Properties

 Label 735.4.a.p.1.1 Level $735$ Weight $4$ Character 735.1 Self dual yes Analytic conductor $43.366$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 735.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: $$43.3664038542$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-3.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 735.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.53113 q^{2} -3.00000 q^{3} +4.46887 q^{4} -5.00000 q^{5} +10.5934 q^{6} +12.4689 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.53113 q^{2} -3.00000 q^{3} +4.46887 q^{4} -5.00000 q^{5} +10.5934 q^{6} +12.4689 q^{8} +9.00000 q^{9} +17.6556 q^{10} -2.93774 q^{11} -13.4066 q^{12} +19.0623 q^{13} +15.0000 q^{15} -79.7802 q^{16} -122.498 q^{17} -31.7802 q^{18} -107.436 q^{19} -22.3444 q^{20} +10.3735 q^{22} +210.623 q^{23} -37.4066 q^{24} +25.0000 q^{25} -67.3113 q^{26} -27.0000 q^{27} +95.4942 q^{29} -52.9669 q^{30} +94.3074 q^{31} +181.963 q^{32} +8.81323 q^{33} +432.556 q^{34} +40.2198 q^{36} +97.1206 q^{37} +379.370 q^{38} -57.1868 q^{39} -62.3444 q^{40} +491.113 q^{41} -43.0039 q^{43} -13.1284 q^{44} -45.0000 q^{45} -743.735 q^{46} -473.494 q^{47} +239.340 q^{48} -88.2782 q^{50} +367.494 q^{51} +85.1868 q^{52} -183.677 q^{53} +95.3405 q^{54} +14.6887 q^{55} +322.307 q^{57} -337.202 q^{58} +760.615 q^{59} +67.0331 q^{60} +198.747 q^{61} -333.012 q^{62} -4.29373 q^{64} -95.3113 q^{65} -31.1206 q^{66} -309.992 q^{67} -547.428 q^{68} -631.868 q^{69} +665.693 q^{71} +112.220 q^{72} -621.288 q^{73} -342.945 q^{74} -75.0000 q^{75} -480.117 q^{76} +201.934 q^{78} -24.7626 q^{79} +398.901 q^{80} +81.0000 q^{81} -1734.18 q^{82} +406.724 q^{83} +612.490 q^{85} +151.852 q^{86} -286.483 q^{87} -36.6303 q^{88} -261.751 q^{89} +158.901 q^{90} +941.245 q^{92} -282.922 q^{93} +1671.97 q^{94} +537.179 q^{95} -545.889 q^{96} +1004.77 q^{97} -26.4397 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 6 q^{3} + 17 q^{4} - 10 q^{5} - 3 q^{6} + 33 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 - 6 * q^3 + 17 * q^4 - 10 * q^5 - 3 * q^6 + 33 * q^8 + 18 * q^9 $$2 q + q^{2} - 6 q^{3} + 17 q^{4} - 10 q^{5} - 3 q^{6} + 33 q^{8} + 18 q^{9} - 5 q^{10} - 22 q^{11} - 51 q^{12} + 22 q^{13} + 30 q^{15} - 87 q^{16} - 116 q^{17} + 9 q^{18} - 102 q^{19} - 85 q^{20} - 76 q^{22} + 260 q^{23} - 99 q^{24} + 50 q^{25} - 54 q^{26} - 54 q^{27} - 196 q^{29} + 15 q^{30} - 150 q^{31} - 15 q^{32} + 66 q^{33} + 462 q^{34} + 153 q^{36} - 96 q^{37} + 404 q^{38} - 66 q^{39} - 165 q^{40} + 176 q^{41} - 344 q^{43} - 252 q^{44} - 90 q^{45} - 520 q^{46} - 560 q^{47} + 261 q^{48} + 25 q^{50} + 348 q^{51} + 122 q^{52} + 326 q^{53} - 27 q^{54} + 110 q^{55} + 306 q^{57} - 1658 q^{58} + 844 q^{59} + 255 q^{60} + 204 q^{61} - 1440 q^{62} - 839 q^{64} - 110 q^{65} + 228 q^{66} - 104 q^{67} - 466 q^{68} - 780 q^{69} + 1670 q^{71} + 297 q^{72} + 386 q^{73} - 1218 q^{74} - 150 q^{75} - 412 q^{76} + 162 q^{78} - 888 q^{79} + 435 q^{80} + 162 q^{81} - 3162 q^{82} - 928 q^{83} + 580 q^{85} - 1212 q^{86} + 588 q^{87} - 428 q^{88} - 588 q^{89} - 45 q^{90} + 1560 q^{92} + 450 q^{93} + 1280 q^{94} + 510 q^{95} + 45 q^{96} - 522 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 - 6 * q^3 + 17 * q^4 - 10 * q^5 - 3 * q^6 + 33 * q^8 + 18 * q^9 - 5 * q^10 - 22 * q^11 - 51 * q^12 + 22 * q^13 + 30 * q^15 - 87 * q^16 - 116 * q^17 + 9 * q^18 - 102 * q^19 - 85 * q^20 - 76 * q^22 + 260 * q^23 - 99 * q^24 + 50 * q^25 - 54 * q^26 - 54 * q^27 - 196 * q^29 + 15 * q^30 - 150 * q^31 - 15 * q^32 + 66 * q^33 + 462 * q^34 + 153 * q^36 - 96 * q^37 + 404 * q^38 - 66 * q^39 - 165 * q^40 + 176 * q^41 - 344 * q^43 - 252 * q^44 - 90 * q^45 - 520 * q^46 - 560 * q^47 + 261 * q^48 + 25 * q^50 + 348 * q^51 + 122 * q^52 + 326 * q^53 - 27 * q^54 + 110 * q^55 + 306 * q^57 - 1658 * q^58 + 844 * q^59 + 255 * q^60 + 204 * q^61 - 1440 * q^62 - 839 * q^64 - 110 * q^65 + 228 * q^66 - 104 * q^67 - 466 * q^68 - 780 * q^69 + 1670 * q^71 + 297 * q^72 + 386 * q^73 - 1218 * q^74 - 150 * q^75 - 412 * q^76 + 162 * q^78 - 888 * q^79 + 435 * q^80 + 162 * q^81 - 3162 * q^82 - 928 * q^83 + 580 * q^85 - 1212 * q^86 + 588 * q^87 - 428 * q^88 - 588 * q^89 - 45 * q^90 + 1560 * q^92 + 450 * q^93 + 1280 * q^94 + 510 * q^95 + 45 * q^96 - 522 * q^97 - 198 * q^99

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.53113 −1.24844 −0.624221 0.781248i $$-0.714584\pi$$
−0.624221 + 0.781248i $$0.714584\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 4.46887 0.558609
$$5$$ −5.00000 −0.447214
$$6$$ 10.5934 0.720789
$$7$$ 0 0
$$8$$ 12.4689 0.551051
$$9$$ 9.00000 0.333333
$$10$$ 17.6556 0.558320
$$11$$ −2.93774 −0.0805239 −0.0402619 0.999189i $$-0.512819\pi$$
−0.0402619 + 0.999189i $$0.512819\pi$$
$$12$$ −13.4066 −0.322513
$$13$$ 19.0623 0.406686 0.203343 0.979108i $$-0.434819\pi$$
0.203343 + 0.979108i $$0.434819\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ −79.7802 −1.24656
$$17$$ −122.498 −1.74766 −0.873828 0.486236i $$-0.838370\pi$$
−0.873828 + 0.486236i $$0.838370\pi$$
$$18$$ −31.7802 −0.416148
$$19$$ −107.436 −1.29723 −0.648617 0.761115i $$-0.724652\pi$$
−0.648617 + 0.761115i $$0.724652\pi$$
$$20$$ −22.3444 −0.249817
$$21$$ 0 0
