Embedded newform 405.4.e.i.136.1

• Label: 405.4.e.i.136.1
• Level: $405$
• Weight: $4$
• Character: 405.136
• Analytic conductor: $23.896$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.8957735523\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			136.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	405.136
Dual form			405.4.e.i.271.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} +(2.50000 + 4.33013i) q^{5} +(12.0000 - 20.7846i) q^{7} +15.0000 q^{8} +5.00000 q^{10} +(26.0000 - 45.0333i) q^{11} +(-11.0000 - 19.0526i) q^{13} +(-12.0000 - 20.7846i) q^{14} +(-20.5000 + 35.5070i) q^{16} +14.0000 q^{17} -20.0000 q^{19} +(-17.5000 + 30.3109i) q^{20} +(-26.0000 - 45.0333i) q^{22} +(-84.0000 - 145.492i) q^{23} +(-12.5000 + 21.6506i) q^{25} -22.0000 q^{26} +168.000 q^{28} +(115.000 - 199.186i) q^{29} +(144.000 + 249.415i) q^{31} +(80.5000 + 139.430i) q^{32} +(7.00000 - 12.1244i) q^{34} +120.000 q^{35} -34.0000 q^{37} +(-10.0000 + 17.3205i) q^{38} +(37.5000 + 64.9519i) q^{40} +(61.0000 + 105.655i) q^{41} +(94.0000 - 162.813i) q^{43} +364.000 q^{44} -168.000 q^{46} +(128.000 - 221.703i) q^{47} +(-116.500 - 201.784i) q^{49} +(12.5000 + 21.6506i) q^{50} +(77.0000 - 133.368i) q^{52} +338.000 q^{53} +260.000 q^{55} +(180.000 - 311.769i) q^{56} +(-115.000 - 199.186i) q^{58} +(50.0000 + 86.6025i) q^{59} +(-371.000 + 642.591i) q^{61} +288.000 q^{62} -167.000 q^{64} +(55.0000 - 95.2628i) q^{65} +(42.0000 + 72.7461i) q^{67} +(49.0000 + 84.8705i) q^{68} +(60.0000 - 103.923i) q^{70} +328.000 q^{71} -38.0000 q^{73} +(-17.0000 + 29.4449i) q^{74} +(-70.0000 - 121.244i) q^{76} +(-624.000 - 1080.80i) q^{77} +(120.000 - 207.846i) q^{79} -205.000 q^{80} +122.000 q^{82} +(606.000 - 1049.62i) q^{83} +(35.0000 + 60.6218i) q^{85} +(-94.0000 - 162.813i) q^{86} +(390.000 - 675.500i) q^{88} -330.000 q^{89} -528.000 q^{91} +(588.000 - 1018.45i) q^{92} +(-128.000 - 221.703i) q^{94} +(-50.0000 - 86.6025i) q^{95} +(-433.000 + 749.978i) q^{97} -233.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 + 7 * q^4 + 5 * q^5 + 24 * q^7 + 30 * q^8 + 10 * q^10 + 52 * q^11 - 22 * q^13 - 24 * q^14 - 41 * q^16 + 28 * q^17 - 40 * q^19 - 35 * q^20 - 52 * q^22 - 168 * q^23 - 25 * q^25 - 44 * q^26 + 336 * q^28 + 230 * q^29 + 288 * q^31 + 161 * q^32 + 14 * q^34 + 240 * q^35 - 68 * q^37 - 20 * q^38 + 75 * q^40 + 122 * q^41 + 188 * q^43 + 728 * q^44 - 336 * q^46 + 256 * q^47 - 233 * q^49 + 25 * q^50 + 154 * q^52 + 676 * q^53 + 520 * q^55 + 360 * q^56 - 230 * q^58 + 100 * q^59 - 742 * q^61 + 576 * q^62 - 334 * q^64 + 110 * q^65 + 84 * q^67 + 98 * q^68 + 120 * q^70 + 656 * q^71 - 76 * q^73 - 34 * q^74 - 140 * q^76 - 1248 * q^77 + 240 * q^79 - 410 * q^80 + 244 * q^82 + 1212 * q^83 + 70 * q^85 - 188 * q^86 + 780 * q^88 - 660 * q^89 - 1056 * q^91 + 1176 * q^92 - 256 * q^94 - 100 * q^95 - 866 * q^97 - 466 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	−	0.866025i	0.176777	−	0.306186i	−0.763998	−	0.645219i	\(-0.776766\pi\)
							0.940775	+	0.339032i	\(0.110100\pi\)
\(3\)		0			0
							\(5\)	2.50000	+	4.33013i	0.223607	+	0.387298i
\(7\)	12.0000	−	20.7846i	0.647939	−	1.12226i								−0.335675	−	0.941978i	\(-0.608964\pi\)
							0.983614	−	0.180286i	\(-0.0577022\pi\)
\(11\)	26.0000	−	45.0333i	0.712663	−	1.23437i	−0.251191	−	0.967938i	\(-0.580822\pi\)
							0.963854	−	0.266431i	\(-0.0858445\pi\)
\(13\)	−11.0000	−	19.0526i	−0.234681	−	0.406479i	0.724499	−	0.689276i	\(-0.242071\pi\)
							−0.959180	+	0.282797i	\(0.908738\pi\)
\(17\)	14.0000			0.199735			0.0998676	−	0.995001i	\(-0.468158\pi\)
							0.0998676	+	0.995001i	\(0.468158\pi\)
\(19\)	−20.0000			−0.241490			−0.120745	−	0.992684i	\(-0.538528\pi\)
