Properties

 Label 7650.2.a.y.1.1 Level $7650$ Weight $2$ Character 7650.1 Self dual yes Analytic conductor $61.086$ 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] = [7650,2,Mod(1,7650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7650, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7650.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: $$61.0855575463$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} -4.00000 q^{11} -2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +4.00000 q^{19} +4.00000 q^{22} +4.00000 q^{23} +2.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} +6.00000 q^{37} -4.00000 q^{38} -8.00000 q^{41} -2.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +14.0000 q^{53} -2.00000 q^{56} +6.00000 q^{58} -6.00000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} +1.00000 q^{68} +10.0000 q^{71} -4.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -8.00000 q^{77} +4.00000 q^{79} +8.00000 q^{82} -16.0000 q^{83} +2.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +4.00000 q^{92} +8.00000 q^{94} +8.00000 q^{97} +3.00000 q^{98} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 8.00000 0.883452
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 20.0000 1.97066 0.985329 0.170664i $$-0.0545913\pi$$
0.985329 + 0.170664i $$0.0545913\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −14.0000 −1.35980
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 8.00000 0.693688
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −10.0000 −0.839181
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ −8.00000 −0.624695
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2.00000 −0.152499
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −12.0000 −0.844317
$$203$$ −12.0000 −0.842235
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −20.0000 −1.39347
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 14.0000 0.961524
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ −2.00000 −0.129641
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ 0 0
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 10.0000 0.593391
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −16.0000 −0.944450
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ 20.0000 1.15857
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −8.00000 −0.445823
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 0 0
$$328$$ 8.00000 0.441726
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 0 0
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 2.00000 0.105703
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 38.0000 1.98358 0.991792 0.127862i $$-0.0408116\pi$$
0.991792 + 0.127862i $$0.0408116\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 28.0000 1.45369
$$372$$ 0 0
$$373$$ −36.0000 −1.86401 −0.932005 0.362446i $$-0.881942\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −20.0000 −1.02329
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 16.0000 0.814379
$$387$$ 0 0
$$388$$ 8.00000 0.406138
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −20.0000 −0.998752 −0.499376 0.866385i $$-0.666437\pi$$
−0.499376 + 0.866385i $$0.666437\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 20.0000 0.985329
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ −32.0000 −1.56330 −0.781651 0.623716i $$-0.785622\pi$$
−0.781651 + 0.623716i $$0.785622\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ −14.0000 −0.679900
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −10.0000 −0.481683 −0.240842 0.970564i $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ 0 0
$$433$$ −6.00000 −0.288342 −0.144171 0.989553i $$-0.546051\pi$$
−0.144171 + 0.989553i $$0.546051\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 16.0000 0.765384
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ 32.0000 1.50682
$$452$$ −18.0000 −0.846649
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 26.0000 1.21490
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −24.0000 −1.11779 −0.558896 0.829238i $$-0.688775\pi$$
−0.558896 + 0.829238i $$0.688775\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −30.0000 −1.35943 −0.679715 0.733476i $$-0.737896\pi$$
−0.679715 + 0.733476i $$0.737896\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 20.0000 0.897123
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −2.00000 −0.0892644
$$503$$ 44.0000 1.96186 0.980932 0.194354i $$-0.0622609\pi$$
0.980932 + 0.194354i $$0.0622609\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ 36.0000 1.59567 0.797836 0.602875i $$-0.205978\pi$$
0.797836 + 0.602875i $$0.205978\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 32.0000 1.40736
$$518$$ −12.0000 −0.527250
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ −30.0000 −1.31181 −0.655904 0.754844i $$-0.727712\pi$$
−0.655904 + 0.754844i $$0.727712\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 8.00000 0.346844
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ −2.00000 −0.0862261
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −36.0000 −1.53925 −0.769624 0.638497i $$-0.779557\pi$$
−0.769624 + 0.638497i $$0.779557\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ 40.0000 1.68580 0.842900 0.538071i $$-0.180847\pi$$
0.842900 + 0.538071i $$0.180847\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ −10.0000 −0.419591
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 16.0000 0.667827
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −32.0000 −1.32758
$$582$$ 0 0
$$583$$ −56.0000 −2.31928
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ 46.0000 1.88899 0.944497 0.328521i $$-0.106550\pi$$
0.944497 + 0.328521i $$0.106550\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 36.0000 1.45403 0.727013 0.686624i $$-0.240908\pi$$
0.727013 + 0.686624i $$0.240908\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ −12.0000 −0.482321 −0.241160 0.970485i $$-0.577528\pi$$
