Embedded newform 630.2.be.b.341.3

• Label: 630.2.be.b.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			\(0.258819 - 0.965926i\) of defining polynomial
Character	\(\chi\)	\(=\)	630.341
Dual form			630.2.be.b.521.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-0.189469 + 2.63896i) q^{7} +1.00000i q^{8} +(0.866025 - 0.500000i) q^{10} +(3.44829 - 1.99087i) q^{11} -0.0681483i q^{13} +(-1.48356 + 2.19067i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(3.66390 + 6.34607i) q^{17} +(-1.76260 - 1.01764i) q^{19} +1.00000 q^{20} +3.98174 q^{22} +(3.23205 + 1.86603i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(0.0340742 - 0.0590182i) q^{26} +(-2.38014 + 1.15539i) q^{28} -0.898979i q^{29} +(-4.18154 + 2.41421i) q^{31} +(-0.866025 + 0.500000i) q^{32} +7.32780i q^{34} +(2.19067 + 1.48356i) q^{35} +(2.03407 - 3.52312i) q^{37} +(-1.01764 - 1.76260i) q^{38} +(0.866025 + 0.500000i) q^{40} +1.68921 q^{41} -0.964724 q^{43} +(3.44829 + 1.99087i) q^{44} +(1.86603 + 3.23205i) q^{46} +(-0.830749 + 1.43890i) q^{47} +(-6.92820 - 1.00000i) q^{49} -1.00000i q^{50} +(0.0590182 - 0.0340742i) q^{52} +(11.4547 - 6.61339i) q^{53} -3.98174i q^{55} +(-2.63896 - 0.189469i) q^{56} +(0.449490 - 0.778539i) q^{58} +(-5.32112 - 9.21645i) q^{59} +(6.51299 + 3.76028i) q^{61} -4.82843 q^{62} -1.00000 q^{64} +(-0.0590182 - 0.0340742i) q^{65} +(-5.33145 - 9.23435i) q^{67} +(-3.66390 + 6.34607i) q^{68} +(1.15539 + 2.38014i) q^{70} -9.93426i q^{71} +(-10.0951 + 5.82843i) q^{73} +(3.52312 - 2.03407i) q^{74} -2.03528i q^{76} +(4.60048 + 9.47710i) q^{77} +(-8.77489 + 15.1986i) q^{79} +(0.500000 + 0.866025i) q^{80} +(1.46290 + 0.844605i) q^{82} -14.3490 q^{83} +7.32780 q^{85} +(-0.835475 - 0.482362i) q^{86} +(1.99087 + 3.44829i) q^{88} +(-0.913956 + 1.58302i) q^{89} +(0.179841 + 0.0129120i) q^{91} +3.73205i q^{92} +(-1.43890 + 0.830749i) q^{94} +(-1.76260 + 1.01764i) q^{95} -17.1502i q^{97} +(-5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
							\(7\)	−0.189469	+	2.63896i	−0.0716124	+	0.997433i
\(11\)	3.44829	−	1.99087i	1.03970	−	0.600270i								0.119950	−	0.992780i	\(-0.461727\pi\)
							0.919748	+	0.392510i	\(0.128393\pi\)
\(13\)		−	0.0681483i		−	0.0189010i	−0.999955	−	0.00945048i	\(-0.996992\pi\)
							0.999955	−	0.00945048i	\(-0.00300822\pi\)
\(17\)	3.66390	+	6.34607i	0.888627	+	1.53915i	0.841499	+	0.540258i	\(0.181673\pi\)
							0.0471274	+	0.998889i	\(0.484993\pi\)
\(19\)	−1.76260	−	1.01764i	−0.404368	−	0.233462i	0.283999	−	0.958825i	\(-0.408339\pi\)
							−0.688367	+	0.725362i	\(0.741672\pi\)
\(23\)	3.23205	+	1.86603i	0.673929	+	0.389093i	0.797564	−	0.603235i	\(-0.206122\pi\)
							−0.123635	+	0.992328i	\(0.539455\pi\)
\(29\)		−	0.898979i		−	0.166936i	−0.996510	−	0.0834681i	\(-0.973400\pi\)
							0.996510	−	0.0834681i	\(-0.0265997\pi\)
\(31\)	−4.18154	+	2.41421i	−0.751027	+	0.433606i	−0.826065	−	0.563575i	\(-0.809426\pi\)
							0.0750380	+	0.997181i	\(0.476092\pi\)
\(37\)	2.03407	−	3.52312i	0.334400	−	0.579197i	−0.648970	−	0.760814i	\(-0.724800\pi\)
							0.983369	+	0.181617i	\(0.0581331\pi\)
