# Properties

 Label 289.4.a.g.1.9 Level $289$ Weight $4$ Character 289.1 Self dual yes Analytic conductor $17.052$ Analytic rank $1$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(1,289)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("289.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$17.0515519917$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156$$ x^12 - 4*x^11 - 58*x^10 + 204*x^9 + 1191*x^8 - 3456*x^7 - 10364*x^6 + 21448*x^5 + 38476*x^4 - 32336*x^3 - 57024*x^2 - 15776*x + 1156 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.9 Root $$1.83828$$ of defining polynomial Character $$\chi$$ $$=$$ 289.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.68604 q^{2} -4.33137 q^{3} -0.785167 q^{4} +2.08666 q^{5} -11.6343 q^{6} +24.9985 q^{7} -23.5973 q^{8} -8.23920 q^{9} +O(q^{10})$$ $$q+2.68604 q^{2} -4.33137 q^{3} -0.785167 q^{4} +2.08666 q^{5} -11.6343 q^{6} +24.9985 q^{7} -23.5973 q^{8} -8.23920 q^{9} +5.60485 q^{10} -3.82158 q^{11} +3.40085 q^{12} +17.6726 q^{13} +67.1470 q^{14} -9.03809 q^{15} -57.1022 q^{16} -22.1309 q^{18} -160.915 q^{19} -1.63837 q^{20} -108.278 q^{21} -10.2649 q^{22} -99.9676 q^{23} +102.209 q^{24} -120.646 q^{25} +47.4693 q^{26} +152.634 q^{27} -19.6280 q^{28} -200.583 q^{29} -24.2767 q^{30} +76.5382 q^{31} +35.3998 q^{32} +16.5527 q^{33} +52.1632 q^{35} +6.46915 q^{36} -244.759 q^{37} -432.224 q^{38} -76.5465 q^{39} -49.2395 q^{40} -54.1132 q^{41} -290.839 q^{42} +142.087 q^{43} +3.00058 q^{44} -17.1924 q^{45} -268.517 q^{46} -468.451 q^{47} +247.331 q^{48} +281.924 q^{49} -324.060 q^{50} -13.8759 q^{52} -96.5673 q^{53} +409.982 q^{54} -7.97432 q^{55} -589.898 q^{56} +696.981 q^{57} -538.776 q^{58} +364.484 q^{59} +7.09641 q^{60} +707.872 q^{61} +205.585 q^{62} -205.967 q^{63} +551.903 q^{64} +36.8766 q^{65} +44.4612 q^{66} -304.454 q^{67} +432.997 q^{69} +140.113 q^{70} -470.003 q^{71} +194.423 q^{72} -142.056 q^{73} -657.432 q^{74} +522.562 q^{75} +126.345 q^{76} -95.5336 q^{77} -205.607 q^{78} +717.865 q^{79} -119.153 q^{80} -438.657 q^{81} -145.350 q^{82} -367.639 q^{83} +85.0162 q^{84} +381.653 q^{86} +868.802 q^{87} +90.1791 q^{88} +1042.28 q^{89} -46.1795 q^{90} +441.788 q^{91} +78.4913 q^{92} -331.516 q^{93} -1258.28 q^{94} -335.773 q^{95} -153.330 q^{96} +903.534 q^{97} +757.260 q^{98} +31.4867 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10})$$ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 $$12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100})$$ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.68604 0.949660 0.474830 0.880078i $$-0.342510\pi$$
0.474830 + 0.880078i $$0.342510\pi$$
$$3$$ −4.33137 −0.833573 −0.416787 0.909004i $$-0.636844\pi$$
−0.416787 + 0.909004i $$0.636844\pi$$
$$4$$ −0.785167 −0.0981459
$$5$$ 2.08666 0.186636 0.0933181 0.995636i $$-0.470253\pi$$
0.0933181 + 0.995636i $$0.470253\pi$$
$$6$$ −11.6343 −0.791611
$$7$$ 24.9985 1.34979 0.674895 0.737913i $$-0.264189\pi$$
0.674895 + 0.737913i $$0.264189\pi$$
$$8$$ −23.5973 −1.04287
$$9$$ −8.23920 −0.305156
$$10$$ 5.60485 0.177241
$$11$$ −3.82158 −0.104750 −0.0523749 0.998627i $$-0.516679\pi$$
−0.0523749 + 0.998627i $$0.516679\pi$$
$$12$$ 3.40085 0.0818118
$$13$$ 17.6726 0.377038 0.188519 0.982070i $$-0.439631\pi$$
0.188519 + 0.982070i $$0.439631\pi$$
$$14$$ 67.1470 1.28184
$$15$$ −9.03809 −0.155575
$$16$$ −57.1022 −0.892221
$$17$$ 0 0
$$18$$ −22.1309 −0.289794
$$19$$ −160.915 −1.94297 −0.971483 0.237111i $$-0.923799\pi$$
−0.971483 + 0.237111i $$0.923799\pi$$
$$20$$ −1.63837 −0.0183176
$$21$$ −108.278 −1.12515
$$22$$ −10.2649 −0.0994768
$$23$$ −99.9676 −0.906291 −0.453145 0.891437i $$-0.649698\pi$$
−0.453145 + 0.891437i $$0.649698\pi$$
$$24$$ 102.209 0.869305
$$25$$ −120.646 −0.965167
$$26$$ 47.4693 0.358058
$$27$$ 152.634 1.08794
$$28$$ −19.6280 −0.132476
$$29$$ −200.583 −1.28439 −0.642197 0.766540i $$-0.721977\pi$$
−0.642197 + 0.766540i $$0.721977\pi$$
$$30$$ −24.2767 −0.147743
$$31$$ 76.5382 0.443441 0.221720 0.975110i $$-0.428833\pi$$
0.221720 + 0.975110i $$0.428833\pi$$
$$32$$ 35.3998 0.195558
$$33$$ 16.5527 0.0873167
$$34$$ 0 0
$$35$$ 52.1632 0.251920
$$36$$ 6.46915 0.0299498
$$37$$ −244.759 −1.08752 −0.543758 0.839242i $$-0.682999\pi$$
−0.543758 + 0.839242i $$0.682999\pi$$
$$38$$ −432.224 −1.84516
$$39$$ −76.5465 −0.314289
$$40$$ −49.2395 −0.194636
$$41$$ −54.1132 −0.206123 −0.103062 0.994675i $$-0.532864\pi$$
−0.103062 + 0.994675i $$0.532864\pi$$
$$42$$ −290.839 −1.06851
$$43$$ 142.087 0.503909 0.251955 0.967739i $$-0.418927\pi$$
0.251955 + 0.967739i $$0.418927\pi$$
$$44$$ 3.00058 0.0102808
$$45$$ −17.1924 −0.0569531
$$46$$ −268.517 −0.860668
$$47$$ −468.451 −1.45384 −0.726921 0.686721i $$-0.759049\pi$$
−0.726921 + 0.686721i $$0.759049\pi$$
$$48$$ 247.331 0.743732
$$49$$ 281.924 0.821935
$$50$$ −324.060 −0.916580
$$51$$ 0 0
$$52$$ −13.8759 −0.0370047
$$53$$ −96.5673 −0.250274 −0.125137 0.992139i $$-0.539937\pi$$
−0.125137 + 0.992139i $$0.539937\pi$$
$$54$$ 409.982 1.03318
$$55$$ −7.97432 −0.0195501
$$56$$ −589.898 −1.40765
$$57$$ 696.981 1.61960
$$58$$ −538.776 −1.21974
$$59$$ 364.484 0.804268 0.402134 0.915581i $$-0.368269\pi$$
