Properties

Label 800.2.o.h
Level $800$
Weight $2$
Character orbit 800.o
Analytic conductor $6.388$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.5
Defining polynomial: \(x^{8} - 7 x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{6} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{6} q^{7} + q^{11} -\beta_{4} q^{13} -3 \beta_{1} q^{17} -3 \beta_{3} q^{19} + \beta_{7} q^{21} -3 \beta_{1} q^{27} + \beta_{5} q^{29} -\beta_{2} q^{33} + \beta_{6} q^{37} + \beta_{5} q^{39} - q^{41} + 2 \beta_{2} q^{43} + \beta_{6} q^{47} -13 \beta_{3} q^{49} + 9 q^{51} + 2 \beta_{4} q^{53} + 3 \beta_{1} q^{57} -4 \beta_{3} q^{59} -\beta_{7} q^{61} + 3 \beta_{1} q^{67} + \beta_{7} q^{71} + \beta_{2} q^{73} + \beta_{6} q^{77} + \beta_{5} q^{79} + 9 q^{81} -\beta_{2} q^{83} -3 \beta_{6} q^{87} + 13 \beta_{3} q^{89} -20 q^{91} -4 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} - 8q^{41} + 72q^{51} + 72q^{81} - 160q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 7 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + \nu^{3} \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 5 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{2} \)\()/4\)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 11 \nu^{2} \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 13 \nu^{3} \)\()/4\)
\(\beta_{7}\)\(=\)\( 4 \nu^{4} - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{3}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6} + 6 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} + 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{4} + 2 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(3 \beta_{5} + 22 \beta_{3}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{6} + 26 \beta_{1}\)\()/4\)

Character values

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

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{3}\)

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
−0.178197 + 1.40294i
1.40294 0.178197i
−1.40294 + 0.178197i
0.178197 1.40294i
−0.178197 1.40294i
1.40294 + 0.178197i
−1.40294 0.178197i
0.178197 + 1.40294i
0 −1.22474 1.22474i 0 0 0 −3.16228 3.16228i 0 0 0
143.2 0 −1.22474 1.22474i 0 0 0 3.16228 + 3.16228i 0 0 0
143.3 0 1.22474 + 1.22474i 0 0 0 −3.16228 3.16228i 0 0 0
143.4 0 1.22474 + 1.22474i 0 0 0 3.16228 + 3.16228i 0 0 0
207.1 0 −1.22474 + 1.22474i 0 0 0 −3.16228 + 3.16228i 0 0 0
207.2 0 −1.22474 + 1.22474i 0 0 0 3.16228 3.16228i 0 0 0
207.3 0 1.22474 1.22474i 0 0 0 −3.16228 + 3.16228i 0 0 0
207.4 0 1.22474 1.22474i 0 0 0 3.16228 3.16228i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.o.h 8
4.b odd 2 1 200.2.k.g 8
5.b even 2 1 inner 800.2.o.h 8
5.c odd 4 2 inner 800.2.o.h 8
8.b even 2 1 200.2.k.g 8
8.d odd 2 1 inner 800.2.o.h 8
20.d odd 2 1 200.2.k.g 8
20.e even 4 2 200.2.k.g 8
40.e odd 2 1 inner 800.2.o.h 8
40.f even 2 1 200.2.k.g 8
40.i odd 4 2 200.2.k.g 8
40.k even 4 2 inner 800.2.o.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.g 8 4.b odd 2 1
200.2.k.g 8 8.b even 2 1
200.2.k.g 8 20.d odd 2 1
200.2.k.g 8 20.e even 4 2
200.2.k.g 8 40.f even 2 1
200.2.k.g 8 40.i odd 4 2
800.2.o.h 8 1.a even 1 1 trivial
800.2.o.h 8 5.b even 2 1 inner
800.2.o.h 8 5.c odd 4 2 inner
800.2.o.h 8 8.d odd 2 1 inner
800.2.o.h 8 40.e odd 2 1 inner
800.2.o.h 8 40.k even 4 2 inner

Hecke kernels

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

\( T_{3}^{4} + 9 \)
\( T_{7}^{4} + 400 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 400 + T^{4} )^{2} \)
$11$ \( ( -1 + T )^{8} \)
$13$ \( ( 400 + T^{4} )^{2} \)
$17$ \( ( 729 + T^{4} )^{2} \)
$19$ \( ( 9 + T^{2} )^{4} \)
$23$ \( T^{8} \)
$29$ \( ( -60 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( 400 + T^{4} )^{2} \)
$41$ \( ( 1 + T )^{8} \)
$43$ \( ( 144 + T^{4} )^{2} \)
$47$ \( ( 400 + T^{4} )^{2} \)
$53$ \( ( 6400 + T^{4} )^{2} \)
$59$ \( ( 16 + T^{2} )^{4} \)
$61$ \( ( 60 + T^{2} )^{4} \)
$67$ \( ( 729 + T^{4} )^{2} \)
$71$ \( ( 60 + T^{2} )^{4} \)
$73$ \( ( 9 + T^{4} )^{2} \)
$79$ \( ( -60 + T^{2} )^{4} \)
$83$ \( ( 9 + T^{4} )^{2} \)
$89$ \( ( 169 + T^{2} )^{4} \)
$97$ \( ( 2304 + T^{4} )^{2} \)
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