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Properties

Label 1332.2.k.b
Level $1332$
Weight $2$
Character orbit 1332.k
Analytic conductor $10.636$
Analytic rank $0$
Dimension $74$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1332,2,Mod(565,1332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1332, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1332.565");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.k (of order \(3\), degree \(2\), 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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(74\)
Relative dimension: \(37\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 74 q - q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q - q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + q^{11} + 2 q^{13} - 2 q^{15} + 3 q^{17} + q^{19} + 6 q^{21} - 17 q^{23} - 48 q^{25} + 17 q^{27} + 6 q^{29} + 3 q^{31} - 13 q^{33} - 6 q^{35} + 5 q^{37} - 16 q^{39} - 27 q^{41} + 3 q^{43} - 4 q^{45} - 21 q^{47} + 92 q^{49} + 11 q^{51} - 11 q^{53} - 11 q^{55} - 16 q^{57} - 26 q^{59} - 24 q^{61} + 5 q^{63} - 19 q^{65} - 19 q^{67} + 16 q^{69} + 18 q^{73} - 7 q^{75} - 46 q^{77} + 2 q^{79} - 35 q^{81} - 20 q^{83} + 9 q^{85} + 15 q^{87} + 13 q^{89} - 11 q^{91} + 3 q^{93} + 26 q^{95} + 17 q^{97} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
565.1 0 −1.70195 + 0.321527i 0 1.39769 − 2.42087i 0 −2.59645 0 2.79324 − 1.09444i 0
565.2 0 −1.69700 + 0.346674i 0 −0.398961 + 0.691021i 0 0.298130 0 2.75963 − 1.17661i 0
565.3 0 −1.68625 − 0.395681i 0 0.421623 − 0.730273i 0 1.60357 0 2.68687 + 1.33443i 0
565.4 0 −1.62077 + 0.610831i 0 −1.20018 + 2.07878i 0 −4.91770 0 2.25377 − 1.98003i 0
565.5 0 −1.60614 − 0.648326i 0 −1.90844 + 3.30552i 0 −0.631224 0 2.15935 + 2.08260i 0
565.6 0 −1.55080 + 0.771367i 0 −1.36923 + 2.37158i 0 2.16592 0 1.80999 − 2.39248i 0
565.7 0 −1.54803 + 0.776920i 0 1.89552 − 3.28313i 0 4.29372 0 1.79279 − 2.40539i 0
565.8 0 −1.30250 − 1.14171i 0 1.98281 − 3.43433i 0 −1.30559 0 0.392998 + 2.97415i 0
565.9 0 −1.25904 − 1.18946i 0 −0.551370 + 0.955001i 0 −4.51094 0 0.170353 + 2.99516i 0
565.10 0 −1.18904 + 1.25943i 0 −0.382364 + 0.662274i 0 −0.0819613 0 −0.172348 − 2.99505i 0
565.11 0 −1.18071 − 1.26725i 0 −1.57975 + 2.73621i 0 4.49716 0 −0.211861 + 2.99251i 0
565.12 0 −1.06751 − 1.36398i 0 0.968299 − 1.67714i 0 −1.61535 0 −0.720857 + 2.91211i 0
565.13 0 −0.964961 + 1.43835i 0 1.27315 − 2.20517i 0 2.17746 0 −1.13770 − 2.77590i 0
565.14 0 −0.807245 + 1.53243i 0 1.15554 − 2.00145i 0 −2.73463 0 −1.69671 − 2.47410i 0
565.15 0 −0.662248 − 1.60045i 0 0.666011 − 1.15356i 0 4.50984 0 −2.12285 + 2.11978i 0
565.16 0 −0.529854 + 1.64902i 0 −2.14112 + 3.70854i 0 1.45487 0 −2.43851 − 1.74748i 0
565.17 0 −0.246851 − 1.71437i 0 −1.08099 + 1.87232i 0 −0.397929 0 −2.87813 + 0.846387i 0
565.18 0 −0.219624 + 1.71807i 0 −0.399638 + 0.692193i 0 −0.658410 0 −2.90353 − 0.754660i 0
565.19 0 −0.168007 + 1.72388i 0 0.00203953 − 0.00353258i 0 4.25820 0 −2.94355 − 0.579250i 0
565.20 0 0.0963829 − 1.72937i 0 −1.38835 + 2.40469i 0 −0.0814174 0 −2.98142 − 0.333363i 0
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 565.37
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.k.b ✓ 74
3.b odd 2 1 3996.2.k.b 74
9.c even 3 1 1332.2.l.b yes 74
9.d odd 6 1 3996.2.l.b 74
37.c even 3 1 1332.2.l.b yes 74
111.i odd 6 1 3996.2.l.b 74
333.g even 3 1 inner 1332.2.k.b ✓ 74
333.u odd 6 1 3996.2.k.b 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.k.b ✓ 74 1.a even 1 1 trivial
1332.2.k.b ✓ 74 333.g even 3 1 inner
1332.2.l.b yes 74 9.c even 3 1
1332.2.l.b yes 74 37.c even 3 1
3996.2.k.b 74 3.b odd 2 1
3996.2.k.b 74 333.u odd 6 1
3996.2.l.b 74 9.d odd 6 1
3996.2.l.b 74 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{74} + 3 T_{5}^{73} + 121 T_{5}^{72} + 322 T_{5}^{71} + 7867 T_{5}^{70} + 19237 T_{5}^{69} + \cdots + 18524553728256 \) acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\). Copy content Toggle raw display