Embedded newform 225.2.q.a.196.22

• Label: 225.2.q.a.196.22
• 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			196.22
Character	\(\chi\)	\(=\)	225.196
Dual form			225.2.q.a.31.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.956880 + 1.06272i) q^{2} +(0.819476 - 1.52593i) q^{3} +(-0.00470356 + 0.0447514i) q^{4} +(-2.07212 + 0.840433i) q^{5} +(2.40578 - 0.589254i) q^{6} +(1.92019 - 3.32586i) q^{7} +(2.26178 - 1.64328i) q^{8} +(-1.65692 - 2.50092i) q^{9} +(-2.87591 - 1.39789i) q^{10} +(3.49787 + 3.88478i) q^{11} +(0.0644329 + 0.0438500i) q^{12} +(-2.25011 + 2.49900i) q^{13} +(5.37185 - 1.14182i) q^{14} +(-0.415610 + 3.85062i) q^{15} +(3.99864 + 0.849937i) q^{16} +(-3.90998 + 2.84077i) q^{17} +(1.07232 - 4.15393i) q^{18} +(-1.90903 + 1.38699i) q^{19} +(-0.0278642 - 0.0966831i) q^{20} +(-3.50148 - 5.65553i) q^{21} +(-0.781401 + 7.43453i) q^{22} +(4.56147 - 0.969570i) q^{23} +(-0.654054 - 4.79795i) q^{24} +(3.58734 - 3.48295i) q^{25} -4.80884 q^{26} +(-5.17404 + 0.478889i) q^{27} +(0.139805 + 0.101574i) q^{28} +(-6.71254 + 2.98862i) q^{29} +(-4.48983 + 3.24290i) q^{30} +(0.593962 + 0.264449i) q^{31} +(0.127249 + 0.220402i) q^{32} +(8.79432 - 2.15402i) q^{33} +(-6.76033 - 1.43695i) q^{34} +(-1.18369 + 8.50536i) q^{35} +(0.119713 - 0.0623860i) q^{36} +(0.315262 + 0.970276i) q^{37} +(-3.30069 - 0.701584i) q^{38} +(1.96939 + 5.48139i) q^{39} +(-3.30561 + 5.30596i) q^{40} +(-2.50903 + 2.78656i) q^{41} +(2.65977 - 9.13276i) q^{42} +(-2.34851 + 4.06773i) q^{43} +(-0.190302 + 0.138262i) q^{44} +(5.53519 + 3.78968i) q^{45} +(5.39516 + 3.91981i) q^{46} +(10.9118 - 4.85827i) q^{47} +(4.57373 - 5.40514i) q^{48} +(-3.87423 - 6.71036i) q^{49} +(7.13407 + 0.479583i) q^{50} +(1.13067 + 8.29429i) q^{51} +(-0.101250 - 0.112450i) q^{52} +(2.92011 + 2.12158i) q^{53} +(-5.45986 - 5.04033i) q^{54} +(-10.5129 - 5.10999i) q^{55} +(-1.12228 - 10.6778i) q^{56} +(0.552044 + 4.04964i) q^{57} +(-9.59917 - 4.27383i) q^{58} +(-5.20768 + 5.78372i) q^{59} +(-0.170366 - 0.0367107i) q^{60} +(-4.40843 - 4.89605i) q^{61} +(0.287314 + 0.884263i) q^{62} +(-11.4993 + 0.708433i) q^{63} +(2.41404 - 7.42965i) q^{64} +(2.56225 - 7.06930i) q^{65} +(10.7042 + 7.28479i) q^{66} +(-0.930565 - 0.414314i) q^{67} +(-0.108737 - 0.188339i) q^{68} +(2.25852 - 7.75502i) q^{69} +(-10.1715 + 6.88068i) q^{70} +(-2.82082 - 2.04945i) q^{71} +(-7.85732 - 2.93377i) q^{72} +(-1.38140 + 4.25150i) q^{73} +(-0.729467 + 1.26347i) q^{74} +(-2.37500 - 8.32823i) q^{75} +(-0.0530904 - 0.0919553i) q^{76} +(19.6368 - 4.17393i) q^{77} +(-3.94073 + 7.33794i) q^{78} +(1.49826 - 0.667070i) q^{79} +(-8.99997 + 1.59942i) q^{80} +(-3.50925 + 8.28765i) q^{81} -5.36219 q^{82} +(-1.18152 - 11.2414i) q^{83} +(0.269562 - 0.130095i) q^{84} +(5.71446 - 9.17248i) q^{85} +(-6.57011 + 1.39652i) q^{86} +(-0.940354 + 12.6920i) q^{87} +(14.2952 + 3.03854i) q^{88} +(-0.935178 + 