Embedded newform 810.4.e.m.541.1

• Label: 810.4.e.m.541.1
• Level: $810$
• Weight: $4$
• Character: 810.541
• Analytic conductor: $47.792$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(47.7915471046\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	810.541
Dual form			810.4.e.m.271.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-16.0000 + 27.7128i) q^{7} -8.00000 q^{8} -10.0000 q^{10} +(30.0000 - 51.9615i) q^{11} +(17.0000 + 29.4449i) q^{13} +(32.0000 + 55.4256i) q^{14} +(-8.00000 + 13.8564i) q^{16} +42.0000 q^{17} -76.0000 q^{19} +(-10.0000 + 17.3205i) q^{20} +(-60.0000 - 103.923i) q^{22} +(-12.5000 + 21.6506i) q^{25} +68.0000 q^{26} +128.000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(116.000 + 200.918i) q^{31} +(16.0000 + 27.7128i) q^{32} +(42.0000 - 72.7461i) q^{34} +160.000 q^{35} +134.000 q^{37} +(-76.0000 + 131.636i) q^{38} +(20.0000 + 34.6410i) q^{40} +(-117.000 - 202.650i) q^{41} +(206.000 - 356.802i) q^{43} -240.000 q^{44} +(180.000 - 311.769i) q^{47} +(-340.500 - 589.763i) q^{49} +(25.0000 + 43.3013i) q^{50} +(68.0000 - 117.779i) q^{52} +222.000 q^{53} -300.000 q^{55} +(128.000 - 221.703i) q^{56} +(6.00000 + 10.3923i) q^{58} +(-330.000 - 571.577i) q^{59} +(245.000 - 424.352i) q^{61} +464.000 q^{62} +64.0000 q^{64} +(85.0000 - 147.224i) q^{65} +(-406.000 - 703.213i) q^{67} +(-84.0000 - 145.492i) q^{68} +(160.000 - 277.128i) q^{70} +120.000 q^{71} +746.000 q^{73} +(134.000 - 232.095i) q^{74} +(152.000 + 263.272i) q^{76} +(960.000 + 1662.77i) q^{77} +(-76.0000 + 131.636i) q^{79} +80.0000 q^{80} -468.000 q^{82} +(402.000 - 696.284i) q^{83} +(-105.000 - 181.865i) q^{85} +(-412.000 - 713.605i) q^{86} +(-240.000 + 415.692i) q^{88} -678.000 q^{89} -1088.00 q^{91} +(-360.000 - 623.538i) q^{94} +(190.000 + 329.090i) q^{95} +(-97.0000 + 168.009i) q^{97} -1362.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 - 4 * q^4 - 5 * q^5 - 32 * q^7 - 16 * q^8 - 20 * q^10 + 60 * q^11 + 34 * q^13 + 64 * q^14 - 16 * q^16 + 84 * q^17 - 152 * q^19 - 20 * q^20 - 120 * q^22 - 25 * q^25 + 136 * q^26 + 256 * q^28 - 6 * q^29 + 232 * q^31 + 32 * q^32 + 84 * q^34 + 320 * q^35 + 268 * q^37 - 152 * q^38 + 40 * q^40 - 234 * q^41 + 412 * q^43 - 480 * q^44 + 360 * q^47 - 681 * q^49 + 50 * q^50 + 136 * q^52 + 444 * q^53 - 600 * q^55 + 256 * q^56 + 12 * q^58 - 660 * q^59 + 490 * q^61 + 928 * q^62 + 128 * q^64 + 170 * q^65 - 812 * q^67 - 168 * q^68 + 320 * q^70 + 240 * q^71 + 1492 * q^73 + 268 * q^74 + 304 * q^76 + 1920 * q^77 - 152 * q^79 + 160 * q^80 - 936 * q^82 + 804 * q^83 - 210 * q^85 - 824 * q^86 - 480 * q^88 - 1356 * q^89 - 2176 * q^91 - 720 * q^94 + 380 * q^95 - 194 * q^97 - 2724 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\)	\(487\)	\(731\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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	0.353553	−	0.612372i
							\(3\)		0			0
\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
							\(7\)	−16.0000	+	27.7128i	−0.863919	+	1.49635i	0.00419795	+	0.999991i	\(0.498664\pi\)
−0.868117	+	0.496360i	\(0.834670\pi\)
\(11\)	30.0000	−	51.9615i	0.822304	−	1.42427i	−0.0816590	−	0.996660i	\(-0.526022\pi\)
							0.903963	−	0.427611i	\(-0.140645\pi\)
\(13\)	17.0000	+	29.4449i	0.362689	+	0.628195i	0.988402	−	0.151858i	\(-0.0485255\pi\)
							−0.625714	+	0.780053i	\(0.715192\pi\)
\(17\)	42.0000			0.599206			0.299603	−	0.954064i	\(-0.403146\pi\)
							0.299603	+	0.954064i	\(0.403146\pi\)
