Embedded newform 103.2.g.a.15.2

• Label: 103.2.g.a.15.2
• 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			15.2
Character	\(\chi\)	\(=\)	103.15
Dual form			103.2.g.a.55.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25145 - 1.29060i) q^{2} +(0.904384 + 1.19760i) q^{3} +(-0.0379322 + 1.23118i) q^{4} +(1.26417 + 3.58746i) q^{5} +(0.413828 - 2.66593i) q^{6} +(-0.234585 + 1.07115i) q^{7} +(-1.02062 + 0.930421i) q^{8} +(0.204658 - 0.719299i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25145	−	1.29060i	−0.884907	−	0.912590i	0.111796	−	0.993731i	\(-0.464340\pi\)
							−0.996703	+	0.0811410i	\(0.974144\pi\)
\(3\)	0.904384	+	1.19760i	0.522146	+	0.691434i	0.980317	−	0.197431i	\(-0.0632598\pi\)
							−0.458170	+	0.888864i	\(0.651495\pi\)
\(5\)	1.26417	+	3.58746i	0.565355	+	1.60436i	0.778709	+	0.627385i	\(0.215875\pi\)
							−0.213354	+	0.976975i	\(0.568439\pi\)
\(7\)	−0.234585	+	1.07115i	−0.0886648	+	0.404856i	−0.999976	−	0.00686142i	\(-0.997816\pi\)
							0.911312	+	0.411717i	\(0.135071\pi\)
\(11\)	3.76059	−	0.945825i	1.13386	−	0.285177i	0.369035	−	0.929416i	\(-0.379688\pi\)
							0.764825	+	0.644239i	\(0.222826\pi\)
\(13\)	−2.96800	−	2.70569i	−0.823174	−	0.750422i	0.147951	−	0.988995i	\(-0.452732\pi\)
							−0.971125	+	0.238572i	\(0.923321\pi\)
\(17\)	−0.113461	−	0.730928i	−0.0275183	−	0.177276i	0.970423	−	0.241411i	\(-0.0776103\pi\)
							−0.997941	+	0.0641353i	\(0.979571\pi\)
\(19\)	−1.73447	−	4.09784i	−0.397914	−	0.940108i	−0.990863	−	0.134869i	\(-0.956939\pi\)
							0.592950	−	0.805239i	\(-0.297963\pi\)
\(23\)	0.958121	+	3.36745i	0.199782	+	0.702161i	0.995352	+	0.0962998i	\(0.0307008\pi\)
							−0.795570	+	0.605861i	\(0.792829\pi\)
\(29\)	−4.54965	+	5.31037i	−0.844849	+	0.986111i	−0.999998	−	0.00177954i	\(-0.999434\pi\)
							0.155150	+	0.987891i	\(0.450414\pi\)
\(31\)	−1.93757	−	3.89117i	−0.347998	−	0.698875i	0.650143	−	0.759812i	\(-0.274709\pi\)
							−0.998142	+	0.0609369i	\(0.980591\pi\)
\(37\)	−0.817186	−	8.81884i	−0.134345	−	1.44981i	−0.749814	−	0.661648i	\(-0.769857\pi\)
							0.615470	−	0.788161i	\(-0.288966\pi\)
\(41\)	3.24723	−	9.21498i	0.507133	−	1.43914i	−0.356155	−	0.934427i	\(-0.615913\pi\)
							0.863288	−	0.504711i	\(-0.168401\pi\)
\(43\)	5.01261	+	3.54835i	0.764416	+	0.541119i	0.892718	−	0.450616i	\(-0.148796\pi\)
							−0.128302	+	0.991735i	\(0.540953\pi\)
\(47\)	1.65123	+	2.86001i	0.240856	+	0.417175i	0.960958	−	0.276693i	\(-0.0892384\pi\)
							−0.720102	+	0.693868i	\(0.755905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.15.2	✓	256
3.2	odd	2		927.2.ba.c.118.7		256
103.55	even	51	inner	103.2.g.a.55.2	yes	256
309.158	odd	102		927.2.ba.c.55.7		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.15.2	✓	256	1.1	even	1	trivial
103.2.g.a.55.2	yes	256	103.55	even	51	inner
927.2.ba.c.55.7		256	309.158	odd	102
927.2.ba.c.118.7		256	3.2	odd	2