Embedded newform 403.2.bt.a.82.18

• Label: 403.2.bt.a.82.18
• Level: $403$
• Weight: $2$
• Character: 403.82
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bt (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			82.18
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0316136 - 0.148730i) q^{2} +(1.83195 - 0.389394i) q^{3} +(1.80597 + 0.804069i) q^{4} +(1.52776 - 0.882054i) q^{5} -0.284777i q^{6} +(-0.353168 - 0.486094i) q^{7} +(0.355432 - 0.489210i) q^{8} +(0.463790 - 0.206493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0316136	−	0.148730i	0.0223542	−	0.105168i	−0.965556	−	0.260195i	\(-0.916213\pi\)
							0.987910	+	0.155027i	\(0.0495464\pi\)
\(3\)	1.83195	−	0.389394i	1.05768	−	0.224817i	0.353935	−	0.935270i	\(-0.384843\pi\)
							0.703744	+	0.710453i	\(0.251510\pi\)
\(5\)	1.52776	−	0.882054i	0.683236	−	0.394466i	−0.117837	−	0.993033i	\(-0.537596\pi\)
							0.801073	+	0.598567i	\(0.204263\pi\)
\(7\)	−0.353168	−	0.486094i	−0.133485	−	0.183726i	0.737042	−	0.675847i	\(-0.236222\pi\)
							−0.870527	+	0.492121i	\(0.836222\pi\)
\(11\)	−3.69813	−	5.09004i	−1.11503	−	1.53471i	−0.813792	−	0.581157i	\(-0.802600\pi\)
							−0.301237	−	0.953549i	\(-0.597400\pi\)
\(13\)	2.51155	+	2.58692i	0.696577	+	0.717482i
							\(17\)	−0.750762	−	0.545461i	−0.182087	−	0.132294i	0.493008	−	0.870025i	\(-0.335897\pi\)
−0.675095	+	0.737731i	\(0.735897\pi\)
\(19\)	−6.31592	+	2.05217i	−1.44897	+	0.470799i	−0.924682	−	0.380740i	\(-0.875669\pi\)
							−0.524290	+	0.851540i	\(0.675669\pi\)
\(23\)	−4.75791	+	2.11836i	−0.992092	+	0.441708i	−0.837599	−	0.546286i	\(-0.816041\pi\)
							−0.154494	+	0.987994i	\(0.549375\pi\)
\(29\)	6.40080	+	1.36053i	1.18860	+	0.252644i	0.759435	−	0.650583i	\(-0.225475\pi\)
							0.429163	+	0.903227i	\(0.358809\pi\)
\(31\)	−1.86905	+	5.24468i	−0.335691	+	0.941972i
							\(37\)	5.74618	−	3.31756i	0.944666	−	0.545403i	0.0532461	−	0.998581i	\(-0.483043\pi\)
0.891420	+	0.453178i	\(0.149710\pi\)
\(41\)	1.73552	−	0.563903i	0.271042	−	0.0880669i	−0.170343	−	0.985385i	\(-0.554487\pi\)
							0.441385	+	0.897318i	\(0.354487\pi\)
\(43\)	−0.249255	−	0.767128i	−0.0380110	−	0.116986i	0.930251	−	0.366925i	\(-0.119589\pi\)
							−0.968262	+	0.249939i	\(0.919589\pi\)
\(47\)	5.36353	+	1.74272i	0.782351	+	0.254201i	0.672843	−	0.739785i	\(-0.265073\pi\)
							0.109507	+	0.993986i	\(0.465073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.18	yes	280
13.10	even	6		403.2.bp.a.361.18	yes	280
31.14	even	15		403.2.bp.a.355.18	✓	280
403.231	even	30	inner	403.2.bt.a.231.18	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.18	✓	280	31.14	even	15
403.2.bp.a.361.18	yes	280	13.10	even	6
403.2.bt.a.82.18	yes	280	1.1	even	1	trivial
403.2.bt.a.231.18	yes	280	403.231	even	30	inner