Newform orbit 432.2.y.e

• Label: 432.2.y.e
• Level: $432$
• Weight: $2$
• Character orbit: 432.y
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 144)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{2} - 2 q^{4} - 4 q^{5} + 8 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.41364	−	0.0402314i		0		1.99676	+	0.113746i	2.21121	−	0.592492i		0		2.67054	−	1.54184i	−2.81813	−	0.241128i		0		−3.14970	+	0.748611i
37.2	−1.30266	−	0.550520i		0		1.39386	+	1.43428i	−2.73721	+	0.733432i		0		−1.14487	+	0.660988i	−1.02612	−	2.63573i		0		3.96942	+	0.551472i
37.3	−1.11640	−	0.868135i		0		0.492685	+	1.93837i	2.41406	−	0.646846i		0		2.82197	−	1.62927i	1.13273	−	2.59170i		0		−3.25660	−	1.37359i
37.4	−1.10904	+	0.877514i		0		0.459938	−	1.94640i	1.98415	−	0.531653i		0		−1.54969	+	0.894715i	1.19790	+	2.56223i		0		−1.73397	+	2.33075i
37.5	−1.05869	−	0.937649i		0		0.241628	+	1.98535i	0.491749	−	0.131764i		0		−2.40518	+	1.38863i	1.60575	−	2.32842i		0		−0.644156	−	0.321592i
37.6	−0.902916	+	1.08846i		0		−0.369486	−	1.96557i	−1.21694	+	0.326078i		0		−0.707732	+	0.408609i	2.47306	+	1.37258i		0		0.743871	−	1.61901i
37.7	−0.297230	−	1.38263i		0		−1.82331	+	0.821916i	0.0112878	−	0.00302457i		0		1.05753	−	0.610563i	1.67835	+	2.27665i		0		−0.00753693	−	0.0147079i
37.8	−0.268956	−	1.38840i		0		−1.85533	+	0.746838i	−2.97932	+	0.798307i		0		1.78208	−	1.02889i	1.53591	+	2.37507i		0		1.90968	+	3.92179i
37.9	0.0882790	+	1.41146i		0		−1.98441	+	0.249204i	−2.90072	+	0.777246i		0		−1.04527	+	0.603486i	−0.526922	−	2.77891i		0		−1.35312	−	4.02562i
37.10	0.174715	−	1.40338i		0		−1.93895	−	0.490384i	−0.174734	+	0.0468197i		0		−4.04791	+	2.33706i	−1.02696	+	2.63540i		0		0.0351772	+	0.253398i
37.11	0.704761	+	1.22610i		0		−1.00662	+	1.72821i	−2.53632	+	0.679606i		0		0.614293	−	0.354662i	−2.82838	−	0.0162452i		0		−2.62076	−	2.63082i
37.12	0.751042	−	1.19831i		0		−0.871873	−	1.79995i	1.60558	−	0.430214i		0		3.62762	−	2.09441i	−2.81171	−	0.307071i		0		0.690330	−	2.24708i
37.13	0.812713	+	1.15737i		0		−0.678997	+	1.88121i	2.69708	−	0.722679i		0		−2.89314	+	1.67035i	−2.72908	+	0.743037i		0		3.02835	+	2.53418i
37.14	1.18142	−	0.777339i		0		0.791487	−	1.83672i	−3.30105	+	0.884514i		0		2.63210	−	1.51965i	−0.492680	−	2.78519i		0		−3.21235	+	3.61102i
37.15	1.20757	−	0.736049i		0		0.916464	−	1.77767i	4.10876	−	1.10094i		0		−1.63313	+	0.942891i	−0.201751	−	2.82122i		0		4.15129	−	4.35372i
37.16	1.35510	+	0.404605i		0		1.67259	+	1.09656i	−3.32531	+	0.891015i		0		−3.95817	+	2.28525i	1.82285	+	2.16269i		0		−4.86664	−	0.138025i
37.17	1.38700	−	0.276075i		0		1.84757	−	0.765835i	0.0691269	−	0.0185225i		0		1.28192	−	0.740118i	2.35115	−	1.57228i		0		0.0907658	−	0.0447750i
37.18	1.40500	+	0.161164i		0		1.94805	+	0.452871i	0.846545	−	0.226831i		0		−0.567074	+	0.327400i	2.66403	+	0.950239i		0		1.22595	−	0.182265i
181.1	−1.41328	−	0.0512684i		0		1.99474	+	0.144914i	−0.430214	+	1.60558i		0		−3.62762	−	2.09441i	−2.81171	−	0.307071i		0		0.690330	−	2.24708i