$$22$$ 10.3735 0.100529
$$23$$ 210.623 1.90947 0.954736 0.297455i $$-0.0961379\pi$$
0.954736 + 0.297455i $$0.0961379\pi$$
$$24$$ −37.4066 −0.318150
$$25$$ 25.0000 0.200000
$$26$$ −67.3113 −0.507724
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 95.4942 0.611477 0.305738 0.952116i $$-0.401097\pi$$
0.305738 + 0.952116i $$0.401097\pi$$
$$30$$ −52.9669 −0.322346
$$31$$ 94.3074 0.546391 0.273195 0.961959i $$-0.411919\pi$$
0.273195 + 0.961959i $$0.411919\pi$$
$$32$$ 181.963 1.00521
$$33$$ 8.81323 0.0464905
$$34$$ 432.556 2.18185
$$35$$ 0 0
$$36$$ 40.2198 0.186203
$$37$$ 97.1206 0.431528 0.215764 0.976446i $$-0.430776\pi$$
0.215764 + 0.976446i $$0.430776\pi$$
$$38$$ 379.370 1.61952
$$39$$ −57.1868 −0.234800
$$40$$ −62.3444 −0.246438
$$41$$ 491.113 1.87071 0.935353 0.353716i $$-0.115082\pi$$
0.935353 + 0.353716i $$0.115082\pi$$
$$42$$ 0 0
$$43$$ −43.0039 −0.152512 −0.0762562 0.997088i $$-0.524297\pi$$
−0.0762562 + 0.997088i $$0.524297\pi$$
$$44$$ −13.1284 −0.0449814
$$45$$ −45.0000 −0.149071
$$46$$ −743.735 −2.38387
$$47$$ −473.494 −1.46949 −0.734747 0.678341i $$-0.762699\pi$$
−0.734747 + 0.678341i $$0.762699\pi$$
$$48$$ 239.340 0.719705
$$49$$ 0 0
$$50$$ −88.2782 −0.249689
$$51$$ 367.494 1.00901
$$52$$ 85.1868 0.227178
$$53$$ −183.677 −0.476038 −0.238019 0.971261i $$-0.576498\pi$$
−0.238019 + 0.971261i $$0.576498\pi$$
$$54$$ 95.3405 0.240263
$$55$$ 14.6887 0.0360114
$$56$$ 0 0
$$57$$ 322.307 0.748959
$$58$$ −337.202 −0.763394
$$59$$ 760.615 1.67837 0.839183 0.543849i $$-0.183034\pi$$
0.839183 + 0.543849i $$0.183034\pi$$
$$60$$ 67.0331 0.144232
$$61$$ 198.747 0.417163 0.208582 0.978005i $$-0.433115\pi$$
0.208582 + 0.978005i $$0.433115\pi$$
$$62$$ −333.012 −0.682137
$$63$$ 0 0
$$64$$ −4.29373 −0.00838618
$$65$$ −95.3113 −0.181876
$$66$$ −31.1206 −0.0580407
$$67$$ −309.992 −0.565247 −0.282624 0.959231i $$-0.591205\pi$$
−0.282624 + 0.959231i $$0.591205\pi$$
$$68$$ −547.428 −0.976256
$$69$$ −631.868 −1.10243
$$70$$ 0 0
$$71$$ 665.693 1.11272 0.556360 0.830941i $$-0.312197\pi$$
0.556360 + 0.830941i $$0.312197\pi$$
$$72$$ 112.220 0.183684
$$73$$ −621.288 −0.996113 −0.498057 0.867145i $$-0.665953\pi$$
−0.498057 + 0.867145i $$0.665953\pi$$
$$74$$ −342.945 −0.538738
$$75$$ −75.0000 −0.115470
$$76$$ −480.117 −0.724647
$$77$$ 0 0
$$78$$ 201.934 0.293135
$$79$$ −24.7626 −0.0352659 −0.0176330 0.999845i $$-0.505613\pi$$
−0.0176330 + 0.999845i $$0.505613\pi$$
$$80$$ 398.901 0.557481
$$81$$ 81.0000 0.111111
$$82$$ −1734.18 −2.33547
$$83$$ 406.724 0.537876 0.268938 0.963157i $$-0.413327\pi$$
0.268938 + 0.963157i $$0.413327\pi$$
$$84$$ 0 0
$$85$$ 612.490 0.781575
$$86$$ 151.852 0.190403
$$87$$ −286.483 −0.353036
$$88$$ −36.6303 −0.0443728
$$89$$ −261.751 −0.311748 −0.155874 0.987777i $$-0.549819\pi$$
−0.155874 + 0.987777i $$0.549819\pi$$
$$90$$ 158.901 0.186107
$$91$$ 0 0
$$92$$ 941.245 1.06665
$$93$$ −282.922 −0.315459
$$94$$ 1671.97 1.83458
$$95$$ 537.179 0.580141
$$96$$ −545.889 −0.580360
$$97$$ 1004.77 1.05175 0.525873 0.850563i $$-0.323739\pi$$
0.525873 + 0.850563i $$0.323739\pi$$
$$98$$ 0 0
$$99$$ −26.4397 −0.0268413
$$100$$ 111.722 0.111722
$$101$$ 128.872 0.126962 0.0634812 0.997983i $$-0.479780\pi$$
0.0634812 + 0.997983i $$0.479780\pi$$
$$102$$ −1297.67 −1.25969
$$103$$ −806.008 −0.771051 −0.385526 0.922697i $$-0.625980\pi$$
−0.385526 + 0.922697i $$0.625980\pi$$
$$104$$ 237.685 0.224105
$$105$$ 0 0
$$106$$ 648.587 0.594305
$$107$$ −769.712 −0.695429 −0.347714 0.937600i $$-0.613042\pi$$
−0.347714 + 0.937600i $$0.613042\pi$$
$$108$$ −120.660 −0.107504
$$109$$ −780.856 −0.686169 −0.343085 0.939304i $$-0.611472\pi$$
−0.343085 + 0.939304i $$0.611472\pi$$
$$110$$ −51.8677 −0.0449581
$$111$$ −291.362 −0.249143
$$112$$ 0 0
$$113$$ −1115.65 −0.928771 −0.464386 0.885633i $$-0.653725\pi$$
−0.464386 + 0.885633i $$0.653725\pi$$
$$114$$ −1138.11 −0.935032
$$115$$ −1053.11 −0.853942
$$116$$ 426.751 0.341576
$$117$$ 171.560 0.135562
$$118$$ −2685.83 −2.09534
$$119$$ 0 0
$$120$$ 187.033 0.142281
$$121$$ −1322.37 −0.993516
$$122$$ −701.802 −0.520804
$$123$$ −1473.34 −1.08005
$$124$$ 421.448 0.305219
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −1875.98 −1.31076 −0.655381 0.755299i $$-0.727492\pi$$
−0.655381 + 0.755299i $$0.727492\pi$$
$$128$$ −1440.54 −0.994744
$$129$$ 129.012 0.0880530
$$130$$ 336.556 0.227061
$$131$$ −364.203 −0.242905 −0.121452 0.992597i $$-0.538755\pi$$
−0.121452 + 0.992597i $$0.538755\pi$$
$$132$$ 39.3852 0.0259700
$$133$$ 0 0
$$134$$ 1094.62 0.705679
$$135$$ 135.000 0.0860663
$$136$$ −1527.41 −0.963048
$$137$$ 1603.13 0.999743 0.499872 0.866099i $$-0.333380\pi$$
0.499872 + 0.866099i $$0.333380\pi$$
$$138$$ 2231.21 1.37633
$$139$$ −2431.12 −1.48349 −0.741746 0.670681i $$-0.766002\pi$$
−0.741746 + 0.670681i $$0.766002\pi$$
$$140$$ 0 0
$$141$$ 1420.48 0.848413
$$142$$ −2350.65 −1.38917
$$143$$ −56.0000 −0.0327479
$$144$$ −718.021 −0.415522
$$145$$ −477.471 −0.273461
$$146$$ 2193.85 1.24359
$$147$$ 0 0
$$148$$ 434.020 0.241055
$$149$$ 2341.57 1.28744 0.643722 0.765260i $$-0.277389\pi$$
0.643722 + 0.765260i $$0.277389\pi$$
$$150$$ 264.835 0.144158
$$151$$ −2104.07 −1.13395 −0.566976 0.823734i $$-0.691887\pi$$
−0.566976 + 0.823734i $$0.691887\pi$$
