Properties

Label 800.2.o.a
Level $800$
Weight $2$
Character orbit 800.o
Analytic conductor $6.388$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 2 i ) q^{3} -5 i q^{9} +O(q^{10})\) \( q + ( -2 + 2 i ) q^{3} -5 i q^{9} + 6 q^{11} + ( 4 + 4 i ) q^{17} + 2 i q^{19} + ( 4 + 4 i ) q^{27} + ( -12 + 12 i ) q^{33} -6 q^{41} + ( -6 + 6 i ) q^{43} + 7 i q^{49} -16 q^{51} + ( -4 - 4 i ) q^{57} + 6 i q^{59} + ( 6 + 6 i ) q^{67} + ( 12 - 12 i ) q^{73} - q^{81} + ( -2 + 2 i ) q^{83} + 18 i q^{89} + ( 12 + 12 i ) q^{97} -30 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + O(q^{10}) \) \( 2q - 4q^{3} + 12q^{11} + 8q^{17} + 8q^{27} - 24q^{33} - 12q^{41} - 12q^{43} - 32q^{51} - 8q^{57} + 12q^{67} + 24q^{73} - 2q^{81} - 4q^{83} + 24q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.00000i
1.00000i
0 −2.00000 2.00000i 0 0 0 0 0 5.00000i 0
207.1 0 −2.00000 + 2.00000i 0 0 0 0 0 5.00000i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.c odd 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.o.a 2
4.b odd 2 1 200.2.k.d yes 2
5.b even 2 1 800.2.o.d 2
5.c odd 4 1 inner 800.2.o.a 2
5.c odd 4 1 800.2.o.d 2
8.b even 2 1 200.2.k.d yes 2
8.d odd 2 1 CM 800.2.o.a 2
20.d odd 2 1 200.2.k.a 2
20.e even 4 1 200.2.k.a 2
20.e even 4 1 200.2.k.d yes 2
40.e odd 2 1 800.2.o.d 2
40.f even 2 1 200.2.k.a 2
40.i odd 4 1 200.2.k.a 2
40.i odd 4 1 200.2.k.d yes 2
40.k even 4 1 inner 800.2.o.a 2
40.k even 4 1 800.2.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.a 2 20.d odd 2 1
200.2.k.a 2 20.e even 4 1
200.2.k.a 2 40.f even 2 1
200.2.k.a 2 40.i odd 4 1
200.2.k.d yes 2 4.b odd 2 1
200.2.k.d yes 2 8.b even 2 1
200.2.k.d yes 2 20.e even 4 1
200.2.k.d yes 2 40.i odd 4 1
800.2.o.a 2 1.a even 1 1 trivial
800.2.o.a 2 5.c odd 4 1 inner
800.2.o.a 2 8.d odd 2 1 CM
800.2.o.a 2 40.k even 4 1 inner
800.2.o.d 2 5.b even 2 1
800.2.o.d 2 5.c odd 4 1
800.2.o.d 2 40.e odd 2 1
800.2.o.d 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 \(S_{2}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 4 T_{3} + 8 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 8 + 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 32 - 8 T + T^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 72 + 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 72 - 12 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 288 - 24 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 8 + 4 T + T^{2} \)
$89$ \( 324 + T^{2} \)
$97$ \( 288 - 24 T + T^{2} \)
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