Embedded newform 2475.2.c.k.199.3

• Label: 2475.2.c.k.199.3
• 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			199.3
Root			\(1.30278i\) of defining polynomial
Character	\(\chi\)	\(=\)	2475.199
Dual form			2475.2.c.k.199.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833i q^{17} +1.00000 q^{19} +1.30278i q^{22} -7.30278i q^{23} +6.51388 q^{26} +0.211103i q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722i q^{32} +9.00000 q^{34} +2.39445i q^{37} +1.30278i q^{38} +5.60555 q^{41} -7.21110i q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000i q^{47} +6.51388 q^{49} -1.51388i q^{52} -1.30278i q^{53} -2.09167 q^{56} +1.18335i q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028i q^{62} -8.81665 q^{64} -4.00000i q^{67} -2.09167i q^{68} +2.60555 q^{71} +7.90833i q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224i q^{77} +10.9083 q^{79} +7.30278i q^{82} -3.51388i q^{83} +9.39445 q^{86} +3.00000i q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110i q^{92} -3.90833 q^{94} +15.3028i q^{97} +8.48612i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30278i	0.921201i	0.887607	+	0.460601i	\(0.152366\pi\)
			−0.887607	+	0.460601i	\(0.847634\pi\)
\(3\)	0	0
			\(5\)	0	0
\(7\)	0.697224i	0.263526i				0.991281	+	0.131763i	\(0.0420638\pi\)
			−0.991281	+	0.131763i	\(0.957936\pi\)
\(11\)	1.00000	0.301511
			\(13\)	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	\(-0.756123\pi\)
0.720577	−	0.693375i				\(-0.243877\pi\)
\(17\)	− 6.90833i	− 1.67552i	−0.546042	−	0.837758i	\(-0.683866\pi\)
			0.546042	−	0.837758i	\(-0.316134\pi\)
\(19\)	1.00000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	− 7.30278i	− 1.52273i	−0.648321	−	0.761367i	\(-0.724529\pi\)
			0.648321	−	0.761367i	\(-0.275471\pi\)
\(29\)	0.908327	0.168672	0.0843360	−	0.996437i	\(-0.473123\pi\)
			0.0843360	+	0.996437i	\(0.473123\pi\)
\(31\)	10.2111	1.83397	0.916984	−	0.398924i	\(-0.130616\pi\)
			0.916984	+	0.398924i	\(0.130616\pi\)
\(37\)	2.39445i	0.393645i	0.980439	+	0.196822i	\(0.0630623\pi\)
			−0.980439	+	0.196822i	\(0.936938\pi\)
\(41\)	5.60555	0.875440	0.437720	−	0.899111i	\(-0.355786\pi\)
			0.437720	+	0.899111i	\(0.355786\pi\)
\(43\)	− 7.21110i	− 1.09968i	−0.835269	−	0.549841i	\(-0.814688\pi\)
			0.835269	−	0.549841i	\(-0.185312\pi\)
\(47\)	3.00000i	0.437595i	0.975770	+	0.218797i	\(0.0702134\pi\)
			−0.975770	+	0.218797i	\(0.929787\pi\)
\(53\)	− 1.30278i	− 0.178950i	−0.995989	−	0.0894750i	\(-0.971481\pi\)
			0.995989	−	0.0894750i	\(-0.0285189\pi\)
\(59\)	−14.2111	−1.85013	−0.925064	−	0.379811i	\(-0.875989\pi\)
			−0.925064	+	0.379811i	\(0.875989\pi\)
\(61\)	−7.90833	−1.01256	−0.506279	−	0.862370i	\(-0.668979\pi\)
			−0.506279	+	0.862370i	\(0.668979\pi\)
\(67\)	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	\(-0.921429\pi\)
			0.969690	−	0.244339i	\(-0.0785709\pi\)
\(71\)	2.60555	0.309222	0.154611	−	0.987975i	\(-0.450588\pi\)
			0.154611	+	0.987975i	\(0.450588\pi\)
\(73\)	7.90833i	0.925600i	0.886463	+	0.462800i	\(0.153155\pi\)
			−0.886463	+	0.462800i	\(0.846845\pi\)
\(79\)	10.9083	1.22728	0.613641	−	0.789585i	\(-0.289704\pi\)
			0.613641	+	0.789585i	\(0.289704\pi\)
\(83\)	− 3.51388i	− 0.385698i	−0.981228	−	0.192849i	\(-0.938227\pi\)
			0.981228	−	0.192849i	\(-0.0617728\pi\)
\(89\)	1.69722	0.179905	0.0899527	−	0.995946i	\(-0.471328\pi\)
			0.0899527	+	0.995946i	\(0.471328\pi\)
\(97\)	15.3028i	1.55376i	0.629648	+	0.776881i	\(0.283199\pi\)
			−0.629648	+	0.776881i	\(0.716801\pi\)

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.199.3		4
3.2	odd	2		275.2.b.c.199.2		4
5.2	odd	4		2475.2.a.t.1.1		2
5.3	odd	4		2475.2.a.o.1.2		2
5.4	even	2	inner	2475.2.c.k.199.2		4
12.11	even	2		4400.2.b.y.4049.4		4
15.2	even	4		275.2.a.e.1.2	✓	2
15.8	even	4		275.2.a.f.1.1	yes	2
15.14	odd	2		275.2.b.c.199.3		4
60.23	odd	4		4400.2.a.bh.1.1		2
60.47	odd	4		4400.2.a.bs.1.2		2
60.59	even	2		4400.2.b.y.4049.1		4
165.32	odd	4		3025.2.a.n.1.1		2
165.98	odd	4		3025.2.a.h.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.2	✓	2	15.2	even	4
275.2.a.f.1.1	yes	2	15.8	even	4
275.2.b.c.199.2		4	3.2	odd	2
275.2.b.c.199.3		4	15.14	odd	2
2475.2.a.o.1.2		2	5.3	odd	4
2475.2.a.t.1.1		2	5.2	odd	4
2475.2.c.k.199.2		4	5.4	even	2	inner
2475.2.c.k.199.3		4	1.1	even	1	trivial
3025.2.a.h.1.2		2	165.98	odd	4
3025.2.a.n.1.1		2	165.32	odd	4
4400.2.a.bh.1.1		2	60.23	odd	4
4400.2.a.bs.1.2		2	60.47	odd	4
4400.2.b.y.4049.1		4	60.59	even	2
4400.2.b.y.4049.4		4	12.11	even	2