Embedded newform 120.2.w.c.53.3

• Label: 120.2.w.c.53.3
• 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			53.3
Character	\(\chi\)	\(=\)	120.53
Dual form			120.2.w.c.77.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30986 + 0.533177i) q^{2} +(-1.59834 + 0.667305i) q^{3} +(1.43144 - 1.39677i) q^{4} +(-0.143028 - 2.23149i) q^{5} +(1.73781 - 1.72627i) q^{6} +(0.582772 - 0.582772i) q^{7} +(-1.13026 + 2.59278i) q^{8} +(2.10941 - 2.13317i) q^{9} +(1.37713 + 2.84667i) q^{10} +3.68607 q^{11} +(-1.35587 + 3.18773i) q^{12} +(3.88771 - 3.88771i) q^{13} +(-0.452626 + 1.07407i) q^{14} +(1.71769 + 3.47124i) q^{15} +(0.0980619 - 3.99880i) q^{16} +(-0.880105 - 0.880105i) q^{17} +(-1.62567 + 3.91883i) q^{18} -6.32919 q^{19} +(-3.32162 - 2.99447i) q^{20} +(-0.542584 + 1.32036i) q^{21} +(-4.82821 + 1.96533i) q^{22} +(2.06626 - 2.06626i) q^{23} +(0.0763660 - 4.89838i) q^{24} +(-4.95909 + 0.638332i) q^{25} +(-3.01950 + 7.16518i) q^{26} +(-1.94809 + 4.81715i) q^{27} +(0.0202063 - 1.64820i) q^{28} -1.37122i q^{29} +(-4.10072 - 3.63099i) q^{30} +3.32075 q^{31} +(2.00362 + 5.29013i) q^{32} +(-5.89160 + 2.45973i) q^{33} +(1.62206 + 0.683558i) q^{34} +(-1.38380 - 1.21710i) q^{35} +(0.0399575 - 5.99987i) q^{36} +(2.44147 + 2.44147i) q^{37} +(8.29032 - 3.37458i) q^{38} +(-3.61961 + 8.80819i) q^{39} +(5.94742 + 2.15132i) q^{40} +0.648104i q^{41} +(0.00672215 - 2.01877i) q^{42} +(0.819412 - 0.819412i) q^{43} +(5.27640 - 5.14859i) q^{44} +(-5.06184 - 4.40202i) q^{45} +(-1.60482 + 3.80818i) q^{46} +(-6.28508 - 6.28508i) q^{47} +(2.51168 + 6.45689i) q^{48} +6.32075i q^{49} +(6.15534 - 3.48020i) q^{50} +(1.99401 + 0.819412i) q^{51} +(0.134798 - 10.9953i) q^{52} +(5.60782 + 5.60782i) q^{53} +(-0.0166776 - 7.34845i) q^{54} +(-0.527212 - 8.22542i) q^{55} +(0.852318 + 2.16968i) q^{56} +(10.1162 - 4.22349i) q^{57} +(0.731106 + 1.79611i) q^{58} +6.12026i q^{59} +(7.30731 + 2.56967i) q^{60} +5.13471i q^{61} +(-4.34971 + 1.77055i) q^{62} +(-0.0138443 - 2.47245i) q^{63} +(-5.44503 - 5.86102i) q^{64} +(-9.23144 - 8.11933i) q^{65} +(6.40568 - 6.36316i) q^{66} +(4.90636 + 4.90636i) q^{67} +(-2.48912 - 0.0305156i) q^{68} +(-1.92377 + 4.68141i) q^{69} +(2.46151 + 0.856408i) q^{70} +4.13251i q^{71} +(3.14666 + 7.88026i) q^{72} +(4.69820 + 4.69820i) q^{73} +(-4.49972 - 1.89624i) q^{74} +(7.50036 - 4.32950i) q^{75} +(-9.05987 + 8.84042i) q^{76} +(2.14814 - 2.14814i) q^{77} +(0.0448439 - 13.4674i) q^{78} -1.10079i q^{79} +(-8.93730 + 0.353117i) q^{80} +(-0.100786 - 8.99944i) q^{81} +(-0.345554 - 0.848922i) q^{82} +(-6.27439 - 6.27439i) q^{83} +(1.06756 + 2.64788i) q^{84} +(-1.83806 + 2.08982i) q^{85} +(-0.636420 + 1.51020i) q^{86} +(0.915024 + 2.19169i) q^{87} +(-4.16621 + 9.55717i) q^{88} +15.3562 q^{89} +(8.97734 + 3.06715i) q^{90} -4.53130i q^{91} +(0.0716428 - 5.84382i) q^{92} +(-5.30771 + 2.21595i) q^{93} +(11.5836 + 4.88148i) q^{94} +(0.905253 + 14.1235i) q^{95} +(-6.73261 - 7.11843i) q^{96} +(-5.42154 + 5.42154i) q^{97} +(-3.37008 - 8.27927i) q^{98} +(7.77542 - 7.86299i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30986	+	0.533177i	−0.926208	+	0.377013i
							\(3\)	−1.59834	+	0.667305i	−0.922805	+	0.385268i
\(5\)	−0.143028	−	2.23149i	−0.0639642	−	0.997952i
							\(7\)	0.582772	−	0.582772i	0.220267	−	0.220267i	−0.588344	−	0.808611i	\(-0.700220\pi\)
0.808611	+	0.588344i	\(0.200220\pi\)
\(11\)	3.68607			1.11139			0.555695	−	0.831386i	\(-0.312452\pi\)
							0.555695	+	0.831386i	\(0.312452\pi\)
\(13\)	3.88771	−	3.88771i	1.07826	−	1.07826i	0.0815911	−	0.996666i	\(-0.474000\pi\)
							0.996666	−	0.0815911i	\(-0.0260002\pi\)
\(17\)	−0.880105	−	0.880105i	−0.213457	−	0.213457i	0.592277	−	0.805734i	\(-0.298229\pi\)
							−0.805734	+	0.592277i	\(0.798229\pi\)
\(19\)	−6.32919			−1.45201			−0.726007	−	0.687687i	\(-0.758626\pi\)
							−0.726007	+	0.687687i	\(0.758626\pi\)
\(23\)	2.06626	−	2.06626i	0.430844	−	0.430844i	−0.458071	−	0.888916i	\(-0.651460\pi\)
							0.888916	+	0.458071i	\(0.151460\pi\)
\(29\)		−	1.37122i		−	0.254630i	−0.991862	−	0.127315i	\(-0.959364\pi\)
							0.991862	−	0.127315i	\(-0.0406359\pi\)
\(31\)	3.32075			0.596425			0.298212	−	0.954500i	\(-0.403610\pi\)
							0.298212	+	0.954500i	\(0.403610\pi\)
\(37\)	2.44147	+	2.44147i	0.401376	+	0.401376i	0.878718	−	0.477342i	\(-0.158400\pi\)
							−0.477342	+	0.878718i	\(0.658400\pi\)
\(41\)			0.648104i			0.101217i	0.998719	+	0.0506084i	\(0.0161160\pi\)
							−0.998719	+	0.0506084i	\(0.983884\pi\)
\(43\)	0.819412	−	0.819412i	0.124959	−	0.124959i	−0.641861	−	0.766821i	\(-0.721838\pi\)
							0.766821	+	0.641861i	\(0.221838\pi\)
\(47\)	−6.28508	−	6.28508i	−0.916772	−	0.916772i	0.0800208	−	0.996793i	\(-0.474501\pi\)
							−0.996793	+	0.0800208i	\(0.974501\pi\)
\(53\)	5.60782	+	5.60782i	0.770293	+	0.770293i	0.978158	−	0.207864i	\(-0.0666512\pi\)
							−0.207864	+	0.978158i	\(0.566651\pi\)
\(59\)			6.12026i			0.796790i	0.917214	+	0.398395i	\(0.130433\pi\)
							−0.917214	+	0.398395i	\(0.869567\pi\)
\(61\)			5.13471i			0.657432i	0.944429	+	0.328716i	\(0.106616\pi\)
							−0.944429	+	0.328716i	\(0.893384\pi\)
\(67\)	4.90636	+	4.90636i	0.599408	+	0.599408i	0.940155	−	0.340747i	\(-0.110680\pi\)
