Embedded newform 225.2.q.a.196.23

• Label: 225.2.q.a.196.23
• Level: $225$
• Weight: $2$
• Character: 225.196
• Analytic conductor: $1.797$
• Analytic rank: $0$
• Dimension: $224$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			196.23
Character	\(\chi\)	\(=\)	225.196
Dual form			225.2.q.a.31.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.15183 + 1.27924i) q^{2} +(1.47785 - 0.903302i) q^{3} +(-0.100677 + 0.957875i) q^{4} +(-0.0644453 + 2.23514i) q^{5} +(2.85777 + 0.850072i) q^{6} +(-2.11497 + 3.66323i) q^{7} +(1.44394 - 1.04909i) q^{8} +(1.36809 - 2.66989i) q^{9} +(-2.93350 + 2.49206i) q^{10} +(-3.66270 - 4.06784i) q^{11} +(0.716464 + 1.50654i) q^{12} +(1.55472 - 1.72669i) q^{13} +(-7.12222 + 1.51387i) q^{14} +(1.92376 + 3.36142i) q^{15} +(4.88941 + 1.03928i) q^{16} +(-1.13466 + 0.824379i) q^{17} +(4.99123 - 1.32515i) q^{18} +(4.74872 - 3.45015i) q^{19} +(-2.13449 - 0.286757i) q^{20} +(0.183394 + 7.32417i) q^{21} +(0.984923 - 9.37092i) q^{22} +(-3.41812 + 0.726543i) q^{23} +(1.18629 - 2.85471i) q^{24} +(-4.99169 - 0.288088i) q^{25} +3.99961 q^{26} +(-0.389879 - 5.18150i) q^{27} +(-3.29599 - 2.39468i) q^{28} +(-3.94221 + 1.75518i) q^{29} +(-2.08420 + 6.33272i) q^{30} +(-2.89216 - 1.28767i) q^{31} +(2.51747 + 4.36039i) q^{32} +(-9.08742 - 2.70315i) q^{33} +(-2.36151 - 0.501954i) q^{34} +(-8.05154 - 4.96333i) q^{35} +(2.41969 + 1.57926i) q^{36} +(0.930443 + 2.86361i) q^{37} +(9.88326 + 2.10075i) q^{38} +(0.737922 - 3.95617i) q^{39} +(2.25180 + 3.29502i) q^{40} +(3.09385 - 3.43607i) q^{41} +(-9.15810 + 8.67079i) q^{42} +(-2.17619 + 3.76927i) q^{43} +(4.26523 - 3.09887i) q^{44} +(5.87941 + 3.22994i) q^{45} +(-4.86650 - 3.53572i) q^{46} +(-2.62209 + 1.16743i) q^{47} +(8.16460 - 2.88072i) q^{48} +(-5.44619 - 9.43308i) q^{49} +(-5.38104 - 6.71738i) q^{50} +(-0.932197 + 2.24325i) q^{51} +(1.49743 + 1.66306i) q^{52} +(4.61926 + 3.35609i) q^{53} +(6.17929 - 6.46695i) q^{54} +(9.32824 - 7.92450i) q^{55} +(0.789153 + 7.50829i) q^{56} +(3.90138 - 9.38833i) q^{57} +(-6.78604 - 3.02134i) q^{58} +(-7.75922 + 8.61749i) q^{59} +(-3.41350 + 1.50431i) q^{60} +(7.25373 + 8.05609i) q^{61} +(-1.68404 - 5.18293i) q^{62} +(6.88697 + 10.6584i) q^{63} +(0.411065 - 1.26513i) q^{64} +(3.75920 + 3.58629i) q^{65} +(-7.00919 - 14.7385i) q^{66} +(10.3055 + 4.58831i) q^{67} +(-0.675418 - 1.16986i) q^{68} +(-4.39518 + 4.16131i) q^{69} +(-2.92473 - 16.0167i) q^{70} +(-5.34318 - 3.88205i) q^{71} +(-0.825499 - 5.29042i) q^{72} +(1.20170 - 3.69845i) q^{73} +(-2.59152 + 4.48864i) q^{74} +(-7.63721 + 4.08325i) q^{75} +(2.82672 + 4.89603i) q^{76} +(22.6480 - 4.81398i) q^{77} +(5.91083 - 3.61286i) q^{78} +(-2.62186 + 1.16733i) q^{79} +(-2.63803 + 10.8615i) q^{80} +(-5.25665 - 7.30532i) q^{81} +7.95913 q^{82} +(-0.713599 - 6.78944i) q^{83} +(-7.03410 - 0.561705i) q^{84} +(-1.76948 - 2.58925i) q^{85} +(-7.32838 + 1.55770i) q^{86} +(-4.24054 + 6.15490i) q^{87} +(-9.55625 - 2.03124i) q^{88} +(-1.41471 + 4.35403i) q^{89} +(2.64022 + 11.2415i) q^{90} +(3.03709 + 9.34720i) q^{91} +(-0.351813 - 3.34727i) q^{92} +(-5.43734 + 0.709504i) q^{93} +(-4.51362 - 2.00959i) q^{94} +(7.40552 + 10.8364i) q^{95} +(7.65919 + 4.16997i) q^{96} +(12.6316 - 5.62394i) q^{97} +(5.79405 - 17.8322i) q^{98} +(-15.8716 + 4.21384i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	224 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 - 8 * q^5 - 10 * q^6 - 8 * q^7 - 20 * q^8 - 8 * q^9 - 20 * q^10 - 11 * q^11 - 4 * q^12 - 3 * q^13 + q^14 - 48 * q^15 + 23 * q^16 - 24 * q^17 - 