# Properties

 Label 1080.6.a.b.1.1 Level $1080$ Weight $6$ Character 1080.1 Self dual yes Analytic conductor $173.215$ 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] = [1080,6,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1080.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: $$173.214525398$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+25.0000 q^{5} -234.000 q^{7} +O(q^{10})$$ $$q+25.0000 q^{5} -234.000 q^{7} +347.000 q^{11} -33.0000 q^{13} +237.000 q^{17} +1496.00 q^{19} -2811.00 q^{23} +625.000 q^{25} -5513.00 q^{29} +2911.00 q^{31} -5850.00 q^{35} +5602.00 q^{37} +4716.00 q^{41} +10479.0 q^{43} +5963.00 q^{47} +37949.0 q^{49} -17964.0 q^{53} +8675.00 q^{55} +30372.0 q^{59} -35530.0 q^{61} -825.000 q^{65} -12476.0 q^{67} +7520.00 q^{71} +36378.0 q^{73} -81198.0 q^{77} -22727.0 q^{79} +46254.0 q^{83} +5925.00 q^{85} -58832.0 q^{89} +7722.00 q^{91} +37400.0 q^{95} -145906. q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ −234.000 −1.80497 −0.902487 0.430718i $$-0.858260\pi$$
−0.902487 + 0.430718i $$0.858260\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 347.000 0.864665 0.432332 0.901714i $$-0.357691\pi$$
0.432332 + 0.901714i $$0.357691\pi$$
$$12$$ 0 0
$$13$$ −33.0000 −0.0541571 −0.0270786 0.999633i $$-0.508620\pi$$
−0.0270786 + 0.999633i $$0.508620\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 237.000 0.198896 0.0994480 0.995043i $$-0.468292\pi$$
0.0994480 + 0.995043i $$0.468292\pi$$
$$18$$ 0 0
$$19$$ 1496.00 0.950709 0.475354 0.879794i $$-0.342320\pi$$
0.475354 + 0.879794i $$0.342320\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2811.00 −1.10800 −0.554002 0.832515i $$-0.686900\pi$$
−0.554002 + 0.832515i $$0.686900\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −5513.00 −1.21729 −0.608644 0.793444i $$-0.708286\pi$$
−0.608644 + 0.793444i $$0.708286\pi$$
$$30$$ 0 0
$$31$$ 2911.00 0.544049 0.272024 0.962290i $$-0.412307\pi$$
0.272024 + 0.962290i $$0.412307\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −5850.00 −0.807209
$$36$$ 0 0
$$37$$ 5602.00 0.672727 0.336363 0.941732i $$-0.390803\pi$$
0.336363 + 0.941732i $$0.390803\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4716.00 0.438141 0.219071 0.975709i $$-0.429697\pi$$
0.219071 + 0.975709i $$0.429697\pi$$
$$42$$ 0 0
$$43$$ 10479.0 0.864269 0.432134 0.901809i $$-0.357761\pi$$
0.432134 + 0.901809i $$0.357761\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5963.00 0.393750 0.196875 0.980429i $$-0.436921\pi$$
0.196875 + 0.980429i $$0.436921\pi$$
$$48$$ 0 0
$$49$$ 37949.0 2.25793
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −17964.0 −0.878443 −0.439221 0.898379i $$-0.644746\pi$$
−0.439221 + 0.898379i $$0.644746\pi$$
$$54$$ 0 0
$$55$$ 8675.00 0.386690
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 30372.0 1.13591 0.567954 0.823060i $$-0.307735\pi$$
0.567954 + 0.823060i $$0.307735\pi$$
$$60$$ 0 0
$$61$$ −35530.0 −1.22256 −0.611281 0.791414i $$-0.709345\pi$$
−0.611281 + 0.791414i $$0.709345\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −825.000 −0.0242198
$$66$$ 0 0
$$67$$ −12476.0 −0.339538 −0.169769 0.985484i $$-0.554302\pi$$
−0.169769 + 0.985484i $$0.554302\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7520.00 0.177040 0.0885201 0.996074i $$-0.471786\pi$$
0.0885201 + 0.996074i $$0.471786\pi$$
$$72$$ 0 0
$$73$$ 36378.0 0.798972 0.399486 0.916739i $$-0.369189\pi$$
0.399486 + 0.916739i $$0.369189\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −81198.0 −1.56070
$$78$$ 0 0
$$79$$ −22727.0 −0.409708 −0.204854 0.978793i $$-0.565672\pi$$
−0.204854 + 0.978793i $$0.565672\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 46254.0 0.736977 0.368489 0.929632i $$-0.379875\pi$$
0.368489 + 0.929632i $$0.379875\pi$$
$$84$$ 0 0
$$85$$ 5925.00 0.0889490
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −58832.0 −0.787297 −0.393648 0.919261i $$-0.628787\pi$$
−0.393648 + 0.919261i $$0.628787\pi$$
