# Properties

 Label 48.4.a.c.1.1 Level $48$ Weight $4$ Character 48.1 Self dual yes Analytic conductor $2.832$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$2.83209168028$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 48.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +6.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +6.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +38.0000 q^{13} +18.0000 q^{15} -126.000 q^{17} -20.0000 q^{19} +48.0000 q^{21} -168.000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +30.0000 q^{29} +88.0000 q^{31} -36.0000 q^{33} +96.0000 q^{35} +254.000 q^{37} +114.000 q^{39} +42.0000 q^{41} +52.0000 q^{43} +54.0000 q^{45} +96.0000 q^{47} -87.0000 q^{49} -378.000 q^{51} +198.000 q^{53} -72.0000 q^{55} -60.0000 q^{57} +660.000 q^{59} -538.000 q^{61} +144.000 q^{63} +228.000 q^{65} -884.000 q^{67} -504.000 q^{69} -792.000 q^{71} +218.000 q^{73} -267.000 q^{75} -192.000 q^{77} +520.000 q^{79} +81.0000 q^{81} +492.000 q^{83} -756.000 q^{85} +90.0000 q^{87} +810.000 q^{89} +608.000 q^{91} +264.000 q^{93} -120.000 q^{95} +1154.00 q^{97} -108.000 q^{99} +O(q^{100})$$

## 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
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ 0 0
$$7$$ 16.0000 0.863919 0.431959 0.901893i $$-0.357822\pi$$
0.431959 + 0.901893i $$0.357822\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −12.0000 −0.328921 −0.164461 0.986384i $$-0.552588\pi$$
−0.164461 + 0.986384i $$0.552588\pi$$
$$12$$ 0 0
$$13$$ 38.0000 0.810716 0.405358 0.914158i $$-0.367147\pi$$
0.405358 + 0.914158i $$0.367147\pi$$
$$14$$ 0 0
$$15$$ 18.0000 0.309839
$$16$$ 0 0
$$17$$ −126.000 −1.79762 −0.898808 0.438342i $$-0.855566\pi$$
−0.898808 + 0.438342i $$0.855566\pi$$
$$18$$ 0 0
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 0 0
$$21$$ 48.0000 0.498784
$$22$$ 0 0
$$23$$ −168.000 −1.52306 −0.761531 0.648129i $$-0.775552\pi$$
−0.761531 + 0.648129i $$0.775552\pi$$
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 30.0000 0.192099 0.0960493 0.995377i $$-0.469379\pi$$
0.0960493 + 0.995377i $$0.469379\pi$$
$$30$$ 0 0
$$31$$ 88.0000 0.509847 0.254924 0.966961i $$-0.417950\pi$$
0.254924 + 0.966961i $$0.417950\pi$$
$$32$$ 0 0
$$33$$ −36.0000 −0.189903
$$34$$ 0 0
$$35$$ 96.0000 0.463627
$$36$$ 0 0
$$37$$ 254.000 1.12858 0.564288 0.825578i $$-0.309151\pi$$
0.564288 + 0.825578i $$0.309151\pi$$
$$38$$ 0 0
$$39$$ 114.000 0.468067
$$40$$ 0 0
$$41$$ 42.0000 0.159983 0.0799914 0.996796i $$-0.474511\pi$$
0.0799914 + 0.996796i $$0.474511\pi$$
$$42$$ 0 0
$$43$$ 52.0000 0.184417 0.0922084 0.995740i $$-0.470607\pi$$
0.0922084 + 0.995740i $$0.470607\pi$$
$$44$$ 0 0
$$45$$ 54.0000 0.178885
$$46$$ 0 0
$$47$$ 96.0000 0.297937 0.148969 0.988842i $$-0.452405\pi$$
0.148969 + 0.988842i $$0.452405\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ −378.000 −1.03785
$$52$$ 0 0
$$53$$ 198.000 0.513158 0.256579 0.966523i $$-0.417405\pi$$
0.256579 + 0.966523i $$0.417405\pi$$
$$54$$ 0 0
$$55$$ −72.0000 −0.176518
$$56$$ 0 0
$$57$$ −60.0000 −0.139424
$$58$$ 0 0
$$59$$ 660.000 1.45635 0.728175 0.685391i $$-0.240369\pi$$
0.728175 + 0.685391i $$0.240369\pi$$
$$60$$ 0 0
$$61$$ −538.000 −1.12924 −0.564622 0.825350i $$-0.690978\pi$$
−0.564622 + 0.825350i $$0.690978\pi$$
$$62$$ 0 0
$$63$$ 144.000 0.287973
$$64$$ 0 0
$$65$$ 228.000 0.435076
$$66$$ 0 0
$$67$$ −884.000 −1.61191 −0.805954 0.591979i $$-0.798347\pi$$
−0.805954 + 0.591979i $$0.798347\pi$$
$$68$$ 0 0
$$69$$ −504.000 −0.879340
$$70$$ 0 0
$$71$$ −792.000 −1.32385 −0.661923 0.749572i $$-0.730260\pi$$
−0.661923 + 0.749572i $$0.730260\pi$$
$$72$$ 0 0
$$73$$ 218.000 0.349520 0.174760 0.984611i $$-0.444085\pi$$
0.174760 + 0.984611i $$0.444085\pi$$
$$74$$ 0 0
$$75$$ −267.000 −0.411073
$$76$$ 0 0
$$77$$ −192.000 −0.284161
$$78$$ 0 0
$$79$$ 520.000 0.740564 0.370282 0.928919i $$-0.379261\pi$$
0.370282 + 0.928919i $$0.379261\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 492.000 0.650651 0.325325 0.945602i $$-0.394526\pi$$
0.325325 + 0.945602i $$0.394526\pi$$
$$84$$ 0 0
$$85$$ −756.000 −0.964703
$$86$$ 0 0
$$87$$ 90.0000 0.110908
$$88$$ 0 0
$$89$$ 810.000 0.964717 0.482359 0.875974i $$-0.339780\pi$$
0.482359 + 0.875974i $$0.339780\pi$$
$$90$$ 0 0
$$91$$ 608.000 0.700393
$$92$$ 0 0
$$93$$ 264.000 0.294360
$$94$$ 0 0
$$95$$ −120.000 −0.129597
$$96$$ 0 0