0.402134 + 0.915581i $$0.368269\pi$$
$$60$$ 7.09641 0.0152690
$$61$$ 707.872 1.48580 0.742899 0.669404i $$-0.233450\pi$$
0.742899 + 0.669404i $$0.233450\pi$$
$$62$$ 205.585 0.421118
$$63$$ −205.967 −0.411896
$$64$$ 551.903 1.07794
$$65$$ 36.8766 0.0703689
$$66$$ 44.4612 0.0829212
$$67$$ −304.454 −0.555150 −0.277575 0.960704i $$-0.589531\pi$$
−0.277575 + 0.960704i $$0.589531\pi$$
$$68$$ 0 0
$$69$$ 432.997 0.755460
$$70$$ 140.113 0.239238
$$71$$ −470.003 −0.785621 −0.392810 0.919619i $$-0.628497\pi$$
−0.392810 + 0.919619i $$0.628497\pi$$
$$72$$ 194.423 0.318236
$$73$$ −142.056 −0.227759 −0.113880 0.993495i $$-0.536328\pi$$
−0.113880 + 0.993495i $$0.536328\pi$$
$$74$$ −657.432 −1.03277
$$75$$ 522.562 0.804537
$$76$$ 126.345 0.190694
$$77$$ −95.5336 −0.141390
$$78$$ −205.607 −0.298467
$$79$$ 717.865 1.02236 0.511178 0.859475i $$-0.329209\pi$$
0.511178 + 0.859475i $$0.329209\pi$$
$$80$$ −119.153 −0.166521
$$81$$ −438.657 −0.601725
$$82$$ −145.350 −0.195747
$$83$$ −367.639 −0.486188 −0.243094 0.970003i $$-0.578162\pi$$
−0.243094 + 0.970003i $$0.578162\pi$$
$$84$$ 85.0162 0.110429
$$85$$ 0 0
$$86$$ 381.653 0.478542
$$87$$ 868.802 1.07064
$$88$$ 90.1791 0.109240
$$89$$ 1042.28 1.24137 0.620684 0.784061i $$-0.286855\pi$$
0.620684 + 0.784061i $$0.286855\pi$$
$$90$$ −46.1795 −0.0540861
$$91$$ 441.788 0.508922
$$92$$ 78.4913 0.0889488
$$93$$ −331.516 −0.369640
$$94$$ −1258.28 −1.38066
$$95$$ −335.773 −0.362628
$$96$$ −153.330 −0.163012
$$97$$ 903.534 0.945773 0.472887 0.881123i $$-0.343212\pi$$
0.472887 + 0.881123i $$0.343212\pi$$
$$98$$ 757.260 0.780559
$$99$$ 31.4867 0.0319650
$$100$$ 94.7272 0.0947272
$$101$$ 443.233 0.436667 0.218333 0.975874i $$-0.429938\pi$$
0.218333 + 0.975874i $$0.429938\pi$$
$$102$$ 0 0
$$103$$ −1396.36 −1.33580 −0.667901 0.744250i $$-0.732807\pi$$
−0.667901 + 0.744250i $$0.732807\pi$$
$$104$$ −417.026 −0.393200
$$105$$ −225.938 −0.209994
$$106$$ −259.384 −0.237676
$$107$$ 1843.89 1.66594 0.832972 0.553315i $$-0.186637\pi$$
0.832972 + 0.553315i $$0.186637\pi$$
$$108$$ −119.843 −0.106777
$$109$$ 372.960 0.327735 0.163867 0.986482i $$-0.447603\pi$$
0.163867 + 0.986482i $$0.447603\pi$$
$$110$$ −21.4194 −0.0185660
$$111$$ 1060.14 0.906524
$$112$$ −1427.47 −1.20431
$$113$$ −1779.13 −1.48112 −0.740560 0.671990i $$-0.765440\pi$$
−0.740560 + 0.671990i $$0.765440\pi$$
$$114$$ 1872.12 1.53807
$$115$$ −208.598 −0.169147
$$116$$ 157.492 0.126058
$$117$$ −145.608 −0.115055
$$118$$ 979.020 0.763781
$$119$$ 0 0
$$120$$ 213.275 0.162244
$$121$$ −1316.40 −0.989027
$$122$$ 1901.37 1.41100
$$123$$ 234.384 0.171819
$$124$$ −60.0953 −0.0435219
$$125$$ −512.578 −0.366771
$$126$$ −553.238 −0.391161
$$127$$ 1735.00 1.21225 0.606126 0.795368i $$-0.292723\pi$$
0.606126 + 0.795368i $$0.292723\pi$$
$$128$$ 1199.24 0.828114
$$129$$ −615.433 −0.420045
$$130$$ 99.0522 0.0668265
$$131$$ −565.413 −0.377102 −0.188551 0.982063i $$-0.560379\pi$$
−0.188551 + 0.982063i $$0.560379\pi$$
$$132$$ −12.9966 −0.00856978
$$133$$ −4022.62 −2.62260
$$134$$ −817.778 −0.527203
$$135$$ 318.495 0.203050
$$136$$ 0 0
$$137$$ −1150.24 −0.717314 −0.358657 0.933469i $$-0.616765\pi$$
−0.358657 + 0.933469i $$0.616765\pi$$
$$138$$ 1163.05 0.717430
$$139$$ −984.663 −0.600849 −0.300425 0.953806i $$-0.597128\pi$$
−0.300425 + 0.953806i $$0.597128\pi$$
$$140$$ −40.9569 −0.0247249
$$141$$ 2029.04 1.21188
$$142$$ −1262.45 −0.746073
$$143$$ −67.5371 −0.0394947
$$144$$ 470.476 0.272266
$$145$$ −418.549 −0.239714
$$146$$ −381.569 −0.216294
$$147$$ −1221.12 −0.685143
$$148$$ 192.176 0.106735
$$149$$ 1772.18 0.974382 0.487191 0.873295i $$-0.338021\pi$$
0.487191 + 0.873295i $$0.338021\pi$$
$$150$$ 1403.63 0.764037
$$151$$ −1892.09 −1.01971 −0.509855 0.860260i $$-0.670301\pi$$
−0.509855 + 0.860260i $$0.670301\pi$$
$$152$$ 3797.16 2.02625
$$153$$ 0 0
$$154$$ −256.607 −0.134273
$$155$$ 159.709 0.0827621
$$156$$ 60.1019 0.0308462
$$157$$ 289.955 0.147395 0.0736973 0.997281i $$-0.476520\pi$$
0.0736973 + 0.997281i $$0.476520\pi$$
$$158$$ 1928.22 0.970891
$$159$$ 418.269 0.208622
$$160$$ 73.8673 0.0364982
$$161$$ −2499.04 −1.22330
$$162$$ −1178.25 −0.571434
$$163$$ −1691.93 −0.813022 −0.406511 0.913646i $$-0.633255\pi$$
−0.406511 + 0.913646i $$0.633255\pi$$
$$164$$ 42.4879 0.0202302
$$165$$ 34.5397 0.0162965
$$166$$ −987.495 −0.461714
$$167$$ −1255.82 −0.581907 −0.290953 0.956737i $$-0.593972\pi$$
−0.290953 + 0.956737i $$0.593972\pi$$
$$168$$ 2555.07 1.17338
$$169$$ −1884.68 −0.857842
$$170$$ 0 0
$$171$$ 1325.81 0.592907
$$172$$ −111.562 −0.0494566
$$173$$ 401.192 0.176313 0.0881564 0.996107i $$-0.471902\pi$$
0.0881564 + 0.996107i $$0.471902\pi$$
$$174$$ 2333.64 1.01674
$$175$$ −3015.96 −1.30277
$$176$$ 218.220 0.0934601
$$177$$ −1578.72 −0.670416
$$178$$ 2799.62 1.17888
$$179$$ −769.231 −0.321201 −0.160601 0.987019i $$-0.551343\pi$$
−0.160601 + 0.987019i $$0.551343\pi$$
$$180$$ 13.4989 0.00558971
$$181$$ 2627.89 1.07917 0.539585 0.841931i $$-0.318581\pi$$
0.539585 + 0.841931i $$0.318581\pi$$
$$182$$ 1186.66 0.483303
$$183$$ −3066.06 −1.23852
$$184$$ 2358.97 0.945139
$$185$$ −510.727 −0.202970
$$186$$ −890.465 −0.351033
$$187$$ 0 0
$$188$$ 367.812 0.142689