$$90$$ 0 0
$$91$$ 7722.00 0.0977521
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 37400.0 0.425170
$$96$$ 0 0
$$97$$ −145906. −1.57450 −0.787252 0.616631i $$-0.788497\pi$$
−0.787252 + 0.616631i $$0.788497\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 89421.0 0.872240 0.436120 0.899888i $$-0.356352\pi$$
0.436120 + 0.899888i $$0.356352\pi$$
$$102$$ 0 0
$$103$$ −134718. −1.25122 −0.625608 0.780137i $$-0.715149\pi$$
−0.625608 + 0.780137i $$0.715149\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 66002.0 0.557311 0.278656 0.960391i $$-0.410111\pi$$
0.278656 + 0.960391i $$0.410111\pi$$
$$108$$ 0 0
$$109$$ −8460.00 −0.0682031 −0.0341016 0.999418i $$-0.510857\pi$$
−0.0341016 + 0.999418i $$0.510857\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −255183. −1.87999 −0.939995 0.341188i $$-0.889171\pi$$
−0.939995 + 0.341188i $$0.889171\pi$$
$$114$$ 0 0
$$115$$ −70275.0 −0.495514
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −55458.0 −0.359002
$$120$$ 0 0
$$121$$ −40642.0 −0.252355
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ 63790.0 0.350948 0.175474 0.984484i $$-0.443854\pi$$
0.175474 + 0.984484i $$0.443854\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 237541. 1.20937 0.604687 0.796464i $$-0.293298\pi$$
0.604687 + 0.796464i $$0.293298\pi$$
$$132$$ 0 0
$$133$$ −350064. −1.71600
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −137062. −0.623901 −0.311950 0.950098i $$-0.600982\pi$$
−0.311950 + 0.950098i $$0.600982\pi$$
$$138$$ 0 0
$$139$$ 189754. 0.833017 0.416509 0.909132i $$-0.363254\pi$$
0.416509 + 0.909132i $$0.363254\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −11451.0 −0.0468278
$$144$$ 0 0
$$145$$ −137825. −0.544387
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −50689.0 −0.187046 −0.0935229 0.995617i $$-0.529813\pi$$
−0.0935229 + 0.995617i $$0.529813\pi$$
$$150$$ 0 0
$$151$$ −322103. −1.14961 −0.574807 0.818289i $$-0.694923\pi$$
−0.574807 + 0.818289i $$0.694923\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 72775.0 0.243306
$$156$$ 0 0
$$157$$ −513807. −1.66361 −0.831804 0.555070i $$-0.812692\pi$$
−0.831804 + 0.555070i $$0.812692\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 657774. 1.99992
$$162$$ 0 0
$$163$$ 410491. 1.21014 0.605069 0.796173i $$-0.293146\pi$$
0.605069 + 0.796173i $$0.293146\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −362204. −1.00499 −0.502495 0.864580i $$-0.667585\pi$$
−0.502495 + 0.864580i $$0.667585\pi$$
$$168$$ 0 0
$$169$$ −370204. −0.997067
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −4690.00 −0.0119140 −0.00595700 0.999982i $$-0.501896\pi$$
−0.00595700 + 0.999982i $$0.501896\pi$$
$$174$$ 0 0
$$175$$ −146250. −0.360995
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −280980. −0.655455 −0.327727 0.944772i $$-0.606283\pi$$
−0.327727 + 0.944772i $$0.606283\pi$$
$$180$$ 0 0
$$181$$ 721880. 1.63783 0.818915 0.573915i $$-0.194576\pi$$
0.818915 + 0.573915i $$0.194576\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 140050. 0.300853
$$186$$ 0 0
$$187$$ 82239.0 0.171978
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −104538. −0.207344 −0.103672 0.994612i $$-0.533059\pi$$
−0.103672 + 0.994612i $$0.533059\pi$$
$$192$$ 0 0
$$193$$ 695302. 1.34363 0.671816 0.740718i $$-0.265515\pi$$
0.671816 + 0.740718i $$0.265515\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −369268. −0.677916 −0.338958 0.940801i $$-0.610075\pi$$
−0.338958 + 0.940801i $$0.610075\pi$$
$$198$$ 0 0
$$199$$ −398589. −0.713498 −0.356749 0.934200i $$-0.616115\pi$$
−0.356749 + 0.934200i $$0.616115\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.29004e6 2.19717
$$204$$ 0 0
$$205$$ 117900. 0.195943
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 519112. 0.822045
$$210$$ 0 0
$$211$$ −142182. −0.219856 −0.109928 0.993940i $$-0.535062\pi$$
−0.109928 + 0.993940i $$0.535062\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 261975. 0.386513
