Properties

Label 624.2.c.e
Level $624$
Weight $2$
Character orbit 624.c
Analytic conductor $4.983$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -2 + 4 \zeta_{6} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -2 + 4 \zeta_{6} ) q^{7} + q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -3 + 4 \zeta_{6} ) q^{13} -6 q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + ( -2 + 4 \zeta_{6} ) q^{21} + 5 q^{25} + q^{27} + 6 q^{29} + ( -2 + 4 \zeta_{6} ) q^{31} + ( -2 + 4 \zeta_{6} ) q^{33} + ( -4 + 8 \zeta_{6} ) q^{37} + ( -3 + 4 \zeta_{6} ) q^{39} + ( 4 - 8 \zeta_{6} ) q^{41} + 4 q^{43} + ( 2 - 4 \zeta_{6} ) q^{47} -5 q^{49} -6 q^{51} + 6 q^{53} + ( 2 - 4 \zeta_{6} ) q^{57} + ( 6 - 12 \zeta_{6} ) q^{59} -2 q^{61} + ( -2 + 4 \zeta_{6} ) q^{63} + ( -6 + 12 \zeta_{6} ) q^{67} + ( 2 - 4 \zeta_{6} ) q^{71} + 5 q^{75} -12 q^{77} + 8 q^{79} + q^{81} + ( -2 + 4 \zeta_{6} ) q^{83} + 6 q^{87} + ( 4 - 8 \zeta_{6} ) q^{89} + ( -10 - 4 \zeta_{6} ) q^{91} + ( -2 + 4 \zeta_{6} ) q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{9} - 2q^{13} - 12q^{17} + 10q^{25} + 2q^{27} + 12q^{29} - 2q^{39} + 8q^{43} - 10q^{49} - 12q^{51} + 12q^{53} - 4q^{61} + 10q^{75} - 24q^{77} + 16q^{79} + 2q^{81} + 12q^{87} - 24q^{91} + O(q^{100}) \)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 0 0 0 3.46410i 0 1.00000 0
337.2 0 1.00000 0 0 0 3.46410i 0 1.00000 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 624.2.c.e 2
3.b odd 2 1 1872.2.c.e 2
4.b odd 2 1 39.2.b.a 2
8.b even 2 1 2496.2.c.d 2
8.d odd 2 1 2496.2.c.k 2
12.b even 2 1 117.2.b.a 2
13.b even 2 1 inner 624.2.c.e 2
13.d odd 4 2 8112.2.a.bv 2
20.d odd 2 1 975.2.b.d 2
20.e even 4 2 975.2.h.f 4
28.d even 2 1 1911.2.c.d 2
39.d odd 2 1 1872.2.c.e 2
52.b odd 2 1 39.2.b.a 2
52.f even 4 2 507.2.a.f 2
52.i odd 6 1 507.2.j.a 2
52.i odd 6 1 507.2.j.c 2
52.j odd 6 1 507.2.j.a 2
52.j odd 6 1 507.2.j.c 2
52.l even 12 4 507.2.e.e 4
104.e even 2 1 2496.2.c.d 2
104.h odd 2 1 2496.2.c.k 2
156.h even 2 1 117.2.b.a 2
156.l odd 4 2 1521.2.a.l 2
260.g odd 2 1 975.2.b.d 2
260.p even 4 2 975.2.h.f 4
364.h even 2 1 1911.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 4.b odd 2 1
39.2.b.a 2 52.b odd 2 1
117.2.b.a 2 12.b even 2 1
117.2.b.a 2 156.h even 2 1
507.2.a.f 2 52.f even 4 2
507.2.e.e 4 52.l even 12 4
507.2.j.a 2 52.i odd 6 1
507.2.j.a 2 52.j odd 6 1
507.2.j.c 2 52.i odd 6 1
507.2.j.c 2 52.j odd 6 1
624.2.c.e 2 1.a even 1 1 trivial
624.2.c.e 2 13.b even 2 1 inner
975.2.b.d 2 20.d odd 2 1
975.2.b.d 2 260.g odd 2 1
975.2.h.f 4 20.e even 4 2
975.2.h.f 4 260.p even 4 2
1521.2.a.l 2 156.l odd 4 2
1872.2.c.e 2 3.b odd 2 1
1872.2.c.e 2 39.d odd 2 1
1911.2.c.d 2 28.d even 2 1
1911.2.c.d 2 364.h even 2 1
2496.2.c.d 2 8.b even 2 1
2496.2.c.d 2 104.e even 2 1
2496.2.c.k 2 8.d odd 2 1
2496.2.c.k 2 104.h odd 2 1
8112.2.a.bv 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 12 \)
\( T_{11}^{2} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 12 + T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 12 + T^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( 48 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 12 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 108 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 108 + T^{2} \)
$71$ \( 12 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 48 + T^{2} \)
$97$ \( 192 + T^{2} \)
show more
show less