Embedded newform 164.2.o.b.15.10

• Label: 164.2.o.b.15.10
• 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			15.10
Character	\(\chi\)	\(=\)	164.15
Dual form			164.2.o.b.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0433699 - 1.41355i) q^{2} +(0.739847 + 1.78615i) q^{3} +(-1.99624 - 0.122611i) q^{4} +(1.08360 + 2.12669i) q^{5} +(2.55689 - 0.968344i) q^{6} +(-0.938602 + 3.90956i) q^{7} +(-0.259893 + 2.81646i) q^{8} +(-0.521631 + 0.521631i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0433699	−	1.41355i	0.0306671	−	0.999530i
							\(3\)	0.739847	+	1.78615i	0.427151	+	1.03123i	0.980187	+	0.198076i	\(0.0634692\pi\)
−0.553036	+	0.833157i	\(0.686531\pi\)
\(5\)	1.08360	+	2.12669i	0.484601	+	0.951084i	0.995794	+	0.0916177i	\(0.0292038\pi\)
							−0.511193	+	0.859466i	\(0.670796\pi\)
\(7\)	−0.938602	+	3.90956i	−0.354758	+	1.47767i	0.454778	+	0.890605i	\(0.349719\pi\)
							−0.809536	+	0.587070i	\(0.800281\pi\)
\(11\)	−2.35255	−	2.75448i	−0.709321	−	0.830508i	0.282349	−	0.959312i	\(-0.408886\pi\)
							−0.991670	+	0.128804i	\(0.958886\pi\)
\(13\)	2.54584	−	4.15443i	0.706088	−	1.15223i	−0.275515	−	0.961297i	\(-0.588848\pi\)
							0.981603	−	0.190934i	\(-0.0611516\pi\)
\(17\)	0.0520733	−	0.661654i	0.0126296	−	0.160475i	−0.987368	−	0.158442i	\(-0.949353\pi\)
							0.999998	−	0.00203275i	\(-0.000647046\pi\)
\(19\)	0.534152	−	0.327329i	0.122543	−	0.0750944i	−0.459876	−	0.887983i	\(-0.652106\pi\)
							0.582419	+	0.812889i	\(0.302106\pi\)
\(23\)	3.84791	−	2.79567i	0.802345	−	0.582938i	−0.109256	−	0.994014i	\(-0.534847\pi\)
							0.911601	+	0.411076i	\(0.134847\pi\)
\(29\)	−0.0823757	−	1.04668i	−0.0152968	−	0.194364i	−0.999818	−	0.0190966i	\(-0.993921\pi\)
							0.984521	−	0.175267i	\(-0.0560790\pi\)
\(31\)	−1.06140	−	3.26666i	−0.190634	−	0.586710i	0.809366	−	0.587304i	\(-0.199811\pi\)
							−1.00000		0.000594276i	\(0.999811\pi\)
\(37\)	−3.25281	+	10.0111i	−0.534758	+	1.64582i	0.209414	+	0.977827i	\(0.432845\pi\)
							−0.744172	+	0.667989i	\(0.767155\pi\)
\(41\)	−5.78985	−	2.73452i	−0.904223	−	0.427061i
							\(43\)	−1.20002	+	7.57663i	−0.183001	+	1.15543i	0.709609	+	0.704596i	\(0.248872\pi\)
−0.892610	+	0.450829i	\(0.851128\pi\)
\(47\)	−0.561062	−	2.33699i	−0.0818393	−	0.340885i	0.916396	−	0.400274i	\(-0.131085\pi\)
							−0.998235	+	0.0593886i	\(0.981085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.15.10	yes	288
4.3	odd	2	inner	164.2.o.b.15.7	yes	288
41.11	odd	40	inner	164.2.o.b.11.7	✓	288
164.11	even	40	inner	164.2.o.b.11.10	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.7	✓	288	41.11	odd	40	inner
164.2.o.b.11.10	yes	288	164.11	even	40	inner
164.2.o.b.15.7	yes	288	4.3	odd	2	inner
164.2.o.b.15.10	yes	288	1.1	even	1	trivial