2.87818i) q^{89} +(1.26913 + 9.50864i) q^{90} +(3.99070 + 12.2821i) q^{91} +(0.0219345 + 0.208692i) q^{92} +(0.890268 - 0.689634i) q^{93} +(15.6043 + 6.94749i) q^{94} +(2.79006 - 4.47841i) q^{95} +(0.440596 - 0.0135589i) q^{96} +(14.8057 - 6.59193i) q^{97} +(3.42408 - 10.5382i) q^{98} +(3.91986 - 15.1847i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.956880	+	1.06272i	0.676616	+	0.751458i	0.979472	−	0.201578i	\(-0.0646069\pi\)
							−0.302856	+	0.953036i	\(0.597940\pi\)
\(3\)	0.819476	−	1.52593i	0.473125	−	0.880995i
							\(5\)	−2.07212	+	0.840433i	−0.926679	+	0.375853i
\(7\)	1.92019	−	3.32586i	0.725762	−	1.25706i								−0.232897	−	0.972501i	\(-0.574821\pi\)
							0.958660	−	0.284556i	\(-0.0918460\pi\)
\(11\)	3.49787	+	3.88478i	1.05465	+	1.17131i	0.984790	+	0.173748i	\(0.0555879\pi\)
							0.0698578	+	0.997557i	\(0.477745\pi\)
\(13\)	−2.25011	+	2.49900i	−0.624069	+	0.693099i	−0.969429	−	0.245371i	\(-0.921090\pi\)
							0.345360	+	0.938470i	\(0.387757\pi\)
\(17\)	−3.90998	+	2.84077i	−0.948309	+	0.688987i	−0.950406	−	0.311011i	\(-0.899332\pi\)
							0.00209712	+	0.999998i	\(0.499332\pi\)
\(19\)	−1.90903	+	1.38699i	−0.437961	+	0.318197i	−0.784824	−	0.619719i	\(-0.787247\pi\)
							0.346863	+	0.937916i	\(0.387247\pi\)
\(23\)	4.56147	−	0.969570i	0.951132	−	0.202169i	0.293882	−	0.955842i	\(-0.405053\pi\)
							0.657250	+	0.753672i	\(0.271719\pi\)
\(29\)	−6.71254	+	2.98862i	−1.24649	+	0.554972i	−0.920626	−	0.390445i	\(-0.872321\pi\)
							−0.325862	+	0.945417i	\(0.605654\pi\)
\(31\)	0.593962	+	0.264449i	0.106679	+	0.0474964i	0.459382	−	0.888239i	\(-0.348071\pi\)
							−0.352703	+	0.935735i	\(0.614738\pi\)
\(37\)	0.315262	+	0.970276i	0.0518287	+	0.159512i	0.973621	−	0.228172i	\(-0.0732749\pi\)
							−0.921792	+	0.387685i	\(0.873275\pi\)
\(41\)	−2.50903	+	2.78656i	−0.391845	+	0.435188i	−0.906498	−	0.422211i	\(-0.861254\pi\)
							0.514653	+	0.857399i	\(0.327921\pi\)
\(43\)	−2.34851	+	4.06773i	−0.358144	+	0.620324i	−0.987651	−	0.156671i	\(-0.949924\pi\)
							0.629507	+	0.776995i	\(0.283257\pi\)
\(47\)	10.9118	−	4.85827i	1.59166	−	0.708651i	0.596108	−	0.802904i	\(-0.296713\pi\)
							0.995548	+	0.0942531i	\(0.0300463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.q.a.196.22	yes	224
3.2	odd	2		675.2.r.a.46.7		224
9.4	even	3	inner	225.2.q.a.121.7	yes	224
9.5	odd	6		675.2.r.a.496.22		224
25.6	even	5	inner	225.2.q.a.106.7	yes	224
75.56	odd	10		675.2.r.a.181.22		224
225.31	even	15	inner	225.2.q.a.31.22	✓	224
225.131	odd	30		675.2.r.a.631.7		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.q.a.31.22	✓	224	225.31	even	15	inner
225.2.q.a.106.7	yes	224	25.6	even	5	inner
225.2.q.a.121.7	yes	224	9.4	even	3	inner
225.2.q.a.196.22	yes	224	1.1	even	1	trivial
675.2.r.a.46.7		224	3.2	odd	2
675.2.r.a.181.22		224	75.56	odd	10
675.2.r.a.496.22		224	9.5	odd	6
675.2.r.a.631.7		224	225.131	odd	30