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Properties

Label 168.9.f.a
Level $168$
Weight $9$
Character orbit 168.f
Analytic conductor $68.440$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,9,Mod(97,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, 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("168.97");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 168.f (of order \(2\), degree \(1\), 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: \(68.4396064903\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, 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 − 1184.35i 0 −2341.67 − 530.467i 0 −2187.00 0
97.2 0 − 46.7654i 0 − 1163.39i 0 2137.40 + 1093.77i 0 −2187.00 0
97.3 0 − 46.7654i 0 − 806.986i 0 973.349 + 2194.86i 0 −2187.00 0
97.4 0 − 46.7654i 0 − 711.295i 0 686.692 − 2300.71i 0 −2187.00 0
97.5 0 − 46.7654i 0 − 536.804i 0 −2222.29 + 908.969i 0 −2187.00 0
97.6 0 − 46.7654i 0 − 374.286i 0 −257.869 − 2387.11i 0 −2187.00 0
97.7 0 − 46.7654i 0 − 353.390i 0 2348.55 − 499.133i 0 −2187.00 0
97.8 0 − 46.7654i 0 − 36.6522i 0 −1277.06 + 2033.20i 0 −2187.00 0
97.9 0 − 46.7654i 0 − 22.6386i 0 −2345.61 + 512.762i 0 −2187.00 0
97.10 0 − 46.7654i 0 283.443i 0 1230.42 − 2061.76i 0 −2187.00 0
97.11 0 − 46.7654i 0 316.880i 0 2304.31 + 674.491i 0 −2187.00 0
97.12 0 − 46.7654i 0 440.856i 0 2395.21 − 166.588i 0 −2187.00 0
97.13 0 − 46.7654i 0 747.150i 0 −1652.59 − 1741.76i 0 −2187.00 0
97.14 0 − 46.7654i 0 888.957i 0 −1900.67 − 1467.06i 0 −2187.00 0
97.15 0 − 46.7654i 0 901.819i 0 −596.194 + 2325.80i 0 −2187.00 0
97.16 0 − 46.7654i 0 973.289i 0 1230.02 + 2062.00i 0 −2187.00 0
97.17 0 46.7654i 0 − 973.289i 0 1230.02 − 2062.00i 0 −2187.00 0
97.18 0 46.7654i 0 − 901.819i 0 −596.194 − 2325.80i 0 −2187.00 0
97.19 0 46.7654i 0 − 888.957i 0 −1900.67 + 1467.06i 0 −2187.00 0
97.20 0 46.7654i 0 − 747.150i 0 −1652.59 + 1741.76i 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 168.9.f.a ✓ 32
4.b odd 2 1 336.9.f.d 32
7.b odd 2 1 inner 168.9.f.a ✓ 32
28.d even 2 1 336.9.f.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.9.f.a ✓ 32 1.a even 1 1 trivial
168.9.f.a ✓ 32 7.b odd 2 1 inner
336.9.f.d 32 4.b odd 2 1
336.9.f.d 32 28.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(168, [\chi])\).