Properties

Label 800.4.f.a
Level $800$
Weight $4$
Character orbit 800.f
Analytic conductor $47.202$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.2015280046\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 8)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - 4 \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - 4 \beta_1 q^{7} + q^{9} - 3 \beta_{3} q^{11} + 10 \beta_{2} q^{13} + 7 \beta_1 q^{17} + 7 \beta_{3} q^{19} + 8 \beta_{3} q^{21} + 76 \beta_1 q^{23} + 26 \beta_{2} q^{27} - 30 \beta_{3} q^{29} - 224 q^{31} + 42 \beta_1 q^{33} - 46 \beta_{2} q^{37} - 280 q^{39} - 70 q^{41} + 83 \beta_{2} q^{43} + 168 \beta_1 q^{47} + 279 q^{49} - 14 \beta_{3} q^{51} + 6 \beta_{2} q^{53} - 98 \beta_1 q^{57} - 101 \beta_{3} q^{59} - 18 \beta_{3} q^{61} - 4 \beta_1 q^{63} + 33 \beta_{2} q^{67} - 152 \beta_{3} q^{69} + 72 q^{71} - 147 \beta_1 q^{73} - 24 \beta_{2} q^{77} - 464 q^{79} - 755 q^{81} + 103 \beta_{2} q^{83} + 420 \beta_1 q^{87} - 266 q^{89} - 80 \beta_{3} q^{91} + 224 \beta_{2} q^{93} - 497 \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 896 q^{31} - 1120 q^{39} - 280 q^{41} + 1116 q^{49} + 288 q^{71} - 1856 q^{79} - 3020 q^{81} - 1064 q^{89}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 4 \) 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 −5.29150 0 0 0 8.00000i 0 1.00000 0
49.2 0 −5.29150 0 0 0 8.00000i 0 1.00000 0
49.3 0 5.29150 0 0 0 8.00000i 0 1.00000 0
49.4 0 5.29150 0 0 0 8.00000i 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 800.4.f.a 4
4.b odd 2 1 200.4.f.a 4
5.b even 2 1 inner 800.4.f.a 4
5.c odd 4 1 32.4.b.a 2
5.c odd 4 1 800.4.d.a 2
8.b even 2 1 inner 800.4.f.a 4
8.d odd 2 1 200.4.f.a 4
15.e even 4 1 288.4.d.a 2
20.d odd 2 1 200.4.f.a 4
20.e even 4 1 8.4.b.a 2
20.e even 4 1 200.4.d.a 2
40.e odd 2 1 200.4.f.a 4
40.f even 2 1 inner 800.4.f.a 4
40.i odd 4 1 32.4.b.a 2
40.i odd 4 1 800.4.d.a 2
40.k even 4 1 8.4.b.a 2
40.k even 4 1 200.4.d.a 2
60.l odd 4 1 72.4.d.b 2
80.i odd 4 1 256.4.a.j 2
80.j even 4 1 256.4.a.l 2
80.s even 4 1 256.4.a.l 2
80.t odd 4 1 256.4.a.j 2
120.q odd 4 1 72.4.d.b 2
120.w even 4 1 288.4.d.a 2
240.z odd 4 1 2304.4.a.bn 2
240.bb even 4 1 2304.4.a.v 2
240.bd odd 4 1 2304.4.a.bn 2
240.bf even 4 1 2304.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 20.e even 4 1
8.4.b.a 2 40.k even 4 1
32.4.b.a 2 5.c odd 4 1
32.4.b.a 2 40.i odd 4 1
72.4.d.b 2 60.l odd 4 1
72.4.d.b 2 120.q odd 4 1
200.4.d.a 2 20.e even 4 1
200.4.d.a 2 40.k even 4 1
200.4.f.a 4 4.b odd 2 1
200.4.f.a 4 8.d odd 2 1
200.4.f.a 4 20.d odd 2 1
200.4.f.a 4 40.e odd 2 1
256.4.a.j 2 80.i odd 4 1
256.4.a.j 2 80.t odd 4 1
256.4.a.l 2 80.j even 4 1
256.4.a.l 2 80.s even 4 1
288.4.d.a 2 15.e even 4 1
288.4.d.a 2 120.w even 4 1
800.4.d.a 2 5.c odd 4 1
800.4.d.a 2 40.i odd 4 1
800.4.f.a 4 1.a even 1 1 trivial
800.4.f.a 4 5.b even 2 1 inner
800.4.f.a 4 8.b even 2 1 inner
800.4.f.a 4 40.f even 2 1 inner
2304.4.a.v 2 240.bb even 4 1
2304.4.a.v 2 240.bf even 4 1
2304.4.a.bn 2 240.z odd 4 1
2304.4.a.bn 2 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 28 \) acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 252)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2800)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1372)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 25200)^{2} \) Copy content Toggle raw display
$31$ \( (T + 224)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 59248)^{2} \) Copy content Toggle raw display
$41$ \( (T + 70)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 192892)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 112896)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1008)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 285628)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 9072)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 30492)^{2} \) Copy content Toggle raw display
$71$ \( (T - 72)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 86436)^{2} \) Copy content Toggle raw display
$79$ \( (T + 464)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 297052)^{2} \) Copy content Toggle raw display
$89$ \( (T + 266)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 988036)^{2} \) Copy content Toggle raw display
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