Embedded newform 400.2.a.d.1.1

• Label: 400.2.a.d.1.1
• Level: $400$
• Weight: $2$
• Character: 400.1
• Self dual: yes
• Analytic conductor: $3.194$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -2.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} -4.00000 q^{13} -3.00000 q^{17} -5.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} +5.00000 q^{27} -2.00000 q^{31} -3.00000 q^{33} +2.00000 q^{37} +4.00000 q^{39} -3.00000 q^{41} +4.00000 q^{43} -12.0000 q^{47} -3.00000 q^{49} +3.00000 q^{51} +6.00000 q^{53} +5.00000 q^{57} +2.00000 q^{61} +4.00000 q^{63} +13.0000 q^{67} +6.00000 q^{69} -12.0000 q^{71} +11.0000 q^{73} -6.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +9.00000 q^{83} +15.0000 q^{89} +8.00000 q^{91} +2.00000 q^{93} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
			\(3\)	−1.00000	−0.577350	−0.288675	−	0.957427i	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	0	0
			\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
−0.377964	+	0.925820i				\(0.623376\pi\)
\(11\)	3.00000	0.904534	0.452267	−	0.891883i	\(-0.350615\pi\)
			0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	−4.00000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
			−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
			−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	−5.00000	−1.14708	−0.573539	−	0.819178i	\(-0.694430\pi\)
			−0.573539	+	0.819178i	\(0.694430\pi\)
\(23\)	−6.00000	−1.25109	−0.625543	−	0.780189i	\(-0.715123\pi\)
			−0.625543	+	0.780189i	\(0.715123\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−2.00000	−0.359211	−0.179605	−	0.983739i	\(-0.557482\pi\)
			−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	−3.00000	−0.468521	−0.234261	−	0.972174i	\(-0.575267\pi\)
			−0.234261	+	0.972174i	\(0.575267\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−12.0000	−1.75038	−0.875190	−	0.483779i	\(-0.839264\pi\)
			−0.875190	+	0.483779i	\(0.839264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.a.d.1.1		1
3.2	odd	2		3600.2.a.l.1.1		1
4.3	odd	2		50.2.a.a.1.1	✓	1
5.2	odd	4		400.2.c.c.49.2		2
5.3	odd	4		400.2.c.c.49.1		2
5.4	even	2		400.2.a.f.1.1		1
8.3	odd	2		1600.2.a.j.1.1		1
8.5	even	2		1600.2.a.p.1.1		1
12.11	even	2		450.2.a.g.1.1		1
15.2	even	4		3600.2.f.f.2449.1		2
15.8	even	4		3600.2.f.f.2449.2		2
15.14	odd	2		3600.2.a.bc.1.1		1
20.3	even	4		50.2.b.a.49.2		2
20.7	even	4		50.2.b.a.49.1		2
20.19	odd	2		50.2.a.b.1.1	yes	1
28.27	even	2		2450.2.a.g.1.1		1
40.3	even	4		1600.2.c.i.449.1		2
40.13	odd	4		1600.2.c.h.449.2		2
40.19	odd	2		1600.2.a.q.1.1		1
40.27	even	4		1600.2.c.i.449.2		2
40.29	even	2		1600.2.a.i.1.1		1
40.37	odd	4		1600.2.c.h.449.1		2
44.43	even	2		6050.2.a.bi.1.1		1
52.51	odd	2		8450.2.a.v.1.1		1
60.23	odd	4		450.2.c.c.199.1		2
60.47	odd	4		450.2.c.c.199.2		2
60.59	even	2		450.2.a.c.1.1		1
140.27	odd	4		2450.2.c.m.99.1		2
140.83	odd	4		2450.2.c.m.99.2		2
140.139	even	2		2450.2.a.bd.1.1		1
220.219	even	2		6050.2.a.h.1.1		1
260.259	odd	2		8450.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.a.a.1.1	✓	1	4.3	odd	2
50.2.a.b.1.1	yes	1	20.19	odd	2
50.2.b.a.49.1		2	20.7	even	4
50.2.b.a.49.2		2	20.3	even	4
400.2.a.d.1.1		1	1.1	even	1	trivial
400.2.a.f.1.1		1	5.4	even	2
400.2.c.c.49.1		2	5.3	odd	4
400.2.c.c.49.2		2	5.2	odd	4
450.2.a.c.1.1		1	60.59	even	2
450.2.a.g.1.1		1	12.11	even	2
450.2.c.c.199.1		2	60.23	odd	4
450.2.c.c.199.2		2	60.47	odd	4
1600.2.a.i.1.1		1	40.29	even	2
1600.2.a.j.1.1		1	8.3	odd	2
1600.2.a.p.1.1		1	8.5	even	2
1600.2.a.q.1.1		1	40.19	odd	2
1600.2.c.h.449.1		2	40.37	odd	4
1600.2.c.h.449.2		2	40.13	odd	4
1600.2.c.i.449.1		2	40.3	even	4
1600.2.c.i.449.2		2	40.27	even	4
2450.2.a.g.1.1		1	28.27	even	2
2450.2.a.bd.1.1		1	140.139	even	2
2450.2.c.m.99.1		2	140.27	odd	4
2450.2.c.m.99.2		2	140.83	odd	4
3600.2.a.l.1.1		1	3.2	odd	2
3600.2.a.bc.1.1		1	15.14	odd	2
3600.2.f.f.2449.1		2	15.2	even	4
3600.2.f.f.2449.2		2	15.8	even	4
6050.2.a.h.1.1		1	220.219	even	2
6050.2.a.bi.1.1		1	44.43	even	2
8450.2.a.d.1.1		1	260.259	odd	2
8450.2.a.v.1.1		1	52.51	odd	2