Embedded newform 120.2.w.c.77.15

• Label: 120.2.w.c.77.15
• 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			77.15
Character	\(\chi\)	\(=\)	120.77
Dual form			120.2.w.c.53.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39193 - 0.250043i) q^{2} +(1.01856 + 1.40091i) q^{3} +(1.87496 - 0.696087i) q^{4} +(-2.23305 + 0.116202i) q^{5} +(1.76805 + 1.69529i) q^{6} +(-2.29041 - 2.29041i) q^{7} +(2.43576 - 1.43773i) q^{8} +(-0.925085 + 2.85381i) q^{9} +(-3.07920 + 0.720103i) q^{10} -2.28378 q^{11} +(2.88490 + 1.91764i) q^{12} +(1.05635 + 1.05635i) q^{13} +(-3.76080 - 2.61540i) q^{14} +(-2.43727 - 3.00993i) q^{15} +(3.03093 - 2.61026i) q^{16} +(3.04391 - 3.04391i) q^{17} +(-0.574081 + 4.20362i) q^{18} -3.36831 q^{19} +(-4.10598 + 1.77227i) q^{20} +(0.875741 - 5.54157i) q^{21} +(-3.17887 + 0.571043i) q^{22} +(3.68785 + 3.68785i) q^{23} +(4.49508 + 1.94787i) q^{24} +(4.97299 - 0.518967i) q^{25} +(1.73449 + 1.20623i) q^{26} +(-4.94017 + 1.61081i) q^{27} +(-5.88875 - 2.70010i) q^{28} +2.71461i q^{29} +(-4.14513 - 3.58020i) q^{30} -6.49196 q^{31} +(3.56617 - 4.39118i) q^{32} +(-2.32616 - 3.19937i) q^{33} +(3.47581 - 4.99803i) q^{34} +(5.38074 + 4.84844i) q^{35} +(0.252003 + 5.99471i) q^{36} +(2.31197 - 2.31197i) q^{37} +(-4.68847 + 0.842223i) q^{38} +(-0.403895 + 2.55579i) q^{39} +(-5.27211 + 3.49355i) q^{40} +10.8056i q^{41} +(-0.166657 - 7.93246i) q^{42} +(-1.16384 - 1.16384i) q^{43} +(-4.28199 + 1.58971i) q^{44} +(1.73414 - 6.48018i) q^{45} +(6.05536 + 4.21112i) q^{46} +(-1.83768 + 1.83768i) q^{47} +(6.74391 + 1.58735i) q^{48} +3.49196i q^{49} +(6.79231 - 1.96583i) q^{50} +(7.36463 + 1.16384i) q^{51} +(2.71591 + 1.24529i) q^{52} +(5.82856 - 5.82856i) q^{53} +(-6.47362 + 3.47739i) q^{54} +(5.09979 - 0.265379i) q^{55} +(-8.87188 - 2.28592i) q^{56} +(-3.43082 - 4.71870i) q^{57} +(0.678770 + 3.77856i) q^{58} -7.41311i q^{59} +(-6.66496 - 3.94694i) q^{60} +8.97044i q^{61} +(-9.03638 + 1.62327i) q^{62} +(8.65522 - 4.41757i) q^{63} +(3.86589 - 7.00392i) q^{64} +(-2.48162 - 2.23612i) q^{65} +(-4.03784 - 3.87166i) q^{66} +(8.66367 - 8.66367i) q^{67} +(3.58837 - 7.82602i) q^{68} +(-1.41005 + 8.92262i) q^{69} +(8.70196 + 5.40329i) q^{70} -7.37570i q^{71} +(1.84971 + 8.28122i) q^{72} +(1.83441 - 1.83441i) q^{73} +(2.64001 - 3.79620i) q^{74} +(5.79230 + 6.43811i) q^{75} +(-6.31544 + 2.34464i) q^{76} +(5.23080 + 5.23080i) q^{77} +(0.0768630 + 3.65848i) q^{78} +8.28844i q^{79} +(-6.46488 + 6.18104i) q^{80} +(-7.28844 - 5.28003i) q^{81} +(2.70185 + 15.0406i) q^{82} +(-5.27928 + 5.27928i) q^{83} +(-2.21543 - 