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Properties

Label 1116.2.g.j
Level $1116$
Weight $2$
Character orbit 1116.g
Analytic conductor $8.911$
Analytic rank $0$
Dimension $16$
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(991,1116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1116, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1116.991");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1116.g (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} - x^{13} - 2 x^{12} + 5 x^{11} + 4 x^{10} - 10 x^{9} - 20 x^{7} + 16 x^{6} + 40 x^{5} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 372)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{5} q^{4} - \beta_{6} q^{5} + \beta_{3} q^{7} - \beta_{8} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{5} q^{4} - \beta_{6} q^{5} + \beta_{3} q^{7} - \beta_{8} q^{8} + \beta_{13} q^{10} - \beta_{4} q^{11} - \beta_{14} q^{13} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{14}+ \cdots + (\beta_{15} + 2 \beta_{14} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{4} - 3 q^{8} - 5 q^{10} + 4 q^{11} - 5 q^{14} + 10 q^{16} + q^{20} - 16 q^{23} + 12 q^{25} - 10 q^{26} + 3 q^{28} + 6 q^{31} + 20 q^{32} - 28 q^{34} - q^{38} - 26 q^{40} + 20 q^{43} - 14 q^{44} + 4 q^{46} - 8 q^{49} + 11 q^{50} + 4 q^{52} - 8 q^{55} + 40 q^{56} - 36 q^{58} - 18 q^{62} - 7 q^{64} - 6 q^{68} + 9 q^{70} - 52 q^{74} - 3 q^{76} + 12 q^{79} + 13 q^{80} - 11 q^{82} - 28 q^{83} + 8 q^{86} + 16 q^{88} - 16 q^{91} + 26 q^{94} - 40 q^{97} + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} - x^{13} - 2 x^{12} + 5 x^{11} + 4 x^{10} - 10 x^{9} - 20 x^{7} + 16 x^{6} + 40 x^{5} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} - \nu^{12} + 3 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 10 \nu^{7} + 12 \nu^{6} - 4 \nu^{5} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 3 \nu^{13} - 9 \nu^{12} + 4 \nu^{11} + 3 \nu^{10} - 10 \nu^{9} + 14 \nu^{8} + \cdots + 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{14} + \nu^{12} + \nu^{11} + 2 \nu^{10} - 5 \nu^{9} - 4 \nu^{8} + 10 \nu^{7} + 20 \nu^{5} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 7 \nu^{12} - 12 \nu^{11} - 13 \nu^{10} + 22 \nu^{9} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + \nu^{13} + \nu^{12} + 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 10 \nu^{8} + 20 \nu^{6} + \cdots + 64 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} - 3 \nu^{13} + \nu^{12} + 6 \nu^{11} - \nu^{10} + 4 \nu^{9} - 10 \nu^{8} - 16 \nu^{7} + \cdots + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} + \nu^{13} - 5 \nu^{12} + 4 \nu^{11} + 7 \nu^{10} - 6 \nu^{9} + 18 \nu^{8} + \cdots + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} + 11 \nu^{13} + 5 \nu^{12} - 16 \nu^{11} + 13 \nu^{10} + 14 \nu^{9} + 10 \nu^{8} + \cdots - 896 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{14} + 2 \nu^{13} - \nu^{12} + \nu^{11} - 4 \nu^{10} + 5 \nu^{9} + 10 \nu^{8} - 2 \nu^{7} + \cdots - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{15} + 2 \nu^{14} - 7 \nu^{13} + 3 \nu^{12} + 12 \nu^{11} + 7 \nu^{10} + 6 \nu^{9} - 10 \nu^{8} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{15} - 2 \nu^{14} + \nu^{13} - 9 \nu^{12} + 8 \nu^{11} - 9 \nu^{10} - 14 \nu^{9} + 14 \nu^{8} + \cdots + 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5 \nu^{15} - 2 \nu^{14} + \nu^{13} - \nu^{12} + 16 \nu^{11} - 41 \nu^{10} + 18 \nu^{9} + 6 \nu^{8} + \cdots - 640 ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5 \nu^{15} - 6 \nu^{14} - \nu^{13} - 7 \nu^{12} - 8 \nu^{11} + 41 \nu^{10} - 42 \nu^{9} + 26 \nu^{8} + \cdots + 896 ) / 256 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} + 2\beta_{8} - 2\beta_{7} - 2\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} + 2\beta_{13} + 2\beta_{9} + 2\beta_{6} + 2\beta_{5} - \beta_{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - \beta_{15} - 2 \beta_{14} + 5 \beta_{13} - \beta_{12} - 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3 \beta_{15} + 4 \beta_{14} + \beta_{13} + \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + \cdots + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8 \beta_{15} + 4 \beta_{14} - 12 \beta_{13} + 4 \beta_{12} - 2 \beta_{11} + 8 \beta_{10} + 6 \beta_{8} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10 \beta_{15} + 14 \beta_{14} - 22 \beta_{13} + 12 \beta_{12} + 20 \beta_{11} - 8 \beta_{10} - 10 \beta_{9} + \cdots - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 5 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} + 23 \beta_{12} - 3 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 20 \beta_{15} - 22 \beta_{14} + 32 \beta_{13} - 26 \beta_{12} + 4 \beta_{11} + 26 \beta_{10} + \cdots + 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 34 \beta_{15} - 40 \beta_{14} - 30 \beta_{13} + 10 \beta_{12} + 22 \beta_{11} - 22 \beta_{10} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 23 \beta_{15} - 4 \beta_{14} + 81 \beta_{13} + 29 \beta_{12} - 41 \beta_{11} + 29 \beta_{10} - 31 \beta_{9} + \cdots + 17 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times\).

