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Properties

Label 1116.2.m.g
Level $1116$
Weight $2$
Character orbit 1116.m
Analytic conductor $8.911$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1116,2,Mod(109,1116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1116, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1116.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1116.m (of order \(5\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 30 x^{14} + 589 x^{12} + 10440 x^{10} + 183661 x^{8} + 1534500 x^{6} + 5607019 x^{4} + \cdots + 14070001 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{8} + 2 \beta_{7} - \beta_{5} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{8} + 2 \beta_{7} - 5 \beta_{5} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 14 q^{7} - 6 q^{13} + 12 q^{19} + 100 q^{25} + 30 q^{31} + 8 q^{37} - 26 q^{43} - 2 q^{49} + 32 q^{55} + 88 q^{61} + 24 q^{67} + 14 q^{73} - 10 q^{79} - 36 q^{85} + 40 q^{91} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30 x^{14} + 589 x^{12} + 10440 x^{10} + 183661 x^{8} + 1534500 x^{6} + 5607019 x^{4} + \cdots + 14070001 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!34 \nu^{14} + \cdots + 68\!\cdots\!55 ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!34 \nu^{15} + \cdots - 68\!\cdots\!55 \nu ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 56\!\cdots\!78 \nu^{14} + \cdots - 71\!\cdots\!43 ) / 45\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\!\cdots\!66 \nu^{14} + \cdots + 23\!\cdots\!00 ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 57\!\cdots\!66 \nu^{15} + \cdots + 23\!\cdots\!00 \nu ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 63\!\cdots\!24 \nu^{14} + \cdots + 65\!\cdots\!76 ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 60\!\cdots\!77 \nu^{14} + \cdots - 25\!\cdots\!06 ) / 83\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!54 \nu^{14} + \cdots - 46\!\cdots\!42 ) / 45\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!24 \nu^{15} + \cdots + 19\!\cdots\!49 \nu ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!71 \nu^{15} + \cdots - 35\!\cdots\!05 \nu ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60\!\cdots\!10 \nu^{15} + \cdots - 13\!\cdots\!76 \nu ) / 29\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!57 \nu^{14} + \cdots + 43\!\cdots\!02 ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36\!\cdots\!03 \nu^{15} + \cdots + 14\!\cdots\!90 \nu ) / 91\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 37\!\cdots\!47 \nu^{15} + \cdots + 33\!\cdots\!84 \nu ) / 91\!\cdots\!82 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + 3\beta_{8} + 3\beta_{7} + 11\beta_{5} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{15} - \beta_{14} + 3\beta_{12} - \beta_{10} + 14\beta_{6} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -81\beta_{9} + 12\beta_{8} - 160\beta_{7} - 60\beta_{5} - 12\beta_{4} + 60\beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{15} - 69\beta_{14} - 69\beta_{11} - 253\beta_{10} - 220\beta_{6} - 220\beta_{3} - 253\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -148\beta_{13} - 148\beta_{9} - 730\beta_{5} - 1483\beta_{4} - 2170\beta_{2} - 582 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1335\beta_{14} - 1335\beta_{12} + 1187\beta_{11} - 730\beta_{6} + 3653\beta_{3} - 730\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -24447\beta_{13} + 2121\beta_{9} - 2121\beta_{8} + 48019\beta_{7} - 48019\beta_{5} + 35836\beta_{2} - 35836 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2121 \beta_{15} + 2121 \beta_{14} + 22326 \beta_{12} - 22326 \beta_{11} + 76708 \beta_{10} + \cdots + 14304 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 437502 \beta_{13} + 472060 \beta_{9} - 437502 \beta_{8} - 204696 \beta_{7} + 472060 \beta_{5} + \cdots + 1081415 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -437502\beta_{15} - 402944\beta_{12} + 437502\beta_{11} + 267364\beta_{10} + 267364\beta_{6} + 1348779\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 608211\beta_{13} + 7748397\beta_{8} + 