Properties

Label 448.2.p.a
Level $448$
Weight $2$
Character orbit 448.p
Analytic conductor $3.577$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{15} + ( -3 - 3 \zeta_{6} ) q^{17} -7 \zeta_{6} q^{19} + ( 3 - \zeta_{6} ) q^{21} + ( -10 + 5 \zeta_{6} ) q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} -5 q^{27} + 6 q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + ( 2 - \zeta_{6} ) q^{33} + ( 4 + \zeta_{6} ) q^{35} -5 \zeta_{6} q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( -2 - 2 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 6 - 3 \zeta_{6} ) q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} + 3 q^{55} + 7 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} + ( -10 + 5 \zeta_{6} ) q^{61} + ( 4 - 6 \zeta_{6} ) q^{63} + ( 3 + 3 \zeta_{6} ) q^{67} + ( 5 - 10 \zeta_{6} ) q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + ( -1 + 5 \zeta_{6} ) q^{77} + ( 6 - 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + 9 q^{85} + ( -6 + 6 \zeta_{6} ) q^{87} + ( -14 + 7 \zeta_{6} ) q^{89} + 5 \zeta_{6} q^{93} + ( 7 + 7 \zeta_{6} ) q^{95} + ( 4 - 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} - 3q^{11} - 9q^{17} - 7q^{19} + 5q^{21} - 15q^{23} - 2q^{25} - 10q^{27} + 12q^{29} + 5q^{31} + 3q^{33} + 9q^{35} - 5q^{37} - 6q^{45} + 3q^{47} + 2q^{49} + 9q^{51} - 9q^{53} + 6q^{55} + 14q^{57} - 9q^{59} - 15q^{61} + 2q^{63} + 9q^{67} + 3q^{73} - 2q^{75} + 3q^{77} + 9q^{79} - q^{81} + 24q^{83} + 18q^{85} - 6q^{87} - 21q^{89} + 5q^{93} + 21q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 −1.50000 + 0.866025i 0 −2.00000 1.73205i 0 1.00000 + 1.73205i 0
383.1 0 −0.500000 0.866025i 0 −1.50000 0.866025i 0 −2.00000 + 1.73205i 0 1.00000 1.73205i 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
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.a 2
4.b odd 2 1 448.2.p.b 2
7.c even 3 1 3136.2.f.b 2
7.d odd 6 1 448.2.p.b 2
7.d odd 6 1 3136.2.f.a 2
8.b even 2 1 112.2.p.b yes 2
8.d odd 2 1 112.2.p.a 2
24.f even 2 1 1008.2.cs.f 2
24.h odd 2 1 1008.2.cs.c 2
28.f even 6 1 inner 448.2.p.a 2
28.f even 6 1 3136.2.f.b 2
28.g odd 6 1 3136.2.f.a 2
56.e even 2 1 784.2.p.d 2
56.h odd 2 1 784.2.p.c 2
56.j odd 6 1 112.2.p.a 2
56.j odd 6 1 784.2.f.b 2
56.k odd 6 1 784.2.f.b 2
56.k odd 6 1 784.2.p.c 2
56.m even 6 1 112.2.p.b yes 2
56.m even 6 1 784.2.f.a 2
56.p even 6 1 784.2.f.a 2
56.p even 6 1 784.2.p.d 2
168.s odd 6 1 7056.2.b.b 2
168.v even 6 1 7056.2.b.m 2
168.ba even 6 1 1008.2.cs.f 2
168.ba even 6 1 7056.2.b.m 2
168.be odd 6 1 1008.2.cs.c 2
168.be odd 6 1 7056.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 8.d odd 2 1
112.2.p.a 2 56.j odd 6 1
112.2.p.b yes 2 8.b even 2 1
112.2.p.b yes 2 56.m even 6 1
448.2.p.a 2 1.a even 1 1 trivial
448.2.p.a 2 28.f even 6 1 inner
448.2.p.b 2 4.b odd 2 1
448.2.p.b 2 7.d odd 6 1
784.2.f.a 2 56.m even 6 1
784.2.f.a 2 56.p even 6 1
784.2.f.b 2 56.j odd 6 1
784.2.f.b 2 56.k odd 6 1
784.2.p.c 2 56.h odd 2 1
784.2.p.c 2 56.k odd 6 1
784.2.p.d 2 56.e even 2 1
784.2.p.d 2 56.p even 6 1
1008.2.cs.c 2 24.h odd 2 1
1008.2.cs.c 2 168.be odd 6 1
1008.2.cs.f 2 24.f even 2 1
1008.2.cs.f 2 168.ba even 6 1
3136.2.f.a 2 7.d odd 6 1
3136.2.f.a 2 28.g odd 6 1
3136.2.f.b 2 7.c even 3 1
3136.2.f.b 2 28.f even 6 1
7056.2.b.b 2 168.s odd 6 1
7056.2.b.b 2 168.be odd 6 1
7056.2.b.m 2 168.v even 6 1
7056.2.b.m 2 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 3 + 3 T + T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 3 + 3 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 27 + 9 T + T^{2} \)
$19$ \( 49 + 7 T + T^{2} \)
$23$ \( 75 + 15 T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 25 - 5 T + T^{2} \)
$37$ \( 25 + 5 T + T^{2} \)
$41$ \( 48 + T^{2} \)
$43$ \( 12 + T^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( 81 + 9 T + T^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( 75 + 15 T + T^{2} \)
$67$ \( 27 - 9 T + T^{2} \)
$71$ \( 12 + T^{2} \)
$73$ \( 3 - 3 T + T^{2} \)
$79$ \( 27 - 9 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 147 + 21 T + T^{2} \)
$97$ \( 48 + T^{2} \)
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