# Properties

 Label 1225.4.a.a.1.1 Level $1225$ Weight $4$ Character 1225.1 Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$72.2773397570$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +6.00000 q^{6} +21.0000 q^{8} -23.0000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +6.00000 q^{6} +21.0000 q^{8} -23.0000 q^{9} -45.0000 q^{11} -2.00000 q^{12} +59.0000 q^{13} -71.0000 q^{16} -54.0000 q^{17} +69.0000 q^{18} +121.000 q^{19} +135.000 q^{22} -69.0000 q^{23} -42.0000 q^{24} -177.000 q^{26} +100.000 q^{27} -162.000 q^{29} +88.0000 q^{31} +45.0000 q^{32} +90.0000 q^{33} +162.000 q^{34} -23.0000 q^{36} +259.000 q^{37} -363.000 q^{38} -118.000 q^{39} -195.000 q^{41} +286.000 q^{43} -45.0000 q^{44} +207.000 q^{46} +45.0000 q^{47} +142.000 q^{48} +108.000 q^{51} +59.0000 q^{52} -597.000 q^{53} -300.000 q^{54} -242.000 q^{57} +486.000 q^{58} +360.000 q^{59} -392.000 q^{61} -264.000 q^{62} +433.000 q^{64} -270.000 q^{66} +280.000 q^{67} -54.0000 q^{68} +138.000 q^{69} +48.0000 q^{71} -483.000 q^{72} +668.000 q^{73} -777.000 q^{74} +121.000 q^{76} +354.000 q^{78} +782.000 q^{79} +421.000 q^{81} +585.000 q^{82} +768.000 q^{83} -858.000 q^{86} +324.000 q^{87} -945.000 q^{88} +1194.00 q^{89} -69.0000 q^{92} -176.000 q^{93} -135.000 q^{94} -90.0000 q^{96} +902.000 q^{97} +1035.00 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ −2.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 1.00000 0.125000
$$5$$ 0 0
$$6$$ 6.00000 0.408248
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ −23.0000 −0.851852
$$10$$ 0 0
$$11$$ −45.0000 −1.23346 −0.616728 0.787177i $$-0.711542\pi$$
−0.616728 + 0.787177i $$0.711542\pi$$
$$12$$ −2.00000 −0.0481125
$$13$$ 59.0000 1.25874 0.629371 0.777105i $$-0.283312\pi$$
0.629371 + 0.777105i $$0.283312\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ 69.0000 0.903525
$$19$$ 121.000 1.46102 0.730508 0.682904i $$-0.239283\pi$$
0.730508 + 0.682904i $$0.239283\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 135.000 1.30828
$$23$$ −69.0000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ −42.0000 −0.357217
$$25$$ 0 0
$$26$$ −177.000 −1.33510
$$27$$ 100.000 0.712778
$$28$$ 0 0
$$29$$ −162.000 −1.03733 −0.518666 0.854977i $$-0.673571\pi$$
−0.518666 + 0.854977i $$0.673571\pi$$
$$30$$ 0 0
$$31$$ 88.0000 0.509847 0.254924 0.966961i $$-0.417950\pi$$
0.254924 + 0.966961i $$0.417950\pi$$
$$32$$ 45.0000 0.248592
$$33$$ 90.0000 0.474757
$$34$$ 162.000 0.817140
$$35$$ 0 0
$$36$$ −23.0000 −0.106481
$$37$$ 259.000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ −363.000 −1.54964
$$39$$ −118.000 −0.484490
$$40$$ 0 0
$$41$$ −195.000 −0.742778 −0.371389 0.928477i $$-0.621118\pi$$
−0.371389 + 0.928477i $$0.621118\pi$$
$$42$$ 0 0
$$43$$ 286.000 1.01429 0.507146 0.861860i $$-0.330700\pi$$
0.507146 + 0.861860i $$0.330700\pi$$
$$44$$ −45.0000 −0.154182
$$45$$ 0 0
$$46$$ 207.000 0.663489
$$47$$ 45.0000 0.139658 0.0698290 0.997559i $$-0.477755\pi$$
0.0698290 + 0.997559i $$0.477755\pi$$
$$48$$ 142.000 0.426999
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 108.000 0.296530
$$52$$ 59.0000 0.157343
$$53$$ −597.000 −1.54725 −0.773625 0.633644i $$-0.781559\pi$$
−0.773625 + 0.633644i $$0.781559\pi$$
$$54$$ −300.000 −0.756015
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −242.000 −0.562345
$$58$$ 486.000 1.10026
$$59$$ 360.000 0.794373 0.397187 0.917738i $$-0.369987\pi$$
0.397187 + 0.917738i $$0.369987\pi$$
$$60$$ 0 0
$$61$$ −392.000 −0.822794 −0.411397 0.911456i $$-0.634959\pi$$
−0.411397 + 0.911456i $$0.634959\pi$$
$$62$$ −264.000 −0.540775
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ 0 0
$$66$$ −270.000 −0.503556
$$67$$ 280.000 0.510559 0.255279 0.966867i $$-0.417833\pi$$
0.255279 + 0.966867i $$0.417833\pi$$
$$68$$ −54.0000 −0.0963009
$$69$$ 138.000 0.240772
$$70$$ 0 0
$$71$$ 48.0000 0.0802331 0.0401166 0.999195i $$-0.487227\pi$$
0.0401166 + 0.999195i $$0.487227\pi$$
$$72$$ −483.000 −0.790585
$$73$$ 668.000 1.07101 0.535503 0.844533i $$-0.320122\pi$$
0.535503 + 0.844533i $$0.320122\pi$$
$$74$$ −777.000 −1.22060
$$75$$ 0 0
$$76$$ 121.000 0.182627
$$77$$ 0 0
$$78$$ 354.000 0.513880
$$79$$ 782.000 1.11369 0.556847 0.830615i $$-0.312011\pi$$
0.556847 + 0.830615i $$0.312011\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ 585.000 0.787835
$$83$$ 768.000 1.01565 0.507825 0.861460i $$-0.330450\pi$$
0.507825 + 0.861460i $$0.330450\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −858.000 −1.07582
$$87$$ 324.000 0.399269
$$88$$ −945.000 −1.14474
$$89$$ 1194.00 1.42206 0.711032 0.703159i $$-0.248228\pi$$
0.711032 + 0.703159i $$0.248228\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −69.0000 −0.0781929
