Embedded newform 350.2.a.f.1.1

• Label: 350.2.a.f.1.1
• Level: $350$
• Weight: $2$
• Character: 350.1
• Self dual: yes
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$2.79476407074$
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	$\chi$	$=$	350.1

$q$-expansion

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{21} +2.00000 q^{24} +4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{38} +8.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} -8.00000 q^{43} +12.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} -12.0000 q^{51} +4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +2.00000 q^{76} +8.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} -12.0000 q^{87} -6.00000 q^{89} -4.00000 q^{91} -8.00000 q^{93} +12.0000 q^{94} +2.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
			$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
0.577350	+	0.816497i				$0.304087\pi$
$5$	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	$-0.759367\pi$
			−0.727607	+	0.685994i	$0.759367\pi$
$19$	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	$-0.688081\pi$
			−0.557086	+	0.830455i	$0.688081\pi$
$31$	−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	$-0.344787\pi$
			0.468521	+	0.883452i	$0.344787\pi$
$43$	−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	$-0.160736\pi$
			0.875190	+	0.483779i	$0.160736\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.a.f.1.1		1
3.2	odd	2		3150.2.a.i.1.1		1
4.3	odd	2		2800.2.a.g.1.1		1
5.2	odd	4		350.2.c.d.99.2		2
5.3	odd	4		350.2.c.d.99.1		2
5.4	even	2		14.2.a.a.1.1	✓	1
7.6	odd	2		2450.2.a.t.1.1		1
15.2	even	4		3150.2.g.j.2899.1		2
15.8	even	4		3150.2.g.j.2899.2		2
15.14	odd	2		126.2.a.b.1.1		1
20.3	even	4		2800.2.g.h.449.1		2
20.7	even	4		2800.2.g.h.449.2		2
20.19	odd	2		112.2.a.c.1.1		1
35.4	even	6		98.2.c.b.79.1		2
35.9	even	6		98.2.c.b.67.1		2
35.13	even	4		2450.2.c.c.99.1		2
35.19	odd	6		98.2.c.a.67.1		2
35.24	odd	6		98.2.c.a.79.1		2
35.27	even	4		2450.2.c.c.99.2		2
35.34	odd	2		98.2.a.a.1.1		1
40.19	odd	2		448.2.a.a.1.1		1
40.29	even	2		448.2.a.g.1.1		1
45.4	even	6		1134.2.f.l.379.1		2
45.14	odd	6		1134.2.f.f.379.1		2
45.29	odd	6		1134.2.f.f.757.1		2
45.34	even	6		1134.2.f.l.757.1		2
55.54	odd	2		1694.2.a.e.1.1		1
60.59	even	2		1008.2.a.h.1.1		1
65.34	odd	4		2366.2.d.b.337.1		2
65.44	odd	4		2366.2.d.b.337.2		2
65.64	even	2		2366.2.a.j.1.1		1
80.19	odd	4		1792.2.b.g.897.2		2
80.29	even	4		1792.2.b.c.897.1		2
80.59	odd	4		1792.2.b.g.897.1		2
80.69	even	4		1792.2.b.c.897.2		2
85.84	even	2		4046.2.a.f.1.1		1
95.94	odd	2		5054.2.a.c.1.1		1
105.44	odd	6		882.2.g.c.361.1		2
105.59	even	6		882.2.g.d.667.1		2
105.74	odd	6		882.2.g.c.667.1		2
105.89	even	6		882.2.g.d.361.1		2
105.104	even	2		882.2.a.i.1.1		1
115.114	odd	2		7406.2.a.a.1.1		1
120.29	odd	2		4032.2.a.w.1.1		1
120.59	even	2		4032.2.a.r.1.1		1
140.19	even	6		784.2.i.i.753.1		2
140.39	odd	6		784.2.i.c.177.1		2
140.59	even	6		784.2.i.i.177.1		2
140.79	odd	6		784.2.i.c.753.1		2
140.139	even	2		784.2.a.b.1.1		1
280.69	odd	2		3136.2.a.e.1.1		1
280.139	even	2		3136.2.a.z.1.1		1
420.419	odd	2		7056.2.a.bd.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	5.4	even	2
98.2.a.a.1.1		1	35.34	odd	2
98.2.c.a.67.1		2	35.19	odd	6
98.2.c.a.79.1		2	35.24	odd	6
98.2.c.b.67.1		2	35.9	even	6
98.2.c.b.79.1		2	35.4	even	6
112.2.a.c.1.1		1	20.19	odd	2
126.2.a.b.1.1		1	15.14	odd	2
350.2.a.f.1.1		1	1.1	even	1	trivial
350.2.c.d.99.1		2	5.3	odd	4
350.2.c.d.99.2		2	5.2	odd	4
448.2.a.a.1.1		1	40.19	odd	2
448.2.a.g.1.1		1	40.29	even	2
784.2.a.b.1.1		1	140.139	even	2
784.2.i.c.177.1		2	140.39	odd	6
784.2.i.c.753.1		2	140.79	odd	6
784.2.i.i.177.1		2	140.59	even	6
784.2.i.i.753.1		2	140.19	even	6
882.2.a.i.1.1		1	105.104	even	2
882.2.g.c.361.1		2	105.44	odd	6
882.2.g.c.667.1		2	105.74	odd	6
882.2.g.d.361.1		2	105.89	even	6
882.2.g.d.667.1		2	105.59	even	6
1008.2.a.h.1.1		1	60.59	even	2
1134.2.f.f.379.1		2	45.14	odd	6
1134.2.f.f.757.1		2	45.29	odd	6
1134.2.f.l.379.1		2	45.4	even	6
1134.2.f.l.757.1		2	45.34	even	6
1694.2.a.e.1.1		1	55.54	odd	2
1792.2.b.c.897.1		2	80.29	even	4
1792.2.b.c.897.2		2	80.69	even	4
1792.2.b.g.897.1		2	80.59	odd	4
1792.2.b.g.897.2		2	80.19	odd	4
2366.2.a.j.1.1		1	65.64	even	2
2366.2.d.b.337.1		2	65.34	odd	4
2366.2.d.b.337.2		2	65.44	odd	4
2450.2.a.t.1.1		1	7.6	odd	2
2450.2.c.c.99.1		2	35.13	even	4
2450.2.c.c.99.2		2	35.27	even	4
2800.2.a.g.1.1		1	4.3	odd	2
2800.2.g.h.449.1		2	20.3	even	4
2800.2.g.h.449.2		2	20.7	even	4
3136.2.a.e.1.1		1	280.69	odd	2
3136.2.a.z.1.1		1	280.139	even	2
3150.2.a.i.1.1		1	3.2	odd	2
3150.2.g.j.2899.1		2	15.2	even	4
3150.2.g.j.2899.2		2	15.8	even	4
4032.2.a.r.1.1		1	120.59	even	2
4032.2.a.w.1.1		1	120.29	odd	2
4046.2.a.f.1.1		1	85.84	even	2
5054.2.a.c.1.1		1	95.94	odd	2
7056.2.a.bd.1.1		1	420.419	odd	2
7406.2.a.a.1.1		1	115.114	odd	2