Properties

Label 1600.4.a.s
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 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} + 6q^{7} - 23q^{9} + O(q^{10}) \) \( q - 2q^{3} + 6q^{7} - 23q^{9} + 32q^{11} - 38q^{13} - 26q^{17} + 100q^{19} - 12q^{21} - 78q^{23} + 100q^{27} + 50q^{29} + 108q^{31} - 64q^{33} + 266q^{37} + 76q^{39} + 22q^{41} - 442q^{43} - 514q^{47} - 307q^{49} + 52q^{51} + 2q^{53} - 200q^{57} + 500q^{59} + 518q^{61} - 138q^{63} - 126q^{67} + 156q^{69} - 412q^{71} + 878q^{73} + 192q^{77} - 600q^{79} + 421q^{81} - 282q^{83} - 100q^{87} - 150q^{89} - 228q^{91} - 216q^{93} - 386q^{97} - 736q^{99} + O(q^{100}) \)

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 6.00000 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.s 1
4.b odd 2 1 1600.4.a.bi 1
5.b even 2 1 320.4.a.h 1
8.b even 2 1 400.4.a.m 1
8.d odd 2 1 25.4.a.c 1
20.d odd 2 1 320.4.a.g 1
24.f even 2 1 225.4.a.b 1
40.e odd 2 1 5.4.a.a 1
40.f even 2 1 80.4.a.d 1
40.i odd 4 2 400.4.c.k 2
40.k even 4 2 25.4.b.a 2
56.e even 2 1 1225.4.a.k 1
80.k odd 4 2 1280.4.d.e 2
80.q even 4 2 1280.4.d.l 2
120.i odd 2 1 720.4.a.u 1
120.m even 2 1 45.4.a.d 1
120.q odd 4 2 225.4.b.c 2
280.n even 2 1 245.4.a.a 1
280.ba even 6 2 245.4.e.g 2
280.bi odd 6 2 245.4.e.f 2
360.z odd 6 2 405.4.e.l 2
360.bd even 6 2 405.4.e.c 2
440.c even 2 1 605.4.a.d 1
520.b odd 2 1 845.4.a.b 1
680.k odd 2 1 1445.4.a.a 1
760.p even 2 1 1805.4.a.h 1
840.b odd 2 1 2205.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 40.e odd 2 1
25.4.a.c 1 8.d odd 2 1
25.4.b.a 2 40.k even 4 2
45.4.a.d 1 120.m even 2 1
80.4.a.d 1 40.f even 2 1
225.4.a.b 1 24.f even 2 1
225.4.b.c 2 120.q odd 4 2
245.4.a.a 1 280.n even 2 1
245.4.e.f 2 280.bi odd 6 2
245.4.e.g 2 280.ba even 6 2
320.4.a.g 1 20.d odd 2 1
320.4.a.h 1 5.b even 2 1
400.4.a.m 1 8.b even 2 1
400.4.c.k 2 40.i odd 4 2
405.4.e.c 2 360.bd even 6 2
405.4.e.l 2 360.z odd 6 2
605.4.a.d 1 440.c even 2 1
720.4.a.u 1 120.i odd 2 1
845.4.a.b 1 520.b odd 2 1
1225.4.a.k 1 56.e even 2 1
1280.4.d.e 2 80.k odd 4 2
1280.4.d.l 2 80.q even 4 2
1445.4.a.a 1 680.k odd 2 1
1600.4.a.s 1 1.a even 1 1 trivial
1600.4.a.bi 1 4.b odd 2 1
1805.4.a.h 1 760.p even 2 1
2205.4.a.q 1 840.b odd 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{7} - 6 \)
\( T_{11} - 32 \)
\( T_{13} + 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 27 T^{2} \)
$5$ 1
$7$ \( 1 - 6 T + 343 T^{2} \)
$11$ \( 1 - 32 T + 1331 T^{2} \)
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 + 26 T + 4913 T^{2} \)
$19$ \( 1 - 100 T + 6859 T^{2} \)
$23$ \( 1 + 78 T + 12167 T^{2} \)
$29$ \( 1 - 50 T + 24389 T^{2} \)
$31$ \( 1 - 108 T + 29791 T^{2} \)
$37$ \( 1 - 266 T + 50653 T^{2} \)
$41$ \( 1 - 22 T + 68921 T^{2} \)
$43$ \( 1 + 442 T + 79507 T^{2} \)
$47$ \( 1 + 514 T + 103823 T^{2} \)
$53$ \( 1 - 2 T + 148877 T^{2} \)
$59$ \( 1 - 500 T + 205379 T^{2} \)
$61$ \( 1 - 518 T + 226981 T^{2} \)
$67$ \( 1 + 126 T + 300763 T^{2} \)
$71$ \( 1 + 412 T + 357911 T^{2} \)
$73$ \( 1 - 878 T + 389017 T^{2} \)
$79$ \( 1 + 600 T + 493039 T^{2} \)
$83$ \( 1 + 282 T + 571787 T^{2} \)
$89$ \( 1 + 150 T + 704969 T^{2} \)
$97$ \( 1 + 386 T + 912673 T^{2} \)
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