Embedded newform 855.2.be.a.64.1

• Label: 855.2.be.a.64.1
• Level: $855$
• Weight: $2$
• Character: 855.64
• Analytic conductor: $6.827$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.82720937282\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			64.1
Root			\(-0.866025 + 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	855.64
Dual form			855.2.be.a.334.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(1.23205 - 1.86603i) q^{5} -4.00000i q^{7} +(-0.267949 + 4.46410i) q^{10} +1.00000 q^{11} +(-1.73205 - 1.00000i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(1.73205 - 1.00000i) q^{17} +(3.50000 - 2.59808i) q^{19} +(-2.00000 - 4.00000i) q^{20} +(-1.73205 + 1.00000i) q^{22} +(-5.19615 - 3.00000i) q^{23} +(-1.96410 - 4.59808i) q^{25} +4.00000 q^{26} +(-6.92820 - 4.00000i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(-6.92820 - 4.00000i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(-7.46410 - 4.92820i) q^{35} +2.00000i q^{37} +(-3.46410 + 8.00000i) q^{38} +(1.00000 + 1.73205i) q^{41} +(-1.73205 + 1.00000i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(5.19615 + 3.00000i) q^{47} -9.00000 q^{49} +(8.00000 + 6.00000i) q^{50} +(-3.46410 + 2.00000i) q^{52} +(-3.46410 - 2.00000i) q^{53} +(1.23205 - 1.86603i) q^{55} -18.0000i q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(12.1244 - 7.00000i) q^{62} +8.00000 q^{64} +(-4.00000 + 2.00000i) q^{65} +(8.66025 + 5.00000i) q^{67} -4.00000i q^{68} +(17.8564 + 1.07180i) q^{70} +(0.500000 + 0.866025i) q^{71} +(-8.66025 + 5.00000i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} -4.00000i q^{77} +(0.500000 + 0.866025i) q^{79} +(8.92820 + 0.535898i) q^{80} +(-3.46410 - 2.00000i) q^{82} -6.00000i q^{83} +(0.267949 - 4.46410i) q^{85} +(2.00000 - 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(-10.3923 + 6.00000i) q^{92} -12.0000 q^{94} +(-0.535898 - 9.73205i) q^{95} +(-5.19615 + 3.00000i) q^{97} +(15.5885 - 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q + 4 * q^4 - 2 * q^5 - 8 * q^10 + 4 * q^11 + 16 * q^14 + 8 * q^16 + 14 * q^19 - 8 * q^20 + 6 * q^25 + 16 * q^26 - 18 * q^29 - 28 * q^31 - 8 * q^34 - 16 * q^35 + 4 * q^41 + 4 * q^44 + 48 * q^46 - 36 * q^49 + 32 * q^50 - 2 * q^55 - 18 * q^59 + 14 * q^61 + 32 * q^64 - 16 * q^65 + 16 * q^70 + 2 * q^71 - 8 * q^74 - 4 * q^76 + 2 * q^79 + 8 * q^80 + 8 * q^85 + 8 * q^86 + 22 * q^89 - 16 * q^91 - 48 * q^94 - 16 * q^95

