Embedded newform 207.4.o.a.11.30

• Label: 207.4.o.a.11.30
• 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			11.30
Character	$\chi$	$=$	207.11
Dual form			207.4.o.a.113.30

$q$-expansion

$f(q)$	$=$	$q+(-1.20601 + 0.0574491i) q^{2} +(-3.19171 + 4.10037i) q^{3} +(-6.51263 + 0.621880i) q^{4} +(-10.2821 - 9.80400i) q^{5} +(3.61366 - 5.12843i) q^{6} +(-0.946655 - 4.91172i) q^{7} +(17.3792 - 2.49875i) q^{8} +(-6.62600 - 26.1743i) q^{9} +(12.9636 + 11.2330i) q^{10} +(-7.00185 + 3.60971i) q^{11} +(18.2365 - 28.6890i) q^{12} +(-78.3694 - 15.1045i) q^{13} +(1.42385 + 5.86917i) q^{14} +(73.0176 - 10.8690i) q^{15} +(30.5763 - 5.89310i) q^{16} +(-1.14356 - 2.50405i) q^{17} +(9.49469 + 31.1857i) q^{18} +(29.7207 + 13.5730i) q^{19} +(73.0606 + 57.4555i) q^{20} +(23.1613 + 11.7951i) q^{21} +(8.23690 - 4.75558i) q^{22} +(17.4070 + 108.922i) q^{23} +(-45.2236 + 79.2364i) q^{24} +(3.65623 + 76.7538i) q^{25} +(95.3817 + 13.7138i) q^{26} +(128.473 + 56.3718i) q^{27} +(9.21971 + 31.3995i) q^{28} +(-10.1462 + 106.256i) q^{29} +(-87.4352 + 17.3029i) q^{30} +(-12.6298 + 9.93218i) q^{31} +(-173.041 + 41.9792i) q^{32} +(7.54675 - 40.2313i) q^{33} +(1.52300 + 2.95420i) q^{34} +(-38.4208 + 59.7840i) q^{35} +(59.4300 + 166.343i) q^{36} +(-4.93156 + 16.7953i) q^{37} +(-36.6230 - 14.6617i) q^{38} +(312.066 - 273.134i) q^{39} +(-203.193 - 144.693i) q^{40} +(-113.025 + 118.537i) q^{41} +(-28.6103 - 12.8944i) q^{42} +(-4.28908 + 5.45401i) q^{43} +(43.3556 - 27.8630i) q^{44} +(-188.484 + 334.090i) q^{45} +(-27.2504 - 130.361i) q^{46} +(547.319 + 315.995i) q^{47} +(-73.4268 + 144.183i) q^{48} +(295.201 - 118.181i) q^{49} +(-8.81888 - 92.3555i) q^{50} +(13.9174 + 3.30318i) q^{51} +(519.784 + 49.6333i) q^{52} +(-228.695 - 263.928i) q^{53} +(-158.177 - 60.6041i) q^{54} +(107.384 + 31.5307i) q^{55} +(-28.7253 - 82.9963i) q^{56} +(-150.514 + 78.5446i) q^{57} +(6.13207 - 128.728i) q^{58} +(66.5220 - 345.149i) q^{59} +(-468.777 + 116.194i) q^{60} +(167.262 - 417.799i) q^{61} +(14.6610 - 12.7038i) q^{62} +(-122.288 + 57.3231i) q^{63} +(-32.7452 + 9.61485i) q^{64} +(657.721 + 923.640i) q^{65} +(-6.79018 + 48.9527i) q^{66} +(319.621 - 619.979i) q^{67} +(9.00481 + 15.5968i) q^{68} +(-502.178 - 276.272i) q^{69} +(42.9012 - 74.3070i) q^{70} +(322.328 + 501.552i) q^{71} +(-180.558 - 438.333i) q^{72} +(243.951 - 534.178i) q^{73} +(4.98261 - 20.5386i) q^{74} +(-326.388 - 229.984i) q^{75} +(-202.000 - 69.9130i) q^{76} +(24.3582 + 30.9740i) q^{77} +(-360.662 + 347.329i) q^{78} +(182.019 - 62.9974i) q^{79} +(-372.166 - 239.176i) q^{80} +(-641.192 + 346.862i) q^{81} +(129.499 - 149.450i) q^{82} +(-46.5653 + 44.3999i) q^{83} +(-158.176 - 62.4137i) q^{84} +(-12.7915 + 