Embedded newform 3450.2.d.j.2899.1

• Label: 3450.2.d.j.2899.1
• Level: $3450$
• Weight: $2$
• Character: 3450.2899
• Analytic conductor: $27.548$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(27.5483886973\)
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 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	3450.2899
Dual form			3450.2.d.j.2899.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +2.00000 q^{21} +6.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +1.00000 q^{36} -2.00000 q^{39} +10.0000 q^{41} -2.00000i q^{42} +12.0000i q^{43} +6.00000 q^{44} +1.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +6.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} -8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -12.0000i q^{67} -1.00000 q^{69} -1.00000i q^{72} +10.0000i q^{73} +12.0000i q^{77} +2.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -14.0000i q^{83} -2.00000 q^{84} +12.0000 q^{86} -6.00000i q^{87} -6.00000i q^{88} +4.00000 q^{91} -1.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} -6.00000i q^{97} -3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^21 - 2 * q^24 + 4 * q^26 - 12 * q^29 + 16 * q^31 + 2 * q^36 - 4 * q^39 + 20 * q^41 + 12 * q^44 + 2 * q^46 + 6 * q^49 - 2 * q^54 + 4 * q^56 + 24 * q^59 + 8 * q^61 - 2 * q^64 - 12 * q^66 - 2 * q^69 + 12 * q^79 + 2 * q^81 - 4 * q^84 + 24 * q^86 + 8 * q^91 - 16 * q^94 + 2 * q^96 + 12 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\)	\(277\)	\(1151\)	\(1201\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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
			\(3\)	1.00000i	0.577350i
\(5\)	0	0
			\(7\)	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	\(-0.876624\pi\)
0.925820	−	0.377964i				\(-0.123376\pi\)
\(11\)	−6.00000	−1.80907	−0.904534	−	0.426401i	\(-0.859781\pi\)
			−0.904534	+	0.426401i	\(0.859781\pi\)
\(13\)	2.00000i	0.554700i	0.960769	+	0.277350i	\(0.0894562\pi\)
			−0.960769	+	0.277350i	\(0.910544\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	1.00000i	0.208514i
			\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
−0.557086	+	0.830455i				\(0.688081\pi\)
\(31\)	8.00000	1.43684	0.718421	−	0.695608i	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	10.0000	1.56174	0.780869	−	0.624695i	\(-0.214777\pi\)
			0.780869	+	0.624695i	\(0.214777\pi\)
\(43\)	12.0000i	1.82998i	0.403473	+	0.914991i	\(0.367803\pi\)
			−0.403473	+	0.914991i	\(0.632197\pi\)
\(47\)	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	\(-0.801699\pi\)
			0.812142	−	0.583460i	\(-0.198301\pi\)
\(53\)	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	\(-0.956138\pi\)
			0.990521	−	0.137361i	\(-0.0438619\pi\)
\(59\)	12.0000	1.56227	0.781133	−	0.624364i	\(-0.214642\pi\)
			0.781133	+	0.624364i	\(0.214642\pi\)
\(61\)	4.00000	0.512148	0.256074	−	0.966657i	\(-0.417571\pi\)
			0.256074	+	0.966657i	\(0.417571\pi\)
\(67\)	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	\(-0.738112\pi\)
			0.680211	−	0.733017i	\(-0.261888\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	10.0000i	1.17041i	0.810885	+	0.585206i	\(0.198986\pi\)
			−0.810885	+	0.585206i	\(0.801014\pi\)
\(79\)	6.00000	0.675053	0.337526	−	0.941316i	\(-0.390410\pi\)
			0.337526	+	0.941316i	\(0.390410\pi\)
\(83\)	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	\(-0.721078\pi\)
			0.640030	−	0.768350i	\(-0.278922\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	\(-0.901476\pi\)
			0.952479	−	0.304604i	\(-0.0985241\pi\)

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	3450.2.d.j.2899.1		2
5.2	odd	4		3450.2.a.y.1.1		1
5.3	odd	4		138.2.a.a.1.1	✓	1
5.4	even	2	inner	3450.2.d.j.2899.2		2
15.8	even	4		414.2.a.d.1.1		1
20.3	even	4		1104.2.a.e.1.1		1
35.13	even	4		6762.2.a.q.1.1		1
40.3	even	4		4416.2.a.m.1.1		1
40.13	odd	4		4416.2.a.z.1.1		1
60.23	odd	4		3312.2.a.n.1.1		1
115.68	even	4		3174.2.a.b.1.1		1
345.68	odd	4		9522.2.a.i.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.a.1.1	✓	1	5.3	odd	4
414.2.a.d.1.1		1	15.8	even	4
1104.2.a.e.1.1		1	20.3	even	4
3174.2.a.b.1.1		1	115.68	even	4
3312.2.a.n.1.1		1	60.23	odd	4
3450.2.a.y.1.1		1	5.2	odd	4
3450.2.d.j.2899.1		2	1.1	even	1	trivial
3450.2.d.j.2899.2		2	5.4	even	2	inner
4416.2.a.m.1.1		1	40.3	even	4
4416.2.a.z.1.1		1	40.13	odd	4
6762.2.a.q.1.1		1	35.13	even	4
9522.2.a.i.1.1		1	345.68	odd	4