# Properties

 Label 2205.4.a.e.1.1 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ 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] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.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: $$130.099211563$$ 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 $$\chi$$ $$=$$ 2205.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +21.0000 q^{8} +O(q^{10})$$ $$q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +21.0000 q^{8} +15.0000 q^{10} +45.0000 q^{11} +59.0000 q^{13} -71.0000 q^{16} +54.0000 q^{17} -121.000 q^{19} -5.00000 q^{20} -135.000 q^{22} -69.0000 q^{23} +25.0000 q^{25} -177.000 q^{26} +162.000 q^{29} -88.0000 q^{31} +45.0000 q^{32} -162.000 q^{34} -259.000 q^{37} +363.000 q^{38} -105.000 q^{40} -195.000 q^{41} -286.000 q^{43} +45.0000 q^{44} +207.000 q^{46} -45.0000 q^{47} -75.0000 q^{50} +59.0000 q^{52} -597.000 q^{53} -225.000 q^{55} -486.000 q^{58} +360.000 q^{59} +392.000 q^{61} +264.000 q^{62} +433.000 q^{64} -295.000 q^{65} -280.000 q^{67} +54.0000 q^{68} -48.0000 q^{71} +668.000 q^{73} +777.000 q^{74} -121.000 q^{76} +782.000 q^{79} +355.000 q^{80} +585.000 q^{82} -768.000 q^{83} -270.000 q^{85} +858.000 q^{86} +945.000 q^{88} +1194.00 q^{89} -69.0000 q^{92} +135.000 q^{94} +605.000 q^{95} +902.000 q^{97} +O(q^{100})$$

## 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 $$0.677932\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.125000
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ 0 0
$$10$$ 15.0000 0.474342
$$11$$ 45.0000 1.23346 0.616728 0.787177i $$-0.288458\pi$$
0.616728 + 0.787177i $$0.288458\pi$$
$$12$$ 0 0
$$13$$ 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 $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ −121.000 −1.46102 −0.730508 0.682904i $$-0.760717\pi$$
−0.730508 + 0.682904i $$0.760717\pi$$
$$20$$ −5.00000 −0.0559017
$$21$$ 0 0
$$22$$ −135.000 −1.30828
$$23$$ −69.0000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −177.000 −1.33510
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 162.000 1.03733 0.518666 0.854977i $$-0.326429\pi$$
0.518666 + 0.854977i $$0.326429\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −162.000 −0.817140
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −259.000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 363.000 1.54964
$$39$$ 0 0
$$40$$ −105.000 −0.415049
$$41$$ −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 $$-0.669300\pi$$
−0.507146 + 0.861860i $$0.669300\pi$$
$$44$$ 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 $$0.522245\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −75.0000 −0.212132
$$51$$ 0 0
$$52$$ 59.0000 0.157343
$$53$$ −597.000 −1.54725 −0.773625 0.633644i $$-0.781559\pi$$
−0.773625 + 0.633644i $$0.781559\pi$$
$$54$$ 0 0
$$55$$ −225.000 −0.551618
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −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 $$-0.365041\pi$$
0.411397 + 0.911456i $$0.365041\pi$$
$$62$$ 264.000 0.540775
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ −295.000 −0.562927
$$66$$ 0 0
$$67$$ −280.000 −0.510559 −0.255279 0.966867i $$-0.582167\pi$$
−0.255279 + 0.966867i $$0.582167\pi$$
$$68$$ 54.0000 0.0963009
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −48.0000 −0.0802331 −0.0401166 0.999195i $$-0.512773\pi$$
−0.0401166 + 0.999195i $$0.512773\pi$$
$$72$$ 0 0
$$73$$ 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
$$78$$ 0 0
$$79$$ 782.000 1.11369 0.556847 0.830615i $$-0.312011\pi$$
0.556847 + 0.830615i $$0.312011\pi$$
$$80$$ 355.000 0.496128
$$81$$ 0 0
$$82$$ 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
