Newform orbit 65.5.g.a

• Label: 65.5.g.a
• Level: $65$
• Weight: $5$
• Character orbit: 65.g
• Analytic conductor: $6.719$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	65.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.71904760045\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 58 q^{5} + 60 q^{6} - 1196 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
34.1	−5.15266	+	5.15266i		−	15.6320i		−	37.0999i	22.4978	+	10.9017i	80.5463	+	80.5463i	−43.9050	−	43.9050i	108.720	+	108.720i	−163.359			−172.097	+	59.7510i
34.2	−4.99838	+	4.99838i			5.28046i		−	33.9676i	24.9466	−	1.63388i	−26.3937	−	26.3937i	65.3572	+	65.3572i	89.8089	+	89.8089i	53.1168			−116.526	+	132.859i
34.3	−4.94726	+	4.94726i		−	3.40815i		−	32.9507i	−22.9591	−	9.89351i	16.8610	+	16.8610i	9.32308	+	9.32308i	83.8595	+	83.8595i	69.3845			162.530	−	64.6387i
34.4	−4.82935	+	4.82935i			11.4208i		−	30.6453i	−1.40781	+	24.9603i	−55.1549	−	55.1549i	−48.2799	−	48.2799i	70.7273	+	70.7273i	−49.4340			−113.743	−	127.341i
34.5	−3.80385	+	3.80385i			13.5404i		−	12.9386i	−6.62279	−	24.1068i	−51.5057	−	51.5057i	−14.0955	−	14.0955i	−11.6451	−	11.6451i	−102.342			116.891	+	66.5067i
34.6	−3.49855	+	3.49855i		−	7.86015i		−	8.47972i	−17.4903	+	17.8631i	27.4991	+	27.4991i	−0.666849	−	0.666849i	−26.3101	−	26.3101i	19.2180			−1.30419	−	123.686i
34.7	−2.95432	+	2.95432i		−	14.1332i		−	1.45602i	−0.166336	−	24.9994i	41.7539	+	41.7539i	27.9563	+	27.9563i	−42.9676	−	42.9676i	−118.747			74.3478	+	73.3650i
34.8	−2.80141	+	2.80141i		−	1.58015i			0.304183i	20.5382	−	14.2543i	4.42666	+	4.42666i	−31.3088	−	31.3088i	−45.6747	−	45.6747i	78.5031			−17.6038	+	97.4679i
34.9	−2.29180	+	2.29180i			1.09931i			5.49535i	14.5107	+	20.3578i	−2.51939	−	2.51939i	7.15078	+	7.15078i	−49.2629	−	49.2629i	79.7915			−79.9114	−	13.4005i
34.10	−1.96266	+	1.96266i			11.7087i			8.29597i	−20.0793	+	14.8937i	−22.9802	−	22.9802i	49.2380	+	49.2380i	−47.6846	−	47.6846i	−56.0939			10.1776	−	68.6399i
34.11	−0.479269	+	0.479269i			2.62394i			15.5406i	−22.0613	−	11.7601i	−1.25757	−	1.25757i	−51.9642	−	51.9642i	−15.1164	−	15.1164i	74.1150			16.2095	−	4.93705i
34.12	−0.212553	+	0.212553i		−	13.8459i			15.9096i	19.7160	+	15.3714i	2.94299	+	2.94299i	38.8793	+	38.8793i	−6.78250	−	6.78250i	−110.709			−7.45794	+	0.923458i
34.13	−0.0630863	+	0.0630863i		−	12.3873i			15.9920i	−17.7387	+	17.6164i	0.781468	+	0.781468i	−27.4701	−	27.4701i	−2.01826	−	2.01826i	−72.4449			0.00771715	−	2.23042i
34.14	0.0630863	−	0.0630863i			12.3873i			15.9920i	17.6164	−	17.7387i	0.781468	+	0.781468i	27.4701	+	27.4701i	2.01826	+	2.01826i	−72.4449			−0.00771715	−	2.23042i
34.15	0.212553	−	0.212553i			13.8459i			15.9096i	15.3714	+	19.7160i	2.94299	+	2.94299i	−38.8793	−	38.8793i	6.78250	+	6.78250i	−110.709			7.45794	+	0.923458i
34.16	0.479269	−	0.479269i		−	2.62394i			15.5406i	−11.7601	−	22.0613i	−1.25757	−	1.25757i	51.9642	+	51.9642i	15.1164	+	15.1164i	74.1150			−16.2095	−	4.93705i
34.17	1.96266	−	1.96266i		−	11.7087i			8.29597i	14.8937	−	20.0793i	−22.9802	−	22.9802i	−49.2380	−	49.2380i	47.6846	+	47.6846i	−56.0939			−10.1776	−	68.6399i
34.18	2.29180	−	2.29180i		−	1.09931i			5.49535i	20.3578	+	14.5107i	−2.51939	−	2.51939i	−7.15078	−	7.15078i	49.2629	+	49.2629i	79.7915			79.9114	−	13.4005i
34.19	2.80141	−	2.80141i			1.58015i			0.304183i	−14.2543	+	20.5382i	4.42666	+	4.42666i	31.3088	+	31.3088i	45.6747	+	45.6747i	78.5031			17.6038	+	97.4679i
34.20	2.95432	−	2.95432i			14.1332i		−	1.45602i	−24.9994	−	0.166336i	41.7539	+	41.7539i	−27.9563	−	27.9563i	42.9676	+	42.9676i	−118.747			−74.3478	+	73.3650i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	65.5.g.a	✓	52
5.b	even	2	1	inner	65.5.g.a	✓	52
13.d	odd	4	1	inner	65.5.g.a	✓	52
65.g	odd	4	1	inner	65.5.g.a	✓	52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.5.g.a	✓	52	1.a	even	1	1	trivial
65.5.g.a	✓	52	5.b	even	2	1	inner
65.5.g.a	✓	52	13.d	odd	4	1	inner
65.5.g.a	✓	52	65.g	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(65, [\chi])\).