Embedded newform 925.2.y.b.193.5

• Label: 925.2.y.b.193.5
• Level: $925$
• Weight: $2$
• Character: 925.193
• Analytic conductor: $7.386$
• Analytic rank: $0$
• Dimension: $68$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.38616218697\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 185)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			193.5
Character	\(\chi\)	\(=\)	925.193
Dual form			925.2.y.b.532.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.971183 + 0.560713i) q^{2} +(-0.168506 - 0.628874i) q^{3} +(-0.371203 + 0.642942i) q^{4} +(0.516268 + 0.516268i) q^{6} +(0.377247 + 1.40791i) q^{7} -3.07540i q^{8} +(2.23099 - 1.28806i) q^{9} +0.304712i q^{11} +(0.466879 + 0.125100i) q^{12} +(0.187550 + 0.108282i) q^{13} +(-1.15581 - 1.15581i) q^{14} +(0.982011 + 1.70089i) q^{16} +(-0.836960 - 1.44966i) q^{17} +(-1.44446 + 2.50189i) q^{18} +(-1.91513 - 7.14737i) q^{19} +(0.821827 - 0.474482i) q^{21} +(-0.170856 - 0.295931i) q^{22} +5.69268i q^{23} +(-1.93404 + 0.518224i) q^{24} -0.242860 q^{26} +(-2.56707 - 2.56707i) q^{27} +(-1.04524 - 0.280071i) q^{28} +(-1.05734 - 1.05734i) q^{29} +(2.64214 - 2.64214i) q^{31} +(3.41933 + 1.97415i) q^{32} +(0.191625 - 0.0513459i) q^{33} +(1.62568 + 0.938587i) q^{34} +1.91253i q^{36} +(6.04562 - 0.671196i) q^{37} +(5.86756 + 5.86756i) q^{38} +(0.0364924 - 0.136191i) q^{39} +(2.07096 + 1.19567i) q^{41} +(-0.532096 + 0.921617i) q^{42} -7.13995i q^{43} +(-0.195912 - 0.113110i) q^{44} +(-3.19196 - 5.52863i) q^{46} +(3.92274 - 3.92274i) q^{47} +(0.904172 - 0.904172i) q^{48} +(4.22229 - 2.43774i) q^{49} +(-0.770618 + 0.770618i) q^{51} +(-0.139238 + 0.0803891i) q^{52} +(-2.13924 + 7.98376i) q^{53} +(3.93248 + 1.05370i) q^{54} +(4.32988 - 1.16019i) q^{56} +(-4.17208 + 2.40875i) q^{57} +(1.61973 + 0.434004i) q^{58} +(6.39295 + 1.71299i) q^{59} +(2.09439 + 7.81639i) q^{61} +(-1.08452 + 4.04748i) q^{62} +(2.65511 + 2.65511i) q^{63} -8.35577 q^{64} +(-0.157313 + 0.157313i) q^{66} +(7.56615 - 2.02734i) q^{67} +1.24273 q^{68} +(3.57998 - 0.959252i) q^{69} +(5.55820 - 9.62708i) q^{71} +(-3.96131 - 6.86119i) q^{72} +(8.42960 - 8.42960i) q^{73} +(-5.49505 + 4.04171i) q^{74} +(5.30625 + 1.42180i) q^{76} +(-0.429006 + 0.114952i) q^{77} +(0.0409234 + 0.152728i) q^{78} +(-1.51908 - 5.66927i) q^{79} +(2.68239 - 4.64604i) q^{81} -2.68171 q^{82} +(-1.06045 + 3.95767i) q^{83} +0.704516i q^{84} +(4.00346 + 6.93419i) q^{86} +(-0.486763 + 0.843098i) q^{87} +0.937113 q^{88} +(0.842850 - 