# Properties

 Label 1815.2.a.x.1.3 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.737640$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.45589 q^{2} +1.00000 q^{3} +4.03138 q^{4} -1.00000 q^{5} +2.45589 q^{6} +3.28684 q^{7} +4.98884 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.45589 q^{2} +1.00000 q^{3} +4.03138 q^{4} -1.00000 q^{5} +2.45589 q^{6} +3.28684 q^{7} +4.98884 q^{8} +1.00000 q^{9} -2.45589 q^{10} +4.03138 q^{12} -0.313133 q^{13} +8.07211 q^{14} -1.00000 q^{15} +4.18926 q^{16} +5.00000 q^{17} +2.45589 q^{18} -7.45408 q^{19} -4.03138 q^{20} +3.28684 q^{21} +1.07392 q^{23} +4.98884 q^{24} +1.00000 q^{25} -0.769020 q^{26} +1.00000 q^{27} +13.2505 q^{28} -5.03647 q^{29} -2.45589 q^{30} +3.44899 q^{31} +0.310680 q^{32} +12.2794 q^{34} -3.28684 q^{35} +4.03138 q^{36} +2.63428 q^{37} -18.3064 q^{38} -0.313133 q^{39} -4.98884 q^{40} +10.8472 q^{41} +8.07211 q^{42} -5.51468 q^{43} -1.00000 q^{45} +2.63743 q^{46} -11.9982 q^{47} +4.18926 q^{48} +3.80333 q^{49} +2.45589 q^{50} +5.00000 q^{51} -1.26236 q^{52} +4.93543 q^{53} +2.45589 q^{54} +16.3975 q^{56} -7.45408 q^{57} -12.3690 q^{58} -9.16409 q^{59} -4.03138 q^{60} +9.18431 q^{61} +8.47033 q^{62} +3.28684 q^{63} -7.61553 q^{64} +0.313133 q^{65} -15.2739 q^{67} +20.1569 q^{68} +1.07392 q^{69} -8.07211 q^{70} +3.07211 q^{71} +4.98884 q^{72} -8.65269 q^{73} +6.46950 q^{74} +1.00000 q^{75} -30.0502 q^{76} -0.769020 q^{78} +5.41446 q^{79} -4.18926 q^{80} +1.00000 q^{81} +26.6395 q^{82} +16.2454 q^{83} +13.2505 q^{84} -5.00000 q^{85} -13.5434 q^{86} -5.03647 q^{87} +1.62118 q^{89} -2.45589 q^{90} -1.02922 q^{91} +4.32938 q^{92} +3.44899 q^{93} -29.4662 q^{94} +7.45408 q^{95} +0.310680 q^{96} +0.224082 q^{97} +9.34054 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + 5 * q^2 + 4 * q^3 + 9 * q^4 - 4 * q^5 + 5 * q^6 - 2 * q^7 + 15 * q^8 + 4 * q^9 $$4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9} - 5 q^{10} + 9 q^{12} + 3 q^{13} - 5 q^{14} - 4 q^{15} + 15 q^{16} + 20 q^{17} + 5 q^{18} + 3 q^{19} - 9 q^{20} - 2 q^{21} - 5 q^{23} + 15 q^{24} + 4 q^{25} + 6 q^{26} + 4 q^{27} + 3 q^{28} + 5 q^{29} - 5 q^{30} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} + 9 q^{36} - 7 q^{37} + q^{38} + 3 q^{39} - 15 q^{40} + 20 q^{41} - 5 q^{42} - 2 q^{43} - 4 q^{45} + 7 q^{46} - 20 q^{47} + 15 q^{48} + 8 q^{49} + 5 q^{50} + 20 q^{51} - 7 q^{52} + 6 q^{53} + 5 q^{54} + 10 q^{56} + 3 q^{57} - 21 q^{58} - 5 q^{59} - 9 q^{60} - 7 q^{61} - 12 q^{62} - 2 q^{63} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} - 5 q^{69} + 5 q^{70} - 25 q^{71} + 15 q^{72} + 23 q^{73} - 7 q^{74} + 4 q^{75} - 7 q^{76} + 6 q^{78} - 15 q^{80} + 4 q^{81} + 11 q^{82} + 33 q^{83} + 3 q^{84} - 20 q^{85} - 12 q^{86} + 5 q^{87} + 16 q^{89} - 5 q^{90} - 24 q^{91} - q^{93} - 17 q^{94} - 3 q^{95} + 30 q^{96} + 25 q^{98}+O(q^{100})$$ 4 * q + 5 * q^2 + 4 * q^3 + 9 * q^4 - 4 * q^5 + 5 * q^6 - 2 * q^7 + 15 * q^8 + 4 * q^9 - 5 * q^10 + 9 * q^12 + 3 * q^13 - 5 * q^14 - 4 * q^15 + 15 * q^16 + 20 * q^17 + 5 * q^18 + 3 * q^19 - 9 * q^20 - 2 * q^21 - 5 * q^23 + 15 * q^24 + 4 * q^25 + 6 * q^26 + 4 * q^27 + 3 * q^28 + 5 * q^29 - 5 * q^30 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 + 9 * q^36 - 7 * q^37 + q^38 + 3 * q^39 - 15 * q^40 + 20 * q^41 - 5 * q^42 - 2 * q^43 - 4 * q^45 + 7 * q^46 - 20 * q^47 + 15 * q^48 + 8 * q^49 + 5 * q^50 + 20 * q^51 - 7 * q^52 + 6 * q^53 + 5 * q^54 + 10 * q^56 + 3 * q^57 - 21 * q^58 - 5 * q^59 - 9 * q^60 - 7 * q^61 - 12 * q^62 - 2 * q^63 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 - 5 * q^69 + 5 * q^70 - 25 * q^71 + 15 * q^72 + 23 * q^73 - 7 * q^74 + 4 * q^75 - 7 * q^76 + 6 * q^78 - 15 * q^80 + 4 * q^81 + 11 * q^82 + 33 * q^83 + 3 * q^84 - 20 * q^85 - 12 * q^86 + 5 * q^87 + 16 * q^89 - 5 * q^90 - 24 * q^91 - q^93 - 17 * q^94 - 3 * q^95 + 30 * q^96 + 25 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.45589 1.73657 0.868287 0.496062i $$-0.165221\pi$$
0.868287 + 0.496062i $$0.165221\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 4.03138 2.01569
$$5$$ −1.00000 −0.447214
$$6$$ 2.45589 1.00261
$$7$$ 3.28684 1.24231 0.621155 0.783688i $$-0.286664\pi$$
0.621155 + 0.783688i $$0.286664\pi$$
$$8$$ 4.98884 1.76382
$$9$$ 1.00000 0.333333
$$10$$ −2.45589 −0.776620
$$11$$ 0 0
$$12$$ 4.03138 1.16376
$$13$$ −0.313133 −0.0868476 −0.0434238 0.999057i $$-0.513827\pi$$
−0.0434238 + 0.999057i $$0.513827\pi$$
$$14$$ 8.07211 2.15736
$$15$$ −1.00000 −0.258199
$$16$$ 4.18926 1.04732
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 2.45589 0.578858
$$19$$ −7.45408 −1.71008 −0.855041 0.518560i $$-0.826468\pi$$
−0.855041 + 0.518560i $$0.826468\pi$$
$$20$$ −4.03138 −0.901444
$$21$$ 3.28684 0.717248
$$22$$ 0 0
$$23$$ 1.07392 0.223928 0.111964 0.993712i $$-0.464286\pi$$
0.111964 + 0.993712i $$0.464286\pi$$
$$24$$ 4.98884 1.01834
$$25$$ 1.00000 0.200000
$$26$$ −0.769020 −0.150817
$$27$$ 1.00000 0.192450
$$28$$ 13.2505 2.50411
$$29$$ −5.03647 −0.935249 −0.467624 0.883927i $$-0.654890\pi$$
−0.467624 + 0.883927i $$0.654890\pi$$
$$30$$ −2.45589 −0.448382
$$31$$ 3.44899 0.619457 0.309728 0.950825i $$-0.399762\pi$$
0.309728 + 0.950825i $$0.399762\pi$$
$$32$$ 0.310680 0.0549210
$$33$$ 0 0
$$34$$ 12.2794 2.10591
$$35$$ −3.28684 −0.555578
$$36$$ 4.03138 0.671897
