Properties

Label 800.4.d.a
Level $800$
Weight $4$
Character orbit 800.d
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 8 q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 8 q^{7} - q^{9} - 3 \beta q^{11} + 10 \beta q^{13} + 14 q^{17} - 7 \beta q^{19} + 8 \beta q^{21} - 152 q^{23} - 26 \beta q^{27} + 30 \beta q^{29} - 224 q^{31} - 84 q^{33} + 46 \beta q^{37} + 280 q^{39} - 70 q^{41} + 83 \beta q^{43} + 336 q^{47} - 279 q^{49} - 14 \beta q^{51} + 6 \beta q^{53} - 196 q^{57} + 101 \beta q^{59} - 18 \beta q^{61} + 8 q^{63} - 33 \beta q^{67} + 152 \beta q^{69} + 72 q^{71} + 294 q^{73} + 24 \beta q^{77} + 464 q^{79} - 755 q^{81} + 103 \beta q^{83} + 840 q^{87} + 266 q^{89} - 80 \beta q^{91} + 224 \beta q^{93} - 994 q^{97} + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{7} - 2 q^{9} + 28 q^{17} - 304 q^{23} - 448 q^{31} - 168 q^{33} + 560 q^{39} - 140 q^{41} + 672 q^{47} - 558 q^{49} - 392 q^{57} + 16 q^{63} + 144 q^{71} + 588 q^{73} + 928 q^{79} - 1510 q^{81} + 1680 q^{87} + 532 q^{89} - 1988 q^{97}+O(q^{100}) \) 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
0.500000 + 1.32288i
0.500000 1.32288i
0 5.29150i 0 0 0 −8.00000 0 −1.00000 0
401.2 0 5.29150i 0 0 0 −8.00000 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
8.b 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.d.a 2
4.b odd 2 1 200.4.d.a 2
5.b even 2 1 32.4.b.a 2
5.c odd 4 2 800.4.f.a 4
8.b even 2 1 inner 800.4.d.a 2
8.d odd 2 1 200.4.d.a 2
15.d odd 2 1 288.4.d.a 2
20.d odd 2 1 8.4.b.a 2
20.e even 4 2 200.4.f.a 4
40.e odd 2 1 8.4.b.a 2
40.f even 2 1 32.4.b.a 2
40.i odd 4 2 800.4.f.a 4
40.k even 4 2 200.4.f.a 4
60.h even 2 1 72.4.d.b 2
80.k odd 4 2 256.4.a.l 2
80.q even 4 2 256.4.a.j 2
120.i odd 2 1 288.4.d.a 2
120.m even 2 1 72.4.d.b 2
240.t even 4 2 2304.4.a.bn 2
240.bm odd 4 2 2304.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 20.d odd 2 1
8.4.b.a 2 40.e odd 2 1
32.4.b.a 2 5.b even 2 1
32.4.b.a 2 40.f even 2 1
72.4.d.b 2 60.h even 2 1
72.4.d.b 2 120.m even 2 1
200.4.d.a 2 4.b odd 2 1
200.4.d.a 2 8.d odd 2 1
200.4.f.a 4 20.e even 4 2
200.4.f.a 4 40.k even 4 2
256.4.a.j 2 80.q even 4 2
256.4.a.l 2 80.k odd 4 2
288.4.d.a 2 15.d odd 2 1
288.4.d.a 2 120.i odd 2 1
800.4.d.a 2 1.a even 1 1 trivial
800.4.d.a 2 8.b even 2 1 inner
800.4.f.a 4 5.c odd 4 2
800.4.f.a 4 40.i odd 4 2
2304.4.a.v 2 240.bm odd 4 2
2304.4.a.bn 2 240.t even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 28 \) Copy content Toggle raw display
\( T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 28 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 252 \) Copy content Toggle raw display
$13$ \( T^{2} + 2800 \) Copy content Toggle raw display
$17$ \( (T - 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1372 \) Copy content Toggle raw display
$23$ \( (T + 152)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 25200 \) Copy content Toggle raw display
$31$ \( (T + 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 59248 \) Copy content Toggle raw display
$41$ \( (T + 70)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 192892 \) Copy content Toggle raw display
$47$ \( (T - 336)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 1008 \) Copy content Toggle raw display
$59$ \( T^{2} + 285628 \) Copy content Toggle raw display
$61$ \( T^{2} + 9072 \) Copy content Toggle raw display
$67$ \( T^{2} + 30492 \) Copy content Toggle raw display
$71$ \( (T - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T - 294)^{2} \) Copy content Toggle raw display
$79$ \( (T - 464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 297052 \) Copy content Toggle raw display
$89$ \( (T - 266)^{2} \) Copy content Toggle raw display
$97$ \( (T + 994)^{2} \) Copy content Toggle raw display
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