Embedded newform 207.4.o.a.11.63

• Label: 207.4.o.a.11.63
• Level: $207$
• Weight: $4$
• Character: 207.11
• Analytic conductor: $12.213$
• Analytic rank: $0$
• Dimension: $1400$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$207 = 3^{2} \cdot 23$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	207.o (of order $66$, degree $20$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$12.2133953712$
Analytic rank:	$0$
Dimension:	$1400$
Relative dimension:	$70$ over $\Q(\zeta_{66})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			11.63
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.63

$q$-expansion

$f(q)$	$=$	$q+(4.58851 - 0.218578i) q^{2} +(-2.75063 + 4.40840i) q^{3} +(13.0429 - 1.24545i) q^{4} +(-3.38925 - 3.23164i) q^{5} +(-11.6577 + 20.8292i) q^{6} +(6.95340 + 36.0777i) q^{7} +(23.1996 - 3.33560i) q^{8} +(-11.8681 - 24.2518i) q^{9} +(-16.2580 - 14.0876i) q^{10} +(-17.4921 + 9.01781i) q^{11} +(-30.3858 + 60.9241i) q^{12} +(28.4217 + 5.47783i) q^{13} +(39.7916 + 164.023i) q^{14} +(23.5689 - 6.05212i) q^{15} +(2.79901 - 0.539465i) q^{16} +(39.4982 + 86.4889i) q^{17} +(-59.7576 - 108.686i) q^{18} +(-55.6562 - 25.4173i) q^{19} +(-48.2304 - 37.9288i) q^{20} +(-178.171 - 68.5830i) q^{21} +(-78.2917 + 45.2017i) q^{22} +(76.7094 + 79.2632i) q^{23} +(-49.1089 + 111.448i) q^{24} +(-4.90424 - 102.953i) q^{25} +(131.611 + 18.9227i) q^{26} +(139.556 + 14.3886i) q^{27} +(135.625 + 461.897i) q^{28} +(-13.0262 + 136.416i) q^{29} +(106.824 - 32.9219i) q^{30} +(189.713 - 149.192i) q^{31} +(-169.495 + 41.1189i) q^{32} +(8.36020 - 101.917i) q^{33} +(200.142 + 388.222i) q^{34} +(93.0233 - 144.747i) q^{35} +(-184.998 - 301.533i) q^{36} +(74.5307 - 253.828i) q^{37} +(-260.935 - 104.463i) q^{38} +(-102.326 + 110.227i) q^{39} +(-89.4087 - 63.6676i) q^{40} +(95.9013 - 100.578i) q^{41} +(-832.532 - 275.750i) q^{42} +(80.5338 - 102.407i) q^{43} +(-216.917 + 139.404i) q^{44} +(-38.1493 + 120.549i) q^{45} +(369.307 + 346.933i) q^{46} +(112.956 + 65.2150i) q^{47} +(-5.32087 + 13.8230i) q^{48} +(-934.819 + 374.245i) q^{49} +(-45.0064 - 471.328i) q^{50} +(-489.923 - 63.7753i) q^{51} +(377.523 + 36.0491i) q^{52} +(222.515 + 256.797i) q^{53} +(643.501 + 35.5182i) q^{54} +(88.4274 + 25.9646i) q^{55} +(281.657 + 813.794i) q^{56} +(265.140 - 175.441i) q^{57} +(-29.9532 + 628.794i) q^{58} +(147.224 - 763.872i) q^{59} +(299.870 - 108.291i) q^{60} +(82.3559 - 205.715i) q^{61} +(837.893 - 726.038i) q^{62} +(792.425 - 596.804i) q^{63} +(-790.621 + 232.147i) q^{64} +(-78.6257 - 110.414i) q^{65} +(16.0841 - 469.475i) q^{66} +(161.382 - 313.037i) q^{67} +(622.888 + 1078.87i) q^{68} +(-560.424 + 120.142i) q^{69} +(395.200 - 684.506i) q^{70} +(385.271 + 599.494i) q^{71} +(-356.229 - 523.045i) q^{72} +(118.171 - 258.757i) q^{73} +(286.504 - 1180.98i) q^{74} +(467.347 + 261.565i) q^{75} +(-757.574 - 262.199i) q^{76} +(-446.971 - 568.370i) q^{77} +(-445.431 + 528.143i) q^{78} +(735.145 - 254.436i) q^{79} +(-11.2299 - 7.21702i) q^{80} +(-447.299 + 575.643i) q^{81} +(418.060 - 482.467i) q^{82} +(70.4876 - 67.2098i) q^{83} +(-2409.29 - 672.618i) q^{84} +(145.632 - 420.776i) q^{85} +(347.147 - 487.499i) q^{86} +(-565.547 - 432.655i) q^{87} +(-375.730 + 267.556i) q^{88} +(-96.1220 + 668.543i) q^{89} +(-148.699 + 561.477i) q^{90} +1063.48i q^{91} +(1099.23 + 938.285i) q^{92} +(135.868 + 1246.71i) q^{93} +(532.553 + 274.550i) q^{94} +(106.493 + 266.006i) q^{95} +(284.948 - 860.304i) q^{96} +(-738.452 - 179.147i) q^{97} +(-4207.63 + 1921.56i) q^{98} +(426.295 + 317.191i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	1400 * q - 27 * q^2 - 22 * q^3 - 281 * q^4 - 33 * q^5 + 29 * q^6 - 11 * q^7 - 22 * q^9 - 44 * q^10 - 33 * q^11 + 86 * q^12 - 9 * q^13 - 33 * q^14 + 726 * q^15 + 999 * q^16 + 51 * q^18 - 44 * q^19 - 33 * q^20 + 638 * q^21 + 252 * q^23 - 2564 * q^24 + 1397 * q^25 - 178 * q^27 - 44 * q^28 - 135 * q^29 - 1518 * q^30 - 45 * q^31 - 339 * q^32 + 1914 * q^33 - 11 * q^34 - 2137 * q^36 - 44 * q^37 - 33 * q^38 + 374 * q^39 + 979 * q^40 + 441 * q^41 - 22 * q^42 - 11 * q^43 + 516 * q^46 - 1992 * q^47 - 20 * q^48 - 4633 * q^49 - 6231 * q^50 - 22 * q^51 + 178 * q^52 + 2640 * q^54 + 256 * q^55 + 17391 * q^56 + 2090 * q^57 + 320 * q^58 - 2061 * q^59 + 418 * q^60 - 11 * q^61 - 2112 * q^63 + 8546 * q^64 - 12969 * q^65 - 5775 * q^66 - 11 * q^67 - 519 * q^69 - 1582 * q^70 - 4156 * q^72 - 36 * q^73 - 10065 * q^74 - 7876 * q^75 - 11 * q^76 + 1317 * q^77 - 10053 * q^78 - 11 * q^79 + 12442 * q^81 + 698 * q^82 + 14025 * q^83 + 16720 * q^84 + 241 * q^85 - 33 * q^86 + 11672 * q^87 - 1419 * q^88 - 37136 * q^90 + 3141 * q^92 + 1138 * q^93 + 396 * q^94 + 5883 * q^95 + 9218 * q^96 + 6325 * q^97 - 9262 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	4.58851	−	0.218578i	1.62228	−	0.0772789i	0.783375	−	0.621550i	$-0.213497\pi$
							0.838910	+	0.544271i	$0.183194\pi$
$3$	−2.75063	+	4.40840i	−0.529359	+	0.848398i
							$5$	−3.38925	−	3.23164i	−0.303143	−	0.289047i	0.523204	−	0.852208i	$-0.324737\pi$
−0.826347	+	0.563161i	$0.809585\pi$
$7$	6.95340	+	36.0777i	0.375448	+	1.94801i	0.312209	+	0.950013i	$0.398931\pi$
							0.0632391	+	0.997998i	$0.479857\pi$
$11$	−17.4921	+	9.01781i	−0.479461	+	0.247179i	−0.680975	−	0.732306i	$-0.738444\pi$
							0.201515	+	0.979486i	$0.435414\pi$
$13$	28.4217	+	5.47783i	0.606366	+	0.116867i	0.483187	−	0.875517i	$-0.339479\pi$
							0.123179	+	0.992385i	$0.460691\pi$
$17$	39.4982	+	86.4889i	0.563513	+	1.23392i	0.950180	+	0.311701i	$0.100899\pi$
							−0.386668	+	0.922219i	$0.626374\pi$
$19$	−55.6562	−	25.4173i	−0.672022	−	0.306902i	0.0500234	−	0.998748i	$-0.484070\pi$
							−0.722045	+	0.691846i	$0.756798\pi$
$23$	76.7094	+	79.2632i	0.695436	+	0.718588i
							$29$	−13.0262	+	136.416i	−0.0834103	+	0.873512i	0.852839	+	0.522175i	$0.174879\pi$
−0.936249	+	0.351337i	$0.885727\pi$
$31$	189.713	−	149.192i	1.09915	−	0.864379i	0.107608	−	0.994193i	$-0.465681\pi$
							0.991538	+	0.129815i	$0.0414383\pi$
$37$	74.5307	−	253.828i	0.331156	−	1.12781i	−0.610718	−	0.791848i	$-0.709119\pi$
							0.941874	−	0.335966i	$-0.109063\pi$
$41$	95.9013	−	100.578i	0.365299	−	0.383115i	−0.515274	−	0.857026i	$-0.672310\pi$
							0.880573	+	0.473911i	$0.157158\pi$
$43$	80.5338	−	102.407i	0.285612	−	0.363185i	−0.621984	−	0.783030i	$-0.713673\pi$
							0.907596	+	0.419845i	$0.137916\pi$
$47$	112.956	+	65.2150i	0.350559	+	0.202396i	0.664932	−	0.746904i	$-0.268461\pi$
							−0.314372	+	0.949300i	$0.601794\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.63	✓	1400
9.5	odd	6	inner	207.4.o.a.149.63	yes	1400
23.21	odd	22	inner	207.4.o.a.182.63	yes	1400
207.113	even	66	inner	207.4.o.a.113.63	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.63	✓	1400	1.1	even	1	trivial
207.4.o.a.113.63	yes	1400	207.113	even	66	inner
207.4.o.a.149.63	yes	1400	9.5	odd	6	inner
207.4.o.a.182.63	yes	1400	23.21	odd	22	inner