Embedded newform 4312.2.a.s.1.2

• Label: 4312.2.a.s.1.2
• Level: $4312$
• Weight: $2$
• Character: 4312.1
• Self dual: yes
• Analytic conductor: $34.431$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4312 = 2^{3} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4312.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(34.4314933516\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	4312.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} +1.00000 q^{11} +2.00000 q^{15} -5.65685 q^{17} -6.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} -6.00000 q^{29} -7.07107 q^{31} +1.41421 q^{33} +6.00000 q^{37} -8.00000 q^{43} -1.41421 q^{45} -1.41421 q^{47} -8.00000 q^{51} +1.41421 q^{55} -9.89949 q^{59} -8.48528 q^{61} +14.0000 q^{67} -8.48528 q^{69} +2.00000 q^{71} +2.82843 q^{73} -4.24264 q^{75} -5.00000 q^{81} +5.65685 q^{83} -8.00000 q^{85} -8.48528 q^{87} +18.3848 q^{89} -10.0000 q^{93} +9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 2 q^{11} + 4 q^{15} - 12 q^{23} - 6 q^{25} - 12 q^{29} + 12 q^{37} - 16 q^{43} - 16 q^{51} + 28 q^{67} + 4 q^{71} - 10 q^{81} - 16 q^{85} - 20 q^{93} - 2 q^{99}+O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
			\(3\)	1.41421	0.816497	0.408248	−	0.912871i	\(-0.366140\pi\)
0.408248	+	0.912871i				\(0.366140\pi\)
\(5\)	1.41421	0.632456	0.316228	−	0.948683i	\(-0.397584\pi\)
			0.316228	+	0.948683i	\(0.397584\pi\)
\(7\)	0	0
			\(11\)	1.00000	0.301511
\(13\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	−5.65685	−1.37199	−0.685994	−	0.727607i	\(-0.740633\pi\)
			−0.685994	+	0.727607i	\(0.740633\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−6.00000	−1.25109	−0.625543	−	0.780189i	\(-0.715123\pi\)
			−0.625543	+	0.780189i	\(0.715123\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−7.07107	−1.27000	−0.635001	−	0.772512i	\(-0.719000\pi\)
			−0.635001	+	0.772512i	\(0.719000\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−1.41421	−0.206284	−0.103142	−	0.994667i	\(-0.532890\pi\)
			−0.103142	+	0.994667i	\(0.532890\pi\)
\(53\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(59\)	−9.89949	−1.28880	−0.644402	−	0.764687i	\(-0.722894\pi\)
			−0.644402	+	0.764687i	\(0.722894\pi\)
\(61\)	−8.48528	−1.08643	−0.543214	−	0.839594i	\(-0.682793\pi\)
			−0.543214	+	0.839594i	\(0.682793\pi\)
\(67\)	14.0000	1.71037	0.855186	−	0.518321i	\(-0.173443\pi\)
			0.855186	+	0.518321i	\(0.173443\pi\)
\(71\)	2.00000	0.237356	0.118678	−	0.992933i	\(-0.462134\pi\)
			0.118678	+	0.992933i	\(0.462134\pi\)
\(73\)	2.82843	0.331042	0.165521	−	0.986206i	\(-0.447069\pi\)
			0.165521	+	0.986206i	\(0.447069\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	5.65685	0.620920	0.310460	−	0.950586i	\(-0.399517\pi\)
			0.310460	+	0.950586i	\(0.399517\pi\)
\(89\)	18.3848	1.94878	0.974391	−	0.224860i	\(-0.0721923\pi\)
			0.974391	+	0.224860i	\(0.0721923\pi\)
\(97\)	9.89949	1.00514	0.502571	−	0.864536i	\(-0.332388\pi\)
			0.502571	+	0.864536i	\(0.332388\pi\)

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	4312.2.a.s.1.2	yes	2
4.3	odd	2		8624.2.a.bq.1.1		2
7.6	odd	2	inner	4312.2.a.s.1.1	✓	2
28.27	even	2		8624.2.a.bq.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4312.2.a.s.1.1	✓	2	7.6	odd	2	inner
4312.2.a.s.1.2	yes	2	1.1	even	1	trivial
8624.2.a.bq.1.1		2	4.3	odd	2
8624.2.a.bq.1.2		2	28.27	even	2