Newform orbit 158.5.f.a

• Label: 158.5.f.a
• Level: $158$
• Weight: $5$
• Character orbit: 158.f
• Analytic conductor: $16.332$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 158 = 2 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	158.f (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.3324541672\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 224 q^{4} + 728 q^{9}+O(q^{10}) \)

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	−2.11711	−	1.87559i	−14.0416	+	9.69224i	0.964293	+	7.94167i	16.7079	+	4.11813i	47.9063	+	5.81688i	−32.8173	+	22.6522i	12.8538	−	18.6220i	74.5049	−	196.453i	−27.6485	−	40.0557i
15.2	−2.11711	−	1.87559i	−13.9748	+	9.64614i	0.964293	+	7.94167i	−38.3319	−	9.44797i	47.6785	+	5.78922i	49.3199	−	34.0431i	12.8538	−	18.6220i	73.5253	−	193.870i	63.4323	+	91.8975i
15.3	−2.11711	−	1.87559i	−8.02312	+	5.53796i	0.964293	+	7.94167i	31.9338	+	7.87097i	27.3728	+	3.32366i	53.7142	−	37.0763i	12.8538	−	18.6220i	4.97848	−	13.1272i	−52.8445	−	76.5585i
15.4	−2.11711	−	1.87559i	−7.89972	+	5.45278i	0.964293	+	7.94167i	−10.9900	−	2.70879i	26.9518	+	3.27254i	−32.3603	+	22.3367i	12.8538	−	18.6220i	3.94974	−	10.4146i	18.1864	+	26.3476i
15.5	−2.11711	−	1.87559i	−6.75585	+	4.66323i	0.964293	+	7.94167i	−21.4889	−	5.29655i	23.0492	+	2.79868i	37.8790	−	26.1460i	12.8538	−	18.6220i	−4.82717	+	12.7282i	35.5602	+	51.5179i
15.6	−2.11711	−	1.87559i	−2.48042	+	1.71211i	0.964293	+	7.94167i	32.4292	+	7.99309i	8.46256	+	1.02754i	−76.0209	+	52.4735i	12.8538	−	18.6220i	−25.5018	+	67.2428i	−53.6644	−	77.7463i
15.7	−2.11711	−	1.87559i	−1.32033	+	0.911360i	0.964293	+	7.94167i	−35.3477	−	8.71242i	4.50463	+	0.546961i	−35.9071	+	24.7849i	12.8538	−	18.6220i	−27.8103	+	73.3297i	58.4939	+	84.7431i
15.8	−2.11711	−	1.87559i	0.829826	−	0.572788i	0.964293	+	7.94167i	4.61412	+	1.13728i	−2.83115	−	0.343764i	7.89964	−	5.45273i	12.8538	−	18.6220i	−28.3625	+	74.7857i	−7.63552	−	11.0620i
15.9	−2.11711	−	1.87559i	2.67821	−	1.84863i	0.964293	+	7.94167i	46.6181	+	11.4903i	−9.13733	−	1.10947i	17.0896	−	11.7961i	12.8538	−	18.6220i	−24.9677	+	65.8343i	−77.1443	−	111.763i
15.10	−2.11711	−	1.87559i	6.28485	−	4.33812i	0.964293	+	7.94167i	−1.28008	−	0.315512i	−21.4423	−	2.60356i	61.2988	−	42.3115i	12.8538	−	18.6220i	−8.04292	+	21.2074i	2.11830	+	3.06889i
15.11	−2.11711	−	1.87559i	7.78771	−	5.37547i	0.964293	+	7.94167i	−2.95654	−	0.728722i	−26.5696	−	3.22613i	−13.4878	+	9.30995i	12.8538	−	18.6220i	3.02975	−	7.98878i	4.89253	+	7.08805i
15.12	−2.11711	−	1.87559i	9.75877	−	6.73599i	0.964293	+	7.94167i	−8.31114	−	2.04851i	−33.2943	−	4.04267i	−78.2990	+	54.0459i	12.8538	−	18.6220i	21.1370	−	55.7336i	13.7534	+	19.9252i
15.13	−2.11711	−	1.87559i	13.1671	−	9.08861i	0.964293	+	7.94167i	−44.4957	−	10.9672i	−44.9228	−	5.45461i	22.6829	−	15.6569i	12.8538	−	18.6220i	62.0474	−	163.606i	73.6322	+	106.675i
15.14	−2.11711	−	1.87559i	13.9894	−	9.65621i	0.964293	+	7.94167i	41.5647	+	10.2448i	−47.7283	−	5.79526i	−0.672097	+	0.463915i	12.8538	−	18.6220i	73.7389	−	194.433i	−68.7819	−	99.6477i
15.15	2.11711	+	1.87559i	−12.9144	+	8.91419i	0.964293	+	7.94167i	20.3119	+	5.00644i	−44.0606	−	5.34993i	73.7058	−	50.8755i	−12.8538	+	18.6220i	58.5967	−	154.507i	33.6125	+	48.6961i
15.16	2.11711	+	1.87559i	−12.3158	+	8.50097i	0.964293	+	7.94167i	3.82693	+	0.943253i	−42.0182	−	5.10193i	−32.8757	+	22.6925i	−12.8538	+	18.6220i	50.6889	−	133.656i	6.33286	+	9.17473i
15.17	2.11711	+	1.87559i	−9.44597	+	6.52008i	0.964293	+	7.94167i	−38.2121	−	9.41843i	−32.2272	−	3.91308i	−63.8026	+	44.0398i	−12.8538	+	18.6220i	17.9918	−	47.4406i	−63.2340	−	91.6102i
15.18	2.11711	+	1.87559i	−7.04877	+	4.86541i	0.964293	+	7.94167i	32.2596	+	7.95128i	−24.0485	−	2.92002i	−40.4891	+	27.9476i	−12.8538	+	18.6220i	−2.71011	+	7.14597i	53.3837	+	77.3397i
15.19	2.11711	+	1.87559i	−6.61651	+	4.56705i	0.964293	+	7.94167i	−43.9969	−	10.8443i	−22.5738	−	2.74096i	59.8762	−	41.3295i	−12.8538	+	18.6220i	−5.80269	+	15.3004i	−72.8067	−	105.479i
15.20	2.11711	+	1.87559i	−4.01794	+	2.77338i	0.964293	+	7.94167i	3.03952	+	0.749174i	−13.7082	−	1.66447i	32.5136	−	22.4425i	−12.8538	+	18.6220i	−20.2708	+	53.4497i	5.02984	+	7.28698i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
79.f	odd	26	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	158.5.f.a	✓	336
79.f	odd	26	1	inner	158.5.f.a	✓	336

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
158.5.f.a	✓	336	1.a	even	1	1	trivial
158.5.f.a	✓	336	79.f	odd	26	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(158, [\chi])\).