Embedded newform 120.2.w.c.77.12

• Label: 120.2.w.c.77.12
• 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			77.12
Character	\(\chi\)	\(=\)	120.77
Dual form			120.2.w.c.53.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.864261 - 1.11940i) q^{2} +(-0.170116 + 1.72368i) q^{3} +(-0.506107 - 1.93490i) q^{4} +(1.36575 - 1.77052i) q^{5} +(1.78246 + 1.68013i) q^{6} +(2.06963 + 2.06963i) q^{7} +(-2.60334 - 1.10573i) q^{8} +(-2.94212 - 0.586449i) q^{9} +(-0.801547 - 3.05901i) q^{10} +0.510276 q^{11} +(3.42125 - 0.543207i) q^{12} +(-0.750647 - 0.750647i) q^{13} +(4.10544 - 0.528041i) q^{14} +(2.81946 + 2.65531i) q^{15} +(-3.48771 + 1.95854i) q^{16} +(-3.14698 + 3.14698i) q^{17} +(-3.19923 + 2.78656i) q^{18} -6.01198 q^{19} +(-4.11699 - 1.74653i) q^{20} +(-3.91945 + 3.21529i) q^{21} +(0.441012 - 0.571203i) q^{22} +(2.54575 + 2.54575i) q^{23} +(2.34878 - 4.29921i) q^{24} +(-1.26945 - 4.83617i) q^{25} +(-1.48903 + 0.191519i) q^{26} +(1.51135 - 4.97150i) q^{27} +(2.95708 - 5.05199i) q^{28} +5.10739i q^{29} +(5.40909 - 0.861223i) q^{30} -4.56672 q^{31} +(-0.821906 + 5.59683i) q^{32} +(-0.0868061 + 0.879551i) q^{33} +(0.802913 + 6.24253i) q^{34} +(6.49090 - 0.837710i) q^{35} +(0.354305 + 5.98953i) q^{36} +(6.76263 - 6.76263i) q^{37} +(-5.19592 + 6.72981i) q^{38} +(1.42157 - 1.16618i) q^{39} +(-5.51322 + 3.09910i) q^{40} -4.24355i q^{41} +(0.211772 + 7.16627i) q^{42} +(5.95972 + 5.95972i) q^{43} +(-0.258254 - 0.987336i) q^{44} +(-5.05652 + 4.40813i) q^{45} +(5.04990 - 0.649518i) q^{46} +(3.33849 - 3.33849i) q^{47} +(-2.78257 - 6.34486i) q^{48} +1.56672i q^{49} +(-6.51073 - 2.75869i) q^{50} +(-4.88902 - 5.95972i) q^{51} +(-1.07252 + 1.83234i) q^{52} +(5.75871 - 5.75871i) q^{53} +(-4.25889 - 5.98848i) q^{54} +(0.696910 - 0.903452i) q^{55} +(-3.09950 - 7.67638i) q^{56} +(1.02273 - 10.3627i) q^{57} +(5.71720 + 4.41411i) q^{58} +1.16514i q^{59} +(3.71082 - 6.79925i) q^{60} -4.92929i q^{61} +(-3.94684 + 5.11198i) q^{62} +(-4.87536 - 7.30283i) q^{63} +(5.55474 + 5.75716i) q^{64} +(-2.35423 + 0.303835i) q^{65} +(0.909545 + 0.857332i) q^{66} +(-7.98415 + 7.98415i) q^{67} +(7.68180 + 4.49639i) q^{68} +(-4.82112 + 3.95497i) q^{69} +(4.67210 - 7.98991i) q^{70} -5.09150i q^{71} +(7.01088 + 4.77991i) q^{72} +(-3.20654 + 3.20654i) q^{73} +(-1.72540 - 13.4148i) q^{74} +(8.55194 - 1.36541i) q^{75} +(3.04271 + 11.6326i) q^{76} +(1.05608 + 1.05608i) q^{77} +(-0.0768090 - 2.59918i) q^{78} -7.31215i q^{79} +(-1.29572 + 8.84992i) q^{80} +(8.31215 + 3.45081i) q^{81} +(-4.75023 - 3.66754i) q^{82} +(-4.77995 + 4.77995i) q^{83} +(8.20494 + 5.95647i) q^{84} +(1.27378 + 9.86975i) q^{85} +(11.8220 - 1.52055i) q^{86} +(-8.80349 - 0.868848i) q^{87} +(-1.32842 - 0.564226i) q^{88} +12.6431 q^{89} +(0.564297 + 9.47004i) q^{90} -3.10712i q^{91} +(3.63736 - 6.21420i) q^{92} +(0.776871 - 7.87155i) q^{93} +(-0.851777 - 6.62243i) q^{94} +(-8.21087 + 10.6443i) q^{95} +(-9.50730 - 2.36881i) q^{96} +(10.8789 + 10.8789i) q^{97} +(1.75378 + 1.35405i) q^{98} +(-1.50129 - 0.299251i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.864261	−	1.11940i	0.611125	−	0.791534i
							\(3\)	−0.170116	+	1.72368i	−0.0982164	+	0.995165i
\(5\)	1.36575	−	1.77052i	0.610783	−	0.791798i
							\(7\)	2.06963	+	2.06963i	0.782246	+	0.782246i	0.980209	−	0.197964i	\(-0.0634328\pi\)
−0.197964	+	0.980209i	\(0.563433\pi\)
\(11\)	0.510276			0.153854			0.0769270	−	0.997037i	\(-0.475489\pi\)
							0.0769270	+	0.997037i	\(0.475489\pi\)
\(13\)	−0.750647	−	0.750647i	−0.208192	−	0.208192i	0.595307	−	0.803499i	\(-0.297031\pi\)
							−0.803499	+	0.595307i	\(0.797031\pi\)
\(17\)	−3.14698	+	3.14698i	−0.763254	+	0.763254i	−0.976909	−	0.213656i	\(-0.931463\pi\)
							0.213656	+	0.976909i	\(0.431463\pi\)
\(19\)	−6.01198			−1.37924			−0.689622	−	0.724170i	\(-0.742223\pi\)
							−0.689622	+	0.724170i	\(0.742223\pi\)
\(23\)	2.54575	+	2.54575i	0.530825	+	0.530825i	0.920818	−	0.389993i	\(-0.127522\pi\)
							−0.389993	+	0.920818i	\(0.627522\pi\)
\(29\)			5.10739i			0.948418i	0.880412	+	0.474209i	\(0.157266\pi\)
							−0.880412	+	0.474209i	\(0.842734\pi\)
\(31\)	−4.56672			−0.820207			−0.410104	−	0.912039i	\(-0.634507\pi\)
							−0.410104	+	0.912039i	\(0.634507\pi\)
\(37\)	6.76263	−	6.76263i	1.11177	−	1.11177i	0.118858	−	0.992911i	\(-0.462077\pi\)
							0.992911	−	0.118858i	\(-0.0379234\pi\)
\(41\)		−	4.24355i		−	0.662732i	−0.943502	−	0.331366i	\(-0.892491\pi\)
							0.943502	−	0.331366i	\(-0.107509\pi\)
\(43\)	5.95972	+	5.95972i	0.908848	+	0.908848i	0.996179	−	0.0873310i	\(-0.0278338\pi\)
							−0.0873310	+	0.996179i	\(0.527834\pi\)
\(47\)	3.33849	−	3.33849i	0.486969	−	0.486969i	−0.420379	−	0.907349i	\(-0.638103\pi\)
							0.907349	+	0.420379i	\(0.138103\pi\)
\(53\)	5.75871	−	5.75871i	0.791019	−	0.791019i	−0.190641	−	0.981660i	\(-0.561057\pi\)
							0.981660	+	0.190641i	\(0.0610565\pi\)
\(59\)			1.16514i			0.151689i	0.997120	+	0.0758444i	\(0.0241652\pi\)
							−0.997120	+	0.0758444i	\(0.975835\pi\)
\(61\)		−	4.92929i		−	0.631131i	−0.948904	−	0.315565i	\(-0.897806\pi\)
							0.948904	−	0.315565i	\(-0.102194\pi\)
\(67\)	−7.98415	+	7.98415i	−0.975419	+	0.975419i	−0.999705	−	0.0242864i	\(-0.992269\pi\)
