Embedded newform 403.2.bl.b.16.15

• Label: 403.2.bl.b.16.15
• Level: $403$
• Weight: $2$
• Character: 403.16
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.15
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.596810 - 0.265717i) q^{2} +(0.0996818 - 0.948409i) q^{3} +(-1.05268 - 1.16913i) q^{4} +0.954231 q^{5} +(-0.311499 + 0.539532i) q^{6} +(2.48894 + 2.76425i) q^{7} +(0.721351 + 2.22009i) q^{8} +(2.04490 + 0.434657i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.596810	−	0.265717i	−0.422008	−	0.187890i	0.184735	−	0.982788i	\(-0.440857\pi\)
							−0.606743	+	0.794898i	\(0.707524\pi\)
\(3\)	0.0996818	−	0.948409i	0.0575513	−	0.547564i	−0.927319	−	0.374272i	\(-0.877893\pi\)
							0.984870	−	0.173292i	\(-0.0554406\pi\)
\(5\)	0.954231			0.426745			0.213373	−	0.976971i	\(-0.431555\pi\)
							0.213373	+	0.976971i	\(0.431555\pi\)
\(7\)	2.48894	+	2.76425i	0.940731	+	1.04479i	0.998919	+	0.0464817i	\(0.0148009\pi\)
							−0.0581884	+	0.998306i	\(0.518532\pi\)
\(11\)	−1.06042	+	0.225399i	−0.319728	+	0.0679603i	−0.364980	−	0.931015i	\(-0.618924\pi\)
							0.0452520	+	0.998976i	\(0.485591\pi\)
\(13\)	3.59428	+	0.284818i	0.996875	+	0.0789942i
							\(17\)	2.63237	+	0.559528i	0.638444	+	0.135705i	0.515750	−	0.856739i	\(-0.327513\pi\)
0.122694	+	0.992445i	\(0.460847\pi\)
\(19\)	−0.658770	−	6.26777i	−0.151132	−	1.43793i	−0.762709	−	0.646741i	\(-0.776131\pi\)
							0.611577	−	0.791185i	\(-0.290535\pi\)
\(23\)	−1.28119	+	1.42290i	−0.267146	+	0.296695i	−0.861760	−	0.507315i	\(-0.830638\pi\)
							0.594615	+	0.804011i	\(0.297305\pi\)
\(29\)	−3.07533	−	1.36922i	−0.571074	−	0.254258i	0.100822	−	0.994905i	\(-0.467853\pi\)
							−0.671895	+	0.740646i	\(0.734520\pi\)
\(31\)	4.76747	−	2.87598i	0.856262	−	0.516541i
							\(37\)	0.530160	+	0.918263i	0.0871577	+	0.150962i	0.906309	−	0.422616i	\(-0.138888\pi\)
−0.819151	+	0.573578i	\(0.805555\pi\)
\(41\)	3.69118	+	1.64342i	0.576466	+	0.256659i	0.674194	−	0.738554i	\(-0.264491\pi\)
							−0.0977285	+	0.995213i	\(0.531158\pi\)
\(43\)	0.845648	+	8.04580i	0.128960	+	1.22697i	0.847237	+	0.531216i	\(0.178265\pi\)
							−0.718276	+	0.695758i	\(0.755069\pi\)
\(47\)	−10.3317	−	7.50639i	−1.50703	−	1.09492i	−0.967474	−	0.252969i	\(-0.918593\pi\)
							−0.539554	−	0.841951i	\(-0.681407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.15	✓	280
13.9	even	3	inner	403.2.bl.b.295.21	yes	280
31.2	even	5	inner	403.2.bl.b.250.21	yes	280
403.126	even	15	inner	403.2.bl.b.126.15	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.15	✓	280	1.1	even	1	trivial
403.2.bl.b.126.15	yes	280	403.126	even	15	inner
403.2.bl.b.250.21	yes	280	31.2	even	5	inner
403.2.bl.b.295.21	yes	280	13.9	even	3	inner