Embedded newform 17.12.d.a.15.8

• Label: 17.12.d.a.15.8
• Level: $17$
• Weight: $12$
• Character: 17.15
• Analytic conductor: $13.062$
• Analytic rank: $0$
• Dimension: $60$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0618340695\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			15.8
Character	\(\chi\)	\(=\)	17.15
Dual form			17.12.d.a.8.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.07850 + 2.07850i) q^{2} +(220.048 + 531.244i) q^{3} -2039.36i q^{4} +(-4868.28 + 2016.51i) q^{5} +(-646.820 + 1561.56i) q^{6} +(-46517.1 - 19268.0i) q^{7} +(8495.59 - 8495.59i) q^{8} +(-108537. + 108537. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 4 q^{3} + 10120 q^{5} - 44716 q^{6} - 4 q^{7} + 8188 q^{8} + 621728 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.07850	+	2.07850i	0.0459288	+	0.0459288i	0.729698	−	0.683769i	\(-0.239661\pi\)
							−0.683769	+	0.729698i	\(0.739661\pi\)
\(3\)	220.048	+	531.244i	0.522819	+	1.26220i	0.936145	+	0.351614i	\(0.114367\pi\)
							−0.413326	+	0.910583i	\(0.635633\pi\)
\(5\)	−4868.28	+	2016.51i	−0.696691	+	0.288579i	−0.702785	−	0.711402i	\(-0.748060\pi\)
							0.00609355	+	0.999981i	\(0.498060\pi\)
\(7\)	−46517.1	−	19268.0i	−1.04610	−	0.433309i	−0.207602	−	0.978213i	\(-0.566566\pi\)
							−0.838498	+	0.544905i	\(0.816566\pi\)
\(11\)	111377.	−	268889.i	0.208515	−	0.503399i	−0.784675	−	0.619908i	\(-0.787170\pi\)
							0.993190	+	0.116508i	\(0.0371701\pi\)
\(13\)		−	574946.i		−	0.429476i	−0.976672	−	0.214738i	\(-0.931110\pi\)
							0.976672	−	0.214738i	\(-0.0688898\pi\)
\(17\)	−5.27001e6	−	2.54929e6i	−0.900207	−	0.435462i
							\(19\)	−4.86433e6	−	4.86433e6i	−0.450690	−	0.450690i	0.444893	−	0.895584i	\(-0.353242\pi\)
−0.895584	+	0.444893i	\(0.853242\pi\)
\(23\)	1.71794e7	−	4.14749e7i	0.556552	−	1.34364i	−0.355927	−	0.934514i	\(-0.615835\pi\)
							0.912479	−	0.409123i	\(-0.134165\pi\)
\(29\)	−8.88530e7	+	3.68041e7i	−0.804420	+	0.333202i	−0.746725	−	0.665132i	\(-0.768375\pi\)
							−0.0576950	+	0.998334i	\(0.518375\pi\)
\(31\)	7.76143e6	+	1.87377e7i	0.0486914	+	0.117551i	0.946354	−	0.323133i	\(-0.104736\pi\)
							−0.897662	+	0.440684i	\(0.854736\pi\)
\(37\)	2.67818e8	+	6.46569e8i	0.634935	+	1.53287i	0.833346	+	0.552751i	\(0.186422\pi\)
							−0.198411	+	0.980119i	\(0.563578\pi\)
\(41\)	−9.55973e8	−	3.95977e8i	−1.28865	−	0.533775i	−0.370066	−	0.929006i	\(-0.620665\pi\)
							−0.918582	+	0.395230i	\(0.870665\pi\)
\(43\)	−5.11563e8	+	5.11563e8i	−0.530667	+	0.530667i	−0.920771	−	0.390104i	\(-0.872439\pi\)
							0.390104	+	0.920771i	\(0.372439\pi\)
\(47\)			9.09538e8i			0.578472i	0.957258	+	0.289236i	\(0.0934012\pi\)
							−0.957258	+	0.289236i	\(0.906599\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.12.d.a.15.8	yes	60
17.8	even	8	inner	17.12.d.a.8.8	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.12.d.a.8.8	✓	60	17.8	even	8	inner
17.12.d.a.15.8	yes	60	1.1	even	1	trivial