Properties

 Label 960.4.a.t.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ 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] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.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: $$56.6418336055$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -5.00000 q^{5} -32.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -5.00000 q^{5} -32.0000 q^{7} +9.00000 q^{9} +64.0000 q^{11} +6.00000 q^{13} -15.0000 q^{15} +38.0000 q^{17} -116.000 q^{19} -96.0000 q^{21} +120.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +122.000 q^{29} -164.000 q^{31} +192.000 q^{33} +160.000 q^{35} -146.000 q^{37} +18.0000 q^{39} -238.000 q^{41} -148.000 q^{43} -45.0000 q^{45} +184.000 q^{47} +681.000 q^{49} +114.000 q^{51} -470.000 q^{53} -320.000 q^{55} -348.000 q^{57} -216.000 q^{59} -806.000 q^{61} -288.000 q^{63} -30.0000 q^{65} -732.000 q^{67} +360.000 q^{69} -264.000 q^{71} -638.000 q^{73} +75.0000 q^{75} -2048.00 q^{77} -596.000 q^{79} +81.0000 q^{81} -884.000 q^{83} -190.000 q^{85} +366.000 q^{87} +930.000 q^{89} -192.000 q^{91} -492.000 q^{93} +580.000 q^{95} +322.000 q^{97} +576.000 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −32.0000 −1.72784 −0.863919 0.503631i $$-0.831997\pi$$
−0.863919 + 0.503631i $$0.831997\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 64.0000 1.75425 0.877124 0.480264i $$-0.159459\pi$$
0.877124 + 0.480264i $$0.159459\pi$$
$$12$$ 0 0
$$13$$ 6.00000 0.128008 0.0640039 0.997950i $$-0.479613\pi$$
0.0640039 + 0.997950i $$0.479613\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ 38.0000 0.542138 0.271069 0.962560i $$-0.412623\pi$$
0.271069 + 0.962560i $$0.412623\pi$$
$$18$$ 0 0
$$19$$ −116.000 −1.40064 −0.700322 0.713827i $$-0.746960\pi$$
−0.700322 + 0.713827i $$0.746960\pi$$
$$20$$ 0 0
$$21$$ −96.0000 −0.997567
$$22$$ 0 0
$$23$$ 120.000 1.08790 0.543951 0.839117i $$-0.316928\pi$$
0.543951 + 0.839117i $$0.316928\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 122.000 0.781201 0.390601 0.920560i $$-0.372267\pi$$
0.390601 + 0.920560i $$0.372267\pi$$
$$30$$ 0 0
$$31$$ −164.000 −0.950170 −0.475085 0.879940i $$-0.657583\pi$$
−0.475085 + 0.879940i $$0.657583\pi$$
$$32$$ 0 0
$$33$$ 192.000 1.01282
$$34$$ 0 0
$$35$$ 160.000 0.772712
$$36$$ 0 0
$$37$$ −146.000 −0.648710 −0.324355 0.945936i $$-0.605147\pi$$
−0.324355 + 0.945936i $$0.605147\pi$$
$$38$$ 0 0
$$39$$ 18.0000 0.0739053
$$40$$ 0 0
$$41$$ −238.000 −0.906570 −0.453285 0.891366i $$-0.649748\pi$$
−0.453285 + 0.891366i $$0.649748\pi$$
$$42$$ 0 0
$$43$$ −148.000 −0.524879 −0.262439 0.964948i $$-0.584527\pi$$
−0.262439 + 0.964948i $$0.584527\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ 184.000 0.571046 0.285523 0.958372i $$-0.407833\pi$$
0.285523 + 0.958372i $$0.407833\pi$$
$$48$$ 0 0
$$49$$ 681.000 1.98542
$$50$$ 0 0
$$51$$ 114.000 0.313004
$$52$$ 0 0
$$53$$ −470.000 −1.21810 −0.609052 0.793131i $$-0.708450\pi$$
−0.609052 + 0.793131i $$0.708450\pi$$
$$54$$ 0 0
$$55$$ −320.000 −0.784523
$$56$$ 0 0
$$57$$ −348.000 −0.808662
$$58$$ 0 0
$$59$$ −216.000 −0.476624 −0.238312 0.971189i $$-0.576594\pi$$
−0.238312 + 0.971189i $$0.576594\pi$$
$$60$$ 0 0
$$61$$ −806.000 −1.69177 −0.845883 0.533369i $$-0.820926\pi$$
−0.845883 + 0.533369i $$0.820926\pi$$
$$62$$ 0 0
$$63$$ −288.000 −0.575946
$$64$$ 0 0
$$65$$ −30.0000 −0.0572468
$$66$$ 0 0
$$67$$ −732.000 −1.33475 −0.667373 0.744723i $$-0.732581\pi$$
−0.667373 + 0.744723i $$0.732581\pi$$
$$68$$ 0 0
$$69$$ 360.000 0.628100
$$70$$ 0 0
$$71$$ −264.000 −0.441282 −0.220641 0.975355i $$-0.570815\pi$$
−0.220641 + 0.975355i $$0.570815\pi$$
$$72$$ 0 0
$$73$$ −638.000 −1.02291 −0.511454 0.859311i $$-0.670893\pi$$
−0.511454 + 0.859311i $$0.670893\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ −2048.00 −3.03106
$$78$$ 0 0
$$79$$ −596.000 −0.848800 −0.424400 0.905475i $$-0.639515\pi$$
−0.424400 + 0.905475i $$0.639515\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −884.000 −1.16906 −0.584528 0.811374i $$-0.698720\pi$$
−0.584528 + 0.811374i $$0.698720\pi$$
$$84$$ 0 0
$$85$$ −190.000 −0.242452
$$86$$ 0 0