12 * q^19 + q^20 + 15 * q^21 - 11 * q^22 + q^23 - 30 * q^24 - 16 * q^25 - 136 * q^26 + 7 * q^27 + 4 * q^28 - 15 * q^29 - 24 * q^30 + 3 * q^31 + 12 * q^32 - 5 * q^33 + q^34 + 14 * q^35 + 38 * q^36 - 24 * q^37 + 55 * q^38 + 20 * q^39 + q^40 - 19 * q^41 - 38 * q^42 - 8 * q^43 + 4 * q^44 - 38 * q^45 - 20 * q^46 - 10 * q^47 - 25 * q^48 - 72 * q^49 - 3 * q^50 - 26 * q^51 - 25 * q^52 - 12 * q^53 + 53 * q^54 - 20 * q^55 - 60 * q^56 + 38 * q^57 - 23 * q^58 - 30 * q^59 - 33 * q^60 - 3 * q^61 - 44 * q^62 + 46 * q^63 - 44 * q^64 + 51 * q^65 - 134 * q^66 - 12 * q^67 - 156 * q^68 + 4 * q^69 - 16 * q^70 + 42 * q^71 + 74 * q^72 - 12 * q^73 + 90 * q^74 + 67 * q^75 - 8 * q^76 + 31 * q^77 - 92 * q^78 - 15 * q^79 + 298 * q^80 - 104 * q^81 + 8 * q^82 + 59 * q^83 + 115 * q^84 - 11 * q^85 + 9 * q^86 - 59 * q^87 - 23 * q^88 + 106 * q^89 + 107 * q^90 + 30 * q^91 + 11 * q^92 + 32 * q^93 + 25 * q^94 + 7 * q^95 + 35 * q^96 - 21 * q^97 + 146 * q^98 - 20 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.15183	+	1.27924i	0.814466	+	0.904556i	0.996901	−	0.0786620i	\(-0.0250648\pi\)
							−0.182436	+	0.983218i	\(0.558398\pi\)
\(3\)	1.47785	−	0.903302i	0.853238	−	0.521521i
							\(5\)	−0.0644453	+	2.23514i	−0.0288208	+	0.999585i
\(7\)	−2.11497	+	3.66323i	−0.799383	+	1.38457i								0.120635	+	0.992697i	\(0.461507\pi\)
							−0.920018	+	0.391876i	\(0.871826\pi\)
\(11\)	−3.66270	−	4.06784i	−1.10435	−	1.22650i	−0.971921	−	0.235308i	\(-0.924390\pi\)
							−0.132426	−	0.991193i	\(-0.542277\pi\)
\(13\)	1.55472	−	1.72669i	0.431201	−	0.478898i	−0.487911	−	0.872893i	\(-0.662241\pi\)
							0.919112	+	0.393996i	\(0.128908\pi\)
\(17\)	−1.13466	+	0.824379i	−0.275195	+	0.199941i	−0.716819	−	0.697259i	\(-0.754403\pi\)
							0.441624	+	0.897200i	\(0.354403\pi\)
\(19\)	4.74872	−	3.45015i	1.08943	−	0.791518i	0.110128	−	0.993917i	\(-0.464874\pi\)
							0.979303	+	0.202400i	\(0.0648740\pi\)
\(23\)	−3.41812	+	0.726543i	−0.712727	+	0.151495i	−0.549986	−	0.835174i	\(-0.685367\pi\)
							−0.162741	+	0.986669i	\(0.552034\pi\)
\(29\)	−3.94221	+	1.75518i	−0.732050	+	0.325929i	−0.738687	−	0.674049i	\(-0.764554\pi\)
							0.00663753	+	0.999978i	\(0.497887\pi\)
\(31\)	−2.89216	−	1.28767i	−0.519448	−	0.231273i	0.130229	−	0.991484i	\(-0.458429\pi\)
							−0.649676	+	0.760211i	\(0.725096\pi\)
\(37\)	0.930443	+	2.86361i	0.152964	+	0.470775i	0.997949	−	0.0640154i	\(-0.0203907\pi\)
							−0.844985	+	0.534790i	\(0.820391\pi\)
\(41\)	3.09385	−	3.43607i	0.483178	−	0.536624i	−0.451428	−	0.892308i	\(-0.649085\pi\)
							0.934606	+	0.355684i	\(0.115752\pi\)
\(43\)	−2.17619	+	3.76927i	−0.331866	+	0.574809i	−0.982878	−	0.184259i	\(-0.941011\pi\)
							0.651012	+	0.759068i	\(0.274345\pi\)
\(47\)	−2.62209	+	1.16743i	−0.382471	+	0.170287i	−0.588961	−	0.808162i	\(-0.700463\pi\)
							0.206489	+	0.978449i	\(0.433796\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.q.a.196.23	yes	224
3.2	odd	2		675.2.r.a.46.6		224
9.4	even	3	inner	225.2.q.a.121.6	yes	224
9.5	odd	6		675.2.r.a.496.23		224
25.6	even	5	inner	225.2.q.a.106.6	yes	224
75.56	odd	10		675.2.r.a.181.23		224
225.31	even	15	inner	225.2.q.a.31.23	✓	224
225.131	odd	30		675.2.r.a.631.6		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.q.a.31.23	✓	224	225.31	even	15	inner
225.2.q.a.106.6	yes	224	25.6	even	5	inner
225.2.q.a.121.6	yes	224	9.4	even	3	inner
225.2.q.a.196.23	yes	224	1.1	even	1	trivial
675.2.r.a.46.6		224	3.2	odd	2
675.2.r.a.181.23		224	75.56	odd	10
675.2.r.a.496.23		224	9.5	odd	6
675.2.r.a.631.6		224	225.131	odd	30