$$189$$ 3815.62 1.46850
$$190$$ −901.902 −0.344373
$$191$$ 207.856 0.0787430 0.0393715 0.999225i $$-0.487464\pi$$
0.0393715 + 0.999225i $$0.487464\pi$$
$$192$$ −2390.50 −0.898538
$$193$$ 4577.63 1.70728 0.853640 0.520864i $$-0.174390\pi$$
0.853640 + 0.520864i $$0.174390\pi$$
$$194$$ 2426.93 0.898163
$$195$$ −159.726 −0.0586576
$$196$$ −221.357 −0.0806696
$$197$$ 2640.99 0.955140 0.477570 0.878594i $$-0.341518\pi$$
0.477570 + 0.878594i $$0.341518\pi$$
$$198$$ 84.5748 0.0303559
$$199$$ 1650.96 0.588106 0.294053 0.955789i $$-0.404996\pi$$
0.294053 + 0.955789i $$0.404996\pi$$
$$200$$ 2846.92 1.00654
$$201$$ 1318.71 0.462758
$$202$$ 1190.54 0.414685
$$203$$ −5014.28 −1.73366
$$204$$ 0 0
$$205$$ −112.916 −0.0384701
$$206$$ −3750.69 −1.26856
$$207$$ 823.653 0.276560
$$208$$ −1009.14 −0.336401
$$209$$ 614.947 0.203525
$$210$$ −606.881 −0.199423
$$211$$ −1104.55 −0.360381 −0.180191 0.983632i $$-0.557672\pi$$
−0.180191 + 0.983632i $$0.557672\pi$$
$$212$$ 75.8215 0.0245634
$$213$$ 2035.76 0.654873
$$214$$ 4952.78 1.58208
$$215$$ 296.487 0.0940477
$$216$$ −3601.76 −1.13458
$$217$$ 1913.34 0.598552
$$218$$ 1001.79 0.311237
$$219$$ 615.299 0.189854
$$220$$ 6.26117 0.00191876
$$221$$ 0 0
$$222$$ 2847.59 0.860889
$$223$$ 2164.79 0.650067 0.325033 0.945703i $$-0.394624\pi$$
0.325033 + 0.945703i $$0.394624\pi$$
$$224$$ 884.942 0.263963
$$225$$ 994.025 0.294526
$$226$$ −4778.83 −1.40656
$$227$$ −2605.70 −0.761879 −0.380940 0.924600i $$-0.624399\pi$$
−0.380940 + 0.924600i $$0.624399\pi$$
$$228$$ −547.247 −0.158958
$$229$$ −1128.94 −0.325774 −0.162887 0.986645i $$-0.552081\pi$$
−0.162887 + 0.986645i $$0.552081\pi$$
$$230$$ −560.304 −0.160632
$$231$$ 413.792 0.117859
$$232$$ 4733.24 1.33945
$$233$$ −4315.97 −1.21351 −0.606756 0.794888i $$-0.707530\pi$$
−0.606756 + 0.794888i $$0.707530\pi$$
$$234$$ −391.109 −0.109263
$$235$$ −977.496 −0.271340
$$236$$ −286.181 −0.0789356
$$237$$ −3109.34 −0.852209
$$238$$ 0 0
$$239$$ 788.197 0.213323 0.106662 0.994295i $$-0.465984\pi$$
0.106662 + 0.994295i $$0.465984\pi$$
$$240$$ 516.094 0.138807
$$241$$ −3622.86 −0.968335 −0.484167 0.874975i $$-0.660877\pi$$
−0.484167 + 0.874975i $$0.660877\pi$$
$$242$$ −3535.90 −0.939240
$$243$$ −2221.13 −0.586361
$$244$$ −555.798 −0.145825
$$245$$ 588.278 0.153403
$$246$$ 629.567 0.163170
$$247$$ −2843.78 −0.732571
$$248$$ −1806.10 −0.462449
$$249$$ 1592.38 0.405274
$$250$$ −1376.81 −0.348308
$$251$$ −5974.99 −1.50254 −0.751271 0.659994i $$-0.770559\pi$$
−0.751271 + 0.659994i $$0.770559\pi$$
$$252$$ 161.719 0.0404259
$$253$$ 382.034 0.0949339
$$254$$ 4660.28 1.15123
$$255$$ 0 0
$$256$$ −1194.02 −0.291509
$$257$$ −3750.91 −0.910411 −0.455205 0.890386i $$-0.650434\pi$$
−0.455205 + 0.890386i $$0.650434\pi$$
$$258$$ −1653.08 −0.398900
$$259$$ −6118.59 −1.46792
$$260$$ −28.9543 −0.00690642
$$261$$ 1652.65 0.391940
$$262$$ −1518.73 −0.358119
$$263$$ −7566.43 −1.77401 −0.887007 0.461755i $$-0.847220\pi$$
−0.887007 + 0.461755i $$0.847220\pi$$
$$264$$ −390.599 −0.0910596
$$265$$ −201.503 −0.0467103
$$266$$ −10804.9 −2.49057
$$267$$ −4514.52 −1.03477
$$268$$ 239.048 0.0544857
$$269$$ 6990.63 1.58449 0.792243 0.610206i $$-0.208913\pi$$
0.792243 + 0.610206i $$0.208913\pi$$
$$270$$ 855.492 0.192828
$$271$$ −1356.64 −0.304097 −0.152049 0.988373i $$-0.548587\pi$$
−0.152049 + 0.988373i $$0.548587\pi$$
$$272$$ 0 0
$$273$$ −1913.55 −0.424224
$$274$$ −3089.61 −0.681204
$$275$$ 461.057 0.101101
$$276$$ −339.975 −0.0741453
$$277$$ 7174.85 1.55630 0.778150 0.628078i $$-0.216158\pi$$
0.778150 + 0.628078i $$0.216158\pi$$
$$278$$ −2644.85 −0.570603
$$279$$ −630.613 −0.135318
$$280$$ −1230.91 −0.262718
$$281$$ 6611.42 1.40357 0.701787 0.712387i $$-0.252386\pi$$
0.701787 + 0.712387i $$0.252386\pi$$
$$282$$ 5450.08 1.15088
$$283$$ 2310.88 0.485398 0.242699 0.970102i $$-0.421967\pi$$
0.242699 + 0.970102i $$0.421967\pi$$
$$284$$ 369.031 0.0771055
$$285$$ 1454.36 0.302277
$$286$$ −181.408 −0.0375065
$$287$$ −1352.75 −0.278223
$$288$$ −291.666 −0.0596757
$$289$$ 0 0
$$290$$ −1124.24 −0.227647
$$291$$ −3913.55 −0.788371
$$292$$ 111.538 0.0223537
$$293$$ −6445.85 −1.28522 −0.642612 0.766192i $$-0.722149\pi$$
−0.642612 + 0.766192i $$0.722149\pi$$
$$294$$ −3279.98 −0.650653
$$295$$ 760.553 0.150105
$$296$$ 5775.65 1.13413
$$297$$ −583.303 −0.113962
$$298$$ 4760.16 0.925332
$$299$$ −1766.69 −0.341706
$$300$$ −410.299 −0.0789621
$$301$$ 3551.96 0.680172
$$302$$ −5082.24 −0.968378
$$303$$ −1919.81 −0.363994
$$304$$ 9188.57 1.73356
$$305$$ 1477.08 0.277304
$$306$$ 0 0
$$307$$ 221.421 0.0411634 0.0205817 0.999788i $$-0.493448\pi$$
0.0205817 + 0.999788i $$0.493448\pi$$
$$308$$ 75.0099 0.0138769
$$309$$ 6048.17 1.11349
$$310$$ 428.985 0.0785959
$$311$$ 1293.87 0.235912 0.117956 0.993019i $$-0.462366\pi$$
0.117956 + 0.993019i $$0.462366\pi$$
$$312$$ 1806.30 0.327761
$$313$$ −385.448 −0.0696064 −0.0348032 0.999394i $$-0.511080\pi$$
−0.0348032 + 0.999394i $$0.511080\pi$$
$$314$$ 778.833 0.139975
$$315$$ −429.783 −0.0768747
$$316$$ −563.644 −0.100340
$$317$$ −649.047 −0.114997 −0.0574986 0.998346i $$-0.518312\pi$$
−0.0574986 + 0.998346i $$0.518312\pi$$
$$318$$ 1123.49 0.198120
