Newform orbit 624.6.a.d

• Label: 624.6.a.d
• Level: $624$
• Weight: $6$
• Character orbit: 624.a
• Self dual: yes
• Analytic conductor: $100.080$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$624 = 2^{4} \cdot 3 \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$100.079503563$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 9 * q^3 + 76 * q^5 - 100 * q^7 + 81 * q^9 + 106 * q^11 - 169 * q^13 - 684 * q^15 + 234 * q^17 + 276 * q^19 + 900 * q^21 - 2548 * q^23 + 2651 * q^25 - 729 * q^27 + 8266 * q^29 + 608 * q^31 - 954 * q^33 - 7600 * q^35 - 2010 * q^37 + 1521 * q^39 + 8844 * q^41 + 17636 * q^43 + 6156 * q^45 - 18770 * q^47 - 6807 * q^49 - 2106 * q^51 - 26970 * q^53 + 8056 * q^55 - 2484 * q^57 + 41966 * q^59 + 778 * q^61 - 8100 * q^63 - 12844 * q^65 + 12632 * q^67 + 22932 * q^69 - 40466 * q^71 + 54302 * q^73 - 23859 * q^75 - 10600 * q^77 + 44656 * q^79 + 6561 * q^81 - 69918 * q^83 + 17784 * q^85 - 74394 * q^87 - 44520 * q^89 + 16900 * q^91 - 5472 * q^93 + 20976 * q^95 - 86026 * q^97 + 8586 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

0

0

−9.00000

0

76.0000

0

−100.000

0

81.0000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$13$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	624.6.a.d		1
4.b	odd	2	1		78.6.a.c	✓	1
12.b	even	2	1		234.6.a.d		1
52.b	odd	2	1		1014.6.a.f		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.6.a.c	✓	1	4.b	odd	2	1
234.6.a.d		1	12.b	even	2	1
624.6.a.d		1	1.a	even	1	1	trivial
1014.6.a.f		1	52.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(624))$:

$T_{5} - 76$

$T_{7} + 100$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T + 9$
$5$	$T - 76$
$7$	$T + 100$
$11$	$T - 106$