$$37$$ 2.63428 0.433073 0.216537 0.976274i $$-0.430524\pi$$
0.216537 + 0.976274i $$0.430524\pi$$
$$38$$ −18.3064 −2.96969
$$39$$ −0.313133 −0.0501415
$$40$$ −4.98884 −0.788805
$$41$$ 10.8472 1.69405 0.847024 0.531554i $$-0.178392\pi$$
0.847024 + 0.531554i $$0.178392\pi$$
$$42$$ 8.07211 1.24555
$$43$$ −5.51468 −0.840980 −0.420490 0.907297i $$-0.638142\pi$$
−0.420490 + 0.907297i $$0.638142\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 2.63743 0.388868
$$47$$ −11.9982 −1.75012 −0.875058 0.484018i $$-0.839177\pi$$
−0.875058 + 0.484018i $$0.839177\pi$$
$$48$$ 4.18926 0.604668
$$49$$ 3.80333 0.543333
$$50$$ 2.45589 0.347315
$$51$$ 5.00000 0.700140
$$52$$ −1.26236 −0.175058
$$53$$ 4.93543 0.677934 0.338967 0.940798i $$-0.389923\pi$$
0.338967 + 0.940798i $$0.389923\pi$$
$$54$$ 2.45589 0.334204
$$55$$ 0 0
$$56$$ 16.3975 2.19121
$$57$$ −7.45408 −0.987317
$$58$$ −12.3690 −1.62413
$$59$$ −9.16409 −1.19306 −0.596531 0.802590i $$-0.703455\pi$$
−0.596531 + 0.802590i $$0.703455\pi$$
$$60$$ −4.03138 −0.520449
$$61$$ 9.18431 1.17593 0.587965 0.808886i $$-0.299929\pi$$
0.587965 + 0.808886i $$0.299929\pi$$
$$62$$ 8.47033 1.07573
$$63$$ 3.28684 0.414103
$$64$$ −7.61553 −0.951942
$$65$$ 0.313133 0.0388394
$$66$$ 0 0
$$67$$ −15.2739 −1.86600 −0.933000 0.359876i $$-0.882819\pi$$
−0.933000 + 0.359876i $$0.882819\pi$$
$$68$$ 20.1569 2.44438
$$69$$ 1.07392 0.129285
$$70$$ −8.07211 −0.964802
$$71$$ 3.07211 0.364593 0.182296 0.983244i $$-0.441647\pi$$
0.182296 + 0.983244i $$0.441647\pi$$
$$72$$ 4.98884 0.587940
$$73$$ −8.65269 −1.01272 −0.506361 0.862322i $$-0.669009\pi$$
−0.506361 + 0.862322i $$0.669009\pi$$
$$74$$ 6.46950 0.752064
$$75$$ 1.00000 0.115470
$$76$$ −30.0502 −3.44700
$$77$$ 0 0
$$78$$ −0.769020 −0.0870744
$$79$$ 5.41446 0.609175 0.304587 0.952484i $$-0.401481\pi$$
0.304587 + 0.952484i $$0.401481\pi$$
$$80$$ −4.18926 −0.468374
$$81$$ 1.00000 0.111111
$$82$$ 26.6395 2.94184
$$83$$ 16.2454 1.78317 0.891583 0.452857i $$-0.149595\pi$$
0.891583 + 0.452857i $$0.149595\pi$$
$$84$$ 13.2505 1.44575
$$85$$ −5.00000 −0.542326
$$86$$ −13.5434 −1.46042
$$87$$ −5.03647 −0.539966
$$88$$ 0 0
$$89$$ 1.62118 0.171845 0.0859223 0.996302i $$-0.472616\pi$$
0.0859223 + 0.996302i $$0.472616\pi$$
$$90$$ −2.45589 −0.258873
$$91$$ −1.02922 −0.107892
$$92$$ 4.32938 0.451369
$$93$$ 3.44899 0.357643
$$94$$ −29.4662 −3.03921
$$95$$ 7.45408 0.764772
$$96$$ 0.310680 0.0317086
$$97$$ 0.224082 0.0227521 0.0113760 0.999935i $$-0.496379\pi$$
0.0113760 + 0.999935i $$0.496379\pi$$
$$98$$ 9.34054 0.943537
$$99$$ 0 0
$$100$$ 4.03138 0.403138
$$101$$ −0.505326 −0.0502818 −0.0251409 0.999684i $$-0.508003\pi$$
−0.0251409 + 0.999684i $$0.508003\pi$$
$$102$$ 12.2794 1.21585
$$103$$ −6.40197 −0.630805 −0.315402 0.948958i $$-0.602139\pi$$
−0.315402 + 0.948958i $$0.602139\pi$$
$$104$$ −1.56217 −0.153184
$$105$$ −3.28684 −0.320763
$$106$$ 12.1209 1.17728
$$107$$ 2.09249 0.202289 0.101144 0.994872i $$-0.467750\pi$$
0.101144 + 0.994872i $$0.467750\pi$$
$$108$$ 4.03138 0.387920
$$109$$ −6.69278 −0.641052 −0.320526 0.947240i $$-0.603860\pi$$
−0.320526 + 0.947240i $$0.603860\pi$$
$$110$$ 0 0
$$111$$ 2.63428 0.250035
$$112$$ 13.7694 1.30109
$$113$$ −10.7941 −1.01542 −0.507712 0.861527i $$-0.669509\pi$$
−0.507712 + 0.861527i $$0.669509\pi$$
$$114$$ −18.3064 −1.71455
$$115$$ −1.07392 −0.100144
$$116$$ −20.3039 −1.88517
$$117$$ −0.313133 −0.0289492
$$118$$ −22.5060 −2.07184
$$119$$ 16.4342 1.50652
$$120$$ −4.98884 −0.455417
$$121$$ 0 0
$$122$$ 22.5556 2.04209
$$123$$ 10.8472 0.978059
$$124$$ 13.9042 1.24863
$$125$$ −1.00000 −0.0894427
$$126$$ 8.07211 0.719121
$$127$$ −17.0033 −1.50880 −0.754398 0.656417i $$-0.772071\pi$$
−0.754398 + 0.656417i $$0.772071\pi$$
$$128$$ −19.3242 −1.70804
$$129$$ −5.51468 −0.485540
$$130$$ 0.769020 0.0674475
$$131$$ −0.0430508 −0.00376136 −0.00188068 0.999998i $$-0.500599\pi$$
−0.00188068 + 0.999998i $$0.500599\pi$$
$$132$$ 0 0
$$133$$ −24.5004 −2.12445
$$134$$ −37.5109 −3.24045
$$135$$ −1.00000 −0.0860663
$$136$$ 24.9442 2.13895
$$137$$ 7.36257 0.629027 0.314513 0.949253i $$-0.398159\pi$$
0.314513 + 0.949253i $$0.398159\pi$$
$$138$$ 2.63743 0.224513
$$139$$ 13.2393 1.12295 0.561473 0.827495i $$-0.310235\pi$$
0.561473 + 0.827495i $$0.310235\pi$$
$$140$$ −13.2505 −1.11987
$$141$$ −11.9982 −1.01043
$$142$$ 7.54476 0.633142
$$143$$ 0 0
$$144$$ 4.18926 0.349105
$$145$$ 5.03647 0.418256
$$146$$ −21.2500 −1.75867
$$147$$ 3.80333 0.313693
$$148$$ 10.6198 0.872942
$$149$$ −5.87858 −0.481592 −0.240796 0.970576i $$-0.577409\pi$$
−0.240796 + 0.970576i $$0.577409\pi$$
$$150$$ 2.45589 0.200522
$$151$$ 7.62821 0.620775 0.310387 0.950610i $$-0.399541\pi$$
0.310387 + 0.950610i $$0.399541\pi$$
$$152$$ −37.1872 −3.01628
$$153$$ 5.00000 0.404226
$$154$$ 0 0
$$155$$ −3.44899 −0.277029
$$156$$ −1.26236 −0.101070
$$157$$ −10.1332 −0.808719 −0.404360 0.914600i $$-0.632506\pi$$
−0.404360 + 0.914600i $$0.632506\pi$$
$$158$$ 13.2973 1.05788
$$159$$ 4.93543 0.391405
$$160$$ −0.310680 −0.0245614
$$161$$ 3.52981 0.278188
$$162$$ 2.45589 0.192953
$$163$$ −5.02906 −0.393906 −0.196953 0.980413i $$-0.563105\pi$$
−0.196953 + 0.980413i $$0.563105\pi$$
$$164$$ 43.7292 3.41468
$$165$$ 0 0
$$166$$ 39.8969 3.09660
$$167$$ 5.79105 0.448125 0.224062 0.974575i $$-0.428068\pi$$
