Embedded newform 3150.2.bp.f.1349.3

• Label: 3150.2.bp.f.1349.3
• Level: $3150$
• Weight: $2$
• Character: 3150.1349
• 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.bp (of order \(6\), degree \(2\), not 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			1349.3
Root			\(0.258819 - 0.965926i\) of defining polynomial
Character	\(\chi\)	\(=\)	3150.1349
Dual form			3150.2.bp.f.899.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.189469 - 2.63896i) q^{7} -1.00000 q^{8} +(4.67303 - 2.69798i) q^{11} +2.51764 q^{13} +(-2.19067 - 1.48356i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(3.89658 - 2.24969i) q^{17} +(2.48004 + 1.43185i) q^{19} -5.39595i q^{22} +(-0.133975 + 0.232051i) q^{23} +(1.25882 - 2.18034i) q^{26} +(-2.38014 + 1.15539i) q^{28} +8.89898i q^{29} +(4.18154 - 2.41421i) q^{31} +(0.500000 + 0.866025i) q^{32} -4.49938i q^{34} +(5.64444 + 3.25882i) q^{37} +(2.48004 - 1.43185i) q^{38} -0.760279 q^{41} +5.86370i q^{43} +(-4.67303 - 2.69798i) q^{44} +(0.133975 + 0.232051i) q^{46} +(-6.92418 - 3.99768i) q^{47} +(-6.92820 - 1.00000i) q^{49} +(-1.25882 - 2.18034i) q^{52} +(-4.19918 - 7.27319i) q^{53} +(-0.189469 + 2.63896i) q^{56} +(7.70674 + 4.44949i) q^{58} +(6.33573 + 10.9738i) q^{59} +(-2.27035 - 1.31079i) q^{61} -4.82843i q^{62} +1.00000 q^{64} +(8.50643 - 4.91119i) q^{67} +(-3.89658 - 2.24969i) q^{68} -4.76268i q^{71} +(-5.82843 - 10.0951i) q^{73} +(5.64444 - 3.25882i) q^{74} -2.86370i q^{76} +(-6.23445 - 12.8431i) q^{77} +(4.29618 - 7.44120i) q^{79} +(-0.380139 + 0.658421i) q^{82} +9.45001i q^{83} +(5.07812 + 2.93185i) q^{86} +(-4.67303 + 2.69798i) q^{88} +(-3.98502 + 6.90226i) q^{89} +(0.477014 - 6.64394i) q^{91} +0.267949 q^{92} +(-6.92418 + 3.99768i) q^{94} -6.16353 q^{97} +(-4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 + 24 * q^11 + 16 * q^13 - 4 * q^16 + 24 * q^17 - 8 * q^23 + 8 * q^26 + 4 * q^32 + 32 * q^41 - 24 * q^44 + 8 * q^46 + 12 * q^47 - 8 * q^52 + 4 * q^53 + 24 * q^59 + 8 * q^64 + 48 * q^67 - 24 * q^68 - 24 * q^73 - 4 * q^77 + 24 * q^79 + 16 * q^82 - 24 * q^88 + 16 * q^89 - 20 * q^91 + 16 * q^92 + 12 * q^94 - 48 * q^97

