Embedded newform 345.2.x.a.2.10

• Label: 345.2.x.a.2.10
• Level: $345$
• Weight: $2$
• Character: 345.2
• Analytic conductor: $2.755$
• Analytic rank: $0$
• Dimension: $880$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	345.x (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.75483886973\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(44\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	345.2
Dual form			345.2.x.a.173.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61252 + 0.350783i) q^{2} +(-0.312005 - 1.70372i) q^{3} +(0.657916 - 0.300460i) q^{4} +(1.52233 + 1.63784i) q^{5} +(1.10075 + 2.63784i) q^{6} +(-2.22999 + 0.159492i) q^{7} +(1.68665 - 1.26261i) q^{8} +(-2.80531 + 1.06314i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q - 22 q^{3} - 44 q^{6} - 36 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/345\mathbb{Z}\right)^\times\).

\(n\)	\(116\)	\(166\)	\(277\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61252	+	0.350783i	−1.14023	+	0.248041i	−0.742748	−	0.669571i	\(-0.766478\pi\)
							−0.397478	+	0.917612i	\(0.630114\pi\)
\(3\)	−0.312005	−	1.70372i	−0.180136	−	0.983642i
							\(5\)	1.52233	+	1.63784i	0.680805	+	0.732464i
\(7\)	−2.22999	+	0.159492i	−0.842856	+	0.0602822i								−0.486087	−	0.873910i	\(-0.661576\pi\)
							−0.356769	+	0.934193i	\(0.616122\pi\)
\(11\)	1.47389	+	2.29341i	0.444394	+	0.691491i	0.989118	−	0.147125i	\(-0.0470019\pi\)
							−0.544724	+	0.838615i	\(0.683366\pi\)
\(13\)	−5.02684	−	0.359526i	−1.39419	−	0.0997147i	−0.646072	−	0.763276i	\(-0.723589\pi\)
							−0.748122	+	0.663562i	\(0.769044\pi\)
\(17\)	−0.119076	−	0.319256i	−0.0288802	−	0.0774308i	0.921709	−	0.387881i	\(-0.126793\pi\)
							−0.950590	+	0.310450i	\(0.899520\pi\)
\(19\)	1.74918	−	0.798822i	0.401289	−	0.183262i	−0.204539	−	0.978858i	\(-0.565569\pi\)
							0.605827	+	0.795596i	\(0.292842\pi\)
\(23\)	0.522525	+	4.76728i	0.108954	+	0.994047i
							\(29\)	−2.65094	+	5.80475i	−0.492268	+	1.07792i	0.486638	+	0.873604i	\(0.338223\pi\)
−0.978906	+	0.204312i	\(0.934504\pi\)
\(31\)	−0.819732	+	5.70136i	−0.147228	+	1.02399i	0.773502	+	0.633794i	\(0.218503\pi\)
							−0.920730	+	0.390200i	\(0.872406\pi\)
\(37\)	6.31906	+	3.45047i	1.03885	+	0.567253i	0.905733	−	0.423848i	\(-0.139321\pi\)
							0.133113	+	0.991101i	\(0.457503\pi\)
\(41\)	−2.11070	+	7.18839i	−0.329636	+	1.12264i	0.613351	+	0.789810i	\(0.289821\pi\)
							−0.942988	+	0.332828i	\(0.891997\pi\)
\(43\)	−4.08526	−	3.05819i	−0.622997	−	0.466369i	0.240551	−	0.970636i	\(-0.422672\pi\)
							−0.863548	+	0.504267i	\(0.831763\pi\)
\(47\)	5.09085	−	5.09085i	0.742576	−	0.742576i	−0.230497	−	0.973073i	\(-0.574035\pi\)
							0.973073	+	0.230497i	\(0.0740351\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	345.2.x.a.2.10	✓	880
3.2	odd	2	inner	345.2.x.a.2.35	yes	880
5.3	odd	4	inner	345.2.x.a.278.35	yes	880
15.8	even	4	inner	345.2.x.a.278.10	yes	880
23.12	even	11	inner	345.2.x.a.242.10	yes	880
69.35	odd	22	inner	345.2.x.a.242.35	yes	880
115.58	odd	44	inner	345.2.x.a.173.35	yes	880
345.173	even	44	inner	345.2.x.a.173.10	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.x.a.2.10	✓	880	1.1	even	1	trivial
345.2.x.a.2.35	yes	880	3.2	odd	2	inner
345.2.x.a.173.10	yes	880	345.173	even	44	inner
345.2.x.a.173.35	yes	880	115.58	odd	44	inner
345.2.x.a.242.10	yes	880	23.12	even	11	inner
345.2.x.a.242.35	yes	880	69.35	odd	22	inner
345.2.x.a.278.10	yes	880	15.8	even	4	inner
345.2.x.a.278.35	yes	880	5.3	odd	4	inner