$$87$$ 366.000 0.451027
$$88$$ 0 0
$$89$$ 930.000 1.10764 0.553819 0.832637i $$-0.313170\pi$$
0.553819 + 0.832637i $$0.313170\pi$$
$$90$$ 0 0
$$91$$ −192.000 −0.221177
$$92$$ 0 0
$$93$$ −492.000 −0.548581
$$94$$ 0 0
$$95$$ 580.000 0.626387
$$96$$ 0 0
$$97$$ 322.000 0.337053 0.168527 0.985697i $$-0.446099\pi$$
0.168527 + 0.985697i $$0.446099\pi$$
$$98$$ 0 0
$$99$$ 576.000 0.584749
$$100$$ 0 0
$$101$$ 946.000 0.931985 0.465993 0.884789i $$-0.345697\pi$$
0.465993 + 0.884789i $$0.345697\pi$$
$$102$$ 0 0
$$103$$ −424.000 −0.405611 −0.202806 0.979219i $$-0.565006\pi$$
−0.202806 + 0.979219i $$0.565006\pi$$
$$104$$ 0 0
$$105$$ 480.000 0.446126
$$106$$ 0 0
$$107$$ −1668.00 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$108$$ 0 0
$$109$$ 1634.00 1.43586 0.717930 0.696115i $$-0.245090\pi$$
0.717930 + 0.696115i $$0.245090\pi$$
$$110$$ 0 0
$$111$$ −438.000 −0.374533
$$112$$ 0 0
$$113$$ 1686.00 1.40359 0.701794 0.712380i $$-0.252383\pi$$
0.701794 + 0.712380i $$0.252383\pi$$
$$114$$ 0 0
$$115$$ −600.000 −0.486524
$$116$$ 0 0
$$117$$ 54.0000 0.0426692
$$118$$ 0 0
$$119$$ −1216.00 −0.936727
$$120$$ 0 0
$$121$$ 2765.00 2.07739
$$122$$ 0 0
$$123$$ −714.000 −0.523408
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −2640.00 −1.84458 −0.922292 0.386494i $$-0.873686\pi$$
−0.922292 + 0.386494i $$0.873686\pi$$
$$128$$ 0 0
$$129$$ −444.000 −0.303039
$$130$$ 0 0
$$131$$ −2536.00 −1.69138 −0.845692 0.533671i $$-0.820812\pi$$
−0.845692 + 0.533671i $$0.820812\pi$$
$$132$$ 0 0
$$133$$ 3712.00 2.42008
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ −634.000 −0.395374 −0.197687 0.980265i $$-0.563343\pi$$
−0.197687 + 0.980265i $$0.563343\pi$$
$$138$$ 0 0
$$139$$ 2980.00 1.81842 0.909210 0.416338i $$-0.136687\pi$$
0.909210 + 0.416338i $$0.136687\pi$$
$$140$$ 0 0
$$141$$ 552.000 0.329694
$$142$$ 0 0
$$143$$ 384.000 0.224557
$$144$$ 0 0
$$145$$ −610.000 −0.349364
$$146$$ 0 0
$$147$$ 2043.00 1.14628
$$148$$ 0 0
$$149$$ 338.000 0.185839 0.0929196 0.995674i $$-0.470380\pi$$
0.0929196 + 0.995674i $$0.470380\pi$$
$$150$$ 0 0
$$151$$ 428.000 0.230663 0.115332 0.993327i $$-0.463207\pi$$
0.115332 + 0.993327i $$0.463207\pi$$
$$152$$ 0 0
$$153$$ 342.000 0.180713
$$154$$ 0 0
$$155$$ 820.000 0.424929
$$156$$ 0 0
$$157$$ −3010.00 −1.53009 −0.765045 0.643977i $$-0.777283\pi$$
−0.765045 + 0.643977i $$0.777283\pi$$
$$158$$ 0 0
$$159$$ −1410.00 −0.703272
$$160$$ 0 0
$$161$$ −3840.00 −1.87972
$$162$$ 0 0
$$163$$ −132.000 −0.0634297 −0.0317148 0.999497i $$-0.510097\pi$$
−0.0317148 + 0.999497i $$0.510097\pi$$
$$164$$ 0 0
$$165$$ −960.000 −0.452945
$$166$$ 0 0
$$167$$ −2176.00 −1.00829 −0.504144 0.863620i $$-0.668192\pi$$
−0.504144 + 0.863620i $$0.668192\pi$$
$$168$$ 0 0
$$169$$ −2161.00 −0.983614
$$170$$ 0 0
$$171$$ −1044.00 −0.466881
$$172$$ 0 0
$$173$$ 2778.00 1.22085 0.610426 0.792073i $$-0.290998\pi$$
0.610426 + 0.792073i $$0.290998\pi$$
$$174$$ 0 0
$$175$$ −800.000 −0.345568
$$176$$ 0 0
$$177$$ −648.000 −0.275179
$$178$$ 0 0
$$179$$ −3888.00 −1.62348 −0.811740 0.584020i $$-0.801479\pi$$
−0.811740 + 0.584020i $$0.801479\pi$$
$$180$$ 0 0
$$181$$ −1350.00 −0.554391 −0.277195 0.960814i $$-0.589405\pi$$
−0.277195 + 0.960814i $$0.589405\pi$$
$$182$$ 0 0
$$183$$ −2418.00 −0.976742
$$184$$ 0 0
$$185$$ 730.000 0.290112
$$186$$ 0 0
$$187$$ 2432.00 0.951045
$$188$$ 0 0
$$189$$ −864.000 −0.332522
$$190$$ 0 0
$$191$$ 4760.00 1.80325 0.901627 0.432514i $$-0.142374\pi$$
0.901627 + 0.432514i $$0.142374\pi$$
$$192$$ 0 0
$$193$$ 1034.00 0.385642 0.192821 0.981234i $$-0.438236\pi$$
0.192821 + 0.981234i $$0.438236\pi$$
$$194$$ 0 0
$$195$$ −90.0000 −0.0330515
$$196$$ 0 0
$$197$$ 1354.00 0.489688 0.244844 0.969563i $$-0.421263\pi$$
0.244844 + 0.969563i $$0.421263\pi$$
$$198$$ 0 0
$$199$$ 2324.00 0.827859 0.413930 0.910309i $$-0.364156\pi$$
0.413930 + 0.910309i $$0.364156\pi$$
$$200$$ 0 0
$$201$$ −2196.00 −0.770616
$$202$$ 0 0
$$203$$ −3904.00 −1.34979
$$204$$ 0 0
$$205$$ 1190.00 0.405430
$$206$$ 0 0
$$207$$ 1080.00 0.362634
$$208$$ 0 0
$$209$$ −7424.00 −2.45708
$$210$$ 0 0
$$211$$ −3220.00 −1.05059 −0.525294 0.850921i $$-0.676045\pi$$
