# Properties

 Label 1225.4.a.b.1.1 Level $1225$ Weight $4$ Character 1225.1 Self dual yes Analytic conductor $72.277$ Analytic rank $0$ 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: $$0$$ 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 $$-0.438357\pi$$
0.192450 + 0.981307i $$0.438357\pi$$
$$4$$ 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 $$-0.716688\pi$$
−0.629371 + 0.777105i $$0.716688\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 69.0000 0.903525
$$19$$ −121.000 −1.46102 −0.730508 0.682904i $$-0.760717\pi$$
−0.730508 + 0.682904i $$0.760717\pi$$
$$20$$ 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 $$-0.582050\pi$$
−0.254924 + 0.966961i $$0.582050\pi$$
$$32$$ 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 $$-0.378882\pi$$
0.371389 + 0.928477i $$0.378882\pi$$
$$42$$ 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 $$-0.522245\pi$$
−0.0698290 + 0.997559i $$0.522245\pi$$
$$48$$ −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 $$-0.630013\pi$$
−0.397187 + 0.917738i $$0.630013\pi$$
$$60$$ 0 0
$$61$$ 392.000 0.822794 0.411397 0.911456i $$-0.365041\pi$$
0.411397 + 0.911456i $$0.365041\pi$$
$$62$$ 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 $$-0.679878\pi$$
−0.535503 + 0.844533i $$0.679878\pi$$
$$74$$ −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 $$-0.669550\pi$$
−0.507825 + 0.861460i $$0.669550\pi$$
$$84$$ 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 $$-0.751772\pi$$
−0.711032 + 0.703159i $$0.751772\pi$$
$$90$$ 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 $$-0.656498\pi$$
−0.472084 + 0.881554i $$0.656498\pi$$
$$98$$ 0 0
$$99$$ 1035.00 1.05072
$$100$$ 0 0
$$101$$ 684.000 0.673867 0.336933 0.941528i $$-0.390610\pi$$
0.336933 + 0.941528i $$0.390610\pi$$
$$102$$ −324.000 −0.314517
$$103$$ 1516.00 1.45025 0.725126 0.688616i $$-0.241782\pi$$
0.725126 + 0.688616i $$0.241782\pi$$
$$104$$ −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 $$-0.264882\pi$$
0.673286 + 0.739382i $$0.264882\pi$$
$$132$$ −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 $$-0.674485\pi$$
−0.521118 + 0.853485i $$0.674485\pi$$
$$140$$ 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 $$-0.481462\pi$$
0.0582044 + 0.998305i $$0.481462\pi$$
$$158$$ −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 $$-0.352522\pi$$
0.446918 + 0.894575i $$0.352522\pi$$
$$168$$ 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 $$-0.450915\pi$$
0.153595 + 0.988134i $$0.450915\pi$$
$$174$$ 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 $$-0.620362\pi$$
−0.369183 + 0.929357i $$0.620362\pi$$
$$182$$ 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 $$-0.418375\pi$$
0.253630 + 0.967301i $$0.418375\pi$$
$$200$$ 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 $$-0.232582\pi$$
0.744722 + 0.667374i $$0.232582\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −4176.00 −1.22913
$$227$$ 1500.00 0.438584 0.219292 0.975659i $$-0.429625\pi$$
0.219292 + 0.975659i $$0.429625\pi$$
$$228$$ −242.000 −0.0702932
$$229$$ 6092.00 1.75795 0.878975 0.476867i $$-0.158228\pi$$
0.878975 + 0.476867i $$0.158228\pi$$
$$230$$ 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 $$-0.342606\pi$$
0.474564 + 0.880221i $$0.342606\pi$$
$$242$$ −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 $$-0.151870\pi$$
0.888324 + 0.459216i $$0.151870\pi$$
$$252$$ 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 $$-0.335116\pi$$
0.495143 + 0.868812i $$0.335116\pi$$
$$258$$ −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 $$-0.620610\pi$$
−0.369906 + 0.929069i $$0.620610\pi$$
$$270$$ 0 0
$$271$$ −2752.00 −0.616871 −0.308436 0.951245i $$-0.599805\pi$$
−0.308436 + 0.951245i $$0.599805\pi$$
$$272$$ −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 $$-0.477985\pi$$
0.0691061 + 0.997609i $$0.477985\pi$$
$$284$$ 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 $$-0.296016\pi$$
