Newform orbit 342.3.z.b

• Label: 342.3.z.b
• Level: $342$
• Weight: $3$
• Character orbit: 342.z
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	342.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.31882504112\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 18 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
91.1	−0.483690	+	1.32893i		0		−1.53209	−	1.28558i	−6.04513	+	5.07246i		0		−3.82954	+	6.63297i	2.44949	−	1.41421i		0		−3.81697	−	10.4870i
91.2	−0.483690	+	1.32893i		0		−1.53209	−	1.28558i	6.81117	−	5.71525i		0		2.55329	−	4.42243i	2.44949	−	1.41421i		0		4.30065	+	11.8160i
91.3	0.483690	−	1.32893i		0		−1.53209	−	1.28558i	−0.297798	+	0.249882i		0		−2.96431	+	5.13434i	−2.44949	+	1.41421i		0		0.188033	+	0.516617i
91.4	0.483690	−	1.32893i		0		−1.53209	−	1.28558i	1.06384	−	0.892670i		0		−4.39759	+	7.61684i	−2.44949	+	1.41421i		0		−0.671723	−	1.84554i
109.1	−0.483690	−	1.32893i		0		−1.53209	+	1.28558i	−6.04513	−	5.07246i		0		−3.82954	−	6.63297i	2.44949	+	1.41421i		0		−3.81697	+	10.4870i
109.2	−0.483690	−	1.32893i		0		−1.53209	+	1.28558i	6.81117	+	5.71525i		0		2.55329	+	4.42243i	2.44949	+	1.41421i		0		4.30065	−	11.8160i
109.3	0.483690	+	1.32893i		0		−1.53209	+	1.28558i	−0.297798	−	0.249882i		0		−2.96431	−	5.13434i	−2.44949	−	1.41421i		0		0.188033	−	0.516617i
109.4	0.483690	+	1.32893i		0		−1.53209	+	1.28558i	1.06384	+	0.892670i		0		−4.39759	−	7.61684i	−2.44949	−	1.41421i		0		−0.671723	+	1.84554i
127.1	−0.909039	−	1.08335i		0		−0.347296	+	1.96962i	−0.0233547	−	0.132451i		0		4.79341	−	8.30244i	2.44949	−	1.41421i		0		−0.122261	+	0.145705i
127.2	−0.909039	−	1.08335i		0		−0.347296	+	1.96962i	0.197003	+	1.11726i		0		−5.55599	+	9.62326i	2.44949	−	1.41421i		0		1.03130	−	1.22906i
127.3	0.909039	+	1.08335i		0		−0.347296	+	1.96962i	−0.678914	−	3.85031i		0		1.77574	−	3.07567i	−2.44949	+	1.41421i		0		3.55408	−	4.23559i
127.4	0.909039	+	1.08335i		0		−0.347296	+	1.96962i	0.852562	+	4.83512i		0		0.583107	−	1.00997i	−2.44949	+	1.41421i		0		−4.46312	+	5.31894i
181.1	−1.39273	+	0.245576i		0		1.87939	−	0.684040i	−5.30025	−	1.92913i		0		−0.990297	+	1.71524i	−2.44949	+	1.41421i		0		7.85555	+	1.38515i
181.2	−1.39273	+	0.245576i		0		1.87939	−	0.684040i	4.36056	+	1.58711i		0		−2.18088	+	3.77740i	−2.44949	+	1.41421i		0		−6.46283	−	1.13957i
181.3	1.39273	−	0.245576i		0		1.87939	−	0.684040i	−7.98876	−	2.90767i		0		−2.58545	+	4.47813i	2.44949	−	1.41421i		0		−11.8402	−	2.08775i
181.4	1.39273	−	0.245576i		0		1.87939	−	0.684040i	7.04907	+	2.56565i		0		3.79852	−	6.57923i	2.44949	−	1.41421i		0		10.4475	+	1.84218i
307.1	−0.909039	+	1.08335i		0		−0.347296	−	1.96962i	−0.0233547	+	0.132451i		0		4.79341	+	8.30244i	2.44949	+	1.41421i		0		−0.122261	−	0.145705i
307.2	−0.909039	+	1.08335i		0		−0.347296	−	1.96962i	0.197003	−	1.11726i		0		−5.55599	−	9.62326i	2.44949	+	1.41421i		0		1.03130	+	1.22906i
307.3	0.909039	−	1.08335i		0		−0.347296	−	1.96962i	−0.678914	+	3.85031i		0		1.77574	+	3.07567i	−2.44949	−	1.41421i		0		3.55408	+	4.23559i
307.4	0.909039	−	1.08335i		0		−0.347296	−	1.96962i	0.852562	−	4.83512i		0		0.583107	+	1.00997i	−2.44949	−	1.41421i		0		−4.46312	−	5.31894i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.f	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	342.3.z.b		24
3.b	odd	2	1		38.3.f.a	✓	24
12.b	even	2	1		304.3.z.c		24
19.f	odd	18	1	inner	342.3.z.b		24
57.j	even	18	1		38.3.f.a	✓	24
57.j	even	18	1		722.3.b.f		24
57.l	odd	18	1		722.3.b.f		24
228.u	odd	18	1		304.3.z.c		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.3.f.a	✓	24	3.b	odd	2	1
38.3.f.a	✓	24	57.j	even	18	1
304.3.z.c		24	12.b	even	2	1
304.3.z.c		24	228.u	odd	18	1
342.3.z.b		24	1.a	even	1	1	trivial
342.3.z.b		24	19.f	odd	18	1	inner
722.3.b.f		24	57.j	even	18	1
722.3.b.f		24	57.l	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^24 - 126*T5^22 + 182*T5^21 + 11529*T5^20 - 15822*T5^19 - 544119*T5^18 + 654174*T5^17 + 14968620*T5^16 - 37014352*T5^15 + 308413296*T5^14 + 1122199542*T5^13 - 12709400570*T5^12 - 4131993366*T5^11 + 90703937871*T5^10 - 494498437470*T5^9 + 5845525761942*T5^8 - 10444822811694*T5^7 + 13761686660553*T5^6 - 7195607667684*T5^5 + 4740643818657*T5^4 + 4771464680646*T5^3 + 2202382936638*T5^2 + 173287312416*T5 + 34296447249