Embedded newform 429.2.bs.a.184.1

• Label: 429.2.bs.a.184.1
• Level: $429$
• Weight: $2$
• Character: 429.184
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(14\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			184.1
Character	\(\chi\)	\(=\)	429.184
Dual form			429.2.bs.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.954534 - 2.48665i) q^{2} +(-0.669131 + 0.743145i) q^{3} +(-3.78598 + 3.40891i) q^{4} +(-2.07220 + 0.328204i) q^{5} +(2.48665 + 0.954534i) q^{6} +(3.34409 + 0.175256i) q^{7} +(7.34411 + 3.74201i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 28 q^{3} - 6 q^{5} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.954534	−	2.48665i	−0.674957	−	1.75832i	−0.651559	−	0.758598i	\(-0.725885\pi\)
							−0.0233981	−	0.999726i	\(-0.507449\pi\)
\(3\)	−0.669131	+	0.743145i	−0.386323	+	0.429055i
							\(5\)	−2.07220	+	0.328204i	−0.926715	+	0.146777i	−0.601509	−	0.798866i	\(-0.705434\pi\)
−0.325206	+	0.945643i	\(0.605434\pi\)
\(7\)	3.34409	+	0.175256i	1.26395	+	0.0662406i	0.672496	−	0.740101i	\(-0.265222\pi\)
							0.591450	+	0.806342i	\(0.298556\pi\)
\(11\)	0.0646015	−	3.31600i	0.0194781	−	0.999810i
							\(13\)	3.38654	−	1.23746i	0.939259	−	0.343210i
\(17\)	−5.41215	+	2.40964i	−1.31264	+	0.584424i								−0.939245	−	0.343248i	\(-0.888473\pi\)
							−0.373394	+	0.927673i	\(0.621806\pi\)
\(19\)	0.660383	+	1.01690i	0.151502	+	0.233293i	0.906270	−	0.422699i	\(-0.138917\pi\)
							−0.754768	+	0.655992i	\(0.772251\pi\)
\(23\)	1.31127	−	0.757065i	0.273420	−	0.157859i	−0.357021	−	0.934096i	\(-0.616208\pi\)
							0.630441	+	0.776237i	\(0.282874\pi\)
\(29\)	−1.69899	−	7.99310i	−0.315494	−	1.48428i	−0.794928	−	0.606703i	\(-0.792492\pi\)
							0.479435	−	0.877578i	\(-0.340842\pi\)
\(31\)	1.24224	−	7.84318i	0.223113	−	1.40868i	−0.580856	−	0.814006i	\(-0.697282\pi\)
							0.803969	−	0.594671i	\(-0.202718\pi\)
\(37\)	−3.96179	−	2.57282i	−0.651315	−	0.422969i	0.176245	−	0.984346i	\(-0.443605\pi\)
							−0.827560	+	0.561378i	\(0.810272\pi\)
\(41\)	9.03876	−	0.473701i	1.41162	−	0.0739797i	0.668784	−	0.743456i	\(-0.266815\pi\)
							0.742833	+	0.669477i	\(0.233482\pi\)
\(43\)	4.99458	−	8.65087i	0.761666	−	1.31925i	−0.180325	−	0.983607i	\(-0.557715\pi\)
							0.941991	−	0.335638i	\(-0.108952\pi\)
\(47\)	4.23370	−	8.30910i	0.617548	−	1.21201i	−0.344412	−	0.938819i	\(-0.611922\pi\)
							0.961961	−	0.273188i	\(-0.0880783\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bs.a.184.1	yes	224
11.7	odd	10	inner	429.2.bs.a.106.1	yes	224
13.7	odd	12	inner	429.2.bs.a.85.1	yes	224
143.7	even	60	inner	429.2.bs.a.7.1	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bs.a.7.1	✓	224	143.7	even	60	inner
429.2.bs.a.85.1	yes	224	13.7	odd	12	inner
429.2.bs.a.106.1	yes	224	11.7	odd	10	inner
429.2.bs.a.184.1	yes	224	1.1	even	1	trivial