Embedded newform 120.2.w.c.77.8

• Label: 120.2.w.c.77.8
• Level: $120$
• Weight: $2$
• Character: 120.77
• 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			77.8
Character	\(\chi\)	\(=\)	120.77
Dual form			120.2.w.c.53.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0941764 + 1.41107i) q^{2} +(-0.416519 + 1.68122i) q^{3} +(-1.98226 - 0.265780i) q^{4} +(1.62104 + 1.54020i) q^{5} +(-2.33310 - 0.746071i) 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} +(1.27249 - 3.22192i) 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} +(2.22609 - 3.61172i) 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} +(4.42653 + 2.09900i) q^{24} +(0.255538 + 4.99347i) q^{25} +(-5.25899 + 4.60093i) q^{26} +(3.45963 - 3.87698i) q^{27} +(0.621348 + 0.813767i) q^{28} -4.76080i q^{29} +(-2.63295 - 4.80287i) q^{30} +3.73793 q^{31} +(-1.85024 + 5.34571i) q^{32} +(1.09703 - 4.42800i) q^{33} +(4.75847 + 5.43906i) q^{34} +(-0.0292613 - 1.14434i) q^{35} +(4.88676 + 3.48132i) 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} +(0.574489 + 1.11463i) 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} +(-3.37872 + 6.04849i) 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} +(5.14489 + 5.24691i) q^{54} +(-4.26949 - 4.05659i) q^{55} +(-1.20680 + 0.800131i) q^{56} +(0.280012 - 1.13023i) q^{57} +(6.71784 + 0.448355i) q^{58} +4.92247i q^{59} +(7.02517 - 3.26298i) q^{60} -6.07190i q^{61} +(-0.352025 + 5.27449i) q^{62} +(0.453392 + 1.46734i) q^{63} +(-7.36895 - 3.11426i) q^{64} +(0.282417 + 11.0446i) q^{65} +(6.14492 + 1.96500i) 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} +(-5.37262 + 6.56772i) 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} +(-5.54472 - 10.7579i) 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} +(-1.62693 + 0.705675i) q^{84} +(11.4228 - 0.292087i) q^{85} +(11.4097 - 9.98203i) q^{86} +(8.00397 + 1.98296i) q^{87} +(-1.47945 + 7.30111i) q^{88} -6.58584 q^{89} +(9.17137 - 2.42609i) q^{90} -2.52941i q^{91} +(-7.41264 - 9.70819i) q^{92} +(-1.55692 + 6.28429i) q^{93} +(1.30049 + 1.48649i) q^{94} +(-1.08977 - 1.03543i) q^{95} +(-8.21668 - 5.33725i) 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 \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

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

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
							\(3\)	−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	0.635380	−	0.772199i	\(-0.280843\pi\)
−0.772199	+	0.635380i	\(0.780843\pi\)
\(11\)	−2.63380			−0.794119			−0.397060	−	0.917793i	\(-0.629969\pi\)
							−0.397060	+	0.917793i	\(0.629969\pi\)
\(13\)	3.49376	+	3.49376i	0.968995	+	0.968995i	0.999534	−	0.0305386i	\(-0.00972224\pi\)
							−0.0305386	+	0.999534i	\(0.509722\pi\)
\(17\)	3.61339	−	3.61339i	0.876376	−	0.876376i	−0.116782	−	0.993158i	\(-0.537258\pi\)
							0.993158	+	0.116782i	\(0.0372578\pi\)
\(19\)	−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	\(-0.0302868\pi\)
							−0.0950052	+	0.995477i	\(0.530287\pi\)
\(29\)		−	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	\(-0.856372\pi\)
							0.436064	+	0.899915i	\(0.356372\pi\)
\(41\)		−	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	−0.985342	−	0.170591i	\(-0.945432\pi\)
							−0.170591	−	0.985342i	\(-0.554568\pi\)
\(47\)	0.987537	−	0.987537i	0.144047	−	0.144047i	−0.631406	−	0.775453i	\(-0.717522\pi\)
							0.775453	+	0.631406i	\(0.217522\pi\)
\(53\)	−0.646149	+	0.646149i	−0.0887554	+	0.0887554i	−0.750091	−	0.661335i	\(-0.769990\pi\)
							0.661335	+	0.750091i	\(0.269990\pi\)
\(59\)			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	\(-0.759601\pi\)
							0.685461	+	0.728109i	\(0.259601\pi\)
\(71\)		−	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	−0.416465	+	0.909152i	\(0.636731\pi\)
							−0.909152	+	0.416465i	\(0.863269\pi\)
\(79\)		−	4.07707i		−	0.458706i	−0.973343	−	0.229353i	\(-0.926339\pi\)
							0.973343	−	0.229353i	\(-0.0736610\pi\)
\(83\)	8.53893	−	8.53893i	0.937270	−	0.937270i	−0.0608758	−	0.998145i	\(-0.519389\pi\)
							0.998145	+	0.0608758i	\(0.0193894\pi\)
\(89\)	−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	0.672760	−	0.739860i	\(-0.265108\pi\)
							−0.739860	+	0.672760i	\(0.765108\pi\)

