Embedded newform 447.2.i.a.100.2

• Label: 447.2.i.a.100.2
• Level: $447$
• Weight: $2$
• Character: 447.100
• Analytic conductor: $3.569$
• Analytic rank: $0$
• Dimension: $864$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 447 = 3 \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	447.i (of order \(74\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.56931297035\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(24\) over \(\Q(\zeta_{74})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{74}]$

Embedding invariants

Embedding label			100.2
Character	\(\chi\)	\(=\)	447.100
Dual form			447.2.i.a.76.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.575237 - 2.20923i) q^{2} +(0.169001 + 0.985616i) q^{3} +(-2.80375 + 1.56627i) q^{4} +(0.115399 - 0.535454i) q^{5} +(2.08023 - 0.940324i) q^{6} +(0.140392 + 3.30494i) q^{7} +(1.91382 + 1.99685i) q^{8} +(-0.942877 + 0.333140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q + 18 q^{4} + 8 q^{5} + 2 q^{6} - 4 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/447\mathbb{Z}\right)^\times\).

\(n\)	\(151\)	\(299\)
\(\chi(n)\)	\(e\left(\frac{31}{74}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.575237	−	2.20923i	−0.406754	−	1.56216i	−0.774711	−	0.632315i	\(-0.782105\pi\)
							0.367957	−	0.929843i	\(-0.380057\pi\)
\(3\)	0.169001	+	0.985616i	0.0975727	+	0.569046i
							\(5\)	0.115399	−	0.535454i	0.0516081	−	0.239462i	−0.944391	−	0.328824i	\(-0.893348\pi\)
0.995999	+	0.0893618i	\(0.0284828\pi\)
\(7\)	0.140392	+	3.30494i	0.0530633	+	1.24915i	0.806342	+	0.591450i	\(0.201444\pi\)
							−0.753279	+	0.657701i	\(0.771529\pi\)
\(11\)	0.215546	−	0.385845i	0.0649895	−	0.116337i	−0.838579	−	0.544779i	\(-0.816613\pi\)
							0.903569	+	0.428443i	\(0.140938\pi\)
\(13\)	1.29061	+	0.954533i	0.357952	+	0.264740i	0.758689	−	0.651453i	\(-0.225840\pi\)
							−0.400737	+	0.916193i	\(0.631246\pi\)
\(17\)	4.12045	−	0.706522i	0.999356	−	0.171357i	0.354005	−	0.935243i	\(-0.384819\pi\)
							0.645350	+	0.763887i	\(0.276711\pi\)
\(19\)	6.49869	+	4.39235i	1.49090	+	1.00767i	0.989828	+	0.142269i	\(0.0454400\pi\)
							0.501074	+	0.865404i	\(0.332938\pi\)
\(23\)	−0.807315	+	0.597087i	−0.168337	+	0.124501i	−0.675534	−	0.737329i	\(-0.736087\pi\)
							0.507197	+	0.861830i	\(0.330682\pi\)
\(29\)	2.11149	+	2.39912i	0.392095	+	0.445506i	0.912909	−	0.408164i	\(-0.133831\pi\)
							−0.520814	+	0.853670i	\(0.674371\pi\)
\(31\)	−6.26052	−	8.46478i	−1.12442	−	1.52032i	−0.824067	−	0.566493i	\(-0.808300\pi\)
							−0.300356	−	0.953827i	\(-0.597106\pi\)
\(37\)	3.90335	+	2.18054i	0.641707	+	0.358479i	0.774417	−	0.632676i	\(-0.218043\pi\)
							−0.132710	+	0.991155i	\(0.542368\pi\)
\(41\)	6.96635	+	4.28935i	1.08796	+	0.669885i	0.947356	−	0.320181i	\(-0.103744\pi\)
							0.140605	+	0.990066i	\(0.455095\pi\)
\(43\)	−9.42967	+	1.20752i	−1.43801	+	0.184145i	−0.806506	−	0.591226i	\(-0.798644\pi\)
							−0.631506	+	0.775371i	\(0.717563\pi\)
\(47\)	−6.31519	+	4.26833i	−0.921166	+	0.622599i	−0.926055	−	0.377389i	\(-0.876822\pi\)
							0.00488924	+	0.999988i	\(0.498444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	447.2.i.a.100.2	yes	864
149.76	even	74	inner	447.2.i.a.76.2	✓	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
447.2.i.a.76.2	✓	864	149.76	even	74	inner
447.2.i.a.100.2	yes	864	1.1	even	1	trivial