Character values

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

\(n\)	\(172\)	\(191\)	\(496\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73205	+	1.00000i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	\(-0.916667\pi\)
							−0.258819	+	0.965926i	\(0.583333\pi\)
\(3\)		0			0
							\(5\)	1.23205	−	1.86603i	0.550990	−	0.834512i
\(7\)		−	4.00000i		−	1.51186i								−0.654654	−	0.755929i	\(-0.727186\pi\)
							0.654654	−	0.755929i	\(-0.272814\pi\)
\(11\)	1.00000			0.301511			0.150756	−	0.988571i	\(-0.451829\pi\)
							0.150756	+	0.988571i	\(0.451829\pi\)
\(13\)	−1.73205	−	1.00000i	−0.480384	−	0.277350i	0.240192	−	0.970725i	\(-0.422790\pi\)
							−0.720577	+	0.693375i	\(0.756123\pi\)
\(17\)	1.73205	−	1.00000i	0.420084	−	0.242536i	−0.275029	−	0.961436i	\(-0.588688\pi\)
							0.695113	+	0.718900i	\(0.255354\pi\)
\(19\)	3.50000	−	2.59808i	0.802955	−	0.596040i
							\(23\)	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.151642	−	0.988436i	\(-0.548456\pi\)
−0.931831	+	0.362892i	\(0.881789\pi\)
\(29\)	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	\(0.481563\pi\)
							−0.893517	+	0.449029i	\(0.851770\pi\)
\(31\)	−7.00000			−1.25724			−0.628619	−	0.777714i	\(-0.716379\pi\)
							−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)			2.00000i			0.328798i	0.986394	+	0.164399i	\(0.0525685\pi\)
							−0.986394	+	0.164399i	\(0.947432\pi\)
\(41\)	1.00000	+	1.73205i	0.156174	+	0.270501i	0.933486	−	0.358614i	\(-0.116751\pi\)
							−0.777312	+	0.629115i	\(0.783417\pi\)
\(43\)	−1.73205	+	1.00000i	−0.264135	+	0.152499i	−0.626219	−	0.779647i	\(-0.715399\pi\)
							0.362084	+	0.932145i	\(0.382065\pi\)
\(47\)	5.19615	+	3.00000i	0.757937	+	0.437595i	0.828554	−	0.559908i	\(-0.189164\pi\)
							−0.0706177	+	0.997503i	\(0.522497\pi\)
\(53\)	−3.46410	−	2.00000i	−0.475831	−	0.274721i	0.242846	−	0.970065i	\(-0.421919\pi\)
							−0.718677	+	0.695344i	\(0.755252\pi\)
\(59\)	−4.50000	−	7.79423i	−0.585850	−	1.01472i	−0.994769	−	0.102151i	\(-0.967427\pi\)
							0.408919	−	0.912571i	\(-0.365906\pi\)
\(61\)	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	\(-0.685424\pi\)
							0.998264	+	0.0588933i	\(0.0187572\pi\)
\(67\)	8.66025	+	5.00000i	1.05802	+	0.610847i	0.924883	−	0.380251i	\(-0.124162\pi\)
							0.133135	+	0.991098i	\(0.457496\pi\)
\(71\)	0.500000	+	0.866025i	0.0593391	+	0.102778i	0.894169	−	0.447730i	\(-0.147767\pi\)
							−0.834830	+	0.550508i	\(0.814434\pi\)
\(73\)	−8.66025	+	5.00000i	−1.01361	+	0.585206i	−0.912245	−	0.409644i	\(-0.865653\pi\)
							−0.101361	+	0.994850i	\(0.532320\pi\)
\(79\)	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	\(-0.148751\pi\)
							−0.836527	+	0.547926i	\(0.815418\pi\)
\(83\)		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	\(-0.893190\pi\)
							0.944228	−	0.329293i	\(-0.106810\pi\)
\(89\)	5.50000	−	9.52628i	0.582999	−	1.00978i	−0.412123	−	0.911128i	\(-0.635213\pi\)
							0.995122	−	0.0986553i	\(-0.0314541\pi\)
\(97\)	−5.19615	+	3.00000i	−0.527589	+	0.304604i	−0.740034	−	0.672569i	\(-0.765191\pi\)
							0.212445	+	0.977173i	\(0.431857\pi\)

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.be.a.64.1		4
3.2	odd	2		95.2.i.a.64.2	yes	4
5.4	even	2	inner	855.2.be.a.64.2		4
15.2	even	4		475.2.e.c.26.1		2
15.8	even	4		475.2.e.a.26.1		2
15.14	odd	2		95.2.i.a.64.1	yes	4
19.11	even	3	inner	855.2.be.a.334.2		4
57.11	odd	6		95.2.i.a.49.1	✓	4
57.26	odd	6		1805.2.b.b.1084.2		2
57.50	even	6		1805.2.b.a.1084.1		2
95.49	even	6	inner	855.2.be.a.334.1		4
285.68	even	12		475.2.e.a.201.1		2
285.83	even	12		9025.2.a.i.1.1		1
285.107	odd	12		9025.2.a.j.1.1		1
285.164	even	6		1805.2.b.a.1084.2		2
285.182	even	12		475.2.e.c.201.1		2
285.197	even	12		9025.2.a.b.1.1		1
285.239	odd	6		95.2.i.a.49.2	yes	4
285.254	odd	6		1805.2.b.b.1084.1		2
285.278	odd	12		9025.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.i.a.49.1	✓	4	57.11	odd	6
95.2.i.a.49.2	yes	4	285.239	odd	6
95.2.i.a.64.1	yes	4	15.14	odd	2
95.2.i.a.64.2	yes	4	3.2	odd	2
475.2.e.a.26.1		2	15.8	even	4
475.2.e.a.201.1		2	285.68	even	12
475.2.e.c.26.1		2	15.2	even	4
475.2.e.c.201.1		2	285.182	even	12
855.2.be.a.64.1		4	1.1	even	1	trivial
855.2.be.a.64.2		4	5.4	even	2	inner
855.2.be.a.334.1		4	95.49	even	6	inner
855.2.be.a.334.2		4	19.11	even	3	inner
1805.2.b.a.1084.1		2	57.50	even	6
1805.2.b.a.1084.2		2	285.164	even	6
1805.2.b.b.1084.1		2	285.254	odd	6
1805.2.b.b.1084.2		2	57.26	odd	6
9025.2.a.a.1.1		1	285.278	odd	12
9025.2.a.b.1.1		1	285.197	even	12
9025.2.a.i.1.1		1	285.83	even	12
9025.2.a.j.1.1		1	285.107	odd	12