# Properties

 Label 1815.2.a.o.1.2 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3 x^{2} + x + 1$$ 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 1.2 Root $$0.737640$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

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

## 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 $$-0.834779\pi$$
−0.868287 + 0.496062i $$0.834779\pi$$
$$3$$ 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 $$-0.713336\pi$$
−0.621155 + 0.783688i $$0.713336\pi$$
$$8$$ −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 $$-0.486173\pi$$
0.0434238 + 0.999057i $$0.486173\pi$$
$$14$$ 8.07211 2.15736
$$15$$ −1.00000 −0.258199
$$16$$ 4.18926 1.04732
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ −2.45589 −0.578858
$$19$$ 7.45408 1.71008 0.855041 0.518560i $$-0.173532\pi$$
0.855041 + 0.518560i $$0.173532\pi$$
$$20$$ −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 $$-0.345110\pi$$
0.467624 + 0.883927i $$0.345110\pi$$
$$30$$ 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 $$-0.821608\pi$$
−0.847024 + 0.531554i $$0.821608\pi$$
$$42$$ 8.07211 1.24555
$$43$$ 5.51468 0.840980 0.420490 0.907297i $$-0.361858\pi$$
0.420490 + 0.907297i $$0.361858\pi$$
$$44$$ 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 $$-0.700071\pi$$
−0.587965 + 0.808886i $$0.700071\pi$$
$$62$$ −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 $$-0.330991\pi$$
0.506361 + 0.862322i $$0.330991\pi$$
$$74$$ −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 $$-0.598519\pi$$
−0.304587 + 0.952484i $$0.598519\pi$$
$$80$$ −4.18926 −0.468374
$$81$$ 1.00000 0.111111
$$82$$ 26.6395 2.94184
$$83$$ −16.2454 −1.78317 −0.891583 0.452857i $$-0.850405\pi$$
−0.891583 + 0.452857i $$0.850405\pi$$
$$84$$ −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 $$-0.491997\pi$$
0.0251409 + 0.999684i $$0.491997\pi$$
$$102$$ 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 $$-0.532250\pi$$
−0.101144 + 0.994872i $$0.532250\pi$$
$$108$$ 4.03138 0.387920
$$109$$ 6.69278 0.641052 0.320526 0.947240i $$-0.396140\pi$$
0.320526 + 0.947240i $$0.396140\pi$$
$$110$$ 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 $$-0.227929\pi$$
0.754398 + 0.656417i $$0.227929\pi$$
$$128$$ 19.3242 1.70804
$$129$$ 5.51468 0.485540
$$130$$ 0.769020 0.0674475
$$131$$ 0.0430508 0.00376136 0.00188068 0.999998i $$-0.499401\pi$$
0.00188068 + 0.999998i $$0.499401\pi$$
$$132$$ 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 $$-0.689765\pi$$
−0.561473 + 0.827495i $$0.689765\pi$$
$$140$$ 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 $$-0.422591\pi$$
0.240796 + 0.970576i $$0.422591\pi$$
$$150$$ −2.45589 −0.200522
$$151$$ −7.62821 −0.620775 −0.310387 0.950610i $$-0.600459\pi$$
−0.310387 + 0.950610i $$0.600459\pi$$
$$152$$ −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 $$-0.571932\pi$$
−0.224062 + 0.974575i $$0.571932\pi$$
$$168$$ 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 $$-0.709115\pi$$
−0.610708 + 0.791856i $$0.709115\pi$$
$$174$$ −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 $$-0.691865\pi$$
−0.566919 + 0.823773i $$0.691865\pi$$
$$194$$ −0.550320 −0.0395107
$$195$$ −0.313133 −0.0224239
$$196$$ 15.3327 1.09519
$$197$$ −16.3940 −1.16802 −0.584010 0.811746i $$-0.698517\pi$$
−0.584010 + 0.811746i $$0.698517\pi$$
$$198$$ 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 $$-0.259008\pi$$
0.686814 + 0.726833i $$0.259008\pi$$
$$212$$ 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 $$-0.505641\pi$$
−0.0177193 + 0.999843i $$0.505641\pi$$
$$228$$ 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 $$-0.452216\pi$$
0.149553 + 0.988754i $$0.452216\pi$$
$$234$$ −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 $$-0.439299\pi$$
0.189543 + 0.981872i $$0.439299\pi$$
$$240$$ −4.18926 −0.270416
$$241$$ −9.96074 −0.641628 −0.320814 0.947142i $$-0.603956\pi$$
−0.320814 + 0.947142i $$0.603956\pi$$
$$242$$ 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 $$-0.189184\pi$$
