Newform orbit 115.4.e.a

• Label: 115.4.e.a
• Level: $115$
• Weight: $4$
• Character orbit: 115.e
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{2} + 4 q^{3} + 32 q^{6} + 28 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
22.1	−3.71645	−	3.71645i	3.49413	−	3.49413i			19.6239i	−10.7999	+	2.89187i	−25.9715			2.69551	−	2.69551i	43.1997	−	43.1997i			2.58205i	50.8846	+	29.3896i
22.2	−3.71645	−	3.71645i	3.49413	−	3.49413i			19.6239i	10.7999	−	2.89187i	−25.9715			−2.69551	+	2.69551i	43.1997	−	43.1997i			2.58205i	−50.8846	−	29.3896i
22.3	−3.67169	−	3.67169i	−4.81114	+	4.81114i			18.9626i	−0.398194	−	11.1732i	35.3300			20.1910	−	20.1910i	40.2512	−	40.2512i		−	19.2941i	−39.5626	+	42.4867i
22.4	−3.67169	−	3.67169i	−4.81114	+	4.81114i			18.9626i	0.398194	+	11.1732i	35.3300			−20.1910	+	20.1910i	40.2512	−	40.2512i		−	19.2941i	39.5626	−	42.4867i
22.5	−2.78118	−	2.78118i	1.69288	−	1.69288i			7.46996i	−9.46680	−	5.94809i	−9.41642			1.06728	−	1.06728i	−1.47414	+	1.47414i			21.2683i	9.78619	+	42.8716i
22.6	−2.78118	−	2.78118i	1.69288	−	1.69288i			7.46996i	9.46680	+	5.94809i	−9.41642			−1.06728	+	1.06728i	−1.47414	+	1.47414i			21.2683i	−9.78619	−	42.8716i
22.7	−2.66509	−	2.66509i	−2.23283	+	2.23283i			6.20541i	−7.41137	+	8.37088i	11.9014			10.9048	−	10.9048i	−4.78275	+	4.78275i			17.0290i	42.0611	−	2.55718i
22.8	−2.66509	−	2.66509i	−2.23283	+	2.23283i			6.20541i	7.41137	−	8.37088i	11.9014			−10.9048	+	10.9048i	−4.78275	+	4.78275i			17.0290i	−42.0611	+	2.55718i
22.9	−2.37561	−	2.37561i	6.86048	−	6.86048i			3.28704i	−2.55138	−	10.8853i	−32.5957			−19.7347	+	19.7347i	−11.1962	+	11.1962i		−	67.1325i	−19.7982	+	31.9204i
22.10	−2.37561	−	2.37561i	6.86048	−	6.86048i			3.28704i	2.55138	+	10.8853i	−32.5957			19.7347	−	19.7347i	−11.1962	+	11.1962i		−	67.1325i	19.7982	−	31.9204i
22.11	−1.84990	−	1.84990i	−5.85544	+	5.85544i		−	1.15573i	−10.6129	−	3.51653i	21.6640			−7.25929	+	7.25929i	−16.9372	+	16.9372i		−	41.5723i	13.1276	+	26.1381i
22.12	−1.84990	−	1.84990i	−5.85544	+	5.85544i		−	1.15573i	10.6129	+	3.51653i	21.6640			7.25929	−	7.25929i	−16.9372	+	16.9372i		−	41.5723i	−13.1276	−	26.1381i
22.13	−1.32617	−	1.32617i	2.49169	−	2.49169i		−	4.48253i	−3.09011	+	10.7448i	−6.60883			−19.9254	+	19.9254i	−16.5540	+	16.5540i			14.5829i	18.3475	−	10.1515i
22.14	−1.32617	−	1.32617i	2.49169	−	2.49169i		−	4.48253i	3.09011	−	10.7448i	−6.60883			19.9254	−	19.9254i	−16.5540	+	16.5540i			14.5829i	−18.3475	+	10.1515i
22.15	−0.386420	−	0.386420i	−0.970983	+	0.970983i		−	7.70136i	−8.97655	−	6.66495i	0.750415			−6.23903	+	6.23903i	−6.06732	+	6.06732i			25.1144i	0.893251	+	6.04419i
22.16	−0.386420	−	0.386420i	−0.970983	+	0.970983i		−	7.70136i	8.97655	+	6.66495i	0.750415			6.23903	−	6.23903i	−6.06732	+	6.06732i			25.1144i	−0.893251	−	6.04419i
22.17	−0.152517	−	0.152517i	−2.82764	+	2.82764i		−	7.95348i	−5.56672	+	9.69596i	0.862527			20.1728	−	20.1728i	−2.43318	+	2.43318i			11.0089i	2.32782	−	0.629778i
22.18	−0.152517	−	0.152517i	−2.82764	+	2.82764i		−	7.95348i	5.56672	−	9.69596i	0.862527			−20.1728	+	20.1728i	−2.43318	+	2.43318i			11.0089i	−2.32782	+	0.629778i
22.19	0.304528	+	0.304528i	5.13977	−	5.13977i		−	7.81453i	−11.1201	+	1.15901i	3.13040			2.31330	−	2.31330i	4.81596	−	4.81596i		−	25.8345i	−3.73933	−	3.03343i
22.20	0.304528	+	0.304528i	5.13977	−	5.13977i		−	7.81453i	11.1201	−	1.15901i	3.13040			−2.31330	+	2.31330i	4.81596	−	4.81596i		−	25.8345i	3.73933	+	3.03343i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
23.b	odd	2	1	inner
115.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	115.4.e.a	✓	68
5.c	odd	4	1	inner	115.4.e.a	✓	68
23.b	odd	2	1	inner	115.4.e.a	✓	68
115.e	even	4	1	inner	115.4.e.a	✓	68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.4.e.a	✓	68	1.a	even	1	1	trivial
115.4.e.a	✓	68	5.c	odd	4	1	inner
115.4.e.a	✓	68	23.b	odd	2	1	inner
115.4.e.a	✓	68	115.e	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(115, [\chi])\).