Properties

Label 800.4.c.h
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{2} q^{7} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{2} q^{7} - 25 q^{9} + 3 \beta_{3} q^{11} - 17 \beta_1 q^{13} + 57 \beta_1 q^{17} + 52 q^{21} + 29 \beta_{2} q^{23} - 2 \beta_{2} q^{27} + 26 q^{29} + 7 \beta_{3} q^{31} - 156 \beta_1 q^{33} - 75 \beta_1 q^{37} - 17 \beta_{3} q^{39} + 342 q^{41} + 63 \beta_{2} q^{43} + 81 \beta_{2} q^{47} + 291 q^{49} + 57 \beta_{3} q^{51} + 131 \beta_1 q^{53} + 34 \beta_{3} q^{59} - 262 q^{61} - 25 \beta_{2} q^{63} - 69 \beta_{2} q^{67} + 1508 q^{69} - 73 \beta_{3} q^{71} - 341 \beta_1 q^{73} + 156 \beta_1 q^{77} + 14 \beta_{3} q^{79} - 779 q^{81} - 21 \beta_{2} q^{83} - 26 \beta_{2} q^{87} + 630 q^{89} + 17 \beta_{3} q^{91} - 364 \beta_1 q^{93} - 483 \beta_1 q^{97} - 75 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 100 q^{9} + 208 q^{21} + 104 q^{29} + 1368 q^{41} + 1164 q^{49} - 1048 q^{61} + 6032 q^{69} - 3116 q^{81} + 2520 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 20\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} + 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 28 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
2.30278i
1.30278i
2.30278i
1.30278i
0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
449.2 0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
449.3 0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
449.4 0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d 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.c.h 4
4.b odd 2 1 inner 800.4.c.h 4
5.b even 2 1 inner 800.4.c.h 4
5.c odd 4 1 160.4.a.e 2
5.c odd 4 1 800.4.a.q 2
15.e even 4 1 1440.4.a.bd 2
20.d odd 2 1 inner 800.4.c.h 4
20.e even 4 1 160.4.a.e 2
20.e even 4 1 800.4.a.q 2
40.i odd 4 1 320.4.a.r 2
40.i odd 4 1 1600.4.a.ci 2
40.k even 4 1 320.4.a.r 2
40.k even 4 1 1600.4.a.ci 2
60.l odd 4 1 1440.4.a.bd 2
80.i odd 4 1 1280.4.d.t 4
80.j even 4 1 1280.4.d.t 4
80.s even 4 1 1280.4.d.t 4
80.t odd 4 1 1280.4.d.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.e 2 5.c odd 4 1
160.4.a.e 2 20.e even 4 1
320.4.a.r 2 40.i odd 4 1
320.4.a.r 2 40.k even 4 1
800.4.a.q 2 5.c odd 4 1
800.4.a.q 2 20.e even 4 1
800.4.c.h 4 1.a even 1 1 trivial
800.4.c.h 4 4.b odd 2 1 inner
800.4.c.h 4 5.b even 2 1 inner
800.4.c.h 4 20.d odd 2 1 inner
1280.4.d.t 4 80.i odd 4 1
1280.4.d.t 4 80.j even 4 1
1280.4.d.t 4 80.s even 4 1
1280.4.d.t 4 80.t odd 4 1
1440.4.a.bd 2 15.e even 4 1
1440.4.a.bd 2 60.l odd 4 1
1600.4.a.ci 2 40.i odd 4 1
1600.4.a.ci 2 40.k even 4 1

Hecke kernels

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

\( T_{3}^{2} + 52 \) Copy content Toggle raw display
\( T_{11}^{2} - 1872 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 1872)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12996)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 43732)^{2} \) Copy content Toggle raw display
$29$ \( (T - 26)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 22500)^{2} \) Copy content Toggle raw display
$41$ \( (T - 342)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 206388)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 341172)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 68644)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 240448)^{2} \) Copy content Toggle raw display
$61$ \( (T + 262)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 247572)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1108432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 465124)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 40768)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 22932)^{2} \) Copy content Toggle raw display
$89$ \( (T - 630)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 933156)^{2} \) Copy content Toggle raw display
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