Embedded newform 630.2.be.a.341.3

• Label: 630.2.be.a.341.3
• Level: $630$
• Weight: $2$
• Character: 630.341
• Analytic conductor: $5.031$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.03057532734\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			341.3
Root			\(0.965926 - 0.258819i\) of defining polynomial
Character	\(\chi\)	\(=\)	630.341
Dual form			630.2.be.a.521.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(-2.63896 - 0.189469i) q^{7} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{10} +(-4.67303 + 2.69798i) q^{11} +2.51764i q^{13} +(-2.19067 - 1.48356i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.24969 + 3.89658i) q^{17} +(-2.48004 - 1.43185i) q^{19} -1.00000 q^{20} -5.39595 q^{22} +(0.232051 + 0.133975i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.25882 + 2.18034i) q^{26} +(-1.15539 - 2.38014i) q^{28} +8.89898i q^{29} +(4.18154 - 2.41421i) q^{31} +(-0.866025 + 0.500000i) q^{32} +4.49938i q^{34} +(1.48356 - 2.19067i) q^{35} +(3.25882 - 5.64444i) q^{37} +(-1.43185 - 2.48004i) q^{38} +(-0.866025 - 0.500000i) q^{40} +0.760279 q^{41} -5.86370 q^{43} +(-4.67303 - 2.69798i) q^{44} +(0.133975 + 0.232051i) q^{46} +(3.99768 - 6.92418i) q^{47} +(6.92820 + 1.00000i) q^{49} -1.00000i q^{50} +(-2.18034 + 1.25882i) q^{52} +(-7.27319 + 4.19918i) q^{53} -5.39595i q^{55} +(0.189469 - 2.63896i) q^{56} +(-4.44949 + 7.70674i) q^{58} +(6.33573 + 10.9738i) q^{59} +(-2.27035 - 1.31079i) q^{61} +4.82843 q^{62} -1.00000 q^{64} +(-2.18034 - 1.25882i) q^{65} +(-4.91119 - 8.50643i) q^{67} +(-2.24969 + 3.89658i) q^{68} +(2.38014 - 1.15539i) q^{70} +4.76268i q^{71} +(10.0951 - 5.82843i) q^{73} +(5.64444 - 3.25882i) q^{74} -2.86370i q^{76} +(12.8431 - 6.23445i) q^{77} +(-4.29618 + 7.44120i) q^{79} +(-0.500000 - 0.866025i) q^{80} +(0.658421 + 0.380139i) q^{82} +9.45001 q^{83} -4.49938 q^{85} +(-5.07812 - 2.93185i) q^{86} +(-2.69798 - 4.67303i) q^{88} +(-3.98502 + 6.90226i) q^{89} +(0.477014 - 6.64394i) q^{91} +0.267949i q^{92} +(6.92418 - 3.99768i) q^{94} +(2.48004 - 1.43185i) q^{95} +6.16353i q^{97} +(5.50000 + 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 4 * q^4 - 4 * q^5 - 24 * q^11 - 4 * q^16 - 8 * q^20 - 12 * q^23 - 4 * q^25 - 8 * q^26 + 24 * q^37 + 4 * q^38 - 32 * q^41 - 16 * q^43 - 24 * q^44 + 8 * q^46 + 8 * q^47 - 24 * q^53 - 16 * q^58 + 24 * q^59 + 16 * q^62 - 8 * q^64 - 24 * q^67 + 16 * q^77 - 24 * q^79 - 4 * q^80 + 16 * q^83 + 16 * q^89 - 20 * q^91 - 12 * q^94 + 44 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(281\)	\(451\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

