Properties

Label 624.4.c.e
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: 4.0.5054412.1
Defining polynomial: \( x^{4} + 29x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + 3 q^{3} - \beta_{2} q^{5} + 3 \beta_1 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - \beta_{2} q^{5} + 3 \beta_1 q^{7} + 9 q^{9} + ( - \beta_{2} + \beta_1) q^{11} + (\beta_{3} + 3 \beta_{2} - 18) q^{13} - 3 \beta_{2} q^{15} + 54 q^{17} + ( - 6 \beta_{2} + 3 \beta_1) q^{19} + 9 \beta_1 q^{21} + (6 \beta_{3} - 30) q^{23} + ( - 4 \beta_{3} - 11) q^{25} + 27 q^{27} + (6 \beta_{3} - 12) q^{29} + ( - 6 \beta_{2} - 9 \beta_1) q^{31} + ( - 3 \beta_{2} + 3 \beta_1) q^{33} + (6 \beta_{3} - 30) q^{35} + ( - 24 \beta_{2} - 24 \beta_1) q^{37} + (3 \beta_{3} + 9 \beta_{2} - 54) q^{39} + (13 \beta_{2} + 2 \beta_1) q^{41} + ( - 6 \beta_{3} - 266) q^{43} - 9 \beta_{2} q^{45} + ( - 21 \beta_{2} - 61 \beta_1) q^{47} + (18 \beta_{3} - 179) q^{49} + 162 q^{51} + ( - 18 \beta_{3} - 216) q^{53} + ( - 2 \beta_{3} - 146) q^{55} + ( - 18 \beta_{2} + 9 \beta_1) q^{57} + ( - 15 \beta_{2} + 17 \beta_1) q^{59} + ( - 4 \beta_{3} - 570) q^{61} + 27 \beta_1 q^{63} + (12 \beta_{3} - 3 \beta_{2} - 26 \beta_1 + 408) q^{65} + ( - 12 \beta_{2} + 75 \beta_1) q^{67} + (18 \beta_{3} - 90) q^{69} + ( - 23 \beta_{2} + 3 \beta_1) q^{71} - 54 \beta_{2} q^{73} + ( - 12 \beta_{3} - 33) q^{75} + (12 \beta_{3} - 204) q^{77} + (16 \beta_{3} - 72) q^{79} + 81 q^{81} + (25 \beta_{2} + 69 \beta_1) q^{83} - 54 \beta_{2} q^{85} + (18 \beta_{3} - 36) q^{87} + ( - 15 \beta_{2} - 2 \beta_1) q^{89} + ( - 18 \beta_{3} + 24 \beta_{2} - 117 \beta_1 + 90) q^{91} + ( - 18 \beta_{2} - 27 \beta_1) q^{93} + ( - 18 \beta_{3} - 846) q^{95} + (54 \beta_{2} + 168 \beta_1) q^{97} + ( - 9 \beta_{2} + 9 \beta_1) q^{99}+O(q^{100}) \) 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 \( 4 q + 12 q^{3} + 36 q^{9} - 72 q^{13} + 216 q^{17} - 120 q^{23} - 44 q^{25} + 108 q^{27} - 48 q^{29} - 120 q^{35} - 216 q^{39} - 1064 q^{43} - 716 q^{49} + 648 q^{51} - 864 q^{53} - 584 q^{55} - 2280 q^{61} + 1632 q^{65} - 360 q^{69} - 132 q^{75} - 816 q^{77} - 288 q^{79} + 324 q^{81} - 144 q^{87} + 360 q^{91} - 3384 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 29x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 25\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 29 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{2} - 25\beta_1 ) / 2 \) 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
1.32750i
5.21898i
5.21898i
1.32750i
0 3.00000 0 15.4241i 0 7.96501i 0 9.00000 0
337.2 0 3.00000 0 5.83936i 0 31.3139i 0 9.00000 0
337.3 0 3.00000 0 5.83936i 0 31.3139i 0 9.00000 0
337.4 0 3.00000 0 15.4241i 0 7.96501i 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.e 4
4.b odd 2 1 39.4.b.a 4
12.b even 2 1 117.4.b.d 4
13.b even 2 1 inner 624.4.c.e 4
52.b odd 2 1 39.4.b.a 4
52.f even 4 2 507.4.a.j 4
156.h even 2 1 117.4.b.d 4
156.l odd 4 2 1521.4.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.a 4 4.b odd 2 1
39.4.b.a 4 52.b odd 2 1
117.4.b.d 4 12.b even 2 1
117.4.b.d 4 156.h even 2 1
507.4.a.j 4 52.f even 4 2
624.4.c.e 4 1.a even 1 1 trivial
624.4.c.e 4 13.b even 2 1 inner
1521.4.a.x 4 156.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 272T_{5}^{2} + 8112 \) 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
$5$ \( T^{4} + 272T^{2} + 8112 \) Copy content Toggle raw display
$7$ \( T^{4} + 1044 T^{2} + 62208 \) Copy content Toggle raw display
$11$ \( T^{4} + 428 T^{2} + 43200 \) Copy content Toggle raw display
$13$ \( T^{4} + 72 T^{3} + 3094 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T - 54)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 11556 T^{2} + \cdots + 31492800 \) Copy content Toggle raw display
$23$ \( (T^{2} + 60 T - 22464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T - 23220)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 17028 T^{2} + \cdots + 47044800 \) Copy content Toggle raw display
$37$ \( T^{4} + 200448 T^{2} + \cdots + 2293235712 \) Copy content Toggle raw display
$41$ \( T^{4} + 45392 T^{2} + \cdots + 128314800 \) Copy content Toggle raw display
$43$ \( (T^{2} + 532 T + 47392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 500348 T^{2} + \cdots + 62387841792 \) Copy content Toggle raw display
$53$ \( (T^{2} + 432 T - 163620)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 104924 T^{2} + \cdots + 2436066048 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1140 T + 314516)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 727668 T^{2} + \cdots + 143327232 \) Copy content Toggle raw display
$71$ \( T^{4} + 147692 T^{2} + \cdots + 3298756800 \) Copy content Toggle raw display
$73$ \( T^{4} + 793152 T^{2} + \cdots + 68976790272 \) Copy content Toggle raw display
$79$ \( (T^{2} + 144 T - 160960)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 653276 T^{2} + \cdots + 106682723328 \) Copy content Toggle raw display
$89$ \( T^{4} + 60464 T^{2} + \cdots + 249304368 \) Copy content Toggle raw display
$97$ \( T^{4} + 3704256 T^{2} + \cdots + 3383532000000 \) Copy content Toggle raw display
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