$$93$$ −176.000 −0.196240
$$94$$ −135.000 −0.148130
$$95$$ 0 0
$$96$$ −90.0000 −0.0956832
$$97$$ 902.000 0.944167 0.472084 0.881554i $$-0.343502\pi$$
0.472084 + 0.881554i $$0.343502\pi$$
$$98$$ 0 0
$$99$$ 1035.00 1.05072
$$100$$ 0 0
$$101$$ −684.000 −0.673867 −0.336933 0.941528i $$-0.609390\pi$$
−0.336933 + 0.941528i $$0.609390\pi$$
$$102$$ −324.000 −0.314517
$$103$$ −1516.00 −1.45025 −0.725126 0.688616i $$-0.758218\pi$$
−0.725126 + 0.688616i $$0.758218\pi$$
$$104$$ 1239.00 1.16821
$$105$$ 0 0
$$106$$ 1791.00 1.64111
$$107$$ 732.000 0.661356 0.330678 0.943744i $$-0.392723\pi$$
0.330678 + 0.943744i $$0.392723\pi$$
$$108$$ 100.000 0.0890973
$$109$$ −1600.00 −1.40598 −0.702992 0.711198i $$-0.748153\pi$$
−0.702992 + 0.711198i $$0.748153\pi$$
$$110$$ 0 0
$$111$$ −518.000 −0.442940
$$112$$ 0 0
$$113$$ 1392.00 1.15883 0.579417 0.815031i $$-0.303280\pi$$
0.579417 + 0.815031i $$0.303280\pi$$
$$114$$ 726.000 0.596457
$$115$$ 0 0
$$116$$ −162.000 −0.129667
$$117$$ −1357.00 −1.07226
$$118$$ −1080.00 −0.842560
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 694.000 0.521412
$$122$$ 1176.00 0.872705
$$123$$ 390.000 0.285895
$$124$$ 88.0000 0.0637309
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −803.000 −0.561061 −0.280530 0.959845i $$-0.590510\pi$$
−0.280530 + 0.959845i $$0.590510\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ −572.000 −0.390401
$$130$$ 0 0
$$131$$ −2019.00 −1.34657 −0.673286 0.739382i $$-0.735118\pi$$
−0.673286 + 0.739382i $$0.735118\pi$$
$$132$$ 90.0000 0.0593447
$$133$$ 0 0
$$134$$ −840.000 −0.541529
$$135$$ 0 0
$$136$$ −1134.00 −0.714998
$$137$$ −60.0000 −0.0374171 −0.0187086 0.999825i $$-0.505955\pi$$
−0.0187086 + 0.999825i $$0.505955\pi$$
$$138$$ −414.000 −0.255377
$$139$$ 1708.00 1.04224 0.521118 0.853485i $$-0.325515\pi$$
0.521118 + 0.853485i $$0.325515\pi$$
$$140$$ 0 0
$$141$$ −90.0000 −0.0537544
$$142$$ −144.000 −0.0851001
$$143$$ −2655.00 −1.55260
$$144$$ 1633.00 0.945023
$$145$$ 0 0
$$146$$ −2004.00 −1.13597
$$147$$ 0 0
$$148$$ 259.000 0.143849
$$149$$ −1086.00 −0.597105 −0.298552 0.954393i $$-0.596504\pi$$
−0.298552 + 0.954393i $$0.596504\pi$$
$$150$$ 0 0
$$151$$ −2866.00 −1.54458 −0.772291 0.635269i $$-0.780889\pi$$
−0.772291 + 0.635269i $$0.780889\pi$$
$$152$$ 2541.00 1.35594
$$153$$ 1242.00 0.656273
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −118.000 −0.0605613
$$157$$ −229.000 −0.116409 −0.0582044 0.998305i $$-0.518538\pi$$
−0.0582044 + 0.998305i $$0.518538\pi$$
$$158$$ −2346.00 −1.18125
$$159$$ 1194.00 0.595537
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1263.00 −0.612535
$$163$$ 1228.00 0.590088 0.295044 0.955484i $$-0.404666\pi$$
0.295044 + 0.955484i $$0.404666\pi$$
$$164$$ −195.000 −0.0928472
$$165$$ 0 0
$$166$$ −2304.00 −1.07726
$$167$$ −1929.00 −0.893835 −0.446918 0.894575i $$-0.647478\pi$$
−0.446918 + 0.894575i $$0.647478\pi$$
$$168$$ 0 0
$$169$$ 1284.00 0.584433
$$170$$ 0 0
$$171$$ −2783.00 −1.24457
$$172$$ 286.000 0.126787
$$173$$ −699.000 −0.307191 −0.153595 0.988134i $$-0.549085\pi$$
−0.153595 + 0.988134i $$0.549085\pi$$
$$174$$ −972.000 −0.423489
$$175$$ 0 0
$$176$$ 3195.00 1.36836
$$177$$ −720.000 −0.305754
$$178$$ −3582.00 −1.50833
$$179$$ 3117.00 1.30154 0.650770 0.759275i $$-0.274446\pi$$
0.650770 + 0.759275i $$0.274446\pi$$
$$180$$ 0 0
$$181$$ 1798.00 0.738366 0.369183 0.929357i $$-0.379638\pi$$
0.369183 + 0.929357i $$0.379638\pi$$
$$182$$ 0 0
$$183$$ 784.000 0.316694
$$184$$ −1449.00 −0.580553
$$185$$ 0 0
$$186$$ 528.000 0.208144
$$187$$ 2430.00 0.950263
$$188$$ 45.0000 0.0174572
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2388.00 −0.904658 −0.452329 0.891851i $$-0.649407\pi$$
−0.452329 + 0.891851i $$0.649407\pi$$
$$192$$ −866.000 −0.325511
$$193$$ −272.000 −0.101446 −0.0507228 0.998713i $$-0.516152\pi$$
−0.0507228 + 0.998713i $$0.516152\pi$$
$$194$$ −2706.00 −1.00144
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2109.00 0.762741 0.381371 0.924422i $$-0.375452\pi$$
0.381371 + 0.924422i $$0.375452\pi$$
$$198$$ −3105.00 −1.11446
$$199$$ −1424.00 −0.507260 −0.253630 0.967301i $$-0.581625\pi$$
−0.253630 + 0.967301i $$0.581625\pi$$
$$200$$ 0 0
$$201$$ −560.000 −0.196514
$$202$$ 2052.00 0.714744
$$203$$ 0 0
$$204$$ 108.000 0.0370662
$$205$$ 0 0
$$206$$ 4548.00 1.53822
$$207$$ 1587.00 0.532870
$$208$$ −4189.00 −1.39642
$$209$$ −5445.00 −1.80210
$$210$$ 0 0
$$211$$ −3625.00 −1.18273 −0.591363 0.806405i $$-0.701410\pi$$
−0.591363 + 0.806405i $$0.701410\pi$$
$$212$$ −597.000 −0.193406
$$213$$ −96.0000 −0.0308817
$$214$$ −2196.00 −0.701474
$$215$$ 0 0
$$216$$ 2100.00 0.661513
$$217$$ 0 0
$$218$$ 4800.00 1.49127
$$219$$ −1336.00 −0.412231
$$220$$ 0 0
$$221$$ −3186.00 −0.969745
$$222$$ 1554.00 0.469809
