# Properties

 Label 441.6.a.j.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ 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] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$70.7292645375$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.00000 q^{2} +4.00000 q^{4} +78.0000 q^{5} -168.000 q^{8} +O(q^{10})$$ $$q+6.00000 q^{2} +4.00000 q^{4} +78.0000 q^{5} -168.000 q^{8} +468.000 q^{10} -444.000 q^{11} +442.000 q^{13} -1136.00 q^{16} -126.000 q^{17} -2684.00 q^{19} +312.000 q^{20} -2664.00 q^{22} -4200.00 q^{23} +2959.00 q^{25} +2652.00 q^{26} +5442.00 q^{29} -80.0000 q^{31} -1440.00 q^{32} -756.000 q^{34} -5434.00 q^{37} -16104.0 q^{38} -13104.0 q^{40} +7962.00 q^{41} -11524.0 q^{43} -1776.00 q^{44} -25200.0 q^{46} -13920.0 q^{47} +17754.0 q^{50} +1768.00 q^{52} +9594.00 q^{53} -34632.0 q^{55} +32652.0 q^{58} +27492.0 q^{59} -49478.0 q^{61} -480.000 q^{62} +27712.0 q^{64} +34476.0 q^{65} -59356.0 q^{67} -504.000 q^{68} -32040.0 q^{71} +61846.0 q^{73} -32604.0 q^{74} -10736.0 q^{76} -65776.0 q^{79} -88608.0 q^{80} +47772.0 q^{82} +40188.0 q^{83} -9828.00 q^{85} -69144.0 q^{86} +74592.0 q^{88} -7974.00 q^{89} -16800.0 q^{92} -83520.0 q^{94} -209352. q^{95} +143662. q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 6.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ 0 0
$$4$$ 4.00000 0.125000
$$5$$ 78.0000 1.39531 0.697653 0.716436i $$-0.254228\pi$$
0.697653 + 0.716436i $$0.254228\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −168.000 −0.928078
$$9$$ 0 0
$$10$$ 468.000 1.47995
$$11$$ −444.000 −1.10637 −0.553186 0.833058i $$-0.686588\pi$$
−0.553186 + 0.833058i $$0.686588\pi$$
$$12$$ 0 0
$$13$$ 442.000 0.725377 0.362689 0.931910i $$-0.381859\pi$$
0.362689 + 0.931910i $$0.381859\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1136.00 −1.10938
$$17$$ −126.000 −0.105742 −0.0528711 0.998601i $$-0.516837\pi$$
−0.0528711 + 0.998601i $$0.516837\pi$$
$$18$$ 0 0
$$19$$ −2684.00 −1.70568 −0.852842 0.522169i $$-0.825123\pi$$
−0.852842 + 0.522169i $$0.825123\pi$$
$$20$$ 312.000 0.174413
$$21$$ 0 0
$$22$$ −2664.00 −1.17348
$$23$$ −4200.00 −1.65550 −0.827751 0.561096i $$-0.810380\pi$$
−0.827751 + 0.561096i $$0.810380\pi$$
$$24$$ 0 0
$$25$$ 2959.00 0.946880
$$26$$ 2652.00 0.769379
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 5442.00 1.20161 0.600805 0.799396i $$-0.294847\pi$$
0.600805 + 0.799396i $$0.294847\pi$$
$$30$$ 0 0
$$31$$ −80.0000 −0.0149515 −0.00747577 0.999972i $$-0.502380\pi$$
−0.00747577 + 0.999972i $$0.502380\pi$$
$$32$$ −1440.00 −0.248592
$$33$$ 0 0
$$34$$ −756.000 −0.112157
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5434.00 −0.652552 −0.326276 0.945274i $$-0.605794\pi$$
−0.326276 + 0.945274i $$0.605794\pi$$
$$38$$ −16104.0 −1.80915
$$39$$ 0 0
$$40$$ −13104.0 −1.29495
$$41$$ 7962.00 0.739712 0.369856 0.929089i $$-0.379407\pi$$
0.369856 + 0.929089i $$0.379407\pi$$
$$42$$ 0 0
$$43$$ −11524.0 −0.950456 −0.475228 0.879863i $$-0.657634\pi$$
−0.475228 + 0.879863i $$0.657634\pi$$
$$44$$ −1776.00 −0.138297
$$45$$ 0 0
$$46$$ −25200.0 −1.75592
$$47$$ −13920.0 −0.919167 −0.459584 0.888134i $$-0.652001\pi$$
−0.459584 + 0.888134i $$0.652001\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 17754.0 1.00432
$$51$$ 0 0
$$52$$ 1768.00 0.0906721
$$53$$ 9594.00 0.469148 0.234574 0.972098i $$-0.424630\pi$$
0.234574 + 0.972098i $$0.424630\pi$$
$$54$$ 0 0
$$55$$ −34632.0 −1.54373
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 32652.0 1.27450
$$59$$ 27492.0 1.02820 0.514098 0.857731i $$-0.328127\pi$$
0.514098 + 0.857731i $$0.328127\pi$$
$$60$$ 0 0
$$61$$ −49478.0 −1.70250 −0.851251 0.524759i $$-0.824155\pi$$
−0.851251 + 0.524759i $$0.824155\pi$$
$$62$$ −480.000 −0.0158585
$$63$$ 0 0
$$64$$ 27712.0 0.845703
$$65$$ 34476.0 1.01212
$$66$$ 0 0
$$67$$ −59356.0 −1.61539 −0.807695 0.589600i $$-0.799285\pi$$
−0.807695 + 0.589600i $$0.799285\pi$$
$$68$$ −504.000 −0.0132178
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −32040.0 −0.754304 −0.377152 0.926151i $$-0.623097\pi$$
−0.377152 + 0.926151i $$0.623097\pi$$
$$72$$ 0 0
$$73$$ 61846.0 1.35833 0.679164 0.733987i $$-0.262343\pi$$
0.679164 + 0.733987i $$0.262343\pi$$
$$74$$ −32604.0 −0.692136
$$75$$ 0 0
$$76$$ −10736.0 −0.213210
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −65776.0 −1.18577 −0.592884 0.805288i $$-0.702011\pi$$
−0.592884 + 0.805288i $$0.702011\pi$$
$$80$$ −88608.0 −1.54792
$$81$$ 0 0
$$82$$ 47772.0 0.784583
$$83$$ 40188.0 0.640326 0.320163 0.947362i $$-0.396262\pi$$
0.320163 + 0.947362i $$0.396262\pi$$
$$84$$ 0 0
$$85$$ −9828.00 −0.147543
$$86$$ −69144.0 −1.00811
$$87$$ 0 0
