Embedded newform 103.2.g.a.17.4

• Label: 103.2.g.a.17.4
• Level: $103$
• Weight: $2$
• Character: 103.17
• 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			17.4
Character	\(\chi\)	\(=\)	103.17
Dual form			103.2.g.a.97.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.312258 - 0.221043i) q^{2} +(-1.81120 + 1.65113i) q^{3} +(-0.616064 - 1.74826i) q^{4} +(-1.15484 + 1.74283i) q^{5} +(0.930533 - 0.115225i) q^{6} +(-2.29789 + 2.36978i) q^{7} +(-0.403465 + 1.41803i) q^{8} +(0.277422 - 2.99386i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{35}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.312258	−	0.221043i	−0.220800	−	0.156301i	0.462064	−	0.886847i	\(-0.347109\pi\)
							−0.682864	+	0.730546i	\(0.739266\pi\)
\(3\)	−1.81120	+	1.65113i	−1.04570	+	0.953280i	−0.998959	−	0.0456270i	\(-0.985471\pi\)
							−0.0467399	+	0.998907i	\(0.514883\pi\)
\(5\)	−1.15484	+	1.74283i	−0.516462	+	0.779419i	−0.995129	−	0.0985774i	\(-0.968571\pi\)
							0.478668	+	0.877996i	\(0.341120\pi\)
\(7\)	−2.29789	+	2.36978i	−0.868521	+	0.895692i	−0.995359	−	0.0962319i	\(-0.969321\pi\)
							0.126838	+	0.991923i	\(0.459517\pi\)
\(11\)	−2.43918	+	1.12220i	−0.735442	+	0.338357i	−0.749911	−	0.661538i	\(-0.769904\pi\)
							0.0144696	+	0.999895i	\(0.495394\pi\)
\(13\)	−0.829830	−	2.91655i	−0.230153	−	0.808905i	−0.987732	−	0.156161i	\(-0.950088\pi\)
							0.757578	−	0.652744i	\(-0.226382\pi\)
\(17\)	4.92719	+	0.610119i	1.19502	+	0.147975i	0.695629	−	0.718401i	\(-0.255126\pi\)
							0.499391	+	0.866377i	\(0.333557\pi\)
\(19\)	5.05333	−	1.60758i	1.15931	−	0.368804i	0.339038	−	0.940773i	\(-0.389899\pi\)
							0.820276	+	0.571969i	\(0.193820\pi\)
\(23\)	0.694236	+	7.49199i	0.144758	+	1.56219i	0.689937	+	0.723869i	\(0.257638\pi\)
							−0.545179	+	0.838320i	\(0.683538\pi\)
\(29\)	−3.52326	−	0.217307i	−0.654253	−	0.0403529i	−0.268966	−	0.963150i	\(-0.586682\pi\)
							−0.385287	+	0.922797i	\(0.625898\pi\)
\(31\)	−8.37076	−	3.24285i	−1.50343	−	0.582433i	−0.537967	−	0.842966i	\(-0.680808\pi\)
							−0.965465	+	0.260533i	\(0.916102\pi\)
\(37\)	−5.13498	+	3.17945i	−0.844186	+	0.522698i	−0.879070	−	0.476694i	\(-0.841835\pi\)
							0.0348836	+	0.999391i	\(0.488894\pi\)
\(41\)	0.662194	+	0.999352i	0.103417	+	0.156073i	0.881650	−	0.471903i	\(-0.156433\pi\)
							−0.778233	+	0.627976i	\(0.783884\pi\)
\(43\)	−4.23855	+	2.27605i	−0.646372	+	0.347095i	−0.763298	−	0.646046i	\(-0.776421\pi\)
							0.116926	+	0.993141i	\(0.462696\pi\)
\(47\)	6.24098	+	10.8097i	0.910341	+	1.57676i	0.813583	+	0.581448i	\(0.197514\pi\)
							0.0967575	+	0.995308i	\(0.469153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.17.4	✓	256
3.2	odd	2		927.2.ba.c.532.5		256
103.97	even	51	inner	103.2.g.a.97.4	yes	256
309.200	odd	102		927.2.ba.c.406.5		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.17.4	✓	256	1.1	even	1	trivial
103.2.g.a.97.4	yes	256	103.97	even	51	inner
927.2.ba.c.406.5		256	309.200	odd	102
927.2.ba.c.532.5		256	3.2	odd	2