Embedded newform 403.2.bi.a.14.2

• Label: 403.2.bi.a.14.2
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			14.2
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.a.144.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.717728 + 2.20894i) q^{2} +(1.96095 - 2.17786i) q^{3} +(-2.74625 - 1.99527i) q^{4} +(1.43599 - 2.48720i) q^{5} +(3.40333 + 5.89474i) q^{6} +(-0.518982 + 4.93778i) q^{7} +(2.62042 - 1.90385i) q^{8} +(-0.584148 - 5.55780i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.717728	+	2.20894i	−0.507511	+	1.56196i	0.288998	+	0.957330i	\(0.406678\pi\)
							−0.796508	+	0.604627i	\(0.793322\pi\)
\(3\)	1.96095	−	2.17786i	1.13216	−	1.25739i	0.169836	−	0.985472i	\(-0.445676\pi\)
							0.962321	−	0.271915i	\(-0.0876571\pi\)
\(5\)	1.43599	−	2.48720i	0.642192	−	1.11231i	−0.342750	−	0.939427i	\(-0.611358\pi\)
							0.984942	−	0.172883i	\(-0.0553083\pi\)
\(7\)	−0.518982	+	4.93778i	−0.196157	+	1.86631i	0.245867	+	0.969303i	\(0.420927\pi\)
							−0.442024	+	0.897003i	\(0.645739\pi\)
\(11\)	2.96281	−	1.31913i	0.893321	−	0.397732i	0.0918549	−	0.995772i	\(-0.470720\pi\)
							0.801466	+	0.598040i	\(0.204054\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	−1.75852	−	0.782942i	−0.426503	−	0.189891i	0.182248	−	0.983253i	\(-0.441663\pi\)
−0.608751	+	0.793361i	\(0.708329\pi\)
\(19\)	4.45514	−	0.946969i	1.02208	−	0.217250i	0.333760	−	0.942658i	\(-0.391682\pi\)
							0.688319	+	0.725408i	\(0.258349\pi\)
\(23\)	4.37279	−	3.17702i	0.911789	−	0.662454i	−0.0296775	−	0.999560i	\(-0.509448\pi\)
							0.941467	+	0.337106i	\(0.109448\pi\)
\(29\)	−0.762869	+	2.34787i	−0.141661	+	0.435988i	−0.996567	−	0.0827957i	\(-0.973615\pi\)
							0.854905	+	0.518784i	\(0.173615\pi\)
\(31\)	−4.53539	−	3.22959i	−0.814580	−	0.580051i
							\(37\)	2.23745	+	3.87538i	0.367835	+	0.637108i	0.989227	−	0.146392i	\(-0.0467659\pi\)
−0.621392	+	0.783500i	\(0.713433\pi\)
\(41\)	−3.75167	−	4.16665i	−0.585913	−	0.650722i	0.375179	−	0.926952i	\(-0.377581\pi\)
							−0.961091	+	0.276231i	\(0.910915\pi\)
\(43\)	−5.15972	+	1.09673i	−0.786849	+	0.167250i	−0.583777	−	0.811914i	\(-0.698426\pi\)
							−0.203072	+	0.979164i	\(0.565093\pi\)
\(47\)	0.790315	+	2.43234i	0.115279	+	0.354793i	0.992005	−	0.126197i	\(-0.0402772\pi\)
							−0.876726	+	0.480990i	\(0.840277\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.2	✓	120
31.20	even	15	inner	403.2.bi.a.144.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.2	✓	120	1.1	even	1	trivial
403.2.bi.a.144.2	yes	120	31.20	even	15	inner