Embedded newform 605.2.o.a.34.12

• Label: 605.2.o.a.34.12
• Level: $605$
• Weight: $2$
• Character: 605.34
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.12
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96271 + 0.282195i) q^{2} +0.434811i q^{3} +(1.85361 - 0.544269i) q^{4} +(1.01315 - 1.99337i) q^{5} +(-0.122702 - 0.853408i) q^{6} +(0.433841 - 0.675070i) q^{7} +(0.122895 - 0.0561242i) q^{8} +2.81094 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96271	+	0.282195i	−1.38785	+	0.199542i	−0.795408	−	0.606074i	\(-0.792743\pi\)
							−0.592438	+	0.805616i	\(0.701834\pi\)
\(3\)			0.434811i			0.251038i	0.992091	+	0.125519i	\(0.0400596\pi\)
							−0.992091	+	0.125519i	\(0.959940\pi\)
\(5\)	1.01315	−	1.99337i	0.453095	−	0.891462i
							\(7\)	0.433841	−	0.675070i	0.163977	−	0.255153i	−0.749535	−	0.661964i	\(-0.769723\pi\)
0.913512	+	0.406812i	\(0.133359\pi\)
\(11\)	0.0177971	−	3.31658i	0.00536602	−	0.999986i
							\(13\)	0.127395	+	0.433868i	0.0353331	+	0.120333i	0.975271	−	0.221011i	\(-0.0709358\pi\)
−0.939938	+	0.341345i	\(0.889118\pi\)
\(17\)	−1.52065	+	1.31765i	−0.368812	+	0.319578i	−0.819473	−	0.573117i	\(-0.805734\pi\)
							0.450661	+	0.892695i	\(0.351188\pi\)
\(19\)	1.91190	−	2.20645i	0.438619	−	0.506193i	−0.492800	−	0.870143i	\(-0.664026\pi\)
							0.931419	+	0.363949i	\(0.118572\pi\)
\(23\)	−3.59810	−	5.59875i	−0.750255	−	1.16742i	−0.980925	−	0.194387i	\(-0.937728\pi\)
							0.230670	−	0.973032i	\(-0.425908\pi\)
\(29\)	−4.64747	+	5.36347i	−0.863014	+	0.995971i	0.136972	+	0.990575i	\(0.456263\pi\)
							−0.999985	+	0.00539591i	\(0.998282\pi\)
\(31\)	1.01802	+	0.298917i	0.182841	+	0.0536870i	0.371871	−	0.928284i	\(-0.378716\pi\)
							−0.189030	+	0.981971i	\(0.560534\pi\)
\(37\)	1.49930	−	5.10614i	0.246483	−	0.839445i	−0.739579	−	0.673070i	\(-0.764975\pi\)
							0.986062	−	0.166375i	\(-0.0532063\pi\)
\(41\)	−0.417878	−	2.90641i	−0.0652616	−	0.453904i	−0.996083	−	0.0884288i	\(-0.971815\pi\)
							0.930821	−	0.365476i	\(-0.119094\pi\)
\(43\)	−7.42373	+	3.39030i	−1.13211	+	0.517016i	−0.891236	−	0.453540i	\(-0.850161\pi\)
							−0.240872	+	0.970557i	\(0.577434\pi\)
\(47\)	7.48776	+	1.07658i	1.09220	+	0.157035i	0.664796	−	0.747025i	\(-0.268518\pi\)
							0.427406	+	0.904060i	\(0.359428\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.12	✓	640
5.4	even	2	inner	605.2.o.a.34.53	yes	640
121.89	even	11	inner	605.2.o.a.89.53	yes	640
605.89	even	22	inner	605.2.o.a.89.12	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.12	✓	640	1.1	even	1	trivial
605.2.o.a.34.53	yes	640	5.4	even	2	inner
605.2.o.a.89.12	yes	640	605.89	even	22	inner
605.2.o.a.89.53	yes	640	121.89	even	11	inner