$$88$$ 74592.0 1.02680
$$89$$ −7974.00 −0.106709 −0.0533545 0.998576i $$-0.516991\pi$$
−0.0533545 + 0.998576i $$0.516991\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −16800.0 −0.206938
$$93$$ 0 0
$$94$$ −83520.0 −0.974924
$$95$$ −209352. −2.37995
$$96$$ 0 0
$$97$$ 143662. 1.55029 0.775144 0.631784i $$-0.217677\pi$$
0.775144 + 0.631784i $$0.217677\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 11836.0 0.118360
$$101$$ −2706.00 −0.0263952 −0.0131976 0.999913i $$-0.504201\pi$$
−0.0131976 + 0.999913i $$0.504201\pi$$
$$102$$ 0 0
$$103$$ −131768. −1.22382 −0.611909 0.790928i $$-0.709598\pi$$
−0.611909 + 0.790928i $$0.709598\pi$$
$$104$$ −74256.0 −0.673206
$$105$$ 0 0
$$106$$ 57564.0 0.497607
$$107$$ 128916. 1.08855 0.544274 0.838908i $$-0.316805\pi$$
0.544274 + 0.838908i $$0.316805\pi$$
$$108$$ 0 0
$$109$$ −100978. −0.814068 −0.407034 0.913413i $$-0.633437\pi$$
−0.407034 + 0.913413i $$0.633437\pi$$
$$110$$ −207792. −1.63737
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −220146. −1.62186 −0.810932 0.585140i $$-0.801040\pi$$
−0.810932 + 0.585140i $$0.801040\pi$$
$$114$$ 0 0
$$115$$ −327600. −2.30993
$$116$$ 21768.0 0.150201
$$117$$ 0 0
$$118$$ 164952. 1.09057
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 36085.0 0.224059
$$122$$ −296868. −1.80578
$$123$$ 0 0
$$124$$ −320.000 −0.00186894
$$125$$ −12948.0 −0.0741187
$$126$$ 0 0
$$127$$ −74320.0 −0.408880 −0.204440 0.978879i $$-0.565537\pi$$
−0.204440 + 0.978879i $$0.565537\pi$$
$$128$$ 212352. 1.14560
$$129$$ 0 0
$$130$$ 206856. 1.07352
$$131$$ −155316. −0.790748 −0.395374 0.918520i $$-0.629385\pi$$
−0.395374 + 0.918520i $$0.629385\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −356136. −1.71338
$$135$$ 0 0
$$136$$ 21168.0 0.0981369
$$137$$ 264246. 1.20284 0.601419 0.798934i $$-0.294602\pi$$
0.601419 + 0.798934i $$0.294602\pi$$
$$138$$ 0 0
$$139$$ −224612. −0.986043 −0.493022 0.870017i $$-0.664108\pi$$
−0.493022 + 0.870017i $$0.664108\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −192240. −0.800061
$$143$$ −196248. −0.802537
$$144$$ 0 0
$$145$$ 424476. 1.67661
$$146$$ 371076. 1.44072
$$147$$ 0 0
$$148$$ −21736.0 −0.0815690
$$149$$ 82074.0 0.302859 0.151429 0.988468i $$-0.451612\pi$$
0.151429 + 0.988468i $$0.451612\pi$$
$$150$$ 0 0
$$151$$ −287032. −1.02444 −0.512222 0.858853i $$-0.671177\pi$$
−0.512222 + 0.858853i $$0.671177\pi$$
$$152$$ 450912. 1.58301
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6240.00 −0.0208620
$$156$$ 0 0
$$157$$ −129878. −0.420520 −0.210260 0.977646i $$-0.567431\pi$$
−0.210260 + 0.977646i $$0.567431\pi$$
$$158$$ −394656. −1.25770
$$159$$ 0 0
$$160$$ −112320. −0.346862
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 555284. 1.63699 0.818495 0.574513i $$-0.194809\pi$$
0.818495 + 0.574513i $$0.194809\pi$$
$$164$$ 31848.0 0.0924640
$$165$$ 0 0
$$166$$ 241128. 0.679168
$$167$$ 43512.0 0.120731 0.0603654 0.998176i $$-0.480773\pi$$
0.0603654 + 0.998176i $$0.480773\pi$$
$$168$$ 0 0
$$169$$ −175929. −0.473828
$$170$$ −58968.0 −0.156493
$$171$$ 0 0
$$172$$ −46096.0 −0.118807
$$173$$ −18330.0 −0.0465637 −0.0232818 0.999729i $$-0.507412\pi$$
−0.0232818 + 0.999729i $$0.507412\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 504384. 1.22738
$$177$$ 0 0
$$178$$ −47844.0 −0.113182
$$179$$ 153324. 0.357666 0.178833 0.983879i $$-0.442768\pi$$
0.178833 + 0.983879i $$0.442768\pi$$
$$180$$ 0 0
$$181$$ 382066. 0.866846 0.433423 0.901191i $$-0.357306\pi$$
0.433423 + 0.901191i $$0.357306\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 705600. 1.53643
$$185$$ −423852. −0.910510
$$186$$ 0 0
$$187$$ 55944.0 0.116990
$$188$$ −55680.0 −0.114896
$$189$$ 0 0
$$190$$ −1.25611e6 −2.52432
$$191$$ 273408. 0.542285 0.271143 0.962539i $$-0.412598\pi$$
0.271143 + 0.962539i $$0.412598\pi$$
$$192$$ 0 0
$$193$$ 153602. 0.296827 0.148414 0.988925i $$-0.452583\pi$$
0.148414 + 0.988925i $$0.452583\pi$$
$$194$$ 861972. 1.64433
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −154422. −0.283494 −0.141747 0.989903i $$-0.545272\pi$$
−0.141747 + 0.989903i $$0.545272\pi$$
$$198$$ 0 0
$$199$$ 366856. 0.656694 0.328347 0.944557i $$-0.393508\pi$$
0.328347 + 0.944557i $$0.393508\pi$$
$$200$$ −497112. −0.878778
$$201$$ 0 0
$$202$$ −16236.0 −0.0279963
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 621036. 1.03212
$$206$$ −790608. −1.29806
$$207$$ 0 0
$$208$$ −502112. −0.804715
$$209$$ 1.19170e6 1.88712
$$210$$ 0 0
$$211$$ 520244. 0.804453 0.402227 0.915540i $$-0.368236\pi$$
0.402227 + 0.915540i $$0.368236\pi$$
$$212$$ 38376.0 0.0586435
$$213$$ 0 0
$$214$$ 773496. 1.15458
$$215$$ −898872. −1.32618
