Embedded newform 207.4.o.a.11.15

• Label: 207.4.o.a.11.15
• 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			11.15
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.15

$q$-expansion

$f(q)$	$=$	$q+(-3.78164 + 0.180142i) q^{2} +(2.93051 + 4.29093i) q^{3} +(6.30456 - 0.602012i) q^{4} +(-11.8152 - 11.2658i) q^{5} +(-11.8551 - 15.6989i) q^{6} +(3.18592 + 16.5301i) q^{7} +(6.24601 - 0.898041i) q^{8} +(-9.82423 + 25.1492i) q^{9} +(46.7102 + 40.4746i) q^{10} +(-4.52737 + 2.33402i) q^{11} +(21.0588 + 25.2882i) q^{12} +(-31.9587 - 6.15954i) q^{13} +(-15.0258 - 61.9370i) q^{14} +(13.7161 - 83.7126i) q^{15} +(-73.2087 + 14.1098i) q^{16} +(-33.5526 - 73.4699i) q^{17} +(32.6212 - 96.8751i) q^{18} +(24.4178 + 11.1512i) q^{19} +(-81.2717 - 63.9127i) q^{20} +(-61.5933 + 62.1123i) q^{21} +(16.7004 - 9.64199i) q^{22} +(88.5501 - 65.7714i) q^{23} +(22.1574 + 24.1695i) q^{24} +(6.73359 + 141.356i) q^{25} +(121.966 + 17.5360i) q^{26} +(-136.704 + 31.5450i) q^{27} +(30.0372 + 102.297i) q^{28} +(-2.38152 + 24.9404i) q^{29} +(-36.7892 + 319.041i) q^{30} +(134.446 - 105.730i) q^{31} +(225.248 - 54.6446i) q^{32} +(-23.2826 - 12.5868i) q^{33} +(140.119 + 271.792i) q^{34} +(148.582 - 231.198i) q^{35} +(-46.7973 + 164.469i) q^{36} +(29.7276 - 101.243i) q^{37} +(-94.3481 - 37.7713i) q^{38} +(-67.2252 - 155.183i) q^{39} +(-83.9149 - 59.7555i) q^{40} +(242.458 - 254.283i) q^{41} +(221.735 - 245.982i) q^{42} +(159.967 - 203.415i) q^{43} +(-27.1380 + 17.4405i) q^{44} +(399.400 - 186.466i) q^{45} +(-323.016 + 264.675i) q^{46} +(177.047 + 102.218i) q^{47} +(-275.083 - 272.785i) q^{48} +(55.3352 - 22.1529i) q^{49} +(-50.9280 - 533.342i) q^{50} +(216.928 - 359.276i) q^{51} +(-205.194 - 19.5936i) q^{52} +(-28.5060 - 32.8977i) q^{53} +(511.281 - 143.918i) q^{54} +(79.7862 + 23.4274i) q^{55} +(34.7440 + 100.386i) q^{56} +(23.7074 + 137.454i) q^{57} +(4.51325 - 94.7447i) q^{58} +(140.776 - 730.414i) q^{59} +(36.0780 - 536.028i) q^{60} +(-148.000 + 369.687i) q^{61} +(-489.380 + 424.050i) q^{62} +(-447.019 - 82.2722i) q^{63} +(-269.675 + 79.1837i) q^{64} +(308.206 + 432.815i) q^{65} +(90.3139 + 43.4045i) q^{66} +(-196.460 + 381.079i) q^{67} +(-255.764 - 442.996i) q^{68} +(541.718 + 187.219i) q^{69} +(-520.235 + 901.074i) q^{70} +(-607.259 - 944.914i) q^{71} +(-38.7772 + 165.905i) q^{72} +(295.148 - 646.284i) q^{73} +(-94.1809 + 388.219i) q^{74} +(-586.814 + 443.137i) q^{75} +(160.657 + 55.6038i) q^{76} +(-53.0055 - 67.4020i) q^{77} +(282.176 + 574.737i) q^{78} +(1012.54 - 350.443i) q^{79} +(1023.93 + 658.041i) q^{80} +(-535.969 - 494.144i) q^{81} +(-871.081 + 1005.28i) q^{82} +(-997.646 + 951.254i) q^{83} +(-350.926 + 428.670i) q^{84} +(-431.264 + 1246.06i) q^{85} +(-568.294 + 798.058i) q^{86} +(-113.997 + 62.8692i) q^{87} +(-26.1820 + 18.6441i) q^{88} +(-88.4310 + 615.051i) q^{89} +(-1476.80 + 777.094i) q^{90} -547.906i q^{91} +(518.674 - 467.968i) q^{92} +(847.674 + 267.058i) q^{93} +(-687.940 - 354.657i) q^{94} +(-162.874 - 406.839i) q^{95} +(894.568 + 806.388i) q^{96} +(235.492 + 57.1296i) q^{97} +(-205.267 + 93.7423i) q^{98} +(-14.2210 - 136.790i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−3.78164	+	0.180142i	−1.33701	+	0.0636897i	−0.703860	−	0.710339i	$-0.748542\pi$
							−0.633151	+	0.774028i	$0.718239\pi$
$3$	2.93051	+	4.29093i	0.563977	+	0.825791i
							$5$	−11.8152	−	11.2658i	−1.05678	−	1.00764i	−0.999957	−	0.00925562i	$-0.997054\pi$
−0.0568250	−	0.998384i	$-0.518098\pi$
$7$	3.18592	+	16.5301i	0.172024	+	0.892543i	0.960669	+	0.277695i	$0.0895705\pi$
							−0.788646	+	0.614848i	$0.789217\pi$
$11$	−4.52737	+	2.33402i	−0.124096	+	0.0639758i	−0.519157	−	0.854679i	$-0.673754\pi$
							0.395061	+	0.918655i	$0.370724\pi$
$13$	−31.9587	−	6.15954i	−0.681827	−	0.131411i	−0.163435	−	0.986554i	$-0.552257\pi$
							−0.518393	+	0.855143i	$0.673469\pi$
$17$	−33.5526	−	73.4699i	−0.478688	−	1.04818i	−0.982822	−	0.184555i	$-0.940915\pi$
							0.504134	−	0.863626i	$-0.331812\pi$
$19$	24.4178	+	11.1512i	0.294833	+	0.134646i	0.557340	−	0.830284i	$-0.311822\pi$
							−0.262507	+	0.964930i	$0.584549\pi$
$23$	88.5501	−	65.7714i	0.802781	−	0.596273i
							$29$	−2.38152	+	24.9404i	−0.0152496	+	0.159701i	−0.999889	−	0.0148994i	$-0.995257\pi$
0.984639	+	0.174600i	$0.0558633\pi$
$31$	134.446	−	105.730i	0.778943	−	0.612567i	−0.147475	−	0.989066i	$-0.547114\pi$
							0.926417	+	0.376499i	$0.122872\pi$
$37$	29.7276	−	101.243i	0.132086	−	0.449844i	−0.866715	−	0.498804i	$-0.833773\pi$
							0.998801	+	0.0489604i	$0.0155908\pi$
$41$	242.458	−	254.283i	0.923550	−	0.968592i	−0.0760432	−	0.997105i	$-0.524229\pi$
							0.999593	+	0.0285129i	$0.00907716\pi$
$43$	159.967	−	203.415i	0.567320	−	0.721406i	−0.414747	−	0.909937i	$-0.636130\pi$
							0.982067	+	0.188531i	$0.0603725\pi$
$47$	177.047	+	102.218i	0.549466	+	0.317234i	0.748906	−	0.662676i	$-0.230579\pi$
							−0.199441	+	0.979910i	$0.563913\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.15	✓	1400
9.5	odd	6	inner	207.4.o.a.149.15	yes	1400
23.21	odd	22	inner	207.4.o.a.182.15	yes	1400
207.113	even	66	inner	207.4.o.a.113.15	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.15	✓	1400	1.1	even	1	trivial
207.4.o.a.113.15	yes	1400	207.113	even	66	inner
207.4.o.a.149.15	yes	1400	9.5	odd	6	inner
207.4.o.a.182.15	yes	1400	23.21	odd	22	inner