Embedded newform 4608.2.a.c.1.2

• Label: 4608.2.a.c.1.2
• Level: $4608$
• Weight: $2$
• Character: 4608.1
• Self dual: yes
• Analytic conductor: $36.795$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(36.7950652514\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	4608.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} +2.82843 q^{7} +4.24264 q^{11} +6.00000 q^{13} +4.24264 q^{19} +8.48528 q^{23} -1.00000 q^{25} -2.00000 q^{29} -5.65685 q^{31} -5.65685 q^{35} +6.00000 q^{37} -6.00000 q^{41} +4.24264 q^{43} +1.00000 q^{49} +2.00000 q^{53} -8.48528 q^{55} +1.41421 q^{59} +6.00000 q^{61} -12.0000 q^{65} -12.7279 q^{67} -8.48528 q^{71} -12.0000 q^{73} +12.0000 q^{77} +5.65685 q^{79} +4.24264 q^{83} +12.0000 q^{89} +16.9706 q^{91} -8.48528 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 12 q^{13} - 2 q^{25} - 4 q^{29} + 12 q^{37} - 12 q^{41} + 2 q^{49} + 4 q^{53} + 12 q^{61} - 24 q^{65} - 24 q^{73} + 24 q^{77} + 24 q^{89} - 16 q^{97}+O(q^{100}) \)

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	0
			\(3\)	0	0
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	2.82843	1.06904	0.534522	−	0.845154i	\(-0.320491\pi\)
			0.534522	+	0.845154i	\(0.320491\pi\)
\(11\)	4.24264	1.27920	0.639602	−	0.768706i	\(-0.279099\pi\)
			0.639602	+	0.768706i	\(0.279099\pi\)
\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
			0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	4.24264	0.973329	0.486664	−	0.873589i	\(-0.338214\pi\)
			0.486664	+	0.873589i	\(0.338214\pi\)
\(23\)	8.48528	1.76930	0.884652	−	0.466252i	\(-0.154396\pi\)
			0.884652	+	0.466252i	\(0.154396\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	−5.65685	−1.01600	−0.508001	−	0.861357i	\(-0.669615\pi\)
			−0.508001	+	0.861357i	\(0.669615\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	4.24264	0.646997	0.323498	−	0.946229i	\(-0.395141\pi\)
			0.323498	+	0.946229i	\(0.395141\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	2.00000	0.274721	0.137361	−	0.990521i	\(-0.456138\pi\)
			0.137361	+	0.990521i	\(0.456138\pi\)
\(59\)	1.41421	0.184115	0.0920575	−	0.995754i	\(-0.470656\pi\)
			0.0920575	+	0.995754i	\(0.470656\pi\)
\(61\)	6.00000	0.768221	0.384111	−	0.923287i	\(-0.374508\pi\)
			0.384111	+	0.923287i	\(0.374508\pi\)
\(67\)	−12.7279	−1.55496	−0.777482	−	0.628906i	\(-0.783503\pi\)
			−0.777482	+	0.628906i	\(0.783503\pi\)
\(71\)	−8.48528	−1.00702	−0.503509	−	0.863990i	\(-0.667958\pi\)
			−0.503509	+	0.863990i	\(0.667958\pi\)
\(73\)	−12.0000	−1.40449	−0.702247	−	0.711934i	\(-0.747820\pi\)
			−0.702247	+	0.711934i	\(0.747820\pi\)
\(79\)	5.65685	0.636446	0.318223	−	0.948016i	\(-0.396914\pi\)
			0.318223	+	0.948016i	\(0.396914\pi\)
\(83\)	4.24264	0.465690	0.232845	−	0.972514i	\(-0.425196\pi\)
			0.232845	+	0.972514i	\(0.425196\pi\)
\(89\)	12.0000	1.27200	0.635999	−	0.771690i	\(-0.280588\pi\)
			0.635999	+	0.771690i	\(0.280588\pi\)
\(97\)	−8.00000	−0.812277	−0.406138	−	0.913812i	\(-0.633125\pi\)
			−0.406138	+	0.913812i	\(0.633125\pi\)

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	4608.2.a.c.1.2		2
3.2	odd	2		512.2.a.e.1.2	yes	2
4.3	odd	2	inner	4608.2.a.c.1.1		2
8.3	odd	2		4608.2.a.p.1.1		2
8.5	even	2		4608.2.a.p.1.2		2
12.11	even	2		512.2.a.e.1.1	yes	2
16.3	odd	4		4608.2.d.j.2305.4		4
16.5	even	4		4608.2.d.j.2305.1		4
16.11	odd	4		4608.2.d.j.2305.2		4
16.13	even	4		4608.2.d.j.2305.3		4
24.5	odd	2		512.2.a.b.1.1	✓	2
24.11	even	2		512.2.a.b.1.2	yes	2
48.5	odd	4		512.2.b.e.257.2		4
48.11	even	4		512.2.b.e.257.4		4
48.29	odd	4		512.2.b.e.257.3		4
48.35	even	4		512.2.b.e.257.1		4
96.5	odd	8		1024.2.e.h.769.2		4
96.11	even	8		1024.2.e.h.769.1		4
96.29	odd	8		1024.2.e.n.257.1		4
96.35	even	8		1024.2.e.h.257.1		4
96.53	odd	8		1024.2.e.n.769.1		4
96.59	even	8		1024.2.e.n.769.2		4
96.77	odd	8		1024.2.e.h.257.2		4
96.83	even	8		1024.2.e.n.257.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.b.1.1	✓	2	24.5	odd	2
512.2.a.b.1.2	yes	2	24.11	even	2
512.2.a.e.1.1	yes	2	12.11	even	2
512.2.a.e.1.2	yes	2	3.2	odd	2
512.2.b.e.257.1		4	48.35	even	4
512.2.b.e.257.2		4	48.5	odd	4
512.2.b.e.257.3		4	48.29	odd	4
512.2.b.e.257.4		4	48.11	even	4
1024.2.e.h.257.1		4	96.35	even	8
1024.2.e.h.257.2		4	96.77	odd	8
1024.2.e.h.769.1		4	96.11	even	8
1024.2.e.h.769.2		4	96.5	odd	8
1024.2.e.n.257.1		4	96.29	odd	8
1024.2.e.n.257.2		4	96.83	even	8
1024.2.e.n.769.1		4	96.53	odd	8
1024.2.e.n.769.2		4	96.59	even	8
4608.2.a.c.1.1		2	4.3	odd	2	inner
4608.2.a.c.1.2		2	1.1	even	1	trivial
4608.2.a.p.1.1		2	8.3	odd	2
4608.2.a.p.1.2		2	8.5	even	2
4608.2.d.j.2305.1		4	16.5	even	4
4608.2.d.j.2305.2		4	16.11	odd	4
4608.2.d.j.2305.3		4	16.13	even	4
4608.2.d.j.2305.4		4	16.3	odd	4