Properties

Label 448.2.b.a
Level $448$
Weight $2$
Character orbit 448.b
Analytic conductor $3.577$
Analytic rank $0$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} -4 i q^{5} - q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} -4 i q^{5} - q^{7} - q^{9} -2 i q^{11} -4 i q^{13} + 8 q^{15} + 2 q^{17} -6 i q^{19} -2 i q^{21} -11 q^{25} + 4 i q^{27} + 8 i q^{29} + 8 q^{31} + 4 q^{33} + 4 i q^{35} -8 i q^{37} + 8 q^{39} + 10 q^{41} -2 i q^{43} + 4 i q^{45} -8 q^{47} + q^{49} + 4 i q^{51} -8 q^{55} + 12 q^{57} + 10 i q^{59} + 4 i q^{61} + q^{63} -16 q^{65} -2 i q^{67} -8 q^{71} -6 q^{73} -22 i q^{75} + 2 i q^{77} -8 q^{79} -11 q^{81} -6 i q^{83} -8 i q^{85} -16 q^{87} + 10 q^{89} + 4 i q^{91} + 16 i q^{93} -24 q^{95} + 2 q^{97} + 2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{7} - 2 q^{9} + 16 q^{15} + 4 q^{17} - 22 q^{25} + 16 q^{31} + 8 q^{33} + 16 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} - 16 q^{55} + 24 q^{57} + 2 q^{63} - 32 q^{65} - 16 q^{71} - 12 q^{73} - 16 q^{79} - 22 q^{81} - 32 q^{87} + 20 q^{89} - 48 q^{95} + 4 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
1.00000i
1.00000i
0 2.00000i 0 4.00000i 0 −1.00000 0 −1.00000 0
225.2 0 2.00000i 0 4.00000i 0 −1.00000 0 −1.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.b.a 2
3.b odd 2 1 4032.2.c.b 2
4.b odd 2 1 448.2.b.b yes 2
7.b odd 2 1 3136.2.b.c 2
8.b even 2 1 inner 448.2.b.a 2
8.d odd 2 1 448.2.b.b yes 2
12.b even 2 1 4032.2.c.f 2
16.e even 4 1 1792.2.a.a 1
16.e even 4 1 1792.2.a.h 1
16.f odd 4 1 1792.2.a.d 1
16.f odd 4 1 1792.2.a.e 1
24.f even 2 1 4032.2.c.f 2
24.h odd 2 1 4032.2.c.b 2
28.d even 2 1 3136.2.b.a 2
56.e even 2 1 3136.2.b.a 2
56.h odd 2 1 3136.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.a 2 1.a even 1 1 trivial
448.2.b.a 2 8.b even 2 1 inner
448.2.b.b yes 2 4.b odd 2 1
448.2.b.b yes 2 8.d odd 2 1
1792.2.a.a 1 16.e even 4 1
1792.2.a.d 1 16.f odd 4 1
1792.2.a.e 1 16.f odd 4 1
1792.2.a.h 1 16.e even 4 1
3136.2.b.a 2 28.d even 2 1
3136.2.b.a 2 56.e even 2 1
3136.2.b.c 2 7.b odd 2 1
3136.2.b.c 2 56.h odd 2 1
4032.2.c.b 2 3.b odd 2 1
4032.2.c.b 2 24.h odd 2 1
4032.2.c.f 2 12.b even 2 1
4032.2.c.f 2 24.f even 2 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( 100 + T^{2} \)
$61$ \( 16 + T^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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