$$152$$ −1339.60 −0.714843
$$153$$ −1102.48 −0.582552
$$154$$ 0 0
$$155$$ −471.537 −0.244353
$$156$$ −255.560 −0.131162
$$157$$ 593.467 0.301680 0.150840 0.988558i $$-0.451802\pi$$
0.150840 + 0.988558i $$0.451802\pi$$
$$158$$ 87.4399 0.0440275
$$159$$ 551.031 0.274840
$$160$$ −909.815 −0.449545
$$161$$ 0 0
$$162$$ −286.021 −0.138716
$$163$$ 2178.71 1.04693 0.523465 0.852047i $$-0.324639\pi$$
0.523465 + 0.852047i $$0.324639\pi$$
$$164$$ 2194.72 1.04499
$$165$$ −44.0661 −0.0207912
$$166$$ −1436.19 −0.671508
$$167$$ −799.502 −0.370463 −0.185231 0.982695i $$-0.559303\pi$$
−0.185231 + 0.982695i $$0.559303\pi$$
$$168$$ 0 0
$$169$$ −1833.63 −0.834606
$$170$$ −2162.78 −0.975752
$$171$$ −966.922 −0.432412
$$172$$ −192.179 −0.0851947
$$173$$ 1444.36 0.634754 0.317377 0.948299i $$-0.397198\pi$$
0.317377 + 0.948299i $$0.397198\pi$$
$$174$$ 1011.61 0.440745
$$175$$ 0 0
$$176$$ 234.374 0.100378
$$177$$ −2281.84 −0.969005
$$178$$ 924.276 0.389199
$$179$$ 3343.49 1.39611 0.698056 0.716043i $$-0.254048\pi$$
0.698056 + 0.716043i $$0.254048\pi$$
$$180$$ −201.099 −0.0832725
$$181$$ −2251.81 −0.924729 −0.462365 0.886690i $$-0.652999\pi$$
−0.462365 + 0.886690i $$0.652999\pi$$
$$182$$ 0 0
$$183$$ −596.241 −0.240849
$$184$$ 2626.23 1.05222
$$185$$ −485.603 −0.192985
$$186$$ 999.035 0.393832
$$187$$ 359.868 0.140728
$$188$$ −2115.98 −0.820873
$$189$$ 0 0
$$190$$ −1896.85 −0.724273
$$191$$ −1001.93 −0.379565 −0.189782 0.981826i $$-0.560778\pi$$
−0.189782 + 0.981826i $$0.560778\pi$$
$$192$$ 12.8812 0.00484177
$$193$$ −4054.97 −1.51235 −0.756173 0.654372i $$-0.772933\pi$$
−0.756173 + 0.654372i $$0.772933\pi$$
$$194$$ −3547.99 −1.31304
$$195$$ 285.934 0.105006
$$196$$ 0 0
$$197$$ −5140.23 −1.85902 −0.929508 0.368802i $$-0.879768\pi$$
−0.929508 + 0.368802i $$0.879768\pi$$
$$198$$ 93.3619 0.0335098
$$199$$ −585.631 −0.208614 −0.104307 0.994545i $$-0.533263\pi$$
−0.104307 + 0.994545i $$0.533263\pi$$
$$200$$ 311.722 0.110210
$$201$$ 929.977 0.326346
$$202$$ −455.062 −0.158505
$$203$$ 0 0
$$204$$ 1642.28 0.563642
$$205$$ −2455.56 −0.836605
$$206$$ 2846.12 0.962614
$$207$$ 1895.60 0.636490
$$208$$ −1520.79 −0.506961
$$209$$ 315.619 0.104458
$$210$$ 0 0
$$211$$ −1055.16 −0.344266 −0.172133 0.985074i $$-0.555066\pi$$
−0.172133 + 0.985074i $$0.555066\pi$$
$$212$$ −820.829 −0.265919
$$213$$ −1997.08 −0.642430
$$214$$ 2717.95 0.868203
$$215$$ 215.019 0.0682056
$$216$$ −336.660 −0.106050
$$217$$ 0 0
$$218$$ 2757.30 0.856643
$$219$$ 1863.86 0.575106
$$220$$ 65.6420 0.0201163
$$221$$ −2335.09 −0.710747
$$222$$ 1028.84 0.311040
$$223$$ −4675.85 −1.40412 −0.702059 0.712119i $$-0.747736\pi$$
−0.702059 + 0.712119i $$0.747736\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 3939.49 1.15952
$$227$$ −5443.11 −1.59151 −0.795754 0.605621i $$-0.792925\pi$$
−0.795754 + 0.605621i $$0.792925\pi$$
$$228$$ 1440.35 0.418375
$$229$$ 536.303 0.154759 0.0773797 0.997002i $$-0.475345\pi$$
0.0773797 + 0.997002i $$0.475345\pi$$
$$230$$ 3718.68 1.06610
$$231$$ 0 0
$$232$$ 1190.70 0.336955
$$233$$ −183.490 −0.0515916 −0.0257958 0.999667i $$-0.508212\pi$$
−0.0257958 + 0.999667i $$0.508212\pi$$
$$234$$ −605.802 −0.169241
$$235$$ 2367.47 0.657178
$$236$$ 3399.09 0.937550
$$237$$ 74.2878 0.0203608
$$238$$ 0 0
$$239$$ 643.218 0.174085 0.0870425 0.996205i $$-0.472258\pi$$
0.0870425 + 0.996205i $$0.472258\pi$$
$$240$$ −1196.70 −0.321862
$$241$$ 5755.61 1.53839 0.769194 0.639015i $$-0.220658\pi$$
0.769194 + 0.639015i $$0.220658\pi$$
$$242$$ 4669.46 1.24035
$$243$$ −243.000 −0.0641500
$$244$$ 888.175 0.233031
$$245$$ 0 0
$$246$$ 5202.55 1.34838
$$247$$ −2047.97 −0.527567
$$248$$ 1175.91 0.301089
$$249$$ −1220.17 −0.310543
$$250$$ 441.391 0.111664
$$251$$ 5132.27 1.29062 0.645311 0.763920i $$-0.276728\pi$$
0.645311 + 0.763920i $$0.276728\pi$$
$$252$$ 0 0
$$253$$ −618.755 −0.153758
$$254$$ 6624.34 1.63641
$$255$$ −1837.47 −0.451243
$$256$$ 5121.09 1.25027
$$257$$ −5041.74 −1.22372 −0.611859 0.790967i $$-0.709578\pi$$
−0.611859 + 0.790967i $$0.709578\pi$$
$$258$$ −455.557 −0.109929
$$259$$ 0 0
$$260$$ −425.934 −0.101597
$$261$$ 859.448 0.203826
$$262$$ 1286.05 0.303253
$$263$$ 7577.00 1.77649 0.888246 0.459367i $$-0.151924\pi$$
0.888246 + 0.459367i $$0.151924\pi$$
$$264$$ 109.891 0.0256186
$$265$$ 918.385 0.212890
$$266$$ 0 0
$$267$$ 785.253 0.179988
$$268$$ −1385.32 −0.315752
$$269$$ −1023.10 −0.231893 −0.115947 0.993255i $$-0.536990\pi$$
−0.115947 + 0.993255i $$0.536990\pi$$
$$270$$ −476.702 −0.107449
$$271$$ 2251.98 0.504790 0.252395 0.967624i $$-0.418782\pi$$
0.252395 + 0.967624i $$0.418782\pi$$
$$272$$ 9772.91 2.17857
$$273$$ 0 0
$$274$$ −5660.87 −1.24812
$$275$$ −73.4436 −0.0161048
$$276$$ −2823.74 −0.615829
$$277$$ −8630.72 −1.87209 −0.936047 0.351875i $$-0.885544\pi$$
−0.936047 + 0.351875i $$0.885544\pi$$
$$278$$ 8584.61 1.85205
$$279$$ 848.767 0.182130
$$280$$ 0 0
$$281$$ −7521.62 −1.59680 −0.798402 0.602124i $$-0.794321\pi$$
−0.798402 + 0.602124i $$0.794321\pi$$
$$282$$ −5015.91 −1.05919
$$283$$ −14.8169 −0.00311226 −0.00155613 0.999999i $$-0.500495\pi$$
−0.00155613 + 0.999999i $$0.500495\pi$$
$$284$$ 2974.89 0.621576
$$285$$ −1611.54 −0.334945
$$286$$ 197.743 0.0408839
$$287$$ 0 0
$$288$$ 1637.67 0.335071
