Embedded newform 475.2.e.c.26.1

• Label: 475.2.e.c.26.1
• Level: $475$
• Weight: $2$
• Character: 475.26
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			26.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	475.26
Dual form			475.2.e.c.201.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{4} +4.00000 q^{7} +(1.50000 - 2.59808i) q^{9} -1.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(-1.00000 - 1.73205i) q^{17} +6.00000 q^{18} +(-3.50000 + 2.59808i) q^{19} +(-1.00000 - 1.73205i) q^{22} +(3.00000 - 5.19615i) q^{23} -4.00000 q^{26} +(-4.00000 + 6.92820i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(-4.00000 + 6.92820i) q^{32} +(2.00000 - 3.46410i) q^{34} +(3.00000 + 5.19615i) q^{36} -2.00000 q^{37} +(-8.00000 - 3.46410i) q^{38} +(-1.00000 - 1.73205i) q^{41} +(1.00000 + 1.73205i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(3.00000 - 5.19615i) q^{47} +9.00000 q^{49} +(-2.00000 - 3.46410i) q^{52} +(2.00000 - 3.46410i) q^{53} -18.0000 q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-7.00000 - 12.1244i) q^{62} +(6.00000 - 10.3923i) q^{63} -8.00000 q^{64} +(-5.00000 + 8.66025i) q^{67} +4.00000 q^{68} +(-0.500000 - 0.866025i) q^{71} +(5.00000 + 8.66025i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} -4.00000 q^{77} +(-0.500000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(2.00000 - 3.46410i) q^{82} +6.00000 q^{83} +(-2.00000 + 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(6.00000 + 10.3923i) q^{92} +12.0000 q^{94} +(-3.00000 - 5.19615i) q^{97} +(9.00000 + 15.5885i) q^{98} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 - 2 * q^4 + 8 * q^7 + 3 * q^9 - 2 * q^11 - 2 * q^13 + 8 * q^14 + 4 * q^16 - 2 * q^17 + 12 * q^18 - 7 * q^19 - 2 * q^22 + 6 * q^23 - 8 * q^26 - 8 * q^28 - 9 * q^29 - 14 * q^31 - 8 * q^32 + 4 * q^34 + 6 * q^36 - 4 * q^37 - 16 * q^38 - 2 * q^41 + 2 * q^43 + 2 * q^44 + 24 * q^46 + 6 * q^47 + 18 * q^49 - 4 * q^52 + 4 * q^53 - 36 * q^58 - 9 * q^59 + 7 * q^61 - 14 * q^62 + 12 * q^63 - 16 * q^64 - 10 * q^67 + 8 * q^68 - q^71 + 10 * q^73 - 4 * q^74 - 2 * q^76 - 8 * q^77 - q^79 - 9 * q^81 + 4 * q^82 + 12 * q^83 - 4 * q^86 + 11 * q^89 - 8 * q^91 + 12 * q^92 + 24 * q^94 - 6 * q^97 + 18 * q^98 - 3 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	+	1.73205i	0.707107	+	1.22474i	0.965926	+	0.258819i	\(0.0833333\pi\)
							−0.258819	+	0.965926i	\(0.583333\pi\)
\(3\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(5\)		0			0
							\(7\)	4.00000			1.51186			0.755929	−	0.654654i	\(-0.227186\pi\)
0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	−1.00000			−0.301511			−0.150756	−	0.988571i	\(-0.548171\pi\)
							−0.150756	+	0.988571i	\(0.548171\pi\)
\(13\)	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	\(-0.244646\pi\)
							−0.961436	+	0.275029i	\(0.911312\pi\)
\(19\)	−3.50000	+	2.59808i	−0.802955	+	0.596040i
							\(23\)	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	\(-0.618211\pi\)
0.988436	−	0.151642i	\(-0.0484560\pi\)
\(29\)	−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	\(0.716379\pi\)