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

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

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.866025	+	0.500000i	0.612372	+	0.353553i
							\(3\)		0			0
\(5\)	−0.500000	+	0.866025i	−0.223607	+	0.387298i
							\(7\)	−2.63896	−	0.189469i	−0.997433	−	0.0716124i
\(11\)	−4.67303	+	2.69798i	−1.40897	+	0.813471i								−0.995289	−	0.0969504i	\(-0.969091\pi\)
							−0.413683	+	0.910421i	\(0.635758\pi\)
\(13\)			2.51764i			0.698267i	0.937073	+	0.349134i	\(0.113524\pi\)
							−0.937073	+	0.349134i	\(0.886476\pi\)
\(17\)	2.24969	+	3.89658i	0.545630	+	0.945058i	0.998567	+	0.0535160i	\(0.0170428\pi\)
							−0.452937	+	0.891542i	\(0.649624\pi\)
\(19\)	−2.48004	−	1.43185i	−0.568960	−	0.328489i	0.187774	−	0.982212i	\(-0.439873\pi\)
							−0.756734	+	0.653723i	\(0.773206\pi\)
\(23\)	0.232051	+	0.133975i	0.0483859	+	0.0279356i	0.523998	−	0.851720i	\(-0.324440\pi\)
							−0.475612	+	0.879655i	\(0.657773\pi\)
\(29\)			8.89898i			1.65250i	0.563304	+	0.826250i	\(0.309530\pi\)
							−0.563304	+	0.826250i	\(0.690470\pi\)
\(31\)	4.18154	−	2.41421i	0.751027	−	0.433606i	−0.0750380	−	0.997181i	\(-0.523908\pi\)
							0.826065	+	0.563575i	\(0.190574\pi\)
\(37\)	3.25882	−	5.64444i	0.535747	−	0.927940i	−0.463380	−	0.886160i	\(-0.653364\pi\)
							0.999127	−	0.0417807i	\(-0.0133031\pi\)
\(41\)	0.760279			0.118736			0.0593678	−	0.998236i	\(-0.481092\pi\)
							0.0593678	+	0.998236i	\(0.481092\pi\)
\(43\)	−5.86370			−0.894206			−0.447103	−	0.894482i	\(-0.647544\pi\)
							−0.447103	+	0.894482i	\(0.647544\pi\)
\(47\)	3.99768	−	6.92418i	0.583121	−	1.01000i	−0.411986	−	0.911190i	\(-0.635165\pi\)
							0.995107	−	0.0988053i	\(-0.0315021\pi\)
\(53\)	−7.27319	+	4.19918i	−0.999050	+	0.576802i	−0.907967	−	0.419042i	\(-0.862366\pi\)
							−0.0910826	+	0.995843i	\(0.529033\pi\)
\(59\)	6.33573	+	10.9738i	0.824842	+	1.42867i	0.902040	+	0.431653i	\(0.142069\pi\)
							−0.0771977	+	0.997016i	\(0.524597\pi\)
\(61\)	−2.27035	−	1.31079i	−0.290689	−	0.167829i	0.347564	−	0.937656i	\(-0.387009\pi\)
							−0.638253	+	0.769827i	\(0.720342\pi\)
\(67\)	−4.91119	−	8.50643i	−0.599997	−	1.03923i	−0.992821	−	0.119612i	\(-0.961835\pi\)
							0.392824	−	0.919614i	\(-0.371498\pi\)
\(71\)			4.76268i			0.565226i	0.959234	+	0.282613i	\(0.0912013\pi\)
							−0.959234	+	0.282613i	\(0.908799\pi\)
\(73\)	10.0951	−	5.82843i	1.18155	−	0.682166i	0.225174	−	0.974319i	\(-0.427705\pi\)
							0.956372	+	0.292153i	\(0.0943716\pi\)
\(79\)	−4.29618	+	7.44120i	−0.483358	+	0.837200i	−0.999817	−	0.0191114i	\(-0.993916\pi\)
							0.516460	+	0.856312i	\(0.327250\pi\)
\(83\)	9.45001			1.03727			0.518636	−	0.854995i	\(-0.326440\pi\)
							0.518636	+	0.854995i	\(0.326440\pi\)
\(89\)	−3.98502	+	6.90226i	−0.422412	+	0.731638i	−0.996175	−	0.0873828i	\(-0.972150\pi\)
							0.573763	+	0.819021i	\(0.305483\pi\)
\(97\)			6.16353i			0.625812i	0.949784	+	0.312906i	\(0.101302\pi\)
							−0.949784	+	0.312906i	\(0.898698\pi\)

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	630.2.be.a.341.3	✓	8
3.2	odd	2		630.2.be.b.341.1	yes	8
5.2	odd	4		3150.2.bp.a.1349.3		8
5.3	odd	4		3150.2.bp.d.1349.2		8
5.4	even	2		3150.2.bf.b.1601.2		8
7.2	even	3		4410.2.b.e.881.1		8
7.3	odd	6		630.2.be.b.521.1	yes	8
7.5	odd	6		4410.2.b.b.881.1		8
15.2	even	4		3150.2.bp.f.1349.3		8
15.8	even	4		3150.2.bp.c.1349.2		8
15.14	odd	2		3150.2.bf.c.1601.4		8
21.2	odd	6		4410.2.b.b.881.8		8
21.5	even	6		4410.2.b.e.881.8		8
21.17	even	6	inner	630.2.be.a.521.3	yes	8
35.3	even	12		3150.2.bp.f.899.3		8
35.17	even	12		3150.2.bp.c.899.2		8
35.24	odd	6		3150.2.bf.c.1151.4		8
105.17	odd	12		3150.2.bp.d.899.2		8
105.38	odd	12		3150.2.bp.a.899.3		8
105.59	even	6		3150.2.bf.b.1151.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.3	✓	8	1.1	even	1	trivial
630.2.be.a.521.3	yes	8	21.17	even	6	inner
630.2.be.b.341.1	yes	8	3.2	odd	2
630.2.be.b.521.1	yes	8	7.3	odd	6
3150.2.bf.b.1151.2		8	105.59	even	6
3150.2.bf.b.1601.2		8	5.4	even	2
3150.2.bf.c.1151.4		8	35.24	odd	6
3150.2.bf.c.1601.4		8	15.14	odd	2
3150.2.bp.a.899.3		8	105.38	odd	12
3150.2.bp.a.1349.3		8	5.2	odd	4
3150.2.bp.c.899.2		8	35.17	even	12
3150.2.bp.c.1349.2		8	15.8	even	4
3150.2.bp.d.899.2		8	105.17	odd	12
3150.2.bp.d.1349.2		8	5.3	odd	4
3150.2.bp.f.899.3		8	35.3	even	12
3150.2.bp.f.1349.3		8	15.2	even	4
4410.2.b.b.881.1		8	7.5	odd	6
4410.2.b.b.881.8		8	21.2	odd	6
4410.2.b.e.881.1		8	7.2	even	3
4410.2.b.e.881.8		8	21.5	even	6