$$223$$ −4960.00 −1.48944 −0.744722 0.667374i $$-0.767418\pi$$
−0.744722 + 0.667374i $$0.767418\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −4176.00 −1.22913
$$227$$ −1500.00 −0.438584 −0.219292 0.975659i $$-0.570375\pi$$
−0.219292 + 0.975659i $$0.570375\pi$$
$$228$$ −242.000 −0.0702932
$$229$$ −6092.00 −1.75795 −0.878975 0.476867i $$-0.841772\pi$$
−0.878975 + 0.476867i $$0.841772\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −3402.00 −0.962725
$$233$$ −138.000 −0.0388012 −0.0194006 0.999812i $$-0.506176\pi$$
−0.0194006 + 0.999812i $$0.506176\pi$$
$$234$$ 4071.00 1.13731
$$235$$ 0 0
$$236$$ 360.000 0.0992966
$$237$$ −1564.00 −0.428661
$$238$$ 0 0
$$239$$ −5502.00 −1.48910 −0.744550 0.667567i $$-0.767336\pi$$
−0.744550 + 0.667567i $$0.767336\pi$$
$$240$$ 0 0
$$241$$ −3551.00 −0.949129 −0.474564 0.880221i $$-0.657394\pi$$
−0.474564 + 0.880221i $$0.657394\pi$$
$$242$$ −2082.00 −0.553041
$$243$$ −3542.00 −0.935059
$$244$$ −392.000 −0.102849
$$245$$ 0 0
$$246$$ −1170.00 −0.303238
$$247$$ 7139.00 1.83904
$$248$$ 1848.00 0.473178
$$249$$ −1536.00 −0.390924
$$250$$ 0 0
$$251$$ −7065.00 −1.77665 −0.888324 0.459216i $$-0.848130\pi$$
−0.888324 + 0.459216i $$0.848130\pi$$
$$252$$ 0 0
$$253$$ 3105.00 0.771580
$$254$$ 2409.00 0.595095
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ −4080.00 −0.990286 −0.495143 0.868812i $$-0.664884\pi$$
−0.495143 + 0.868812i $$0.664884\pi$$
$$258$$ 1716.00 0.414083
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 3726.00 0.883654
$$262$$ 6057.00 1.42825
$$263$$ 3288.00 0.770900 0.385450 0.922729i $$-0.374046\pi$$
0.385450 + 0.922729i $$0.374046\pi$$
$$264$$ 1890.00 0.440612
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2388.00 −0.547353
$$268$$ 280.000 0.0638199
$$269$$ 3264.00 0.739813 0.369906 0.929069i $$-0.379390\pi$$
0.369906 + 0.929069i $$0.379390\pi$$
$$270$$ 0 0
$$271$$ 2752.00 0.616871 0.308436 0.951245i $$-0.400195\pi$$
0.308436 + 0.951245i $$0.400195\pi$$
$$272$$ 3834.00 0.854671
$$273$$ 0 0
$$274$$ 180.000 0.0396869
$$275$$ 0 0
$$276$$ 138.000 0.0300965
$$277$$ 4690.00 1.01731 0.508655 0.860971i $$-0.330143\pi$$
0.508655 + 0.860971i $$0.330143\pi$$
$$278$$ −5124.00 −1.10546
$$279$$ −2024.00 −0.434314
$$280$$ 0 0
$$281$$ 7821.00 1.66036 0.830181 0.557494i $$-0.188237\pi$$
0.830181 + 0.557494i $$0.188237\pi$$
$$282$$ 270.000 0.0570151
$$283$$ −658.000 −0.138212 −0.0691061 0.997609i $$-0.522015\pi$$
−0.0691061 + 0.997609i $$0.522015\pi$$
$$284$$ 48.0000 0.0100291
$$285$$ 0 0
$$286$$ 7965.00 1.64678
$$287$$ 0 0
$$288$$ −1035.00 −0.211764
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ −1804.00 −0.363410
$$292$$ 668.000 0.133876
$$293$$ −5997.00 −1.19573 −0.597864 0.801597i $$-0.703984\pi$$
−0.597864 + 0.801597i $$0.703984\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 5439.00 1.06803
$$297$$ −4500.00 −0.879180
$$298$$ 3258.00 0.633325
$$299$$ −4071.00 −0.787398
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8598.00 1.63828
$$303$$ 1368.00 0.259371
$$304$$ −8591.00 −1.62081
$$305$$ 0 0
$$306$$ −3726.00 −0.696082
$$307$$ −6226.00 −1.15745 −0.578724 0.815523i $$-0.696449\pi$$
−0.578724 + 0.815523i $$0.696449\pi$$
$$308$$ 0 0
$$309$$ 3032.00 0.558202
$$310$$ 0 0
$$311$$ −4680.00 −0.853307 −0.426653 0.904415i $$-0.640308\pi$$
−0.426653 + 0.904415i $$0.640308\pi$$
$$312$$ −2478.00 −0.449645
$$313$$ 1028.00 0.185642 0.0928211 0.995683i $$-0.470412\pi$$
0.0928211 + 0.995683i $$0.470412\pi$$
$$314$$ 687.000 0.123470
$$315$$ 0 0
$$316$$ 782.000 0.139212
$$317$$ −8622.00 −1.52763 −0.763817 0.645433i $$-0.776677\pi$$
−0.763817 + 0.645433i $$0.776677\pi$$
$$318$$ −3582.00 −0.631662
$$319$$ 7290.00 1.27950
$$320$$ 0 0
$$321$$ −1464.00 −0.254556
$$322$$ 0 0
$$323$$ −6534.00 −1.12558
$$324$$ 421.000 0.0721879
$$325$$ 0 0
$$326$$ −3684.00 −0.625883
$$327$$ 3200.00 0.541163
$$328$$ −4095.00 −0.689355
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1999.00 −0.331949 −0.165974 0.986130i $$-0.553077\pi$$
−0.165974 + 0.986130i $$0.553077\pi$$
$$332$$ 768.000 0.126956
$$333$$ −5957.00 −0.980305
$$334$$ 5787.00 0.948056
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −5114.00 −0.826639 −0.413319 0.910586i $$-0.635631\pi$$
−0.413319 + 0.910586i $$0.635631\pi$$
$$338$$ −3852.00 −0.619885
$$339$$ −2784.00 −0.446036
$$340$$ 0 0
$$341$$ −3960.00 −0.628874
$$342$$ 8349.00 1.32006
$$343$$ 0 0
$$344$$ 6006.00 0.941342
$$345$$ 0 0
$$346$$ 2097.00 0.325825
$$347$$ −4320.00 −0.668328 −0.334164 0.942515i $$-0.608454\pi$$
−0.334164 + 0.942515i $$0.608454\pi$$
$$348$$ 324.000 0.0499087
$$349$$ −7922.00 −1.21506 −0.607529 0.794298i $$-0.707839\pi$$
−0.607529 + 0.794298i $$0.707839\pi$$
$$350$$ 0 0
