Embedded newform 504.2.r.c.169.2

• Label: 504.2.r.c.169.2
• Level: $504$
• Weight: $2$
• Character: 504.169
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			169.2
Root			\(-0.766044 - 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	504.169
Dual form			504.2.r.c.337.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.592396 + 1.62760i) q^{3} +(0.326352 + 0.565258i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-2.29813 + 1.92836i) q^{9} +(-1.70574 + 2.95442i) q^{11} +(0.152704 + 0.264490i) q^{13} +(-0.726682 + 0.866025i) q^{15} -0.226682 q^{17} -2.16250 q^{19} +(-1.70574 - 0.300767i) q^{21} +(3.35117 + 5.80439i) q^{23} +(2.28699 - 3.96118i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(0.254900 - 0.441500i) q^{29} +(2.85844 + 4.95096i) q^{31} +(-5.81908 - 1.02606i) q^{33} -0.652704 q^{35} -2.28312 q^{37} +(-0.340022 + 0.405223i) q^{39} +(-0.479055 - 0.829748i) q^{41} +(3.85844 - 6.68302i) q^{43} +(-1.84002 - 0.669713i) q^{45} +(-4.14543 + 7.18009i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(-0.134285 - 0.368946i) q^{51} -3.18479 q^{53} -2.22668 q^{55} +(-1.28106 - 3.51968i) q^{57} +(6.07785 + 10.5271i) q^{59} +(1.75490 - 3.03958i) q^{61} +(-0.520945 - 2.95442i) q^{63} +(-0.0996702 + 0.172634i) q^{65} +(-2.87211 - 4.97464i) q^{67} +(-7.46198 + 8.89284i) q^{69} +14.0273 q^{71} +12.4192 q^{73} +(7.80200 + 1.37570i) q^{75} +(-1.70574 - 2.95442i) q^{77} +(5.98293 - 10.3627i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-2.13563 + 3.69902i) q^{83} +(-0.0739780 - 0.128134i) q^{85} +(0.869585 + 0.153331i) q^{87} -1.19934 q^{89} -0.305407 q^{91} +(-6.36484 + 7.58532i) q^{93} +(-0.705737 - 1.22237i) q^{95} +(7.48932 - 12.9719i) q^{97} +(-1.77719 - 10.0789i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 3 * q^5 - 3 * q^7 + 3 * q^13 + 9 * q^15 + 12 * q^17 - 18 * q^19 - 6 * q^23 + 6 * q^25 - 27 * q^27 + 3 * q^29 + 9 * q^31 - 18 * q^33 - 6 * q^35 - 30 * q^37 + 18 * q^39 - 6 * q^41 + 15 * q^43 + 9 * q^45 - 9 * q^47 - 3 * q^49 + 9 * q^51 - 12 * q^53 + 27 * q^57 - 3 * q^59 + 12 * q^61 - 15 * q^65 + 12 * q^67 - 27 * q^69 + 42 * q^71 + 6 * q^73 + 9 * q^75 + 15 * q^79 + 6 * q^83 + 15 * q^85 - 9 * q^87 - 36 * q^89 - 6 * q^91 + 9 * q^93 + 6 * q^95 - 15 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(127\)	\(253\)	\(281\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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			0
							\(3\)	0.592396	+	1.62760i	0.342020	+	0.939693i
\(5\)	0.326352	+	0.565258i	0.145949	+	0.252791i								0.929727	−	0.368251i	\(-0.120043\pi\)
							−0.783778	+	0.621042i	\(0.786710\pi\)
\(7\)	−0.500000	+	0.866025i	−0.188982	+	0.327327i
							\(11\)	−1.70574	+	2.95442i	−0.514299	+	0.890792i	0.485563	+	0.874202i	\(0.338615\pi\)
−0.999862	+	0.0165906i	\(0.994719\pi\)
\(13\)	0.152704	+	0.264490i	0.0423524	+	0.0733565i	0.886425	−	0.462873i	\(-0.153181\pi\)
							−0.844072	+	0.536230i	\(0.819848\pi\)
\(17\)	−0.226682			−0.0549784			−0.0274892	−	0.999622i	\(-0.508751\pi\)
							−0.0274892	+	0.999622i	\(0.508751\pi\)
\(19\)	−2.16250			−0.496112			−0.248056	−	0.968746i	\(-0.579792\pi\)
