Properties

Label 800.2.o.f
Level $800$
Weight $2$
Character orbit 800.o
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
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 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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( -3 + \beta_{1} - \beta_{3} ) q^{11} + ( -2 + 2 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{19} + ( -5 + 5 \beta_{2} + 3 \beta_{3} ) q^{27} -\beta_{1} q^{33} + ( 3 + 4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -6 - 6 \beta_{2} ) q^{43} -7 \beta_{2} q^{49} + ( 5 + \beta_{1} - \beta_{3} ) q^{51} + ( -10 + 10 \beta_{2} + 7 \beta_{3} ) q^{57} -6 \beta_{2} q^{59} + ( -3 + 3 \beta_{2} - 7 \beta_{3} ) q^{67} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{73} + ( -13 - 2 \beta_{1} + 2 \beta_{3} ) q^{81} + ( 1 - 9 \beta_{1} + \beta_{2} ) q^{83} + ( 2 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 12 - 12 \beta_{2} ) q^{97} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + O(q^{10}) \) \( 4q + 4q^{3} - 12q^{11} - 8q^{17} - 20q^{27} + 12q^{41} - 24q^{43} + 20q^{51} - 40q^{57} - 12q^{67} - 24q^{73} - 52q^{81} + 4q^{83} + 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

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

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{2}\)

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
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
0 −0.224745 0.224745i 0 0 0 0 0 2.89898i 0
143.2 0 2.22474 + 2.22474i 0 0 0 0 0 6.89898i 0
207.1 0 −0.224745 + 0.224745i 0 0 0 0 0 2.89898i 0
207.2 0 2.22474 2.22474i 0 0 0 0 0 6.89898i 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.f 4
4.b odd 2 1 200.2.k.f yes 4
5.b even 2 1 800.2.o.e 4
5.c odd 4 1 800.2.o.e 4
5.c odd 4 1 inner 800.2.o.f 4
8.b even 2 1 200.2.k.f yes 4
8.d odd 2 1 CM 800.2.o.f 4
20.d odd 2 1 200.2.k.e 4
20.e even 4 1 200.2.k.e 4
20.e even 4 1 200.2.k.f yes 4
40.e odd 2 1 800.2.o.e 4
40.f even 2 1 200.2.k.e 4
40.i odd 4 1 200.2.k.e 4
40.i odd 4 1 200.2.k.f yes 4
40.k even 4 1 800.2.o.e 4
40.k even 4 1 inner 800.2.o.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.e 4 20.d odd 2 1
200.2.k.e 4 20.e even 4 1
200.2.k.e 4 40.f even 2 1
200.2.k.e 4 40.i odd 4 1
200.2.k.f yes 4 4.b odd 2 1
200.2.k.f yes 4 8.b even 2 1
200.2.k.f yes 4 20.e even 4 1
200.2.k.f yes 4 40.i odd 4 1
800.2.o.e 4 5.b even 2 1
800.2.o.e 4 5.c odd 4 1
800.2.o.e 4 40.e odd 2 1
800.2.o.e 4 40.k even 4 1
800.2.o.f 4 1.a even 1 1 trivial
800.2.o.f 4 5.c odd 4 1 inner
800.2.o.f 4 8.d odd 2 1 CM
800.2.o.f 4 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 \(S_{2}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} - 4 T_{3}^{3} + 8 T_{3}^{2} + 4 T_{3} + 1 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 4 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 3 + 6 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 361 - 152 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 2809 + 110 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -87 - 6 T + T^{2} )^{2} \)
$43$ \( ( 72 + 12 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( 16641 - 1548 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 4761 + 1656 T + 288 T^{2} + 24 T^{3} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( 58081 + 964 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( 3249 + 210 T^{2} + T^{4} \)
$97$ \( ( 288 - 24 T + T^{2} )^{2} \)
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