$$351$$ 5900.00 0.897204
$$352$$ −2025.00 −0.306627
$$353$$ 828.000 0.124844 0.0624221 0.998050i $$-0.480118\pi$$
0.0624221 + 0.998050i $$0.480118\pi$$
$$354$$ 2160.00 0.324301
$$355$$ 0 0
$$356$$ 1194.00 0.177758
$$357$$ 0 0
$$358$$ −9351.00 −1.38049
$$359$$ −1350.00 −0.198469 −0.0992344 0.995064i $$-0.531639\pi$$
−0.0992344 + 0.995064i $$0.531639\pi$$
$$360$$ 0 0
$$361$$ 7782.00 1.13457
$$362$$ −5394.00 −0.783156
$$363$$ −1388.00 −0.200692
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2352.00 −0.335904
$$367$$ 2801.00 0.398395 0.199198 0.979959i $$-0.436166\pi$$
0.199198 + 0.979959i $$0.436166\pi$$
$$368$$ 4899.00 0.693962
$$369$$ 4485.00 0.632737
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −176.000 −0.0245300
$$373$$ −6602.00 −0.916457 −0.458229 0.888834i $$-0.651516\pi$$
−0.458229 + 0.888834i $$0.651516\pi$$
$$374$$ −7290.00 −1.00791
$$375$$ 0 0
$$376$$ 945.000 0.129613
$$377$$ −9558.00 −1.30573
$$378$$ 0 0
$$379$$ −8305.00 −1.12559 −0.562796 0.826596i $$-0.690274\pi$$
−0.562796 + 0.826596i $$0.690274\pi$$
$$380$$ 0 0
$$381$$ 1606.00 0.215952
$$382$$ 7164.00 0.959534
$$383$$ 945.000 0.126076 0.0630382 0.998011i $$-0.479921\pi$$
0.0630382 + 0.998011i $$0.479921\pi$$
$$384$$ 3318.00 0.440940
$$385$$ 0 0
$$386$$ 816.000 0.107599
$$387$$ −6578.00 −0.864027
$$388$$ 902.000 0.118021
$$389$$ 12036.0 1.56876 0.784382 0.620278i $$-0.212980\pi$$
0.784382 + 0.620278i $$0.212980\pi$$
$$390$$ 0 0
$$391$$ 3726.00 0.481923
$$392$$ 0 0
$$393$$ 4038.00 0.518296
$$394$$ −6327.00 −0.809009
$$395$$ 0 0
$$396$$ 1035.00 0.131340
$$397$$ −2698.00 −0.341080 −0.170540 0.985351i $$-0.554551\pi$$
−0.170540 + 0.985351i $$0.554551\pi$$
$$398$$ 4272.00 0.538030
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7053.00 0.878329 0.439165 0.898407i $$-0.355274\pi$$
0.439165 + 0.898407i $$0.355274\pi$$
$$402$$ 1680.00 0.208435
$$403$$ 5192.00 0.641767
$$404$$ −684.000 −0.0842333
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −11655.0 −1.41945
$$408$$ 2268.00 0.275203
$$409$$ 10870.0 1.31415 0.657074 0.753826i $$-0.271794\pi$$
0.657074 + 0.753826i $$0.271794\pi$$
$$410$$ 0 0
$$411$$ 120.000 0.0144019
$$412$$ −1516.00 −0.181281
$$413$$ 0 0
$$414$$ −4761.00 −0.565194
$$415$$ 0 0
$$416$$ 2655.00 0.312914
$$417$$ −3416.00 −0.401156
$$418$$ 16335.0 1.91141
$$419$$ 9729.00 1.13435 0.567175 0.823597i $$-0.308036\pi$$
0.567175 + 0.823597i $$0.308036\pi$$
$$420$$ 0 0
$$421$$ −12550.0 −1.45285 −0.726425 0.687246i $$-0.758819\pi$$
−0.726425 + 0.687246i $$0.758819\pi$$
$$422$$ 10875.0 1.25447
$$423$$ −1035.00 −0.118968
$$424$$ −12537.0 −1.43597
$$425$$ 0 0
$$426$$ 288.000 0.0327550
$$427$$ 0 0
$$428$$ 732.000 0.0826695
$$429$$ 5310.00 0.597597
$$430$$ 0 0
$$431$$ 2988.00 0.333937 0.166969 0.985962i $$-0.446602\pi$$
0.166969 + 0.985962i $$0.446602\pi$$
$$432$$ −7100.00 −0.790738
$$433$$ 16616.0 1.84414 0.922072 0.387019i $$-0.126495\pi$$
0.922072 + 0.387019i $$0.126495\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1600.00 −0.175748
$$437$$ −8349.00 −0.913929
$$438$$ 4008.00 0.437237
$$439$$ −7346.00 −0.798646 −0.399323 0.916810i $$-0.630755\pi$$
−0.399323 + 0.916810i $$0.630755\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 9558.00 1.02857
$$443$$ −12.0000 −0.00128699 −0.000643496 1.00000i $$-0.500205\pi$$
−0.000643496 1.00000i $$0.500205\pi$$
$$444$$ −518.000 −0.0553675
$$445$$ 0 0
$$446$$ 14880.0 1.57979
$$447$$ 2172.00 0.229826
$$448$$ 0 0
$$449$$ 9669.00 1.01628 0.508138 0.861275i $$-0.330334\pi$$
0.508138 + 0.861275i $$0.330334\pi$$
$$450$$ 0 0
$$451$$ 8775.00 0.916183
$$452$$ 1392.00 0.144854
$$453$$ 5732.00 0.594510
$$454$$ 4500.00 0.465188
$$455$$ 0 0
$$456$$ −5082.00 −0.521900
$$457$$ 9634.00 0.986126 0.493063 0.869994i $$-0.335877\pi$$
0.493063 + 0.869994i $$0.335877\pi$$
$$458$$ 18276.0 1.86459
$$459$$ −5400.00 −0.549129
$$460$$ 0 0
$$461$$ 342.000 0.0345521 0.0172761 0.999851i $$-0.494501\pi$$
0.0172761 + 0.999851i $$0.494501\pi$$
$$462$$ 0 0
$$463$$ −2411.00 −0.242006 −0.121003 0.992652i $$-0.538611\pi$$
−0.121003 + 0.992652i $$0.538611\pi$$
$$464$$ 11502.0 1.15079
$$465$$ 0 0
$$466$$ 414.000 0.0411549
$$467$$ −1206.00 −0.119501 −0.0597506 0.998213i $$-0.519031\pi$$
−0.0597506 + 0.998213i $$0.519031\pi$$
$$468$$ −1357.00 −0.134033
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 458.000 0.0448058
$$472$$ 7560.00 0.737240
$$473$$ −12870.0 −1.25109
$$474$$ 4692.00 0.454664
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 13731.0 1.31803
$$478$$ 16506.0 1.57943
$$479$$ 432.000 0.0412079 0.0206039 0.999788i $$-0.493441\pi$$
0.0206039 + 0.999788i $$0.493441\pi$$
$$480$$ 0 0
$$481$$ 15281.0 1.44855
$$482$$ 10653.0 1.00670
$$483$$ 0 0
$$484$$ 694.000 0.0651766
