Embedded newform 207.4.o.a.11.46

• Label: 207.4.o.a.11.46
• 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			11.46
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.46

$q$-expansion

$f(q)$	$=$	$q+(1.84994 - 0.0881234i) q^{2} +(4.64817 + 2.32262i) q^{3} +(-4.54927 + 0.434403i) q^{4} +(-12.8457 - 12.2483i) q^{5} +(8.80350 + 3.88709i) q^{6} +(3.83084 + 19.8763i) q^{7} +(-23.0431 + 3.31309i) q^{8} +(16.2109 + 21.5918i) q^{9} +(-24.8431 - 21.5267i) q^{10} +(-5.26614 + 2.71489i) q^{11} +(-22.1547 - 8.54703i) q^{12} +(-71.3381 - 13.7493i) q^{13} +(8.83838 + 36.4323i) q^{14} +(-31.2607 - 86.7679i) q^{15} +(-6.43727 + 1.24068i) q^{16} +(53.4500 + 117.039i) q^{17} +(31.8919 + 38.5150i) q^{18} +(-144.304 - 65.9013i) q^{19} +(63.7592 + 50.1408i) q^{20} +(-28.3586 + 101.286i) q^{21} +(-9.50280 + 5.48644i) q^{22} +(-109.469 + 13.5446i) q^{23} +(-114.803 - 38.1204i) q^{24} +(9.04214 + 189.818i) q^{25} +(-133.183 - 19.1488i) q^{26} +(25.2015 + 138.014i) q^{27} +(-26.0618 - 88.7584i) q^{28} +(4.63171 - 48.5055i) q^{29} +(-65.4766 - 157.760i) q^{30} +(89.7019 - 70.5423i) q^{31} +(169.191 - 41.0453i) q^{32} +(-30.7836 + 0.388009i) q^{33} +(109.193 + 211.805i) q^{34} +(194.242 - 302.246i) q^{35} +(-83.1273 - 91.1849i) q^{36} +(25.9859 - 88.5000i) q^{37} +(-272.761 - 109.197i) q^{38} +(-299.657 - 229.600i) q^{39} +(336.584 + 239.680i) q^{40} +(-2.09017 + 2.19210i) q^{41} +(-43.5360 + 189.872i) q^{42} +(82.5340 - 104.950i) q^{43} +(22.7778 - 14.6384i) q^{44} +(56.2236 - 475.918i) q^{45} +(-201.318 + 34.7035i) q^{46} +(188.118 + 108.610i) q^{47} +(-32.8031 - 9.18442i) q^{48} +(-61.9611 + 24.8055i) q^{49} +(33.4548 + 350.355i) q^{50} +(-23.3926 + 668.161i) q^{51} +(330.509 + 31.5598i) q^{52} +(-68.7272 - 79.3154i) q^{53} +(58.7834 + 253.097i) q^{54} +(100.900 + 29.6269i) q^{55} +(-154.126 - 445.319i) q^{56} +(-517.685 - 641.483i) q^{57} +(4.29391 - 90.1403i) q^{58} +(-146.231 + 758.719i) q^{59} +(179.905 + 381.151i) q^{60} +(112.135 - 280.100i) q^{61} +(159.727 - 138.404i) q^{62} +(-367.064 + 404.927i) q^{63} +(359.698 - 105.617i) q^{64} +(747.981 + 1050.39i) q^{65} +(-56.9135 + 3.43054i) q^{66} +(-394.896 + 765.992i) q^{67} +(-294.000 - 509.224i) q^{68} +(-540.291 - 191.298i) q^{69} +(332.700 - 576.253i) q^{70} +(72.0616 + 112.130i) q^{71} +(-445.085 - 443.833i) q^{72} +(131.272 - 287.446i) q^{73} +(40.2735 - 166.009i) q^{74} +(-398.845 + 903.307i) q^{75} +(685.105 + 237.117i) q^{76} +(-74.1356 - 94.2711i) q^{77} +(-574.580 - 398.339i) q^{78} +(-419.763 + 145.281i) q^{79} +(97.8874 + 62.9084i) q^{80} +(-203.413 + 700.046i) q^{81} +(-3.67350 + 4.23945i) q^{82} +(247.913 - 