$$319$$ 766.545 0.134540
$$320$$ 1151.63 0.201182
$$321$$ −7986.60 −1.38869
$$322$$ −6712.53 −1.16172
$$323$$ 0 0
$$324$$ 344.419 0.0590568
$$325$$ −2132.12 −0.363904
$$326$$ −4544.61 −0.772094
$$327$$ −1615.43 −0.273191
$$328$$ 1276.93 0.214959
$$329$$ −11710.6 −1.96238
$$330$$ 92.7753 0.0154761
$$331$$ 1429.71 0.237415 0.118707 0.992929i $$-0.462125\pi$$
0.118707 + 0.992929i $$0.462125\pi$$
$$332$$ 288.658 0.0477174
$$333$$ 2016.61 0.331861
$$334$$ −3373.19 −0.552614
$$335$$ −635.292 −0.103611
$$336$$ 6182.89 1.00388
$$337$$ −565.364 −0.0913867 −0.0456934 0.998956i $$-0.514550\pi$$
−0.0456934 + 0.998956i $$0.514550\pi$$
$$338$$ −5062.33 −0.814659
$$339$$ 7706.08 1.23462
$$340$$ 0 0
$$341$$ −292.497 −0.0464504
$$342$$ 3561.18 0.563060
$$343$$ −1526.81 −0.240350
$$344$$ −3352.88 −0.525509
$$345$$ 903.516 0.140996
$$346$$ 1077.62 0.167437
$$347$$ 3788.72 0.586135 0.293068 0.956092i $$-0.405324\pi$$
0.293068 + 0.956092i $$0.405324\pi$$
$$348$$ −682.155 −0.105079
$$349$$ −6387.30 −0.979669 −0.489834 0.871816i $$-0.662943\pi$$
−0.489834 + 0.871816i $$0.662943\pi$$
$$350$$ −8101.01 −1.23719
$$351$$ 2697.44 0.410196
$$352$$ −135.283 −0.0204847
$$353$$ 6291.34 0.948595 0.474298 0.880365i $$-0.342702\pi$$
0.474298 + 0.880365i $$0.342702\pi$$
$$354$$ −4240.50 −0.636667
$$355$$ −980.735 −0.146625
$$356$$ −818.367 −0.121835
$$357$$ 0 0
$$358$$ −2066.19 −0.305032
$$359$$ −3369.51 −0.495364 −0.247682 0.968841i $$-0.579669\pi$$
−0.247682 + 0.968841i $$0.579669\pi$$
$$360$$ 405.694 0.0593944
$$361$$ 19034.5 2.77511
$$362$$ 7058.63 1.02484
$$363$$ 5701.80 0.824427
$$364$$ −346.877 −0.0499486
$$365$$ −296.423 −0.0425081
$$366$$ −8235.56 −1.17617
$$367$$ 997.517 0.141880 0.0709400 0.997481i $$-0.477400\pi$$
0.0709400 + 0.997481i $$0.477400\pi$$
$$368$$ 5708.37 0.808612
$$369$$ 445.849 0.0628997
$$370$$ −1371.84 −0.192752
$$371$$ −2414.04 −0.337818
$$372$$ 260.295 0.0362787
$$373$$ 3180.59 0.441514 0.220757 0.975329i $$-0.429147\pi$$
0.220757 + 0.975329i $$0.429147\pi$$
$$374$$ 0 0
$$375$$ 2220.17 0.305731
$$376$$ 11054.2 1.51616
$$377$$ −3544.83 −0.484265
$$378$$ 10248.9 1.39457
$$379$$ −10833.9 −1.46834 −0.734170 0.678965i $$-0.762429\pi$$
−0.734170 + 0.678965i $$0.762429\pi$$
$$380$$ 263.638 0.0355904
$$381$$ −7514.92 −1.01050
$$382$$ 558.310 0.0747791
$$383$$ −7567.93 −1.00967 −0.504834 0.863216i $$-0.668446\pi$$
−0.504834 + 0.863216i $$0.668446\pi$$
$$384$$ −5194.34 −0.690294
$$385$$ −199.346 −0.0263886
$$386$$ 12295.7 1.62134
$$387$$ −1170.69 −0.153771
$$388$$ −709.426 −0.0928238
$$389$$ 9599.36 1.25117 0.625587 0.780154i $$-0.284859\pi$$
0.625587 + 0.780154i $$0.284859\pi$$
$$390$$ −429.032 −0.0557048
$$391$$ 0 0
$$392$$ −6652.65 −0.857168
$$393$$ 2449.02 0.314342
$$394$$ 7093.81 0.907058
$$395$$ 1497.94 0.190809
$$396$$ −24.7224 −0.00313724
$$397$$ 5272.03 0.666487 0.333244 0.942841i $$-0.391857\pi$$
0.333244 + 0.942841i $$0.391857\pi$$
$$398$$ 4434.54 0.558501
$$399$$ 17423.5 2.18613
$$400$$ 6889.14 0.861143
$$401$$ −3600.34 −0.448360 −0.224180 0.974548i $$-0.571970\pi$$
−0.224180 + 0.974548i $$0.571970\pi$$
$$402$$ 3542.10 0.439463
$$403$$ 1352.63 0.167194
$$404$$ −348.012 −0.0428571
$$405$$ −915.327 −0.112304
$$406$$ −13468.6 −1.64639
$$407$$ 935.364 0.113917
$$408$$ 0 0
$$409$$ 9516.13 1.15047 0.575235 0.817988i $$-0.304911\pi$$
0.575235 + 0.817988i $$0.304911\pi$$
$$410$$ −303.296 −0.0365335
$$411$$ 4982.14 0.597933
$$412$$ 1096.38 0.131104
$$413$$ 9111.55 1.08559
$$414$$ 2212.37 0.262638
$$415$$ −767.137 −0.0907404
$$416$$ 625.606 0.0737329
$$417$$ 4264.95 0.500852
$$418$$ 1651.78 0.193280
$$419$$ −6310.52 −0.735774 −0.367887 0.929871i $$-0.619919\pi$$
−0.367887 + 0.929871i $$0.619919\pi$$
$$420$$ 177.400 0.0206100
$$421$$ −1544.81 −0.178835 −0.0894177 0.995994i $$-0.528501\pi$$
−0.0894177 + 0.995994i $$0.528501\pi$$
$$422$$ −2966.88 −0.342240
$$423$$ 3859.66 0.443648
$$424$$ 2278.73 0.261002
$$425$$ 0 0
$$426$$ 5468.14 0.621906
$$427$$ 17695.7 2.00552
$$428$$ −1447.77 −0.163506
$$429$$ 292.528 0.0329217
$$430$$ 796.378 0.0893134
$$431$$ 897.334 0.100286 0.0501428 0.998742i $$-0.484032\pi$$
0.0501428 + 0.998742i $$0.484032\pi$$
$$432$$ −8715.74 −0.970686
$$433$$ 4896.47 0.543440 0.271720 0.962376i $$-0.412408\pi$$
0.271720 + 0.962376i $$0.412408\pi$$
$$434$$ 5139.31 0.568421
$$435$$ 1812.89 0.199819
$$436$$ −292.836 −0.0321659
$$437$$ 16086.2 1.76089
$$438$$ 1652.72 0.180297
$$439$$ −1404.01 −0.152642 −0.0763210 0.997083i $$-0.524317\pi$$
−0.0763210 + 0.997083i $$0.524317\pi$$
$$440$$ 188.173 0.0203881
$$441$$ −2322.83 −0.250818
$$442$$ 0 0
$$443$$ −6453.57 −0.692141 −0.346070 0.938209i $$-0.612484\pi$$
−0.346070 + 0.938209i $$0.612484\pi$$
$$444$$ −832.388 −0.0889716
$$445$$ 2174.89 0.231684
$$446$$ 5814.71 0.617342
$$447$$ −7675.99 −0.812219
$$448$$ 13796.7 1.45499
$$449$$ −4409.11 −0.463427 −0.231714 0.972784i $$-0.574433\pi$$
−0.231714 + 0.972784i $$0.574433\pi$$
$$450$$ 2670.00 0.279700
$$451$$ 206.798 0.0215914
$$452$$ 1396.92 0.145366
$$453$$ 8195.36 0.850003
$$454$$ −6999.03 −0.723526
$$455$$ 921.859 0.0949833
$$456$$ −16446.9 −1.68903
$$457$$ 12571.6 1.28681 0.643406 0.765525i $$-0.277521\pi$$