0.224062 + 0.974575i $$0.428068\pi$$
$$168$$ 16.3975 1.26510
$$169$$ −12.9019 −0.992457
$$170$$ −12.2794 −0.941790
$$171$$ −7.45408 −0.570028
$$172$$ −22.2318 −1.69516
$$173$$ 16.0652 1.22142 0.610708 0.791856i $$-0.290885\pi$$
0.610708 + 0.791856i $$0.290885\pi$$
$$174$$ −12.3690 −0.937691
$$175$$ 3.28684 0.248462
$$176$$ 0 0
$$177$$ −9.16409 −0.688815
$$178$$ 3.98143 0.298421
$$179$$ −8.30309 −0.620602 −0.310301 0.950638i $$-0.600430\pi$$
−0.310301 + 0.950638i $$0.600430\pi$$
$$180$$ −4.03138 −0.300481
$$181$$ −6.46425 −0.480484 −0.240242 0.970713i $$-0.577227\pi$$
−0.240242 + 0.970713i $$0.577227\pi$$
$$182$$ −2.52765 −0.187362
$$183$$ 9.18431 0.678924
$$184$$ 5.35762 0.394969
$$185$$ −2.63428 −0.193676
$$186$$ 8.47033 0.621074
$$187$$ 0 0
$$188$$ −48.3693 −3.52769
$$189$$ 3.28684 0.239083
$$190$$ 18.3064 1.32808
$$191$$ 15.3693 1.11208 0.556041 0.831155i $$-0.312320\pi$$
0.556041 + 0.831155i $$0.312320\pi$$
$$192$$ −7.61553 −0.549604
$$193$$ 15.7518 1.13384 0.566919 0.823773i $$-0.308135\pi$$
0.566919 + 0.823773i $$0.308135\pi$$
$$194$$ 0.550320 0.0395107
$$195$$ 0.313133 0.0224239
$$196$$ 15.3327 1.09519
$$197$$ 16.3940 1.16802 0.584010 0.811746i $$-0.301483\pi$$
0.584010 + 0.811746i $$0.301483\pi$$
$$198$$ 0 0
$$199$$ 6.96500 0.493736 0.246868 0.969049i $$-0.420599\pi$$
0.246868 + 0.969049i $$0.420599\pi$$
$$200$$ 4.98884 0.352764
$$201$$ −15.2739 −1.07734
$$202$$ −1.24102 −0.0873180
$$203$$ −16.5541 −1.16187
$$204$$ 20.1569 1.41127
$$205$$ −10.8472 −0.757602
$$206$$ −15.7225 −1.09544
$$207$$ 1.07392 0.0746427
$$208$$ −1.31180 −0.0909569
$$209$$ 0 0
$$210$$ −8.07211 −0.557029
$$211$$ −19.9531 −1.37363 −0.686814 0.726833i $$-0.740992\pi$$
−0.686814 + 0.726833i $$0.740992\pi$$
$$212$$ 19.8966 1.36650
$$213$$ 3.07211 0.210498
$$214$$ 5.13892 0.351289
$$215$$ 5.51468 0.376098
$$216$$ 4.98884 0.339447
$$217$$ 11.3363 0.769557
$$218$$ −16.4367 −1.11323
$$219$$ −8.65269 −0.584695
$$220$$ 0 0
$$221$$ −1.56567 −0.105318
$$222$$ 6.46950 0.434204
$$223$$ −20.1466 −1.34912 −0.674559 0.738221i $$-0.735666\pi$$
−0.674559 + 0.738221i $$0.735666\pi$$
$$224$$ 1.02116 0.0682289
$$225$$ 1.00000 0.0666667
$$226$$ −26.5091 −1.76336
$$227$$ 0.533937 0.0354386 0.0177193 0.999843i $$-0.494359\pi$$
0.0177193 + 0.999843i $$0.494359\pi$$
$$228$$ −30.0502 −1.99012
$$229$$ −22.1931 −1.46656 −0.733279 0.679928i $$-0.762011\pi$$
−0.733279 + 0.679928i $$0.762011\pi$$
$$230$$ −2.63743 −0.173907
$$231$$ 0 0
$$232$$ −25.1261 −1.64961
$$233$$ −4.56567 −0.299107 −0.149553 0.988754i $$-0.547784\pi$$
−0.149553 + 0.988754i $$0.547784\pi$$
$$234$$ −0.769020 −0.0502724
$$235$$ 11.9982 0.782676
$$236$$ −36.9439 −2.40485
$$237$$ 5.41446 0.351707
$$238$$ 40.3606 2.61619
$$239$$ −5.86053 −0.379086 −0.189543 0.981872i $$-0.560701\pi$$
−0.189543 + 0.981872i $$0.560701\pi$$
$$240$$ −4.18926 −0.270416
$$241$$ 9.96074 0.641628 0.320814 0.947142i $$-0.396044\pi$$
0.320814 + 0.947142i $$0.396044\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 37.0254 2.37031
$$245$$ −3.80333 −0.242986
$$246$$ 26.6395 1.69847
$$247$$ 2.33412 0.148517
$$248$$ 17.2065 1.09261
$$249$$ 16.2454 1.02951
$$250$$ −2.45589 −0.155324
$$251$$ −16.8788 −1.06538 −0.532690 0.846310i $$-0.678819\pi$$
−0.532690 + 0.846310i $$0.678819\pi$$
$$252$$ 13.2505 0.834704
$$253$$ 0 0
$$254$$ −41.7581 −2.62014
$$255$$ −5.00000 −0.313112
$$256$$ −32.2271 −2.01419
$$257$$ 11.1436 0.695117 0.347559 0.937658i $$-0.387011\pi$$
0.347559 + 0.937658i $$0.387011\pi$$
$$258$$ −13.5434 −0.843177
$$259$$ 8.65847 0.538011
$$260$$ 1.26236 0.0782882
$$261$$ −5.03647 −0.311750
$$262$$ −0.105728 −0.00653189
$$263$$ −26.8726 −1.65704 −0.828519 0.559961i $$-0.810816\pi$$
−0.828519 + 0.559961i $$0.810816\pi$$
$$264$$ 0 0
$$265$$ −4.93543 −0.303181
$$266$$ −60.1701 −3.68927
$$267$$ 1.62118 0.0992145
$$268$$ −61.5748 −3.76128
$$269$$ −10.0629 −0.613545 −0.306773 0.951783i $$-0.599249\pi$$
−0.306773 + 0.951783i $$0.599249\pi$$
$$270$$ −2.45589 −0.149461
$$271$$ 10.5441 0.640509 0.320255 0.947331i $$-0.396232\pi$$
0.320255 + 0.947331i $$0.396232\pi$$
$$272$$ 20.9463 1.27006
$$273$$ −1.02922 −0.0622912
$$274$$ 18.0816 1.09235
$$275$$ 0 0
$$276$$ 4.32938 0.260598
$$277$$ −17.9376 −1.07777 −0.538883 0.842381i $$-0.681153\pi$$
−0.538883 + 0.842381i $$0.681153\pi$$
$$278$$ 32.5143 1.95008
$$279$$ 3.44899 0.206486
$$280$$ −16.3975 −0.979939
$$281$$ 7.41103 0.442105 0.221052 0.975262i $$-0.429051\pi$$
0.221052 + 0.975262i $$0.429051\pi$$
$$282$$ −29.4662 −1.75469
$$283$$ 5.38684 0.320214 0.160107 0.987100i $$-0.448816\pi$$
0.160107 + 0.987100i $$0.448816\pi$$
$$284$$ 12.3848 0.734905
$$285$$ 7.45408 0.441541
$$286$$ 0 0
$$287$$ 35.6530 2.10453
$$288$$ 0.310680 0.0183070
$$289$$ 8.00000 0.470588
$$290$$ 12.3690 0.726332
$$291$$ 0.224082 0.0131359
$$292$$ −34.8823 −2.04133
$$293$$ −1.74006 −0.101655 −0.0508277 0.998707i $$-0.516186\pi$$
−0.0508277 + 0.998707i $$0.516186\pi$$
$$294$$ 9.34054 0.544752
$$295$$ 9.16409 0.533554
$$296$$ 13.1420 0.763864
$$297$$ 0 0
$$298$$ −14.4371 −0.836321
$$299$$ −0.336280 −0.0194476
$$300$$ 4.03138 0.232752
$$301$$ −18.1259 −1.04476
$$302$$ 18.7340 1.07802
$$303$$ −0.505326 −0.0290302
$$304$$ −31.2271 −1.79100
$$305$$ −9.18431 −0.525892
$$306$$ 12.2794 0.701969
