Properties

Label 1600.4.a.bt
Level $1600$
Weight $4$
Character orbit 1600.a
Self dual yes
Analytic conductor $94.403$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 7 q^{3} - 6 q^{7} + 22 q^{9} + O(q^{10}) \) \( q + 7 q^{3} - 6 q^{7} + 22 q^{9} + 43 q^{11} - 28 q^{13} - 91 q^{17} + 35 q^{19} - 42 q^{21} - 162 q^{23} - 35 q^{27} - 160 q^{29} + 42 q^{31} + 301 q^{33} - 314 q^{37} - 196 q^{39} - 203 q^{41} + 92 q^{43} - 196 q^{47} - 307 q^{49} - 637 q^{51} + 82 q^{53} + 245 q^{57} + 280 q^{59} + 518 q^{61} - 132 q^{63} + 141 q^{67} - 1134 q^{69} + 412 q^{71} + 763 q^{73} - 258 q^{77} + 510 q^{79} - 839 q^{81} + 777 q^{83} - 1120 q^{87} - 945 q^{89} + 168 q^{91} + 294 q^{93} - 1246 q^{97} + 946 q^{99} + O(q^{100}) \)

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 7.00000 0 0 0 −6.00000 0 22.0000 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.bt 1
4.b odd 2 1 1600.4.a.h 1
5.b even 2 1 1600.4.a.i 1
8.b even 2 1 25.4.a.a 1
8.d odd 2 1 400.4.a.s 1
20.d odd 2 1 1600.4.a.bs 1
24.h odd 2 1 225.4.a.e 1
40.e odd 2 1 400.4.a.c 1
40.f even 2 1 25.4.a.b yes 1
40.i odd 4 2 25.4.b.b 2
40.k even 4 2 400.4.c.e 2
56.h odd 2 1 1225.4.a.h 1
120.i odd 2 1 225.4.a.c 1
120.w even 4 2 225.4.b.f 2
280.c odd 2 1 1225.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 8.b even 2 1
25.4.a.b yes 1 40.f even 2 1
25.4.b.b 2 40.i odd 4 2
225.4.a.c 1 120.i odd 2 1
225.4.a.e 1 24.h odd 2 1
225.4.b.f 2 120.w even 4 2
400.4.a.c 1 40.e odd 2 1
400.4.a.s 1 8.d odd 2 1
400.4.c.e 2 40.k even 4 2
1225.4.a.h 1 56.h odd 2 1
1225.4.a.i 1 280.c odd 2 1
1600.4.a.h 1 4.b odd 2 1
1600.4.a.i 1 5.b even 2 1
1600.4.a.bs 1 20.d odd 2 1
1600.4.a.bt 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3} - 7 \)
\( T_{7} + 6 \)
\( T_{11} - 43 \)
\( T_{13} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -7 + T \)
$5$ \( T \)
$7$ \( 6 + T \)
$11$ \( -43 + T \)
$13$ \( 28 + T \)
$17$ \( 91 + T \)
$19$ \( -35 + T \)
$23$ \( 162 + T \)
$29$ \( 160 + T \)
$31$ \( -42 + T \)
$37$ \( 314 + T \)
$41$ \( 203 + T \)
$43$ \( -92 + T \)
$47$ \( 196 + T \)
$53$ \( -82 + T \)
$59$ \( -280 + T \)
$61$ \( -518 + T \)
$67$ \( -141 + T \)
$71$ \( -412 + T \)
$73$ \( -763 + T \)
$79$ \( -510 + T \)
$83$ \( -777 + T \)
$89$ \( 945 + T \)
$97$ \( 1246 + T \)
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