Embedded newform 605.2.o.a.34.4

• Label: 605.2.o.a.34.4
• 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			34.4
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48873 + 0.357826i) q^{2} +1.71261i q^{3} +(4.14677 - 1.21760i) q^{4} +(-2.15526 - 0.595682i) q^{5} +(-0.612817 - 4.26224i) q^{6} +(1.27993 - 1.99162i) q^{7} +(-5.31030 + 2.42513i) q^{8} +0.0669571 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48873	+	0.357826i	−1.75980	+	0.253021i	−0.945082	−	0.326834i	\(-0.894018\pi\)
							−0.814719	+	0.579855i	\(0.803109\pi\)
\(3\)			1.71261i			0.988778i	0.869241	+	0.494389i	\(0.164608\pi\)
							−0.869241	+	0.494389i	\(0.835392\pi\)
\(5\)	−2.15526	−	0.595682i	−0.963863	−	0.266397i
							\(7\)	1.27993	−	1.99162i	0.483770	−	0.752761i	−0.510475	−	0.859893i	\(-0.670530\pi\)
0.994245	+	0.107132i	\(0.0341668\pi\)
\(11\)	−2.75824	−	1.84176i	−0.831642	−	0.555312i
							\(13\)	−0.108910	−	0.370914i	−0.0302062	−	0.102873i	0.943010	−	0.332764i	\(-0.107981\pi\)
−0.973216	+	0.229891i	\(0.926163\pi\)
\(17\)	0.114761	−	0.0994410i	0.0278336	−	0.0241180i	−0.640833	−	0.767680i	\(-0.721411\pi\)
							0.668667	+	0.743562i	\(0.266865\pi\)
\(19\)	−2.59828	+	2.99858i	−0.596087	+	0.687921i	−0.970984	−	0.239146i	\(-0.923133\pi\)
							0.374897	+	0.927066i	\(0.377678\pi\)
\(23\)	3.61228	+	5.62081i	0.753212	+	1.17202i	0.980178	+	0.198118i	\(0.0634829\pi\)
							−0.226966	+	0.973903i	\(0.572881\pi\)
\(29\)	−4.72527	+	5.45325i	−0.877460	+	1.01264i	0.122336	+	0.992489i	\(0.460961\pi\)
							−0.999797	+	0.0201546i	\(0.993584\pi\)
\(31\)	0.266441	+	0.0782341i	0.0478542	+	0.0140513i	0.305572	−	0.952169i	\(-0.401152\pi\)
							−0.257718	+	0.966220i	\(0.582970\pi\)
\(37\)	−0.684933	+	2.33267i	−0.112602	+	0.383488i	−0.996440	−	0.0843057i	\(-0.973133\pi\)
							0.883838	+	0.467794i	\(0.154951\pi\)
\(41\)	−0.742363	−	5.16325i	−0.115938	−	0.806363i	−0.961955	−	0.273208i	\(-0.911915\pi\)
							0.846017	−	0.533155i	\(-0.178994\pi\)
\(43\)	2.79148	−	1.27483i	0.425697	−	0.194409i	−0.191034	−	0.981583i	\(-0.561184\pi\)
							0.616731	+	0.787174i	\(0.288457\pi\)
\(47\)	−0.575221	−	0.0827043i	−0.0839047	−	0.0120637i	0.100235	−	0.994964i	\(-0.468041\pi\)
							−0.184139	+	0.982900i	\(0.558950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.4	✓	640
5.4	even	2	inner	605.2.o.a.34.61	yes	640
121.89	even	11	inner	605.2.o.a.89.61	yes	640
605.89	even	22	inner	605.2.o.a.89.4	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.4	✓	640	1.1	even	1	trivial
605.2.o.a.34.61	yes	640	5.4	even	2	inner
605.2.o.a.89.4	yes	640	605.89	even	22	inner
605.2.o.a.89.61	yes	640	121.89	even	11	inner