Embedded newform 103.2.e.a.14.6

• Label: 103.2.e.a.14.6
• Level: $103$
• Weight: $2$
• Character: 103.14
• Analytic conductor: $0.822$
• Analytic rank: $0$
• Dimension: $112$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	103.e (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.822459140819\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(7\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			14.6
Character	\(\chi\)	\(=\)	103.14
Dual form			103.2.e.a.81.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.906203 - 1.20001i) q^{2} +(0.748937 + 1.50407i) q^{3} +(-0.0714856 - 0.251246i) q^{4} +(-0.199214 - 2.14986i) q^{5} +(2.48358 + 0.464261i) q^{6} +(-3.19253 + 1.23679i) q^{7} +(2.43810 + 0.944527i) q^{8} +(0.106590 - 0.141148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{8}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.906203	−	1.20001i	0.640782	−	0.848532i	−0.355501	−	0.934676i	\(-0.615690\pi\)
							0.996283	+	0.0861435i	\(0.0274544\pi\)
\(3\)	0.748937	+	1.50407i	0.432399	+	0.868374i	0.999014	+	0.0443916i	\(0.0141349\pi\)
							−0.566615	+	0.823982i	\(0.691747\pi\)
\(5\)	−0.199214	−	2.14986i	−0.0890912	−	0.961448i	−0.916550	−	0.399921i	\(-0.869038\pi\)
							0.827459	−	0.561527i	\(-0.189786\pi\)
\(7\)	−3.19253	+	1.23679i	−1.20666	+	0.467464i	−0.878768	−	0.477250i	\(-0.841634\pi\)
							−0.327896	+	0.944714i	\(0.606339\pi\)
\(11\)	1.56769	−	2.07596i	0.472676	−	0.625925i	−0.497750	−	0.867320i	\(-0.665840\pi\)
							0.970427	+	0.241396i	\(0.0776051\pi\)
\(13\)	−6.11077	+	2.36732i	−1.69482	+	0.656577i	−0.998149	−	0.0608124i	\(-0.980631\pi\)
							−0.696672	+	0.717390i	\(0.745337\pi\)
\(17\)	−6.23637	+	1.16578i	−1.51254	+	0.282743i	−0.873650	−	0.486555i	\(-0.838253\pi\)
							−0.638891	+	0.769298i	\(0.720606\pi\)
\(19\)	1.20029	−	2.41050i	0.275364	−	0.553006i	−0.713707	−	0.700445i	\(-0.752985\pi\)
							0.989071	+	0.147438i	\(0.0471028\pi\)
\(23\)	0.369025	+	0.488669i	0.0769471	+	0.101894i	0.834863	−	0.550458i	\(-0.185547\pi\)
							−0.757916	+	0.652352i	\(0.773782\pi\)
\(29\)	0.693385	+	7.48281i	0.128758	+	1.38952i	0.777676	+	0.628665i	\(0.216398\pi\)
							−0.648918	+	0.760858i	\(0.724778\pi\)
\(31\)	2.82850	−	1.75133i	0.508014	−	0.314549i	−0.248383	−	0.968662i	\(-0.579899\pi\)
							0.756397	+	0.654113i	\(0.226958\pi\)
\(37\)	−0.372895	+	0.339938i	−0.0613035	+	0.0558855i	−0.703768	−	0.710430i	\(-0.748500\pi\)
							0.642464	+	0.766316i	\(0.277912\pi\)
\(41\)	0.956734	−	10.3248i	0.149417	−	1.61246i	−0.509651	−	0.860381i	\(-0.670226\pi\)
							0.659068	−	0.752083i	\(-0.270951\pi\)
\(43\)	2.78098	+	2.53520i	0.424096	+	0.386615i	0.857415	−	0.514626i	\(-0.172069\pi\)
							−0.433319	+	0.901241i	\(0.642657\pi\)
\(47\)	5.46631			0.797343			0.398671	−	0.917094i	\(-0.369471\pi\)
							0.398671	+	0.917094i	\(0.369471\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.14.6	✓	112
3.2	odd	2		927.2.u.a.838.2		112
103.81	even	17	inner	103.2.e.a.81.6	yes	112
309.287	odd	34		927.2.u.a.802.2		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.14.6	✓	112	1.1	even	1	trivial
103.2.e.a.81.6	yes	112	103.81	even	17	inner
927.2.u.a.802.2		112	309.287	odd	34
927.2.u.a.838.2		112	3.2	odd	2