Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -\beta_{2} + \beta_{7} ) q^{3} + \beta_{6} q^{5} + ( 2 \beta_{1} - \beta_{5} ) q^{7} -2 \beta_{4} q^{9} +O(q^{10})\) \( q + ( -\beta_{2} + \beta_{7} ) q^{3} + \beta_{6} q^{5} + ( 2 \beta_{1} - \beta_{5} ) q^{7} -2 \beta_{4} q^{9} + ( -\beta_{2} - \beta_{7} ) q^{11} -5 \beta_{5} q^{15} + ( 3 - 3 \beta_{4} ) q^{17} + 3 \beta_{2} q^{19} + ( -3 \beta_{3} + \beta_{6} ) q^{21} + 3 \beta_{1} q^{23} -\beta_{7} q^{27} + ( 2 \beta_{3} - 4 \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{5} ) q^{31} + ( 10 + 5 \beta_{4} ) q^{33} + ( 3 \beta_{2} - 2 \beta_{7} ) q^{35} + ( 6 \beta_{3} - 3 \beta_{6} ) q^{37} + ( -6 - 12 \beta_{4} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{6} ) q^{45} + ( 3 \beta_{1} + 6 \beta_{5} ) q^{47} + ( 3 + 8 \beta_{4} ) q^{49} + ( -3 \beta_{2} + 6 \beta_{7} ) q^{51} + ( -\beta_{3} - \beta_{6} ) q^{53} + ( -10 \beta_{1} - 5 \beta_{5} ) q^{55} -15 q^{57} + ( -\beta_{2} + \beta_{7} ) q^{59} + 3 \beta_{6} q^{61} + ( 6 \beta_{1} + 4 \beta_{5} ) q^{63} + ( -3 \beta_{2} - 3 \beta_{7} ) q^{67} -3 \beta_{3} q^{69} -6 \beta_{5} q^{71} + ( -3 + 3 \beta_{4} ) q^{73} + ( -\beta_{3} + 5 \beta_{6} ) q^{77} -13 \beta_{1} q^{79} + ( 11 + 11 \beta_{4} ) q^{81} -2 \beta_{7} q^{83} + ( -3 \beta_{3} + 6 \beta_{6} ) q^{85} + ( -10 \beta_{1} + 10 \beta_{5} ) q^{87} + ( -6 - 3 \beta_{4} ) q^{89} + ( -2 \beta_{3} + \beta_{6} ) q^{93} + ( 15 \beta_{1} + 15 \beta_{5} ) q^{95} + ( -2 - 4 \beta_{4} ) q^{97} + ( -4 \beta_{2} + 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} + 36q^{17} + 60q^{33} - 8q^{49} - 120q^{57} - 36q^{73} + 44q^{81} - 36q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 29 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 9 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 9 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 21 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} - 8 \nu^{5} + 22 \nu^{3} - \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{4} - \beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{5}\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{6} + 7 \beta_{4}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} + 11 \beta_{5} - 5 \beta_{2} + 11 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{3} - 9\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{2} + 29 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.40126 0.809017i
0.535233 0.309017i
−1.40126 + 0.809017i
−0.535233 + 0.309017i
1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 0.809017i
−0.535233 0.309017i
0 −1.93649 1.11803i 0 −1.11803 1.93649i 0 −1.73205 + 2.00000i 0 1.00000 + 1.73205i 0
31.2 0 −1.93649 1.11803i 0 1.11803 + 1.93649i 0 1.73205 2.00000i 0 1.00000 + 1.73205i 0
31.3 0 1.93649 + 1.11803i 0 −1.11803 1.93649i 0 1.73205 2.00000i 0 1.00000 + 1.73205i 0
31.4 0 1.93649 + 1.11803i 0 1.11803 + 1.93649i 0 −1.73205 + 2.00000i 0 1.00000 + 1.73205i 0
159.1 0 −1.93649 + 1.11803i 0 −1.11803 + 1.93649i 0 −1.73205 2.00000i 0 1.00000 1.73205i 0
159.2 0 −1.93649 + 1.11803i 0 1.11803 1.93649i 0 1.73205 + 2.00000i 0 1.00000 1.73205i 0
159.3 0 1.93649 1.11803i 0 −1.11803 + 1.93649i 0 1.73205 + 2.00000i 0 1.00000 1.73205i 0
159.4 0 1.93649 1.11803i 0 1.11803 1.93649i 0 −1.73205 2.00000i 0 1.00000 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.f even 6 1 inner
56.j odd 6 1 inner
56.m 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.q.a 8
4.b odd 2 1 inner 448.2.q.a 8
7.c even 3 1 3136.2.e.a 8
7.d odd 6 1 inner 448.2.q.a 8
7.d odd 6 1 3136.2.e.a 8
8.b even 2 1 inner 448.2.q.a 8
8.d odd 2 1 inner 448.2.q.a 8
28.f even 6 1 inner 448.2.q.a 8
28.f even 6 1 3136.2.e.a 8
28.g odd 6 1 3136.2.e.a 8
56.j odd 6 1 inner 448.2.q.a 8
56.j odd 6 1 3136.2.e.a 8
56.k odd 6 1 3136.2.e.a 8
56.m even 6 1 inner 448.2.q.a 8
56.m even 6 1 3136.2.e.a 8
56.p even 6 1 3136.2.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.q.a 8 1.a even 1 1 trivial
448.2.q.a 8 4.b odd 2 1 inner
448.2.q.a 8 7.d odd 6 1 inner
448.2.q.a 8 8.b even 2 1 inner
448.2.q.a 8 8.d odd 2 1 inner
448.2.q.a 8 28.f even 6 1 inner
448.2.q.a 8 56.j odd 6 1 inner
448.2.q.a 8 56.m even 6 1 inner
3136.2.e.a 8 7.c even 3 1
3136.2.e.a 8 7.d odd 6 1
3136.2.e.a 8 28.f even 6 1
3136.2.e.a 8 28.g odd 6 1
3136.2.e.a 8 56.j odd 6 1
3136.2.e.a 8 56.k odd 6 1
3136.2.e.a 8 56.m even 6 1
3136.2.e.a 8 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 \(S_{2}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{4} - 5 T_{3}^{2} + 25 \)
\( T_{5}^{4} + 5 T_{5}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 25 - 5 T^{2} + T^{4} )^{2} \)
$5$ \( ( 25 + 5 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + 2 T^{2} + T^{4} )^{2} \)
$11$ \( ( 225 + 15 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 27 - 9 T + T^{2} )^{4} \)
$19$ \( ( 2025 - 45 T^{2} + T^{4} )^{2} \)
$23$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
$29$ \( ( 60 + T^{2} )^{4} \)
$31$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$37$ \( ( 18225 - 135 T^{2} + T^{4} )^{2} \)
$41$ \( ( 108 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( ( 729 + 27 T^{2} + T^{4} )^{2} \)
$53$ \( ( 225 - 15 T^{2} + T^{4} )^{2} \)
$59$ \( ( 25 - 5 T^{2} + T^{4} )^{2} \)
$61$ \( ( 2025 + 45 T^{2} + T^{4} )^{2} \)
$67$ \( ( 18225 + 135 T^{2} + T^{4} )^{2} \)
$71$ \( ( 36 + T^{2} )^{4} \)
$73$ \( ( 27 + 9 T + T^{2} )^{4} \)
$79$ \( ( 28561 - 169 T^{2} + T^{4} )^{2} \)
$83$ \( ( 20 + T^{2} )^{4} \)
$89$ \( ( 27 + 9 T + T^{2} )^{4} \)
$97$ \( ( 12 + T^{2} )^{4} \)
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