Properties

Label 200.6.a.b
Level $200$
Weight $6$
Character orbit 200.a
Self dual yes
Analytic conductor $32.077$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} + 62 q^{7} - 239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 62 q^{7} - 239 q^{9} - 144 q^{11} + 654 q^{13} + 1190 q^{17} + 556 q^{19} + 124 q^{21} - 2182 q^{23} - 964 q^{27} - 1578 q^{29} + 9660 q^{31} - 288 q^{33} + 3534 q^{37} + 1308 q^{39} + 7462 q^{41} + 7114 q^{43} + 28294 q^{47} - 12963 q^{49} + 2380 q^{51} + 13046 q^{53} + 1112 q^{57} - 37092 q^{59} + 39570 q^{61} - 14818 q^{63} + 56734 q^{67} - 4364 q^{69} + 45588 q^{71} - 11842 q^{73} - 8928 q^{77} + 94216 q^{79} + 56149 q^{81} + 31482 q^{83} - 3156 q^{87} - 94054 q^{89} + 40548 q^{91} + 19320 q^{93} - 23714 q^{97} + 34416 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 2.00000 0 0 0 62.0000 0 −239.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.6.a.b 1
4.b odd 2 1 400.6.a.h 1
5.b even 2 1 40.6.a.c 1
5.c odd 4 2 200.6.c.d 2
15.d odd 2 1 360.6.a.f 1
20.d odd 2 1 80.6.a.d 1
20.e even 4 2 400.6.c.k 2
40.e odd 2 1 320.6.a.h 1
40.f even 2 1 320.6.a.i 1
60.h even 2 1 720.6.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.c 1 5.b even 2 1
80.6.a.d 1 20.d odd 2 1
200.6.a.b 1 1.a even 1 1 trivial
200.6.c.d 2 5.c odd 4 2
320.6.a.h 1 40.e odd 2 1
320.6.a.i 1 40.f even 2 1
360.6.a.f 1 15.d odd 2 1
400.6.a.h 1 4.b odd 2 1
400.6.c.k 2 20.e even 4 2
720.6.a.t 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 62 \) Copy content Toggle raw display
$11$ \( T + 144 \) Copy content Toggle raw display
$13$ \( T - 654 \) Copy content Toggle raw display
$17$ \( T - 1190 \) Copy content Toggle raw display
$19$ \( T - 556 \) Copy content Toggle raw display
$23$ \( T + 2182 \) Copy content Toggle raw display
$29$ \( T + 1578 \) Copy content Toggle raw display
$31$ \( T - 9660 \) Copy content Toggle raw display
$37$ \( T - 3534 \) Copy content Toggle raw display
$41$ \( T - 7462 \) Copy content Toggle raw display
$43$ \( T - 7114 \) Copy content Toggle raw display
$47$ \( T - 28294 \) Copy content Toggle raw display
$53$ \( T - 13046 \) Copy content Toggle raw display
$59$ \( T + 37092 \) Copy content Toggle raw display
$61$ \( T - 39570 \) Copy content Toggle raw display
$67$ \( T - 56734 \) Copy content Toggle raw display
$71$ \( T - 45588 \) Copy content Toggle raw display
$73$ \( T + 11842 \) Copy content Toggle raw display
$79$ \( T - 94216 \) Copy content Toggle raw display
$83$ \( T - 31482 \) Copy content Toggle raw display
$89$ \( T + 94054 \) Copy content Toggle raw display
$97$ \( T + 23714 \) Copy content Toggle raw display
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