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Properties

Label 1224.2.f.f
Level $1224$
Weight $2$
Character orbit 1224.f
Analytic conductor $9.774$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + x^{12} + 2 x^{11} - 10 x^{10} + 8 x^{9} + 10 x^{8} - 20 x^{7} + 20 x^{6} + \cdots + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 408)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{2} q^{4} - \beta_{3} q^{5} + ( - \beta_{13} - \beta_{8} + \beta_1) q^{7} - \beta_{11} q^{8} + ( - \beta_{5} + \beta_{3} - 1) q^{10} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_{3}) q^{11}+ \cdots + ( - 2 \beta_{13} + 2 \beta_{12} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{2} - 2 q^{4} + 12 q^{7} + 2 q^{8} - 8 q^{10} - 4 q^{14} + 26 q^{16} + 14 q^{17} + 16 q^{22} + 24 q^{23} + 6 q^{25} + 24 q^{26} - 16 q^{28} + 4 q^{31} + 6 q^{32} - 4 q^{34} - 4 q^{38} + 28 q^{40}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + x^{12} + 2 x^{11} - 10 x^{10} + 8 x^{9} + 10 x^{8} - 20 x^{7} + 20 x^{6} + \cdots + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{12} + 3\nu^{10} + 2\nu^{8} + 8\nu^{7} - 6\nu^{6} - 16\nu^{5} + 4\nu^{4} - 16\nu^{3} - 8\nu^{2} + 32\nu + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{13} + \nu^{12} + \nu^{11} - 3 \nu^{10} + 8 \nu^{9} + 2 \nu^{8} - 18 \nu^{7} + 10 \nu^{6} + \cdots + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{13} + \nu^{12} - 7 \nu^{10} + 10 \nu^{9} + 6 \nu^{8} - 20 \nu^{7} + 18 \nu^{6} + \cdots - 128 \nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{13} + \nu^{12} - 3 \nu^{11} + 5 \nu^{10} + 6 \nu^{9} - 14 \nu^{8} + 6 \nu^{7} + 18 \nu^{6} + \cdots + 160 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{12} + 7 \nu^{10} - 12 \nu^{9} + 14 \nu^{8} + 36 \nu^{7} - 42 \nu^{6} - 16 \nu^{5} + 52 \nu^{4} + \cdots - 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{13} - 2 \nu^{12} + \nu^{11} + 10 \nu^{10} - 24 \nu^{9} - 8 \nu^{8} + 26 \nu^{7} - 44 \nu^{6} + \cdots - 224 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{13} + \nu^{11} + 4 \nu^{10} - 13 \nu^{9} - 8 \nu^{8} + 6 \nu^{7} - 20 \nu^{6} + 14 \nu^{5} + \cdots - 176 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{13} - 2 \nu^{12} + \nu^{11} + 6 \nu^{10} - 30 \nu^{9} - 12 \nu^{8} + 18 \nu^{7} - 36 \nu^{6} + \cdots - 384 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{13} + 2 \nu^{11} - 4 \nu^{10} + 15 \nu^{9} + 16 \nu^{8} - 12 \nu^{7} + 4 \nu^{6} - 10 \nu^{5} + \cdots + 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2 \nu^{13} - 7 \nu^{12} + 6 \nu^{11} + 13 \nu^{10} - 36 \nu^{9} + 18 \nu^{8} + 40 \nu^{7} - 102 \nu^{6} + \cdots - 416 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6 \nu^{13} + 7 \nu^{12} + 8 \nu^{11} - 13 \nu^{10} + 38 \nu^{9} + 22 \nu^{8} - 76 \nu^{7} + \cdots + 352 ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8 \nu^{13} - 9 \nu^{12} + 6 \nu^{11} + 27 \nu^{10} - 70 \nu^{9} - 2 \nu^{8} + 80 \nu^{7} - 162 \nu^{6} + \cdots - 768 ) / 32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4 \nu^{13} - 5 \nu^{12} + 13 \nu^{10} - 36 \nu^{9} - 4 \nu^{8} + 48 \nu^{7} - 62 \nu^{6} + 8 \nu^{5} + \cdots - 400 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{12} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{13} - \beta_{12} + \beta_{10} - \beta_{8} + 3 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{13} - \beta_{10} + 2\beta_{9} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{13} - 3 \beta_{12} + 7 \beta_{10} - 4 \beta_{9} + \beta_{8} + 5 \beta_{7} - 3 \beta_{6} + \cdots + 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4 \beta_{12} + \beta_{10} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3 \beta_{13} + 3 \beta_{12} - 4 \beta_{11} - 7 \beta_{10} - 9 \beta_{8} + 15 \beta_{7} - 13 \beta_{6} + \cdots - 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12 \beta_{13} - 8 \beta_{12} + 8 \beta_{11} + 3 \beta_{10} + 6 \beta_{8} + \beta_{7} - 18 \beta_{6} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5 \beta_{13} + 5 \beta_{12} - 4 \beta_{11} + 7 \beta_{10} + 16 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + \cdots + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4 \beta_{13} - 12 \beta_{12} + 8 \beta_{11} + 17 \beta_{10} - 20 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + \cdots - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27 \beta_{13} + 11 \beta_{12} - 36 \beta_{11} + 9 \beta_{10} + 24 \beta_{9} - 25 \beta_{8} + \cdots - 159 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 24 \beta_{13} + 24 \beta_{11} - 21 \beta_{10} - 20 \beta_{9} - 14 \beta_{8} + \beta_{7} - 34 \beta_{6} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11 \beta_{13} - 27 \beta_{12} + 36 \beta_{11} + 71 \beta_{10} + 24 \beta_{9} + 121 \beta_{8} - 63 \beta_{7} + \cdots + 63 ) / 4 \) 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
