Embedded newform 441.4.e.n.361.1

• Label: 441.4.e.n.361.1
• Level: $441$
• Weight: $4$
• Character: 441.361
• Analytic conductor: $26.020$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.0198423125\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	441.361
Dual form			441.4.e.n.226.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(-4.00000 + 6.92820i) q^{4} +(2.00000 + 3.46410i) q^{5} +(-8.00000 + 13.8564i) q^{10} +(31.0000 - 53.6936i) q^{11} +62.0000 q^{13} +(32.0000 + 55.4256i) q^{16} +(-42.0000 + 72.7461i) q^{17} +(50.0000 + 86.6025i) q^{19} -32.0000 q^{20} +248.000 q^{22} +(-21.0000 - 36.3731i) q^{23} +(54.5000 - 94.3968i) q^{25} +(124.000 + 214.774i) q^{26} +10.0000 q^{29} +(-24.0000 + 41.5692i) q^{31} +(-128.000 + 221.703i) q^{32} -336.000 q^{34} +(123.000 + 213.042i) q^{37} +(-200.000 + 346.410i) q^{38} -248.000 q^{41} +68.0000 q^{43} +(248.000 + 429.549i) q^{44} +(84.0000 - 145.492i) q^{46} +(-162.000 - 280.592i) q^{47} +436.000 q^{50} +(-248.000 + 429.549i) q^{52} +(129.000 - 223.435i) q^{53} +248.000 q^{55} +(20.0000 + 34.6410i) q^{58} +(-60.0000 + 103.923i) q^{59} +(311.000 + 538.668i) q^{61} -192.000 q^{62} -512.000 q^{64} +(124.000 + 214.774i) q^{65} +(-452.000 + 782.887i) q^{67} +(-336.000 - 581.969i) q^{68} +678.000 q^{71} +(-321.000 + 555.988i) q^{73} +(-492.000 + 852.169i) q^{74} -800.000 q^{76} +(-370.000 - 640.859i) q^{79} +(-128.000 + 221.703i) q^{80} +(-496.000 - 859.097i) q^{82} +468.000 q^{83} -336.000 q^{85} +(136.000 + 235.559i) q^{86} +(-100.000 - 173.205i) q^{89} +336.000 q^{92} +(648.000 - 1122.37i) q^{94} +(-200.000 + 346.410i) q^{95} +1266.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 - 8 * q^4 + 4 * q^5 - 16 * q^10 + 62 * q^11 + 124 * q^13 + 64 * q^16 - 84 * q^17 + 100 * q^19 - 64 * q^20 + 496 * q^22 - 42 * q^23 + 109 * q^25 + 248 * q^26 + 20 * q^29 - 48 * q^31 - 256 * q^32 - 672 * q^34 + 246 * q^37 - 400 * q^38 - 496 * q^41 + 136 * q^43 + 496 * q^44 + 168 * q^46 - 324 * q^47 + 872 * q^50 - 496 * q^52 + 258 * q^53 + 496 * q^55 + 40 * q^58 - 120 * q^59 + 622 * q^61 - 384 * q^62 - 1024 * q^64 + 248 * q^65 - 904 * q^67 - 672 * q^68 + 1356 * q^71 - 642 * q^73 - 984 * q^74 - 1600 * q^76 - 740 * q^79 - 256 * q^80 - 992 * q^82 + 936 * q^83 - 672 * q^85 + 272 * q^86 - 200 * q^89 + 672 * q^92 + 1296 * q^94 - 400 * q^95 + 2532 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	+	3.46410i	0.707107	+	1.22474i	0.965926	+	0.258819i	\(0.0833333\pi\)
							−0.258819	+	0.965926i	\(0.583333\pi\)
\(3\)		0			0
							\(5\)	2.00000	+	3.46410i	0.178885	+	0.309839i	0.941499	−	0.337016i	\(-0.109418\pi\)
−0.762614	+	0.646854i	\(0.776084\pi\)
\(7\)		0			0
							\(11\)	31.0000	−	53.6936i	0.849714	−	1.47175i	−0.0317500	−	0.999496i	\(-0.510108\pi\)
0.881464	−	0.472252i	\(-0.156559\pi\)
\(13\)	62.0000			1.32275			0.661373	−	0.750057i	\(-0.269974\pi\)
							0.661373	+	0.750057i	\(0.269974\pi\)
\(17\)	−42.0000	+	72.7461i	−0.599206	+	1.03785i	0.393733	+	0.919225i	\(0.371183\pi\)
							−0.992939	+	0.118630i	\(0.962150\pi\)
\(19\)	50.0000	+	86.6025i	0.603726	+	1.04568i	0.992251	+	0.124246i	\(0.0396511\pi\)
							−0.388526	+	0.921438i	\(0.627016\pi\)
\(23\)	−21.0000	−	36.3731i	−0.190383	−	0.329753i	0.754994	−	0.655731i	\(-0.227640\pi\)
							−0.945377	+	0.325979i	\(0.894306\pi\)
\(29\)	10.0000			0.0640329			0.0320164	−	0.999487i	\(-0.489807\pi\)