$$485$$ 0 0
$$486$$ 10626.0 0.991780
$$487$$ 11896.0 1.10690 0.553449 0.832883i $$-0.313311\pi$$
0.553449 + 0.832883i $$0.313311\pi$$
$$488$$ −8232.00 −0.763617
$$489$$ −2456.00 −0.227125
$$490$$ 0 0
$$491$$ −12276.0 −1.12833 −0.564163 0.825663i $$-0.690801\pi$$
−0.564163 + 0.825663i $$0.690801\pi$$
$$492$$ 390.000 0.0357369
$$493$$ 8748.00 0.799169
$$494$$ −21417.0 −1.95060
$$495$$ 0 0
$$496$$ −6248.00 −0.565612
$$497$$ 0 0
$$498$$ 4608.00 0.414637
$$499$$ −10876.0 −0.975705 −0.487852 0.872926i $$-0.662220\pi$$
−0.487852 + 0.872926i $$0.662220\pi$$
$$500$$ 0 0
$$501$$ 3858.00 0.344037
$$502$$ 21195.0 1.88442
$$503$$ 12000.0 1.06372 0.531862 0.846831i $$-0.321492\pi$$
0.531862 + 0.846831i $$0.321492\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −9315.00 −0.818384
$$507$$ −2568.00 −0.224948
$$508$$ −803.000 −0.0701326
$$509$$ 11682.0 1.01728 0.508640 0.860979i $$-0.330148\pi$$
0.508640 + 0.860979i $$0.330148\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 8733.00 0.753804
$$513$$ 12100.0 1.04138
$$514$$ 12240.0 1.05036
$$515$$ 0 0
$$516$$ −572.000 −0.0488002
$$517$$ −2025.00 −0.172262
$$518$$ 0 0
$$519$$ 1398.00 0.118238
$$520$$ 0 0
$$521$$ −9609.00 −0.808019 −0.404010 0.914755i $$-0.632384\pi$$
−0.404010 + 0.914755i $$0.632384\pi$$
$$522$$ −11178.0 −0.937256
$$523$$ 21188.0 1.77148 0.885742 0.464177i $$-0.153650\pi$$
0.885742 + 0.464177i $$0.153650\pi$$
$$524$$ −2019.00 −0.168321
$$525$$ 0 0
$$526$$ −9864.00 −0.817663
$$527$$ −4752.00 −0.392790
$$528$$ −6390.00 −0.526684
$$529$$ −7406.00 −0.608696
$$530$$ 0 0
$$531$$ −8280.00 −0.676688
$$532$$ 0 0
$$533$$ −11505.0 −0.934966
$$534$$ 7164.00 0.580555
$$535$$ 0 0
$$536$$ 5880.00 0.473838
$$537$$ −6234.00 −0.500963
$$538$$ −9792.00 −0.784690
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8072.00 0.641483 0.320742 0.947167i $$-0.396068\pi$$
0.320742 + 0.947167i $$0.396068\pi$$
$$542$$ −8256.00 −0.654291
$$543$$ −3596.00 −0.284197
$$544$$ −2430.00 −0.191517
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −344.000 −0.0268892 −0.0134446 0.999910i $$-0.504280\pi$$
−0.0134446 + 0.999910i $$0.504280\pi$$
$$548$$ −60.0000 −0.00467714
$$549$$ 9016.00 0.700899
$$550$$ 0 0
$$551$$ −19602.0 −1.51556
$$552$$ 2898.00 0.223455
$$553$$ 0 0
$$554$$ −14070.0 −1.07902
$$555$$ 0 0
$$556$$ 1708.00 0.130279
$$557$$ −18363.0 −1.39689 −0.698443 0.715666i $$-0.746123\pi$$
−0.698443 + 0.715666i $$0.746123\pi$$
$$558$$ 6072.00 0.460660
$$559$$ 16874.0 1.27673
$$560$$ 0 0
$$561$$ −4860.00 −0.365756
$$562$$ −23463.0 −1.76108
$$563$$ −6294.00 −0.471155 −0.235578 0.971856i $$-0.575698\pi$$
−0.235578 + 0.971856i $$0.575698\pi$$
$$564$$ −90.0000 −0.00671930
$$565$$ 0 0
$$566$$ 1974.00 0.146596
$$567$$ 0 0
$$568$$ 1008.00 0.0744626
$$569$$ 11733.0 0.864452 0.432226 0.901765i $$-0.357728\pi$$
0.432226 + 0.901765i $$0.357728\pi$$
$$570$$ 0 0
$$571$$ 1052.00 0.0771013 0.0385506 0.999257i $$-0.487726\pi$$
0.0385506 + 0.999257i $$0.487726\pi$$
$$572$$ −2655.00 −0.194075
$$573$$ 4776.00 0.348203
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −9959.00 −0.720414
$$577$$ −13156.0 −0.949205 −0.474603 0.880200i $$-0.657408\pi$$
−0.474603 + 0.880200i $$0.657408\pi$$
$$578$$ 5991.00 0.431129
$$579$$ 544.000 0.0390464
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 5412.00 0.385455
$$583$$ 26865.0 1.90846
$$584$$ 14028.0 0.993977
$$585$$ 0 0
$$586$$ 17991.0 1.26826
$$587$$ −13368.0 −0.939960 −0.469980 0.882677i $$-0.655739\pi$$
−0.469980 + 0.882677i $$0.655739\pi$$
$$588$$ 0 0
$$589$$ 10648.0 0.744895
$$590$$ 0 0
$$591$$ −4218.00 −0.293579
$$592$$ −18389.0 −1.27666
$$593$$ 26664.0 1.84647 0.923237 0.384231i $$-0.125533\pi$$
0.923237 + 0.384231i $$0.125533\pi$$
$$594$$ 13500.0 0.932511
$$595$$ 0 0
$$596$$ −1086.00 −0.0746381
$$597$$ 2848.00 0.195244
$$598$$ 12213.0 0.835162
$$599$$ 7614.00 0.519365 0.259682 0.965694i $$-0.416382\pi$$
0.259682 + 0.965694i $$0.416382\pi$$
$$600$$ 0 0
$$601$$ −6410.00 −0.435057 −0.217529 0.976054i $$-0.569800\pi$$
−0.217529 + 0.976054i $$0.569800\pi$$
$$602$$ 0 0
$$603$$ −6440.00 −0.434921
$$604$$ −2866.00 −0.193073
$$605$$ 0 0
$$606$$ −4104.00 −0.275105
$$607$$ −21469.0 −1.43558 −0.717792 0.696257i $$-0.754847\pi$$
−0.717792 + 0.696257i $$0.754847\pi$$
$$608$$ 5445.00 0.363197
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2655.00 0.175793
$$612$$ 1242.00 0.0820341
$$613$$ −3737.00 −0.246225 −0.123113 0.992393i $$-0.539288\pi$$
−0.123113 + 0.992393i $$0.539288\pi$$
$$614$$ 18678.0 1.22766
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18078.0 −1.17957 −0.589784 0.807561i $$-0.700787\pi$$
−0.589784 + 0.807561i $$0.700787\pi$$
$$618$$ −9096.00 −0.592063
