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Properties

Label 1224.2.f.e
Level $1224$
Weight $2$
Character orbit 1224.f
Analytic conductor $9.774$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: 12.0.4767670494822400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{8} q^{4} + ( - \beta_{8} - \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{8} q^{4} + ( - \beta_{8} - \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} + 16 q^{10} + 8 q^{14} + 2 q^{16} - 12 q^{17} - 24 q^{20} - 4 q^{22} - 16 q^{23} - 12 q^{25} + 4 q^{26} + 32 q^{28} - 32 q^{31} + 2 q^{32} - 2 q^{34} - 36 q^{40} - 28 q^{44} - 4 q^{46} + 32 q^{47} + 12 q^{49} - 2 q^{50} + 4 q^{52} - 48 q^{55} - 16 q^{56} + 28 q^{58} + 24 q^{62} - 38 q^{64} + 24 q^{65} + 2 q^{68} - 40 q^{70} - 32 q^{71} + 8 q^{73} + 44 q^{74} + 56 q^{76} - 16 q^{79} - 20 q^{80} + 24 q^{82} + 8 q^{86} - 16 q^{88} - 8 q^{89} + 20 q^{92} - 40 q^{94} + 32 q^{95} + 24 q^{97} - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - \nu^{10} - \nu^{9} + \nu^{8} - 6\nu^{5} - 2\nu^{4} + 8\nu^{3} - 32\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 3\nu^{10} - \nu^{9} - 3\nu^{8} + 2\nu^{7} + 6\nu^{5} - 10\nu^{4} - 12\nu^{3} + 32\nu^{2} + 32\nu - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 3\nu^{10} + \nu^{9} + 3\nu^{8} - 2\nu^{7} - 6\nu^{5} + 10\nu^{4} + 12\nu^{3} - 32\nu^{2} + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + \nu^{9} + 7 \nu^{8} - 10 \nu^{7} + 12 \nu^{6} - 14 \nu^{5} + 10 \nu^{4} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 3\nu^{9} + \nu^{8} + 7\nu^{7} - 10\nu^{6} + 12\nu^{5} - 14\nu^{4} + 10\nu^{3} + 28\nu^{2} - 72\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} + 6 \nu^{10} - 11 \nu^{9} + 2 \nu^{8} + 12 \nu^{7} - 24 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + \cdots + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{11} + 7 \nu^{10} - 10 \nu^{9} + \nu^{8} + 20 \nu^{7} - 40 \nu^{6} + 52 \nu^{5} - 58 \nu^{4} + \cdots + 144 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{11} - 9 \nu^{10} + 14 \nu^{9} - 3 \nu^{8} - 24 \nu^{7} + 44 \nu^{6} - 68 \nu^{5} + 86 \nu^{4} + \cdots - 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4 \nu^{11} - 11 \nu^{10} + 12 \nu^{9} + 7 \nu^{8} - 32 \nu^{7} + 52 \nu^{6} - 72 \nu^{5} + 82 \nu^{4} + \cdots - 112 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 8 \nu^{9} + \nu^{8} + 17 \nu^{7} - 30 \nu^{6} + 38 \nu^{5} - 46 \nu^{4} + \cdots + 120 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} - 2\beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{11} + 2\beta_{10} - 3\beta_{9} + \beta_{7} + \beta_{6} - 4\beta_{5} - 3\beta_{3} - 4\beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{11} - \beta_{9} - \beta_{8} + 2\beta_{6} - 2\beta_{4} - \beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} + \cdots - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{11} - \beta_{9} + 3 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14 \beta_{11} - 2 \beta_{10} - 9 \beta_{9} - 16 \beta_{8} - 13 \beta_{7} - 17 \beta_{6} + 4 \beta_{5} + \cdots - 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7 \beta_{11} + 6 \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 10 \beta_{4} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2 \beta_{11} + 26 \beta_{10} - 27 \beta_{9} - 7 \beta_{7} - 11 \beta_{6} - 36 \beta_{5} + 2 \beta_{4} + \cdots + 1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
0.618969 + 1.27156i
0.618969 − 1.27156i
1.22306 + 0.710021i
1.22306 − 0.710021i
−1.41313 + 0.0554252i
−1.41313 − 0.0554252i
1.35979 − 0.388551i
1.35979 + 0.388551i
0.719139 − 1.21772i
0.719139 + 1.21772i
−0.507829 + 1.31989i
−0.507829 − 1.31989i
−1.27156 − 0.618969i 0 1.23375 + 1.57412i 0.843047i 0 2.62446 −0.594467 − 2.76525i 0 0.521820 − 1.07199i
613.2 −1.27156 + 0.618969i 0 1.23375 − 1.57412i − 0.843047i 0 2.62446 −0.594467 + 2.76525i 0 0.521820 + 1.07199i
