Embedded newform 45.4.l.a.2.13

• Label: 45.4.l.a.2.13
• Level: $45$
• Weight: $4$
• Character: 45.2
• Analytic conductor: $2.655$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.65508595026\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	45.2
Dual form			45.4.l.a.23.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.956489 + 3.56967i) q^{2} +(3.31660 - 4.00002i) q^{3} +(-4.89944 + 2.82870i) q^{4} +(6.37460 + 9.18501i) q^{5} +(17.4510 + 8.01320i) q^{6} +(-4.17492 + 1.11867i) q^{7} +(6.12165 + 6.12165i) q^{8} +(-5.00028 - 26.5329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{2} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.956489	+	3.56967i	0.338170	+	1.26207i	0.900391	+	0.435081i	\(0.143280\pi\)
							−0.562221	+	0.826987i	\(0.690053\pi\)
\(3\)	3.31660	−	4.00002i	0.638281	−	0.769804i
							\(5\)	6.37460	+	9.18501i	0.570162	+	0.821532i
\(7\)	−4.17492	+	1.11867i	−0.225425	+	0.0604023i								−0.369763	−	0.929126i	\(-0.620561\pi\)
							0.144339	+	0.989528i	\(0.453895\pi\)
\(11\)	−19.8083	−	11.4363i	−0.542947	−	0.313470i	0.203326	−	0.979111i	\(-0.434825\pi\)
							−0.746272	+	0.665641i	\(0.768158\pi\)
\(13\)	−39.8912	−	10.6888i	−0.851064	−	0.228042i	−0.193182	−	0.981163i	\(-0.561881\pi\)
							−0.657882	+	0.753121i	\(0.728547\pi\)
\(17\)	53.3704	−	53.3704i	0.761425	−	0.761425i	−0.215155	−	0.976580i	\(-0.569026\pi\)
							0.976580	+	0.215155i	\(0.0690256\pi\)
\(19\)		−	63.9989i		−	0.772755i	−0.922341	−	0.386378i	\(-0.873726\pi\)
							0.922341	−	0.386378i	\(-0.126274\pi\)
\(23\)	37.9948	−	141.799i	0.344455	−	1.28552i	−0.548792	−	0.835959i	\(-0.684912\pi\)
							0.893247	−	0.449566i	\(-0.148421\pi\)
\(29\)	−145.682	+	252.328i	−0.932843	+	1.61573i	−0.154408	+	0.988007i	\(0.549347\pi\)
							−0.778435	+	0.627725i	\(0.783986\pi\)
\(31\)	60.4920	+	104.775i	0.350474	+	0.607038i	0.986333	−	0.164767i	\(-0.0526872\pi\)
							−0.635859	+	0.771805i	\(0.719354\pi\)
\(37\)	−132.295	−	132.295i	−0.587816	−	0.587816i	0.349224	−	0.937039i	\(-0.386445\pi\)
							−0.937039	+	0.349224i	\(0.886445\pi\)
\(41\)	169.696	−	97.9741i	0.646392	−	0.373195i	−0.140680	−	0.990055i	\(-0.544929\pi\)
							0.787073	+	0.616860i	\(0.211596\pi\)
\(43\)	−49.1037	−	183.258i	−0.174145	−	0.649919i	−0.996696	−	0.0812268i	\(-0.974116\pi\)
							0.822550	−	0.568692i	\(-0.192550\pi\)
\(47\)	74.4360	+	277.799i	0.231013	+	0.862152i	0.979906	+	0.199461i	\(0.0639190\pi\)
							−0.748893	+	0.662691i	\(0.769414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.l.a.2.13	✓	64
3.2	odd	2		135.4.m.a.62.4		64
5.2	odd	4		225.4.p.b.218.4		64
5.3	odd	4	inner	45.4.l.a.38.13	yes	64
5.4	even	2		225.4.p.b.182.4		64
9.4	even	3		135.4.m.a.17.4		64
9.5	odd	6	inner	45.4.l.a.32.13	yes	64
15.8	even	4		135.4.m.a.8.4		64
45.13	odd	12		135.4.m.a.98.4		64
45.14	odd	6		225.4.p.b.32.4		64
45.23	even	12	inner	45.4.l.a.23.13	yes	64
45.32	even	12		225.4.p.b.68.4		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.l.a.2.13	✓	64	1.1	even	1	trivial
45.4.l.a.23.13	yes	64	45.23	even	12	inner
45.4.l.a.32.13	yes	64	9.5	odd	6	inner
45.4.l.a.38.13	yes	64	5.3	odd	4	inner
135.4.m.a.8.4		64	15.8	even	4
135.4.m.a.17.4		64	9.4	even	3
135.4.m.a.62.4		64	3.2	odd	2
135.4.m.a.98.4		64	45.13	odd	12
225.4.p.b.32.4		64	45.14	odd	6
225.4.p.b.68.4		64	45.32	even	12
225.4.p.b.182.4		64	5.4	even	2
225.4.p.b.218.4		64	5.2	odd	4