\(37\)	−2.00000			−0.328798			−0.164399	−	0.986394i	\(-0.552568\pi\)
							−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−1.00000	−	1.73205i	−0.156174	−	0.270501i	0.777312	−	0.629115i	\(-0.216583\pi\)
							−0.933486	+	0.358614i	\(0.883249\pi\)
\(43\)	1.00000	+	1.73205i	0.152499	+	0.264135i	0.932145	−	0.362084i	\(-0.117935\pi\)
							−0.779647	+	0.626219i	\(0.784601\pi\)
\(47\)	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	\(-0.689164\pi\)
							0.997503	+	0.0706177i	\(0.0224970\pi\)
\(53\)	2.00000	−	3.46410i	0.274721	−	0.475831i	−0.695344	−	0.718677i	\(-0.744748\pi\)
							0.970065	+	0.242846i	\(0.0780811\pi\)
\(59\)	−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	\(0.0187572\pi\)
\(67\)	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	\(0.375838\pi\)
							−0.991098	+	0.133135i	\(0.957496\pi\)
\(71\)	−0.500000	−	0.866025i	−0.0593391	−	0.102778i	0.834830	−	0.550508i	\(-0.185566\pi\)
							−0.894169	+	0.447730i	\(0.852233\pi\)
\(73\)	5.00000	+	8.66025i	0.585206	+	1.01361i	0.994850	+	0.101361i	\(0.0323196\pi\)
							−0.409644	+	0.912245i	\(0.634347\pi\)
\(79\)	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	\(-0.184582\pi\)
							−0.892781	+	0.450490i	\(0.851249\pi\)
\(83\)	6.00000			0.658586			0.329293	−	0.944228i	\(-0.393190\pi\)
							0.329293	+	0.944228i	\(0.393190\pi\)
\(89\)	5.50000	−	9.52628i	0.582999	−	1.00978i	−0.412123	−	0.911128i	\(-0.635213\pi\)
							0.995122	−	0.0986553i	\(-0.0314541\pi\)
\(97\)	−3.00000	−	5.19615i	−0.304604	−	0.527589i	0.672569	−	0.740034i	\(-0.265191\pi\)
							−0.977173	+	0.212445i	\(0.931857\pi\)

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	475.2.e.c.26.1		2
5.2	odd	4		95.2.i.a.64.1	yes	4
5.3	odd	4		95.2.i.a.64.2	yes	4
5.4	even	2		475.2.e.a.26.1		2
15.2	even	4		855.2.be.a.64.2		4
15.8	even	4		855.2.be.a.64.1		4
19.7	even	3		9025.2.a.b.1.1		1
19.11	even	3	inner	475.2.e.c.201.1		2
19.12	odd	6		9025.2.a.j.1.1		1
95.7	odd	12		1805.2.b.b.1084.1		2
95.12	even	12		1805.2.b.a.1084.2		2
95.49	even	6		475.2.e.a.201.1		2
95.64	even	6		9025.2.a.i.1.1		1
95.68	odd	12		95.2.i.a.49.1	✓	4
95.69	odd	6		9025.2.a.a.1.1		1
95.83	odd	12		1805.2.b.b.1084.2		2
95.87	odd	12		95.2.i.a.49.2	yes	4
95.88	even	12		1805.2.b.a.1084.1		2
285.68	even	12		855.2.be.a.334.2		4
285.182	even	12		855.2.be.a.334.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.i.a.49.1	✓	4	95.68	odd	12
95.2.i.a.49.2	yes	4	95.87	odd	12
95.2.i.a.64.1	yes	4	5.2	odd	4
95.2.i.a.64.2	yes	4	5.3	odd	4
475.2.e.a.26.1		2	5.4	even	2
475.2.e.a.201.1		2	95.49	even	6
475.2.e.c.26.1		2	1.1	even	1	trivial
475.2.e.c.201.1		2	19.11	even	3	inner
855.2.be.a.64.1		4	15.8	even	4
855.2.be.a.64.2		4	15.2	even	4
855.2.be.a.334.1		4	285.182	even	12
855.2.be.a.334.2		4	285.68	even	12
1805.2.b.a.1084.1		2	95.88	even	12
1805.2.b.a.1084.2		2	95.12	even	12
1805.2.b.b.1084.1		2	95.7	odd	12
1805.2.b.b.1084.2		2	95.83	odd	12
9025.2.a.a.1.1		1	95.69	odd	6
9025.2.a.b.1.1		1	19.7	even	3
9025.2.a.i.1.1		1	95.64	even	6
9025.2.a.j.1.1		1	19.12	odd	6