Properties

Label 800.6.a.e
Level $800$
Weight $6$
Character orbit 800.a
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 8q^{3} + 208q^{7} - 179q^{9} + O(q^{10}) \) \( q + 8q^{3} + 208q^{7} - 179q^{9} - 536q^{11} - 694q^{13} + 1278q^{17} + 1112q^{19} + 1664q^{21} - 3216q^{23} - 3376q^{27} + 2918q^{29} - 2624q^{31} - 4288q^{33} + 9458q^{37} - 5552q^{39} + 170q^{41} + 19928q^{43} - 32q^{47} + 26457q^{49} + 10224q^{51} + 22178q^{53} + 8896q^{57} + 41480q^{59} + 15462q^{61} - 37232q^{63} + 20744q^{67} - 25728q^{69} + 28592q^{71} + 53670q^{73} - 111488q^{77} - 69152q^{79} + 16489q^{81} + 37800q^{83} + 23344q^{87} - 126806q^{89} - 144352q^{91} - 20992q^{93} - 62290q^{97} + 95944q^{99} + O(q^{100}) \)

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 8.00000 0 0 0 208.000 0 −179.000 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 800.6.a.e 1
4.b odd 2 1 800.6.a.a 1
5.b even 2 1 32.6.a.a 1
5.c odd 4 2 800.6.c.a 2
15.d odd 2 1 288.6.a.d 1
20.d odd 2 1 32.6.a.c yes 1
20.e even 4 2 800.6.c.b 2
40.e odd 2 1 64.6.a.c 1
40.f even 2 1 64.6.a.e 1
60.h even 2 1 288.6.a.e 1
80.k odd 4 2 256.6.b.b 2
80.q even 4 2 256.6.b.h 2
120.i odd 2 1 576.6.a.u 1
120.m even 2 1 576.6.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.a 1 5.b even 2 1
32.6.a.c yes 1 20.d odd 2 1
64.6.a.c 1 40.e odd 2 1
64.6.a.e 1 40.f even 2 1
256.6.b.b 2 80.k odd 4 2
256.6.b.h 2 80.q even 4 2
288.6.a.d 1 15.d odd 2 1
288.6.a.e 1 60.h even 2 1
576.6.a.u 1 120.i odd 2 1
576.6.a.v 1 120.m even 2 1
800.6.a.a 1 4.b odd 2 1
800.6.a.e 1 1.a even 1 1 trivial
800.6.c.a 2 5.c odd 4 2
800.6.c.b 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3} - 8 \)
\( T_{11} + 536 \)
\( T_{13} + 694 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 8 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 - 208 T + 16807 T^{2} \)
$11$ \( 1 + 536 T + 161051 T^{2} \)
$13$ \( 1 + 694 T + 371293 T^{2} \)
$17$ \( 1 - 1278 T + 1419857 T^{2} \)
$19$ \( 1 - 1112 T + 2476099 T^{2} \)
$23$ \( 1 + 3216 T + 6436343 T^{2} \)
$29$ \( 1 - 2918 T + 20511149 T^{2} \)
$31$ \( 1 + 2624 T + 28629151 T^{2} \)
$37$ \( 1 - 9458 T + 69343957 T^{2} \)
$41$ \( 1 - 170 T + 115856201 T^{2} \)
$43$ \( 1 - 19928 T + 147008443 T^{2} \)
$47$ \( 1 + 32 T + 229345007 T^{2} \)
$53$ \( 1 - 22178 T + 418195493 T^{2} \)
$59$ \( 1 - 41480 T + 714924299 T^{2} \)
$61$ \( 1 - 15462 T + 844596301 T^{2} \)
$67$ \( 1 - 20744 T + 1350125107 T^{2} \)
$71$ \( 1 - 28592 T + 1804229351 T^{2} \)
$73$ \( 1 - 53670 T + 2073071593 T^{2} \)
$79$ \( 1 + 69152 T + 3077056399 T^{2} \)
$83$ \( 1 - 37800 T + 3939040643 T^{2} \)
$89$ \( 1 + 126806 T + 5584059449 T^{2} \)
$97$ \( 1 + 62290 T + 8587340257 T^{2} \)
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