Embedded newform 207.4.o.a.11.23

• Label: 207.4.o.a.11.23
• 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			11.23
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.23

$q$-expansion

$f(q)$	$=$	$q+(-2.39090 + 0.113893i) q^{2} +(-3.55848 - 3.78645i) q^{3} +(-2.26035 + 0.215838i) q^{4} +(-2.67766 - 2.55314i) q^{5} +(8.93922 + 8.64773i) q^{6} +(0.426954 + 2.21525i) q^{7} +(24.3337 - 3.49865i) q^{8} +(-1.67440 + 26.9480i) q^{9} +(6.69278 + 5.79933i) q^{10} +(-58.3228 + 30.0675i) q^{11} +(8.86069 + 7.79066i) q^{12} +(-59.2808 - 11.4254i) q^{13} +(-1.27310 - 5.24781i) q^{14} +(-0.138942 + 19.2241i) q^{15} +(-39.9440 + 7.69858i) q^{16} +(13.7876 + 30.1905i) q^{17} +(0.934140 - 64.6207i) q^{18} +(80.3466 + 36.6931i) q^{19} +(6.60351 + 5.19306i) q^{20} +(6.86862 - 9.49957i) q^{21} +(136.019 - 78.5308i) q^{22} +(82.5951 - 73.1098i) q^{23} +(-99.8384 - 79.6883i) q^{24} +(-5.29642 - 111.186i) q^{25} +(143.036 + 20.5654i) q^{26} +(107.996 - 89.5541i) q^{27} +(-1.44320 - 4.91510i) q^{28} +(-11.5448 + 120.903i) q^{29} +(-1.85729 - 45.9787i) q^{30} +(31.8964 - 25.0836i) q^{31} +(-96.5020 + 23.4111i) q^{32} +(321.390 + 113.842i) q^{33} +(-36.4031 - 70.6122i) q^{34} +(4.51260 - 7.02175i) q^{35} +(-2.03166 - 61.2735i) q^{36} +(56.2997 - 191.739i) q^{37} +(-196.280 - 78.5785i) q^{38} +(167.688 + 265.121i) q^{39} +(-74.0897 - 52.7590i) q^{40} +(-22.7383 + 23.8472i) q^{41} +(-15.3402 + 23.4948i) q^{42} +(279.183 - 355.010i) q^{43} +(125.340 - 80.5514i) q^{44} +(73.2855 - 67.8826i) q^{45} +(-189.150 + 184.205i) q^{46} +(-434.469 - 250.841i) q^{47} +(171.290 + 123.851i) q^{48} +(313.705 - 125.589i) q^{49} +(25.3264 + 265.230i) q^{50} +(65.2521 - 159.638i) q^{51} +(136.462 + 13.0305i) q^{52} +(466.297 + 538.135i) q^{53} +(-248.007 + 226.415i) q^{54} +(232.935 + 68.3958i) q^{55} +(18.1397 + 52.4114i) q^{56} +(-146.976 - 434.800i) q^{57} +(13.8326 - 290.382i) q^{58} +(-118.115 + 612.837i) q^{59} +(-3.83523 - 43.4833i) q^{60} +(100.549 - 251.160i) q^{61} +(-73.4041 + 63.6051i) q^{62} +(-60.4115 + 7.79637i) q^{63} +(540.311 - 158.650i) q^{64} +(129.563 + 181.946i) q^{65} +(-781.376 - 235.580i) q^{66} +(-335.028 + 649.863i) q^{67} +(-37.6810 - 65.2654i) q^{68} +(-570.740 - 52.5823i) q^{69} +(-9.98945 + 17.3022i) q^{70} +(98.2662 + 152.905i) q^{71} +(53.5375 + 661.602i) q^{72} +(372.956 - 816.660i) q^{73} +(-112.769 + 464.841i) q^{74} +(-402.151 + 415.707i) q^{75} +(-189.532 - 65.5975i) q^{76} +(-91.5082 - 116.362i) q^{77} +(-431.120 - 614.779i) q^{78} +(-495.491 + 171.491i) q^{79} +(126.612 + 81.3685i) q^{80} +(-723.393 - 90.2436i) q^{81} +(51.6488 - 59.6059i) q^{82} +(540.073 - 514.958i) q^{83} +(-13.4751 + 22.9549i) q^{84} +(40.1623 - 116.041i) q^{85} +(-627.065 + 880.590i) q^{86} +(498.876 - 386.518i) q^{87} +(-1314.01 + 935.703i) q^{88} +(115.401 - 802.630i) q^{89} +(-167.487 + 170.647i) q^{90} -136.200i q^{91} +(-170.914 + 183.081i) q^{92} +(-208.480 - 31.5145i) q^{93} +(1067.34 + 550.251i) q^{94} +(-121.458 - 303.387i) q^{95} +(432.046 + 282.092i) q^{96} +(294.414 + 71.4240i) q^{97} +(-735.733 + 335.998i) q^{98} +(-712.604 - 1622.03i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−2.39090	+	0.113893i	−0.845310	+	0.0402671i	−0.465767	−	0.884907i	$-0.654222\pi$
							−0.379543	+	0.925174i	$0.623919\pi$
$3$	−3.55848	−	3.78645i	−0.684830	−	0.728703i
							$5$	−2.67766	−	2.55314i	−0.239497	−	0.228360i	0.560819	−	0.827938i	$-0.310486\pi$
−0.800316	+	0.599579i	$0.795335\pi$
$7$	0.426954	+	2.21525i	0.0230534	+	0.119612i	0.991632	−	0.129094i	$-0.0412068\pi$
							−0.968579	+	0.248706i	$0.919995\pi$
$11$	−58.3228	+	30.0675i	−1.59863	+	0.824154i	−0.598907	+	0.800818i	$0.704398\pi$
							−0.999728	+	0.0233350i	$0.992572\pi$
$13$	−59.2808	−	11.4254i	−1.26473	−	0.243757i	−0.487586	−	0.873075i	$-0.662122\pi$
							−0.777148	+	0.629318i	$0.783334\pi$
$17$	13.7876	+	30.1905i	0.196704	+	0.430722i	0.982122	−	0.188243i	$-0.0602793\pi$
							−0.785418	+	0.618966i	$0.787552\pi$
$19$	80.3466	+	36.6931i	0.970146	+	0.443051i	0.836493	−	0.547977i	$-0.184602\pi$
							0.133653	+	0.991028i	$0.457329\pi$
$23$	82.5951	−	73.1098i	0.748795	−	0.662802i
							$29$	−11.5448	+	120.903i	−0.0739250	+	0.774178i	0.880167	+	0.474664i	$0.157430\pi$
−0.954092	+	0.299514i	$0.903176\pi$
$31$	31.8964	−	25.0836i	0.184799	−	0.145327i	−0.521474	−	0.853267i	$-0.674618\pi$
							0.706273	+	0.707940i	$0.250375\pi$
$37$	56.2997	−	191.739i	0.250152	−	0.851939i	−0.734678	−	0.678416i	$-0.762667\pi$
							0.984830	−	0.173523i	$-0.0555151\pi$
$41$	−22.7383	+	23.8472i	−0.0866127	+	0.0908367i	−0.765614	−	0.643300i	$-0.777565\pi$
							0.679001	+	0.734137i	$0.262413\pi$
$43$	279.183	−	355.010i	0.990117	−	1.25904i	0.0243285	−	0.999704i	$-0.492255\pi$
							0.965788	−	0.259332i	$-0.0835023\pi$
$47$	−434.469	−	250.841i	−1.34838	−	0.778486i	−0.360358	−	0.932814i	$-0.617345\pi$
							−0.988020	+	0.154328i	$0.950679\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.23	✓	1400
9.5	odd	6	inner	207.4.o.a.149.23	yes	1400
23.21	odd	22	inner	207.4.o.a.182.23	yes	1400
207.113	even	66	inner	207.4.o.a.113.23	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.23	✓	1400	1.1	even	1	trivial
207.4.o.a.113.23	yes	1400	207.113	even	66	inner
207.4.o.a.149.23	yes	1400	9.5	odd	6	inner
207.4.o.a.182.23	yes	1400	23.21	odd	22	inner