$$85$$ −270.000 −0.344537
$$86$$ 858.000 1.07582
$$87$$ 0 0
$$88$$ 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
$$93$$ 0 0
$$94$$ 135.000 0.148130
$$95$$ 605.000 0.653386
$$96$$ 0 0
$$97$$ 902.000 0.944167 0.472084 0.881554i $$-0.343502\pi$$
0.472084 + 0.881554i $$0.343502\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 25.0000 0.0250000
$$101$$ −684.000 −0.673867 −0.336933 0.941528i $$-0.609390\pi$$
−0.336933 + 0.941528i $$0.609390\pi$$
$$102$$ 0 0
$$103$$ −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 $$0.392723\pi$$
$$108$$ 0 0
$$109$$ −1600.00 −1.40598 −0.702992 0.711198i $$-0.748153\pi$$
−0.702992 + 0.711198i $$0.748153\pi$$
$$110$$ 675.000 0.585079
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1392.00 1.15883 0.579417 0.815031i $$-0.303280\pi$$
0.579417 + 0.815031i $$0.303280\pi$$
$$114$$ 0 0
$$115$$ 345.000 0.279751
$$116$$ 162.000 0.129667
$$117$$ 0 0
$$118$$ −1080.00 −0.842560
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 694.000 0.521412
$$122$$ −1176.00 −0.872705
$$123$$ 0 0
$$124$$ −88.0000 −0.0637309
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 803.000 0.561061 0.280530 0.959845i $$-0.409490\pi$$
0.280530 + 0.959845i $$0.409490\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ 0 0
$$130$$ 885.000 0.597074
$$131$$ −2019.00 −1.34657 −0.673286 0.739382i $$-0.735118\pi$$
−0.673286 + 0.739382i $$0.735118\pi$$
$$132$$ 0 0
$$133$$ 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 $$0.505955\pi$$
$$138$$ 0 0
$$139$$ −1708.00 −1.04224 −0.521118 0.853485i $$-0.674485\pi$$
−0.521118 + 0.853485i $$0.674485\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 144.000 0.0851001
$$143$$ 2655.00 1.55260
$$144$$ 0 0
$$145$$ −810.000 −0.463909
$$146$$ −2004.00 −1.13597
$$147$$ 0 0
$$148$$ −259.000 −0.143849
$$149$$ 1086.00 0.597105 0.298552 0.954393i $$-0.403496\pi$$
0.298552 + 0.954393i $$0.403496\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 440.000 0.228011
$$156$$ 0 0
$$157$$ −229.000 −0.116409 −0.0582044 0.998305i $$-0.518538\pi$$
−0.0582044 + 0.998305i $$0.518538\pi$$
$$158$$ −2346.00 −1.18125
$$159$$ 0 0
$$160$$ −225.000 −0.111174
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1228.00 −0.590088 −0.295044 0.955484i $$-0.595334\pi$$
−0.295044 + 0.955484i $$0.595334\pi$$
$$164$$ −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
$$170$$ 810.000 0.365436
$$171$$ 0 0
$$172$$ −286.000 −0.126787
$$173$$ 699.000 0.307191 0.153595 0.988134i $$-0.450915\pi$$
0.153595 + 0.988134i $$0.450915\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3195.00 −1.36836
$$177$$ 0 0
$$178$$ −3582.00 −1.50833
$$179$$ −3117.00 −1.30154 −0.650770 0.759275i $$-0.725554\pi$$
−0.650770 + 0.759275i $$0.725554\pi$$
$$180$$ 0 0
$$181$$ −1798.00 −0.738366 −0.369183 0.929357i $$-0.620362\pi$$
−0.369183 + 0.929357i $$0.620362\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1449.00 −0.580553
$$185$$ 1295.00 0.514650
$$186$$ 0 0
$$187$$ 2430.00 0.950263
$$188$$ −45.0000 −0.0174572
$$189$$ 0 0
$$190$$ −1815.00 −0.693021
$$191$$ 2388.00 0.904658 0.452329 0.891851i $$-0.350593\pi$$
0.452329 + 0.891851i $$0.350593\pi$$
$$192$$ 0 0
$$193$$ 272.000 0.101446 0.0507228 0.998713i $$-0.483848\pi$$
0.0507228 + 0.998713i $$0.483848\pi$$
$$194$$ −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 $$0.375452\pi$$
$$198$$ 0 0
$$199$$ 1424.00 0.507260 0.253630 0.967301i $$-0.418375\pi$$
0.253630 + 0.967301i $$0.418375\pi$$
$$200$$ 525.000 0.185616
$$201$$ 0 0
$$202$$ 2052.00 0.714744
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 975.000 0.332180
$$206$$ 4548.00 1.53822
