Embedded newform 441.4.e.p.361.2

• Label: 441.4.e.p.361.2
• Level: $441$
• Weight: $4$
• Character: 441.361
• Analytic conductor: $26.020$
• Analytic rank: $0$
• Dimension: $4$
• 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:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial:	\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			\(-1.63746 - 1.52274i\) of defining polynomial
Character	\(\chi\)	\(=\)	441.361
Dual form			441.4.e.p.226.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13746 + 1.97014i) q^{2} +(1.41238 - 2.44631i) q^{4} +(2.27492 + 3.94027i) q^{5} +24.6254 q^{8} +(-5.17525 + 8.96379i) q^{10} +(-20.3746 + 35.2898i) q^{11} -53.2990 q^{13} +(16.7114 + 28.9450i) q^{16} +(-2.27492 + 3.94027i) q^{17} +(61.2990 + 106.173i) q^{19} +12.8522 q^{20} -92.7010 q^{22} +(65.6736 + 113.750i) q^{23} +(52.1495 - 90.3256i) q^{25} +(-60.6254 - 105.006i) q^{26} +216.598 q^{29} +(-125.897 + 218.060i) q^{31} +(60.4846 - 104.762i) q^{32} -10.3505 q^{34} +(-5.94851 - 10.3031i) q^{37} +(-139.450 + 241.535i) q^{38} +(56.0208 + 97.0308i) q^{40} -111.752 q^{41} +369.196 q^{43} +(57.5531 + 99.6850i) q^{44} +(-149.402 + 258.772i) q^{46} +(131.347 + 227.500i) q^{47} +237.272 q^{50} +(-75.2782 + 130.386i) q^{52} +(-283.550 + 491.123i) q^{53} -185.402 q^{55} +(246.371 + 426.728i) q^{58} +(-419.945 + 727.366i) q^{59} +(-242.897 - 420.710i) q^{61} -572.811 q^{62} +542.577 q^{64} +(-121.251 - 210.013i) q^{65} +(166.846 - 288.985i) q^{67} +(6.42608 + 11.1303i) q^{68} -590.248 q^{71} +(245.350 - 424.960i) q^{73} +(13.5324 - 23.4387i) q^{74} +346.309 q^{76} +(-60.8455 - 105.388i) q^{79} +(-76.0340 + 131.695i) q^{80} +(-127.114 - 220.168i) q^{82} +609.608 q^{83} -20.7010 q^{85} +(419.945 + 727.366i) q^{86} +(-501.733 + 869.026i) q^{88} +(-359.519 - 622.705i) q^{89} +371.023 q^{92} +(-298.804 + 517.544i) q^{94} +(-278.900 + 483.070i) q^{95} +637.877 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 3 * q^2 - 17 * q^4 - 6 * q^5 + 174 * q^8 - 66 * q^10 - 6 * q^11 - 32 * q^13 - 137 * q^16 + 6 * q^17 + 64 * q^19 + 444 * q^20 - 552 * q^22 + 6 * q^23 + 118 * q^25 - 318 * q^26 + 504 * q^29 + 40 * q^31 - 279 * q^32 - 132 * q^34 + 248 * q^37 - 588 * q^38 - 546 * q^40 - 900 * q^41 + 752 * q^43 + 804 * q^44 - 960 * q^46 + 12 * q^47 + 330 * q^50 - 890 * q^52 - 1104 * q^53 - 1104 * q^55 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 4224 * q^62 + 2578 * q^64 - 636 * q^65 - 148 * q^67 + 222 * q^68 - 1908 * q^71 + 1072 * q^73 + 1398 * q^74 + 3016 * q^76 + 572 * q^79 - 1950 * q^80 + 1530 * q^82 + 3888 * q^83 - 264 * q^85 + 804 * q^86 + 1164 * q^88 - 366 * q^89 + 5712 * q^92 - 1920 * q^94 - 1176 * q^95 - 1616 * 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13746	+	1.97014i	0.402152	+	0.696548i	0.993985	−	0.109512i	\(-0.0349289\pi\)
							−0.591833	+	0.806061i	\(0.701596\pi\)
\(3\)		0			0
							\(5\)	2.27492	+	3.94027i	0.203475	+	0.352429i	0.949646	−	0.313326i	\(-0.101443\pi\)
−0.746171	+	0.665754i	\(0.768110\pi\)
\(7\)		0			0
							\(11\)	−20.3746	+	35.2898i	−0.558470	+	0.967298i	0.439155	+	0.898412i	\(0.355278\pi\)
−0.997624	+	0.0688867i	\(0.978055\pi\)
\(13\)	−53.2990			−1.13711			−0.568557	−	0.822644i	\(-0.692498\pi\)
							−0.568557	+	0.822644i	\(0.692498\pi\)
\(17\)	−2.27492	+	3.94027i	−0.0324558	+	0.0562151i	−0.881797	−	0.471629i	\(-0.843666\pi\)
							0.849341	+	0.527844i	\(0.176999\pi\)
\(19\)	61.2990	+	106.173i	0.740156	+	1.28199i	0.952424	+	0.304776i	\(0.0985815\pi\)
							−0.212269	+	0.977211i	\(0.568085\pi\)
