Embedded newform 135.4.b.a.109.1

• Label: 135.4.b.a.109.1
• Level: $135$
• Weight: $4$
• Character: 135.109
• Analytic conductor: $7.965$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -15
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.96525785077\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{5})\)
Defining polynomial:	\( x^{4} + 3x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.1
Root			\(-0.618034i\) of defining polynomial
Character	\(\chi\)	\(=\)	135.109
Dual form			135.4.b.a.109.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-5.61803i q^{2} -23.5623 q^{4} +11.1803i q^{5} +87.4296i q^{8} +62.8115 q^{10} +302.684 q^{16} +51.3181i q^{17} -61.8754 q^{19} -263.435i q^{20} +220.193i q^{23} -125.000 q^{25} -105.371 q^{31} -1001.05i q^{32} +288.307 q^{34} +347.618i q^{38} -977.492 q^{40} +1237.05 q^{46} +545.601i q^{47} +343.000 q^{49} +702.254i q^{50} -85.1866i q^{53} -585.735 q^{61} +591.976i q^{62} -3202.47 q^{64} -1209.17i q^{68} +1457.93 q^{76} -1035.35 q^{79} +3384.11i q^{80} +75.9567i q^{83} -573.754 q^{85} -5188.27i q^{92} +3065.20 q^{94} -691.788i q^{95} -1926.99i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 54 q^{4} + 50 q^{10} + 446 q^{16} - 328 q^{19} - 500 q^{25} + 464 q^{31} - 14 q^{34} - 2300 q^{40} + 3298 q^{46} + 1372 q^{49} + 716 q^{61} - 6692 q^{64} + 3618 q^{76} + 608 q^{79} - 3100 q^{85} + 2440 q^{94}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	− 5.61803i	− 1.98627i	−0.116953	−	0.993137i	\(-0.537313\pi\)
			0.116953	−	0.993137i	\(-0.462687\pi\)
\(3\)	0	0
			\(5\)	11.1803i	1.00000i
\(7\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	51.3181i	0.732145i	0.930586	+	0.366073i	\(0.119298\pi\)
			−0.930586	+	0.366073i	\(0.880702\pi\)
\(19\)	−61.8754	−0.747115	−0.373558	−	0.927607i	\(-0.621862\pi\)
			−0.373558	+	0.927607i	\(0.621862\pi\)
\(23\)	220.193i	1.99624i	0.0612908	+	0.998120i	\(0.480478\pi\)
			−0.0612908	+	0.998120i	\(0.519522\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−105.371	−0.610488	−0.305244	−	0.952274i	\(-0.598738\pi\)
			−0.305244	+	0.952274i	\(0.598738\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	545.601i	1.69328i	0.532168	+	0.846639i	\(0.321377\pi\)
			−0.532168	+	0.846639i	\(0.678623\pi\)
\(53\)	− 85.1866i	− 0.220779i	−0.993888	−	0.110389i	\(-0.964790\pi\)
			0.993888	−	0.110389i	\(-0.0352098\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−585.735	−1.22944	−0.614719	−	0.788746i	\(-0.710731\pi\)
			−0.614719	+	0.788746i	\(0.710731\pi\)
\(67\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(79\)	−1035.35	−1.47451	−0.737255	−	0.675615i	\(-0.763878\pi\)
			−0.737255	+	0.675615i	\(0.763878\pi\)
\(83\)	75.9567i	0.100450i	0.998738	+	0.0502249i	\(0.0159938\pi\)
			−0.998738	+	0.0502249i	\(0.984006\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

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	135.4.b.a.109.1	✓	4
3.2	odd	2	inner	135.4.b.a.109.4	yes	4
5.2	odd	4		675.4.a.o.1.2		2
5.3	odd	4		675.4.a.k.1.1		2
5.4	even	2	inner	135.4.b.a.109.4	yes	4
15.2	even	4		675.4.a.k.1.1		2
15.8	even	4		675.4.a.o.1.2		2
15.14	odd	2	CM	135.4.b.a.109.1	✓	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.a.109.1	✓	4	1.1	even	1	trivial
135.4.b.a.109.1	✓	4	15.14	odd	2	CM
135.4.b.a.109.4	yes	4	3.2	odd	2	inner
135.4.b.a.109.4	yes	4	5.4	even	2	inner
675.4.a.k.1.1		2	5.3	odd	4
675.4.a.k.1.1		2	15.2	even	4
675.4.a.o.1.2		2	5.2	odd	4
675.4.a.o.1.2		2	15.8	even	4