−0.525294 + 0.850921i $$0.676045\pi$$
$$212$$ 0 0
$$213$$ −792.000 −0.254774
$$214$$ 0 0
$$215$$ 740.000 0.234733
$$216$$ 0 0
$$217$$ 5248.00 1.64174
$$218$$ 0 0
$$219$$ −1914.00 −0.590576
$$220$$ 0 0
$$221$$ 228.000 0.0693979
$$222$$ 0 0
$$223$$ 32.0000 0.00960932 0.00480466 0.999988i $$-0.498471\pi$$
0.00480466 + 0.999988i $$0.498471\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −3996.00 −1.16839 −0.584193 0.811614i $$-0.698589\pi$$
−0.584193 + 0.811614i $$0.698589\pi$$
$$228$$ 0 0
$$229$$ 3010.00 0.868587 0.434293 0.900771i $$-0.356998\pi$$
0.434293 + 0.900771i $$0.356998\pi$$
$$230$$ 0 0
$$231$$ −6144.00 −1.74998
$$232$$ 0 0
$$233$$ −2698.00 −0.758592 −0.379296 0.925275i $$-0.623834\pi$$
−0.379296 + 0.925275i $$0.623834\pi$$
$$234$$ 0 0
$$235$$ −920.000 −0.255380
$$236$$ 0 0
$$237$$ −1788.00 −0.490055
$$238$$ 0 0
$$239$$ 2200.00 0.595423 0.297712 0.954656i $$-0.403777\pi$$
0.297712 + 0.954656i $$0.403777\pi$$
$$240$$ 0 0
$$241$$ −6174.00 −1.65022 −0.825109 0.564974i $$-0.808886\pi$$
−0.825109 + 0.564974i $$0.808886\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −3405.00 −0.887908
$$246$$ 0 0
$$247$$ −696.000 −0.179293
$$248$$ 0 0
$$249$$ −2652.00 −0.674955
$$250$$ 0 0
$$251$$ 4808.00 1.20908 0.604538 0.796576i $$-0.293358\pi$$
0.604538 + 0.796576i $$0.293358\pi$$
$$252$$ 0 0
$$253$$ 7680.00 1.90845
$$254$$ 0 0
$$255$$ −570.000 −0.139980
$$256$$ 0 0
$$257$$ 3974.00 0.964558 0.482279 0.876018i $$-0.339809\pi$$
0.482279 + 0.876018i $$0.339809\pi$$
$$258$$ 0 0
$$259$$ 4672.00 1.12086
$$260$$ 0 0
$$261$$ 1098.00 0.260400
$$262$$ 0 0
$$263$$ −1408.00 −0.330118 −0.165059 0.986284i $$-0.552781\pi$$
−0.165059 + 0.986284i $$0.552781\pi$$
$$264$$ 0 0
$$265$$ 2350.00 0.544752
$$266$$ 0 0
$$267$$ 2790.00 0.639495
$$268$$ 0 0
$$269$$ 3258.00 0.738453 0.369226 0.929340i $$-0.379623\pi$$
0.369226 + 0.929340i $$0.379623\pi$$
$$270$$ 0 0
$$271$$ −8612.00 −1.93041 −0.965206 0.261490i $$-0.915786\pi$$
−0.965206 + 0.261490i $$0.915786\pi$$
$$272$$ 0 0
$$273$$ −576.000 −0.127696
$$274$$ 0 0
$$275$$ 1600.00 0.350850
$$276$$ 0 0
$$277$$ 4006.00 0.868943 0.434472 0.900686i $$-0.356935\pi$$
0.434472 + 0.900686i $$0.356935\pi$$
$$278$$ 0 0
$$279$$ −1476.00 −0.316723
$$280$$ 0 0
$$281$$ 6194.00 1.31496 0.657479 0.753473i $$-0.271623\pi$$
0.657479 + 0.753473i $$0.271623\pi$$
$$282$$ 0 0
$$283$$ 1724.00 0.362124 0.181062 0.983472i $$-0.442046\pi$$
0.181062 + 0.983472i $$0.442046\pi$$
$$284$$ 0 0
$$285$$ 1740.00 0.361645
$$286$$ 0 0
$$287$$ 7616.00 1.56641
$$288$$ 0 0
$$289$$ −3469.00 −0.706086
$$290$$ 0 0
$$291$$ 966.000 0.194598
$$292$$ 0 0
$$293$$ −2502.00 −0.498868 −0.249434 0.968392i $$-0.580245\pi$$
−0.249434 + 0.968392i $$0.580245\pi$$
$$294$$ 0 0
$$295$$ 1080.00 0.213153
$$296$$ 0 0
$$297$$ 1728.00 0.337605
$$298$$ 0 0
$$299$$ 720.000 0.139260
$$300$$ 0 0
$$301$$ 4736.00 0.906905
$$302$$ 0 0
$$303$$ 2838.00 0.538082
$$304$$ 0 0
$$305$$ 4030.00 0.756581
$$306$$ 0 0
$$307$$ −6404.00 −1.19054 −0.595270 0.803526i $$-0.702955\pi$$
−0.595270 + 0.803526i $$0.702955\pi$$
$$308$$ 0 0
$$309$$ −1272.00 −0.234180
$$310$$ 0 0
$$311$$ −896.000 −0.163368 −0.0816841 0.996658i $$-0.526030\pi$$
−0.0816841 + 0.996658i $$0.526030\pi$$
$$312$$ 0 0
$$313$$ −4110.00 −0.742207 −0.371104 0.928591i $$-0.621021\pi$$
−0.371104 + 0.928591i $$0.621021\pi$$
$$314$$ 0 0
$$315$$ 1440.00 0.257571
$$316$$ 0 0
$$317$$ −6926.00 −1.22714 −0.613569 0.789641i $$-0.710267\pi$$
−0.613569 + 0.789641i $$0.710267\pi$$
$$318$$ 0 0
$$319$$ 7808.00 1.37042
$$320$$ 0 0
$$321$$ −5004.00 −0.870081
$$322$$ 0 0
$$323$$ −4408.00 −0.759343
$$324$$ 0 0
$$325$$ 150.000 0.0256015
$$326$$ 0 0
$$327$$ 4902.00 0.828995
$$328$$ 0 0
$$329$$ −5888.00 −0.986675
$$330$$ 0 0
$$331$$ −2692.00 −0.447026 −0.223513 0.974701i $$-0.571753\pi$$
−0.223513 + 0.974701i $$0.571753\pi$$
$$332$$ 0 0
$$333$$ −1314.00 −0.216237
$$334$$ 0 0
$$335$$ 3660.00 0.596917
$$336$$ 0 0
$$337$$ 11914.0 1.92581 0.962903 0.269846i $$-0.0869728\pi$$
0.962903 + 0.269846i $$0.0869728\pi$$
$$338$$ 0 0
$$339$$ 5058.00 0.810362
$$340$$ 0 0
$$341$$ −10496.0 −1.66683
$$342$$ 0 0
$$343$$ −10816.0 −1.70265
$$344$$ 0 0