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ −2196.00 −0.701474
$$215$$ 1430.00 0.453606
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4800.00 1.49127
$$219$$ 0 0
$$220$$ −225.000 −0.0689523
$$221$$ 3186.00 0.969745
$$222$$ 0 0
$$223$$ −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 $$-0.429625\pi$$
0.219292 + 0.975659i $$0.429625\pi$$
$$228$$ 0 0
$$229$$ 6092.00 1.75795 0.878975 0.476867i $$-0.158228\pi$$
0.878975 + 0.476867i $$0.158228\pi$$
$$230$$ −1035.00 −0.296721
$$231$$ 0 0
$$232$$ 3402.00 0.962725
$$233$$ −138.000 −0.0388012 −0.0194006 0.999812i $$-0.506176\pi$$
−0.0194006 + 0.999812i $$0.506176\pi$$
$$234$$ 0 0
$$235$$ 225.000 0.0624569
$$236$$ 360.000 0.0992966
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 5502.00 1.48910 0.744550 0.667567i $$-0.232664\pi$$
0.744550 + 0.667567i $$0.232664\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ 392.000 0.102849
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −7139.00 −1.83904
$$248$$ −1848.00 −0.473178
$$249$$ 0 0
$$250$$ 375.000 0.0948683
$$251$$ −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 $$-0.335116\pi$$
0.495143 + 0.868812i $$0.335116\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −295.000 −0.0703659
$$261$$ 0 0
$$262$$ 6057.00 1.42825
$$263$$ 3288.00 0.770900 0.385450 0.922729i $$-0.374046\pi$$
0.385450 + 0.922729i $$0.374046\pi$$
$$264$$ 0 0
$$265$$ 2985.00 0.691951
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −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 $$-0.599805\pi$$
−0.308436 + 0.951245i $$0.599805\pi$$
$$272$$ −3834.00 −0.854671
$$273$$ 0 0
$$274$$ 180.000 0.0396869
$$275$$ 1125.00 0.246691
$$276$$ 0 0
$$277$$ −4690.00 −1.01731 −0.508655 0.860971i $$-0.669857\pi$$
−0.508655 + 0.860971i $$0.669857\pi$$
$$278$$ 5124.00 1.10546
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −7821.00 −1.66036 −0.830181 0.557494i $$-0.811763\pi$$
−0.830181 + 0.557494i $$0.811763\pi$$
$$282$$ 0 0
$$283$$ −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
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 2430.00 0.492050
$$291$$ 0 0
$$292$$ 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
$$295$$ −1800.00 −0.355254
$$296$$ −5439.00 −1.06803
$$297$$ 0 0
$$298$$ −3258.00 −0.633325
$$299$$ −4071.00 −0.787398
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8598.00 1.63828
$$303$$ 0 0
$$304$$ 8591.00 1.62081
$$305$$ −1960.00 −0.367965
$$306$$ 0 0
$$307$$ −6226.00 −1.15745 −0.578724 0.815523i $$-0.696449\pi$$
−0.578724 + 0.815523i $$0.696449\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1320.00 −0.241842
$$311$$ −4680.00 −0.853307 −0.426653 0.904415i $$-0.640308\pi$$
−0.426653 + 0.904415i $$0.640308\pi$$
$$312$$ 0 0
$$313$$ 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 $$0.776677\pi$$
$$318$$ 0 0
$$319$$ 7290.00 1.27950
$$320$$ −2165.00 −0.378210
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6534.00 −1.12558
$$324$$ 0 0
$$325$$ 1475.00 0.251749
$$326$$ 3684.00 0.625883
$$327$$ 0 0
$$328$$ −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
$$333$$ 0 0
$$334$$ −5787.00 −0.948056
$$335$$ 1400.00 0.228329
$$336$$ 0 0
$$337$$ 5114.00 0.826639 0.413319 0.910586i $$-0.364369\pi$$
0.413319 + 0.910586i $$0.364369\pi$$
$$338$$ −3852.00 −0.619885
$$339$$ 0 0
$$340$$ −270.000 −0.0430671
$$341$$ −3960.00 −0.628874
$$342$$ 0 0
$$343$$ 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 $$0.608454\pi$$
$$348$$ 0 0
$$349$$ 7922.00 1.21506 0.607529 0.794298i $$-0.292161\pi$$
0.607529 + 0.794298i $$0.292161\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2025.00 0.306627
$$353$$ −828.000 −0.124844 −0.0624221 0.998050i $$-0.519882\pi$$
