Newform orbit 117.3.n.a

• Label: 117.3.n.a
• Level: $117$
• Weight: $3$
• Character orbit: 117.n
• Analytic conductor: $3.188$
• Analytic rank: $0$
• Dimension: $52$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	117.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.18801909302\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{3} - 50 q^{4} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
38.1	−1.92341	+	3.33144i	−2.97788	−	0.363638i	−5.39901	−	9.35136i	0.506509	+	0.877299i	6.93912	−	9.22121i	−3.89680	−	2.24982i	26.1508			8.73553	+	2.16574i	−3.89690
38.2	−1.77860	+	3.08062i	2.89850	−	0.773763i	−4.32683	−	7.49428i	−4.26663	−	7.39002i	−2.77159	+	10.3054i	7.99720	+	4.61719i	16.5540			7.80258	−	4.48550i	30.3545
38.3	−1.68517	+	2.91881i	2.36131	+	1.85046i	−3.67962	−	6.37330i	3.61500	+	6.26136i	−9.38036	+	3.77386i	−2.20723	−	1.27434i	11.3218			2.15158	+	8.73903i	−24.3676
38.4	−1.55290	+	2.68970i	0.846137	−	2.87820i	−2.82298	−	4.88955i	0.611466	+	1.05909i	6.42753	+	6.74541i	−8.52943	−	4.92447i	5.11202			−7.56810	−	4.87071i	−3.79817
38.5	−1.38552	+	2.39980i	0.210571	+	2.99260i	−1.83936	−	3.18586i	−2.12220	−	3.67577i	−7.47339	−	3.64100i	−4.71279	−	2.72093i	−0.890291			−8.91132	+	1.26031i	11.7615
38.6	−1.33789	+	2.31729i	−1.30160	−	2.70293i	−1.57990	−	2.73646i	0.194524	+	0.336926i	8.00488	+	0.600028i	8.30078	+	4.79246i	−2.24820			−5.61166	+	7.03628i	−1.04101
38.7	−1.11386	+	1.92926i	−2.21197	+	2.02662i	−0.481371	−	0.833760i	4.12144	+	7.13855i	−1.44605	−	6.52485i	8.40220	+	4.85102i	−6.76616			0.785636	−	8.96564i	−18.3629
38.8	−1.04133	+	1.80365i	−2.79053	+	1.10134i	−0.168757	−	0.292296i	−2.59043	−	4.48676i	0.919442	−	6.17999i	−1.68126	−	0.970675i	−7.62775			6.57409	−	6.14666i	10.7900
38.9	−0.647986	+	1.12234i	2.50362	−	1.65285i	1.16023	+	2.00957i	2.25417	+	3.90433i	0.232759	+	3.88094i	0.162124	+	0.0936021i	−8.19114			3.53618	−	8.27620i	−5.84268
38.10	−0.641309	+	1.11078i	2.61442	+	1.47132i	1.17745	+	2.03940i	−0.763734	−	1.32283i	−3.31097	+	1.96048i	5.50560	+	3.17866i	−8.15090			4.67043	+	7.69331i	1.95916
38.11	−0.433498	+	0.750840i	−0.162662	−	2.99559i	1.62416	+	2.81313i	−4.23652	−	7.33786i	2.31972	+	1.17645i	−3.54375	−	2.04598i	−6.28426			−8.94708	+	0.974539i	7.34608
38.12	−0.245962	+	0.426018i	−2.80287	−	1.06954i	1.87901	+	3.25453i	1.35391	+	2.34503i	1.14504	−	0.931009i	−8.51494	−	4.91611i	−3.81635			6.71217	+	5.99556i	−1.33204
38.13	−0.146647	+	0.254001i	−0.187042	+	2.99416i	1.95699	+	3.38960i	2.67644	+	4.63573i	−0.733091	−	0.486595i	−8.20362	−	4.73636i	−2.32113			−8.93003	−	1.12007i	−1.56997
38.14	0.146647	−	0.254001i	−0.187042	+	2.99416i	1.95699	+	3.38960i	−2.67644	−	4.63573i	0.733091	+	0.486595i	8.20362	+	4.73636i	2.32113			−8.93003	−	1.12007i	−1.56997
38.15	0.245962	−	0.426018i	−2.80287	−	1.06954i	1.87901	+	3.25453i	−1.35391	−	2.34503i	−1.14504	+	0.931009i	8.51494	+	4.91611i	3.81635			6.71217	+	5.99556i	−1.33204
38.16	0.433498	−	0.750840i	−0.162662	−	2.99559i	1.62416	+	2.81313i	4.23652	+	7.33786i	−2.31972	−	1.17645i	3.54375	+	2.04598i	6.28426			−8.94708	+	0.974539i	7.34608
38.17	0.641309	−	1.11078i	2.61442	+	1.47132i	1.17745	+	2.03940i	0.763734	+	1.32283i	3.31097	−	1.96048i	−5.50560	−	3.17866i	8.15090			4.67043	+	7.69331i	1.95916
38.18	0.647986	−	1.12234i	2.50362	−	1.65285i	1.16023	+	2.00957i	−2.25417	−	3.90433i	−0.232759	−	3.88094i	−0.162124	−	0.0936021i	8.19114			3.53618	−	8.27620i	−5.84268
38.19	1.04133	−	1.80365i	−2.79053	+	1.10134i	−0.168757	−	0.292296i	2.59043	+	4.48676i	−0.919442	+	6.17999i	1.68126	+	0.970675i	7.62775			6.57409	−	6.14666i	10.7900
38.20	1.11386	−	1.92926i	−2.21197	+	2.02662i	−0.481371	−	0.833760i	−4.12144	−	7.13855i	1.44605	+	6.52485i	−8.40220	−	4.85102i	6.76616			0.785636	−	8.96564i	−18.3629

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
13.b	even	2	1	inner
117.n	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.3.n.a	✓	52
3.b	odd	2	1		351.3.n.a		52
9.c	even	3	1		351.3.n.a		52
9.d	odd	6	1	inner	117.3.n.a	✓	52
13.b	even	2	1	inner	117.3.n.a	✓	52
39.d	odd	2	1		351.3.n.a		52
117.n	odd	6	1	inner	117.3.n.a	✓	52
117.t	even	6	1		351.3.n.a		52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.3.n.a	✓	52	1.a	even	1	1	trivial
117.3.n.a	✓	52	9.d	odd	6	1	inner
117.3.n.a	✓	52	13.b	even	2	1	inner
117.3.n.a	✓	52	117.n	odd	6	1	inner
351.3.n.a		52	3.b	odd	2	1
351.3.n.a		52	9.c	even	3	1
351.3.n.a		52	39.d	odd	2	1
351.3.n.a		52	117.t	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).