\(23\)	65.6736	+	113.750i	0.595387	+	1.03124i	0.993492	+	0.113900i	\(0.0363344\pi\)
							−0.398106	+	0.917340i	\(0.630332\pi\)
\(29\)	216.598			1.38694			0.693470	−	0.720486i	\(-0.256081\pi\)
							0.693470	+	0.720486i	\(0.256081\pi\)
\(31\)	−125.897	+	218.060i	−0.729412	+	1.26338i	0.227720	+	0.973727i	\(0.426873\pi\)
							−0.957132	+	0.289652i	\(0.906460\pi\)
\(37\)	−5.94851	−	10.3031i	−0.0264305	−	0.0457790i	0.852508	−	0.522715i	\(-0.175081\pi\)
							−0.878938	+	0.476936i	\(0.841747\pi\)
\(41\)	−111.752			−0.425678			−0.212839	−	0.977087i	\(-0.568271\pi\)
							−0.212839	+	0.977087i	\(0.568271\pi\)
\(43\)	369.196			1.30935			0.654673	−	0.755912i	\(-0.272806\pi\)
							0.654673	+	0.755912i	\(0.272806\pi\)
\(47\)	131.347	+	227.500i	0.407637	+	0.706049i	0.994624	−	0.103548i	\(-0.0330194\pi\)
							−0.586987	+	0.809596i	\(0.699686\pi\)
\(53\)	−283.550	+	491.123i	−0.734879	+	1.27285i	0.219898	+	0.975523i	\(0.429428\pi\)
							−0.954777	+	0.297324i	\(0.903906\pi\)
\(59\)	−419.945	+	727.366i	−0.926648	+	1.60500i	−0.137758	+	0.990466i	\(0.543990\pi\)
							−0.788890	+	0.614535i	\(0.789344\pi\)
\(61\)	−242.897	−	420.710i	−0.509832	−	0.883056i	−0.999935	−	0.0113909i	\(-0.996374\pi\)
							0.490103	−	0.871665i	\(-0.336959\pi\)
\(67\)	166.846	−	288.985i	0.304230	−	0.526942i	−0.672859	−	0.739770i	\(-0.734934\pi\)
							0.977090	+	0.212828i	\(0.0682675\pi\)
\(71\)	−590.248			−0.986613			−0.493306	−	0.869856i	\(-0.664212\pi\)
							−0.493306	+	0.869856i	\(0.664212\pi\)
\(73\)	245.350	−	424.960i	0.393371	−	0.681339i	−0.599521	−	0.800359i	\(-0.704642\pi\)
							0.992892	+	0.119020i	\(0.0379754\pi\)
\(79\)	−60.8455	−	105.388i	−0.0866539	−	0.150089i	0.819441	−	0.573164i	\(-0.194284\pi\)
							−0.906095	+	0.423075i	\(0.860951\pi\)
\(83\)	609.608			0.806183			0.403091	−	0.915160i	\(-0.367936\pi\)
							0.403091	+	0.915160i	\(0.367936\pi\)
\(89\)	−359.519	−	622.705i	−0.428190	−	0.741648i	0.568522	−	0.822668i	\(-0.307515\pi\)
							−0.996712	+	0.0810204i	\(0.974182\pi\)
\(97\)	637.877			0.667697			0.333849	−	0.942627i	\(-0.391653\pi\)
							0.333849	+	0.942627i	\(0.391653\pi\)

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.p.361.2		4
3.2	odd	2		147.4.e.m.67.1		4
7.2	even	3	inner	441.4.e.p.226.2		4
7.3	odd	6		63.4.a.e.1.1		2
7.4	even	3		441.4.a.r.1.1		2
7.5	odd	6		441.4.e.q.226.2		4
7.6	odd	2		441.4.e.q.361.2		4
21.2	odd	6		147.4.e.m.79.1		4
21.5	even	6		147.4.e.l.79.1		4
21.11	odd	6		147.4.a.i.1.2		2
21.17	even	6		21.4.a.c.1.2	✓	2
21.20	even	2		147.4.e.l.67.1		4
28.3	even	6		1008.4.a.ba.1.2		2
35.24	odd	6		1575.4.a.p.1.2		2
84.11	even	6		2352.4.a.bz.1.2		2
84.59	odd	6		336.4.a.m.1.1		2
105.17	odd	12		525.4.d.g.274.3		4
105.38	odd	12		525.4.d.g.274.2		4
105.59	even	6		525.4.a.n.1.1		2
168.59	odd	6		1344.4.a.bo.1.2		2
168.101	even	6		1344.4.a.bg.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.2	✓	2	21.17	even	6
63.4.a.e.1.1		2	7.3	odd	6
147.4.a.i.1.2		2	21.11	odd	6
147.4.e.l.67.1		4	21.20	even	2
147.4.e.l.79.1		4	21.5	even	6
147.4.e.m.67.1		4	3.2	odd	2
147.4.e.m.79.1		4	21.2	odd	6
336.4.a.m.1.1		2	84.59	odd	6
441.4.a.r.1.1		2	7.4	even	3
441.4.e.p.226.2		4	7.2	even	3	inner
441.4.e.p.361.2		4	1.1	even	1	trivial
441.4.e.q.226.2		4	7.5	odd	6
441.4.e.q.361.2		4	7.6	odd	2
525.4.a.n.1.1		2	105.59	even	6
525.4.d.g.274.2		4	105.38	odd	12
525.4.d.g.274.3		4	105.17	odd	12
1008.4.a.ba.1.2		2	28.3	even	6
1344.4.a.bg.1.2		2	168.101	even	6
1344.4.a.bo.1.2		2	168.59	odd	6
1575.4.a.p.1.2		2	35.24	odd	6
2352.4.a.bz.1.2		2	84.11	even	6