0.828519 + 0.559961i $$0.189184\pi$$
$$264$$ 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 $$-0.603768\pi$$
−0.320255 + 0.947331i $$0.603768\pi$$
$$272$$ −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 $$-0.318847\pi$$
0.538883 + 0.842381i $$0.318847\pi$$
$$278$$ 32.5143 1.95008
$$279$$ 3.44899 0.206486
$$280$$ −16.3975 −0.979939
$$281$$ −7.41103 −0.442105 −0.221052 0.975262i $$-0.570949\pi$$
−0.221052 + 0.975262i $$0.570949\pi$$
$$282$$ 29.4662 1.75469
$$283$$ −5.38684 −0.320214 −0.160107 0.987100i $$-0.551184\pi$$
−0.160107 + 0.987100i $$0.551184\pi$$
$$284$$ 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 $$-0.483814\pi$$
0.0508277 + 0.998707i $$0.483814\pi$$
$$294$$ −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 $$-0.708607\pi$$
−0.609444 + 0.792829i $$0.708607\pi$$
$$308$$ 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 $$-0.643039\pi$$
−0.434398 + 0.900721i $$0.643039\pi$$
$$338$$ 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 $$-0.793400\pi$$
−0.796656 + 0.604433i $$0.793400\pi$$
$$348$$ 20.3039 1.08840
$$349$$ −31.6937 −1.69653 −0.848263 0.529574i $$-0.822352\pi$$
−0.848263 + 0.529574i $$0.822352\pi$$
$$350$$ 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 $$-0.400450\pi$$
0.307671 + 0.951493i $$0.400450\pi$$
$$360$$ 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 $$-0.502653\pi$$
−0.00833561 + 0.999965i $$0.502653\pi$$
$$374$$ 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 $$-0.808020\pi$$
−0.823567 + 0.567218i $$0.808020\pi$$
$$410$$ −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 $$-0.596138\pi$$
−0.297455 + 0.954736i $$0.596138\pi$$
$$432$$ 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 $$-0.442062\pi$$
0.181014 + 0.983481i $$0.442062\pi$$
$$440$$ 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 $$-0.498589\pi$$
0.00443407 + 0.999990i $$0.498589\pi$$
$$458$$ 54.5036 2.54679
$$459$$ −5.00000 −0.233380
$$460$$ −4.32938 −0.201859
$$461$$ 26.6198 1.23981 0.619904 0.784678i $$-0.287172\pi$$
0.619904 + 0.784678i $$0.287172\pi$$
$$462$$ 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 $$-0.866398\pi$$
−0.913201 + 0.407509i $$0.866398\pi$$
$$480$$ 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 $$-0.535798\pi$$
−0.112225 + 0.993683i $$0.535798\pi$$
$$492$$ −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 $$-0.544729\pi$$
−0.140058 + 0.990143i $$0.544729\pi$$
$$504$$ 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 $$-0.301146\pi$$
0.584869 + 0.811128i $$0.301146\pi$$
$$524$$ 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 $$-0.600960\pi$$
−0.311883 + 0.950120i $$0.600960\pi$$
$$542$$ 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 $$-0.306345\pi$$
0.571543 + 0.820572i $$0.306345\pi$$
$$548$$ 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 $$-0.619046\pi$$
−0.365335 + 0.930876i $$0.619046\pi$$
$$558$$ −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 $$-0.494420\pi$$
0.0175305 + 0.999846i $$0.494420\pi$$
$$564$$ −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 $$-0.578153\pi$$
−0.243066 + 0.970010i $$0.578153\pi$$
$$570$$ 18.3064 0.766769
$$571$$ −21.8414 −0.914034 −0.457017 0.889458i $$-0.651082\pi$$
−0.457017 + 0.889458i $$0.651082\pi$$
$$572$$ 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 $$-0.701253\pi$$
−0.590965 + 0.806697i $$0.701253\pi$$
$$594$$ 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 $$-0.456492\pi$$
0.136259 + 0.990673i $$0.456492\pi$$
$$602$$ 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 $$-0.167471\pi$$
0.864759 + 0.502188i $$0.167471\pi$$
$$608$$ −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 $$-0.462426\pi$$
0.117767 + 0.993041i $$0.462426\pi$$
$$614$$ 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 $$-0.380823\pi$$
0.365721 + 0.930725i $$0.380823\pi$$
$$660$$ 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 $$-0.648261\pi$$