$$345$$ −1800.00 −0.280895
$$346$$ 0 0
$$347$$ −6660.00 −1.03034 −0.515169 0.857088i $$-0.672271\pi$$
−0.515169 + 0.857088i $$0.672271\pi$$
$$348$$ 0 0
$$349$$ −3046.00 −0.467188 −0.233594 0.972334i $$-0.575049\pi$$
−0.233594 + 0.972334i $$0.575049\pi$$
$$350$$ 0 0
$$351$$ 162.000 0.0246351
$$352$$ 0 0
$$353$$ −3522.00 −0.531040 −0.265520 0.964105i $$-0.585544\pi$$
−0.265520 + 0.964105i $$0.585544\pi$$
$$354$$ 0 0
$$355$$ 1320.00 0.197347
$$356$$ 0 0
$$357$$ −3648.00 −0.540820
$$358$$ 0 0
$$359$$ 8656.00 1.27255 0.636276 0.771461i $$-0.280474\pi$$
0.636276 + 0.771461i $$0.280474\pi$$
$$360$$ 0 0
$$361$$ 6597.00 0.961802
$$362$$ 0 0
$$363$$ 8295.00 1.19938
$$364$$ 0 0
$$365$$ 3190.00 0.457458
$$366$$ 0 0
$$367$$ −936.000 −0.133130 −0.0665651 0.997782i $$-0.521204\pi$$
−0.0665651 + 0.997782i $$0.521204\pi$$
$$368$$ 0 0
$$369$$ −2142.00 −0.302190
$$370$$ 0 0
$$371$$ 15040.0 2.10468
$$372$$ 0 0
$$373$$ −11578.0 −1.60720 −0.803601 0.595169i $$-0.797085\pi$$
−0.803601 + 0.595169i $$0.797085\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ 732.000 0.0999998
$$378$$ 0 0
$$379$$ 9948.00 1.34827 0.674135 0.738608i $$-0.264517\pi$$
0.674135 + 0.738608i $$0.264517\pi$$
$$380$$ 0 0
$$381$$ −7920.00 −1.06497
$$382$$ 0 0
$$383$$ −8336.00 −1.11214 −0.556070 0.831135i $$-0.687691\pi$$
−0.556070 + 0.831135i $$0.687691\pi$$
$$384$$ 0 0
$$385$$ 10240.0 1.35553
$$386$$ 0 0
$$387$$ −1332.00 −0.174960
$$388$$ 0 0
$$389$$ 6370.00 0.830262 0.415131 0.909762i $$-0.363736\pi$$
0.415131 + 0.909762i $$0.363736\pi$$
$$390$$ 0 0
$$391$$ 4560.00 0.589793
$$392$$ 0 0
$$393$$ −7608.00 −0.976521
$$394$$ 0 0
$$395$$ 2980.00 0.379595
$$396$$ 0 0
$$397$$ −10394.0 −1.31400 −0.657002 0.753888i $$-0.728176\pi$$
−0.657002 + 0.753888i $$0.728176\pi$$
$$398$$ 0 0
$$399$$ 11136.0 1.39724
$$400$$ 0 0
$$401$$ −7470.00 −0.930259 −0.465130 0.885243i $$-0.653992\pi$$
−0.465130 + 0.885243i $$0.653992\pi$$
$$402$$ 0 0
$$403$$ −984.000 −0.121629
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −9344.00 −1.13800
$$408$$ 0 0
$$409$$ 2810.00 0.339720 0.169860 0.985468i $$-0.445668\pi$$
0.169860 + 0.985468i $$0.445668\pi$$
$$410$$ 0 0
$$411$$ −1902.00 −0.228269
$$412$$ 0 0
$$413$$ 6912.00 0.823529
$$414$$ 0 0
$$415$$ 4420.00 0.522818
$$416$$ 0 0
$$417$$ 8940.00 1.04986
$$418$$ 0 0
$$419$$ −4320.00 −0.503689 −0.251845 0.967768i $$-0.581037\pi$$
−0.251845 + 0.967768i $$0.581037\pi$$
$$420$$ 0 0
$$421$$ 15122.0 1.75060 0.875298 0.483583i $$-0.160665\pi$$
0.875298 + 0.483583i $$0.160665\pi$$
$$422$$ 0 0
$$423$$ 1656.00 0.190349
$$424$$ 0 0
$$425$$ 950.000 0.108428
$$426$$ 0 0
$$427$$ 25792.0 2.92310
$$428$$ 0 0
$$429$$ 1152.00 0.129648
$$430$$ 0 0
$$431$$ 12616.0 1.40996 0.704978 0.709229i $$-0.250957\pi$$
0.704978 + 0.709229i $$0.250957\pi$$
$$432$$ 0 0
$$433$$ 15098.0 1.67567 0.837833 0.545926i $$-0.183822\pi$$
0.837833 + 0.545926i $$0.183822\pi$$
$$434$$ 0 0
$$435$$ −1830.00 −0.201705
$$436$$ 0 0
$$437$$ −13920.0 −1.52376
$$438$$ 0 0
$$439$$ 2372.00 0.257880 0.128940 0.991652i $$-0.458843\pi$$
0.128940 + 0.991652i $$0.458843\pi$$
$$440$$ 0 0
$$441$$ 6129.00 0.661808
$$442$$ 0 0
$$443$$ −36.0000 −0.00386097 −0.00193049 0.999998i $$-0.500614\pi$$
−0.00193049 + 0.999998i $$0.500614\pi$$
$$444$$ 0 0
$$445$$ −4650.00 −0.495351
$$446$$ 0 0
$$447$$ 1014.00 0.107294
$$448$$ 0 0
$$449$$ 7330.00 0.770432 0.385216 0.922826i $$-0.374127\pi$$
0.385216 + 0.922826i $$0.374127\pi$$
$$450$$ 0 0
$$451$$ −15232.0 −1.59035
$$452$$ 0 0
$$453$$ 1284.00 0.133173
$$454$$ 0 0
$$455$$ 960.000 0.0989132
$$456$$ 0 0
$$457$$ 9642.00 0.986945 0.493472 0.869761i $$-0.335727\pi$$
0.493472 + 0.869761i $$0.335727\pi$$
$$458$$ 0 0
$$459$$ 1026.00 0.104335
$$460$$ 0 0
$$461$$ −9654.00 −0.975340 −0.487670 0.873028i $$-0.662153\pi$$
−0.487670 + 0.873028i $$0.662153\pi$$
$$462$$ 0 0
$$463$$ 13960.0 1.40124 0.700622 0.713532i $$-0.252906\pi$$
0.700622 + 0.713532i $$0.252906\pi$$
$$464$$ 0 0
$$465$$ 2460.00 0.245333
$$466$$ 0 0
$$467$$ 9996.00 0.990492 0.495246 0.868753i $$-0.335078\pi$$
0.495246 + 0.868753i $$0.335078\pi$$
$$468$$ 0 0
$$469$$ 23424.0 2.30623
$$470$$ 0 0
