Embedded newform 225.2.q.a.106.9

• Label: 225.2.q.a.106.9
• Level: $225$
• Weight: $2$
• Character: 225.106
• 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			106.9
Character	\(\chi\)	\(=\)	225.106
Dual form			225.2.q.a.121.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16617 - 0.247876i) q^{2} +(-1.18208 + 1.26597i) q^{3} +(-0.528590 - 0.235343i) q^{4} +(0.918757 + 2.03860i) q^{5} +(1.69231 - 1.18332i) q^{6} +(0.157578 - 0.272934i) q^{7} +(2.48714 + 1.80701i) q^{8} +(-0.205352 - 2.99296i) q^{9} +(-0.566103 - 2.60508i) q^{10} +(-2.82788 - 0.601085i) q^{11} +(0.922776 - 0.390983i) q^{12} +(-6.38538 + 1.35725i) q^{13} +(-0.251417 + 0.279226i) q^{14} +(-3.66685 - 1.24668i) q^{15} +(-1.67816 - 1.86379i) q^{16} +(0.794350 + 0.577129i) q^{17} +(-0.502410 + 3.54119i) q^{18} +(-5.88399 - 4.27497i) q^{19} +(-0.00587474 - 1.29381i) q^{20} +(0.159255 + 0.522120i) q^{21} +(3.14879 + 1.40193i) q^{22} +(-0.876336 + 0.973270i) q^{23} +(-5.22763 + 1.01260i) q^{24} +(-3.31177 + 3.74595i) q^{25} +7.78284 q^{26} +(4.03174 + 3.27797i) q^{27} +(-0.147528 + 0.107185i) q^{28} +(0.450873 + 4.28977i) q^{29} +(3.96713 + 2.36276i) q^{30} +(0.905916 - 8.61921i) q^{31} +(-1.57924 - 2.73533i) q^{32} +(4.10375 - 2.86948i) q^{33} +(-0.783287 - 0.869928i) q^{34} +(0.701179 + 0.0704794i) q^{35} +(-0.595827 + 1.63038i) q^{36} +(0.00738727 - 0.0227357i) q^{37} +(5.80205 + 6.44383i) q^{38} +(5.82981 - 9.68807i) q^{39} +(-1.39870 + 6.73048i) q^{40} +(-4.48136 + 0.952542i) q^{41} +(-0.0562962 - 0.648355i) q^{42} +(-1.34599 + 2.33132i) q^{43} +(1.35333 + 0.983252i) q^{44} +(5.91278 - 3.16844i) q^{45} +(1.26320 - 0.917772i) q^{46} +(0.544941 + 5.18477i) q^{47} +(4.34323 + 0.0786545i) q^{48} +(3.45034 + 5.97616i) q^{49} +(4.79061 - 3.54749i) q^{50} +(-1.66962 + 0.323407i) q^{51} +(3.69467 + 0.785326i) q^{52} +(-3.44104 + 2.50006i) q^{53} +(-3.88915 - 4.82203i) q^{54} +(-1.37277 - 6.31717i) q^{55} +(0.885114 - 0.394078i) q^{56} +(12.3674 - 2.39557i) q^{57} +(0.537539 - 5.11434i) q^{58} +(-5.06713 + 1.07705i) q^{59} +(1.64486 + 1.52195i) q^{60} +(7.24968 + 1.54097i) q^{61} +(-3.19295 + 9.82688i) q^{62} +(-0.849240 - 0.415579i) q^{63} +(2.71365 + 8.35176i) q^{64} +(-8.63350 - 11.7702i) q^{65} +(-5.49693 + 2.32907i) q^{66} +(0.260529 - 2.47877i) q^{67} +(-0.284062 - 0.492010i) q^{68} +(-0.196226 - 2.25990i) q^{69} +(-0.800221 - 0.255996i) q^{70} +(9.63629 - 7.00118i) q^{71} +(4.89758 - 7.81499i) q^{72} +(-0.283594 - 0.872814i) q^{73} +(-0.0142504 + 0.0246825i) q^{74} +(-0.827464 - 8.62063i) q^{75} +(2.10413 + 3.64447i) q^{76} +(-0.609670 + 0.677108i) q^{77} +(-9.19997 + 9.85283i) q^{78} +(1.18974 + 11.3196i) q^{79} +(2.25770 - 5.13348i) q^{80} +(-8.91566 + 1.22922i) q^{81} +5.46212 q^{82} +(-9.08283 + 4.04394i) q^{83} +(0.0386972 - 0.313467i) q^{84} +(-0.446720 + 2.14960i) q^{85} +(2.14753 - 2.38507i) q^{86} +(-5.96368 - 4.50007i) q^{87} +(-5.94717 - 6.60500i) q^{88} +(3.78718 + 