# Properties

 Label 21.6.a.a.1.1 Level $21$ Weight $6$ Character 21.1 Self dual yes Analytic conductor $3.368$ 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] = [21,6,Mod(1,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, 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("21.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 21.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: $$3.36806021607$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 21.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +78.0000 q^{5} +54.0000 q^{6} +49.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +78.0000 q^{5} +54.0000 q^{6} +49.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} -468.000 q^{10} +444.000 q^{11} -36.0000 q^{12} -442.000 q^{13} -294.000 q^{14} -702.000 q^{15} -1136.00 q^{16} -126.000 q^{17} -486.000 q^{18} +2684.00 q^{19} +312.000 q^{20} -441.000 q^{21} -2664.00 q^{22} +4200.00 q^{23} -1512.00 q^{24} +2959.00 q^{25} +2652.00 q^{26} -729.000 q^{27} +196.000 q^{28} -5442.00 q^{29} +4212.00 q^{30} +80.0000 q^{31} +1440.00 q^{32} -3996.00 q^{33} +756.000 q^{34} +3822.00 q^{35} +324.000 q^{36} -5434.00 q^{37} -16104.0 q^{38} +3978.00 q^{39} +13104.0 q^{40} +7962.00 q^{41} +2646.00 q^{42} -11524.0 q^{43} +1776.00 q^{44} +6318.00 q^{45} -25200.0 q^{46} -13920.0 q^{47} +10224.0 q^{48} +2401.00 q^{49} -17754.0 q^{50} +1134.00 q^{51} -1768.00 q^{52} -9594.00 q^{53} +4374.00 q^{54} +34632.0 q^{55} +8232.00 q^{56} -24156.0 q^{57} +32652.0 q^{58} +27492.0 q^{59} -2808.00 q^{60} +49478.0 q^{61} -480.000 q^{62} +3969.00 q^{63} +27712.0 q^{64} -34476.0 q^{65} +23976.0 q^{66} -59356.0 q^{67} -504.000 q^{68} -37800.0 q^{69} -22932.0 q^{70} +32040.0 q^{71} +13608.0 q^{72} -61846.0 q^{73} +32604.0 q^{74} -26631.0 q^{75} +10736.0 q^{76} +21756.0 q^{77} -23868.0 q^{78} -65776.0 q^{79} -88608.0 q^{80} +6561.00 q^{81} -47772.0 q^{82} +40188.0 q^{83} -1764.00 q^{84} -9828.00 q^{85} +69144.0 q^{86} +48978.0 q^{87} +74592.0 q^{88} -7974.00 q^{89} -37908.0 q^{90} -21658.0 q^{91} +16800.0 q^{92} -720.000 q^{93} +83520.0 q^{94} +209352. q^{95} -12960.0 q^{96} -143662. q^{97} -14406.0 q^{98} +35964.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −6.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 4.00000 0.125000
$$5$$ 78.0000 1.39531 0.697653 0.716436i $$-0.254228\pi$$
0.697653 + 0.716436i $$0.254228\pi$$
$$6$$ 54.0000 0.612372
$$7$$ 49.0000 0.377964
$$8$$ 168.000 0.928078
$$9$$ 81.0000 0.333333
$$10$$ −468.000 −1.47995
$$11$$ 444.000 1.10637 0.553186 0.833058i $$-0.313412\pi$$