$$471$$ −9030.00 −0.883398
$$472$$ 0 0
$$473$$ −9472.00 −0.920767
$$474$$ 0 0
$$475$$ −2900.00 −0.280129
$$476$$ 0 0
$$477$$ −4230.00 −0.406034
$$478$$ 0 0
$$479$$ 8736.00 0.833315 0.416658 0.909063i $$-0.363201\pi$$
0.416658 + 0.909063i $$0.363201\pi$$
$$480$$ 0 0
$$481$$ −876.000 −0.0830398
$$482$$ 0 0
$$483$$ −11520.0 −1.08525
$$484$$ 0 0
$$485$$ −1610.00 −0.150735
$$486$$ 0 0
$$487$$ −6712.00 −0.624537 −0.312269 0.949994i $$-0.601089\pi$$
−0.312269 + 0.949994i $$0.601089\pi$$
$$488$$ 0 0
$$489$$ −396.000 −0.0366211
$$490$$ 0 0
$$491$$ −2512.00 −0.230886 −0.115443 0.993314i $$-0.536829\pi$$
−0.115443 + 0.993314i $$0.536829\pi$$
$$492$$ 0 0
$$493$$ 4636.00 0.423519
$$494$$ 0 0
$$495$$ −2880.00 −0.261508
$$496$$ 0 0
$$497$$ 8448.00 0.762464
$$498$$ 0 0
$$499$$ −5708.00 −0.512074 −0.256037 0.966667i $$-0.582417\pi$$
−0.256037 + 0.966667i $$0.582417\pi$$
$$500$$ 0 0
$$501$$ −6528.00 −0.582135
$$502$$ 0 0
$$503$$ 5440.00 0.482222 0.241111 0.970498i $$-0.422488\pi$$
0.241111 + 0.970498i $$0.422488\pi$$
$$504$$ 0 0
$$505$$ −4730.00 −0.416797
$$506$$ 0 0
$$507$$ −6483.00 −0.567890
$$508$$ 0 0
$$509$$ −3942.00 −0.343273 −0.171637 0.985160i $$-0.554905\pi$$
−0.171637 + 0.985160i $$0.554905\pi$$
$$510$$ 0 0
$$511$$ 20416.0 1.76742
$$512$$ 0 0
$$513$$ −3132.00 −0.269554
$$514$$ 0 0
$$515$$ 2120.00 0.181395
$$516$$ 0 0
$$517$$ 11776.0 1.00176
$$518$$ 0 0
$$519$$ 8334.00 0.704859
$$520$$ 0 0
$$521$$ −2310.00 −0.194247 −0.0971237 0.995272i $$-0.530964\pi$$
−0.0971237 + 0.995272i $$0.530964\pi$$
$$522$$ 0 0
$$523$$ −2956.00 −0.247145 −0.123573 0.992336i $$-0.539435\pi$$
−0.123573 + 0.992336i $$0.539435\pi$$
$$524$$ 0 0
$$525$$ −2400.00 −0.199513
$$526$$ 0 0
$$527$$ −6232.00 −0.515124
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ −1944.00 −0.158875
$$532$$ 0 0
$$533$$ −1428.00 −0.116048
$$534$$ 0 0
$$535$$ 8340.00 0.673962
$$536$$ 0 0
$$537$$ −11664.0 −0.937316
$$538$$ 0 0
$$539$$ 43584.0 3.48292
$$540$$ 0 0
$$541$$ −2078.00 −0.165139 −0.0825695 0.996585i $$-0.526313\pi$$
−0.0825695 + 0.996585i $$0.526313\pi$$
$$542$$ 0 0
$$543$$ −4050.00 −0.320078
$$544$$ 0 0
$$545$$ −8170.00 −0.642136
$$546$$ 0 0
$$547$$ 6164.00 0.481816 0.240908 0.970548i $$-0.422555\pi$$
0.240908 + 0.970548i $$0.422555\pi$$
$$548$$ 0 0
$$549$$ −7254.00 −0.563922
$$550$$ 0 0
$$551$$ −14152.0 −1.09418
$$552$$ 0 0
$$553$$ 19072.0 1.46659
$$554$$ 0 0
$$555$$ 2190.00 0.167496
$$556$$ 0 0
$$557$$ −13526.0 −1.02893 −0.514466 0.857511i $$-0.672010\pi$$
−0.514466 + 0.857511i $$0.672010\pi$$
$$558$$ 0 0
$$559$$ −888.000 −0.0671885
$$560$$ 0 0
$$561$$ 7296.00 0.549086
$$562$$ 0 0
$$563$$ 276.000 0.0206608 0.0103304 0.999947i $$-0.496712\pi$$
0.0103304 + 0.999947i $$0.496712\pi$$
$$564$$ 0 0
$$565$$ −8430.00 −0.627704
$$566$$ 0 0
$$567$$ −2592.00 −0.191982
$$568$$ 0 0
$$569$$ −774.000 −0.0570260 −0.0285130 0.999593i $$-0.509077\pi$$
−0.0285130 + 0.999593i $$0.509077\pi$$
$$570$$ 0 0
$$571$$ −6676.00 −0.489285 −0.244643 0.969613i $$-0.578671\pi$$
−0.244643 + 0.969613i $$0.578671\pi$$
$$572$$ 0 0
$$573$$ 14280.0 1.04111
$$574$$ 0 0
$$575$$ 3000.00 0.217580
$$576$$ 0 0
$$577$$ −11774.0 −0.849494 −0.424747 0.905312i $$-0.639637\pi$$
−0.424747 + 0.905312i $$0.639637\pi$$
$$578$$ 0 0
$$579$$ 3102.00 0.222651
$$580$$ 0 0
$$581$$ 28288.0 2.01994
$$582$$ 0 0
$$583$$ −30080.0 −2.13685
$$584$$ 0 0
$$585$$ −270.000 −0.0190823
$$586$$ 0 0
$$587$$ −12292.0 −0.864302 −0.432151 0.901801i $$-0.642245\pi$$
−0.432151 + 0.901801i $$0.642245\pi$$
$$588$$ 0 0
$$589$$ 19024.0 1.33085
$$590$$ 0 0
$$591$$ 4062.00 0.282721
$$592$$ 0 0
$$593$$ 7110.00 0.492365 0.246183 0.969223i $$-0.420824\pi$$
0.246183 + 0.969223i $$0.420824\pi$$
$$594$$ 0 0
$$595$$ 6080.00 0.418917
$$596$$ 0 0
$$597$$ 6972.00 0.477965
$$598$$ 0 0
$$599$$ −4416.00 −0.301223 −0.150612 0.988593i $$-0.548124\pi$$
−0.150612 + 0.988593i $$0.548124\pi$$
$$600$$ 0 0
$$601$$ 3850.00 0.261306 0.130653 0.991428i $$-0.458293\pi$$
0.130653 + 0.991428i $$0.458293\pi$$
$$602$$ 0 0
$$603$$ −6588.00 −0.444916
$$604$$ 0 0
$$605$$ −13825.0 −0.929035
$$606$$ 0 0
$$607$$ 21880.0 1.46307 0.731534 0.681805i $$-0.238805\pi$$
