Embedded newform 203.3.i.a.115.18

• Label: 203.3.i.a.115.18
• Level: $203$
• Weight: $3$
• Character: 203.115
• Analytic conductor: $5.531$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 203 = 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	203.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53134936651\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			115.18
Character	\(\chi\)	\(=\)	203.115
Dual form			203.3.i.a.173.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.345652 + 0.199562i) q^{2} +(2.20751 - 3.82351i) q^{3} +(-1.92035 + 3.32614i) q^{4} +(-1.32876 + 0.767159i) q^{5} +1.76214i q^{6} +(-6.23313 - 3.18561i) q^{7} -3.12941i q^{8} +(-5.24616 - 9.08662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 64 q^{4} - 6 q^{5} - 12 q^{7} - 116 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/203\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(176\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.345652	+	0.199562i	−0.172826	+	0.0997810i	−0.583918	−	0.811813i	\(-0.698481\pi\)
							0.411092	+	0.911594i	\(0.365148\pi\)
\(3\)	2.20751	−	3.82351i	0.735835	−	1.27450i	−0.218521	−	0.975832i	\(-0.570123\pi\)
							0.954356	−	0.298672i	\(-0.0965435\pi\)
\(5\)	−1.32876	+	0.767159i	−0.265752	+	0.153432i	−0.626955	−	0.779055i	\(-0.715699\pi\)
							0.361204	+	0.932487i	\(0.382366\pi\)
\(7\)	−6.23313	−	3.18561i	−0.890447	−	0.455087i
							\(11\)	−12.5609	−	7.25204i	−1.14190	−	0.659276i	−0.195000	−	0.980803i	\(-0.562471\pi\)
−0.946900	+	0.321527i	\(0.895804\pi\)
\(13\)		−	15.5879i		−	1.19907i	−0.800348	−	0.599536i	\(-0.795352\pi\)
							0.800348	−	0.599536i	\(-0.204648\pi\)
\(17\)	3.21883	−	5.57518i	0.189343	−	0.327952i	−0.755688	−	0.654931i	\(-0.772698\pi\)
							0.945031	+	0.326980i	\(0.106031\pi\)
\(19\)	3.09525	+	5.36113i	0.162908	+	0.282165i	0.935910	−	0.352238i	\(-0.114579\pi\)
							−0.773003	+	0.634403i	\(0.781246\pi\)
\(23\)	−0.204954	−	0.354991i	−0.00891104	−	0.0154344i	0.861535	−	0.507697i	\(-0.169503\pi\)
							−0.870447	+	0.492263i	\(0.836170\pi\)
\(29\)	−28.7533	−	3.77442i	−0.991494	−	0.130152i
							\(31\)	−14.7561	+	25.5583i	−0.476004	+	0.824462i	−0.999622	−	0.0274904i	\(-0.991248\pi\)
0.523618	+	0.851953i	\(0.324582\pi\)
\(37\)	52.1975	−	30.1362i	1.41074	−	0.814492i	0.415284	−	0.909692i	\(-0.363682\pi\)
							0.995458	+	0.0951993i	\(0.0303488\pi\)
\(41\)	18.7087			0.456309			0.228155	−	0.973625i	\(-0.426731\pi\)
							0.228155	+	0.973625i	\(0.426731\pi\)
\(43\)		−	59.4240i		−	1.38195i	−0.722878	−	0.690976i	\(-0.757181\pi\)
							0.722878	−	0.690976i	\(-0.242819\pi\)
\(47\)	3.74811	+	6.49191i	0.0797469	+	0.138126i	0.903141	−	0.429345i	\(-0.141255\pi\)
							−0.823394	+	0.567470i	\(0.807922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	203.3.i.a.115.18	✓	76
7.5	odd	6	inner	203.3.i.a.173.21	yes	76
29.28	even	2	inner	203.3.i.a.115.21	yes	76
203.173	odd	6	inner	203.3.i.a.173.18	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
203.3.i.a.115.18	✓	76	1.1	even	1	trivial
203.3.i.a.115.21	yes	76	29.28	even	2	inner
203.3.i.a.173.18	yes	76	203.173	odd	6	inner
203.3.i.a.173.21	yes	76	7.5	odd	6	inner