$$97$$ 1154.00 1.20795 0.603974 0.797004i $$-0.293583\pi$$
0.603974 + 0.797004i $$0.293583\pi$$
$$98$$ 0 0
$$99$$ −108.000 −0.109640
$$100$$ 0 0
$$101$$ −618.000 −0.608845 −0.304422 0.952537i $$-0.598463\pi$$
−0.304422 + 0.952537i $$0.598463\pi$$
$$102$$ 0 0
$$103$$ −128.000 −0.122449 −0.0612243 0.998124i $$-0.519501\pi$$
−0.0612243 + 0.998124i $$0.519501\pi$$
$$104$$ 0 0
$$105$$ 288.000 0.267675
$$106$$ 0 0
$$107$$ 1476.00 1.33355 0.666777 0.745257i $$-0.267673\pi$$
0.666777 + 0.745257i $$0.267673\pi$$
$$108$$ 0 0
$$109$$ 1190.00 1.04570 0.522850 0.852425i $$-0.324869\pi$$
0.522850 + 0.852425i $$0.324869\pi$$
$$110$$ 0 0
$$111$$ 762.000 0.651584
$$112$$ 0 0
$$113$$ −462.000 −0.384613 −0.192307 0.981335i $$-0.561597\pi$$
−0.192307 + 0.981335i $$0.561597\pi$$
$$114$$ 0 0
$$115$$ −1008.00 −0.817361
$$116$$ 0 0
$$117$$ 342.000 0.270239
$$118$$ 0 0
$$119$$ −2016.00 −1.55300
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ 0 0
$$123$$ 126.000 0.0923662
$$124$$ 0 0
$$125$$ −1284.00 −0.918756
$$126$$ 0 0
$$127$$ 2536.00 1.77192 0.885959 0.463763i $$-0.153501\pi$$
0.885959 + 0.463763i $$0.153501\pi$$
$$128$$ 0 0
$$129$$ 156.000 0.106473
$$130$$ 0 0
$$131$$ −2292.00 −1.52865 −0.764324 0.644832i $$-0.776927\pi$$
−0.764324 + 0.644832i $$0.776927\pi$$
$$132$$ 0 0
$$133$$ −320.000 −0.208628
$$134$$ 0 0
$$135$$ 162.000 0.103280
$$136$$ 0 0
$$137$$ −726.000 −0.452747 −0.226374 0.974041i $$-0.572687\pi$$
−0.226374 + 0.974041i $$0.572687\pi$$
$$138$$ 0 0
$$139$$ −380.000 −0.231879 −0.115939 0.993256i $$-0.536988\pi$$
−0.115939 + 0.993256i $$0.536988\pi$$
$$140$$ 0 0
$$141$$ 288.000 0.172014
$$142$$ 0 0
$$143$$ −456.000 −0.266662
$$144$$ 0 0
$$145$$ 180.000 0.103091
$$146$$ 0 0
$$147$$ −261.000 −0.146442
$$148$$ 0 0
$$149$$ 1590.00 0.874214 0.437107 0.899410i $$-0.356003\pi$$
0.437107 + 0.899410i $$0.356003\pi$$
$$150$$ 0 0
$$151$$ −2432.00 −1.31068 −0.655342 0.755332i $$-0.727476\pi$$
−0.655342 + 0.755332i $$0.727476\pi$$
$$152$$ 0 0
$$153$$ −1134.00 −0.599206
$$154$$ 0 0
$$155$$ 528.000 0.273613
$$156$$ 0 0
$$157$$ 614.000 0.312118 0.156059 0.987748i $$-0.450121\pi$$
0.156059 + 0.987748i $$0.450121\pi$$
$$158$$ 0 0
$$159$$ 594.000 0.296272
$$160$$ 0 0
$$161$$ −2688.00 −1.31580
$$162$$ 0 0
$$163$$ 1852.00 0.889938 0.444969 0.895546i $$-0.353215\pi$$
0.444969 + 0.895546i $$0.353215\pi$$
$$164$$ 0 0
$$165$$ −216.000 −0.101913
$$166$$ 0 0
$$167$$ 2136.00 0.989752 0.494876 0.868964i $$-0.335213\pi$$
0.494876 + 0.868964i $$0.335213\pi$$
$$168$$ 0 0
$$169$$ −753.000 −0.342740
$$170$$ 0 0
$$171$$ −180.000 −0.0804967
$$172$$ 0 0
$$173$$ 1758.00 0.772591 0.386296 0.922375i $$-0.373754\pi$$
0.386296 + 0.922375i $$0.373754\pi$$
$$174$$ 0 0
$$175$$ −1424.00 −0.615110
$$176$$ 0 0
$$177$$ 1980.00 0.840824
$$178$$ 0 0
$$179$$ 540.000 0.225483 0.112742 0.993624i $$-0.464037\pi$$
0.112742 + 0.993624i $$0.464037\pi$$
$$180$$ 0 0
$$181$$ 1982.00 0.813928 0.406964 0.913444i $$-0.366588\pi$$
0.406964 + 0.913444i $$0.366588\pi$$
$$182$$ 0 0
$$183$$ −1614.00 −0.651969
$$184$$ 0 0
$$185$$ 1524.00 0.605658
$$186$$ 0 0
$$187$$ 1512.00 0.591275
$$188$$ 0 0
$$189$$ 432.000 0.166261
$$190$$ 0 0
$$191$$ 2688.00 1.01831 0.509154 0.860675i $$-0.329958\pi$$
0.509154 + 0.860675i $$0.329958\pi$$
$$192$$ 0 0
$$193$$ −2302.00 −0.858557 −0.429279 0.903172i $$-0.641232\pi$$
−0.429279 + 0.903172i $$0.641232\pi$$
$$194$$ 0 0
$$195$$ 684.000 0.251191
$$196$$ 0 0
$$197$$ 4374.00 1.58190 0.790951 0.611880i $$-0.209586\pi$$
0.790951 + 0.611880i $$0.209586\pi$$
$$198$$ 0 0
$$199$$ 1600.00 0.569955 0.284977 0.958534i $$-0.408014\pi$$
0.284977 + 0.958534i $$0.408014\pi$$
$$200$$ 0 0
$$201$$ −2652.00 −0.930635
$$202$$ 0 0
$$203$$ 480.000 0.165958
$$204$$ 0 0
$$205$$ 252.000 0.0858558
$$206$$ 0 0
$$207$$ −1512.00 −0.507687
$$208$$ 0 0
$$209$$ 240.000 0.0794313
$$210$$ 0 0
$$211$$ −3332.00 −1.08713 −0.543565 0.839367i $$-0.682926\pi$$
−0.543565 + 0.839367i $$0.682926\pi$$
$$212$$ 0 0
$$213$$ −2376.00 −0.764323
$$214$$ 0 0
$$215$$ 312.000 0.0989685
$$216$$ 0 0
$$217$$ 1408.00 0.440467
$$218$$ 0 0
$$219$$ 654.000 0.201796
$$220$$ 0 0
$$221$$ −4788.00 −1.45736
$$222$$ 0 0
$$223$$ −2648.00 −0.795171 −0.397586 0.917565i $$-0.630152\pi$$
−0.397586 + 0.917565i $$0.630152\pi$$
$$224$$ 0 0
$$225$$ −801.000 −0.237333
$$226$$ 0 0
$$227$$ −2244.00 −0.656121 −0.328061 0.944657i $$-0.606395\pi$$
−0.328061 + 0.944657i $$0.606395\pi$$
$$228$$ 0 0
