Embedded newform 1274.2.f.i.1145.1

• Label: 1274.2.f.i.1145.1
• Level: $1274$
• Weight: $2$
• Character: 1274.1145
• Analytic conductor: $10.173$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1274.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1729412175\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1145.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1274.1145
Dual form			1274.2.f.i.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +1.00000 q^{8} +(1.00000 + 1.73205i) q^{9} +(1.50000 - 2.59808i) q^{11} +(0.500000 + 0.866025i) q^{12} -1.00000 q^{13} +(-0.500000 - 0.866025i) q^{16} +(1.00000 - 1.73205i) q^{18} +(1.00000 + 1.73205i) q^{19} -3.00000 q^{22} +(1.50000 + 2.59808i) q^{23} +(0.500000 - 0.866025i) q^{24} +(2.50000 - 4.33013i) q^{25} +(0.500000 + 0.866025i) q^{26} +5.00000 q^{27} +(2.50000 - 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-1.50000 - 2.59808i) q^{33} -2.00000 q^{36} +(3.50000 + 6.06218i) q^{37} +(1.00000 - 1.73205i) q^{38} +(-0.500000 + 0.866025i) q^{39} -3.00000 q^{41} +8.00000 q^{43} +(1.50000 + 2.59808i) q^{44} +(1.50000 - 2.59808i) q^{46} +(-1.50000 - 2.59808i) q^{47} -1.00000 q^{48} -5.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(6.00000 - 10.3923i) q^{53} +(-2.50000 - 4.33013i) q^{54} +2.00000 q^{57} +(3.00000 - 5.19615i) q^{59} +(-0.500000 - 0.866025i) q^{61} -5.00000 q^{62} +1.00000 q^{64} +(-1.50000 + 2.59808i) q^{66} +(-2.50000 + 4.33013i) q^{67} +3.00000 q^{69} +12.0000 q^{71} +(1.00000 + 1.73205i) q^{72} +(5.50000 - 9.52628i) q^{73} +(3.50000 - 6.06218i) q^{74} +(-2.50000 - 4.33013i) q^{75} -2.00000 q^{76} +1.00000 q^{78} +(0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(1.50000 + 2.59808i) q^{82} -12.0000 q^{83} +(-4.00000 - 6.92820i) q^{86} +(1.50000 - 2.59808i) q^{88} +(-9.00000 - 15.5885i) q^{89} -3.00000 q^{92} +(-2.50000 - 4.33013i) q^{93} +(-1.50000 + 2.59808i) q^{94} +(0.500000 + 0.866025i) q^{96} -17.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 + q^3 - q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9 + 3 * q^11 + q^12 - 2 * q^13 - q^16 + 2 * q^18 + 2 * q^19 - 6 * q^22 + 3 * q^23 + q^24 + 5 * q^25 + q^26 + 10 * q^27 + 5 * q^31 - q^32 - 3 * q^33 - 4 * q^36 + 7 * q^37 + 2 * q^38 - q^39 - 6 * q^41 + 16 * q^43 + 3 * q^44 + 3 * q^46 - 3 * q^47 - 2 * q^48 - 10 * q^50 + q^52 + 12 * q^53 - 5 * q^54 + 4 * q^57 + 6 * q^59 - q^61 - 10 * q^62 + 2 * q^64 - 3 * q^66 - 5 * q^67 + 6 * q^69 + 24 * q^71 + 2 * q^72 + 11 * q^73 + 7 * q^74 - 5 * q^75 - 4 * q^76 + 2 * q^78 + q^79 - q^81 + 3 * q^82 - 24 * q^83 - 8 * q^86 + 3 * q^88 - 18 * q^89 - 6 * q^92 - 5 * q^93 - 3 * q^94 + q^96 - 34 * q^97 + 12 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(885\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500000	−	0.866025i	−0.353553	−	0.612372i
							\(3\)	0.500000	−	0.866025i	0.288675	−	0.500000i	−0.684819	−	0.728714i	\(-0.740119\pi\)
0.973494	+	0.228714i	\(0.0734519\pi\)
\(5\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(7\)		0			0
							\(11\)	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	\(-0.683949\pi\)
0.998526	+	0.0542666i	\(0.0172821\pi\)
\(13\)	−1.00000			−0.277350
