Newform orbit 115.3.f.a

• Label: 115.3.f.a
• Level: $115$
• Weight: $3$
• Character orbit: 115.f
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{2} - 8 q^{5} - 16 q^{7} + 12 q^{8}+O(q^{10}) \)

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	−2.74685	−	2.74685i	3.72491	−	3.72491i			11.0904i	4.51806	−	2.14175i	−20.4635			−2.87693	−	2.87693i	19.4763	−	19.4763i		−	18.7498i	−18.2935	−	6.52737i
47.2	−2.67250	−	2.67250i	−2.93217	+	2.93217i			10.2845i	−1.24988	+	4.84126i	15.6725			−7.73999	−	7.73999i	16.7954	−	16.7954i		−	8.19524i	16.2786	−	9.59798i
47.3	−2.33506	−	2.33506i	1.23717	−	1.23717i			6.90499i	−4.37439	+	2.42172i	−5.77773			9.05038	+	9.05038i	6.78332	−	6.78332i			5.93882i	15.8693	+	4.55958i
47.4	−2.11083	−	2.11083i	−0.751612	+	0.751612i			4.91121i	−1.47659	−	4.77700i	3.17305			−3.30771	−	3.30771i	1.92342	−	1.92342i			7.87016i	−6.96661	+	13.2003i
47.5	−1.95143	−	1.95143i	−0.210269	+	0.210269i			3.61618i	4.62528	+	1.89915i	0.820651			2.37158	+	2.37158i	−0.749007	+	0.749007i			8.91157i	−5.31987	−	12.7320i
47.6	−1.46415	−	1.46415i	3.27120	−	3.27120i			0.287471i	−4.76493	+	1.51507i	−9.57906			−5.58012	−	5.58012i	−5.43570	+	5.43570i		−	12.4015i	9.19486	+	4.75829i
47.7	−1.45626	−	1.45626i	−3.93198	+	3.93198i			0.241409i	−4.31815	−	2.52063i	11.4520			5.37896	+	5.37896i	−5.47350	+	5.47350i		−	21.9209i	2.61766	+	9.95907i
47.8	−1.06417	−	1.06417i	−3.13999	+	3.13999i		−	1.73510i	4.95922	+	0.637278i	6.68293			−2.16662	−	2.16662i	−6.10310	+	6.10310i		−	10.7190i	−4.59926	−	5.95560i
47.9	−0.888159	−	0.888159i	1.93967	−	1.93967i		−	2.42235i	1.78108	−	4.67202i	−3.44547			0.600352	+	0.600352i	−5.70407	+	5.70407i			1.47535i	−5.73138	+	2.56762i
47.10	−0.616305	−	0.616305i	−1.36474	+	1.36474i		−	3.24034i	−1.00227	+	4.89852i	1.68219			2.04545	+	2.04545i	−4.46226	+	4.46226i			5.27497i	3.63668	−	2.40128i
47.11	−0.306853	−	0.306853i	3.69350	−	3.69350i		−	3.81168i	2.97062	+	4.02187i	−2.26672			8.01534	+	8.01534i	−2.39704	+	2.39704i		−	18.2839i	0.322578	−	2.14567i
47.12	−0.275364	−	0.275364i	−0.547869	+	0.547869i		−	3.84835i	−4.58859	+	1.98617i	0.301727			−4.59988	−	4.59988i	−2.16115	+	2.16115i			8.39968i	1.81045	+	0.716611i
47.13	0.605975	+	0.605975i	1.58126	−	1.58126i		−	3.26559i	4.51316	+	2.15207i	1.91641			−8.09437	−	8.09437i	4.40277	−	4.40277i			3.99921i	1.43076	+	4.03896i
47.14	0.694377	+	0.694377i	−1.86203	+	1.86203i		−	3.03568i	3.90318	−	3.12494i	−2.58590			7.40287	+	7.40287i	4.88542	−	4.88542i			2.06569i	4.88016	+	0.540392i
47.15	0.701445	+	0.701445i	−2.49569	+	2.49569i		−	3.01595i	−1.50802	−	4.76717i	−3.50118			−7.98271	−	7.98271i	4.92130	−	4.92130i		−	3.45692i	2.28611	−	4.40170i
47.16	1.03401	+	1.03401i	2.52164	−	2.52164i		−	1.86167i	−4.44006	−	2.29910i	5.21478			1.69314	+	1.69314i	6.06099	−	6.06099i		−	3.71736i	−2.21377	−	6.96833i
47.17	1.56641	+	1.56641i	0.265917	−	0.265917i			0.907288i	−0.310831	+	4.99033i	0.833069			3.53378	+	3.53378i	4.84446	−	4.84446i			8.85858i	−8.30380	+	7.33002i
47.18	1.71185	+	1.71185i	−3.25485	+	3.25485i			1.86083i	−4.79289	+	1.42415i	−11.1436			2.88120	+	2.88120i	3.66193	−	3.66193i		−	12.1881i	−10.6426	−	5.76677i
47.19	2.13032	+	2.13032i	1.62695	−	1.62695i			5.07649i	2.93890	−	4.04510i	6.93185			−0.365078	−	0.365078i	−2.29326	+	2.29326i			3.70603i	14.8781	−	2.35655i
47.20	2.29778	+	2.29778i	−2.60211	+	2.60211i			6.55958i	4.64561	+	1.84886i	−11.9581			−4.55541	−	4.55541i	−5.88134	+	5.88134i		−	4.54191i	6.42633	+	14.9229i

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	115.3.f.a	✓	44
5.c	odd	4	1	inner	115.3.f.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.3.f.a	✓	44	1.a	even	1	1	trivial
115.3.f.a	✓	44	5.c	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(115, [\chi])\).