0.553186 + 0.833058i $$0.313412\pi$$
$$12$$ −36.0000 −0.0721688
$$13$$ −442.000 −0.725377 −0.362689 0.931910i $$-0.618141\pi$$
−0.362689 + 0.931910i $$0.618141\pi$$
$$14$$ −294.000 −0.400892
$$15$$ −702.000 −0.805581
$$16$$ −1136.00 −1.10938
$$17$$ −126.000 −0.105742 −0.0528711 0.998601i $$-0.516837\pi$$
−0.0528711 + 0.998601i $$0.516837\pi$$
$$18$$ −486.000 −0.353553
$$19$$ 2684.00 1.70568 0.852842 0.522169i $$-0.174877\pi$$
0.852842 + 0.522169i $$0.174877\pi$$
$$20$$ 312.000 0.174413
$$21$$ −441.000 −0.218218
$$22$$ −2664.00 −1.17348
$$23$$ 4200.00 1.65550 0.827751 0.561096i $$-0.189620\pi$$
0.827751 + 0.561096i $$0.189620\pi$$
$$24$$ −1512.00 −0.535826
$$25$$ 2959.00 0.946880
$$26$$ 2652.00 0.769379
$$27$$ −729.000 −0.192450
$$28$$ 196.000 0.0472456
$$29$$ −5442.00 −1.20161 −0.600805 0.799396i $$-0.705153\pi$$
−0.600805 + 0.799396i $$0.705153\pi$$
$$30$$ 4212.00 0.854447
$$31$$ 80.0000 0.0149515 0.00747577 0.999972i $$-0.497620\pi$$
0.00747577 + 0.999972i $$0.497620\pi$$
$$32$$ 1440.00 0.248592
$$33$$ −3996.00 −0.638764
$$34$$ 756.000 0.112157
$$35$$ 3822.00 0.527376
$$36$$ 324.000 0.0416667
$$37$$ −5434.00 −0.652552 −0.326276 0.945274i $$-0.605794\pi$$
−0.326276 + 0.945274i $$0.605794\pi$$
$$38$$ −16104.0 −1.80915
$$39$$ 3978.00 0.418797
$$40$$ 13104.0 1.29495
$$41$$ 7962.00 0.739712 0.369856 0.929089i $$-0.379407\pi$$
0.369856 + 0.929089i $$0.379407\pi$$
$$42$$ 2646.00 0.231455
$$43$$ −11524.0 −0.950456 −0.475228 0.879863i $$-0.657634\pi$$
−0.475228 + 0.879863i $$0.657634\pi$$
$$44$$ 1776.00 0.138297
$$45$$ 6318.00 0.465102
$$46$$ −25200.0 −1.75592
$$47$$ −13920.0 −0.919167 −0.459584 0.888134i $$-0.652001\pi$$
−0.459584 + 0.888134i $$0.652001\pi$$
$$48$$ 10224.0 0.640498
$$49$$ 2401.00 0.142857
$$50$$ −17754.0 −1.00432
$$51$$ 1134.00 0.0610503
$$52$$ −1768.00 −0.0906721
$$53$$ −9594.00 −0.469148 −0.234574 0.972098i $$-0.575370\pi$$
−0.234574 + 0.972098i $$0.575370\pi$$
$$54$$ 4374.00 0.204124
$$55$$ 34632.0 1.54373
$$56$$ 8232.00 0.350780
$$57$$ −24156.0 −0.984777
$$58$$ 32652.0 1.27450
$$59$$ 27492.0 1.02820 0.514098 0.857731i $$-0.328127\pi$$
0.514098 + 0.857731i $$0.328127\pi$$
$$60$$ −2808.00 −0.100698
$$61$$ 49478.0 1.70250 0.851251 0.524759i $$-0.175845\pi$$
0.851251 + 0.524759i $$0.175845\pi$$
$$62$$ −480.000 −0.0158585
$$63$$ 3969.00 0.125988
$$64$$ 27712.0 0.845703
$$65$$ −34476.0 −1.01212
$$66$$ 23976.0 0.677512
$$67$$ −59356.0 −1.61539 −0.807695 0.589600i $$-0.799285\pi$$
−0.807695 + 0.589600i $$0.799285\pi$$
$$68$$ −504.000 −0.0132178
$$69$$ −37800.0 −0.955805