$$307$$ 21.3566 1.21889 0.609444 0.792829i $$-0.291393\pi$$
0.609444 + 0.792829i $$0.291393\pi$$
$$308$$ 0 0
$$309$$ −6.40197 −0.364195
$$310$$ −8.47033 −0.481082
$$311$$ 32.8096 1.86046 0.930231 0.366975i $$-0.119607\pi$$
0.930231 + 0.366975i $$0.119607\pi$$
$$312$$ −1.56217 −0.0884406
$$313$$ −3.45852 −0.195487 −0.0977436 0.995212i $$-0.531163\pi$$
−0.0977436 + 0.995212i $$0.531163\pi$$
$$314$$ −24.8860 −1.40440
$$315$$ −3.28684 −0.185193
$$316$$ 21.8278 1.22791
$$317$$ −2.87566 −0.161513 −0.0807565 0.996734i $$-0.525734\pi$$
−0.0807565 + 0.996734i $$0.525734\pi$$
$$318$$ 12.1209 0.679704
$$319$$ 0 0
$$320$$ 7.61553 0.425721
$$321$$ 2.09249 0.116791
$$322$$ 8.66881 0.483094
$$323$$ −37.2704 −2.07378
$$324$$ 4.03138 0.223966
$$325$$ −0.313133 −0.0173695
$$326$$ −12.3508 −0.684048
$$327$$ −6.69278 −0.370112
$$328$$ 54.1150 2.98800
$$329$$ −39.4362 −2.17419
$$330$$ 0 0
$$331$$ −14.1221 −0.776219 −0.388109 0.921613i $$-0.626872\pi$$
−0.388109 + 0.921613i $$0.626872\pi$$
$$332$$ 65.4915 3.59431
$$333$$ 2.63428 0.144358
$$334$$ 14.2222 0.778202
$$335$$ 15.2739 0.834501
$$336$$ 13.7694 0.751185
$$337$$ 15.9490 0.868796 0.434398 0.900721i $$-0.356961\pi$$
0.434398 + 0.900721i $$0.356961\pi$$
$$338$$ −31.6857 −1.72348
$$339$$ −10.7941 −0.586256
$$340$$ −20.1569 −1.09316
$$341$$ 0 0
$$342$$ −18.3064 −0.989895
$$343$$ −10.5070 −0.567322
$$344$$ −27.5118 −1.48334
$$345$$ −1.07392 −0.0578180
$$346$$ 39.4543 2.12108
$$347$$ 29.6801 1.59331 0.796656 0.604433i $$-0.206600\pi$$
0.796656 + 0.604433i $$0.206600\pi$$
$$348$$ −20.3039 −1.08840
$$349$$ 31.6937 1.69653 0.848263 0.529574i $$-0.177648\pi$$
0.848263 + 0.529574i $$0.177648\pi$$
$$350$$ 8.07211 0.431472
$$351$$ −0.313133 −0.0167138
$$352$$ 0 0
$$353$$ 1.20189 0.0639703 0.0319852 0.999488i $$-0.489817\pi$$
0.0319852 + 0.999488i $$0.489817\pi$$
$$354$$ −22.5060 −1.19618
$$355$$ −3.07211 −0.163051
$$356$$ 6.53559 0.346385
$$357$$ 16.4342 0.869791
$$358$$ −20.3915 −1.07772
$$359$$ −11.6591 −0.615343 −0.307671 0.951493i $$-0.599550\pi$$
−0.307671 + 0.951493i $$0.599550\pi$$
$$360$$ −4.98884 −0.262935
$$361$$ 36.5633 1.92438
$$362$$ −15.8755 −0.834396
$$363$$ 0 0
$$364$$ −4.14918 −0.217476
$$365$$ 8.65269 0.452903
$$366$$ 22.5556 1.17900
$$367$$ 15.9860 0.834465 0.417232 0.908800i $$-0.363000\pi$$
0.417232 + 0.908800i $$0.363000\pi$$
$$368$$ 4.49894 0.234523
$$369$$ 10.8472 0.564683
$$370$$ −6.46950 −0.336333
$$371$$ 16.2220 0.842203
$$372$$ 13.9042 0.720898
$$373$$ 0.321975 0.0166712 0.00833561 0.999965i $$-0.497347\pi$$
0.00833561 + 0.999965i $$0.497347\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ −59.8570 −3.08689
$$377$$ 1.57709 0.0812241
$$378$$ 8.07211 0.415185
$$379$$ 11.4174 0.586475 0.293237 0.956040i $$-0.405267\pi$$
0.293237 + 0.956040i $$0.405267\pi$$
$$380$$ 30.0502 1.54154
$$381$$ −17.0033 −0.871104
$$382$$ 37.7452 1.93121
$$383$$ 28.3673 1.44950 0.724750 0.689012i $$-0.241955\pi$$
0.724750 + 0.689012i $$0.241955\pi$$
$$384$$ −19.3242 −0.986136
$$385$$ 0 0
$$386$$ 38.6846 1.96899
$$387$$ −5.51468 −0.280327
$$388$$ 0.903359 0.0458611
$$389$$ 15.1802 0.769666 0.384833 0.922986i $$-0.374259\pi$$
0.384833 + 0.922986i $$0.374259\pi$$
$$390$$ 0.769020 0.0389409
$$391$$ 5.36960 0.271553
$$392$$ 18.9742 0.958341
$$393$$ −0.0430508 −0.00217163
$$394$$ 40.2617 2.02835
$$395$$ −5.41446 −0.272431
$$396$$ 0 0
$$397$$ 5.22461 0.262216 0.131108 0.991368i $$-0.458147\pi$$
0.131108 + 0.991368i $$0.458147\pi$$
$$398$$ 17.1053 0.857409
$$399$$ −24.5004 −1.22655
$$400$$ 4.18926 0.209463
$$401$$ 14.0007 0.699160 0.349580 0.936907i $$-0.386324\pi$$
0.349580 + 0.936907i $$0.386324\pi$$
$$402$$ −37.5109 −1.87087
$$403$$ −1.07999 −0.0537983
$$404$$ −2.03716 −0.101352
$$405$$ −1.00000 −0.0496904
$$406$$ −40.6549 −2.01767
$$407$$ 0 0
$$408$$ 24.9442 1.23492
$$409$$ 33.3112 1.64713 0.823567 0.567218i $$-0.191980\pi$$
0.823567 + 0.567218i $$0.191980\pi$$
$$410$$ −26.6395 −1.31563
$$411$$ 7.36257 0.363169
$$412$$ −25.8088 −1.27151
$$413$$ −30.1209 −1.48215
$$414$$ 2.63743 0.129623
$$415$$ −16.2454 −0.797456
$$416$$ −0.0972843 −0.00476975
$$417$$ 13.2393 0.648334
$$418$$ 0 0
$$419$$ −5.28460 −0.258170 −0.129085 0.991634i $$-0.541204\pi$$
−0.129085 + 0.991634i $$0.541204\pi$$
$$420$$ −13.2505 −0.646559
$$421$$ −30.7810 −1.50017 −0.750087 0.661340i $$-0.769988\pi$$
−0.750087 + 0.661340i $$0.769988\pi$$
$$422$$ −49.0026 −2.38541
$$423$$ −11.9982 −0.583372
$$424$$ 24.6221 1.19575
$$425$$ 5.00000 0.242536
$$426$$ 7.54476 0.365545
$$427$$ 30.1874 1.46087
$$428$$ 8.43562 0.407751
$$429$$ 0 0
$$430$$ 13.5434 0.653122
$$431$$ 12.3506 0.594910 0.297455 0.954736i $$-0.403862\pi$$
0.297455 + 0.954736i $$0.403862\pi$$
$$432$$ 4.18926 0.201556
$$433$$ 1.41287 0.0678983 0.0339491 0.999424i $$-0.489192\pi$$
0.0339491 + 0.999424i $$0.489192\pi$$
$$434$$ 27.8406 1.33639
$$435$$ 5.03647 0.241480
$$436$$ −26.9811 −1.29216
$$437$$ −8.00509 −0.382935
$$438$$ −21.2500 −1.01537
$$439$$ −7.58532 −0.362028 −0.181014 0.983481i $$-0.557938\pi$$
−0.181014 + 0.983481i $$0.557938\pi$$
$$440$$ 0 0
$$441$$ 3.80333 0.181111
$$442$$ −3.84510 −0.182893
$$443$$ 11.0662 0.525771 0.262885 0.964827i $$-0.415326\pi$$
0.262885 + 0.964827i $$0.415326\pi$$
$$444$$ 10.6198 0.503993
$$445$$ −1.62118 −0.0768512
$$446$$ −49.4779 −2.34284