11.6557i) q^{89} +(-7.68067 + 2.22928i) q^{90} +(-0.635757 + 1.95666i) q^{91} +(0.692276 - 0.308221i) q^{92} +(9.84078 + 11.3355i) q^{93} +(0.649689 - 6.18138i) q^{94} +(3.30899 - 15.9228i) q^{95} +(5.32964 + 1.23412i) q^{96} +(0.772754 + 7.35226i) q^{97} +(-2.54232 - 7.82445i) q^{98} +(-1.21831 + 8.58719i) 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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16617	−	0.247876i	−0.824604	−	0.175275i	−0.223763	−	0.974644i	\(-0.571834\pi\)
							−0.600841	+	0.799369i	\(0.705168\pi\)
\(3\)	−1.18208	+	1.26597i	−0.682477	+	0.730907i
							\(5\)	0.918757	+	2.03860i	0.410881	+	0.911689i
\(7\)	0.157578	−	0.272934i	0.0595591	−	0.103159i								−0.834708	−	0.550692i	\(-0.814364\pi\)
							0.894268	+	0.447533i	\(0.147697\pi\)
\(11\)	−2.82788	−	0.601085i	−0.852639	−	0.181234i	−0.239195	−	0.970972i	\(-0.576883\pi\)
							−0.613445	+	0.789738i	\(0.710217\pi\)
\(13\)	−6.38538	+	1.35725i	−1.77098	+	0.376434i	−0.973819	−	0.227326i	\(-0.927002\pi\)
							−0.797166	+	0.603761i	\(0.793668\pi\)
\(17\)	0.794350	+	0.577129i	0.192658	+	0.139974i	0.679933	−	0.733274i	\(-0.262009\pi\)
							−0.487275	+	0.873249i	\(0.662009\pi\)
\(19\)	−5.88399	−	4.27497i	−1.34988	−	0.980746i	−0.999017	−	0.0443190i	\(-0.985888\pi\)
							−0.350863	−	0.936427i	\(-0.614112\pi\)
\(23\)	−0.876336	+	0.973270i	−0.182729	+	0.202941i	−0.827549	−	0.561393i	\(-0.810266\pi\)
							0.644820	+	0.764334i	\(0.276932\pi\)
\(29\)	0.450873	+	4.28977i	0.0837249	+	0.796589i	0.953145	+	0.302515i	\(0.0978263\pi\)
							−0.869420	+	0.494074i	\(0.835507\pi\)
\(31\)	0.905916	−	8.61921i	0.162707	−	1.54806i	−0.543110	−	0.839662i	\(-0.682753\pi\)
							0.705817	−	0.708394i	\(-0.250580\pi\)
\(37\)	0.00738727	−	0.0227357i	0.00121446	−	0.00373772i	−0.950447	−	0.310885i	\(-0.899375\pi\)
							0.951662	+	0.307148i	\(0.0993745\pi\)
\(41\)	−4.48136	+	0.952542i	−0.699871	+	0.148762i	−0.544087	−	0.839029i	\(-0.683124\pi\)
							−0.155784	+	0.987791i	\(0.549790\pi\)
\(43\)	−1.34599	+	2.33132i	−0.205261	+	0.355523i	−0.950216	−	0.311592i	\(-0.899138\pi\)
							0.744955	+	0.667115i	\(0.232471\pi\)
\(47\)	0.544941	+	5.18477i	0.0794878	+	0.756276i	0.959573	+	0.281460i	\(0.0908187\pi\)
							−0.880085	+	0.474816i	\(0.842515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.q.a.106.9	yes	224
3.2	odd	2		675.2.r.a.181.20		224
9.4	even	3	inner	225.2.q.a.31.20	✓	224
9.5	odd	6		675.2.r.a.631.9		224
25.21	even	5	inner	225.2.q.a.196.20	yes	224
75.71	odd	10		675.2.r.a.46.9		224
225.121	even	15	inner	225.2.q.a.121.9	yes	224
225.221	odd	30		675.2.r.a.496.20		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.q.a.31.20	✓	224	9.4	even	3	inner
225.2.q.a.106.9	yes	224	1.1	even	1	trivial
225.2.q.a.121.9	yes	224	225.121	even	15	inner
225.2.q.a.196.20	yes	224	25.21	even	5	inner
675.2.r.a.46.9		224	75.71	odd	10
675.2.r.a.181.20		224	3.2	odd	2
675.2.r.a.496.20		224	225.221	odd	30
675.2.r.a.631.9		224	9.5	odd	6