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Properties

Label 336.7.f.d
Level $336$
Weight $7$
Character orbit 336.f
Analytic conductor $77.298$
Analytic rank $0$
Dimension $24$
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] = [336,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.97");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 564 q^{7} - 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 564 q^{7} - 5832 q^{9} + 3248 q^{11} + 4536 q^{15} - 4860 q^{21} - 35616 q^{23} - 30720 q^{25} + 51520 q^{29} - 177240 q^{35} - 76728 q^{37} + 136080 q^{39} - 57816 q^{43} - 91824 q^{49} + 42120 q^{51} + 703184 q^{53} + 44712 q^{57} + 137052 q^{63} - 27248 q^{65} - 401448 q^{67} - 224048 q^{71} + 155232 q^{77} - 1029432 q^{79} + 1417176 q^{81} - 1278264 q^{85} + 576 q^{91} + 1745712 q^{93} + 3602640 q^{95} - 789264 q^{99}+O(q^{100}) \) 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 − 15.5885i 0 − 200.741i 0 −340.588 − 40.6080i 0 −243.000 0
97.2 0 − 15.5885i 0 − 200.659i 0 −142.921 − 311.805i 0 −243.000 0
97.3 0 − 15.5885i 0 − 101.639i 0 321.387 + 119.830i 0 −243.000 0
97.4 0 − 15.5885i 0 − 41.5226i 0 324.697 − 110.547i 0 −243.000 0
97.5 0 − 15.5885i 0 − 38.3025i 0 −224.706 + 259.145i 0 −243.000 0
97.6 0 − 15.5885i 0 7.72491i 0 43.3396 − 340.251i 0 −243.000 0
97.7 0 − 15.5885i 0 46.5357i 0 94.5964 + 329.698i 0 −243.000 0
97.8 0 − 15.5885i 0 47.5479i 0 −292.228 − 179.587i 0 −243.000 0
97.9 0 − 15.5885i 0 115.395i 0 303.058 − 160.639i 0 −243.000 0
97.10 0 − 15.5885i 0 124.470i 0 −251.080 + 233.683i 0 −243.000 0
97.11 0 − 15.5885i 0 174.350i 0 −179.776 − 292.112i 0 −243.000 0
97.12 0 − 15.5885i 0 212.332i 0 62.2206 + 337.309i 0 −243.000 0
97.13 0 15.5885i 0 − 212.332i 0 62.2206 − 337.309i 0 −243.000 0
97.14 0 15.5885i 0 − 174.350i 0 −179.776 + 292.112i 0 −243.000 0
97.15 0 15.5885i 0 − 124.470i 0 −251.080 − 233.683i 0 −243.000 0
97.16 0 15.5885i 0 − 115.395i 0 303.058 + 160.639i 0 −243.000 0
97.17 0 15.5885i 0 − 47.5479i 0 −292.228 + 179.587i 0 −243.000 0
97.18 0 15.5885i 0 − 46.5357i 0 94.5964 − 329.698i 0 −243.000 0
97.19 0 15.5885i 0 − 7.72491i 0 43.3396 + 340.251i 0 −243.000 0
97.20 0 15.5885i 0 38.3025i 0 −224.706 − 259.145i 0 −243.000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.24
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 336.7.f.d 24
4.b odd 2 1 168.7.f.a ✓ 24
7.b odd 2 1 inner 336.7.f.d 24
12.b even 2 1 504.7.f.c 24
28.d even 2 1 168.7.f.a ✓ 24
84.h odd 2 1 504.7.f.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.7.f.a ✓ 24 4.b odd 2 1
168.7.f.a ✓ 24 28.d even 2 1
336.7.f.d 24 1.a even 1 1 trivial
336.7.f.d 24 7.b odd 2 1 inner
504.7.f.c 24 12.b even 2 1
504.7.f.c 24 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 202860 T_{5}^{22} + 17205743196 T_{5}^{20} + 793531283637280 T_{5}^{18} + \cdots + 35\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display