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Properties

Label 1116.2.l.b
Level $1116$
Weight $2$
Character orbit 1116.l
Analytic conductor $8.911$
Analytic rank $0$
Dimension $62$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1116,2,Mod(25,1116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1116, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1116.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1116.l (of order \(3\), 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: \(8.91130486557\)
Analytic rank: \(0\)
Dimension: \(62\)
Relative dimension: \(31\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 62 q + 5 q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 62 q + 5 q^{3} - q^{9} - 4 q^{11} - 2 q^{13} + q^{15} - 5 q^{17} + 9 q^{19} + 13 q^{21} - 6 q^{23} + 66 q^{25} + 11 q^{27} - 9 q^{29} - 16 q^{31} + 12 q^{33} + 14 q^{35} + 12 q^{37} + q^{39} + 30 q^{41} + 2 q^{43} + 15 q^{45} + 10 q^{47} + 78 q^{49} + 34 q^{51} + 5 q^{53} + 9 q^{55} + 12 q^{57} + 17 q^{59} + 5 q^{63} + 48 q^{65} + 10 q^{67} - 21 q^{69} + 14 q^{71} - 4 q^{73} + 6 q^{75} - 14 q^{77} + 32 q^{79} - 13 q^{81} + 14 q^{83} - 3 q^{85} - 6 q^{87} + 120 q^{89} + 12 q^{91} + 18 q^{93} - 37 q^{95} - 3 q^{97} + 11 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −1.73202 − 0.0109011i 0 −3.97165 0 −3.84466 0 2.99976 + 0.0377618i 0
25.2 0 −1.71712 + 0.226928i 0 1.74053 0 −3.79236 0 2.89701 − 0.779325i 0
25.3 0 −1.67590 − 0.437446i 0 0.895844 0 3.23284 0 2.61728 + 1.46623i 0
25.4 0 −1.66961 + 0.460868i 0 1.97781 0 4.16481 0 2.57520 − 1.53894i 0
25.5 0 −1.40082 + 1.01867i 0 −0.000556757 0 0 0.142167 0 0.924608 − 2.85396i 0
25.6 0 −1.26957 − 1.17822i 0 −2.32205 0 1.74697 0 0.223604 + 2.99166i 0
25.7 0 −1.25549 − 1.19320i 0 4.34820 0 −0.812710 0 0.152535 + 2.99612i 0
25.8 0 −1.16074 + 1.28557i 0 −0.0217289 0 −2.04095 0 −0.305371 − 2.98442i 0
25.9 0 −1.10892 + 1.33053i 0 −2.95928 0 4.35978 0 −0.540596 − 2.95089i 0
25.10 0 −0.724994 − 1.57302i 0 0.788071 0 −2.40400 0 −1.94877 + 2.28086i 0
25.11 0 −0.576633 + 1.63325i 0 2.76668 0 −2.08710 0 −2.33499 − 1.88357i 0
25.12 0 −0.572335 − 1.63476i 0 −2.43402 0 −4.86531 0 −2.34487 + 1.87126i 0
25.13 0 −0.486187 + 1.66241i 0 −2.87645 0 −2.31566 0 −2.52724 − 1.61649i 0
25.14 0 −0.451382 − 1.67220i 0 1.80005 0 −1.14748 0 −2.59251 + 1.50960i 0
25.15 0 −0.0623871 + 1.73093i 0 2.17711 0 1.90295 0 −2.99222 − 0.215975i 0
25.16 0 0.0578120 − 1.73109i 0 −3.12320 0 3.60812 0 −2.99332 − 0.200155i 0
25.17 0 0.231645 − 1.71649i 0 2.68942 0 3.78671 0 −2.89268 − 0.795232i 0
25.18 0 0.503597 + 1.65722i 0 −2.11402 0 −0.413019 0 −2.49278 + 1.66915i 0
25.19 0 0.769519 − 1.55172i 0 1.88877 0 1.37521 0 −1.81568 − 2.38816i 0
25.20 0 0.897248 − 1.48153i 0 −3.67847 0 −0.242672 0 −1.38989 − 2.65861i 0
See all 62 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.31
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
279.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1116.2.l.b yes 62
3.b odd 2 1 3348.2.l.b 62
9.c even 3 1 1116.2.k.b ✓ 62
9.d odd 6 1 3348.2.k.b 62
31.c even 3 1 1116.2.k.b ✓ 62
93.h odd 6 1 3348.2.k.b 62
279.g even 3 1 inner 1116.2.l.b yes 62
279.p odd 6 1 3348.2.l.b 62
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1116.2.k.b ✓ 62 9.c even 3 1
1116.2.k.b ✓ 62 31.c even 3 1
1116.2.l.b yes 62 1.a even 1 1 trivial
1116.2.l.b yes 62 279.g even 3 1 inner
3348.2.k.b 62 9.d odd 6 1
3348.2.k.b 62 93.h odd 6 1
3348.2.l.b 62 3.b odd 2 1
3348.2.l.b 62 279.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{31} - 94 T_{5}^{29} + 11 T_{5}^{28} + 3906 T_{5}^{27} - 926 T_{5}^{26} - 94842 T_{5}^{25} + \cdots - 4374 \) acting on \(S_{2}^{\mathrm{new}}(1116, [\chi])\). Copy content Toggle raw display