$$619$$ −12287.0 −0.797829 −0.398915 0.916988i $$-0.630613\pi$$
−0.398915 + 0.916988i $$0.630613\pi$$
$$620$$ 0 0
$$621$$ −6900.00 −0.445874
$$622$$ 14040.0 0.905069
$$623$$ 0 0
$$624$$ 8378.00 0.537481
$$625$$ 0 0
$$626$$ −3084.00 −0.196903
$$627$$ 10890.0 0.693628
$$628$$ −229.000 −0.0145511
$$629$$ −13986.0 −0.886579
$$630$$ 0 0
$$631$$ −9580.00 −0.604396 −0.302198 0.953245i $$-0.597720\pi$$
−0.302198 + 0.953245i $$0.597720\pi$$
$$632$$ 16422.0 1.03360
$$633$$ 7250.00 0.455232
$$634$$ 25866.0 1.62030
$$635$$ 0 0
$$636$$ 1194.00 0.0744421
$$637$$ 0 0
$$638$$ −21870.0 −1.35712
$$639$$ −1104.00 −0.0683467
$$640$$ 0 0
$$641$$ 10779.0 0.664189 0.332094 0.943246i $$-0.392245\pi$$
0.332094 + 0.943246i $$0.392245\pi$$
$$642$$ 4392.00 0.269998
$$643$$ 8882.00 0.544746 0.272373 0.962192i $$-0.412191\pi$$
0.272373 + 0.962192i $$0.412191\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 19602.0 1.19386
$$647$$ −11019.0 −0.669554 −0.334777 0.942297i $$-0.608661\pi$$
−0.334777 + 0.942297i $$0.608661\pi$$
$$648$$ 8841.00 0.535968
$$649$$ −16200.0 −0.979824
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 1228.00 0.0737610
$$653$$ −22323.0 −1.33777 −0.668887 0.743364i $$-0.733229\pi$$
−0.668887 + 0.743364i $$0.733229\pi$$
$$654$$ −9600.00 −0.573990
$$655$$ 0 0
$$656$$ 13845.0 0.824019
$$657$$ −15364.0 −0.912339
$$658$$ 0 0
$$659$$ −11856.0 −0.700826 −0.350413 0.936595i $$-0.613959\pi$$
−0.350413 + 0.936595i $$0.613959\pi$$
$$660$$ 0 0
$$661$$ 33244.0 1.95619 0.978095 0.208158i $$-0.0667470\pi$$
0.978095 + 0.208158i $$0.0667470\pi$$
$$662$$ 5997.00 0.352085
$$663$$ 6372.00 0.373255
$$664$$ 16128.0 0.942602
$$665$$ 0 0
$$666$$ 17871.0 1.03977
$$667$$ 11178.0 0.648896
$$668$$ −1929.00 −0.111729
$$669$$ 9920.00 0.573288
$$670$$ 0 0
$$671$$ 17640.0 1.01488
$$672$$ 0 0
$$673$$ 12322.0 0.705763 0.352881 0.935668i $$-0.385202\pi$$
0.352881 + 0.935668i $$0.385202\pi$$
$$674$$ 15342.0 0.876783
$$675$$ 0 0
$$676$$ 1284.00 0.0730542
$$677$$ −12597.0 −0.715129 −0.357564 0.933889i $$-0.616393\pi$$
−0.357564 + 0.933889i $$0.616393\pi$$
$$678$$ 8352.00 0.473092
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 3000.00 0.168811
$$682$$ 11880.0 0.667022
$$683$$ 8340.00 0.467235 0.233617 0.972329i $$-0.424944\pi$$
0.233617 + 0.972329i $$0.424944\pi$$
$$684$$ −2783.00 −0.155571
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 12184.0 0.676636
$$688$$ −20306.0 −1.12523
$$689$$ −35223.0 −1.94759
$$690$$ 0 0
$$691$$ 20200.0 1.11208 0.556038 0.831157i $$-0.312321\pi$$
0.556038 + 0.831157i $$0.312321\pi$$
$$692$$ −699.000 −0.0383988
$$693$$ 0 0
$$694$$ 12960.0 0.708869
$$695$$ 0 0
$$696$$ 6804.00 0.370553
$$697$$ 10530.0 0.572241
$$698$$ 23766.0 1.28876
$$699$$ 276.000 0.0149346
$$700$$ 0 0
$$701$$ 474.000 0.0255388 0.0127694 0.999918i $$-0.495935\pi$$
0.0127694 + 0.999918i $$0.495935\pi$$
$$702$$ −17700.0 −0.951629
$$703$$ 31339.0 1.68133
$$704$$ −19485.0 −1.04314
$$705$$ 0 0
$$706$$ −2484.00 −0.132417
$$707$$ 0 0
$$708$$ −720.000 −0.0382193
$$709$$ −25126.0 −1.33093 −0.665463 0.746431i $$-0.731766\pi$$
−0.665463 + 0.746431i $$0.731766\pi$$
$$710$$ 0 0
$$711$$ −17986.0 −0.948703
$$712$$ 25074.0 1.31979
$$713$$ −6072.00 −0.318932
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3117.00 0.162692
$$717$$ 11004.0 0.573155
$$718$$ 4050.00 0.210508
$$719$$ 7296.00 0.378435 0.189218 0.981935i $$-0.439405\pi$$
0.189218 + 0.981935i $$0.439405\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −23346.0 −1.20339
$$723$$ 7102.00 0.365320
$$724$$ 1798.00 0.0922958
$$725$$ 0 0
$$726$$ 4164.00 0.212866
$$727$$ −15421.0 −0.786703 −0.393352 0.919388i $$-0.628684\pi$$
−0.393352 + 0.919388i $$0.628684\pi$$
$$728$$ 0 0
$$729$$ −4283.00 −0.217599
$$730$$ 0 0
$$731$$ −15444.0 −0.781419
$$732$$ 784.000 0.0395867
$$733$$ −29167.0 −1.46972 −0.734862 0.678217i $$-0.762753\pi$$
−0.734862 + 0.678217i $$0.762753\pi$$
$$734$$ −8403.00 −0.422562
$$735$$ 0 0
$$736$$ −3105.00 −0.155505
$$737$$ −12600.0 −0.629752
$$738$$ −13455.0 −0.671118
$$739$$ −13381.0 −0.666073 −0.333037 0.942914i $$-0.608073\pi$$
−0.333037 + 0.942914i $$0.608073\pi$$
$$740$$ 0 0
$$741$$ −14278.0 −0.707848
$$742$$ 0 0
$$743$$ −5487.00 −0.270927 −0.135463 0.990782i $$-0.543252\pi$$
−0.135463 + 0.990782i $$0.543252\pi$$
$$744$$ −3696.00 −0.182126
$$745$$ 0 0
$$746$$ 19806.0 0.972050
$$747$$ −17664.0 −0.865183
$$748$$ 2430.00 0.118783
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 6638.00 0.322535 0.161268 0.986911i $$-0.448442\pi$$
0.161268 + 0.986911i $$0.448442\pi$$
$$752$$ −3195.00 −0.154933
$$753$$ 14130.0 0.683832
$$754$$ 28674.0 1.38494
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −14846.0 −0.712797 −0.356398 0.934334i $$-0.615995\pi$$