$$229$$ −5650.00 −1.63040 −0.815202 0.579177i $$-0.803374\pi$$
−0.815202 + 0.579177i $$0.803374\pi$$
$$230$$ 0 0
$$231$$ −576.000 −0.164061
$$232$$ 0 0
$$233$$ 4698.00 1.32093 0.660464 0.750858i $$-0.270360\pi$$
0.660464 + 0.750858i $$0.270360\pi$$
$$234$$ 0 0
$$235$$ 576.000 0.159890
$$236$$ 0 0
$$237$$ 1560.00 0.427565
$$238$$ 0 0
$$239$$ 1200.00 0.324776 0.162388 0.986727i $$-0.448080\pi$$
0.162388 + 0.986727i $$0.448080\pi$$
$$240$$ 0 0
$$241$$ −718.000 −0.191911 −0.0959553 0.995386i $$-0.530591\pi$$
−0.0959553 + 0.995386i $$0.530591\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −522.000 −0.136120
$$246$$ 0 0
$$247$$ −760.000 −0.195780
$$248$$ 0 0
$$249$$ 1476.00 0.375653
$$250$$ 0 0
$$251$$ −6012.00 −1.51185 −0.755924 0.654659i $$-0.772812\pi$$
−0.755924 + 0.654659i $$0.772812\pi$$
$$252$$ 0 0
$$253$$ 2016.00 0.500968
$$254$$ 0 0
$$255$$ −2268.00 −0.556971
$$256$$ 0 0
$$257$$ −2046.00 −0.496599 −0.248300 0.968683i $$-0.579872\pi$$
−0.248300 + 0.968683i $$0.579872\pi$$
$$258$$ 0 0
$$259$$ 4064.00 0.974999
$$260$$ 0 0
$$261$$ 270.000 0.0640329
$$262$$ 0 0
$$263$$ 6072.00 1.42363 0.711817 0.702365i $$-0.247873\pi$$
0.711817 + 0.702365i $$0.247873\pi$$
$$264$$ 0 0
$$265$$ 1188.00 0.275390
$$266$$ 0 0
$$267$$ 2430.00 0.556980
$$268$$ 0 0
$$269$$ −6930.00 −1.57074 −0.785371 0.619025i $$-0.787528\pi$$
−0.785371 + 0.619025i $$0.787528\pi$$
$$270$$ 0 0
$$271$$ −1352.00 −0.303056 −0.151528 0.988453i $$-0.548419\pi$$
−0.151528 + 0.988453i $$0.548419\pi$$
$$272$$ 0 0
$$273$$ 1824.00 0.404372
$$274$$ 0 0
$$275$$ 1068.00 0.234192
$$276$$ 0 0
$$277$$ −1186.00 −0.257256 −0.128628 0.991693i $$-0.541057\pi$$
−0.128628 + 0.991693i $$0.541057\pi$$
$$278$$ 0 0
$$279$$ 792.000 0.169949
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 0 0
$$283$$ −2828.00 −0.594018 −0.297009 0.954875i $$-0.595989\pi$$
−0.297009 + 0.954875i $$0.595989\pi$$
$$284$$ 0 0
$$285$$ −360.000 −0.0748230
$$286$$ 0 0
$$287$$ 672.000 0.138212
$$288$$ 0 0
$$289$$ 10963.0 2.23143
$$290$$ 0 0
$$291$$ 3462.00 0.697409
$$292$$ 0 0
$$293$$ 4758.00 0.948687 0.474344 0.880340i $$-0.342685\pi$$
0.474344 + 0.880340i $$0.342685\pi$$
$$294$$ 0 0
$$295$$ 3960.00 0.781560
$$296$$ 0 0
$$297$$ −324.000 −0.0633010
$$298$$ 0 0
$$299$$ −6384.00 −1.23477
$$300$$ 0 0
$$301$$ 832.000 0.159321
$$302$$ 0 0
$$303$$ −1854.00 −0.351517
$$304$$ 0 0
$$305$$ −3228.00 −0.606016
$$306$$ 0 0
$$307$$ 8476.00 1.57574 0.787868 0.615844i $$-0.211185\pi$$
0.787868 + 0.615844i $$0.211185\pi$$
$$308$$ 0 0
$$309$$ −384.000 −0.0706958
$$310$$ 0 0
$$311$$ −4632.00 −0.844555 −0.422278 0.906467i $$-0.638769\pi$$
−0.422278 + 0.906467i $$0.638769\pi$$
$$312$$ 0 0
$$313$$ −4822.00 −0.870785 −0.435392 0.900241i $$-0.643390\pi$$
−0.435392 + 0.900241i $$0.643390\pi$$
$$314$$ 0 0
$$315$$ 864.000 0.154542
$$316$$ 0 0
$$317$$ −3426.00 −0.607014 −0.303507 0.952829i $$-0.598158\pi$$
−0.303507 + 0.952829i $$0.598158\pi$$
$$318$$ 0 0
$$319$$ −360.000 −0.0631854
$$320$$ 0 0
$$321$$ 4428.00 0.769928
$$322$$ 0 0
$$323$$ 2520.00 0.434107
$$324$$ 0 0
$$325$$ −3382.00 −0.577230
$$326$$ 0 0
$$327$$ 3570.00 0.603735
$$328$$ 0 0
$$329$$ 1536.00 0.257393
$$330$$ 0 0
$$331$$ 2788.00 0.462968 0.231484 0.972839i $$-0.425642\pi$$
0.231484 + 0.972839i $$0.425642\pi$$
$$332$$ 0 0
$$333$$ 2286.00 0.376192
$$334$$ 0 0
$$335$$ −5304.00 −0.865040
$$336$$ 0 0
$$337$$ 434.000 0.0701528 0.0350764 0.999385i $$-0.488833\pi$$
0.0350764 + 0.999385i $$0.488833\pi$$
$$338$$ 0 0
$$339$$ −1386.00 −0.222057
$$340$$ 0 0
$$341$$ −1056.00 −0.167700
$$342$$ 0 0
$$343$$ −6880.00 −1.08305
$$344$$ 0 0
$$345$$ −3024.00 −0.471903
$$346$$ 0 0
$$347$$ −6684.00 −1.03405 −0.517026 0.855970i $$-0.672961\pi$$
−0.517026 + 0.855970i $$0.672961\pi$$
$$348$$ 0 0
$$349$$ 2630.00 0.403383 0.201692 0.979449i $$-0.435356\pi$$
0.201692 + 0.979449i $$0.435356\pi$$
$$350$$ 0 0
$$351$$ 1026.00 0.156022
$$352$$ 0 0
$$353$$ −7422.00 −1.11907 −0.559537 0.828805i $$-0.689021\pi$$
−0.559537 + 0.828805i $$0.689021\pi$$
$$354$$ 0 0
$$355$$ −4752.00 −0.710451
$$356$$ 0 0
$$357$$ −6048.00 −0.896622
$$358$$ 0 0
$$359$$ 10440.0 1.53482 0.767412 0.641154i $$-0.221544\pi$$
0.767412 + 0.641154i $$0.221544\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 0 0
$$363$$ −3561.00 −0.514887
$$364$$ 0 0
$$365$$ 1308.00 0.187572
$$366$$ 0 0
$$367$$ −10424.0 −1.48264 −0.741319 0.671153i $$-0.765800\pi$$
