Newform orbit 117.3.y.a

• Label: 117.3.y.a
• Level: $117$
• Weight: $3$
• Character orbit: 117.y
• Analytic conductor: $3.188$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.18801909302\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 8 q^{3} - 2 q^{5} + 10 q^{6} - 60 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−3.62659	−	0.971741i	−2.94231	−	0.585489i	8.74375	+	5.04820i	−1.52926	−	5.70728i	10.1016	+	4.98249i	2.26965	−	8.47046i	−16.1850	−	16.1850i	8.31441	+	3.44538i			22.1840i
31.2	−3.62452	−	0.971189i	1.24318	−	2.73029i	8.72987	+	5.04019i	0.305966	+	1.14188i	−7.15758	+	8.68865i	−3.11346	+	11.6196i	−16.1333	−	16.1333i	−5.90899	−	6.78851i		−	4.43593i
31.3	−3.32897	−	0.891995i	−0.797857	+	2.89196i	6.82228	+	3.93885i	2.07873	+	7.75792i	5.23566	−	8.91556i	1.39866	−	5.21987i	−9.44985	−	9.44985i	−7.72685	−	4.61474i		−	27.6801i
31.4	−3.18343	−	0.852998i	2.65511	+	1.39656i	5.94252	+	3.43092i	−1.23046	−	4.59214i	−7.26110	−	6.71066i	0.411953	−	1.53743i	−6.66931	−	6.66931i	5.09923	+	7.41605i			15.6684i
31.5	−2.62588	−	0.703603i	−1.51101	+	2.59169i	2.93610	+	1.69516i	−0.871516	−	3.25254i	5.79125	−	5.74232i	−3.16613	+	11.8161i	1.17198	+	1.17198i	−4.43370	−	7.83213i			9.15399i
31.6	−2.48463	−	0.665755i	−1.77959	−	2.41518i	2.26606	+	1.30831i	1.74973	+	6.53009i	2.81370	+	7.18559i	0.485382	−	1.81147i	2.51620	+	2.51620i	−2.66615	+	8.59602i		−	17.3897i
31.7	−2.32589	−	0.623221i	2.30915	−	1.91516i	1.55728	+	0.899094i	0.326628	+	1.21899i	−6.56440	+	3.01534i	2.50428	−	9.34611i	3.74897	+	3.74897i	1.66434	−	8.84477i		−	3.03881i
31.8	−1.66139	−	0.445168i	−0.924780	−	2.85391i	−0.902057	−	0.520803i	−1.70724	−	6.37150i	0.265953	+	5.15314i	−0.446010	+	1.66453i	6.13171	+	6.13171i	−7.28956	+	5.27847i			11.3456i
31.9	−1.60696	−	0.430585i	1.68465	+	2.48233i	−1.06717	−	0.616132i	0.359970	+	1.34342i	−1.63832	−	4.71440i	−0.198185	+	0.739637i	6.15513	+	6.15513i	−3.32390	+	8.36371i		−	2.31383i
31.10	−1.33961	−	0.358947i	−2.93992	+	0.597373i	−1.79839	−	1.03830i	−0.613360	−	2.28909i	4.15277	+	0.255030i	0.0472052	−	0.176172i	5.95909	+	5.95909i	8.28629	−	3.51246i			3.28665i
31.11	−1.06825	−	0.286237i	2.91836	−	0.695104i	−2.40487	−	1.38845i	1.81354	+	6.76822i	−3.31650	−	0.0927971i	−3.04303	+	11.3567i	5.29964	+	5.29964i	8.03366	−	4.05713i		−	7.74925i
31.12	−0.514097	−	0.137752i	−2.55437	+	1.57328i	−3.21878	−	1.85836i	1.42653	+	5.32387i	1.52991	−	0.456949i	2.51662	−	9.39214i	2.90415	+	2.90415i	4.04958	−	8.03747i		−	2.93349i
31.13	0.149774	+	0.0401318i	0.0310137	+	2.99984i	−3.44328	−	1.98798i	−1.98994	−	7.42657i	−0.115744	+	0.450542i	1.37301	−	5.12415i	−0.874500	−	0.874500i	−8.99808	+	0.186072i		−	1.19217i
31.14	0.181674	+	0.0486794i	2.99854	+	0.0936789i	−3.43347	−	1.98231i	−1.04393	−	3.89601i	0.540196	+	0.162986i	1.72376	−	6.43316i	−1.05925	−	1.05925i	8.98245	+	0.561799i		−	0.758622i
31.15	0.531767	+	0.142486i	−2.42608	−	1.76470i	−3.20163	−	1.84846i	1.12819	+	4.21046i	−1.03866	−	1.28409i	−2.13692	+	7.97511i	−2.99626	−	2.99626i	2.77170	+	8.56257i			2.39973i
31.16	0.583000	+	0.156214i	1.09433	−	2.79329i	−3.14862	−	1.81785i	−0.592695	−	2.21197i	1.07434	−	1.45754i	−0.518405	+	1.93471i	−3.25881	−	3.25881i	−6.60490	−	6.11353i		−	1.38217i
31.17	1.07169	+	0.287157i	−0.645473	+	2.92974i	−2.39805	−	1.38451i	0.951873	+	3.55244i	−1.53304	+	2.95441i	−2.20661	+	8.23517i	−5.31050	−	5.31050i	−8.16673	−	3.78213i			4.08044i
31.18	1.55517	+	0.416705i	2.60419	+	1.48937i	−1.21920	−	0.703908i	2.23994	+	8.35955i	3.42932	+	3.40139i	2.18521	−	8.15533i	−6.15658	−	6.15658i	4.56358	+	7.75718i			13.9339i
31.19	1.88597	+	0.505345i	−2.06210	−	2.17894i	−0.162580	−	0.0938654i	−0.0385330	−	0.143807i	−2.78794	−	5.15148i	3.35702	−	12.5286i	−5.78170	−	5.78170i	−0.495516	+	8.98635i		−	0.290689i
31.20	1.91711	+	0.513688i	−2.98190	+	0.329042i	−0.0526611	−	0.0304039i	−2.49739	−	9.32037i	−5.88566	−	0.900957i	−1.04281	+	3.89184i	−5.69903	−	5.69903i	8.78346	−	1.96234i		−	19.1511i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
13.d	odd	4	1	inner
117.y	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.3.y.a	✓	104
3.b	odd	2	1		351.3.bb.a		104
9.c	even	3	1	inner	117.3.y.a	✓	104
9.d	odd	6	1		351.3.bb.a		104
13.d	odd	4	1	inner	117.3.y.a	✓	104
39.f	even	4	1		351.3.bb.a		104
117.y	odd	12	1	inner	117.3.y.a	✓	104
117.z	even	12	1		351.3.bb.a		104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.3.y.a	✓	104	1.a	even	1	1	trivial
117.3.y.a	✓	104	9.c	even	3	1	inner
117.3.y.a	✓	104	13.d	odd	4	1	inner
117.3.y.a	✓	104	117.y	odd	12	1	inner
351.3.bb.a		104	3.b	odd	2	1
351.3.bb.a		104	9.d	odd	6	1
351.3.bb.a		104	39.f	even	4	1
351.3.bb.a		104	117.z	even	12	1

Hecke kernels

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