$$447$$ −5.87858 −0.278047
$$448$$ −25.0311 −1.18261
$$449$$ 6.32856 0.298663 0.149332 0.988787i $$-0.452288\pi$$
0.149332 + 0.988787i $$0.452288\pi$$
$$450$$ 2.45589 0.115772
$$451$$ 0 0
$$452$$ −43.5152 −2.04678
$$453$$ 7.62821 0.358405
$$454$$ 1.31129 0.0615418
$$455$$ 1.02922 0.0482506
$$456$$ −37.1872 −1.74145
$$457$$ −0.189579 −0.00886814 −0.00443407 0.999990i $$-0.501411\pi$$
−0.00443407 + 0.999990i $$0.501411\pi$$
$$458$$ −54.5036 −2.54679
$$459$$ 5.00000 0.233380
$$460$$ −4.32938 −0.201859
$$461$$ −26.6198 −1.23981 −0.619904 0.784678i $$-0.712828\pi$$
−0.619904 + 0.784678i $$0.712828\pi$$
$$462$$ 0 0
$$463$$ 20.9935 0.975652 0.487826 0.872941i $$-0.337790\pi$$
0.487826 + 0.872941i $$0.337790\pi$$
$$464$$ −21.0991 −0.979501
$$465$$ −3.44899 −0.159943
$$466$$ −11.2128 −0.519421
$$467$$ 7.89989 0.365563 0.182782 0.983154i $$-0.441490\pi$$
0.182782 + 0.983154i $$0.441490\pi$$
$$468$$ −1.26236 −0.0583526
$$469$$ −50.2028 −2.31815
$$470$$ 29.4662 1.35917
$$471$$ −10.1332 −0.466914
$$472$$ −45.7182 −2.10435
$$473$$ 0 0
$$474$$ 13.2973 0.610766
$$475$$ −7.45408 −0.342017
$$476$$ 66.2525 3.03668
$$477$$ 4.93543 0.225978
$$478$$ −14.3928 −0.658311
$$479$$ 39.9728 1.82640 0.913201 0.407509i $$-0.133602\pi$$
0.913201 + 0.407509i $$0.133602\pi$$
$$480$$ −0.310680 −0.0141805
$$481$$ −0.824882 −0.0376114
$$482$$ 24.4624 1.11423
$$483$$ 3.52981 0.160612
$$484$$ 0 0
$$485$$ −0.224082 −0.0101750
$$486$$ 2.45589 0.111401
$$487$$ −9.93556 −0.450223 −0.225112 0.974333i $$-0.572275\pi$$
−0.225112 + 0.974333i $$0.572275\pi$$
$$488$$ 45.8190 2.07413
$$489$$ −5.02906 −0.227422
$$490$$ −9.34054 −0.421963
$$491$$ 4.97349 0.224451 0.112225 0.993683i $$-0.464202\pi$$
0.112225 + 0.993683i $$0.464202\pi$$
$$492$$ 43.7292 1.97146
$$493$$ −25.1823 −1.13416
$$494$$ 5.73234 0.257910
$$495$$ 0 0
$$496$$ 14.4487 0.648767
$$497$$ 10.0975 0.452937
$$498$$ 39.8969 1.78782
$$499$$ 43.7757 1.95967 0.979834 0.199812i $$-0.0640332\pi$$
0.979834 + 0.199812i $$0.0640332\pi$$
$$500$$ −4.03138 −0.180289
$$501$$ 5.79105 0.258725
$$502$$ −41.4524 −1.85011
$$503$$ 6.28236 0.280117 0.140058 0.990143i $$-0.455271\pi$$
0.140058 + 0.990143i $$0.455271\pi$$
$$504$$ 16.3975 0.730404
$$505$$ 0.505326 0.0224867
$$506$$ 0 0
$$507$$ −12.9019 −0.572996
$$508$$ −68.5467 −3.04127
$$509$$ 24.8381 1.10093 0.550465 0.834858i $$-0.314450\pi$$
0.550465 + 0.834858i $$0.314450\pi$$
$$510$$ −12.2794 −0.543742
$$511$$ −28.4400 −1.25811
$$512$$ −40.4976 −1.78976
$$513$$ −7.45408 −0.329106
$$514$$ 27.3674 1.20712
$$515$$ 6.40197 0.282104
$$516$$ −22.2318 −0.978699
$$517$$ 0 0
$$518$$ 21.2642 0.934296
$$519$$ 16.0652 0.705185
$$520$$ 1.56217 0.0685058
$$521$$ −6.94869 −0.304428 −0.152214 0.988348i $$-0.548640\pi$$
−0.152214 + 0.988348i $$0.548640\pi$$
$$522$$ −12.3690 −0.541376
$$523$$ −26.7510 −1.16974 −0.584869 0.811128i $$-0.698854\pi$$
−0.584869 + 0.811128i $$0.698854\pi$$
$$524$$ −0.173554 −0.00758175
$$525$$ 3.28684 0.143450
$$526$$ −65.9962 −2.87757
$$527$$ 17.2449 0.751202
$$528$$ 0 0
$$529$$ −21.8467 −0.949856
$$530$$ −12.1209 −0.526496
$$531$$ −9.16409 −0.397688
$$532$$ −98.7703 −4.28224
$$533$$ −3.39662 −0.147124
$$534$$ 3.98143 0.172293
$$535$$ −2.09249 −0.0904662
$$536$$ −76.1989 −3.29129
$$537$$ −8.30309 −0.358305
$$538$$ −24.7133 −1.06547
$$539$$ 0 0
$$540$$ −4.03138 −0.173483
$$541$$ 14.5084 0.623767 0.311883 0.950120i $$-0.399040\pi$$
0.311883 + 0.950120i $$0.399040\pi$$
$$542$$ 25.8951 1.11229
$$543$$ −6.46425 −0.277408
$$544$$ 1.55340 0.0666015
$$545$$ 6.69278 0.286687
$$546$$ −2.52765 −0.108173
$$547$$ −26.7346 −1.14309 −0.571543 0.820572i $$-0.693655\pi$$
−0.571543 + 0.820572i $$0.693655\pi$$
$$548$$ 29.6813 1.26792
$$549$$ 9.18431 0.391977
$$550$$ 0 0
$$551$$ 37.5422 1.59935
$$552$$ 5.35762 0.228035
$$553$$ 17.7965 0.756784
$$554$$ −44.0527 −1.87162
$$555$$ −2.63428 −0.111819
$$556$$ 53.3728 2.26351
$$557$$ 17.2444 0.730670 0.365335 0.930876i $$-0.380954\pi$$
0.365335 + 0.930876i $$0.380954\pi$$
$$558$$ 8.47033 0.358578
$$559$$ 1.72683 0.0730371
$$560$$ −13.7694 −0.581865
$$561$$ 0 0
$$562$$ 18.2006 0.767748
$$563$$ −0.831914 −0.0350610 −0.0175305 0.999846i $$-0.505580\pi$$
−0.0175305 + 0.999846i $$0.505580\pi$$
$$564$$ −48.3693 −2.03671
$$565$$ 10.7941 0.454112
$$566$$ 13.2295 0.556076
$$567$$ 3.28684 0.138034
$$568$$ 15.3263 0.643076
$$569$$ 11.5961 0.486132 0.243066 0.970010i $$-0.421847\pi$$
0.243066 + 0.970010i $$0.421847\pi$$
$$570$$ 18.3064 0.766769
$$571$$ 21.8414 0.914034 0.457017 0.889458i $$-0.348918\pi$$
0.457017 + 0.889458i $$0.348918\pi$$
$$572$$ 0 0
$$573$$ 15.3693 0.642061
$$574$$ 87.5598 3.65468
$$575$$ 1.07392 0.0447856
$$576$$ −7.61553 −0.317314
$$577$$ 9.74587 0.405726 0.202863 0.979207i $$-0.434975\pi$$
0.202863 + 0.979207i $$0.434975\pi$$
$$578$$ 19.6471 0.817211
$$579$$ 15.7518 0.654622
$$580$$ 20.3039 0.843074
$$581$$ 53.3961 2.21524
$$582$$ 0.550320 0.0228115
$$583$$ 0 0
$$584$$ −43.1669 −1.78626
$$585$$ 0.313133 0.0129465
$$586$$ −4.27339 −0.176532
$$587$$ 22.9441 0.947005 0.473502 0.880793i $$-0.342990\pi$$
0.473502 + 0.880793i $$0.342990\pi$$
$$588$$ 15.3327 0.632308
$$589$$ −25.7090 −1.05932
$$590$$ 22.5060 0.926556
$$591$$ 16.3940 0.674357
$$592$$ 11.0357 0.453565
$$593$$ 28.7819 1.18193 0.590965 0.806697i $$-0.298747\pi$$