−0.741319 + 0.671153i $$0.765800\pi$$
$$368$$ 0 0
$$369$$ 378.000 0.0533276
$$370$$ 0 0
$$371$$ 3168.00 0.443327
$$372$$ 0 0
$$373$$ 3278.00 0.455036 0.227518 0.973774i $$-0.426939\pi$$
0.227518 + 0.973774i $$0.426939\pi$$
$$374$$ 0 0
$$375$$ −3852.00 −0.530444
$$376$$ 0 0
$$377$$ 1140.00 0.155737
$$378$$ 0 0
$$379$$ −6140.00 −0.832165 −0.416083 0.909327i $$-0.636597\pi$$
−0.416083 + 0.909327i $$0.636597\pi$$
$$380$$ 0 0
$$381$$ 7608.00 1.02302
$$382$$ 0 0
$$383$$ 3072.00 0.409848 0.204924 0.978778i $$-0.434305\pi$$
0.204924 + 0.978778i $$0.434305\pi$$
$$384$$ 0 0
$$385$$ −1152.00 −0.152497
$$386$$ 0 0
$$387$$ 468.000 0.0614723
$$388$$ 0 0
$$389$$ 6150.00 0.801587 0.400794 0.916168i $$-0.368734\pi$$
0.400794 + 0.916168i $$0.368734\pi$$
$$390$$ 0 0
$$391$$ 21168.0 2.73788
$$392$$ 0 0
$$393$$ −6876.00 −0.882566
$$394$$ 0 0
$$395$$ 3120.00 0.397428
$$396$$ 0 0
$$397$$ −106.000 −0.0134005 −0.00670024 0.999978i $$-0.502133\pi$$
−0.00670024 + 0.999978i $$0.502133\pi$$
$$398$$ 0 0
$$399$$ −960.000 −0.120451
$$400$$ 0 0
$$401$$ −1758.00 −0.218929 −0.109464 0.993991i $$-0.534914\pi$$
−0.109464 + 0.993991i $$0.534914\pi$$
$$402$$ 0 0
$$403$$ 3344.00 0.413341
$$404$$ 0 0
$$405$$ 486.000 0.0596285
$$406$$ 0 0
$$407$$ −3048.00 −0.371213
$$408$$ 0 0
$$409$$ −3670.00 −0.443691 −0.221846 0.975082i $$-0.571208\pi$$
−0.221846 + 0.975082i $$0.571208\pi$$
$$410$$ 0 0
$$411$$ −2178.00 −0.261394
$$412$$ 0 0
$$413$$ 10560.0 1.25817
$$414$$ 0 0
$$415$$ 2952.00 0.349176
$$416$$ 0 0
$$417$$ −1140.00 −0.133875
$$418$$ 0 0
$$419$$ 9660.00 1.12631 0.563153 0.826353i $$-0.309588\pi$$
0.563153 + 0.826353i $$0.309588\pi$$
$$420$$ 0 0
$$421$$ 8462.00 0.979602 0.489801 0.871834i $$-0.337069\pi$$
0.489801 + 0.871834i $$0.337069\pi$$
$$422$$ 0 0
$$423$$ 864.000 0.0993123
$$424$$ 0 0
$$425$$ 11214.0 1.27990
$$426$$ 0 0
$$427$$ −8608.00 −0.975575
$$428$$ 0 0
$$429$$ −1368.00 −0.153957
$$430$$ 0 0
$$431$$ −9792.00 −1.09435 −0.547174 0.837019i $$-0.684296\pi$$
−0.547174 + 0.837019i $$0.684296\pi$$
$$432$$ 0 0
$$433$$ −7342.00 −0.814859 −0.407430 0.913237i $$-0.633575\pi$$
−0.407430 + 0.913237i $$0.633575\pi$$
$$434$$ 0 0
$$435$$ 540.000 0.0595196
$$436$$ 0 0
$$437$$ 3360.00 0.367805
$$438$$ 0 0
$$439$$ −10640.0 −1.15676 −0.578382 0.815766i $$-0.696316\pi$$
−0.578382 + 0.815766i $$0.696316\pi$$
$$440$$ 0 0
$$441$$ −783.000 −0.0845481
$$442$$ 0 0
$$443$$ 17412.0 1.86742 0.933712 0.358024i $$-0.116549\pi$$
0.933712 + 0.358024i $$0.116549\pi$$
$$444$$ 0 0
$$445$$ 4860.00 0.517722
$$446$$ 0 0
$$447$$ 4770.00 0.504728
$$448$$ 0 0
$$449$$ −1710.00 −0.179732 −0.0898662 0.995954i $$-0.528644\pi$$
−0.0898662 + 0.995954i $$0.528644\pi$$
$$450$$ 0 0
$$451$$ −504.000 −0.0526218
$$452$$ 0 0
$$453$$ −7296.00 −0.756724
$$454$$ 0 0
$$455$$ 3648.00 0.375870
$$456$$ 0 0
$$457$$ −646.000 −0.0661239 −0.0330619 0.999453i $$-0.510526\pi$$
−0.0330619 + 0.999453i $$0.510526\pi$$
$$458$$ 0 0
$$459$$ −3402.00 −0.345952
$$460$$ 0 0
$$461$$ −6018.00 −0.607996 −0.303998 0.952673i $$-0.598322\pi$$
−0.303998 + 0.952673i $$0.598322\pi$$
$$462$$ 0 0
$$463$$ 6712.00 0.673722 0.336861 0.941554i $$-0.390635\pi$$
0.336861 + 0.941554i $$0.390635\pi$$
$$464$$ 0 0
$$465$$ 1584.00 0.157970
$$466$$ 0 0
$$467$$ −5364.00 −0.531512 −0.265756 0.964040i $$-0.585622\pi$$
−0.265756 + 0.964040i $$0.585622\pi$$
$$468$$ 0 0
$$469$$ −14144.0 −1.39256
$$470$$ 0 0
$$471$$ 1842.00 0.180201
$$472$$ 0 0
$$473$$ −624.000 −0.0606587
$$474$$ 0 0
$$475$$ 1780.00 0.171941
$$476$$ 0 0
$$477$$ 1782.00 0.171053
$$478$$ 0 0
$$479$$ −9840.00 −0.938624 −0.469312 0.883032i $$-0.655498\pi$$
−0.469312 + 0.883032i $$0.655498\pi$$
$$480$$ 0 0
$$481$$ 9652.00 0.914955
$$482$$ 0 0
$$483$$ −8064.00 −0.759678
$$484$$ 0 0
$$485$$ 6924.00 0.648253
$$486$$ 0 0
$$487$$ −1424.00 −0.132500 −0.0662501 0.997803i $$-0.521104\pi$$
−0.0662501 + 0.997803i $$0.521104\pi$$
$$488$$ 0 0
$$489$$ 5556.00 0.513806
$$490$$ 0 0
$$491$$ 4548.00 0.418021 0.209011 0.977913i $$-0.432976\pi$$
0.209011 + 0.977913i $$0.432976\pi$$
$$492$$ 0 0
$$493$$ −3780.00 −0.345320
$$494$$ 0 0
$$495$$ −648.000 −0.0588393
$$496$$ 0 0
$$497$$ −12672.0 −1.14370
$$498$$ 0 0
$$499$$ −6500.00 −0.583126 −0.291563 0.956552i $$-0.594175\pi$$
−0.291563 + 0.956552i $$0.594175\pi$$
$$500$$ 0 0
$$501$$ 6408.00 0.571434
$$502$$ 0 0
$$503$$ −12168.0 −1.07862 −0.539308 0.842108i $$-0.681314\pi$$
−0.539308 + 0.842108i $$0.681314\pi$$