0.731534 + 0.681805i $$0.238805\pi$$
$$608$$ 0 0
$$609$$ −11712.0 −0.779301
$$610$$ 0 0
$$611$$ 1104.00 0.0730983
$$612$$ 0 0
$$613$$ −2786.00 −0.183565 −0.0917826 0.995779i $$-0.529256\pi$$
−0.0917826 + 0.995779i $$0.529256\pi$$
$$614$$ 0 0
$$615$$ 3570.00 0.234075
$$616$$ 0 0
$$617$$ 1014.00 0.0661622 0.0330811 0.999453i $$-0.489468\pi$$
0.0330811 + 0.999453i $$0.489468\pi$$
$$618$$ 0 0
$$619$$ 10708.0 0.695300 0.347650 0.937624i $$-0.386980\pi$$
0.347650 + 0.937624i $$0.386980\pi$$
$$620$$ 0 0
$$621$$ 3240.00 0.209367
$$622$$ 0 0
$$623$$ −29760.0 −1.91382
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −22272.0 −1.41859
$$628$$ 0 0
$$629$$ −5548.00 −0.351690
$$630$$ 0 0
$$631$$ 6660.00 0.420175 0.210087 0.977683i $$-0.432625\pi$$
0.210087 + 0.977683i $$0.432625\pi$$
$$632$$ 0 0
$$633$$ −9660.00 −0.606557
$$634$$ 0 0
$$635$$ 13200.0 0.824923
$$636$$ 0 0
$$637$$ 4086.00 0.254149
$$638$$ 0 0
$$639$$ −2376.00 −0.147094
$$640$$ 0 0
$$641$$ 90.0000 0.00554569 0.00277284 0.999996i $$-0.499117\pi$$
0.00277284 + 0.999996i $$0.499117\pi$$
$$642$$ 0 0
$$643$$ −21684.0 −1.32991 −0.664956 0.746882i $$-0.731550\pi$$
−0.664956 + 0.746882i $$0.731550\pi$$
$$644$$ 0 0
$$645$$ 2220.00 0.135523
$$646$$ 0 0
$$647$$ 1344.00 0.0816663 0.0408331 0.999166i $$-0.486999\pi$$
0.0408331 + 0.999166i $$0.486999\pi$$
$$648$$ 0 0
$$649$$ −13824.0 −0.836116
$$650$$ 0 0
$$651$$ 15744.0 0.947859
$$652$$ 0 0
$$653$$ 9210.00 0.551937 0.275969 0.961167i $$-0.411001\pi$$
0.275969 + 0.961167i $$0.411001\pi$$
$$654$$ 0 0
$$655$$ 12680.0 0.756410
$$656$$ 0 0
$$657$$ −5742.00 −0.340969
$$658$$ 0 0
$$659$$ 184.000 0.0108765 0.00543826 0.999985i $$-0.498269\pi$$
0.00543826 + 0.999985i $$0.498269\pi$$
$$660$$ 0 0
$$661$$ 12866.0 0.757079 0.378540 0.925585i $$-0.376426\pi$$
0.378540 + 0.925585i $$0.376426\pi$$
$$662$$ 0 0
$$663$$ 684.000 0.0400669
$$664$$ 0 0
$$665$$ −18560.0 −1.08229
$$666$$ 0 0
$$667$$ 14640.0 0.849870
$$668$$ 0 0
$$669$$ 96.0000 0.00554794
$$670$$ 0 0
$$671$$ −51584.0 −2.96778
$$672$$ 0 0
$$673$$ −26534.0 −1.51978 −0.759889 0.650053i $$-0.774747\pi$$
−0.759889 + 0.650053i $$0.774747\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −30062.0 −1.70661 −0.853306 0.521410i $$-0.825406\pi$$
−0.853306 + 0.521410i $$0.825406\pi$$
$$678$$ 0 0
$$679$$ −10304.0 −0.582373
$$680$$ 0 0
$$681$$ −11988.0 −0.674569
$$682$$ 0 0
$$683$$ 12724.0 0.712841 0.356420 0.934326i $$-0.383997\pi$$
0.356420 + 0.934326i $$0.383997\pi$$
$$684$$ 0 0
$$685$$ 3170.00 0.176817
$$686$$ 0 0
$$687$$ 9030.00 0.501479
$$688$$ 0 0
$$689$$ −2820.00 −0.155927
$$690$$ 0 0
$$691$$ −21972.0 −1.20963 −0.604815 0.796366i $$-0.706753\pi$$
−0.604815 + 0.796366i $$0.706753\pi$$
$$692$$ 0 0
$$693$$ −18432.0 −1.01035
$$694$$ 0 0
$$695$$ −14900.0 −0.813222
$$696$$ 0 0
$$697$$ −9044.00 −0.491486
$$698$$ 0 0
$$699$$ −8094.00 −0.437973
$$700$$ 0 0
$$701$$ 15642.0 0.842782 0.421391 0.906879i $$-0.361542\pi$$
0.421391 + 0.906879i $$0.361542\pi$$
$$702$$ 0 0
$$703$$ 16936.0 0.908611
$$704$$ 0 0
$$705$$ −2760.00 −0.147443
$$706$$ 0 0
$$707$$ −30272.0 −1.61032
$$708$$ 0 0
$$709$$ −36398.0 −1.92801 −0.964003 0.265893i $$-0.914333\pi$$
−0.964003 + 0.265893i $$0.914333\pi$$
$$710$$ 0 0
$$711$$ −5364.00 −0.282933
$$712$$ 0 0
$$713$$ −19680.0 −1.03369
$$714$$ 0 0
$$715$$ −1920.00 −0.100425
$$716$$ 0 0
$$717$$ 6600.00 0.343768
$$718$$ 0 0
$$719$$ 36248.0 1.88014 0.940071 0.340978i $$-0.110758\pi$$
0.940071 + 0.340978i $$0.110758\pi$$
$$720$$ 0 0
$$721$$ 13568.0 0.700830
$$722$$ 0 0
$$723$$ −18522.0 −0.952753
$$724$$ 0 0
$$725$$ 3050.00 0.156240
$$726$$ 0 0
$$727$$ 4936.00 0.251810 0.125905 0.992042i $$-0.459816\pi$$
0.125905 + 0.992042i $$0.459816\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −5624.00 −0.284557
$$732$$ 0 0
$$733$$ 23838.0 1.20120 0.600598 0.799551i $$-0.294929\pi$$
0.600598 + 0.799551i $$0.294929\pi$$
$$734$$ 0 0
$$735$$ −10215.0 −0.512634
$$736$$ 0 0
$$737$$ −46848.0 −2.34148
$$738$$ 0 0
$$739$$ 31700.0 1.57795 0.788974 0.614427i $$-0.210613\pi$$
0.788974 + 0.614427i $$0.210613\pi$$
$$740$$ 0 0
$$741$$ −2088.00 −0.103515
$$742$$ 0 0