−0.356398 + 0.934334i $$0.615995\pi$$
$$758$$ 24915.0 1.19387
$$759$$ −6210.00 −0.296981
$$760$$ 0 0
$$761$$ 3651.00 0.173914 0.0869571 0.996212i $$-0.472286\pi$$
0.0869571 + 0.996212i $$0.472286\pi$$
$$762$$ −4818.00 −0.229052
$$763$$ 0 0
$$764$$ −2388.00 −0.113082
$$765$$ 0 0
$$766$$ −2835.00 −0.133724
$$767$$ 21240.0 0.999911
$$768$$ −3026.00 −0.142176
$$769$$ −29855.0 −1.40000 −0.699999 0.714144i $$-0.746816\pi$$
−0.699999 + 0.714144i $$0.746816\pi$$
$$770$$ 0 0
$$771$$ 8160.00 0.381161
$$772$$ −272.000 −0.0126807
$$773$$ −6519.00 −0.303327 −0.151664 0.988432i $$-0.548463\pi$$
−0.151664 + 0.988432i $$0.548463\pi$$
$$774$$ 19734.0 0.916439
$$775$$ 0 0
$$776$$ 18942.0 0.876261
$$777$$ 0 0
$$778$$ −36108.0 −1.66393
$$779$$ −23595.0 −1.08521
$$780$$ 0 0
$$781$$ −2160.00 −0.0989640
$$782$$ −11178.0 −0.511157
$$783$$ −16200.0 −0.739388
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −12114.0 −0.549735
$$787$$ 35114.0 1.59044 0.795222 0.606319i $$-0.207354\pi$$
0.795222 + 0.606319i $$0.207354\pi$$
$$788$$ 2109.00 0.0953427
$$789$$ −6576.00 −0.296720
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 21735.0 0.975151
$$793$$ −23128.0 −1.03569
$$794$$ 8094.00 0.361770
$$795$$ 0 0
$$796$$ −1424.00 −0.0634075
$$797$$ 20910.0 0.929323 0.464661 0.885488i $$-0.346176\pi$$
0.464661 + 0.885488i $$0.346176\pi$$
$$798$$ 0 0
$$799$$ −2430.00 −0.107594
$$800$$ 0 0
$$801$$ −27462.0 −1.21139
$$802$$ −21159.0 −0.931609
$$803$$ −30060.0 −1.32104
$$804$$ −560.000 −0.0245643
$$805$$ 0 0
$$806$$ −15576.0 −0.680696
$$807$$ −6528.00 −0.284754
$$808$$ −14364.0 −0.625401
$$809$$ 4431.00 0.192566 0.0962829 0.995354i $$-0.469305\pi$$
0.0962829 + 0.995354i $$0.469305\pi$$
$$810$$ 0 0
$$811$$ 9577.00 0.414666 0.207333 0.978270i $$-0.433522\pi$$
0.207333 + 0.978270i $$0.433522\pi$$
$$812$$ 0 0
$$813$$ −5504.00 −0.237434
$$814$$ 34965.0 1.50556
$$815$$ 0 0
$$816$$ −7668.00 −0.328963
$$817$$ 34606.0 1.48190
$$818$$ −32610.0 −1.39387
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −10938.0 −0.464968 −0.232484 0.972600i $$-0.574685\pi$$
−0.232484 + 0.972600i $$0.574685\pi$$
$$822$$ −360.000 −0.0152755
$$823$$ −11540.0 −0.488772 −0.244386 0.969678i $$-0.578586\pi$$
−0.244386 + 0.969678i $$0.578586\pi$$
$$824$$ −31836.0 −1.34595
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 18762.0 0.788898 0.394449 0.918918i $$-0.370935\pi$$
0.394449 + 0.918918i $$0.370935\pi$$
$$828$$ 1587.00 0.0666088
$$829$$ 39610.0 1.65948 0.829742 0.558147i $$-0.188488\pi$$
0.829742 + 0.558147i $$0.188488\pi$$
$$830$$ 0 0
$$831$$ −9380.00 −0.391563
$$832$$ 25547.0 1.06452
$$833$$ 0 0
$$834$$ 10248.0 0.425491
$$835$$ 0 0
$$836$$ −5445.00 −0.225262
$$837$$ 8800.00 0.363408
$$838$$ −29187.0 −1.20316
$$839$$ −39162.0 −1.61147 −0.805734 0.592277i $$-0.798229\pi$$
−0.805734 + 0.592277i $$0.798229\pi$$
$$840$$ 0 0
$$841$$ 1855.00 0.0760589
$$842$$ 37650.0 1.54098
$$843$$ −15642.0 −0.639074
$$844$$ −3625.00 −0.147841
$$845$$ 0 0
$$846$$ 3105.00 0.126185
$$847$$ 0 0
$$848$$ 42387.0 1.71648
$$849$$ 1316.00 0.0531979
$$850$$ 0 0
$$851$$ −17871.0 −0.719871
$$852$$ −96.0000 −0.00386022
$$853$$ −11527.0 −0.462693 −0.231346 0.972871i $$-0.574313\pi$$
−0.231346 + 0.972871i $$0.574313\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 15372.0 0.613790
$$857$$ −41826.0 −1.66715 −0.833576 0.552405i $$-0.813710\pi$$
−0.833576 + 0.552405i $$0.813710\pi$$
$$858$$ −15930.0 −0.633848
$$859$$ −35192.0 −1.39783 −0.698915 0.715205i $$-0.746333\pi$$
−0.698915 + 0.715205i $$0.746333\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −8964.00 −0.354194
$$863$$ 9063.00 0.357483 0.178742 0.983896i $$-0.442797\pi$$
0.178742 + 0.983896i $$0.442797\pi$$
$$864$$ 4500.00 0.177191
$$865$$ 0 0
$$866$$ −49848.0 −1.95601
$$867$$ 3994.00 0.156451
$$868$$ 0 0
$$869$$ −35190.0 −1.37369
$$870$$ 0 0
$$871$$ 16520.0 0.642662
$$872$$ −33600.0 −1.30486
$$873$$ −20746.0 −0.804291
$$874$$ 25047.0 0.969368
$$875$$ 0 0
$$876$$ −1336.00 −0.0515288
$$877$$ −28439.0 −1.09500 −0.547501 0.836805i $$-0.684421\pi$$
−0.547501 + 0.836805i $$0.684421\pi$$
$$878$$ 22038.0 0.847092
$$879$$ 11994.0 0.460236
$$880$$ 0 0
$$881$$ 9303.00 0.355762 0.177881 0.984052i $$-0.443076\pi$$
0.177881 + 0.984052i $$0.443076\pi$$
$$882$$ 0 0
$$883$$ 14728.0 0.561310 0.280655 0.959809i $$-0.409448\pi$$
0.280655 + 0.959809i $$0.409448\pi$$
$$884$$ −3186.00 −0.121218
$$885$$ 0 0
$$886$$ 36.0000 0.00136506
$$887$$ −17016.0 −0.644128 −0.322064 0.946718i $$-0.604377\pi$$
−0.322064 + 0.946718i $$0.604377\pi$$
$$888$$ −10878.0 −0.411083
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −18945.0 −0.712325
$$892$$ −4960.00 −0.186181
$$893$$ 5445.00 0.204043
$$894$$ −6516.00 −0.243767
