Embedded newform 1320.2.d.a.529.6

• Label: 1320.2.d.a.529.6
• Level: $1320$
• Weight: $2$
• Character: 1320.529
• Analytic conductor: $10.540$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5402530668\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.350464.1
Defining polynomial:	\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.6
Root			\(0.403032 + 0.403032i\) of defining polynomial
Character	\(\chi\)	\(=\)	1320.529
Dual form			1320.2.d.a.529.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} +0.806063i q^{7} -1.00000 q^{9} -1.00000 q^{11} +2.15633i q^{13} +(-1.67513 + 1.48119i) q^{15} +2.54420i q^{17} +0.387873 q^{19} -0.806063 q^{21} +(-0.612127 + 4.96239i) q^{25} -1.00000i q^{27} +0.649738 q^{29} -4.96239 q^{31} -1.00000i q^{33} +(-1.35026 + 1.19394i) q^{35} +8.31265i q^{37} -2.15633 q^{39} -2.57452 q^{41} -0.806063i q^{43} +(-1.48119 - 1.67513i) q^{45} -1.61213i q^{47} +6.35026 q^{49} -2.54420 q^{51} -0.649738i q^{53} +(-1.48119 - 1.67513i) q^{55} +0.387873i q^{57} +0.649738 q^{59} -2.00000 q^{61} -0.806063i q^{63} +(-3.61213 + 3.19394i) q^{65} -3.22425i q^{67} -4.64974 q^{71} -0.231548i q^{73} +(-4.96239 - 0.612127i) q^{75} -0.806063i q^{77} -5.53690 q^{79} +1.00000 q^{81} -4.08110i q^{83} +(-4.26187 + 3.76845i) q^{85} +0.649738i q^{87} -4.88717 q^{89} -1.73813 q^{91} -4.96239i q^{93} +(0.574515 + 0.649738i) q^{95} +13.9248i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 2 * q^5 - 6 * q^9 - 6 * q^11 + 4 * q^19 - 4 * q^21 - 2 * q^25 + 24 * q^29 - 8 * q^31 + 12 * q^35 + 8 * q^39 + 8 * q^41 + 2 * q^45 + 18 * q^49 + 4 * q^51 + 2 * q^55 + 24 * q^59 - 12 * q^61 - 20 * q^65 - 48 * q^71 - 8 * q^75 + 12 * q^79 + 6 * q^81 - 44 * q^85 + 36 * q^89 + 8 * q^91 - 20 * q^95 + 6 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).

\(n\)	\(661\)	\(881\)	\(991\)	\(1057\)	\(1201\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)

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			0
							\(3\)			1.00000i			0.577350i
\(5\)	1.48119	+	1.67513i	0.662410	+	0.749141i
							\(7\)			0.806063i			0.304663i	0.988329	+	0.152332i	\(0.0486782\pi\)
−0.988329	+	0.152332i	\(0.951322\pi\)
\(11\)	−1.00000			−0.301511
							\(13\)			2.15633i			0.598057i	0.954244	+	0.299028i	\(0.0966626\pi\)
−0.954244	+	0.299028i	\(0.903337\pi\)
\(17\)			2.54420i			0.617059i	0.951215	+	0.308529i	\(0.0998368\pi\)
							−0.951215	+	0.308529i	\(0.900163\pi\)
\(19\)	0.387873			0.0889842			0.0444921	−	0.999010i	\(-0.485833\pi\)
							0.0444921	+	0.999010i	\(0.485833\pi\)
\(23\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(29\)	0.649738			0.120653			0.0603267	−	0.998179i	\(-0.480786\pi\)
							0.0603267	+	0.998179i	\(0.480786\pi\)
\(31\)	−4.96239			−0.891271			−0.445636	−	0.895214i	\(-0.647022\pi\)
							−0.445636	+	0.895214i	\(0.647022\pi\)
\(37\)			8.31265i			1.36659i	0.730142	+	0.683296i	\(0.239454\pi\)
							−0.730142	+	0.683296i	\(0.760546\pi\)
\(41\)	−2.57452			−0.402072			−0.201036	−	0.979584i	\(-0.564431\pi\)
							−0.201036	+	0.979584i	\(0.564431\pi\)
\(43\)		−	0.806063i		−	0.122924i	−0.998109	−	0.0614618i	\(-0.980424\pi\)
							0.998109	−	0.0614618i	\(-0.0195762\pi\)
\(47\)		−	1.61213i		−	0.235153i	−0.993064	−	0.117576i	\(-0.962487\pi\)
							0.993064	−	0.117576i	\(-0.0375125\pi\)
\(53\)		−	0.649738i		−	0.0892484i	−0.999004	−	0.0446242i	\(-0.985791\pi\)
							0.999004	−	0.0446242i	\(-0.0142090\pi\)
\(59\)	0.649738			0.0845887			0.0422944	−	0.999105i	\(-0.486533\pi\)
							0.0422944	+	0.999105i	\(0.486533\pi\)
\(61\)	−2.00000			−0.256074			−0.128037	−	0.991769i	\(-0.540868\pi\)
							−0.128037	+	0.991769i	\(0.540868\pi\)
\(67\)		−	3.22425i		−	0.393905i	−0.980413	−	0.196953i	\(-0.936895\pi\)
							0.980413	−	0.196953i	\(-0.0631045\pi\)
\(71\)	−4.64974			−0.551822			−0.275911	−	0.961183i	\(-0.588980\pi\)
							−0.275911	+	0.961183i	\(0.588980\pi\)
\(73\)		−	0.231548i		−	0.0271006i	−0.999908	−	0.0135503i	\(-0.995687\pi\)
							0.999908	−	0.0135503i	\(-0.00431333\pi\)
\(79\)	−5.53690			−0.622950			−0.311475	−	0.950254i	\(-0.600823\pi\)
							−0.311475	+	0.950254i	\(0.600823\pi\)
\(83\)		−	4.08110i		−	0.447959i	−0.974594	−	0.223980i	\(-0.928095\pi\)
							0.974594	−	0.223980i	\(-0.0719049\pi\)
\(89\)	−4.88717			−0.518039			−0.259019	−	0.965872i	\(-0.583399\pi\)
							−0.259019	+	0.965872i	\(0.583399\pi\)
\(97\)			13.9248i			1.41385i	0.707290	+	0.706923i	\(0.249917\pi\)
							−0.707290	+	0.706923i	\(0.750083\pi\)

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	1320.2.d.a.529.6	yes	6
3.2	odd	2		3960.2.d.e.3169.1		6
4.3	odd	2		2640.2.d.g.529.3		6
5.2	odd	4		6600.2.a.bt.1.2		3
5.3	odd	4		6600.2.a.bp.1.2		3
5.4	even	2	inner	1320.2.d.a.529.3	✓	6
15.14	odd	2		3960.2.d.e.3169.2		6
20.19	odd	2		2640.2.d.g.529.6		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.d.a.529.3	✓	6	5.4	even	2	inner
1320.2.d.a.529.6	yes	6	1.1	even	1	trivial
2640.2.d.g.529.3		6	4.3	odd	2
2640.2.d.g.529.6		6	20.19	odd	2
3960.2.d.e.3169.1		6	3.2	odd	2
3960.2.d.e.3169.2		6	15.14	odd	2
6600.2.a.bp.1.2		3	5.3	odd	4
6600.2.a.bt.1.2		3	5.2	odd	4