Embedded newform 3150.2.a.i.1.1

• Label: 3150.2.a.i.1.1
• Level: $3150$
• Weight: $2$
• Character: 3150.1
• Self dual: yes
• Analytic conductor: $25.153$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.1528766367\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3150.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} -4.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -2.00000 q^{37} -2.00000 q^{38} -6.00000 q^{41} -8.00000 q^{43} -12.0000 q^{47} +1.00000 q^{49} +4.00000 q^{52} +6.00000 q^{53} +1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{67} +6.00000 q^{68} -2.00000 q^{73} +2.00000 q^{74} +2.00000 q^{76} +8.00000 q^{79} +6.00000 q^{82} -6.00000 q^{83} +8.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} +12.0000 q^{94} +10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	0	0
\(5\)	0	0
			\(7\)	−1.00000	−0.377964
\(11\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−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	3150.2.a.i.1.1		1
3.2	odd	2		350.2.a.f.1.1		1
5.2	odd	4		3150.2.g.j.2899.1		2
5.3	odd	4		3150.2.g.j.2899.2		2
5.4	even	2		126.2.a.b.1.1		1
12.11	even	2		2800.2.a.g.1.1		1
15.2	even	4		350.2.c.d.99.2		2
15.8	even	4		350.2.c.d.99.1		2
15.14	odd	2		14.2.a.a.1.1	✓	1
20.19	odd	2		1008.2.a.h.1.1		1
21.20	even	2		2450.2.a.t.1.1		1
35.4	even	6		882.2.g.c.667.1		2
35.9	even	6		882.2.g.c.361.1		2
35.19	odd	6		882.2.g.d.361.1		2
35.24	odd	6		882.2.g.d.667.1		2
35.34	odd	2		882.2.a.i.1.1		1
40.19	odd	2		4032.2.a.r.1.1		1
40.29	even	2		4032.2.a.w.1.1		1
45.4	even	6		1134.2.f.f.379.1		2
45.14	odd	6		1134.2.f.l.379.1		2
45.29	odd	6		1134.2.f.l.757.1		2
45.34	even	6		1134.2.f.f.757.1		2
60.23	odd	4		2800.2.g.h.449.1		2
60.47	odd	4		2800.2.g.h.449.2		2
60.59	even	2		112.2.a.c.1.1		1
105.44	odd	6		98.2.c.b.67.1		2
105.59	even	6		98.2.c.a.79.1		2
105.62	odd	4		2450.2.c.c.99.2		2
105.74	odd	6		98.2.c.b.79.1		2
105.83	odd	4		2450.2.c.c.99.1		2
105.89	even	6		98.2.c.a.67.1		2
105.104	even	2		98.2.a.a.1.1		1
120.29	odd	2		448.2.a.g.1.1		1
120.59	even	2		448.2.a.a.1.1		1
140.139	even	2		7056.2.a.bd.1.1		1
165.164	even	2		1694.2.a.e.1.1		1
195.44	even	4		2366.2.d.b.337.2		2
195.164	even	4		2366.2.d.b.337.1		2
195.194	odd	2		2366.2.a.j.1.1		1
240.29	odd	4		1792.2.b.c.897.1		2
240.59	even	4		1792.2.b.g.897.1		2
240.149	odd	4		1792.2.b.c.897.2		2
240.179	even	4		1792.2.b.g.897.2		2
255.254	odd	2		4046.2.a.f.1.1		1
285.284	even	2		5054.2.a.c.1.1		1
345.344	even	2		7406.2.a.a.1.1		1
420.59	odd	6		784.2.i.i.177.1		2
420.179	even	6		784.2.i.c.177.1		2
420.299	odd	6		784.2.i.i.753.1		2
420.359	even	6		784.2.i.c.753.1		2
420.419	odd	2		784.2.a.b.1.1		1
840.419	odd	2		3136.2.a.z.1.1		1
840.629	even	2		3136.2.a.e.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	15.14	odd	2
98.2.a.a.1.1		1	105.104	even	2
98.2.c.a.67.1		2	105.89	even	6
98.2.c.a.79.1		2	105.59	even	6
98.2.c.b.67.1		2	105.44	odd	6
98.2.c.b.79.1		2	105.74	odd	6
112.2.a.c.1.1		1	60.59	even	2
126.2.a.b.1.1		1	5.4	even	2
350.2.a.f.1.1		1	3.2	odd	2
350.2.c.d.99.1		2	15.8	even	4
350.2.c.d.99.2		2	15.2	even	4
448.2.a.a.1.1		1	120.59	even	2
448.2.a.g.1.1		1	120.29	odd	2
784.2.a.b.1.1		1	420.419	odd	2
784.2.i.c.177.1		2	420.179	even	6
784.2.i.c.753.1		2	420.359	even	6
784.2.i.i.177.1		2	420.59	odd	6
784.2.i.i.753.1		2	420.299	odd	6
882.2.a.i.1.1		1	35.34	odd	2
882.2.g.c.361.1		2	35.9	even	6
882.2.g.c.667.1		2	35.4	even	6
882.2.g.d.361.1		2	35.19	odd	6
882.2.g.d.667.1		2	35.24	odd	6
1008.2.a.h.1.1		1	20.19	odd	2
1134.2.f.f.379.1		2	45.4	even	6
1134.2.f.f.757.1		2	45.34	even	6
1134.2.f.l.379.1		2	45.14	odd	6
1134.2.f.l.757.1		2	45.29	odd	6
1694.2.a.e.1.1		1	165.164	even	2
1792.2.b.c.897.1		2	240.29	odd	4
1792.2.b.c.897.2		2	240.149	odd	4
1792.2.b.g.897.1		2	240.59	even	4
1792.2.b.g.897.2		2	240.179	even	4
2366.2.a.j.1.1		1	195.194	odd	2
2366.2.d.b.337.1		2	195.164	even	4
2366.2.d.b.337.2		2	195.44	even	4
2450.2.a.t.1.1		1	21.20	even	2
2450.2.c.c.99.1		2	105.83	odd	4
2450.2.c.c.99.2		2	105.62	odd	4
2800.2.a.g.1.1		1	12.11	even	2
2800.2.g.h.449.1		2	60.23	odd	4
2800.2.g.h.449.2		2	60.47	odd	4
3136.2.a.e.1.1		1	840.629	even	2
3136.2.a.z.1.1		1	840.419	odd	2
3150.2.a.i.1.1		1	1.1	even	1	trivial
3150.2.g.j.2899.1		2	5.2	odd	4
3150.2.g.j.2899.2		2	5.3	odd	4
4032.2.a.r.1.1		1	40.19	odd	2
4032.2.a.w.1.1		1	40.29	even	2
4046.2.a.f.1.1		1	255.254	odd	2
5054.2.a.c.1.1		1	285.284	even	2
7056.2.a.bd.1.1		1	140.139	even	2
7406.2.a.a.1.1		1	345.344	even	2