$$216$$ 0 0
$$217$$ −681174. −0.981994
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −7821.00 −0.0107716
$$222$$ 0 0
$$223$$ −287548. −0.387211 −0.193606 0.981079i $$-0.562018\pi$$
−0.193606 + 0.981079i $$0.562018\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −1.07446e6 −1.38397 −0.691983 0.721914i $$-0.743263\pi$$
−0.691983 + 0.721914i $$0.743263\pi$$
$$228$$ 0 0
$$229$$ 832856. 1.04950 0.524749 0.851257i $$-0.324159\pi$$
0.524749 + 0.851257i $$0.324159\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 249566. 0.301159 0.150579 0.988598i $$-0.451886\pi$$
0.150579 + 0.988598i $$0.451886\pi$$
$$234$$ 0 0
$$235$$ 149075. 0.176090
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1.43083e6 −1.62029 −0.810144 0.586231i $$-0.800611\pi$$
−0.810144 + 0.586231i $$0.800611\pi$$
$$240$$ 0 0
$$241$$ −854747. −0.947971 −0.473985 0.880533i $$-0.657185\pi$$
−0.473985 + 0.880533i $$0.657185\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 948725. 1.00978
$$246$$ 0 0
$$247$$ −49368.0 −0.0514877
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1.77159e6 1.77492 0.887461 0.460883i $$-0.152467\pi$$
0.887461 + 0.460883i $$0.152467\pi$$
$$252$$ 0 0
$$253$$ −975417. −0.958052
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.50216e6 −1.41868 −0.709338 0.704869i $$-0.751006\pi$$
−0.709338 + 0.704869i $$0.751006\pi$$
$$258$$ 0 0
$$259$$ −1.31087e6 −1.21425
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2.09514e6 −1.86777 −0.933886 0.357572i $$-0.883605\pi$$
−0.933886 + 0.357572i $$0.883605\pi$$
$$264$$ 0 0
$$265$$ −449100. −0.392851
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −1035.00 −0.000872087 0 −0.000436043 1.00000i $$-0.500139\pi$$
−0.000436043 1.00000i $$0.500139\pi$$
$$270$$ 0 0
$$271$$ −1.46544e6 −1.21212 −0.606059 0.795420i $$-0.707250\pi$$
−0.606059 + 0.795420i $$0.707250\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 216875. 0.172933
$$276$$ 0 0
$$277$$ −911462. −0.713739 −0.356869 0.934154i $$-0.616156\pi$$
−0.356869 + 0.934154i $$0.616156\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 333494. 0.251955 0.125977 0.992033i $$-0.459793\pi$$
0.125977 + 0.992033i $$0.459793\pi$$
$$282$$ 0 0
$$283$$ −2.27522e6 −1.68872 −0.844358 0.535780i $$-0.820018\pi$$
−0.844358 + 0.535780i $$0.820018\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.10354e6 −0.790833
$$288$$ 0 0
$$289$$ −1.36369e6 −0.960440
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1.14585e6 −0.779753 −0.389877 0.920867i $$-0.627482\pi$$
−0.389877 + 0.920867i $$0.627482\pi$$
$$294$$ 0 0
$$295$$ 759300. 0.507994
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 92763.0 0.0600063
$$300$$ 0 0
$$301$$ −2.45209e6 −1.55998
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −888250. −0.546746
$$306$$ 0 0
$$307$$ −1.61298e6 −0.976750 −0.488375 0.872634i $$-0.662410\pi$$
−0.488375 + 0.872634i $$0.662410\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.42846e6 1.42374 0.711869 0.702312i $$-0.247849\pi$$
0.711869 + 0.702312i $$0.247849\pi$$
$$312$$ 0 0
$$313$$ −336764. −0.194296 −0.0971482 0.995270i $$-0.530972\pi$$
−0.0971482 + 0.995270i $$0.530972\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −743008. −0.415284 −0.207642 0.978205i $$-0.566579\pi$$
−0.207642 + 0.978205i $$0.566579\pi$$
$$318$$ 0 0
$$319$$ −1.91301e6 −1.05255
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 354552. 0.189092
$$324$$ 0 0
$$325$$ −20625.0 −0.0108314
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −1.39534e6 −0.710708
$$330$$ 0 0
$$331$$ −200254. −0.100464 −0.0502321 0.998738i $$-0.515996\pi$$
−0.0502321 + 0.998738i $$0.515996\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −311900. −0.151846
$$336$$ 0 0
$$337$$ −1.14969e6 −0.551449 −0.275724 0.961237i $$-0.588918\pi$$
−0.275724 + 0.961237i $$0.588918\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.01012e6 0.470420
$$342$$ 0 0
$$343$$ −4.94723e6 −2.27053