0.590965 + 0.806697i $$0.298747\pi$$
$$594$$ 0 0
$$595$$ −16.4342 −0.673737
$$596$$ −23.6988 −0.970741
$$597$$ 6.96500 0.285059
$$598$$ −0.825867 −0.0337722
$$599$$ −29.1951 −1.19288 −0.596440 0.802657i $$-0.703419\pi$$
−0.596440 + 0.802657i $$0.703419\pi$$
$$600$$ 4.98884 0.203668
$$601$$ −6.68087 −0.272518 −0.136259 0.990673i $$-0.543508\pi$$
−0.136259 + 0.990673i $$0.543508\pi$$
$$602$$ −44.5151 −1.81430
$$603$$ −15.2739 −0.622000
$$604$$ 30.7522 1.25129
$$605$$ 0 0
$$606$$ −1.24102 −0.0504131
$$607$$ −42.6108 −1.72952 −0.864759 0.502188i $$-0.832529\pi$$
−0.864759 + 0.502188i $$0.832529\pi$$
$$608$$ −2.31583 −0.0939194
$$609$$ −16.5541 −0.670805
$$610$$ −22.5556 −0.913251
$$611$$ 3.75703 0.151993
$$612$$ 20.1569 0.814794
$$613$$ −5.83156 −0.235535 −0.117767 0.993041i $$-0.537574\pi$$
−0.117767 + 0.993041i $$0.537574\pi$$
$$614$$ 52.4495 2.11669
$$615$$ −10.8472 −0.437401
$$616$$ 0 0
$$617$$ −33.6386 −1.35424 −0.677119 0.735874i $$-0.736772\pi$$
−0.677119 + 0.735874i $$0.736772\pi$$
$$618$$ −15.7225 −0.632452
$$619$$ 42.5616 1.71070 0.855349 0.518053i $$-0.173343\pi$$
0.855349 + 0.518053i $$0.173343\pi$$
$$620$$ −13.9042 −0.558405
$$621$$ 1.07392 0.0430950
$$622$$ 80.5766 3.23083
$$623$$ 5.32856 0.213484
$$624$$ −1.31180 −0.0525140
$$625$$ 1.00000 0.0400000
$$626$$ −8.49374 −0.339478
$$627$$ 0 0
$$628$$ −40.8509 −1.63013
$$629$$ 13.1714 0.525179
$$630$$ −8.07211 −0.321601
$$631$$ −8.89989 −0.354299 −0.177149 0.984184i $$-0.556688\pi$$
−0.177149 + 0.984184i $$0.556688\pi$$
$$632$$ 27.0119 1.07448
$$633$$ −19.9531 −0.793065
$$634$$ −7.06228 −0.280479
$$635$$ 17.0033 0.674755
$$636$$ 19.8966 0.788951
$$637$$ −1.19095 −0.0471871
$$638$$ 0 0
$$639$$ 3.07211 0.121531
$$640$$ 19.3242 0.763858
$$641$$ −6.16806 −0.243624 −0.121812 0.992553i $$-0.538870\pi$$
−0.121812 + 0.992553i $$0.538870\pi$$
$$642$$ 5.13892 0.202817
$$643$$ 4.35335 0.171680 0.0858398 0.996309i $$-0.472643\pi$$
0.0858398 + 0.996309i $$0.472643\pi$$
$$644$$ 14.2300 0.560740
$$645$$ 5.51468 0.217140
$$646$$ −91.5318 −3.60127
$$647$$ 13.4933 0.530478 0.265239 0.964183i $$-0.414549\pi$$
0.265239 + 0.964183i $$0.414549\pi$$
$$648$$ 4.98884 0.195980
$$649$$ 0 0
$$650$$ −0.769020 −0.0301635
$$651$$ 11.3363 0.444304
$$652$$ −20.2741 −0.793993
$$653$$ 27.4481 1.07413 0.537064 0.843541i $$-0.319533\pi$$
0.537064 + 0.843541i $$0.319533\pi$$
$$654$$ −16.4367 −0.642726
$$655$$ 0.0430508 0.00168213
$$656$$ 45.4418 1.77420
$$657$$ −8.65269 −0.337574
$$658$$ −96.8507 −3.77563
$$659$$ −18.7768 −0.731441 −0.365721 0.930725i $$-0.619177\pi$$
−0.365721 + 0.930725i $$0.619177\pi$$
$$660$$ 0 0
$$661$$ −21.6525 −0.842184 −0.421092 0.907018i $$-0.638353\pi$$
−0.421092 + 0.907018i $$0.638353\pi$$
$$662$$ −34.6822 −1.34796
$$663$$ −1.56567 −0.0608055
$$664$$ 81.0458 3.14519
$$665$$ 24.5004 0.950084
$$666$$ 6.46950 0.250688
$$667$$ −5.40877 −0.209428
$$668$$ 23.3459 0.903281
$$669$$ −20.1466 −0.778914
$$670$$ 37.5109 1.44917
$$671$$ 0 0
$$672$$ 1.02116 0.0393919
$$673$$ 23.3021 0.898232 0.449116 0.893474i $$-0.351739\pi$$
0.449116 + 0.893474i $$0.351739\pi$$
$$674$$ 39.1689 1.50873
$$675$$ 1.00000 0.0384900
$$676$$ −52.0127 −2.00049
$$677$$ 33.2808 1.27909 0.639543 0.768756i $$-0.279124\pi$$
0.639543 + 0.768756i $$0.279124\pi$$
$$678$$ −26.5091 −1.01808
$$679$$ 0.736522 0.0282651
$$680$$ −24.9442 −0.956566
$$681$$ 0.533937 0.0204605
$$682$$ 0 0
$$683$$ 16.9244 0.647593 0.323796 0.946127i $$-0.395041\pi$$
0.323796 + 0.946127i $$0.395041\pi$$
$$684$$ −30.0502 −1.14900
$$685$$ −7.36257 −0.281309
$$686$$ −25.8039 −0.985197
$$687$$ −22.1931 −0.846718
$$688$$ −23.1024 −0.880772
$$689$$ −1.54545 −0.0588769
$$690$$ −2.63743 −0.100405
$$691$$ −48.4335 −1.84250 −0.921249 0.388973i $$-0.872830\pi$$
−0.921249 + 0.388973i $$0.872830\pi$$
$$692$$ 64.7650 2.46200
$$693$$ 0 0
$$694$$ 72.8910 2.76690
$$695$$ −13.2393 −0.502197
$$696$$ −25.1261 −0.952403
$$697$$ 54.2360 2.05434
$$698$$ 77.8362 2.94614
$$699$$ −4.56567 −0.172689
$$700$$ 13.2505 0.500822
$$701$$ −45.4161 −1.71534 −0.857672 0.514197i $$-0.828090\pi$$
−0.857672 + 0.514197i $$0.828090\pi$$
$$702$$ −0.769020 −0.0290248
$$703$$ −19.6361 −0.740591
$$704$$ 0 0
$$705$$ 11.9982 0.451878
$$706$$ 2.95171 0.111089
$$707$$ −1.66093 −0.0624655
$$708$$ −36.9439 −1.38844
$$709$$ −1.76497 −0.0662848 −0.0331424 0.999451i $$-0.510551\pi$$
−0.0331424 + 0.999451i $$0.510551\pi$$
$$710$$ −7.54476 −0.283150
$$711$$ 5.41446 0.203058
$$712$$ 8.08780 0.303103
$$713$$ 3.70394 0.138714
$$714$$ 40.3606 1.51046
$$715$$ 0 0
$$716$$ −33.4729 −1.25094
$$717$$ −5.86053 −0.218865
$$718$$ −28.6334 −1.06859
$$719$$ −3.29998 −0.123069 −0.0615343 0.998105i $$-0.519599\pi$$
−0.0615343 + 0.998105i $$0.519599\pi$$
$$720$$ −4.18926 −0.156125
$$721$$ −21.0423 −0.783655
$$722$$ 89.7952 3.34183
$$723$$ 9.96074 0.370444
$$724$$ −26.0599 −0.968507
$$725$$ −5.03647 −0.187050
$$726$$ 0 0
$$727$$ 11.7838 0.437037 0.218519 0.975833i $$-0.429878\pi$$
0.218519 + 0.975833i $$0.429878\pi$$
$$728$$ −5.13461 −0.190301
$$729$$ 1.00000 0.0370370
$$730$$ 21.2500 0.786499
$$731$$ −27.5734 −1.01984
$$732$$ 37.0254 1.36850
$$733$$ 5.73108 0.211682 0.105841 0.994383i $$-0.466246\pi$$
0.105841 + 0.994383i $$0.466246\pi$$
$$734$$ 39.2599 1.44911
$$735$$ −3.80333 −0.140288
