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Properties

Label 1224.2.f.g
Level $1224$
Weight $2$
Character orbit 1224.f
Analytic conductor $9.774$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,2,Mod(613,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.613");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.f (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: \(9.77368920740\)
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: \( x^{16} - x^{14} - 2x^{13} - 2x^{10} + 16x^{8} - 8x^{6} - 64x^{3} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + \beta_{5} q^{7} - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + \beta_{5} q^{7} - \beta_{3} q^{8} + (\beta_{13} - \beta_{6} - \beta_{4} + \cdots + 1) q^{10}+ \cdots + (2 \beta_{15} - 2 \beta_{9} - 2 \beta_{6} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4} - 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{4} - 4 q^{7} - 6 q^{8} + 10 q^{10} - 8 q^{14} + 2 q^{16} + 16 q^{17} - 10 q^{22} - 28 q^{23} - 16 q^{25} + 14 q^{26} - 4 q^{28} + 12 q^{31} - 10 q^{32} - 6 q^{38} + 4 q^{40} + 28 q^{44} - 2 q^{46} + 32 q^{47} + 24 q^{49} + 42 q^{50} + 24 q^{52} - 16 q^{55} - 32 q^{56} + 8 q^{58} - 44 q^{62} + 26 q^{64} + 24 q^{65} + 2 q^{68} + 16 q^{70} - 36 q^{71} - 8 q^{73} + 28 q^{74} + 16 q^{76} + 28 q^{79} - 60 q^{80} - 10 q^{82} - 54 q^{86} - 20 q^{88} + 28 q^{92} + 8 q^{94} + 48 q^{95} + 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} - 2x^{13} - 2x^{10} + 16x^{8} - 8x^{6} - 64x^{3} - 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} - \nu^{13} - 4 \nu^{11} - 10 \nu^{9} - 12 \nu^{8} + 32 \nu^{6} + 40 \nu^{5} + \cdots - 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} - 9 \nu^{13} + 8 \nu^{12} + 4 \nu^{11} - 8 \nu^{10} + 6 \nu^{9} + 36 \nu^{8} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} + 7 \nu^{13} + 4 \nu^{11} - 8 \nu^{10} - 26 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 8 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} - 8 \nu^{7} + 32 \nu^{6} + \cdots + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} - \nu^{13} - 8 \nu^{12} + 12 \nu^{11} - 42 \nu^{9} + 36 \nu^{8} + 32 \nu^{6} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} - 5 \nu^{13} + 4 \nu^{12} - 4 \nu^{10} - 2 \nu^{9} + 12 \nu^{8} + 8 \nu^{7} + \cdots + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 8 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{15} - 2 \nu^{14} - 5 \nu^{13} + 4 \nu^{11} + 8 \nu^{10} - 18 \nu^{9} + 36 \nu^{8} + \cdots + 640 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} - \nu^{14} - \nu^{13} - \nu^{12} + 6\nu^{9} + 2\nu^{8} - 4\nu^{7} - 8\nu^{6} + 24\nu^{4} + 64\nu + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + \nu^{14} + \nu^{13} + \nu^{12} - 6 \nu^{9} - 2 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + 40 \nu^{4} + \cdots - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} - \nu^{13} - 2\nu^{12} - 2\nu^{9} + 16\nu^{7} - 8\nu^{5} - 64\nu^{2} - 64\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} + \nu^{14} - \nu^{13} - \nu^{12} + 2 \nu^{11} - 6 \nu^{10} - 10 \nu^{9} + 2 \nu^{8} + \cdots - 128 \nu ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{11} - 2\beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{15} + 2\beta_{14} - \beta_{13} + \beta_{11} - \beta_{8} + 3\beta_{6} + \beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{15} + 3 \beta_{13} + 3 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{15} + 2 \beta_{14} + 5 \beta_{13} + 4 \beta_{12} - \beta_{11} - 4 \beta_{9} - 3 \beta_{8} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7 \beta_{15} + 4 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5 \beta_{15} + 2 \beta_{14} + \beta_{13} + 4 \beta_{12} - 5 \beta_{11} + 12 \beta_{10} - 8 \beta_{9} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3 \beta_{15} - 8 \beta_{14} - \beta_{13} - 10 \beta_{12} + \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5 \beta_{15} - 2 \beta_{14} + 21 \beta_{13} + 15 \beta_{11} - 12 \beta_{9} - 11 \beta_{8} - 16 \beta_{7} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11 \beta_{15} - 4 \beta_{14} - \beta_{13} + 6 \beta_{12} - 19 \beta_{11} - 10 \beta_{10} + 14 \beta_{9} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 9 \beta_{15} + 18 \beta_{14} + 37 \beta_{13} - 12 \beta_{12} - 9 \beta_{11} + 4 \beta_{10} - 16 \beta_{9} + \cdots + 72 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
613.1
1.41193 + 0.0803908i
1.41193 − 0.0803908i
1.30172 + 0.552751i
1.30172 − 0.552751i
0.786103 + 1.17560i
0.786103 − 1.17560i
0.312143 + 1.37934i
