Properties

Label 800.2.o.c
Level $800$
Weight $2$
Character orbit 800.o
Analytic conductor $6.388$
Analytic rank $0$
Dimension $2$
CM discriminant -40
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 i ) q^{7} + 3 i q^{9} +O(q^{10})\) \( q + ( 2 - 2 i ) q^{7} + 3 i q^{9} -2 q^{11} + ( 4 + 4 i ) q^{13} -6 i q^{19} + ( 6 + 6 i ) q^{23} + ( 8 - 8 i ) q^{37} + 2 q^{41} + ( 2 - 2 i ) q^{47} -i q^{49} + ( 4 + 4 i ) q^{53} + 14 i q^{59} + ( 6 + 6 i ) q^{63} + ( -4 + 4 i ) q^{77} -9 q^{81} -14 i q^{89} + 16 q^{91} -6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} + O(q^{10}) \) \( 2q + 4q^{7} - 4q^{11} + 8q^{13} + 12q^{23} + 16q^{37} + 4q^{41} + 4q^{47} + 8q^{53} + 12q^{63} - 8q^{77} - 18q^{81} + 32q^{91} + O(q^{100}) \)

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 0 0 0 0 2.00000 + 2.00000i 0 3.00000i 0
207.1 0 0 0 0 0 2.00000 2.00000i 0 3.00000i 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
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.c 2
4.b odd 2 1 200.2.k.b 2
5.b even 2 1 800.2.o.b 2
5.c odd 4 1 800.2.o.b 2
5.c odd 4 1 inner 800.2.o.c 2
8.b even 2 1 200.2.k.c yes 2
8.d odd 2 1 800.2.o.b 2
20.d odd 2 1 200.2.k.c yes 2
20.e even 4 1 200.2.k.b 2
20.e even 4 1 200.2.k.c yes 2
40.e odd 2 1 CM 800.2.o.c 2
40.f even 2 1 200.2.k.b 2
40.i odd 4 1 200.2.k.b 2
40.i odd 4 1 200.2.k.c yes 2
40.k even 4 1 800.2.o.b 2
40.k even 4 1 inner 800.2.o.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.b 2 4.b odd 2 1
200.2.k.b 2 20.e even 4 1
200.2.k.b 2 40.f even 2 1
200.2.k.b 2 40.i odd 4 1
200.2.k.c yes 2 8.b even 2 1
200.2.k.c yes 2 20.d odd 2 1
200.2.k.c yes 2 20.e even 4 1
200.2.k.c yes 2 40.i odd 4 1
800.2.o.b 2 5.b even 2 1
800.2.o.b 2 5.c odd 4 1
800.2.o.b 2 8.d odd 2 1
800.2.o.b 2 40.k even 4 1
800.2.o.c 2 1.a even 1 1 trivial
800.2.o.c 2 5.c odd 4 1 inner
800.2.o.c 2 40.e odd 2 1 CM
800.2.o.c 2 40.k even 4 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 8 - 4 T + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 32 - 8 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 72 - 12 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 128 - 16 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( 8 - 4 T + T^{2} \)
$53$ \( 32 - 8 T + T^{2} \)
$59$ \( 196 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 196 + T^{2} \)
$97$ \( T^{2} \)
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