Embedded newform 273.2.bt.b.145.3

• Label: 273.2.bt.b.145.3
• Level: $273$
• Weight: $2$
• Character: 273.145
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.bt (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			145.3
Character	\(\chi\)	\(=\)	273.145
Dual form			273.2.bt.b.241.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22934 + 1.22934i) q^{2} +(0.866025 + 0.500000i) q^{3} -1.02253i q^{4} +(-1.95691 + 0.524353i) q^{5} +(-1.67930 + 0.449968i) q^{6} +(-1.82011 - 1.92021i) q^{7} +(-1.20163 - 1.20163i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22934	+	1.22934i	−0.869272	+	0.869272i	−0.992392	−	0.123120i	\(-0.960710\pi\)
							0.123120	+	0.992392i	\(0.460710\pi\)
\(3\)	0.866025	+	0.500000i	0.500000	+	0.288675i
							\(5\)	−1.95691	+	0.524353i	−0.875158	+	0.234498i	−0.668317	−	0.743877i	\(-0.732985\pi\)
−0.206841	+	0.978375i	\(0.566318\pi\)
\(7\)	−1.82011	−	1.92021i	−0.687936	−	0.725771i
							\(11\)	−3.50126	+	0.938159i	−1.05567	+	0.282866i	−0.744592	−	0.667519i	\(-0.767356\pi\)
−0.311076	+	0.950385i	\(0.600689\pi\)
\(13\)	−1.78841	−	3.13075i	−0.496016	−	0.868313i
							\(17\)	−1.50579			−0.365208			−0.182604	−	0.983187i	\(-0.558453\pi\)
−0.182604	+	0.983187i	\(0.558453\pi\)
\(19\)	−1.07053	+	3.99526i	−0.245595	+	0.916575i	0.727488	+	0.686121i	\(0.240688\pi\)
							−0.973083	+	0.230454i	\(0.925979\pi\)
\(23\)			3.71137i			0.773873i	0.922106	+	0.386937i	\(0.126467\pi\)
							−0.922106	+	0.386937i	\(0.873533\pi\)
\(29\)	−1.84505	−	3.19572i	−0.342617	−	0.593430i	0.642301	−	0.766452i	\(-0.277980\pi\)
							−0.984918	+	0.173023i	\(0.944647\pi\)
\(31\)	1.89117	−	7.05794i	0.339664	−	1.26764i	−0.559059	−	0.829128i	\(-0.688838\pi\)
							0.898723	−	0.438516i	\(-0.144496\pi\)
\(37\)	1.85150	+	1.85150i	0.304385	+	0.304385i	0.842727	−	0.538342i	\(-0.180949\pi\)
							−0.538342	+	0.842727i	\(0.680949\pi\)
\(41\)	−1.33248	+	4.97288i	−0.208098	+	0.776634i	0.780384	+	0.625300i	\(0.215023\pi\)
							−0.988483	+	0.151334i	\(0.951643\pi\)
\(43\)	−4.51217	−	2.60510i	−0.688099	−	0.397274i	0.114800	−	0.993389i	\(-0.463377\pi\)
							−0.802900	+	0.596114i	\(0.796711\pi\)
\(47\)	0.0684756	+	0.255554i	0.00998819	+	0.0372764i	0.970740	−	0.240133i	\(-0.0771909\pi\)
							−0.960752	+	0.277409i	\(0.910524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.145.3	✓	40
3.2	odd	2		819.2.et.d.145.8		40
7.3	odd	6		273.2.cg.b.262.3	yes	40
13.7	odd	12		273.2.cg.b.124.3	yes	40
21.17	even	6		819.2.gh.d.262.8		40
39.20	even	12		819.2.gh.d.397.8		40
91.59	even	12	inner	273.2.bt.b.241.3	yes	40
273.59	odd	12		819.2.et.d.514.8		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.145.3	✓	40	1.1	even	1	trivial
273.2.bt.b.241.3	yes	40	91.59	even	12	inner
273.2.cg.b.124.3	yes	40	13.7	odd	12
273.2.cg.b.262.3	yes	40	7.3	odd	6
819.2.et.d.145.8		40	3.2	odd	2
819.2.et.d.514.8		40	273.59	odd	12
819.2.gh.d.262.8		40	21.17	even	6
819.2.gh.d.397.8		40	39.20	even	12