Embedded newform 123.3.h.a.55.2

• Label: 123.3.h.a.55.2
• Level: $123$
• Weight: $3$
• Character: 123.55
• Analytic conductor: $3.352$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	123.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35150725163\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			55.2
Character	\(\chi\)	\(=\)	123.55
Dual form			123.3.h.a.85.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.07159 - 2.07159i) q^{2} +(-0.662827 - 1.60021i) q^{3} +4.58295i q^{4} +(-2.20300 + 2.20300i) q^{5} +(-1.94186 + 4.68807i) q^{6} +(2.89751 + 6.99522i) q^{7} +(1.20764 - 1.20764i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{2} - 48 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(88\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.07159	−	2.07159i	−1.03579	−	1.03579i	−0.999335	−	0.0364588i	\(-0.988392\pi\)
							−0.0364588	−	0.999335i	\(-0.511608\pi\)
\(3\)	−0.662827	−	1.60021i	−0.220942	−	0.533402i
							\(5\)	−2.20300	+	2.20300i	−0.440600	+	0.440600i	−0.892214	−	0.451613i	\(-0.850849\pi\)
0.451613	+	0.892214i	\(0.350849\pi\)
\(7\)	2.89751	+	6.99522i	0.413931	+	0.999317i	0.984072	+	0.177770i	\(0.0568884\pi\)
							−0.570141	+	0.821547i	\(0.693112\pi\)
\(11\)	−3.88003	+	1.60716i	−0.352730	+	0.146106i	−0.552011	−	0.833837i	\(-0.686139\pi\)
							0.199281	+	0.979942i	\(0.436139\pi\)
\(13\)	3.06397	+	7.39708i	0.235690	+	0.569006i	0.996828	−	0.0795836i	\(-0.0253591\pi\)
							−0.761138	+	0.648590i	\(0.775359\pi\)
\(17\)	2.04988	−	4.94886i	0.120581	−	0.291109i	−0.852051	−	0.523459i	\(-0.824641\pi\)
							0.972632	+	0.232350i	\(0.0746413\pi\)
\(19\)	−0.822429	+	1.98552i	−0.0432857	+	0.104501i	−0.944044	−	0.329820i	\(-0.893012\pi\)
							0.900758	+	0.434321i	\(0.143012\pi\)
\(23\)			0.809595i			0.0351998i	0.999845	+	0.0175999i	\(0.00560251\pi\)
							−0.999845	+	0.0175999i	\(0.994397\pi\)
\(29\)	12.5895	+	30.3938i	0.434121	+	1.04806i	0.977945	+	0.208862i	\(0.0669758\pi\)
							−0.543824	+	0.839199i	\(0.683024\pi\)
\(31\)			24.8986i			0.803180i	0.915819	+	0.401590i	\(0.131542\pi\)
							−0.915819	+	0.401590i	\(0.868458\pi\)
\(37\)	−48.9406			−1.32272			−0.661360	−	0.750069i	\(-0.730020\pi\)
							−0.661360	+	0.750069i	\(0.730020\pi\)
\(41\)	−30.3141	+	27.6054i	−0.739367	+	0.673302i
							\(43\)	−12.7965	−	12.7965i	−0.297594	−	0.297594i	0.542477	−	0.840071i	\(-0.317487\pi\)
−0.840071	+	0.542477i	\(0.817487\pi\)
\(47\)	−20.7103	+	49.9990i	−0.440644	+	1.06381i	0.535080	+	0.844802i	\(0.320282\pi\)
							−0.975723	+	0.219006i	\(0.929718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.3.h.a.55.2	✓	56
3.2	odd	2		369.3.l.c.55.13		56
41.3	odd	8	inner	123.3.h.a.85.2	yes	56
123.44	even	8		369.3.l.c.208.13		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.3.h.a.55.2	✓	56	1.1	even	1	trivial
123.3.h.a.85.2	yes	56	41.3	odd	8	inner
369.3.l.c.55.13		56	3.2	odd	2
369.3.l.c.208.13		56	123.44	even	8