$$504$$ 0 0
$$505$$ −3708.00 −0.326740
$$506$$ 0 0
$$507$$ −2259.00 −0.197881
$$508$$ 0 0
$$509$$ −21090.0 −1.83654 −0.918269 0.395957i $$-0.870413\pi$$
−0.918269 + 0.395957i $$0.870413\pi$$
$$510$$ 0 0
$$511$$ 3488.00 0.301957
$$512$$ 0 0
$$513$$ −540.000 −0.0464748
$$514$$ 0 0
$$515$$ −768.000 −0.0657129
$$516$$ 0 0
$$517$$ −1152.00 −0.0979979
$$518$$ 0 0
$$519$$ 5274.00 0.446056
$$520$$ 0 0
$$521$$ −5238.00 −0.440462 −0.220231 0.975448i $$-0.570681\pi$$
−0.220231 + 0.975448i $$0.570681\pi$$
$$522$$ 0 0
$$523$$ −8588.00 −0.718025 −0.359012 0.933333i $$-0.616886\pi$$
−0.359012 + 0.933333i $$0.616886\pi$$
$$524$$ 0 0
$$525$$ −4272.00 −0.355134
$$526$$ 0 0
$$527$$ −11088.0 −0.916510
$$528$$ 0 0
$$529$$ 16057.0 1.31972
$$530$$ 0 0
$$531$$ 5940.00 0.485450
$$532$$ 0 0
$$533$$ 1596.00 0.129701
$$534$$ 0 0
$$535$$ 8856.00 0.715660
$$536$$ 0 0
$$537$$ 1620.00 0.130183
$$538$$ 0 0
$$539$$ 1044.00 0.0834291
$$540$$ 0 0
$$541$$ 3062.00 0.243338 0.121669 0.992571i $$-0.461175\pi$$
0.121669 + 0.992571i $$0.461175\pi$$
$$542$$ 0 0
$$543$$ 5946.00 0.469921
$$544$$ 0 0
$$545$$ 7140.00 0.561182
$$546$$ 0 0
$$547$$ 8476.00 0.662537 0.331268 0.943537i $$-0.392523\pi$$
0.331268 + 0.943537i $$0.392523\pi$$
$$548$$ 0 0
$$549$$ −4842.00 −0.376414
$$550$$ 0 0
$$551$$ −600.000 −0.0463899
$$552$$ 0 0
$$553$$ 8320.00 0.639787
$$554$$ 0 0
$$555$$ 4572.00 0.349677
$$556$$ 0 0
$$557$$ −12546.0 −0.954383 −0.477191 0.878799i $$-0.658345\pi$$
−0.477191 + 0.878799i $$0.658345\pi$$
$$558$$ 0 0
$$559$$ 1976.00 0.149510
$$560$$ 0 0
$$561$$ 4536.00 0.341373
$$562$$ 0 0
$$563$$ 12.0000 0.000898294 0 0.000449147 1.00000i $$-0.499857\pi$$
0.000449147 1.00000i $$0.499857\pi$$
$$564$$ 0 0
$$565$$ −2772.00 −0.206405
$$566$$ 0 0
$$567$$ 1296.00 0.0959910
$$568$$ 0 0
$$569$$ 19290.0 1.42123 0.710614 0.703582i $$-0.248417\pi$$
0.710614 + 0.703582i $$0.248417\pi$$
$$570$$ 0 0
$$571$$ 12148.0 0.890329 0.445165 0.895449i $$-0.353145\pi$$
0.445165 + 0.895449i $$0.353145\pi$$
$$572$$ 0 0
$$573$$ 8064.00 0.587920
$$574$$ 0 0
$$575$$ 14952.0 1.08442
$$576$$ 0 0
$$577$$ −10366.0 −0.747907 −0.373953 0.927447i $$-0.621998\pi$$
−0.373953 + 0.927447i $$0.621998\pi$$
$$578$$ 0 0
$$579$$ −6906.00 −0.495688
$$580$$ 0 0
$$581$$ 7872.00 0.562109
$$582$$ 0 0
$$583$$ −2376.00 −0.168789
$$584$$ 0 0
$$585$$ 2052.00 0.145025
$$586$$ 0 0
$$587$$ −7644.00 −0.537482 −0.268741 0.963213i $$-0.586607\pi$$
−0.268741 + 0.963213i $$0.586607\pi$$
$$588$$ 0 0
$$589$$ −1760.00 −0.123123
$$590$$ 0 0
$$591$$ 13122.0 0.913311
$$592$$ 0 0
$$593$$ 8658.00 0.599564 0.299782 0.954008i $$-0.403086\pi$$
0.299782 + 0.954008i $$0.403086\pi$$
$$594$$ 0 0
$$595$$ −12096.0 −0.833425
$$596$$ 0 0
$$597$$ 4800.00 0.329064
$$598$$ 0 0
$$599$$ −25800.0 −1.75987 −0.879933 0.475098i $$-0.842413\pi$$
−0.879933 + 0.475098i $$0.842413\pi$$
$$600$$ 0 0
$$601$$ 16202.0 1.09966 0.549828 0.835278i $$-0.314693\pi$$
0.549828 + 0.835278i $$0.314693\pi$$
$$602$$ 0 0
$$603$$ −7956.00 −0.537302
$$604$$ 0 0
$$605$$ −7122.00 −0.478596
$$606$$ 0 0
$$607$$ 24136.0 1.61392 0.806960 0.590605i $$-0.201111\pi$$
0.806960 + 0.590605i $$0.201111\pi$$
$$608$$ 0 0
$$609$$ 1440.00 0.0958157
$$610$$ 0 0
$$611$$ 3648.00 0.241542
$$612$$ 0 0
$$613$$ −4642.00 −0.305854 −0.152927 0.988237i $$-0.548870\pi$$
−0.152927 + 0.988237i $$0.548870\pi$$
$$614$$ 0 0
$$615$$ 756.000 0.0495689
$$616$$ 0 0
$$617$$ −6726.00 −0.438863 −0.219432 0.975628i $$-0.570420\pi$$
−0.219432 + 0.975628i $$0.570420\pi$$
$$618$$ 0 0
$$619$$ 21220.0 1.37787 0.688937 0.724821i $$-0.258078\pi$$
0.688937 + 0.724821i $$0.258078\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ 0 0
$$623$$ 12960.0 0.833437
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ 720.000 0.0458597
$$628$$ 0 0
$$629$$ −32004.0 −2.02875
$$630$$ 0 0
$$631$$ −29792.0 −1.87956 −0.939779 0.341783i $$-0.888969\pi$$
−0.939779 + 0.341783i $$0.888969\pi$$
$$632$$ 0 0
$$633$$ −9996.00 −0.627655
$$634$$ 0 0
$$635$$ 15216.0 0.950911
$$636$$ 0 0
$$637$$ −3306.00 −0.205633
$$638$$ 0 0
$$639$$ −7128.00 −0.441282
$$640$$ 0 0
$$641$$ −10158.0 −0.625923 −0.312962 0.949766i $$-0.601321\pi$$
−0.312962 + 0.949766i $$0.601321\pi$$
$$642$$ 0 0
$$643$$ −29828.0 −1.82940 −0.914698 0.404138i $$-0.867571\pi$$
−0.914698 + 0.404138i $$0.867571\pi$$
$$644$$ 0 0
$$645$$ 936.000 0.0571395
$$646$$ 0 0
$$647$$ −1944.00 −0.118124 −0.0590622 0.998254i $$-0.518811\pi$$