$$289$$ 10092.8 2.05430
$$290$$ 1686.01 0.341400
$$291$$ −3014.32 −0.607226
$$292$$ −2776.46 −0.556438
$$293$$ −6913.39 −1.37844 −0.689222 0.724550i $$-0.742048\pi$$
−0.689222 + 0.724550i $$0.742048\pi$$
$$294$$ 0 0
$$295$$ −3803.07 −0.750588
$$296$$ 1210.98 0.237794
$$297$$ 79.3190 0.0154968
$$298$$ −8268.39 −1.60730
$$299$$ 4014.94 0.776555
$$300$$ −335.165 −0.0645026
$$301$$ 0 0
$$302$$ 7429.74 1.41567
$$303$$ −386.615 −0.0733018
$$304$$ 8571.25 1.61709
$$305$$ −993.735 −0.186561
$$306$$ 3893.01 0.727283
$$307$$ 7644.12 1.42108 0.710542 0.703655i $$-0.248450\pi$$
0.710542 + 0.703655i $$0.248450\pi$$
$$308$$ 0 0
$$309$$ 2418.02 0.445167
$$310$$ 1665.06 0.305061
$$311$$ −7593.99 −1.38462 −0.692308 0.721602i $$-0.743406\pi$$
−0.692308 + 0.721602i $$0.743406\pi$$
$$312$$ −713.055 −0.129387
$$313$$ 9127.84 1.64836 0.824179 0.566329i $$-0.191637\pi$$
0.824179 + 0.566329i $$0.191637\pi$$
$$314$$ −2095.61 −0.376631
$$315$$ 0 0
$$316$$ −110.661 −0.0196999
$$317$$ −4929.81 −0.873456 −0.436728 0.899593i $$-0.643863\pi$$
−0.436728 + 0.899593i $$0.643863\pi$$
$$318$$ −1945.76 −0.343122
$$319$$ −280.537 −0.0492385
$$320$$ 21.4686 0.00375042
$$321$$ 2309.14 0.401506
$$322$$ 0 0
$$323$$ 13160.7 2.26712
$$324$$ 361.979 0.0620677
$$325$$ 476.556 0.0813372
$$326$$ −7693.30 −1.30703
$$327$$ 2342.57 0.396160
$$328$$ 6123.62 1.03086
$$329$$ 0 0
$$330$$ 155.603 0.0259566
$$331$$ 1221.67 0.202867 0.101433 0.994842i $$-0.467657\pi$$
0.101433 + 0.994842i $$0.467657\pi$$
$$332$$ 1817.60 0.300463
$$333$$ 874.086 0.143843
$$334$$ 2823.14 0.462502
$$335$$ 1549.96 0.252786
$$336$$ 0 0
$$337$$ −8744.83 −1.41354 −0.706768 0.707446i $$-0.749847\pi$$
−0.706768 + 0.707446i $$0.749847\pi$$
$$338$$ 6474.79 1.04196
$$339$$ 3346.94 0.536226
$$340$$ 2737.14 0.436595
$$341$$ −277.051 −0.0439975
$$342$$ 3414.33 0.539841
$$343$$ 0 0
$$344$$ −536.210 −0.0840421
$$345$$ 3159.34 0.493023
$$346$$ −5100.21 −0.792454
$$347$$ 4589.56 0.710031 0.355015 0.934860i $$-0.384476\pi$$
0.355015 + 0.934860i $$0.384476\pi$$
$$348$$ −1280.25 −0.197209
$$349$$ 3989.89 0.611960 0.305980 0.952038i $$-0.401016\pi$$
0.305980 + 0.952038i $$0.401016\pi$$
$$350$$ 0 0
$$351$$ −514.681 −0.0782668
$$352$$ −534.561 −0.0809437
$$353$$ −2416.35 −0.364333 −0.182166 0.983268i $$-0.558311\pi$$
−0.182166 + 0.983268i $$0.558311\pi$$
$$354$$ 8057.49 1.20975
$$355$$ −3328.46 −0.497624
$$356$$ −1169.73 −0.174145
$$357$$ 0 0
$$358$$ −11806.3 −1.74297
$$359$$ −2756.24 −0.405206 −0.202603 0.979261i $$-0.564940\pi$$
−0.202603 + 0.979261i $$0.564940\pi$$
$$360$$ −561.099 −0.0821459
$$361$$ 4683.45 0.682818
$$362$$ 7951.44 1.15447
$$363$$ 3967.11 0.573607
$$364$$ 0 0
$$365$$ 3106.44 0.445475
$$366$$ 2105.40 0.300687
$$367$$ −11112.8 −1.58061 −0.790307 0.612711i $$-0.790079\pi$$
−0.790307 + 0.612711i $$0.790079\pi$$
$$368$$ −16803.5 −2.38028
$$369$$ 4420.02 0.623569
$$370$$ 1714.73 0.240931
$$371$$ 0 0
$$372$$ −1264.34 −0.176218
$$373$$ 6091.09 0.845535 0.422768 0.906238i $$-0.361059\pi$$
0.422768 + 0.906238i $$0.361059\pi$$
$$374$$ −1270.74 −0.175691
$$375$$ 375.000 0.0516398
$$376$$ −5903.94 −0.809767
$$377$$ 1820.33 0.248679
$$378$$ 0 0
$$379$$ 3984.29 0.539998 0.269999 0.962861i $$-0.412977\pi$$
0.269999 + 0.962861i $$0.412977\pi$$
$$380$$ 2400.58 0.324072
$$381$$ 5627.95 0.756768
$$382$$ 3537.93 0.473865
$$383$$ −318.475 −0.0424890 −0.0212445 0.999774i $$-0.506763\pi$$
−0.0212445 + 0.999774i $$0.506763\pi$$
$$384$$ 4321.63 0.574316
$$385$$ 0 0
$$386$$ 14318.6 1.88808
$$387$$ −387.035 −0.0508374
$$388$$ 4490.21 0.587515
$$389$$ 3885.46 0.506429 0.253214 0.967410i $$-0.418512\pi$$
0.253214 + 0.967410i $$0.418512\pi$$
$$390$$ −1009.67 −0.131094
$$391$$ −25800.9 −3.33710
$$392$$ 0 0
$$393$$ 1092.61 0.140241
$$394$$ 18150.8 2.32088
$$395$$ 123.813 0.0157714
$$396$$ −118.156 −0.0149938
$$397$$ −4806.04 −0.607578 −0.303789 0.952739i $$-0.598252\pi$$
−0.303789 + 0.952739i $$0.598252\pi$$
$$398$$ 2067.94 0.260443
$$399$$ 0 0
$$400$$ −1994.50 −0.249313
$$401$$ 3618.59 0.450633 0.225316 0.974286i $$-0.427658\pi$$
0.225316 + 0.974286i $$0.427658\pi$$
$$402$$ −3283.87 −0.407424
$$403$$ 1797.71 0.222209
$$404$$ 575.911 0.0709223
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −285.315 −0.0347483
$$408$$ 4582.24 0.556016
$$409$$ 2109.05 0.254978 0.127489 0.991840i $$-0.459308\pi$$
0.127489 + 0.991840i $$0.459308\pi$$
$$410$$ 8670.91 1.04445
$$411$$ −4809.40 −0.577202
$$412$$ −3601.94 −0.430716
$$413$$ 0 0
$$414$$ −6693.62 −0.794622
$$415$$ −2033.62 −0.240546
$$416$$ 3468.63 0.408806
$$417$$ 7293.37 0.856494
$$418$$ −1114.49 −0.130410
$$419$$ 6905.91 0.805193 0.402597 0.915377i $$-0.368108\pi$$
0.402597 + 0.915377i $$0.368108\pi$$
$$420$$ 0 0
$$421$$ −9647.54 −1.11685 −0.558423 0.829556i $$-0.688593\pi$$
−0.558423 + 0.829556i $$0.688593\pi$$
$$422$$ 3725.91 0.429797
$$423$$ −4261.45 −0.489831
$$424$$ −2290.25 −0.262321
$$425$$ −3062.45 −0.349531
$$426$$ 7051.94 0.802037
$$427$$ 0 0
$$428$$ −3439.74 −0.388473
$$429$$ 168.000 0.0189070
$$430$$ −759.261 −0.0851508
$$431$$ −13002.7 −1.45318 −0.726589 0.687073i $$-0.758895\pi$$
−0.726589 + 0.687073i $$0.758895\pi$$
$$432$$ 2154.06 0.239902
$$433$$ −7356.07 −0.816420 −0.408210 0.912888i $$-0.633847\pi$$
