Properties

Label 624.2.bv.e
Level $624$
Weight $2$
Character orbit 624.bv
Analytic conductor $4.983$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{3} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12}^{2} q^{3} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( 2 - 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{11} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{13} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{15} + ( -4 - \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{17} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{19} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{21} + ( -3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{25} - q^{27} + ( 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{29} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( 1 - 3 \zeta_{12} + \zeta_{12}^{2} ) q^{33} + ( -5 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{35} + ( -4 + 7 \zeta_{12} + 2 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{37} + ( \zeta_{12} - 4 \zeta_{12}^{3} ) q^{39} + ( 12 - \zeta_{12} - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{41} + ( 1 + 5 \zeta_{12} - \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{43} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{45} + ( 3 - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{47} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{49} + ( -4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{55} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{57} + 8 \zeta_{12} q^{59} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( 2 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{63} + ( -8 - 5 \zeta_{12} + 6 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{65} + ( 14 + \zeta_{12} - 7 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{67} + ( -1 + 3 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{69} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{71} + ( -1 + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{73} + ( -4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{75} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} + ( 3 - 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{83} + ( 6 + 11 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{85} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{87} + ( -4 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{89} + ( -1 + 2 \zeta_{12} + 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{91} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{93} + ( -9 - 5 \zeta_{12} + 9 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{95} -6 \zeta_{12} q^{97} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 6q^{7} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 6q^{7} - 2q^{9} + 6q^{11} + 6q^{15} - 8q^{17} - 6q^{19} + 2q^{23} - 8q^{25} - 4q^{27} + 2q^{29} + 6q^{33} - 10q^{35} - 12q^{37} + 36q^{41} + 2q^{43} + 6q^{45} - 6q^{49} - 16q^{51} - 12q^{53} + 6q^{55} + 8q^{61} + 6q^{63} - 20q^{65} + 42q^{67} - 2q^{69} - 6q^{71} - 4q^{75} + 24q^{79} - 2q^{81} + 36q^{85} - 2q^{87} - 12q^{89} + 4q^{91} - 12q^{93} - 18q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(1 - \zeta_{12}^{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0.500000 + 0.866025i 0 3.73205i 0 −2.36603 1.36603i 0 −0.500000 + 0.866025i 0
49.2 0 0.500000 + 0.866025i 0 0.267949i 0 −0.633975 0.366025i 0 −0.500000 + 0.866025i 0
433.1 0 0.500000 0.866025i 0 0.267949i 0 −0.633975 + 0.366025i 0 −0.500000 0.866025i 0
433.2 0 0.500000 0.866025i 0 3.73205i 0 −2.36603 + 1.36603i 0 −0.500000 0.866025i 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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bv.e 4
3.b odd 2 1 1872.2.by.h 4
4.b odd 2 1 78.2.i.a 4
12.b even 2 1 234.2.l.c 4
13.e even 6 1 inner 624.2.bv.e 4
13.f odd 12 1 8112.2.a.bj 2
13.f odd 12 1 8112.2.a.bp 2
20.d odd 2 1 1950.2.bc.d 4
20.e even 4 1 1950.2.y.b 4
20.e even 4 1 1950.2.y.g 4
39.h odd 6 1 1872.2.by.h 4
52.b odd 2 1 1014.2.i.a 4
52.f even 4 1 1014.2.e.g 4
52.f even 4 1 1014.2.e.i 4
52.i odd 6 1 78.2.i.a 4
52.i odd 6 1 1014.2.b.e 4
52.j odd 6 1 1014.2.b.e 4
52.j odd 6 1 1014.2.i.a 4
52.l even 12 1 1014.2.a.i 2
52.l even 12 1 1014.2.a.k 2
52.l even 12 1 1014.2.e.g 4
52.l even 12 1 1014.2.e.i 4
156.p even 6 1 3042.2.b.i 4
156.r even 6 1 234.2.l.c 4
156.r even 6 1 3042.2.b.i 4
156.v odd 12 1 3042.2.a.p 2
156.v odd 12 1 3042.2.a.y 2
260.w odd 6 1 1950.2.bc.d 4
260.bg even 12 1 1950.2.y.b 4
260.bg even 12 1 1950.2.y.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 4.b odd 2 1
78.2.i.a 4 52.i odd 6 1
234.2.l.c 4 12.b even 2 1
234.2.l.c 4 156.r even 6 1
624.2.bv.e 4 1.a even 1 1 trivial
624.2.bv.e 4 13.e even 6 1 inner
1014.2.a.i 2 52.l even 12 1
1014.2.a.k 2 52.l even 12 1
1014.2.b.e 4 52.i odd 6 1
1014.2.b.e 4 52.j odd 6 1
1014.2.e.g 4 52.f even 4 1
1014.2.e.g 4 52.l even 12 1
1014.2.e.i 4 52.f even 4 1
1014.2.e.i 4 52.l even 12 1
1014.2.i.a 4 52.b odd 2 1
1014.2.i.a 4 52.j odd 6 1
1872.2.by.h 4 3.b odd 2 1
1872.2.by.h 4 39.h odd 6 1
1950.2.y.b 4 20.e even 4 1
1950.2.y.b 4 260.bg even 12 1
1950.2.y.g 4 20.e even 4 1
1950.2.y.g 4 260.bg even 12 1
1950.2.bc.d 4 20.d odd 2 1
1950.2.bc.d 4 260.w odd 6 1
3042.2.a.p 2 156.v odd 12 1
3042.2.a.y 2 156.v odd 12 1
3042.2.b.i 4 156.p even 6 1
3042.2.b.i 4 156.r even 6 1
8112.2.a.bj 2 13.f odd 12 1
8112.2.a.bp 2 13.f odd 12 1

Hecke kernels

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

\( T_{5}^{4} + 14 T_{5}^{2} + 1 \)
\( T_{7}^{4} + 6 T_{7}^{3} + 14 T_{7}^{2} + 12 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 1 + 14 T^{2} + T^{4} \)
$7$ \( 4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( 36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4} \)
$13$ \( 169 - T^{2} + T^{4} \)
$17$ \( 169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( 121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( 64 + 32 T^{2} + T^{4} \)
$37$ \( 1369 - 444 T + 11 T^{2} + 12 T^{3} + T^{4} \)
$41$ \( 11449 - 3852 T + 539 T^{2} - 36 T^{3} + T^{4} \)
$43$ \( 5476 + 148 T + 78 T^{2} - 2 T^{3} + T^{4} \)
$47$ \( 324 + 72 T^{2} + T^{4} \)
$53$ \( ( -3 + 6 T + T^{2} )^{2} \)
$59$ \( 4096 - 64 T^{2} + T^{4} \)
$61$ \( 121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 21316 - 6132 T + 734 T^{2} - 42 T^{3} + T^{4} \)
$71$ \( 36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( 3721 + 134 T^{2} + T^{4} \)
$79$ \( ( 24 - 12 T + T^{2} )^{2} \)
$83$ \( 4 + 104 T^{2} + T^{4} \)
$89$ \( 576 - 288 T + 24 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 1296 - 36 T^{2} + T^{4} \)
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