Embedded newform 207.4.o.a.11.34

• Label: 207.4.o.a.11.34
• 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			11.34
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.34

$q$-expansion

$f(q)$	$=$	$q+(-0.492214 + 0.0234470i) q^{2} +(4.87654 + 1.79425i) q^{3} +(-7.72205 + 0.737366i) q^{4} +(-5.71829 - 5.45238i) q^{5} +(-2.44237 - 0.768814i) q^{6} +(-0.637075 - 3.30546i) q^{7} +(7.68566 - 1.10503i) q^{8} +(20.5613 + 17.4995i) q^{9} +(2.94246 + 2.54966i) q^{10} +(8.97987 - 4.62944i) q^{11} +(-38.9799 - 10.2595i) q^{12} +(31.9152 + 6.15114i) q^{13} +(0.391080 + 1.61205i) q^{14} +(-18.1026 - 36.8488i) q^{15} +(57.1789 - 11.0203i) q^{16} +(23.1941 + 50.7881i) q^{17} +(-10.5309 - 8.13138i) q^{18} +(126.301 + 57.6797i) q^{19} +(48.1774 + 37.8871i) q^{20} +(2.82410 - 17.2623i) q^{21} +(-4.31146 + 2.48923i) q^{22} +(91.0811 - 62.2193i) q^{23} +(39.4621 + 8.40127i) q^{24} +(-2.97733 - 62.5018i) q^{25} +(-15.8533 - 2.27936i) q^{26} +(68.8697 + 122.229i) q^{27} +(7.35686 + 25.0552i) q^{28} +(-2.54939 + 26.6984i) q^{29} +(9.77432 + 17.7130i) q^{30} +(-185.912 + 146.203i) q^{31} +(-88.2523 + 21.4098i) q^{32} +(52.0971 - 6.46354i) q^{33} +(-12.6073 - 24.4547i) q^{34} +(-14.3796 + 22.3752i) q^{35} +(-171.679 - 119.971i) q^{36} +(-61.9613 + 211.021i) q^{37} +(-63.5195 - 25.4294i) q^{38} +(144.599 + 87.2601i) q^{39} +(-49.9739 - 35.5862i) q^{40} +(291.120 - 305.318i) q^{41} +(-0.985311 + 8.56295i) q^{42} +(22.1013 - 28.1041i) q^{43} +(-65.9294 + 42.3702i) q^{44} +(-22.1619 - 212.175i) q^{45} +(-43.3725 + 32.7608i) q^{46} +(232.321 + 134.130i) q^{47} +(298.608 + 48.8522i) q^{48} +(307.910 - 123.269i) q^{49} +(2.93096 + 30.6944i) q^{50} +(21.9807 + 289.286i) q^{51} +(-250.986 - 23.9663i) q^{52} +(-92.4001 - 106.635i) q^{53} +(-36.7645 - 58.5480i) q^{54} +(-76.5910 - 22.4891i) q^{55} +(-8.54898 - 24.7006i) q^{56} +(512.420 + 507.893i) q^{57} +(0.628846 - 13.2011i) q^{58} +(90.6320 - 470.243i) q^{59} +(166.960 + 271.200i) q^{60} +(-0.792270 + 1.97899i) q^{61} +(88.0804 - 76.3221i) q^{62} +(44.7447 - 79.1131i) q^{63} +(-404.042 + 118.638i) q^{64} +(-148.962 - 209.188i) q^{65} +(-25.4913 + 4.40296i) q^{66} +(156.105 - 302.802i) q^{67} +(-216.556 - 375.086i) q^{68} +(555.798 - 139.993i) q^{69} +(6.55323 - 11.3505i) q^{70} +(331.880 + 516.415i) q^{71} +(177.365 + 111.774i) q^{72} +(-387.598 + 848.722i) q^{73} +(25.5504 - 105.320i) q^{74} +(97.6248 - 310.135i) q^{75} +(-1017.83 - 352.276i) q^{76} +(-21.0233 - 26.7333i) q^{77} +(-73.2196 - 39.5602i) q^{78} +(-1047.53 + 362.555i) q^{79} +(-387.053 - 248.744i) q^{80} +(116.537 + 719.625i) q^{81} +(-136.135 + 157.108i) q^{82} +(703.577 - 670.859i) q^{83} +(-9.07921 + 135.383i) q^{84} +(144.285 - 416.884i) q^{85} +(-10.2196 + 14.3514i) q^{86} +(-60.3359 + 125.622i) q^{87} +(63.9005 - 45.5033i) q^{88} +(145.267 - 1010.35i) q^{89} +(15.8833 + 103.916i) q^{90} -109.413i q^{91} +(-657.454 + 547.621i) q^{92} +(-1168.93 + 379.391i) q^{93} +(-117.496 - 60.5736i) q^{94} +(-407.734 - 1018.47i) q^{95} +(-468.781 - 53.9411i) q^{96} +(-664.866 - 161.295i) q^{97} +(-148.667 + 67.8940i) q^{98} +(265.651 + 61.9555i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−0.492214	+	0.0234470i	−0.174024	+	0.00828977i	−0.134413	−	0.990925i	$-0.542915\pi$
							−0.0396108	+	0.999215i	$0.512612\pi$
$3$	4.87654	+	1.79425i	0.938491	+	0.345304i
							$5$	−5.71829	−	5.45238i	−0.511460	−	0.487676i	0.389834	−	0.920885i	$-0.372533\pi$
−0.901293	+	0.433210i	$0.857381\pi$
$7$	−0.637075	−	3.30546i	−0.0343988	−	0.178478i	0.960666	−	0.277707i	$-0.0895745\pi$
							−0.995065	+	0.0992290i	$0.968362\pi$
$11$	8.97987	−	4.62944i	0.246139	−	0.126894i	−0.330743	−	0.943721i	$-0.607299\pi$
							0.576882	+	0.816827i	$0.304269\pi$
$13$	31.9152	+	6.15114i	0.680898	+	0.131232i	0.517961	−	0.855404i	$-0.326691\pi$
							0.162937	+	0.986636i	$0.447903\pi$
$17$	23.1941	+	50.7881i	0.330906	+	0.724584i	0.999824	−	0.0187527i	$-0.00596953\pi$
							−0.668918	+	0.743336i	$0.733242\pi$
$19$	126.301	+	57.6797i	1.52502	+	0.696455i	0.989023	−	0.147763i	$-0.0472074\pi$
							0.536000	+	0.844218i	$0.319935\pi$
$23$	91.0811	−	62.2193i	0.825727	−	0.564070i
							$29$	−2.54939	+	26.6984i	−0.0163245	+	0.170958i	−0.999957	−	0.00922611i	$-0.997063\pi$
0.983633	+	0.180184i	$0.0576693\pi$
$31$	−185.912	+	146.203i	−1.07712	+	0.847058i	−0.988832	−	0.149037i	$-0.952383\pi$
							−0.0882902	+	0.996095i	$0.528140\pi$
$37$	−61.9613	+	211.021i	−0.275307	+	0.937611i	0.699513	+	0.714620i	$0.253400\pi$
							−0.974820	+	0.222991i	$0.928418\pi$
$41$	291.120	−	305.318i	1.10891	−	1.16299i	0.123581	−	0.992334i	$-0.460562\pi$
							0.985330	−	0.170658i	$-0.0545894\pi$
$43$	22.1013	−	28.1041i	0.0783818	−	0.0996706i	−0.745281	−	0.666751i	$-0.767685\pi$
							0.823663	+	0.567080i	$0.191927\pi$
$47$	232.321	+	134.130i	0.721010	+	0.416275i	0.815124	−	0.579286i	$-0.196669\pi$
							−0.0941144	+	0.995561i	$0.530002\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.34	✓	1400
9.5	odd	6	inner	207.4.o.a.149.34	yes	1400
23.21	odd	22	inner	207.4.o.a.182.34	yes	1400
207.113	even	66	inner	207.4.o.a.113.34	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.34	✓	1400	1.1	even	1	trivial
207.4.o.a.113.34	yes	1400	207.113	even	66	inner
207.4.o.a.149.34	yes	1400	9.5	odd	6	inner
207.4.o.a.182.34	yes	1400	23.21	odd	22	inner