Embedded newform 320.2.bf.a.29.18

• Label: 320.2.bf.a.29.18
• Level: $320$
• Weight: $2$
• Character: 320.29
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bf (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			29.18
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561336 + 1.29804i) q^{2} +(-0.279704 + 1.40617i) q^{3} +(-1.36980 - 1.45727i) q^{4} +(2.09531 - 0.780827i) q^{5} +(-1.66825 - 1.15240i) q^{6} +(0.431062 + 1.04068i) q^{7} +(2.66051 - 0.960039i) q^{8} +(0.872568 + 0.361429i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561336	+	1.29804i	−0.396924	+	0.917851i
							\(3\)	−0.279704	+	1.40617i	−0.161487	+	0.811851i	0.812097	+	0.583522i	\(0.198326\pi\)
−0.973584	+	0.228328i	\(0.926674\pi\)
\(5\)	2.09531	−	0.780827i	0.937050	−	0.349196i
							\(7\)	0.431062	+	1.04068i	0.162926	+	0.393339i	0.984167	−	0.177243i	\(-0.0567178\pi\)
−0.821241	+	0.570582i	\(0.806718\pi\)
\(11\)	0.655007	+	3.29294i	0.197492	+	0.992860i	0.944617	+	0.328176i	\(0.106434\pi\)
							−0.747125	+	0.664684i	\(0.768566\pi\)
\(13\)	−0.600637	+	0.401333i	−0.166587	+	0.111310i	−0.636069	−	0.771632i	\(-0.719441\pi\)
							0.469482	+	0.882942i	\(0.344441\pi\)
\(17\)	−0.881787	+	0.881787i	−0.213865	+	0.213865i	−0.805907	−	0.592042i	\(-0.798322\pi\)
							0.592042	+	0.805907i	\(0.298322\pi\)
\(19\)	−0.892274	+	0.596199i	−0.204702	+	0.136777i	−0.653696	−	0.756757i	\(-0.726782\pi\)
							0.448994	+	0.893535i	\(0.351782\pi\)
\(23\)	−1.02184	−	0.423258i	−0.213068	−	0.0882555i	0.273597	−	0.961844i	\(-0.411787\pi\)
							−0.486664	+	0.873589i	\(0.661787\pi\)
\(29\)	1.74723	−	8.78390i	0.324452	−	1.63113i	−0.382561	−	0.923930i	\(-0.624958\pi\)
							0.707013	−	0.707200i	\(-0.250042\pi\)
\(31\)			2.16476i			0.388802i	0.980922	+	0.194401i	\(0.0622762\pi\)
							−0.980922	+	0.194401i	\(0.937724\pi\)
\(37\)	−4.14426	+	6.20232i	−0.681312	+	1.01966i	0.316168	+	0.948703i	\(0.397604\pi\)
							−0.997480	+	0.0709520i	\(0.977396\pi\)
\(41\)	−2.52652	+	6.09956i	−0.394576	+	0.952591i	0.594353	+	0.804204i	\(0.297408\pi\)
							−0.988929	+	0.148387i	\(0.952592\pi\)
\(43\)	0.207666	+	1.04401i	0.0316687	+	0.159209i	0.993384	−	0.114842i	\(-0.0366362\pi\)
							−0.961715	+	0.274051i	\(0.911636\pi\)
\(47\)	7.71113	−	7.71113i	1.12478	−	1.12478i	0.133773	−	0.991012i	\(-0.457291\pi\)
							0.991012	−	0.133773i	\(-0.0427092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.18	✓	368
5.4	even	2	inner	320.2.bf.a.29.29	yes	368
64.53	even	16	inner	320.2.bf.a.309.29	yes	368
320.309	even	16	inner	320.2.bf.a.309.18	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.18	✓	368	1.1	even	1	trivial
320.2.bf.a.29.29	yes	368	5.4	even	2	inner
320.2.bf.a.309.18	yes	368	320.309	even	16	inner
320.2.bf.a.309.29	yes	368	64.53	even	16	inner