Embedded newform 103.2.g.a.15.4

• Label: 103.2.g.a.15.4
• Level: $103$
• Weight: $2$
• Character: 103.15
• Analytic conductor: $0.822$
• Analytic rank: $0$
• Dimension: $256$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	103.g (of order \(51\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.822459140819\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(8\) over \(\Q(\zeta_{51})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{51}]$

Embedding invariants

Embedding label			15.4
Character	\(\chi\)	\(=\)	103.15
Dual form			103.2.g.a.55.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.136937 - 0.141221i) q^{2} +(1.95152 + 2.58423i) q^{3} +(0.0603985 - 1.96037i) q^{4} +(-0.282653 - 0.802111i) q^{5} +(0.0977118 - 0.629470i) q^{6} +(0.255911 - 1.16852i) q^{7} +(-0.575857 + 0.524963i) q^{8} +(-2.04882 + 7.20086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{20}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.136937	−	0.141221i	−0.0968288	−	0.0998580i	0.667759	−	0.744377i	\(-0.267254\pi\)
							−0.764588	+	0.644519i	\(0.777058\pi\)
\(3\)	1.95152	+	2.58423i	1.12671	+	1.49200i	0.846748	+	0.531993i	\(0.178557\pi\)
							0.279961	+	0.960011i	\(0.409678\pi\)
\(5\)	−0.282653	−	0.802111i	−0.126406	−	0.358715i	0.862740	−	0.505647i	\(-0.168746\pi\)
							−0.989147	+	0.146932i	\(0.953060\pi\)
\(7\)	0.255911	−	1.16852i	0.0967252	−	0.441661i	−0.903197	−	0.429225i	\(-0.858787\pi\)
							0.999923	−	0.0124355i	\(-0.00395845\pi\)
\(11\)	−4.29900	+	1.08124i	−1.29620	+	0.326007i	−0.829548	−	0.558435i	\(-0.811402\pi\)
							−0.466650	+	0.884442i	\(0.654539\pi\)
\(13\)	1.87819	+	1.71219i	0.520915	+	0.474877i	0.890925	−	0.454151i	\(-0.150057\pi\)
							−0.370009	+	0.929028i	\(0.620646\pi\)
\(17\)	−1.00408	−	6.46842i	−0.243526	−	1.56882i	−0.722737	−	0.691123i	\(-0.757116\pi\)
							0.479211	−	0.877700i	\(-0.340923\pi\)
\(19\)	−1.34465	−	3.17687i	−0.308484	−	0.728823i	−0.999979	−	0.00644181i	\(-0.997949\pi\)
							0.691495	−	0.722381i	\(-0.256952\pi\)
\(23\)	1.79640	+	6.31368i	0.374574	+	1.31649i	0.888586	+	0.458710i	\(0.151688\pi\)
							−0.514012	+	0.857783i	\(0.671841\pi\)
\(29\)	−0.764211	+	0.891991i	−0.141910	+	0.165639i	−0.825831	−	0.563918i	\(-0.809293\pi\)
							0.683920	+	0.729557i	\(0.260274\pi\)
\(31\)	−0.0100166	−	0.0201161i	−0.00179904	−	0.00361296i	0.894262	−	0.447544i	\(-0.147701\pi\)
							−0.896061	+	0.443931i	\(0.853584\pi\)
\(37\)	−0.537432	−	5.79982i	−0.0883533	−	0.953484i	−0.918364	−	0.395737i	\(-0.870489\pi\)
							0.830011	−	0.557747i	\(-0.188334\pi\)
\(41\)	−0.750181	+	2.12886i	−0.117159	+	0.332472i	−0.986968	−	0.160916i	\(-0.948555\pi\)
							0.869810	+	0.493388i	\(0.164242\pi\)
\(43\)	2.58621	+	1.83074i	0.394394	+	0.279186i	0.757854	−	0.652424i	\(-0.226248\pi\)
							−0.363460	+	0.931610i	\(0.618405\pi\)
\(47\)	3.98742	+	6.90641i	0.581624	+	1.00740i	0.995287	+	0.0969727i	\(0.0309160\pi\)
							−0.413663	+	0.910430i	\(0.635751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.15.4	✓	256
3.2	odd	2		927.2.ba.c.118.5		256
103.55	even	51	inner	103.2.g.a.55.4	yes	256
309.158	odd	102		927.2.ba.c.55.5		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.15.4	✓	256	1.1	even	1	trivial
103.2.g.a.55.4	yes	256	103.55	even	51	inner
927.2.ba.c.55.5		256	309.158	odd	102
927.2.ba.c.118.5		256	3.2	odd	2