Embedded newform 3150.2.bf.b.1601.1

• Label: 3150.2.bf.b.1601.1
• 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			1601.1
Root			\(0.258819 - 0.965926i\) of defining polynomial
Character	\(\chi\)	\(=\)	3150.1601
Dual form			3150.2.bf.b.1151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.63896 - 0.189469i) q^{7} -1.00000i q^{8} +(-1.32697 + 0.766125i) q^{11} -1.48236i q^{13} +(2.19067 + 1.48356i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.21441 - 2.10342i) q^{17} +(4.21209 + 2.43185i) q^{19} +1.53225 q^{22} +(-0.232051 - 0.133975i) q^{23} +(-0.741181 + 1.28376i) q^{26} +(-1.15539 - 2.38014i) q^{28} -0.898979i q^{29} +(-0.717439 + 0.414214i) q^{31} +(0.866025 - 0.500000i) q^{32} +2.42883i q^{34} +(-2.74118 + 4.74786i) q^{37} +(-2.43185 - 4.21209i) q^{38} -8.76028 q^{41} -1.86370 q^{43} +(-1.32697 - 0.766125i) q^{44} +(0.133975 + 0.232051i) q^{46} +(3.72973 - 6.46008i) q^{47} +(6.92820 + 1.00000i) q^{49} +(1.28376 - 0.741181i) q^{52} +(-3.00524 + 1.73508i) q^{53} +(-0.189469 + 2.63896i) q^{56} +(-0.449490 + 0.778539i) q^{58} +(3.12837 + 5.41849i) q^{59} +(5.73445 + 3.31079i) q^{61} +0.828427 q^{62} -1.00000 q^{64} +(8.01702 + 13.8859i) q^{67} +(1.21441 - 2.10342i) q^{68} -12.7627i q^{71} +(-0.297173 + 0.171573i) q^{73} +(4.74786 - 2.74118i) q^{74} +4.86370i q^{76} +(3.64697 - 1.77035i) q^{77} +(5.22438 - 9.04889i) q^{79} +(7.58662 + 4.38014i) q^{82} +5.45001 q^{83} +(1.61401 + 0.931852i) q^{86} +(0.766125 + 1.32697i) q^{88} +(7.98502 - 13.8305i) q^{89} +(-0.280861 + 3.91189i) q^{91} -0.267949i q^{92} +(-6.46008 + 3.72973i) q^{94} +14.9481i q^{97} +(-5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
\(11\)	−1.32697	+	0.766125i	−0.400096	+	0.230995i								−0.686525	−	0.727106i	\(-0.740865\pi\)
							0.286430	+	0.958101i	\(0.407532\pi\)
\(13\)		−	1.48236i		−	0.411133i	−0.978643	−	0.205567i	\(-0.934096\pi\)
							0.978643	−	0.205567i	\(-0.0659037\pi\)
\(17\)	−1.21441	−	2.10342i	−0.294538	−	0.510155i	0.680339	−	0.732898i	\(-0.261833\pi\)
							−0.974877	+	0.222742i	\(0.928499\pi\)
\(19\)	4.21209	+	2.43185i	0.966320	+	0.557905i	0.898112	−	0.439766i	\(-0.144939\pi\)
							0.0682075	+	0.997671i	\(0.478272\pi\)
\(23\)	−0.232051	−	0.133975i	−0.0483859	−	0.0279356i	0.475612	−	0.879655i	\(-0.342227\pi\)
							−0.523998	+	0.851720i	\(0.675560\pi\)
\(29\)		−	0.898979i		−	0.166936i	−0.996510	−	0.0834681i	\(-0.973400\pi\)
							0.996510	−	0.0834681i	\(-0.0265997\pi\)
\(31\)	−0.717439	+	0.414214i	−0.128856	+	0.0743950i	−0.563042	−	0.826428i	\(-0.690369\pi\)
							0.434187	+	0.900823i	\(0.357036\pi\)
\(37\)	−2.74118	+	4.74786i	−0.450647	+	0.780544i	−0.998426	−	0.0560790i	\(-0.982140\pi\)
							0.547779	+	0.836623i	\(0.315473\pi\)
\(41\)	−8.76028			−1.36813			−0.684063	−	0.729423i	\(-0.739789\pi\)
							−0.684063	+	0.729423i	\(0.739789\pi\)
\(43\)	−1.86370			−0.284212			−0.142106	−	0.989851i	\(-0.545387\pi\)
