Properties

Label 800.4.c.l
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: \(\Q(i, \sqrt{5})\)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + 7 \beta_{2} q^{7} + 7 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + 7 \beta_{2} q^{7} + 7 q^{9} -\beta_{3} q^{11} -31 \beta_{1} q^{13} + 23 \beta_{1} q^{17} -12 \beta_{3} q^{19} -140 q^{21} + 43 \beta_{2} q^{23} + 34 \beta_{2} q^{27} + 90 q^{29} -17 \beta_{3} q^{31} -20 \beta_{1} q^{33} + 107 \beta_{1} q^{37} + 31 \beta_{3} q^{39} -10 q^{41} -15 \beta_{2} q^{43} -89 \beta_{2} q^{47} -637 q^{49} -23 \beta_{3} q^{51} -339 \beta_{1} q^{53} -240 \beta_{1} q^{57} + 46 \beta_{3} q^{59} + 250 q^{61} + 49 \beta_{2} q^{63} -11 \beta_{2} q^{67} -860 q^{69} -41 \beta_{3} q^{71} + 261 \beta_{1} q^{73} -140 \beta_{1} q^{77} -98 \beta_{3} q^{79} -491 q^{81} + 85 \beta_{2} q^{83} + 90 \beta_{2} q^{87} -970 q^{89} + 217 \beta_{3} q^{91} -340 \beta_{1} q^{93} + 467 \beta_{1} q^{97} -7 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 28q^{9} + O(q^{10}) \) \( 4q + 28q^{9} - 560q^{21} + 360q^{29} - 40q^{41} - 2548q^{49} + 1000q^{61} - 3440q^{69} - 1964q^{81} - 3880q^{89} + O(q^{100}) \)

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 4.47214i 0 0 0 31.3050i 0 7.00000 0
449.2 0 4.47214i 0 0 0 31.3050i 0 7.00000 0
449.3 0 4.47214i 0 0 0 31.3050i 0 7.00000 0
449.4 0 4.47214i 0 0 0 31.3050i 0 7.00000 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.l 4
4.b odd 2 1 inner 800.4.c.l 4
5.b even 2 1 inner 800.4.c.l 4
5.c odd 4 1 160.4.a.d 2
5.c odd 4 1 800.4.a.o 2
15.e even 4 1 1440.4.a.bb 2
20.d odd 2 1 inner 800.4.c.l 4
20.e even 4 1 160.4.a.d 2
20.e even 4 1 800.4.a.o 2
40.i odd 4 1 320.4.a.q 2
40.i odd 4 1 1600.4.a.cg 2
40.k even 4 1 320.4.a.q 2
40.k even 4 1 1600.4.a.cg 2
60.l odd 4 1 1440.4.a.bb 2
80.i odd 4 1 1280.4.d.v 4
80.j even 4 1 1280.4.d.v 4
80.s even 4 1 1280.4.d.v 4
80.t odd 4 1 1280.4.d.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 5.c odd 4 1
160.4.a.d 2 20.e even 4 1
320.4.a.q 2 40.i odd 4 1
320.4.a.q 2 40.k even 4 1
800.4.a.o 2 5.c odd 4 1
800.4.a.o 2 20.e even 4 1
800.4.c.l 4 1.a even 1 1 trivial
800.4.c.l 4 4.b odd 2 1 inner
800.4.c.l 4 5.b even 2 1 inner
800.4.c.l 4 20.d odd 2 1 inner
1280.4.d.v 4 80.i odd 4 1
1280.4.d.v 4 80.j even 4 1
1280.4.d.v 4 80.s even 4 1
1280.4.d.v 4 80.t odd 4 1
1440.4.a.bb 2 15.e even 4 1
1440.4.a.bb 2 60.l odd 4 1
1600.4.a.cg 2 40.i odd 4 1
1600.4.a.cg 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 \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 20 \)
\( T_{11}^{2} - 80 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 20 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 980 + T^{2} )^{2} \)
$11$ \( ( -80 + T^{2} )^{2} \)
$13$ \( ( 3844 + T^{2} )^{2} \)
$17$ \( ( 2116 + T^{2} )^{2} \)
$19$ \( ( -11520 + T^{2} )^{2} \)
$23$ \( ( 36980 + T^{2} )^{2} \)
$29$ \( ( -90 + T )^{4} \)
$31$ \( ( -23120 + T^{2} )^{2} \)
$37$ \( ( 45796 + T^{2} )^{2} \)
$41$ \( ( 10 + T )^{4} \)
$43$ \( ( 4500 + T^{2} )^{2} \)
$47$ \( ( 158420 + T^{2} )^{2} \)
$53$ \( ( 459684 + T^{2} )^{2} \)
$59$ \( ( -169280 + T^{2} )^{2} \)
$61$ \( ( -250 + T )^{4} \)
$67$ \( ( 2420 + T^{2} )^{2} \)
$71$ \( ( -134480 + T^{2} )^{2} \)
$73$ \( ( 272484 + T^{2} )^{2} \)
$79$ \( ( -768320 + T^{2} )^{2} \)
$83$ \( ( 144500 + T^{2} )^{2} \)
$89$ \( ( 970 + T )^{4} \)
$97$ \( ( 872356 + T^{2} )^{2} \)
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