Embedded newform 605.2.o.a.34.3

• Label: 605.2.o.a.34.3
• 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			34.3
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.62964 + 0.378084i) q^{2} -1.51252i q^{3} +(4.85305 - 1.42498i) q^{4} +(-0.110182 + 2.23335i) q^{5} +(0.571859 + 3.97737i) q^{6} +(1.36099 - 2.11774i) q^{7} +(-7.38979 + 3.37480i) q^{8} +0.712295 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.62964	+	0.378084i	−1.85943	+	0.267346i	−0.978553	−	0.205994i	\(-0.933957\pi\)
							−0.880880	+	0.473340i	\(0.843048\pi\)
\(3\)		−	1.51252i		−	0.873252i	−0.899643	−	0.436626i	\(-0.856173\pi\)
							0.899643	−	0.436626i	\(-0.143827\pi\)
\(5\)	−0.110182	+	2.23335i	−0.0492748	+	0.998785i
							\(7\)	1.36099	−	2.11774i	0.514404	−	0.800429i	−0.482754	−	0.875756i	\(-0.660364\pi\)
0.997159	+	0.0753268i	\(0.0240000\pi\)
\(11\)	−1.33608	+	3.03561i	−0.402842	+	0.915269i
							\(13\)	1.44393	+	4.91758i	0.400475	+	1.36389i	0.875202	+	0.483758i	\(0.160729\pi\)
−0.474727	+	0.880133i	\(0.657453\pi\)
\(17\)	−4.95688	+	4.29516i	−1.20222	+	1.04173i	−0.204193	+	0.978931i	\(0.565457\pi\)
							−0.998026	+	0.0627978i	\(0.979998\pi\)
\(19\)	−0.591211	+	0.682294i	−0.135633	+	0.156529i	−0.819503	−	0.573075i	\(-0.805750\pi\)
							0.683870	+	0.729604i	\(0.260296\pi\)
\(23\)	−1.23222	−	1.91737i	−0.256935	−	0.399799i	0.688691	−	0.725055i	\(-0.258186\pi\)
							−0.945626	+	0.325256i	\(0.894549\pi\)
\(29\)	−2.30170	+	2.65630i	−0.427415	+	0.493263i	−0.928082	−	0.372377i	\(-0.878543\pi\)
							0.500667	+	0.865640i	\(0.333088\pi\)
\(31\)	−1.17454	−	0.344877i	−0.210954	−	0.0619418i	0.174548	−	0.984649i	\(-0.444154\pi\)
							−0.385502	+	0.922707i	\(0.625972\pi\)
\(37\)	−2.54061	+	8.65253i	−0.417674	+	1.42247i	0.435192	+	0.900338i	\(0.356680\pi\)
							−0.852866	+	0.522129i	\(0.825138\pi\)
\(41\)	−1.66055	−	11.5494i	−0.259334	−	1.80371i	−0.537598	−	0.843201i	\(-0.680668\pi\)
							0.278264	−	0.960505i	\(-0.410241\pi\)
\(43\)	−10.7211	+	4.89614i	−1.63495	+	0.746654i	−0.999671	−	0.0256324i	\(-0.991840\pi\)
							−0.635274	+	0.772287i	\(0.719113\pi\)
\(47\)	11.5736	+	1.66403i	1.68818	+	0.242723i	0.918419	−	0.395610i	\(-0.129467\pi\)
							0.769760	+	0.638333i	\(0.220376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.3	✓	640
5.4	even	2	inner	605.2.o.a.34.62	yes	640
121.89	even	11	inner	605.2.o.a.89.62	yes	640
605.89	even	22	inner	605.2.o.a.89.3	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.3	✓	640	1.1	even	1	trivial
605.2.o.a.34.62	yes	640	5.4	even	2	inner
605.2.o.a.89.3	yes	640	605.89	even	22	inner
605.2.o.a.89.62	yes	640	121.89	even	11	inner