−0.0590622 + 0.998254i $$0.518811\pi$$
$$648$$ 0 0
$$649$$ −7920.00 −0.479025
$$650$$ 0 0
$$651$$ 4224.00 0.254304
$$652$$ 0 0
$$653$$ 26718.0 1.60116 0.800579 0.599227i $$-0.204525\pi$$
0.800579 + 0.599227i $$0.204525\pi$$
$$654$$ 0 0
$$655$$ −13752.0 −0.820359
$$656$$ 0 0
$$657$$ 1962.00 0.116507
$$658$$ 0 0
$$659$$ −4260.00 −0.251815 −0.125907 0.992042i $$-0.540184\pi$$
−0.125907 + 0.992042i $$0.540184\pi$$
$$660$$ 0 0
$$661$$ 22862.0 1.34528 0.672639 0.739971i $$-0.265161\pi$$
0.672639 + 0.739971i $$0.265161\pi$$
$$662$$ 0 0
$$663$$ −14364.0 −0.841405
$$664$$ 0 0
$$665$$ −1920.00 −0.111962
$$666$$ 0 0
$$667$$ −5040.00 −0.292578
$$668$$ 0 0
$$669$$ −7944.00 −0.459092
$$670$$ 0 0
$$671$$ 6456.00 0.371432
$$672$$ 0 0
$$673$$ −32542.0 −1.86390 −0.931948 0.362592i $$-0.881892\pi$$
−0.931948 + 0.362592i $$0.881892\pi$$
$$674$$ 0 0
$$675$$ −2403.00 −0.137024
$$676$$ 0 0
$$677$$ 14214.0 0.806925 0.403463 0.914996i $$-0.367807\pi$$
0.403463 + 0.914996i $$0.367807\pi$$
$$678$$ 0 0
$$679$$ 18464.0 1.04357
$$680$$ 0 0
$$681$$ −6732.00 −0.378812
$$682$$ 0 0
$$683$$ 7092.00 0.397317 0.198659 0.980069i $$-0.436341\pi$$
0.198659 + 0.980069i $$0.436341\pi$$
$$684$$ 0 0
$$685$$ −4356.00 −0.242970
$$686$$ 0 0
$$687$$ −16950.0 −0.941314
$$688$$ 0 0
$$689$$ 7524.00 0.416026
$$690$$ 0 0
$$691$$ 13228.0 0.728244 0.364122 0.931351i $$-0.381369\pi$$
0.364122 + 0.931351i $$0.381369\pi$$
$$692$$ 0 0
$$693$$ −1728.00 −0.0947205
$$694$$ 0 0
$$695$$ −2280.00 −0.124439
$$696$$ 0 0
$$697$$ −5292.00 −0.287588
$$698$$ 0 0
$$699$$ 14094.0 0.762638
$$700$$ 0 0
$$701$$ 28062.0 1.51196 0.755982 0.654592i $$-0.227160\pi$$
0.755982 + 0.654592i $$0.227160\pi$$
$$702$$ 0 0
$$703$$ −5080.00 −0.272540
$$704$$ 0 0
$$705$$ 1728.00 0.0923124
$$706$$ 0 0
$$707$$ −9888.00 −0.525992
$$708$$ 0 0
$$709$$ −27250.0 −1.44343 −0.721717 0.692188i $$-0.756647\pi$$
−0.721717 + 0.692188i $$0.756647\pi$$
$$710$$ 0 0
$$711$$ 4680.00 0.246855
$$712$$ 0 0
$$713$$ −14784.0 −0.776529
$$714$$ 0 0
$$715$$ −2736.00 −0.143106
$$716$$ 0 0
$$717$$ 3600.00 0.187510
$$718$$ 0 0
$$719$$ 14400.0 0.746912 0.373456 0.927648i $$-0.378173\pi$$
0.373456 + 0.927648i $$0.378173\pi$$
$$720$$ 0 0
$$721$$ −2048.00 −0.105786
$$722$$ 0 0
$$723$$ −2154.00 −0.110800
$$724$$ 0 0
$$725$$ −2670.00 −0.136774
$$726$$ 0 0
$$727$$ −17984.0 −0.917455 −0.458727 0.888577i $$-0.651695\pi$$
−0.458727 + 0.888577i $$0.651695\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −6552.00 −0.331511
$$732$$ 0 0
$$733$$ 16598.0 0.836373 0.418186 0.908361i $$-0.362666\pi$$
0.418186 + 0.908361i $$0.362666\pi$$
$$734$$ 0 0
$$735$$ −1566.00 −0.0785888
$$736$$ 0 0
$$737$$ 10608.0 0.530191
$$738$$ 0 0
$$739$$ −1460.00 −0.0726752 −0.0363376 0.999340i $$-0.511569\pi$$
−0.0363376 + 0.999340i $$0.511569\pi$$
$$740$$ 0 0
$$741$$ −2280.00 −0.113034
$$742$$ 0 0
$$743$$ 30072.0 1.48484 0.742419 0.669936i $$-0.233678\pi$$
0.742419 + 0.669936i $$0.233678\pi$$
$$744$$ 0 0
$$745$$ 9540.00 0.469152
$$746$$ 0 0
$$747$$ 4428.00 0.216884
$$748$$ 0 0
$$749$$ 23616.0 1.15208
$$750$$ 0 0
$$751$$ 18088.0 0.878882 0.439441 0.898271i $$-0.355177\pi$$
0.439441 + 0.898271i $$0.355177\pi$$
$$752$$ 0 0
$$753$$ −18036.0 −0.872866
$$754$$ 0 0
$$755$$ −14592.0 −0.703387
$$756$$ 0 0
$$757$$ 24734.0 1.18755 0.593773 0.804633i $$-0.297638\pi$$
0.593773 + 0.804633i $$0.297638\pi$$
$$758$$ 0 0
$$759$$ 6048.00 0.289234
$$760$$ 0 0
$$761$$ −22278.0 −1.06120 −0.530602 0.847621i $$-0.678034\pi$$
−0.530602 + 0.847621i $$0.678034\pi$$
$$762$$ 0 0
$$763$$ 19040.0 0.903400
$$764$$ 0 0
$$765$$ −6804.00 −0.321568
$$766$$ 0 0
$$767$$ 25080.0 1.18069
$$768$$ 0 0
$$769$$ 16130.0 0.756388 0.378194 0.925726i $$-0.376545\pi$$
0.378194 + 0.925726i $$0.376545\pi$$
$$770$$ 0 0
$$771$$ −6138.00 −0.286712
$$772$$ 0 0
$$773$$ 29718.0 1.38277 0.691386 0.722486i $$-0.257001\pi$$
0.691386 + 0.722486i $$0.257001\pi$$
$$774$$ 0 0
$$775$$ −7832.00 −0.363011
$$776$$ 0 0
$$777$$ 12192.0 0.562916
$$778$$ 0 0
$$779$$ −840.000 −0.0386343
$$780$$ 0 0
$$781$$ 9504.00 0.435442
$$782$$ 0 0
$$783$$ 810.000 0.0369694
$$784$$ 0 0
$$785$$ 3684.00 0.167500
$$786$$ 0 0
$$787$$ −9524.00 −0.431377 −0.215689 0.976462i $$-0.569200\pi$$
−0.215689 + 0.976462i $$0.569200\pi$$
$$788$$ 0 0
$$789$$ 18216.0 0.821935
$$790$$ 0 0
$$791$$ −7392.00 −0.332275
$$792$$ 0 0
$$793$$ −20444.0 −0.915495
$$794$$ 0 0
$$795$$ 3564.00 0.158996
$$796$$ 0 0
