Embedded newform 1425.2.c.f.799.1

• Label: 1425.2.c.f.799.1
• Level: $1425$
• Weight: $2$
• Character: 1425.799
• Analytic conductor: $11.379$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3786822880\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	1425.799
Dual form			1425.2.c.f.799.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -4.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} -3.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +2.00000 q^{29} -5.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -1.00000i q^{38} +2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} -14.0000i q^{53} +1.00000 q^{54} -12.0000 q^{56} -1.00000i q^{57} -2.00000i q^{58} -4.00000 q^{59} +14.0000 q^{61} +4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +3.00000i q^{72} -14.0000i q^{73} +6.00000 q^{74} +1.00000 q^{76} -16.0000i q^{77} -2.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -4.00000 q^{84} +8.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{91} -4.00000i q^{92} +12.0000 q^{94} -5.00000 q^{96} +10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 8 * q^14 - 2 * q^16 + 2 * q^19 - 8 * q^21 - 6 * q^24 + 4 * q^26 + 4 * q^29 - 4 * q^34 - 2 * q^36 + 4 * q^39 - 12 * q^41 + 8 * q^44 - 8 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 24 * q^56 - 8 * q^59 + 28 * q^61 - 14 * q^64 - 8 * q^66 - 8 * q^69 + 12 * q^74 + 2 * q^76 - 32 * q^79 + 2 * q^81 - 8 * q^84 + 16 * q^86 + 12 * q^89 + 16 * q^91 + 24 * q^94 - 10 * q^96 - 8 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\)	\(476\)	\(1027\)	\(1351\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	\(-0.884973\pi\)
			0.935414	−	0.353553i	\(-0.115027\pi\)
\(3\)	− 1.00000i	− 0.577350i
			\(5\)	0	0
\(7\)	− 4.00000i	− 1.51186i				−0.654654	−	0.755929i	\(-0.727186\pi\)
			0.654654	−	0.755929i	\(-0.272814\pi\)
\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	2.00000i	0.554700i	0.960769	+	0.277350i	\(0.0894562\pi\)
			−0.960769	+	0.277350i	\(0.910544\pi\)
\(17\)	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	\(-0.922021\pi\)
			0.970143	−	0.242536i	\(-0.0779791\pi\)
\(19\)	1.00000	0.229416
			\(23\)	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	\(-0.863071\pi\)
0.908893	−	0.417029i				\(-0.136929\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	6.00000i	0.986394i	0.869918	+	0.493197i	\(0.164172\pi\)
			−0.869918	+	0.493197i	\(0.835828\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	8.00000i	1.21999i	0.792406	+	0.609994i	\(0.208828\pi\)
			−0.792406	+	0.609994i	\(0.791172\pi\)
\(47\)	12.0000i	1.75038i	0.483779	+	0.875190i	\(0.339264\pi\)
			−0.483779	+	0.875190i	\(0.660736\pi\)
\(53\)	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	\(-0.588586\pi\)
			0.274721	−	0.961524i	\(-0.411414\pi\)
\(59\)	−4.00000	−0.520756	−0.260378	−	0.965507i	\(-0.583847\pi\)
			−0.260378	+	0.965507i	\(0.583847\pi\)
\(61\)	14.0000	1.79252	0.896258	−	0.443533i	\(-0.146275\pi\)
			0.896258	+	0.443533i	\(0.146275\pi\)
\(67\)	4.00000i	0.488678i	0.969690	+	0.244339i	\(0.0785709\pi\)
			−0.969690	+	0.244339i	\(0.921429\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	\(-0.694369\pi\)
			0.573382	−	0.819288i	\(-0.305631\pi\)
\(79\)	−16.0000	−1.80014	−0.900070	−	0.435745i	\(-0.856485\pi\)
			−0.900070	+	0.435745i	\(0.856485\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	6.00000	0.635999	0.317999	−	0.948091i	\(-0.396989\pi\)
			0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)	10.0000i	1.01535i	0.861550	+	0.507673i	\(0.169494\pi\)
			−0.861550	+	0.507673i	\(0.830506\pi\)

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	1425.2.c.f.799.1		2
5.2	odd	4		285.2.a.c.1.1	✓	1
5.3	odd	4		1425.2.a.c.1.1		1
5.4	even	2	inner	1425.2.c.f.799.2		2
15.2	even	4		855.2.a.a.1.1		1
15.8	even	4		4275.2.a.j.1.1		1
20.7	even	4		4560.2.a.w.1.1		1
95.37	even	4		5415.2.a.e.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.c.1.1	✓	1	5.2	odd	4
855.2.a.a.1.1		1	15.2	even	4
1425.2.a.c.1.1		1	5.3	odd	4
1425.2.c.f.799.1		2	1.1	even	1	trivial
1425.2.c.f.799.2		2	5.4	even	2	inner
4275.2.a.j.1.1		1	15.8	even	4
4560.2.a.w.1.1		1	20.7	even	4
5415.2.a.e.1.1		1	95.37	even	4