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Properties

Label 616.2.bw.a
Level $616$
Weight $2$
Character orbit 616.bw
Analytic conductor $4.919$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [616,2,Mod(9,616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(616, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 0, 10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("616.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.bw (of order \(15\), degree \(8\), 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: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} - 6 q^{7} + 18 q^{9} - 2 q^{11} + 18 q^{13} - 2 q^{15} - 9 q^{17} + 11 q^{19} - 14 q^{21} + 24 q^{25} + 16 q^{27} + 34 q^{29} + 2 q^{31} - 17 q^{33} + 40 q^{35} + 6 q^{37} - 6 q^{39} - 10 q^{41}+ \cdots - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.58064 − 1.14897i 0 0.790449 + 0.168015i 0 −2.48787 + 0.900269i 0 3.33215 + 3.70073i 0
9.2 0 −2.48929 − 1.10830i 0 4.09501 + 0.870421i 0 2.53615 − 0.753612i 0 2.96084 + 3.28835i 0
9.3 0 −2.16291 − 0.962991i 0 −3.79797 − 0.807283i 0 −1.08776 − 2.41180i 0 1.74345 + 1.93630i 0
9.4 0 −1.63732 − 0.728981i 0 −1.76997 − 0.376220i 0 2.36549 + 1.18510i 0 0.142008 + 0.157716i 0
9.5 0 −1.46878 − 0.653941i 0 0.236112 + 0.0501871i 0 −1.54893 + 2.14495i 0 −0.277730 − 0.308450i 0
9.6 0 −0.258235 − 0.114974i 0 0.279761 + 0.0594650i 0 2.23433 − 1.41697i 0 −1.95393 − 2.17005i 0
9.7 0 0.171513 + 0.0763627i 0 3.90687 + 0.830432i 0 −2.12786 − 1.57232i 0 −1.98381 − 2.20324i 0
9.8 0 1.09424 + 0.487186i 0 2.77006 + 0.588795i 0 1.03405 + 2.43531i 0 −1.04739 − 1.16324i 0
9.9 0 1.13046 + 0.503313i 0 −1.64736 − 0.350157i 0 −1.84936 + 1.89205i 0 −0.982776 − 1.09148i 0
9.10 0 1.65728 + 0.737868i 0 −2.98976 − 0.635493i 0 −1.95619 − 1.78138i 0 0.194734 + 0.216274i 0
9.11 0 2.03125 + 0.904372i 0 −0.801161 − 0.170292i 0 2.64018 + 0.171552i 0 1.30071 + 1.44458i 0
9.12 0 2.68534 + 1.19559i 0 2.46693 + 0.524361i 0 −0.432037 − 2.61024i 0 3.77421 + 4.19168i 0
25.1 0 −2.92414 − 0.621545i 0 0.424764 − 4.04136i 0 2.33990 + 1.23485i 0 5.42364 + 2.41476i 0
25.2 0 −2.51342 − 0.534245i 0 −0.226421 + 2.15425i 0 2.52533 − 0.789122i 0 3.29125 + 1.46536i 0
25.3 0 −2.25684 − 0.479705i 0 −0.100588 + 0.957030i 0 −2.09175 − 1.62005i 0 2.12255 + 0.945022i 0
25.4 0 −1.33634 − 0.284048i 0 −0.00194692 + 0.0185237i 0 −0.886493 + 2.49282i 0 −1.03551 − 0.461039i 0
25.5 0 −0.227792 − 0.0484186i 0 0.298475 − 2.83980i 0 1.34842 − 2.27635i 0 −2.69109 − 1.19815i 0
25.6 0 −0.0275529 − 0.00585654i 0 0.0415095 − 0.394937i 0 0.188898 + 2.63900i 0 −2.73991 − 1.21989i 0
25.7 0 0.337058 + 0.0716438i 0 −0.393273 + 3.74174i 0 −2.63607 − 0.226126i 0 −2.63216 − 1.17191i 0
25.8 0 1.04060 + 0.221187i 0 −0.00568803 + 0.0541180i 0 2.19956 − 1.47036i 0 −1.70670 − 0.759873i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 616.2.bw.a ✓ 96
7.c even 3 1 inner 616.2.bw.a ✓ 96
11.c even 5 1 inner 616.2.bw.a ✓ 96
77.m even 15 1 inner 616.2.bw.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.bw.a ✓ 96 1.a even 1 1 trivial
616.2.bw.a ✓ 96 7.c even 3 1 inner
616.2.bw.a ✓ 96 11.c even 5 1 inner
616.2.bw.a ✓ 96 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} + 2 T_{3}^{95} - 25 T_{3}^{94} - 66 T_{3}^{93} + 230 T_{3}^{92} + 756 T_{3}^{91} + \cdots + 96059601 \) acting on \(S_{2}^{\mathrm{new}}(616, [\chi])\). Copy content Toggle raw display