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Properties

Label 1224.4.c.c
Level $1224$
Weight $4$
Character orbit 1224.c
Analytic conductor $72.218$
Analytic rank $0$
Dimension $6$
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] = [1224,4,Mod(577,1224)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1224.c (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: \(72.2183378470\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 81x^{4} + 222x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 136)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} + \beta_1) q^{11} + (\beta_{5} - 12) q^{13} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 21) q^{17}+ \cdots + ( - 70 \beta_{4} + 8 \beta_{2} + 116 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 72 q^{13} + 126 q^{17} - 24 q^{19} - 114 q^{25} - 408 q^{35} + 192 q^{43} + 72 q^{47} - 18 q^{49} - 924 q^{53} - 456 q^{55} - 1680 q^{59} - 312 q^{67} + 1668 q^{77} - 1296 q^{83} + 1344 q^{85} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 81x^{4} + 222x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 85\nu^{3} + 562\nu ) / 68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 85\nu^{3} - 426\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{4} + 391\nu^{2} + 362 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{5} + 1207\nu^{3} + 2650\nu ) / 68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{4} - 731\nu^{2} - 1386 ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{5} - 9\beta_{3} - 216 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{4} - 85\beta_{2} - 110\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 391\beta_{5} + 731\beta_{3} + 16312 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 340\beta_{4} + 6663\beta_{2} + 8498\beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
− 0.572006i
1.58186i
8.84141i
− 8.84141i
− 1.58186i
0.572006i
0 0 0 − 18.9829i 0 − 7.78768i 0 0 0
577.2 0 0 0 − 6.42821i 0 − 24.9151i 0 0 0
577.3 0 0 0 − 5.50702i 0 18.8836i 0 0 0
577.4 0 0 0 5.50702i 0 − 18.8836i 0 0 0
577.5 0 0 0 6.42821i 0 24.9151i 0 0 0
577.6 0 0 0 18.9829i 0 7.78768i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.4.c.c 6
3.b odd 2 1 136.4.b.a ✓ 6
12.b even 2 1 272.4.b.e 6
17.b even 2 1 inner 1224.4.c.c 6
51.c odd 2 1 136.4.b.a ✓ 6
51.f odd 4 2 2312.4.a.h 6
204.h even 2 1 272.4.b.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.a ✓ 6 3.b odd 2 1
136.4.b.a ✓ 6 51.c odd 2 1
272.4.b.e 6 12.b even 2 1
272.4.b.e 6 204.h even 2 1
1224.4.c.c 6 1.a even 1 1 trivial
1224.4.c.c 6 17.b even 2 1 inner
2312.4.a.h 6 51.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{6} + 432T_{5}^{4} + 27072T_{5}^{2} + 451584 \) Copy content Toggle raw display
\( T_{47}^{3} - 36T_{47}^{2} - 92928T_{47} + 11699200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 432 T^{4} + \cdots + 451584 \) Copy content Toggle raw display
$7$ \( T^{6} + 1038 T^{4} + \cdots + 13424896 \) Copy content Toggle raw display
$11$ \( T^{6} + 1062 T^{4} + \cdots + 25482304 \) Copy content Toggle raw display
$13$ \( (T^{3} + 36 T^{2} + \cdots + 28016)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 118587876497 \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} + \cdots + 178496)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 3953894400 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 1770986054656 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 10987474749696 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 7252378264576 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 8454416891904 \) Copy content Toggle raw display
$43$ \( (T^{3} - 96 T^{2} + \cdots - 20641152)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 36 T^{2} + \cdots + 11699200)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 462 T^{2} + \cdots - 1630776)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 840 T^{2} + \cdots - 116758016)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{3} + 156 T^{2} + \cdots - 49226176)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{3} + 648 T^{2} + \cdots + 10110464)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} + \cdots - 2334896)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
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