Embedded newform 855.2.be.a.64.2

• Label: 855.2.be.a.64.2
• 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			64.2
Root			\(0.866025 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	855.64
Dual form			855.2.be.a.334.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.23205 + 0.133975i) q^{5} +4.00000i q^{7} +(-3.73205 + 2.46410i) q^{10} +1.00000 q^{11} +(1.73205 + 1.00000i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(-1.73205 + 1.00000i) q^{17} +(3.50000 - 2.59808i) q^{19} +(-2.00000 + 4.00000i) q^{20} +(1.73205 - 1.00000i) q^{22} +(5.19615 + 3.00000i) q^{23} +(4.96410 - 0.598076i) q^{25} +4.00000 q^{26} +(6.92820 + 4.00000i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(6.92820 + 4.00000i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(-0.535898 - 8.92820i) q^{35} -2.00000i q^{37} +(3.46410 - 8.00000i) q^{38} +(1.00000 + 1.73205i) q^{41} +(1.73205 - 1.00000i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(-5.19615 - 3.00000i) q^{47} -9.00000 q^{49} +(8.00000 - 6.00000i) q^{50} +(3.46410 - 2.00000i) q^{52} +(3.46410 + 2.00000i) q^{53} +(-2.23205 + 0.133975i) q^{55} +18.0000i q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-12.1244 + 7.00000i) q^{62} +8.00000 q^{64} +(-4.00000 - 2.00000i) q^{65} +(-8.66025 - 5.00000i) q^{67} +4.00000i q^{68} +(-9.85641 - 14.9282i) q^{70} +(0.500000 + 0.866025i) q^{71} +(8.66025 - 5.00000i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} +4.00000i q^{77} +(0.500000 + 0.866025i) q^{79} +(-4.92820 - 7.46410i) q^{80} +(3.46410 + 2.00000i) q^{82} +6.00000i q^{83} +(3.73205 - 2.46410i) q^{85} +(2.00000 - 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(10.3923 - 6.00000i) q^{92} -12.0000 q^{94} +(-7.46410 + 6.26795i) q^{95} +(5.19615 - 3.00000i) q^{97} +(-15.5885 + 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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	0.258819	−	0.965926i	\(-0.416667\pi\)
							0.965926	+	0.258819i	\(0.0833333\pi\)
\(3\)		0			0
							\(5\)	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
\(7\)			4.00000i			1.51186i								0.654654	+	0.755929i	\(0.272814\pi\)
							−0.654654	+	0.755929i	\(0.727186\pi\)
\(11\)	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	0.720577	−	0.693375i	\(-0.243877\pi\)
							−0.240192	+	0.970725i	\(0.577210\pi\)
\(17\)	−1.73205	+	1.00000i	−0.420084	+	0.242536i	−0.695113	−	0.718900i	\(-0.744646\pi\)
							0.275029	+	0.961436i	\(0.411312\pi\)
\(19\)	3.50000	−	2.59808i	0.802955	−	0.596040i
							\(23\)	5.19615	+	3.00000i	1.08347	+	0.625543i	0.931831	−	0.362892i	\(-0.118211\pi\)
0.151642	+	0.988436i	\(0.451544\pi\)
\(29\)	−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	\(-0.947432\pi\)
							0.986394	−	0.164399i	\(-0.0525685\pi\)
\(41\)	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	−0.362084	−	0.932145i	\(-0.617935\pi\)
							0.626219	+	0.779647i	\(0.284601\pi\)
\(47\)	−5.19615	−	3.00000i	−0.757937	−	0.437595i	0.0706177	−	0.997503i	\(-0.477503\pi\)
							−0.828554	+	0.559908i	\(0.810836\pi\)
\(53\)	3.46410	+	2.00000i	0.475831	+	0.274721i	0.718677	−	0.695344i	\(-0.244748\pi\)
							−0.242846	+	0.970065i	\(0.578081\pi\)
\(59\)	−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	−0.133135	−	0.991098i	\(-0.542504\pi\)
							−0.924883	+	0.380251i	\(0.875838\pi\)
\(71\)	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	0.101361	−	0.994850i	\(-0.467680\pi\)
							0.912245	+	0.409644i	\(0.134347\pi\)
\(79\)	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	\(0.106810\pi\)
							−0.944228	+	0.329293i	\(0.893190\pi\)
\(89\)	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	−0.212445	−	0.977173i	\(-0.568143\pi\)
							0.740034	+	0.672569i	\(0.234809\pi\)

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

    

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