\(n\) \(497\) \(559\) \(685\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
991.1
1.40848 − 0.127164i
1.40848 + 0.127164i
1.22918 − 0.699371i
1.22918 + 0.699371i
0.939419 − 1.05712i
0.939419 + 1.05712i
0.0932801 − 1.41113i
0.0932801 + 1.41113i
−0.369764 − 1.36502i
−0.369764 + 1.36502i
−0.620810 − 1.27067i
−0.620810 + 1.27067i
−1.31420 − 0.522370i
−1.31420 + 0.522370i
−1.36558 − 0.367666i
−1.36558 + 0.367666i
−1.40848 − 0.127164i 0 1.96766 + 0.358218i 0.0263015 0 − 3.60417i −2.72586 − 0.754760i 0 −0.0370452 − 0.00334461i
991.2 −1.40848 + 0.127164i 0 1.96766 − 0.358218i 0.0263015 0 3.60417i −2.72586 + 0.754760i 0 −0.0370452 + 0.00334461i
991.3 −1.22918 − 0.699371i 0 1.02176 + 1.71930i 3.14632 0 2.51381i −0.0534949 − 2.82792i 0 −3.86740 − 2.20045i
991.4 −1.22918 + 0.699371i 0 1.02176 − 1.71930i 3.14632 0 − 2.51381i −0.0534949 + 2.82792i 0 −3.86740 + 2.20045i
991.5 −0.939419 − 1.05712i 0 −0.234985 + 1.98615i −2.51794 0 0.373492i 2.32034 − 1.61742i 0 2.36540 + 2.66176i
991.6 −0.939419 + 1.05712i 0 −0.234985 − 1.98615i −2.51794 0 − 0.373492i 2.32034 + 1.61742i 0 2.36540 − 2.66176i
991.7 −0.0932801 − 1.41113i 0 −1.98260 + 0.263261i −1.86118 0 − 4.60176i 0.556434 + 2.77315i 0 0.173611 + 2.62637i
991.8 −0.0932801 + 1.41113i 0 −1.98260 − 0.263261i −1.86118 0 4.60176i 0.556434 − 2.77315i 0 0.173611 − 2.62637i
991.9 0.369764 − 1.36502i 0 −1.72655 − 1.00947i 3.34926 0 − 0.954099i −2.01636 + 1.98350i 0 1.23844 − 4.57180i
991.10 0.369764 + 1.36502i 0 −1.72655 + 1.00947i 3.34926 0 0.954099i −2.01636 − 1.98350i 0 1.23844 + 4.57180i
991.11 0.620810 − 1.27067i 0 −1.22919 − 1.57769i −0.507103 0 2.46926i −2.76781 + 0.582449i 0 −0.314814 + 0.644359i
991.12 0.620810 + 1.27067i 0 −1.22919 + 1.57769i −0.507103 0 − 2.46926i −2.76781 − 0.582449i 0 −0.314814 − 0.644359i
991.13 1.31420 − 0.522370i 0 1.45426 − 1.37300i −3.41432 0 2.57432i 1.19398 − 2.56406i 0 −4.48711 + 1.78354i
991.14 1.31420 + 0.522370i 0 1.45426 + 1.37300i −3.41432 0 − 2.57432i 1.19398 + 2.56406i 0 −4.48711 − 1.78354i
991.15 1.36558 − 0.367666i 0 1.72964 − 1.00416i 1.77866 0 − 2.39590i 1.99278 − 2.00719i 0 2.42891 − 0.653952i
991.16 1.36558 + 0.367666i 0 1.72964 + 1.00416i 1.77866 0 2.39590i 1.99278 + 2.00719i 0 2.42891 + 0.653952i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
124.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1116.2.g.j 16
3.b odd 2 1 372.2.g.a ✓ 16
4.b odd 2 1 1116.2.g.i 16
12.b even 2 1 372.2.g.b yes 16
31.b odd 2 1 1116.2.g.i 16
93.c even 2 1 372.2.g.b yes 16
124.d even 2 1 inner 1116.2.g.j 16
372.b odd 2 1 372.2.g.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
372.2.g.a ✓ 16 3.b odd 2 1
372.2.g.a ✓ 16 372.b odd 2 1
372.2.g.b yes 16 12.b even 2 1
372.2.g.b yes 16 93.c even 2 1
1116.2.g.i 16 4.b odd 2 1
1116.2.g.i 16 31.b odd 2 1
1116.2.g.j 16 1.a even 1 1 trivial
1116.2.g.j 16 124.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1116, [\chi])\):

\( T_{5}^{8} - 23T_{5}^{6} - 4T_{5}^{5} + 159T_{5}^{4} + 60T_{5}^{3} - 309T_{5}^{2} - 144T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} - 2T_{11}^{7} - 49T_{11}^{6} + 60T_{11}^{5} + 730T_{11}^{4} - 248T_{11}^{3} - 3200T_{11}^{2} - 896T_{11} + 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 23 T^{6} - 4 T^{5} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 60 T^{14} + \cdots + 51200 \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{7} + \cdots + 512)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 110 T^{14} + \cdots + 131072 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 838860800 \) Copy content Toggle raw display
$19$ \( T^{16} + 122 T^{14} + \cdots + 32 \) Copy content Toggle raw display
$23$ \( (T^{8} + 8 T^{7} + \cdots + 57344)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 969408512 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1369155043328 \) Copy content Toggle raw display
$41$ \( (T^{8} - 108 T^{6} + \cdots + 5536)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 10 T^{7} + \cdots - 1865728)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10404896768 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 6855304675328 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11396876288 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 335544320000 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 467935232 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 723051225800 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 26043641495552 \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} + \cdots - 1146880)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 14 T^{7} + \cdots - 7045120)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( (T^{8} + 20 T^{7} + \cdots - 19336)^{2} \) Copy content Toggle raw display
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