Embedded newform 3025.2.a.n.1.2

• Label: 3025.2.a.n.1.2
• Level: $3025$
• Weight: $2$
• Character: 3025.1
• Self dual: yes
• Analytic conductor: $24.155$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(24.1547466114\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(2.30278\) of defining polynomial
Character	\(\chi\)	\(=\)	3025.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} +1.30278 q^{3} +3.30278 q^{4} +3.00000 q^{6} +4.30278 q^{7} +3.00000 q^{8} -1.30278 q^{9} +4.30278 q^{12} +5.00000 q^{13} +9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -3.00000 q^{18} +1.00000 q^{19} +5.60555 q^{21} +3.69722 q^{23} +3.90833 q^{24} +11.5139 q^{26} -5.60555 q^{27} +14.2111 q^{28} +9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} -9.00000 q^{34} -4.30278 q^{36} -9.60555 q^{37} +2.30278 q^{38} +6.51388 q^{39} -1.60555 q^{41} +12.9083 q^{42} -7.21110 q^{43} +8.51388 q^{46} +3.00000 q^{47} +0.394449 q^{48} +11.5139 q^{49} -5.09167 q^{51} +16.5139 q^{52} -2.30278 q^{53} -12.9083 q^{54} +12.9083 q^{56} +1.30278 q^{57} +22.8167 q^{58} +0.211103 q^{59} -2.90833 q^{61} -9.69722 q^{62} -5.60555 q^{63} -12.8167 q^{64} +4.00000 q^{67} -12.9083 q^{68} +4.81665 q^{69} +4.60555 q^{71} -3.90833 q^{72} +2.90833 q^{73} -22.1194 q^{74} +3.30278 q^{76} +15.0000 q^{78} +0.0916731 q^{79} -3.39445 q^{81} -3.69722 q^{82} +14.5139 q^{83} +18.5139 q^{84} -16.6056 q^{86} +12.9083 q^{87} +5.30278 q^{89} +21.5139 q^{91} +12.2111 q^{92} -5.48612 q^{93} +6.90833 q^{94} -6.90833 q^{96} -11.6972 q^{97} +26.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^3 + 3 * q^4 + 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 + 5 * q^12 + 10 * q^13 + 9 * q^14 - 3 * q^16 + 3 * q^17 - 6 * q^18 + 2 * q^19 + 4 * q^21 + 11 * q^23 - 3 * q^24 + 5 * q^26 - 4 * q^27 + 14 * q^28 + 9 * q^29 + 6 * q^31 - 7 * q^32 - 18 * q^34 - 5 * q^36 - 12 * q^37 + q^38 - 5 * q^39 + 4 * q^41 + 15 * q^42 - q^46 + 6 * q^47 + 8 * q^48 + 5 * q^49 - 21 * q^51 + 15 * q^52 - q^53 - 15 * q^54 + 15 * q^56 - q^57 + 24 * q^58 - 14 * q^59 + 5 * q^61 - 23 * q^62 - 4 * q^63 - 4 * q^64 + 8 * q^67 - 15 * q^68 - 12 * q^69 + 2 * q^71 + 3 * q^72 - 5 * q^73 - 19 * q^74 + 3 * q^76 + 30 * q^78 + 11 * q^79 - 14 * q^81 - 11 * q^82 + 11 * q^83 + 19 * q^84 - 26 * q^86 + 15 * q^87 + 7 * q^89 + 25 * q^91 + 10 * q^92 - 29 * q^93 + 3 * q^94 - 3 * q^96 - 27 * q^97 + 35 * q^98

