Embedded newform 32.4.g.a.21.4

• Label: 32.4.g.a.21.4
• Level: $32$
• Weight: $4$
• Character: 32.21
• Analytic conductor: $1.888$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.88806112018\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			21.4
Character	\(\chi\)	\(=\)	32.21
Dual form			32.4.g.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98147 - 2.01836i) q^{2} +(5.53310 - 2.29188i) q^{3} +(-0.147593 + 7.99864i) q^{4} +(4.22177 - 10.1923i) q^{5} +(-15.5895 - 6.62652i) q^{6} +(-11.6451 - 11.6451i) q^{7} +(16.4366 - 15.5511i) q^{8} +(6.27055 - 6.27055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98147	−	2.01836i	−0.700554	−	0.713600i
							\(3\)	5.53310	−	2.29188i	1.06485	−	0.441073i	0.219676	−	0.975573i	\(-0.429500\pi\)
0.845169	+	0.534499i	\(0.179500\pi\)
\(5\)	4.22177	−	10.1923i	0.377607	−	0.911624i	−0.614807	−	0.788678i	\(-0.710766\pi\)
							0.992413	−	0.122946i	\(-0.0392341\pi\)
\(7\)	−11.6451	−	11.6451i	−0.628775	−	0.628775i	0.318985	−	0.947760i	\(-0.396658\pi\)
							−0.947760	+	0.318985i	\(0.896658\pi\)
\(11\)	27.7531	+	11.4957i	0.760717	+	0.315099i	0.729106	−	0.684401i	\(-0.239936\pi\)
							0.0316109	+	0.999500i	\(0.489936\pi\)
\(13\)	15.9612	+	38.5338i	0.340527	+	0.822104i	0.997663	+	0.0683319i	\(0.0217677\pi\)
							−0.657136	+	0.753772i	\(0.728232\pi\)
\(17\)			131.437i			1.87518i	0.347736	+	0.937592i	\(0.386951\pi\)
							−0.347736	+	0.937592i	\(0.613049\pi\)
\(19\)	−13.7982	−	33.3119i	−0.166607	−	0.402224i	0.818421	−	0.574619i	\(-0.194850\pi\)
							−0.985028	+	0.172394i	\(0.944850\pi\)
\(23\)	128.594	−	128.594i	1.16582	−	1.16582i	0.182634	−	0.983181i	\(-0.441538\pi\)
							0.983181	−	0.182634i	\(-0.0584624\pi\)
\(29\)	−192.157	+	79.5938i	−1.23043	+	0.509662i	−0.900712	−	0.434416i	\(-0.856955\pi\)
							−0.329721	+	0.944078i	\(0.606955\pi\)
\(31\)	−215.350			−1.24768			−0.623839	−	0.781553i	\(-0.714428\pi\)
							−0.623839	+	0.781553i	\(0.714428\pi\)
\(37\)	−9.39705	+	22.6865i	−0.0417531	+	0.100801i	−0.943380	−	0.331713i	\(-0.892373\pi\)
							0.901627	+	0.432514i	\(0.142373\pi\)
\(41\)	257.295	−	257.295i	0.980067	−	0.980067i	−0.0197381	−	0.999805i	\(-0.506283\pi\)
							0.999805	+	0.0197381i	\(0.00628325\pi\)
\(43\)	−81.5685	−	33.7868i	−0.289281	−	0.119824i	0.233324	−	0.972399i	\(-0.425040\pi\)
							−0.522605	+	0.852575i	\(0.675040\pi\)
\(47\)		−	113.432i		−	0.352037i	−0.984387	−	0.176018i	\(-0.943678\pi\)
							0.984387	−	0.176018i	\(-0.0563218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.21.4	✓	44
4.3	odd	2		128.4.g.a.49.3		44
8.3	odd	2		256.4.g.a.97.9		44
8.5	even	2		256.4.g.b.97.3		44
32.3	odd	8		128.4.g.a.81.3		44
32.13	even	8		256.4.g.b.161.3		44
32.19	odd	8		256.4.g.a.161.9		44
32.29	even	8	inner	32.4.g.a.29.4	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.21.4	✓	44	1.1	even	1	trivial
32.4.g.a.29.4	yes	44	32.29	even	8	inner
128.4.g.a.49.3		44	4.3	odd	2
128.4.g.a.81.3		44	32.3	odd	8
256.4.g.a.97.9		44	8.3	odd	2
256.4.g.a.161.9		44	32.19	odd	8
256.4.g.b.97.3		44	8.5	even	2
256.4.g.b.161.3		44	32.13	even	8