$$797$$ −33906.0 −1.50692 −0.753458 0.657496i $$-0.771616\pi$$
−0.753458 + 0.657496i $$0.771616\pi$$
$$798$$ 0 0
$$799$$ −12096.0 −0.535577
$$800$$ 0 0
$$801$$ 7290.00 0.321572
$$802$$ 0 0
$$803$$ −2616.00 −0.114965
$$804$$ 0 0
$$805$$ −16128.0 −0.706133
$$806$$ 0 0
$$807$$ −20790.0 −0.906868
$$808$$ 0 0
$$809$$ −630.000 −0.0273790 −0.0136895 0.999906i $$-0.504358\pi$$
−0.0136895 + 0.999906i $$0.504358\pi$$
$$810$$ 0 0
$$811$$ 20788.0 0.900081 0.450040 0.893008i $$-0.351410\pi$$
0.450040 + 0.893008i $$0.351410\pi$$
$$812$$ 0 0
$$813$$ −4056.00 −0.174969
$$814$$ 0 0
$$815$$ 11112.0 0.477591
$$816$$ 0 0
$$817$$ −1040.00 −0.0445349
$$818$$ 0 0
$$819$$ 5472.00 0.233464
$$820$$ 0 0
$$821$$ −43098.0 −1.83207 −0.916036 0.401097i $$-0.868629\pi$$
−0.916036 + 0.401097i $$0.868629\pi$$
$$822$$ 0 0
$$823$$ 14272.0 0.604484 0.302242 0.953231i $$-0.402265\pi$$
0.302242 + 0.953231i $$0.402265\pi$$
$$824$$ 0 0
$$825$$ 3204.00 0.135211
$$826$$ 0 0
$$827$$ −13644.0 −0.573698 −0.286849 0.957976i $$-0.592608\pi$$
−0.286849 + 0.957976i $$0.592608\pi$$
$$828$$ 0 0
$$829$$ −2410.00 −0.100968 −0.0504842 0.998725i $$-0.516076\pi$$
−0.0504842 + 0.998725i $$0.516076\pi$$
$$830$$ 0 0
$$831$$ −3558.00 −0.148527
$$832$$ 0 0
$$833$$ 10962.0 0.455955
$$834$$ 0 0
$$835$$ 12816.0 0.531157
$$836$$ 0 0
$$837$$ 2376.00 0.0981202
$$838$$ 0 0
$$839$$ −23160.0 −0.953006 −0.476503 0.879173i $$-0.658096\pi$$
−0.476503 + 0.879173i $$0.658096\pi$$
$$840$$ 0 0
$$841$$ −23489.0 −0.963098
$$842$$ 0 0
$$843$$ 7326.00 0.299313
$$844$$ 0 0
$$845$$ −4518.00 −0.183934
$$846$$ 0 0
$$847$$ −18992.0 −0.770452
$$848$$ 0 0
$$849$$ −8484.00 −0.342957
$$850$$ 0 0
$$851$$ −42672.0 −1.71889
$$852$$ 0 0
$$853$$ 32078.0 1.28761 0.643804 0.765190i $$-0.277355\pi$$
0.643804 + 0.765190i $$0.277355\pi$$
$$854$$ 0 0
$$855$$ −1080.00 −0.0431991
$$856$$ 0 0
$$857$$ −14406.0 −0.574212 −0.287106 0.957899i $$-0.592693\pi$$
−0.287106 + 0.957899i $$0.592693\pi$$
$$858$$ 0 0
$$859$$ −30620.0 −1.21623 −0.608115 0.793849i $$-0.708074\pi$$
−0.608115 + 0.793849i $$0.708074\pi$$
$$860$$ 0 0
$$861$$ 2016.00 0.0797969
$$862$$ 0 0
$$863$$ −17568.0 −0.692957 −0.346478 0.938058i $$-0.612623\pi$$
−0.346478 + 0.938058i $$0.612623\pi$$
$$864$$ 0 0
$$865$$ 10548.0 0.414616
$$866$$ 0 0
$$867$$ 32889.0 1.28831
$$868$$ 0 0
$$869$$ −6240.00 −0.243587
$$870$$ 0 0
$$871$$ −33592.0 −1.30680
$$872$$ 0 0
$$873$$ 10386.0 0.402649
$$874$$ 0 0
$$875$$ −20544.0 −0.793730
$$876$$ 0 0
$$877$$ −21706.0 −0.835758 −0.417879 0.908503i $$-0.637226\pi$$
−0.417879 + 0.908503i $$0.637226\pi$$
$$878$$ 0 0
$$879$$ 14274.0 0.547725
$$880$$ 0 0
$$881$$ −14958.0 −0.572018 −0.286009 0.958227i $$-0.592329\pi$$
−0.286009 + 0.958227i $$0.592329\pi$$
$$882$$ 0 0
$$883$$ 32812.0 1.25052 0.625261 0.780415i $$-0.284992\pi$$
0.625261 + 0.780415i $$0.284992\pi$$
$$884$$ 0 0
$$885$$ 11880.0 0.451234
$$886$$ 0 0
$$887$$ 38856.0 1.47086 0.735432 0.677598i $$-0.236979\pi$$
0.735432 + 0.677598i $$0.236979\pi$$
$$888$$ 0 0
$$889$$ 40576.0 1.53079
$$890$$ 0 0
$$891$$ −972.000 −0.0365468
$$892$$ 0 0
$$893$$ −1920.00 −0.0719489
$$894$$ 0 0
$$895$$ 3240.00 0.121007
$$896$$ 0 0
$$897$$ −19152.0 −0.712895
$$898$$ 0 0
$$899$$ 2640.00 0.0979410
$$900$$ 0 0
$$901$$ −24948.0 −0.922462
$$902$$ 0 0
$$903$$ 2496.00 0.0919841
$$904$$ 0 0
$$905$$ 11892.0 0.436799
$$906$$ 0 0
$$907$$ 28276.0 1.03516 0.517579 0.855635i $$-0.326833\pi$$
0.517579 + 0.855635i $$0.326833\pi$$
$$908$$ 0 0
$$909$$ −5562.00 −0.202948
$$910$$ 0 0
$$911$$ −8112.00 −0.295019 −0.147510 0.989061i $$-0.547126\pi$$
−0.147510 + 0.989061i $$0.547126\pi$$
$$912$$ 0 0
$$913$$ −5904.00 −0.214013
$$914$$ 0 0
$$915$$ −9684.00 −0.349883
$$916$$ 0 0
$$917$$ −36672.0 −1.32063
$$918$$ 0 0
$$919$$ 26080.0 0.936126 0.468063 0.883695i $$-0.344952\pi$$
0.468063 + 0.883695i $$0.344952\pi$$
$$920$$ 0 0
$$921$$ 25428.0 0.909751
$$922$$ 0 0
$$923$$ −30096.0 −1.07326
$$924$$ 0 0
$$925$$ −22606.0 −0.803547
$$926$$ 0 0
$$927$$ −1152.00 −0.0408162
$$928$$ 0 0
$$929$$ 49170.0 1.73651 0.868254 0.496120i $$-0.165243\pi$$
0.868254 + 0.496120i $$0.165243\pi$$
$$930$$ 0 0
$$931$$ 1740.00 0.0612526
$$932$$ 0 0
$$933$$ −13896.0 −0.487604
$$934$$ 0 0
$$935$$ 9072.00 0.317311
$$936$$ 0 0
$$937$$ 48314.0 1.68447 0.842236 0.539110i $$-0.181239\pi$$
0.842236 + 0.539110i $$0.181239\pi$$
$$938$$ 0 0
$$939$$ −14466.0 −0.502748
$$940$$ 0 0