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1.35551e6 0.604336 0.302168 0.953255i $$-0.402290\pi$$
0.302168 + 0.953255i $$0.402290\pi$$
$$348$$ 0 0
$$349$$ 1.24421e6 0.546802 0.273401 0.961900i $$-0.411851\pi$$
0.273401 + 0.961900i $$0.411851\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 492467. 0.210349 0.105174 0.994454i $$-0.466460\pi$$
0.105174 + 0.994454i $$0.466460\pi$$
$$354$$ 0 0
$$355$$ 188000. 0.0791748
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −524400. −0.214747 −0.107373 0.994219i $$-0.534244\pi$$
−0.107373 + 0.994219i $$0.534244\pi$$
$$360$$ 0 0
$$361$$ −238083. −0.0961525
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 909450. 0.357311
$$366$$ 0 0
$$367$$ 3.15695e6 1.22349 0.611747 0.791053i $$-0.290467\pi$$
0.611747 + 0.791053i $$0.290467\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.20358e6 1.58557
$$372$$ 0 0
$$373$$ 416509. 0.155007 0.0775037 0.996992i $$-0.475305\pi$$
0.0775037 + 0.996992i $$0.475305\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 181929. 0.0659248
$$378$$ 0 0
$$379$$ −3.96054e6 −1.41630 −0.708151 0.706061i $$-0.750470\pi$$
−0.708151 + 0.706061i $$0.750470\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.38757e6 1.18003 0.590013 0.807394i $$-0.299123\pi$$
0.590013 + 0.807394i $$0.299123\pi$$
$$384$$ 0 0
$$385$$ −2.02995e6 −0.697965
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 248475. 0.0832547 0.0416273 0.999133i $$-0.486746\pi$$
0.0416273 + 0.999133i $$0.486746\pi$$
$$390$$ 0 0
$$391$$ −666207. −0.220378
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −568175. −0.183227
$$396$$ 0 0
$$397$$ −3.40110e6 −1.08304 −0.541518 0.840689i $$-0.682150\pi$$
−0.541518 + 0.840689i $$0.682150\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.22917e6 −1.93450 −0.967251 0.253820i $$-0.918313\pi$$
−0.967251 + 0.253820i $$0.918313\pi$$
$$402$$ 0 0
$$403$$ −96063.0 −0.0294641
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.94389e6 0.581683
$$408$$ 0 0
$$409$$ 510359. 0.150858 0.0754289 0.997151i $$-0.475967\pi$$
0.0754289 + 0.997151i $$0.475967\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −7.10705e6 −2.05028
$$414$$ 0 0
$$415$$ 1.15635e6 0.329586
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 6.40576e6 1.78253 0.891263 0.453486i $$-0.149820\pi$$
0.891263 + 0.453486i $$0.149820\pi$$
$$420$$ 0 0
$$421$$ 1.01408e6 0.278848 0.139424 0.990233i $$-0.455475\pi$$
0.139424 + 0.990233i $$0.455475\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 148125. 0.0397792
$$426$$ 0 0
$$427$$ 8.31402e6 2.20669
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −144184. −0.0373873 −0.0186936 0.999825i $$-0.505951\pi$$
−0.0186936 + 0.999825i $$0.505951\pi$$
$$432$$ 0 0
$$433$$ −501426. −0.128525 −0.0642624 0.997933i $$-0.520469\pi$$
−0.0642624 + 0.997933i $$0.520469\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.20526e6 −1.05339
$$438$$ 0 0
$$439$$ −4.62831e6 −1.14620 −0.573101 0.819485i $$-0.694260\pi$$
−0.573101 + 0.819485i $$0.694260\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 5.89143e6 1.42630 0.713151 0.701010i $$-0.247267\pi$$
0.713151 + 0.701010i $$0.247267\pi$$
$$444$$ 0 0
$$445$$ −1.47080e6 −0.352090
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3.08619e6 −0.722448 −0.361224 0.932479i $$-0.617641\pi$$
−0.361224 + 0.932479i $$0.617641\pi$$
$$450$$ 0 0
$$451$$ 1.63645e6 0.378845
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 193050. 0.0437161
$$456$$ 0 0
$$457$$ −5.26306e6 −1.17882 −0.589410 0.807834i $$-0.700640\pi$$
−0.589410 + 0.807834i $$0.700640\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −4.92047e6 −1.07834 −0.539168 0.842198i $$-0.681261\pi$$
−0.539168 + 0.842198i $$0.681261\pi$$
$$462$$ 0 0
$$463$$ 3.85093e6 0.834860 0.417430 0.908709i $$-0.362931\pi$$
0.417430 + 0.908709i $$0.362931\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 7.75105e6 1.64463 0.822315 0.569033i $$-0.192682\pi$$
