Newform orbit 300.2.x.a

• Label: 300.2.x.a
• Level: $300$
• Weight: $2$
• Character orbit: 300.x
• Analytic conductor: $2.396$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.x (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39551206064\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(10\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80q - 2q^{3} + 4q^{7} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0		−1.73066	−	0.0694365i		0		0.354250	+	2.20783i		0		−1.67035	−	1.67035i		0		2.99036	+	0.240342i		0
17.2		0		−1.58605	+	0.696021i		0		−1.99640	−	1.00718i		0		0.814380	+	0.814380i		0		2.03111	−	2.20785i		0
17.3		0		−1.13130	+	1.31155i		0		2.14441	−	0.633635i		0		0.907947	+	0.907947i		0		−0.440321	−	2.96751i		0
17.4		0		−0.719024	−	1.57576i		0		−1.13602	+	1.92599i		0		3.00936	+	3.00936i		0		−1.96601	+	2.26601i		0
17.5		0		−0.351313	−	1.69605i		0		2.22846	−	0.184289i		0		−2.84010	−	2.84010i		0		−2.75316	+	1.19169i		0
17.6		0		0.670639	+	1.59695i		0		−2.14441	+	0.633635i		0		0.907947	+	0.907947i		0		−2.10049	+	2.14195i		0
17.7		0		0.858227	−	1.50448i		0		−2.22846	+	0.184289i		0		−2.84010	−	2.84010i		0		−1.52689	−	2.58236i		0
17.8		0		1.17077	−	1.27644i		0		1.13602	−	1.92599i		0		3.00936	+	3.00936i		0		−0.258608	−	2.98883i		0
17.9		0		1.29334	+	1.15207i		0		1.99640	+	1.00718i		0		0.814380	+	0.814380i		0		0.345460	+	2.98004i		0
17.10		0		1.66741	+	0.468765i		0		−0.354250	−	2.20783i		0		−1.67035	−	1.67035i		0		2.56052	+	1.56325i		0
53.1		0		−1.73066	+	0.0694365i		0		0.354250	−	2.20783i		0		−1.67035	+	1.67035i		0		2.99036	−	0.240342i		0
53.2		0		−1.58605	−	0.696021i		0		−1.99640	+	1.00718i		0		0.814380	−	0.814380i		0		2.03111	+	2.20785i		0
53.3		0		−1.13130	−	1.31155i		0		2.14441	+	0.633635i		0		0.907947	−	0.907947i		0		−0.440321	+	2.96751i		0
53.4		0		−0.719024	+	1.57576i		0		−1.13602	−	1.92599i		0		3.00936	−	3.00936i		0		−1.96601	−	2.26601i		0
53.5		0		−0.351313	+	1.69605i		0		2.22846	+	0.184289i		0		−2.84010	+	2.84010i		0		−2.75316	−	1.19169i		0
53.6		0		0.670639	−	1.59695i		0		−2.14441	−	0.633635i		0		0.907947	−	0.907947i		0		−2.10049	−	2.14195i		0
53.7		0		0.858227	+	1.50448i		0		−2.22846	−	0.184289i		0		−2.84010	+	2.84010i		0		−1.52689	+	2.58236i		0
53.8		0		1.17077	+	1.27644i		0		1.13602	+	1.92599i		0		3.00936	−	3.00936i		0		−0.258608	+	2.98883i		0
53.9		0		1.29334	−	1.15207i		0		1.99640	−	1.00718i		0		0.814380	−	0.814380i		0		0.345460	−	2.98004i		0
53.10		0		1.66741	−	0.468765i		0		−0.354250	+	2.20783i		0		−1.67035	+	1.67035i		0		2.56052	−	1.56325i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
25.f	odd	20	1	inner
75.l	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.2.x.a	✓	80
3.b	odd	2	1	inner	300.2.x.a	✓	80
25.f	odd	20	1	inner	300.2.x.a	✓	80
75.l	even	20	1	inner	300.2.x.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.2.x.a	✓	80	1.a	even	1	1	trivial
300.2.x.a	✓	80	3.b	odd	2	1	inner
300.2.x.a	✓	80	25.f	odd	20	1	inner
300.2.x.a	✓	80	75.l	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).