Properties

Label 800.2.c.f
Level $800$
Weight $2$
Character orbit 800.c
Analytic conductor $6.388$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} -5 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} -5 q^{9} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + 2 \zeta_{8}^{2} q^{13} + 2 \zeta_{8}^{2} q^{17} -8 q^{21} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} -6 q^{29} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{31} + 16 \zeta_{8}^{2} q^{33} -10 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + 2 q^{41} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{43} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} - q^{49} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51} -6 \zeta_{8}^{2} q^{53} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{63} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{67} -8 q^{69} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + 6 \zeta_{8}^{2} q^{73} + 16 \zeta_{8}^{2} q^{77} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{79} + q^{81} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{87} -10 q^{89} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} -16 \zeta_{8}^{2} q^{93} + 2 \zeta_{8}^{2} q^{97} + ( -20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} + O(q^{10}) \) \( 4 q - 20 q^{9} - 32 q^{21} - 24 q^{29} + 8 q^{41} - 4 q^{49} - 8 q^{61} - 32 q^{69} + 4 q^{81} - 40 q^{89} + O(q^{100}) \)

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
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.2 0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.3 0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.4 0 2.82843i 0 0 0 2.82843i 0 −5.00000 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 800.2.c.f 4
3.b odd 2 1 7200.2.f.bh 4
4.b odd 2 1 inner 800.2.c.f 4
5.b even 2 1 inner 800.2.c.f 4
5.c odd 4 1 160.2.a.c 2
5.c odd 4 1 800.2.a.m 2
8.b even 2 1 1600.2.c.n 4
8.d odd 2 1 1600.2.c.n 4
12.b even 2 1 7200.2.f.bh 4
15.d odd 2 1 7200.2.f.bh 4
15.e even 4 1 1440.2.a.o 2
15.e even 4 1 7200.2.a.cm 2
20.d odd 2 1 inner 800.2.c.f 4
20.e even 4 1 160.2.a.c 2
20.e even 4 1 800.2.a.m 2
35.f even 4 1 7840.2.a.bf 2
40.e odd 2 1 1600.2.c.n 4
40.f even 2 1 1600.2.c.n 4
40.i odd 4 1 320.2.a.g 2
40.i odd 4 1 1600.2.a.bc 2
40.k even 4 1 320.2.a.g 2
40.k even 4 1 1600.2.a.bc 2
60.h even 2 1 7200.2.f.bh 4
60.l odd 4 1 1440.2.a.o 2
60.l odd 4 1 7200.2.a.cm 2
80.i odd 4 1 1280.2.d.l 4
80.j even 4 1 1280.2.d.l 4
80.s even 4 1 1280.2.d.l 4
80.t odd 4 1 1280.2.d.l 4
120.q odd 4 1 2880.2.a.bk 2
120.w even 4 1 2880.2.a.bk 2
140.j odd 4 1 7840.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 5.c odd 4 1
160.2.a.c 2 20.e even 4 1
320.2.a.g 2 40.i odd 4 1
320.2.a.g 2 40.k even 4 1
800.2.a.m 2 5.c odd 4 1
800.2.a.m 2 20.e even 4 1
800.2.c.f 4 1.a even 1 1 trivial
800.2.c.f 4 4.b odd 2 1 inner
800.2.c.f 4 5.b even 2 1 inner
800.2.c.f 4 20.d odd 2 1 inner
1280.2.d.l 4 80.i odd 4 1
1280.2.d.l 4 80.j even 4 1
1280.2.d.l 4 80.s even 4 1
1280.2.d.l 4 80.t odd 4 1
1440.2.a.o 2 15.e even 4 1
1440.2.a.o 2 60.l odd 4 1
1600.2.a.bc 2 40.i odd 4 1
1600.2.a.bc 2 40.k even 4 1
1600.2.c.n 4 8.b even 2 1
1600.2.c.n 4 8.d odd 2 1
1600.2.c.n 4 40.e odd 2 1
1600.2.c.n 4 40.f even 2 1
2880.2.a.bk 2 120.q odd 4 1
2880.2.a.bk 2 120.w even 4 1
7200.2.a.cm 2 15.e even 4 1
7200.2.a.cm 2 60.l odd 4 1
7200.2.f.bh 4 3.b odd 2 1
7200.2.f.bh 4 12.b even 2 1
7200.2.f.bh 4 15.d odd 2 1
7200.2.f.bh 4 60.h even 2 1
7840.2.a.bf 2 35.f even 4 1
7840.2.a.bf 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 8 \)
\( T_{11}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 8 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 8 + T^{2} )^{2} \)
$11$ \( ( -32 + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( 4 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 8 + T^{2} )^{2} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( ( 100 + T^{2} )^{2} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( ( 72 + T^{2} )^{2} \)
$47$ \( ( 8 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( ( 8 + T^{2} )^{2} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( ( 36 + T^{2} )^{2} \)
$79$ \( ( -128 + T^{2} )^{2} \)
$83$ \( ( 8 + T^{2} )^{2} \)
$89$ \( ( 10 + T )^{4} \)
$97$ \( ( 4 + T^{2} )^{2} \)
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