$$70$$ −22932.0 −0.559367
$$71$$ 32040.0 0.754304 0.377152 0.926151i $$-0.376903\pi$$
0.377152 + 0.926151i $$0.376903\pi$$
$$72$$ 13608.0 0.309359
$$73$$ −61846.0 −1.35833 −0.679164 0.733987i $$-0.737657\pi$$
−0.679164 + 0.733987i $$0.737657\pi$$
$$74$$ 32604.0 0.692136
$$75$$ −26631.0 −0.546681
$$76$$ 10736.0 0.213210
$$77$$ 21756.0 0.418169
$$78$$ −23868.0 −0.444201
$$79$$ −65776.0 −1.18577 −0.592884 0.805288i $$-0.702011\pi$$
−0.592884 + 0.805288i $$0.702011\pi$$
$$80$$ −88608.0 −1.54792
$$81$$ 6561.00 0.111111
$$82$$ −47772.0 −0.784583
$$83$$ 40188.0 0.640326 0.320163 0.947362i $$-0.396262\pi$$
0.320163 + 0.947362i $$0.396262\pi$$
$$84$$ −1764.00 −0.0272772
$$85$$ −9828.00 −0.147543
$$86$$ 69144.0 1.00811
$$87$$ 48978.0 0.693750
$$88$$ 74592.0 1.02680
$$89$$ −7974.00 −0.106709 −0.0533545 0.998576i $$-0.516991\pi$$
−0.0533545 + 0.998576i $$0.516991\pi$$
$$90$$ −37908.0 −0.493315
$$91$$ −21658.0 −0.274167
$$92$$ 16800.0 0.206938
$$93$$ −720.000 −0.00863227
$$94$$ 83520.0 0.974924
$$95$$ 209352. 2.37995
$$96$$ −12960.0 −0.143525
$$97$$ −143662. −1.55029 −0.775144 0.631784i $$-0.782323\pi$$
−0.775144 + 0.631784i $$0.782323\pi$$
$$98$$ −14406.0 −0.151523
$$99$$ 35964.0 0.368791
$$100$$ 11836.0 0.118360
$$101$$ −2706.00 −0.0263952 −0.0131976 0.999913i $$-0.504201\pi$$
−0.0131976 + 0.999913i $$0.504201\pi$$
$$102$$ −6804.00 −0.0647536
$$103$$ 131768. 1.22382 0.611909 0.790928i $$-0.290402\pi$$
0.611909 + 0.790928i $$0.290402\pi$$
$$104$$ −74256.0 −0.673206
$$105$$ −34398.0 −0.304481
$$106$$ 57564.0 0.497607
$$107$$ −128916. −1.08855 −0.544274 0.838908i $$-0.683195\pi$$
−0.544274 + 0.838908i $$0.683195\pi$$
$$108$$ −2916.00 −0.0240563
$$109$$ −100978. −0.814068 −0.407034 0.913413i $$-0.633437\pi$$
−0.407034 + 0.913413i $$0.633437\pi$$
$$110$$ −207792. −1.63737
$$111$$ 48906.0 0.376751
$$112$$ −55664.0 −0.419304
$$113$$ 220146. 1.62186 0.810932 0.585140i $$-0.198960\pi$$
0.810932 + 0.585140i $$0.198960\pi$$
$$114$$ 144936. 1.04451
$$115$$ 327600. 2.30993
$$116$$ −21768.0 −0.150201
$$117$$ −35802.0 −0.241792
$$118$$ −164952. −1.09057
$$119$$ −6174.00 −0.0399668
$$120$$ −117936. −0.747641
$$121$$ 36085.0 0.224059
$$122$$ −296868. −1.80578
$$123$$ −71658.0 −0.427073
$$124$$ 320.000 0.00186894
$$125$$ −12948.0 −0.0741187
$$126$$ −23814.0 −0.133631
$$127$$ −74320.0 −0.408880 −0.204440 0.978879i $$-0.565537\pi$$
−0.204440 + 0.978879i $$0.565537\pi$$
$$128$$ −212352. −1.14560
$$129$$ 103716. 0.548746
$$130$$ 206856. 1.07352
$$131$$ −155316. −0.790748 −0.395374 0.918520i $$-0.629385\pi$$
