Properties

Label 800.2.f.d
Level $800$
Weight $2$
Character orbit 800.f
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + 4 \beta_1 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + 4 \beta_1 q^{7} + 4 q^{9} - \beta_{3} q^{11} + 3 \beta_1 q^{17} + \beta_{3} q^{19} - 4 \beta_{3} q^{21} - 4 \beta_1 q^{23} - \beta_{2} q^{27} - 4 q^{31} + 7 \beta_1 q^{33} - 4 \beta_{2} q^{37} - 5 q^{41} - 2 \beta_{2} q^{43} - 8 \beta_1 q^{47} - 9 q^{49} - 3 \beta_{3} q^{51} - 4 \beta_{2} q^{53} - 7 \beta_1 q^{57} + 2 \beta_{3} q^{59} + 4 \beta_{3} q^{61} + 16 \beta_1 q^{63} - 3 \beta_{2} q^{67} + 4 \beta_{3} q^{69} - 8 q^{71} - 7 \beta_1 q^{73} + 4 \beta_{2} q^{77} + 4 q^{79} - 5 q^{81} + 3 \beta_{2} q^{83} + q^{89} + 4 \beta_{2} q^{93} + 2 \beta_1 q^{97} - 4 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{9} - 16 q^{31} - 20 q^{41} - 36 q^{49} - 32 q^{71} + 16 q^{79} - 20 q^{81} + 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i
−1.32288 + 0.500000i
0 −2.64575 0 0 0 4.00000i 0 4.00000 0
49.2 0 −2.64575 0 0 0 4.00000i 0 4.00000 0
49.3 0 2.64575 0 0 0 4.00000i 0 4.00000 0
49.4 0 2.64575 0 0 0 4.00000i 0 4.00000 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.f.d 4
3.b odd 2 1 7200.2.d.m 4
4.b odd 2 1 200.2.f.d 4
5.b even 2 1 inner 800.2.f.d 4
5.c odd 4 1 800.2.d.a 2
5.c odd 4 1 800.2.d.d 2
8.b even 2 1 inner 800.2.f.d 4
8.d odd 2 1 200.2.f.d 4
12.b even 2 1 1800.2.d.m 4
15.d odd 2 1 7200.2.d.m 4
15.e even 4 1 7200.2.k.b 2
15.e even 4 1 7200.2.k.i 2
20.d odd 2 1 200.2.f.d 4
20.e even 4 1 200.2.d.b 2
20.e even 4 1 200.2.d.c yes 2
24.f even 2 1 1800.2.d.m 4
24.h odd 2 1 7200.2.d.m 4
40.e odd 2 1 200.2.f.d 4
40.f even 2 1 inner 800.2.f.d 4
40.i odd 4 1 800.2.d.a 2
40.i odd 4 1 800.2.d.d 2
40.k even 4 1 200.2.d.b 2
40.k even 4 1 200.2.d.c yes 2
60.h even 2 1 1800.2.d.m 4
60.l odd 4 1 1800.2.k.d 2
60.l odd 4 1 1800.2.k.f 2
80.i odd 4 1 6400.2.a.bh 2
80.i odd 4 1 6400.2.a.cb 2
80.j even 4 1 6400.2.a.bg 2
80.j even 4 1 6400.2.a.cc 2
80.s even 4 1 6400.2.a.bg 2
80.s even 4 1 6400.2.a.cc 2
80.t odd 4 1 6400.2.a.bh 2
80.t odd 4 1 6400.2.a.cb 2
120.i odd 2 1 7200.2.d.m 4
120.m even 2 1 1800.2.d.m 4
120.q odd 4 1 1800.2.k.d 2
120.q odd 4 1 1800.2.k.f 2
120.w even 4 1 7200.2.k.b 2
120.w even 4 1 7200.2.k.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 20.e even 4 1
200.2.d.b 2 40.k even 4 1
200.2.d.c yes 2 20.e even 4 1
200.2.d.c yes 2 40.k even 4 1
200.2.f.d 4 4.b odd 2 1
200.2.f.d 4 8.d odd 2 1
200.2.f.d 4 20.d odd 2 1
200.2.f.d 4 40.e odd 2 1
800.2.d.a 2 5.c odd 4 1
800.2.d.a 2 40.i odd 4 1
800.2.d.d 2 5.c odd 4 1
800.2.d.d 2 40.i odd 4 1
800.2.f.d 4 1.a even 1 1 trivial
800.2.f.d 4 5.b even 2 1 inner
800.2.f.d 4 8.b even 2 1 inner
800.2.f.d 4 40.f even 2 1 inner
1800.2.d.m 4 12.b even 2 1
1800.2.d.m 4 24.f even 2 1
1800.2.d.m 4 60.h even 2 1
1800.2.d.m 4 120.m even 2 1
1800.2.k.d 2 60.l odd 4 1
1800.2.k.d 2 120.q odd 4 1
1800.2.k.f 2 60.l odd 4 1
1800.2.k.f 2 120.q odd 4 1
6400.2.a.bg 2 80.j even 4 1
6400.2.a.bg 2 80.s even 4 1
6400.2.a.bh 2 80.i odd 4 1
6400.2.a.bh 2 80.t odd 4 1
6400.2.a.cb 2 80.i odd 4 1
6400.2.a.cb 2 80.t odd 4 1
6400.2.a.cc 2 80.j even 4 1
6400.2.a.cc 2 80.s even 4 1
7200.2.d.m 4 3.b odd 2 1
7200.2.d.m 4 15.d odd 2 1
7200.2.d.m 4 24.h odd 2 1
7200.2.d.m 4 120.i odd 2 1
7200.2.k.b 2 15.e even 4 1
7200.2.k.b 2 120.w even 4 1
7200.2.k.i 2 15.e even 4 1
7200.2.k.i 2 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 7 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$41$ \( (T + 5)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$71$ \( (T + 8)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$89$ \( (T - 1)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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