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Properties

Label 592.2.bc.g
Level $592$
Weight $2$
Character orbit 592.bc
Analytic conductor $4.727$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [592,2,Mod(33,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bc (of order \(9\), degree \(6\), not 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 296)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{5} + 3 q^{7} + 3 q^{11} + 9 q^{13} - 3 q^{15} - 27 q^{17} - 9 q^{19} + 15 q^{21} + 3 q^{23} - 12 q^{25} + 3 q^{27} + 9 q^{29} + 6 q^{31} - 6 q^{33} - 9 q^{35} + 9 q^{37} + 12 q^{39} + 48 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −0.550728 − 3.12333i 0 −1.88866 + 1.58478i 0 −1.02296 + 0.858367i 0 −6.63284 + 2.41415i 0
33.2 0 −0.183544 − 1.04093i 0 2.29685 − 1.92729i 0 2.00963 − 1.68628i 0 1.76923 − 0.643948i 0
33.3 0 −0.152613 − 0.865514i 0 0.753113 − 0.631937i 0 −2.17602 + 1.82590i 0 2.09325 − 0.761882i 0
33.4 0 0.260200 + 1.47567i 0 −1.75197 + 1.47008i 0 3.54112 − 2.97135i 0 0.709192 − 0.258125i 0
33.5 0 0.453038 + 2.56930i 0 0.356716 − 0.299320i 0 −1.67812 + 1.40811i 0 −3.57700 + 1.30192i 0
49.1 0 −2.29049 − 1.92195i 0 1.16457 − 0.423869i 0 −1.37884 + 0.501855i 0 1.03151 + 5.84999i 0
49.2 0 −1.26432 − 1.06089i 0 −3.29954 + 1.20094i 0 2.65716 − 0.967128i 0 −0.0479306 − 0.271828i 0
49.3 0 −0.149659 − 0.125579i 0 2.90613 − 1.05775i 0 0.434234 − 0.158048i 0 −0.514317 − 2.91684i 0
49.4 0 0.714590 + 0.599612i 0 −2.46699 + 0.897912i 0 −3.18160 + 1.15801i 0 −0.369841 − 2.09747i 0
49.5 0 2.22383 + 1.86602i 0 −0.243860 + 0.0887576i 0 2.73508 − 0.995489i 0 0.942466 + 5.34499i 0
81.1 0 −2.77384 − 1.00960i 0 −0.559739 − 3.17444i 0 −0.344404 − 1.95321i 0 4.37680 + 3.67257i 0
81.2 0 −0.675801 − 0.245972i 0 −0.280258 − 1.58942i 0 0.580938 + 3.29466i 0 −1.90193 − 1.59591i 0
81.3 0 −0.160935 − 0.0585756i 0 0.269863 + 1.53047i 0 −0.837999 − 4.75253i 0 −2.27566 − 1.90951i 0
81.4 0 2.19103 + 0.797470i 0 0.401866 + 2.27909i 0 0.354126 + 2.00835i 0 1.86652 + 1.56620i 0
81.5 0 2.35924 + 0.858695i 0 −0.658084 − 3.73218i 0 −0.192353 − 1.09089i 0 2.53054 + 2.12338i 0
145.1 0 −2.29049 + 1.92195i 0 1.16457 + 0.423869i 0 −1.37884 − 0.501855i 0 1.03151 − 5.84999i 0
145.2 0 −1.26432 + 1.06089i 0 −3.29954 − 1.20094i 0 2.65716 + 0.967128i 0 −0.0479306 + 0.271828i 0
145.3 0 −0.149659 + 0.125579i 0 2.90613 + 1.05775i 0 0.434234 + 0.158048i 0 −0.514317 + 2.91684i 0
145.4 0 0.714590 − 0.599612i 0 −2.46699 − 0.897912i 0 −3.18160 − 1.15801i 0 −0.369841 + 2.09747i 0
145.5 0 2.22383 − 1.86602i 0 −0.243860 − 0.0887576i 0 2.73508 + 0.995489i 0 0.942466 − 5.34499i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.bc.g 30
4.b odd 2 1 296.2.u.b ✓ 30
37.f even 9 1 inner 592.2.bc.g 30
148.p odd 18 1 296.2.u.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.u.b ✓ 30 4.b odd 2 1
296.2.u.b ✓ 30 148.p odd 18 1
592.2.bc.g 30 1.a even 1 1 trivial
592.2.bc.g 30 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 7 T_{3}^{27} + 3 T_{3}^{26} + 39 T_{3}^{25} + 650 T_{3}^{24} - 849 T_{3}^{23} - 1077 T_{3}^{22} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\). Copy content Toggle raw display