−0.395374 + 0.918520i $$0.629385\pi$$
$$132$$ −15984.0 −0.0798455
$$133$$ 131516. 0.644688
$$134$$ 356136. 1.71338
$$135$$ −56862.0 −0.268527
$$136$$ −21168.0 −0.0981369
$$137$$ −264246. −1.20284 −0.601419 0.798934i $$-0.705398\pi$$
−0.601419 + 0.798934i $$0.705398\pi$$
$$138$$ 226800. 1.01378
$$139$$ 224612. 0.986043 0.493022 0.870017i $$-0.335892\pi$$
0.493022 + 0.870017i $$0.335892\pi$$
$$140$$ 15288.0 0.0659220
$$141$$ 125280. 0.530682
$$142$$ −192240. −0.800061
$$143$$ −196248. −0.802537
$$144$$ −92016.0 −0.369792
$$145$$ −424476. −1.67661
$$146$$ 371076. 1.44072
$$147$$ −21609.0 −0.0824786
$$148$$ −21736.0 −0.0815690
$$149$$ −82074.0 −0.302859 −0.151429 0.988468i $$-0.548388\pi$$
−0.151429 + 0.988468i $$0.548388\pi$$
$$150$$ 159786. 0.579843
$$151$$ −287032. −1.02444 −0.512222 0.858853i $$-0.671177\pi$$
−0.512222 + 0.858853i $$0.671177\pi$$
$$152$$ 450912. 1.58301
$$153$$ −10206.0 −0.0352474
$$154$$ −130536. −0.443536
$$155$$ 6240.00 0.0208620
$$156$$ 15912.0 0.0523496
$$157$$ 129878. 0.420520 0.210260 0.977646i $$-0.432569\pi$$
0.210260 + 0.977646i $$0.432569\pi$$
$$158$$ 394656. 1.25770
$$159$$ 86346.0 0.270863
$$160$$ 112320. 0.346862
$$161$$ 205800. 0.625721
$$162$$ −39366.0 −0.117851
$$163$$ 555284. 1.63699 0.818495 0.574513i $$-0.194809\pi$$
0.818495 + 0.574513i $$0.194809\pi$$
$$164$$ 31848.0 0.0924640
$$165$$ −311688. −0.891272
$$166$$ −241128. −0.679168
$$167$$ 43512.0 0.120731 0.0603654 0.998176i $$-0.480773\pi$$
0.0603654 + 0.998176i $$0.480773\pi$$
$$168$$ −74088.0 −0.202523
$$169$$ −175929. −0.473828
$$170$$ 58968.0 0.156493
$$171$$ 217404. 0.568561
$$172$$ −46096.0 −0.118807
$$173$$ −18330.0 −0.0465637 −0.0232818 0.999729i $$-0.507412\pi$$
−0.0232818 + 0.999729i $$0.507412\pi$$
$$174$$ −293868. −0.735833
$$175$$ 144991. 0.357887
$$176$$ −504384. −1.22738
$$177$$ −247428. −0.593630
$$178$$ 47844.0 0.113182
$$179$$ −153324. −0.357666 −0.178833 0.983879i $$-0.557232\pi$$
−0.178833 + 0.983879i $$0.557232\pi$$
$$180$$ 25272.0 0.0581378
$$181$$ −382066. −0.866846 −0.433423 0.901191i $$-0.642694\pi$$
−0.433423 + 0.901191i $$0.642694\pi$$
$$182$$ 129948. 0.290798
$$183$$ −445302. −0.982940
$$184$$ 705600. 1.53643
$$185$$ −423852. −0.910510
$$186$$ 4320.00 0.00915591
$$187$$ −55944.0 −0.116990
$$188$$ −55680.0 −0.114896
$$189$$ −35721.0 −0.0727393
$$190$$ −1.25611e6 −2.52432
$$191$$ −273408. −0.542285 −0.271143 0.962539i $$-0.587402\pi$$
−0.271143 + 0.962539i $$0.587402\pi$$
$$192$$ −249408. −0.488267
$$193$$ 153602. 0.296827 0.148414 0.988925i $$-0.452583\pi$$
