Newform orbit 120.2.w.c

• Label: 120.2.w.c
• Level: $120$
• Weight: $2$
• Character orbit: 120.w
• Analytic conductor: $0.958$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	120.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.958204824255\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.41107	−	0.0941764i	0.416519	+	1.68122i	1.98226	+	0.265780i	−1.62104	+	1.54020i	−0.429408	−	2.41156i	−0.361989	+	0.361989i	−2.77209	−	0.561717i	−2.65302	+	1.40052i	2.43246	−	2.02068i
53.2	−1.39193	−	0.250043i	1.40091	−	1.01856i	1.87496	+	0.696087i	2.23305	+	0.116202i	−2.20465	+	1.06748i	−2.29041	+	2.29041i	−2.43576	−	1.43773i	0.925085	−	2.85381i	−3.07920	−	0.720103i
53.3	−1.30986	+	0.533177i	−1.59834	+	0.667305i	1.43144	−	1.39677i	−0.143028	−	2.23149i	1.73781	−	1.72627i	0.582772	−	0.582772i	−1.13026	+	2.59278i	2.10941	−	2.13317i	1.37713	+	2.84667i
53.4	−1.11940	−	0.864261i	−1.72368	−	0.170116i	0.506107	+	1.93490i	1.36575	+	1.77052i	1.78246	+	1.68013i	2.06963	−	2.06963i	1.10573	−	2.60334i	2.94212	+	0.586449i	0.00136610	−	3.16228i
53.5	−0.864261	−	1.11940i	1.72368	+	0.170116i	−0.506107	+	1.93490i	−1.36575	−	1.77052i	−1.29928	−	2.07651i	2.06963	−	2.06963i	2.60334	−	1.10573i	2.94212	+	0.586449i	−0.801547	+	3.05901i
53.6	−0.533177	+	1.30986i	0.667305	−	1.59834i	−1.43144	−	1.39677i	−0.143028	−	2.23149i	1.73781	+	1.72627i	0.582772	−	0.582772i	2.59278	−	1.13026i	−2.10941	−	2.13317i	2.99919	+	1.00243i
53.7	−0.250043	−	1.39193i	−1.40091	+	1.01856i	−1.87496	+	0.696087i	−2.23305	−	0.116202i	1.76805	+	1.69529i	−2.29041	+	2.29041i	1.43773	+	2.43576i	0.925085	−	2.85381i	0.396613	+	3.13731i
53.8	−0.0941764	−	1.41107i	−0.416519	−	1.68122i	−1.98226	+	0.265780i	1.62104	−	1.54020i	−2.33310	+	0.746071i	−0.361989	+	0.361989i	0.561717	+	2.77209i	−2.65302	+	1.40052i	−2.32601	−	2.14236i
53.9	0.0941764	+	1.41107i	1.68122	+	0.416519i	−1.98226	+	0.265780i	−1.62104	+	1.54020i	−0.429408	+	2.41156i	−0.361989	+	0.361989i	−0.561717	−	2.77209i	2.65302	+	1.40052i	−2.32601	−	2.14236i
53.10	0.250043	+	1.39193i	−1.01856	+	1.40091i	−1.87496	+	0.696087i	2.23305	+	0.116202i	−2.20465	−	1.06748i	−2.29041	+	2.29041i	−1.43773	−	2.43576i	−0.925085	−	2.85381i	0.396613	+	3.13731i
53.11	0.533177	−	1.30986i	1.59834	−	0.667305i	−1.43144	−	1.39677i	0.143028	+	2.23149i	−0.0218716	−	2.44939i	0.582772	−	0.582772i	−2.59278	+	1.13026i	2.10941	−	2.13317i	2.99919	+	1.00243i
53.12	0.864261	+	1.11940i	−0.170116	−	1.72368i	−0.506107	+	1.93490i	1.36575	+	1.77052i	1.78246	−	1.68013i	2.06963	−	2.06963i	−2.60334	+	1.10573i	−2.94212	+	0.586449i	−0.801547	+	3.05901i
53.13	1.11940	+	0.864261i	0.170116	+	1.72368i	0.506107	+	1.93490i	−1.36575	−	1.77052i	−1.29928	+	2.07651i	2.06963	−	2.06963i	−1.10573	+	2.60334i	−2.94212	+	0.586449i	0.00136610	−	3.16228i
53.14	1.30986	−	0.533177i	−0.667305	+	1.59834i	1.43144	−	1.39677i	0.143028	+	2.23149i	−0.0218716	+	2.44939i	0.582772	−	0.582772i	1.13026	−	2.59278i	−2.10941	−	2.13317i	1.37713	+	2.84667i
