Embedded newform 403.2.ch.a.11.2

• Label: 403.2.ch.a.11.2
• Level: $403$
• Weight: $2$
• Character: 403.11
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.ch (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(35\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	403.11
Dual form			403.2.ch.a.110.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.52766 + 0.132469i) q^{2} +(2.14081 + 0.695592i) q^{3} +(4.38247 - 0.460616i) q^{4} +(-0.253940 + 0.947716i) q^{5} +(-5.50339 - 1.47463i) q^{6} +(0.297869 - 0.114341i) q^{7} +(-6.01643 + 0.952909i) q^{8} +(1.67218 + 1.21491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52766	+	0.132469i	−1.78732	+	0.0936697i	−0.917019	−	0.398843i	\(-0.869412\pi\)
							−0.870305	+	0.492513i	\(0.836078\pi\)
\(3\)	2.14081	+	0.695592i	1.23600	+	0.401600i	0.852884	−	0.522100i	\(-0.174851\pi\)
							0.383115	+	0.923701i	\(0.374851\pi\)
\(5\)	−0.253940	+	0.947716i	−0.113565	+	0.423832i	−0.999176	−	0.0405976i	\(-0.987074\pi\)
							0.885610	+	0.464429i	\(0.153740\pi\)
\(7\)	0.297869	−	0.114341i	0.112584	−	0.0432169i	−0.301424	−	0.953490i	\(-0.597462\pi\)
							0.414008	+	0.910273i	\(0.364129\pi\)
\(11\)	−3.24705	+	4.00977i	−0.979022	+	1.20899i	−0.00110479	+	0.999999i	\(0.500352\pi\)
							−0.977917	+	0.208992i	\(0.932982\pi\)
\(13\)	−0.283655	+	3.59438i	−0.0786718	+	0.996901i
							\(17\)	−0.237161	+	2.25644i	−0.0575200	+	0.547266i	0.927377	+	0.374127i	\(0.122058\pi\)
−0.984897	+	0.173139i	\(0.944609\pi\)
\(19\)	−0.765414	+	0.0401136i	−0.175598	+	0.00920270i	−0.139932	−	0.990161i	\(-0.544688\pi\)
							−0.0356657	+	0.999364i	\(0.511355\pi\)
\(23\)	0.0323778	−	0.308054i	0.00675124	−	0.0642338i	−0.990628	−	0.136591i	\(-0.956385\pi\)
							0.997379	+	0.0723571i	\(0.0230521\pi\)
\(29\)	−0.902382	−	0.812509i	−0.167568	−	0.150879i	0.581080	−	0.813846i	\(-0.302630\pi\)
							−0.748648	+	0.662967i	\(0.769297\pi\)
\(31\)	4.57338	−	3.17556i	0.821403	−	0.570348i
							\(37\)	0.441776	+	0.441776i	0.0726275	+	0.0726275i	0.742487	−	0.669860i	\(-0.233646\pi\)
−0.669860	+	0.742487i	\(0.733646\pi\)
\(41\)	1.09780	+	1.69047i	0.171448	+	0.264007i	0.913941	−	0.405847i	\(-0.133023\pi\)
							−0.742493	+	0.669854i	\(0.766357\pi\)
\(43\)	−0.552173	+	0.613250i	−0.0842056	+	0.0935198i	−0.783775	−	0.621045i	\(-0.786708\pi\)
							0.699569	+	0.714565i	\(0.253375\pi\)
\(47\)	3.94207	+	7.73675i	0.575010	+	1.12852i	0.977072	+	0.212908i	\(0.0682935\pi\)
							−0.402062	+	0.915612i	\(0.631706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.11.2	yes	560
13.6	odd	12		403.2.cc.a.383.34	yes	560
31.17	odd	30		403.2.cc.a.141.34	✓	560
403.110	even	60	inner	403.2.ch.a.110.2	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.34	✓	560	31.17	odd	30
403.2.cc.a.383.34	yes	560	13.6	odd	12
403.2.ch.a.11.2	yes	560	1.1	even	1	trivial
403.2.ch.a.110.2	yes	560	403.110	even	60	inner