Properties

Label 800.2.n.a
Level $800$
Weight $2$
Character orbit 800.n
Analytic conductor $6.388$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.n (of order \(4\), degree \(2\), 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 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 - 2 i ) q^{3} + ( -2 + 2 i ) q^{7} + 5 i q^{9} +O(q^{10})\) \( q + ( -2 - 2 i ) q^{3} + ( -2 + 2 i ) q^{7} + 5 i q^{9} + ( 1 - i ) q^{13} + ( 5 + 5 i ) q^{17} + 4 q^{19} + 8 q^{21} + ( -2 - 2 i ) q^{23} + ( 4 - 4 i ) q^{27} -4 i q^{29} -4 i q^{31} + ( -1 - i ) q^{37} -4 q^{39} + ( 6 + 6 i ) q^{43} + ( 2 - 2 i ) q^{47} -i q^{49} -20 i q^{51} + ( 7 - 7 i ) q^{53} + ( -8 - 8 i ) q^{57} + 4 q^{59} -4 q^{61} + ( -10 - 10 i ) q^{63} + ( 10 - 10 i ) q^{67} + 8 i q^{69} + 12 i q^{71} + ( 3 - 3 i ) q^{73} + 16 q^{79} - q^{81} + ( 2 + 2 i ) q^{83} + ( -8 + 8 i ) q^{87} + 4 i q^{91} + ( -8 + 8 i ) q^{93} + ( 3 + 3 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{3} - 4q^{7} + 2q^{13} + 10q^{17} + 8q^{19} + 16q^{21} - 4q^{23} + 8q^{27} - 2q^{37} - 8q^{39} + 12q^{43} + 4q^{47} + 14q^{53} - 16q^{57} + 8q^{59} - 8q^{61} - 20q^{63} + 20q^{67} + 6q^{73} + 32q^{79} - 2q^{81} + 4q^{83} - 16q^{87} - 16q^{93} + 6q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
543.1
1.00000i
1.00000i
0 −2.00000 + 2.00000i 0 0 0 −2.00000 2.00000i 0 5.00000i 0
607.1 0 −2.00000 2.00000i 0 0 0 −2.00000 + 2.00000i 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
20.e 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.n.a 2
4.b odd 2 1 800.2.n.j 2
5.b even 2 1 160.2.n.f yes 2
5.c odd 4 1 160.2.n.a 2
5.c odd 4 1 800.2.n.j 2
8.b even 2 1 1600.2.n.n 2
8.d odd 2 1 1600.2.n.a 2
15.d odd 2 1 1440.2.x.j 2
15.e even 4 1 1440.2.x.i 2
20.d odd 2 1 160.2.n.a 2
20.e even 4 1 160.2.n.f yes 2
20.e even 4 1 inner 800.2.n.a 2
40.e odd 2 1 320.2.n.h 2
40.f even 2 1 320.2.n.a 2
40.i odd 4 1 320.2.n.h 2
40.i odd 4 1 1600.2.n.a 2
40.k even 4 1 320.2.n.a 2
40.k even 4 1 1600.2.n.n 2
60.h even 2 1 1440.2.x.i 2
60.l odd 4 1 1440.2.x.j 2
80.i odd 4 1 1280.2.o.p 2
80.j even 4 1 1280.2.o.o 2
80.k odd 4 1 1280.2.o.a 2
80.k odd 4 1 1280.2.o.p 2
80.q even 4 1 1280.2.o.b 2
80.q even 4 1 1280.2.o.o 2
80.s even 4 1 1280.2.o.b 2
80.t odd 4 1 1280.2.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.a 2 5.c odd 4 1
160.2.n.a 2 20.d odd 2 1
160.2.n.f yes 2 5.b even 2 1
160.2.n.f yes 2 20.e even 4 1
320.2.n.a 2 40.f even 2 1
320.2.n.a 2 40.k even 4 1
320.2.n.h 2 40.e odd 2 1
320.2.n.h 2 40.i odd 4 1
800.2.n.a 2 1.a even 1 1 trivial
800.2.n.a 2 20.e even 4 1 inner
800.2.n.j 2 4.b odd 2 1
800.2.n.j 2 5.c odd 4 1
1280.2.o.a 2 80.k odd 4 1
1280.2.o.a 2 80.t odd 4 1
1280.2.o.b 2 80.q even 4 1
1280.2.o.b 2 80.s even 4 1
1280.2.o.o 2 80.j even 4 1
1280.2.o.o 2 80.q even 4 1
1280.2.o.p 2 80.i odd 4 1
1280.2.o.p 2 80.k odd 4 1
1440.2.x.i 2 15.e even 4 1
1440.2.x.i 2 60.h even 2 1
1440.2.x.j 2 15.d odd 2 1
1440.2.x.j 2 60.l odd 4 1
1600.2.n.a 2 8.d odd 2 1
1600.2.n.a 2 40.i odd 4 1
1600.2.n.n 2 8.b even 2 1
1600.2.n.n 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 \(S_{2}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 4 T_{3} + 8 \)
\( T_{7}^{2} + 4 T_{7} + 8 \)
\( T_{11} \)
\( T_{13}^{2} - 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 8 + 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 8 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 2 - 2 T + T^{2} \)
$17$ \( 50 - 10 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 8 + 4 T + T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 2 + 2 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 72 - 12 T + T^{2} \)
$47$ \( 8 - 4 T + T^{2} \)
$53$ \( 98 - 14 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( 200 - 20 T + T^{2} \)
$71$ \( 144 + T^{2} \)
$73$ \( 18 - 6 T + T^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( 8 - 4 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 18 - 6 T + T^{2} \)
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