\(41\)	1.68921			0.263810			0.131905	−	0.991262i	\(-0.457891\pi\)
							0.131905	+	0.991262i	\(0.457891\pi\)
\(43\)	−0.964724			−0.147119			−0.0735595	−	0.997291i	\(-0.523436\pi\)
							−0.0735595	+	0.997291i	\(0.523436\pi\)
\(47\)	−0.830749	+	1.43890i	−0.121177	+	0.209885i	−0.920232	−	0.391373i	\(-0.872000\pi\)
							0.799055	+	0.601258i	\(0.205334\pi\)
\(53\)	11.4547	−	6.61339i	1.57343	−	0.908419i	0.577684	−	0.816260i	\(-0.303957\pi\)
							0.995744	−	0.0921588i	\(-0.0293767\pi\)
\(59\)	−5.32112	−	9.21645i	−0.692751	−	1.19988i	−0.970933	−	0.239352i	\(-0.923065\pi\)
							0.278182	−	0.960528i	\(-0.410268\pi\)
\(61\)	6.51299	+	3.76028i	0.833903	+	0.481454i	0.855187	−	0.518319i	\(-0.173442\pi\)
							−0.0212839	+	0.999773i	\(0.506775\pi\)
\(67\)	−5.33145	−	9.23435i	−0.651341	−	1.12816i	−0.982798	−	0.184685i	\(-0.940874\pi\)
							0.331457	−	0.943470i	\(-0.392460\pi\)
\(71\)		−	9.93426i		−	1.17898i	−0.807776	−	0.589490i	\(-0.799329\pi\)
							0.807776	−	0.589490i	\(-0.200671\pi\)
\(73\)	−10.0951	+	5.82843i	−1.18155	+	0.682166i	−0.956372	−	0.292153i	\(-0.905628\pi\)
							−0.225174	+	0.974319i	\(0.572295\pi\)
\(79\)	−8.77489	+	15.1986i	−0.987252	+	1.70997i	−0.355787	+	0.934567i	\(0.615787\pi\)
							−0.631465	+	0.775404i	\(0.717546\pi\)
\(83\)	−14.3490			−1.57501			−0.787503	−	0.616311i	\(-0.788626\pi\)
							−0.787503	+	0.616311i	\(0.788626\pi\)
\(89\)	−0.913956	+	1.58302i	−0.0968791	+	0.167800i	−0.910391	−	0.413748i	\(-0.864219\pi\)
							0.813512	+	0.581548i	\(0.197553\pi\)
\(97\)		−	17.1502i		−	1.74134i	−0.491870	−	0.870668i	\(-0.663687\pi\)
							0.491870	−	0.870668i	\(-0.336313\pi\)

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.b.341.3	yes	8
3.2	odd	2		630.2.be.a.341.1	✓	8
5.2	odd	4		3150.2.bp.c.1349.1		8
5.3	odd	4		3150.2.bp.f.1349.4		8
5.4	even	2		3150.2.bf.c.1601.2		8
7.2	even	3		4410.2.b.b.881.4		8
7.3	odd	6		630.2.be.a.521.1	yes	8
7.5	odd	6		4410.2.b.e.881.4		8
15.2	even	4		3150.2.bp.d.1349.1		8
15.8	even	4		3150.2.bp.a.1349.4		8
15.14	odd	2		3150.2.bf.b.1601.4		8
21.2	odd	6		4410.2.b.e.881.5		8
21.5	even	6		4410.2.b.b.881.5		8
21.17	even	6	inner	630.2.be.b.521.3	yes	8
35.3	even	12		3150.2.bp.d.899.1		8
35.17	even	12		3150.2.bp.a.899.4		8
35.24	odd	6		3150.2.bf.b.1151.4		8
105.17	odd	12		3150.2.bp.f.899.4		8
105.38	odd	12		3150.2.bp.c.899.1		8
105.59	even	6		3150.2.bf.c.1151.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.1	✓	8	3.2	odd	2
630.2.be.a.521.1	yes	8	7.3	odd	6
630.2.be.b.341.3	yes	8	1.1	even	1	trivial
630.2.be.b.521.3	yes	8	21.17	even	6	inner
3150.2.bf.b.1151.4		8	35.24	odd	6
3150.2.bf.b.1601.4		8	15.14	odd	2
3150.2.bf.c.1151.2		8	105.59	even	6
3150.2.bf.c.1601.2		8	5.4	even	2
3150.2.bp.a.899.4		8	35.17	even	12
3150.2.bp.a.1349.4		8	15.8	even	4
3150.2.bp.c.899.1		8	105.38	odd	12
3150.2.bp.c.1349.1		8	5.2	odd	4
3150.2.bp.d.899.1		8	35.3	even	12
3150.2.bp.d.1349.1		8	15.2	even	4
3150.2.bp.f.899.4		8	105.17	odd	12
3150.2.bp.f.1349.4		8	5.3	odd	4
4410.2.b.b.881.4		8	7.2	even	3
4410.2.b.b.881.5		8	21.5	even	6
4410.2.b.e.881.4		8	7.5	odd	6
4410.2.b.e.881.5		8	21.2	odd	6