Newform orbit 68.5.d.e

• Label: 68.5.d.e
• Level: $68$
• Weight: $5$
• Character orbit: 68.d
• Analytic conductor: $7.029$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	68.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.02915748970\)
Analytic rank:	\(0\)
Dimension:	\(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 10 q^{2} - 126 q^{4} - 130 q^{8} + 756 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−3.68242	−	1.56199i	−7.07390			11.1204	+	11.5038i		−	33.9165i	26.0490	+	11.0493i	46.9299			−22.9811	−	59.7316i	−30.9600			−52.9772	+	124.895i
67.2	−3.68242	−	1.56199i	7.07390			11.1204	+	11.5038i			33.9165i	−26.0490	−	11.0493i	−46.9299			−22.9811	−	59.7316i	−30.9600			52.9772	−	124.895i
67.3	−3.68242	+	1.56199i	−7.07390			11.1204	−	11.5038i			33.9165i	26.0490	−	11.0493i	46.9299			−22.9811	+	59.7316i	−30.9600			−52.9772	−	124.895i
67.4	−3.68242	+	1.56199i	7.07390			11.1204	−	11.5038i		−	33.9165i	−26.0490	+	11.0493i	−46.9299			−22.9811	+	59.7316i	−30.9600			52.9772	+	124.895i
67.5	−3.00319	−	2.64213i	−13.6848			2.03833	+	15.8696i			24.6088i	41.0982	+	36.1570i	−2.33921			35.8081	−	53.0451i	106.274			65.0196	−	73.9050i
67.6	−3.00319	−	2.64213i	13.6848			2.03833	+	15.8696i		−	24.6088i	−41.0982	−	36.1570i	2.33921			35.8081	−	53.0451i	106.274			−65.0196	+	73.9050i
67.7	−3.00319	+	2.64213i	−13.6848			2.03833	−	15.8696i		−	24.6088i	41.0982	−	36.1570i	−2.33921			35.8081	+	53.0451i	106.274			65.0196	+	73.9050i
67.8	−3.00319	+	2.64213i	13.6848			2.03833	−	15.8696i			24.6088i	−41.0982	+	36.1570i	2.33921			35.8081	+	53.0451i	106.274			−65.0196	−	73.9050i
67.9	−1.58200	−	3.67386i	−3.67150			−10.9945	+	11.6241i			9.06588i	5.80833	+	13.4886i	36.5470			60.0988	+	22.0030i	−67.5201			33.3068	−	14.3422i
67.10	−1.58200	−	3.67386i	3.67150			−10.9945	+	11.6241i		−	9.06588i	−5.80833	−	13.4886i	−36.5470			60.0988	+	22.0030i	−67.5201			−33.3068	+	14.3422i
67.11	−1.58200	+	3.67386i	−3.67150			−10.9945	−	11.6241i		−	9.06588i	5.80833	−	13.4886i	36.5470			60.0988	−	22.0030i	−67.5201			33.3068	+	14.3422i
67.12	−1.58200	+	3.67386i	3.67150			−10.9945	−	11.6241i			9.06588i	−5.80833	+	13.4886i	−36.5470			60.0988	−	22.0030i	−67.5201			−33.3068	−	14.3422i
67.13	−0.329312	−	3.98642i	−16.0485			−15.7831	+	2.62556i		−	39.8749i	5.28497	+	63.9761i	−63.7500			15.6641	+	62.0535i	176.554			−158.958	+	13.1313i
67.14	−0.329312	−	3.98642i	16.0485			−15.7831	+	2.62556i			39.8749i	−5.28497	−	63.9761i	63.7500			15.6641	+	62.0535i	176.554			158.958	−	13.1313i
67.15	−0.329312	+	3.98642i	−16.0485			−15.7831	−	2.62556i			39.8749i	5.28497	−	63.9761i	−63.7500			15.6641	−	62.0535i	176.554			−158.958	−	13.1313i
67.16	−0.329312	+	3.98642i	16.0485			−15.7831	−	2.62556i		−	39.8749i	−5.28497	+	63.9761i	63.7500			15.6641	−	62.0535i	176.554			158.958	+	13.1313i
67.17	1.13928	−	3.83432i	−0.417207			−13.4041	−	8.73677i		−	31.1542i	−0.475317	+	1.59970i	68.1200			−48.7706	+	41.4418i	−80.8259			−119.455	−	35.4935i
67.18	1.13928	−	3.83432i	0.417207			−13.4041	−	8.73677i			31.1542i	0.475317	−	1.59970i	−68.1200			−48.7706	+	41.4418i	−80.8259			119.455	+	35.4935i
67.19	1.13928	+	3.83432i	−0.417207			−13.4041	+	8.73677i			31.1542i	−0.475317	−	1.59970i	68.1200			−48.7706	−	41.4418i	−80.8259			−119.455	+	35.4935i
67.20	1.13928	+	3.83432i	0.417207			−13.4041	+	8.73677i		−	31.1542i	0.475317	+	1.59970i	−68.1200			−48.7706	−	41.4418i	−80.8259			119.455	−	35.4935i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	68.5.d.e	✓	28
4.b	odd	2	1	inner	68.5.d.e	✓	28
17.b	even	2	1	inner	68.5.d.e	✓	28
68.d	odd	2	1	inner	68.5.d.e	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.5.d.e	✓	28	1.a	even	1	1	trivial
68.5.d.e	✓	28	4.b	odd	2	1	inner
68.5.d.e	✓	28	17.b	even	2	1	inner
68.5.d.e	✓	28	68.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^14 - 756*T3^12 + 215732*T3^10 - 28919168*T3^8 + 1844787456*T3^6 - 50864972544*T3^4 + 419792212224*T3^2 - 71538204672