0.822315 + 0.569033i $$0.192682\pi$$
$$468$$ 0 0
$$469$$ 2.91938e6 0.612857
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 3.63621e6 0.747303
$$474$$ 0 0
$$475$$ 935000. 0.190142
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 603352. 0.120152 0.0600761 0.998194i $$-0.480866\pi$$
0.0600761 + 0.998194i $$0.480866\pi$$
$$480$$ 0 0
$$481$$ −184866. −0.0364330
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −3.64765e6 −0.704140
$$486$$ 0 0
$$487$$ 1.12268e6 0.214502 0.107251 0.994232i $$-0.465795\pi$$
0.107251 + 0.994232i $$0.465795\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7.26616e6 −1.36019 −0.680097 0.733122i $$-0.738062\pi$$
−0.680097 + 0.733122i $$0.738062\pi$$
$$492$$ 0 0
$$493$$ −1.30658e6 −0.242114
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.75968e6 −0.319553
$$498$$ 0 0
$$499$$ −6.28754e6 −1.13039 −0.565196 0.824956i $$-0.691200\pi$$
−0.565196 + 0.824956i $$0.691200\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6.62421e6 −1.16739 −0.583693 0.811975i $$-0.698393\pi$$
−0.583693 + 0.811975i $$0.698393\pi$$
$$504$$ 0 0
$$505$$ 2.23552e6 0.390078
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 7.55534e6 1.29259 0.646293 0.763089i $$-0.276318\pi$$
0.646293 + 0.763089i $$0.276318\pi$$
$$510$$ 0 0
$$511$$ −8.51245e6 −1.44212
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −3.36795e6 −0.559561
$$516$$ 0 0
$$517$$ 2.06916e6 0.340461
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7.75865e6 1.25225 0.626126 0.779722i $$-0.284640\pi$$
0.626126 + 0.779722i $$0.284640\pi$$
$$522$$ 0 0
$$523$$ 407729. 0.0651805 0.0325902 0.999469i $$-0.489624\pi$$
0.0325902 + 0.999469i $$0.489624\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 689907. 0.108209
$$528$$ 0 0
$$529$$ 1.46538e6 0.227672
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −155628. −0.0237285
$$534$$ 0 0
$$535$$ 1.65005e6 0.249237
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1.31683e7 1.95235
$$540$$ 0 0
$$541$$ 53682.0 0.00788561 0.00394281 0.999992i $$-0.498745\pi$$
0.00394281 + 0.999992i $$0.498745\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −211500. −0.0305014
$$546$$ 0 0
$$547$$ −1.21855e7 −1.74131 −0.870656 0.491893i $$-0.836305\pi$$
−0.870656 + 0.491893i $$0.836305\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.24745e6 −1.15729
$$552$$ 0 0
$$553$$ 5.31812e6 0.739512
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.83593e6 −0.250737 −0.125368 0.992110i $$-0.540011\pi$$
−0.125368 + 0.992110i $$0.540011\pi$$
$$558$$ 0 0
$$559$$ −345807. −0.0468063
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.14405e7 1.52116 0.760581 0.649243i $$-0.224914\pi$$
0.760581 + 0.649243i $$0.224914\pi$$
$$564$$ 0 0
$$565$$ −6.37958e6 −0.840757
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.89543e6 −0.763370 −0.381685 0.924293i $$-0.624656\pi$$
−0.381685 + 0.924293i $$0.624656\pi$$
$$570$$ 0 0
$$571$$ 2.71925e6 0.349027 0.174514 0.984655i $$-0.444165\pi$$
0.174514 + 0.984655i $$0.444165\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.75688e6 −0.221601
$$576$$ 0 0
$$577$$ −1.29749e7 −1.62243 −0.811215 0.584748i $$-0.801193\pi$$
−0.811215 + 0.584748i $$0.801193\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.08234e7 −1.33022
$$582$$ 0 0
$$583$$ −6.23351e6 −0.759558
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1.13612e6 −0.136091 −0.0680455 0.997682i $$-0.521676\pi$$
−0.0680455 + 0.997682i $$0.521676\pi$$
$$588$$ 0 0
$$589$$ 4.35486e6 0.517232
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.91577e6 −0.807614 −0.403807 0.914844i $$-0.632313\pi$$
−0.403807 + 0.914844i $$0.632313\pi$$
$$594$$ 0 0
$$595$$ −1.38645e6 −0.160551
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −1.17157e7 −1.33414 −0.667072 0.744993i $$-0.732453\pi$$
−0.667072 + 0.744993i $$0.732453\pi$$
$$600$$ 0 0
$$601$$ −1.45503e7 −1.64318 −0.821589 0.570080i $$-0.806912\pi$$