236.385i) q^{83} +(85.0121 - 473.096i) q^{84} +(746.933 - 2158.12i) q^{85} +(143.434 - 201.425i) q^{86} +(134.189 - 214.704i) q^{87} +(112.353 - 80.0065i) q^{88} +(29.7959 - 207.235i) q^{89} +(62.0707 - 885.374i) q^{90} -1470.61i q^{91} +(492.122 - 109.172i) q^{92} +(580.792 - 119.549i) q^{93} +(357.578 + 184.344i) q^{94} +(1046.50 + 2614.03i) q^{95} +(881.761 + 202.181i) q^{96} +(-1037.12 - 251.603i) q^{97} +(-112.438 + 51.3488i) q^{98} +(-143.988 - 69.6949i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	1.84994	−	0.0881234i	0.654052	−	0.0311563i	0.282066	−	0.959395i	$-0.408980\pi$
							0.371986	+	0.928239i	$0.378677\pi$
$3$	4.64817	+	2.32262i	0.894540	+	0.446988i
							$5$	−12.8457	−	12.2483i	−1.14895	−	1.09552i	−0.994197	−	0.107571i	$-0.965693\pi$
−0.154755	−	0.987953i	$-0.549459\pi$
$7$	3.83084	+	19.8763i	0.206846	+	1.07322i	0.925758	+	0.378117i	$0.123428\pi$
							−0.718912	+	0.695101i	$0.755360\pi$
$11$	−5.26614	+	2.71489i	−0.144346	+	0.0744154i	−0.528886	−	0.848693i	$-0.677390\pi$
							0.384541	+	0.923108i	$0.374360\pi$
$13$	−71.3381	−	13.7493i	−1.52197	−	0.293336i	−0.641384	−	0.767220i	$-0.721639\pi$
							−0.880587	+	0.473884i	$0.842852\pi$
$17$	53.4500	+	117.039i	0.762560	+	1.66977i	0.742382	+	0.669977i	$0.233696\pi$
							0.0201784	+	0.999796i	$0.493577\pi$
$19$	−144.304	−	65.9013i	−1.74240	−	0.795727i	−0.990675	−	0.136248i	$-0.956496\pi$
							−0.751723	−	0.659479i	$-0.770777\pi$
$23$	−109.469	+	13.5446i	−0.992432	+	0.122793i
							$29$	4.63171	−	48.5055i	0.0296582	−	0.310595i	−0.968455	−	0.249190i	$-0.919836\pi$
0.998113	−	0.0614051i	$-0.0195581\pi$
$31$	89.7019	−	70.5423i	0.519708	−	0.408703i	−0.323551	−	0.946211i	$-0.604877\pi$
							0.843259	+	0.537508i	$0.180634\pi$
$37$	25.9859	−	88.5000i	0.115461	−	0.393224i	−0.881402	−	0.472366i	$-0.843400\pi$
							0.996863	+	0.0791421i	$0.0252181\pi$
$41$	−2.09017	+	2.19210i	−0.00796168	+	0.00834997i	−0.727703	−	0.685893i	$-0.759412\pi$
							0.719741	+	0.694243i	$0.244260\pi$
$43$	82.5340	−	104.950i	0.292705	−	0.372205i	−0.617353	−	0.786687i	$-0.711795\pi$
							0.910058	+	0.414482i	$0.136037\pi$
$47$	188.118	+	108.610i	0.583826	+	0.337072i	0.762652	−	0.646809i	$-0.223897\pi$
							−0.178826	+	0.983881i	$0.557230\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.46	✓	1400
9.5	odd	6	inner	207.4.o.a.149.46	yes	1400
23.21	odd	22	inner	207.4.o.a.182.46	yes	1400
207.113	even	66	inner	207.4.o.a.113.46	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.46	✓	1400	1.1	even	1	trivial
207.4.o.a.113.46	yes	1400	207.113	even	66	inner
207.4.o.a.149.46	yes	1400	9.5	odd	6	inner
207.4.o.a.182.46	yes	1400	23.21	odd	22	inner