3460329\beta_{7} + 15412303\beta_{5} + 608211\beta_{4} - 3460329 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7748397 \beta_{15} - 608211 \beta_{14} + 7748397 \beta_{12} - 4896279 \beta_{10} + \cdots - 4896279 \beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 147678727 \beta_{9} + 11113426 \beta_{8} - 260525836 \beta_{7} - 69966256 \beta_{5} - 11113426 \beta_{4} + \cdots - 11113426 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 11113426 \beta_{15} - 136565301 \beta_{14} - 136565301 \beta_{11} - 419317989 \beta_{10} + \cdots - 419317989 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(497\) \(559\) \(685\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{2} - \beta_{5} + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
3.40345 − 2.47275i
0.973505 − 0.707293i
−0.973505 + 0.707293i
−3.40345 + 2.47275i
−1.29199 − 3.97635i
−0.894203 − 2.75207i
0.894203 + 2.75207i
1.29199 + 3.97635i
−1.29199 + 3.97635i
−0.894203 + 2.75207i
0.894203 − 2.75207i
1.29199 − 3.97635i
3.40345 + 2.47275i
0.973505 + 0.707293i
−0.973505 − 0.707293i
−3.40345 − 2.47275i
0 0 0 −4.20689 0 −0.0518067 + 0.159445i 0 0 0
109.2 0 0 0 −1.20332 0 −1.13918 + 3.50602i 0 0 0
109.3 0 0 0 1.20332 0 −1.13918 + 3.50602i 0 0 0
109.4 0 0 0 4.20689 0 −0.0518067 + 0.159445i 0 0 0
469.1 0 0 0 −4.18098 0 −2.70109 + 1.96245i 0 0 0
469.2 0 0 0 −2.89370 0 0.392071 − 0.284856i 0 0 0
469.3 0 0 0 2.89370 0 0.392071 − 0.284856i 0 0 0
469.4 0 0 0 4.18098 0 −2.70109 + 1.96245i 0 0 0
721.1 0 0 0 −4.18098 0 −2.70109 − 1.96245i 0 0 0
721.2 0 0 0 −2.89370 0 0.392071 + 0.284856i 0 0 0
721.3 0 0 0 2.89370 0 0.392071 + 0.284856i 0 0 0
721.4 0 0 0 4.18098 0 −2.70109 − 1.96245i 0 0 0
901.1 0 0 0 −4.20689 0 −0.0518067 − 0.159445i 0 0 0
901.2 0 0 0 −1.20332 0 −1.13918 − 3.50602i 0 0 0
901.3 0 0 0 1.20332 0 −1.13918 − 3.50602i 0 0 0
901.4 0 0 0 4.20689 0 −0.0518067 − 0.159445i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.d even 5 1 inner
93.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1116.2.m.g ✓ 16
3.b odd 2 1 inner 1116.2.m.g ✓ 16
31.d even 5 1 inner 1116.2.m.g ✓ 16
93.l odd 10 1 inner 1116.2.m.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1116.2.m.g ✓ 16 1.a even 1 1 trivial
1116.2.m.g ✓ 16 3.b odd 2 1 inner
1116.2.m.g ✓ 16 31.d even 5 1 inner
1116.2.m.g ✓ 16 93.l odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 45T_{5}^{6} + 667T_{5}^{4} - 3465T_{5}^{2} + 3751 \) acting on \(S_{2}^{\mathrm{new}}(1116, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 45 T^{6} + \cdots + 3751)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 7 T^{7} + 32 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 8793750625 \) Copy content Toggle raw display
$13$ \( (T^{8} + 3 T^{7} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 8793750625 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 8793750625 \) Copy content Toggle raw display
$29$ \( T^{16} + 30 T^{14} + \cdots + 14070001 \) Copy content Toggle raw display
$31$ \( (T^{8} - 15 T^{7} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 304)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 12993941393521 \) Copy content Toggle raw display
$43$ \( (T^{8} + 13 T^{7} + \cdots + 1946025)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 92313276561 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 5496094140625 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 92313276561 \) Copy content Toggle raw display
$61$ \( (T^{4} - 22 T^{3} + \cdots - 2025)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots + 144)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 57695797850625 \) Copy content Toggle raw display
$73$ \( (T^{8} - 7 T^{7} + \cdots + 17161)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 5 T^{7} + \cdots + 31640625)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 57695797850625 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1833616600321 \) Copy content Toggle raw display
$97$ \( (T^{8} + 5 T^{7} + \cdots + 11881)^{2} \) Copy content Toggle raw display
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