−0.449116 + 0.893474i $$0.648261\pi$$
$$674$$ 39.1689 1.50873
$$675$$ 1.00000 0.0384900
$$676$$ −52.0127 −2.00049
$$677$$ −33.2808 −1.27909 −0.639543 0.768756i $$-0.720876\pi$$
−0.639543 + 0.768756i $$0.720876\pi$$
$$678$$ 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 $$-0.171910\pi$$
0.857672 + 0.514197i $$0.171910\pi$$
$$702$$ −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 $$-0.533754\pi$$
−0.105841 + 0.994383i $$0.533754\pi$$
$$734$$ −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 $$-0.373421\pi$$
0.387262 + 0.921970i $$0.373421\pi$$
$$740$$ −10.6198 −0.390391
$$741$$ 2.33412 0.0857461
$$742$$ 39.8393 1.46255
$$743$$ −13.1283 −0.481630 −0.240815 0.970571i $$-0.577415\pi$$
−0.240815 + 0.970571i $$0.577415\pi$$
$$744$$ −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 $$-0.566107\pi$$
−0.206191 + 0.978512i $$0.566107\pi$$
$$762$$ −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 $$-0.560008\pi$$
−0.187405 + 0.982283i $$0.560008\pi$$
$$770$$ 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 $$-0.420794\pi$$
0.246273 + 0.969201i $$0.420794\pi$$
$$788$$ −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 $$-0.637247\pi$$
−0.417938 + 0.908476i $$0.637247\pi$$
$$810$$ 2.45589 0.0862911
$$811$$ −9.46335 −0.332303 −0.166152 0.986100i $$-0.553134\pi$$
−0.166152 + 0.986100i $$0.553134\pi$$
$$812$$ −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 $$-0.441804\pi$$
0.181811 + 0.983334i $$0.441804\pi$$
$$822$$ −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 $$-0.375057\pi$$
0.382519 + 0.923948i $$0.375057\pi$$
$$828$$ 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 $$-0.435456\pi$$
0.201383 + 0.979513i $$0.435456\pi$$
$$854$$ −74.1368 −2.53691
$$855$$ −7.45408 −0.254924
$$856$$ 10.4391 0.356801
$$857$$ −1.61311 −0.0551026 −0.0275513 0.999620i $$-0.508771\pi$$
−0.0275513 + 0.999620i $$0.508771\pi$$
$$858$$ 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 $$-0.644694\pi$$
−0.439076 + 0.898450i $$0.644694\pi$$
$$878$$ −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 $$-0.261101\pi$$
0.682021 + 0.731333i $$0.261101\pi$$
$$888$$ −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 $$-0.375404\pi$$
0.381511 + 0.924364i $$0.375404\pi$$
$$920$$ 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 $$-0.624847\pi$$
−0.382240 + 0.924063i $$0.624847\pi$$
$$938$$ −123.292 −4.02564
$$939$$ −3.45852 −0.112865
$$940$$ 48.3693 1.57763
$$941$$ 10.9687 0.357570 0.178785 0.983888i $$-0.442783\pi$$
0.178785 + 0.983888i $$0.442783\pi$$
$$942$$ 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 $$-0.384999\pi$$
0.353478 + 0.935443i $$0.384999\pi$$
$$954$$ −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 $$-0.586321\pi$$
−0.267875 + 0.963454i $$0.586321\pi$$
$$968$$ 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 $$-0.573989\pi$$
−0.230355 + 0.973107i $$0.573989\pi$$
$$998$$ −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.o.1.2 4
3.2 odd 2 5445.2.a.bv.1.3 4
5.4 even 2 9075.2.a.dj.1.3 4
11.7 odd 10 165.2.m.a.16.2 8
11.8 odd 10 165.2.m.a.31.2 yes 8
11.10 odd 2 1815.2.a.x.1.3 4
33.8 even 10 495.2.n.d.361.1 8
33.29 even 10 495.2.n.d.181.1 8
33.32 even 2 5445.2.a.be.1.2 4
55.7 even 20 825.2.bx.h.49.1 16
55.8 even 20 825.2.bx.h.724.1 16
55.18 even 20 825.2.bx.h.49.4 16
55.19 odd 10 825.2.n.k.526.1 8
55.29 odd 10 825.2.n.k.676.1 8
55.52 even 20 825.2.bx.h.724.4 16
55.54 odd 2 9075.2.a.cl.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.16.2 8 11.7 odd 10
165.2.m.a.31.2 yes 8 11.8 odd 10
495.2.n.d.181.1 8 33.29 even 10
495.2.n.d.361.1 8 33.8 even 10
825.2.n.k.526.1 8 55.19 odd 10
825.2.n.k.676.1 8 55.29 odd 10
825.2.bx.h.49.1 16 55.7 even 20
825.2.bx.h.49.4 16 55.18 even 20
825.2.bx.h.724.1 16 55.8 even 20
825.2.bx.h.724.4 16 55.52 even 20
1815.2.a.o.1.2 4 1.1 even 1 trivial
1815.2.a.x.1.3 4 11.10 odd 2
5445.2.a.be.1.2 4 33.32 even 2
5445.2.a.bv.1.3 4 3.2 odd 2
9075.2.a.cl.1.2 4 55.54 odd 2
9075.2.a.dj.1.3 4 5.4 even 2