Embedded newform 4275.2.a.g.1.1

• Label: 4275.2.a.g.1.1
• Level: $4275$
• Weight: $2$
• Character: 4275.1
• Self dual: yes
• Analytic conductor: $34.136$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(34.1360468641\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 855)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4275.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} +3.00000 q^{8} +2.00000 q^{11} -2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{22} +2.00000 q^{26} -2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{37} -1.00000 q^{38} +2.00000 q^{41} -10.0000 q^{43} -2.00000 q^{44} -3.00000 q^{49} +2.00000 q^{52} +10.0000 q^{53} +6.00000 q^{56} +6.00000 q^{58} -10.0000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{71} +4.00000 q^{73} -2.00000 q^{74} -1.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} -2.00000 q^{82} +12.0000 q^{83} +10.0000 q^{86} +6.00000 q^{88} +10.0000 q^{89} -4.00000 q^{91} -18.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107	−0.353553	−	0.935414i	\(-0.615027\pi\)
			−0.353553	+	0.935414i	\(0.615027\pi\)
\(3\)	0	0
			\(5\)	0	0
\(7\)	2.00000	0.755929				0.377964	−	0.925820i	\(-0.376624\pi\)
			0.377964	+	0.925820i	\(0.376624\pi\)
\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
			0.301511	+	0.953463i	\(0.402509\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	1.00000	0.229416
			\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	10.0000	1.37361	0.686803	−	0.726844i	\(-0.259014\pi\)
			0.686803	+	0.726844i	\(0.259014\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−10.0000	−1.28037	−0.640184	−	0.768221i	\(-0.721142\pi\)
			−0.640184	+	0.768221i	\(0.721142\pi\)
\(67\)	−4.00000	−0.488678	−0.244339	−	0.969690i	\(-0.578571\pi\)
			−0.244339	+	0.969690i	\(0.578571\pi\)
\(71\)	−8.00000	−0.949425	−0.474713	−	0.880141i	\(-0.657448\pi\)
			−0.474713	+	0.880141i	\(0.657448\pi\)
\(73\)	4.00000	0.468165	0.234082	−	0.972217i	\(-0.424791\pi\)
			0.234082	+	0.972217i	\(0.424791\pi\)
\(79\)	−8.00000	−0.900070	−0.450035	−	0.893011i	\(-0.648589\pi\)
			−0.450035	+	0.893011i	\(0.648589\pi\)
\(83\)	12.0000	1.31717	0.658586	−	0.752506i	\(-0.271155\pi\)
			0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	10.0000	1.06000	0.529999	−	0.847998i	\(-0.322192\pi\)
			0.529999	+	0.847998i	\(0.322192\pi\)
\(97\)	−18.0000	−1.82762	−0.913812	−	0.406138i	\(-0.866875\pi\)
			−0.913812	+	0.406138i	\(0.866875\pi\)

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	4275.2.a.g.1.1		1
3.2	odd	2		4275.2.a.n.1.1		1
5.2	odd	4		855.2.c.c.514.1	yes	2
5.3	odd	4		855.2.c.c.514.2	yes	2
5.4	even	2		4275.2.a.k.1.1		1
15.2	even	4		855.2.c.a.514.2	yes	2
15.8	even	4		855.2.c.a.514.1	✓	2
15.14	odd	2		4275.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.c.a.514.1	✓	2	15.8	even	4
855.2.c.a.514.2	yes	2	15.2	even	4
855.2.c.c.514.1	yes	2	5.2	odd	4
855.2.c.c.514.2	yes	2	5.3	odd	4
4275.2.a.d.1.1		1	15.14	odd	2
4275.2.a.g.1.1		1	1.1	even	1	trivial
4275.2.a.k.1.1		1	5.4	even	2
4275.2.a.n.1.1		1	3.2	odd	2