Embedded newform 103.4.c.a.46.13

• Label: 103.4.c.a.46.13
• Level: $103$
• Weight: $4$
• Character: 103.46
• Analytic conductor: $6.077$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	103.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.07719673059\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			46.13
Character	\(\chi\)	\(=\)	103.46
Dual form			103.4.c.a.56.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.189714 - 0.328594i) q^{2} -4.93781 q^{3} +(3.92802 - 6.80353i) q^{4} +(3.76703 - 6.52469i) q^{5} +(0.936770 + 1.62253i) q^{6} +(-6.95955 + 12.0543i) q^{7} -6.01622 q^{8} -2.61807 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - q^{2} + 4 q^{3} - 97 q^{4} - 3 q^{5} - 17 q^{6} + 13 q^{7} - 30 q^{8} + 534 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.189714	−	0.328594i	−0.0670739	−	0.116175i	0.830538	−	0.556962i	\(-0.188033\pi\)
							−0.897612	+	0.440786i	\(0.854700\pi\)
\(3\)	−4.93781			−0.950281			−0.475141	−	0.879910i	\(-0.657603\pi\)
							−0.475141	+	0.879910i	\(0.657603\pi\)
\(5\)	3.76703	−	6.52469i	0.336934	−	0.583586i	−0.646921	−	0.762557i	\(-0.723943\pi\)
							0.983854	+	0.178971i	\(0.0572768\pi\)
\(7\)	−6.95955	+	12.0543i	−0.375780	+	0.650871i	−0.990444	−	0.137919i	\(-0.955959\pi\)
							0.614663	+	0.788790i	\(0.289292\pi\)
\(11\)	9.43790	−	16.3469i	0.258694	−	0.448071i	−0.707198	−	0.707015i	\(-0.750041\pi\)
							0.965892	+	0.258944i	\(0.0833745\pi\)
\(13\)	−57.9896			−1.23719			−0.618593	−	0.785712i	\(-0.712297\pi\)
							−0.618593	+	0.785712i	\(0.712297\pi\)
\(17\)	−31.1604	+	53.9714i	−0.444559	+	0.770000i	−0.998021	−	0.0628749i	\(-0.979973\pi\)
							0.553462	+	0.832874i	\(0.313306\pi\)
\(19\)	−63.3700	−	109.760i	−0.765162	−	1.32530i	−0.940161	−	0.340731i	\(-0.889325\pi\)
							0.174999	−	0.984569i	\(-0.444008\pi\)
\(23\)	−170.244			−1.54340			−0.771702	−	0.635984i	\(-0.780595\pi\)
							−0.771702	+	0.635984i	\(0.780595\pi\)
\(29\)	−9.79939	−	16.9730i	−0.0627483	−	0.108683i	0.832945	−	0.553356i	\(-0.186653\pi\)
							−0.895693	+	0.444673i	\(0.853320\pi\)
\(31\)	217.746			1.26156			0.630780	−	0.775962i	\(-0.282735\pi\)
							0.630780	+	0.775962i	\(0.282735\pi\)
\(37\)	75.6637			0.336190			0.168095	−	0.985771i	\(-0.446238\pi\)
							0.168095	+	0.985771i	\(0.446238\pi\)
\(41\)	−104.051	−	180.221i	−0.396342	−	0.686484i	0.596930	−	0.802294i	\(-0.296387\pi\)
							−0.993271	+	0.115809i	\(0.963054\pi\)
\(43\)	−73.0985	−	126.610i	−0.259242	−	0.449021i	0.706797	−	0.707416i	\(-0.250139\pi\)
							−0.966039	+	0.258396i	\(0.916806\pi\)
\(47\)	221.078	−	382.919i	0.686118	−	1.18839i	−0.286965	−	0.957941i	\(-0.592646\pi\)
							0.973084	−	0.230451i	\(-0.0740202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.4.c.a.46.13	✓	50
103.56	even	3	inner	103.4.c.a.56.13	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.4.c.a.46.13	✓	50	1.1	even	1	trivial
103.4.c.a.56.13	yes	50	103.56	even	3	inner