0.148414 + 0.988925i $$0.452583\pi$$
$$194$$ 861972. 1.64433
$$195$$ 310284. 0.584350
$$196$$ 9604.00 0.0178571
$$197$$ 154422. 0.283494 0.141747 0.989903i $$-0.454728\pi$$
0.141747 + 0.989903i $$0.454728\pi$$
$$198$$ −215784. −0.391162
$$199$$ −366856. −0.656694 −0.328347 0.944557i $$-0.606492\pi$$
−0.328347 + 0.944557i $$0.606492\pi$$
$$200$$ 497112. 0.878778
$$201$$ 534204. 0.932646
$$202$$ 16236.0 0.0279963
$$203$$ −266658. −0.454166
$$204$$ 4536.00 0.00763128
$$205$$ 621036. 1.03212
$$206$$ −790608. −1.29806
$$207$$ 340200. 0.551834
$$208$$ 502112. 0.804715
$$209$$ 1.19170e6 1.88712
$$210$$ 206388. 0.322951
$$211$$ 520244. 0.804453 0.402227 0.915540i $$-0.368236\pi$$
0.402227 + 0.915540i $$0.368236\pi$$
$$212$$ −38376.0 −0.0586435
$$213$$ −288360. −0.435498
$$214$$ 773496. 1.15458
$$215$$ −898872. −1.32618
$$216$$ −122472. −0.178609
$$217$$ 3920.00 0.00565115
$$218$$ 605868. 0.863449
$$219$$ 556614. 0.784231
$$220$$ 138528. 0.192966
$$221$$ 55692.0 0.0767030
$$222$$ −293436. −0.399605
$$223$$ 304736. 0.410357 0.205178 0.978725i $$-0.434223\pi$$
0.205178 + 0.978725i $$0.434223\pi$$
$$224$$ 70560.0 0.0939590
$$225$$ 239679. 0.315627
$$226$$ −1.32088e6 −1.72025
$$227$$ 288588. 0.371718 0.185859 0.982576i $$-0.440493\pi$$
0.185859 + 0.982576i $$0.440493\pi$$
$$228$$ −96624.0 −0.123097
$$229$$ 772190. 0.973051 0.486525 0.873666i $$-0.338264\pi$$
0.486525 + 0.873666i $$0.338264\pi$$
$$230$$ −1.96560e6 −2.45005
$$231$$ −195804. −0.241430
$$232$$ −914256. −1.11519
$$233$$ 252234. 0.304378 0.152189 0.988351i $$-0.451368\pi$$
0.152189 + 0.988351i $$0.451368\pi$$
$$234$$ 214812. 0.256460
$$235$$ −1.08576e6 −1.28252
$$236$$ 109968. 0.128525
$$237$$ 591984. 0.684603
$$238$$ 37044.0 0.0423912
$$239$$ −1.45114e6 −1.64329 −0.821643 0.570002i $$-0.806942\pi$$
−0.821643 + 0.570002i $$0.806942\pi$$
$$240$$ 797472. 0.893691
$$241$$ −146398. −0.162365 −0.0811825 0.996699i $$-0.525870\pi$$
−0.0811825 + 0.996699i $$0.525870\pi$$
$$242$$ −216510. −0.237651
$$243$$ −59049.0 −0.0641500
$$244$$ 197912. 0.212813
$$245$$ 187278. 0.199329
$$246$$ 429948. 0.452979
$$247$$ −1.18633e6 −1.23726
$$248$$ 13440.0 0.0138762
$$249$$ −361692. −0.369692
$$250$$ 77688.0 0.0786147
$$251$$ 607860. 0.609003 0.304501 0.952512i $$-0.401510\pi$$
0.304501 + 0.952512i $$0.401510\pi$$
$$252$$ 15876.0 0.0157485
$$253$$ 1.86480e6 1.83160
$$254$$ 445920. 0.433683
$$255$$ 88452.0 0.0851838
$$256$$ 387328. 0.369385
$$257$$ 95586.0 0.0902737 0.0451369 0.998981i $$-0.485628\pi$$
0.0451369 + 0.998981i $$0.485628\pi$$
$$258$$ −622296. −0.582033
