Newform orbit 115.5.f.a

• Label: 115.5.f.a
• Level: $115$
• Weight: $5$
• Character orbit: 115.f
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 48 q^{5} + 160 q^{7} + 180 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−5.45879	−	5.45879i	0.249588	−	0.249588i			43.5967i	18.7380	+	16.5495i	−2.72490			19.4401	+	19.4401i	150.645	−	150.645i			80.8754i	−11.9468	−	192.627i
47.2	−5.33319	−	5.33319i	−7.23554	+	7.23554i			40.8858i	−24.4965	−	4.99232i	77.1770			29.3451	+	29.3451i	132.721	−	132.721i		−	23.7062i	104.019	+	157.269i
47.3	−4.97771	−	4.97771i	7.01798	−	7.01798i			33.5552i	−23.4643	+	8.62722i	−69.8670			−45.7084	−	45.7084i	87.3849	−	87.3849i		−	17.5042i	159.742	+	73.8545i
47.4	−4.97086	−	4.97086i	2.56662	−	2.56662i			33.4190i	6.66462	−	24.0953i	−25.5166			17.2314	+	17.2314i	86.5873	−	86.5873i			67.8250i	−152.903	+	86.6455i
47.5	−4.73356	−	4.73356i	−10.2373	+	10.2373i			28.8132i	11.3791	−	22.2602i	96.9182			−46.2071	−	46.2071i	60.6523	−	60.6523i		−	128.606i	−159.234	+	51.5063i
47.6	−4.33895	−	4.33895i	−0.812388	+	0.812388i			21.6530i	24.9780	+	1.04909i	7.04982			−60.2203	−	60.2203i	24.5280	−	24.5280i			79.6801i	−103.826	−	112.930i
47.7	−4.30136	−	4.30136i	10.0667	−	10.0667i			21.0033i	7.80587	+	23.7501i	−86.6013			36.0694	+	36.0694i	21.5211	−	21.5211i		−	121.679i	68.5819	−	135.734i
47.8	−4.20795	−	4.20795i	9.21285	−	9.21285i			19.4137i	−8.64878	−	23.4563i	−77.5344			16.5187	+	16.5187i	14.3646	−	14.3646i		−	88.7533i	−62.3093	+	135.097i
47.9	−4.08601	−	4.08601i	−10.8237	+	10.8237i			17.3909i	16.5685	+	18.7212i	88.4518			35.7857	+	35.7857i	5.68317	−	5.68317i		−	153.307i	8.79606	−	144.194i
47.10	−3.82088	−	3.82088i	−3.62979	+	3.62979i			13.1983i	−11.2115	+	22.3451i	27.7380			−9.87428	−	9.87428i	−10.7050	+	10.7050i			54.6493i	128.216	−	42.5403i
47.11	−3.55253	−	3.55253i	−5.80472	+	5.80472i			9.24088i	−21.4504	−	12.8406i	41.2428			2.81474	+	2.81474i	−24.0119	+	24.0119i			13.6105i	30.5864	+	121.820i
47.12	−2.95338	−	2.95338i	0.0137449	−	0.0137449i			1.44491i	21.7668	−	12.2965i	−0.0811878			52.4860	+	52.4860i	−42.9867	+	42.9867i			80.9996i	−100.602	−	27.9694i
47.13	−2.86409	−	2.86409i	9.27176	−	9.27176i			0.406037i	24.7873	−	3.25454i	−53.1103			−31.1339	−	31.1339i	−44.6625	+	44.6625i		−	90.9310i	−80.3142	−	61.6717i
47.14	−2.39187	−	2.39187i	3.09239	−	3.09239i		−	4.55787i	−24.9722	+	1.17796i	−14.7932			52.6653	+	52.6653i	−49.1719	+	49.1719i			61.8743i	62.5480	+	56.9129i
47.15	−2.19710	−	2.19710i	−7.89548	+	7.89548i		−	6.34549i	12.9495	−	21.3848i	34.6943			18.7701	+	18.7701i	−49.0953	+	49.0953i		−	43.6771i	−75.4360	+	18.5332i
47.16	−2.04029	−	2.04029i	2.48006	−	2.48006i		−	7.67444i	−14.0131	−	20.7035i	−10.1201			−49.5729	−	49.5729i	−48.3027	+	48.3027i			68.6986i	−13.6504	+	70.8318i
47.17	−1.82478	−	1.82478i	3.91359	−	3.91359i		−	9.34035i	−0.467365	+	24.9956i	−14.2829			−5.47905	−	5.47905i	−46.2406	+	46.2406i			50.3677i	46.4644	−	44.7587i
47.18	−1.60759	−	1.60759i	−11.9434	+	11.9434i		−	10.8313i	−22.7256	+	10.4187i	38.4002			−60.8173	−	60.8173i	−43.1337	+	43.1337i		−	204.290i	53.2823	+	19.7844i
47.19	−1.05609	−	1.05609i	11.5301	−	11.5301i		−	13.7694i	−24.8276	+	2.93067i	−24.3536			12.8452	+	12.8452i	−31.4390	+	31.4390i		−	184.888i	29.3151	+	23.1251i
47.20	−0.148728	−	0.148728i	10.5922	−	10.5922i		−	15.9558i	9.50761	−	23.1215i	−3.15072			0.0244290	+	0.0244290i	−4.75271	+	4.75271i		−	143.391i	−4.85286	+	2.02477i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	115.5.f.a	✓	88
5.c	odd	4	1	inner	115.5.f.a	✓	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.5.f.a	✓	88	1.a	even	1	1	trivial
115.5.f.a	✓	88	5.c	odd	4	1	inner

Hecke kernels

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