0.643406 + 0.765525i $$0.277521\pi$$
$$458$$ −3032.37 −0.309374
$$459$$ 0 0
$$460$$ 163.784 0.0166011
$$461$$ −5115.00 −0.516767 −0.258383 0.966042i $$-0.583190\pi$$
−0.258383 + 0.966042i $$0.583190\pi$$
$$462$$ 1111.46 0.111926
$$463$$ 5280.96 0.530080 0.265040 0.964237i $$-0.414615\pi$$
0.265040 + 0.964237i $$0.414615\pi$$
$$464$$ 11453.8 1.14596
$$465$$ −691.759 −0.0689883
$$466$$ −11592.9 −1.15242
$$467$$ −4430.45 −0.439008 −0.219504 0.975612i $$-0.570444\pi$$
−0.219504 + 0.975612i $$0.570444\pi$$
$$468$$ 114.327 0.0112922
$$469$$ −7610.90 −0.749336
$$470$$ −2625.60 −0.257680
$$471$$ −1255.91 −0.122864
$$472$$ −8600.86 −0.838743
$$473$$ −542.997 −0.0527844
$$474$$ −8351.83 −0.809308
$$475$$ 19413.7 1.87529
$$476$$ 0 0
$$477$$ 795.637 0.0763726
$$478$$ 2117.13 0.202584
$$479$$ 8697.09 0.829604 0.414802 0.909912i $$-0.363851\pi$$
0.414802 + 0.909912i $$0.363851\pi$$
$$480$$ −319.947 −0.0304240
$$481$$ −4325.52 −0.410034
$$482$$ −9731.15 −0.919589
$$483$$ 10824.3 1.01971
$$484$$ 1033.59 0.0970690
$$485$$ 1885.37 0.176516
$$486$$ −5966.06 −0.556844
$$487$$ −13031.1 −1.21252 −0.606258 0.795268i $$-0.707330\pi$$
−0.606258 + 0.795268i $$0.707330\pi$$
$$488$$ −16703.9 −1.54949
$$489$$ 7328.40 0.677713
$$490$$ 1580.14 0.145681
$$491$$ −11760.3 −1.08093 −0.540464 0.841367i $$-0.681751\pi$$
−0.540464 + 0.841367i $$0.681751\pi$$
$$492$$ −184.031 −0.0168633
$$493$$ 0 0
$$494$$ −7638.51 −0.695694
$$495$$ 65.7020 0.00596583
$$496$$ −4370.50 −0.395647
$$497$$ −11749.4 −1.06042
$$498$$ 4277.21 0.384872
$$499$$ −10724.6 −0.962118 −0.481059 0.876688i $$-0.659748\pi$$
−0.481059 + 0.876688i $$0.659748\pi$$
$$500$$ 402.460 0.0359971
$$501$$ 5439.43 0.485062
$$502$$ −16049.1 −1.42690
$$503$$ −3904.96 −0.346150 −0.173075 0.984909i $$-0.555370\pi$$
−0.173075 + 0.984909i $$0.555370\pi$$
$$504$$ 4860.29 0.429552
$$505$$ 924.875 0.0814978
$$506$$ 1026.16 0.0901549
$$507$$ 8163.25 0.715075
$$508$$ −1362.26 −0.118978
$$509$$ 15132.3 1.31774 0.658870 0.752257i $$-0.271035\pi$$
0.658870 + 0.752257i $$0.271035\pi$$
$$510$$ 0 0
$$511$$ −3551.19 −0.307427
$$512$$ −12801.1 −1.10495
$$513$$ −24561.1 −2.11383
$$514$$ −10075.1 −0.864581
$$515$$ −2913.73 −0.249309
$$516$$ 483.218 0.0412257
$$517$$ 1790.22 0.152290
$$518$$ −16434.8 −1.39402
$$519$$ −1737.71 −0.146970
$$520$$ −870.190 −0.0733853
$$521$$ −16534.9 −1.39041 −0.695207 0.718810i $$-0.744687\pi$$
−0.695207 + 0.718810i $$0.744687\pi$$
$$522$$ 4439.08 0.372210
$$523$$ −8724.36 −0.729426 −0.364713 0.931120i $$-0.618833\pi$$
−0.364713 + 0.931120i $$0.618833\pi$$
$$524$$ 443.944 0.0370111
$$525$$ 13063.3 1.08596
$$526$$ −20323.8 −1.68471
$$527$$ 0 0
$$528$$ −945.194 −0.0779058
$$529$$ −2173.47 −0.178637
$$530$$ −541.245 −0.0443589
$$531$$ −3003.06 −0.245427
$$532$$ 3158.43 0.257397
$$533$$ −956.319 −0.0777163
$$534$$ −12126.2 −0.982681
$$535$$ 3847.57 0.310925
$$536$$ 7184.32 0.578946
$$537$$ 3331.83 0.267745
$$538$$ 18777.2 1.50472
$$539$$ −1077.39 −0.0860976
$$540$$ −250.072 −0.0199285
$$541$$ 8246.12 0.655320 0.327660 0.944796i $$-0.393740\pi$$
0.327660 + 0.944796i $$0.393740\pi$$
$$542$$ −3644.01 −0.288789
$$543$$ −11382.4 −0.899567
$$544$$ 0 0
$$545$$ 778.240 0.0611672
$$546$$ −5139.87 −0.402868
$$547$$ 23842.2 1.86366 0.931829 0.362898i $$-0.118213\pi$$
0.931829 + 0.362898i $$0.118213\pi$$
$$548$$ 903.134 0.0704014
$$549$$ −5832.30 −0.453399
$$550$$ 1238.42 0.0960117
$$551$$ 32276.8 2.49553
$$552$$ −10217.6 −0.787843
$$553$$ 17945.5 1.37997
$$554$$ 19272.0 1.47796
$$555$$ 2212.15 0.169190
$$556$$ 773.126 0.0589709
$$557$$ 19341.5 1.47132 0.735661 0.677350i $$-0.236872\pi$$
0.735661 + 0.677350i $$0.236872\pi$$
$$558$$ −1693.86 −0.128506
$$559$$ 2511.05 0.189993
$$560$$ −2978.63 −0.224768
$$561$$ 0 0
$$562$$ 17758.6 1.33292
$$563$$ −14210.3 −1.06376 −0.531878 0.846821i $$-0.678513\pi$$
−0.531878 + 0.846821i $$0.678513\pi$$
$$564$$ −1593.13 −0.118942
$$565$$ −3712.44 −0.276431
$$566$$ 6207.12 0.460963
$$567$$ −10965.8 −0.812202
$$568$$ 11090.8 0.819297
$$569$$ −9576.15 −0.705542 −0.352771 0.935710i $$-0.614761\pi$$
−0.352771 + 0.935710i $$0.614761\pi$$
$$570$$ 3906.48 0.287060
$$571$$ 14785.0 1.08359 0.541796 0.840510i $$-0.317745\pi$$
0.541796 + 0.840510i $$0.317745\pi$$
$$572$$ 53.0279 0.00387624
$$573$$ −900.301 −0.0656381
$$574$$ −3633.54 −0.264218
$$575$$ 12060.7 0.874722
$$576$$ −4547.24 −0.328938
$$577$$ −11438.5 −0.825284 −0.412642 0.910893i $$-0.635394\pi$$
−0.412642 + 0.910893i $$0.635394\pi$$
$$578$$ 0 0
$$579$$ −19827.4 −1.42314
$$580$$ 328.631 0.0235270
$$581$$ −9190.42 −0.656253
$$582$$ −10512.0 −0.748685
$$583$$ 369.039 0.0262162
$$584$$ 3352.15 0.237522
$$585$$ −303.834 −0.0214735
$$586$$ −17313.8 −1.22053
$$587$$ −7892.09 −0.554925 −0.277463 0.960736i $$-0.589493\pi$$
−0.277463 + 0.960736i $$0.589493\pi$$
$$588$$ 958.782 0.0672440
$$589$$ −12316.1 −0.861590
$$590$$ 2042.88 0.142549
$$591$$ −11439.1 −0.796179
$$592$$ 13976.2 0.970304
$$593$$ 7829.44 0.542186 0.271093 0.962553i $$-0.412615\pi$$
0.271093 + 0.962553i $$0.412615\pi$$
$$594$$ −1566.78 −0.108225
$$595$$ 0 0
$$596$$ −1391.46 −0.0956317
$$597$$ −7150.91 −0.490230
