Embedded newform 147.4.o.a.101.7

• Label: 147.4.o.a.101.7
• Level: $147$
• Weight: $4$
• Character: 147.101
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.7
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.20199 - 1.64916i) q^{2} +(-2.59318 + 4.50283i) q^{3} +(9.07260 + 8.41814i) q^{4} +(15.6024 + 10.6376i) q^{5} +(18.3224 - 14.6443i) q^{6} +(16.0425 - 9.25417i) q^{7} +(-8.57160 - 17.7991i) q^{8} +(-13.5509 - 23.3532i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.20199	−	1.64916i	−1.48563	−	0.583066i	−0.522627	−	0.852561i	\(-0.675048\pi\)
							−0.963002	+	0.269495i	\(0.913143\pi\)
\(3\)	−2.59318	+	4.50283i	−0.499057	+	0.866569i
							\(5\)	15.6024	+	10.6376i	1.39552	+	0.951452i	0.999448	+	0.0332347i	\(0.0105809\pi\)
0.396077	+	0.918218i	\(0.370371\pi\)
\(7\)	16.0425	−	9.25417i	0.866211	−	0.499678i
							\(11\)	4.73214	+	31.3957i	0.129709	+	0.860560i	0.955161	+	0.296088i	\(0.0956822\pi\)
−0.825452	+	0.564472i	\(0.809080\pi\)
\(13\)	13.7855	−	10.9936i	0.294109	−	0.234544i	−0.465307	−	0.885149i	\(-0.654056\pi\)
							0.759416	+	0.650605i	\(0.225485\pi\)
\(17\)	50.1250	+	15.4615i	0.715124	+	0.220586i	0.630904	−	0.775861i	\(-0.282684\pi\)
							0.0842195	+	0.996447i	\(0.473160\pi\)
\(19\)	87.2935	−	50.3989i	1.05403	−	0.608542i	0.130252	−	0.991481i	\(-0.458421\pi\)
							0.923774	+	0.382939i	\(0.125088\pi\)
\(23\)	−35.9587	−	116.575i	−0.325996	−	1.05685i	−0.959167	−	0.282842i	\(-0.908723\pi\)
							0.633170	−	0.774012i	\(-0.281753\pi\)
\(29\)	5.99707	−	1.36879i	0.0384009	−	0.00876476i	−0.203277	−	0.979121i	\(-0.565159\pi\)
							0.241678	+	0.970356i	\(0.422302\pi\)
\(31\)	95.2515	+	54.9935i	0.551861	+	0.318617i	0.749872	−	0.661583i	\(-0.230115\pi\)
							−0.198012	+	0.980200i	\(0.563448\pi\)
\(37\)	−296.879	+	275.464i	−1.31910	+	1.22395i	−0.362692	+	0.931909i	\(0.618142\pi\)
							−0.956407	+	0.292037i	\(0.905667\pi\)
\(41\)	44.3470	−	21.3564i	0.168923	−	0.0813490i	−0.347512	−	0.937676i	\(-0.612973\pi\)
							0.516435	+	0.856327i	\(0.327259\pi\)
\(43\)	452.662	+	217.991i	1.60536	+	0.773099i	0.999743	−	0.0226597i	\(-0.00721342\pi\)
							0.605614	+	0.795759i	\(0.292928\pi\)
\(47\)	−120.772	+	307.721i	−0.374816	+	0.955016i	0.611661	+	0.791120i	\(0.290502\pi\)
							−0.986478	+	0.163896i	\(0.947594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.7	✓	648
3.2	odd	2	inner	147.4.o.a.101.48	yes	648
49.33	odd	42	inner	147.4.o.a.131.48	yes	648
147.131	even	42	inner	147.4.o.a.131.7	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.7	✓	648	1.1	even	1	trivial
147.4.o.a.101.48	yes	648	3.2	odd	2	inner
147.4.o.a.131.7	yes	648	147.131	even	42	inner
147.4.o.a.131.48	yes	648	49.33	odd	42	inner