181.2	−1.30272	+	0.550382i		0		1.39416	−	1.43399i	0.0468197	−	0.174734i		0		4.04791	+	2.33706i	−1.02696	+	2.63540i		0		0.0351772	+	0.253398i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
16.e	even	4	1	inner
144.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	432.2.y.e		72
3.b	odd	2	1		144.2.x.e	✓	72
4.b	odd	2	1		1728.2.bc.e		72
9.c	even	3	1	inner	432.2.y.e		72
9.d	odd	6	1		144.2.x.e	✓	72
12.b	even	2	1		576.2.bb.e		72
16.e	even	4	1	inner	432.2.y.e		72
16.f	odd	4	1		1728.2.bc.e		72
36.f	odd	6	1		1728.2.bc.e		72
36.h	even	6	1		576.2.bb.e		72
48.i	odd	4	1		144.2.x.e	✓	72
48.k	even	4	1		576.2.bb.e		72
144.u	even	12	1		576.2.bb.e		72
144.v	odd	12	1		1728.2.bc.e		72
144.w	odd	12	1		144.2.x.e	✓	72
144.x	even	12	1	inner	432.2.y.e		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.2.x.e	✓	72	3.b	odd	2	1
144.2.x.e	✓	72	9.d	odd	6	1
144.2.x.e	✓	72	48.i	odd	4	1
144.2.x.e	✓	72	144.w	odd	12	1
432.2.y.e		72	1.a	even	1	1	trivial
432.2.y.e		72	9.c	even	3	1	inner
432.2.y.e		72	16.e	even	4	1	inner
432.2.y.e		72	144.x	even	12	1	inner
576.2.bb.e		72	12.b	even	2	1
576.2.bb.e		72	36.h	even	6	1
576.2.bb.e		72	48.k	even	4	1
576.2.bb.e		72	144.u	even	12	1
1728.2.bc.e		72	4.b	odd	2	1
1728.2.bc.e		72	16.f	odd	4	1
1728.2.bc.e		72	36.f	odd	6	1
1728.2.bc.e		72	144.v	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^72 + 4*T5^71 + 8*T5^70 + 48*T5^69 - 362*T5^68 - 1784*T5^67 - 3088*T5^66 - 23768*T5^65 + 94237*T5^64 + 647604*T5^63 + 1169128*T5^62 + 9220904*T5^61 + 4482466*T5^60 - 87574008*T5^59 - 102190800*T5^58 - 1185653784*T5^57 - 2169959238*T5^56 + 8254656280*T5^55 + 6905641872*T5^54 + 101695957504*T5^53 + 295350309958*T5^52 - 533946292408*T5^51 - 175025802064*T5^50 - 5628465266968*T5^49 - 23026889487723*T5^48 + 28260485297716*T5^47 + 3035870852136*T5^46 + 232390112675416*T5^45 + 1224421682900894*T5^44 - 1240988593682056*T5^43 + 172437944203344*T5^42 - 7250050997426568*T5^41 - 46279845911145158*T5^40 + 48670605365446072*T5^39 - 6925557804744304*T5^38 + 189546581537084768*T5^37 + 1216211875310614698*T5^36 - 1563571872169877000*T5^35 + 499101228449455952*T5^34 - 4344279007693992008*T5^33 - 20879107223791719475*T5^32 + 37086914580080592148*T5^31 - 14921726447356849304*T5^30 + 78613589983483996552*T5^29 + 164870016580306565158*T5^28 - 538599259122490202056*T5^27 + 479791665250293780688*T5^26 - 989958955532405566152*T5^25 + 473404950447331448209*T5^24 + 1237114307880567719028*T5^23 - 1455235273623812252472*T5^22 + 2561931499336123469440*T5^21 - 633893128674998962120*T5^20 - 2898867428941814917632*T5^19 + 3008293621653808728640*T5^18 - 3760234108568108074112*T5^17 + 3973109811737667405360*T5^16 - 1816677854618774416960*T5^15 + 706395466379395084160*T5^14 - 362817429438562370560*T5^13 + 52519669916909275520*T5^12 + 17411820000057848320*T5^11 - 1270940444129055744*T5^10 + 522867693995038720*T5^9 + 68383728502827264*T5^8 - 12099310841701376*T5^7 + 638304994052096*T5^6 - 65234148261888*T5^5 + 4529605668864*T5^4 - 66961801216*T5^3 + 615022592*T5^2 - 8978432*T5 + 65536