Embedded newform 273.2.ba.c.38.20

• Label: 273.2.ba.c.38.20
• Level: $273$
• Weight: $2$
• Character: 273.38
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.20
Character	\(\chi\)	\(=\)	273.38
Dual form			273.2.ba.c.194.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.387119 + 0.670510i) q^{2} +(0.557417 - 1.63990i) q^{3} +(0.700278 - 1.21292i) q^{4} +(1.11726 - 0.645050i) q^{5} +(1.31536 - 0.261085i) q^{6} +(-0.867295 + 2.49956i) q^{7} +2.63284 q^{8} +(-2.37857 - 1.82822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 12 q^{3} - 40 q^{4} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.387119	+	0.670510i	0.273735	+	0.474122i	0.969815	−	0.243842i	\(-0.0784077\pi\)
							−0.696081	+	0.717964i	\(0.745074\pi\)
\(3\)	0.557417	−	1.63990i	0.321825	−	0.946799i
							\(5\)	1.11726	−	0.645050i	0.499653	−	0.288475i	−0.228917	−	0.973446i	\(-0.573518\pi\)
0.728570	+	0.684971i	\(0.240185\pi\)
\(7\)	−0.867295	+	2.49956i	−0.327807	+	0.944745i
							\(11\)	0.781451	−	1.35351i	0.235616	−	0.408100i	−0.723835	−	0.689973i	\(-0.757622\pi\)
0.959452	+	0.281873i	\(0.0909558\pi\)
\(13\)	−2.18348	+	2.86922i	−0.605589	+	0.795778i
							\(17\)	2.00435	−	3.47163i	0.486126	−	0.841995i	−0.513747	−	0.857942i	\(-0.671743\pi\)
0.999873	+	0.0159468i	\(0.00507624\pi\)
\(19\)	−0.235259	−	0.407481i	−0.0539721	−	0.0934825i	0.837777	−	0.546012i	\(-0.183855\pi\)
							−0.891749	+	0.452530i	\(0.850522\pi\)
\(23\)	1.78716	−	1.03182i	0.372650	−	0.215149i	−0.301966	−	0.953319i	\(-0.597643\pi\)
							0.674615	+	0.738169i	\(0.264310\pi\)
\(29\)			2.04471i			0.379694i	0.981814	+	0.189847i	\(0.0607992\pi\)
							−0.981814	+	0.189847i	\(0.939201\pi\)
\(31\)	−4.16626	+	7.21617i	−0.748282	+	1.29606i	0.200364	+	0.979722i	\(0.435788\pi\)
							−0.948646	+	0.316341i	\(0.897546\pi\)
\(37\)	3.40583	−	1.96636i	0.559915	−	0.323267i	−0.193196	−	0.981160i	\(-0.561885\pi\)
							0.753112	+	0.657893i	\(0.228552\pi\)
\(41\)			11.1312i			1.73840i	0.494462	+	0.869199i	\(0.335365\pi\)
							−0.494462	+	0.869199i	\(0.664635\pi\)
\(43\)	7.08155			1.07993			0.539963	−	0.841689i	\(-0.318438\pi\)
							0.539963	+	0.841689i	\(0.318438\pi\)
\(47\)	−4.15367	+	2.39812i	−0.605875	+	0.349802i	−0.771349	−	0.636412i	\(-0.780418\pi\)
							0.165474	+	0.986214i	\(0.447084\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.ba.c.38.20	yes	64
3.2	odd	2	inner	273.2.ba.c.38.13	✓	64
7.5	odd	6	inner	273.2.ba.c.194.19	yes	64
13.12	even	2	inner	273.2.ba.c.38.14	yes	64
21.5	even	6	inner	273.2.ba.c.194.14	yes	64
39.38	odd	2	inner	273.2.ba.c.38.19	yes	64
91.12	odd	6	inner	273.2.ba.c.194.13	yes	64
273.194	even	6	inner	273.2.ba.c.194.20	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.ba.c.38.13	✓	64	3.2	odd	2	inner
273.2.ba.c.38.14	yes	64	13.12	even	2	inner
273.2.ba.c.38.19	yes	64	39.38	odd	2	inner
273.2.ba.c.38.20	yes	64	1.1	even	1	trivial
273.2.ba.c.194.13	yes	64	91.12	odd	6	inner
273.2.ba.c.194.14	yes	64	21.5	even	6	inner
273.2.ba.c.194.19	yes	64	7.5	odd	6	inner
273.2.ba.c.194.20	yes	64	273.194	even	6	inner