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −605868. −0.863449
$$219$$ 0 0
$$220$$ −138528. −0.192966
$$221$$ −55692.0 −0.0767030
$$222$$ 0 0
$$223$$ −304736. −0.410357 −0.205178 0.978725i $$-0.565777\pi$$
−0.205178 + 0.978725i $$0.565777\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −1.32088e6 −1.72025
$$227$$ 288588. 0.371718 0.185859 0.982576i $$-0.440493\pi$$
0.185859 + 0.982576i $$0.440493\pi$$
$$228$$ 0 0
$$229$$ −772190. −0.973051 −0.486525 0.873666i $$-0.661736\pi$$
−0.486525 + 0.873666i $$0.661736\pi$$
$$230$$ −1.96560e6 −2.45005
$$231$$ 0 0
$$232$$ −914256. −1.11519
$$233$$ −252234. −0.304378 −0.152189 0.988351i $$-0.548632\pi$$
−0.152189 + 0.988351i $$0.548632\pi$$
$$234$$ 0 0
$$235$$ −1.08576e6 −1.28252
$$236$$ 109968. 0.128525
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.45114e6 1.64329 0.821643 0.570002i $$-0.193058\pi$$
0.821643 + 0.570002i $$0.193058\pi$$
$$240$$ 0 0
$$241$$ 146398. 0.162365 0.0811825 0.996699i $$-0.474130\pi$$
0.0811825 + 0.996699i $$0.474130\pi$$
$$242$$ 216510. 0.237651
$$243$$ 0 0
$$244$$ −197912. −0.212813
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.18633e6 −1.23726
$$248$$ 13440.0 0.0138762
$$249$$ 0 0
$$250$$ −77688.0 −0.0786147
$$251$$ 607860. 0.609003 0.304501 0.952512i $$-0.401510\pi$$
0.304501 + 0.952512i $$0.401510\pi$$
$$252$$ 0 0
$$253$$ 1.86480e6 1.83160
$$254$$ −445920. −0.433683
$$255$$ 0 0
$$256$$ 387328. 0.369385
$$257$$ 95586.0 0.0902737 0.0451369 0.998981i $$-0.485628\pi$$
0.0451369 + 0.998981i $$0.485628\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 137904. 0.126515
$$261$$ 0 0
$$262$$ −931896. −0.838715
$$263$$ 2.20034e6 1.96156 0.980779 0.195121i $$-0.0625100\pi$$
0.980779 + 0.195121i $$0.0625100\pi$$
$$264$$ 0 0
$$265$$ 748332. 0.654605
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −237424. −0.201924
$$269$$ 1.77025e6 1.49160 0.745801 0.666169i $$-0.232067\pi$$
0.745801 + 0.666169i $$0.232067\pi$$
$$270$$ 0 0
$$271$$ 223504. 0.184868 0.0924341 0.995719i $$-0.470535\pi$$
0.0924341 + 0.995719i $$0.470535\pi$$
$$272$$ 143136. 0.117308
$$273$$ 0 0
$$274$$ 1.58548e6 1.27580
$$275$$ −1.31380e6 −1.04760
$$276$$ 0 0
$$277$$ −342778. −0.268419 −0.134210 0.990953i $$-0.542850\pi$$
−0.134210 + 0.990953i $$0.542850\pi$$
$$278$$ −1.34767e6 −1.04586
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −480378. −0.362925 −0.181463 0.983398i $$-0.558083\pi$$
−0.181463 + 0.983398i $$0.558083\pi$$
$$282$$ 0 0
$$283$$ 29980.0 0.0222518 0.0111259 0.999938i $$-0.496458\pi$$
0.0111259 + 0.999938i $$0.496458\pi$$
$$284$$ −128160. −0.0942880
$$285$$ 0 0
$$286$$ −1.17749e6 −0.851219
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.40398e6 −0.988819
$$290$$ 2.54686e6 1.77832
$$291$$ 0 0
$$292$$ 247384. 0.169791
$$293$$ −198066. −0.134785 −0.0673924 0.997727i $$-0.521468\pi$$
−0.0673924 + 0.997727i $$0.521468\pi$$
$$294$$ 0 0
$$295$$ 2.14438e6 1.43465
$$296$$ 912912. 0.605619
$$297$$ 0 0
$$298$$ 492444. 0.321230
$$299$$ −1.85640e6 −1.20086
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −1.72219e6 −1.08659
$$303$$ 0 0
$$304$$ 3.04902e6 1.89224
$$305$$ −3.85928e6 −2.37551
$$306$$ 0 0
$$307$$ 1.04564e6 0.633191 0.316595 0.948561i $$-0.397460\pi$$
0.316595 + 0.948561i $$0.397460\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −37440.0 −0.0221275
$$311$$ 1.83718e6 1.07708 0.538542 0.842598i $$-0.318975\pi$$
0.538542 + 0.842598i $$0.318975\pi$$
$$312$$ 0 0
$$313$$ 365494. 0.210872 0.105436 0.994426i $$-0.466376\pi$$
0.105436 + 0.994426i $$0.466376\pi$$
$$314$$ −779268. −0.446029
$$315$$ 0 0
$$316$$ −263104. −0.148221
$$317$$ 28338.0 0.0158388 0.00791938 0.999969i $$-0.497479\pi$$
0.00791938 + 0.999969i $$0.497479\pi$$
$$318$$ 0 0
$$319$$ −2.41625e6 −1.32943
$$320$$ 2.16154e6 1.18001
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 338184. 0.180363
$$324$$ 0 0
$$325$$ 1.30788e6 0.686845
$$326$$ 3.33170e6 1.73629
$$327$$ 0 0
$$328$$ −1.33762e6 −0.686510
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.93392e6 0.970214 0.485107 0.874455i $$-0.338781\pi$$
0.485107 + 0.874455i $$0.338781\pi$$
$$332$$ 160752. 0.0800408
$$333$$ 0 0
$$334$$ 261072. 0.128054
$$335$$ −4.62977e6 −2.25397
$$336$$ 0 0
$$337$$ −1.88817e6 −0.905664 −0.452832 0.891596i $$-0.649586\pi$$
−0.452832 + 0.891596i $$0.649586\pi$$
$$338$$ −1.05557e6 −0.502570
$$339$$ 0 0
$$340$$ −39312.0 −0.0184428
$$341$$ 35520.0 0.0165420
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 1.93603e6 0.882097
$$345$$ 0 0
$$346$$ −109980. −0.0493882
$$347$$ −2.91937e6 −1.30156 −0.650782 0.759264i $$-0.725559\pi$$
−0.650782 + 0.759264i $$0.725559\pi$$
$$348$$ 0 0
