Embedded newform 605.2.r.a.32.16

• Label: 605.2.r.a.32.16
• Level: $605$
• Weight: $2$
• Character: 605.32
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			32.16
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02585 - 1.37037i) q^{2} +(-1.69135 + 1.69135i) q^{3} +(-0.262091 + 0.892598i) q^{4} +(1.54512 - 1.61636i) q^{5} +(4.05284 + 0.582710i) q^{6} +(-0.361306 + 1.66090i) q^{7} +(-1.71570 + 0.639922i) q^{8} -2.72132i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02585	−	1.37037i	−0.725384	−	0.968999i	−0.999972	−	0.00743733i	\(-0.997633\pi\)
							0.274589	−	0.961562i	\(-0.411458\pi\)
\(3\)	−1.69135	+	1.69135i	−0.976500	+	0.976500i	−0.999730	−	0.0232300i	\(-0.992605\pi\)
							0.0232300	+	0.999730i	\(0.492605\pi\)
\(5\)	1.54512	−	1.61636i	0.690997	−	0.722858i
							\(7\)	−0.361306	+	1.66090i	−0.136561	+	0.627760i	0.857082	+	0.515180i	\(0.172275\pi\)
−0.993643	+	0.112580i	\(0.964089\pi\)
\(11\)	0.707000	+	3.24039i	0.213168	+	0.977015i
							\(13\)	−2.86197	−	5.24131i	−0.793768	−	1.45368i	−0.888845	−	0.458208i	\(-0.848492\pi\)
0.0950772	−	0.995470i	\(-0.469690\pi\)
\(17\)	−0.172077	−	2.40595i	−0.0417348	−	0.583528i	−0.974694	−	0.223545i	\(-0.928237\pi\)
							0.932959	−	0.359983i	\(-0.117218\pi\)
\(19\)	1.44967	+	1.67300i	0.332576	+	0.383814i	0.897266	−	0.441489i	\(-0.145550\pi\)
							−0.564690	+	0.825303i	\(0.691004\pi\)
\(23\)	−2.61925	+	0.569783i	−0.546151	+	0.118808i	−0.477172	−	0.878810i	\(-0.658338\pi\)
							−0.0689797	+	0.997618i	\(0.521974\pi\)
\(29\)	−0.387726	−	0.447459i	−0.0719988	−	0.0830911i	0.718607	−	0.695416i	\(-0.244780\pi\)
							−0.790606	+	0.612325i	\(0.790234\pi\)
\(31\)	−6.07370	+	1.78340i	−1.09087	+	0.320308i	−0.777219	−	0.629231i	\(-0.783370\pi\)
							−0.313650	+	0.949539i	\(0.601552\pi\)
\(37\)	−0.769522	−	0.420191i	−0.126509	−	0.0690789i	0.414748	−	0.909936i	\(-0.363870\pi\)
							−0.541257	+	0.840857i	\(0.682051\pi\)
\(41\)	−10.3375	−	1.48631i	−1.61445	−	0.232123i	−0.724865	−	0.688890i	\(-0.758098\pi\)
							−0.889586	+	0.456767i	\(0.849007\pi\)
\(43\)	−5.11508	+	1.90782i	−0.780042	+	0.290941i	−0.707769	−	0.706444i	\(-0.750298\pi\)
							−0.0722731	+	0.997385i	\(0.523025\pi\)
\(47\)	−4.34500	+	5.80424i	−0.633783	+	0.846635i	−0.996354	−	0.0853211i	\(-0.972808\pi\)
							0.362570	+	0.931956i	\(0.381899\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.16	✓	1280
5.3	odd	4	inner	605.2.r.a.153.16	yes	1280
121.87	odd	22	inner	605.2.r.a.87.16	yes	1280
605.208	even	44	inner	605.2.r.a.208.16	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.16	✓	1280	1.1	even	1	trivial
605.2.r.a.87.16	yes	1280	121.87	odd	22	inner
605.2.r.a.153.16	yes	1280	5.3	odd	4	inner
605.2.r.a.208.16	yes	1280	605.208	even	44	inner