−0.0624221 + 0.998050i $$0.519882\pi$$
$$354$$ 0 0
$$355$$ 240.000 0.0358813
$$356$$ 1194.00 0.177758
$$357$$ 0 0
$$358$$ 9351.00 1.38049
$$359$$ 1350.00 0.198469 0.0992344 0.995064i $$-0.468361\pi$$
0.0992344 + 0.995064i $$0.468361\pi$$
$$360$$ 0 0
$$361$$ 7782.00 1.13457
$$362$$ 5394.00 0.783156
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3340.00 −0.478969
$$366$$ 0 0
$$367$$ 2801.00 0.398395 0.199198 0.979959i $$-0.436166\pi$$
0.199198 + 0.979959i $$0.436166\pi$$
$$368$$ 4899.00 0.693962
$$369$$ 0 0
$$370$$ −3885.00 −0.545869
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6602.00 0.916457 0.458229 0.888834i $$-0.348484\pi$$
0.458229 + 0.888834i $$0.348484\pi$$
$$374$$ −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 $$0.690274\pi$$
$$380$$ 605.000 0.0816733
$$381$$ 0 0
$$382$$ −7164.00 −0.959534
$$383$$ −945.000 −0.126076 −0.0630382 0.998011i $$-0.520079\pi$$
−0.0630382 + 0.998011i $$0.520079\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −816.000 −0.107599
$$387$$ 0 0
$$388$$ 902.000 0.118021
$$389$$ −12036.0 −1.56876 −0.784382 0.620278i $$-0.787020\pi$$
−0.784382 + 0.620278i $$0.787020\pi$$
$$390$$ 0 0
$$391$$ −3726.00 −0.481923
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −6327.00 −0.809009
$$395$$ −3910.00 −0.498059
$$396$$ 0 0
$$397$$ −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
$$400$$ −1775.00 −0.221875
$$401$$ −7053.00 −0.878329 −0.439165 0.898407i $$-0.644726\pi$$
−0.439165 + 0.898407i $$0.644726\pi$$
$$402$$ 0 0
$$403$$ −5192.00 −0.641767
$$404$$ −684.000 −0.0842333
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −11655.0 −1.41945
$$408$$ 0 0
$$409$$ −10870.0 −1.31415 −0.657074 0.753826i $$-0.728206\pi$$
−0.657074 + 0.753826i $$0.728206\pi$$
$$410$$ −2925.00 −0.352330
$$411$$ 0 0
$$412$$ −1516.00 −0.181281
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3840.00 0.454212
$$416$$ 2655.00 0.312914
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ −12537.0 −1.43597
$$425$$ 1350.00 0.154081
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 732.000 0.0826695
$$429$$ 0 0
$$430$$ −4290.00 −0.481121
$$431$$ −2988.00 −0.333937 −0.166969 0.985962i $$-0.553398\pi$$
−0.166969 + 0.985962i $$0.553398\pi$$
$$432$$ 0 0
$$433$$ 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
$$438$$ 0 0
$$439$$ 7346.00 0.798646 0.399323 0.916810i $$-0.369245\pi$$
0.399323 + 0.916810i $$0.369245\pi$$
$$440$$ −4725.00 −0.511944
$$441$$ 0 0
$$442$$ −9558.00 −1.02857
$$443$$ −12.0000 −0.00128699 −0.000643496 1.00000i $$-0.500205\pi$$
−0.000643496 1.00000i $$0.500205\pi$$
$$444$$ 0 0
$$445$$ −5970.00 −0.635967
$$446$$ 14880.0 1.57979
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9669.00 −1.01628 −0.508138 0.861275i $$-0.669666\pi$$
−0.508138 + 0.861275i $$0.669666\pi$$
$$450$$ 0 0
$$451$$ −8775.00 −0.916183
$$452$$ 1392.00 0.144854
$$453$$ 0 0
$$454$$ −4500.00 −0.465188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9634.00 −0.986126 −0.493063 0.869994i $$-0.664123\pi$$
−0.493063 + 0.869994i $$0.664123\pi$$
$$458$$ −18276.0 −1.86459
$$459$$ 0 0
$$460$$ 345.000 0.0349689
$$461$$ 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 $$-0.461389\pi$$
0.121003 + 0.992652i $$0.461389\pi$$
$$464$$ −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 $$0.480969\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −675.000 −0.0662456
$$471$$ 0 0
$$472$$ 7560.00 0.737240
$$473$$ −12870.0 −1.25109
$$474$$ 0 0
$$475$$ −3025.00 −0.292203
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −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
$$485$$ −4510.00 −0.422244
