Embedded newform 103.2.g.a.16.6

• Label: 103.2.g.a.16.6
• Level: $103$
• Weight: $2$
• Character: 103.16
• 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			16.6
Character	\(\chi\)	\(=\)	103.16
Dual form			103.2.g.a.58.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.775703 - 0.195097i) q^{2} +(0.868708 + 1.15036i) q^{3} +(-1.19837 + 0.643512i) q^{4} +(-1.15470 + 1.34778i) q^{5} +(0.898291 + 0.722852i) q^{6} +(4.84843 - 1.54240i) q^{7} +(-1.98624 + 1.81070i) q^{8} +(0.252325 - 0.886832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{37}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.775703	−	0.195097i	0.548505	−	0.137955i	0.0402952	−	0.999188i	\(-0.487170\pi\)
							0.508210	+	0.861233i	\(0.330307\pi\)
\(3\)	0.868708	+	1.15036i	0.501549	+	0.664158i	0.976416	−	0.215900i	\(-0.0692685\pi\)
							−0.474867	+	0.880058i	\(0.657504\pi\)
\(5\)	−1.15470	+	1.34778i	−0.516400	+	0.602744i	−0.955244	−	0.295819i	\(-0.904408\pi\)
							0.438844	+	0.898563i	\(0.355388\pi\)
\(7\)	4.84843	−	1.54240i	1.83253	−	0.582971i	0.832974	−	0.553312i	\(-0.186636\pi\)
							0.999560	−	0.0296596i	\(-0.00944234\pi\)
\(11\)	−2.07240	−	2.13723i	−0.624852	−	0.644400i	0.329123	−	0.944287i	\(-0.393247\pi\)
							−0.953975	+	0.299887i	\(0.903051\pi\)
\(13\)	−3.35788	−	3.06111i	−0.931308	−	0.848999i	0.0574770	−	0.998347i	\(-0.481694\pi\)
							−0.988785	+	0.149348i	\(0.952283\pi\)
\(17\)	−1.17676	+	0.946939i	−0.285407	+	0.229666i	−0.759021	−	0.651066i	\(-0.774322\pi\)
							0.473613	+	0.880733i	\(0.342949\pi\)
\(19\)	0.00273750		0.000338976i	0.000628026		7.77665e-5i	0.123202	−	0.992382i	\(-0.460684\pi\)
							−0.122574	+	0.992459i	\(0.539115\pi\)
\(23\)	1.69364	+	5.95254i	0.353149	+	1.24119i	0.911401	+	0.411518i	\(0.135001\pi\)
							−0.558253	+	0.829671i	\(0.688528\pi\)
\(29\)	−0.968579	−	2.74863i	−0.179861	−	0.510407i	0.818041	−	0.575160i	\(-0.195060\pi\)
							−0.997901	+	0.0647530i	\(0.979374\pi\)
\(31\)	−3.29443	−	6.61611i	−0.591698	−	1.18829i	−0.965408	−	0.260744i	\(-0.916032\pi\)
							0.373711	−	0.927545i	\(-0.378085\pi\)
\(37\)	0.425709	+	4.59413i	0.0699861	+	0.755271i	0.956339	+	0.292260i	\(0.0944073\pi\)
							−0.886353	+	0.463011i	\(0.846769\pi\)
\(41\)	2.43346	+	2.84035i	0.380043	+	0.443588i	0.916316	−	0.400457i	\(-0.131149\pi\)
							−0.536273	+	0.844045i	\(0.680168\pi\)
\(43\)	−8.47306	+	3.89823i	−1.29213	+	0.594474i	−0.939903	−	0.341441i	\(-0.889085\pi\)
							−0.352226	+	0.935915i	\(0.614575\pi\)
\(47\)	−2.61955	+	4.53719i	−0.382100	+	0.661817i	−0.991362	−	0.131152i	\(-0.958133\pi\)
							0.609262	+	0.792969i	\(0.291466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.16.6	✓	256
3.2	odd	2		927.2.ba.c.325.3		256
103.58	even	51	inner	103.2.g.a.58.6	yes	256
309.161	odd	102		927.2.ba.c.676.3		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.16.6	✓	256	1.1	even	1	trivial
103.2.g.a.58.6	yes	256	103.58	even	51	inner
927.2.ba.c.325.3		256	3.2	odd	2
927.2.ba.c.676.3		256	309.161	odd	102