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Properties

Label 448.2.t.a
Level $448$
Weight $2$
Character orbit 448.t
Analytic conductor $3.577$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(289,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.t (of order \(6\), degree \(2\), 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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{6} q^{5} + ( - \beta_{7} + \beta_{5}) q^{7} + ( - 2 \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_{6} q^{5} + ( - \beta_{7} + \beta_{5}) q^{7} + ( - 2 \beta_{3} - 2) q^{9} + \beta_{4} q^{11} + (2 \beta_{6} + 2 \beta_1) q^{13} + ( - \beta_{7} + \beta_{5}) q^{15} + 5 \beta_{3} q^{17} - 3 \beta_{2} q^{19} - \beta_1 q^{21} - 3 \beta_{7} q^{23} - 2 \beta_{3} q^{25} + (5 \beta_{4} + 5 \beta_{2}) q^{27} + (2 \beta_{6} + 2 \beta_1) q^{29} - \beta_{5} q^{31} + ( - \beta_{3} - 1) q^{33} + 7 \beta_{2} q^{35} - \beta_{6} q^{37} + 2 \beta_{5} q^{39} + 4 q^{41} + ( - 10 \beta_{4} - 10 \beta_{2}) q^{43} + 2 \beta_1 q^{45} - \beta_{7} q^{47} + 7 q^{49} - 5 \beta_{2} q^{51} - 3 \beta_1 q^{53} + (\beta_{7} - \beta_{5}) q^{55} - 3 q^{57} - 5 \beta_{4} q^{59} - \beta_{6} q^{61} + 2 \beta_{7} q^{63} + ( - 14 \beta_{3} - 14) q^{65} + 7 \beta_{4} q^{67} + ( - 3 \beta_{6} - 3 \beta_1) q^{69} + (4 \beta_{7} - 4 \beta_{5}) q^{71} + 15 \beta_{3} q^{73} + 2 \beta_{2} q^{75} + \beta_1 q^{77} + 5 \beta_{7} q^{79} + \beta_{3} q^{81} + (6 \beta_{4} + 6 \beta_{2}) q^{83} + ( - 5 \beta_{6} - 5 \beta_1) q^{85} + 2 \beta_{5} q^{87} + (5 \beta_{3} + 5) q^{89} + (14 \beta_{4} + 14 \beta_{2}) q^{91} - \beta_{6} q^{93} - 3 \beta_{5} q^{95} + 8 q^{97} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 20 q^{17} + 8 q^{25} - 4 q^{33} + 32 q^{41} + 56 q^{49} - 24 q^{57} - 56 q^{65} - 60 q^{73} - 4 q^{81} + 20 q^{89} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 7\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 13\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 3\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} + 5\beta_{4} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 11\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} - 5\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{5} - 13\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.228425 + 1.39564i
−1.09445 − 0.895644i
−0.228425 − 1.39564i
1.09445 + 0.895644i
0.228425 − 1.39564i
−1.09445 + 0.895644i
−0.228425 + 1.39564i
1.09445 − 0.895644i
0 −0.866025 + 0.500000i 0 −2.29129 − 1.32288i 0 2.64575 0 −1.00000 + 1.73205i 0
289.2 0 −0.866025 + 0.500000i 0 2.29129 + 1.32288i 0 −2.64575 0 −1.00000 + 1.73205i 0
289.3 0 0.866025 − 0.500000i 0 −2.29129 − 1.32288i 0 −2.64575 0 −1.00000 + 1.73205i 0
289.4 0 0.866025 − 0.500000i 0 2.29129 + 1.32288i 0 2.64575 0 −1.00000 + 1.73205i 0
417.1 0 −0.866025 − 0.500000i 0 −2.29129 + 1.32288i 0 2.64575 0 −1.00000 − 1.73205i 0
417.2 0 −0.866025 − 0.500000i 0 2.29129 − 1.32288i 0 −2.64575 0 −1.00000 − 1.73205i 0
417.3 0 0.866025 + 0.500000i 0 −2.29129 + 1.32288i 0 −2.64575 0 −1.00000 − 1.73205i 0
417.4 0 0.866025 + 0.500000i 0 2.29129 − 1.32288i 0 2.64575 0 −1.00000 − 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.g odd 6 1 inner
56.k odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.t.a ✓ 8
4.b odd 2 1 inner 448.2.t.a ✓ 8
7.c even 3 1 inner 448.2.t.a ✓ 8
7.c even 3 1 3136.2.b.j 4
7.d odd 6 1 3136.2.b.i 4
8.b even 2 1 inner 448.2.t.a ✓ 8
8.d odd 2 1 inner 448.2.t.a ✓ 8
28.f even 6 1 3136.2.b.i 4
28.g odd 6 1 inner 448.2.t.a ✓ 8
28.g odd 6 1 3136.2.b.j 4
56.j odd 6 1 3136.2.b.i 4
56.k odd 6 1 inner 448.2.t.a ✓ 8
56.k odd 6 1 3136.2.b.j 4
56.m even 6 1 3136.2.b.i 4
56.p even 6 1 inner 448.2.t.a ✓ 8
56.p even 6 1 3136.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.t.a ✓ 8 1.a even 1 1 trivial
448.2.t.a ✓ 8 4.b odd 2 1 inner
448.2.t.a ✓ 8 7.c even 3 1 inner
448.2.t.a ✓ 8 8.b even 2 1 inner
448.2.t.a ✓ 8 8.d odd 2 1 inner
448.2.t.a ✓ 8 28.g odd 6 1 inner
448.2.t.a ✓ 8 56.k odd 6 1 inner
448.2.t.a ✓ 8 56.p even 6 1 inner
3136.2.b.i 4 7.d odd 6 1
3136.2.b.i 4 28.f even 6 1
3136.2.b.i 4 56.j odd 6 1
3136.2.b.i 4 56.m even 6 1
3136.2.b.j 4 7.c even 3 1
3136.2.b.j 4 28.g odd 6 1
3136.2.b.j 4 56.k odd 6 1
3136.2.b.j 4 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 7T_{5}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 63 T^{2} + 3969)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$41$ \( (T - 4)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 63 T^{2} + 3969)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 25 T^{2} + 625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 49 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 112)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 15 T + 225)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 175 T^{2} + 30625)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$97$ \( (T - 8)^{8} \) Copy content Toggle raw display
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