−0.241160 + 0.970485i $$0.577528\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 10.0000 0.400963
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −24.0000 −0.950169
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −28.0000 −1.10593 −0.552967 0.833203i $$-0.686504\pi$$
−0.552967 + 0.833203i $$0.686504\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −8.00000 −0.313304
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ 0 0
$$658$$ 16.0000 0.623745
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −24.0000 −0.929284
$$668$$ −16.0000 −0.619059
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 20.0000 0.770371
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −32.0000 −1.22534
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ −2.00000 −0.0762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ 14.0000 0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8.00000 0.302156 0.151078 0.988522i $$-0.451726\pi$$
0.151078 + 0.988522i $$0.451726\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 24.0000 0.902613
$$708$$ 0 0
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.00000 −0.0747435
$$717$$ 0 0
$$718$$ −4.00000 −0.149279
$$719$$ −14.0000 −0.522112 −0.261056 0.965324i $$-0.584071\pi$$
−0.261056 + 0.965324i $$0.584071\pi$$
$$720$$ 0 0
$$721$$ 40.0000 1.48968
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.00000 −0.0739727
$$732$$ 0 0
$$733$$ 44.0000 1.62518 0.812589 0.582838i $$-0.198058\pi$$
0.812589 + 0.582838i $$0.198058\pi$$
$$734$$ −38.0000 −1.40261
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ −32.0000 −1.17714 −0.588570 0.808447i $$-0.700309\pi$$
−0.588570 + 0.808447i $$0.700309\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −28.0000 −1.02791
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 36.0000 1.31805
$$747$$ 0 0
$$748$$ −4.00000 −0.146254
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 36.0000 1.31366 0.656829 0.754039i $$-0.271897\pi$$
0.656829 + 0.754039i $$0.271897\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 46.0000 1.66750 0.833749 0.552143i $$-0.186190\pi$$
0.833749 + 0.552143i $$0.186190\pi$$
$$762$$ 0 0
$$763$$ −4.00000 −0.144810
$$764$$ 20.0000 0.723575
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −50.0000 −1.80305 −0.901523 0.432731i $$-0.857550\pi$$
−0.901523 + 0.432731i $$0.857550\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −16.0000 −0.575853
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −8.00000 −0.287183
$$777$$ 0 0
$$778$$ −36.0000 −1.29066
$$779$$ −32.0000 −1.14652
$$780$$ 0 0
$$781$$ −40.0000 −1.43131
$$782$$ −4.00000 −0.143040
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −36.0000 −1.28001
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 34.0000 1.20661
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 20.0000 0.706225
$$803$$ 16.0000 0.564628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ −20.0000 −0.696733
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −3.00000 −0.103944
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ 32.0000 1.10542
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 14.0000 0.480762
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ 26.0000 0.888143 0.444072 0.895991i $$-0.353534\pi$$
0.444072 + 0.895991i $$0.353534\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10.0000 0.340601
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 6.00000 0.203888
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −44.0000 −1.48240 −0.741199 0.671286i $$-0.765742\pi$$
−0.741199 + 0.671286i $$0.765742\pi$$
$$882$$ 0 0
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 14.0000 0.466408
$$902$$ −32.0000 −1.06548
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 16.0000 0.531271 0.265636 0.964073i $$-0.414418\pi$$
0.265636 + 0.964073i $$0.414418\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38.0000 −1.25900 −0.629498 0.777002i $$-0.716739\pi$$
−0.629498 + 0.777002i $$0.716739\pi$$
$$912$$ 0 0
$$913$$ 64.0000 2.11809
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 24.0000 0.790398
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ 6.00000 0.196960
$$929$$ 32.0000 1.04989 0.524943 0.851137i $$-0.324087\pi$$
0.524943 + 0.851137i $$0.324087\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 10.0000 0.327561
$$933$$ 0 0
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ −32.0000 −1.04206
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −2.00000 −0.0648204
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 30.0000 0.969256
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −50.0000 −1.60458 −0.802288 0.596937i $$-0.796384\pi$$
−0.802288 + 0.596937i $$0.796384\pi$$
$$972$$ 0 0
$$973$$ −8.00000 −0.256468
$$974$$ 30.0000 0.961262
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −6.00000 −0.191468
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 6.00000 0.191079
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 48.0000 1.52477 0.762385 0.647124i $$-0.224028\pi$$
0.762385 + 0.647124i $$0.224028\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 0 0
$$994$$ −20.0000 −0.634361
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 62.0000 1.96356 0.981780 0.190022i $$-0.0608559\pi$$
0.981780 + 0.190022i $$0.0608559\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ 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 7650.2.a.y.1.1 1
3.2 odd 2 2550.2.a.y.1.1 1
5.4 even 2 1530.2.a.i.1.1 1
15.2 even 4 2550.2.d.j.2449.2 2
15.8 even 4 2550.2.d.j.2449.1 2
15.14 odd 2 510.2.a.b.1.1 1
60.59 even 2 4080.2.a.n.1.1 1
255.254 odd 2 8670.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.b.1.1 1 15.14 odd 2
1530.2.a.i.1.1 1 5.4 even 2
2550.2.a.y.1.1 1 3.2 odd 2
2550.2.d.j.2449.1 2 15.8 even 4
2550.2.d.j.2449.2 2 15.2 even 4
4080.2.a.n.1.1 1 60.59 even 2
7650.2.a.y.1.1 1 1.1 even 1 trivial
8670.2.a.c.1.1 1 255.254 odd 2