−0.408210 + 0.912888i $$0.633847\pi$$
$$434$$ 0 0
$$435$$ 1432.41 0.157883
$$436$$ −3489.55 −0.383300
$$437$$ −22628.4 −2.47703
$$438$$ −6581.54 −0.717987
$$439$$ −6909.21 −0.751159 −0.375579 0.926790i $$-0.622556\pi$$
−0.375579 + 0.926790i $$0.622556\pi$$
$$440$$ 183.152 0.0198441
$$441$$ 0 0
$$442$$ 8245.50 0.887327
$$443$$ −14812.6 −1.58864 −0.794318 0.607502i $$-0.792172\pi$$
−0.794318 + 0.607502i $$0.792172\pi$$
$$444$$ −1302.06 −0.139173
$$445$$ 1308.75 0.139418
$$446$$ 16511.0 1.75296
$$447$$ −7024.72 −0.743306
$$448$$ 0 0
$$449$$ −10654.5 −1.11986 −0.559932 0.828538i $$-0.689173\pi$$
−0.559932 + 0.828538i $$0.689173\pi$$
$$450$$ −794.504 −0.0832295
$$451$$ −1442.76 −0.150636
$$452$$ −4985.68 −0.518820
$$453$$ 6312.21 0.654688
$$454$$ 19220.3 1.98691
$$455$$ 0 0
$$456$$ 4018.81 0.412715
$$457$$ −5855.16 −0.599328 −0.299664 0.954045i $$-0.596875\pi$$
−0.299664 + 0.954045i $$0.596875\pi$$
$$458$$ −1893.76 −0.193208
$$459$$ 3307.45 0.336336
$$460$$ −4706.23 −0.477019
$$461$$ −3204.74 −0.323774 −0.161887 0.986809i $$-0.551758\pi$$
−0.161887 + 0.986809i $$0.551758\pi$$
$$462$$ 0 0
$$463$$ 371.658 0.0373054 0.0186527 0.999826i $$-0.494062\pi$$
0.0186527 + 0.999826i $$0.494062\pi$$
$$464$$ −7618.54 −0.762245
$$465$$ 1414.61 0.141077
$$466$$ 647.927 0.0644091
$$467$$ 19752.3 1.95723 0.978614 0.205703i $$-0.0659482\pi$$
0.978614 + 0.205703i $$0.0659482\pi$$
$$468$$ 766.681 0.0757262
$$469$$ 0 0
$$470$$ −8359.84 −0.820449
$$471$$ −1780.40 −0.174175
$$472$$ 9484.01 0.924866
$$473$$ 126.334 0.0122809
$$474$$ −262.320 −0.0254193
$$475$$ −2685.90 −0.259447
$$476$$ 0 0
$$477$$ −1653.09 −0.158679
$$478$$ −2271.28 −0.217335
$$479$$ 20762.0 1.98046 0.990232 0.139433i $$-0.0445279\pi$$
0.990232 + 0.139433i $$0.0445279\pi$$
$$480$$ 2729.45 0.259545
$$481$$ 1851.34 0.175496
$$482$$ −20323.8 −1.92059
$$483$$ 0 0
$$484$$ −5909.50 −0.554987
$$485$$ −5023.87 −0.470355
$$486$$ 858.064 0.0800876
$$487$$ 17647.6 1.64207 0.821035 0.570878i $$-0.193397\pi$$
0.821035 + 0.570878i $$0.193397\pi$$
$$488$$ 2478.15 0.229878
$$489$$ −6536.12 −0.604445
$$490$$ 0 0
$$491$$ −5637.46 −0.518157 −0.259078 0.965856i $$-0.583419\pi$$
−0.259078 + 0.965856i $$0.583419\pi$$
$$492$$ −6584.16 −0.603327
$$493$$ −11697.9 −1.06865
$$494$$ 7231.64 0.658638
$$495$$ 132.198 0.0120038
$$496$$ −7523.86 −0.681112
$$497$$ 0 0
$$498$$ 4308.58 0.387695
$$499$$ −17474.1 −1.56764 −0.783818 0.620991i $$-0.786730\pi$$
−0.783818 + 0.620991i $$0.786730\pi$$
$$500$$ −558.609 −0.0499635
$$501$$ 2398.51 0.213887
$$502$$ −18122.7 −1.61127
$$503$$ −7444.81 −0.659936 −0.329968 0.943992i $$-0.607038\pi$$
−0.329968 + 0.943992i $$0.607038\pi$$
$$504$$ 0 0
$$505$$ −644.358 −0.0567793
$$506$$ 2184.90 0.191958
$$507$$ 5500.89 0.481860
$$508$$ −8383.53 −0.732203
$$509$$ 3384.48 0.294724 0.147362 0.989083i $$-0.452922\pi$$
0.147362 + 0.989083i $$0.452922\pi$$
$$510$$ 6488.35 0.563351
$$511$$ 0 0
$$512$$ −6558.89 −0.566142
$$513$$ 2900.77 0.249653
$$514$$ 17803.0 1.52774
$$515$$ 4030.04 0.344825
$$516$$ 576.536 0.0491872
$$517$$ 1391.00 0.118329
$$518$$ 0 0
$$519$$ −4333.07 −0.366476
$$520$$ −1188.42 −0.100223
$$521$$ −2973.12 −0.250009 −0.125005 0.992156i $$-0.539895\pi$$
−0.125005 + 0.992156i $$0.539895\pi$$
$$522$$ −3034.82 −0.254465
$$523$$ −2689.02 −0.224823 −0.112412 0.993662i $$-0.535858\pi$$
−0.112412 + 0.993662i $$0.535858\pi$$
$$524$$ −1627.57 −0.135689
$$525$$ 0 0
$$526$$ −26755.3 −2.21785
$$527$$ −11552.5 −0.954903
$$528$$ −703.121 −0.0579534
$$529$$ 32194.9 2.64608
$$530$$ −3242.94 −0.265781
$$531$$ 6845.53 0.559455
$$532$$ 0 0
$$533$$ 9361.72 0.760790
$$534$$ −2772.83 −0.224704
$$535$$ 3848.56 0.311005
$$536$$ −3865.25 −0.311480
$$537$$ −10030.5 −0.806046
$$538$$ 3612.69 0.289506
$$539$$ 0 0
$$540$$ 603.298 0.0480774
$$541$$ −14429.5 −1.14671 −0.573356 0.819306i $$-0.694359\pi$$
−0.573356 + 0.819306i $$0.694359\pi$$
$$542$$ −7952.03 −0.630201
$$543$$ 6755.44 0.533893
$$544$$ −22290.1 −1.75677
$$545$$ 3904.28 0.306864
$$546$$ 0 0
$$547$$ 13811.2 1.07957 0.539784 0.841804i $$-0.318506\pi$$
0.539784 + 0.841804i $$0.318506\pi$$
$$548$$ 7164.19 0.558466
$$549$$ 1788.72 0.139054
$$550$$ 259.339 0.0201059
$$551$$ −10259.5 −0.793229
$$552$$ −7878.68 −0.607498
$$553$$ 0 0
$$554$$ 30476.2 2.33720
$$555$$ 1456.81 0.111420
$$556$$ −10864.4 −0.828692
$$557$$ −6033.26 −0.458954 −0.229477 0.973314i $$-0.573702\pi$$
−0.229477 + 0.973314i $$0.573702\pi$$
$$558$$ −2997.10 −0.227379
$$559$$ −819.751 −0.0620246
$$560$$ 0 0
$$561$$ −1079.60 −0.0812493
$$562$$ 26559.8 1.99352
$$563$$ 6958.47 0.520896 0.260448 0.965488i $$-0.416130\pi$$
0.260448 + 0.965488i $$0.416130\pi$$
$$564$$ 6347.95 0.473931
$$565$$ 5578.23 0.415359
$$566$$ 52.3202 0.00388548
$$567$$ 0 0
$$568$$ 8300.44 0.613166
$$569$$ −13396.4 −0.987009 −0.493505 0.869743i $$-0.664284\pi$$
−0.493505 + 0.869743i $$0.664284\pi$$
$$570$$ 5690.55 0.418159
$$571$$ −8055.84 −0.590414 −0.295207 0.955433i $$-0.595389\pi$$
−0.295207 + 0.955433i $$0.595389\pi$$
$$572$$ −250.257 −0.0182933
$$573$$ 3005.78 0.219142
$$574$$ 0 0
$$575$$ 5265.56 0.381894
$$576$$ −38.6435 −0.00279539
$$577$$ 21456.9 1.54812 0.774059 0.633114i $$-0.218223\pi$$