$$349$$ 780682. 0.343092 0.171546 0.985176i $$-0.445124\pi$$
0.171546 + 0.985176i $$0.445124\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 639360. 0.275036
$$353$$ 1.33437e6 0.569954 0.284977 0.958534i $$-0.408014\pi$$
0.284977 + 0.958534i $$0.408014\pi$$
$$354$$ 0 0
$$355$$ −2.49912e6 −1.05249
$$356$$ −31896.0 −0.0133386
$$357$$ 0 0
$$358$$ 919944. 0.379362
$$359$$ −1.01743e6 −0.416648 −0.208324 0.978060i $$-0.566801\pi$$
−0.208324 + 0.978060i $$0.566801\pi$$
$$360$$ 0 0
$$361$$ 4.72776e6 1.90936
$$362$$ 2.29240e6 0.919429
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.82399e6 1.89528
$$366$$ 0 0
$$367$$ −837680. −0.324648 −0.162324 0.986737i $$-0.551899\pi$$
−0.162324 + 0.986737i $$0.551899\pi$$
$$368$$ 4.77120e6 1.83657
$$369$$ 0 0
$$370$$ −2.54311e6 −0.965742
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.51993e6 −0.565655 −0.282827 0.959171i $$-0.591272\pi$$
−0.282827 + 0.959171i $$0.591272\pi$$
$$374$$ 335664. 0.124087
$$375$$ 0 0
$$376$$ 2.33856e6 0.853059
$$377$$ 2.40536e6 0.871620
$$378$$ 0 0
$$379$$ 2.64465e6 0.945737 0.472869 0.881133i $$-0.343219\pi$$
0.472869 + 0.881133i $$0.343219\pi$$
$$380$$ −837408. −0.297494
$$381$$ 0 0
$$382$$ 1.64045e6 0.575180
$$383$$ 2.01336e6 0.701333 0.350667 0.936500i $$-0.385955\pi$$
0.350667 + 0.936500i $$0.385955\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 921612. 0.314833
$$387$$ 0 0
$$388$$ 574648. 0.193786
$$389$$ 726234. 0.243334 0.121667 0.992571i $$-0.461176\pi$$
0.121667 + 0.992571i $$0.461176\pi$$
$$390$$ 0 0
$$391$$ 529200. 0.175056
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −926532. −0.300691
$$395$$ −5.13053e6 −1.65451
$$396$$ 0 0
$$397$$ −4.57578e6 −1.45710 −0.728549 0.684993i $$-0.759805\pi$$
−0.728549 + 0.684993i $$0.759805\pi$$
$$398$$ 2.20114e6 0.696529
$$399$$ 0 0
$$400$$ −3.36142e6 −1.05045
$$401$$ 33870.0 0.0105185 0.00525926 0.999986i $$-0.498326\pi$$
0.00525926 + 0.999986i $$0.498326\pi$$
$$402$$ 0 0
$$403$$ −35360.0 −0.0108455
$$404$$ −10824.0 −0.00329940
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.41270e6 0.721966
$$408$$ 0 0
$$409$$ 5.86178e6 1.73269 0.866346 0.499444i $$-0.166462\pi$$
0.866346 + 0.499444i $$0.166462\pi$$
$$410$$ 3.72622e6 1.09473
$$411$$ 0 0
$$412$$ −527072. −0.152977
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3.13466e6 0.893451
$$416$$ −636480. −0.180323
$$417$$ 0 0
$$418$$ 7.15018e6 2.00159
$$419$$ 302748. 0.0842454 0.0421227 0.999112i $$-0.486588\pi$$
0.0421227 + 0.999112i $$0.486588\pi$$
$$420$$ 0 0
$$421$$ −5.36708e6 −1.47582 −0.737909 0.674900i $$-0.764187\pi$$
−0.737909 + 0.674900i $$0.764187\pi$$
$$422$$ 3.12146e6 0.853252
$$423$$ 0 0
$$424$$ −1.61179e6 −0.435406
$$425$$ −372834. −0.100125
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 515664. 0.136068
$$429$$ 0 0
$$430$$ −5.39323e6 −1.40662
$$431$$ −1.17706e6 −0.305214 −0.152607 0.988287i $$-0.548767\pi$$
−0.152607 + 0.988287i $$0.548767\pi$$
$$432$$ 0 0
$$433$$ 3.66249e6 0.938766 0.469383 0.882995i $$-0.344476\pi$$
0.469383 + 0.882995i $$0.344476\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −403912. −0.101758
$$437$$ 1.12728e7 2.82376
$$438$$ 0 0
$$439$$ 2.53674e6 0.628225 0.314113 0.949386i $$-0.398293\pi$$
0.314113 + 0.949386i $$0.398293\pi$$
$$440$$ 5.81818e6 1.43270
$$441$$ 0 0
$$442$$ −334152. −0.0813558
$$443$$ −6.01504e6 −1.45623 −0.728113 0.685457i $$-0.759603\pi$$
−0.728113 + 0.685457i $$0.759603\pi$$
$$444$$ 0 0
$$445$$ −621972. −0.148892
$$446$$ −1.82842e6 −0.435249
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −5.65965e6 −1.32487 −0.662436 0.749119i $$-0.730477\pi$$
−0.662436 + 0.749119i $$0.730477\pi$$
$$450$$ 0 0
$$451$$ −3.53513e6 −0.818397
$$452$$ −880584. −0.202733
$$453$$ 0 0
$$454$$ 1.73153e6 0.394267
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.46159e6 −1.44727 −0.723634 0.690184i $$-0.757530\pi$$
−0.723634 + 0.690184i $$0.757530\pi$$
$$458$$ −4.63314e6 −1.03208
$$459$$ 0 0
$$460$$ −1.31040e6 −0.288742
$$461$$ −3.37353e6 −0.739320 −0.369660 0.929167i $$-0.620526\pi$$
−0.369660 + 0.929167i $$0.620526\pi$$
$$462$$ 0 0
$$463$$ −4.54974e6 −0.986358 −0.493179 0.869928i $$-0.664165\pi$$
−0.493179 + 0.869928i $$0.664165\pi$$
$$464$$ −6.18211e6 −1.33304
$$465$$ 0 0
$$466$$ −1.51340e6 −0.322842
$$467$$ 2.01136e6 0.426773 0.213386 0.976968i $$-0.431551\pi$$
0.213386 + 0.976968i $$0.431551\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −6.51456e6 −1.36032
$$471$$ 0 0
$$472$$ −4.61866e6 −0.954247
$$473$$ 5.11666e6 1.05156
$$474$$ 0 0
$$475$$ −7.94196e6 −1.61508
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 8.70682e6 1.74297
