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Properties

Label 336.4.bj.f
Level $336$
Weight $4$
Character orbit 336.bj
Analytic conductor $19.825$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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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: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 38 q^{7} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 38 q^{7} - 70 q^{9} + 124 q^{13} - 462 q^{19} + 500 q^{21} + 566 q^{25} + 1266 q^{31} + 64 q^{33} + 338 q^{37} + 1254 q^{39} - 488 q^{45} - 206 q^{49} + 522 q^{51} + 2324 q^{57} - 340 q^{61} - 840 q^{63} + 2934 q^{67} - 776 q^{69} + 2050 q^{73} - 1458 q^{75} + 2070 q^{79} + 910 q^{81} + 2400 q^{85} + 3132 q^{87} - 5170 q^{91} + 714 q^{93} - 3344 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −5.18844 + 0.283083i 0 −8.90809 + 5.14309i 0 9.06303 + 16.1512i 0 26.8397 − 2.93751i 0
95.2 0 −5.04533 + 1.24284i 0 11.5701 − 6.68000i 0 −16.4792 + 8.45200i 0 23.9107 − 12.5411i 0
95.3 0 −4.15916 + 3.11470i 0 −17.2240 + 9.94429i 0 −5.66612 − 17.6322i 0 7.59723 − 25.9091i 0
95.4 0 −2.63380 + 4.47918i 0 10.1094 − 5.83669i 0 17.9159 − 4.69245i 0 −13.1262 − 23.5946i 0
95.5 0 −2.34906 − 4.63486i 0 8.90809 − 5.14309i 0 9.06303 + 16.1512i 0 −15.9638 + 21.7751i 0
95.6 0 −1.44633 − 4.99080i 0 −11.5701 + 6.68000i 0 −16.4792 + 8.45200i 0 −22.8163 + 14.4367i 0
95.7 0 −0.692259 + 5.14983i 0 4.30532 − 2.48568i 0 −14.4573 − 11.5752i 0 −26.0416 − 7.13004i 0
95.8 0 0.538038 + 5.16822i 0 −7.60125 + 4.38858i 0 3.39378 + 18.2067i 0 −26.4210 + 5.56140i 0
95.9 0 0.617833 − 5.15929i 0 17.2240 − 9.94429i 0 −5.66612 − 17.6322i 0 −26.2366 − 6.37516i 0
95.10 0 2.56219 − 4.52053i 0 −10.1094 + 5.83669i 0 17.9159 − 4.69245i 0 −13.8704 − 23.1649i 0
95.11 0 4.11376 − 3.17443i 0 −4.30532 + 2.48568i 0 −14.4573 − 11.5752i 0 6.84598 − 26.1177i 0
95.12 0 4.16378 + 3.10853i 0 13.4200 − 7.74804i 0 15.7299 − 9.77601i 0 7.67408 + 25.8865i 0
95.13 0 4.74483 − 2.11816i 0 7.60125 − 4.38858i 0 3.39378 + 18.2067i 0 18.0268 − 20.1006i 0
95.14 0 4.77395 + 2.05167i 0 −13.4200 + 7.74804i 0 15.7299 − 9.77601i 0 18.5813 + 19.5892i 0
191.1 0 −5.18844 − 0.283083i 0 −8.90809 − 5.14309i 0 9.06303 − 16.1512i 0 26.8397 + 2.93751i 0
191.2 0 −5.04533 − 1.24284i 0 11.5701 + 6.68000i 0 −16.4792 − 8.45200i 0 23.9107 + 12.5411i 0
191.3 0 −4.15916 − 3.11470i 0 −17.2240 − 9.94429i 0 −5.66612 + 17.6322i 0 7.59723 + 25.9091i 0
191.4 0 −2.63380 − 4.47918i 0 10.1094 + 5.83669i 0 17.9159 + 4.69245i 0 −13.1262 + 23.5946i 0
191.5 0 −2.34906 + 4.63486i 0 8.90809 + 5.14309i 0 9.06303 − 16.1512i 0 −15.9638 − 21.7751i 0
191.6 0 −1.44633 + 4.99080i 0 −11.5701 − 6.68000i 0 −16.4792 − 8.45200i 0 −22.8163 − 14.4367i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 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.f yes 28
3.b odd 2 1 inner 336.4.bj.f yes 28
4.b odd 2 1 336.4.bj.e ✓ 28
7.c even 3 1 336.4.bj.e ✓ 28
12.b even 2 1 336.4.bj.e ✓ 28
21.h odd 6 1 336.4.bj.e ✓ 28
28.g odd 6 1 inner 336.4.bj.f yes 28
84.n even 6 1 inner 336.4.bj.f yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.bj.e ✓ 28 4.b odd 2 1
336.4.bj.e ✓ 28 7.c even 3 1
336.4.bj.e ✓ 28 12.b even 2 1
336.4.bj.e ✓ 28 21.h odd 6 1
336.4.bj.f yes 28 1.a even 1 1 trivial
336.4.bj.f yes 28 3.b odd 2 1 inner
336.4.bj.f yes 28 28.g odd 6 1 inner
336.4.bj.f yes 28 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 \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{28} - 1158 T_{5}^{26} + 811629 T_{5}^{24} - 365180866 T_{5}^{22} + 120762762282 T_{5}^{20} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
\( T_{13}^{7} - 31 T_{13}^{6} - 7296 T_{13}^{5} + 191240 T_{13}^{4} + 8458096 T_{13}^{3} + \cdots + 5757261056 \) Copy content Toggle raw display
\( T_{19}^{14} + 231 T_{19}^{13} + 1995 T_{19}^{12} - 3647952 T_{19}^{11} - 13805967 T_{19}^{10} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display