Properties

Label 624.4.a.t
Level $624$
Weight $4$
Character orbit 624.a
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta_{2} + 1) q^{5} + ( - 3 \beta_1 - 11) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta_{2} + 1) q^{5} + ( - 3 \beta_1 - 11) q^{7} + 9 q^{9} + ( - 3 \beta_{2} - \beta_1 + 4) q^{11} + 13 q^{13} + (3 \beta_{2} - 3) q^{15} + ( - 4 \beta_1 - 50) q^{17} + ( - 8 \beta_{2} + 3 \beta_1 - 33) q^{19} + (9 \beta_1 + 33) q^{21} + (4 \beta_{2} - 16 \beta_1 + 12) q^{23} + ( - 10 \beta_{2} - 12 \beta_1 + 41) q^{25} - 27 q^{27} + (4 \beta_{2} + 10 \beta_1 + 4) q^{29} + (2 \beta_{2} + 27 \beta_1 - 91) q^{31} + (9 \beta_{2} + 3 \beta_1 - 12) q^{33} + (2 \beta_{2} + 18 \beta_1 - 20) q^{35} + ( - 14 \beta_{2} - 24 \beta_1 + 112) q^{37} - 39 q^{39} + (17 \beta_{2} - 2 \beta_1 + 165) q^{41} + ( - 2 \beta_{2} + 30 \beta_1 + 96) q^{43} + ( - 9 \beta_{2} + 9) q^{45} + (21 \beta_{2} - 27 \beta_1 + 6) q^{47} + (18 \beta_{2} + 84 \beta_1 + 183) q^{49} + (12 \beta_1 + 150) q^{51} + ( - 6 \beta_{2} - 54 \beta_1 - 246) q^{53} + ( - 34 \beta_{2} - 30 \beta_1 + 496) q^{55} + (24 \beta_{2} - 9 \beta_1 + 99) q^{57} + ( - \beta_{2} - 41 \beta_1 + 582) q^{59} + ( - 14 \beta_{2} + 12 \beta_1 + 76) q^{61} + ( - 27 \beta_1 - 99) q^{63} + ( - 13 \beta_{2} + 13) q^{65} + (38 \beta_{2} - 21 \beta_1 - 19) q^{67} + ( - 12 \beta_{2} + 48 \beta_1 - 36) q^{69} + ( - 7 \beta_{2} + 67 \beta_1 + 336) q^{71} + (6 \beta_{2} - 120 \beta_1 - 112) q^{73} + (30 \beta_{2} + 36 \beta_1 - 123) q^{75} + (12 \beta_{2} + 68 \beta_1 + 64) q^{77} + (12 \beta_{2} - 24 \beta_1 + 4) q^{79} + 81 q^{81} + (5 \beta_{2} - 15 \beta_1 + 262) q^{83} + (38 \beta_{2} + 24 \beta_1 - 62) q^{85} + ( - 12 \beta_{2} - 30 \beta_1 - 12) q^{87} + (15 \beta_{2} + 58 \beta_1 + 503) q^{89} + ( - 39 \beta_1 - 143) q^{91} + ( - 6 \beta_{2} - 81 \beta_1 + 273) q^{93} + ( - 30 \beta_{2} - 114 \beta_1 + 1296) q^{95} + ( - 2 \beta_{2} - 24 \beta_1 + 1072) q^{97} + ( - 27 \beta_{2} - 9 \beta_1 + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9} + 16 q^{11} + 39 q^{13} - 12 q^{15} - 146 q^{17} - 94 q^{19} + 90 q^{21} + 48 q^{23} + 145 q^{25} - 81 q^{27} - 2 q^{29} - 302 q^{31} - 48 q^{33} - 80 q^{35} + 374 q^{37} - 117 q^{39} + 480 q^{41} + 260 q^{43} + 36 q^{45} + 24 q^{47} + 447 q^{49} + 438 q^{51} - 678 q^{53} + 1552 q^{55} + 282 q^{57} + 1788 q^{59} + 230 q^{61} - 270 q^{63} + 52 q^{65} - 74 q^{67} - 144 q^{69} + 948 q^{71} - 222 q^{73} - 435 q^{75} + 112 q^{77} + 24 q^{79} + 243 q^{81} + 796 q^{83} - 248 q^{85} + 6 q^{87} + 1436 q^{89} - 390 q^{91} + 906 q^{93} + 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 16x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 23 ) / 2 \) 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
−3.20905
4.73549
−0.526440
0 −3.00000 0 −11.4322 0 11.2543 0 9.00000 0
1.2 0 −3.00000 0 −3.90776 0 −36.4129 0 9.00000 0
1.3 0 −3.00000 0 19.3400 0 −4.84136 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(-1\)