0.105804 + 1.41025i
0.105804 − 1.41025i
0.548965 + 1.30332i
0.548965 − 1.30332i
−1.17656 − 0.784673i
−1.17656 + 0.784673i
1.37100 + 0.346935i
1.37100 − 0.346935i
−1.40561 + 0.155777i
−1.40561 − 0.155777i
1.38507 − 0.285600i
1.38507 + 0.285600i
0.171324 − 1.40380i
0.171324 + 1.40380i
−1.41025 − 0.105804i 0 1.97761 + 0.298419i − 2.49626i 0 −1.77008 −2.75735 − 0.630084i 0 −0.264113 + 3.52034i
613.2 −1.41025 + 0.105804i 0 1.97761 − 0.298419i 2.49626i 0 −1.77008 −2.75735 + 0.630084i 0 −0.264113 − 3.52034i
613.3 −1.30332 − 0.548965i 0 1.39727 + 1.43095i 3.97294i 0 3.04833 −1.03555 − 2.63204i 0 2.18101 − 5.17800i
613.4 −1.30332 + 0.548965i 0 1.39727 − 1.43095i − 3.97294i 0 3.04833 −1.03555 + 2.63204i 0 2.18101 + 5.17800i
613.5 −0.784673 − 1.17656i 0 −0.768577 + 1.84643i − 2.31278i 0 2.94745 2.77551 − 0.544565i 0 −2.72112 + 1.81478i
613.6 −0.784673 + 1.17656i 0 −0.768577 − 1.84643i 2.31278i 0 2.94745 2.77551 + 0.544565i 0 −2.72112 − 1.81478i
613.7 −0.346935 − 1.37100i 0 −1.75927 + 0.951293i − 1.24794i 0 −3.59705 1.91457 + 2.08192i 0 −1.71093 + 0.432954i
613.8 −0.346935 + 1.37100i 0 −1.75927 − 0.951293i 1.24794i 0 −3.59705 1.91457 − 2.08192i 0 −1.71093 − 0.432954i
613.9 0.155777 − 1.40561i 0 −1.95147 − 0.437924i 0.191498i 0 1.22930 −0.919543 + 2.67478i 0 0.269171 + 0.0298310i
613.10 0.155777 + 1.40561i 0 −1.95147 + 0.437924i − 0.191498i 0 1.22930 −0.919543 − 2.67478i 0 0.269171 − 0.0298310i
613.11 0.285600 − 1.38507i 0 −1.83687 − 0.791155i − 1.39571i 0 4.88695 −1.62042 + 2.31824i 0 −1.93317 − 0.398616i
613.12 0.285600 + 1.38507i 0 −1.83687 + 0.791155i 1.39571i 0 4.88695 −1.62042 − 2.31824i 0 −1.93317 + 0.398616i
613.13 1.40380 − 0.171324i 0 1.94130 − 0.481007i 1.04568i 0 −0.744897 2.64278 − 1.00783i 0 0.179150 + 1.46793i
613.14 1.40380 + 0.171324i 0 1.94130 + 0.481007i − 1.04568i 0 −0.744897 2.64278 + 1.00783i 0 0.179150 − 1.46793i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.14
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.f 14
3.b odd 2 1 408.2.f.d ✓ 14
4.b odd 2 1 4896.2.f.f 14
8.b even 2 1 inner 1224.2.f.f 14
8.d odd 2 1 4896.2.f.f 14
12.b even 2 1 1632.2.f.d 14
24.f even 2 1 1632.2.f.d 14
24.h odd 2 1 408.2.f.d ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.f.d ✓ 14 3.b odd 2 1
408.2.f.d ✓ 14 24.h odd 2 1
1224.2.f.f 14 1.a even 1 1 trivial
1224.2.f.f 14 8.b even 2 1 inner
1632.2.f.d 14 12.b even 2 1
1632.2.f.d 14 24.f even 2 1
4896.2.f.f 14 4.b odd 2 1
4896.2.f.f 14 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 \(S_{2}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{14} + 32T_{5}^{12} + 350T_{5}^{10} + 1724T_{5}^{8} + 4057T_{5}^{6} + 4476T_{5}^{4} + 1904T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{23}^{7} - 12T_{23}^{6} - 42T_{23}^{5} + 720T_{23}^{4} + 373T_{23}^{3} - 12444T_{23}^{2} - 1088T_{23} + 62848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 4 T^{13} + \cdots + 128 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 32 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{7} - 6 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + 92 T^{12} + \cdots + 262144 \) Copy content Toggle raw display
$13$ \( T^{14} + 108 T^{12} + \cdots + 11075584 \) Copy content Toggle raw display
$17$ \( (T - 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 125798656 \) Copy content Toggle raw display
$23$ \( (T^{7} - 12 T^{6} + \cdots + 62848)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 177209344 \) Copy content Toggle raw display
$31$ \( (T^{7} - 2 T^{6} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 268435456 \) Copy content Toggle raw display
$41$ \( (T^{7} + 2 T^{6} + \cdots - 170584)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 172 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{7} + 28 T^{6} + \cdots + 26624)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 48020586496 \) Copy content Toggle raw display
$59$ \( T^{14} + 248 T^{12} + \cdots + 11075584 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 3743481856 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 84727164829696 \) Copy content Toggle raw display
$71$ \( (T^{7} - 20 T^{6} + \cdots + 26624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + 2 T^{6} + \cdots - 8768)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} - 2 T^{6} + \cdots + 730624)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 6964903936 \) Copy content Toggle raw display
$89$ \( (T^{7} + 2 T^{6} + \cdots + 1280704)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} - 10 T^{6} + \cdots + 11072)^{2} \) Copy content Toggle raw display
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