0.774059 + 0.633114i $$0.218223\pi$$
$$578$$ −35638.9 −2.56468
$$579$$ 12164.9 0.873153
$$580$$ −2133.76 −0.152758
$$581$$ 0 0
$$582$$ 10644.0 0.758087
$$583$$ 539.596 0.0383324
$$584$$ −7746.76 −0.548910
$$585$$ −857.802 −0.0606252
$$586$$ 24412.1 1.72091
$$587$$ −20156.3 −1.41728 −0.708638 0.705572i $$-0.750690\pi$$
−0.708638 + 0.705572i $$0.750690\pi$$
$$588$$ 0 0
$$589$$ −10132.0 −0.708797
$$590$$ 13429.1 0.937066
$$591$$ 15420.7 1.07330
$$592$$ −7748.30 −0.537928
$$593$$ −599.307 −0.0415018 −0.0207509 0.999785i $$-0.506606\pi$$
−0.0207509 + 0.999785i $$0.506606\pi$$
$$594$$ −280.086 −0.0193469
$$595$$ 0 0
$$596$$ 10464.2 0.719177
$$597$$ 1756.89 0.120444
$$598$$ −14177.3 −0.969485
$$599$$ −5493.05 −0.374691 −0.187346 0.982294i $$-0.559988\pi$$
−0.187346 + 0.982294i $$0.559988\pi$$
$$600$$ −935.165 −0.0636299
$$601$$ −24292.8 −1.64879 −0.824396 0.566014i $$-0.808485\pi$$
−0.824396 + 0.566014i $$0.808485\pi$$
$$602$$ 0 0
$$603$$ −2789.93 −0.188416
$$604$$ −9402.82 −0.633436
$$605$$ 6611.85 0.444314
$$606$$ 1365.19 0.0915131
$$607$$ −3029.50 −0.202576 −0.101288 0.994857i $$-0.532296\pi$$
−0.101288 + 0.994857i $$0.532296\pi$$
$$608$$ −19549.3 −1.30400
$$609$$ 0 0
$$610$$ 3509.01 0.232911
$$611$$ −9025.87 −0.597623
$$612$$ −4926.85 −0.325419
$$613$$ −19339.6 −1.27426 −0.637129 0.770757i $$-0.719878\pi$$
−0.637129 + 0.770757i $$0.719878\pi$$
$$614$$ −26992.4 −1.77414
$$615$$ 7366.69 0.483014
$$616$$ 0 0
$$617$$ −5743.91 −0.374783 −0.187391 0.982285i $$-0.560003\pi$$
−0.187391 + 0.982285i $$0.560003\pi$$
$$618$$ −8538.35 −0.555765
$$619$$ 8243.35 0.535264 0.267632 0.963521i $$-0.413759\pi$$
0.267632 + 0.963521i $$0.413759\pi$$
$$620$$ −2107.24 −0.136498
$$621$$ −5686.81 −0.367478
$$622$$ 26815.4 1.72861
$$623$$ 0 0
$$624$$ 4562.37 0.292694
$$625$$ 625.000 0.0400000
$$626$$ −32231.6 −2.05788
$$627$$ −946.856 −0.0603091
$$628$$ 2652.13 0.168521
$$629$$ −11897.1 −0.754162
$$630$$ 0 0
$$631$$ −4376.56 −0.276114 −0.138057 0.990424i $$-0.544086\pi$$
−0.138057 + 0.990424i $$0.544086\pi$$
$$632$$ −308.762 −0.0194334
$$633$$ 3165.48 0.198762
$$634$$ 17407.8 1.09046
$$635$$ 9379.92 0.586190
$$636$$ 2462.49 0.153528
$$637$$ 0 0
$$638$$ 990.613 0.0614714
$$639$$ 5991.23 0.370907
$$640$$ 7202.71 0.444863
$$641$$ 11836.6 0.729357 0.364678 0.931133i $$-0.381179\pi$$
0.364678 + 0.931133i $$0.381179\pi$$
$$642$$ −8153.86 −0.501257
$$643$$ −1448.21 −0.0888209 −0.0444104 0.999013i $$-0.514141\pi$$
−0.0444104 + 0.999013i $$0.514141\pi$$
$$644$$ 0 0
$$645$$ −645.058 −0.0393785
$$646$$ −46472.0 −2.83037
$$647$$ 8732.95 0.530646 0.265323 0.964160i $$-0.414521\pi$$
0.265323 + 0.964160i $$0.414521\pi$$
$$648$$ 1009.98 0.0612279
$$649$$ −2234.49 −0.135149
$$650$$ −1682.78 −0.101545
$$651$$ 0 0
$$652$$ 9736.37 0.584824
$$653$$ −21978.4 −1.31712 −0.658562 0.752527i $$-0.728835\pi$$
−0.658562 + 0.752527i $$0.728835\pi$$
$$654$$ −8271.91 −0.494583
$$655$$ 1821.01 0.108630
$$656$$ −39181.1 −2.33196
$$657$$ −5591.59 −0.332038
$$658$$ 0 0
$$659$$ −27761.7 −1.64103 −0.820516 0.571623i $$-0.806314\pi$$
−0.820516 + 0.571623i $$0.806314\pi$$
$$660$$ −196.926 −0.0116141
$$661$$ 8573.72 0.504507 0.252254 0.967661i $$-0.418828\pi$$
0.252254 + 0.967661i $$0.418828\pi$$
$$662$$ −4313.86 −0.253267
$$663$$ 7005.27 0.410350
$$664$$ 5071.39 0.296398
$$665$$ 0 0
$$666$$ −3086.51 −0.179579
$$667$$ 20113.2 1.16760
$$668$$ −3572.87 −0.206944
$$669$$ 14027.6 0.810668
$$670$$ −5473.11 −0.315589
$$671$$ −583.868 −0.0335916
$$672$$ 0 0
$$673$$ 27159.2 1.55559 0.777795 0.628518i $$-0.216338\pi$$
0.777795 + 0.628518i $$0.216338\pi$$
$$674$$ 30879.1 1.76472
$$675$$ −675.000 −0.0384900
$$676$$ −8194.26 −0.466219
$$677$$ 1392.30 0.0790404 0.0395202 0.999219i $$-0.487417\pi$$
0.0395202 + 0.999219i $$0.487417\pi$$
$$678$$ −11818.5 −0.669448
$$679$$ 0 0
$$680$$ 7637.06 0.430688
$$681$$ 16329.3 0.918857
$$682$$ 978.302 0.0549283
$$683$$ −8675.09 −0.486007 −0.243004 0.970025i $$-0.578133\pi$$
−0.243004 + 0.970025i $$0.578133\pi$$
$$684$$ −4321.05 −0.241549
$$685$$ −8015.66 −0.447099
$$686$$ 0 0
$$687$$ −1608.91 −0.0893504
$$688$$ 3430.86 0.190117
$$689$$ −3501.30 −0.193598
$$690$$ −11156.0 −0.615511
$$691$$ 21426.0 1.17957 0.589785 0.807561i $$-0.299213\pi$$
0.589785 + 0.807561i $$0.299213\pi$$
$$692$$ 6454.65 0.354579
$$693$$ 0 0
$$694$$ −16206.3 −0.886433
$$695$$ 12155.6 0.663438
$$696$$ −3572.11 −0.194541
$$697$$ −60160.4 −3.26935
$$698$$ −14088.8 −0.763997
$$699$$ 550.470 0.0297864
$$700$$ 0 0
$$701$$ 24840.5 1.33839 0.669197 0.743085i $$-0.266638\pi$$
0.669197 + 0.743085i $$0.266638\pi$$
$$702$$ 1817.40 0.0977116
$$703$$ −10434.2 −0.559793
$$704$$ 12.6139 0.000675288 0
$$705$$ −7102.41 −0.379422
$$706$$ 8532.45 0.454848
$$707$$ 0 0
$$708$$ −10197.3 −0.541295
$$709$$ 12525.0 0.663450 0.331725 0.943376i $$-0.392369\pi$$
0.331725 + 0.943376i $$0.392369\pi$$
$$710$$ 11753.2 0.621255
$$711$$ −222.863 −0.0117553
$$712$$ −3263.74 −0.171789
$$713$$ 19863.3 1.04332
$$714$$ 0 0
$$715$$ 280.000 0.0146453
$$716$$ 14941.6 0.779881
$$717$$ −1929.65 −0.100508
$$718$$ 9732.66 0.505877
$$719$$ −28085.0 −1.45674 −0.728369 0.685185i $$-0.759721\pi$$
−0.728369 + 0.685185i $$0.759721\pi$$
$$720$$ 3590.11 0.185827
$$721$$ 0 0
