Properties

Label 624.2.cn.b
Level $624$
Weight $2$
Character orbit 624.cn
Analytic conductor $4.983$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.cn (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(4\)
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 39)
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 + 2 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 + 2 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 3 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + ( -5 + 5 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( 4 + \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{21} -5 \zeta_{12}^{3} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 1 - 5 \zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{31} + ( -4 - 3 \zeta_{12} + 7 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{37} + ( 5 + 2 \zeta_{12}^{2} ) q^{39} + ( 1 + \zeta_{12}^{2} ) q^{43} + ( 6 + 7 \zeta_{12} - 3 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} + ( -1 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{57} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{61} + ( -9 + 3 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{63} + ( -7 - 9 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( -9 + \zeta_{12} + \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{73} + ( 10 - 5 \zeta_{12}^{2} ) q^{75} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 5 + 10 \zeta_{12} - 11 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{91} + ( 7 - 4 \zeta_{12} - 11 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{93} + ( -8 + 8 \zeta_{12} - 3 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 10q^{7} - 6q^{9} - 14q^{19} + 18q^{21} + 14q^{31} - 2q^{37} + 24q^{39} + 6q^{43} + 18q^{49} + 12q^{57} - 24q^{63} - 32q^{67} - 34q^{73} + 30q^{75} - 18q^{81} - 2q^{91} + 6q^{93} - 38q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 −0.866025 + 1.50000i 0 0 0 0.767949 2.86603i 0 −1.50000 2.59808i 0
353.1 0 0.866025 + 1.50000i 0 0 0 4.23205 1.13397i 0 −1.50000 + 2.59808i 0
401.1 0 −0.866025 1.50000i 0 0 0 0.767949 + 2.86603i 0 −1.50000 + 2.59808i 0
449.1 0 0.866025 1.50000i 0 0 0 4.23205 + 1.13397i 0 −1.50000 2.59808i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cn.b 4
3.b odd 2 1 CM 624.2.cn.b 4
4.b odd 2 1 39.2.k.a 4
12.b even 2 1 39.2.k.a 4
13.f odd 12 1 inner 624.2.cn.b 4
20.d odd 2 1 975.2.bo.c 4
20.e even 4 1 975.2.bp.a 4
20.e even 4 1 975.2.bp.d 4
39.k even 12 1 inner 624.2.cn.b 4
52.b odd 2 1 507.2.k.c 4
52.f even 4 1 507.2.k.a 4
52.f even 4 1 507.2.k.b 4
52.i odd 6 1 507.2.f.b 4
52.i odd 6 1 507.2.k.a 4
52.j odd 6 1 507.2.f.c 4
52.j odd 6 1 507.2.k.b 4
52.l even 12 1 39.2.k.a 4
52.l even 12 1 507.2.f.b 4
52.l even 12 1 507.2.f.c 4
52.l even 12 1 507.2.k.c 4
60.h even 2 1 975.2.bo.c 4
60.l odd 4 1 975.2.bp.a 4
60.l odd 4 1 975.2.bp.d 4
156.h even 2 1 507.2.k.c 4
156.l odd 4 1 507.2.k.a 4
156.l odd 4 1 507.2.k.b 4
156.p even 6 1 507.2.f.c 4
156.p even 6 1 507.2.k.b 4
156.r even 6 1 507.2.f.b 4
156.r even 6 1 507.2.k.a 4
156.v odd 12 1 39.2.k.a 4
156.v odd 12 1 507.2.f.b 4
156.v odd 12 1 507.2.f.c 4
156.v odd 12 1 507.2.k.c 4
260.bc even 12 1 975.2.bo.c 4
260.be odd 12 1 975.2.bp.a 4
260.bl odd 12 1 975.2.bp.d 4
780.cf even 12 1 975.2.bp.a 4
780.cr odd 12 1 975.2.bo.c 4
780.cy even 12 1 975.2.bp.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 4.b odd 2 1
39.2.k.a 4 12.b even 2 1
39.2.k.a 4 52.l even 12 1
39.2.k.a 4 156.v odd 12 1
507.2.f.b 4 52.i odd 6 1
507.2.f.b 4 52.l even 12 1
507.2.f.b 4 156.r even 6 1
507.2.f.b 4 156.v odd 12 1
507.2.f.c 4 52.j odd 6 1
507.2.f.c 4 52.l even 12 1
507.2.f.c 4 156.p even 6 1
507.2.f.c 4 156.v odd 12 1
507.2.k.a 4 52.f even 4 1
507.2.k.a 4 52.i odd 6 1
507.2.k.a 4 156.l odd 4 1
507.2.k.a 4 156.r even 6 1
507.2.k.b 4 52.f even 4 1
507.2.k.b 4 52.j odd 6 1
507.2.k.b 4 156.l odd 4 1
507.2.k.b 4 156.p even 6 1
507.2.k.c 4 52.b odd 2 1
507.2.k.c 4 52.l even 12 1
507.2.k.c 4 156.h even 2 1
507.2.k.c 4 156.v odd 12 1
624.2.cn.b 4 1.a even 1 1 trivial
624.2.cn.b 4 3.b odd 2 1 CM
624.2.cn.b 4 13.f odd 12 1 inner
624.2.cn.b 4 39.k even 12 1 inner
975.2.bo.c 4 20.d odd 2 1
975.2.bo.c 4 60.h even 2 1
975.2.bo.c 4 260.bc even 12 1
975.2.bo.c 4 780.cr odd 12 1
975.2.bp.a 4 20.e even 4 1
975.2.bp.a 4 60.l odd 4 1
975.2.bp.a 4 260.be odd 12 1
975.2.bp.a 4 780.cf even 12 1
975.2.bp.d 4 20.e even 4 1
975.2.bp.d 4 60.l odd 4 1
975.2.bp.d 4 260.bl odd 12 1
975.2.bp.d 4 780.cy even 12 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{4} - 10 T_{7}^{3} + 41 T_{7}^{2} - 104 T_{7} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 169 - 104 T + 41 T^{2} - 10 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 169 - T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 676 + 52 T + 50 T^{2} + 14 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 169 + 182 T + 98 T^{2} - 14 T^{3} + T^{4} \)
$37$ \( 676 - 572 T + 122 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 3 - 3 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( 5625 + 75 T^{2} + T^{4} \)
$67$ \( 169 + 130 T + 281 T^{2} + 32 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 20449 + 4862 T + 578 T^{2} + 34 T^{3} + T^{4} \)
$79$ \( ( -147 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 28561 + 4732 T + 557 T^{2} + 38 T^{3} + T^{4} \)
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