Properties

Label 1600.4.a.bg
Level $1600$
Weight $4$
Character orbit 1600.a
Self dual yes
Analytic conductor $94.403$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(94.4030560092\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 26q^{7} - 23q^{9} + O(q^{10}) \) \( q + 2q^{3} - 26q^{7} - 23q^{9} - 28q^{11} - 12q^{13} - 64q^{17} - 60q^{19} - 52q^{21} + 58q^{23} - 100q^{27} - 90q^{29} + 128q^{31} - 56q^{33} - 236q^{37} - 24q^{39} + 242q^{41} + 362q^{43} - 226q^{47} + 333q^{49} - 128q^{51} + 108q^{53} - 120q^{57} - 20q^{59} - 542q^{61} + 598q^{63} - 434q^{67} + 116q^{69} + 1128q^{71} + 632q^{73} + 728q^{77} + 720q^{79} + 421q^{81} - 478q^{83} - 180q^{87} - 490q^{89} + 312q^{91} + 256q^{93} + 1456q^{97} + 644q^{99} + O(q^{100}) \)

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 −26.0000 0 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 1600.4.a.bg 1
4.b odd 2 1 1600.4.a.u 1
5.b even 2 1 1600.4.a.t 1
5.c odd 4 2 320.4.c.c 2
8.b even 2 1 400.4.a.h 1
8.d odd 2 1 50.4.a.d 1
20.d odd 2 1 1600.4.a.bh 1
20.e even 4 2 320.4.c.d 2
24.f even 2 1 450.4.a.j 1
40.e odd 2 1 50.4.a.b 1
40.f even 2 1 400.4.a.n 1
40.i odd 4 2 80.4.c.a 2
40.k even 4 2 10.4.b.a 2
56.e even 2 1 2450.4.a.bb 1
120.m even 2 1 450.4.a.k 1
120.q odd 4 2 90.4.c.b 2
120.w even 4 2 720.4.f.f 2
280.n even 2 1 2450.4.a.o 1
280.y odd 4 2 490.4.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 40.k even 4 2
50.4.a.b 1 40.e odd 2 1
50.4.a.d 1 8.d odd 2 1
80.4.c.a 2 40.i odd 4 2
90.4.c.b 2 120.q odd 4 2
320.4.c.c 2 5.c odd 4 2
320.4.c.d 2 20.e even 4 2
400.4.a.h 1 8.b even 2 1
400.4.a.n 1 40.f even 2 1
450.4.a.j 1 24.f even 2 1
450.4.a.k 1 120.m even 2 1
490.4.c.b 2 280.y odd 4 2
720.4.f.f 2 120.w even 4 2
1600.4.a.t 1 5.b even 2 1
1600.4.a.u 1 4.b odd 2 1
1600.4.a.bg 1 1.a even 1 1 trivial
1600.4.a.bh 1 20.d odd 2 1
2450.4.a.o 1 280.n even 2 1
2450.4.a.bb 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3} - 2 \)
\( T_{7} + 26 \)
\( T_{11} + 28 \)
\( T_{13} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( 26 + T \)
$11$ \( 28 + T \)
$13$ \( 12 + T \)
$17$ \( 64 + T \)
$19$ \( 60 + T \)
$23$ \( -58 + T \)
$29$ \( 90 + T \)
$31$ \( -128 + T \)
$37$ \( 236 + T \)
$41$ \( -242 + T \)
$43$ \( -362 + T \)
$47$ \( 226 + T \)
$53$ \( -108 + T \)
$59$ \( 20 + T \)
$61$ \( 542 + T \)
$67$ \( 434 + T \)
$71$ \( -1128 + T \)
$73$ \( -632 + T \)
$79$ \( -720 + T \)
$83$ \( 478 + T \)
$89$ \( 490 + T \)
$97$ \( -1456 + T \)
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