i 0 0.0272207 + 2.23590i 0 4.31390 + 1.97009i 0 −0.221856 + 1.54304i 0
9.12 0 −0.881059 + 0.763442i 0 −0.765027 − 2.10113i 0 2.05736 + 0.939563i 0 −0.233523 + 1.62419i 0
9.13 0 −0.861219 + 0.746251i 0 −1.32062 − 1.80443i 0 −1.27431 − 0.581956i 0 −0.242136 + 1.68409i 0
9.14 0 −0.769082 + 0.666413i 0 −2.20350 + 0.380244i 0 −0.811886 − 0.370776i 0 −0.279564 + 1.94441i 0
9.15 0 −0.553283 + 0.479422i 0 2.04052 − 0.914477i 0 −3.64366 − 1.66400i 0 −0.350668 + 2.43895i 0
9.16 0 −0.438011 + 0.379538i 0 −1.74103 + 1.40314i 0 −1.41552 − 0.646448i 0 −0.379141 + 2.63698i 0
9.17 0 −0.120110 + 0.104076i 0 −1.45823 − 1.69516i 0 −3.88737 − 1.77530i 0 −0.423350 + 2.94446i 0
9.18 0 −0.0402939 + 0.0349149i 0 0.969361 + 2.01503i 0 −2.92618 − 1.33634i 0 −0.426540 + 2.96665i 0
9.19 0 0.0402939 − 0.0349149i 0 −0.362396 + 2.20651i 0 2.92618 + 1.33634i 0 −0.426540 + 2.96665i 0
9.20 0 0.120110 − 0.104076i 0 0.921574 − 2.03733i 0 3.88737 + 1.77530i 0 −0.423350 + 2.94446i 0
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.2.bk.a ✓ 360
5.b even 2 1 inner 920.2.bk.a ✓ 360
23.c even 11 1 inner 920.2.bk.a ✓ 360
115.j even 22 1 inner 920.2.bk.a ✓ 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.bk.a ✓ 360 1.a even 1 1 trivial
920.2.bk.a ✓ 360 5.b even 2 1 inner
920.2.bk.a ✓ 360 23.c even 11 1 inner
920.2.bk.a ✓ 360 115.j even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(920, [\chi])\).