$$259$$ −266266. −0.246642
$$260$$ −137904. −0.126515
$$261$$ −440802. −0.400537
$$262$$ 931896. 0.838715
$$263$$ −2.20034e6 −1.96156 −0.980779 0.195121i $$-0.937490\pi$$
−0.980779 + 0.195121i $$0.937490\pi$$
$$264$$ −671328. −0.592823
$$265$$ −748332. −0.654605
$$266$$ −789096. −0.683795
$$267$$ 71766.0 0.0616085
$$268$$ −237424. −0.201924
$$269$$ 1.77025e6 1.49160 0.745801 0.666169i $$-0.232067\pi$$
0.745801 + 0.666169i $$0.232067\pi$$
$$270$$ 341172. 0.284816
$$271$$ −223504. −0.184868 −0.0924341 0.995719i $$-0.529465\pi$$
−0.0924341 + 0.995719i $$0.529465\pi$$
$$272$$ 143136. 0.117308
$$273$$ 194922. 0.158290
$$274$$ 1.58548e6 1.27580
$$275$$ 1.31380e6 1.04760
$$276$$ −151200. −0.119476
$$277$$ −342778. −0.268419 −0.134210 0.990953i $$-0.542850\pi$$
−0.134210 + 0.990953i $$0.542850\pi$$
$$278$$ −1.34767e6 −1.04586
$$279$$ 6480.00 0.00498384
$$280$$ 642096. 0.489446
$$281$$ 480378. 0.362925 0.181463 0.983398i $$-0.441917\pi$$
0.181463 + 0.983398i $$0.441917\pi$$
$$282$$ −751680. −0.562873
$$283$$ −29980.0 −0.0222518 −0.0111259 0.999938i $$-0.503542\pi$$
−0.0111259 + 0.999938i $$0.503542\pi$$
$$284$$ 128160. 0.0942880
$$285$$ −1.88417e6 −1.37407
$$286$$ 1.17749e6 0.851219
$$287$$ 390138. 0.279585
$$288$$ 116640. 0.0828641
$$289$$ −1.40398e6 −0.988819
$$290$$ 2.54686e6 1.77832
$$291$$ 1.29296e6 0.895060
$$292$$ −247384. −0.169791
$$293$$ −198066. −0.134785 −0.0673924 0.997727i $$-0.521468\pi$$
−0.0673924 + 0.997727i $$0.521468\pi$$
$$294$$ 129654. 0.0874818
$$295$$ 2.14438e6 1.43465
$$296$$ −912912. −0.605619
$$297$$ −323676. −0.212921
$$298$$ 492444. 0.321230
$$299$$ −1.85640e6 −1.20086
$$300$$ −106524. −0.0683352
$$301$$ −564676. −0.359239
$$302$$ 1.72219e6 1.08659
$$303$$ 24354.0 0.0152393
$$304$$ −3.04902e6 −1.89224
$$305$$ 3.85928e6 2.37551
$$306$$ 61236.0 0.0373855
$$307$$ −1.04564e6 −0.633191 −0.316595 0.948561i $$-0.602540\pi$$
−0.316595 + 0.948561i $$0.602540\pi$$
$$308$$ 87024.0 0.0522712
$$309$$ −1.18591e6 −0.706572
$$310$$ −37440.0 −0.0221275
$$311$$ 1.83718e6 1.07708 0.538542 0.842598i $$-0.318975\pi$$
0.538542 + 0.842598i $$0.318975\pi$$
$$312$$ 668304. 0.388676
$$313$$ −365494. −0.210872 −0.105436 0.994426i $$-0.533624\pi$$
−0.105436 + 0.994426i $$0.533624\pi$$
$$314$$ −779268. −0.446029
$$315$$ 309582. 0.175792
$$316$$ −263104. −0.148221
$$317$$ −28338.0 −0.0158388 −0.00791938 0.999969i $$-0.502521\pi$$
−0.00791938 + 0.999969i $$0.502521\pi$$
$$318$$ −518076. −0.287293
$$319$$ −2.41625e6 −1.32943
$$320$$ 2.16154e6 1.18001
$$321$$ 1.16024e6 0.628473
$$322$$ −1.23480e6 −0.663677
