Newform orbit 624.2.cn.f

• Label: 624.2.cn.f
• Level: $624$
• Weight: $2$
• Character orbit: 624.cn
• Analytic conductor: $4.983$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

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:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
305.1		0		−1.68186	−	0.413931i		0		−1.44076	−	1.44076i		0		−0.918532	+	3.42801i		0		2.65732	+	1.39235i		0
305.2		0		−1.55432	−	0.764251i		0		−0.504580	−	0.504580i		0		0.836809	−	3.12301i		0		1.83184	+	2.37579i		0
305.3		0		−1.52267	+	0.825508i		0		1.33870	+	1.33870i		0		−0.566234	+	2.11322i		0		1.63707	−	2.51396i		0
305.4		0		−1.31722	+	1.12469i		0		1.72616	+	1.72616i		0		0.574711	−	2.14485i		0		0.470140	−	2.96293i		0
305.5		0		−1.19063	−	1.25794i		0		2.36656	+	2.36656i		0		0.332282	−	1.24009i		0		−0.164807	+	2.99547i		0
305.6		0		−0.315401	+	1.70309i		0		−1.72616	−	1.72616i		0		0.574711	−	2.14485i		0		−2.80104	−	1.07431i		0
305.7		0		0.0464265	+	1.73143i		0		−1.33870	−	1.33870i		0		−0.566234	+	2.11322i		0		−2.99569	+	0.160768i		0
305.8		0		0.121290	−	1.72780i		0		−2.19967	−	2.19967i		0		−1.17763	+	4.39498i		0		−2.97058	−	0.419131i		0
305.9		0		0.386983	−	1.68827i		0		1.56802	+	1.56802i		0		0.418596	−	1.56222i		0		−2.70049	−	1.30666i		0
305.10		0		1.19941	+	1.24957i		0		1.44076	+	1.44076i		0		−0.918532	+	3.42801i		0		−0.122851	+	2.99748i		0
305.11		0		1.26859	−	1.17927i		0		−1.56802	−	1.56802i		0		0.418596	−	1.56222i		0		0.218643	−	2.99202i		0
305.12		0		1.43567	−	0.968940i		0		2.19967	+	2.19967i		0		−1.17763	+	4.39498i		0		1.12231	−	2.78216i		0
305.13		0		1.43902	+	0.963958i		0		0.504580	+	0.504580i		0		0.836809	−	3.12301i		0		1.14157	+	2.77431i		0
305.14		0		1.68472	+	0.402146i		0		−2.36656	−	2.36656i		0		0.332282	−	1.24009i		0		2.67656	+	1.35501i		0
353.1		0		−1.72861	−	0.109042i		0		−0.190181	−	0.190181i		0		−3.71203	+	0.994635i		0		2.97622	+	0.376984i		0
353.2		0		−1.55814	−	0.756439i		0		−3.14772	−	3.14772i		0		−2.26593	+	0.607154i		0		1.85560	+	2.35728i		0
353.3		0		−1.51931	−	0.831691i		0		0.741654	+	0.741654i		0		4.39168	−	1.17675i		0		1.61658	+	2.52719i		0
353.4		0		−1.49406	+	0.876234i		0		1.23702	+	1.23702i		0		−0.738861	+	0.197977i		0		1.46443	−	2.61829i		0
353.5		0		−0.852797	+	1.50756i		0		0.477579	+	0.477579i		0		−0.988569	+	0.264886i		0		−1.54548	−	2.57128i		0
353.6		0		−0.411861	+	1.68237i		0		−1.70770	−	1.70770i		0		3.27107	−	0.876480i		0		−2.66074	−	1.38580i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
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.f		56
3.b	odd	2	1	inner	624.2.cn.f		56
4.b	odd	2	1		312.2.bp.a	✓	56
12.b	even	2	1		312.2.bp.a	✓	56
13.f	odd	12	1	inner	624.2.cn.f		56
39.k	even	12	1	inner	624.2.cn.f		56
52.l	even	12	1		312.2.bp.a	✓	56
156.v	odd	12	1		312.2.bp.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.bp.a	✓	56	4.b	odd	2	1
312.2.bp.a	✓	56	12.b	even	2	1
312.2.bp.a	✓	56	52.l	even	12	1
312.2.bp.a	✓	56	156.v	odd	12	1
624.2.cn.f		56	1.a	even	1	1	trivial
624.2.cn.f		56	3.b	odd	2	1	inner
624.2.cn.f		56	13.f	odd	12	1	inner
624.2.cn.f		56	39.k	even	12	1	inner

Hecke kernels

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

T5^56 + 840*T5^52 + 257308*T5^48 + 39993864*T5^44 + 3564261030*T5^40 + 192504391400*T5^36 + 6462932619964*T5^32 + 135083727890104*T5^28 + 1713686471440033*T5^24 + 12348325923944336*T5^20 + 43446127204687456*T5^16 + 50140526998507776*T5^12 + 17065654164930816*T5^8 + 1791854982070272*T5^4 + 8916100448256

T7^28 + 2*T7^27 + 5*T7^26 - 68*T7^25 - 530*T7^24 - 470*T7^23 + 1697*T7^22 + 25116*T7^21 + 137540*T7^20 + 204510*T7^19 + 351485*T7^18 - 2381184*T7^17 - 13035585*T7^16 - 24832044*T7^15 - 26434980*T7^14 + 144453200*T7^13 + 915125414*T7^12 + 2916289584*T7^11 + 7870274592*T7^10 + 16308578688*T7^9 + 28327747468*T7^8 + 43859146704*T7^7 + 56231985120*T7^6 + 59866181184*T7^5 + 55601218320*T7^4 + 41482606464*T7^3 + 20964115584*T7^2 + 6023115648*T7 + 734193216