$$486$$ 0 0
$$487$$ −11896.0 −1.10690 −0.553449 0.832883i $$-0.686689\pi$$
−0.553449 + 0.832883i $$0.686689\pi$$
$$488$$ 8232.00 0.763617
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12276.0 1.12833 0.564163 0.825663i $$-0.309199\pi$$
0.564163 + 0.825663i $$0.309199\pi$$
$$492$$ 0 0
$$493$$ 8748.00 0.799169
$$494$$ 21417.0 1.95060
$$495$$ 0 0
$$496$$ 6248.00 0.565612
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −10876.0 −0.975705 −0.487852 0.872926i $$-0.662220\pi$$
−0.487852 + 0.872926i $$0.662220\pi$$
$$500$$ −125.000 −0.0111803
$$501$$ 0 0
$$502$$ 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
$$505$$ 3420.00 0.301362
$$506$$ 9315.00 0.818384
$$507$$ 0 0
$$508$$ 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
$$513$$ 0 0
$$514$$ −12240.0 −1.05036
$$515$$ 7580.00 0.648572
$$516$$ 0 0
$$517$$ −2025.00 −0.172262
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6195.00 −0.522440
$$521$$ −9609.00 −0.808019 −0.404010 0.914755i $$-0.632384\pi$$
−0.404010 + 0.914755i $$0.632384\pi$$
$$522$$ 0 0
$$523$$ 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
$$528$$ 0 0
$$529$$ −7406.00 −0.608696
$$530$$ −8955.00 −0.733925
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −11505.0 −0.934966
$$534$$ 0 0
$$535$$ −3660.00 −0.295767
$$536$$ −5880.00 −0.473838
$$537$$ 0 0
$$538$$ −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
$$543$$ 0 0
$$544$$ 2430.00 0.191517
$$545$$ 8000.00 0.628775
$$546$$ 0 0
$$547$$ 344.000 0.0268892 0.0134446 0.999910i $$-0.495720\pi$$
0.0134446 + 0.999910i $$0.495720\pi$$
$$548$$ −60.0000 −0.00467714
$$549$$ 0 0
$$550$$ −3375.00 −0.261655
$$551$$ −19602.0 −1.51556
$$552$$ 0 0
$$553$$ 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 $$0.746123\pi$$
$$558$$ 0 0
$$559$$ −16874.0 −1.27673
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 23463.0 1.76108
$$563$$ 6294.00 0.471155 0.235578 0.971856i $$-0.424302\pi$$
0.235578 + 0.971856i $$0.424302\pi$$
$$564$$ 0 0
$$565$$ −6960.00 −0.518247
$$566$$ 1974.00 0.146596
$$567$$ 0 0
$$568$$ −1008.00 −0.0744626
$$569$$ −11733.0 −0.864452 −0.432226 0.901765i $$-0.642272\pi$$
−0.432226 + 0.901765i $$0.642272\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1725.00 −0.125109
$$576$$ 0 0
$$577$$ −13156.0 −0.949205 −0.474603 0.880200i $$-0.657408\pi$$
−0.474603 + 0.880200i $$0.657408\pi$$
$$578$$ 5991.00 0.431129
$$579$$ 0 0
$$580$$ −810.000 −0.0579887
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −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
$$590$$ 5400.00 0.376804
$$591$$ 0 0
$$592$$ 18389.0 1.27666
$$593$$ −26664.0 −1.84647 −0.923237 0.384231i $$-0.874467\pi$$
−0.923237 + 0.384231i $$0.874467\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1086.00 0.0746381
$$597$$ 0 0
$$598$$ 12213.0 0.835162
$$599$$ −7614.00 −0.519365 −0.259682 0.965694i $$-0.583618\pi$$
−0.259682 + 0.965694i $$0.583618\pi$$
$$600$$ 0 0
$$601$$ 6410.00 0.435057 0.217529 0.976054i $$-0.430200\pi$$
0.217529 + 0.976054i $$0.430200\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −2866.00 −0.193073
$$605$$ −3470.00 −0.233183
$$606$$ 0 0
$$607$$ −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
$$610$$ 5880.00 0.390286
$$611$$ −2655.00 −0.175793
$$612$$ 0 0
$$613$$ 3737.00 0.246225 0.123113 0.992393i $$-0.460712\pi$$
0.123113 + 0.992393i $$0.460712\pi$$
$$614$$ 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 $$0.700787\pi$$
$$618$$ 0 0
$$619$$ 12287.0 0.797829 0.398915 0.916988i $$-0.369387\pi$$
0.398915 + 0.916988i $$0.369387\pi$$
$$620$$ 440.000 0.0285013
$$621$$ 0 0
$$622$$ 14040.0 0.905069
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −3084.00 −0.196903
