Embedded newform 225.2.q.a.121.9

• Label: 225.2.q.a.121.9
• Level: $225$
• Weight: $2$
• Character: 225.121
• 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			121.9
Character	\(\chi\)	\(=\)	225.121
Dual form			225.2.q.a.106.9

$q$-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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.q.a.121.9	yes	224
3.2	odd	2		675.2.r.a.496.20		224
9.2	odd	6		675.2.r.a.46.9		224
9.7	even	3	inner	225.2.q.a.196.20	yes	224
25.6	even	5	inner	225.2.q.a.31.20	✓	224
75.56	odd	10		675.2.r.a.631.9		224
225.56	odd	30		675.2.r.a.181.20		224
225.106	even	15	inner	225.2.q.a.106.9	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.q.a.31.20	✓	224	25.6	even	5	inner
225.2.q.a.106.9	yes	224	225.106	even	15	inner
225.2.q.a.121.9	yes	224	1.1	even	1	trivial
225.2.q.a.196.20	yes	224	9.7	even	3	inner
675.2.r.a.46.9		224	9.2	odd	6
675.2.r.a.181.20		224	225.56	odd	30
675.2.r.a.496.20		224	3.2	odd	2
675.2.r.a.631.9		224	75.56	odd	10