Embedded newform 875.2.bb.b.857.10

• Label: 875.2.bb.b.857.10
• Level: $875$
• Weight: $2$
• Character: 875.857
• Analytic conductor: $6.987$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 875 = 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	875.bb (of order \(60\), degree \(16\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.98691017686\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{60})\)
Twist minimal:	no (minimal twist has level 175)
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			857.10
Character	\(\chi\)	\(=\)	875.857
Dual form			875.2.bb.b.243.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0171512 + 0.000898858i) q^{2} +(0.711812 - 0.879015i) q^{3} +(-1.98875 + 0.209026i) q^{4} +(-0.0114183 + 0.0157160i) q^{6} +(-0.822022 - 2.51481i) q^{7} +(0.0678483 - 0.0107461i) q^{8} +(0.357745 + 1.68306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q + 2 q^{2} + 6 q^{3} + 10 q^{4} - 10 q^{7} + 64 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/875\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(626\)
\(\chi(n)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0171512		0.000898858i	−0.0121278		0.000635588i	−0.0583989	−	0.998293i	\(-0.518600\pi\)
							0.0462711	+	0.998929i	\(0.485266\pi\)
\(3\)	0.711812	−	0.879015i	0.410965	−	0.507499i	−0.529070	−	0.848578i	\(-0.677459\pi\)
							0.940035	+	0.341079i	\(0.110792\pi\)
\(5\)		0			0
							\(7\)	−0.822022	−	2.51481i	−0.310695	−	0.950510i
\(11\)	−4.96090	−	1.05447i	−1.49577	−	0.317935i								−0.613882	−	0.789398i	\(-0.710393\pi\)
							−0.881886	+	0.471463i	\(0.843726\pi\)
\(13\)	0.286147	−	0.145799i	0.0793630	−	0.0404375i	−0.413859	−	0.910341i	\(-0.635819\pi\)
							0.493222	+	0.869904i	\(0.335819\pi\)
\(17\)	−1.33924	+	3.48884i	−0.324814	+	0.846169i	0.669802	+	0.742540i	\(0.266379\pi\)
							−0.994616	+	0.103629i	\(0.966954\pi\)
\(19\)	−0.698358	+	6.64443i	−0.160214	+	1.52434i	0.558777	+	0.829318i	\(0.311271\pi\)
							−0.718991	+	0.695019i	\(0.755396\pi\)
\(23\)	0.178896	+	3.41354i	0.0373024	+	0.711772i	0.951714	+	0.306986i	\(0.0993206\pi\)
							−0.914412	+	0.404786i	\(0.867346\pi\)
\(29\)	−1.99592	−	2.74715i	−0.370633	−	0.510133i	0.582440	−	0.812874i	\(-0.302098\pi\)
							−0.953073	+	0.302741i	\(0.902098\pi\)
\(31\)	0.305449	+	0.686049i	0.0548602	+	0.123218i	0.938896	−	0.344202i	\(-0.111850\pi\)
							−0.884036	+	0.467420i	\(0.845184\pi\)
\(37\)	2.76416	+	1.79507i	0.454426	+	0.295107i	0.751483	−	0.659752i	\(-0.229339\pi\)
							−0.297058	+	0.954860i	\(0.596005\pi\)
\(41\)	−7.88534	−	2.56210i	−1.23148	−	0.400133i	−0.380232	−	0.924891i	\(-0.624156\pi\)
							−0.851252	+	0.524758i	\(0.824156\pi\)
\(43\)	−3.99562	+	3.99562i	−0.609326	+	0.609326i	−0.942770	−	0.333444i	\(-0.891789\pi\)
							0.333444	+	0.942770i	\(0.391789\pi\)
\(47\)	−5.87681	+	2.25590i	−0.857221	+	0.329056i	−0.746978	−	0.664849i	\(-0.768496\pi\)
							−0.110242	+	0.993905i	\(0.535163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	875.2.bb.b.857.10		288
5.2	odd	4		875.2.bb.c.143.10		288
5.3	odd	4		175.2.x.a.108.9	yes	288
5.4	even	2		875.2.bb.a.857.9		288
7.5	odd	6	inner	875.2.bb.b.607.9		288
25.3	odd	20	inner	875.2.bb.b.493.9		288
25.4	even	10		875.2.bb.c.507.10		288
25.21	even	5		175.2.x.a.122.9	yes	288
25.22	odd	20		875.2.bb.a.493.10		288
35.12	even	12		875.2.bb.c.768.10		288
35.19	odd	6		875.2.bb.a.607.10		288
35.33	even	12		175.2.x.a.33.9	✓	288
175.47	even	60		875.2.bb.a.243.9		288
175.54	odd	30		875.2.bb.c.257.10		288
175.96	odd	30		175.2.x.a.47.9	yes	288
175.103	even	60	inner	875.2.bb.b.243.10		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.x.a.33.9	✓	288	35.33	even	12
175.2.x.a.47.9	yes	288	175.96	odd	30
175.2.x.a.108.9	yes	288	5.3	odd	4
175.2.x.a.122.9	yes	288	25.21	even	5
875.2.bb.a.243.9		288	175.47	even	60
875.2.bb.a.493.10		288	25.22	odd	20
875.2.bb.a.607.10		288	35.19	odd	6
875.2.bb.a.857.9		288	5.4	even	2
875.2.bb.b.243.10		288	175.103	even	60	inner
875.2.bb.b.493.9		288	25.3	odd	20	inner
875.2.bb.b.607.9		288	7.5	odd	6	inner
875.2.bb.b.857.10		288	1.1	even	1	trivial
875.2.bb.c.143.10		288	5.2	odd	4
875.2.bb.c.257.10		288	175.54	odd	30
875.2.bb.c.507.10		288	25.4	even	10
875.2.bb.c.768.10		288	35.12	even	12