Properties

Label 800.6.c.h
Level 800
Weight 6
Character orbit 800.c
Analytic conductor 128.307
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q -3 \beta_{2} q^{3} + 31 \beta_{2} q^{7} + 63 q^{9} +O(q^{10})\) \( q -3 \beta_{2} q^{3} + 31 \beta_{2} q^{7} + 63 q^{9} -29 \beta_{3} q^{11} + 77 \beta_{1} q^{13} -89 \beta_{1} q^{17} + 108 \beta_{3} q^{19} + 1860 q^{21} -589 \beta_{2} q^{23} -918 \beta_{2} q^{27} -4110 q^{29} -353 \beta_{3} q^{31} + 1740 \beta_{1} q^{33} -3721 \beta_{1} q^{37} + 231 \beta_{3} q^{39} + 7270 q^{41} + 4005 \beta_{2} q^{43} -1657 \beta_{2} q^{47} -2413 q^{49} -267 \beta_{3} q^{51} + 16113 \beta_{1} q^{53} -6480 \beta_{1} q^{57} + 3806 \beta_{3} q^{59} + 26770 q^{61} + 1953 \beta_{2} q^{63} + 11137 \beta_{2} q^{67} -35340 q^{69} -6049 \beta_{3} q^{71} -9267 \beta_{1} q^{73} -17980 \beta_{1} q^{77} -9698 \beta_{3} q^{79} -39771 q^{81} + 17585 \beta_{2} q^{83} + 12330 \beta_{2} q^{87} + 107590 q^{89} -2387 \beta_{3} q^{91} + 21180 \beta_{1} q^{93} + 54419 \beta_{1} q^{97} -1827 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 252q^{9} + O(q^{10}) \) \( 4q + 252q^{9} + 7440q^{21} - 16440q^{29} + 29080q^{41} - 9652q^{49} + 107080q^{61} - 141360q^{69} - 159084q^{81} + 430360q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 12\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1}\)\()/2\)

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
0.618034i
1.61803i
0.618034i
1.61803i
0 13.4164i 0 0 0 138.636i 0 63.0000 0
449.2 0 13.4164i 0 0 0 138.636i 0 63.0000 0
449.3 0 13.4164i 0 0 0 138.636i 0 63.0000 0
449.4 0 13.4164i 0 0 0 138.636i 0 63.0000 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 800.6.c.h 4
4.b odd 2 1 inner 800.6.c.h 4
5.b even 2 1 inner 800.6.c.h 4
5.c odd 4 1 160.6.a.b 2
5.c odd 4 1 800.6.a.i 2
20.d odd 2 1 inner 800.6.c.h 4
20.e even 4 1 160.6.a.b 2
20.e even 4 1 800.6.a.i 2
40.i odd 4 1 320.6.a.t 2
40.k even 4 1 320.6.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 5.c odd 4 1
160.6.a.b 2 20.e even 4 1
320.6.a.t 2 40.i odd 4 1
320.6.a.t 2 40.k even 4 1
800.6.a.i 2 5.c odd 4 1
800.6.a.i 2 20.e even 4 1
800.6.c.h 4 1.a even 1 1 trivial
800.6.c.h 4 4.b odd 2 1 inner
800.6.c.h 4 5.b even 2 1 inner
800.6.c.h 4 20.d odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 180 \)
\( T_{11}^{2} - 67280 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 306 T^{2} + 59049 T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 - 14394 T^{2} + 282475249 T^{4} )^{2} \)
$11$ \( ( 1 + 254822 T^{2} + 25937424601 T^{4} )^{2} \)
$13$ \( ( 1 - 718870 T^{2} + 137858491849 T^{4} )^{2} \)
$17$ \( ( 1 - 2808030 T^{2} + 2015993900449 T^{4} )^{2} \)
$19$ \( ( 1 + 4019078 T^{2} + 6131066257801 T^{4} )^{2} \)
$23$ \( ( 1 - 5934266 T^{2} + 41426511213649 T^{4} )^{2} \)
$29$ \( ( 1 + 4110 T + 20511149 T^{2} )^{4} \)
$31$ \( ( 1 + 47289582 T^{2} + 819628286980801 T^{4} )^{2} \)
$37$ \( ( 1 - 83304550 T^{2} + 4808584372417849 T^{4} )^{2} \)
$41$ \( ( 1 - 7270 T + 115856201 T^{2} )^{4} \)
$43$ \( ( 1 + 26783614 T^{2} + 21611482313284249 T^{4} )^{2} \)
$47$ \( ( 1 - 403777034 T^{2} + 52599132235830049 T^{4} )^{2} \)
$53$ \( ( 1 + 202124090 T^{2} + 174887470365513049 T^{4} )^{2} \)
$59$ \( ( 1 + 270997718 T^{2} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 - 26770 T + 844596301 T^{2} )^{4} \)
$67$ \( ( 1 - 219594834 T^{2} + 1822837804551761449 T^{4} )^{2} \)
$71$ \( ( 1 + 681226622 T^{2} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 - 3802634030 T^{2} + 4297625829703557649 T^{4} )^{2} \)
$79$ \( ( 1 - 1369983522 T^{2} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( ( 1 - 1693436786 T^{2} + 15516041187205853449 T^{4} )^{2} \)
$89$ \( ( 1 - 107590 T + 5584059449 T^{2} )^{4} \)
$97$ \( ( 1 - 5328970270 T^{2} + 73742412689492826049 T^{4} )^{2} \)
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