53.15	1.39193	+	0.250043i	1.01856	−	1.40091i	1.87496	+	0.696087i	−2.23305	−	0.116202i	1.76805	−	1.69529i	−2.29041	+	2.29041i	2.43576	+	1.43773i	−0.925085	−	2.85381i	−3.07920	−	0.720103i
53.16	1.41107	+	0.0941764i	−1.68122	−	0.416519i	1.98226	+	0.265780i	1.62104	−	1.54020i	−2.33310	−	0.746071i	−0.361989	+	0.361989i	2.77209	+	0.561717i	2.65302	+	1.40052i	2.43246	−	2.02068i
77.1	−1.41107	+	0.0941764i	0.416519	−	1.68122i	1.98226	−	0.265780i	−1.62104	−	1.54020i	−0.429408	+	2.41156i	−0.361989	−	0.361989i	−2.77209	+	0.561717i	−2.65302	−	1.40052i	2.43246	+	2.02068i
77.2	−1.39193	+	0.250043i	1.40091	+	1.01856i	1.87496	−	0.696087i	2.23305	−	0.116202i	−2.20465	−	1.06748i	−2.29041	−	2.29041i	−2.43576	+	1.43773i	0.925085	+	2.85381i	−3.07920	+	0.720103i
77.3	−1.30986	−	0.533177i	−1.59834	−	0.667305i	1.43144	+	1.39677i	−0.143028	+	2.23149i	1.73781	+	1.72627i	0.582772	+	0.582772i	−1.13026	−	2.59278i	2.10941	+	2.13317i	1.37713	−	2.84667i
77.4	−1.11940	+	0.864261i	−1.72368	+	0.170116i	0.506107	−	1.93490i	1.36575	−	1.77052i	1.78246	−	1.68013i	2.06963	+	2.06963i	1.10573	+	2.60334i	2.94212	−	0.586449i	0.00136610	+	3.16228i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
8.b	even	2	1	inner
15.e	even	4	1	inner
24.h	odd	2	1	inner
40.i	odd	4	1	inner
120.w	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	120.2.w.c	✓	32
3.b	odd	2	1	inner	120.2.w.c	✓	32
4.b	odd	2	1		480.2.bi.c		32
5.b	even	2	1		600.2.w.j		32
5.c	odd	4	1	inner	120.2.w.c	✓	32
5.c	odd	4	1		600.2.w.j		32
8.b	even	2	1	inner	120.2.w.c	✓	32
8.d	odd	2	1		480.2.bi.c		32
12.b	even	2	1		480.2.bi.c		32
15.d	odd	2	1		600.2.w.j		32
15.e	even	4	1	inner	120.2.w.c	✓	32
15.e	even	4	1		600.2.w.j		32
20.e	even	4	1		480.2.bi.c		32
24.f	even	2	1		480.2.bi.c		32
24.h	odd	2	1	inner	120.2.w.c	✓	32
40.f	even	2	1		600.2.w.j		32
40.i	odd	4	1	inner	120.2.w.c	✓	32
40.i	odd	4	1		600.2.w.j		32
40.k	even	4	1		480.2.bi.c		32
60.l	odd	4	1		480.2.bi.c		32
120.i	odd	2	1		600.2.w.j		32
120.q	odd	4	1		480.2.bi.c		32
120.w	even	4	1	inner	120.2.w.c	✓	32
120.w	even	4	1		600.2.w.j		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.w.c	✓	32	1.a	even	1	1	trivial
120.2.w.c	✓	32	3.b	odd	2	1	inner
120.2.w.c	✓	32	5.c	odd	4	1	inner
120.2.w.c	✓	32	8.b	even	2	1	inner
120.2.w.c	✓	32	15.e	even	4	1	inner
120.2.w.c	✓	32	24.h	odd	2	1	inner
120.2.w.c	✓	32	40.i	odd	4	1	inner
120.2.w.c	✓	32	120.w	even	4	1	inner
480.2.bi.c		32	4.b	odd	2	1
480.2.bi.c		32	8.d	odd	2	1
480.2.bi.c		32	12.b	even	2	1
480.2.bi.c		32	20.e	even	4	1
480.2.bi.c		32	24.f	even	2	1
480.2.bi.c		32	40.k	even	4	1
480.2.bi.c		32	60.l	odd	4	1
480.2.bi.c		32	120.q	odd	4	1
600.2.w.j		32	5.b	even	2	1
600.2.w.j		32	5.c	odd	4	1
600.2.w.j		32	15.d	odd	2	1
600.2.w.j		32	15.e	even	4	1
600.2.w.j		32	40.f	even	2	1
600.2.w.j		32	40.i	odd	4	1
600.2.w.j		32	120.i	odd	2	1
600.2.w.j		32	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\):

\( T_{7}^{8} - 4T_{7}^{5} + 92T_{7}^{4} - 40T_{7}^{3} + 8T_{7}^{2} + 16T_{7} + 16 \)

\( T_{11}^{8} - 26T_{11}^{6} + 208T_{11}^{4} - 544T_{11}^{2} + 128 \)