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 8142.00 0.303070
$$898$$ −29007.0 −1.07792
$$899$$ −14256.0 −0.528881
$$900$$ 0 0
$$901$$ 32238.0 1.19201
$$902$$ −26325.0 −0.971759
$$903$$ 0 0
$$904$$ 29232.0 1.07549
$$905$$ 0 0
$$906$$ −17196.0 −0.630573
$$907$$ 24922.0 0.912372 0.456186 0.889884i $$-0.349215\pi$$
0.456186 + 0.889884i $$0.349215\pi$$
$$908$$ −1500.00 −0.0548230
$$909$$ 15732.0 0.574035
$$910$$ 0 0
$$911$$ 30714.0 1.11701 0.558507 0.829500i $$-0.311374\pi$$
0.558507 + 0.829500i $$0.311374\pi$$
$$912$$ 17182.0 0.623852
$$913$$ −34560.0 −1.25276
$$914$$ −28902.0 −1.04594
$$915$$ 0 0
$$916$$ −6092.00 −0.219744
$$917$$ 0 0
$$918$$ 16200.0 0.582440
$$919$$ 17426.0 0.625496 0.312748 0.949836i $$-0.398750\pi$$
0.312748 + 0.949836i $$0.398750\pi$$
$$920$$ 0 0
$$921$$ 12452.0 0.445502
$$922$$ −1026.00 −0.0366481
$$923$$ 2832.00 0.100993
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 7233.00 0.256686
$$927$$ 34868.0 1.23540
$$928$$ −7290.00 −0.257873
$$929$$ −26649.0 −0.941147 −0.470573 0.882361i $$-0.655953\pi$$
−0.470573 + 0.882361i $$0.655953\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −138.000 −0.00485015
$$933$$ 9360.00 0.328438
$$934$$ 3618.00 0.126750
$$935$$ 0 0
$$936$$ −28497.0 −0.995143
$$937$$ 27686.0 0.965274 0.482637 0.875820i $$-0.339679\pi$$
0.482637 + 0.875820i $$0.339679\pi$$
$$938$$ 0 0
$$939$$ −2056.00 −0.0714537
$$940$$ 0 0
$$941$$ 17808.0 0.616923 0.308461 0.951237i $$-0.400186\pi$$
0.308461 + 0.951237i $$0.400186\pi$$
$$942$$ −1374.00 −0.0475237
$$943$$ 13455.0 0.464640
$$944$$ −25560.0 −0.881258
$$945$$ 0 0
$$946$$ 38610.0 1.32698
$$947$$ 6906.00 0.236974 0.118487 0.992956i $$-0.462196\pi$$
0.118487 + 0.992956i $$0.462196\pi$$
$$948$$ −1564.00 −0.0535827
$$949$$ 39412.0 1.34812
$$950$$ 0 0
$$951$$ 17244.0 0.587986
$$952$$ 0 0
$$953$$ 20940.0 0.711766 0.355883 0.934530i $$-0.384180\pi$$
0.355883 + 0.934530i $$0.384180\pi$$
$$954$$ −41193.0 −1.39798
$$955$$ 0 0
$$956$$ −5502.00 −0.186137
$$957$$ −14580.0 −0.492481
$$958$$ −1296.00 −0.0437076
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22047.0 −0.740056
$$962$$ −45843.0 −1.53642
$$963$$ −16836.0 −0.563377
$$964$$ −3551.00 −0.118641
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −9176.00 −0.305150 −0.152575 0.988292i $$-0.548757\pi$$
−0.152575 + 0.988292i $$0.548757\pi$$
$$968$$ 14574.0 0.483911
$$969$$ 13068.0 0.433235
$$970$$ 0 0
$$971$$ −29763.0 −0.983666 −0.491833 0.870689i $$-0.663673\pi$$
−0.491833 + 0.870689i $$0.663673\pi$$
$$972$$ −3542.00 −0.116882
$$973$$ 0 0
$$974$$ −35688.0 −1.17404
$$975$$ 0 0
$$976$$ 27832.0 0.912788
$$977$$ 38490.0 1.26039 0.630197 0.776436i $$-0.282974\pi$$
0.630197 + 0.776436i $$0.282974\pi$$
$$978$$ 7368.00 0.240903
$$979$$ −53730.0 −1.75405
$$980$$ 0 0
$$981$$ 36800.0 1.19769
$$982$$ 36828.0 1.19677
$$983$$ −12609.0 −0.409120 −0.204560 0.978854i $$-0.565576\pi$$
−0.204560 + 0.978854i $$0.565576\pi$$
$$984$$ 8190.00 0.265333
$$985$$ 0 0
$$986$$ −26244.0 −0.847646
$$987$$ 0 0
$$988$$ 7139.00 0.229880
$$989$$ −19734.0 −0.634484
$$990$$ 0 0
$$991$$ 19820.0 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 3960.00 0.126744
$$993$$ 3998.00 0.127767
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −1536.00 −0.0488655
$$997$$ 46034.0 1.46230 0.731149 0.682218i $$-0.238984\pi$$
0.731149 + 0.682218i $$0.238984\pi$$
$$998$$ 32628.0 1.03489
$$999$$ 25900.0 0.820260
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 1225.4.a.a.1.1 1
5.4 even 2 245.4.a.f.1.1 1
7.3 odd 6 175.4.e.b.51.1 2
7.5 odd 6 175.4.e.b.151.1 2
7.6 odd 2 1225.4.a.b.1.1 1
15.14 odd 2 2205.4.a.g.1.1 1
35.3 even 12 175.4.k.b.149.1 4
35.4 even 6 245.4.e.a.226.1 2
35.9 even 6 245.4.e.a.116.1 2
35.12 even 12 175.4.k.b.74.1 4
35.17 even 12 175.4.k.b.149.2 4
35.19 odd 6 35.4.e.a.11.1 2
35.24 odd 6 35.4.e.a.16.1 yes 2
35.33 even 12 175.4.k.b.74.2 4
35.34 odd 2 245.4.a.e.1.1 1
105.59 even 6 315.4.j.b.226.1 2
105.89 even 6 315.4.j.b.46.1 2
105.104 even 2 2205.4.a.e.1.1 1
140.19 even 6 560.4.q.b.81.1 2
140.59 even 6 560.4.q.b.401.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.a.11.1 2 35.19 odd 6
35.4.e.a.16.1 yes 2 35.24 odd 6
175.4.e.b.51.1 2 7.3 odd 6
175.4.e.b.151.1 2 7.5 odd 6
175.4.k.b.74.1 4 35.12 even 12
175.4.k.b.74.2 4 35.33 even 12
175.4.k.b.149.1 4 35.3 even 12
175.4.k.b.149.2 4 35.17 even 12
245.4.a.e.1.1 1 35.34 odd 2
245.4.a.f.1.1 1 5.4 even 2
245.4.e.a.116.1 2 35.9 even 6
245.4.e.a.226.1 2 35.4 even 6
315.4.j.b.46.1 2 105.89 even 6
315.4.j.b.226.1 2 105.59 even 6
560.4.q.b.81.1 2 140.19 even 6
560.4.q.b.401.1 2 140.59 even 6
1225.4.a.a.1.1 1 1.1 even 1 trivial
1225.4.a.b.1.1 1 7.6 odd 2
2205.4.a.e.1.1 1 105.104 even 2
2205.4.a.g.1.1 1 15.14 odd 2