Properties

Label 400.6.a.s
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1535279252\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 11 \beta q^{7} - 119 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 11 \beta q^{7} - 119 q^{9} + 100 q^{11} + 66 \beta q^{13} - 88 \beta q^{17} + 2244 q^{19} - 1364 q^{21} + 307 \beta q^{23} - 362 \beta q^{27} - 7854 q^{29} + 2144 q^{31} + 100 \beta q^{33} - 934 \beta q^{37} + 8184 q^{39} - 7414 q^{41} - 1595 \beta q^{43} - 847 \beta q^{47} - 1803 q^{49} - 10912 q^{51} - 2178 \beta q^{53} + 2244 \beta q^{57} - 25972 q^{59} - 3058 q^{61} + 1309 \beta q^{63} + 5279 \beta q^{67} + 38068 q^{69} - 37608 q^{71} + 2156 \beta q^{73} - 1100 \beta q^{77} - 79728 q^{79} - 15971 q^{81} - 1463 \beta q^{83} - 7854 \beta q^{87} - 826 q^{89} - 90024 q^{91} + 2144 \beta q^{93} + 3376 \beta q^{97} - 11900 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 238 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 238 q^{9} + 200 q^{11} + 4488 q^{19} - 2728 q^{21} - 15708 q^{29} + 4288 q^{31} + 16368 q^{39} - 14828 q^{41} - 3606 q^{49} - 21824 q^{51} - 51944 q^{59} - 6116 q^{61} + 76136 q^{69} - 75216 q^{71} - 159456 q^{79} - 31942 q^{81} - 1652 q^{89} - 180048 q^{91} - 23800 q^{99}+O(q^{100}) \) 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
−5.56776
5.56776
0 −11.1355 0 0 0 122.491 0 −119.000 0
1.2 0 11.1355 0 0 0 −122.491 0 −119.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.s 2
4.b odd 2 1 100.6.a.d 2
5.b even 2 1 inner 400.6.a.s 2
5.c odd 4 2 80.6.c.b 2
12.b even 2 1 900.6.a.q 2
15.e even 4 2 720.6.f.d 2
20.d odd 2 1 100.6.a.d 2
20.e even 4 2 20.6.c.a 2
40.i odd 4 2 320.6.c.d 2
40.k even 4 2 320.6.c.e 2
60.h even 2 1 900.6.a.q 2
60.l odd 4 2 180.6.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.c.a 2 20.e even 4 2
80.6.c.b 2 5.c odd 4 2
100.6.a.d 2 4.b odd 2 1
100.6.a.d 2 20.d odd 2 1
180.6.d.b 2 60.l odd 4 2
320.6.c.d 2 40.i odd 4 2
320.6.c.e 2 40.k even 4 2
400.6.a.s 2 1.a even 1 1 trivial
400.6.a.s 2 5.b even 2 1 inner
720.6.f.d 2 15.e even 4 2
900.6.a.q 2 12.b even 2 1
900.6.a.q 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 124 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 124 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 15004 \) Copy content Toggle raw display
$11$ \( (T - 100)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 540144 \) Copy content Toggle raw display
$17$ \( T^{2} - 960256 \) Copy content Toggle raw display
$19$ \( (T - 2244)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 11686876 \) Copy content Toggle raw display
$29$ \( (T + 7854)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2144)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 108172144 \) Copy content Toggle raw display
$41$ \( (T + 7414)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 315459100 \) Copy content Toggle raw display
$47$ \( T^{2} - 88958716 \) Copy content Toggle raw display
$53$ \( T^{2} - 588216816 \) Copy content Toggle raw display
$59$ \( (T + 25972)^{2} \) Copy content Toggle raw display
$61$ \( (T + 3058)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 3455612284 \) Copy content Toggle raw display
$71$ \( (T + 37608)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 576393664 \) Copy content Toggle raw display
$79$ \( (T + 79728)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 265405756 \) Copy content Toggle raw display
$89$ \( (T + 826)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1413274624 \) Copy content Toggle raw display
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