Embedded newform 3150.2.bf.c.1601.4

• Label: 3150.2.bf.c.1601.4
• Level: $3150$
• Weight: $2$
• Character: 3150.1601
• Analytic conductor: $25.153$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(25.1528766367\)
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:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.4
Root			\(0.258819 - 0.965926i\) of defining polynomial
Character	\(\chi\)	\(=\)	3150.1601
Dual form			3150.2.bf.c.1151.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(2.63896 + 0.189469i) q^{7} +1.00000i q^{8} +(4.67303 - 2.69798i) q^{11} -2.51764i q^{13} +(2.19067 + 1.48356i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.24969 + 3.89658i) q^{17} +(-2.48004 - 1.43185i) q^{19} +5.39595 q^{22} +(0.232051 + 0.133975i) q^{23} +(1.25882 - 2.18034i) q^{26} +(1.15539 + 2.38014i) q^{28} -8.89898i q^{29} +(4.18154 - 2.41421i) q^{31} +(-0.866025 + 0.500000i) q^{32} +4.49938i q^{34} +(-3.25882 + 5.64444i) q^{37} +(-1.43185 - 2.48004i) q^{38} -0.760279 q^{41} +5.86370 q^{43} +(4.67303 + 2.69798i) q^{44} +(0.133975 + 0.232051i) q^{46} +(3.99768 - 6.92418i) q^{47} +(6.92820 + 1.00000i) q^{49} +(2.18034 - 1.25882i) q^{52} +(-7.27319 + 4.19918i) q^{53} +(-0.189469 + 2.63896i) q^{56} +(4.44949 - 7.70674i) q^{58} +(-6.33573 - 10.9738i) q^{59} +(-2.27035 - 1.31079i) q^{61} +4.82843 q^{62} -1.00000 q^{64} +(4.91119 + 8.50643i) q^{67} +(-2.24969 + 3.89658i) q^{68} -4.76268i q^{71} +(-10.0951 + 5.82843i) q^{73} +(-5.64444 + 3.25882i) q^{74} -2.86370i q^{76} +(12.8431 - 6.23445i) q^{77} +(-4.29618 + 7.44120i) q^{79} +(-0.658421 - 0.380139i) q^{82} +9.45001 q^{83} +(5.07812 + 2.93185i) q^{86} +(2.69798 + 4.67303i) q^{88} +(3.98502 - 6.90226i) q^{89} +(0.477014 - 6.64394i) q^{91} +0.267949i q^{92} +(6.92418 - 3.99768i) q^{94} -6.16353i q^{97} +(5.50000 + 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 4 * q^4 + 24 * q^11 - 4 * q^16 - 12 * q^23 + 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 + 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 \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(451\)	\(2801\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.866025	+	0.500000i	0.612372	+	0.353553i
							\(3\)		0			0
\(5\)		0			0
							\(7\)	2.63896	+	0.189469i	0.997433	+	0.0716124i
\(11\)	4.67303	−	2.69798i	1.40897	−	0.813471i								0.413683	−	0.910421i	\(-0.364242\pi\)
							0.995289	+	0.0969504i	\(0.0309088\pi\)
\(13\)		−	2.51764i		−	0.698267i	−0.937073	−	0.349134i	\(-0.886476\pi\)
							0.937073	−	0.349134i	\(-0.113524\pi\)
\(17\)	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	\(-0.690470\pi\)
							0.563304	−	0.826250i	\(-0.309530\pi\)
\(31\)	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	\(0.346636\pi\)
							−0.999127	+	0.0417807i	\(0.986697\pi\)
\(41\)	−0.760279			−0.118736			−0.0593678	−	0.998236i	\(-0.518908\pi\)
							−0.0593678	+	0.998236i	\(0.518908\pi\)
\(43\)	5.86370			0.894206			0.447103	−	0.894482i	\(-0.352456\pi\)
							0.447103	+	0.894482i	\(0.352456\pi\)
\(47\)	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	\(-0.857931\pi\)
							0.0771977	−	0.997016i	\(-0.475403\pi\)
\(61\)	−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	\(0.0381649\pi\)
							−0.392824	+	0.919614i	\(0.628502\pi\)
\(71\)		−	4.76268i		−	0.565226i	−0.959234	−	0.282613i	\(-0.908799\pi\)
							0.959234	−	0.282613i	\(-0.0912013\pi\)
\(73\)	−10.0951	+	5.82843i	−1.18155	+	0.682166i	−0.956372	−	0.292153i	\(-0.905628\pi\)
							−0.225174	+	0.974319i	\(0.572295\pi\)
\(79\)	−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	−0.573763	−	0.819021i	\(-0.694517\pi\)
							0.996175	+	0.0873828i	\(0.0278503\pi\)
\(97\)		−	6.16353i		−	0.625812i	−0.949784	−	0.312906i	\(-0.898698\pi\)
							0.949784	−	0.312906i	\(-0.101302\pi\)

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.3	✓	8	15.14	odd	2
630.2.be.a.521.3	yes	8	35.24	odd	6
630.2.be.b.341.1	yes	8	5.4	even	2
630.2.be.b.521.1	yes	8	105.59	even	6
3150.2.bf.b.1151.2		8	7.3	odd	6
3150.2.bf.b.1601.2		8	3.2	odd	2
3150.2.bf.c.1151.4		8	21.17	even	6	inner
3150.2.bf.c.1601.4		8	1.1	even	1	trivial
3150.2.bp.a.899.3		8	35.17	even	12
3150.2.bp.a.1349.3		8	15.8	even	4
3150.2.bp.c.899.2		8	105.38	odd	12
3150.2.bp.c.1349.2		8	5.2	odd	4
3150.2.bp.d.899.2		8	35.3	even	12
3150.2.bp.d.1349.2		8	15.2	even	4
3150.2.bp.f.899.3		8	105.17	odd	12
3150.2.bp.f.1349.3		8	5.3	odd	4
4410.2.b.b.881.1		8	105.89	even	6
4410.2.b.b.881.8		8	35.9	even	6
4410.2.b.e.881.1		8	105.44	odd	6
4410.2.b.e.881.8		8	35.19	odd	6