Embedded newform 403.2.bi.a.14.11

• Label: 403.2.bi.a.14.11
• 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			14.11
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.a.144.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.362743 - 1.11641i) q^{2} +(-0.166092 + 0.184464i) q^{3} +(0.503249 + 0.365632i) q^{4} +(2.00419 - 3.47136i) q^{5} +(0.145688 + 0.252339i) q^{6} +(0.284216 - 2.70413i) q^{7} +(2.49009 - 1.80916i) q^{8} +(0.307145 + 2.92229i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.362743	−	1.11641i	0.256498	−	0.789420i	−0.737033	−	0.675857i	\(-0.763774\pi\)
							0.993531	−	0.113563i	\(-0.0362263\pi\)
\(3\)	−0.166092	+	0.184464i	−0.0958931	+	0.106500i	−0.789181	−	0.614161i	\(-0.789495\pi\)
							0.693288	+	0.720661i	\(0.256161\pi\)
\(5\)	2.00419	−	3.47136i	0.896302	−	1.55244i	0.0641164	−	0.997942i	\(-0.479577\pi\)
							0.832185	−	0.554498i	\(-0.187090\pi\)
\(7\)	0.284216	−	2.70413i	0.107424	−	1.02207i	−0.799470	−	0.600706i	\(-0.794886\pi\)
							0.906893	−	0.421360i	\(-0.138447\pi\)
\(11\)	−5.01050	+	2.23082i	−1.51072	+	0.672617i	−0.984122	−	0.177495i	\(-0.943201\pi\)
							−0.526601	+	0.850112i	\(0.676534\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	3.97551	+	1.77001i	0.964202	+	0.429291i	0.827589	−	0.561334i	\(-0.189712\pi\)
0.136613	+	0.990625i	\(0.456378\pi\)
\(19\)	−4.16993	+	0.886347i	−0.956649	+	0.203342i	−0.659681	−	0.751546i	\(-0.729309\pi\)
							−0.296968	+	0.954888i	\(0.595975\pi\)
\(23\)	−1.80875	+	1.31413i	−0.377150	+	0.274016i	−0.760170	−	0.649725i	\(-0.774884\pi\)
							0.383019	+	0.923740i	\(0.374884\pi\)
\(29\)	0.374456	−	1.15246i	0.0695348	−	0.214006i	−0.910251	−	0.414058i	\(-0.864111\pi\)
							0.979785	+	0.200052i	\(0.0641111\pi\)
\(31\)	5.14576	−	2.12630i	0.924206	−	0.381895i
							\(37\)	−1.17808	−	2.04049i	−0.193675	−	0.335455i	0.752790	−	0.658260i	\(-0.228707\pi\)
−0.946465	+	0.322806i	\(0.895374\pi\)
\(41\)	5.12400	+	5.69078i	0.800234	+	0.888750i	0.995764	−	0.0919475i	\(-0.0293092\pi\)
							−0.195530	+	0.980698i	\(0.562643\pi\)
\(43\)	−0.444111	+	0.0943988i	−0.0677264	+	0.0143957i	−0.241650	−	0.970363i	\(-0.577689\pi\)
							0.173924	+	0.984759i	\(0.444355\pi\)
\(47\)	0.411245	+	1.26568i	0.0599863	+	0.184619i	0.976559	−	0.215249i	\(-0.0690562\pi\)
							−0.916573	+	0.399867i	\(0.869056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.11	✓	120
31.20	even	15	inner	403.2.bi.a.144.11	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.11	✓	120	1.1	even	1	trivial
403.2.bi.a.144.11	yes	120	31.20	even	15	inner