$$941$$ 34782.0 1.20495 0.602477 0.798137i $$-0.294181\pi$$
0.602477 + 0.798137i $$0.294181\pi$$
$$942$$ 0 0
$$943$$ −7056.00 −0.243664
$$944$$ 0 0
$$945$$ 2592.00 0.0892251
$$946$$ 0 0
$$947$$ 25116.0 0.861838 0.430919 0.902391i $$-0.358190\pi$$
0.430919 + 0.902391i $$0.358190\pi$$
$$948$$ 0 0
$$949$$ 8284.00 0.283361
$$950$$ 0 0
$$951$$ −10278.0 −0.350460
$$952$$ 0 0
$$953$$ −15462.0 −0.525565 −0.262782 0.964855i $$-0.584640\pi$$
−0.262782 + 0.964855i $$0.584640\pi$$
$$954$$ 0 0
$$955$$ 16128.0 0.546481
$$956$$ 0 0
$$957$$ −1080.00 −0.0364801
$$958$$ 0 0
$$959$$ −11616.0 −0.391137
$$960$$ 0 0
$$961$$ −22047.0 −0.740056
$$962$$ 0 0
$$963$$ 13284.0 0.444518
$$964$$ 0 0
$$965$$ −13812.0 −0.460750
$$966$$ 0 0
$$967$$ 736.000 0.0244759 0.0122379 0.999925i $$-0.496104\pi$$
0.0122379 + 0.999925i $$0.496104\pi$$
$$968$$ 0 0
$$969$$ 7560.00 0.250632
$$970$$ 0 0
$$971$$ 29268.0 0.967307 0.483653 0.875260i $$-0.339310\pi$$
0.483653 + 0.875260i $$0.339310\pi$$
$$972$$ 0 0
$$973$$ −6080.00 −0.200325
$$974$$ 0 0
$$975$$ −10146.0 −0.333264
$$976$$ 0 0
$$977$$ 16674.0 0.546007 0.273003 0.962013i $$-0.411983\pi$$
0.273003 + 0.962013i $$0.411983\pi$$
$$978$$ 0 0
$$979$$ −9720.00 −0.317316
$$980$$ 0 0
$$981$$ 10710.0 0.348567
$$982$$ 0 0
$$983$$ 31272.0 1.01467 0.507336 0.861749i $$-0.330630\pi$$
0.507336 + 0.861749i $$0.330630\pi$$
$$984$$ 0 0
$$985$$ 26244.0 0.848937
$$986$$ 0 0
$$987$$ 4608.00 0.148606
$$988$$ 0 0
$$989$$ −8736.00 −0.280878
$$990$$ 0 0
$$991$$ 15928.0 0.510565 0.255282 0.966867i $$-0.417832\pi$$
0.255282 + 0.966867i $$0.417832\pi$$
$$992$$ 0 0
$$993$$ 8364.00 0.267295
$$994$$ 0 0
$$995$$ 9600.00 0.305870
$$996$$ 0 0
$$997$$ 42014.0 1.33460 0.667300 0.744789i $$-0.267450\pi$$
0.667300 + 0.744789i $$0.267450\pi$$
$$998$$ 0 0
$$999$$ 6858.00 0.217195
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 48.4.a.c.1.1 1
3.2 odd 2 144.4.a.c.1.1 1
4.3 odd 2 6.4.a.a.1.1 1
5.2 odd 4 1200.4.f.j.49.1 2
5.3 odd 4 1200.4.f.j.49.2 2
5.4 even 2 1200.4.a.b.1.1 1
7.6 odd 2 2352.4.a.e.1.1 1
8.3 odd 2 192.4.a.i.1.1 1
8.5 even 2 192.4.a.c.1.1 1
12.11 even 2 18.4.a.a.1.1 1
16.3 odd 4 768.4.d.n.385.1 2
16.5 even 4 768.4.d.c.385.1 2
16.11 odd 4 768.4.d.n.385.2 2
16.13 even 4 768.4.d.c.385.2 2
20.3 even 4 150.4.c.d.49.2 2
20.7 even 4 150.4.c.d.49.1 2
20.19 odd 2 150.4.a.i.1.1 1
24.5 odd 2 576.4.a.r.1.1 1
24.11 even 2 576.4.a.q.1.1 1
28.3 even 6 294.4.e.g.79.1 2
28.11 odd 6 294.4.e.h.79.1 2
28.19 even 6 294.4.e.g.67.1 2
28.23 odd 6 294.4.e.h.67.1 2
28.27 even 2 294.4.a.e.1.1 1
36.7 odd 6 162.4.c.f.109.1 2
36.11 even 6 162.4.c.c.109.1 2
36.23 even 6 162.4.c.c.55.1 2
36.31 odd 6 162.4.c.f.55.1 2
44.43 even 2 726.4.a.f.1.1 1
52.31 even 4 1014.4.b.d.337.2 2
52.47 even 4 1014.4.b.d.337.1 2
52.51 odd 2 1014.4.a.g.1.1 1
60.23 odd 4 450.4.c.e.199.1 2
60.47 odd 4 450.4.c.e.199.2 2
60.59 even 2 450.4.a.h.1.1 1
68.67 odd 2 1734.4.a.d.1.1 1
76.75 even 2 2166.4.a.i.1.1 1
84.11 even 6 882.4.g.i.667.1 2
84.23 even 6 882.4.g.i.361.1 2
84.47 odd 6 882.4.g.f.361.1 2
84.59 odd 6 882.4.g.f.667.1 2
84.83 odd 2 882.4.a.n.1.1 1
132.131 odd 2 2178.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 4.3 odd 2
18.4.a.a.1.1 1 12.11 even 2
48.4.a.c.1.1 1 1.1 even 1 trivial
144.4.a.c.1.1 1 3.2 odd 2
150.4.a.i.1.1 1 20.19 odd 2
150.4.c.d.49.1 2 20.7 even 4
150.4.c.d.49.2 2 20.3 even 4
162.4.c.c.55.1 2 36.23 even 6
162.4.c.c.109.1 2 36.11 even 6
162.4.c.f.55.1 2 36.31 odd 6
162.4.c.f.109.1 2 36.7 odd 6
192.4.a.c.1.1 1 8.5 even 2
192.4.a.i.1.1 1 8.3 odd 2
294.4.a.e.1.1 1 28.27 even 2
294.4.e.g.67.1 2 28.19 even 6
294.4.e.g.79.1 2 28.3 even 6
294.4.e.h.67.1 2 28.23 odd 6
294.4.e.h.79.1 2 28.11 odd 6
450.4.a.h.1.1 1 60.59 even 2
450.4.c.e.199.1 2 60.23 odd 4
450.4.c.e.199.2 2 60.47 odd 4
576.4.a.q.1.1 1 24.11 even 2
576.4.a.r.1.1 1 24.5 odd 2
726.4.a.f.1.1 1 44.43 even 2
768.4.d.c.385.1 2 16.5 even 4
768.4.d.c.385.2 2 16.13 even 4
768.4.d.n.385.1 2 16.3 odd 4
768.4.d.n.385.2 2 16.11 odd 4
882.4.a.n.1.1 1 84.83 odd 2
882.4.g.f.361.1 2 84.47 odd 6
882.4.g.f.667.1 2 84.59 odd 6
882.4.g.i.361.1 2 84.23 even 6
882.4.g.i.667.1 2 84.11 even 6
1014.4.a.g.1.1 1 52.51 odd 2
1014.4.b.d.337.1 2 52.47 even 4
1014.4.b.d.337.2 2 52.31 even 4
1200.4.a.b.1.1 1 5.4 even 2
1200.4.f.j.49.1 2 5.2 odd 4
1200.4.f.j.49.2 2 5.3 odd 4
1734.4.a.d.1.1 1 68.67 odd 2
2166.4.a.i.1.1 1 76.75 even 2
2178.4.a.e.1.1 1 132.131 odd 2
2352.4.a.e.1.1 1 7.6 odd 2