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 624.4.a.t 3
3.b odd 2 1 1872.4.a.bk 3
4.b odd 2 1 39.4.a.c 3
8.b even 2 1 2496.4.a.bp 3
8.d odd 2 1 2496.4.a.bl 3
12.b even 2 1 117.4.a.f 3
20.d odd 2 1 975.4.a.l 3
28.d even 2 1 1911.4.a.k 3
52.b odd 2 1 507.4.a.h 3
52.f even 4 2 507.4.b.g 6
156.h even 2 1 1521.4.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 4.b odd 2 1
117.4.a.f 3 12.b even 2 1
507.4.a.h 3 52.b odd 2 1
507.4.b.g 6 52.f even 4 2
624.4.a.t 3 1.a even 1 1 trivial
975.4.a.l 3 20.d odd 2 1
1521.4.a.u 3 156.h even 2 1
1872.4.a.bk 3 3.b odd 2 1
1911.4.a.k 3 28.d even 2 1
2496.4.a.bl 3 8.d odd 2 1
2496.4.a.bp 3 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(624))\):

\( T_{5}^{3} - 4T_{5}^{2} - 252T_{5} - 864 \) Copy content Toggle raw display
\( T_{7}^{3} + 30T_{7}^{2} - 288T_{7} - 1984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 4 T^{2} - 252 T - 864 \) Copy content Toggle raw display
$7$ \( T^{3} + 30 T^{2} - 288 T - 1984 \) Copy content Toggle raw display
$11$ \( T^{3} - 16 T^{2} - 2256 T - 30336 \) Copy content Toggle raw display
$13$ \( (T - 13)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 146 T^{2} + 6060 T + 71256 \) Copy content Toggle raw display
$19$ \( T^{3} + 94 T^{2} - 14432 T - 779616 \) Copy content Toggle raw display
$23$ \( T^{3} - 48 T^{2} - 20928 T - 534528 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 10116 T - 199176 \) Copy content Toggle raw display
$31$ \( T^{3} + 302 T^{2} - 17536 T - 7197248 \) Copy content Toggle raw display
$37$ \( T^{3} - 374 T^{2} - 36964 T + 7758104 \) Copy content Toggle raw display
$41$ \( T^{3} - 480 T^{2} + \cdots + 12919824 \) Copy content Toggle raw display
$43$ \( T^{3} - 260 T^{2} - 38096 T + 3663168 \) Copy content Toggle raw display
$47$ \( T^{3} - 24 T^{2} - 168480 T - 18102528 \) Copy content Toggle raw display
$53$ \( T^{3} + 678 T^{2} - 42228 T - 1471608 \) Copy content Toggle raw display
$59$ \( T^{3} - 1788 T^{2} + \cdots - 137423808 \) Copy content Toggle raw display
$61$ \( T^{3} - 230 T^{2} - 44452 T + 6279512 \) Copy content Toggle raw display
$67$ \( T^{3} + 74 T^{2} - 409216 T - 4260896 \) Copy content Toggle raw display
$71$ \( T^{3} - 948 T^{2} + \cdots + 70464384 \) Copy content Toggle raw display
$73$ \( T^{3} + 222 T^{2} + \cdots + 22780552 \) Copy content Toggle raw display
$79$ \( T^{3} - 24 T^{2} - 78336 T - 7757824 \) Copy content Toggle raw display
$83$ \( T^{3} - 796 T^{2} + \cdots - 13963968 \) Copy content Toggle raw display
$89$ \( T^{3} - 1436 T^{2} + \cdots - 30129888 \) Copy content Toggle raw display
$97$ \( T^{3} - 3242 T^{2} + \cdots - 1218481048 \) Copy content Toggle raw display
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