Embedded newform 605.2.k.b.56.11

• Label: 605.2.k.b.56.11
• Level: $605$
• Weight: $2$
• Character: 605.56
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			56.11
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.b.551.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.127482 - 0.0374322i) q^{2} -3.30973 q^{3} +(-1.66766 - 1.07174i) q^{4} +(0.654861 - 0.755750i) q^{5} +(0.421933 + 0.123891i) q^{6} +(1.07269 - 2.34886i) q^{7} +(0.346495 + 0.399876i) q^{8} +7.95431 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.127482	−	0.0374322i	−0.0901437	−	0.0264686i	0.236350	−	0.971668i	\(-0.424049\pi\)
							−0.326493	+	0.945199i	\(0.605867\pi\)
\(3\)	−3.30973			−1.91087			−0.955437	−	0.295196i	\(-0.904615\pi\)
							−0.955437	+	0.295196i	\(0.904615\pi\)
\(5\)	0.654861	−	0.755750i	0.292863	−	0.337981i
							\(7\)	1.07269	−	2.34886i	0.405439	−	0.887787i	−0.591251	−	0.806488i	\(-0.701366\pi\)
0.996690	−	0.0812995i	\(-0.0259070\pi\)
\(11\)	−0.579616	−	3.26558i	−0.174761	−	0.984611i
							\(13\)	−0.259129	−	0.166532i	−0.0718696	−	0.0461878i	0.504213	−	0.863580i	\(-0.331783\pi\)
−0.576082	+	0.817392i	\(0.695419\pi\)
\(17\)	−0.0360756	−	0.250912i	−0.00874963	−	0.0608550i	0.984979	−	0.172675i	\(-0.0552411\pi\)
							−0.993728	+	0.111820i	\(0.964332\pi\)
\(19\)	1.19247	−	8.29382i	0.273572	−	1.90273i	−0.136433	−	0.990649i	\(-0.543564\pi\)
							0.410005	−	0.912083i	\(-0.365527\pi\)
\(23\)	1.44763	+	3.16986i	0.301851	+	0.660962i	0.998400	−	0.0565478i	\(-0.0180093\pi\)
							−0.696549	+	0.717509i	\(0.745282\pi\)
\(29\)	0.636511	−	4.42703i	0.118197	−	0.822079i	−0.841342	−	0.540503i	\(-0.818234\pi\)
							0.959539	−	0.281575i	\(-0.0908570\pi\)
\(31\)	−4.24465	+	2.72787i	−0.762361	+	0.489940i	−0.863137	−	0.504969i	\(-0.831504\pi\)
							0.100776	+	0.994909i	\(0.467867\pi\)
\(37\)	−5.20440	+	3.34467i	−0.855598	+	0.549860i	−0.893316	−	0.449429i	\(-0.851628\pi\)
							0.0377177	+	0.999288i	\(0.487991\pi\)
\(41\)	−2.41911	−	0.710316i	−0.377802	−	0.110933i	0.0873175	−	0.996181i	\(-0.472171\pi\)
							−0.465120	+	0.885248i	\(0.653989\pi\)
\(43\)	−6.80732	−	7.85607i	−1.03811	−	1.19804i	−0.979849	−	0.199738i	\(-0.935991\pi\)
							−0.0582579	−	0.998302i	\(-0.518555\pi\)
\(47\)	−5.96034	+	1.75011i	−0.869405	+	0.255280i	−0.685862	−	0.727732i	\(-0.740575\pi\)
							−0.183542	+	0.983012i	\(0.558756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.b.56.11	✓	220
121.67	even	11	inner	605.2.k.b.551.11	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.b.56.11	✓	220	1.1	even	1	trivial
605.2.k.b.551.11	yes	220	121.67	even	11	inner