$$479$$ −7.60402e6 −1.51427 −0.757137 0.653257i $$-0.773402\pi$$
−0.757137 + 0.653257i $$0.773402\pi$$
$$480$$ 0 0
$$481$$ −2.40183e6 −0.473347
$$482$$ 878388. 0.172214
$$483$$ 0 0
$$484$$ 144340. 0.0280074
$$485$$ 1.12056e7 2.16313
$$486$$ 0 0
$$487$$ 673112. 0.128607 0.0643035 0.997930i $$-0.479517\pi$$
0.0643035 + 0.997930i $$0.479517\pi$$
$$488$$ 8.31230e6 1.58005
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2.47170e6 0.462692 0.231346 0.972872i $$-0.425687\pi$$
0.231346 + 0.972872i $$0.425687\pi$$
$$492$$ 0 0
$$493$$ −685692. −0.127061
$$494$$ −7.11797e6 −1.31232
$$495$$ 0 0
$$496$$ 90880.0 0.0165869
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 6.08152e6 1.09335 0.546677 0.837343i $$-0.315892\pi$$
0.546677 + 0.837343i $$0.315892\pi$$
$$500$$ −51792.0 −0.00926483
$$501$$ 0 0
$$502$$ 3.64716e6 0.645945
$$503$$ −846216. −0.149129 −0.0745644 0.997216i $$-0.523757\pi$$
−0.0745644 + 0.997216i $$0.523757\pi$$
$$504$$ 0 0
$$505$$ −211068. −0.0368293
$$506$$ 1.11888e7 1.94271
$$507$$ 0 0
$$508$$ −297280. −0.0511101
$$509$$ −7.66785e6 −1.31183 −0.655917 0.754833i $$-0.727718\pi$$
−0.655917 + 0.754833i $$0.727718\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −4.47130e6 −0.753804
$$513$$ 0 0
$$514$$ 573516. 0.0957498
$$515$$ −1.02779e7 −1.70760
$$516$$ 0 0
$$517$$ 6.18048e6 1.01694
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −5.79197e6 −0.939329
$$521$$ −9.68938e6 −1.56387 −0.781937 0.623357i $$-0.785768\pi$$
−0.781937 + 0.623357i $$0.785768\pi$$
$$522$$ 0 0
$$523$$ 7.51678e6 1.20165 0.600824 0.799381i $$-0.294839\pi$$
0.600824 + 0.799381i $$0.294839\pi$$
$$524$$ −621264. −0.0988435
$$525$$ 0 0
$$526$$ 1.32021e7 2.08055
$$527$$ 10080.0 0.00158101
$$528$$ 0 0
$$529$$ 1.12037e7 1.74069
$$530$$ 4.48999e6 0.694314
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 3.51920e6 0.536570
$$534$$ 0 0
$$535$$ 1.00554e7 1.51886
$$536$$ 9.97181e6 1.49921
$$537$$ 0 0
$$538$$ 1.06215e7 1.58208
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 7.34325e6 1.07869 0.539343 0.842086i $$-0.318673\pi$$
0.539343 + 0.842086i $$0.318673\pi$$
$$542$$ 1.34102e6 0.196082
$$543$$ 0 0
$$544$$ 181440. 0.0262867
$$545$$ −7.87628e6 −1.13587
$$546$$ 0 0
$$547$$ 2.18296e6 0.311945 0.155973 0.987761i $$-0.450149\pi$$
0.155973 + 0.987761i $$0.450149\pi$$
$$548$$ 1.05698e6 0.150355
$$549$$ 0 0
$$550$$ −7.88278e6 −1.11115
$$551$$ −1.46063e7 −2.04957
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.05667e6 −0.284702
$$555$$ 0 0
$$556$$ −898448. −0.123255
$$557$$ −1.25466e7 −1.71351 −0.856755 0.515724i $$-0.827523\pi$$
−0.856755 + 0.515724i $$0.827523\pi$$
$$558$$ 0 0
$$559$$ −5.09361e6 −0.689439
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2.88227e6 −0.384940
$$563$$ 5.15972e6 0.686050 0.343025 0.939326i $$-0.388549\pi$$
0.343025 + 0.939326i $$0.388549\pi$$
$$564$$ 0 0
$$565$$ −1.71714e7 −2.26300
$$566$$ 179880. 0.0236016
$$567$$ 0 0
$$568$$ 5.38272e6 0.700053
$$569$$ −1.17452e7 −1.52083 −0.760414 0.649439i $$-0.775004\pi$$
−0.760414 + 0.649439i $$0.775004\pi$$
$$570$$ 0 0
$$571$$ −7.54728e6 −0.968725 −0.484362 0.874867i $$-0.660948\pi$$
−0.484362 + 0.874867i $$0.660948\pi$$
$$572$$ −784992. −0.100317
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.24278e7 −1.56756
$$576$$ 0 0
$$577$$ −9.28483e6 −1.16101 −0.580503 0.814258i $$-0.697144\pi$$
−0.580503 + 0.814258i $$0.697144\pi$$
$$578$$ −8.42389e6 −1.04880
$$579$$ 0 0
$$580$$ 1.69790e6 0.209577
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.25974e6 −0.519053
$$584$$ −1.03901e7 −1.26063
$$585$$ 0 0
$$586$$ −1.18840e6 −0.142961
$$587$$ 1.47623e6 0.176831 0.0884155 0.996084i $$-0.471820\pi$$
0.0884155 + 0.996084i $$0.471820\pi$$
$$588$$ 0 0
$$589$$ 214720. 0.0255026
$$590$$ 1.28663e7 1.52168
$$591$$ 0 0
$$592$$ 6.17302e6 0.723925
$$593$$ −1.24007e7 −1.44813 −0.724067 0.689729i $$-0.757730\pi$$
−0.724067 + 0.689729i $$0.757730\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 328296. 0.0378573
$$597$$ 0 0
$$598$$ −1.11384e7 −1.27371
$$599$$ 3.69127e6 0.420348 0.210174 0.977664i $$-0.432597\pi$$
0.210174 + 0.977664i $$0.432597\pi$$
$$600$$ 0 0
$$601$$ −9.12223e6 −1.03018 −0.515092 0.857135i $$-0.672242\pi$$
−0.515092 + 0.857135i $$0.672242\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −1.14813e6 −0.128055
$$605$$ 2.81463e6 0.312632
$$606$$ 0 0
$$607$$ 5.67914e6 0.625620 0.312810 0.949816i $$-0.398730\pi$$
0.312810 + 0.949816i $$0.398730\pi$$
$$608$$ 3.86496e6 0.424020
$$609$$ 0 0
$$610$$ −2.31557e7 −2.51961
$$611$$ −6.15264e6 −0.666743
$$612$$ 0 0
$$613$$ −1.40106e7 −1.50593 −0.752966 0.658060i $$-0.771377\pi$$
−0.752966 + 0.658060i $$0.771377\pi$$
