Embedded newform 120.2.w.c.53.1

• Label: 120.2.w.c.53.1
• Level: $120$
• Weight: $2$
• Character: 120.53
• Analytic conductor: $0.958$
• Analytic rank: $0$
• Dimension: $32$
• 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:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.1
Character	\(\chi\)	\(=\)	120.53
Dual form			120.2.w.c.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41107 - 0.0941764i) q^{2} +(0.416519 + 1.68122i) q^{3} +(1.98226 + 0.265780i) q^{4} +(-1.62104 + 1.54020i) q^{5} +(-0.429408 - 2.41156i) q^{6} +(-0.361989 + 0.361989i) q^{7} +(-2.77209 - 0.561717i) q^{8} +(-2.65302 + 1.40052i) q^{9} +(2.43246 - 2.02068i) q^{10} +2.63380 q^{11} +(0.378815 + 3.44333i) q^{12} +(-3.49376 + 3.49376i) q^{13} +(0.544885 - 0.476703i) q^{14} +(-3.26462 - 2.08380i) q^{15} +(3.85872 + 1.05369i) q^{16} +(3.61339 + 3.61339i) q^{17} +(3.87551 - 1.72639i) q^{18} +0.672266 q^{19} +(-3.62268 + 2.62225i) q^{20} +(-0.759360 - 0.457809i) q^{21} +(-3.71648 - 0.248041i) q^{22} +(4.31851 - 4.31851i) q^{23} +(-0.210256 - 4.89447i) q^{24} +(0.255538 - 4.99347i) q^{25} +(5.25899 - 4.60093i) q^{26} +(-3.45963 - 3.87698i) q^{27} +(-0.813767 + 0.621348i) q^{28} -4.76080i q^{29} +(4.41038 + 3.24785i) q^{30} +3.73793 q^{31} +(-5.34571 - 1.85024i) q^{32} +(1.09703 + 4.42800i) q^{33} +(-4.75847 - 5.43906i) q^{34} +(0.0292613 - 1.14434i) q^{35} +(-5.63122 + 2.07108i) q^{36} +(2.82150 + 2.82150i) q^{37} +(-0.948617 - 0.0633116i) q^{38} +(-7.32901 - 4.41857i) q^{39} +(5.35882 - 3.35902i) q^{40} +4.10027i q^{41} +(1.02840 + 0.717517i) q^{42} +(7.57996 - 7.57996i) q^{43} +(5.22087 + 0.700010i) q^{44} +(2.14356 - 6.35650i) q^{45} +(-6.50044 + 5.68704i) q^{46} +(0.987537 + 0.987537i) q^{47} +(-0.164257 + 6.92626i) q^{48} +6.73793i q^{49} +(-0.830850 + 7.02209i) q^{50} +(-4.56987 + 7.57996i) q^{51} +(-7.85412 + 5.99698i) q^{52} +(0.646149 + 0.646149i) q^{53} +(4.51667 + 5.79652i) q^{54} +(-4.26949 + 4.05659i) q^{55} +(1.20680 - 0.800131i) q^{56} +(0.280012 + 1.13023i) q^{57} +(-0.448355 + 6.71784i) q^{58} +4.92247i q^{59} +(-5.91750 - 4.99832i) q^{60} -6.07190i q^{61} +(-5.27449 - 0.352025i) q^{62} +(0.453392 - 1.46734i) q^{63} +(7.36895 + 3.11426i) q^{64} +(0.282417 - 11.0446i) q^{65} +(-1.13097 - 6.35155i) q^{66} +(0.349085 + 0.349085i) q^{67} +(6.20232 + 8.12305i) q^{68} +(9.05912 + 5.46164i) q^{69} +(-0.149059 + 1.61199i) q^{70} +8.63702i q^{71} +(8.14111 - 2.39213i) q^{72} +(-11.3261 - 11.3261i) q^{73} +(-3.71562 - 4.24706i) q^{74} +(8.50157 - 1.65026i) q^{75} +(1.33261 + 0.178675i) q^{76} +(-0.953406 + 0.953406i) q^{77} +(9.92565 + 6.92516i) q^{78} +4.07707i q^{79} +(-7.87804 + 4.23515i) q^{80} +(5.07707 - 7.43124i) q^{81} +(0.386149 - 5.78579i) q^{82} +(-8.53893 - 8.53893i) q^{83} +(-1.38357 - 1.10932i) q^{84} +(-11.4228 - 0.292087i) q^{85} +(-11.4097 + 