Embedded newform 4900.2.e.p.2549.3

• Label: 4900.2.e.p.2549.3
• Level: $4900$
• Weight: $2$
• Character: 4900.2549
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
Defining polynomial:	$x^{4} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 196)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.p.2549.1

$q$-expansion

$f(q)$	$=$	$q+2.82843i q^{3} -5.00000 q^{9} +4.00000 q^{11} -4.24264i q^{13} +1.41421i q^{17} -2.82843 q^{19} +4.00000i q^{23} -5.65685i q^{27} -8.00000 q^{29} +11.3137i q^{33} -8.00000i q^{37} +12.0000 q^{39} -7.07107 q^{41} +4.00000i q^{43} +5.65685i q^{47} -4.00000 q^{51} -10.0000i q^{53} -8.00000i q^{57} -14.1421 q^{59} -7.07107 q^{61} -11.3137 q^{69} +7.07107i q^{73} -8.00000 q^{79} +1.00000 q^{81} +14.1421i q^{83} -22.6274i q^{87} -7.07107 q^{89} -1.41421i q^{97} -20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 20 q^{9} + 16 q^{11} - 32 q^{29} + 48 q^{39} - 16 q^{51} - 32 q^{79} + 4 q^{81} - 80 q^{99}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

(See $a_n$ instead)

$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$
$2$	0	0
			$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
−0.577350	+	0.816497i				$0.695913\pi$
$5$	0	0
			$7$	0	0
$11$	4.00000	1.20605				0.603023	−	0.797724i	$-0.293963\pi$
			0.603023	+	0.797724i	$0.293963\pi$
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
			0.808608	−	0.588348i	$-0.200222\pi$
$17$	1.41421i	0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
			−0.985184	+	0.171499i	$0.945139\pi$
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
			−0.324443	+	0.945905i	$0.605177\pi$
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
			−0.908893	+	0.417029i	$0.863071\pi$
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
			−0.742781	+	0.669534i	$0.766494\pi$
$31$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
			0.753371	−	0.657596i	$-0.228427\pi$
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
			−0.552158	+	0.833740i	$0.686195\pi$
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
			−0.952353	+	0.304997i	$0.901344\pi$
$47$	5.65685i	0.825137i	0.910927	+	0.412568i	$0.135368\pi$
			−0.910927	+	0.412568i	$0.864632\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.p.2549.3		4
5.2	odd	4		4900.2.a.y.1.2		2
5.3	odd	4		196.2.a.c.1.1	✓	2
5.4	even	2	inner	4900.2.e.p.2549.1		4
7.6	odd	2	inner	4900.2.e.p.2549.2		4
15.8	even	4		1764.2.a.l.1.1		2
20.3	even	4		784.2.a.m.1.2		2
35.3	even	12		196.2.e.b.177.1		4
35.13	even	4		196.2.a.c.1.2	yes	2
35.18	odd	12		196.2.e.b.177.2		4
35.23	odd	12		196.2.e.b.165.2		4
35.27	even	4		4900.2.a.y.1.1		2
35.33	even	12		196.2.e.b.165.1		4
35.34	odd	2	inner	4900.2.e.p.2549.4		4
40.3	even	4		3136.2.a.bs.1.1		2
40.13	odd	4		3136.2.a.br.1.2		2
60.23	odd	4		7056.2.a.cr.1.1		2
105.23	even	12		1764.2.k.l.361.2		4
105.38	odd	12		1764.2.k.l.1549.1		4
105.53	even	12		1764.2.k.l.1549.2		4
105.68	odd	12		1764.2.k.l.361.1		4
105.83	odd	4		1764.2.a.l.1.2		2
140.3	odd	12		784.2.i.l.177.2		4
140.23	even	12		784.2.i.l.753.1		4
140.83	odd	4		784.2.a.m.1.1		2
140.103	odd	12		784.2.i.l.753.2		4
140.123	even	12		784.2.i.l.177.1		4
280.13	even	4		3136.2.a.br.1.1		2
280.83	odd	4		3136.2.a.bs.1.2		2
420.83	even	4		7056.2.a.cr.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.a.c.1.1	✓	2	5.3	odd	4
196.2.a.c.1.2	yes	2	35.13	even	4
196.2.e.b.165.1		4	35.33	even	12
196.2.e.b.165.2		4	35.23	odd	12
196.2.e.b.177.1		4	35.3	even	12
196.2.e.b.177.2		4	35.18	odd	12
784.2.a.m.1.1		2	140.83	odd	4
784.2.a.m.1.2		2	20.3	even	4
784.2.i.l.177.1		4	140.123	even	12
784.2.i.l.177.2		4	140.3	odd	12
784.2.i.l.753.1		4	140.23	even	12
784.2.i.l.753.2		4	140.103	odd	12
1764.2.a.l.1.1		2	15.8	even	4
1764.2.a.l.1.2		2	105.83	odd	4
1764.2.k.l.361.1		4	105.68	odd	12
1764.2.k.l.361.2		4	105.23	even	12
1764.2.k.l.1549.1		4	105.38	odd	12
1764.2.k.l.1549.2		4	105.53	even	12
3136.2.a.br.1.1		2	280.13	even	4
3136.2.a.br.1.2		2	40.13	odd	4
3136.2.a.bs.1.1		2	40.3	even	4
3136.2.a.bs.1.2		2	280.83	odd	4
4900.2.a.y.1.1		2	35.27	even	4
4900.2.a.y.1.2		2	5.2	odd	4
4900.2.e.p.2549.1		4	5.4	even	2	inner
4900.2.e.p.2549.2		4	7.6	odd	2	inner
4900.2.e.p.2549.3		4	1.1	even	1	trivial
4900.2.e.p.2549.4		4	35.34	odd	2	inner
7056.2.a.cr.1.1		2	60.23	odd	4
7056.2.a.cr.1.2		2	420.83	even	4