$$736$$ 0.333646 0.0122983
$$737$$ 0 0
$$738$$ 26.6395 0.980614
$$739$$ −21.0551 −0.774524 −0.387262 0.921970i $$-0.626579\pi$$
−0.387262 + 0.921970i $$0.626579\pi$$
$$740$$ −10.6198 −0.390391
$$741$$ 2.33412 0.0857461
$$742$$ 39.8393 1.46255
$$743$$ 13.1283 0.481630 0.240815 0.970571i $$-0.422585\pi$$
0.240815 + 0.970571i $$0.422585\pi$$
$$744$$ 17.2065 0.630819
$$745$$ 5.87858 0.215375
$$746$$ 0.790734 0.0289508
$$747$$ 16.2454 0.594389
$$748$$ 0 0
$$749$$ 6.87768 0.251305
$$750$$ −2.45589 −0.0896763
$$751$$ 25.6251 0.935073 0.467537 0.883974i $$-0.345142\pi$$
0.467537 + 0.883974i $$0.345142\pi$$
$$752$$ −50.2636 −1.83292
$$753$$ −16.8788 −0.615098
$$754$$ 3.87315 0.141052
$$755$$ −7.62821 −0.277619
$$756$$ 13.2505 0.481916
$$757$$ 31.1970 1.13387 0.566936 0.823762i $$-0.308129\pi$$
0.566936 + 0.823762i $$0.308129\pi$$
$$758$$ 28.0400 1.01846
$$759$$ 0 0
$$760$$ 37.1872 1.34892
$$761$$ 11.3761 0.412382 0.206191 0.978512i $$-0.433893\pi$$
0.206191 + 0.978512i $$0.433893\pi$$
$$762$$ −41.7581 −1.51274
$$763$$ −21.9981 −0.796385
$$764$$ 61.9594 2.24161
$$765$$ −5.00000 −0.180775
$$766$$ 69.6668 2.51717
$$767$$ 2.86958 0.103615
$$768$$ −32.2271 −1.16290
$$769$$ 10.3938 0.374811 0.187405 0.982283i $$-0.439992\pi$$
0.187405 + 0.982283i $$0.439992\pi$$
$$770$$ 0 0
$$771$$ 11.1436 0.401326
$$772$$ 63.5014 2.28547
$$773$$ 14.0348 0.504796 0.252398 0.967623i $$-0.418781\pi$$
0.252398 + 0.967623i $$0.418781\pi$$
$$774$$ −13.5434 −0.486808
$$775$$ 3.44899 0.123891
$$776$$ 1.11791 0.0401306
$$777$$ 8.65847 0.310621
$$778$$ 37.2808 1.33658
$$779$$ −80.8559 −2.89696
$$780$$ 1.26236 0.0451997
$$781$$ 0 0
$$782$$ 13.1871 0.471571
$$783$$ −5.03647 −0.179989
$$784$$ 15.9331 0.569041
$$785$$ 10.1332 0.361670
$$786$$ −0.105728 −0.00377119
$$787$$ −13.8176 −0.492545 −0.246273 0.969201i $$-0.579206\pi$$
−0.246273 + 0.969201i $$0.579206\pi$$
$$788$$ 66.0902 2.35437
$$789$$ −26.8726 −0.956691
$$790$$ −13.2973 −0.473097
$$791$$ −35.4785 −1.26147
$$792$$ 0 0
$$793$$ −2.87591 −0.102127
$$794$$ 12.8311 0.455357
$$795$$ −4.93543 −0.175042
$$796$$ 28.0786 0.995218
$$797$$ −5.38594 −0.190780 −0.0953898 0.995440i $$-0.530410\pi$$
−0.0953898 + 0.995440i $$0.530410\pi$$
$$798$$ −60.1701 −2.13000
$$799$$ −59.9910 −2.12233
$$800$$ 0.310680 0.0109842
$$801$$ 1.62118 0.0572815
$$802$$ 34.3840 1.21414
$$803$$ 0 0
$$804$$ −61.5748 −2.17157
$$805$$ −3.52981 −0.124409
$$806$$ −2.65234 −0.0934248
$$807$$ −10.0629 −0.354231
$$808$$ −2.52099 −0.0886880
$$809$$ 23.7748 0.835876 0.417938 0.908476i $$-0.362753\pi$$
0.417938 + 0.908476i $$0.362753\pi$$
$$810$$ −2.45589 −0.0862911
$$811$$ 9.46335 0.332303 0.166152 0.986100i $$-0.446866\pi$$
0.166152 + 0.986100i $$0.446866\pi$$
$$812$$ −66.7358 −2.34197
$$813$$ 10.5441 0.369798
$$814$$ 0 0
$$815$$ 5.02906 0.176160
$$816$$ 20.9463 0.733268
$$817$$ 41.1068 1.43815
$$818$$ 81.8086 2.86037
$$819$$ −1.02922 −0.0359639
$$820$$ −43.7292 −1.52709
$$821$$ −10.4189 −0.363622 −0.181811 0.983334i $$-0.558196\pi$$
−0.181811 + 0.983334i $$0.558196\pi$$
$$822$$ 18.0816 0.630670
$$823$$ 24.3540 0.848928 0.424464 0.905445i $$-0.360463\pi$$
0.424464 + 0.905445i $$0.360463\pi$$
$$824$$ −31.9384 −1.11263
$$825$$ 0 0
$$826$$ −73.9736 −2.57387
$$827$$ −22.0006 −0.765037 −0.382519 0.923948i $$-0.624943\pi$$
−0.382519 + 0.923948i $$0.624943\pi$$
$$828$$ 4.32938 0.150456
$$829$$ 3.03289 0.105337 0.0526683 0.998612i $$-0.483227\pi$$
0.0526683 + 0.998612i $$0.483227\pi$$
$$830$$ −39.8969 −1.38484
$$831$$ −17.9376 −0.622248
$$832$$ 2.38468 0.0826738
$$833$$ 19.0166 0.658888
$$834$$ 32.5143 1.12588
$$835$$ −5.79105 −0.200408
$$836$$ 0 0
$$837$$ 3.44899 0.119214
$$838$$ −12.9784 −0.448331
$$839$$ −48.5383 −1.67573 −0.837864 0.545879i $$-0.816196\pi$$
−0.837864 + 0.545879i $$0.816196\pi$$
$$840$$ −16.3975 −0.565768
$$841$$ −3.63399 −0.125310
$$842$$ −75.5946 −2.60516
$$843$$ 7.41103 0.255249
$$844$$ −80.4386 −2.76881
$$845$$ 12.9019 0.443840
$$846$$ −29.4662 −1.01307
$$847$$ 0 0
$$848$$ 20.6758 0.710011
$$849$$ 5.38684 0.184876
$$850$$ 12.2794 0.421181
$$851$$ 2.82901 0.0969773
$$852$$ 12.3848 0.424298
$$853$$ −11.7632 −0.402766 −0.201383 0.979513i $$-0.564544\pi$$
−0.201383 + 0.979513i $$0.564544\pi$$
$$854$$ 74.1368 2.53691
$$855$$ 7.45408 0.254924
$$856$$ 10.4391 0.356801
$$857$$ 1.61311 0.0551026 0.0275513 0.999620i $$-0.491229\pi$$
0.0275513 + 0.999620i $$0.491229\pi$$
$$858$$ 0 0
$$859$$ 47.3263 1.61475 0.807376 0.590038i $$-0.200887\pi$$
0.807376 + 0.590038i $$0.200887\pi$$
$$860$$ 22.2318 0.758097
$$861$$ 35.6530 1.21505
$$862$$ 30.3318 1.03310
$$863$$ −9.97233 −0.339462 −0.169731 0.985490i $$-0.554290\pi$$
−0.169731 + 0.985490i $$0.554290\pi$$
$$864$$ 0.310680 0.0105695
$$865$$ −16.0652 −0.546234
$$866$$ 3.46985 0.117910
$$867$$ 8.00000 0.271694
$$868$$ 45.7009 1.55119
$$869$$ 0 0
$$870$$ 12.3690 0.419348
$$871$$ 4.78276 0.162058
$$872$$ −33.3892 −1.13070
$$873$$ 0.224082 0.00758402
$$874$$ −19.6596 −0.664996
$$875$$ −3.28684 −0.111116
$$876$$ −34.8823 −1.17856
$$877$$ 26.0057 0.878151 0.439076 0.898450i $$-0.355306\pi$$
0.439076 + 0.898450i $$0.355306\pi$$
$$878$$ −18.6287 −0.628688
$$879$$ −1.74006 −0.0586907
$$880$$ 0 0
$$881$$ 10.4081 0.350657 0.175329 0.984510i $$-0.443901\pi$$
0.175329 + 0.984510i $$0.443901\pi$$