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	2.30278	1.62831	0.814154	−	0.580649i	\(-0.197201\pi\)
			0.814154	+	0.580649i	\(0.197201\pi\)
\(3\)	1.30278	0.752158	0.376079	−	0.926588i	\(-0.377272\pi\)
			0.376079	+	0.926588i	\(0.377272\pi\)
\(5\)	0	0
			\(7\)	4.30278	1.62630	0.813148	−	0.582057i	\(-0.197752\pi\)
0.813148	+	0.582057i				\(0.197752\pi\)
\(11\)	0	0
			\(13\)	5.00000	1.38675	0.693375	−	0.720577i	\(-0.256123\pi\)
0.693375	+	0.720577i				\(0.256123\pi\)
\(17\)	−3.90833	−0.947909	−0.473954	−	0.880549i	\(-0.657174\pi\)
			−0.473954	+	0.880549i	\(0.657174\pi\)
\(19\)	1.00000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	3.69722	0.770925	0.385462	−	0.922724i	\(-0.374042\pi\)
			0.385462	+	0.922724i	\(0.374042\pi\)
\(29\)	9.90833	1.83993	0.919965	−	0.392000i	\(-0.128217\pi\)
			0.919965	+	0.392000i	\(0.128217\pi\)
\(31\)	−4.21110	−0.756336	−0.378168	−	0.925737i	\(-0.623446\pi\)
			−0.378168	+	0.925737i	\(0.623446\pi\)
\(37\)	−9.60555	−1.57914	−0.789571	−	0.613659i	\(-0.789697\pi\)
			−0.789571	+	0.613659i	\(0.789697\pi\)
\(41\)	−1.60555	−0.250745	−0.125372	−	0.992110i	\(-0.540013\pi\)
			−0.125372	+	0.992110i	\(0.540013\pi\)
\(43\)	−7.21110	−1.09968	−0.549841	−	0.835269i	\(-0.685312\pi\)
			−0.549841	+	0.835269i	\(0.685312\pi\)
\(47\)	3.00000	0.437595	0.218797	−	0.975770i	\(-0.429787\pi\)
			0.218797	+	0.975770i	\(0.429787\pi\)
\(53\)	−2.30278	−0.316311	−0.158155	−	0.987414i	\(-0.550555\pi\)
			−0.158155	+	0.987414i	\(0.550555\pi\)
\(59\)	0.211103	0.0274832	0.0137416	−	0.999906i	\(-0.495626\pi\)
			0.0137416	+	0.999906i	\(0.495626\pi\)
\(61\)	−2.90833	−0.372373	−0.186187	−	0.982514i	\(-0.559613\pi\)
			−0.186187	+	0.982514i	\(0.559613\pi\)
\(67\)	4.00000	0.488678	0.244339	−	0.969690i	\(-0.421429\pi\)
			0.244339	+	0.969690i	\(0.421429\pi\)
\(71\)	4.60555	0.546578	0.273289	−	0.961932i	\(-0.411888\pi\)
			0.273289	+	0.961932i	\(0.411888\pi\)
\(73\)	2.90833	0.340394	0.170197	−	0.985410i	\(-0.445560\pi\)
			0.170197	+	0.985410i	\(0.445560\pi\)
\(79\)	0.0916731	0.0103140	0.00515701	−	0.999987i	\(-0.498358\pi\)
			0.00515701	+	0.999987i	\(0.498358\pi\)
\(83\)	14.5139	1.59311	0.796553	−	0.604569i	\(-0.206655\pi\)
			0.796553	+	0.604569i	\(0.206655\pi\)
\(89\)	5.30278	0.562093	0.281047	−	0.959694i	\(-0.409318\pi\)
			0.281047	+	0.959694i	\(0.409318\pi\)
\(97\)	−11.6972	−1.18767	−0.593837	−	0.804586i	\(-0.702387\pi\)
			−0.593837	+	0.804586i	\(0.702387\pi\)

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	3025.2.a.n.1.2		2
5.4	even	2		3025.2.a.h.1.1		2
11.10	odd	2		275.2.a.e.1.1	✓	2
33.32	even	2		2475.2.a.t.1.2		2
44.43	even	2		4400.2.a.bs.1.1		2
55.32	even	4		275.2.b.c.199.1		4
55.43	even	4		275.2.b.c.199.4		4
55.54	odd	2		275.2.a.f.1.2	yes	2
165.32	odd	4		2475.2.c.k.199.4		4
165.98	odd	4		2475.2.c.k.199.1		4
165.164	even	2		2475.2.a.o.1.1		2
220.43	odd	4		4400.2.b.y.4049.2		4
220.87	odd	4		4400.2.b.y.4049.3		4
220.219	even	2		4400.2.a.bh.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	11.10	odd	2
275.2.a.f.1.2	yes	2	55.54	odd	2
275.2.b.c.199.1		4	55.32	even	4
275.2.b.c.199.4		4	55.43	even	4
2475.2.a.o.1.1		2	165.164	even	2
2475.2.a.t.1.2		2	33.32	even	2
2475.2.c.k.199.1		4	165.98	odd	4
2475.2.c.k.199.4		4	165.32	odd	4
3025.2.a.h.1.1		2	5.4	even	2
3025.2.a.n.1.2		2	1.1	even	1	trivial
4400.2.a.bh.1.2		2	220.219	even	2
4400.2.a.bs.1.1		2	44.43	even	2
4400.2.b.y.4049.2		4	220.43	odd	4
4400.2.b.y.4049.3		4	220.87	odd	4