10.9998i) q^{84} +(-6.44348 + 7.15090i) q^{85} +(-1.91100 - 1.32898i) q^{86} +(-3.80292 + 2.76499i) q^{87} +(-5.56275 + 3.28345i) q^{88} +11.5311 q^{89} +(0.793483 - 9.45359i) q^{90} -4.83893i q^{91} +(9.48162 + 4.34749i) q^{92} +(-6.61243 - 9.09464i) q^{93} +(-2.09843 + 3.01742i) q^{94} +(7.52160 - 0.391403i) q^{95} +(9.78398 + 0.523212i) q^{96} +(-2.79647 - 2.79647i) q^{97} +(0.873141 + 4.86058i) q^{98} +(2.11269 - 6.51747i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39193	−	0.250043i	0.984246	−	0.176807i
							\(3\)	1.01856	+	1.40091i	0.588064	+	0.808814i
\(5\)	−2.23305	+	0.116202i	−0.998649	+	0.0519669i
							\(7\)	−2.29041	−	2.29041i	−0.865694	−	0.865694i	0.126298	−	0.991992i	\(-0.459690\pi\)
−0.991992	+	0.126298i	\(0.959690\pi\)
\(11\)	−2.28378			−0.688586			−0.344293	−	0.938862i	\(-0.611881\pi\)
							−0.344293	+	0.938862i	\(0.611881\pi\)
\(13\)	1.05635	+	1.05635i	0.292977	+	0.292977i	0.838255	−	0.545278i	\(-0.183576\pi\)
							−0.545278	+	0.838255i	\(0.683576\pi\)
\(17\)	3.04391	−	3.04391i	0.738256	−	0.738256i	−0.233984	−	0.972240i	\(-0.575176\pi\)
							0.972240	+	0.233984i	\(0.0751764\pi\)
\(19\)	−3.36831			−0.772744			−0.386372	−	0.922343i	\(-0.626272\pi\)
							−0.386372	+	0.922343i	\(0.626272\pi\)
\(23\)	3.68785	+	3.68785i	0.768969	+	0.768969i	0.977925	−	0.208956i	\(-0.0670065\pi\)
							−0.208956	+	0.977925i	\(0.567006\pi\)
\(29\)			2.71461i			0.504091i	0.967715	+	0.252045i	\(0.0811032\pi\)
							−0.967715	+	0.252045i	\(0.918897\pi\)
\(31\)	−6.49196			−1.16599			−0.582995	−	0.812475i	\(-0.698119\pi\)
							−0.582995	+	0.812475i	\(0.698119\pi\)
\(37\)	2.31197	−	2.31197i	0.380085	−	0.380085i	−0.491048	−	0.871133i	\(-0.663386\pi\)
							0.871133	+	0.491048i	\(0.163386\pi\)
\(41\)			10.8056i			1.68754i	0.536702	+	0.843772i	\(0.319670\pi\)
							−0.536702	+	0.843772i	\(0.680330\pi\)
\(43\)	−1.16384	−	1.16384i	−0.177484	−	0.177484i	0.612774	−	0.790258i	\(-0.290054\pi\)
							−0.790258	+	0.612774i	\(0.790054\pi\)
\(47\)	−1.83768	+	1.83768i	−0.268053	+	0.268053i	−0.828315	−	0.560262i	\(-0.810700\pi\)
							0.560262	+	0.828315i	\(0.310700\pi\)
\(53\)	5.82856	−	5.82856i	0.800615	−	0.800615i	−0.182577	−	0.983192i	\(-0.558444\pi\)
							0.983192	+	0.182577i	\(0.0584439\pi\)
\(59\)		−	7.41311i		−	0.965104i	−0.875867	−	0.482552i	\(-0.839710\pi\)
							0.875867	−	0.482552i	\(-0.160290\pi\)
\(61\)			8.97044i			1.14855i	0.818663	+	0.574274i	\(0.194716\pi\)
							−0.818663	+	0.574274i	\(0.805284\pi\)
\(67\)	8.66367	−	8.66367i	1.05844	−	1.05844i	0.0602525	−	0.998183i	\(-0.480809\pi\)
