Embedded newform 120.2.w.c.53.6

• Label: 120.2.w.c.53.6
• 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			53.6
Character	\(\chi\)	\(=\)	120.53
Dual form			120.2.w.c.77.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.533177 + 1.30986i) q^{2} +(0.667305 - 1.59834i) 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} +(2.59278 - 1.13026i) q^{8} +(-2.10941 - 2.13317i) q^{9} +(2.99919 + 1.00243i) q^{10} +3.68607 q^{11} +(-3.18773 + 1.35587i) q^{12} +(-3.88771 + 3.88771i) q^{13} +(0.452626 + 1.07407i) q^{14} +(-3.66213 - 1.26047i) q^{15} +(0.0980619 + 3.99880i) q^{16} +(0.880105 + 0.880105i) q^{17} +(3.91883 - 1.62567i) q^{18} +6.32919 q^{19} +(-2.91214 + 3.39403i) q^{20} +(-0.542584 - 1.32036i) q^{21} +(-1.96533 + 4.82821i) 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} +(-4.81715 + 1.94809i) q^{27} +(-1.64820 + 0.0202063i) q^{28} -1.37122i q^{29} +(3.60361 - 4.12481i) q^{30} +3.32075 q^{31} +(-5.29013 - 2.00362i) q^{32} +(2.45973 - 5.89160i) 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} +(-3.37458 + 8.29032i) q^{38} +(3.61961 + 8.80819i) q^{39} +(-2.89300 - 5.62410i) q^{40} -0.648104i q^{41} +(2.01877 - 0.00672215i) q^{42} +(-0.819412 + 0.819412i) q^{43} +(-5.27640 - 5.14859i) q^{44} +(-4.45843 + 5.01223i) q^{45} +(-1.60482 - 3.80818i) q^{46} +(6.28508 + 6.28508i) q^{47} +(6.45689 + 2.51168i) q^{48} +6.32075i q^{49} +(1.80795 - 6.83603i) q^{50} +(1.99401 - 0.819412i) q^{51} +(10.9953 - 0.134798i) 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} +(4.22349 - 10.1162i) q^{57} +(1.79611 + 0.731106i) q^{58} +6.12026i q^{59} +(3.48154 + 6.91946i) q^{60} -5.13471i q^{61} +(-1.77055 + 4.34971i) q^{62} +(-2.47245 - 0.0138443i) 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} +(-0.0305156 - 2.48912i) q^{68} +(1.92377 + 4.68141i) q^{69} +(2.33203 - 1.16365i) q^{70} -4.13251i q^{71} +(-7.88026 - 3.14666i) q^{72} +(4.69820 + 4.69820i) q^{73} +(4.49972 - 1.89624i) q^{74} +(-2.28895 + 8.35229i) q^{75} +(-9.05987 - 8.84042i) q^{76} +(2.14814 - 2.14814i) q^{77} +(-13.4674 + 0.0448439i) q^{78} -1.10079i q^{79} +(8.90925 - 0.790765i) q^{80} +(-0.100786 + 8.99944i) q^{81} +(0.848922 + 0.345554i) 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} +(-2.19169 - 0.915024i) q^{87} +(9.55717 - 4.16621i) q^{88} -15.3562 q^{89} +(-4.18816 - 8.51230i) q^{90} +4.53130i q^{91} +(5.84382 - 0.0716428i) q^{92} +(2.21595 - 5.30771i) 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} +(-8.27927 - 3.37008i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.533177	+	1.30986i	−0.377013	+	0.926208i
							\(3\)	0.667305	−	1.59834i	0.385268	−	0.922805i
\(5\)	−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	\(0.526000\pi\)
							−0.996666	+	0.0815911i	\(0.974000\pi\)
\(17\)	0.880105	+	0.880105i	0.213457	+	0.213457i	0.805734	−	0.592277i	\(-0.201771\pi\)
							−0.592277	+	0.805734i	\(0.701771\pi\)
\(19\)	6.32919			1.45201			0.726007	−	0.687687i	\(-0.241374\pi\)
							0.726007	+	0.687687i	\(0.241374\pi\)
\(23\)	−2.06626	+	2.06626i	−0.430844	+	0.430844i	−0.888916	−	0.458071i	\(-0.848540\pi\)
							0.458071	+	0.888916i	\(0.348540\pi\)
\(29\)		−	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	0.477342	−	0.878718i	\(-0.341600\pi\)
							−0.878718	+	0.477342i	\(0.841600\pi\)
\(41\)		−	0.648104i		−	0.101217i	−0.998719	−	0.0506084i	\(-0.983884\pi\)
							0.998719	−	0.0506084i	\(-0.0161160\pi\)
\(43\)	−0.819412	+	0.819412i	−0.124959	+	0.124959i	−0.766821	−	0.641861i	\(-0.778162\pi\)
							0.641861	+	0.766821i	\(0.278162\pi\)
\(47\)	6.28508	+	6.28508i	0.916772	+	0.916772i	0.996793	−	0.0800208i	\(-0.0254987\pi\)
							−0.0800208	+	0.996793i	\(0.525499\pi\)
\(53\)	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	\(-0.893384\pi\)
							0.944429	−	0.328716i	\(-0.106616\pi\)
\(67\)	−4.90636	−	4.90636i	−0.599408	−	0.599408i	0.340747	−	0.940155i	\(-0.389320\pi\)
							−0.940155	+	0.340747i	\(0.889320\pi\)
\(71\)		−	4.13251i		−	0.490439i	−0.969468	−	0.245220i	\(-0.921140\pi\)
							0.969468	−	0.245220i	\(-0.0788601\pi\)
\(73\)	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	\(-0.802646\pi\)
							−0.813875	+	0.581040i	\(0.802646\pi\)
\(97\)	−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.53.6	yes	32
3.2	odd	2	inner	120.2.w.c.53.11	yes	32
4.3	odd	2		480.2.bi.c.113.6		32
5.2	odd	4	inner	120.2.w.c.77.3	yes	32
5.3	odd	4		600.2.w.j.557.14		32
5.4	even	2		600.2.w.j.293.11		32
8.3	odd	2		480.2.bi.c.113.11		32
8.5	even	2	inner	120.2.w.c.53.14	yes	32
12.11	even	2		480.2.bi.c.113.3		32
15.2	even	4	inner	120.2.w.c.77.14	yes	32
15.8	even	4		600.2.w.j.557.3		32
15.14	odd	2		600.2.w.j.293.6		32
20.7	even	4		480.2.bi.c.17.14		32
24.5	odd	2	inner	120.2.w.c.53.3	✓	32
24.11	even	2		480.2.bi.c.113.14		32
40.13	odd	4		600.2.w.j.557.6		32
40.27	even	4		480.2.bi.c.17.3		32
40.29	even	2		600.2.w.j.293.3		32
40.37	odd	4	inner	120.2.w.c.77.11	yes	32
60.47	odd	4		480.2.bi.c.17.11		32
120.29	odd	2		600.2.w.j.293.14		32
120.53	even	4		600.2.w.j.557.11		32
120.77	even	4	inner	120.2.w.c.77.6	yes	32
120.107	odd	4		480.2.bi.c.17.6		32

    

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