$$614$$ 6.27382e6 0.671600
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 253686. 0.0268277 0.0134139 0.999910i $$-0.495730\pi$$
0.0134139 + 0.999910i $$0.495730\pi$$
$$618$$ 0 0
$$619$$ −4.30034e6 −0.451103 −0.225552 0.974231i $$-0.572418\pi$$
−0.225552 + 0.974231i $$0.572418\pi$$
$$620$$ −24960.0 −0.00260775
$$621$$ 0 0
$$622$$ 1.10231e7 1.14242
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.02568e7 −1.05030
$$626$$ 2.19296e6 0.223664
$$627$$ 0 0
$$628$$ −519512. −0.0525650
$$629$$ 684684. 0.0690023
$$630$$ 0 0
$$631$$ 1.04150e7 1.04132 0.520662 0.853763i $$-0.325685\pi$$
0.520662 + 0.853763i $$0.325685\pi$$
$$632$$ 1.10504e7 1.10048
$$633$$ 0 0
$$634$$ 170028. 0.0167995
$$635$$ −5.79696e6 −0.570514
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −1.44975e7 −1.41007
$$639$$ 0 0
$$640$$ 1.65635e7 1.59846
$$641$$ −4.52714e6 −0.435190 −0.217595 0.976039i $$-0.569821\pi$$
−0.217595 + 0.976039i $$0.569821\pi$$
$$642$$ 0 0
$$643$$ −1.49687e7 −1.42776 −0.713882 0.700266i $$-0.753065\pi$$
−0.713882 + 0.700266i $$0.753065\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 2.02910e6 0.191304
$$647$$ −1.73020e7 −1.62493 −0.812465 0.583010i $$-0.801875\pi$$
−0.812465 + 0.583010i $$0.801875\pi$$
$$648$$ 0 0
$$649$$ −1.22064e7 −1.13757
$$650$$ 7.84727e6 0.728509
$$651$$ 0 0
$$652$$ 2.22114e6 0.204624
$$653$$ −4.07470e6 −0.373949 −0.186975 0.982365i $$-0.559868\pi$$
−0.186975 + 0.982365i $$0.559868\pi$$
$$654$$ 0 0
$$655$$ −1.21146e7 −1.10334
$$656$$ −9.04483e6 −0.820618
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 3.79475e6 0.340384 0.170192 0.985411i $$-0.445561\pi$$
0.170192 + 0.985411i $$0.445561\pi$$
$$660$$ 0 0
$$661$$ −1.64261e7 −1.46228 −0.731142 0.682225i $$-0.761012\pi$$
−0.731142 + 0.682225i $$0.761012\pi$$
$$662$$ 1.16035e7 1.02907
$$663$$ 0 0
$$664$$ −6.75158e6 −0.594272
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.28564e7 −1.98927
$$668$$ 174048. 0.0150913
$$669$$ 0 0
$$670$$ −2.77786e7 −2.39069
$$671$$ 2.19682e7 1.88360
$$672$$ 0 0
$$673$$ 5.50675e6 0.468660 0.234330 0.972157i $$-0.424710\pi$$
0.234330 + 0.972157i $$0.424710\pi$$
$$674$$ −1.13290e7 −0.960602
$$675$$ 0 0
$$676$$ −703716. −0.0592285
$$677$$ 1.83957e7 1.54257 0.771286 0.636488i $$-0.219614\pi$$
0.771286 + 0.636488i $$0.219614\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1.65110e6 0.136931
$$681$$ 0 0
$$682$$ 213120. 0.0175454
$$683$$ −1.75835e6 −0.144229 −0.0721146 0.997396i $$-0.522975\pi$$
−0.0721146 + 0.997396i $$0.522975\pi$$
$$684$$ 0 0
$$685$$ 2.06112e7 1.67833
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 1.30913e7 1.05441
$$689$$ 4.24055e6 0.340309
$$690$$ 0 0
$$691$$ 5.36314e6 0.427291 0.213646 0.976911i $$-0.431466\pi$$
0.213646 + 0.976911i $$0.431466\pi$$
$$692$$ −73320.0 −0.00582046
$$693$$ 0 0
$$694$$ −1.75162e7 −1.38052
$$695$$ −1.75197e7 −1.37583
$$696$$ 0 0
$$697$$ −1.00321e6 −0.0782187
$$698$$ 4.68409e6 0.363904
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.12606e7 1.63411 0.817054 0.576561i $$-0.195606\pi$$
0.817054 + 0.576561i $$0.195606\pi$$
$$702$$ 0 0
$$703$$ 1.45849e7 1.11305
$$704$$ −1.23041e7 −0.935662
$$705$$ 0 0
$$706$$ 8.00622e6 0.604527
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.07729e6 0.155196 0.0775980 0.996985i $$-0.475275\pi$$
0.0775980 + 0.996985i $$0.475275\pi$$
$$710$$ −1.49947e7 −1.11633
$$711$$ 0 0
$$712$$ 1.33963e6 0.0990343
$$713$$ 336000. 0.0247523
$$714$$ 0 0
$$715$$ −1.53073e7 −1.11979
$$716$$ 613296. 0.0447082
$$717$$ 0 0
$$718$$ −6.10459e6 −0.441922
$$719$$ 4.23619e6 0.305600 0.152800 0.988257i $$-0.451171\pi$$
0.152800 + 0.988257i $$0.451171\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2.83665e7 2.02518
$$723$$ 0 0
$$724$$ 1.52826e6 0.108356
$$725$$ 1.61029e7 1.13778
$$726$$ 0 0
$$727$$ −2.14524e7 −1.50536 −0.752678 0.658389i $$-0.771238\pi$$
−0.752678 + 0.658389i $$0.771238\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 2.89439e7 2.01025
$$731$$ 1.45202e6 0.100503
$$732$$ 0 0
$$733$$ 1.48892e7 1.02355 0.511777 0.859118i $$-0.328987\pi$$
0.511777 + 0.859118i $$0.328987\pi$$
$$734$$ −5.02608e6 −0.344341
$$735$$ 0 0
$$736$$ 6.04800e6 0.411545
$$737$$ 2.63541e7 1.78722
$$738$$ 0 0
$$739$$ 6.99324e6 0.471050 0.235525 0.971868i $$-0.424319\pi$$
0.235525 + 0.971868i $$0.424319\pi$$
$$740$$ −1.69541e6 −0.113814
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.90428e6 −0.126549 −0.0632745 0.997996i $$-0.520154\pi$$
−0.0632745 + 0.997996i $$0.520154\pi$$
$$744$$ 0 0
$$745$$ 6.40177e6 0.422581
$$746$$ −9.11958e6 −0.599968
$$747$$ 0 0
$$748$$ 223776. 0.0146238