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ 25866.0 1.62030
$$635$$ −4015.00 −0.250914
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −21870.0 −1.35712
$$639$$ 0 0
$$640$$ 8295.00 0.512326
$$641$$ −10779.0 −0.664189 −0.332094 0.943246i $$-0.607755\pi$$
−0.332094 + 0.943246i $$0.607755\pi$$
$$642$$ 0 0
$$643$$ 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 $$-0.391339\pi$$
0.334777 + 0.942297i $$0.391339\pi$$
$$648$$ 0 0
$$649$$ 16200.0 0.979824
$$650$$ −4425.00 −0.267020
$$651$$ 0 0
$$652$$ −1228.00 −0.0737610
$$653$$ −22323.0 −1.33777 −0.668887 0.743364i $$-0.733229\pi$$
−0.668887 + 0.743364i $$0.733229\pi$$
$$654$$ 0 0
$$655$$ 10095.0 0.602205
$$656$$ 13845.0 0.824019
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 11856.0 0.700826 0.350413 0.936595i $$-0.386041\pi$$
0.350413 + 0.936595i $$0.386041\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ −16128.0 −0.942602
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −11178.0 −0.648896
$$668$$ 1929.00 0.111729
$$669$$ 0 0
$$670$$ −4200.00 −0.242179
$$671$$ 17640.0 1.01488
$$672$$ 0 0
$$673$$ −12322.0 −0.705763 −0.352881 0.935668i $$-0.614798\pi$$
−0.352881 + 0.935668i $$0.614798\pi$$
$$674$$ −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 $$0.383607\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −5670.00 −0.319757
$$681$$ 0 0
$$682$$ 11880.0 0.667022
$$683$$ 8340.00 0.467235 0.233617 0.972329i $$-0.424944\pi$$
0.233617 + 0.972329i $$0.424944\pi$$
$$684$$ 0 0
$$685$$ 300.000 0.0167334
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$695$$ 8540.00 0.466102
$$696$$ 0 0
$$697$$ −10530.0 −0.572241
$$698$$ −23766.0 −1.28876
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −474.000 −0.0255388 −0.0127694 0.999918i $$-0.504065\pi$$
−0.0127694 + 0.999918i $$0.504065\pi$$
$$702$$ 0 0
$$703$$ 31339.0 1.68133
$$704$$ 19485.0 1.04314
$$705$$ 0 0
$$706$$ 2484.00 0.132417
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −25126.0 −1.33093 −0.665463 0.746431i $$-0.731766\pi$$
−0.665463 + 0.746431i $$0.731766\pi$$
$$710$$ −720.000 −0.0380579
$$711$$ 0 0
$$712$$ 25074.0 1.31979
$$713$$ 6072.00 0.318932
$$714$$ 0 0
$$715$$ −13275.0 −0.694345
$$716$$ −3117.00 −0.162692
$$717$$ 0 0
$$718$$ −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
$$723$$ 0 0
$$724$$ −1798.00 −0.0922958
$$725$$ 4050.00 0.207467
$$726$$ 0 0
$$727$$ −15421.0 −0.786703 −0.393352 0.919388i $$-0.628684\pi$$
−0.393352 + 0.919388i $$0.628684\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 10020.0 0.508023
$$731$$ −15444.0 −0.781419
$$732$$ 0 0
$$733$$ −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
$$738$$ 0 0
$$739$$ −13381.0 −0.666073 −0.333037 0.942914i $$-0.608073\pi$$
−0.333037 + 0.942914i $$0.608073\pi$$
$$740$$ 1295.00 0.0643313
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5487.00 −0.270927 −0.135463 0.990782i $$-0.543252\pi$$
−0.135463 + 0.990782i $$0.543252\pi$$
$$744$$ 0 0
$$745$$ −5430.00 −0.267033
$$746$$ −19806.0 −0.972050
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ −28674.0 −1.38494
$$755$$ 14330.0 0.690758
$$756$$ 0 0
$$757$$ 14846.0 0.712797 0.356398 0.934334i $$-0.384005\pi$$
0.356398 + 0.934334i $$0.384005\pi$$
$$758$$ 24915.0 1.19387
$$759$$ 0 0
$$760$$ 12705.0 0.606393
$$761$$ 3651.00 0.173914 0.0869571 0.996212i $$-0.472286\pi$$
0.0869571 + 0.996212i $$0.472286\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 2388.00 0.113082
$$765$$ 0 0
$$766$$ 2835.00 0.133724
$$767$$ 21240.0 0.999911
$$768$$ 0 0
$$769$$ 29855.0 1.40000 0.699999 0.714144i $$-0.253184\pi$$
