Embedded newform 403.2.y.a.64.20

• Label: 403.2.y.a.64.20
• Level: $403$
• Weight: $2$
• Character: 403.64
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			64.20
Character	\(\chi\)	\(=\)	403.64
Dual form			403.2.y.a.233.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.169061 - 0.232692i) q^{2} +(-0.823965 + 0.598646i) q^{3} +(0.592470 + 1.82343i) q^{4} -4.12859i q^{5} +0.292938i q^{6} +(-4.35350 + 1.41454i) q^{7} +(1.07156 + 0.348169i) q^{8} +(-0.606509 + 1.86664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.169061	−	0.232692i	0.119544	−	0.164538i	−0.745051	−	0.667007i	\(-0.767575\pi\)
							0.864595	+	0.502469i	\(0.167575\pi\)
\(3\)	−0.823965	+	0.598646i	−0.475716	+	0.345628i	−0.799665	−	0.600447i	\(-0.794990\pi\)
							0.323949	+	0.946075i	\(0.394990\pi\)
\(5\)		−	4.12859i		−	1.84636i	−0.384366	−	0.923181i	\(-0.625580\pi\)
							0.384366	−	0.923181i	\(-0.374420\pi\)
\(7\)	−4.35350	+	1.41454i	−1.64547	+	0.534645i	−0.977751	−	0.209769i	\(-0.932729\pi\)
							−0.667718	+	0.744414i	\(0.732729\pi\)
\(11\)	−4.19272	+	1.36230i	−1.26415	+	0.410748i	−0.862972	−	0.505251i	\(-0.831400\pi\)
							−0.401180	+	0.915999i	\(0.631400\pi\)
\(13\)	2.56078	+	2.53819i	0.710234	+	0.703966i
							\(17\)	−0.765160	+	2.35492i	−0.185578	+	0.571152i	−0.999958	−	0.00918105i	\(-0.997078\pi\)
0.814379	+	0.580333i	\(0.197078\pi\)
\(19\)	−0.541586	+	0.745429i	−0.124248	+	0.171013i	−0.866610	−	0.498986i	\(-0.833706\pi\)
							0.742361	+	0.670000i	\(0.233706\pi\)
\(23\)	1.03855	−	3.19632i	0.216552	−	0.666480i	−0.782487	−	0.622667i	\(-0.786049\pi\)
							0.999040	−	0.0438133i	\(-0.0139507\pi\)
\(29\)	2.74201	+	1.99219i	0.509179	+	0.369940i	0.812512	−	0.582944i	\(-0.198099\pi\)
							−0.303333	+	0.952885i	\(0.598099\pi\)
\(31\)	−4.07494	+	3.79405i	−0.731881	+	0.681432i
							\(37\)			3.06541i			0.503950i	0.967734	+	0.251975i	\(0.0810801\pi\)
−0.967734	+	0.251975i	\(0.918920\pi\)
\(41\)	2.32211	−	3.19610i	0.362652	−	0.499148i	−0.588233	−	0.808691i	\(-0.700176\pi\)
							0.950885	+	0.309544i	\(0.100176\pi\)
\(43\)	−7.43820	−	5.40417i	−1.13432	−	0.824128i	−0.147998	−	0.988988i	\(-0.547283\pi\)
							−0.986317	+	0.164860i	\(0.947283\pi\)
\(47\)	−2.72337	−	3.74840i	−0.397245	−	0.546761i	0.562805	−	0.826590i	\(-0.309722\pi\)
							−0.960050	+	0.279829i	\(0.909722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.y.a.64.20	yes	136
13.12	even	2	inner	403.2.y.a.64.15	✓	136
31.16	even	5	inner	403.2.y.a.233.15	yes	136
403.233	even	10	inner	403.2.y.a.233.20	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.y.a.64.15	✓	136	13.12	even	2	inner
403.2.y.a.64.20	yes	136	1.1	even	1	trivial
403.2.y.a.233.15	yes	136	31.16	even	5	inner
403.2.y.a.233.20	yes	136	403.233	even	10	inner