0.597864 + 0.801597i $$0.296016\pi$$
$$294$$ 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 $$-0.303551\pi$$
0.578724 + 0.815523i $$0.303551\pi$$
$$308$$ 0 0
$$309$$ 3032.00 0.558202
$$310$$ 0 0
$$311$$ 4680.00 0.853307 0.426653 0.904415i $$-0.359692\pi$$
0.426653 + 0.904415i $$0.359692\pi$$
$$312$$ −2478.00 −0.449645
$$313$$ −1028.00 −0.185642 −0.0928211 0.995683i $$-0.529588\pi$$
−0.0928211 + 0.995683i $$0.529588\pi$$
$$314$$ −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 $$-0.292161\pi$$
0.607529 + 0.794298i $$0.292161\pi$$
$$350$$ 0 0
$$351$$ 5900.00 0.897204
$$352$$ −2025.00 −0.306627
$$353$$ −828.000 −0.124844 −0.0624221 0.998050i $$-0.519882\pi$$
−0.0624221 + 0.998050i $$0.519882\pi$$
$$354$$ 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 $$-0.563834\pi$$
−0.199198 + 0.979959i $$0.563834\pi$$
$$368$$ 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 $$-0.520079\pi$$
−0.0630382 + 0.998011i $$0.520079\pi$$
$$384$$ −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 $$-0.445449\pi$$
0.170540 + 0.985351i $$0.445449\pi$$
$$398$$ −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 $$-0.728206\pi$$
−0.657074 + 0.753826i $$0.728206\pi$$
$$410$$ 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 $$-0.691964\pi$$
−0.567175 + 0.823597i $$0.691964\pi$$
$$420$$ 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 $$-0.873505\pi$$
−0.922072 + 0.387019i $$0.873505\pi$$
$$434$$ 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 $$-0.369245\pi$$
0.399323 + 0.916810i $$0.369245\pi$$
$$440$$ 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 $$-0.505499\pi$$
−0.0172761 + 0.999851i $$0.505499\pi$$
$$462$$ 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 $$-0.480969\pi$$
0.0597506 + 0.998213i $$0.480969\pi$$
$$468$$ 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 $$-0.506559\pi$$
−0.0206039 + 0.999788i $$0.506559\pi$$
$$480$$ 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 $$-0.678508\pi$$
−0.531862 + 0.846831i $$0.678508\pi$$
$$504$$ 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 $$-0.669852\pi$$
−0.508640 + 0.860979i $$0.669852\pi$$
$$510$$ 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 $$-0.367616\pi$$
0.404010 + 0.914755i $$0.367616\pi$$
$$522$$ −11178.0 −0.937256
$$523$$ −21188.0 −1.77148 −0.885742 0.464177i $$-0.846350\pi$$
−0.885742 + 0.464177i $$0.846350\pi$$
$$524$$ 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 $$-0.424302\pi$$
0.235578 + 0.971856i $$0.424302\pi$$
$$564$$ −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 $$-0.342592\pi$$
0.474603 + 0.880200i $$0.342592\pi$$
$$578$$ 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 $$-0.344261\pi$$
0.469980 + 0.882677i $$0.344261\pi$$
$$588$$ 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 $$-0.874467\pi$$
−0.923237 + 0.384231i $$0.874467\pi$$
$$594$$ −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 $$-0.430200\pi$$
0.217529 + 0.976054i $$0.430200\pi$$
$$602$$ 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 $$-0.245153\pi$$
0.717792 + 0.696257i $$0.245153\pi$$
$$608$$ −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 $$-0.369387\pi$$
0.398915 + 0.916988i $$0.369387\pi$$
$$620$$ 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 $$-0.587809\pi$$
−0.272373 + 0.962192i $$0.587809\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 19602.0 1.19386
$$647$$ 11019.0 0.669554 0.334777 0.942297i $$-0.391339\pi$$
0.334777 + 0.942297i $$0.391339\pi$$
$$648$$ 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 $$-0.933253\pi$$
−0.978095 + 0.208158i $$0.933253\pi$$
$$662$$ 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 $$-0.383607\pi$$
0.357564 + 0.933889i $$0.383607\pi$$
$$678$$ −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 $$-0.687679\pi$$
−0.556038 + 0.831157i $$0.687679\pi$$
$$692$$ 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 $$-0.560595\pi$$
