Properties

Label 800.4.a.z
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.37485.2
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 12x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + ( - 3 \beta_{2} + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + ( - 3 \beta_{2} + 19) q^{9} + 2 \beta_{3} q^{11} + ( - 2 \beta_{2} + 52) q^{13} - 8 \beta_{2} q^{17} + (2 \beta_{3} - 16 \beta_1) q^{19} + (17 \beta_{2} - 34) q^{21} + (\beta_{3} + 11 \beta_1) q^{23} + ( - 6 \beta_{3} - 10 \beta_1) q^{27} + ( - 16 \beta_{2} - 78) q^{29} - 16 \beta_1 q^{31} + (28 \beta_{2} + 24) q^{33} + (14 \beta_{2} + 68) q^{37} + ( - 4 \beta_{3} - 64 \beta_1) q^{39} + ( - 13 \beta_{2} - 24) q^{41} + ( - 6 \beta_{3} - 49 \beta_1) q^{43} + (9 \beta_{3} + 37 \beta_1) q^{47} + (17 \beta_{2} + 303) q^{49} + ( - 16 \beta_{3} - 48 \beta_1) q^{51} + ( - 30 \beta_{2} + 12) q^{53} + ( - 20 \beta_{2} + 760) q^{57} + ( - 10 \beta_{3} - 48 \beta_1) q^{59} + ( - 27 \beta_{2} + 264) q^{61} + (7 \beta_{3} + 109 \beta_1) q^{63} + (10 \beta_{3} - 51 \beta_1) q^{67} + (47 \beta_{2} - 494) q^{69} + ( - 4 \beta_{3} + 16 \beta_1) q^{71} + ( - 44 \beta_{2} + 360) q^{73} + (68 \beta_{2} + 1224) q^{77} + (24 \beta_{3} + 80 \beta_1) q^{79} + ( - 33 \beta_{2} - 125) q^{81} + (22 \beta_{3} - 117 \beta_1) q^{83} + ( - 32 \beta_{3} - 18 \beta_1) q^{87} + (80 \beta_{2} - 10) q^{89} + (44 \beta_{3} + 112 \beta_1) q^{91} + ( - 48 \beta_{2} + 736) q^{93} + 1136 q^{97} + (2 \beta_{3} + 144 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 76 q^{9} + 208 q^{13} - 136 q^{21} - 312 q^{29} + 96 q^{33} + 272 q^{37} - 96 q^{41} + 1212 q^{49} + 48 q^{53} + 3040 q^{57} + 1056 q^{61} - 1976 q^{69} + 1440 q^{73} + 4896 q^{77} - 500 q^{81} - 40 q^{89} + 2944 q^{93} + 4544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + 12x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} - 18\nu + 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{3} - 28\nu + 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} + 8\nu - 36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + 2\beta _1 + 34 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{2} - 14\beta _1 - 22 ) / 8 \) Copy content Toggle raw display

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
−3.03888
0.320221
2.47107
1.24759
0 −8.57295 0 0 0 22.1403 0 46.4955 0
1.2 0 −4.30169 0 0 0 −28.3162 0 −8.49545 0
1.3 0 4.30169 0 0 0 28.3162 0 −8.49545 0
1.4 0 8.57295 0 0 0 −22.1403 0 46.4955 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.a.z 4
4.b odd 2 1 inner 800.4.a.z 4
5.b even 2 1 800.4.a.y 4
5.c odd 4 2 160.4.c.d 8
8.b even 2 1 1600.4.a.cu 4
8.d odd 2 1 1600.4.a.cu 4
15.e even 4 2 1440.4.f.k 8
20.d odd 2 1 800.4.a.y 4
20.e even 4 2 160.4.c.d 8
40.e odd 2 1 1600.4.a.cv 4
40.f even 2 1 1600.4.a.cv 4
40.i odd 4 2 320.4.c.j 8
40.k even 4 2 320.4.c.j 8
60.l odd 4 2 1440.4.f.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.d 8 5.c odd 4 2
160.4.c.d 8 20.e even 4 2
320.4.c.j 8 40.i odd 4 2
320.4.c.j 8 40.k even 4 2
800.4.a.y 4 5.b even 2 1
800.4.a.y 4 20.d odd 2 1
800.4.a.z 4 1.a even 1 1 trivial
800.4.a.z 4 4.b odd 2 1 inner
1440.4.f.k 8 15.e even 4 2
1440.4.f.k 8 60.l odd 4 2
1600.4.a.cu 4 8.b even 2 1
1600.4.a.cu 4 8.d odd 2 1
1600.4.a.cv 4 40.e odd 2 1
1600.4.a.cv 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{4} - 92T_{3}^{2} + 1360 \) Copy content Toggle raw display
\( T_{11}^{4} - 4992T_{11}^{2} + 3133440 \) Copy content Toggle raw display
\( T_{13}^{2} - 104T_{13} + 2368 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 92T^{2} + 1360 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 1292 T^{2} + 393040 \) Copy content Toggle raw display
$11$ \( T^{4} - 4992 T^{2} + \cdots + 3133440 \) Copy content Toggle raw display
$13$ \( (T^{2} - 104 T + 2368)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 5376)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 30080 T^{2} + \cdots + 217600000 \) Copy content Toggle raw display
$23$ \( T^{4} - 11852 T^{2} + \cdots + 2514640 \) Copy content Toggle raw display
$29$ \( (T^{2} + 156 T - 15420)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 23552 T^{2} + \cdots + 89128960 \) Copy content Toggle raw display
$37$ \( (T^{2} - 136 T - 11840)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 48 T - 13620)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 251708 T^{2} + \cdots + 57154000 \) Copy content Toggle raw display
$47$ \( T^{4} - 211052 T^{2} + \cdots + 3485953360 \) Copy content Toggle raw display
$53$ \( (T^{2} - 24 T - 75456)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 313728 T^{2} + \cdots + 4289679360 \) Copy content Toggle raw display
$61$ \( (T^{2} - 528 T + 8460)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 388572 T^{2} + \cdots + 27064708560 \) Copy content Toggle raw display
$71$ \( T^{4} - 46592 T^{2} + \cdots + 272957440 \) Copy content Toggle raw display
$73$ \( (T^{2} - 720 T - 33024)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1215488 T^{2} + \cdots + 196885872640 \) Copy content Toggle raw display
$83$ \( T^{4} - 1986972 T^{2} + \cdots + 739915650000 \) Copy content Toggle raw display
$89$ \( (T^{2} + 20 T - 537500)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1136)^{4} \) Copy content Toggle raw display
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