							0.998183	−	0.0602525i	\(-0.0191906\pi\)
\(71\)		−	7.37570i		−	0.875334i	−0.899137	−	0.437667i	\(-0.855805\pi\)
							0.899137	−	0.437667i	\(-0.144195\pi\)
\(73\)	1.83441	−	1.83441i	0.214701	−	0.214701i	−0.591560	−	0.806261i	\(-0.701488\pi\)
							0.806261	+	0.591560i	\(0.201488\pi\)
\(79\)			8.28844i			0.932522i	0.884647	+	0.466261i	\(0.154399\pi\)
							−0.884647	+	0.466261i	\(0.845601\pi\)
\(83\)	−5.27928	+	5.27928i	−0.579476	+	0.579476i	−0.934759	−	0.355283i	\(-0.884384\pi\)
							0.355283	+	0.934759i	\(0.384384\pi\)
\(89\)	11.5311			1.22230			0.611149	−	0.791515i	\(-0.290708\pi\)
							0.611149	+	0.791515i	\(0.290708\pi\)
\(97\)	−2.79647	−	2.79647i	−0.283939	−	0.283939i	0.550739	−	0.834678i	\(-0.314346\pi\)
							−0.834678	+	0.550739i	\(0.814346\pi\)

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.77.15	yes	32
3.2	odd	2	inner	120.2.w.c.77.2	yes	32
4.3	odd	2		480.2.bi.c.17.5		32
5.2	odd	4		600.2.w.j.293.10		32
5.3	odd	4	inner	120.2.w.c.53.7	yes	32
5.4	even	2		600.2.w.j.557.2		32
8.3	odd	2		480.2.bi.c.17.12		32
8.5	even	2	inner	120.2.w.c.77.10	yes	32
12.11	even	2		480.2.bi.c.17.4		32
15.2	even	4		600.2.w.j.293.7		32
15.8	even	4	inner	120.2.w.c.53.10	yes	32
15.14	odd	2		600.2.w.j.557.15		32
20.3	even	4		480.2.bi.c.113.13		32
24.5	odd	2	inner	120.2.w.c.77.7	yes	32
24.11	even	2		480.2.bi.c.17.13		32
40.3	even	4		480.2.bi.c.113.4		32
40.13	odd	4	inner	120.2.w.c.53.2	✓	32
40.29	even	2		600.2.w.j.557.7		32
40.37	odd	4		600.2.w.j.293.15		32
60.23	odd	4		480.2.bi.c.113.12		32
120.29	odd	2		600.2.w.j.557.10		32
120.53	even	4	inner	120.2.w.c.53.15	yes	32
120.77	even	4		600.2.w.j.293.2		32
120.83	odd	4		480.2.bi.c.113.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.2	✓	32	40.13	odd	4	inner
120.2.w.c.53.7	yes	32	5.3	odd	4	inner
120.2.w.c.53.10	yes	32	15.8	even	4	inner
120.2.w.c.53.15	yes	32	120.53	even	4	inner
120.2.w.c.77.2	yes	32	3.2	odd	2	inner
120.2.w.c.77.7	yes	32	24.5	odd	2	inner
120.2.w.c.77.10	yes	32	8.5	even	2	inner
120.2.w.c.77.15	yes	32	1.1	even	1	trivial
480.2.bi.c.17.4		32	12.11	even	2
480.2.bi.c.17.5		32	4.3	odd	2
480.2.bi.c.17.12		32	8.3	odd	2
480.2.bi.c.17.13		32	24.11	even	2
480.2.bi.c.113.4		32	40.3	even	4
480.2.bi.c.113.5		32	120.83	odd	4
480.2.bi.c.113.12		32	60.23	odd	4
480.2.bi.c.113.13		32	20.3	even	4
600.2.w.j.293.2		32	120.77	even	4
600.2.w.j.293.7		32	15.2	even	4
600.2.w.j.293.10		32	5.2	odd	4
600.2.w.j.293.15		32	40.37	odd	4
600.2.w.j.557.2		32	5.4	even	2
600.2.w.j.557.7		32	40.29	even	2
600.2.w.j.557.10		32	120.29	odd	2
600.2.w.j.557.15		32	15.14	odd	2