Embedded newform 164.2.o.b.15.16

• Label: 164.2.o.b.15.16
• Level: $164$
• Weight: $2$
• Character: 164.15
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30954659315\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.16
Character	\(\chi\)	\(=\)	164.15
Dual form			164.2.o.b.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23500 - 0.689031i) q^{2} +(-0.900558 - 2.17414i) q^{3} +(1.05047 - 1.70191i) q^{4} +(-0.390234 - 0.765878i) q^{5} +(-2.61024 - 2.06456i) q^{6} +(-0.876658 + 3.65154i) q^{7} +(0.124665 - 2.82568i) q^{8} +(-1.79456 + 1.79456i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.23500	−	0.689031i	0.873280	−	0.487219i
							\(3\)	−0.900558	−	2.17414i	−0.519938	−	1.25524i	−0.937941	−	0.346795i	\(-0.887270\pi\)
0.418003	−	0.908445i	\(-0.362730\pi\)
\(5\)	−0.390234	−	0.765878i	−0.174518	−	0.342511i	0.787135	−	0.616781i	\(-0.211564\pi\)
							−0.961653	+	0.274270i	\(0.911564\pi\)
\(7\)	−0.876658	+	3.65154i	−0.331346	+	1.38015i	0.519380	+	0.854544i	\(0.326163\pi\)
							−0.850725	+	0.525611i	\(0.823837\pi\)
\(11\)	1.26843	+	1.48514i	0.382446	+	0.447787i	0.917920	−	0.396765i	\(-0.129867\pi\)
							−0.535474	+	0.844552i	\(0.679867\pi\)
\(13\)	1.79670	−	2.93195i	0.498316	−	0.813177i	−0.500296	−	0.865854i	\(-0.666775\pi\)
							0.998612	+	0.0526774i	\(0.0167755\pi\)
\(17\)	−0.302571	+	3.84453i	−0.0733843	+	0.932435i	0.844492	+	0.535568i	\(0.179903\pi\)
							−0.917876	+	0.396867i	\(0.870097\pi\)
\(19\)	5.10695	−	3.12954i	1.17161	−	0.717966i	0.206155	−	0.978519i	\(-0.433905\pi\)
							0.965459	+	0.260553i	\(0.0839050\pi\)
\(23\)	−1.43017	+	1.03908i	−0.298210	+	0.216663i	−0.726821	−	0.686827i	\(-0.759003\pi\)
							0.428611	+	0.903489i	\(0.359003\pi\)
\(29\)	0.535913	+	6.80942i	0.0995166	+	1.26448i	0.820549	+	0.571577i	\(0.193668\pi\)
							−0.721032	+	0.692902i	\(0.756332\pi\)
\(31\)	0.437955	+	1.34789i	0.0786590	+	0.242088i	0.982652	−	0.185460i	\(-0.0593776\pi\)
							−0.903993	+	0.427548i	\(0.859378\pi\)
\(37\)	−2.67942	+	8.24640i	−0.440494	+	1.35570i	0.446857	+	0.894605i	\(0.352543\pi\)
							−0.887351	+	0.461095i	\(0.847457\pi\)
\(41\)	−5.03089	−	3.96108i	−0.785693	−	0.618617i
							\(43\)	−1.17766	+	7.43544i	−0.179591	+	1.13389i	0.718968	+	0.695043i	\(0.244615\pi\)
−0.898559	+	0.438852i	\(0.855385\pi\)
\(47\)	1.70845	+	7.11622i	0.249204	+	1.03801i	0.946878	+	0.321593i	\(0.104218\pi\)
							−0.697674	+	0.716415i	\(0.745782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.15.16	yes	288
4.3	odd	2	inner	164.2.o.b.15.14	yes	288
41.11	odd	40	inner	164.2.o.b.11.14	✓	288
164.11	even	40	inner	164.2.o.b.11.16	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.14	✓	288	41.11	odd	40	inner
164.2.o.b.11.16	yes	288	164.11	even	40	inner
164.2.o.b.15.14	yes	288	4.3	odd	2	inner
164.2.o.b.15.16	yes	288	1.1	even	1	trivial