Embedded newform 170.3.e.b.157.5

• Label: 170.3.e.b.157.5
• Level: $170$
• Weight: $3$
• Character: 170.157
• Analytic conductor: $4.632$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	170.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.63216449413\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 80*x^14 + 2532*x^12 + 40532*x^10 + 346464*x^8 + 1518752*x^6 + 2895224*x^4 + 1253728*x^2 + 148996
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2\cdot 5 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			157.5
Root			\(-0.574749i\) of defining polynomial
Character	\(\chi\)	\(=\)	170.157
Dual form			170.3.e.b.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +0.574749i q^{3} -2.00000i q^{4} +(-4.36413 - 2.44015i) q^{5} +(0.574749 + 0.574749i) q^{6} -5.24832i q^{7} +(-2.00000 - 2.00000i) q^{8} +8.66966 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} - 2 q^{5} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\)	\(71\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	−	1.00000i	0.500000	−	0.500000i
							\(3\)			0.574749i			0.191583i	0.995401	+	0.0957915i	\(0.0305382\pi\)
−0.995401	+	0.0957915i	\(0.969462\pi\)
\(5\)	−4.36413	−	2.44015i	−0.872827	−	0.488030i
							\(7\)		−	5.24832i		−	0.749760i	−0.927073	−	0.374880i	\(-0.877684\pi\)
0.927073	−	0.374880i	\(-0.122316\pi\)
\(11\)	−12.9078	−	12.9078i	−1.17344	−	1.17344i	−0.981384	−	0.192053i	\(-0.938485\pi\)
							−0.192053	−	0.981384i	\(-0.561515\pi\)
\(13\)	2.34417	−	2.34417i	0.180321	−	0.180321i	−0.611175	−	0.791496i	\(-0.709303\pi\)
							0.791496	+	0.611175i	\(0.209303\pi\)
\(17\)	−15.4156	−	7.16645i	−0.906803	−	0.421556i
							\(19\)	17.5328			0.922781			0.461391	−	0.887197i	\(-0.347351\pi\)
0.461391	+	0.887197i	\(0.347351\pi\)
\(23\)	15.0470			0.654219			0.327110	−	0.944986i	\(-0.393925\pi\)
							0.327110	+	0.944986i	\(0.393925\pi\)
\(29\)	10.9063	+	10.9063i	0.376079	+	0.376079i	0.869686	−	0.493606i	\(-0.164322\pi\)
							−0.493606	+	0.869686i	\(0.664322\pi\)
\(31\)	5.76255	−	5.76255i	0.185889	−	0.185889i	−0.608027	−	0.793916i	\(-0.708039\pi\)
							0.793916	+	0.608027i	\(0.208039\pi\)
\(37\)	−14.0439			−0.379564			−0.189782	−	0.981826i	\(-0.560778\pi\)
							−0.189782	+	0.981826i	\(0.560778\pi\)
\(41\)	1.12248	+	1.12248i	0.0273777	+	0.0273777i	0.720663	−	0.693285i	\(-0.243837\pi\)
							−0.693285	+	0.720663i	\(0.743837\pi\)
\(43\)	12.5385	+	12.5385i	0.291593	+	0.291593i	0.837709	−	0.546117i	\(-0.183894\pi\)
							−0.546117	+	0.837709i	\(0.683894\pi\)
\(47\)	30.4179	+	30.4179i	0.647190	+	0.647190i	0.952313	−	0.305123i	\(-0.0986976\pi\)
							−0.305123	+	0.952313i	\(0.598698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.e.b.157.5	yes	16
5.3	odd	4		170.3.j.b.123.4	yes	16
17.13	even	4		170.3.j.b.47.4	yes	16
85.13	odd	4	inner	170.3.e.b.13.4	✓	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.e.b.13.4	✓	16	85.13	odd	4	inner
170.3.e.b.157.5	yes	16	1.1	even	1	trivial
170.3.j.b.47.4	yes	16	17.13	even	4
170.3.j.b.123.4	yes	16	5.3	odd	4