							0.0320164	+	0.999487i	\(0.489807\pi\)
\(31\)	−24.0000	+	41.5692i	−0.139049	+	0.240840i	−0.927137	−	0.374723i	\(-0.877738\pi\)
							0.788088	+	0.615563i	\(0.211071\pi\)
\(37\)	123.000	+	213.042i	0.546516	+	0.946593i	0.998510	+	0.0545719i	\(0.0173794\pi\)
							−0.451994	+	0.892021i	\(0.649287\pi\)
\(41\)	−248.000			−0.944661			−0.472330	−	0.881422i	\(-0.656587\pi\)
							−0.472330	+	0.881422i	\(0.656587\pi\)
\(43\)	68.0000			0.241161			0.120580	−	0.992704i	\(-0.461524\pi\)
							0.120580	+	0.992704i	\(0.461524\pi\)
\(47\)	−162.000	−	280.592i	−0.502769	−	0.870821i	−0.999995	−	0.00319997i	\(-0.998981\pi\)
							0.497226	−	0.867621i	\(-0.334352\pi\)
\(53\)	129.000	−	223.435i	0.334330	−	0.579077i	−0.649026	−	0.760767i	\(-0.724823\pi\)
							0.983356	+	0.181689i	\(0.0581565\pi\)
\(59\)	−60.0000	+	103.923i	−0.132396	+	0.229316i	−0.924600	−	0.380941i	\(-0.875600\pi\)
							0.792204	+	0.610256i	\(0.208934\pi\)
\(61\)	311.000	+	538.668i	0.652778	+	1.13064i	0.982446	+	0.186548i	\(0.0597300\pi\)
							−0.329668	+	0.944097i	\(0.606937\pi\)
\(67\)	−452.000	+	782.887i	−0.824188	+	1.42754i	0.0783505	+	0.996926i	\(0.475035\pi\)
							−0.902538	+	0.430609i	\(0.858299\pi\)
\(71\)	678.000			1.13329			0.566646	−	0.823961i	\(-0.308241\pi\)
							0.566646	+	0.823961i	\(0.308241\pi\)
\(73\)	−321.000	+	555.988i	−0.514660	+	0.891418i	0.485195	+	0.874406i	\(0.338749\pi\)
							−0.999855	+	0.0170119i	\(0.994585\pi\)
\(79\)	−370.000	−	640.859i	−0.526940	−	0.912687i	−0.999507	−	0.0313921i	\(-0.990006\pi\)
							0.472567	−	0.881295i	\(-0.343327\pi\)
\(83\)	468.000			0.618912			0.309456	−	0.950914i	\(-0.399853\pi\)
							0.309456	+	0.950914i	\(0.399853\pi\)
\(89\)	−100.000	−	173.205i	−0.119101	−	0.206289i	0.800311	−	0.599585i	\(-0.204668\pi\)
							−0.919412	+	0.393297i	\(0.871335\pi\)
\(97\)	1266.00			1.32518			0.662592	−	0.748981i	\(-0.269456\pi\)
							0.662592	+	0.748981i	\(0.269456\pi\)

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	441.4.e.n.361.1		2
3.2	odd	2		147.4.e.b.67.1		2
7.2	even	3	inner	441.4.e.n.226.1		2
7.3	odd	6		63.4.a.a.1.1		1
7.4	even	3		441.4.a.b.1.1		1
7.5	odd	6		441.4.e.m.226.1		2
7.6	odd	2		441.4.e.m.361.1		2
21.2	odd	6		147.4.e.b.79.1		2
21.5	even	6		147.4.e.c.79.1		2
21.11	odd	6		147.4.a.g.1.1		1
21.17	even	6		21.4.a.b.1.1	✓	1
21.20	even	2		147.4.e.c.67.1		2
28.3	even	6		1008.4.a.m.1.1		1
35.24	odd	6		1575.4.a.k.1.1		1
84.11	even	6		2352.4.a.l.1.1		1
84.59	odd	6		336.4.a.h.1.1		1
105.17	odd	12		525.4.d.b.274.2		2
105.38	odd	12		525.4.d.b.274.1		2
105.59	even	6		525.4.a.b.1.1		1
168.59	odd	6		1344.4.a.i.1.1		1
168.101	even	6		1344.4.a.w.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.b.1.1	✓	1	21.17	even	6
63.4.a.a.1.1		1	7.3	odd	6
147.4.a.g.1.1		1	21.11	odd	6
147.4.e.b.67.1		2	3.2	odd	2
147.4.e.b.79.1		2	21.2	odd	6
147.4.e.c.67.1		2	21.20	even	2
147.4.e.c.79.1		2	21.5	even	6
336.4.a.h.1.1		1	84.59	odd	6
441.4.a.b.1.1		1	7.4	even	3
441.4.e.m.226.1		2	7.5	odd	6
441.4.e.m.361.1		2	7.6	odd	2
441.4.e.n.226.1		2	7.2	even	3	inner
441.4.e.n.361.1		2	1.1	even	1	trivial
525.4.a.b.1.1		1	105.59	even	6
525.4.d.b.274.1		2	105.38	odd	12
525.4.d.b.274.2		2	105.17	odd	12
1008.4.a.m.1.1		1	28.3	even	6
1344.4.a.i.1.1		1	168.59	odd	6
1344.4.a.w.1.1		1	168.101	even	6
1575.4.a.k.1.1		1	35.24	odd	6
2352.4.a.l.1.1		1	84.11	even	6