Newform orbit 540.2.y.a

• Label: 540.2.y.a
• Level: $540$
• Weight: $2$
• Character orbit: 540.y
• Analytic conductor: $4.312$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.31192170915\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 180)
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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 2 q^{2} + 4 q^{5} + 8 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
127.1	−1.41305	+	0.0573585i		0		1.99342	−	0.162101i	1.42527	+	1.72296i		0		−0.891731	−	3.32798i	−2.80750	+	0.343396i		0		−2.11281	−	2.35288i
127.2	−1.41143	+	0.0886778i		0		1.98427	−	0.250325i	−0.395606	−	2.20079i		0		0.331032	+	1.23543i	−2.77846	+	0.529277i		0		0.753532	+	3.07119i
127.3	−1.29311	−	0.572594i		0		1.34427	+	1.48086i	1.88223	−	1.20715i		0		0.361284	+	1.34833i	−0.890364	−	2.68463i		0		−3.12514	+	0.483221i
127.4	−1.26667	+	0.628918i		0		1.20892	−	1.59327i	−0.395606	−	2.20079i		0		−0.331032	−	1.23543i	−0.529277	+	2.77846i		0		1.88522	+	2.53888i
127.5	−1.25242	+	0.656851i		0		1.13709	−	1.64530i	1.42527	+	1.72296i		0		0.891731	+	3.32798i	−0.343396	+	2.80750i		0		−2.91676	−	1.22168i
127.6	−1.25181	−	0.658006i		0		1.13406	+	1.64740i	−1.61454	+	1.54701i		0		−0.797301	−	2.97557i	−0.335623	−	2.80844i		0		3.03905	−	0.874184i
127.7	−1.16262	−	0.805180i		0		0.703370	+	1.87224i	1.28154	+	1.83239i		0		0.600346	+	2.24052i	0.689735	−	2.74304i		0		−0.0145407	−	3.16224i
127.8	−0.984936	−	1.01484i		0		−0.0598006	+	1.99911i	−2.16651	−	0.553382i		0		0.224679	+	0.838513i	2.08767	−	1.90830i		0		1.57228	+	2.74371i
127.9	−0.833570	+	1.14244i		0		−0.610322	−	1.90460i	1.88223	−	1.20715i		0		−0.361284	−	1.34833i	2.68463	+	0.890364i		0		−0.189882	+	3.15657i
127.10	−0.755096	+	1.19576i		0		−0.859661	−	1.80582i	−1.61454	+	1.54701i		0		0.797301	+	2.97557i	2.80844	+	0.335623i		0		−0.630711	−	3.09874i
127.11	−0.641238	−	1.26048i		0		−1.17763	+	1.61654i	−0.993469	−	2.00325i		0		−0.596237	−	2.22519i	2.79276	+	0.447791i		0		−1.88801	+	2.53681i
127.12	−0.604268	+	1.27862i		0		−1.26972	−	1.54525i	1.28154	+	1.83239i		0		−0.600346	−	2.24052i	2.74304	−	0.689735i		0		−3.11732	+	0.531343i
127.13	−0.453270	−	1.33961i		0		−1.58909	+	1.21441i	2.23308	+	0.115576i		0		−0.00100179	−	0.00373871i	2.34712	+	1.57830i		0		−0.857362	−	3.04383i
127.14	−0.345560	+	1.37135i		0		−1.76118	−	0.947764i	−2.16651	−	0.553382i		0		−0.224679	−	0.838513i	1.90830	−	2.08767i		0		1.50754	−	2.77981i
127.15	−0.0534584	−	1.41320i		0		−1.99428	+	0.151095i	−0.795868	+	2.08964i		0		−0.659597	−	2.46165i	0.320139	+	2.81025i		0		2.99563	+	1.01301i