9.98203i) q^{86} +(8.00397 - 1.98296i) q^{87} +(-7.30111 - 1.47945i) q^{88} -6.58584 q^{89} +(-3.62336 + 8.76763i) q^{90} -2.52941i q^{91} +(9.70819 - 7.41264i) q^{92} +(1.55692 + 6.28429i) q^{93} +(-1.30049 - 1.48649i) q^{94} +(-1.08977 + 1.03543i) q^{95} +(0.884069 - 9.75799i) q^{96} +(-0.660859 + 0.660859i) q^{97} +(0.634554 - 9.50772i) q^{98} +(-6.98752 + 3.68869i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q - 4 * q^6 + 4 * q^10 - 8 * q^12 - 28 * q^15 + 28 * q^16 - 20 * q^18 - 52 * q^22 - 8 * q^25 + 12 * q^28 - 32 * q^30 - 32 * q^31 + 8 * q^33 - 20 * q^36 + 24 * q^40 + 16 * q^42 + 24 * q^46 + 44 * q^48 + 8 * q^52 + 8 * q^55 - 16 * q^57 + 28 * q^58 + 56 * q^60 + 48 * q^63 + 16 * q^66 + 20 * q^70 + 32 * q^72 - 64 * q^73 - 88 * q^76 + 64 * q^78 + 48 * q^81 + 64 * q^82 - 8 * q^87 - 52 * q^88 + 84 * q^90 - 52 * q^96 + 16 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(41\)	\(61\)	\(97\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41107	−	0.0941764i	−0.997780	−	0.0665928i
							\(3\)	0.416519	+	1.68122i	0.240477	+	0.970655i
\(5\)	−1.62104	+	1.54020i	−0.724951	+	0.688801i
							\(7\)	−0.361989	+	0.361989i	−0.136819	+	0.136819i	−0.772199	−	0.635380i	\(-0.780843\pi\)
0.635380	+	0.772199i	\(0.280843\pi\)
\(11\)	2.63380			0.794119			0.397060	−	0.917793i	\(-0.370031\pi\)
							0.397060	+	0.917793i	\(0.370031\pi\)
\(13\)	−3.49376	+	3.49376i	−0.968995	+	0.968995i	−0.999534	−	0.0305386i	\(-0.990278\pi\)
							0.0305386	+	0.999534i	\(0.490278\pi\)
\(17\)	3.61339	+	3.61339i	0.876376	+	0.876376i	0.993158	−	0.116782i	\(-0.0372578\pi\)
							−0.116782	+	0.993158i	\(0.537258\pi\)
\(19\)	0.672266			0.154228			0.0771142	−	0.997022i	\(-0.475429\pi\)
							0.0771142	+	0.997022i	\(0.475429\pi\)
\(23\)	4.31851	−	4.31851i	0.900472	−	0.900472i	−0.0950052	−	0.995477i	\(-0.530287\pi\)
							0.995477	+	0.0950052i	\(0.0302868\pi\)
\(29\)		−	4.76080i		−	0.884058i	−0.897001	−	0.442029i	\(-0.854259\pi\)
							0.897001	−	0.442029i	\(-0.145741\pi\)
\(31\)	3.73793			0.671352			0.335676	−	0.941978i	\(-0.391035\pi\)
							0.335676	+	0.941978i	\(0.391035\pi\)
\(37\)	2.82150	+	2.82150i	0.463851	+	0.463851i	0.899915	−	0.436064i	\(-0.143628\pi\)
							−0.436064	+	0.899915i	\(0.643628\pi\)
\(41\)			4.10027i			0.640355i	0.947358	+	0.320177i	\(0.103743\pi\)
							−0.947358	+	0.320177i	\(0.896257\pi\)
\(43\)	7.57996	−	7.57996i	1.15593	−	1.15593i	0.170591	−	0.985342i	\(-0.445432\pi\)
							0.985342	−	0.170591i	\(-0.0545678\pi\)
\(47\)	0.987537	+	0.987537i	0.144047	+	0.144047i	0.775453	−	0.631406i	\(-0.217522\pi\)
							−0.631406	+	0.775453i	\(0.717522\pi\)
\(53\)	0.646149	+	0.646149i	0.0887554	+	0.0887554i	0.750091	−	0.661335i	\(-0.230010\pi\)
							−0.661335	+	0.750091i	\(0.730010\pi\)
\(59\)			4.92247i			0.640851i	0.947274	+	0.320425i	\(0.103826\pi\)