0.312143 − 1.37934i
−0.533517 + 1.30972i
−0.533517 − 1.30972i
−0.714485 + 1.22046i
−0.714485 − 1.22046i
−1.19616 + 0.754447i
−1.19616 − 0.754447i
−1.36772 + 0.359629i
−1.36772 − 0.359629i
−1.41193 − 0.0803908i 0 1.98707 + 0.227012i − 0.772266i 0 2.51223 −2.78735 − 0.480266i 0 −0.0620831 + 1.09038i
613.2 −1.41193 + 0.0803908i 0 1.98707 − 0.227012i 0.772266i 0 2.51223 −2.78735 + 0.480266i 0 −0.0620831 − 1.09038i
613.3 −1.30172 − 0.552751i 0 1.38893 + 1.43905i 4.39996i 0 −2.20903 −1.01256 − 2.64097i 0 2.43208 − 5.72750i
613.4 −1.30172 + 0.552751i 0 1.38893 − 1.43905i − 4.39996i 0 −2.20903 −1.01256 + 2.64097i 0 2.43208 + 5.72750i
613.5 −0.786103 − 1.17560i 0 −0.764084 + 1.84829i − 2.85970i 0 0.945702 2.77350 − 0.554686i 0 −3.36187 + 2.24802i
613.6 −0.786103 + 1.17560i 0 −0.764084 − 1.84829i 2.85970i 0 0.945702 2.77350 + 0.554686i 0 −3.36187 − 2.24802i
613.7 −0.312143 − 1.37934i 0 −1.80513 + 0.861101i 2.58454i 0 −1.15359 1.75121 + 2.22110i 0 3.56495 − 0.806748i
613.8 −0.312143 + 1.37934i 0 −1.80513 − 0.861101i − 2.58454i 0 −1.15359 1.75121 − 2.22110i 0 3.56495 + 0.806748i
613.9 0.533517 − 1.30972i 0 −1.43072 − 1.39751i − 0.341388i 0 −3.88845 −2.59366 + 1.12824i 0 −0.447122 − 0.182136i
613.10 0.533517 + 1.30972i 0 −1.43072 + 1.39751i 0.341388i 0 −3.88845 −2.59366 − 1.12824i 0 −0.447122 + 0.182136i
613.11 0.714485 − 1.22046i 0 −0.979022 − 1.74399i 2.55461i 0 4.87950 −2.82796 − 0.0512048i 0 3.11779 + 1.82523i
613.12 0.714485 + 1.22046i 0 −0.979022 + 1.74399i − 2.55461i 0 4.87950 −2.82796 + 0.0512048i 0 3.11779 − 1.82523i
613.13 1.19616 − 0.754447i 0 0.861619 − 1.80489i 0.830277i 0 0.790410 −0.331052 − 2.80899i 0 0.626400 + 0.993148i
613.14 1.19616 + 0.754447i 0 0.861619 + 1.80489i − 0.830277i 0 0.790410 −0.331052 + 2.80899i 0 0.626400 − 0.993148i
613.15 1.36772 − 0.359629i 0 1.74133 − 0.983747i − 2.41956i 0 −3.87677 2.02788 − 1.97173i 0 −0.870146 − 3.30929i
613.16 1.36772 + 0.359629i 0 1.74133 + 0.983747i 2.41956i 0 −3.87677 2.02788 + 1.97173i 0 −0.870146 + 3.30929i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.f.g ✓ 16
3.b odd 2 1 1224.2.f.h yes 16
4.b odd 2 1 4896.2.f.h 16
8.b even 2 1 inner 1224.2.f.g ✓ 16
8.d odd 2 1 4896.2.f.h 16
12.b even 2 1 4896.2.f.g 16
24.f even 2 1 4896.2.f.g 16
24.h odd 2 1 1224.2.f.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.2.f.g ✓ 16 1.a even 1 1 trivial
1224.2.f.g ✓ 16 8.b even 2 1 inner
1224.2.f.h yes 16 3.b odd 2 1
1224.2.f.h yes 16 24.h odd 2 1
4896.2.f.g 16 12.b even 2 1
4896.2.f.g 16 24.f even 2 1
4896.2.f.h 16 4.b odd 2 1
4896.2.f.h 16 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{16} + 48T_{5}^{14} + 870T_{5}^{12} + 7756T_{5}^{10} + 35881T_{5}^{8} + 80844T_{5}^{6} + 71656T_{5}^{4} + 23920T_{5}^{2} + 1936 \) Copy content Toggle raw display
\( T_{23}^{8} + 14T_{23}^{7} - 14T_{23}^{6} - 684T_{23}^{5} - 111T_{23}^{4} + 11810T_{23}^{3} - 5308T_{23}^{2} - 73964T_{23} + 97956 \) 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
$5$ \( T^{16} + 48 T^{14} + \cdots + 1936 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 352)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 76 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 26873856 \) Copy content Toggle raw display
$17$ \( (T - 1)^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 156 T^{14} + \cdots + 10036224 \) Copy content Toggle raw display
$23$ \( (T^{8} + 14 T^{7} + \cdots + 97956)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 6544162816 \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 671846400 \) Copy content Toggle raw display
$41$ \( (T^{8} - 150 T^{6} + \cdots - 191664)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 428 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$47$ \( (T^{8} - 16 T^{7} + \cdots - 308736)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 29683154944 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 66324791296 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 5912994816 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 27518828544 \) Copy content Toggle raw display
$71$ \( (T^{8} + 18 T^{7} + \cdots - 1603008)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 4 T^{7} + \cdots + 1047680)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 14 T^{7} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1024000000 \) Copy content Toggle raw display
$89$ \( (T^{8} - 436 T^{6} + \cdots - 58140288)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 388 T^{6} + \cdots - 18127232)^{2} \) Copy content Toggle raw display
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