127.16	0.0749121	+	1.41223i		0		−1.98878	+	0.211586i	−0.993469	−	2.00325i		0		0.596237	+	2.22519i	−0.447791	−	2.79276i		0		2.75463	−	1.55307i
127.17	0.168283	−	1.40417i		0		−1.94336	−	0.472596i	−2.09202	−	0.789601i		0		1.04292	+	3.89223i	−0.990638	+	2.64927i		0		−1.46078	+	2.80466i
127.18	0.241682	−	1.39341i		0		−1.88318	−	0.673524i	1.26760	−	1.84206i		0		−1.09131	−	4.07282i	−1.39362	+	2.46126i		0		−2.26039	−	2.21148i
127.19	0.277260	+	1.38677i		0		−1.84625	+	0.768990i	2.23308	+	0.115576i		0		0.00100179	+	0.00373871i	−1.57830	−	2.34712i		0		0.458866	+	3.12881i
127.20	0.608499	−	1.27661i		0		−1.25946	−	1.55363i	−1.09795	+	1.94795i		0		0.0500694	+	0.186862i	−2.74976	+	0.662452i		0		1.81866	+	2.58698i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
9.c	even	3	1	inner
20.e	even	4	1	inner
36.f	odd	6	1	inner
45.k	odd	12	1	inner
180.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	540.2.y.a		128
3.b	odd	2	1		180.2.x.a	✓	128
4.b	odd	2	1	inner	540.2.y.a		128
5.c	odd	4	1	inner	540.2.y.a		128
9.c	even	3	1	inner	540.2.y.a		128
9.d	odd	6	1		180.2.x.a	✓	128
12.b	even	2	1		180.2.x.a	✓	128
15.d	odd	2	1		900.2.bf.e		128
15.e	even	4	1		180.2.x.a	✓	128
15.e	even	4	1		900.2.bf.e		128
20.e	even	4	1	inner	540.2.y.a		128
36.f	odd	6	1	inner	540.2.y.a		128
36.h	even	6	1		180.2.x.a	✓	128
45.h	odd	6	1		900.2.bf.e		128
45.k	odd	12	1	inner	540.2.y.a		128
45.l	even	12	1		180.2.x.a	✓	128
45.l	even	12	1		900.2.bf.e		128
60.h	even	2	1		900.2.bf.e		128
60.l	odd	4	1		180.2.x.a	✓	128
60.l	odd	4	1		900.2.bf.e		128
180.n	even	6	1		900.2.bf.e		128
180.v	odd	12	1		180.2.x.a	✓	128
180.v	odd	12	1		900.2.bf.e		128
180.x	even	12	1	inner	540.2.y.a		128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.2.x.a	✓	128	3.b	odd	2	1
180.2.x.a	✓	128	9.d	odd	6	1
180.2.x.a	✓	128	12.b	even	2	1
180.2.x.a	✓	128	15.e	even	4	1
180.2.x.a	✓	128	36.h	even	6	1
180.2.x.a	✓	128	45.l	even	12	1
180.2.x.a	✓	128	60.l	odd	4	1
180.2.x.a	✓	128	180.v	odd	12	1
540.2.y.a		128	1.a	even	1	1	trivial
540.2.y.a		128	4.b	odd	2	1	inner
540.2.y.a		128	5.c	odd	4	1	inner
540.2.y.a		128	9.c	even	3	1	inner
540.2.y.a		128	20.e	even	4	1	inner
540.2.y.a		128	36.f	odd	6	1	inner
540.2.y.a		128	45.k	odd	12	1	inner
540.2.y.a		128	180.x	even	12	1	inner
900.2.bf.e		128	15.d	odd	2	1
900.2.bf.e		128	15.e	even	4	1
900.2.bf.e		128	45.h	odd	6	1
900.2.bf.e		128	45.l	even	12	1
900.2.bf.e		128	60.h	even	2	1
900.2.bf.e		128	60.l	odd	4	1
900.2.bf.e		128	180.n	even	6	1
900.2.bf.e		128	180.v	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(540, [\chi])\).