Embedded newform 447.2.i.a.100.6

• Label: 447.2.i.a.100.6
• 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			100.6
Character	\(\chi\)	\(=\)	447.100
Dual form			447.2.i.a.76.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.384192 - 1.47551i) q^{2} +(0.169001 + 0.985616i) q^{3} +(-0.283486 + 0.158365i) q^{4} +(0.652141 - 3.02594i) q^{5} +(1.38935 - 0.628027i) q^{6} +(-0.162178 - 3.81779i) q^{7} +(-1.76742 - 1.84410i) q^{8} +(-0.942877 + 0.333140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q + 18 q^{4} + 8 q^{5} + 2 q^{6} - 4 q^{7} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.384192	−	1.47551i	−0.271665	−	1.04334i	−0.950920	−	0.309438i	\(-0.899859\pi\)
							0.679255	−	0.733902i	\(-0.262303\pi\)
\(3\)	0.169001	+	0.985616i	0.0975727	+	0.569046i
							\(5\)	0.652141	−	3.02594i	0.291646	−	1.35324i	−0.559724	−	0.828679i	\(-0.689093\pi\)
0.851371	−	0.524565i	\(-0.175772\pi\)
\(7\)	−0.162178	−	3.81779i	−0.0612974	−	1.44299i	−0.721767	−	0.692136i	\(-0.756670\pi\)
							0.660470	−	0.750853i	\(-0.270357\pi\)
\(11\)	−1.63205	+	2.92151i	−0.492082	+	0.880868i	0.507730	+	0.861516i	\(0.330485\pi\)
							−0.999813	+	0.0193520i	\(0.993840\pi\)
\(13\)	1.71565	+	1.26889i	0.475836	+	0.351926i	0.805919	−	0.592026i	\(-0.201672\pi\)
							−0.330084	+	0.943952i	\(0.607077\pi\)
\(17\)	−2.84531	+	0.487878i	−0.690090	+	0.118328i	−0.503351	−	0.864082i	\(-0.667900\pi\)
							−0.186738	+	0.982410i	\(0.559792\pi\)
\(19\)	3.00045	+	2.02795i	0.688350	+	0.465243i	0.853546	−	0.521017i	\(-0.174447\pi\)
							−0.165196	+	0.986261i	\(0.552826\pi\)
\(23\)	1.35912	−	1.00520i	0.283396	−	0.209598i	−0.443627	−	0.896211i	\(-0.646309\pi\)
							0.727023	+	0.686613i	\(0.240903\pi\)
\(29\)	−7.04703	−	8.00698i	−1.30860	−	1.48686i	−0.758328	−	0.651873i	\(-0.773983\pi\)
							−0.550272	−	0.834985i	\(-0.685476\pi\)
\(31\)	−1.16002	−	1.56845i	−0.208346	−	0.281703i	0.687390	−	0.726288i	\(-0.258756\pi\)
							−0.895736	+	0.444586i	\(0.853351\pi\)
\(37\)	3.18122	+	1.77714i	0.522990	+	0.292160i	0.726798	−	0.686851i	\(-0.241008\pi\)
							−0.203808	+	0.979011i	\(0.565332\pi\)
\(41\)	6.96227	+	4.28684i	1.08732	+	0.669492i	0.947200	−	0.320642i	\(-0.103899\pi\)
							0.140123	+	0.990134i	\(0.455250\pi\)
\(43\)	5.24610	−	0.671789i	0.800023	−	0.102447i	0.283780	−	0.958889i	\(-0.408411\pi\)
							0.516243	+	0.856442i	\(0.327330\pi\)
\(47\)	9.34793	−	6.31810i	1.36354	−	0.921589i	0.363585	−	0.931561i	\(-0.381553\pi\)
							0.999950	+	0.00997210i	\(0.00317427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	447.2.i.a.100.6	yes	864
149.76	even	74	inner	447.2.i.a.76.6	✓	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
447.2.i.a.76.6	✓	864	149.76	even	74	inner
447.2.i.a.100.6	yes	864	1.1	even	1	trivial