36.9585i) q^{85} +(4.85933 - 6.82398i) q^{86} +(-403.304 - 380.741i) q^{87} +(-112.667 + 80.2297i) q^{88} +(-194.868 + 1355.34i) q^{89} +(208.119 - 413.742i) q^{90} +399.227i q^{91} +(-181.101 - 698.543i) q^{92} +(-0.414956 - 83.4874i) q^{93} +(-678.223 - 349.648i) q^{94} +(-172.523 - 430.941i) q^{95} +(380.165 - 843.515i) q^{96} +(-1282.54 - 311.141i) q^{97} +(-349.225 + 159.486i) q^{98} +(140.876 + 159.351i) 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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−1.20601	+	0.0574491i	−0.426387	+	0.0203113i	−0.259679	−	0.965695i	$-0.583617\pi$
							−0.166708	+	0.986006i	$0.553314\pi$
$3$	−3.19171	+	4.10037i	−0.614244	+	0.789116i
							$5$	−10.2821	−	9.80400i	−0.919663	−	0.876897i	0.0730591	−	0.997328i	$-0.476724\pi$
−0.992722	+	0.120431i	$0.961572\pi$
$7$	−0.946655	−	4.91172i	−0.0511146	−	0.265208i	0.947377	−	0.320121i	$-0.103724\pi$
							−0.998491	+	0.0549138i	$0.982512\pi$
$11$	−7.00185	+	3.60971i	−0.191922	+	0.0989425i	−0.551509	−	0.834169i	$-0.685948\pi$
							0.359587	+	0.933111i	$0.382917\pi$
$13$	−78.3694	−	15.1045i	−1.67198	−	0.322248i	−0.736718	−	0.676200i	$-0.763625\pi$
							−0.935264	+	0.353952i	$0.884838\pi$
$17$	−1.14356	−	2.50405i	−0.0163150	−	0.0357248i	0.901299	−	0.433197i	$-0.142615\pi$
							−0.917614	+	0.397473i	$0.869887\pi$
$19$	29.7207	+	13.5730i	0.358862	+	0.163887i	0.586684	−	0.809816i	$-0.300433\pi$
							−0.227822	+	0.973703i	$0.573160\pi$
$23$	17.4070	+	108.922i	0.157809	+	0.987470i
							$29$	−10.1462	+	106.256i	−0.0649690	+	0.680386i	0.903054	+	0.429526i	$0.141319\pi$
−0.968023	+	0.250860i	$0.919287\pi$
$31$	−12.6298	+	9.93218i	−0.0731735	+	0.0575442i	−0.654076	−	0.756429i	$-0.726942\pi$
							0.580902	+	0.813973i	$0.302700\pi$
$37$	−4.93156	+	16.7953i	−0.0219120	+	0.0746253i	−0.969723	−	0.244207i	$-0.921472\pi$
							0.947811	+	0.318832i	$0.103291\pi$
$41$	−113.025	+	118.537i	−0.430526	+	0.451523i	−0.902884	−	0.429884i	$-0.858554\pi$
							0.472358	+	0.881407i	$0.343403\pi$
$43$	−4.28908	+	5.45401i	−0.0152111	+	0.0193425i	−0.793599	−	0.608441i	$-0.791795\pi$
							0.778388	+	0.627783i	$0.216038\pi$
$47$	547.319	+	315.995i	1.69861	+	0.980692i	0.947083	+	0.320989i	$0.104015\pi$
							0.751526	+	0.659703i	$0.229318\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.o.a.11.30	✓	1400
9.5	odd	6	inner	207.4.o.a.149.30	yes	1400
23.21	odd	22	inner	207.4.o.a.182.30	yes	1400
207.113	even	66	inner	207.4.o.a.113.30	yes	1400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.o.a.11.30	✓	1400	1.1	even	1	trivial
207.4.o.a.113.30	yes	1400	207.113	even	66	inner
207.4.o.a.149.30	yes	1400	9.5	odd	6	inner
207.4.o.a.182.30	yes	1400	23.21	odd	22	inner