Embedded newform 207.4.o.a.11.22

• Label: 207.4.o.a.11.22
• 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			11.22
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.22

$q$-expansion

$f(q)$	$=$	$q+(-2.50996 + 0.119564i) q^{2} +(-4.40022 - 2.76370i) q^{3} +(-1.67815 + 0.160244i) q^{4} +(-8.89914 - 8.48531i) q^{5} +(11.3748 + 6.41068i) q^{6} +(4.83275 + 25.0747i) q^{7} +(24.0908 - 3.46374i) q^{8} +(11.7239 + 24.3218i) q^{9} +(23.3511 + 20.2338i) q^{10} +(39.5179 - 20.3729i) q^{11} +(7.82711 + 3.93280i) q^{12} +(21.9694 + 4.23426i) q^{13} +(-15.1281 - 62.3587i) q^{14} +(15.7073 + 61.9318i) q^{15} +(-46.8103 + 9.02195i) q^{16} +(-1.38454 - 3.03172i) q^{17} +(-32.3346 - 59.6451i) q^{18} +(-120.500 - 55.0307i) q^{19} +(16.2938 + 12.8136i) q^{20} +(48.0338 - 123.690i) q^{21} +(-96.7526 + 55.8601i) q^{22} +(6.54965 + 110.110i) q^{23} +(-115.578 - 51.3386i) q^{24} +(1.24641 + 26.1655i) q^{25} +(-55.6488 - 8.00108i) q^{26} +(15.6304 - 139.423i) q^{27} +(-12.1282 - 41.3047i) q^{28} +(-12.3419 + 129.250i) q^{29} +(-46.8296 - 153.569i) q^{30} +(-18.4210 + 14.4864i) q^{31} +(-72.8062 + 17.6626i) q^{32} +(-230.192 - 19.5704i) q^{33} +(3.83764 + 7.44397i) q^{34} +(169.759 - 264.150i) q^{35} +(-23.5719 - 38.9370i) q^{36} +(59.4484 - 202.463i) q^{37} +(309.031 + 123.717i) q^{38} +(-84.9682 - 79.3487i) q^{39} +(-243.778 - 173.594i) q^{40} +(240.732 - 252.472i) q^{41} +(-105.774 + 316.202i) q^{42} +(-115.607 + 147.007i) q^{43} +(-63.0524 + 40.5213i) q^{44} +(102.045 - 315.924i) q^{45} +(-29.6045 - 275.588i) q^{46} +(138.367 + 79.8864i) q^{47} +(230.910 + 89.6712i) q^{48} +(-286.954 + 114.879i) q^{49} +(-6.25691 - 65.5253i) q^{50} +(-2.28649 + 17.1667i) q^{51} +(-37.5466 - 3.58526i) q^{52} +(-408.699 - 471.663i) q^{53} +(-22.5618 + 351.815i) q^{54} +(-524.545 - 154.020i) q^{55} +(203.277 + 587.330i) q^{56} +(378.140 + 575.174i) q^{57} +(15.5240 - 325.889i) q^{58} +(52.3345 - 271.537i) q^{59} +(-36.2835 - 101.414i) q^{60} +(37.3477 - 93.2900i) q^{61} +(44.5039 - 38.5628i) q^{62} +(-553.203 + 411.514i) q^{63} +(546.556 - 160.483i) q^{64} +(-159.580 - 224.099i) q^{65} +(580.114 + 21.5983i) q^{66} +(180.812 - 350.727i) q^{67} +(2.80929 + 4.86583i) q^{68} +(275.490 - 502.608i) q^{69} +(-394.506 + 683.305i) q^{70} +(-272.534 - 424.072i) q^{71} +(366.683 + 545.323i) q^{72} +(6.35885 - 13.9239i) q^{73} +(-125.006 + 515.282i) q^{74} +(66.8290 - 118.579i) q^{75} +(211.036 + 73.0404i) q^{76} +(701.824 + 892.441i) q^{77} +(222.754 + 189.003i) q^{78} +(-1209.26 + 418.530i) q^{79} +(493.126 + 316.913i) q^{80} +(-454.100 + 570.293i) q^{81} +(-574.041 + 662.479i) q^{82} +(862.586 - 822.474i) q^{83} +(-60.7873 + 215.269i) q^{84} +(-13.4039 + 38.7280i) q^{85} +(272.594 - 382.804i) q^{86} +(411.516 - 534.620i) q^{87} +(881.451 - 627.679i) q^{88} +(81.1665 - 564.526i) q^{89} +(-218.357 + 805.159i) q^{90} +571.340i q^{91} +(-28.6357 - 183.731i) q^{92} +(121.092 - 12.8333i) q^{93} +(-356.849 - 183.968i) q^{94} +(605.397 + 1512.21i) q^{95} +(369.178 + 123.495i) q^{96} +(901.558 + 218.716i) q^{97} +(706.509 - 322.652i) q^{98} +(958.809 + 722.296i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−2.50996	+	0.119564i	−0.887406	+	0.0422724i	−0.486333	−	0.873774i	$-0.661666\pi$
							−0.401073	+	0.916046i	$0.631363\pi$
$3$	−4.40022	−	2.76370i	−0.846823	−	0.531875i
							$5$	−8.89914	−	8.48531i	−0.795963	−	0.758949i	0.178393	−	0.983959i	$-0.442910\pi$
−0.974356	+	0.225010i	$0.927759\pi$
$7$	4.83275	+	25.0747i	0.260944	+	1.35391i	0.841752	+	0.539865i	$0.181525\pi$
							−0.580808	+	0.814041i	$0.697263\pi$
$11$	39.5179	−	20.3729i	1.08319	−	0.558423i	0.178260	−	0.983983i	$-0.442953\pi$
							0.904930	+	0.425560i	$0.139923\pi$
$13$	21.9694	+	4.23426i	0.468710	+	0.0903364i	0.418137	−	0.908384i	$-0.362683\pi$
							0.0505727	+	0.998720i	$0.483895\pi$
$17$	−1.38454	−	3.03172i	−0.0197530	−	0.0432530i	0.899498	−	0.436924i	$-0.143932\pi$
							−0.919251	+	0.393671i	$0.871205\pi$
$19$	−120.500	−	55.0307i	−1.45498	−	0.664468i	−0.478112	−	0.878299i	$-0.658679\pi$
							−0.976871	+	0.213831i	$0.931406\pi$
$23$	6.54965	+	110.110i	0.0593781	+	0.998236i
							$29$	−12.3419	+	129.250i	−0.0790287	+	0.827626i	0.865871	+	0.500267i	$0.166765\pi$
−0.944900	+	0.327359i	$0.893841\pi$
$31$	−18.4210	+	14.4864i	−0.106726	+	0.0839301i	−0.670096	−	0.742275i	$-0.733747\pi$
							0.563370	+	0.826205i	$0.309505\pi$
$37$	59.4484	−	202.463i	0.264142	−	0.899586i	−0.715462	−	0.698651i	$-0.753784\pi$
							0.979605	−	0.200935i	$-0.0643979\pi$
$41$	240.732	−	252.472i	0.916975	−	0.961695i	−0.0824161	−	0.996598i	$-0.526264\pi$
							0.999391	+	0.0349027i	$0.0111121\pi$
$43$	−115.607	+	147.007i	−0.409999	+	0.521356i	−0.946570	−	0.322499i	$-0.895477\pi$
							0.536571	+	0.843855i	$0.319719\pi$
$47$	138.367	+	79.8864i	0.429425	+	0.247928i	0.699101	−	0.715023i	$-0.253584\pi$
							−0.269677	+	0.962951i	$0.586917\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.22	✓	1400
9.5	odd	6	inner	207.4.o.a.149.22	yes	1400
23.21	odd	22	inner	207.4.o.a.182.22	yes	1400
207.113	even	66	inner	207.4.o.a.113.22	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.22	✓	1400	1.1	even	1	trivial
207.4.o.a.113.22	yes	1400	207.113	even	66	inner
207.4.o.a.149.22	yes	1400	9.5	odd	6	inner
207.4.o.a.182.22	yes	1400	23.21	odd	22	inner