Embedded newform 403.2.bl.a.295.1

• Label: 403.2.bl.a.295.1
• Level: $403$
• Weight: $2$
• Character: 403.295
• 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			295.1
Root			\(-0.104528 + 0.994522i\) of defining polynomial
Character	\(\chi\)	\(=\)	403.295
Dual form			403.2.bl.a.250.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.273659 + 2.60369i) q^{2} +(-2.04275 + 0.909491i) q^{3} +(-4.74803 + 1.00922i) q^{4} -3.00000 q^{5} +(-2.92705 - 5.06980i) q^{6} +(2.93444 - 0.623735i) q^{7} +(-2.30902 - 7.10642i) q^{8} +(1.33826 - 1.48629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 5 q^{3} - 6 q^{4} - 24 q^{5} - 10 q^{6} - 3 q^{7} - 14 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.273659	+	2.60369i	0.193506	+	1.84109i	0.473147	+	0.880984i	\(0.343118\pi\)
							−0.279641	+	0.960105i	\(0.590215\pi\)
\(3\)	−2.04275	+	0.909491i	−1.17938	+	0.525095i	−0.900341	−	0.435184i	\(-0.856683\pi\)
							−0.279041	+	0.960279i	\(0.590016\pi\)
\(5\)	−3.00000			−1.34164			−0.670820	−	0.741620i	\(-0.734058\pi\)
							−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	2.93444	−	0.623735i	1.10912	−	0.235750i	0.383289	−	0.923628i	\(-0.374791\pi\)
							0.725826	+	0.687879i	\(0.241458\pi\)
\(11\)	1.49622	+	1.66172i	0.451128	+	0.501028i	0.925212	−	0.379452i	\(-0.123887\pi\)
							−0.474084	+	0.880480i	\(0.657221\pi\)
\(13\)	−3.54965	+	0.632425i	−0.984497	+	0.175403i
							\(17\)	−0.924716	+	1.02700i	−0.224277	+	0.249084i	−0.844773	−	0.535125i	\(-0.820264\pi\)
0.620496	+	0.784209i	\(0.286931\pi\)
\(19\)	−5.48127	−	2.44042i	−1.25749	−	0.559871i	−0.333667	−	0.942691i	\(-0.608286\pi\)
							−0.923823	+	0.382820i	\(0.874953\pi\)
\(23\)	3.68168	+	0.782565i	0.767684	+	0.163176i	0.575076	−	0.818100i	\(-0.304972\pi\)
							0.192607	+	0.981276i	\(0.438306\pi\)
\(29\)	0.587244	+	5.58726i	0.109049	+	1.03753i	0.903028	+	0.429583i	\(0.141339\pi\)
							−0.793979	+	0.607945i	\(0.791994\pi\)
\(31\)	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	\(-0.0649052\pi\)
\(41\)	−0.765800	−	7.28610i	−0.119598	−	1.13790i	−0.875502	−	0.483215i	\(-0.839469\pi\)
							0.755904	−	0.654683i	\(-0.227198\pi\)
\(43\)	−5.91259	−	2.63245i	−0.901662	−	0.401446i	−0.0970723	−	0.995277i	\(-0.530948\pi\)
							−0.804589	+	0.593832i	\(0.797614\pi\)
\(47\)	−6.85410	−	4.97980i	−0.999774	−	0.726378i	−0.0377339	−	0.999288i	\(-0.512014\pi\)
							−0.962040	+	0.272910i	\(0.912014\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.a.16.1	✓	8	13.3	even	3	inner
403.2.bl.a.126.1	yes	8	31.2	even	5	inner
403.2.bl.a.250.1	yes	8	403.250	even	15	inner
403.2.bl.a.295.1	yes	8	1.1	even	1	trivial