$$722$$ −16537.9 −0.852460
$$723$$ −17266.8 −0.888189
$$724$$ −10063.1 −0.516562
$$725$$ 2387.35 0.122295
$$726$$ −14008.4 −0.716115
$$727$$ 14326.2 0.730851 0.365426 0.930841i $$-0.380923\pi$$
0.365426 + 0.930841i $$0.380923\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −10969.2 −0.556150
$$731$$ 5267.89 0.266539
$$732$$ −2664.53 −0.134541
$$733$$ −6727.85 −0.339016 −0.169508 0.985529i $$-0.554218\pi$$
−0.169508 + 0.985529i $$0.554218\pi$$
$$734$$ 39240.8 1.97331
$$735$$ 0 0
$$736$$ 38325.5 1.91943
$$737$$ 910.677 0.0455159
$$738$$ −15607.6 −0.778490
$$739$$ −3418.51 −0.170165 −0.0850826 0.996374i $$-0.527115\pi$$
−0.0850826 + 0.996374i $$0.527115\pi$$
$$740$$ −2170.10 −0.107803
$$741$$ 6143.91 0.304591
$$742$$ 0 0
$$743$$ 8095.50 0.399724 0.199862 0.979824i $$-0.435951\pi$$
0.199862 + 0.979824i $$0.435951\pi$$
$$744$$ −3527.72 −0.173834
$$745$$ −11707.9 −0.575762
$$746$$ −21508.4 −1.05560
$$747$$ 3660.51 0.179292
$$748$$ 1608.20 0.0786119
$$749$$ 0 0
$$750$$ −1324.17 −0.0644693
$$751$$ 13446.8 0.653371 0.326686 0.945133i $$-0.394068\pi$$
0.326686 + 0.945133i $$0.394068\pi$$
$$752$$ 37775.4 1.83182
$$753$$ −15396.8 −0.745141
$$754$$ −6427.84 −0.310462
$$755$$ 10520.4 0.507119
$$756$$ 0 0
$$757$$ −2593.24 −0.124508 −0.0622541 0.998060i $$-0.519829\pi$$
−0.0622541 + 0.998060i $$0.519829\pi$$
$$758$$ −14069.0 −0.674156
$$759$$ 1856.26 0.0887722
$$760$$ 6698.02 0.319688
$$761$$ −27079.4 −1.28992 −0.644959 0.764217i $$-0.723125\pi$$
−0.644959 + 0.764217i $$0.723125\pi$$
$$762$$ −19873.0 −0.944782
$$763$$ 0 0
$$764$$ −4477.48 −0.212028
$$765$$ 5512.41 0.260525
$$766$$ 1124.58 0.0530451
$$767$$ 14499.0 0.682568
$$768$$ −15363.3 −0.721842
$$769$$ −2138.72 −0.100292 −0.0501458 0.998742i $$-0.515969\pi$$
−0.0501458 + 0.998742i $$0.515969\pi$$
$$770$$ 0 0
$$771$$ 15125.2 0.706513
$$772$$ −18121.1 −0.844810
$$773$$ −25864.0 −1.20345 −0.601724 0.798704i $$-0.705519\pi$$
−0.601724 + 0.798704i $$0.705519\pi$$
$$774$$ 1366.67 0.0634676
$$775$$ 2357.69 0.109278
$$776$$ 12528.4 0.579566
$$777$$ 0 0
$$778$$ −13720.1 −0.632247
$$779$$ −52763.1 −2.42675
$$780$$ 1277.80 0.0586572
$$781$$ −1955.63 −0.0896006
$$782$$ 91106.2 4.16618
$$783$$ −2578.34 −0.117679
$$784$$ 0 0
$$785$$ −2967.33 −0.134916
$$786$$ −3858.14 −0.175083
$$787$$ 32371.3 1.46621 0.733107 0.680113i $$-0.238069\pi$$
0.733107 + 0.680113i $$0.238069\pi$$
$$788$$ −22971.0 −1.03846
$$789$$ −22731.0 −1.02566
$$790$$ −437.200 −0.0196897
$$791$$ 0 0
$$792$$ −329.673 −0.0147909
$$793$$ 3788.57 0.169654
$$794$$ 16970.8 0.758526
$$795$$ −2755.16 −0.122912
$$796$$ −2617.11 −0.116534
$$797$$ −2024.33 −0.0899691 −0.0449845 0.998988i $$-0.514324\pi$$
−0.0449845 + 0.998988i $$0.514324\pi$$
$$798$$ 0 0
$$799$$ 58002.1 2.56817
$$800$$ 4549.08 0.201043
$$801$$ −2355.76 −0.103916
$$802$$ −12777.7 −0.562589
$$803$$ 1825.18 0.0802109
$$804$$ 4155.95 0.182300
$$805$$ 0 0
$$806$$ −6347.95 −0.277416
$$807$$ 3069.29 0.133884
$$808$$ 1606.88 0.0699628
$$809$$ 12391.7 0.538526 0.269263 0.963067i $$-0.413220\pi$$
0.269263 + 0.963067i $$0.413220\pi$$
$$810$$ 1430.11 0.0620356
$$811$$ −14654.5 −0.634511 −0.317256 0.948340i $$-0.602761\pi$$
−0.317256 + 0.948340i $$0.602761\pi$$
$$812$$ 0 0
$$813$$ −6755.94 −0.291441
$$814$$ 1007.49 0.0433813
$$815$$ −10893.5 −0.468201
$$816$$ −29318.7 −1.25780
$$817$$ 4620.16 0.197844
$$818$$ −7447.33 −0.318325
$$819$$ 0 0
$$820$$ −10973.6 −0.467335
$$821$$ 23887.9 1.01546 0.507731 0.861516i $$-0.330485\pi$$
0.507731 + 0.861516i $$0.330485\pi$$
$$822$$ 16982.6 0.720604
$$823$$ −4008.41 −0.169774 −0.0848871 0.996391i $$-0.527053\pi$$
−0.0848871 + 0.996391i $$0.527053\pi$$
$$824$$ −10050.0 −0.424889
$$825$$ 220.331 0.00929810
$$826$$ 0 0
$$827$$ −45110.4 −1.89679 −0.948394 0.317096i $$-0.897292\pi$$
−0.948394 + 0.317096i $$0.897292\pi$$
$$828$$ 8471.21 0.355549
$$829$$ −16165.4 −0.677260 −0.338630 0.940920i $$-0.609964\pi$$
−0.338630 + 0.940920i $$0.609964\pi$$
$$830$$ 7180.97 0.300307
$$831$$ 25892.2 1.08085
$$832$$ −81.8481 −0.00341054
$$833$$ 0 0
$$834$$ −25753.8 −1.06928
$$835$$ 3997.51 0.165676
$$836$$ 1410.46 0.0583514
$$837$$ −2546.30 −0.105153
$$838$$ −24385.7 −1.00524
$$839$$ 25244.4 1.03878 0.519388 0.854538i $$-0.326160\pi$$
0.519388 + 0.854538i $$0.326160\pi$$
$$840$$ 0 0
$$841$$ −15269.9 −0.626096
$$842$$ 34066.7 1.39432
$$843$$ 22564.9 0.921916
$$844$$ −4715.37 −0.192310
$$845$$ 9168.15 0.373247
$$846$$ 15047.7 0.611526
$$847$$ 0 0
$$848$$ 14653.8 0.593412
$$849$$ 44.4506 0.00179687
$$850$$ 10813.9 0.436370
$$851$$ 20455.8 0.823990
$$852$$ −8924.68 −0.358867
$$853$$ 30168.1 1.21094 0.605472 0.795867i $$-0.292984\pi$$
0.605472 + 0.795867i $$0.292984\pi$$
$$854$$ 0 0
$$855$$ 4834.61 0.193380
$$856$$ −9597.44 −0.383217
$$857$$ 13393.6 0.533857 0.266929 0.963716i $$-0.413991\pi$$
0.266929 + 0.963716i $$0.413991\pi$$
$$858$$ −593.230 −0.0236043
$$859$$ −19060.4 −0.757081 −0.378541 0.925585i $$-0.623574\pi$$
−0.378541 + 0.925585i $$0.623574\pi$$
$$860$$ 960.894 0.0381002
$$861$$ 0 0
$$862$$ 45914.3 1.81421
$$863$$ −9466.86 −0.373413 −0.186707 0.982416i $$-0.559781\pi$$
−0.186707 + 0.982416i $$0.559781\pi$$
$$864$$ −4913.00 −0.193453
$$865$$ −7221.79 −0.283871
$$866$$ 25975.2 1.01925
