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Properties

Label 1332.2.i.d
Level $1332$
Weight $2$
Character orbit 1332.i
Analytic conductor $10.636$
Analytic rank $0$
Dimension $34$
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(445,1332)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1332.445");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.i (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: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + q^{3} + 7 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + q^{3} + 7 q^{5} - q^{7} + 13 q^{9} + 4 q^{11} + 5 q^{13} - 9 q^{15} - 36 q^{17} + 14 q^{19} - 13 q^{21} + 9 q^{23} - 14 q^{25} + 7 q^{27} + 21 q^{29} - q^{31} + 3 q^{33} - 12 q^{35} + 34 q^{37} - 16 q^{39} + 12 q^{41} - 16 q^{43} + 36 q^{45} + 28 q^{47} - 30 q^{49} - 57 q^{51} + 36 q^{55} + 8 q^{57} - 5 q^{59} + 14 q^{61} + 71 q^{63} - 6 q^{65} + 2 q^{67} - 39 q^{69} - 6 q^{71} - 22 q^{73} + 34 q^{75} + 15 q^{77} + 29 q^{79} + 13 q^{81} - 26 q^{83} + 24 q^{85} - 45 q^{87} - 86 q^{89} - 38 q^{91} - 10 q^{93} + 44 q^{95} + 2 q^{97} + 69 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
445.1 0 −1.72973 − 0.0896595i 0 0.676378 − 1.17152i 0 −1.27801 − 2.21358i 0 2.98392 + 0.310173i 0
445.2 0 −1.71767 + 0.222713i 0 −1.85744 + 3.21718i 0 −0.223302 − 0.386770i 0 2.90080 − 0.765097i 0
445.3 0 −1.45052 − 0.946575i 0 −0.245956 + 0.426007i 0 −0.771134 − 1.33564i 0 1.20799 + 2.74604i 0
445.4 0 −1.39829 + 1.02215i 0 1.92085 − 3.32701i 0 2.12180 + 3.67507i 0 0.910435 − 2.85852i 0
445.5 0 −1.38511 − 1.03993i 0 1.38217 − 2.39398i 0 1.28064 + 2.21814i 0 0.837075 + 2.88085i 0
445.6 0 −1.18838 + 1.26006i 0 −0.644687 + 1.11663i 0 0.533014 + 0.923208i 0 −0.175514 − 2.99486i 0
445.7 0 −0.636375 + 1.61091i 0 0.826372 − 1.43132i 0 −1.33266 − 2.30823i 0 −2.19005 − 2.05028i 0
445.8 0 −0.452329 − 1.67194i 0 1.84760 − 3.20014i 0 −1.71498 − 2.97044i 0 −2.59080 + 1.51254i 0
445.9 0 0.239568 + 1.71540i 0 −0.601818 + 1.04238i 0 1.31086 + 2.27047i 0 −2.88521 + 0.821911i 0
445.10 0 0.549470 − 1.64258i 0 −0.664374 + 1.15073i 0 −2.53870 − 4.39716i 0 −2.39617 − 1.80510i 0
445.11 0 0.559011 + 1.63936i 0 −1.59923 + 2.76995i 0 0.659657 + 1.14256i 0 −2.37501 + 1.83284i 0
445.12 0 1.20687 + 1.24237i 0 −0.230726 + 0.399628i 0 −1.11313 − 1.92799i 0 −0.0869511 + 2.99874i 0
445.13 0 1.24253 − 1.20669i 0 −0.779330 + 1.34984i 0 1.99590 + 3.45700i 0 0.0877744 − 2.99872i 0
445.14 0 1.48415 + 0.892914i 0 2.11929 − 3.67071i 0 −1.62952 − 2.82241i 0 1.40541 + 2.65044i 0
445.15 0 1.72166 − 0.189455i 0 1.39939 − 2.42381i 0 1.00951 + 1.74853i 0 2.92821 − 0.652353i 0
445.16 0 1.72392 − 0.167676i 0 0.289753 − 0.501867i 0 2.27209 + 3.93538i 0 2.94377 − 0.578119i 0
445.17 0 1.73123 − 0.0532770i 0 −0.338232 + 0.585835i 0 −1.08204 − 1.87415i 0 2.99432 − 0.184470i 0
889.1 0 −1.72973 + 0.0896595i 0 0.676378 + 1.17152i 0 −1.27801 + 2.21358i 0 2.98392 − 0.310173i 0
889.2 0 −1.71767 − 0.222713i 0 −1.85744 − 3.21718i 0 −0.223302 + 0.386770i 0 2.90080 + 0.765097i 0
889.3 0 −1.45052 + 0.946575i 0 −0.245956 − 0.426007i 0 −0.771134 + 1.33564i 0 1.20799 − 2.74604i 0
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 445.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c 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.i.d ✓ 34
3.b odd 2 1 3996.2.i.c 34
9.c even 3 1 inner 1332.2.i.d ✓ 34
9.d odd 6 1 3996.2.i.c 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.i.d ✓ 34 1.a even 1 1 trivial
1332.2.i.d ✓ 34 9.c even 3 1 inner
3996.2.i.c 34 3.b odd 2 1
3996.2.i.c 34 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{34} - 7 T_{5}^{33} + 74 T_{5}^{32} - 311 T_{5}^{31} + 2181 T_{5}^{30} - 7167 T_{5}^{29} + \cdots + 12510369 \) acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\). Copy content Toggle raw display