Embedded newform 120.2.w.c.53.7

• Label: 120.2.w.c.53.7
• 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			53.7
Character	\(\chi\)	\(=\)	120.53
Dual form			120.2.w.c.77.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.250043 - 1.39193i) q^{2} +(-1.40091 + 1.01856i) 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} +(1.43773 + 2.43576i) q^{8} +(0.925085 - 2.85381i) q^{9} +(0.396613 + 3.13731i) q^{10} -2.28378 q^{11} +(1.91764 - 2.88490i) q^{12} +(-1.05635 + 1.05635i) q^{13} +(3.76080 + 2.61540i) q^{14} +(3.24665 - 2.11170i) q^{15} +(3.03093 - 2.61026i) q^{16} +(-3.04391 - 3.04391i) q^{17} +(-4.20362 - 0.574081i) q^{18} +3.36831 q^{19} +(4.26775 - 1.33652i) q^{20} +(0.875741 - 5.54157i) q^{21} +(0.571043 + 3.17887i) 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} +(1.61081 + 4.94017i) q^{27} +(2.70010 - 5.88875i) q^{28} -2.71461i q^{29} +(-3.75114 - 3.99111i) q^{30} -6.49196 q^{31} +(-4.39118 - 3.56617i) q^{32} +(3.19937 - 2.32616i) 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} +(-0.842223 - 4.68847i) q^{38} +(0.403895 - 2.55579i) q^{39} +(-2.92747 - 5.60624i) q^{40} +10.8056i q^{41} +(-7.93246 + 0.166657i) q^{42} +(1.16384 - 1.16384i) q^{43} +(4.28199 - 1.58971i) q^{44} +(-2.39737 + 6.26519i) q^{45} +(6.05536 + 4.21112i) q^{46} +(1.83768 + 1.83768i) q^{47} +(-1.58735 + 6.74391i) q^{48} -3.49196i q^{49} +(-0.521095 - 7.05184i) q^{50} +(7.36463 + 1.16384i) q^{51} +(1.24529 - 2.71591i) 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} +(-4.71870 + 3.43082i) q^{57} +(-3.77856 + 0.678770i) q^{58} +7.41311i q^{59} +(-4.61741 + 6.21929i) q^{60} +8.97044i q^{61} +(1.62327 + 9.03638i) q^{62} +(4.41757 + 8.65522i) 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} +(7.82602 + 3.58837i) q^{68} +(1.41005 - 8.92262i) q^{69} +(-8.09413 - 6.27732i) q^{70} -7.37570i q^{71} +(8.28122 - 1.84971i) q^{72} +(1.83441 + 1.83441i) q^{73} +(-2.64001 + 3.79620i) q^{74} +(-7.49530 + 4.33825i) q^{75} +(-6.31544 + 2.34464i) q^{76} +(5.23080 - 5.23080i) q^{77} +(-3.65848 + 0.0768630i) q^{78} -8.28844i q^{79} +(-7.07152 + 5.47664i) q^{80} +(-7.28844 - 5.28003i) q^{81} +(15.0406 - 2.70185i) 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} +(2.76499 + 3.80292i) q^{87} +(-3.28345 - 5.56275i) q^{88} -11.5311 q^{89} +(9.32017 + 1.77042i) q^{90} -4.83893i q^{91} +(4.34749 - 9.48162i) q^{92} +(9.09464 - 6.61243i) 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} +(-4.86058 + 0.873141i) 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 \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(41\)	\(61\)	\(97\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.250043	−	1.39193i	−0.176807	−	0.984246i
							\(3\)	−1.40091	+	1.01856i	−0.808814	+	0.588064i
\(5\)	−2.23305	−	0.116202i	−0.998649	−	0.0519669i
							\(7\)	−2.29041	+	2.29041i	−0.865694	+	0.865694i	−0.991992	−	0.126298i	\(-0.959690\pi\)
0.126298	+	0.991992i	\(0.459690\pi\)