$$867$$ −30278.3 −1.18605
$$868$$ 0 0
$$869$$ 72.7461 0.00283975
$$870$$ −5058.03 −0.197107
$$871$$ −5909.15 −0.229878
$$872$$ −9736.39 −0.378115
$$873$$ 9042.97 0.350582
$$874$$ 79903.8 3.09243
$$875$$ 0 0
$$876$$ 8329.37 0.321259
$$877$$ 37740.6 1.45315 0.726573 0.687090i $$-0.241112\pi$$
0.726573 + 0.687090i $$0.241112\pi$$
$$878$$ 24397.3 0.937778
$$879$$ 20740.2 0.795845
$$880$$ −1171.87 −0.0448905
$$881$$ −25991.5 −0.993957 −0.496979 0.867763i $$-0.665557\pi$$
−0.496979 + 0.867763i $$0.665557\pi$$
$$882$$ 0 0
$$883$$ 39420.3 1.50238 0.751189 0.660087i $$-0.229481\pi$$
0.751189 + 0.660087i $$0.229481\pi$$
$$884$$ −10435.2 −0.397030
$$885$$ 11409.2 0.433352
$$886$$ 52305.0 1.98332
$$887$$ −46005.2 −1.74149 −0.870745 0.491735i $$-0.836363\pi$$
−0.870745 + 0.491735i $$0.836363\pi$$
$$888$$ −3632.95 −0.137290
$$889$$ 0 0
$$890$$ −4621.38 −0.174055
$$891$$ −237.957 −0.00894710
$$892$$ −20895.8 −0.784353
$$893$$ 50870.2 1.90628
$$894$$ 24805.2 0.927975
$$895$$ −16717.5 −0.624361
$$896$$ 0 0
$$897$$ −12044.8 −0.448345
$$898$$ 37622.6 1.39809
$$899$$ 9005.81 0.334105
$$900$$ 1005.50 0.0372406
$$901$$ 22500.1 0.831950
$$902$$ 5094.58 0.188061
$$903$$ 0 0
$$904$$ −13910.8 −0.511801
$$905$$ 11259.1 0.413551
$$906$$ −22289.2 −0.817340
$$907$$ −2838.97 −0.103932 −0.0519661 0.998649i $$-0.516549\pi$$
−0.0519661 + 0.998649i $$0.516549\pi$$
$$908$$ −24324.6 −0.889030
$$909$$ 1159.84 0.0423208
$$910$$ 0 0
$$911$$ 39890.9 1.45076 0.725382 0.688347i $$-0.241663\pi$$
0.725382 + 0.688347i $$0.241663\pi$$
$$912$$ −25713.7 −0.933626
$$913$$ −1194.85 −0.0433119
$$914$$ 20675.3 0.748226
$$915$$ 2981.21 0.107711
$$916$$ 2396.67 0.0864500
$$917$$ 0 0
$$918$$ −11679.0 −0.419897
$$919$$ −646.475 −0.0232048 −0.0116024 0.999933i $$-0.503693\pi$$
−0.0116024 + 0.999933i $$0.503693\pi$$
$$920$$ −13131.1 −0.470566
$$921$$ −22932.4 −0.820463
$$922$$ 11316.3 0.404213
$$923$$ 12689.6 0.452528
$$924$$ 0 0
$$925$$ 2428.02 0.0863056
$$926$$ −1312.37 −0.0465737
$$927$$ −7254.07 −0.257017
$$928$$ 17376.4 0.614665
$$929$$ −51188.2 −1.80778 −0.903892 0.427760i $$-0.859303\pi$$
−0.903892 + 0.427760i $$0.859303\pi$$
$$930$$ −4995.17 −0.176127
$$931$$ 0 0
$$932$$ −819.993 −0.0288195
$$933$$ 22782.0 0.799409
$$934$$ −69747.8 −2.44349
$$935$$ −1799.34 −0.0629355
$$936$$ 2139.16 0.0747017
$$937$$ −29786.1 −1.03849 −0.519247 0.854624i $$-0.673788\pi$$
−0.519247 + 0.854624i $$0.673788\pi$$
$$938$$ 0 0
$$939$$ −27383.5 −0.951680
$$940$$ 10579.9 0.367105
$$941$$ 44817.4 1.55261 0.776304 0.630358i $$-0.217092\pi$$
0.776304 + 0.630358i $$0.217092\pi$$
$$942$$ 6286.82 0.217448
$$943$$ 103439. 3.57206
$$944$$ −60682.0 −2.09219
$$945$$ 0 0
$$946$$ −446.103 −0.0153320
$$947$$ 54697.1 1.87689 0.938446 0.345425i $$-0.112265\pi$$
0.938446 + 0.345425i $$0.112265\pi$$
$$948$$ 331.983 0.0113737
$$949$$ −11843.2 −0.405105
$$950$$ 9484.24 0.323905
$$951$$ 14789.4 0.504290
$$952$$ 0 0
$$953$$ −7577.51 −0.257565 −0.128783 0.991673i $$-0.541107\pi$$
−0.128783 + 0.991673i $$0.541107\pi$$
$$954$$ 5837.29 0.198102
$$955$$ 5009.63 0.169746
$$956$$ 2874.46 0.0972454
$$957$$ 841.612 0.0284278
$$958$$ −73313.4 −2.47249
$$959$$ 0 0
$$960$$ −64.4059 −0.00216530
$$961$$ −20897.1 −0.701457
$$962$$ −6537.32 −0.219097
$$963$$ −6927.41 −0.231810
$$964$$ 25721.1 0.859357
$$965$$ 20274.8 0.676342
$$966$$ 0 0
$$967$$ 50779.0 1.68867 0.844334 0.535817i $$-0.179996\pi$$
0.844334 + 0.535817i $$0.179996\pi$$
$$968$$ −16488.5 −0.547478
$$969$$ −39482.0 −1.30892
$$970$$ 17739.9 0.587212
$$971$$ 15313.2 0.506102 0.253051 0.967453i $$-0.418566\pi$$
0.253051 + 0.967453i $$0.418566\pi$$
$$972$$ −1085.94 −0.0358348
$$973$$ 0 0
$$974$$ −62315.9 −2.05003
$$975$$ −1429.67 −0.0469601
$$976$$ −15856.1 −0.520021
$$977$$ 46620.4 1.52663 0.763316 0.646025i $$-0.223570\pi$$
0.763316 + 0.646025i $$0.223570\pi$$
$$978$$ 23079.9 0.754615
$$979$$ 768.957 0.0251031
$$980$$ 0 0
$$981$$ −7027.70 −0.228723
$$982$$ 19906.6 0.646889
$$983$$ 2824.37 0.0916414 0.0458207 0.998950i $$-0.485410\pi$$
0.0458207 + 0.998950i $$0.485410\pi$$
$$984$$ −18370.9 −0.595165
$$985$$ 25701.1 0.831377
$$986$$ 41306.6 1.33415
$$987$$ 0 0
$$988$$ −9152.11 −0.294704
$$989$$ −9057.59 −0.291218
$$990$$ −466.810 −0.0149860
$$991$$ 16951.4 0.543370 0.271685 0.962386i $$-0.412419\pi$$
0.271685 + 0.962386i $$0.412419\pi$$
$$992$$ 17160.5 0.549239
$$993$$ −3665.00 −0.117125
$$994$$ 0 0
$$995$$ 2928.15 0.0932952
$$996$$ −5452.79 −0.173472
$$997$$ −23847.8 −0.757540 −0.378770 0.925491i $$-0.623653\pi$$
−0.378770 + 0.925491i $$0.623653\pi$$
$$998$$ 61703.4 1.95710
$$999$$ −2622.26 −0.0830476
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 735.4.a.p.1.1 2
3.2 odd 2 2205.4.a.z.1.2 2
7.6 odd 2 105.4.a.f.1.1 2
21.20 even 2 315.4.a.i.1.2 2
28.27 even 2 1680.4.a.bg.1.1 2
35.13 even 4 525.4.d.h.274.3 4
35.27 even 4 525.4.d.h.274.2 4
35.34 odd 2 525.4.a.k.1.2 2
105.104 even 2 1575.4.a.w.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 7.6 odd 2
315.4.a.i.1.2 2 21.20 even 2
525.4.a.k.1.2 2 35.34 odd 2
525.4.d.h.274.2 4 35.27 even 4
525.4.d.h.274.3 4 35.13 even 4
735.4.a.p.1.1 2 1.1 even 1 trivial
1575.4.a.w.1.1 2 105.104 even 2
1680.4.a.bg.1.1 2 28.27 even 2
2205.4.a.z.1.2 2 3.2 odd 2