Embedded newform 2475.2.c.k.199.1

• Label: 2475.2.c.k.199.1
• Level: $2475$
• Weight: $2$
• Character: 2475.199
• Analytic conductor: $19.763$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.7629745003\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{13})\)
Defining polynomial:	\( x^{4} + 7x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 275)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			\(-2.30278i\) of defining polynomial
Character	\(\chi\)	\(=\)	2475.199
Dual form			2475.2.c.k.199.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} +9.90833 q^{14} +0.302776 q^{16} +3.90833i q^{17} +1.00000 q^{19} -2.30278i q^{22} -3.69722i q^{23} -11.5139 q^{26} -14.2111i q^{28} -9.90833 q^{29} -4.21110 q^{31} +5.30278i q^{32} +9.00000 q^{34} +9.60555i q^{37} -2.30278i q^{38} -1.60555 q^{41} +7.21110i q^{43} -3.30278 q^{44} -8.51388 q^{46} +3.00000i q^{47} -11.5139 q^{49} +16.5139i q^{52} +2.30278i q^{53} -12.9083 q^{56} +22.8167i q^{58} +0.211103 q^{59} +2.90833 q^{61} +9.69722i q^{62} +12.8167 q^{64} -4.00000i q^{67} -12.9083i q^{68} -4.60555 q^{71} -2.90833i q^{73} +22.1194 q^{74} -3.30278 q^{76} +4.30278i q^{77} +0.0916731 q^{79} +3.69722i q^{82} +14.5139i q^{83} +16.6056 q^{86} +3.00000i q^{88} +5.30278 q^{89} +21.5139 q^{91} +12.2111i q^{92} +6.90833 q^{94} +11.6972i q^{97} +26.5139i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 6 * q^4 + 4 * q^11 + 18 * q^14 - 6 * q^16 + 4 * q^19 - 10 * q^26 - 18 * q^29 + 12 * q^31 + 36 * q^34 + 8 * q^41 - 6 * q^44 + 2 * q^46 - 10 * q^49 - 30 * q^56 - 28 * q^59 - 10 * q^61 + 8 * q^64 - 4 * q^71 + 38 * q^74 - 6 * q^76 + 22 * q^79 + 52 * q^86 + 14 * q^89 + 50 * q^91 + 6 * q^94

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\)	\(551\)	\(2026\)	\(2377\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	− 2.30278i	− 1.62831i	−0.580649	−	0.814154i	\(-0.697201\pi\)
			0.580649	−	0.814154i	\(-0.302799\pi\)
\(3\)	0	0
			\(5\)	0	0
\(7\)	4.30278i	1.62630i				0.582057	+	0.813148i	\(0.302248\pi\)
			−0.582057	+	0.813148i	\(0.697752\pi\)
\(11\)	1.00000	0.301511
			\(13\)	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	\(-0.756123\pi\)
0.720577	−	0.693375i				\(-0.243877\pi\)
\(17\)	3.90833i	0.947909i	0.880549	+	0.473954i	\(0.157174\pi\)
			−0.880549	+	0.473954i	\(0.842826\pi\)
\(19\)	1.00000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	− 3.69722i	− 0.770925i	−0.922724	−	0.385462i	\(-0.874042\pi\)
			0.922724	−	0.385462i	\(-0.125958\pi\)
\(29\)	−9.90833	−1.83993	−0.919965	−	0.392000i	\(-0.871783\pi\)
			−0.919965	+	0.392000i	\(0.871783\pi\)
\(31\)	−4.21110	−0.756336	−0.378168	−	0.925737i	\(-0.623446\pi\)
			−0.378168	+	0.925737i	\(0.623446\pi\)
\(37\)	9.60555i	1.57914i	0.613659	+	0.789571i	\(0.289697\pi\)
			−0.613659	+	0.789571i	\(0.710303\pi\)
\(41\)	−1.60555	−0.250745	−0.125372	−	0.992110i	\(-0.540013\pi\)
			−0.125372	+	0.992110i	\(0.540013\pi\)
\(43\)	7.21110i	1.09968i	0.835269	+	0.549841i	\(0.185312\pi\)
			−0.835269	+	0.549841i	\(0.814688\pi\)
\(47\)	3.00000i	0.437595i	0.975770	+	0.218797i	\(0.0702134\pi\)
			−0.975770	+	0.218797i	\(0.929787\pi\)
\(53\)	2.30278i	0.316311i	0.987414	+	0.158155i	\(0.0505547\pi\)
			−0.987414	+	0.158155i	\(0.949445\pi\)
\(59\)	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	\(-0.440387\pi\)
			0.186187	+	0.982514i	\(0.440387\pi\)
\(67\)	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	\(-0.921429\pi\)
			0.969690	−	0.244339i	\(-0.0785709\pi\)
\(71\)	−4.60555	−0.546578	−0.273289	−	0.961932i	\(-0.588112\pi\)
			−0.273289	+	0.961932i	\(0.588112\pi\)
\(73\)	− 2.90833i	− 0.340394i	−0.985410	−	0.170197i	\(-0.945560\pi\)
			0.985410	−	0.170197i	\(-0.0544404\pi\)
\(79\)	0.0916731	0.0103140	0.00515701	−	0.999987i	\(-0.498358\pi\)
			0.00515701	+	0.999987i	\(0.498358\pi\)
\(83\)	14.5139i	1.59311i	0.604569	+	0.796553i	\(0.293345\pi\)
			−0.604569	+	0.796553i	\(0.706655\pi\)
\(89\)	5.30278	0.562093	0.281047	−	0.959694i	\(-0.409318\pi\)
			0.281047	+	0.959694i	\(0.409318\pi\)
\(97\)	11.6972i	1.18767i	0.804586	+	0.593837i	\(0.202387\pi\)
			−0.804586	+	0.593837i	\(0.797613\pi\)

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	2475.2.c.k.199.1		4
3.2	odd	2		275.2.b.c.199.4		4
5.2	odd	4		2475.2.a.t.1.2		2
5.3	odd	4		2475.2.a.o.1.1		2
5.4	even	2	inner	2475.2.c.k.199.4		4
12.11	even	2		4400.2.b.y.4049.2		4
15.2	even	4		275.2.a.e.1.1	✓	2
15.8	even	4		275.2.a.f.1.2	yes	2
15.14	odd	2		275.2.b.c.199.1		4
60.23	odd	4		4400.2.a.bh.1.2		2
60.47	odd	4		4400.2.a.bs.1.1		2
60.59	even	2		4400.2.b.y.4049.3		4
165.32	odd	4		3025.2.a.n.1.2		2
165.98	odd	4		3025.2.a.h.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	15.2	even	4
275.2.a.f.1.2	yes	2	15.8	even	4
275.2.b.c.199.1		4	15.14	odd	2
275.2.b.c.199.4		4	3.2	odd	2
2475.2.a.o.1.1		2	5.3	odd	4
2475.2.a.t.1.2		2	5.2	odd	4
2475.2.c.k.199.1		4	1.1	even	1	trivial
2475.2.c.k.199.4		4	5.4	even	2	inner
3025.2.a.h.1.1		2	165.98	odd	4
3025.2.a.n.1.2		2	165.32	odd	4
4400.2.a.bh.1.2		2	60.23	odd	4
4400.2.a.bs.1.1		2	60.47	odd	4
4400.2.b.y.4049.2		4	12.11	even	2
4400.2.b.y.4049.3		4	60.59	even	2