							−0.120745	+	0.992684i	\(0.538528\pi\)
\(23\)	−84.0000	−	145.492i	−0.761531	−	1.31901i	−0.942061	−	0.335441i	\(-0.891115\pi\)
							0.180530	−	0.983569i	\(-0.442219\pi\)
\(29\)	115.000	−	199.186i	0.736378	−	1.27544i	−0.217738	−	0.976007i	\(-0.569868\pi\)
							0.954116	−	0.299437i	\(-0.0967988\pi\)
\(31\)	144.000	+	249.415i	0.834296	+	1.44504i	0.894603	+	0.446862i	\(0.147459\pi\)
							−0.0603071	+	0.998180i	\(0.519208\pi\)
\(37\)	−34.0000			−0.151069			−0.0755347	−	0.997143i	\(-0.524066\pi\)
							−0.0755347	+	0.997143i	\(0.524066\pi\)
\(41\)	61.0000	+	105.655i	0.232356	+	0.402453i	0.958501	−	0.285089i	\(-0.0920232\pi\)
							−0.726145	+	0.687542i	\(0.758690\pi\)
\(43\)	94.0000	−	162.813i	0.333369	−	0.577412i	−0.649801	−	0.760104i	\(-0.725148\pi\)
							0.983170	+	0.182692i	\(0.0584812\pi\)
\(47\)	128.000	−	221.703i	0.397249	−	0.688056i	−0.596136	−	0.802883i	\(-0.703298\pi\)
							0.993385	+	0.114827i	\(0.0366315\pi\)
\(53\)	338.000			0.875998			0.437999	−	0.898976i	\(-0.355687\pi\)
							0.437999	+	0.898976i	\(0.355687\pi\)
\(59\)	50.0000	+	86.6025i	0.110330	+	0.191096i	0.915903	−	0.401399i	\(-0.131476\pi\)
							−0.805574	+	0.592496i	\(0.798143\pi\)
\(61\)	−371.000	+	642.591i	−0.778716	+	1.34878i	0.153966	+	0.988076i	\(0.450795\pi\)
							−0.932682	+	0.360700i	\(0.882538\pi\)
\(67\)	42.0000	+	72.7461i	0.0765838	+	0.132647i	0.901774	−	0.432208i	\(-0.142265\pi\)
							−0.825190	+	0.564855i	\(0.808932\pi\)
\(71\)	328.000			0.548260			0.274130	−	0.961693i	\(-0.411610\pi\)
							0.274130	+	0.961693i	\(0.411610\pi\)
\(73\)	−38.0000			−0.0609255			−0.0304628	−	0.999536i	\(-0.509698\pi\)
							−0.0304628	+	0.999536i	\(0.509698\pi\)
\(79\)	120.000	−	207.846i	0.170899	−	0.296006i	−0.767835	−	0.640647i	\(-0.778666\pi\)
							0.938735	+	0.344641i	\(0.111999\pi\)
\(83\)	606.000	−	1049.62i	0.801411	−	1.38809i	−0.117276	−	0.993099i	\(-0.537416\pi\)
							0.918687	−	0.394986i	\(-0.129250\pi\)
\(89\)	−330.000			−0.393033			−0.196516	−	0.980501i	\(-0.562963\pi\)
							−0.196516	+	0.980501i	\(0.562963\pi\)
\(97\)	−433.000	+	749.978i	−0.453242	+	0.785038i	−0.998585	−	0.0531745i	\(-0.983066\pi\)
							0.545343	+	0.838213i	\(0.316399\pi\)

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	405.4.e.i.136.1		2
3.2	odd	2		405.4.e.g.136.1		2
9.2	odd	6		15.4.a.a.1.1	✓	1
9.4	even	3	inner	405.4.e.i.271.1		2
9.5	odd	6		405.4.e.g.271.1		2
9.7	even	3		45.4.a.c.1.1		1
36.7	odd	6		720.4.a.n.1.1		1
36.11	even	6		240.4.a.e.1.1		1
45.2	even	12		75.4.b.b.49.2		2
45.7	odd	12		225.4.b.e.199.1		2
45.29	odd	6		75.4.a.b.1.1		1
45.34	even	6		225.4.a.f.1.1		1
45.38	even	12		75.4.b.b.49.1		2
45.43	odd	12		225.4.b.e.199.2		2
63.20	even	6		735.4.a.e.1.1		1
63.34	odd	6		2205.4.a.l.1.1		1
72.11	even	6		960.4.a.ba.1.1		1
72.29	odd	6		960.4.a.b.1.1		1
99.65	even	6		1815.4.a.e.1.1		1
180.47	odd	12		1200.4.f.b.49.2		2
180.83	odd	12		1200.4.f.b.49.1		2
180.119	even	6		1200.4.a.t.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	9.2	odd	6
45.4.a.c.1.1		1	9.7	even	3
75.4.a.b.1.1		1	45.29	odd	6
75.4.b.b.49.1		2	45.38	even	12
75.4.b.b.49.2		2	45.2	even	12
225.4.a.f.1.1		1	45.34	even	6
225.4.b.e.199.1		2	45.7	odd	12
225.4.b.e.199.2		2	45.43	odd	12
240.4.a.e.1.1		1	36.11	even	6
405.4.e.g.136.1		2	3.2	odd	2
405.4.e.g.271.1		2	9.5	odd	6
405.4.e.i.136.1		2	1.1	even	1	trivial
405.4.e.i.271.1		2	9.4	even	3	inner
720.4.a.n.1.1		1	36.7	odd	6
735.4.a.e.1.1		1	63.20	even	6
960.4.a.b.1.1		1	72.29	odd	6
960.4.a.ba.1.1		1	72.11	even	6
1200.4.a.t.1.1		1	180.119	even	6
1200.4.f.b.49.1		2	180.83	odd	12
1200.4.f.b.49.2		2	180.47	odd	12
1815.4.a.e.1.1		1	99.65	even	6
2205.4.a.l.1.1		1	63.34	odd	6