							0.0242864	+	0.999705i	\(0.492269\pi\)
\(71\)		−	5.09150i		−	0.604250i	−0.953268	−	0.302125i	\(-0.902304\pi\)
							0.953268	−	0.302125i	\(-0.0976959\pi\)
\(73\)	−3.20654	+	3.20654i	−0.375297	+	0.375297i	−0.869402	−	0.494105i	\(-0.835496\pi\)
							0.494105	+	0.869402i	\(0.335496\pi\)
\(79\)		−	7.31215i		−	0.822682i	−0.911482	−	0.411341i	\(-0.865061\pi\)
							0.911482	−	0.411341i	\(-0.134939\pi\)
\(83\)	−4.77995	+	4.77995i	−0.524668	+	0.524668i	−0.918978	−	0.394310i	\(-0.870984\pi\)
							0.394310	+	0.918978i	\(0.370984\pi\)
\(89\)	12.6431			1.34017			0.670083	−	0.742286i	\(-0.266259\pi\)
							0.670083	+	0.742286i	\(0.266259\pi\)
\(97\)	10.8789	+	10.8789i	1.10458	+	1.10458i	0.993850	+	0.110732i	\(0.0353195\pi\)
							0.110732	+	0.993850i	\(0.464680\pi\)

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.77.12	yes	32
3.2	odd	2	inner	120.2.w.c.77.5	yes	32
4.3	odd	2		480.2.bi.c.17.9		32
5.2	odd	4		600.2.w.j.293.13		32
5.3	odd	4	inner	120.2.w.c.53.4	✓	32
5.4	even	2		600.2.w.j.557.5		32
8.3	odd	2		480.2.bi.c.17.8		32
8.5	even	2	inner	120.2.w.c.77.13	yes	32
12.11	even	2		480.2.bi.c.17.1		32
15.2	even	4		600.2.w.j.293.4		32
15.8	even	4	inner	120.2.w.c.53.13	yes	32
15.14	odd	2		600.2.w.j.557.12		32
20.3	even	4		480.2.bi.c.113.16		32
24.5	odd	2	inner	120.2.w.c.77.4	yes	32
24.11	even	2		480.2.bi.c.17.16		32
40.3	even	4		480.2.bi.c.113.1		32
40.13	odd	4	inner	120.2.w.c.53.5	yes	32
40.29	even	2		600.2.w.j.557.4		32
40.37	odd	4		600.2.w.j.293.12		32
60.23	odd	4		480.2.bi.c.113.8		32
120.29	odd	2		600.2.w.j.557.13		32
120.53	even	4	inner	120.2.w.c.53.12	yes	32
120.77	even	4		600.2.w.j.293.5		32
120.83	odd	4		480.2.bi.c.113.9		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.4	✓	32	5.3	odd	4	inner
120.2.w.c.53.5	yes	32	40.13	odd	4	inner
120.2.w.c.53.12	yes	32	120.53	even	4	inner
120.2.w.c.53.13	yes	32	15.8	even	4	inner
120.2.w.c.77.4	yes	32	24.5	odd	2	inner
120.2.w.c.77.5	yes	32	3.2	odd	2	inner
120.2.w.c.77.12	yes	32	1.1	even	1	trivial
120.2.w.c.77.13	yes	32	8.5	even	2	inner
480.2.bi.c.17.1		32	12.11	even	2
480.2.bi.c.17.8		32	8.3	odd	2
480.2.bi.c.17.9		32	4.3	odd	2
480.2.bi.c.17.16		32	24.11	even	2
480.2.bi.c.113.1		32	40.3	even	4
480.2.bi.c.113.8		32	60.23	odd	4
480.2.bi.c.113.9		32	120.83	odd	4
480.2.bi.c.113.16		32	20.3	even	4
600.2.w.j.293.4		32	15.2	even	4
600.2.w.j.293.5		32	120.77	even	4
600.2.w.j.293.12		32	40.37	odd	4
600.2.w.j.293.13		32	5.2	odd	4
600.2.w.j.557.4		32	40.29	even	2
600.2.w.j.557.5		32	5.4	even	2
600.2.w.j.557.12		32	15.14	odd	2
600.2.w.j.557.13		32	120.29	odd	2