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Properties

Label 2116.4.a.j
Level $2116$
Weight $4$
Character orbit 2116.a
Self dual yes
Analytic conductor $124.848$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2116,4,Mod(1,2116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2116, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2116.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2116 = 2^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2116.a (trivial)

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: yes
Analytic conductor: \(124.848041572\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 92)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{3} + 50 q^{5} - 2 q^{7} + 272 q^{9} + 92 q^{11} - 28 q^{13} - 64 q^{15} + 316 q^{17} + 222 q^{19} + 512 q^{21} + 786 q^{25} - 212 q^{27} + 218 q^{29} + 350 q^{31} + 576 q^{33} + 150 q^{35} + 314 q^{37}+ \cdots + 930 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −9.94474 0 12.4605 0 −26.2182 0 71.8979 0
1.2 0 −9.19219 0 −7.93087 0 −7.58084 0 57.4963 0
1.3 0 −8.69590 0 13.9258 0 −29.7076 0 48.6186 0
1.4 0 −7.95437 0 16.6634 0 21.8554 0 36.2721 0
1.5 0 −7.94569 0 −15.5280 0 −9.00951 0 36.1340 0
1.6 0 −7.88030 0 −2.89546 0 7.59476 0 35.0991 0
1.7 0 −6.17580 0 −12.1175 0 13.5754 0 11.1405 0
1.8 0 −4.73733 0 8.82100 0 28.0962 0 −4.55769 0
1.9 0 −4.17691 0 5.12216 0 29.2816 0 −9.55339 0
1.10 0 −3.06951 0 16.5539 0 5.57853 0 −17.5781 0
1.11 0 −2.97381 0 −12.3295 0 −14.6286 0 −18.1565 0
1.12 0 −2.04083 0 −0.768282 0 −21.6878 0 −22.8350 0
1.13 0 −1.94926 0 −3.12092 0 −11.3326 0 −23.2004 0
1.14 0 −1.09089 0 9.43805 0 −34.7650 0 −25.8100 0
1.15 0 −0.780799 0 19.4878 0 32.8146 0 −26.3904 0
1.16 0 0.0220256 0 −3.09464 0 16.7552 0 −26.9995 0
1.17 0 1.06823 0 16.3925 0 −17.0749 0 −25.8589 0
1.18 0 1.60050 0 2.46888 0 −22.7375 0 −24.4384 0
1.19 0 2.93512 0 −18.7922 0 −4.55096 0 −18.3851 0
1.20 0 3.07231 0 −13.8777 0 28.7429 0 −17.5609 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2116.4.a.j 30
23.b odd 2 1 2116.4.a.i 30
23.d odd 22 2 92.4.e.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.4.e.a ✓ 60 23.d odd 22 2
2116.4.a.i 30 23.b odd 2 1
2116.4.a.j 30 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2116))\):

\( T_{3}^{30} + 2 T_{3}^{29} - 539 T_{3}^{28} - 974 T_{3}^{27} + 127882 T_{3}^{26} + 206494 T_{3}^{25} + \cdots - 57\!\cdots\!31 \) Copy content Toggle raw display
\( T_{5}^{30} - 50 T_{5}^{29} - 1018 T_{5}^{28} + 85216 T_{5}^{27} + 63111 T_{5}^{26} + \cdots + 69\!\cdots\!97 \) Copy content Toggle raw display