0.699999 + 0.714144i $$0.253184\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 272.000 0.0126807
$$773$$ 6519.00 0.303327 0.151664 0.988432i $$-0.451537\pi$$
0.151664 + 0.988432i $$0.451537\pi$$
$$774$$ 0 0
$$775$$ −2200.00 −0.101969
$$776$$ 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
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1145.00 0.0520596
$$786$$ 0 0
$$787$$ 35114.0 1.59044 0.795222 0.606319i $$-0.207354\pi$$
0.795222 + 0.606319i $$0.207354\pi$$
$$788$$ 2109.00 0.0953427
$$789$$ 0 0
$$790$$ 11730.0 0.528272
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 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
$$800$$ 1125.00 0.0497184
$$801$$ 0 0
$$802$$ 21159.0 0.931609
$$803$$ 30060.0 1.32104
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 15576.0 0.680696
$$807$$ 0 0
$$808$$ −14364.0 −0.625401
$$809$$ −4431.00 −0.192566 −0.0962829 0.995354i $$-0.530695\pi$$
−0.0962829 + 0.995354i $$0.530695\pi$$
$$810$$ 0 0
$$811$$ −9577.00 −0.414666 −0.207333 0.978270i $$-0.566478\pi$$
−0.207333 + 0.978270i $$0.566478\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 34965.0 1.50556
$$815$$ 6140.00 0.263895
$$816$$ 0 0
$$817$$ 34606.0 1.48190
$$818$$ 32610.0 1.39387
$$819$$ 0 0
$$820$$ 975.000 0.0415225
$$821$$ 10938.0 0.464968 0.232484 0.972600i $$-0.425315\pi$$
0.232484 + 0.972600i $$0.425315\pi$$
$$822$$ 0 0
$$823$$ 11540.0 0.488772 0.244386 0.969678i $$-0.421414\pi$$
0.244386 + 0.969678i $$0.421414\pi$$
$$824$$ −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 $$0.370935\pi$$
$$828$$ 0 0
$$829$$ −39610.0 −1.65948 −0.829742 0.558147i $$-0.811512\pi$$
−0.829742 + 0.558147i $$0.811512\pi$$
$$830$$ −11520.0 −0.481765
$$831$$ 0 0
$$832$$ 25547.0 1.06452
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −9645.00 −0.399735
$$836$$ −5445.00 −0.225262
$$837$$ 0 0
$$838$$ −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
$$843$$ 0 0
$$844$$ −3625.00 −0.147841
$$845$$ −6420.00 −0.261367
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 42387.0 1.71648
$$849$$ 0 0
$$850$$ −4050.00 −0.163428
$$851$$ 17871.0 0.719871
$$852$$ 0 0
$$853$$ −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 $$-0.186290\pi$$
0.833576 + 0.552405i $$0.186290\pi$$
$$858$$ 0 0
$$859$$ 35192.0 1.39783 0.698915 0.715205i $$-0.253667\pi$$
0.698915 + 0.715205i $$0.253667\pi$$
$$860$$ 1430.00 0.0567007
$$861$$ 0 0
$$862$$ 8964.00 0.354194
$$863$$ 9063.00 0.357483 0.178742 0.983896i $$-0.442797\pi$$
0.178742 + 0.983896i $$0.442797\pi$$
$$864$$ 0 0
$$865$$ −3495.00 −0.137380
$$866$$ −49848.0 −1.95601
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 35190.0 1.37369
$$870$$ 0 0
$$871$$ −16520.0 −0.642662
$$872$$ −33600.0 −1.30486
$$873$$ 0 0
$$874$$ −25047.0 −0.969368
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28439.0 1.09500 0.547501 0.836805i $$-0.315579\pi$$
0.547501 + 0.836805i $$0.315579\pi$$
$$878$$ −22038.0 −0.847092
$$879$$ 0 0
$$880$$ 15975.0 0.611951
$$881$$ 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 $$-0.590552\pi$$
−0.280655 + 0.959809i $$0.590552\pi$$
$$884$$ 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 $$0.395623\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 17910.0 0.674544
$$891$$ 0 0
$$892$$ −4960.00 −0.186181
$$893$$ 5445.00 0.204043
$$894$$ 0 0
$$895$$ 15585.0 0.582066
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 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
$$905$$ 8990.00 0.330207
$$906$$ 0 0
$$907$$ −24922.0 −0.912372 −0.456186 0.889884i $$-0.650785\pi$$
−0.456186 + 0.889884i $$0.650785\pi$$
$$908$$ 1500.00 0.0548230
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30714.0 −1.11701 −0.558507 0.829500i $$-0.688626\pi$$