\(11\)	−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	\(-0.816424\pi\)
							0.545278	+	0.838255i	\(0.316424\pi\)
\(17\)	−3.04391	−	3.04391i	−0.738256	−	0.738256i	0.233984	−	0.972240i	\(-0.424824\pi\)
							−0.972240	+	0.233984i	\(0.924824\pi\)
\(19\)	3.36831			0.772744			0.386372	−	0.922343i	\(-0.373728\pi\)
							0.386372	+	0.922343i	\(0.373728\pi\)
\(23\)	−3.68785	+	3.68785i	−0.768969	+	0.768969i	−0.977925	−	0.208956i	\(-0.932994\pi\)
							0.208956	+	0.977925i	\(0.432994\pi\)
\(29\)		−	2.71461i		−	0.504091i	−0.967715	−	0.252045i	\(-0.918897\pi\)
							0.967715	−	0.252045i	\(-0.0811032\pi\)
\(31\)	−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	\(-0.336614\pi\)
							−0.871133	+	0.491048i	\(0.836614\pi\)
\(41\)			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	\(-0.709946\pi\)
							0.790258	+	0.612774i	\(0.209946\pi\)
\(47\)	1.83768	+	1.83768i	0.268053	+	0.268053i	0.828315	−	0.560262i	\(-0.189300\pi\)
							−0.560262	+	0.828315i	\(0.689300\pi\)
\(53\)	5.82856	+	5.82856i	0.800615	+	0.800615i	0.983192	−	0.182577i	\(-0.0584439\pi\)
							−0.182577	+	0.983192i	\(0.558444\pi\)
\(59\)			7.41311i			0.965104i	0.875867	+	0.482552i	\(0.160290\pi\)
							−0.875867	+	0.482552i	\(0.839710\pi\)
\(61\)			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	−0.998183	−	0.0602525i	\(-0.980809\pi\)
							−0.0602525	−	0.998183i	\(-0.519191\pi\)
\(71\)		−	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	0.806261	−	0.591560i	\(-0.201488\pi\)
							−0.591560	+	0.806261i	\(0.701488\pi\)
\(79\)		−	8.28844i		−	0.932522i	−0.884647	−	0.466261i	\(-0.845601\pi\)
							0.884647	−	0.466261i	\(-0.154399\pi\)
\(83\)	−5.27928	−	5.27928i	−0.579476	−	0.579476i	0.355283	−	0.934759i	\(-0.384384\pi\)
							−0.934759	+	0.355283i	\(0.884384\pi\)
\(89\)	−11.5311			−1.22230			−0.611149	−	0.791515i	\(-0.709292\pi\)
							−0.611149	+	0.791515i	\(0.709292\pi\)
\(97\)	−2.79647	+	2.79647i	−0.283939	+	0.283939i	−0.834678	−	0.550739i	\(-0.814346\pi\)
							0.550739	+	0.834678i	\(0.314346\pi\)

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.c.53.2	✓	32	8.5	even	2	inner
120.2.w.c.53.7	yes	32	1.1	even	1	trivial
120.2.w.c.53.10	yes	32	3.2	odd	2	inner
120.2.w.c.53.15	yes	32	24.5	odd	2	inner
120.2.w.c.77.2	yes	32	15.2	even	4	inner
120.2.w.c.77.7	yes	32	120.77	even	4	inner
120.2.w.c.77.10	yes	32	40.37	odd	4	inner
120.2.w.c.77.15	yes	32	5.2	odd	4	inner
480.2.bi.c.17.4		32	60.47	odd	4
480.2.bi.c.17.5		32	20.7	even	4
480.2.bi.c.17.12		32	40.27	even	4
480.2.bi.c.17.13		32	120.107	odd	4
480.2.bi.c.113.4		32	8.3	odd	2
480.2.bi.c.113.5		32	24.11	even	2
480.2.bi.c.113.12		32	12.11	even	2
480.2.bi.c.113.13		32	4.3	odd	2
600.2.w.j.293.2		32	120.29	odd	2
600.2.w.j.293.7		32	15.14	odd	2
600.2.w.j.293.10		32	5.4	even	2
600.2.w.j.293.15		32	40.29	even	2
600.2.w.j.557.2		32	5.3	odd	4
600.2.w.j.557.7		32	40.13	odd	4
600.2.w.j.557.10		32	120.53	even	4
600.2.w.j.557.15		32	15.8	even	4