							\(17\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
0.866025	+	0.500000i	\(0.166667\pi\)
\(19\)	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	\(-0.0929851\pi\)
							−0.728219	+	0.685344i	\(0.759652\pi\)
\(23\)	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	\(-0.0654092\pi\)
							−0.666190	+	0.745782i	\(0.732076\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	\(-0.685109\pi\)
							0.998322	+	0.0579057i	\(0.0184423\pi\)
\(37\)	3.50000	+	6.06218i	0.575396	+	0.996616i	0.995998	+	0.0893706i	\(0.0284856\pi\)
							−0.420602	+	0.907245i	\(0.638181\pi\)
\(41\)	−3.00000			−0.468521			−0.234261	−	0.972174i	\(-0.575267\pi\)
							−0.234261	+	0.972174i	\(0.575267\pi\)
\(43\)	8.00000			1.21999			0.609994	−	0.792406i	\(-0.291172\pi\)
							0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	−1.50000	−	2.59808i	−0.218797	−	0.378968i	0.735643	−	0.677369i	\(-0.236880\pi\)
							−0.954441	+	0.298401i	\(0.903547\pi\)
\(53\)	6.00000	−	10.3923i	0.824163	−	1.42749i	−0.0783936	−	0.996922i	\(-0.524979\pi\)
							0.902557	−	0.430570i	\(-0.141688\pi\)
\(59\)	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	\(-0.705612\pi\)
							0.992524	+	0.122047i	\(0.0389457\pi\)
\(61\)	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	\(-0.187058\pi\)
							−0.896258	+	0.443533i	\(0.853725\pi\)
\(67\)	−2.50000	+	4.33013i	−0.305424	+	0.529009i	−0.977356	−	0.211604i	\(-0.932131\pi\)
							0.671932	+	0.740613i	\(0.265465\pi\)
\(71\)	12.0000			1.42414			0.712069	−	0.702109i	\(-0.247758\pi\)
							0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	5.50000	−	9.52628i	0.643726	−	1.11497i	−0.340868	−	0.940111i	\(-0.610721\pi\)
							0.984594	−	0.174855i	\(-0.0559458\pi\)
\(79\)	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	\(-0.148751\pi\)
							−0.836527	+	0.547926i	\(0.815418\pi\)
\(83\)	−12.0000			−1.31717			−0.658586	−	0.752506i	\(-0.728845\pi\)
							−0.658586	+	0.752506i	\(0.728845\pi\)
\(89\)	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.736644	−	0.676280i	\(-0.763591\pi\)
							−0.217354	−	0.976093i	\(-0.569742\pi\)
\(97\)	−17.0000			−1.72609			−0.863044	−	0.505128i	\(-0.831445\pi\)
							−0.863044	+	0.505128i	\(0.831445\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1274.2.f.i.1145.1		2
7.2	even	3	inner	1274.2.f.i.79.1		2
7.3	odd	6		182.2.a.d.1.1	✓	1
7.4	even	3		1274.2.a.j.1.1		1
7.5	odd	6		1274.2.f.d.79.1		2
7.6	odd	2		1274.2.f.d.1145.1		2
21.17	even	6		1638.2.a.f.1.1		1
28.3	even	6		1456.2.a.d.1.1		1
35.24	odd	6		4550.2.a.c.1.1		1
56.3	even	6		5824.2.a.x.1.1		1
56.45	odd	6		5824.2.a.k.1.1		1
91.31	even	12		2366.2.d.e.337.1		2
91.38	odd	6		2366.2.a.e.1.1		1
91.73	even	12		2366.2.d.e.337.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.d.1.1	✓	1	7.3	odd	6
1274.2.a.j.1.1		1	7.4	even	3
1274.2.f.d.79.1		2	7.5	odd	6
1274.2.f.d.1145.1		2	7.6	odd	2
1274.2.f.i.79.1		2	7.2	even	3	inner
1274.2.f.i.1145.1		2	1.1	even	1	trivial
1456.2.a.d.1.1		1	28.3	even	6
1638.2.a.f.1.1		1	21.17	even	6
2366.2.a.e.1.1		1	91.38	odd	6
2366.2.d.e.337.1		2	91.31	even	12
2366.2.d.e.337.2		2	91.73	even	12
4550.2.a.c.1.1		1	35.24	odd	6
5824.2.a.k.1.1		1	56.45	odd	6
5824.2.a.x.1.1		1	56.3	even	6