							−0.947274	+	0.320425i	\(0.896174\pi\)
\(61\)		−	6.07190i		−	0.777428i	−0.921359	−	0.388714i	\(-0.872919\pi\)
							0.921359	−	0.388714i	\(-0.127081\pi\)
\(67\)	0.349085	+	0.349085i	0.0426476	+	0.0426476i	0.728109	−	0.685461i	\(-0.240399\pi\)
							−0.685461	+	0.728109i	\(0.740399\pi\)
\(71\)			8.63702i			1.02503i	0.858680	+	0.512513i	\(0.171285\pi\)
							−0.858680	+	0.512513i	\(0.828715\pi\)
\(73\)	−11.3261	−	11.3261i	−1.32562	−	1.32562i	−0.909152	−	0.416465i	\(-0.863269\pi\)
							−0.416465	−	0.909152i	\(-0.636731\pi\)
\(79\)			4.07707i			0.458706i	0.973343	+	0.229353i	\(0.0736610\pi\)
							−0.973343	+	0.229353i	\(0.926339\pi\)
\(83\)	−8.53893	−	8.53893i	−0.937270	−	0.937270i	0.0608758	−	0.998145i	\(-0.480611\pi\)
							−0.998145	+	0.0608758i	\(0.980611\pi\)
\(89\)	−6.58584			−0.698097			−0.349049	−	0.937105i	\(-0.613495\pi\)
							−0.349049	+	0.937105i	\(0.613495\pi\)
\(97\)	−0.660859	+	0.660859i	−0.0671001	+	0.0671001i	−0.739860	−	0.672760i	\(-0.765108\pi\)
							0.672760	+	0.739860i	\(0.265108\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.2.w.c.53.1	✓	32
3.2	odd	2	inner	120.2.w.c.53.16	yes	32
4.3	odd	2		480.2.bi.c.113.7		32
5.2	odd	4	inner	120.2.w.c.77.9	yes	32
5.3	odd	4		600.2.w.j.557.8		32
5.4	even	2		600.2.w.j.293.16		32
8.3	odd	2		480.2.bi.c.113.10		32
8.5	even	2	inner	120.2.w.c.53.8	yes	32
12.11	even	2		480.2.bi.c.113.15		32
15.2	even	4	inner	120.2.w.c.77.8	yes	32
15.8	even	4		600.2.w.j.557.9		32
15.14	odd	2		600.2.w.j.293.1		32
20.7	even	4		480.2.bi.c.17.2		32
24.5	odd	2	inner	120.2.w.c.53.9	yes	32
24.11	even	2		480.2.bi.c.113.2		32
40.13	odd	4		600.2.w.j.557.1		32
40.27	even	4		480.2.bi.c.17.15		32
40.29	even	2		600.2.w.j.293.9		32
40.37	odd	4	inner	120.2.w.c.77.16	yes	32
60.47	odd	4		480.2.bi.c.17.10		32
120.29	odd	2		600.2.w.j.293.8		32
120.53	even	4		600.2.w.j.557.16		32
120.77	even	4	inner	120.2.w.c.77.1	yes	32
120.107	odd	4		480.2.bi.c.17.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.1	✓	32	1.1	even	1	trivial
120.2.w.c.53.8	yes	32	8.5	even	2	inner
120.2.w.c.53.9	yes	32	24.5	odd	2	inner
120.2.w.c.53.16	yes	32	3.2	odd	2	inner
120.2.w.c.77.1	yes	32	120.77	even	4	inner
120.2.w.c.77.8	yes	32	15.2	even	4	inner
120.2.w.c.77.9	yes	32	5.2	odd	4	inner
120.2.w.c.77.16	yes	32	40.37	odd	4	inner
480.2.bi.c.17.2		32	20.7	even	4
480.2.bi.c.17.7		32	120.107	odd	4
480.2.bi.c.17.10		32	60.47	odd	4
480.2.bi.c.17.15		32	40.27	even	4
480.2.bi.c.113.2		32	24.11	even	2
480.2.bi.c.113.7		32	4.3	odd	2
480.2.bi.c.113.10		32	8.3	odd	2
480.2.bi.c.113.15		32	12.11	even	2
600.2.w.j.293.1		32	15.14	odd	2
600.2.w.j.293.8		32	120.29	odd	2
600.2.w.j.293.9		32	40.29	even	2
600.2.w.j.293.16		32	5.4	even	2
600.2.w.j.557.1		32	40.13	odd	4
600.2.w.j.557.8		32	5.3	odd	4
600.2.w.j.557.9		32	15.8	even	4
600.2.w.j.557.16		32	120.53	even	4