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	−	0.866025i	0.353553	−	0.612372i
							\(3\)		0			0
\(5\)		0			0
							\(7\)	0.189469	−	2.63896i	0.0716124	−	0.997433i
\(11\)	4.67303	−	2.69798i	1.40897	−	0.813471i								0.413683	−	0.910421i	\(-0.364242\pi\)
							0.995289	+	0.0969504i	\(0.0309088\pi\)
\(13\)	2.51764			0.698267			0.349134	−	0.937073i	\(-0.386476\pi\)
							0.349134	+	0.937073i	\(0.386476\pi\)
\(17\)	3.89658	−	2.24969i	0.945058	−	0.545630i	0.0535160	−	0.998567i	\(-0.482957\pi\)
							0.891542	+	0.452937i	\(0.149624\pi\)
\(19\)	2.48004	+	1.43185i	0.568960	+	0.328489i	0.756734	−	0.653723i	\(-0.226794\pi\)
							−0.187774	+	0.982212i	\(0.560127\pi\)
\(23\)	−0.133975	+	0.232051i	−0.0279356	+	0.0483859i	−0.879655	−	0.475612i	\(-0.842227\pi\)
							0.851720	+	0.523998i	\(0.175560\pi\)
\(29\)			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	\(0.190574\pi\)
\(37\)	5.64444	+	3.25882i	0.927940	+	0.535747i	0.886160	−	0.463380i	\(-0.153364\pi\)
							0.0417807	+	0.999127i	\(0.486697\pi\)
\(41\)	−0.760279			−0.118736			−0.0593678	−	0.998236i	\(-0.518908\pi\)
							−0.0593678	+	0.998236i	\(0.518908\pi\)
\(43\)			5.86370i			0.894206i	0.894482	+	0.447103i	\(0.147544\pi\)
							−0.894482	+	0.447103i	\(0.852456\pi\)
\(47\)	−6.92418	−	3.99768i	−1.01000	−	0.583121i	−0.0988053	−	0.995107i	\(-0.531502\pi\)
							−0.911190	+	0.411986i	\(0.864835\pi\)
\(53\)	−4.19918	−	7.27319i	−0.576802	−	0.999050i	−0.995843	−	0.0910826i	\(-0.970967\pi\)
							0.419042	−	0.907967i	\(-0.362366\pi\)
\(59\)	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	\(0.720342\pi\)
\(67\)	8.50643	−	4.91119i	1.03923	−	0.599997i	0.119612	−	0.992821i	\(-0.461835\pi\)
							0.919614	+	0.392824i	\(0.128502\pi\)
\(71\)		−	4.76268i		−	0.565226i	−0.959234	−	0.282613i	\(-0.908799\pi\)
							0.959234	−	0.282613i	\(-0.0912013\pi\)
\(73\)	−5.82843	−	10.0951i	−0.682166	−	1.18155i	−0.974319	−	0.225174i	\(-0.927705\pi\)
							0.292153	−	0.956372i	\(-0.405628\pi\)
\(79\)	4.29618	−	7.44120i	0.483358	−	0.837200i	−0.516460	−	0.856312i	\(-0.672750\pi\)
							0.999817	+	0.0191114i	\(0.00608373\pi\)
\(83\)			9.45001i			1.03727i	0.854995	+	0.518636i	\(0.173560\pi\)
							−0.854995	+	0.518636i	\(0.826440\pi\)
\(89\)	−3.98502	+	6.90226i	−0.422412	+	0.731638i	−0.996175	−	0.0873828i	\(-0.972150\pi\)
							0.573763	+	0.819021i	\(0.305483\pi\)
\(97\)	−6.16353			−0.625812			−0.312906	−	0.949784i	\(-0.601302\pi\)
							−0.312906	+	0.949784i	\(0.601302\pi\)

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.3	✓	8	15.8	even	4
630.2.be.a.521.3	yes	8	35.3	even	12
630.2.be.b.341.1	yes	8	5.3	odd	4
630.2.be.b.521.1	yes	8	105.38	odd	12
3150.2.bf.b.1151.2		8	35.17	even	12
3150.2.bf.b.1601.2		8	15.2	even	4
3150.2.bf.c.1151.4		8	105.17	odd	12
3150.2.bf.c.1601.4		8	5.2	odd	4
3150.2.bp.a.899.3		8	35.24	odd	6
3150.2.bp.a.1349.3		8	3.2	odd	2
3150.2.bp.c.899.2		8	21.17	even	6
3150.2.bp.c.1349.2		8	5.4	even	2
3150.2.bp.d.899.2		8	7.3	odd	6
3150.2.bp.d.1349.2		8	15.14	odd	2
3150.2.bp.f.899.3		8	105.59	even	6	inner
3150.2.bp.f.1349.3		8	1.1	even	1	trivial
4410.2.b.b.881.1		8	105.68	odd	12
4410.2.b.b.881.8		8	35.23	odd	12
4410.2.b.e.881.1		8	105.23	even	12
4410.2.b.e.881.8		8	35.33	even	12