−0.821589 + 0.570080i $$0.806912\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1.01605e6 −0.112857
$$606$$ 0 0
$$607$$ 7.53972e6 0.830584 0.415292 0.909688i $$-0.363679\pi$$
0.415292 + 0.909688i $$0.363679\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −196779. −0.0213243
$$612$$ 0 0
$$613$$ 8.90986e6 0.957679 0.478839 0.877903i $$-0.341058\pi$$
0.478839 + 0.877903i $$0.341058\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.62134e6 0.911721 0.455860 0.890051i $$-0.349332\pi$$
0.455860 + 0.890051i $$0.349332\pi$$
$$618$$ 0 0
$$619$$ −2.89049e6 −0.303211 −0.151605 0.988441i $$-0.548444\pi$$
−0.151605 + 0.988441i $$0.548444\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 1.37667e7 1.42105
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.32767e6 0.133803
$$630$$ 0 0
$$631$$ 7.04252e6 0.704133 0.352066 0.935975i $$-0.385479\pi$$
0.352066 + 0.935975i $$0.385479\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 1.59475e6 0.156949
$$636$$ 0 0
$$637$$ −1.25232e6 −0.122283
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.58361e7 −1.52231 −0.761157 0.648568i $$-0.775368\pi$$
−0.761157 + 0.648568i $$0.775368\pi$$
$$642$$ 0 0
$$643$$ −7.62712e6 −0.727500 −0.363750 0.931497i $$-0.618504\pi$$
−0.363750 + 0.931497i $$0.618504\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1.11038e7 1.04282 0.521412 0.853305i $$-0.325405\pi$$
0.521412 + 0.853305i $$0.325405\pi$$
$$648$$ 0 0
$$649$$ 1.05391e7 0.982180
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.84407e7 1.69237 0.846183 0.532892i $$-0.178895\pi$$
0.846183 + 0.532892i $$0.178895\pi$$
$$654$$ 0 0
$$655$$ 5.93852e6 0.540848
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.82622e7 1.63810 0.819050 0.573722i $$-0.194501\pi$$
0.819050 + 0.573722i $$0.194501\pi$$
$$660$$ 0 0
$$661$$ 1.44528e7 1.28661 0.643306 0.765609i $$-0.277562\pi$$
0.643306 + 0.765609i $$0.277562\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8.75160e6 −0.767420
$$666$$ 0 0
$$667$$ 1.54970e7 1.34876
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −1.23289e7 −1.05711
$$672$$ 0 0
$$673$$ −1.24675e7 −1.06107 −0.530533 0.847665i $$-0.678008\pi$$
−0.530533 + 0.847665i $$0.678008\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.51168e6 0.546036 0.273018 0.962009i $$-0.411978\pi$$
0.273018 + 0.962009i $$0.411978\pi$$
$$678$$ 0 0
$$679$$ 3.41420e7 2.84194
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.69933e7 −1.39388 −0.696940 0.717129i $$-0.745456\pi$$
−0.696940 + 0.717129i $$0.745456\pi$$
$$684$$ 0 0
$$685$$ −3.42655e6 −0.279017
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 592812. 0.0475739
$$690$$ 0 0
$$691$$ 1.66330e7 1.32518 0.662591 0.748982i $$-0.269457\pi$$
0.662591 + 0.748982i $$0.269457\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.74385e6 0.372537
$$696$$ 0 0
$$697$$ 1.11769e6 0.0871445
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.65476e7 −1.27187 −0.635933 0.771744i $$-0.719384\pi$$
−0.635933 + 0.771744i $$0.719384\pi$$
$$702$$ 0 0
$$703$$ 8.38059e6 0.639567
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.09245e7 −1.57437
$$708$$ 0 0
$$709$$ −837868. −0.0625979 −0.0312990 0.999510i $$-0.509964\pi$$
−0.0312990 + 0.999510i $$0.509964\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −8.18282e6 −0.602808
$$714$$ 0 0
$$715$$ −286275. −0.0209420
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.14252e7 1.54562 0.772809 0.634639i $$-0.218851\pi$$
0.772809 + 0.634639i $$0.218851\pi$$
$$720$$ 0 0
$$721$$ 3.15240e7 2.25841
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −3.44562e6 −0.243457
$$726$$ 0 0
$$727$$ −4.96740e6 −0.348572 −0.174286 0.984695i $$-0.555762\pi$$
−0.174286 + 0.984695i $$0.555762\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2.48352e6 0.171900
$$732$$ 0 0
$$733$$ 2.37177e7 1.63047 0.815233 0.579133i $$-0.196609\pi$$
0.815233 + 0.579133i $$0.196609\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4.32917e6 −0.293587
