Newform orbit 68.5.g.a

• Label: 68.5.g.a
• Level: $68$
• Weight: $5$
• Character orbit: 68.g
• Analytic conductor: $7.029$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	68.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.02915748970\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 4 q^{2} - 8 q^{5} - 76 q^{6} - 4 q^{8} - 232 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
15.1	−3.97976	+	0.401912i	−5.14096	+	2.12946i	15.6769	−	3.19903i	−30.8993	+	12.7989i	19.6039	−	10.5409i	8.44599	−	20.3904i	−61.1047	+	19.0321i	−35.3807	+	35.3807i	117.828	−	63.3554i
15.2	−3.93941	−	0.693552i	12.4503	−	5.15708i	15.0380	+	5.46437i	−33.4720	+	13.8645i	−52.6236	+	11.6810i	−33.8486	+	81.7177i	−55.4510	−	31.9560i	71.1386	−	71.1386i	141.476	−	31.4036i
15.3	−3.92559	+	0.767965i	−14.3470	+	5.94274i	14.8205	−	6.02942i	24.9160	−	10.3205i	51.7567	−	34.3468i	6.13863	−	14.8200i	−53.5486	+	35.0506i	113.246	−	113.246i	−89.8839	+	59.6487i
15.4	−3.88253	−	0.962278i	6.09911	−	2.52633i	14.1480	+	7.47214i	44.1858	−	18.3024i	−26.1110	+	3.93952i	−18.8823	+	45.5859i	−47.7399	−	42.6251i	−26.4589	+	26.4589i	−189.164	+	28.5404i
15.5	−3.73480	+	1.43221i	10.8531	−	4.49551i	11.8975	−	10.6981i	17.1956	−	7.12264i	−34.0957	+	32.3338i	21.5256	−	51.9674i	−29.1130	+	56.9950i	40.3050	−	40.3050i	−54.0210	+	51.2294i
15.6	−3.65611	−	1.62260i	−2.02334	+	0.838095i	10.7344	+	11.8648i	0.220874	−	0.0914891i	8.75745	+	0.218896i	15.3199	−	36.9854i	−19.9942	−	60.7966i	−53.8841	+	53.8841i	−0.955992	−	0.0238954i
15.7	−3.22362	+	2.36818i	−2.41525	+	1.00043i	4.78349	−	15.2682i	14.1064	−	5.84305i	5.41666	−	8.94473i	−18.4314	+	44.4973i	20.7376	+	60.5471i	−52.4431	+	52.4431i	−31.6362	+	52.2421i
15.8	−3.03366	+	2.60708i	5.40922	−	2.24057i	2.40622	−	15.8180i	−17.1875	+	7.11928i	−10.5684	+	20.8994i	−1.03674	+	2.50292i	33.9393	+	54.2598i	−33.0362	+	33.0362i	33.5804	−	66.4067i
15.9	−2.99412	−	2.65240i	−11.9983	+	4.96987i	1.92950	+	15.8832i	−6.12124	+	2.53550i	49.1065	+	16.9440i	−23.3729	+	56.4272i	36.3516	−	52.6741i	61.9846	−	61.9846i	25.0529	+	8.64441i
15.10	−2.65240	−	2.99412i	11.9983	−	4.96987i	−1.92950	+	15.8832i	−6.12124	+	2.53550i	−46.7048	−	22.7423i	23.3729	−	56.4272i	52.6741	−	36.3516i	61.9846	−	61.9846i	23.8276	+	11.6025i
15.11	−2.03932	+	3.44110i	−13.5696	+	5.62070i	−7.68235	−	14.0350i	−37.8294	+	15.6694i	8.33129	−	58.1567i	−20.3620	+	49.1581i	63.9627	+	2.18611i	95.2653	−	95.2653i	23.2260	−	162.130i
15.12	−1.62260	−	3.65611i	2.02334	−	0.838095i	−10.7344	+	11.8648i	0.220874	−	0.0914891i	−6.34724	−	6.03767i	−15.3199	+	36.9854i	60.7966	+	19.9942i	−53.8841	+	53.8841i	−0.692885	−	0.659092i
15.13	−1.50860	+	3.70461i	−6.72667	+	2.78628i	−11.4482	−	11.1776i	23.2295	−	9.62197i	−0.174207	−	29.1231i	17.6038	−	42.4993i	58.6793	−	25.5488i	−19.7909	+	19.7909i	0.601596	+	100.572i
15.14	−1.34120	+	3.76845i	15.7770	−	6.53505i	−12.4024	−	10.1085i	7.80119	−	3.23136i	3.46692	+	68.2196i	−15.2593	+	36.8392i	54.7272	−	33.1803i	148.931	−	148.931i	1.71427	+	33.7323i
15.15	−0.962278	−	3.88253i	−6.09911	+	2.52633i	−14.1480	+	7.47214i	44.1858	−	18.3024i	15.6776	+	21.2489i	18.8823	−	45.5859i	42.6251	+	47.7399i	−26.4589	+	26.4589i	−113.578	−	153.941i
15.16	−0.693552	−	3.93941i	−12.4503	+	5.15708i	−15.0380	+	5.46437i	−33.4720	+	13.8645i	28.9508	+	45.4702i	33.8486	−	81.7177i	31.9560	+	55.4510i	71.1386	−	71.1386i	77.8327	+	122.244i
15.17	−0.583213	+	3.95725i	5.61605	−	2.32624i	−15.3197	−	4.61584i	−34.3663	+	14.2350i	5.93019	+	23.5808i	28.0507	−	67.7204i	27.2007	−	57.9320i	−31.1470	+	31.1470i	−36.2886	−	144.298i
15.18	0.211077	+	3.99443i	1.44231	−	0.597422i	−15.9109	+	1.68627i	11.7050	−	4.84835i	2.69080	+	5.63508i	−27.8893	+	67.3306i	−10.0941	−	63.1990i	−55.5523	+	55.5523i	21.8370	+	45.7312i
15.19	0.401912	−	3.97976i	5.14096	−	2.12946i	−15.6769	−	3.19903i	−30.8993	+	12.7989i	−6.40850	−	21.3156i	−8.44599	+	20.3904i	−19.0321	+	61.1047i	−35.3807	+	35.3807i	38.5177	+	128.116i
15.20	0.767965	−	3.92559i	14.3470	−	5.94274i	−14.8205	−	6.02942i	24.9160	−	10.3205i	−12.3107	−	60.8844i	−6.13863	+	14.8200i	−35.0506	+	53.5486i	113.246	−	113.246i	−21.3795	−	105.736i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
17.d	even	8	1	inner
68.g	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	68.5.g.a	✓	136
4.b	odd	2	1	inner	68.5.g.a	✓	136
17.d	even	8	1	inner	68.5.g.a	✓	136
68.g	odd	8	1	inner	68.5.g.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.5.g.a	✓	136	1.a	even	1	1	trivial
68.5.g.a	✓	136	4.b	odd	2	1	inner
68.5.g.a	✓	136	17.d	even	8	1	inner
68.5.g.a	✓	136	68.g	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(68, [\chi])\).