Newform orbit 162.4.e.b

• Label: 162.4.e.b
• Level: $162$
• Weight: $4$
• Character orbit: 162.e
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 12 q^{5} + 33 q^{7} - 120 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.87939	+	0.684040i		0		3.06418	−	2.57115i	−3.05422	−	17.3214i		0		−19.9833	−	16.7680i	−4.00000	+	6.92820i		0		17.5886	+	30.4643i
19.2	−1.87939	+	0.684040i		0		3.06418	−	2.57115i	−2.39299	−	13.5713i		0		20.6081	+	17.2922i	−4.00000	+	6.92820i		0		13.7807	+	23.8688i
19.3	−1.87939	+	0.684040i		0		3.06418	−	2.57115i	−0.127307	−	0.721992i		0		1.18538	+	0.994655i	−4.00000	+	6.92820i		0		0.733130	+	1.26982i
19.4	−1.87939	+	0.684040i		0		3.06418	−	2.57115i	2.36712	+	13.4246i		0		−10.7136	−	8.98975i	−4.00000	+	6.92820i		0		−13.6317	−	23.6108i
19.5	−1.87939	+	0.684040i		0		3.06418	−	2.57115i	3.01146	+	17.0788i		0		16.7015	+	14.0142i	−4.00000	+	6.92820i		0		−17.3423	−	30.0378i
37.1	1.53209	−	1.28558i		0		0.694593	−	3.93923i	−13.2521	+	4.82337i		0		3.00195	+	17.0249i	−4.00000	−	6.92820i		0		−14.1026	+	24.4264i
37.2	1.53209	−	1.28558i		0		0.694593	−	3.93923i	−11.4993	+	4.18541i		0		3.15689	+	17.9036i	−4.00000	−	6.92820i		0		−12.2373	+	21.1956i
37.3	1.53209	−	1.28558i		0		0.694593	−	3.93923i	−3.40278	+	1.23851i		0		−3.51598	−	19.9401i	−4.00000	−	6.92820i		0		−3.62116	+	6.27204i
37.4	1.53209	−	1.28558i		0		0.694593	−	3.93923i	7.46345	−	2.71647i		0		−2.66046	−	15.0882i	−4.00000	−	6.92820i		0		7.94244	−	13.7567i
37.5	1.53209	−	1.28558i		0		0.694593	−	3.93923i	17.2977	−	6.29584i		0		6.03855	+	34.2463i	−4.00000	−	6.92820i		0		18.4078	−	31.8832i
73.1	0.347296	−	1.96962i		0		−3.75877	−	1.36808i	−7.73816	+	6.49309i		0		7.54967	−	2.74785i	−4.00000	+	6.92820i		0		10.1015	+	17.4962i
73.2	0.347296	−	1.96962i		0		−3.75877	−	1.36808i	−6.77242	+	5.68273i		0		30.7336	−	11.1861i	−4.00000	+	6.92820i		0		8.84076	+	15.3127i
73.3	0.347296	−	1.96962i		0		−3.75877	−	1.36808i	−0.657259	+	0.551505i		0		−20.5288	+	7.47187i	−4.00000	+	6.92820i		0		0.857990	+	1.48608i
73.4	0.347296	−	1.96962i		0		−3.75877	−	1.36808i	9.84993	−	8.26508i		0		−26.8516	+	9.77320i	−4.00000	+	6.92820i		0		−12.8582	−	22.2710i
73.5	0.347296	−	1.96962i		0		−3.75877	−	1.36808i	14.9069	−	12.5084i		0		11.7781	−	4.28689i	−4.00000	+	6.92820i		0		−19.4596	−	33.7050i
91.1	0.347296	+	1.96962i		0		−3.75877	+	1.36808i	−7.73816	−	6.49309i		0		7.54967	+	2.74785i	−4.00000	−	6.92820i		0		10.1015	−	17.4962i
91.2	0.347296	+	1.96962i		0		−3.75877	+	1.36808i	−6.77242	−	5.68273i		0		30.7336	+	11.1861i	−4.00000	−	6.92820i		0		8.84076	−	15.3127i
91.3	0.347296	+	1.96962i		0		−3.75877	+	1.36808i	−0.657259	−	0.551505i		0		−20.5288	−	7.47187i	−4.00000	−	6.92820i		0		0.857990	−	1.48608i
91.4	0.347296	+	1.96962i		0		−3.75877	+	1.36808i	9.84993	+	8.26508i		0		−26.8516	−	9.77320i	−4.00000	−	6.92820i		0		−12.8582	+	22.2710i
91.5	0.347296	+	1.96962i		0		−3.75877	+	1.36808i	14.9069	+	12.5084i		0		11.7781	+	4.28689i	−4.00000	−	6.92820i		0		−19.4596	+	33.7050i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.4.e.b		30
3.b	odd	2	1		54.4.e.b	✓	30
27.e	even	9	1	inner	162.4.e.b		30
27.e	even	9	1		1458.4.a.j		15
27.f	odd	18	1		54.4.e.b	✓	30
27.f	odd	18	1		1458.4.a.i		15

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.4.e.b	✓	30	3.b	odd	2	1
54.4.e.b	✓	30	27.f	odd	18	1
162.4.e.b		30	1.a	even	1	1	trivial
162.4.e.b		30	27.e	even	9	1	inner
1458.4.a.i		15	27.f	odd	18	1
1458.4.a.j		15	27.e	even	9	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^30 - 12*T5^29 + 288*T5^28 - 1077*T5^27 - 7965*T5^26 + 1743363*T5^25 + 7544988*T5^24 + 107176932*T5^23 + 5239574748*T5^22 - 56186887842*T5^21 + 776738951613*T5^20 + 33295905820323*T5^19 + 538031305531971*T5^18 + 9470908894122324*T5^17 + 116831265863124312*T5^16 + 807525497728322793*T5^15 + 12459848342420520558*T5^14 + 196451207835627177288*T5^13 + 2463364103381439322464*T5^12 + 28342032699721311967518*T5^11 + 213914784932151977385018*T5^10 - 1086946898088440305025451*T5^9 - 24851671359647079960049086*T5^8 - 56015892596566334106830763*T5^7 + 1372884370965282518466927669*T5^6 + 10332697078675107661340059632*T5^5 + 28723378260909320654046959340*T5^4 + 35820655458076594122624961272*T5^3 + 29843185477858381469104998624*T5^2 + 15448935323095314214017461472*T5 + 5414906537278760306136568896