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Properties

Label 1224.2.h.b
Level $1224$
Weight $2$
Character orbit 1224.h
Analytic conductor $9.774$
Analytic rank $0$
Dimension $8$
CM discriminant -136
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,2,Mod(611,1224)]
 
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([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.611");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.h (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: \(8\)
Coefficient field: 8.0.1401249857536.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 q^{4} + \beta_{4} q^{5} + \beta_{6} q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 q^{4} + \beta_{4} q^{5} + \beta_{6} q^{7} - 2 \beta_1 q^{8} - \beta_{3} q^{10} + \beta_{7} q^{14} + 4 q^{16} - \beta_{5} q^{17} + ( - \beta_{2} + 2) q^{19} - 2 \beta_{4} q^{20} - \beta_{4} q^{23} + ( - \beta_{2} - 9) q^{25} - 2 \beta_{6} q^{28} + ( - \beta_{7} + \beta_{4}) q^{29} + (\beta_{6} + \beta_{3}) q^{31} + 4 \beta_1 q^{32} + \beta_{2} q^{34} + ( - 4 \beta_{5} - \beta_1) q^{35} - \beta_{6} q^{37} + ( - 2 \beta_{5} + 2 \beta_1) q^{38} + 2 \beta_{3} q^{40} + 2 \beta_{2} q^{43} + \beta_{3} q^{46} + ( - \beta_{2} + 15) q^{49} + ( - 2 \beta_{5} - 9 \beta_1) q^{50} - 2 \beta_{7} q^{56} + (2 \beta_{6} - \beta_{3}) q^{58} + ( - 2 \beta_{5} + 5 \beta_1) q^{59} + ( - \beta_{6} - \beta_{3}) q^{61} + (\beta_{7} + 2 \beta_{4}) q^{62} - 8 q^{64} + ( - \beta_{2} - 10) q^{67} + 2 \beta_{5} q^{68} + (4 \beta_{2} + 2) q^{70} + ( - \beta_{7} - 3 \beta_{4}) q^{71} - \beta_{7} q^{74} + (2 \beta_{2} - 4) q^{76} + ( - \beta_{6} - 3 \beta_{3}) q^{79} + 4 \beta_{4} q^{80} + ( - 2 \beta_{5} - 7 \beta_1) q^{83} + ( - 3 \beta_{6} + \beta_{3}) q^{85} + 4 \beta_{5} q^{86} - 2 \beta_{5} q^{89} + 2 \beta_{4} q^{92} + 3 \beta_{7} q^{95} + ( - 2 \beta_{5} + 15 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 32 q^{16} + 16 q^{19} - 72 q^{25} + 120 q^{49} - 64 q^{64} - 80 q^{67} + 16 q^{70} - 32 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 7\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 25\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 9\nu^{5} - 11\nu^{3} - 117\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - 9\nu^{5} + 5\nu^{3} + 90\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} - 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} + 9\nu^{5} + 37\nu^{3} - 36\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 9\nu^{5} - 53\nu^{3} - 9\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + \beta_{6} - 5\beta_{4} - 4\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\beta_{7} + 13\beta_{6} + \beta_{4} - 4\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7\beta_{2} + 25\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{7} - 11\beta_{6} - 53\beta_{4} - 37\beta_{3} ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
611.1
−1.71981 + 0.205577i
0.205577 − 1.71981i
−0.205577 + 1.71981i
1.71981 − 0.205577i
1.71981 + 0.205577i
−0.205577 − 1.71981i
0.205577 + 1.71981i
−1.71981 − 0.205577i
− 1.41421i 0 −2.00000 − 4.45320i 0 −4.02108 2.82843i 0 −6.29777
611.2 − 1.41421i 0 −2.00000 − 2.85815i 0 5.27550 2.82843i 0 −4.04204
611.3 − 1.41421i 0 −2.00000 2.85815i 0 −5.27550 2.82843i 0 4.04204
611.4 − 1.41421i 0 −2.00000 4.45320i 0 4.02108 2.82843i 0 6.29777
611.5 1.41421i 0 −2.00000 − 4.45320i 0 4.02108 − 2.82843i 0 6.29777
611.6 1.41421i 0 −2.00000 − 2.85815i 0 −5.27550 − 2.82843i 0 4.04204
611.7 1.41421i 0 −2.00000 2.85815i 0 5.27550 − 2.82843i 0 −4.04204
611.8 1.41421i 0 −2.00000 4.45320i 0 −4.02108 − 2.82843i 0 −6.29777
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 611.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
136.e odd 2 1 CM by \(\Q(\sqrt{-34}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
17.b even 2 1 inner
24.f even 2 1 inner
51.c odd 2 1 inner
408.h 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.h.b ✓ 8
3.b odd 2 1 inner 1224.2.h.b ✓ 8
4.b odd 2 1 4896.2.h.a 8
8.b even 2 1 4896.2.h.a 8
8.d odd 2 1 inner 1224.2.h.b ✓ 8
12.b even 2 1 4896.2.h.a 8
17.b even 2 1 inner 1224.2.h.b ✓ 8
24.f even 2 1 inner 1224.2.h.b ✓ 8
24.h odd 2 1 4896.2.h.a 8
51.c odd 2 1 inner 1224.2.h.b ✓ 8
68.d odd 2 1 4896.2.h.a 8
136.e odd 2 1 CM 1224.2.h.b ✓ 8
136.h even 2 1 4896.2.h.a 8
204.h even 2 1 4896.2.h.a 8
408.b odd 2 1 4896.2.h.a 8
408.h even 2 1 inner 1224.2.h.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.2.h.b ✓ 8 1.a even 1 1 trivial
1224.2.h.b ✓ 8 3.b odd 2 1 inner
1224.2.h.b ✓ 8 8.d odd 2 1 inner
1224.2.h.b ✓ 8 17.b even 2 1 inner
1224.2.h.b ✓ 8 24.f even 2 1 inner
1224.2.h.b ✓ 8 51.c odd 2 1 inner
1224.2.h.b ✓ 8 136.e odd 2 1 CM
1224.2.h.b ✓ 8 408.h even 2 1 inner
4896.2.h.a 8 4.b odd 2 1
4896.2.h.a 8 8.b even 2 1
4896.2.h.a 8 12.b even 2 1
4896.2.h.a 8 24.h odd 2 1
4896.2.h.a 8 68.d odd 2 1
4896.2.h.a 8 136.h even 2 1
4896.2.h.a 8 204.h even 2 1
4896.2.h.a 8 408.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 28T_{5}^{2} + 162 \) acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 28 T^{2} + 162)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 44 T^{2} + 450)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 17)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 30)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{2} + 162)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 124 T^{2} + 2178)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 92 T^{2} + 450)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 44 T^{2} + 450)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} - 136)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 236 T^{2} + 324)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 92 T^{2} + 450)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 20 T + 66)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 316 T^{2} + 15138)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 524 T^{2} + 66978)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 332 T^{2} + 900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 68)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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