Embedded newform 103.2.g.a.16.2

• Label: 103.2.g.a.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	103.16
Dual form			103.2.g.a.58.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.06844 + 0.520233i) q^{2} +(-0.732157 - 0.969533i) q^{3} +(2.24577 - 1.20595i) q^{4} +(-1.66156 + 1.93938i) q^{5} +(2.01880 + 1.62453i) q^{6} +(4.05899 - 1.29126i) q^{7} +(-0.865454 + 0.788965i) q^{8} +(0.417049 - 1.46578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06844	+	0.520233i	−1.46261	+	0.367860i	−0.891502	−	0.453017i	\(-0.850348\pi\)
							−0.571104	+	0.820877i	\(0.693485\pi\)
\(3\)	−0.732157	−	0.969533i	−0.422711	−	0.559760i	0.535972	−	0.844236i	\(-0.319945\pi\)
							−0.958683	+	0.284476i	\(0.908181\pi\)
\(5\)	−1.66156	+	1.93938i	−0.743072	+	0.867317i	−0.994943	−	0.100444i	\(-0.967974\pi\)
							0.251871	+	0.967761i	\(0.418954\pi\)
\(7\)	4.05899	−	1.29126i	1.53415	−	0.488050i	0.587206	−	0.809437i	\(-0.300228\pi\)
							0.946948	+	0.321388i	\(0.104149\pi\)
\(11\)	3.81404	+	3.93335i	1.14997	+	1.18595i	0.979682	+	0.200558i	\(0.0642754\pi\)
							0.170293	+	0.985393i	\(0.445529\pi\)
\(13\)	0.792749	+	0.722687i	0.219869	+	0.200437i	0.776134	−	0.630568i	\(-0.217178\pi\)
							−0.556265	+	0.831005i	\(0.687766\pi\)
\(17\)	3.98583	−	3.20739i	0.966706	−	0.777905i	−0.00832723	−	0.999965i	\(-0.502651\pi\)
							0.975033	+	0.222060i	\(0.0712781\pi\)
\(19\)	−2.03728	−	0.252271i	−0.467385	−	0.0578748i	−0.114260	−	0.993451i	\(-0.536450\pi\)
							−0.353125	+	0.935576i	\(0.614881\pi\)
\(23\)	−1.49788	−	5.26452i	−0.312331	−	1.09773i	−0.946011	−	0.324133i	\(-0.894927\pi\)
							0.633681	−	0.773595i	\(-0.281543\pi\)
\(29\)	0.152426	+	0.432554i	0.0283048	+	0.0803232i	0.956451	−	0.291892i	\(-0.0942847\pi\)
							−0.928147	+	0.372215i	\(0.878598\pi\)
\(31\)	3.04581	+	6.11680i	0.547043	+	1.09861i	0.980332	+	0.197355i	\(0.0632352\pi\)
							−0.433289	+	0.901255i	\(0.642647\pi\)
\(37\)	−0.273739	−	2.95411i	−0.0450023	−	0.485653i	−0.988173	−	0.153344i	\(-0.950996\pi\)
							0.943171	−	0.332309i	\(-0.107828\pi\)
\(41\)	1.60335	+	1.87144i	0.250402	+	0.292270i	0.870412	−	0.492323i	\(-0.163852\pi\)
							−0.620011	+	0.784593i	\(0.712872\pi\)
\(43\)	5.43156	−	2.49891i	0.828305	−	0.381081i	0.0421511	−	0.999111i	\(-0.486579\pi\)
							0.786154	+	0.618030i	\(0.212069\pi\)
\(47\)	−5.50228	+	9.53023i	−0.802590	+	1.39013i	0.115316	+	0.993329i	\(0.463212\pi\)
							−0.917906	+	0.396798i	\(0.870121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.16.2	✓	256
3.2	odd	2		927.2.ba.c.325.7		256
103.58	even	51	inner	103.2.g.a.58.2	yes	256
309.161	odd	102		927.2.ba.c.676.7		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.16.2	✓	256	1.1	even	1	trivial
103.2.g.a.58.2	yes	256	103.58	even	51	inner
927.2.ba.c.325.7		256	3.2	odd	2
927.2.ba.c.676.7		256	309.161	odd	102