613.3 −0.710021 − 1.22306i 0 −0.991741 + 1.73679i 3.49966i 0 −4.48314 2.82835 − 0.0202025i 0 4.28029 − 2.48483i
613.4 −0.710021 + 1.22306i 0 −0.991741 − 1.73679i − 3.49966i 0 −4.48314 2.82835 + 0.0202025i 0 4.28029 + 2.48483i
613.5 0.0554252 − 1.41313i 0 −1.99386 − 0.156646i 3.40211i 0 0.414400 −0.331870 + 2.80889i 0 4.80762 + 0.188563i
613.6 0.0554252 + 1.41313i 0 −1.99386 + 0.156646i − 3.40211i 0 0.414400 −0.331870 − 2.80889i 0 4.80762 − 0.188563i
613.7 0.388551 − 1.35979i 0 −1.69806 − 1.05670i 0.383290i 0 −2.77940 −2.09667 + 1.89842i 0 0.521194 + 0.148928i
613.8 0.388551 + 1.35979i 0 −1.69806 + 1.05670i − 0.383290i 0 −2.77940 −2.09667 − 1.89842i 0 0.521194 − 0.148928i
613.9 1.21772 − 0.719139i 0 0.965679 − 1.75142i − 0.629120i 0 3.56048 −0.0835873 − 2.82719i 0 −0.452425 − 0.766092i
613.10 1.21772 + 0.719139i 0 0.965679 + 1.75142i 0.629120i 0 3.56048 −0.0835873 + 2.82719i 0 −0.452425 + 0.766092i
613.11 1.31989 − 0.507829i 0 1.48422 − 1.34056i − 3.30524i 0 0.663205 1.27824 − 2.52312i 0 −1.67849 − 4.36255i
613.12 1.31989 + 0.507829i 0 1.48422 + 1.34056i 3.30524i 0 0.663205 1.27824 + 2.52312i 0 −1.67849 + 4.36255i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.12
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.e 12
3.b odd 2 1 408.2.f.c ✓ 12
4.b odd 2 1 4896.2.f.e 12
8.b even 2 1 inner 1224.2.f.e 12
8.d odd 2 1 4896.2.f.e 12
12.b even 2 1 1632.2.f.c 12
24.f even 2 1 1632.2.f.c 12
24.h odd 2 1 408.2.f.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.f.c ✓ 12 3.b odd 2 1
408.2.f.c ✓ 12 24.h odd 2 1
1224.2.f.e 12 1.a even 1 1 trivial
1224.2.f.e 12 8.b even 2 1 inner
1632.2.f.c 12 12.b even 2 1
1632.2.f.c 12 24.f even 2 1
4896.2.f.e 12 4.b odd 2 1
4896.2.f.e 12 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 \(S_{2}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 446T_{5}^{8} + 2068T_{5}^{6} + 2121T_{5}^{4} + 704T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{23}^{6} + 8T_{23}^{5} + 10T_{23}^{4} - 32T_{23}^{3} - 51T_{23}^{2} + 32T_{23} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 36 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{6} - 24 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 60 T^{10} + \cdots + 25600 \) Copy content Toggle raw display
$13$ \( T^{12} + 60 T^{10} + \cdots + 30976 \) Copy content Toggle raw display
$17$ \( (T + 1)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + 140 T^{10} + \cdots + 5776 \) Copy content Toggle raw display
$23$ \( (T^{6} + 8 T^{5} + 10 T^{4} + \cdots + 52)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 160 T^{10} + \cdots + 1048576 \) Copy content Toggle raw display
$31$ \( (T^{6} + 16 T^{5} + \cdots + 6464)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 288 T^{10} + \cdots + 1048576 \) Copy content Toggle raw display
$41$ \( (T^{6} - 162 T^{4} + \cdots + 7348)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 220 T^{10} + \cdots + 17272336 \) Copy content Toggle raw display
$47$ \( (T^{6} - 16 T^{5} + \cdots + 11392)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1257127936 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 41134329856 \) Copy content Toggle raw display
$61$ \( T^{12} + 216 T^{10} + \cdots + 102400 \) Copy content Toggle raw display
$67$ \( T^{12} + 376 T^{10} + \cdots + 23658496 \) Copy content Toggle raw display
$71$ \( (T^{6} + 16 T^{5} + \cdots - 10688)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 4 T^{5} + \cdots + 10016)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots + 368960)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 400 T^{10} + \cdots + 28558336 \) Copy content Toggle raw display
$89$ \( (T^{6} + 4 T^{5} + \cdots - 3040)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 12 T^{5} + \cdots - 1090912)^{2} \) Copy content Toggle raw display
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