Embedded newform 207.4.o.a.11.42

• Label: 207.4.o.a.11.42
• 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			11.42
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.42

$q$-expansion

$f(q)$	$=$	$q+(1.03007 - 0.0490683i) q^{2} +(-3.48800 + 3.85147i) q^{3} +(-6.90514 + 0.659361i) q^{4} +(4.97939 + 4.74784i) q^{5} +(-3.40390 + 4.13843i) q^{6} +(-5.41053 - 28.0725i) q^{7} +(-15.2463 + 2.19209i) q^{8} +(-2.66766 - 26.8679i) q^{9} +(5.36209 + 4.64628i) q^{10} +(1.06769 - 0.550433i) q^{11} +(21.5456 - 28.8948i) q^{12} +(57.4175 + 11.0663i) q^{13} +(-6.95069 - 28.6511i) q^{14} +(-35.6543 + 2.61750i) q^{15} +(38.8923 - 7.49588i) q^{16} +(21.5544 + 47.1976i) q^{17} +(-4.06624 - 27.5449i) q^{18} +(76.5942 + 34.9794i) q^{19} +(-37.5139 - 29.5013i) q^{20} +(126.992 + 77.0785i) q^{21} +(1.07279 - 0.619374i) q^{22} +(-73.0398 + 82.6570i) q^{23} +(44.7365 - 66.3669i) q^{24} +(-3.69538 - 77.5757i) q^{25} +(59.6870 + 8.58170i) q^{26} +(112.786 + 83.4409i) q^{27} +(55.8704 + 190.277i) q^{28} +(7.95121 - 83.2688i) q^{29} +(-36.5980 + 4.44570i) q^{30} +(103.046 - 81.0366i) q^{31} +(159.446 - 38.6811i) q^{32} +(-1.60413 + 6.03209i) q^{33} +(24.5184 + 47.5591i) q^{34} +(106.343 - 165.472i) q^{35} +(36.1362 + 183.768i) q^{36} +(70.3795 - 239.691i) q^{37} +(80.6137 + 32.2728i) q^{38} +(-242.894 + 182.543i) q^{39} +(-86.3253 - 61.4719i) q^{40} +(325.955 - 341.852i) q^{41} +(134.593 + 73.1649i) q^{42} +(56.0930 - 71.3280i) q^{43} +(-7.00962 + 4.50481i) q^{44} +(114.281 - 146.451i) q^{45} +(-71.1802 + 88.7264i) q^{46} +(-258.871 - 149.460i) q^{47} +(-106.786 + 175.938i) q^{48} +(-440.362 + 176.294i) q^{49} +(-7.61301 - 79.7270i) q^{50} +(-256.962 - 81.6091i) q^{51} +(-403.773 - 38.5556i) q^{52} +(161.799 + 186.726i) q^{53} +(120.271 + 80.4157i) q^{54} +(7.92982 + 2.32840i) q^{55} +(144.028 + 416.143i) q^{56} +(-401.883 + 172.992i) q^{57} +(4.10444 - 86.1628i) q^{58} +(-21.2171 + 110.085i) q^{59} +(244.472 - 41.5833i) q^{60} +(263.775 - 658.879i) q^{61} +(102.169 - 88.5297i) q^{62} +(-739.816 + 220.258i) q^{63} +(-141.688 + 41.6033i) q^{64} +(233.363 + 327.713i) q^{65} +(-1.35638 + 6.29218i) q^{66} +(-75.2973 + 146.056i) q^{67} +(-179.957 - 311.694i) q^{68} +(-63.5881 - 569.619i) q^{69} +(101.421 - 175.666i) q^{70} +(402.379 + 626.114i) q^{71} +(99.5691 + 403.789i) q^{72} +(323.592 - 708.567i) q^{73} +(60.7346 - 250.351i) q^{74} +(311.670 + 256.352i) q^{75} +(-551.957 - 191.034i) q^{76} +(-21.2288 - 26.9946i) q^{77} +(-241.241 + 199.950i) q^{78} +(-367.713 + 127.267i) q^{79} +(229.250 + 147.330i) q^{80} +(-714.767 + 143.349i) q^{81} +(318.983 - 368.125i) q^{82} +(468.958 - 447.151i) q^{83} +(-927.723 - 448.504i) q^{84} +(-116.759 + 337.352i) q^{85} +(54.2797 - 76.2252i) q^{86} +(292.974 + 321.066i) q^{87} +(-15.0718 + 10.7326i) q^{88} +(-98.3404 + 683.972i) q^{89} +(110.531 - 156.463i) q^{90} -1671.73i q^{91} +(449.849 - 618.918i) q^{92} +(-47.3161 + 679.537i) q^{93} +(-273.989 - 141.251i) q^{94} +(215.316 + 537.833i) q^{95} +(-407.167 + 749.019i) q^{96} +(1508.06 + 365.852i) q^{97} +(-444.953 + 203.203i) q^{98} +(-17.6372 - 27.2182i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	1.03007	−	0.0490683i	0.364185	−	0.0173482i	0.135308	−	0.990804i	$-0.456798\pi$
							0.228877	+	0.973455i	$0.426495\pi$
$3$	−3.48800	+	3.85147i	−0.671267	+	0.741216i
							$5$	4.97939	+	4.74784i	0.445370	+	0.424660i	0.879286	−	0.476294i	$-0.158020\pi$
−0.433916	+	0.900953i	$0.642869\pi$
$7$	−5.41053	−	28.0725i	−0.292141	−	1.51577i	−0.770404	−	0.637556i	$-0.779945\pi$
							0.478263	−	0.878217i	$-0.341267\pi$
$11$	1.06769	−	0.550433i	0.0292655	−	0.0150874i	−0.443532	−	0.896259i	$-0.646275\pi$
							0.472797	+	0.881171i	$0.343244\pi$
$13$	57.4175	+	11.0663i	1.22498	+	0.236096i	0.760412	−	0.649441i	$-0.224997\pi$
							0.464569	+	0.885537i	$0.346209\pi$
$17$	21.5544	+	47.1976i	0.307513	+	0.673359i	0.998787	−	0.0492334i	$-0.0156778\pi$
							−0.691275	+	0.722592i	$0.742951\pi$
$19$	76.5942	+	34.9794i	0.924837	+	0.422359i	0.820151	−	0.572146i	$-0.193889\pi$
							0.104686	+	0.994505i	$0.466616\pi$
$23$	−73.0398	+	82.6570i	−0.662168	+	0.749356i
							$29$	7.95121	−	83.2688i	0.0509139	−	0.533194i	−0.933652	−	0.358181i	$-0.883397\pi$
0.984566	−	0.175013i	$-0.0559969\pi$
$31$	103.046	−	81.0366i	0.597022	−	0.469504i	−0.273339	−	0.961918i	$-0.588128\pi$
							0.870361	+	0.492414i	$0.163886\pi$
$37$	70.3795	−	239.691i	0.312711	−	1.06500i	−0.641812	−	0.766862i	$-0.721817\pi$
							0.954524	−	0.298135i	$-0.0963646\pi$
$41$	325.955	−	341.852i	1.24160	−	1.30215i	0.304748	−	0.952433i	$-0.401428\pi$
							0.936854	−	0.349722i	$-0.113724\pi$
$43$	56.0930	−	71.3280i	0.198932	−	0.252963i	−0.676344	−	0.736586i	$-0.736437\pi$
							0.875277	+	0.483622i	$0.160679\pi$
$47$	−258.871	−	149.460i	−0.803410	−	0.463849i	0.0412519	−	0.999149i	$-0.486865\pi$
							−0.844662	+	0.535300i	$0.820199\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.42	✓	1400
9.5	odd	6	inner	207.4.o.a.149.42	yes	1400
23.21	odd	22	inner	207.4.o.a.182.42	yes	1400
207.113	even	66	inner	207.4.o.a.113.42	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.42	✓	1400	1.1	even	1	trivial
207.4.o.a.113.42	yes	1400	207.113	even	66	inner
207.4.o.a.149.42	yes	1400	9.5	odd	6	inner
207.4.o.a.182.42	yes	1400	23.21	odd	22	inner