Embedded newform 450.2.q.c.31.14

• Label: 450.2.q.c.31.14
• Level: $450$
• Weight: $2$
• Character: 450.31
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $120$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			31.14
Character	\(\chi\)	\(=\)	450.31
Dual form			450.2.q.c.421.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669131 + 0.743145i) q^{2} +(1.69823 + 0.340598i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(-2.07884 - 0.823674i) q^{5} +(-1.38945 + 1.03413i) q^{6} +(0.746133 + 1.29234i) q^{7} +(0.809017 + 0.587785i) q^{8} +(2.76799 + 1.15683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.669131	+	0.743145i	−0.473147	+	0.525483i
							\(3\)	1.69823	+	0.340598i	0.980475	+	0.196644i
\(5\)	−2.07884	−	0.823674i	−0.929684	−	0.368358i
							\(7\)	0.746133	+	1.29234i	0.282012	+	0.488458i	0.971880	−	0.235476i	\(-0.0756650\pi\)
−0.689869	+	0.723935i	\(0.742332\pi\)
\(11\)	−0.545196	+	0.605501i	−0.164383	+	0.182565i	−0.819709	−	0.572781i	\(-0.805865\pi\)
							0.655326	+	0.755346i	\(0.272531\pi\)
\(13\)	2.74467	+	3.04826i	0.761233	+	0.845435i	0.991824	−	0.127616i	\(-0.0407324\pi\)
							−0.230590	+	0.973051i	\(0.574066\pi\)
\(17\)	0.388751	+	0.282444i	0.0942860	+	0.0685028i	0.633929	−	0.773391i	\(-0.281441\pi\)
							−0.539643	+	0.841894i	\(0.681441\pi\)
\(19\)	3.79739	+	2.75897i	0.871181	+	0.632950i	0.930904	−	0.365265i	\(-0.119022\pi\)
							−0.0597223	+	0.998215i	\(0.519022\pi\)
\(23\)	4.99327	+	1.06135i	1.04117	+	0.221307i	0.696604	−	0.717456i	\(-0.254693\pi\)
							0.344564	+	0.938763i	\(0.388027\pi\)
\(29\)	−1.24943	−	0.556284i	−0.232014	−	0.103299i	0.287437	−	0.957800i	\(-0.407197\pi\)
							−0.519451	+	0.854500i	\(0.673864\pi\)
\(31\)	−2.86697	+	1.27646i	−0.514924	+	0.229259i	−0.647712	−	0.761886i	\(-0.724274\pi\)
							0.132788	+	0.991144i	\(0.457607\pi\)
\(37\)	1.63213	−	5.02317i	0.268320	−	0.825804i	−0.722590	−	0.691277i	\(-0.757049\pi\)
							0.990910	−	0.134527i	\(-0.0429515\pi\)
\(41\)	−5.82668	−	6.47118i	−0.909974	−	1.01063i	−0.999892	−	0.0146798i	\(-0.995327\pi\)
							0.0899178	−	0.995949i	\(-0.471340\pi\)
\(43\)	−0.371715	−	0.643829i	−0.0566860	−	0.0981830i	0.836290	−	0.548288i	\(-0.184720\pi\)
							−0.892976	+	0.450105i	\(0.851387\pi\)
\(47\)	−3.53986	−	1.57605i	−0.516341	−	0.229890i	0.131986	−	0.991252i	\(-0.457865\pi\)
							−0.648328	+	0.761362i	\(0.724531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.31.14	✓	120
9.7	even	3	inner	450.2.q.c.331.8	yes	120
25.21	even	5	inner	450.2.q.c.121.8	yes	120
225.196	even	15	inner	450.2.q.c.421.14	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.14	✓	120	1.1	even	1	trivial
450.2.q.c.121.8	yes	120	25.21	even	5	inner
450.2.q.c.331.8	yes	120	9.7	even	3	inner
450.2.q.c.421.14	yes	120	225.196	even	15	inner