Properties

Label 400.6.a.x
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.47217.1
Defining polynomial: \( x^{3} - x^{2} - 38x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 + 24) q^{7} + ( - \beta_{2} + 5 \beta_1 + 50) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 + 24) q^{7} + ( - \beta_{2} + 5 \beta_1 + 50) q^{9} + ( - \beta_{2} + 8 \beta_1 + 4) q^{11} + (\beta_{2} + 31 \beta_1 - 76) q^{13} + ( - 5 \beta_{2} + 49 \beta_1 + 393) q^{17} + ( - 6 \beta_{2} - 87 \beta_1 - 376) q^{19} + ( - 5 \beta_{2} - 147 \beta_1 + 370) q^{21} + (6 \beta_{2} - 204 \beta_1 + 896) q^{23} + (\beta_{2} + 40 \beta_1 - 1388) q^{27} + ( - 5 \beta_{2} + 13 \beta_1 + 3968) q^{29} + (2 \beta_{2} - 220 \beta_1 - 2408) q^{31} + (4 \beta_{2} - 172 \beta_1 - 2267) q^{33} + (39 \beta_{2} + 201 \beta_1 + 4842) q^{37} + (35 \beta_{2} + 49 \beta_1 - 9160) q^{39} + (44 \beta_{2} - 292 \beta_1 + 1157) q^{41} + ( - 41 \beta_{2} + 241 \beta_1 + 13620) q^{43} + (31 \beta_{2} + 733 \beta_1 - 9984) q^{47} + (72 \beta_{2} - 888 \beta_1 + 22413) q^{49} + (29 \beta_{2} - 1278 \beta_1 - 13972) q^{51} + ( - 76 \beta_{2} - 1468 \beta_1 - 3778) q^{53} + ( - 111 \beta_{2} + 43 \beta_1 + 25953) q^{57} + ( - 35 \beta_{2} + 151 \beta_1 - 21676) q^{59} + ( - 155 \beta_{2} + 19 \beta_1 + 7538) q^{61} + (76 \beta_{2} - 32 \beta_1 + 37624) q^{63} + ( - 105 \beta_{2} + 162 \beta_1 - 8852) q^{67} + ( - 180 \beta_{2} + 892 \beta_1 + 59310) q^{69} + ( - 135 \beta_{2} + 1299 \beta_1 - 29280) q^{71} + (257 \beta_{2} + 2363 \beta_1 + 1895) q^{73} + (137 \beta_{2} + 455 \beta_1 + 35410) q^{77} + ( - 113 \beta_{2} - 1181 \beta_1 - 6896) q^{79} + (287 \beta_{2} + 101 \beta_1 - 23947) q^{81} + (43 \beta_{2} - 1310 \beta_1 + 71540) q^{83} + ( - 7 \beta_{2} - 4673 \beta_1 - 3424) q^{87} + ( - 41 \beta_{2} + 1141 \beta_1 + 60427) q^{89} + (130 \beta_{2} + 5350 \beta_1 - 51568) q^{91} + ( - 212 \beta_{2} + 3764 \beta_1 + 64306) q^{93} + ( - 82 \beta_{2} - 4846 \beta_1 + 27090) q^{97} + (87 \beta_{2} + 1695 \beta_1 + 49116) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 70 q^{7} + 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 70 q^{7} + 154 q^{9} + 19 q^{11} - 196 q^{13} + 1223 q^{17} - 1221 q^{19} + 958 q^{21} + 2490 q^{23} - 4123 q^{27} + 11912 q^{29} - 7442 q^{31} - 6969 q^{33} + 14766 q^{37} - 27396 q^{39} + 3223 q^{41} + 41060 q^{43} - 29188 q^{47} + 66423 q^{49} - 43165 q^{51} - 12878 q^{53} + 77791 q^{57} - 64912 q^{59} + 22478 q^{61} + 112916 q^{63} - 26499 q^{67} + 178642 q^{69} - 86676 q^{71} + 8305 q^{73} + 106822 q^{77} - 21982 q^{79} - 71453 q^{81} + 213353 q^{83} - 14952 q^{87} + 182381 q^{89} - 149224 q^{91} + 196470 q^{93} + 76342 q^{97} + 149130 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 38x - 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{2} + 3\nu + 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{2} + 31\nu - 87 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 + 12 ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} - 31\beta _1 + 1036 ) / 40 \) 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
−0.649919
6.95752
−5.30760
0 −22.6278 0 0 0 107.252 0 269.020 0
1.2 0 2.53448 0 0 0 −247.370 0 −236.576 0
1.3 0 19.0934 0 0 0 210.117 0 121.557 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 400.6.a.x 3
4.b odd 2 1 200.6.a.i yes 3
5.b even 2 1 400.6.a.y 3
5.c odd 4 2 400.6.c.p 6
20.d odd 2 1 200.6.a.h 3
20.e even 4 2 200.6.c.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.a.h 3 20.d odd 2 1
200.6.a.i yes 3 4.b odd 2 1
200.6.c.g 6 20.e even 4 2
400.6.a.x 3 1.a even 1 1 trivial
400.6.a.y 3 5.b even 2 1
400.6.c.p 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + T_{3}^{2} - 441T_{3} + 1095 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 441 T + 1095 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 70 T^{2} - 55972 T + 5574616 \) Copy content Toggle raw display
$11$ \( T^{3} - 19 T^{2} - 84401 T + 1542819 \) Copy content Toggle raw display
$13$ \( T^{3} + 196 T^{2} + \cdots + 51768768 \) Copy content Toggle raw display
$17$ \( T^{3} - 1223 T^{2} + \cdots + 654046475 \) Copy content Toggle raw display
$19$ \( T^{3} + 1221 T^{2} + \cdots - 7033288181 \) Copy content Toggle raw display
$23$ \( T^{3} - 2490 T^{2} + \cdots + 50437840744 \) Copy content Toggle raw display
$29$ \( T^{3} - 11912 T^{2} + \cdots - 55993024512 \) Copy content Toggle raw display
$31$ \( T^{3} + 7442 T^{2} + \cdots - 14434811400 \) Copy content Toggle raw display
$37$ \( T^{3} - 14766 T^{2} + \cdots + 435216240216 \) Copy content Toggle raw display
$41$ \( T^{3} - 3223 T^{2} + \cdots - 86972031621 \) Copy content Toggle raw display
$43$ \( T^{3} - 41060 T^{2} + \cdots - 660952050112 \) Copy content Toggle raw display
$47$ \( T^{3} + 29188 T^{2} + \cdots - 324350860864 \) Copy content Toggle raw display
$53$ \( T^{3} + 12878 T^{2} + \cdots - 22325036001624 \) Copy content Toggle raw display
$59$ \( T^{3} + 64912 T^{2} + \cdots + 8618337690624 \) Copy content Toggle raw display
$61$ \( T^{3} - 22478 T^{2} + \cdots + 28184724893400 \) Copy content Toggle raw display
$67$ \( T^{3} + 26499 T^{2} + \cdots + 1261173627437 \) Copy content Toggle raw display
$71$ \( T^{3} + 86676 T^{2} + \cdots - 31597978616640 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 160799137840701 \) Copy content Toggle raw display
$79$ \( T^{3} + 21982 T^{2} + \cdots - 26059657326840 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 289154781250911 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 190810675088559 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 166573103686456 \) Copy content Toggle raw display
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