							−0.340747	+	0.940155i	\(0.610680\pi\)
\(71\)			4.13251i			0.490439i	0.969468	+	0.245220i	\(0.0788601\pi\)
							−0.969468	+	0.245220i	\(0.921140\pi\)
\(73\)	4.69820	+	4.69820i	0.549883	+	0.549883i	0.926407	−	0.376524i	\(-0.122881\pi\)
							−0.376524	+	0.926407i	\(0.622881\pi\)
\(79\)		−	1.10079i		−	0.123848i	−0.998081	−	0.0619241i	\(-0.980276\pi\)
							0.998081	−	0.0619241i	\(-0.0197237\pi\)
\(83\)	−6.27439	−	6.27439i	−0.688703	−	0.688703i	0.273242	−	0.961945i	\(-0.411904\pi\)
							−0.961945	+	0.273242i	\(0.911904\pi\)
\(89\)	15.3562			1.62775			0.813875	−	0.581040i	\(-0.197354\pi\)
							0.813875	+	0.581040i	\(0.197354\pi\)
\(97\)	−5.42154	+	5.42154i	−0.550474	+	0.550474i	−0.926578	−	0.376104i	\(-0.877264\pi\)
							0.376104	+	0.926578i	\(0.377264\pi\)

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.53.3	✓	32
3.2	odd	2	inner	120.2.w.c.53.14	yes	32
4.3	odd	2		480.2.bi.c.113.14		32
5.2	odd	4	inner	120.2.w.c.77.6	yes	32
5.3	odd	4		600.2.w.j.557.11		32
5.4	even	2		600.2.w.j.293.14		32
8.3	odd	2		480.2.bi.c.113.3		32
8.5	even	2	inner	120.2.w.c.53.11	yes	32
12.11	even	2		480.2.bi.c.113.11		32
15.2	even	4	inner	120.2.w.c.77.11	yes	32
15.8	even	4		600.2.w.j.557.6		32
15.14	odd	2		600.2.w.j.293.3		32
20.7	even	4		480.2.bi.c.17.6		32
24.5	odd	2	inner	120.2.w.c.53.6	yes	32
24.11	even	2		480.2.bi.c.113.6		32
40.13	odd	4		600.2.w.j.557.3		32
40.27	even	4		480.2.bi.c.17.11		32
40.29	even	2		600.2.w.j.293.6		32
40.37	odd	4	inner	120.2.w.c.77.14	yes	32
60.47	odd	4		480.2.bi.c.17.3		32
120.29	odd	2		600.2.w.j.293.11		32
120.53	even	4		600.2.w.j.557.14		32
120.77	even	4	inner	120.2.w.c.77.3	yes	32
120.107	odd	4		480.2.bi.c.17.14		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.3	✓	32	1.1	even	1	trivial
120.2.w.c.53.6	yes	32	24.5	odd	2	inner
120.2.w.c.53.11	yes	32	8.5	even	2	inner
120.2.w.c.53.14	yes	32	3.2	odd	2	inner
120.2.w.c.77.3	yes	32	120.77	even	4	inner
120.2.w.c.77.6	yes	32	5.2	odd	4	inner
120.2.w.c.77.11	yes	32	15.2	even	4	inner
120.2.w.c.77.14	yes	32	40.37	odd	4	inner
480.2.bi.c.17.3		32	60.47	odd	4
480.2.bi.c.17.6		32	20.7	even	4
480.2.bi.c.17.11		32	40.27	even	4
480.2.bi.c.17.14		32	120.107	odd	4
480.2.bi.c.113.3		32	8.3	odd	2
480.2.bi.c.113.6		32	24.11	even	2
480.2.bi.c.113.11		32	12.11	even	2
480.2.bi.c.113.14		32	4.3	odd	2
600.2.w.j.293.3		32	15.14	odd	2
600.2.w.j.293.6		32	40.29	even	2
600.2.w.j.293.11		32	120.29	odd	2
600.2.w.j.293.14		32	5.4	even	2
600.2.w.j.557.3		32	40.13	odd	4
600.2.w.j.557.6		32	15.8	even	4
600.2.w.j.557.11		32	5.3	odd	4
600.2.w.j.557.14		32	120.53	even	4