Properties

Label 448.2.p.d
Level $448$
Weight $2$
Character orbit 448.p
Analytic conductor $3.577$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 \zeta_{12}^{3} q^{15} + ( -2 + \zeta_{12}^{2} ) q^{17} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{19} + ( -4 + 5 \zeta_{12}^{2} ) q^{21} + \zeta_{12} q^{23} -2 \zeta_{12}^{2} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -4 q^{29} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{31} + ( 1 + \zeta_{12}^{2} ) q^{33} + ( -\zeta_{12} - 4 \zeta_{12}^{3} ) q^{35} + ( 3 - 3 \zeta_{12}^{2} ) q^{37} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{39} + ( 2 - 4 \zeta_{12}^{2} ) q^{41} + 2 \zeta_{12}^{3} q^{43} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} + 3 \zeta_{12} q^{51} -\zeta_{12}^{2} q^{53} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} -9 q^{57} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{59} + ( 3 + 3 \zeta_{12}^{2} ) q^{61} + ( 6 - 6 \zeta_{12}^{2} ) q^{65} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{67} + ( 1 - 2 \zeta_{12}^{2} ) q^{69} + 14 \zeta_{12}^{3} q^{71} + ( 10 - 5 \zeta_{12}^{2} ) q^{73} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{75} + ( 2 + \zeta_{12}^{2} ) q^{77} + 9 \zeta_{12} q^{79} + 9 \zeta_{12}^{2} q^{81} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{83} -3 q^{85} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{87} + ( 9 + 9 \zeta_{12}^{2} ) q^{89} + ( -8 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{91} + ( -3 + 3 \zeta_{12}^{2} ) q^{93} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{95} + ( 10 - 20 \zeta_{12}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} + O(q^{10}) \) \( 4 q + 6 q^{5} - 6 q^{17} - 6 q^{21} - 4 q^{25} - 16 q^{29} + 6 q^{33} + 6 q^{37} - 4 q^{49} - 2 q^{53} - 36 q^{57} + 18 q^{61} + 12 q^{65} + 30 q^{73} + 10 q^{77} + 18 q^{81} - 12 q^{85} + 54 q^{89} - 6 q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 1.50000i 0 1.50000 0.866025i 0 −1.73205 + 2.00000i 0 0 0
255.2 0 0.866025 1.50000i 0 1.50000 0.866025i 0 1.73205 2.00000i 0 0 0
383.1 0 −0.866025 1.50000i 0 1.50000 + 0.866025i 0 −1.73205 2.00000i 0 0 0
383.2 0 0.866025 + 1.50000i 0 1.50000 + 0.866025i 0 1.73205 + 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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
28.f 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.p.d 4
4.b odd 2 1 inner 448.2.p.d 4
7.c even 3 1 3136.2.f.e 4
7.d odd 6 1 inner 448.2.p.d 4
7.d odd 6 1 3136.2.f.e 4
8.b even 2 1 28.2.f.a 4
8.d odd 2 1 28.2.f.a 4
24.f even 2 1 252.2.bf.e 4
24.h odd 2 1 252.2.bf.e 4
28.f even 6 1 inner 448.2.p.d 4
28.f even 6 1 3136.2.f.e 4
28.g odd 6 1 3136.2.f.e 4
40.e odd 2 1 700.2.p.a 4
40.f even 2 1 700.2.p.a 4
40.i odd 4 1 700.2.t.a 4
40.i odd 4 1 700.2.t.b 4
40.k even 4 1 700.2.t.a 4
40.k even 4 1 700.2.t.b 4
56.e even 2 1 196.2.f.a 4
56.h odd 2 1 196.2.f.a 4
56.j odd 6 1 28.2.f.a 4
56.j odd 6 1 196.2.d.b 4
56.k odd 6 1 196.2.d.b 4
56.k odd 6 1 196.2.f.a 4
56.m even 6 1 28.2.f.a 4
56.m even 6 1 196.2.d.b 4
56.p even 6 1 196.2.d.b 4
56.p even 6 1 196.2.f.a 4
168.s odd 6 1 1764.2.b.a 4
168.v even 6 1 1764.2.b.a 4
168.ba even 6 1 252.2.bf.e 4
168.ba even 6 1 1764.2.b.a 4
168.be odd 6 1 252.2.bf.e 4
168.be odd 6 1 1764.2.b.a 4
280.ba even 6 1 700.2.p.a 4
280.bk odd 6 1 700.2.p.a 4
280.bp odd 12 1 700.2.t.a 4
280.bp odd 12 1 700.2.t.b 4
280.bv even 12 1 700.2.t.a 4
280.bv even 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 8.b even 2 1
28.2.f.a 4 8.d odd 2 1
28.2.f.a 4 56.j odd 6 1
28.2.f.a 4 56.m even 6 1
196.2.d.b 4 56.j odd 6 1
196.2.d.b 4 56.k odd 6 1
196.2.d.b 4 56.m even 6 1
196.2.d.b 4 56.p even 6 1
196.2.f.a 4 56.e even 2 1
196.2.f.a 4 56.h odd 2 1
196.2.f.a 4 56.k odd 6 1
196.2.f.a 4 56.p even 6 1
252.2.bf.e 4 24.f even 2 1
252.2.bf.e 4 24.h odd 2 1
252.2.bf.e 4 168.ba even 6 1
252.2.bf.e 4 168.be odd 6 1
448.2.p.d 4 1.a even 1 1 trivial
448.2.p.d 4 4.b odd 2 1 inner
448.2.p.d 4 7.d odd 6 1 inner
448.2.p.d 4 28.f even 6 1 inner
700.2.p.a 4 40.e odd 2 1
700.2.p.a 4 40.f even 2 1
700.2.p.a 4 280.ba even 6 1
700.2.p.a 4 280.bk odd 6 1
700.2.t.a 4 40.i odd 4 1
700.2.t.a 4 40.k even 4 1
700.2.t.a 4 280.bp odd 12 1
700.2.t.a 4 280.bv even 12 1
700.2.t.b 4 40.i odd 4 1
700.2.t.b 4 40.k even 4 1
700.2.t.b 4 280.bp odd 12 1
700.2.t.b 4 280.bv even 12 1
1764.2.b.a 4 168.s odd 6 1
1764.2.b.a 4 168.v even 6 1
1764.2.b.a 4 168.ba even 6 1
1764.2.b.a 4 168.be odd 6 1
3136.2.f.e 4 7.c even 3 1
3136.2.f.e 4 7.d odd 6 1
3136.2.f.e 4 28.f even 6 1
3136.2.f.e 4 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3 T_{3}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( ( 3 - 3 T + T^{2} )^{2} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( ( 3 + 3 T + T^{2} )^{2} \)
$19$ \( 729 + 27 T^{2} + T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( 9 + 3 T^{2} + T^{4} \)
$37$ \( ( 9 - 3 T + T^{2} )^{2} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( 5625 + 75 T^{2} + T^{4} \)
$53$ \( ( 1 + T + T^{2} )^{2} \)
$59$ \( 729 + 27 T^{2} + T^{4} \)
$61$ \( ( 27 - 9 T + T^{2} )^{2} \)
$67$ \( 81 - 9 T^{2} + T^{4} \)
$71$ \( ( 196 + T^{2} )^{2} \)
$73$ \( ( 75 - 15 T + T^{2} )^{2} \)
$79$ \( 6561 - 81 T^{2} + T^{4} \)
$83$ \( ( -192 + T^{2} )^{2} \)
$89$ \( ( 243 - 27 T + T^{2} )^{2} \)
$97$ \( ( 300 + T^{2} )^{2} \)
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