$$743$$ 13128.0 0.648209 0.324105 0.946021i $$-0.394937\pi$$
0.324105 + 0.946021i $$0.394937\pi$$
$$744$$ 0 0
$$745$$ −1690.00 −0.0831098
$$746$$ 0 0
$$747$$ −7956.00 −0.389685
$$748$$ 0 0
$$749$$ 53376.0 2.60389
$$750$$ 0 0
$$751$$ 15676.0 0.761685 0.380842 0.924640i $$-0.375634\pi$$
0.380842 + 0.924640i $$0.375634\pi$$
$$752$$ 0 0
$$753$$ 14424.0 0.698061
$$754$$ 0 0
$$755$$ −2140.00 −0.103156
$$756$$ 0 0
$$757$$ 13238.0 0.635592 0.317796 0.948159i $$-0.397057\pi$$
0.317796 + 0.948159i $$0.397057\pi$$
$$758$$ 0 0
$$759$$ 23040.0 1.10184
$$760$$ 0 0
$$761$$ −5270.00 −0.251035 −0.125517 0.992091i $$-0.540059\pi$$
−0.125517 + 0.992091i $$0.540059\pi$$
$$762$$ 0 0
$$763$$ −52288.0 −2.48093
$$764$$ 0 0
$$765$$ −1710.00 −0.0808172
$$766$$ 0 0
$$767$$ −1296.00 −0.0610115
$$768$$ 0 0
$$769$$ −8526.00 −0.399812 −0.199906 0.979815i $$-0.564064\pi$$
−0.199906 + 0.979815i $$0.564064\pi$$
$$770$$ 0 0
$$771$$ 11922.0 0.556888
$$772$$ 0 0
$$773$$ −13606.0 −0.633084 −0.316542 0.948579i $$-0.602522\pi$$
−0.316542 + 0.948579i $$0.602522\pi$$
$$774$$ 0 0
$$775$$ −4100.00 −0.190034
$$776$$ 0 0
$$777$$ 14016.0 0.647132
$$778$$ 0 0
$$779$$ 27608.0 1.26978
$$780$$ 0 0
$$781$$ −16896.0 −0.774118
$$782$$ 0 0
$$783$$ 3294.00 0.150342
$$784$$ 0 0
$$785$$ 15050.0 0.684277
$$786$$ 0 0
$$787$$ −8836.00 −0.400215 −0.200108 0.979774i $$-0.564129\pi$$
−0.200108 + 0.979774i $$0.564129\pi$$
$$788$$ 0 0
$$789$$ −4224.00 −0.190594
$$790$$ 0 0
$$791$$ −53952.0 −2.42517
$$792$$ 0 0
$$793$$ −4836.00 −0.216559
$$794$$ 0 0
$$795$$ 7050.00 0.314513
$$796$$ 0 0
$$797$$ 29082.0 1.29252 0.646259 0.763118i $$-0.276332\pi$$
0.646259 + 0.763118i $$0.276332\pi$$
$$798$$ 0 0
$$799$$ 6992.00 0.309586
$$800$$ 0 0
$$801$$ 8370.00 0.369213
$$802$$ 0 0
$$803$$ −40832.0 −1.79443
$$804$$ 0 0
$$805$$ 19200.0 0.840635
$$806$$ 0 0
$$807$$ 9774.00 0.426346
$$808$$ 0 0
$$809$$ 26994.0 1.17313 0.586563 0.809904i $$-0.300481\pi$$
0.586563 + 0.809904i $$0.300481\pi$$
$$810$$ 0 0
$$811$$ −9268.00 −0.401287 −0.200643 0.979664i $$-0.564303\pi$$
−0.200643 + 0.979664i $$0.564303\pi$$
$$812$$ 0 0
$$813$$ −25836.0 −1.11452
$$814$$ 0 0
$$815$$ 660.000 0.0283666
$$816$$ 0 0
$$817$$ 17168.0 0.735168
$$818$$ 0 0
$$819$$ −1728.00 −0.0737255
$$820$$ 0 0
$$821$$ −6286.00 −0.267214 −0.133607 0.991034i $$-0.542656\pi$$
−0.133607 + 0.991034i $$0.542656\pi$$
$$822$$ 0 0
$$823$$ −44088.0 −1.86733 −0.933664 0.358150i $$-0.883408\pi$$
−0.933664 + 0.358150i $$0.883408\pi$$
$$824$$ 0 0
$$825$$ 4800.00 0.202563
$$826$$ 0 0
$$827$$ 30100.0 1.26563 0.632817 0.774301i $$-0.281898\pi$$
0.632817 + 0.774301i $$0.281898\pi$$
$$828$$ 0 0
$$829$$ −18254.0 −0.764762 −0.382381 0.924005i $$-0.624896\pi$$
−0.382381 + 0.924005i $$0.624896\pi$$
$$830$$ 0 0
$$831$$ 12018.0 0.501684
$$832$$ 0 0
$$833$$ 25878.0 1.07637
$$834$$ 0 0
$$835$$ 10880.0 0.450920
$$836$$ 0 0
$$837$$ −4428.00 −0.182860
$$838$$ 0 0
$$839$$ −13008.0 −0.535263 −0.267632 0.963521i $$-0.586241\pi$$
−0.267632 + 0.963521i $$0.586241\pi$$
$$840$$ 0 0
$$841$$ −9505.00 −0.389725
$$842$$ 0 0
$$843$$ 18582.0 0.759191
$$844$$ 0 0
$$845$$ 10805.0 0.439886
$$846$$ 0 0
$$847$$ −88480.0 −3.58938
$$848$$ 0 0
$$849$$ 5172.00 0.209073
$$850$$ 0 0
$$851$$ −17520.0 −0.705732
$$852$$ 0 0
$$853$$ 16094.0 0.646012 0.323006 0.946397i $$-0.395307\pi$$
0.323006 + 0.946397i $$0.395307\pi$$
$$854$$ 0 0
$$855$$ 5220.00 0.208796
$$856$$ 0 0
$$857$$ −25938.0 −1.03387 −0.516934 0.856025i $$-0.672927\pi$$
−0.516934 + 0.856025i $$0.672927\pi$$
$$858$$ 0 0
$$859$$ −38564.0 −1.53177 −0.765883 0.642980i $$-0.777698\pi$$
−0.765883 + 0.642980i $$0.777698\pi$$
$$860$$ 0 0
$$861$$ 22848.0 0.904364
$$862$$ 0 0
$$863$$ −10216.0 −0.402963 −0.201481 0.979492i $$-0.564576\pi$$
−0.201481 + 0.979492i $$0.564576\pi$$
$$864$$ 0 0
$$865$$ −13890.0 −0.545982
$$866$$ 0 0
$$867$$ −10407.0 −0.407659
$$868$$ 0 0
$$869$$ −38144.0 −1.48901
$$870$$ 0 0
$$871$$ −4392.00 −0.170858
$$872$$ 0 0
$$873$$ 2898.00 0.112351
$$874$$ 0 0
$$875$$ 4000.00 0.154542
$$876$$ 0 0
$$877$$ 17006.0 0.654791 0.327396 0.944887i $$-0.393829\pi$$
0.327396 + 0.944887i $$0.393829\pi$$
$$878$$ 0 0
$$879$$ −7506.00 −0.288022
$$880$$ 0 0
