Newform orbit 115.5.k.a

• Label: 115.5.k.a
• Level: $115$
• Weight: $5$
• Character orbit: 115.k
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $920$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(920\)
Relative dimension:	\(46\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 18 q^{2} + 2 q^{3} - 54 q^{5} - 92 q^{6} - 78 q^{7} - 82 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.64826	−	7.57694i	6.23582	+	4.66808i	−40.1392	+	18.3310i	−22.2936	+	11.3136i	25.0915	−	54.9427i	86.4073	−	6.17997i	130.702	+	174.598i	−5.72581	−	19.5003i	122.468	+	150.269i
2.2	−1.63117	−	7.49837i	−11.9757	−	8.96489i	−39.0107	+	17.8156i	5.97536	−	24.2754i	−47.6877	+	104.421i	57.2971	−	4.09797i	123.642	+	165.166i	40.2276	+	137.003i	−191.773	−	5.20810i
2.3	−1.59882	−	7.34967i	−5.85949	−	4.38636i	−36.9073	+	16.8550i	11.5839	+	22.1543i	−22.8700	+	50.0783i	−73.5488	+	5.26032i	110.767	+	147.967i	−7.72685	−	26.3152i	144.306	−	120.558i
2.4	−1.54390	−	7.09721i	12.4943	+	9.35310i	−33.4326	+	15.2682i	24.9081	−	2.14186i	47.0910	−	103.115i	−24.4487	+	1.74861i	90.3355	+	120.674i	45.8062	+	156.002i	−53.6569	−	173.471i
2.5	−1.37772	−	6.33327i	1.17608	+	0.880403i	−23.6580	+	10.8043i	21.2005	−	13.2490i	3.95552	−	8.66138i	4.47168	−	0.319821i	38.8740	+	51.9295i	−22.2123	−	75.6481i	−113.118	−	116.015i
2.6	−1.35404	−	6.22442i	3.19389	+	2.39092i	−22.3559	+	10.2096i	−7.43981	−	23.8673i	10.5574	−	23.1175i	−18.9307	+	1.35395i	32.7414	+	43.7374i	−18.3359	−	62.4463i	−138.487	+	78.6259i
2.7	−1.34542	−	6.18479i	−5.45202	−	4.08133i	−21.8874	+	9.99565i	−24.9174	−	2.03060i	−17.9069	+	39.2107i	−30.6871	+	2.19478i	30.5793	+	40.8492i	−9.75309	−	33.2160i	20.9655	+	156.841i
2.8	−1.22929	−	5.65095i	5.38967	+	4.03466i	−15.8679	+	7.24664i	2.29214	+	24.8947i	16.1742	−	35.4165i	−9.04583	+	0.646970i	5.00559	+	6.68669i	−10.0502	−	34.2280i	137.861	−	43.5555i
2.9	−1.17356	−	5.39479i	12.7213	+	9.52306i	−13.1724	+	6.01563i	−24.2240	+	6.18061i	36.4456	−	79.8048i	−85.1936	+	6.09317i	−5.02569	−	6.71354i	48.3230	+	164.573i	61.7715	+	123.430i
2.10	−1.15508	−	5.30983i	−5.53251	−	4.14159i	−12.3060	+	5.61997i	20.5259	+	14.2719i	−15.6006	+	34.1606i	82.0798	−	5.87046i	−8.04819	−	10.7511i	−9.36440	−	31.8922i	52.0721	−	125.475i
2.11	−1.08683	−	4.99609i	−14.0087	−	10.4867i	−9.22556	+	4.21317i	−14.7650	+	20.1741i	−37.1676	+	81.3858i	17.2026	−	1.23035i	−17.9490	−	23.9771i	63.4502	+	216.092i	116.839	+	51.8414i
2.12	−0.959568	−	4.41106i	10.6132	+	7.94498i	−3.98260	+	1.81879i	−7.61693	−	23.8114i	24.8617	−	54.4395i	60.9556	−	4.35963i	−31.4400	−	41.9990i	26.6980	+	90.9250i	−97.7246	+	56.4474i
2.13	−0.865714	−	3.97962i	−8.57523	−	6.41933i	−0.533825	+	0.243790i	−4.26575	−	24.6334i	−18.1228	+	39.6835i	−29.9148	+	2.13955i	−37.6185	−	50.2524i	9.50631	+	32.3755i	−94.3386	+	38.3015i
2.14	−0.819355	−	3.76651i	−9.83410	−	7.36171i	1.03882	−	0.474413i	24.9699	+	1.22702i	−19.6704	+	43.0721i	−43.7948	+	3.13227i	−39.5977	−	52.8963i	19.6943	+	67.0727i	−15.8376	−	95.0548i
2.15	−0.747185	−	3.43475i	4.35604	+	3.26089i	3.31487	−	1.51385i	−8.31625	+	23.5763i	7.94559	−	17.3984i	−11.4939	+	0.822060i	−41.3807	−	55.2781i	−14.4787	−	49.3098i	87.1924	+	10.9484i
2.16	−0.689190	−	3.16816i	−2.52437	−	1.88972i	4.99188	−	2.27971i	−24.5573	+	4.68402i	−4.24716	+	9.29999i	60.8127	−	4.34941i	−41.7510	−	55.7728i	−20.0189	−	68.1782i	31.7644	+	74.5731i
2.17	−0.589685	−	2.71074i	10.7543	+	8.05056i	7.55375	−	3.44968i	21.8879	+	12.0797i	15.4813	−	33.8994i	41.7114	−	2.98326i	−40.4051	−	53.9749i	28.0229	+	95.4373i	19.8380	−	66.4556i
2.18	−0.575345	−	2.64482i	3.92696	+	2.93968i	7.89006	−	3.60327i	21.8414	−	12.1636i	5.51558	−	12.0774i	−60.1392	+	4.30124i	−40.0223	−	53.4636i	−16.0411	−	54.6309i	−44.7369	−	50.7683i
2.19	−0.365393	−	1.67968i	−6.98500	−	5.22890i	11.8663	−	5.41915i	0.0469611	−	25.0000i	−6.23064	+	13.6432i	56.3758	−	4.03207i	−29.9206	−	39.9692i	−1.37158	−	4.67117i	−42.0092	+	9.05593i
2.20	−0.298380	−	1.37163i	5.72026	+	4.28213i	12.7618	−	5.82810i	−18.0552	−	17.2919i	4.16670	−	9.12380i	−68.0162	+	4.86461i	−25.2613	−	33.7451i	−8.43563	−	28.7291i	−18.3308	+	29.9247i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
23.c	even	11	1	inner
115.k	odd	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	115.5.k.a	✓	920
5.c	odd	4	1	inner	115.5.k.a	✓	920
23.c	even	11	1	inner	115.5.k.a	✓	920
115.k	odd	44	1	inner	115.5.k.a	✓	920

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.5.k.a	✓	920	1.a	even	1	1	trivial
115.5.k.a	✓	920	5.c	odd	4	1	inner
115.5.k.a	✓	920	23.c	even	11	1	inner
115.5.k.a	✓	920	115.k	odd	44	1	inner

Hecke kernels

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