3.14556i) q^{89} +(-0.0816982 + 0.304902i) q^{91} +(-3.66007 - 2.11314i) q^{92} +(-2.10679 - 1.21635i) q^{93} +(-1.61017 + 6.00923i) q^{94} +(0.665313 - 2.48298i) q^{96} +4.65513 q^{97} +(-2.73374 + 4.73498i) q^{98} +(0.392488 + 0.679809i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	68 * q + 6 * q^2 + 4 * q^3 + 30 * q^4 - 8 * q^6 + 2 * q^7 + 10 * q^12 + 6 * q^13 - 26 * q^16 + 10 * q^17 + 8 * q^18 - 4 * q^19 - 12 * q^21 + 14 * q^22 - 24 * q^26 - 68 * q^27 - 14 * q^28 - 14 * q^29 - 24 * q^31 - 18 * q^32 - 10 * q^33 + 18 * q^37 + 36 * q^38 - 52 * q^39 - 18 * q^41 + 40 * q^42 + 36 * q^44 - 52 * q^46 + 24 * q^47 - 60 * q^48 + 36 * q^49 - 8 * q^51 + 78 * q^52 + 38 * q^53 - 40 * q^54 + 16 * q^56 - 90 * q^57 - 16 * q^58 + 8 * q^59 + 4 * q^61 + 22 * q^62 + 48 * q^63 + 20 * q^64 + 80 * q^66 + 56 * q^67 + 20 * q^68 - 8 * q^69 + 4 * q^71 - 32 * q^72 - 60 * q^73 + 44 * q^74 + 72 * q^76 - 6 * q^77 + 24 * q^78 - 56 * q^79 - 6 * q^81 + 8 * q^82 - 12 * q^83 - 4 * q^86 + 32 * q^87 + 36 * q^88 + 22 * q^89 + 44 * q^91 - 156 * q^92 + 30 * q^93 + 20 * q^94 - 8 * q^96 - 16 * q^97 - 48 * q^98 - 32 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\)	\(76\)	\(852\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{3}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.971183	+	0.560713i	−0.686730	+	0.396484i	−0.802386	−	0.596806i	\(-0.796436\pi\)
							0.115656	+	0.993289i	\(0.463103\pi\)
\(3\)	−0.168506	−	0.628874i	−0.0972871	−	0.363080i	0.900069	−	0.435747i	\(-0.143516\pi\)
							−0.997356	+	0.0726666i	\(0.976849\pi\)
\(5\)		0			0
							\(7\)	0.377247	+	1.40791i	0.142586	+	0.532139i	0.999851	+	0.0172630i	\(0.00549527\pi\)
−0.857265	+	0.514876i	\(0.827838\pi\)
\(11\)			0.304712i			0.0918742i	0.998944	+	0.0459371i	\(0.0146274\pi\)
							−0.998944	+	0.0459371i	\(0.985373\pi\)
\(13\)	0.187550	+	0.108282i	0.0520170	+	0.0300320i	0.525783	−	0.850619i	\(-0.323772\pi\)
							−0.473766	+	0.880651i	\(0.657106\pi\)
\(17\)	−0.836960	−	1.44966i	−0.202993	−	0.351593i	0.746499	−	0.665387i	\(-0.231733\pi\)
							−0.949491	+	0.313793i	\(0.898400\pi\)
\(19\)	−1.91513	−	7.14737i	−0.439361	−	1.63972i	−0.730409	−	0.683010i	\(-0.760671\pi\)
							0.291048	−	0.956708i	\(-0.405996\pi\)
\(23\)			5.69268i			1.18701i	0.804832	+	0.593503i	\(0.202256\pi\)
							−0.804832	+	0.593503i	\(0.797744\pi\)
\(29\)	−1.05734	−	1.05734i	−0.196342	−	0.196342i	0.602088	−	0.798430i	\(-0.294336\pi\)
							−0.798430	+	0.602088i	\(0.794336\pi\)