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.77.8	yes	32
3.2	odd	2	inner	120.2.w.c.77.9	yes	32
4.3	odd	2		480.2.bi.c.17.10		32
5.2	odd	4		600.2.w.j.293.1		32
5.3	odd	4	inner	120.2.w.c.53.16	yes	32
5.4	even	2		600.2.w.j.557.9		32
8.3	odd	2		480.2.bi.c.17.7		32
8.5	even	2	inner	120.2.w.c.77.1	yes	32
12.11	even	2		480.2.bi.c.17.2		32
15.2	even	4		600.2.w.j.293.16		32
15.8	even	4	inner	120.2.w.c.53.1	✓	32
15.14	odd	2		600.2.w.j.557.8		32
20.3	even	4		480.2.bi.c.113.15		32
24.5	odd	2	inner	120.2.w.c.77.16	yes	32
24.11	even	2		480.2.bi.c.17.15		32
40.3	even	4		480.2.bi.c.113.2		32
40.13	odd	4	inner	120.2.w.c.53.9	yes	32
40.29	even	2		600.2.w.j.557.16		32
40.37	odd	4		600.2.w.j.293.8		32
60.23	odd	4		480.2.bi.c.113.7		32
120.29	odd	2		600.2.w.j.557.1		32
120.53	even	4	inner	120.2.w.c.53.8	yes	32
120.77	even	4		600.2.w.j.293.9		32
120.83	odd	4		480.2.bi.c.113.10		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.1	✓	32	15.8	even	4	inner
120.2.w.c.53.8	yes	32	120.53	even	4	inner
120.2.w.c.53.9	yes	32	40.13	odd	4	inner
120.2.w.c.53.16	yes	32	5.3	odd	4	inner
120.2.w.c.77.1	yes	32	8.5	even	2	inner
120.2.w.c.77.8	yes	32	1.1	even	1	trivial
120.2.w.c.77.9	yes	32	3.2	odd	2	inner
120.2.w.c.77.16	yes	32	24.5	odd	2	inner
480.2.bi.c.17.2		32	12.11	even	2
480.2.bi.c.17.7		32	8.3	odd	2
480.2.bi.c.17.10		32	4.3	odd	2
480.2.bi.c.17.15		32	24.11	even	2
480.2.bi.c.113.2		32	40.3	even	4
480.2.bi.c.113.7		32	60.23	odd	4
480.2.bi.c.113.10		32	120.83	odd	4
480.2.bi.c.113.15		32	20.3	even	4
600.2.w.j.293.1		32	5.2	odd	4
600.2.w.j.293.8		32	40.37	odd	4
600.2.w.j.293.9		32	120.77	even	4
600.2.w.j.293.16		32	15.2	even	4
600.2.w.j.557.1		32	120.29	odd	2
600.2.w.j.557.8		32	15.14	odd	2
600.2.w.j.557.9		32	5.4	even	2
600.2.w.j.557.16		32	40.29	even	2