Embedded newform 605.2.s.a.16.9

• Label: 605.2.s.a.16.9
• Level: $605$
• Weight: $2$
• Character: 605.16
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.s (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(22\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			16.9
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.949888 - 0.108989i) q^{2} +(-1.30238 - 0.946237i) q^{3} +(-1.05762 - 0.245938i) q^{4} +(0.564443 - 0.825472i) q^{5} +(1.13399 + 1.04077i) q^{6} +(-0.316454 - 0.325624i) q^{7} +(2.77884 + 0.991490i) q^{8} +(-0.126212 - 0.388441i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.949888	−	0.108989i	−0.671672	−	0.0770672i	−0.228555	−	0.973531i	\(-0.573400\pi\)
							−0.443117	+	0.896464i	\(0.646127\pi\)
\(3\)	−1.30238	−	0.946237i	−0.751932	−	0.546310i	0.144493	−	0.989506i	\(-0.453845\pi\)
							−0.896425	+	0.443195i	\(0.853845\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	−0.316454	−	0.325624i	−0.119608	−	0.123074i	0.654682	−	0.755904i	\(-0.272802\pi\)
−0.774291	+	0.632830i	\(0.781893\pi\)
\(11\)	0.443105	−	3.28689i	0.133601	−	0.991035i
							\(13\)	1.73440	−	2.87617i	0.481035	−	0.797707i	−0.517275	−	0.855819i	\(-0.673054\pi\)
0.998310	+	0.0581125i	\(0.0185082\pi\)
\(17\)	−1.36918	+	1.11958i	−0.332076	+	0.271537i	−0.784319	−	0.620358i	\(-0.786987\pi\)
							0.452243	+	0.891895i	\(0.350624\pi\)
\(19\)	1.72388	+	0.0985751i	0.395485	+	0.0226147i	0.253701	−	0.967283i	\(-0.418352\pi\)
							0.141785	+	0.989897i	\(0.454716\pi\)
\(23\)	0.503539	−	3.50219i	0.104995	−	0.730257i	−0.867518	−	0.497407i	\(-0.834286\pi\)
							0.972513	−	0.232850i	\(-0.0748052\pi\)
\(29\)	1.12683	−	2.89423i	0.209247	−	0.537446i	−0.787654	−	0.616118i	\(-0.788704\pi\)
							0.996901	+	0.0786723i	\(0.0250681\pi\)
\(31\)	−6.44968	−	2.72566i	−1.15840	−	0.489543i	−0.276513	−	0.961010i	\(-0.589179\pi\)
							−0.881883	+	0.471468i	\(0.843724\pi\)
\(37\)	−0.615284	+	7.16363i	−0.101152	+	1.17769i	0.753126	+	0.657877i	\(0.228545\pi\)
							−0.854278	+	0.519816i	\(0.826000\pi\)
\(41\)	−1.59347	+	0.899874i	−0.248858	+	0.140537i	−0.611115	−	0.791541i	\(-0.709279\pi\)
							0.362257	+	0.932078i	\(0.382006\pi\)
\(43\)	−7.04207	+	2.06774i	−1.07391	+	0.315327i	−0.770438	−	0.637514i	\(-0.779963\pi\)
							−0.303468	+	0.952842i	\(0.598144\pi\)
\(47\)	−2.39201	+	11.8050i	−0.348910	+	1.72194i	0.293832	+	0.955857i	\(0.405069\pi\)
							−0.642742	+	0.766082i	\(0.722203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.9	✓	880
121.53	even	55	inner	605.2.s.a.416.9	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.9	✓	880	1.1	even	1	trivial
605.2.s.a.416.9	yes	880	121.53	even	55	inner