Newform orbit 300.5.f.c

• Label: 300.5.f.c
• Level: $300$
• Weight: $5$
• Character orbit: 300.f
• Analytic conductor: $31.011$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	300.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.0109889252\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{4} + 36 q^{6} + 864 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
199.1	−3.93366	−	0.725488i	5.19615			14.9473	+	5.70765i		0		−20.4399	−	3.76975i	−36.7329			−54.6569	−	33.2960i	27.0000				0
199.2	−3.93366	+	0.725488i	5.19615			14.9473	−	5.70765i		0		−20.4399	+	3.76975i	−36.7329			−54.6569	+	33.2960i	27.0000				0
199.3	−3.65760	−	1.61924i	−5.19615			10.7561	+	11.8451i		0		19.0055	+	8.41384i	95.1090			−20.1614	−	60.7414i	27.0000				0
199.4	−3.65760	+	1.61924i	−5.19615			10.7561	−	11.8451i		0		19.0055	−	8.41384i	95.1090			−20.1614	+	60.7414i	27.0000				0
199.5	−3.53130	−	1.87881i	−5.19615			8.94016	+	13.2693i		0		18.3492	+	9.76257i	−7.86734			−6.63999	−	63.6546i	27.0000				0
199.6	−3.53130	+	1.87881i	−5.19615			8.94016	−	13.2693i		0		18.3492	−	9.76257i	−7.86734			−6.63999	+	63.6546i	27.0000				0
199.7	−2.85111	−	2.80556i	5.19615			0.257663	+	15.9979i		0		−14.8148	−	14.5781i	63.6232			44.1485	−	46.3347i	27.0000				0
199.8	−2.85111	+	2.80556i	5.19615			0.257663	−	15.9979i		0		−14.8148	+	14.5781i	63.6232			44.1485	+	46.3347i	27.0000				0
199.9	−2.65655	−	2.99044i	−5.19615			−1.88545	+	15.8885i		0		13.8039	+	15.5388i	−74.3539			52.5224	−	36.5704i	27.0000				0
199.10	−2.65655	+	2.99044i	−5.19615			−1.88545	−	15.8885i		0		13.8039	−	15.5388i	−74.3539			52.5224	+	36.5704i	27.0000				0
199.11	−2.16777	−	3.36166i	5.19615			−6.60152	+	14.5746i		0		−11.2641	−	17.4677i	−36.6738			63.3056	−	9.40241i	27.0000				0
199.12	−2.16777	+	3.36166i	5.19615			−6.60152	−	14.5746i		0		−11.2641	+	17.4677i	−36.6738			63.3056	+	9.40241i	27.0000				0
199.13	−0.889027	−	3.89995i	−5.19615			−14.4193	+	6.93433i		0		4.61952	+	20.2647i	−44.0991			39.8627	+	50.0696i	27.0000				0
199.14	−0.889027	+	3.89995i	−5.19615			−14.4193	−	6.93433i		0		4.61952	−	20.2647i	−44.0991			39.8627	−	50.0696i	27.0000				0
199.15	−0.0498899	−	3.99969i	5.19615			−15.9950	+	0.399088i		0		−0.259236	−	20.7830i	−35.2842			2.39422	+	63.9552i	27.0000				0
199.16	−0.0498899	+	3.99969i	5.19615			−15.9950	−	0.399088i		0		−0.259236	+	20.7830i	−35.2842			2.39422	−	63.9552i	27.0000				0
199.17	0.0498899	−	3.99969i	−5.19615			−15.9950	−	0.399088i		0		−0.259236	+	20.7830i	35.2842			−2.39422	+	63.9552i	27.0000				0
199.18	0.0498899	+	3.99969i	−5.19615			−15.9950	+	0.399088i		0		−0.259236	−	20.7830i	35.2842			−2.39422	−	63.9552i	27.0000				0
199.19	0.889027	−	3.89995i	5.19615			−14.4193	−	6.93433i		0		4.61952	−	20.2647i	44.0991			−39.8627	+	50.0696i	27.0000				0
199.20	0.889027	+	3.89995i	5.19615			−14.4193	+	6.93433i		0		4.61952	+	20.2647i	44.0991			−39.8627	−	50.0696i	27.0000				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.5.f.c		32
4.b	odd	2	1	inner	300.5.f.c		32
5.b	even	2	1	inner	300.5.f.c		32
5.c	odd	4	1		300.5.c.b	✓	16
5.c	odd	4	1		300.5.c.c	yes	16
20.d	odd	2	1	inner	300.5.f.c		32
20.e	even	4	1		300.5.c.b	✓	16
20.e	even	4	1		300.5.c.c	yes	16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.5.c.b	✓	16	5.c	odd	4	1
300.5.c.b	✓	16	20.e	even	4	1
300.5.c.c	yes	16	5.c	odd	4	1
300.5.c.c	yes	16	20.e	even	4	1
300.5.f.c		32	1.a	even	1	1	trivial
300.5.f.c		32	4.b	odd	2	1	inner
300.5.f.c		32	5.b	even	2	1	inner
300.5.f.c		32	20.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^16 - 24568*T7^14 + 232923164*T7^12 - 1109365466056*T7^10 + 2891061394272838*T7^8 - 4195788114349404104*T7^6 + 3220136517901201939484*T7^4 - 1073357410042249038453752*T7^2 + 55052665246124996150837761