Embedded newform 403.2.bl.b.126.9

• Label: 403.2.bl.b.126.9
• Level: $403$
• Weight: $2$
• Character: 403.126
• 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			126.9
Character	\(\chi\)	\(=\)	403.126
Dual form			403.2.bl.b.16.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41616 + 0.630515i) q^{2} +(-0.113478 - 1.07967i) q^{3} +(0.269698 - 0.299530i) q^{4} -3.63191 q^{5} +(0.841454 + 1.45744i) q^{6} +(-0.969238 + 1.07645i) q^{7} +(0.764987 - 2.35439i) q^{8} +(1.78162 - 0.378696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41616	+	0.630515i	−1.00138	+	0.445841i	−0.840895	−	0.541198i	\(-0.817971\pi\)
							−0.160481	+	0.987039i	\(0.551304\pi\)
\(3\)	−0.113478	−	1.07967i	−0.0655167	−	0.623350i	−0.977180	−	0.212412i	\(-0.931868\pi\)
							0.911663	−	0.410938i	\(-0.134799\pi\)
\(5\)	−3.63191			−1.62424			−0.812120	−	0.583490i	\(-0.801687\pi\)
							−0.812120	+	0.583490i	\(0.801687\pi\)
\(7\)	−0.969238	+	1.07645i	−0.366337	+	0.406859i	−0.897928	−	0.440143i	\(-0.854928\pi\)
							0.531591	+	0.847001i	\(0.321595\pi\)
\(11\)	−4.10052	−	0.871592i	−1.23635	−	0.262795i	−0.457049	−	0.889441i	\(-0.651094\pi\)
							−0.779304	+	0.626647i	\(0.784427\pi\)
\(13\)	3.56235	−	0.556465i	0.988018	−	0.154336i
							\(17\)	0.395127	−	0.0839868i	0.0958323	−	0.0203698i	−0.159746	−	0.987158i	\(-0.551067\pi\)
0.255578	+	0.966788i	\(0.417734\pi\)
\(19\)	−0.434956	+	4.13833i	−0.0997857	+	0.949397i	0.824026	+	0.566551i	\(0.191723\pi\)
							−0.923812	+	0.382846i	\(0.874944\pi\)
\(23\)	4.61258	+	5.12279i	0.961790	+	1.06818i	0.997629	+	0.0688184i	\(0.0219229\pi\)
							−0.0358392	+	0.999358i	\(0.511410\pi\)
\(29\)	3.31584	−	1.47631i	0.615737	−	0.274144i	−0.0750756	−	0.997178i	\(-0.523920\pi\)
							0.690812	+	0.723034i	\(0.257253\pi\)
\(31\)	4.14682	+	3.71536i	0.744791	+	0.667298i
							\(37\)	4.01631	−	6.95646i	0.660278	−	1.14363i	−0.320265	−	0.947328i	\(-0.603772\pi\)
0.980543	−	0.196307i	\(-0.0628948\pi\)
\(41\)	5.25959	−	2.34172i	0.821410	−	0.365715i	0.0473903	−	0.998876i	\(-0.484910\pi\)
							0.774020	+	0.633161i	\(0.218243\pi\)
\(43\)	−0.508398	+	4.83708i	−0.0775299	+	0.737648i	0.884838	+	0.465899i	\(0.154269\pi\)
							−0.962368	+	0.271749i	\(0.912398\pi\)
\(47\)	4.70255	−	3.41660i	0.685937	−	0.498362i	−0.189385	−	0.981903i	\(-0.560649\pi\)
							0.875322	+	0.483540i	\(0.160649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.126.9	yes	280
13.3	even	3	inner	403.2.bl.b.250.27	yes	280
31.16	even	5	inner	403.2.bl.b.295.27	yes	280
403.16	even	15	inner	403.2.bl.b.16.9	✓	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.9	✓	280	403.16	even	15	inner
403.2.bl.b.126.9	yes	280	1.1	even	1	trivial
403.2.bl.b.250.27	yes	280	13.3	even	3	inner
403.2.bl.b.295.27	yes	280	31.16	even	5	inner