Embedded newform 855.2.c.a.514.2

• Label: 855.2.c.a.514.2
• Level: $855$
• Weight: $2$
• Character: 855.514
• Analytic conductor: $6.827$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.82720937282\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			514.2
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	855.514
Dual form			855.2.c.a.514.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +(1.00000 - 2.00000i) q^{10} -2.00000 q^{11} +2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} -2.00000i q^{22} +(3.00000 + 4.00000i) q^{25} -2.00000 q^{26} +2.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +5.00000i q^{32} -2.00000 q^{34} +(2.00000 - 4.00000i) q^{35} +2.00000i q^{37} -1.00000i q^{38} +(3.00000 - 6.00000i) q^{40} -2.00000 q^{41} +10.0000i q^{43} -2.00000 q^{44} +3.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} +2.00000i q^{52} +10.0000i q^{53} +(4.00000 + 2.00000i) q^{55} -6.00000 q^{56} -6.00000i q^{58} -10.0000 q^{61} -4.00000i q^{62} -7.00000 q^{64} +(2.00000 - 4.00000i) q^{65} -4.00000i q^{67} +2.00000i q^{68} +(4.00000 + 2.00000i) q^{70} +8.00000 q^{71} -4.00000i q^{73} -2.00000 q^{74} -1.00000 q^{76} -4.00000i q^{77} +8.00000 q^{79} +(2.00000 + 1.00000i) q^{80} -2.00000i q^{82} +12.0000i q^{83} +(2.00000 - 4.00000i) q^{85} -10.0000 q^{86} -6.00000i q^{88} +10.0000 q^{89} -4.00000 q^{91} +(2.00000 + 1.00000i) q^{95} -18.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^4 - 4 * q^5 + 2 * q^10 - 4 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^19 - 4 * q^20 + 6 * q^25 - 4 * q^26 - 12 * q^29 - 8 * q^31 - 4 * q^34 + 4 * q^35 + 6 * q^40 - 4 * q^41 - 4 * q^44 + 6 * q^49 - 8 * q^50 + 8 * q^55 - 12 * q^56 - 20 * q^61 - 14 * q^64 + 4 * q^65 + 8 * q^70 + 16 * q^71 - 4 * q^74 - 2 * q^76 + 16 * q^79 + 4 * q^80 + 4 * q^85 - 20 * q^86 + 20 * q^89 - 8 * q^91 + 4 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\)	\(172\)	\(191\)	\(496\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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	0.935414	+	0.353553i	\(0.115027\pi\)
							−0.935414	+	0.353553i	\(0.884973\pi\)
\(3\)		0			0
							\(5\)	−2.00000	−	1.00000i	−0.894427	−	0.447214i
\(7\)			2.00000i			0.755929i								0.925820	+	0.377964i	\(0.123376\pi\)
							−0.925820	+	0.377964i	\(0.876624\pi\)
\(11\)	−2.00000			−0.603023			−0.301511	−	0.953463i	\(-0.597491\pi\)
							−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)			2.00000i			0.554700i	0.960769	+	0.277350i	\(0.0894562\pi\)
							−0.960769	+	0.277350i	\(0.910544\pi\)
\(17\)			2.00000i			0.485071i	0.970143	+	0.242536i	\(0.0779791\pi\)
							−0.970143	+	0.242536i	\(0.922021\pi\)
\(19\)	−1.00000			−0.229416
							\(23\)		0			0		1.00000			\(0\)
−1.00000			\(\pi\)
\(29\)	−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	\(0.616954\pi\)
\(37\)			2.00000i			0.328798i	0.986394	+	0.164399i	\(0.0525685\pi\)
							−0.986394	+	0.164399i	\(0.947432\pi\)
\(41\)	−2.00000			−0.312348			−0.156174	−	0.987730i	\(-0.549916\pi\)
							−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)			10.0000i			1.52499i	0.646997	+	0.762493i	\(0.276025\pi\)
							−0.646997	+	0.762493i	\(0.723975\pi\)
\(47\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(53\)			10.0000i			1.37361i	0.726844	+	0.686803i	\(0.240986\pi\)
							−0.726844	+	0.686803i	\(0.759014\pi\)
\(59\)		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	\(0.721142\pi\)
\(67\)		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	\(-0.921429\pi\)
							0.969690	−	0.244339i	\(-0.0785709\pi\)
\(71\)	8.00000			0.949425			0.474713	−	0.880141i	\(-0.342552\pi\)
							0.474713	+	0.880141i	\(0.342552\pi\)
\(73\)		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	\(-0.924791\pi\)
							0.972217	−	0.234082i	\(-0.0752085\pi\)
\(79\)	8.00000			0.900070			0.450035	−	0.893011i	\(-0.351411\pi\)
							0.450035	+	0.893011i	\(0.351411\pi\)
\(83\)			12.0000i			1.31717i	0.752506	+	0.658586i	\(0.228845\pi\)
							−0.752506	+	0.658586i	\(0.771155\pi\)
\(89\)	10.0000			1.06000			0.529999	−	0.847998i	\(-0.322192\pi\)
							0.529999	+	0.847998i	\(0.322192\pi\)
\(97\)		−	18.0000i		−	1.82762i	−0.406138	−	0.913812i	\(-0.633125\pi\)
							0.406138	−	0.913812i	\(-0.366875\pi\)

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

    

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