Embedded newform 120.2.w.c.53.9

• Label: 120.2.w.c.53.9
• 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			53.9
Character	\(\chi\)	\(=\)	120.53
Dual form			120.2.w.c.77.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0941764 + 1.41107i) q^{2} +(1.68122 + 0.416519i) 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} +(-0.561717 - 2.77209i) q^{8} +(2.65302 + 1.40052i) q^{9} +(-2.32601 - 2.14236i) q^{10} +2.63380 q^{11} +(-3.44333 - 0.378815i) q^{12} +(3.49376 - 3.49376i) q^{13} +(-0.544885 - 0.476703i) q^{14} +(-3.36685 + 1.91423i) q^{15} +(3.85872 - 1.05369i) q^{16} +(-3.61339 - 3.61339i) q^{17} +(-1.72639 + 3.87551i) q^{18} -0.672266 q^{19} +(2.80397 - 3.48393i) q^{20} +(-0.759360 + 0.457809i) q^{21} +(0.248041 + 3.71648i) 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} +(3.87698 + 3.45963i) q^{27} +(0.621348 - 0.813767i) q^{28} -4.76080i q^{29} +(-3.01821 - 4.57061i) q^{30} +3.73793 q^{31} +(1.85024 + 5.34571i) q^{32} +(4.42800 + 1.09703i) 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} +(-0.0633116 - 0.948617i) q^{38} +(7.32901 - 4.41857i) q^{39} +(5.18015 + 3.62850i) q^{40} -4.10027i q^{41} +(-0.717517 - 1.02840i) q^{42} +(-7.57996 + 7.57996i) q^{43} +(-5.22087 + 0.700010i) q^{44} +(-6.45775 + 1.81590i) q^{45} +(-6.50044 - 5.68704i) q^{46} +(-0.987537 - 0.987537i) q^{47} +(6.92626 - 0.164257i) q^{48} +6.73793i q^{49} +(7.07022 - 0.109683i) q^{50} +(-4.56987 - 7.57996i) q^{51} +(-5.99698 + 7.85412i) 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} +(-1.13023 - 0.280012i) q^{57} +(6.71784 - 0.448355i) q^{58} +4.92247i q^{59} +(6.16522 - 4.68935i) q^{60} +6.07190i q^{61} +(0.352025 + 5.27449i) q^{62} +(-1.46734 + 0.453392i) 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} +(8.12305 + 6.20232i) q^{68} +(-9.05912 + 5.46164i) q^{69} +(1.61750 - 0.0664796i) q^{70} -8.63702i q^{71} +(2.39213 - 8.14111i) q^{72} +(-11.3261 - 11.3261i) q^{73} +(3.71562 - 4.24706i) q^{74} +(2.50949 - 8.28869i) q^{75} +(1.33261 - 0.178675i) q^{76} +(-0.953406 + 0.953406i) q^{77} +(6.92516 + 9.92565i) q^{78} +4.07707i q^{79} +(-4.63224 + 7.65130i) q^{80} +(5.07707 + 7.43124i) q^{81} +(5.78579 - 0.386149i) 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} +(1.98296 - 8.00397i) q^{87} +(-1.47945 - 7.30111i) q^{88} +6.58584 q^{89} +(-3.17053 - 8.94135i) q^{90} +2.52941i q^{91} +(7.41264 - 9.70819i) q^{92} +(6.28429 + 1.55692i) 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} +(-9.50772 + 0.634554i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0941764	+	1.41107i	0.0665928	+	0.997780i
							\(3\)	1.68122	+	0.416519i	0.970655	+	0.240477i
\(5\)	−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	−0.0305386	−	0.999534i	\(-0.509722\pi\)
							0.999534	+	0.0305386i	\(0.00972224\pi\)
\(17\)	−3.61339	−	3.61339i	−0.876376	−	0.876376i	0.116782	−	0.993158i	\(-0.462742\pi\)
							−0.993158	+	0.116782i	\(0.962742\pi\)
\(19\)	−0.672266			−0.154228			−0.0771142	−	0.997022i	\(-0.524571\pi\)
							−0.0771142	+	0.997022i	\(0.524571\pi\)
\(23\)	−4.31851	+	4.31851i	−0.900472	+	0.900472i	−0.995477	−	0.0950052i	\(-0.969713\pi\)
							0.0950052	+	0.995477i	\(0.469713\pi\)
\(29\)		−	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	0.436064	−	0.899915i	\(-0.356372\pi\)
							−0.899915	+	0.436064i	\(0.856372\pi\)
\(41\)		−	4.10027i		−	0.640355i	−0.947358	−	0.320177i	\(-0.896257\pi\)
							0.947358	−	0.320177i	\(-0.103743\pi\)
\(43\)	−7.57996	+	7.57996i	−1.15593	+	1.15593i	−0.170591	+	0.985342i	\(0.554568\pi\)
							−0.985342	+	0.170591i	\(0.945432\pi\)
\(47\)	−0.987537	−	0.987537i	−0.144047	−	0.144047i	0.631406	−	0.775453i	\(-0.282478\pi\)
							−0.775453	+	0.631406i	\(0.782478\pi\)
\(53\)	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	\(0.127081\pi\)
							−0.921359	+	0.388714i	\(0.872919\pi\)
\(67\)	−0.349085	−	0.349085i	−0.0426476	−	0.0426476i	0.685461	−	0.728109i	\(-0.259601\pi\)
							−0.728109	+	0.685461i	\(0.759601\pi\)
\(71\)		−	8.63702i		−	1.02503i	−0.858680	−	0.512513i	\(-0.828715\pi\)
							0.858680	−	0.512513i	\(-0.171285\pi\)
\(73\)	−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	\(-0.386505\pi\)
							0.349049	+	0.937105i	\(0.386505\pi\)
\(97\)	−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.53.9	yes	32
3.2	odd	2	inner	120.2.w.c.53.8	yes	32
4.3	odd	2		480.2.bi.c.113.2		32
5.2	odd	4	inner	120.2.w.c.77.1	yes	32
5.3	odd	4		600.2.w.j.557.16		32
5.4	even	2		600.2.w.j.293.8		32
8.3	odd	2		480.2.bi.c.113.15		32
8.5	even	2	inner	120.2.w.c.53.16	yes	32
12.11	even	2		480.2.bi.c.113.10		32
15.2	even	4	inner	120.2.w.c.77.16	yes	32
15.8	even	4		600.2.w.j.557.1		32
15.14	odd	2		600.2.w.j.293.9		32
20.7	even	4		480.2.bi.c.17.7		32
24.5	odd	2	inner	120.2.w.c.53.1	✓	32
24.11	even	2		480.2.bi.c.113.7		32
40.13	odd	4		600.2.w.j.557.9		32
40.27	even	4		480.2.bi.c.17.10		32
40.29	even	2		600.2.w.j.293.1		32
40.37	odd	4	inner	120.2.w.c.77.8	yes	32
60.47	odd	4		480.2.bi.c.17.15		32
120.29	odd	2		600.2.w.j.293.16		32
120.53	even	4		600.2.w.j.557.8		32
120.77	even	4	inner	120.2.w.c.77.9	yes	32
120.107	odd	4		480.2.bi.c.17.2		32

    

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