Embedded newform 207.4.o.a.11.49

• Label: 207.4.o.a.11.49
• 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			11.49
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.49

$q$-expansion

$f(q)$	$=$	$q+(2.25210 - 0.107281i) q^{2} +(-1.47393 + 4.98272i) q^{3} +(-2.90332 + 0.277233i) q^{4} +(14.7068 + 14.0229i) q^{5} +(-2.78489 + 11.3797i) q^{6} +(2.50982 + 13.0222i) q^{7} +(-24.3625 + 3.50280i) q^{8} +(-22.6551 - 14.6884i) q^{9} +(34.6256 + 30.0032i) q^{10} +(26.3302 - 13.5741i) q^{11} +(2.89791 - 14.8751i) q^{12} +(-24.0420 - 4.63371i) q^{13} +(7.04941 + 29.0581i) q^{14} +(-91.5490 + 52.6111i) q^{15} +(-31.5805 + 6.08664i) q^{16} +(3.19207 + 6.98966i) q^{17} +(-52.5973 - 30.6492i) q^{18} +(-72.7485 - 33.2231i) q^{19} +(-46.5861 - 36.6357i) q^{20} +(-68.5853 - 6.68804i) q^{21} +(57.8420 - 33.3951i) q^{22} +(95.6852 + 54.8757i) q^{23} +(18.4551 - 126.554i) q^{24} +(13.7003 + 287.605i) q^{25} +(-54.6421 - 7.85634i) q^{26} +(106.580 - 91.2343i) q^{27} +(-10.8970 - 37.1118i) q^{28} +(14.6217 - 153.126i) q^{29} +(-200.533 + 128.307i) q^{30} +(-243.252 + 191.295i) q^{31} +(120.884 - 29.3262i) q^{32} +(28.8274 + 151.203i) q^{33} +(7.93873 + 15.3990i) q^{34} +(-145.697 + 226.710i) q^{35} +(69.8470 + 36.3643i) q^{36} +(-122.035 + 415.614i) q^{37} +(-167.401 - 67.0174i) q^{38} +(58.5246 - 112.965i) q^{39} +(-407.413 - 290.118i) q^{40} +(-2.35981 + 2.47490i) q^{41} +(-155.179 - 7.70426i) q^{42} +(193.329 - 245.838i) q^{43} +(-72.6817 + 46.7097i) q^{44} +(-127.210 - 533.708i) q^{45} +(221.380 + 113.321i) q^{46} +(403.617 + 233.029i) q^{47} +(16.2193 - 166.328i) q^{48} +(155.152 - 62.1134i) q^{49} +(61.7090 + 646.246i) q^{50} +(-39.5324 + 5.60294i) q^{51} +(71.0861 + 6.78790i) q^{52} +(-41.0100 - 47.3281i) q^{53} +(230.241 - 216.903i) q^{54} +(577.581 + 169.593i) q^{55} +(-106.760 - 308.462i) q^{56} +(272.768 - 313.517i) q^{57} +(16.5022 - 346.423i) q^{58} +(-66.6223 + 345.669i) q^{59} +(251.210 - 178.127i) q^{60} +(283.267 - 707.566i) q^{61} +(-527.306 + 456.913i) q^{62} +(134.415 - 331.884i) q^{63} +(515.969 - 151.502i) q^{64} +(-288.602 - 405.285i) q^{65} +(81.1435 + 337.432i) q^{66} +(107.252 - 208.040i) q^{67} +(-11.2054 - 19.4083i) q^{68} +(-414.464 + 395.890i) q^{69} +(-303.804 + 526.204i) q^{70} +(82.9726 + 129.108i) q^{71} +(603.384 + 278.489i) q^{72} +(-243.076 + 532.261i) q^{73} +(-230.248 + 949.097i) q^{74} +(-1453.25 - 355.645i) q^{75} +(220.423 + 76.2891i) q^{76} +(242.849 + 308.808i) q^{77} +(119.685 - 260.687i) q^{78} +(741.740 - 256.719i) q^{79} +(-549.799 - 353.335i) q^{80} +(297.504 + 665.532i) q^{81} +(-5.04902 + 5.82688i) q^{82} +(114.302 - 108.987i) q^{83} +(200.979 + 0.403383i) q^{84} +(-51.0702 + 147.558i) q^{85} +(409.023 - 574.393i) q^{86} +(741.431 + 298.552i) q^{87} +(-593.921 + 422.929i) q^{88} +(-122.617 + 852.820i) q^{89} +(-343.746 - 1188.32i) q^{90} -324.709i q^{91} +(-293.018 - 132.795i) q^{92} +(-594.636 - 1494.01i) q^{93} +(933.987 + 481.504i) q^{94} +(-604.013 - 1508.75i) q^{95} +(-32.0504 + 645.557i) q^{96} +(569.617 + 138.188i) q^{97} +(342.754 - 156.531i) q^{98} +(-795.893 - 79.2238i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	2.25210	−	0.107281i	0.796238	−	0.0379295i	0.354480	−	0.935064i	$-0.384658\pi$
							0.441758	+	0.897134i	$0.354355\pi$
$3$	−1.47393	+	4.98272i	−0.283658	+	0.958926i
							$5$	14.7068	+	14.0229i	1.31542	+	1.25425i	0.945676	+	0.325110i	$0.105401\pi$
0.369739	+	0.929136i	$0.379447\pi$
$7$	2.50982	+	13.0222i	0.135518	+	0.703133i	0.984593	+	0.174861i	$0.0559476\pi$
							−0.849075	+	0.528272i	$0.822840\pi$
$11$	26.3302	−	13.5741i	0.721713	−	0.372069i	−0.0579069	−	0.998322i	$-0.518443\pi$
							0.779620	+	0.626253i	$0.215412\pi$
$13$	−24.0420	−	4.63371i	−0.512926	−	0.0988584i	−0.0737809	−	0.997274i	$-0.523507\pi$
							−0.439145	+	0.898416i	$0.644719\pi$
$17$	3.19207	+	6.98966i	0.0455406	+	0.0997201i	0.931035	−	0.364930i	$-0.118907\pi$
							−0.885494	+	0.464651i	$0.846180\pi$
$19$	−72.7485	−	33.2231i	−0.878403	−	0.401153i	−0.0754177	−	0.997152i	$-0.524029\pi$
							−0.802985	+	0.595999i	$0.796756\pi$
$23$	95.6852	+	54.8757i	0.867467	+	0.497495i
							$29$	14.6217	−	153.126i	0.0936271	−	0.980507i	−0.819567	−	0.572983i	$-0.805786\pi$
0.913194	−	0.407524i	$-0.133608\pi$
$31$	−243.252	+	191.295i	−1.40933	+	1.10831i	−0.430747	+	0.902473i	$0.641750\pi$
							−0.978586	+	0.205838i	$0.934008\pi$
$37$	−122.035	+	415.614i	−0.542229	+	1.84666i	−0.0101793	+	0.999948i	$0.503240\pi$
							−0.532049	+	0.846713i	$0.678578\pi$
$41$	−2.35981	+	2.47490i	−0.00898879	+	0.00942717i	−0.728213	−	0.685351i	$-0.759649\pi$
							0.719224	+	0.694778i	$0.244497\pi$
$43$	193.329	−	245.838i	0.685638	−	0.871859i	−0.311280	−	0.950318i	$-0.600758\pi$
							0.996917	+	0.0784592i	$0.0250000\pi$
$47$	403.617	+	233.029i	1.25263	+	0.723207i	0.971631	−	0.236501i	$-0.0760006\pi$
							0.281000	+	0.959708i	$0.409334\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.49	✓	1400
9.5	odd	6	inner	207.4.o.a.149.49	yes	1400
23.21	odd	22	inner	207.4.o.a.182.49	yes	1400
207.113	even	66	inner	207.4.o.a.113.49	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.49	✓	1400	1.1	even	1	trivial
207.4.o.a.113.49	yes	1400	207.113	even	66	inner
207.4.o.a.149.49	yes	1400	9.5	odd	6	inner
207.4.o.a.182.49	yes	1400	23.21	odd	22	inner