$$738$$ 0 0
$$739$$ 274304. 0.0184766 0.00923828 0.999957i $$-0.497059\pi$$
0.00923828 + 0.999957i $$0.497059\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 7.63188e6 0.507177 0.253588 0.967312i $$-0.418389\pi$$
0.253588 + 0.967312i $$0.418389\pi$$
$$744$$ 0 0
$$745$$ −1.26722e6 −0.0836494
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −1.54445e7 −1.00593
$$750$$ 0 0
$$751$$ −4.28073e6 −0.276961 −0.138480 0.990365i $$-0.544222\pi$$
−0.138480 + 0.990365i $$0.544222\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −8.05258e6 −0.514123
$$756$$ 0 0
$$757$$ −4.74502e6 −0.300952 −0.150476 0.988614i $$-0.548081\pi$$
−0.150476 + 0.988614i $$0.548081\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.08597e7 0.679758 0.339879 0.940469i $$-0.389614\pi$$
0.339879 + 0.940469i $$0.389614\pi$$
$$762$$ 0 0
$$763$$ 1.97964e6 0.123105
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.00228e6 −0.0615175
$$768$$ 0 0
$$769$$ 2.60429e7 1.58808 0.794042 0.607863i $$-0.207973\pi$$
0.794042 + 0.607863i $$0.207973\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −2.60052e7 −1.56535 −0.782676 0.622430i $$-0.786146\pi$$
−0.782676 + 0.622430i $$0.786146\pi$$
$$774$$ 0 0
$$775$$ 1.81938e6 0.108810
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 7.05514e6 0.416545
$$780$$ 0 0
$$781$$ 2.60944e6 0.153080
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −1.28452e7 −0.743988
$$786$$ 0 0
$$787$$ 3.33737e7 1.92074 0.960369 0.278733i $$-0.0899144\pi$$
0.960369 + 0.278733i $$0.0899144\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.97128e7 3.39333
$$792$$ 0 0
$$793$$ 1.17249e6 0.0662104
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −2.51931e7 −1.40487 −0.702434 0.711749i $$-0.747903\pi$$
−0.702434 + 0.711749i $$0.747903\pi$$
$$798$$ 0 0
$$799$$ 1.41323e6 0.0783152
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 1.26232e7 0.690843
$$804$$ 0 0
$$805$$ 1.64444e7 0.894390
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 2.18255e7 1.17245 0.586224 0.810149i $$-0.300614\pi$$
0.586224 + 0.810149i $$0.300614\pi$$
$$810$$ 0 0
$$811$$ −3.44158e7 −1.83741 −0.918704 0.394946i $$-0.870763\pi$$
−0.918704 + 0.394946i $$0.870763\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 1.02623e7 0.541190
$$816$$ 0 0
$$817$$ 1.56766e7 0.821668
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.53637e7 1.31327 0.656637 0.754207i $$-0.271978\pi$$
0.656637 + 0.754207i $$0.271978\pi$$
$$822$$ 0 0
$$823$$ 2.28931e7 1.17816 0.589080 0.808075i $$-0.299490\pi$$
0.589080 + 0.808075i $$0.299490\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.21487e7 1.12612 0.563060 0.826416i $$-0.309624\pi$$
0.563060 + 0.826416i $$0.309624\pi$$
$$828$$ 0 0
$$829$$ −2.61213e7 −1.32010 −0.660051 0.751220i $$-0.729466\pi$$
−0.660051 + 0.751220i $$0.729466\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 8.99391e6 0.449093
$$834$$ 0 0
$$835$$ −9.05510e6 −0.449446
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 2.72008e7 1.33407 0.667033 0.745028i $$-0.267564\pi$$
0.667033 + 0.745028i $$0.267564\pi$$
$$840$$ 0 0
$$841$$ 9.88202e6 0.481788
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −9.25510e6 −0.445902
$$846$$ 0 0
$$847$$ 9.51023e6 0.455494
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.57472e7 −0.745384
$$852$$ 0 0
$$853$$ 1.92152e7 0.904217 0.452109 0.891963i $$-0.350672\pi$$
0.452109 + 0.891963i $$0.350672\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −3.29237e7 −1.53129 −0.765644 0.643264i $$-0.777580\pi$$
−0.765644 + 0.643264i $$0.777580\pi$$
$$858$$ 0 0
$$859$$ −2.72165e6 −0.125849 −0.0629244 0.998018i $$-0.520043\pi$$
−0.0629244 + 0.998018i $$0.520043\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −4.94282e6 −0.225917 −0.112958 0.993600i $$-0.536033\pi$$
−0.112958 + 0.993600i $$0.536033\pi$$
$$864$$ 0 0
$$865$$ −117250. −0.00532810
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −7.88627e6 −0.354260
$$870$$ 0 0
