Newform orbit 117.2.x.a

• Label: 117.2.x.a
• Level: $117$
• Weight: $2$
• Character orbit: 117.x
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} - 6 q^{5} - 8 q^{6} - 4 q^{7} + 30 q^{8} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.90884	−	1.90884i	1.70206	−	0.320913i			5.28736i	0.518864	−	1.93643i	−3.86154	−	2.63639i	0.463844	−	1.73109i	6.27505	−	6.27505i	2.79403	−	1.09243i	−4.68676	+	2.70590i
2.2	−1.68690	−	1.68690i	−1.73170	+	0.0346549i			3.69124i	−0.162372	+	0.605981i	2.97967	+	2.86275i	−0.608844	+	2.27224i	2.85295	−	2.85295i	2.99760	−	0.120024i	1.29613	−	0.748322i
2.3	−1.27835	−	1.27835i	0.905977	+	1.47621i			1.26837i	−0.981766	+	3.66400i	0.728964	−	3.04528i	−0.245704	+	0.916981i	−0.935276	+	0.935276i	−1.35841	+	2.67483i	5.93893	−	3.42884i
2.4	−1.13966	−	1.13966i	−0.338004	−	1.69875i			0.597661i	−0.184916	+	0.690114i	−1.55079	+	2.32121i	1.13668	−	4.24215i	−1.59819	+	1.59819i	−2.77151	+	1.14837i	0.997239	−	0.575756i
2.5	−0.754009	−	0.754009i	0.861374	−	1.50268i		−	0.862941i	0.595426	−	2.22216i	−1.78251	+	0.483547i	−1.34030	+	5.00206i	−2.15868	+	2.15868i	−1.51607	−	2.58873i	−2.12448	+	1.22657i
2.6	−0.530710	−	0.530710i	−1.31776	+	1.12406i		−	1.43669i	0.231013	−	0.862152i	1.29590	+	0.102802i	0.858276	−	3.20313i	−1.82389	+	1.82389i	0.472998	−	2.96248i	−0.580153	+	0.334952i
2.7	0.0604708	+	0.0604708i	1.25573	+	1.19295i		−	1.99269i	0.466078	−	1.73943i	0.00379642	+	0.148074i	0.132635	−	0.495001i	0.241441	−	0.241441i	0.153731	+	2.99606i	0.133369	−	0.0770004i
2.8	0.311551	+	0.311551i	1.53120	−	0.809582i		−	1.80587i	−0.789277	+	2.94562i	0.729272	+	0.224821i	0.0751575	−	0.280491i	1.18572	−	1.18572i	1.68915	−	2.47927i	−1.16361	+	0.671810i
2.9	0.632076	+	0.632076i	−1.57700	−	0.716290i		−	1.20096i	0.753527	−	2.81220i	−0.544035	−	1.44953i	−0.239127	+	0.892433i	2.02325	−	2.02325i	1.97386	+	2.25918i	2.25381	−	1.30124i
2.10	0.954676	+	0.954676i	−0.157210	+	1.72490i		−	0.177187i	−0.0632653	+	0.236109i	−1.79681	+	1.49664i	−0.804746	+	3.00335i	2.07851	−	2.07851i	−2.95057	−	0.542344i	−0.285806	+	0.165010i
2.11	1.36755	+	1.36755i	−1.69413	+	0.360438i			1.74037i	−1.08145	+	4.03603i	−2.80972	−	1.82389i	0.692170	−	2.58321i	0.355058	−	0.355058i	2.74017	−	1.22126i	−6.99840	+	4.04053i
2.12	1.60613	+	1.60613i	0.0594668	−	1.73103i			3.15929i	0.0641657	−	0.239470i	2.87576	−	2.68474i	−0.254018	+	0.948008i	−1.86196	+	1.86196i	−2.99293	−	0.205878i	0.487677	−	0.281560i
11.1	−1.86694	+	1.86694i	0.154854	+	1.72511i		−	4.97091i	−2.60400	−	0.697740i	−3.50978	−	2.93158i	−2.18636	−	0.585835i	5.54651	+	5.54651i	−2.95204	+	0.534283i	6.16415	−	3.55887i
11.2	−1.75651	+	1.75651i	−1.38793	−	1.03618i		−	4.17069i	3.52054	+	0.943327i	4.25797	−	0.617852i	−1.05231	−	0.281966i	3.81284	+	3.81284i	0.852675	+	2.87627i	−7.84085	+	4.52692i
11.3	−1.53591	+	1.53591i	1.40963	−	1.00645i		−	2.71806i	−0.529972	−	0.142006i	−0.619261	+	3.71089i	4.13954	+	1.10919i	1.10288	+	1.10288i	0.974134	−	2.83744i	1.03210	−	0.595883i
11.4	−0.841634	+	0.841634i	−0.947612	−	1.44984i			0.583303i	−2.77680	−	0.744040i	2.01778	+	0.422691i	−1.42554	−	0.381973i	−2.17420	−	2.17420i	−1.20406	+	2.74777i	2.96326	−	1.71084i
11.5	−0.662754	+	0.662754i	1.56154	+	0.749386i			1.12151i	0.101414	+	0.0271738i	−1.53158	+	0.538261i	−0.353095	−	0.0946115i	−2.06880	−	2.06880i	1.87684	+	2.34040i	−0.0852221	+	0.0492030i
11.6	−0.0488315	+	0.0488315i	0.804456	−	1.53390i			1.99523i	2.57746	+	0.690628i	0.0356199	+	0.114186i	−2.39973	−	0.643006i	−0.195093	−	0.195093i	−1.70570	−	2.46791i	−0.159586	+	0.0921368i
11.7	0.286505	−	0.286505i	−1.72577	−	0.147373i			1.83583i	2.00620	+	0.537559i	−0.536665	+	0.452219i	1.90533	+	0.510531i	1.09899	+	1.09899i	2.95656	+	0.508664i	0.728800	−	0.420773i
11.8	0.375702	−	0.375702i	−0.0440912	+	1.73149i			1.71770i	−2.85246	−	0.764314i	0.633958	+	0.667088i	3.85210	+	1.03217i	1.39674	+	1.39674i	−2.99611	−	0.152687i	−1.35883	+	0.784520i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.2.x.a	✓	48
3.b	odd	2	1		351.2.ba.a		48
9.c	even	3	1		351.2.bf.a		48
9.d	odd	6	1		117.2.bc.a	yes	48
13.f	odd	12	1		117.2.bc.a	yes	48
39.k	even	12	1		351.2.bf.a		48
117.w	odd	12	1		351.2.ba.a		48
117.x	even	12	1	inner	117.2.x.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.x.a	✓	48	1.a	even	1	1	trivial
117.2.x.a	✓	48	117.x	even	12	1	inner
117.2.bc.a	yes	48	9.d	odd	6	1
117.2.bc.a	yes	48	13.f	odd	12	1
351.2.ba.a		48	3.b	odd	2	1
351.2.ba.a		48	117.w	odd	12	1
351.2.bf.a		48	9.c	even	3	1
351.2.bf.a		48	39.k	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(117, [\chi])\).