Properties

Label 800.2.d.f
Level $800$
Weight $2$
Character orbit 800.d
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{7} + q^{9} + \beta_{3} q^{11} - 2 \beta_{2} q^{17} + \beta_{3} q^{19} - \beta_{3} q^{21} - \beta_{2} q^{23} + 4 \beta_1 q^{27} - 4 q^{31} - 2 \beta_{2} q^{33} + 6 \beta_1 q^{37} - 3 \beta_1 q^{43} - 3 \beta_{2} q^{47} - q^{49} - 2 \beta_{3} q^{51} + 4 \beta_1 q^{53} - 2 \beta_{2} q^{57} - 3 \beta_{3} q^{59} + \beta_{3} q^{61} - \beta_{2} q^{63} - 3 \beta_1 q^{67} - \beta_{3} q^{69} + 12 q^{71} + 2 \beta_{2} q^{73} - 6 \beta_1 q^{77} - 4 q^{79} - 5 q^{81} + 7 \beta_1 q^{83} - 6 q^{89} - 4 \beta_1 q^{93} + 2 \beta_{2} q^{97} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 16 q^{31} - 4 q^{49} + 48 q^{71} - 16 q^{79} - 20 q^{81} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
0 1.41421i 0 0 0 −2.44949 0 1.00000 0
401.2 0 1.41421i 0 0 0 2.44949 0 1.00000 0
401.3 0 1.41421i 0 0 0 −2.44949 0 1.00000 0
401.4 0 1.41421i 0 0 0 2.44949 0 1.00000 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.d.f 4
3.b odd 2 1 7200.2.k.l 4
4.b odd 2 1 200.2.d.e 4
5.b even 2 1 inner 800.2.d.f 4
5.c odd 4 2 160.2.f.a 4
8.b even 2 1 inner 800.2.d.f 4
8.d odd 2 1 200.2.d.e 4
12.b even 2 1 1800.2.k.m 4
15.d odd 2 1 7200.2.k.l 4
15.e even 4 2 1440.2.d.c 4
16.e even 4 2 6400.2.a.cm 4
16.f odd 4 2 6400.2.a.co 4
20.d odd 2 1 200.2.d.e 4
20.e even 4 2 40.2.f.a 4
24.f even 2 1 1800.2.k.m 4
24.h odd 2 1 7200.2.k.l 4
40.e odd 2 1 200.2.d.e 4
40.f even 2 1 inner 800.2.d.f 4
40.i odd 4 2 160.2.f.a 4
40.k even 4 2 40.2.f.a 4
60.h even 2 1 1800.2.k.m 4
60.l odd 4 2 360.2.d.b 4
80.i odd 4 2 1280.2.c.k 4
80.j even 4 2 1280.2.c.i 4
80.k odd 4 2 6400.2.a.co 4
80.q even 4 2 6400.2.a.cm 4
80.s even 4 2 1280.2.c.i 4
80.t odd 4 2 1280.2.c.k 4
120.i odd 2 1 7200.2.k.l 4
120.m even 2 1 1800.2.k.m 4
120.q odd 4 2 360.2.d.b 4
120.w even 4 2 1440.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 20.e even 4 2
40.2.f.a 4 40.k even 4 2
160.2.f.a 4 5.c odd 4 2
160.2.f.a 4 40.i odd 4 2
200.2.d.e 4 4.b odd 2 1
200.2.d.e 4 8.d odd 2 1
200.2.d.e 4 20.d odd 2 1
200.2.d.e 4 40.e odd 2 1
360.2.d.b 4 60.l odd 4 2
360.2.d.b 4 120.q odd 4 2
800.2.d.f 4 1.a even 1 1 trivial
800.2.d.f 4 5.b even 2 1 inner
800.2.d.f 4 8.b even 2 1 inner
800.2.d.f 4 40.f even 2 1 inner
1280.2.c.i 4 80.j even 4 2
1280.2.c.i 4 80.s even 4 2
1280.2.c.k 4 80.i odd 4 2
1280.2.c.k 4 80.t odd 4 2
1440.2.d.c 4 15.e even 4 2
1440.2.d.c 4 120.w even 4 2
1800.2.k.m 4 12.b even 2 1
1800.2.k.m 4 24.f even 2 1
1800.2.k.m 4 60.h even 2 1
1800.2.k.m 4 120.m even 2 1
6400.2.a.cm 4 16.e even 4 2
6400.2.a.cm 4 80.q even 4 2
6400.2.a.co 4 16.f odd 4 2
6400.2.a.co 4 80.k odd 4 2
7200.2.k.l 4 3.b odd 2 1
7200.2.k.l 4 15.d odd 2 1
7200.2.k.l 4 24.h odd 2 1
7200.2.k.l 4 120.i odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
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