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1.95361e7 1.26398 0.631988 0.774978i $$-0.282239\pi$$
0.631988 + 0.774978i $$0.282239\pi$$
$$752$$ 1.58131e7 1.01970
$$753$$ 0 0
$$754$$ 1.44322e7 0.924493
$$755$$ −2.23885e7 −1.42941
$$756$$ 0 0
$$757$$ 1.25183e6 0.0793973 0.0396986 0.999212i $$-0.487360\pi$$
0.0396986 + 0.999212i $$0.487360\pi$$
$$758$$ 1.58679e7 1.00311
$$759$$ 0 0
$$760$$ 3.51711e7 2.20878
$$761$$ 2.04472e7 1.27989 0.639944 0.768422i $$-0.278958\pi$$
0.639944 + 0.768422i $$0.278958\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 1.09363e6 0.0677857
$$765$$ 0 0
$$766$$ 1.20802e7 0.743876
$$767$$ 1.21515e7 0.745831
$$768$$ 0 0
$$769$$ −2.21064e6 −0.134804 −0.0674020 0.997726i $$-0.521471\pi$$
−0.0674020 + 0.997726i $$0.521471\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 614408. 0.0371034
$$773$$ 1.29151e7 0.777405 0.388703 0.921363i $$-0.372923\pi$$
0.388703 + 0.921363i $$0.372923\pi$$
$$774$$ 0 0
$$775$$ −236720. −0.0141573
$$776$$ −2.41352e7 −1.43879
$$777$$ 0 0
$$778$$ 4.35740e6 0.258095
$$779$$ −2.13700e7 −1.26171
$$780$$ 0 0
$$781$$ 1.42258e7 0.834541
$$782$$ 3.17520e6 0.185675
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −1.01305e7 −0.586754
$$786$$ 0 0
$$787$$ 1.35499e7 0.779830 0.389915 0.920851i $$-0.372504\pi$$
0.389915 + 0.920851i $$0.372504\pi$$
$$788$$ −617688. −0.0354367
$$789$$ 0 0
$$790$$ −3.07832e7 −1.75487
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.18693e7 −1.23496
$$794$$ −2.74547e7 −1.54549
$$795$$ 0 0
$$796$$ 1.46742e6 0.0820867
$$797$$ −2.45956e7 −1.37155 −0.685776 0.727813i $$-0.740537\pi$$
−0.685776 + 0.727813i $$0.740537\pi$$
$$798$$ 0 0
$$799$$ 1.75392e6 0.0971948
$$800$$ −4.26096e6 −0.235387
$$801$$ 0 0
$$802$$ 203220. 0.0111566
$$803$$ −2.74596e7 −1.50282
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −212160. −0.0115034
$$807$$ 0 0
$$808$$ 454608. 0.0244968
$$809$$ −1.55237e7 −0.833920 −0.416960 0.908925i $$-0.636905\pi$$
−0.416960 + 0.908925i $$0.636905\pi$$
$$810$$ 0 0
$$811$$ 2.66262e7 1.42153 0.710766 0.703429i $$-0.248349\pi$$
0.710766 + 0.703429i $$0.248349\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 1.44762e7 0.765760
$$815$$ 4.33122e7 2.28410
$$816$$ 0 0
$$817$$ 3.09304e7 1.62118
$$818$$ 3.51707e7 1.83780
$$819$$ 0 0
$$820$$ 2.48414e6 0.129016
$$821$$ 1.23891e7 0.641477 0.320739 0.947168i $$-0.396069\pi$$
0.320739 + 0.947168i $$0.396069\pi$$
$$822$$ 0 0
$$823$$ −3.65630e6 −0.188166 −0.0940831 0.995564i $$-0.529992\pi$$
−0.0940831 + 0.995564i $$0.529992\pi$$
$$824$$ 2.21370e7 1.13580
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.80463e7 −1.42597 −0.712987 0.701178i $$-0.752658\pi$$
−0.712987 + 0.701178i $$0.752658\pi$$
$$828$$ 0 0
$$829$$ −2.11153e7 −1.06712 −0.533558 0.845763i $$-0.679145\pi$$
−0.533558 + 0.845763i $$0.679145\pi$$
$$830$$ 1.88080e7 0.947648
$$831$$ 0 0
$$832$$ 1.22487e7 0.613454
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3.39394e6 0.168456
$$836$$ 4.76678e6 0.235890
$$837$$ 0 0
$$838$$ 1.81649e6 0.0893557
$$839$$ 1.33947e7 0.656944 0.328472 0.944514i $$-0.393466\pi$$
0.328472 + 0.944514i $$0.393466\pi$$
$$840$$ 0 0
$$841$$ 9.10422e6 0.443867
$$842$$ −3.22025e7 −1.56534
$$843$$ 0 0
$$844$$ 2.08098e6 0.100557
$$845$$ −1.37225e7 −0.661135
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −1.08988e7 −0.520461
$$849$$ 0 0
$$850$$ −2.23700e6 −0.106199
$$851$$ 2.28228e7 1.08030
$$852$$ 0 0
$$853$$ −3.01513e7 −1.41884 −0.709420 0.704786i $$-0.751043\pi$$
−0.709420 + 0.704786i $$0.751043\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −2.16579e7 −1.01026
$$857$$ 2.39894e7 1.11575 0.557875 0.829925i $$-0.311617\pi$$
0.557875 + 0.829925i $$0.311617\pi$$
$$858$$ 0 0
$$859$$ 8.87576e6 0.410414 0.205207 0.978719i $$-0.434213\pi$$
0.205207 + 0.978719i $$0.434213\pi$$
$$860$$ −3.59549e6 −0.165772
$$861$$ 0 0
$$862$$ −7.06234e6 −0.323728
$$863$$ 8.71286e6 0.398230 0.199115 0.979976i $$-0.436193\pi$$
0.199115 + 0.979976i $$0.436193\pi$$
$$864$$ 0 0
$$865$$ −1.42974e6 −0.0649706
$$866$$ 2.19750e7 0.995711
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2.92045e7 1.31190
$$870$$ 0 0
$$871$$ −2.62354e7 −1.17177
$$872$$ 1.69643e7 0.755518
$$873$$ 0 0
$$874$$ 6.76368e7 2.99505
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.95788e7 −1.29862 −0.649310 0.760524i $$-0.724942\pi$$
−0.649310 + 0.760524i $$0.724942\pi$$
$$878$$ 1.52205e7 0.666333
$$879$$ 0 0
$$880$$ 3.93420e7 1.71257
$$881$$ 2.45670e7 1.06638 0.533190 0.845995i $$-0.320993\pi$$
0.533190 + 0.845995i $$0.320993\pi$$
$$882$$ 0 0
$$883$$ 1.45682e7 0.628788 0.314394 0.949293i $$-0.398199\pi$$
0.314394 + 0.949293i $$0.398199\pi$$