$$598$$ −4745.40 −0.324504
$$599$$ −18939.7 −1.29191 −0.645956 0.763375i $$-0.723541\pi$$
−0.645956 + 0.763375i $$0.723541\pi$$
$$600$$ −12331.1 −0.839024
$$601$$ −17777.4 −1.20658 −0.603290 0.797522i $$-0.706144\pi$$
−0.603290 + 0.797522i $$0.706144\pi$$
$$602$$ 9540.73 0.645932
$$603$$ 2508.46 0.169407
$$604$$ 1485.61 0.100080
$$605$$ −2746.86 −0.184588
$$606$$ −5156.69 −0.345670
$$607$$ −17713.3 −1.18445 −0.592223 0.805774i $$-0.701750\pi$$
−0.592223 + 0.805774i $$0.701750\pi$$
$$608$$ −5696.35 −0.379963
$$609$$ 21718.7 1.44513
$$610$$ 3967.51 0.263344
$$611$$ −8278.74 −0.548154
$$612$$ 0 0
$$613$$ 25427.3 1.67537 0.837683 0.546156i $$-0.183910\pi$$
0.837683 + 0.546156i $$0.183910\pi$$
$$614$$ 594.746 0.0390912
$$615$$ 489.080 0.0320676
$$616$$ 2254.34 0.147451
$$617$$ −10028.4 −0.654343 −0.327172 0.944965i $$-0.606096\pi$$
−0.327172 + 0.944965i $$0.606096\pi$$
$$618$$ 16245.6 1.05744
$$619$$ −4460.94 −0.289661 −0.144831 0.989456i $$-0.546264\pi$$
−0.144831 + 0.989456i $$0.546264\pi$$
$$620$$ −125.398 −0.00812276
$$621$$ −15258.5 −0.985993
$$622$$ 3475.39 0.224036
$$623$$ 26055.5 1.67559
$$624$$ 4370.97 0.280415
$$625$$ 14011.2 0.896714
$$626$$ −1035.33 −0.0661024
$$627$$ −2663.57 −0.169653
$$628$$ −227.664 −0.0144662
$$629$$ 0 0
$$630$$ −1154.42 −0.0730049
$$631$$ −1098.49 −0.0693032 −0.0346516 0.999399i $$-0.511032\pi$$
−0.0346516 + 0.999399i $$0.511032\pi$$
$$632$$ −16939.7 −1.06618
$$633$$ 4784.23 0.300404
$$634$$ −1743.37 −0.109208
$$635$$ 3620.34 0.226250
$$636$$ −328.411 −0.0204754
$$637$$ 4982.32 0.309901
$$638$$ 2058.97 0.127767
$$639$$ 3872.45 0.239737
$$640$$ 2502.39 0.154556
$$641$$ −1063.17 −0.0655115 −0.0327557 0.999463i $$-0.510428\pi$$
−0.0327557 + 0.999463i $$0.510428\pi$$
$$642$$ −21452.4 −1.31878
$$643$$ −6571.26 −0.403025 −0.201513 0.979486i $$-0.564586\pi$$
−0.201513 + 0.979486i $$0.564586\pi$$
$$644$$ 1962.16 0.120062
$$645$$ −1284.20 −0.0783957
$$646$$ 0 0
$$647$$ −14783.1 −0.898273 −0.449137 0.893463i $$-0.648268\pi$$
−0.449137 + 0.893463i $$0.648268\pi$$
$$648$$ 10351.1 0.627518
$$649$$ −1392.90 −0.0842469
$$650$$ −5726.98 −0.345585
$$651$$ −8287.38 −0.498937
$$652$$ 1328.45 0.0797948
$$653$$ −25912.0 −1.55285 −0.776427 0.630208i $$-0.782970\pi$$
−0.776427 + 0.630208i $$0.782970\pi$$
$$654$$ −4339.12 −0.259439
$$655$$ −1179.82 −0.0703809
$$656$$ 3089.98 0.183908
$$657$$ 1170.43 0.0695020
$$658$$ −31455.1 −1.86360
$$659$$ 4302.30 0.254315 0.127158 0.991883i $$-0.459415\pi$$
0.127158 + 0.991883i $$0.459415\pi$$
$$660$$ −27.1195 −0.00159943
$$661$$ −7467.66 −0.439423 −0.219711 0.975565i $$-0.570512\pi$$
−0.219711 + 0.975565i $$0.570512\pi$$
$$662$$ 3840.28 0.225463
$$663$$ 0 0
$$664$$ 8675.31 0.507029
$$665$$ −8393.82 −0.489471
$$666$$ 5416.72 0.315155
$$667$$ 20051.9 1.16403
$$668$$ 986.031 0.0571118
$$669$$ −9376.50 −0.541878
$$670$$ −1706.42 −0.0983953
$$671$$ −2705.19 −0.155637
$$672$$ −3833.01 −0.220032
$$673$$ −4236.79 −0.242669 −0.121335 0.992612i $$-0.538717\pi$$
−0.121335 + 0.992612i $$0.538717\pi$$
$$674$$ −1518.59 −0.0867863
$$675$$ −18414.7 −1.05005
$$676$$ 1479.79 0.0841938
$$677$$ −23130.5 −1.31311 −0.656555 0.754278i $$-0.727987\pi$$
−0.656555 + 0.754278i $$0.727987\pi$$
$$678$$ 20698.9 1.17247
$$679$$ 22587.0 1.27660
$$680$$ 0 0
$$681$$ 11286.3 0.635082
$$682$$ −785.659 −0.0441121
$$683$$ 11474.5 0.642839 0.321419 0.946937i $$-0.395840\pi$$
0.321419 + 0.946937i $$0.395840\pi$$
$$684$$ −1040.98 −0.0581914
$$685$$ −2400.16 −0.133877
$$686$$ −4101.08 −0.228251
$$687$$ 4889.84 0.271556
$$688$$ −8113.49 −0.449599
$$689$$ −1706.59 −0.0943629
$$690$$ 2426.88 0.133898
$$691$$ −15221.6 −0.837997 −0.418999 0.907987i $$-0.637619\pi$$
−0.418999 + 0.907987i $$0.637619\pi$$
$$692$$ −315.003 −0.0173044
$$693$$ 787.120 0.0431461
$$694$$ 10176.7 0.556629
$$695$$ −2054.65 −0.112140
$$696$$ −20501.4 −1.11653
$$697$$ 0 0
$$698$$ −17156.6 −0.930352
$$699$$ 18694.1 1.01155
$$700$$ 2368.04 0.127862
$$701$$ −23996.1 −1.29289 −0.646447 0.762959i $$-0.723746\pi$$
−0.646447 + 0.762959i $$0.723746\pi$$
$$702$$ 7245.44 0.389546
$$703$$ 39385.2 2.11300
$$704$$ −2109.14 −0.112914
$$705$$ 4233.90 0.226181
$$706$$ 16898.8 0.900843
$$707$$ 11080.2 0.589409
$$708$$ 1239.56 0.0657986
$$709$$ 22346.8 1.18371 0.591856 0.806044i $$-0.298395\pi$$
0.591856 + 0.806044i $$0.298395\pi$$
$$710$$ −2634.30 −0.139244
$$711$$ −5914.63 −0.311978
$$712$$ −24595.1 −1.29458
$$713$$ −7651.34 −0.401886
$$714$$ 0 0
$$715$$ −140.927 −0.00737113
$$716$$ 603.975 0.0315246
$$717$$ −3413.98 −0.177820
$$718$$ −9050.64 −0.470427
$$719$$ −32468.2 −1.68409 −0.842045 0.539408i $$-0.818648\pi$$
−0.842045 + 0.539408i $$0.818648\pi$$
$$720$$ 981.722 0.0508148
$$721$$ −34906.9 −1.80305
$$722$$ 51127.5 2.63541
$$723$$ 15691.9 0.807178
$$724$$ −2063.34 −0.105916
$$725$$ 24199.6 1.23965
$$726$$ 15315.3 0.782925
$$727$$ −2899.33 −0.147909 −0.0739547 0.997262i $$-0.523562\pi$$
−0.0739547 + 0.997262i $$0.523562\pi$$
$$728$$ −10425.0 −0.530737
$$729$$ 21464.3 1.09050
$$730$$ −796.204 −0.0403683
$$731$$ 0 0
$$732$$ 2407.37 0.121556
$$733$$ 16399.9 0.826390 0.413195 0.910643i $$-0.364413\pi$$
0.413195 + 0.910643i $$0.364413\pi$$