							−0.142106	+	0.989851i	\(0.545387\pi\)
\(47\)	3.72973	−	6.46008i	0.544037	−	0.942299i	−0.454630	−	0.890680i	\(-0.650229\pi\)
							0.998667	−	0.0516191i	\(-0.0164382\pi\)
\(53\)	−3.00524	+	1.73508i	−0.412802	+	0.238331i	−0.691993	−	0.721904i	\(-0.743267\pi\)
							0.279191	+	0.960236i	\(0.409934\pi\)
\(59\)	3.12837	+	5.41849i	0.407279	+	0.705428i	0.994584	−	0.103938i	\(-0.0331444\pi\)
							−0.587305	+	0.809366i	\(0.699811\pi\)
\(61\)	5.73445	+	3.31079i	0.734222	+	0.423903i	0.819965	−	0.572414i	\(-0.193993\pi\)
							−0.0857429	+	0.996317i	\(0.527326\pi\)
\(67\)	8.01702	+	13.8859i	0.979434	+	1.69643i	0.664449	+	0.747334i	\(0.268666\pi\)
							0.314985	+	0.949097i	\(0.398000\pi\)
\(71\)		−	12.7627i		−	1.51465i	−0.653037	−	0.757326i	\(-0.726505\pi\)
							0.653037	−	0.757326i	\(-0.273495\pi\)
\(73\)	−0.297173	+	0.171573i	−0.0347815	+	0.0200811i	−0.517290	−	0.855810i	\(-0.673059\pi\)
							0.482508	+	0.875891i	\(0.339726\pi\)
\(79\)	5.22438	−	9.04889i	0.587789	−	1.01808i	−0.406733	−	0.913547i	\(-0.633332\pi\)
							0.994521	−	0.104533i	\(-0.0333347\pi\)
\(83\)	5.45001			0.598216			0.299108	−	0.954219i	\(-0.403311\pi\)
							0.299108	+	0.954219i	\(0.403311\pi\)
\(89\)	7.98502	−	13.8305i	0.846411	−	1.46603i	−0.0379795	−	0.999279i	\(-0.512092\pi\)
							0.884390	−	0.466748i	\(-0.154574\pi\)
\(97\)			14.9481i			1.51775i	0.651234	+	0.758877i	\(0.274252\pi\)
							−0.651234	+	0.758877i	\(0.725748\pi\)

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.b.1601.1		8
3.2	odd	2		3150.2.bf.c.1601.3		8
5.2	odd	4		3150.2.bp.d.1349.3		8
5.3	odd	4		3150.2.bp.a.1349.2		8
5.4	even	2		630.2.be.a.341.4	✓	8
7.3	odd	6		3150.2.bf.c.1151.3		8
15.2	even	4		3150.2.bp.c.1349.3		8
15.8	even	4		3150.2.bp.f.1349.2		8
15.14	odd	2		630.2.be.b.341.2	yes	8
21.17	even	6	inner	3150.2.bf.b.1151.1		8
35.3	even	12		3150.2.bp.c.899.3		8
35.9	even	6		4410.2.b.e.881.2		8
35.17	even	12		3150.2.bp.f.899.2		8
35.19	odd	6		4410.2.b.b.881.2		8
35.24	odd	6		630.2.be.b.521.2	yes	8
105.17	odd	12		3150.2.bp.a.899.2		8
105.38	odd	12		3150.2.bp.d.899.3		8
105.44	odd	6		4410.2.b.b.881.7		8
105.59	even	6		630.2.be.a.521.4	yes	8
105.89	even	6		4410.2.b.e.881.7		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.4	✓	8	5.4	even	2
630.2.be.a.521.4	yes	8	105.59	even	6
630.2.be.b.341.2	yes	8	15.14	odd	2
630.2.be.b.521.2	yes	8	35.24	odd	6
3150.2.bf.b.1151.1		8	21.17	even	6	inner
3150.2.bf.b.1601.1		8	1.1	even	1	trivial
3150.2.bf.c.1151.3		8	7.3	odd	6
3150.2.bf.c.1601.3		8	3.2	odd	2
3150.2.bp.a.899.2		8	105.17	odd	12
3150.2.bp.a.1349.2		8	5.3	odd	4
3150.2.bp.c.899.3		8	35.3	even	12
3150.2.bp.c.1349.3		8	15.2	even	4
3150.2.bp.d.899.3		8	105.38	odd	12
3150.2.bp.d.1349.3		8	5.2	odd	4
3150.2.bp.f.899.2		8	35.17	even	12
3150.2.bp.f.1349.2		8	15.8	even	4
4410.2.b.b.881.2		8	35.19	odd	6
4410.2.b.b.881.7		8	105.44	odd	6
4410.2.b.e.881.2		8	35.9	even	6
4410.2.b.e.881.7		8	105.89	even	6