−0.558507 + 0.829500i $$0.688626\pi$$
$$912$$ 0 0
$$913$$ −34560.0 −1.25276
$$914$$ 28902.0 1.04594
$$915$$ 0 0
$$916$$ 6092.00 0.219744
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 17426.0 0.625496 0.312748 0.949836i $$-0.398750\pi$$
0.312748 + 0.949836i $$0.398750\pi$$
$$920$$ 7245.00 0.259631
$$921$$ 0 0
$$922$$ −1026.00 −0.0366481
$$923$$ −2832.00 −0.100993
$$924$$ 0 0
$$925$$ −6475.00 −0.230159
$$926$$ −7233.00 −0.256686
$$927$$ 0 0
$$928$$ 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
$$933$$ 0 0
$$934$$ −3618.00 −0.126750
$$935$$ −12150.0 −0.424971
$$936$$ 0 0
$$937$$ 27686.0 0.965274 0.482637 0.875820i $$-0.339679\pi$$
0.482637 + 0.875820i $$0.339679\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 225.000 0.00780712
$$941$$ 17808.0 0.616923 0.308461 0.951237i $$-0.400186\pi$$
0.308461 + 0.951237i $$0.400186\pi$$
$$942$$ 0 0
$$943$$ 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 $$0.462196\pi$$
$$948$$ 0 0
$$949$$ 39412.0 1.34812
$$950$$ 9075.00 0.309928
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 20940.0 0.711766 0.355883 0.934530i $$-0.384180\pi$$
0.355883 + 0.934530i $$0.384180\pi$$
$$954$$ 0 0
$$955$$ −11940.0 −0.404575
$$956$$ 5502.00 0.186137
$$957$$ 0 0
$$958$$ −1296.00 −0.0437076
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22047.0 −0.740056
$$962$$ 45843.0 1.53642
$$963$$ 0 0
$$964$$ 3551.00 0.118641
$$965$$ −1360.00 −0.0453678
$$966$$ 0 0
$$967$$ 9176.00 0.305150 0.152575 0.988292i $$-0.451243\pi$$
0.152575 + 0.988292i $$0.451243\pi$$
$$968$$ 14574.0 0.483911
$$969$$ 0 0
$$970$$ 13530.0 0.447858
$$971$$ −29763.0 −0.983666 −0.491833 0.870689i $$-0.663673\pi$$
−0.491833 + 0.870689i $$0.663673\pi$$
$$972$$ 0 0
$$973$$ 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 $$0.282974\pi$$
$$978$$ 0 0
$$979$$ 53730.0 1.75405
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −36828.0 −1.19677
$$983$$ 12609.0 0.409120 0.204560 0.978854i $$-0.434424\pi$$
0.204560 + 0.978854i $$0.434424\pi$$
$$984$$ 0 0
$$985$$ −10545.0 −0.341108
$$986$$ −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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −7120.00 −0.226853
$$996$$ 0 0
$$997$$ 46034.0 1.46230 0.731149 0.682218i $$-0.238984\pi$$
0.731149 + 0.682218i $$0.238984\pi$$
$$998$$ 32628.0 1.03489
$$999$$ 0 0
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 2205.4.a.e.1.1 1
3.2 odd 2 245.4.a.e.1.1 1
7.2 even 3 315.4.j.b.46.1 2
7.4 even 3 315.4.j.b.226.1 2
7.6 odd 2 2205.4.a.g.1.1 1
15.14 odd 2 1225.4.a.b.1.1 1
21.2 odd 6 35.4.e.a.11.1 2
21.5 even 6 245.4.e.a.116.1 2
21.11 odd 6 35.4.e.a.16.1 yes 2
21.17 even 6 245.4.e.a.226.1 2
21.20 even 2 245.4.a.f.1.1 1
84.11 even 6 560.4.q.b.401.1 2
84.23 even 6 560.4.q.b.81.1 2
105.2 even 12 175.4.k.b.74.2 4
105.23 even 12 175.4.k.b.74.1 4
105.32 even 12 175.4.k.b.149.1 4
105.44 odd 6 175.4.e.b.151.1 2
105.53 even 12 175.4.k.b.149.2 4
105.74 odd 6 175.4.e.b.51.1 2
105.104 even 2 1225.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.a.11.1 2 21.2 odd 6
35.4.e.a.16.1 yes 2 21.11 odd 6
175.4.e.b.51.1 2 105.74 odd 6
175.4.e.b.151.1 2 105.44 odd 6
175.4.k.b.74.1 4 105.23 even 12
175.4.k.b.74.2 4 105.2 even 12
175.4.k.b.149.1 4 105.32 even 12
175.4.k.b.149.2 4 105.53 even 12
245.4.a.e.1.1 1 3.2 odd 2
245.4.a.f.1.1 1 21.20 even 2
245.4.e.a.116.1 2 21.5 even 6
245.4.e.a.226.1 2 21.17 even 6
315.4.j.b.46.1 2 7.2 even 3
315.4.j.b.226.1 2 7.4 even 3
560.4.q.b.81.1 2 84.23 even 6
560.4.q.b.401.1 2 84.11 even 6
1225.4.a.a.1.1 1 105.104 even 2
1225.4.a.b.1.1 1 15.14 odd 2
2205.4.a.e.1.1 1 1.1 even 1 trivial
2205.4.a.g.1.1 1 7.6 odd 2