Embedded newform 103.2.g.a.15.8

• Label: 103.2.g.a.15.8
• 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			15.8
Character	\(\chi\)	\(=\)	103.15
Dual form			103.2.g.a.55.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.86905 + 1.92752i) q^{2} +(-0.109141 - 0.144526i) q^{3} +(-0.160402 + 5.20622i) q^{4} +(-1.21909 - 3.45952i) q^{5} +(0.0745869 - 0.480497i) q^{6} +(0.0557294 - 0.254468i) q^{7} +(-6.36657 + 5.80389i) q^{8} +(0.812013 - 2.85393i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.86905	+	1.92752i	1.32162	+	1.36296i	0.886597	+	0.462544i	\(0.153063\pi\)
							0.435021	+	0.900420i	\(0.356741\pi\)
\(3\)	−0.109141	−	0.144526i	−0.0630124	−	0.0834419i	0.765420	−	0.643531i	\(-0.222531\pi\)
							−0.828432	+	0.560090i	\(0.810767\pi\)
\(5\)	−1.21909	−	3.45952i	−0.545193	−	1.54714i	−0.812158	−	0.583438i	\(-0.801707\pi\)
							0.266965	−	0.963706i	\(-0.413979\pi\)
\(7\)	0.0557294	−	0.254468i	0.0210637	−	0.0961799i	−0.965132	−	0.261764i	\(-0.915696\pi\)
							0.986196	+	0.165584i	\(0.0529508\pi\)
\(11\)	−4.53697	+	1.14109i	−1.36795	+	0.344052i	−0.856892	−	0.515496i	\(-0.827608\pi\)
							−0.511055	+	0.859548i	\(0.670745\pi\)
\(13\)	3.97336	+	3.62220i	1.10201	+	1.00462i	0.999952	+	0.00983867i	\(0.00313180\pi\)
							0.102061	+	0.994778i	\(0.467456\pi\)
\(17\)	−0.175897	−	1.13315i	−0.0426614	−	0.274829i	0.957234	−	0.289316i	\(-0.0934278\pi\)
							−0.999895	+	0.0144868i	\(0.995389\pi\)
\(19\)	0.0976643	+	0.230741i	0.0224057	+	0.0529356i	0.931728	−	0.363157i	\(-0.118301\pi\)
							−0.909322	+	0.416093i	\(0.863399\pi\)
\(23\)	0.861358	+	3.02736i	0.179605	+	0.631248i	0.998368	+	0.0571149i	\(0.0181901\pi\)
							−0.818762	+	0.574133i	\(0.805339\pi\)
\(29\)	1.10029	−	1.28427i	0.204319	−	0.238482i	−0.647825	−	0.761789i	\(-0.724321\pi\)
							0.852144	+	0.523307i	\(0.175302\pi\)
\(31\)	−0.814005	−	1.63474i	−0.146200	−	0.293609i	0.809944	−	0.586508i	\(-0.199498\pi\)
							−0.956143	+	0.292899i	\(0.905380\pi\)
\(37\)	0.00226159	+	0.0244064i	0.000371802	+	0.00401239i	0.995918	−	0.0902620i	\(-0.0287704\pi\)
							−0.995546	+	0.0942744i	\(0.969947\pi\)
\(41\)	1.65289	−	4.69057i	0.258138	−	0.732543i	−0.740060	−	0.672541i	\(-0.765203\pi\)
							0.998198	−	0.0600024i	\(-0.0191108\pi\)
\(43\)	−1.28190	−	0.907441i	−0.195488	−	0.138383i	0.475872	−	0.879515i	\(-0.342133\pi\)
							−0.671360	+	0.741131i	\(0.734290\pi\)
\(47\)	−1.39370	−	2.41395i	−0.203292	−	0.352111i	0.746295	−	0.665615i	\(-0.231831\pi\)
							−0.949587	+	0.313503i	\(0.898497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.15.8	✓	256
3.2	odd	2		927.2.ba.c.118.1		256
103.55	even	51	inner	103.2.g.a.55.8	yes	256
309.158	odd	102		927.2.ba.c.55.1		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.15.8	✓	256	1.1	even	1	trivial
103.2.g.a.55.8	yes	256	103.55	even	51	inner
927.2.ba.c.55.1		256	309.158	odd	102
927.2.ba.c.118.1		256	3.2	odd	2