Embedded newform 8325.2.a.h.1.1

• Label: 8325.2.a.h.1.1
• Level: $8325$
• Weight: $2$
• Character: 8325.1
• Self dual: yes
• Analytic conductor: $66.475$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(66.4754596827\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 333)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	8325.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{7} +3.00000 q^{8} +4.00000 q^{11} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -6.00000 q^{19} -4.00000 q^{22} -8.00000 q^{23} -2.00000 q^{26} -4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} -5.00000 q^{32} -6.00000 q^{34} +1.00000 q^{37} +6.00000 q^{38} +10.0000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} -2.00000 q^{52} -4.00000 q^{53} +12.0000 q^{56} -6.00000 q^{58} -4.00000 q^{59} +10.0000 q^{61} -2.00000 q^{62} +7.00000 q^{64} +4.00000 q^{67} -6.00000 q^{68} -12.0000 q^{71} +10.0000 q^{73} -1.00000 q^{74} +6.00000 q^{76} +16.0000 q^{77} +10.0000 q^{79} -10.0000 q^{86} +12.0000 q^{88} -2.00000 q^{89} +8.00000 q^{91} +8.00000 q^{92} -12.0000 q^{94} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
\(7\)	4.00000	1.51186				0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	−8.00000	−1.66812	−0.834058	−	0.551677i	\(-0.813988\pi\)
			−0.834058	+	0.551677i	\(0.813988\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	2.00000	0.359211	0.179605	−	0.983739i	\(-0.442518\pi\)
			0.179605	+	0.983739i	\(0.442518\pi\)
\(37\)	1.00000	0.164399
			\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(43\)	10.0000	1.52499	0.762493	−	0.646997i	\(-0.223975\pi\)
			0.762493	+	0.646997i	\(0.223975\pi\)
\(47\)	12.0000	1.75038	0.875190	−	0.483779i	\(-0.160736\pi\)
			0.875190	+	0.483779i	\(0.160736\pi\)
\(53\)	−4.00000	−0.549442	−0.274721	−	0.961524i	\(-0.588586\pi\)
			−0.274721	+	0.961524i	\(0.588586\pi\)
\(59\)	−4.00000	−0.520756	−0.260378	−	0.965507i	\(-0.583847\pi\)
			−0.260378	+	0.965507i	\(0.583847\pi\)
\(61\)	10.0000	1.28037	0.640184	−	0.768221i	\(-0.278858\pi\)
			0.640184	+	0.768221i	\(0.278858\pi\)
\(67\)	4.00000	0.488678	0.244339	−	0.969690i	\(-0.421429\pi\)
			0.244339	+	0.969690i	\(0.421429\pi\)
\(71\)	−12.0000	−1.42414	−0.712069	−	0.702109i	\(-0.752242\pi\)
			−0.712069	+	0.702109i	\(0.752242\pi\)
\(73\)	10.0000	1.17041	0.585206	−	0.810885i	\(-0.301014\pi\)
			0.585206	+	0.810885i	\(0.301014\pi\)
\(79\)	10.0000	1.12509	0.562544	−	0.826767i	\(-0.309823\pi\)
			0.562544	+	0.826767i	\(0.309823\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	−2.00000	−0.212000	−0.106000	−	0.994366i	\(-0.533804\pi\)
			−0.106000	+	0.994366i	\(0.533804\pi\)
\(97\)	2.00000	0.203069	0.101535	−	0.994832i	\(-0.467625\pi\)
			0.101535	+	0.994832i	\(0.467625\pi\)

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	8325.2.a.h.1.1		1
3.2	odd	2		8325.2.a.z.1.1		1
5.4	even	2		333.2.a.c.1.1	yes	1
15.14	odd	2		333.2.a.a.1.1	✓	1
20.19	odd	2		5328.2.a.g.1.1		1
60.59	even	2		5328.2.a.u.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.2.a.a.1.1	✓	1	15.14	odd	2
333.2.a.c.1.1	yes	1	5.4	even	2
5328.2.a.g.1.1		1	20.19	odd	2
5328.2.a.u.1.1		1	60.59	even	2
8325.2.a.h.1.1		1	1.1	even	1	trivial
8325.2.a.z.1.1		1	3.2	odd	2