Properties

Label 448.2.a.g
Level 448
Weight 2
Character orbit 448.a
Self dual yes
Analytic conductor 3.577
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + q^{7} + q^{9} + 4q^{13} + 6q^{17} - 2q^{19} + 2q^{21} - 5q^{25} - 4q^{27} + 6q^{29} - 4q^{31} - 2q^{37} + 8q^{39} + 6q^{41} - 8q^{43} - 12q^{47} + q^{49} + 12q^{51} - 6q^{53} - 4q^{57} + 6q^{59} - 8q^{61} + q^{63} + 4q^{67} + 2q^{73} - 10q^{75} + 8q^{79} - 11q^{81} + 6q^{83} + 12q^{87} - 6q^{89} + 4q^{91} - 8q^{93} - 10q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 2.00000 0 0 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.a.g 1
3.b odd 2 1 4032.2.a.w 1
4.b odd 2 1 448.2.a.a 1
7.b odd 2 1 3136.2.a.e 1
8.b even 2 1 14.2.a.a 1
8.d odd 2 1 112.2.a.c 1
12.b even 2 1 4032.2.a.r 1
16.e even 4 2 1792.2.b.c 2
16.f odd 4 2 1792.2.b.g 2
24.f even 2 1 1008.2.a.h 1
24.h odd 2 1 126.2.a.b 1
28.d even 2 1 3136.2.a.z 1
40.e odd 2 1 2800.2.a.g 1
40.f even 2 1 350.2.a.f 1
40.i odd 4 2 350.2.c.d 2
40.k even 4 2 2800.2.g.h 2
56.e even 2 1 784.2.a.b 1
56.h odd 2 1 98.2.a.a 1
56.j odd 6 2 98.2.c.a 2
56.k odd 6 2 784.2.i.c 2
56.m even 6 2 784.2.i.i 2
56.p even 6 2 98.2.c.b 2
72.j odd 6 2 1134.2.f.f 2
72.n even 6 2 1134.2.f.l 2
88.b odd 2 1 1694.2.a.e 1
104.e even 2 1 2366.2.a.j 1
104.j odd 4 2 2366.2.d.b 2
120.i odd 2 1 3150.2.a.i 1
120.w even 4 2 3150.2.g.j 2
136.h even 2 1 4046.2.a.f 1
152.g odd 2 1 5054.2.a.c 1
168.e odd 2 1 7056.2.a.bd 1
168.i even 2 1 882.2.a.i 1
168.s odd 6 2 882.2.g.c 2
168.ba even 6 2 882.2.g.d 2
184.e odd 2 1 7406.2.a.a 1
280.c odd 2 1 2450.2.a.t 1
280.s even 4 2 2450.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 8.b even 2 1
98.2.a.a 1 56.h odd 2 1
98.2.c.a 2 56.j odd 6 2
98.2.c.b 2 56.p even 6 2
112.2.a.c 1 8.d odd 2 1
126.2.a.b 1 24.h odd 2 1
350.2.a.f 1 40.f even 2 1
350.2.c.d 2 40.i odd 4 2
448.2.a.a 1 4.b odd 2 1
448.2.a.g 1 1.a even 1 1 trivial
784.2.a.b 1 56.e even 2 1
784.2.i.c 2 56.k odd 6 2
784.2.i.i 2 56.m even 6 2
882.2.a.i 1 168.i even 2 1
882.2.g.c 2 168.s odd 6 2
882.2.g.d 2 168.ba even 6 2
1008.2.a.h 1 24.f even 2 1
1134.2.f.f 2 72.j odd 6 2
1134.2.f.l 2 72.n even 6 2
1694.2.a.e 1 88.b odd 2 1
1792.2.b.c 2 16.e even 4 2
1792.2.b.g 2 16.f odd 4 2
2366.2.a.j 1 104.e even 2 1
2366.2.d.b 2 104.j odd 4 2
2450.2.a.t 1 280.c odd 2 1
2450.2.c.c 2 280.s even 4 2
2800.2.a.g 1 40.e odd 2 1
2800.2.g.h 2 40.k even 4 2
3136.2.a.e 1 7.b odd 2 1
3136.2.a.z 1 28.d even 2 1
3150.2.a.i 1 120.i odd 2 1
3150.2.g.j 2 120.w even 4 2
4032.2.a.r 1 12.b even 2 1
4032.2.a.w 1 3.b odd 2 1
4046.2.a.f 1 136.h even 2 1
5054.2.a.c 1 152.g odd 2 1
7056.2.a.bd 1 168.e odd 2 1
7406.2.a.a 1 184.e odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} - 2 \)
\( T_{5} \)
\( T_{11} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 + 2 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 + 12 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 - 6 T + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
show more
show less