Embedded newform 4900.2.a.bi.1.4

• Label: 4900.2.a.bi.1.4
• Level: $4900$
• Weight: $2$
• Character: 4900.1
• Self dual: yes
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(39.1266969904\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - 6x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			\(2.28825\) of defining polynomial
Character	\(\chi\)	\(=\)	4900.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.28825 q^{3} +2.23607 q^{9} -5.47214 q^{11} -0.874032 q^{13} +4.57649 q^{17} +5.99070 q^{19} -3.47214 q^{23} -1.74806 q^{27} +0.236068 q^{29} +8.27895 q^{31} -12.5216 q^{33} +4.23607 q^{37} -2.00000 q^{39} +5.11667 q^{41} +3.76393 q^{43} +4.91034 q^{47} +10.4721 q^{51} +11.7082 q^{53} +13.7082 q^{57} -1.95440 q^{59} +7.53244 q^{61} -13.9443 q^{67} -7.94510 q^{69} +16.7082 q^{71} -7.53244 q^{73} -11.4721 q^{79} -10.7082 q^{81} +12.1877 q^{83} +0.540182 q^{87} +5.86319 q^{89} +18.9443 q^{93} +16.3516 q^{97} -12.2361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} + 4 q^{23} - 8 q^{29} + 8 q^{37} - 8 q^{39} + 24 q^{43} + 24 q^{51} + 20 q^{53} + 28 q^{57} - 20 q^{67} + 40 q^{71} - 28 q^{79} - 16 q^{81} + 40 q^{93} - 40 q^{99}+O(q^{100}) \)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	2.28825	1.32112	0.660560	−	0.750774i	\(-0.270319\pi\)
0.660560	+	0.750774i				\(0.270319\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	−5.47214	−1.64991				−0.824956	−	0.565198i	\(-0.808800\pi\)
			−0.824956	+	0.565198i	\(0.808800\pi\)
\(13\)	−0.874032	−0.242413	−0.121206	−	0.992627i	\(-0.538676\pi\)
			−0.121206	+	0.992627i	\(0.538676\pi\)
\(17\)	4.57649	1.10996	0.554981	−	0.831863i	\(-0.312725\pi\)
			0.554981	+	0.831863i	\(0.312725\pi\)
\(19\)	5.99070	1.37436	0.687181	−	0.726486i	\(-0.258848\pi\)
			0.687181	+	0.726486i	\(0.258848\pi\)
\(23\)	−3.47214	−0.723990	−0.361995	−	0.932180i	\(-0.617904\pi\)
			−0.361995	+	0.932180i	\(0.617904\pi\)
\(29\)	0.236068	0.0438367	0.0219184	−	0.999760i	\(-0.493023\pi\)
			0.0219184	+	0.999760i	\(0.493023\pi\)
\(31\)	8.27895	1.48694	0.743472	−	0.668767i	\(-0.233178\pi\)
			0.743472	+	0.668767i	\(0.233178\pi\)
\(37\)	4.23607	0.696405	0.348203	−	0.937419i	\(-0.386792\pi\)
			0.348203	+	0.937419i	\(0.386792\pi\)
\(41\)	5.11667	0.799090	0.399545	−	0.916714i	\(-0.369168\pi\)
			0.399545	+	0.916714i	\(0.369168\pi\)
\(43\)	3.76393	0.573994	0.286997	−	0.957931i	\(-0.407343\pi\)
			0.286997	+	0.957931i	\(0.407343\pi\)
\(47\)	4.91034	0.716247	0.358123	−	0.933674i	\(-0.383417\pi\)
			0.358123	+	0.933674i	\(0.383417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bi.1.4	yes	4
5.2	odd	4		4900.2.e.u.2549.1		8
5.3	odd	4		4900.2.e.u.2549.7		8
5.4	even	2		4900.2.a.bg.1.1	✓	4
7.6	odd	2	inner	4900.2.a.bi.1.1	yes	4
35.13	even	4		4900.2.e.u.2549.2		8
35.27	even	4		4900.2.e.u.2549.8		8
35.34	odd	2		4900.2.a.bg.1.4	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4900.2.a.bg.1.1	✓	4	5.4	even	2
4900.2.a.bg.1.4	yes	4	35.34	odd	2
4900.2.a.bi.1.1	yes	4	7.6	odd	2	inner
4900.2.a.bi.1.4	yes	4	1.1	even	1	trivial
4900.2.e.u.2549.1		8	5.2	odd	4
4900.2.e.u.2549.2		8	35.13	even	4
4900.2.e.u.2549.7		8	5.3	odd	4
4900.2.e.u.2549.8		8	35.27	even	4