Properties

Label 624.4.c.c
Level $624$
Weight $4$
Character orbit 624.c
Analytic conductor $36.817$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1362828.1
Defining polynomial: \( x^{4} + 23x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (\beta_{2} - \beta_1) q^{5} + (2 \beta_{2} - \beta_1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (\beta_{2} - \beta_1) q^{5} + (2 \beta_{2} - \beta_1) q^{7} + 9 q^{9} + ( - 3 \beta_{2} - 4 \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{13} + ( - 3 \beta_{2} + 3 \beta_1) q^{15} + ( - 3 \beta_{3} + 24) q^{17} + ( - 8 \beta_{2} - 5 \beta_1) q^{19} + ( - 6 \beta_{2} + 3 \beta_1) q^{21} + 72 q^{23} + (\beta_{3} - 5) q^{25} - 27 q^{27} + (3 \beta_{3} - 96) q^{29} + (20 \beta_{2} - 19 \beta_1) q^{31} + (9 \beta_{2} + 12 \beta_1) q^{33} - 216 q^{35} + (4 \beta_{2} + 34 \beta_1) q^{37} + (3 \beta_{3} + 3 \beta_{2} + 12 \beta_1 + 9) q^{39} + ( - 13 \beta_{2} - 33 \beta_1) q^{41} + ( - 3 \beta_{3} + 322) q^{43} + (9 \beta_{2} - 9 \beta_1) q^{45} + (\beta_{2} + 18 \beta_1) q^{47} + ( - 3 \beta_{3} - 47) q^{49} + (9 \beta_{3} - 72) q^{51} + (12 \beta_{3} + 246) q^{53} + (11 \beta_{3} + 82) q^{55} + (24 \beta_{2} + 15 \beta_1) q^{57} + ( - \beta_{2} - 26 \beta_1) q^{59} + (7 \beta_{3} + 72) q^{61} + (18 \beta_{2} - 9 \beta_1) q^{63} + (9 \beta_{3} - 17 \beta_{2} - 55 \beta_1 - 90) q^{65} + ( - 54 \beta_{2} - 27 \beta_1) q^{67} - 216 q^{69} + ( - 41 \beta_{2} + 84 \beta_1) q^{71} + ( - 42 \beta_{2} - 78 \beta_1) q^{73} + ( - 3 \beta_{3} + 15) q^{75} + (21 \beta_{3} + 354) q^{77} + ( - 4 \beta_{3} - 1080) q^{79} + 81 q^{81} + (63 \beta_{2} - 94 \beta_1) q^{83} + ( - 18 \beta_{2} - 198 \beta_1) q^{85} + ( - 9 \beta_{3} + 288) q^{87} + ( - 25 \beta_{2} - 69 \beta_1) q^{89} + (15 \beta_{3} - 50 \beta_{2} - 83 \beta_1 + 6) q^{91} + ( - 60 \beta_{2} + 57 \beta_1) q^{93} + (18 \beta_{3} + 468) q^{95} + ( - 14 \beta_{2} + 34 \beta_1) q^{97} + ( - 27 \beta_{2} - 36 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 23x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 46 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{2} - 19\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
4.54739i
1.52356i
1.52356i
4.54739i
0 −3.00000 0 12.9118i 0 16.7289i 0 9.00000 0
337.2 0 −3.00000 0 9.65841i 0 22.3639i 0 9.00000 0
337.3 0 −3.00000 0 9.65841i 0 22.3639i 0 9.00000 0
337.4 0 −3.00000 0 12.9118i 0 16.7289i 0 9.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.c.c 4
4.b odd 2 1 39.4.b.b 4
12.b even 2 1 117.4.b.e 4
13.b even 2 1 inner 624.4.c.c 4
52.b odd 2 1 39.4.b.b 4
52.f even 4 2 507.4.a.l 4
156.h even 2 1 117.4.b.e 4
156.l odd 4 2 1521.4.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.b 4 4.b odd 2 1
39.4.b.b 4 52.b odd 2 1
117.4.b.e 4 12.b even 2 1
117.4.b.e 4 156.h even 2 1
507.4.a.l 4 52.f even 4 2
624.4.c.c 4 1.a even 1 1 trivial
624.4.c.c 4 13.b even 2 1 inner
1521.4.a.w 4 156.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 260T_{5}^{2} + 15552 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 260 T^{2} + 15552 \) Copy content Toggle raw display
$7$ \( T^{4} + 780 T^{2} + 139968 \) Copy content Toggle raw display
$11$ \( T^{4} + 3152 T^{2} + \cdots + 1572528 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} - 962 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{2} - 48 T - 11556)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 13884 T^{2} + \cdots + 3048192 \) Copy content Toggle raw display
$23$ \( (T - 72)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 192 T - 2916)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 100572 T^{2} + \cdots + 2389782528 \) Copy content Toggle raw display
$37$ \( T^{4} + 110256 T^{2} + \cdots + 2060577792 \) Copy content Toggle raw display
$41$ \( T^{4} + 133364 T^{2} + \cdots + 4431055872 \) Copy content Toggle raw display
$43$ \( (T^{2} - 644 T + 91552)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 30128 T^{2} + \cdots + 116663088 \) Copy content Toggle raw display
$53$ \( (T^{2} - 492 T - 133596)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 62576 T^{2} + \cdots + 457419312 \) Copy content Toggle raw display
$61$ \( (T^{2} - 144 T - 60868)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 591948 T^{2} + \cdots + 918330048 \) Copy content Toggle raw display
$71$ \( T^{4} + 917456 T^{2} + \cdots + 59484058032 \) Copy content Toggle raw display
$73$ \( T^{4} + 896400 T^{2} + \cdots + 179358354432 \) Copy content Toggle raw display
$79$ \( (T^{2} + 2160 T + 1144832)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1464080 T^{2} + \cdots + 317023217328 \) Copy content Toggle raw display
$89$ \( T^{4} + 561812 T^{2} + \cdots + 78903164928 \) Copy content Toggle raw display
$97$ \( T^{4} + 137040 T^{2} + \cdots + 725594112 \) Copy content Toggle raw display
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