$$734$$ 2679.37 0.134738
$$735$$ −2548.05 −0.127873
$$736$$ −3538.84 −0.177233
$$737$$ 1163.50 0.0581519
$$738$$ 1197.57 0.0597333
$$739$$ 6470.65 0.322093 0.161047 0.986947i $$-0.448513\pi$$
0.161047 + 0.986947i $$0.448513\pi$$
$$740$$ 401.006 0.0199207
$$741$$ 12317.5 0.610652
$$742$$ −6484.21 −0.320812
$$743$$ −7520.20 −0.371318 −0.185659 0.982614i $$-0.559442\pi$$
−0.185659 + 0.982614i $$0.559442\pi$$
$$744$$ 7822.89 0.385485
$$745$$ 3697.94 0.181855
$$746$$ 8543.20 0.419288
$$747$$ 3029.05 0.148363
$$748$$ 0 0
$$749$$ 46094.6 2.24868
$$750$$ 5963.47 0.290340
$$751$$ 10921.9 0.530685 0.265342 0.964154i $$-0.414515\pi$$
0.265342 + 0.964154i $$0.414515\pi$$
$$752$$ 26749.6 1.29715
$$753$$ 25879.9 1.25248
$$754$$ −9521.56 −0.459887
$$755$$ −3948.14 −0.190315
$$756$$ −2995.90 −0.144127
$$757$$ 17112.9 0.821636 0.410818 0.911717i $$-0.365243\pi$$
0.410818 + 0.911717i $$0.365243\pi$$
$$758$$ −29100.4 −1.39442
$$759$$ −1654.73 −0.0791343
$$760$$ 7923.36 0.378172
$$761$$ −35934.3 −1.71172 −0.855860 0.517207i $$-0.826972\pi$$
−0.855860 + 0.517207i $$0.826972\pi$$
$$762$$ −20185.4 −0.959633
$$763$$ 9323.44 0.442374
$$764$$ −163.202 −0.00772831
$$765$$ 0 0
$$766$$ −20327.8 −0.958842
$$767$$ 6441.37 0.303239
$$768$$ 5171.75 0.242994
$$769$$ 29085.3 1.36390 0.681952 0.731397i $$-0.261131\pi$$
0.681952 + 0.731397i $$0.261131\pi$$
$$770$$ −535.451 −0.0250602
$$771$$ 16246.6 0.758894
$$772$$ −3594.21 −0.167563
$$773$$ −16502.8 −0.767873 −0.383936 0.923360i $$-0.625432\pi$$
−0.383936 + 0.923360i $$0.625432\pi$$
$$774$$ −3144.51 −0.146030
$$775$$ −9234.02 −0.427994
$$776$$ −21321.0 −0.986314
$$777$$ 26501.9 1.22362
$$778$$ 25784.3 1.18819
$$779$$ 8707.60 0.400491
$$780$$ 125.412 0.00575701
$$781$$ 1796.15 0.0822937
$$782$$ 0 0
$$783$$ −30615.9 −1.39735
$$784$$ −16098.5 −0.733348
$$785$$ 605.037 0.0275092
$$786$$ 6578.17 0.298518
$$787$$ 27374.1 1.23987 0.619937 0.784651i $$-0.287158\pi$$
0.619937 + 0.784651i $$0.287158\pi$$
$$788$$ −2073.62 −0.0937431
$$789$$ 32773.0 1.47877
$$790$$ 4023.53 0.181203
$$791$$ −44475.6 −1.99920
$$792$$ −743.003 −0.0333352
$$793$$ 12509.9 0.560202
$$794$$ 14160.9 0.632936
$$795$$ 872.784 0.0389364
$$796$$ −1296.28 −0.0577202
$$797$$ −31776.1 −1.41226 −0.706128 0.708084i $$-0.749560\pi$$
−0.706128 + 0.708084i $$0.749560\pi$$
$$798$$ 46800.2 2.07608
$$799$$ 0 0
$$800$$ −4270.84 −0.188746
$$801$$ −8587.58 −0.378810
$$802$$ −9670.68 −0.425790
$$803$$ 542.879 0.0238578
$$804$$ −1035.41 −0.0454178
$$805$$ −5214.63 −0.228313
$$806$$ 3633.22 0.158777
$$807$$ −30279.0 −1.32078
$$808$$ −10459.1 −0.455385
$$809$$ −34367.2 −1.49355 −0.746777 0.665074i $$-0.768400\pi$$
−0.746777 + 0.665074i $$0.768400\pi$$
$$810$$ −2458.61 −0.106650
$$811$$ −1316.99 −0.0570232 −0.0285116 0.999593i $$-0.509077\pi$$
−0.0285116 + 0.999593i $$0.509077\pi$$
$$812$$ 3937.05 0.170152
$$813$$ 5876.14 0.253487
$$814$$ 2512.43 0.108182
$$815$$ −3530.49 −0.151739
$$816$$ 0 0
$$817$$ −22863.9 −0.979078
$$818$$ 25560.7 1.09256
$$819$$ −3639.98 −0.155300
$$820$$ 88.6577 0.00377568
$$821$$ 33481.1 1.42326 0.711632 0.702552i $$-0.247956\pi$$
0.711632 + 0.702552i $$0.247956\pi$$
$$822$$ 13382.2 0.567833
$$823$$ 11746.8 0.497529 0.248765 0.968564i $$-0.419975\pi$$
0.248765 + 0.968564i $$0.419975\pi$$
$$824$$ 32950.5 1.39306
$$825$$ −1997.01 −0.0842752
$$826$$ 24474.0 1.03094
$$827$$ −4367.84 −0.183658 −0.0918288 0.995775i $$-0.529271\pi$$
−0.0918288 + 0.995775i $$0.529271\pi$$
$$828$$ −646.706 −0.0271432
$$829$$ 19994.7 0.837689 0.418844 0.908058i $$-0.362435\pi$$
0.418844 + 0.908058i $$0.362435\pi$$
$$830$$ −2060.56 −0.0861725
$$831$$ −31077.0 −1.29729
$$832$$ 9753.55 0.406422
$$833$$ 0 0
$$834$$ 11455.8 0.475639
$$835$$ −2620.47 −0.108605
$$836$$ −482.837 −0.0199752
$$837$$ 11682.3 0.482438
$$838$$ −16950.3 −0.698735
$$839$$ −25672.1 −1.05638 −0.528189 0.849127i $$-0.677129\pi$$
−0.528189 + 0.849127i $$0.677129\pi$$
$$840$$ 5331.55 0.218995
$$841$$ 15844.7 0.649667
$$842$$ −4149.44 −0.169833
$$843$$ −28636.5 −1.16998
$$844$$ 867.258 0.0353700
$$845$$ −3932.68 −0.160104
$$846$$ 10367.2 0.421315
$$847$$ −32907.9 −1.33498
$$848$$ 5514.20 0.223300
$$849$$ −10009.3 −0.404614
$$850$$ 0 0
$$851$$ 24467.9 0.985605
$$852$$ −1598.41 −0.0642731
$$853$$ −15504.6 −0.622355 −0.311178 0.950352i $$-0.600723\pi$$
−0.311178 + 0.950352i $$0.600723\pi$$
$$854$$ 47531.5 1.90456
$$855$$ 2766.50 0.110658
$$856$$ −43511.0 −1.73736
$$857$$ −35915.1 −1.43155 −0.715774 0.698332i $$-0.753926\pi$$
−0.715774 + 0.698332i $$0.753926\pi$$
$$858$$ 785.744 0.0312644
$$859$$ −36575.0 −1.45276 −0.726382 0.687292i $$-0.758799\pi$$
−0.726382 + 0.687292i $$0.758799\pi$$
$$860$$ −232.792 −0.00923040
$$861$$ 5859.25 0.231920
$$862$$ 2410.28 0.0952372
$$863$$ 11402.6 0.449768 0.224884 0.974386i $$-0.427800\pi$$
0.224884 + 0.974386i $$0.427800\pi$$
$$864$$ 5403.22 0.212756
$$865$$ 837.151 0.0329063
$$866$$ 13152.1 0.516083
$$867$$ 0 0
$$868$$ −1502.29 −0.0587455
$$869$$ −2743.38 −0.107092
$$870$$ 4869.51 0.189761
$$871$$ −5380.50 −0.209312
$$872$$ −8800.87 −0.341783
$$873$$ −7444.40 −0.288608
$$874$$ 43208.4 1.67225
$$875$$ −12813.7 −0.495065
$$876$$ −483.113 −0.0186334
