Newform orbit 115.5.h.a

• Label: 115.5.h.a
• Level: $115$
• Weight: $5$
• Character orbit: 115.h
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−6.63856	+	4.26634i	0.913905	−	6.35635i	19.2221	−	42.0906i	−3.14987	−	10.7275i	21.0513	+	46.0960i	−67.6414	+	58.6116i	33.9969	+	236.453i	38.1510	+	11.2021i	66.6775	+	57.7764i
11.2	−6.40115	+	4.11377i	−1.77574	+	12.3506i	17.4050	−	38.1116i	−3.14987	−	10.7275i	−39.4406	−	86.3628i	66.0549	−	57.2369i	28.0444	+	195.053i	−71.6639	−	21.0424i	64.2931	+	55.7103i
11.3	−5.95787	+	3.82889i	−1.40129	+	9.74618i	14.1892	−	31.0700i	3.14987	+	10.7275i	−28.9684	−	63.4319i	−16.5348	+	14.3275i	18.3000	+	127.279i	−15.3055	−	4.49411i	−59.8408	−	51.8523i
11.4	−5.30060	+	3.40649i	−0.401632	+	2.79341i	9.84555	−	21.5587i	3.14987	+	10.7275i	−7.38684	−	16.1749i	1.20039	−	1.04014i	6.90507	+	48.0258i	70.0771	+	20.5765i	−53.2391	−	46.1320i
11.5	−5.04053	+	3.23935i	1.96068	−	13.6368i	8.26690	−	18.1020i	3.14987	+	10.7275i	34.2916	+	75.0881i	−27.5585	+	23.8796i	3.32584	+	23.1317i	−104.400	−	30.6545i	−50.6270	−	43.8685i
11.6	−4.84924	+	3.11642i	1.03283	−	7.18350i	7.15645	−	15.6704i	−3.14987	−	10.7275i	17.3783	+	38.0533i	22.5454	−	19.5357i	1.00676	+	7.00216i	27.1830	+	7.98165i	48.7057	+	42.2038i
11.7	−4.32656	+	2.78051i	−0.455362	+	3.16711i	4.34125	−	9.50600i	−3.14987	−	10.7275i	−6.83604	−	14.9688i	−11.2058	+	9.70991i	−4.06190	−	28.2511i	67.8957	+	19.9360i	43.4559	+	37.6548i
11.8	−3.32783	+	2.13867i	2.30957	−	16.0634i	−0.146084	+	0.319880i	−3.14987	−	10.7275i	26.6685	+	58.3958i	−26.7873	+	23.2114i	−9.20547	−	64.0255i	−174.981	−	51.3790i	33.4247	+	28.9626i
11.9	−3.24867	+	2.08779i	−2.37840	+	16.5422i	−0.451668	+	0.989014i	3.14987	+	10.7275i	−26.8100	−	58.7057i	1.96580	−	1.70338i	−9.39077	−	65.3143i	−190.268	−	55.8676i	−32.6296	−	28.2737i
11.10	−2.75822	+	1.77260i	−0.284694	+	1.98009i	−2.18096	+	4.77564i	3.14987	+	10.7275i	−2.72467	−	5.96619i	52.3980	−	45.4031i	−9.91547	−	68.9636i	73.8792	+	21.6929i	−27.7036	−	24.0053i
11.11	−2.63066	+	1.69062i	0.551076	−	3.83282i	−2.58448	+	5.65923i	3.14987	+	10.7275i	5.03015	+	11.0145i	−52.9290	+	45.8632i	−9.88917	−	68.7807i	63.3321	+	18.5960i	−26.4223	−	22.8950i
11.12	−2.57806	+	1.65682i	−1.85705	+	12.9161i	−2.74528	+	6.01133i	−3.14987	−	10.7275i	−16.6120	−	36.3753i	54.7272	−	47.4214i	−9.86028	−	68.5798i	−85.6574	−	25.1513i	25.8940	+	22.4373i
11.13	−1.37551	+	0.883987i	−1.60372	+	11.1541i	−5.53604	+	12.1222i	−3.14987	−	10.7275i	−7.65414	−	16.7602i	−44.1661	+	38.2702i	−6.82414	−	47.4629i	−44.1230	−	12.9557i	13.8156	+	11.9713i
11.14	−1.04486	+	0.671490i	2.02842	−	14.1080i	−6.00581	+	13.1509i	3.14987	+	10.7275i	7.35396	+	16.1029i	35.3618	−	30.6412i	−5.38361	−	37.4438i	−117.202	−	34.4135i	−10.4945	−	9.09358i
11.15	−0.0293825	+	0.0188830i	−0.0510492	+	0.355055i	−6.64613	+	14.5530i	−3.14987	−	10.7275i	−0.00520454	−	0.0113963i	22.4414	−	19.4456i	−0.159054	−	1.10625i	77.5955	+	22.7841i	0.295117	+	0.255720i
11.16	0.562353	−	0.361402i	−1.00918	+	7.01902i	−6.46101	+	14.1476i	3.14987	+	10.7275i	1.96917	+	4.31189i	10.6690	−	9.24478i	3.00175	+	20.8776i	29.4707	+	8.65337i	5.64826	+	4.89425i
11.17	0.777273	−	0.499523i	0.742802	−	5.16630i	−6.29201	+	13.7776i	−3.14987	−	10.7275i	−2.00333	−	4.38667i	−14.1558	+	12.2660i	4.09547	+	28.4846i	51.5800	+	15.1453i	−7.80692	−	6.76473i
11.18	1.06653	−	0.685415i	−1.69116	+	11.7623i	−5.97895	+	13.0921i	3.14987	+	10.7275i	6.25839	+	13.7040i	−44.3476	+	38.4274i	5.48359	+	38.1392i	−57.7727	−	16.9636i	10.7122	+	9.28216i
11.19	1.27422	−	0.818893i	1.58968	−	11.0565i	−5.69358	+	12.4672i	3.14987	+	10.7275i	−7.02847	−	15.3902i	−22.4817	+	19.4805i	6.40338	+	44.5365i	−41.9999	−	12.3323i	12.7983	+	11.0898i
11.20	1.74399	−	1.12079i	2.03039	−	14.1217i	−4.86132	+	10.6448i	−3.14987	−	10.7275i	−12.2865	−	26.9037i	−63.0090	+	54.5976i	8.17303	+	56.8447i	−117.580	−	34.5247i	−17.5166	−	15.1782i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.d	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	115.5.h.a	✓	320
23.d	odd	22	1	inner	115.5.h.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.5.h.a	✓	320	1.a	even	1	1	trivial
115.5.h.a	✓	320	23.d	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(115, [\chi])\).