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Properties

Label 336.9.f.d
Level $336$
Weight $9$
Character orbit 336.f
Analytic conductor $136.879$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,9,Mod(97,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.97");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 336.f (of order \(2\), degree \(1\), 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: \(136.879212981\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 1424 q^{7} - 69984 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 1424 q^{7} - 69984 q^{9} - 23616 q^{11} + 59616 q^{15} + 60912 q^{21} - 455040 q^{23} - 3404928 q^{25} + 632064 q^{29} + 1543200 q^{35} + 4150496 q^{37} + 4162752 q^{39} - 6028000 q^{43} + 15115072 q^{49} + 1340064 q^{51} - 37728576 q^{53} + 7286112 q^{57} + 3114288 q^{63} + 29977536 q^{65} - 68431648 q^{67} + 19788096 q^{71} + 87499392 q^{77} + 86954656 q^{79} + 153055008 q^{81} - 31326496 q^{85} + 24540288 q^{91} - 38833344 q^{93} - 329546688 q^{95} + 51648192 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 − 46.7654i 0 − 973.289i 0 −1230.02 + 2062.00i 0 −2187.00 0
97.2 0 − 46.7654i 0 − 901.819i 0 596.194 + 2325.80i 0 −2187.00 0
97.3 0 − 46.7654i 0 − 888.957i 0 1900.67 − 1467.06i 0 −2187.00 0
97.4 0 − 46.7654i 0 − 747.150i 0 1652.59 − 1741.76i 0 −2187.00 0
97.5 0 − 46.7654i 0 − 440.856i 0 −2395.21 − 166.588i 0 −2187.00 0
97.6 0 − 46.7654i 0 − 316.880i 0 −2304.31 + 674.491i 0 −2187.00 0
97.7 0 − 46.7654i 0 − 283.443i 0 −1230.42 − 2061.76i 0 −2187.00 0
97.8 0 − 46.7654i 0 22.6386i 0 2345.61 + 512.762i 0 −2187.00 0
97.9 0 − 46.7654i 0 36.6522i 0 1277.06 + 2033.20i 0 −2187.00 0
97.10 0 − 46.7654i 0 353.390i 0 −2348.55 − 499.133i 0 −2187.00 0
97.11 0 − 46.7654i 0 374.286i 0 257.869 − 2387.11i 0 −2187.00 0
97.12 0 − 46.7654i 0 536.804i 0 2222.29 + 908.969i 0 −2187.00 0
97.13 0 − 46.7654i 0 711.295i 0 −686.692 − 2300.71i 0 −2187.00 0
97.14 0 − 46.7654i 0 806.986i 0 −973.349 + 2194.86i 0 −2187.00 0
97.15 0 − 46.7654i 0 1163.39i 0 −2137.40 + 1093.77i 0 −2187.00 0
97.16 0 − 46.7654i 0 1184.35i 0 2341.67 − 530.467i 0 −2187.00 0
97.17 0 46.7654i 0 − 1184.35i 0 2341.67 + 530.467i 0 −2187.00 0
97.18 0 46.7654i 0 − 1163.39i 0 −2137.40 − 1093.77i 0 −2187.00 0
97.19 0 46.7654i 0 − 806.986i 0 −973.349 − 2194.86i 0 −2187.00 0
97.20 0 46.7654i 0 − 711.295i 0 −686.692 + 2300.71i 0 −2187.00 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.9.f.d 32
4.b odd 2 1 168.9.f.a ✓ 32
7.b odd 2 1 inner 336.9.f.d 32
28.d even 2 1 168.9.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.9.f.a ✓ 32 4.b odd 2 1
168.9.f.a ✓ 32 28.d even 2 1
336.9.f.d 32 1.a even 1 1 trivial
336.9.f.d 32 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 7952464 T_{5}^{30} + 28047371041904 T_{5}^{28} + \cdots + 11\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display