							−0.248056	+	0.968746i	\(0.579792\pi\)
\(23\)	3.35117	+	5.80439i	0.698767	+	1.21030i	0.968894	+	0.247475i	\(0.0796007\pi\)
							−0.270128	+	0.962824i	\(0.587066\pi\)
\(29\)	0.254900	−	0.441500i	0.0473338	−	0.0819845i	−0.841388	−	0.540432i	\(-0.818261\pi\)
							0.888722	+	0.458447i	\(0.151594\pi\)
\(31\)	2.85844	+	4.95096i	0.513391	+	0.889219i	0.999879	+	0.0155324i	\(0.00494430\pi\)
							−0.486488	+	0.873687i	\(0.661722\pi\)
\(37\)	−2.28312			−0.375342			−0.187671	−	0.982232i	\(-0.560094\pi\)
							−0.187671	+	0.982232i	\(0.560094\pi\)
\(41\)	−0.479055	−	0.829748i	−0.0748159	−	0.129585i	0.826190	−	0.563391i	\(-0.190504\pi\)
							−0.901006	+	0.433806i	\(0.857170\pi\)
\(43\)	3.85844	−	6.68302i	0.588407	−	1.01915i	−0.406035	−	0.913858i	\(-0.633089\pi\)
							0.994441	−	0.105293i	\(-0.0335779\pi\)
\(47\)	−4.14543	+	7.18009i	−0.604673	+	1.04732i	0.387430	+	0.921899i	\(0.373363\pi\)
							−0.992103	+	0.125425i	\(0.959970\pi\)
\(53\)	−3.18479			−0.437465			−0.218732	−	0.975785i	\(-0.570192\pi\)
							−0.218732	+	0.975785i	\(0.570192\pi\)
\(59\)	6.07785	+	10.5271i	0.791268	+	1.37052i	0.925182	+	0.379524i	\(0.123912\pi\)
							−0.133913	+	0.990993i	\(0.542754\pi\)
\(61\)	1.75490	−	3.03958i	0.224692	−	0.389178i	−0.731535	−	0.681804i	\(-0.761196\pi\)
							0.956227	+	0.292626i	\(0.0945291\pi\)
\(67\)	−2.87211	−	4.97464i	−0.350884	−	0.607749i	0.635520	−	0.772084i	\(-0.280786\pi\)
							−0.986405	+	0.164335i	\(0.947452\pi\)
\(71\)	14.0273			1.66474			0.832370	−	0.554221i	\(-0.186984\pi\)
							0.832370	+	0.554221i	\(0.186984\pi\)
\(73\)	12.4192			1.45356			0.726780	−	0.686871i	\(-0.241016\pi\)
							0.726780	+	0.686871i	\(0.241016\pi\)
\(79\)	5.98293	−	10.3627i	0.673132	−	1.16590i	−0.303879	−	0.952710i	\(-0.598282\pi\)
							0.977011	−	0.213188i	\(-0.0683847\pi\)
\(83\)	−2.13563	+	3.69902i	−0.234416	+	0.406020i	−0.959103	−	0.283058i	\(-0.908651\pi\)
							0.724687	+	0.689078i	\(0.241984\pi\)
\(89\)	−1.19934			−0.127130			−0.0635649	−	0.997978i	\(-0.520247\pi\)
							−0.0635649	+	0.997978i	\(0.520247\pi\)
\(97\)	7.48932	−	12.9719i	0.760425	−	1.31710i	−0.182206	−	0.983260i	\(-0.558324\pi\)
							0.942631	−	0.333835i	\(-0.108343\pi\)

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	504.2.r.c.169.2	✓	6
3.2	odd	2		1512.2.r.c.505.2		6
4.3	odd	2		1008.2.r.i.673.2		6
9.2	odd	6		4536.2.a.v.1.2		3
9.4	even	3	inner	504.2.r.c.337.2	yes	6
9.5	odd	6		1512.2.r.c.1009.2		6
9.7	even	3		4536.2.a.s.1.2		3
12.11	even	2		3024.2.r.h.2017.2		6
36.7	odd	6		9072.2.a.br.1.2		3
36.11	even	6		9072.2.a.cc.1.2		3
36.23	even	6		3024.2.r.h.1009.2		6
36.31	odd	6		1008.2.r.i.337.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.c.169.2	✓	6	1.1	even	1	trivial
504.2.r.c.337.2	yes	6	9.4	even	3	inner
1008.2.r.i.337.2		6	36.31	odd	6
1008.2.r.i.673.2		6	4.3	odd	2
1512.2.r.c.505.2		6	3.2	odd	2
1512.2.r.c.1009.2		6	9.5	odd	6
3024.2.r.h.1009.2		6	36.23	even	6
3024.2.r.h.2017.2		6	12.11	even	2
4536.2.a.s.1.2		3	9.7	even	3
4536.2.a.v.1.2		3	9.2	odd	6
9072.2.a.br.1.2		3	36.7	odd	6
9072.2.a.cc.1.2		3	36.11	even	6