$$877$$ −23424.6 −0.901932 −0.450966 0.892541i $$-0.648920\pi$$
−0.450966 + 0.892541i $$0.648920\pi$$
$$878$$ −3771.24 −0.144958
$$879$$ 27919.4 1.07133
$$880$$ 455.351 0.0174430
$$881$$ 2452.52 0.0937882 0.0468941 0.998900i $$-0.485068\pi$$
0.0468941 + 0.998900i $$0.485068\pi$$
$$882$$ −6239.21 −0.238192
$$883$$ 3832.99 0.146082 0.0730409 0.997329i $$-0.476730\pi$$
0.0730409 + 0.997329i $$0.476730\pi$$
$$884$$ 0 0
$$885$$ −3294.24 −0.125124
$$886$$ −17334.6 −0.657298
$$887$$ 13334.4 0.504762 0.252381 0.967628i $$-0.418786\pi$$
0.252381 + 0.967628i $$0.418786\pi$$
$$888$$ −25016.5 −0.945382
$$889$$ 43372.3 1.63629
$$890$$ 5841.84 0.220021
$$891$$ 1676.36 0.0630306
$$892$$ −1699.72 −0.0638014
$$893$$ 75380.6 2.82477
$$894$$ −20618.1 −0.771332
$$895$$ −1605.12 −0.0599478
$$896$$ 29979.1 1.11778
$$897$$ 7652.18 0.284837
$$898$$ −11843.1 −0.440098
$$899$$ −15352.3 −0.569553
$$900$$ −780.476 −0.0289065
$$901$$ 0 0
$$902$$ 555.468 0.0205045
$$903$$ −15384.9 −0.566973
$$904$$ 41982.8 1.54461
$$905$$ 5483.51 0.201412
$$906$$ 22013.1 0.807214
$$907$$ −14432.7 −0.528368 −0.264184 0.964472i $$-0.585103\pi$$
−0.264184 + 0.964472i $$0.585103\pi$$
$$908$$ 2045.91 0.0747753
$$909$$ −3651.89 −0.133251
$$910$$ 2476.15 0.0902018
$$911$$ 50465.5 1.83534 0.917670 0.397343i $$-0.130068\pi$$
0.917670 + 0.397343i $$0.130068\pi$$
$$912$$ −39799.1 −1.44505
$$913$$ 1404.96 0.0509282
$$914$$ 33767.8 1.22203
$$915$$ −6397.81 −0.231153
$$916$$ 886.404 0.0319734
$$917$$ −14134.5 −0.509009
$$918$$ 0 0
$$919$$ −33955.2 −1.21880 −0.609401 0.792862i $$-0.708590\pi$$
−0.609401 + 0.792862i $$0.708590\pi$$
$$920$$ 4922.36 0.176397
$$921$$ −959.057 −0.0343127
$$922$$ −13739.1 −0.490753
$$923$$ −8306.16 −0.296209
$$924$$ −324.896 −0.0115674
$$925$$ 29529.1 1.04963
$$926$$ 14184.9 0.503396
$$927$$ 11504.9 0.407628
$$928$$ −7100.62 −0.251174
$$929$$ 14807.7 0.522954 0.261477 0.965210i $$-0.415790\pi$$
0.261477 + 0.965210i $$0.415790\pi$$
$$930$$ −1858.09 −0.0655154
$$931$$ −45365.6 −1.59699
$$932$$ 3388.76 0.119101
$$933$$ −5604.23 −0.196650
$$934$$ −11900.4 −0.416908
$$935$$ 0 0
$$936$$ 3435.96 0.119987
$$937$$ −5141.33 −0.179253 −0.0896265 0.995975i $$-0.528567\pi$$
−0.0896265 + 0.995975i $$0.528567\pi$$
$$938$$ −20443.2 −0.711614
$$939$$ 1669.52 0.0580220
$$940$$ 767.498 0.0266309
$$941$$ 38958.4 1.34964 0.674819 0.737984i $$-0.264222\pi$$
0.674819 + 0.737984i $$0.264222\pi$$
$$942$$ −3373.42 −0.116679
$$943$$ 5409.57 0.186808
$$944$$ −20812.8 −0.717585
$$945$$ 7961.89 0.274074
$$946$$ −1458.51 −0.0501273
$$947$$ −14191.6 −0.486974 −0.243487 0.969904i $$-0.578291\pi$$
−0.243487 + 0.969904i $$0.578291\pi$$
$$948$$ 2441.35 0.0836408
$$949$$ −2510.50 −0.0858739
$$950$$ 52146.0 1.78088
$$951$$ 2811.27 0.0958586
$$952$$ 0 0
$$953$$ −32646.8 −1.10969 −0.554844 0.831955i $$-0.687222\pi$$
−0.554844 + 0.831955i $$0.687222\pi$$
$$954$$ 2137.12 0.0725280
$$955$$ 433.724 0.0146963
$$956$$ −618.867 −0.0209368
$$957$$ −3320.19 −0.112149
$$958$$ 23360.8 0.787842
$$959$$ −28754.3 −0.968223
$$960$$ −4988.15 −0.167700
$$961$$ −23932.9 −0.803360
$$962$$ −11618.5 −0.389393
$$963$$ −15192.2 −0.508372
$$964$$ 2844.55 0.0950381
$$965$$ 9551.94 0.318640
$$966$$ 29074.5 0.968381
$$967$$ −16183.8 −0.538196 −0.269098 0.963113i $$-0.586726\pi$$
−0.269098 + 0.963113i $$0.586726\pi$$
$$968$$ 31063.4 1.03142
$$969$$ 0 0
$$970$$ 5064.18 0.167630
$$971$$ 28807.7 0.952095 0.476048 0.879420i $$-0.342069\pi$$
0.476048 + 0.879420i $$0.342069\pi$$
$$972$$ 1743.96 0.0575490
$$973$$ −24615.1 −0.811021
$$974$$ −35002.1 −1.15148
$$975$$ 9235.02 0.303341
$$976$$ −40421.0 −1.32566
$$977$$ 3155.66 0.103335 0.0516675 0.998664i $$-0.483546\pi$$
0.0516675 + 0.998664i $$0.483546\pi$$
$$978$$ 19684.4 0.643597
$$979$$ −3983.16 −0.130033
$$980$$ −461.897 −0.0150559
$$981$$ −3072.89 −0.100010
$$982$$ −31588.8 −1.02651
$$983$$ −14818.7 −0.480816 −0.240408 0.970672i $$-0.577281\pi$$
−0.240408 + 0.970672i $$0.577281\pi$$
$$984$$ −5530.85 −0.179184
$$985$$ 5510.83 0.178264
$$986$$ 0 0
$$987$$ 50722.8 1.63579
$$988$$ 2232.84 0.0718989
$$989$$ −14204.1 −0.456688
$$990$$ 176.478 0.00566551
$$991$$ 888.414 0.0284777 0.0142389 0.999899i $$-0.495467\pi$$
0.0142389 + 0.999899i $$0.495467\pi$$
$$992$$ 2709.44 0.0867185
$$993$$ −6192.63 −0.197902
$$994$$ −31559.3 −1.00704
$$995$$ 3444.98 0.109762
$$996$$ −1250.29 −0.0397760
$$997$$ −5116.97 −0.162544 −0.0812718 0.996692i $$-0.525898\pi$$
−0.0812718 + 0.996692i $$0.525898\pi$$
$$998$$ −28806.6 −0.913685
$$999$$ −37358.5 −1.18315
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 289.4.a.g.1.9 12
17.4 even 4 289.4.b.e.288.4 12
17.10 odd 16 17.4.d.a.15.1 yes 12
17.12 odd 16 17.4.d.a.8.1 12
17.13 even 4 289.4.b.e.288.3 12
17.16 even 2 inner 289.4.a.g.1.10 12
51.29 even 16 153.4.l.a.127.3 12
51.44 even 16 153.4.l.a.100.3 12

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.8.1 12 17.12 odd 16
17.4.d.a.15.1 yes 12 17.10 odd 16
153.4.l.a.100.3 12 51.44 even 16
153.4.l.a.127.3 12 51.29 even 16
289.4.a.g.1.9 12 1.1 even 1 trivial
289.4.a.g.1.10 12 17.16 even 2 inner
289.4.b.e.288.3 12 17.13 even 4
289.4.b.e.288.4 12 17.4 even 4