\(31\)	2.64214	−	2.64214i	0.474542	−	0.474542i	−0.428839	−	0.903381i	\(-0.641077\pi\)
							0.903381	+	0.428839i	\(0.141077\pi\)
\(37\)	6.04562	−	0.671196i	0.993893	−	0.110344i
							\(41\)	2.07096	+	1.19567i	0.323430	+	0.186732i	0.652920	−	0.757427i	\(-0.273544\pi\)
−0.329491	+	0.944159i	\(0.606877\pi\)
\(43\)		−	7.13995i		−	1.08883i	−0.838815	−	0.544416i	\(-0.816751\pi\)
							0.838815	−	0.544416i	\(-0.183249\pi\)
\(47\)	3.92274	−	3.92274i	0.572191	−	0.572191i	−0.360549	−	0.932740i	\(-0.617411\pi\)
							0.932740	+	0.360549i	\(0.117411\pi\)
\(53\)	−2.13924	+	7.98376i	−0.293848	+	1.09665i	0.648281	+	0.761401i	\(0.275488\pi\)
							−0.942128	+	0.335253i	\(0.891178\pi\)
\(59\)	6.39295	+	1.71299i	0.832291	+	0.223012i	0.649713	−	0.760180i	\(-0.274889\pi\)
							0.182578	+	0.983191i	\(0.441556\pi\)
\(61\)	2.09439	+	7.81639i	0.268160	+	1.00079i	0.960288	+	0.279011i	\(0.0900066\pi\)
							−0.692128	+	0.721775i	\(0.743327\pi\)
\(67\)	7.56615	−	2.02734i	0.924353	−	0.247680i	0.234908	−	0.972018i	\(-0.424521\pi\)
							0.689445	+	0.724338i	\(0.257855\pi\)
\(71\)	5.55820	−	9.62708i	0.659637	−	1.14252i	−0.321073	−	0.947054i	\(-0.604044\pi\)
							0.980710	−	0.195470i	\(-0.0626231\pi\)
\(73\)	8.42960	−	8.42960i	0.986611	−	0.986611i	−0.0133009	−	0.999912i	\(-0.504234\pi\)
							0.999912	+	0.0133009i	\(0.00423394\pi\)
\(79\)	−1.51908	−	5.66927i	−0.170909	−	0.637843i	−0.997212	−	0.0746166i	\(-0.976227\pi\)
							0.826303	−	0.563226i	\(-0.190440\pi\)
\(83\)	−1.06045	+	3.95767i	−0.116400	+	0.434410i	−0.999388	−	0.0349854i	\(-0.988862\pi\)
							0.882988	+	0.469396i	\(0.155528\pi\)
\(89\)	0.842850	−	3.14556i	0.0893419	−	0.333429i	−0.906759	−	0.421649i	\(-0.861451\pi\)
							0.996101	+	0.0882206i	\(0.0281181\pi\)
\(97\)	4.65513			0.472657			0.236328	−	0.971673i	\(-0.424056\pi\)
							0.236328	+	0.971673i	\(0.424056\pi\)

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	925.2.y.b.193.5		68
5.2	odd	4		925.2.t.b.82.5		68
5.3	odd	4		185.2.p.a.82.13	✓	68
5.4	even	2		185.2.u.a.8.13	yes	68
37.14	odd	12		925.2.t.b.643.5		68
185.14	odd	12		185.2.p.a.88.13	yes	68
185.88	even	12		185.2.u.a.162.13	yes	68
185.162	even	12	inner	925.2.y.b.532.5		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.p.a.82.13	✓	68	5.3	odd	4
185.2.p.a.88.13	yes	68	185.14	odd	12
185.2.u.a.8.13	yes	68	5.4	even	2
185.2.u.a.162.13	yes	68	185.88	even	12
925.2.t.b.82.5		68	5.2	odd	4
925.2.t.b.643.5		68	37.14	odd	12
925.2.y.b.193.5		68	1.1	even	1	trivial
925.2.y.b.532.5		68	185.162	even	12	inner