$$871$$ 411708. 0.0183884
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −3.65625e6 −0.161442
$$876$$ 0 0
$$877$$ 5.57096e6 0.244586 0.122293 0.992494i $$-0.460975\pi$$
0.122293 + 0.992494i $$0.460975\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1.21235e7 0.526243 0.263122 0.964763i $$-0.415248\pi$$
0.263122 + 0.964763i $$0.415248\pi$$
$$882$$ 0 0
$$883$$ 1.21668e7 0.525141 0.262570 0.964913i $$-0.415430\pi$$
0.262570 + 0.964913i $$0.415430\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.76495e6 0.0753223 0.0376612 0.999291i $$-0.488009\pi$$
0.0376612 + 0.999291i $$0.488009\pi$$
$$888$$ 0 0
$$889$$ −1.49269e7 −0.633453
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 8.92065e6 0.374341
$$894$$ 0 0
$$895$$ −7.02450e6 −0.293128
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −1.60483e7 −0.662264
$$900$$ 0 0
$$901$$ −4.25747e6 −0.174719
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1.80470e7 0.732459
$$906$$ 0 0
$$907$$ −7.44128e6 −0.300351 −0.150176 0.988659i $$-0.547984\pi$$
−0.150176 + 0.988659i $$0.547984\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 3.44782e7 1.37641 0.688206 0.725515i $$-0.258398\pi$$
0.688206 + 0.725515i $$0.258398\pi$$
$$912$$ 0 0
$$913$$ 1.60501e7 0.637238
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −5.55846e7 −2.18289
$$918$$ 0 0
$$919$$ −4.55258e6 −0.177815 −0.0889076 0.996040i $$-0.528338\pi$$
−0.0889076 + 0.996040i $$0.528338\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −248160. −0.00958799
$$924$$ 0 0
$$925$$ 3.50125e6 0.134545
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −2.51258e7 −0.955170 −0.477585 0.878585i $$-0.658488\pi$$
−0.477585 + 0.878585i $$0.658488\pi$$
$$930$$ 0 0
$$931$$ 5.67717e7 2.14663
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2.05598e6 0.0769111
$$936$$ 0 0
$$937$$ 1.49058e7 0.554635 0.277318 0.960778i $$-0.410555\pi$$
0.277318 + 0.960778i $$0.410555\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −3.96462e7 −1.45958 −0.729789 0.683673i $$-0.760381\pi$$
−0.729789 + 0.683673i $$0.760381\pi$$
$$942$$ 0 0
$$943$$ −1.32567e7 −0.485462
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −3.81296e7 −1.38161 −0.690807 0.723039i $$-0.742745\pi$$
−0.690807 + 0.723039i $$0.742745\pi$$
$$948$$ 0 0
$$949$$ −1.20047e6 −0.0432700
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1.95418e7 0.697000 0.348500 0.937309i $$-0.386691\pi$$
0.348500 + 0.937309i $$0.386691\pi$$
$$954$$ 0 0
$$955$$ −2.61345e6 −0.0927269
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3.20725e7 1.12612
$$960$$ 0 0
$$961$$ −2.01552e7 −0.704011
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 1.73826e7 0.600890
$$966$$ 0 0
$$967$$ −5.27096e7 −1.81269 −0.906345 0.422539i $$-0.861139\pi$$
−0.906345 + 0.422539i $$0.861139\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.41631e7 0.822440 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$972$$ 0 0
$$973$$ −4.44024e7 −1.50357
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −4.87535e7 −1.63407 −0.817033 0.576591i $$-0.804382\pi$$
−0.817033 + 0.576591i $$0.804382\pi$$
$$978$$ 0 0
$$979$$ −2.04147e7 −0.680748
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1.79029e7 0.590936 0.295468 0.955353i $$-0.404524\pi$$
0.295468 + 0.955353i $$0.404524\pi$$
$$984$$ 0 0
$$985$$ −9.23170e6 −0.303173
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −2.94565e7 −0.957613
$$990$$ 0 0
$$991$$ −5.11153e7 −1.65336 −0.826679 0.562674i $$-0.809773\pi$$
−0.826679 + 0.562674i $$0.809773\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −9.96472e6 −0.319086
$$996$$ 0 0
$$997$$ 4.95876e7 1.57992 0.789961 0.613157i $$-0.210101\pi$$
0.789961 + 0.613157i $$0.210101\pi$$
$$998$$ 0 0
$$999$$ 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 1080.6.a.b.1.1 yes 1
3.2 odd 2 1080.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.6.a.a.1.1 1 3.2 odd 2
1080.6.a.b.1.1 yes 1 1.1 even 1 trivial