$$881$$ 43898.0 1.67873 0.839365 0.543568i $$-0.182927\pi$$
0.839365 + 0.543568i $$0.182927\pi$$
$$882$$ 0 0
$$883$$ 29180.0 1.11210 0.556051 0.831149i $$-0.312316\pi$$
0.556051 + 0.831149i $$0.312316\pi$$
$$884$$ 0 0
$$885$$ 3240.00 0.123064
$$886$$ 0 0
$$887$$ 14752.0 0.558426 0.279213 0.960229i $$-0.409926\pi$$
0.279213 + 0.960229i $$0.409926\pi$$
$$888$$ 0 0
$$889$$ 84480.0 3.18714
$$890$$ 0 0
$$891$$ 5184.00 0.194916
$$892$$ 0 0
$$893$$ −21344.0 −0.799832
$$894$$ 0 0
$$895$$ 19440.0 0.726042
$$896$$ 0 0
$$897$$ 2160.00 0.0804017
$$898$$ 0 0
$$899$$ −20008.0 −0.742274
$$900$$ 0 0
$$901$$ −17860.0 −0.660381
$$902$$ 0 0
$$903$$ 14208.0 0.523602
$$904$$ 0 0
$$905$$ 6750.00 0.247931
$$906$$ 0 0
$$907$$ −12020.0 −0.440041 −0.220021 0.975495i $$-0.570612\pi$$
−0.220021 + 0.975495i $$0.570612\pi$$
$$908$$ 0 0
$$909$$ 8514.00 0.310662
$$910$$ 0 0
$$911$$ 18560.0 0.674995 0.337497 0.941326i $$-0.390420\pi$$
0.337497 + 0.941326i $$0.390420\pi$$
$$912$$ 0 0
$$913$$ −56576.0 −2.05081
$$914$$ 0 0
$$915$$ 12090.0 0.436812
$$916$$ 0 0
$$917$$ 81152.0 2.92244
$$918$$ 0 0
$$919$$ 10244.0 0.367702 0.183851 0.982954i $$-0.441144\pi$$
0.183851 + 0.982954i $$0.441144\pi$$
$$920$$ 0 0
$$921$$ −19212.0 −0.687358
$$922$$ 0 0
$$923$$ −1584.00 −0.0564875
$$924$$ 0 0
$$925$$ −3650.00 −0.129742
$$926$$ 0 0
$$927$$ −3816.00 −0.135204
$$928$$ 0 0
$$929$$ 1266.00 0.0447106 0.0223553 0.999750i $$-0.492884\pi$$
0.0223553 + 0.999750i $$0.492884\pi$$
$$930$$ 0 0
$$931$$ −78996.0 −2.78087
$$932$$ 0 0
$$933$$ −2688.00 −0.0943207
$$934$$ 0 0
$$935$$ −12160.0 −0.425320
$$936$$ 0 0
$$937$$ 47626.0 1.66048 0.830242 0.557403i $$-0.188202\pi$$
0.830242 + 0.557403i $$0.188202\pi$$
$$938$$ 0 0
$$939$$ −12330.0 −0.428514
$$940$$ 0 0
$$941$$ −31958.0 −1.10712 −0.553561 0.832809i $$-0.686731\pi$$
−0.553561 + 0.832809i $$0.686731\pi$$
$$942$$ 0 0
$$943$$ −28560.0 −0.986258
$$944$$ 0 0
$$945$$ 4320.00 0.148709
$$946$$ 0 0
$$947$$ 25196.0 0.864583 0.432291 0.901734i $$-0.357705\pi$$
0.432291 + 0.901734i $$0.357705\pi$$
$$948$$ 0 0
$$949$$ −3828.00 −0.130940
$$950$$ 0 0
$$951$$ −20778.0 −0.708489
$$952$$ 0 0
$$953$$ 51574.0 1.75304 0.876519 0.481367i $$-0.159859\pi$$
0.876519 + 0.481367i $$0.159859\pi$$
$$954$$ 0 0
$$955$$ −23800.0 −0.806440
$$956$$ 0 0
$$957$$ 23424.0 0.791213
$$958$$ 0 0
$$959$$ 20288.0 0.683143
$$960$$ 0 0
$$961$$ −2895.00 −0.0971770
$$962$$ 0 0
$$963$$ −15012.0 −0.502342
$$964$$ 0 0
$$965$$ −5170.00 −0.172464
$$966$$ 0 0
$$967$$ −48296.0 −1.60610 −0.803048 0.595914i $$-0.796790\pi$$
−0.803048 + 0.595914i $$0.796790\pi$$
$$968$$ 0 0
$$969$$ −13224.0 −0.438407
$$970$$ 0 0
$$971$$ −27288.0 −0.901868 −0.450934 0.892557i $$-0.648909\pi$$
−0.450934 + 0.892557i $$0.648909\pi$$
$$972$$ 0 0
$$973$$ −95360.0 −3.14193
$$974$$ 0 0
$$975$$ 450.000 0.0147811
$$976$$ 0 0
$$977$$ 14406.0 0.471739 0.235869 0.971785i $$-0.424206\pi$$
0.235869 + 0.971785i $$0.424206\pi$$
$$978$$ 0 0
$$979$$ 59520.0 1.94307
$$980$$ 0 0
$$981$$ 14706.0 0.478620
$$982$$ 0 0
$$983$$ −18952.0 −0.614929 −0.307464 0.951560i $$-0.599480\pi$$
−0.307464 + 0.951560i $$0.599480\pi$$
$$984$$ 0 0
$$985$$ −6770.00 −0.218995
$$986$$ 0 0
$$987$$ −17664.0 −0.569657
$$988$$ 0 0
$$989$$ −17760.0 −0.571016
$$990$$ 0 0
$$991$$ 44308.0 1.42027 0.710136 0.704064i $$-0.248633\pi$$
0.710136 + 0.704064i $$0.248633\pi$$
$$992$$ 0 0
$$993$$ −8076.00 −0.258091
$$994$$ 0 0
$$995$$ −11620.0 −0.370230
$$996$$ 0 0
$$997$$ −954.000 −0.0303044 −0.0151522 0.999885i $$-0.504823\pi$$
−0.0151522 + 0.999885i $$0.504823\pi$$
$$998$$ 0 0
$$999$$ −3942.00 −0.124844
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 960.4.a.t.1.1 1
4.3 odd 2 960.4.a.i.1.1 1
8.3 odd 2 480.4.a.l.1.1 yes 1
8.5 even 2 480.4.a.c.1.1 1
24.5 odd 2 1440.4.a.a.1.1 1
24.11 even 2 1440.4.a.j.1.1 1
40.19 odd 2 2400.4.a.a.1.1 1
40.29 even 2 2400.4.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.c.1.1 1 8.5 even 2
480.4.a.l.1.1 yes 1 8.3 odd 2
960.4.a.i.1.1 1 4.3 odd 2
960.4.a.t.1.1 1 1.1 even 1 trivial
1440.4.a.a.1.1 1 24.5 odd 2
1440.4.a.j.1.1 1 24.11 even 2
2400.4.a.a.1.1 1 40.19 odd 2
2400.4.a.v.1.1 1 40.29 even 2