Newform orbit 414.3.l.a

• Label: 414.3.l.a
• Level: $414$
• Weight: $3$
• Character orbit: 414.l
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.l (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(4\) over \(\Q(\zeta_{22})\)
Twist minimal:	no (minimal twist has level 46)
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−0.587486	−	1.28641i		0		−1.30972	+	1.51150i	−0.680920	+	1.05953i		0		−2.11717	+	0.304404i	2.71386	+	0.796860i		0		1.76303	+	0.253485i
19.2	−0.587486	−	1.28641i		0		−1.30972	+	1.51150i	3.93626	−	6.12494i		0		12.3131	−	1.77035i	2.71386	+	0.796860i		0		−10.1917	−	1.46534i
19.3	0.587486	+	1.28641i		0		−1.30972	+	1.51150i	1.16762	−	1.81685i		0		−10.6509	+	1.53137i	−2.71386	−	0.796860i		0		3.02319	+	0.434669i
19.4	0.587486	+	1.28641i		0		−1.30972	+	1.51150i	2.08772	−	3.24855i		0		−5.96665	+	0.857874i	−2.71386	−	0.796860i		0		5.40548	+	0.777191i
37.1	−1.35693	+	0.398430i		0		1.68251	−	1.08128i	−3.51835	+	0.505863i		0		0.228956	+	0.104560i	−1.85223	+	2.13758i		0		4.57260	−	2.08824i
37.2	−1.35693	+	0.398430i		0		1.68251	−	1.08128i	3.34128	−	0.480403i		0		−3.29739	−	1.50587i	−1.85223	+	2.13758i		0		−4.34246	+	1.98314i
37.3	1.35693	−	0.398430i		0		1.68251	−	1.08128i	−7.10773	+	1.02194i		0		4.73539	+	2.16258i	1.85223	−	2.13758i		0		−9.23750	+	4.21863i
37.4	1.35693	−	0.398430i		0		1.68251	−	1.08128i	6.93065	−	0.996477i		0		9.13753	+	4.17297i	1.85223	−	2.13758i		0		9.00737	−	4.11353i
109.1	−0.587486	+	1.28641i		0		−1.30972	−	1.51150i	−0.680920	−	1.05953i		0		−2.11717	−	0.304404i	2.71386	−	0.796860i		0		1.76303	−	0.253485i
109.2	−0.587486	+	1.28641i		0		−1.30972	−	1.51150i	3.93626	+	6.12494i		0		12.3131	+	1.77035i	2.71386	−	0.796860i		0		−10.1917	+	1.46534i
109.3	0.587486	−	1.28641i		0		−1.30972	−	1.51150i	1.16762	+	1.81685i		0		−10.6509	−	1.53137i	−2.71386	+	0.796860i		0		3.02319	−	0.434669i
109.4	0.587486	−	1.28641i		0		−1.30972	−	1.51150i	2.08772	+	3.24855i		0		−5.96665	−	0.857874i	−2.71386	+	0.796860i		0		5.40548	−	0.777191i
145.1	−0.926113	+	1.06879i		0		−0.284630	−	1.97964i	−4.73531	+	2.16254i		0		−0.352465	−	1.20038i	2.37942	+	1.52916i		0		2.07412	−	7.06382i
145.2	−0.926113	+	1.06879i		0		−0.284630	−	1.97964i	7.87646	−	3.59706i		0		2.42105	+	8.24534i	2.37942	+	1.52916i		0		−3.44999	+	11.7496i
145.3	0.926113	−	1.06879i		0		−0.284630	−	1.97964i	−3.29959	+	1.50687i		0		1.07230	+	3.65191i	−2.37942	−	1.52916i		0		−1.44526	+	4.92211i
145.4	0.926113	−	1.06879i		0		−0.284630	−	1.97964i	6.44075	−	2.94139i		0		−0.0656098	−	0.223446i	−2.37942	−	1.52916i		0		2.82113	−	9.60787i
181.1	−0.201264	−	1.39982i		0		−1.91899	+	0.563465i	−6.91983	−	5.99606i		0		−4.33066	−	6.73865i	1.17497	+	2.57283i		0		−7.00069	+	10.8933i
181.2	−0.201264	−	1.39982i		0		−1.91899	+	0.563465i	1.05657	+	0.915527i		0		2.42614	+	3.77515i	1.17497	+	2.57283i		0		1.06892	−	1.66328i
181.3	0.201264	+	1.39982i		0		−1.91899	+	0.563465i	−5.47346	−	4.74278i		0		−0.874937	−	1.36143i	−1.17497	−	2.57283i		0		5.53742	−	8.61640i
181.4	0.201264	+	1.39982i		0		−1.91899	+	0.563465i	−0.389792	−	0.337757i		0		−2.12361	−	3.30440i	−1.17497	−	2.57283i		0		0.394348	−	0.613617i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.d	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	414.3.l.a		40
3.b	odd	2	1		46.3.d.a	✓	40
12.b	even	2	1		368.3.p.b		40
23.d	odd	22	1	inner	414.3.l.a		40
69.g	even	22	1		46.3.d.a	✓	40
69.g	even	22	1		1058.3.b.e		40
69.h	odd	22	1		1058.3.b.e		40
276.j	odd	22	1		368.3.p.b		40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
46.3.d.a	✓	40	3.b	odd	2	1
46.3.d.a	✓	40	69.g	even	22	1
368.3.p.b		40	12.b	even	2	1
368.3.p.b		40	276.j	odd	22	1
414.3.l.a		40	1.a	even	1	1	trivial
414.3.l.a		40	23.d	odd	22	1	inner
1058.3.b.e		40	69.g	even	22	1
1058.3.b.e		40	69.h	odd	22	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^40 - 182*T5^38 - 286*T5^37 + 20465*T5^36 + 34144*T5^35 - 1518999*T5^34 - 1054328*T5^33 + 83680980*T5^32 - 27332932*T5^31 - 3813170125*T5^30 + 2587791558*T5^29 + 131374725404*T5^28 + 13513602696*T5^27 - 2507846339829*T5^26 - 7655054507254*T5^25 + 10360034949595*T5^24 + 259734253310070*T5^23 + 19170090656567*T5^22 - 3600543822797064*T5^21 + 33426808666852315*T5^20 + 79338625695616980*T5^19 - 620486083323115237*T5^18 + 277593074125439930*T5^17 + 5772326832195377755*T5^16 - 27699563414639055918*T5^15 - 47115063111288937233*T5^14 + 243848338591110971168*T5^13 + 40771124589812247780*T5^12 - 533910280109446766922*T5^11 + 2115877439730835114383*T5^10 + 1896917629393936517940*T5^9 + 4230069966673305518504*T5^8 - 906955337283824014604*T5^7 + 6151909785947153261173*T5^6 + 8205657208752868548640*T5^5 + 12108683241596833831625*T5^4 + 8725587522891604665410*T5^3 + 5040705919688995071882*T5^2 + 1665455262590699159668*T5 + 391612757977095518161