$$882$$ 9.34054 0.314512
$$883$$ −53.7283 −1.80810 −0.904051 0.427424i $$-0.859421\pi$$
−0.904051 + 0.427424i $$0.859421\pi$$
$$884$$ −6.31180 −0.212289
$$885$$ 9.16409 0.308048
$$886$$ 27.1773 0.913040
$$887$$ −40.6246 −1.36404 −0.682021 0.731333i $$-0.738899\pi$$
−0.682021 + 0.731333i $$0.738899\pi$$
$$888$$ 13.1420 0.441017
$$889$$ −55.8871 −1.87439
$$890$$ −3.98143 −0.133458
$$891$$ 0 0
$$892$$ −81.2188 −2.71941
$$893$$ 89.4354 2.99284
$$894$$ −14.4371 −0.482850
$$895$$ 8.30309 0.277542
$$896$$ −63.5157 −2.12191
$$897$$ −0.336280 −0.0112281
$$898$$ 15.5422 0.518651
$$899$$ −17.3707 −0.579346
$$900$$ 4.03138 0.134379
$$901$$ 24.6772 0.822115
$$902$$ 0 0
$$903$$ −18.1259 −0.603191
$$904$$ −53.8501 −1.79103
$$905$$ 6.46425 0.214879
$$906$$ 18.7340 0.622396
$$907$$ −19.4070 −0.644398 −0.322199 0.946672i $$-0.604422\pi$$
−0.322199 + 0.946672i $$0.604422\pi$$
$$908$$ 2.15250 0.0714333
$$909$$ −0.505326 −0.0167606
$$910$$ 2.52765 0.0837907
$$911$$ 10.7208 0.355195 0.177597 0.984103i $$-0.443168\pi$$
0.177597 + 0.984103i $$0.443168\pi$$
$$912$$ −31.2271 −1.03403
$$913$$ 0 0
$$914$$ −0.465585 −0.0154002
$$915$$ −9.18431 −0.303624
$$916$$ −89.4686 −2.95613
$$917$$ −0.141501 −0.00467278
$$918$$ 12.2794 0.405282
$$919$$ −23.1310 −0.763021 −0.381511 0.924364i $$-0.624596\pi$$
−0.381511 + 0.924364i $$0.624596\pi$$
$$920$$ −5.35762 −0.176635
$$921$$ 21.3566 0.703725
$$922$$ −65.3752 −2.15302
$$923$$ −0.961981 −0.0316640
$$924$$ 0 0
$$925$$ 2.63428 0.0866147
$$926$$ 51.5577 1.69429
$$927$$ −6.40197 −0.210268
$$928$$ −1.56473 −0.0513648
$$929$$ 26.2273 0.860489 0.430245 0.902712i $$-0.358427\pi$$
0.430245 + 0.902712i $$0.358427\pi$$
$$930$$ −8.47033 −0.277753
$$931$$ −28.3503 −0.929144
$$932$$ −18.4059 −0.602907
$$933$$ 32.8096 1.07414
$$934$$ 19.4012 0.634828
$$935$$ 0 0
$$936$$ −1.56217 −0.0510612
$$937$$ 23.4011 0.764480 0.382240 0.924063i $$-0.375153\pi$$
0.382240 + 0.924063i $$0.375153\pi$$
$$938$$ −123.292 −4.02564
$$939$$ −3.45852 −0.112865
$$940$$ 48.3693 1.57763
$$941$$ −10.9687 −0.357570 −0.178785 0.983888i $$-0.557217\pi$$
−0.178785 + 0.983888i $$0.557217\pi$$
$$942$$ −24.8860 −0.810831
$$943$$ 11.6490 0.379345
$$944$$ −38.3908 −1.24951
$$945$$ −3.28684 −0.106921
$$946$$ 0 0
$$947$$ −13.3652 −0.434310 −0.217155 0.976137i $$-0.569678\pi$$
−0.217155 + 0.976137i $$0.569678\pi$$
$$948$$ 21.8278 0.708933
$$949$$ 2.70945 0.0879524
$$950$$ −18.3064 −0.593937
$$951$$ −2.87566 −0.0932495
$$952$$ 81.9876 2.65723
$$953$$ −21.8242 −0.706956 −0.353478 0.935443i $$-0.615001\pi$$
−0.353478 + 0.935443i $$0.615001\pi$$
$$954$$ 12.1209 0.392427
$$955$$ −15.3693 −0.497339
$$956$$ −23.6260 −0.764120
$$957$$ 0 0
$$958$$ 98.1686 3.17168
$$959$$ 24.1996 0.781446
$$960$$ 7.61553 0.245790
$$961$$ −19.1045 −0.616273
$$962$$ −2.02582 −0.0653150
$$963$$ 2.09249 0.0674295
$$964$$ 40.1555 1.29332
$$965$$ −15.7518 −0.507068
$$966$$ 8.66881 0.278914
$$967$$ 16.6600 0.535750 0.267875 0.963454i $$-0.413679\pi$$
0.267875 + 0.963454i $$0.413679\pi$$
$$968$$ 0 0
$$969$$ −37.2704 −1.19730
$$970$$ −0.550320 −0.0176697
$$971$$ 11.1032 0.356320 0.178160 0.984002i $$-0.442986\pi$$
0.178160 + 0.984002i $$0.442986\pi$$
$$972$$ 4.03138 0.129307
$$973$$ 43.5156 1.39505
$$974$$ −24.4006 −0.781846
$$975$$ −0.313133 −0.0100283
$$976$$ 38.4755 1.23157
$$977$$ 18.8144 0.601926 0.300963 0.953636i $$-0.402692\pi$$
0.300963 + 0.953636i $$0.402692\pi$$
$$978$$ −12.3508 −0.394935
$$979$$ 0 0
$$980$$ −15.3327 −0.489784
$$981$$ −6.69278 −0.213684
$$982$$ 12.2143 0.389775
$$983$$ −1.37848 −0.0439667 −0.0219833 0.999758i $$-0.506998\pi$$
−0.0219833 + 0.999758i $$0.506998\pi$$
$$984$$ 54.1150 1.72512
$$985$$ −16.3940 −0.522355
$$986$$ −61.8450 −1.96955
$$987$$ −39.4362 −1.25527
$$988$$ 9.40973 0.299363
$$989$$ −5.92233 −0.188319
$$990$$ 0 0
$$991$$ −46.3186 −1.47136 −0.735680 0.677329i $$-0.763137\pi$$
−0.735680 + 0.677329i $$0.763137\pi$$
$$992$$ 1.07153 0.0340212
$$993$$ −14.1221 −0.448150
$$994$$ 24.7984 0.786558
$$995$$ −6.96500 −0.220805
$$996$$ 65.4915 2.07518
$$997$$ 14.5470 0.460709 0.230355 0.973107i $$-0.426011\pi$$
0.230355 + 0.973107i $$0.426011\pi$$
$$998$$ 107.508 3.40311
$$999$$ 2.63428 0.0833450
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1815.2.a.x.1.3 4
3.2 odd 2 5445.2.a.be.1.2 4
5.4 even 2 9075.2.a.cl.1.2 4
11.3 even 5 165.2.m.a.31.2 yes 8
11.4 even 5 165.2.m.a.16.2 8
11.10 odd 2 1815.2.a.o.1.2 4
33.14 odd 10 495.2.n.d.361.1 8
33.26 odd 10 495.2.n.d.181.1 8
33.32 even 2 5445.2.a.bv.1.3 4
55.3 odd 20 825.2.bx.h.724.1 16
55.4 even 10 825.2.n.k.676.1 8
55.14 even 10 825.2.n.k.526.1 8
55.37 odd 20 825.2.bx.h.49.1 16
55.47 odd 20 825.2.bx.h.724.4 16
55.48 odd 20 825.2.bx.h.49.4 16
55.54 odd 2 9075.2.a.dj.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.16.2 8 11.4 even 5
165.2.m.a.31.2 yes 8 11.3 even 5
495.2.n.d.181.1 8 33.26 odd 10
495.2.n.d.361.1 8 33.14 odd 10
825.2.n.k.526.1 8 55.14 even 10
825.2.n.k.676.1 8 55.4 even 10
825.2.bx.h.49.1 16 55.37 odd 20
825.2.bx.h.49.4 16 55.48 odd 20
825.2.bx.h.724.1 16 55.3 odd 20
825.2.bx.h.724.4 16 55.47 odd 20
1815.2.a.o.1.2 4 11.10 odd 2
1815.2.a.x.1.3 4 1.1 even 1 trivial
5445.2.a.be.1.2 4 3.2 odd 2
5445.2.a.bv.1.3 4 33.32 even 2
9075.2.a.cl.1.2 4 5.4 even 2
9075.2.a.dj.1.3 4 55.54 odd 2