\(19\)	−76.0000			−0.917663			−0.458831	−	0.888523i	\(-0.651732\pi\)
							−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(29\)	−3.00000	+	5.19615i	−0.0192099	+	0.0332725i	−0.875471	−	0.483272i	\(-0.839448\pi\)
							0.856261	+	0.516544i	\(0.172782\pi\)
\(31\)	116.000	+	200.918i	0.672071	+	1.16406i	0.977316	+	0.211788i	\(0.0679286\pi\)
							−0.305244	+	0.952274i	\(0.598738\pi\)
\(37\)	134.000			0.595391			0.297695	−	0.954661i	\(-0.403782\pi\)
							0.297695	+	0.954661i	\(0.403782\pi\)
\(41\)	−117.000	−	202.650i	−0.445667	−	0.771917i	0.552432	−	0.833558i	\(-0.313700\pi\)
							−0.998098	+	0.0616409i	\(0.980367\pi\)
\(43\)	206.000	−	356.802i	0.730575	−	1.26539i	−0.226063	−	0.974113i	\(-0.572586\pi\)
							0.956638	−	0.291280i	\(-0.0940810\pi\)
\(47\)	180.000	−	311.769i	0.558632	−	0.967579i	−0.438979	−	0.898497i	\(-0.644660\pi\)
							0.997611	−	0.0690815i	\(-0.0220069\pi\)
\(53\)	222.000			0.575359			0.287680	−	0.957727i	\(-0.407116\pi\)
							0.287680	+	0.957727i	\(0.407116\pi\)
\(59\)	−330.000	−	571.577i	−0.728175	−	1.26124i	−0.957654	−	0.287923i	\(-0.907035\pi\)
							0.229478	−	0.973314i	\(-0.426298\pi\)
\(61\)	245.000	−	424.352i	0.514246	−	0.890701i	−0.485617	−	0.874172i	\(-0.661405\pi\)
							0.999863	−	0.0165293i	\(-0.00526168\pi\)
\(67\)	−406.000	−	703.213i	−0.740310	−	1.28226i	−0.952354	−	0.304995i	\(-0.901345\pi\)
							0.212044	−	0.977260i	\(-0.431988\pi\)
\(71\)	120.000			0.200583			0.100291	−	0.994958i	\(-0.468022\pi\)
							0.100291	+	0.994958i	\(0.468022\pi\)
\(73\)	746.000			1.19606			0.598032	−	0.801472i	\(-0.295949\pi\)
							0.598032	+	0.801472i	\(0.295949\pi\)
\(79\)	−76.0000	+	131.636i	−0.108236	+	0.187471i	−0.915056	−	0.403327i	\(-0.867854\pi\)
							0.806820	+	0.590798i	\(0.201187\pi\)
\(83\)	402.000	−	696.284i	0.531629	−	0.920809i	−0.467689	−	0.883893i	\(-0.654913\pi\)
							0.999318	−	0.0369159i	\(-0.0117534\pi\)
\(89\)	−678.000			−0.807504			−0.403752	−	0.914868i	\(-0.632294\pi\)
							−0.403752	+	0.914868i	\(0.632294\pi\)
\(97\)	−97.0000	+	168.009i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	\(-0.865709\pi\)
							0.810782	+	0.585348i	\(0.199042\pi\)

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	810.4.e.m.541.1		2
3.2	odd	2		810.4.e.e.541.1		2
9.2	odd	6		90.4.a.d.1.1		1
9.4	even	3	inner	810.4.e.m.271.1		2
9.5	odd	6		810.4.e.e.271.1		2
9.7	even	3		30.4.a.a.1.1	✓	1
36.7	odd	6		240.4.a.c.1.1		1
36.11	even	6		720.4.a.b.1.1		1
45.2	even	12		450.4.c.k.199.2		2
45.7	odd	12		150.4.c.a.49.1		2
45.29	odd	6		450.4.a.b.1.1		1
45.34	even	6		150.4.a.e.1.1		1
45.38	even	12		450.4.c.k.199.1		2
45.43	odd	12		150.4.c.a.49.2		2
63.34	odd	6		1470.4.a.a.1.1		1
72.43	odd	6		960.4.a.s.1.1		1
72.61	even	6		960.4.a.j.1.1		1
180.7	even	12		1200.4.f.u.49.2		2
180.43	even	12		1200.4.f.u.49.1		2
180.79	odd	6		1200.4.a.bk.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	9.7	even	3
90.4.a.d.1.1		1	9.2	odd	6
150.4.a.e.1.1		1	45.34	even	6
150.4.c.a.49.1		2	45.7	odd	12
150.4.c.a.49.2		2	45.43	odd	12
240.4.a.c.1.1		1	36.7	odd	6
450.4.a.b.1.1		1	45.29	odd	6
450.4.c.k.199.1		2	45.38	even	12
450.4.c.k.199.2		2	45.2	even	12
720.4.a.b.1.1		1	36.11	even	6
810.4.e.e.271.1		2	9.5	odd	6
810.4.e.e.541.1		2	3.2	odd	2
810.4.e.m.271.1		2	9.4	even	3	inner
810.4.e.m.541.1		2	1.1	even	1	trivial
960.4.a.j.1.1		1	72.61	even	6
960.4.a.s.1.1		1	72.43	odd	6
1200.4.a.bk.1.1		1	180.79	odd	6
1200.4.f.u.49.1		2	180.43	even	12
1200.4.f.u.49.2		2	180.7	even	12
1470.4.a.a.1.1		1	63.34	odd	6