Properties

Label 336.4.bj.g
Level $336$
Weight $4$
Character orbit 336.bj
Analytic conductor $19.825$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,4,Mod(95,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.95");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.bj (of order \(6\), degree \(2\), 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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{9} - 40 q^{13} - 148 q^{21} + 316 q^{25} - 128 q^{33} - 644 q^{37} + 316 q^{45} + 632 q^{49} + 1136 q^{57} + 328 q^{61} - 1424 q^{69} + 1124 q^{73} + 1564 q^{81} - 912 q^{85} + 24 q^{93} - 3304 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −5.19375 0.158148i 0 17.5520 10.1337i 0 18.5198 + 0.135878i 0 26.9500 + 1.64276i 0
95.2 0 −5.16981 0.522553i 0 −5.91262 + 3.41365i 0 −13.7402 12.4180i 0 26.4539 + 5.40300i 0
95.3 0 −5.06455 + 1.16202i 0 −6.80588 + 3.92938i 0 0.395544 + 18.5160i 0 24.2994 11.7702i 0
95.4 0 −3.03745 4.21591i 0 5.91262 3.41365i 0 −13.7402 12.4180i 0 −8.54780 + 25.6112i 0
95.5 0 −2.89222 + 4.31684i 0 −6.70049 + 3.86853i 0 13.9129 12.2242i 0 −10.2702 24.9705i 0
95.6 0 −2.73383 4.41884i 0 −17.5520 + 10.1337i 0 18.5198 + 0.135878i 0 −12.0523 + 24.1607i 0
95.7 0 −2.29238 + 4.66315i 0 6.70049 3.86853i 0 −13.9129 + 12.2242i 0 −16.4900 21.3795i 0
95.8 0 −1.52594 4.96704i 0 6.80588 3.92938i 0 0.395544 + 18.5160i 0 −22.3430 + 15.1588i 0
95.9 0 1.52594 + 4.96704i 0 6.80588 3.92938i 0 −0.395544 18.5160i 0 −22.3430 + 15.1588i 0
95.10 0 2.29238 4.66315i 0 6.70049 3.86853i 0 13.9129 12.2242i 0 −16.4900 21.3795i 0
95.11 0 2.73383 + 4.41884i 0 −17.5520 + 10.1337i 0 −18.5198 0.135878i 0 −12.0523 + 24.1607i 0
95.12 0 2.89222 4.31684i 0 −6.70049 + 3.86853i 0 −13.9129 + 12.2242i 0 −10.2702 24.9705i 0
95.13 0 3.03745 + 4.21591i 0 5.91262 3.41365i 0 13.7402 + 12.4180i 0 −8.54780 + 25.6112i 0
95.14 0 5.06455 1.16202i 0 −6.80588 + 3.92938i 0 −0.395544 18.5160i 0 24.2994 11.7702i 0
95.15 0 5.16981 + 0.522553i 0 −5.91262 + 3.41365i 0 13.7402 + 12.4180i 0 26.4539 + 5.40300i 0
95.16 0 5.19375 + 0.158148i 0 17.5520 10.1337i 0 −18.5198 0.135878i 0 26.9500 + 1.64276i 0
191.1 0 −5.19375 + 0.158148i 0 17.5520 + 10.1337i 0 18.5198 0.135878i 0 26.9500 1.64276i 0
191.2 0 −5.16981 + 0.522553i 0 −5.91262 3.41365i 0 −13.7402 + 12.4180i 0 26.4539 5.40300i 0
191.3 0 −5.06455 1.16202i 0 −6.80588 3.92938i 0 0.395544 18.5160i 0 24.2994 + 11.7702i 0
191.4 0 −3.03745 + 4.21591i 0 5.91262 + 3.41365i 0 −13.7402 + 12.4180i 0 −8.54780 25.6112i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bj.g 32
3.b odd 2 1 inner 336.4.bj.g 32
4.b odd 2 1 inner 336.4.bj.g 32
7.c even 3 1 inner 336.4.bj.g 32
12.b even 2 1 inner 336.4.bj.g 32
21.h odd 6 1 inner 336.4.bj.g 32
28.g odd 6 1 inner 336.4.bj.g 32
84.n even 6 1 inner 336.4.bj.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.bj.g 32 1.a even 1 1 trivial
336.4.bj.g 32 3.b odd 2 1 inner
336.4.bj.g 32 4.b odd 2 1 inner
336.4.bj.g 32 7.c even 3 1 inner
336.4.bj.g 32 12.b even 2 1 inner
336.4.bj.g 32 21.h odd 6 1 inner
336.4.bj.g 32 28.g odd 6 1 inner
336.4.bj.g 32 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{16} - 579 T_{5}^{14} + 256770 T_{5}^{12} - 37395467 T_{5}^{10} + 3759550482 T_{5}^{8} - 233452565091 T_{5}^{6} + 10602642000841 T_{5}^{4} - 284536107802800 T_{5}^{2} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{4} + 5T_{13}^{3} - 3684T_{13}^{2} + 85684T_{13} - 180880 \) Copy content Toggle raw display
\( T_{19}^{16} - 19023 T_{19}^{14} + 243345570 T_{19}^{12} - 1748942749615 T_{19}^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display