$$884$$ −222768. −0.00958787
$$885$$ 0 0
$$886$$ −3.60902e7 −1.54456
$$887$$ 1.61714e7 0.690141 0.345070 0.938577i $$-0.387855\pi$$
0.345070 + 0.938577i $$0.387855\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −3.73183e6 −0.157924
$$891$$ 0 0
$$892$$ −1.21894e6 −0.0512946
$$893$$ 3.73613e7 1.56781
$$894$$ 0 0
$$895$$ 1.19593e7 0.499054
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −3.39579e7 −1.40524
$$899$$ −435360. −0.0179659
$$900$$ 0 0
$$901$$ −1.20884e6 −0.0496087
$$902$$ −2.12108e7 −0.868041
$$903$$ 0 0
$$904$$ 3.69845e7 1.50522
$$905$$ 2.98011e7 1.20952
$$906$$ 0 0
$$907$$ 3.14446e7 1.26919 0.634596 0.772844i $$-0.281167\pi$$
0.634596 + 0.772844i $$0.281167\pi$$
$$908$$ 1.15435e6 0.0464648
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1.51427e7 −0.604514 −0.302257 0.953227i $$-0.597740\pi$$
−0.302257 + 0.953227i $$0.597740\pi$$
$$912$$ 0 0
$$913$$ −1.78435e7 −0.708439
$$914$$ −3.87695e7 −1.53506
$$915$$ 0 0
$$916$$ −3.08876e6 −0.121631
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 4.14876e7 1.62043 0.810214 0.586134i $$-0.199351\pi$$
0.810214 + 0.586134i $$0.199351\pi$$
$$920$$ 5.50368e7 2.14380
$$921$$ 0 0
$$922$$ −2.02412e7 −0.784167
$$923$$ −1.41617e7 −0.547155
$$924$$ 0 0
$$925$$ −1.60792e7 −0.617889
$$926$$ −2.72985e7 −1.04619
$$927$$ 0 0
$$928$$ −7.83648e6 −0.298711
$$929$$ −1.78495e7 −0.678556 −0.339278 0.940686i $$-0.610183\pi$$
−0.339278 + 0.940686i $$0.610183\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −1.00894e6 −0.0380473
$$933$$ 0 0
$$934$$ 1.20681e7 0.452661
$$935$$ 4.36363e6 0.163237
$$936$$ 0 0
$$937$$ −2.96399e7 −1.10288 −0.551439 0.834215i $$-0.685921\pi$$
−0.551439 + 0.834215i $$0.685921\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −4.34304e6 −0.160315
$$941$$ −3.22282e7 −1.18648 −0.593242 0.805024i $$-0.702152\pi$$
−0.593242 + 0.805024i $$0.702152\pi$$
$$942$$ 0 0
$$943$$ −3.34404e7 −1.22459
$$944$$ −3.12309e7 −1.14066
$$945$$ 0 0
$$946$$ 3.06999e7 1.11535
$$947$$ −4.84885e7 −1.75697 −0.878484 0.477772i $$-0.841444\pi$$
−0.878484 + 0.477772i $$0.841444\pi$$
$$948$$ 0 0
$$949$$ 2.73359e7 0.985300
$$950$$ −4.76517e7 −1.71305
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.03264e7 0.724983 0.362491 0.931987i $$-0.381926\pi$$
0.362491 + 0.931987i $$0.381926\pi$$
$$954$$ 0 0
$$955$$ 2.13258e7 0.756654
$$956$$ 5.80454e6 0.205411
$$957$$ 0 0
$$958$$ −4.56241e7 −1.60613
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.86228e7 −0.999776
$$962$$ −1.44110e7 −0.502060
$$963$$ 0 0
$$964$$ 585592. 0.0202956
$$965$$ 1.19810e7 0.414165
$$966$$ 0 0
$$967$$ −3.66292e6 −0.125968 −0.0629841 0.998015i $$-0.520062\pi$$
−0.0629841 + 0.998015i $$0.520062\pi$$
$$968$$ −6.06228e6 −0.207945
$$969$$ 0 0
$$970$$ 6.72338e7 2.29434
$$971$$ 1.48741e6 0.0506271 0.0253136 0.999680i $$-0.491942\pi$$
0.0253136 + 0.999680i $$0.491942\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 4.03867e6 0.136408
$$975$$ 0 0
$$976$$ 5.62070e7 1.88871
$$977$$ −4.07930e7 −1.36725 −0.683627 0.729831i $$-0.739599\pi$$
−0.683627 + 0.729831i $$0.739599\pi$$
$$978$$ 0 0
$$979$$ 3.54046e6 0.118060
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1.48302e7 0.490759
$$983$$ −9.26326e6 −0.305759 −0.152880 0.988245i $$-0.548855\pi$$
−0.152880 + 0.988245i $$0.548855\pi$$
$$984$$ 0 0
$$985$$ −1.20449e7 −0.395561
$$986$$ −4.11415e6 −0.134768
$$987$$ 0 0
$$988$$ −4.74531e6 −0.154658
$$989$$ 4.84008e7 1.57348
$$990$$ 0 0
$$991$$ −5.22051e7 −1.68861 −0.844303 0.535866i $$-0.819985\pi$$
−0.844303 + 0.535866i $$0.819985\pi$$
$$992$$ 115200. 0.00371684
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.86148e7 0.916289
$$996$$ 0 0
$$997$$ 1.86609e7 0.594560 0.297280 0.954790i $$-0.403921\pi$$
0.297280 + 0.954790i $$0.403921\pi$$
$$998$$ 3.64891e7 1.15968
$$999$$ 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 441.6.a.j.1.1 1
3.2 odd 2 147.6.a.b.1.1 1
7.6 odd 2 63.6.a.d.1.1 1
21.2 odd 6 147.6.e.i.67.1 2
21.5 even 6 147.6.e.j.67.1 2
21.11 odd 6 147.6.e.i.79.1 2
21.17 even 6 147.6.e.j.79.1 2
21.20 even 2 21.6.a.a.1.1 1
28.27 even 2 1008.6.a.c.1.1 1
84.83 odd 2 336.6.a.r.1.1 1
105.62 odd 4 525.6.d.b.274.1 2
105.83 odd 4 525.6.d.b.274.2 2
105.104 even 2 525.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.a.1.1 1 21.20 even 2
63.6.a.d.1.1 1 7.6 odd 2
147.6.a.b.1.1 1 3.2 odd 2
147.6.e.i.67.1 2 21.2 odd 6
147.6.e.i.79.1 2 21.11 odd 6
147.6.e.j.67.1 2 21.5 even 6
147.6.e.j.79.1 2 21.17 even 6
336.6.a.r.1.1 1 84.83 odd 2
441.6.a.j.1.1 1 1.1 even 1 trivial
525.6.a.d.1.1 1 105.104 even 2
525.6.d.b.274.1 2 105.62 odd 4
525.6.d.b.274.2 2 105.83 odd 4
1008.6.a.c.1.1 1 28.27 even 2