Embedded newform 403.2.bl.a.126.1

• Label: 403.2.bl.a.126.1
• Level: $403$
• Weight: $2$
• Character: 403.126
• Analytic conductor: $3.218$
• Analytic rank: $1$
• Dimension: $8$
• 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:	\(1\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			126.1
Root			\(0.913545 + 0.406737i\) of defining polynomial
Character	\(\chi\)	\(=\)	403.126
Dual form			403.2.bl.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.39169 + 1.06485i) q^{2} +(0.233733 + 2.22382i) q^{3} +(3.24803 - 3.60730i) q^{4} -3.00000 q^{5} +(-2.92705 - 5.06980i) q^{6} +(-2.00739 + 2.22943i) q^{7} +(-2.30902 + 7.10642i) q^{8} +(-1.95630 + 0.415823i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 5 q^{3} - 6 q^{4} - 24 q^{5} - 10 q^{6} - 3 q^{7} - 14 q^{8} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39169	+	1.06485i	−1.69118	+	0.752963i	−0.691655	+	0.722228i	\(0.743118\pi\)
							−0.999528	+	0.0307347i	\(0.990215\pi\)
\(3\)	0.233733	+	2.22382i	0.134946	+	1.28392i	0.827051	+	0.562126i	\(0.190016\pi\)
							−0.692106	+	0.721796i	\(0.743317\pi\)
\(5\)	−3.00000			−1.34164			−0.670820	−	0.741620i	\(-0.734058\pi\)
							−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	−2.00739	+	2.22943i	−0.758723	+	0.842647i	−0.991530	−	0.129876i	\(-0.958542\pi\)
							0.232807	+	0.972523i	\(0.425209\pi\)
\(11\)	−2.18720	−	0.464905i	−0.659467	−	0.140174i	−0.133991	−	0.990983i	\(-0.542779\pi\)
							−0.525476	+	0.850809i	\(0.676113\pi\)
\(13\)	−0.495432	+	3.57135i	−0.137408	+	0.990515i
							\(17\)	1.35177	−	0.287327i	0.327852	−	0.0696870i	−0.0410454	−	0.999157i	\(-0.513069\pi\)
0.368897	+	0.929470i	\(0.379736\pi\)
\(19\)	0.627171	−	5.96713i	0.143883	−	1.36895i	−0.649561	−	0.760310i	\(-0.725047\pi\)
							0.793443	−	0.608644i	\(-0.208286\pi\)
\(23\)	−2.51856	−	2.79715i	−0.525157	−	0.583245i	0.420957	−	0.907081i	\(-0.361694\pi\)
							−0.946114	+	0.323835i	\(0.895028\pi\)
\(29\)	−5.13233	+	2.28506i	−0.953050	+	0.424325i	−0.823543	−	0.567254i	\(-0.808006\pi\)
							−0.129506	+	0.991579i	\(0.541339\pi\)
\(31\)	5.28115	+	1.76336i	0.948523	+	0.316708i
							\(37\)	4.04508	−	7.00629i	0.665008	−	1.15183i	−0.314275	−	0.949332i	\(-0.601762\pi\)
0.979283	−	0.202496i	\(-0.0649052\pi\)
\(41\)	6.69285	−	2.97985i	1.04525	−	0.465374i	0.189020	−	0.981973i	\(-0.439469\pi\)
							0.856228	+	0.516599i	\(0.172802\pi\)
\(43\)	0.676522	−	6.43668i	0.103169	−	0.981585i	−0.813399	−	0.581706i	\(-0.802386\pi\)
							0.916568	−	0.399879i	\(-0.130948\pi\)
\(47\)	−6.85410	+	4.97980i	−0.999774	+	0.726378i	−0.962040	−	0.272910i	\(-0.912014\pi\)
							−0.0377339	+	0.999288i	\(0.512014\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.a.126.1	yes	8
13.3	even	3	inner	403.2.bl.a.250.1	yes	8
31.16	even	5	inner	403.2.bl.a.295.1	yes	8
403.16	even	15	inner	403.2.bl.a.16.1	✓	8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.a.16.1	✓	8	403.16	even	15	inner
403.2.bl.a.126.1	yes	8	1.1	even	1	trivial
403.2.bl.a.250.1	yes	8	13.3	even	3	inner
403.2.bl.a.295.1	yes	8	31.16	even	5	inner