−0.189218 + 0.981935i $$0.560595\pi$$
$$720$$ 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 $$-0.371316\pi$$
0.393352 + 0.919388i $$0.371316\pi$$
$$728$$ 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 $$-0.237247\pi$$
0.734862 + 0.678217i $$0.237247\pi$$
$$734$$ 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 $$-0.527714\pi$$
−0.0869571 + 0.996212i $$0.527714\pi$$
$$762$$ 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 $$-0.253184\pi$$
0.699999 + 0.714144i $$0.253184\pi$$
$$770$$ 0 0
$$771$$ 8160.00 0.381161
$$772$$ −272.000 −0.0126807
$$773$$ 6519.00 0.303327 0.151664 0.988432i $$-0.451537\pi$$
0.151664 + 0.988432i $$0.451537\pi$$
$$774$$ 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 $$-0.792646\pi$$
−0.795222 + 0.606319i $$0.792646\pi$$
$$788$$ 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 $$-0.653824\pi$$
−0.464661 + 0.885488i $$0.653824\pi$$
$$798$$ 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 $$-0.566478\pi$$
−0.207333 + 0.978270i $$0.566478\pi$$
$$812$$ 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 $$-0.811512\pi$$
−0.829742 + 0.558147i $$0.811512\pi$$
$$830$$ 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 $$-0.201771\pi$$
0.805734 + 0.592277i $$0.201771\pi$$
$$840$$ 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 $$-0.425687\pi$$
0.231346 + 0.972871i $$0.425687\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 15372.0 0.613790
$$857$$ 41826.0 1.66715 0.833576 0.552405i $$-0.186290\pi$$
0.833576 + 0.552405i $$0.186290\pi$$
$$858$$ −15930.0 −0.633848
$$859$$ 35192.0 1.39783 0.698915 0.715205i $$-0.253667\pi$$
0.698915 + 0.715205i $$0.253667\pi$$
$$860$$ 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 $$-0.556924\pi$$
−0.177881 + 0.984052i $$0.556924\pi$$
$$882$$ 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 $$-0.395623\pi$$
0.322064 + 0.946718i $$0.395623\pi$$
$$888$$ 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 $$-0.344047\pi$$
0.470573 + 0.882361i $$0.344047\pi$$
$$930$$ 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 $$-0.660321\pi$$
−0.482637 + 0.875820i $$0.660321\pi$$
$$938$$ 0 0
$$939$$ −2056.00 −0.0714537
$$940$$ 0 0
$$941$$ −17808.0 −0.616923 −0.308461 0.951237i $$-0.599814\pi$$
−0.308461 + 0.951237i $$0.599814\pi$$
$$942$$ −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 $$-0.336327\pi$$
0.491833 + 0.870689i $$0.336327\pi$$
$$972$$ 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 $$-0.434424\pi$$
0.204560 + 0.978854i $$0.434424\pi$$
$$984$$ 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 $$-0.761016\pi$$
−0.731149 + 0.682218i $$0.761016\pi$$
$$998$$ 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.b.1.1 1
5.4 even 2 245.4.a.e.1.1 1
7.2 even 3 175.4.e.b.151.1 2
7.4 even 3 175.4.e.b.51.1 2
7.6 odd 2 1225.4.a.a.1.1 1
15.14 odd 2 2205.4.a.e.1.1 1
35.2 odd 12 175.4.k.b.74.1 4
35.4 even 6 35.4.e.a.16.1 yes 2
35.9 even 6 35.4.e.a.11.1 2
35.18 odd 12 175.4.k.b.149.1 4
35.19 odd 6 245.4.e.a.116.1 2
35.23 odd 12 175.4.k.b.74.2 4
35.24 odd 6 245.4.e.a.226.1 2
35.32 odd 12 175.4.k.b.149.2 4
35.34 odd 2 245.4.a.f.1.1 1
105.44 odd 6 315.4.j.b.46.1 2
105.74 odd 6 315.4.j.b.226.1 2
105.104 even 2 2205.4.a.g.1.1 1
140.39 odd 6 560.4.q.b.401.1 2
140.79 odd 6 560.4.q.b.81.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.a.11.1 2 35.9 even 6
35.4.e.a.16.1 yes 2 35.4 even 6
175.4.e.b.51.1 2 7.4 even 3
175.4.e.b.151.1 2 7.2 even 3
175.4.k.b.74.1 4 35.2 odd 12
175.4.k.b.74.2 4 35.23 odd 12
175.4.k.b.149.1 4 35.18 odd 12
175.4.k.b.149.2 4 35.32 odd 12
245.4.a.e.1.1 1 5.4 even 2
245.4.a.f.1.1 1 35.34 odd 2
245.4.e.a.116.1 2 35.19 odd 6
245.4.e.a.226.1 2 35.24 odd 6
315.4.j.b.46.1 2 105.44 odd 6
315.4.j.b.226.1 2 105.74 odd 6
560.4.q.b.81.1 2 140.79 odd 6
560.4.q.b.401.1 2 140.39 odd 6
1225.4.a.a.1.1 1 7.6 odd 2
1225.4.a.b.1.1 1 1.1 even 1 trivial
2205.4.a.e.1.1 1 15.14 odd 2
2205.4.a.g.1.1 1 105.104 even 2