$$323$$ −338184. −0.180363
$$324$$ 26244.0 0.0138889
$$325$$ −1.30788e6 −0.686845
$$326$$ −3.33170e6 −1.73629
$$327$$ 908802. 0.470002
$$328$$ 1.33762e6 0.686510
$$329$$ −682080. −0.347413
$$330$$ 1.87013e6 0.945337
$$331$$ 1.93392e6 0.970214 0.485107 0.874455i $$-0.338781\pi$$
0.485107 + 0.874455i $$0.338781\pi$$
$$332$$ 160752. 0.0800408
$$333$$ −440154. −0.217517
$$334$$ −261072. −0.128054
$$335$$ −4.62977e6 −2.25397
$$336$$ 500976. 0.242085
$$337$$ −1.88817e6 −0.905664 −0.452832 0.891596i $$-0.649586\pi$$
−0.452832 + 0.891596i $$0.649586\pi$$
$$338$$ 1.05557e6 0.502570
$$339$$ −1.98131e6 −0.936384
$$340$$ −39312.0 −0.0184428
$$341$$ 35520.0 0.0165420
$$342$$ −1.30442e6 −0.603050
$$343$$ 117649. 0.0539949
$$344$$ −1.93603e6 −0.882097
$$345$$ −2.94840e6 −1.33364
$$346$$ 109980. 0.0493882
$$347$$ 2.91937e6 1.30156 0.650782 0.759264i $$-0.274441\pi$$
0.650782 + 0.759264i $$0.274441\pi$$
$$348$$ 195912. 0.0867187
$$349$$ −780682. −0.343092 −0.171546 0.985176i $$-0.554876\pi$$
−0.171546 + 0.985176i $$0.554876\pi$$
$$350$$ −869946. −0.379596
$$351$$ 322218. 0.139599
$$352$$ 639360. 0.275036
$$353$$ 1.33437e6 0.569954 0.284977 0.958534i $$-0.408014\pi$$
0.284977 + 0.958534i $$0.408014\pi$$
$$354$$ 1.48457e6 0.629639
$$355$$ 2.49912e6 1.05249
$$356$$ −31896.0 −0.0133386
$$357$$ 55566.0 0.0230748
$$358$$ 919944. 0.379362
$$359$$ 1.01743e6 0.416648 0.208324 0.978060i $$-0.433199\pi$$
0.208324 + 0.978060i $$0.433199\pi$$
$$360$$ 1.06142e6 0.431651
$$361$$ 4.72776e6 1.90936
$$362$$ 2.29240e6 0.919429
$$363$$ −324765. −0.129361
$$364$$ −86632.0 −0.0342709
$$365$$ −4.82399e6 −1.89528
$$366$$ 2.67181e6 1.04257
$$367$$ 837680. 0.324648 0.162324 0.986737i $$-0.448101\pi$$
0.162324 + 0.986737i $$0.448101\pi$$
$$368$$ −4.77120e6 −1.83657
$$369$$ 644922. 0.246571
$$370$$ 2.54311e6 0.965742
$$371$$ −470106. −0.177321
$$372$$ −2880.00 −0.00107903
$$373$$ −1.51993e6 −0.565655 −0.282827 0.959171i $$-0.591272\pi$$
−0.282827 + 0.959171i $$0.591272\pi$$
$$374$$ 335664. 0.124087
$$375$$ 116532. 0.0427924
$$376$$ −2.33856e6 −0.853059
$$377$$ 2.40536e6 0.871620
$$378$$ 214326. 0.0771517
$$379$$ 2.64465e6 0.945737 0.472869 0.881133i $$-0.343219\pi$$
0.472869 + 0.881133i $$0.343219\pi$$
$$380$$ 837408. 0.297494
$$381$$ 668880. 0.236067
$$382$$ 1.64045e6 0.575180
$$383$$ 2.01336e6 0.701333 0.350667 0.936500i $$-0.385955\pi$$
0.350667 + 0.936500i $$0.385955\pi$$
$$384$$ 1.91117e6 0.661410
$$385$$ 1.69697e6 0.583474
$$386$$ −921612. −0.314833
$$387$$ −933444. −0.316819
$$388$$ −574648. −0.193786
$$389$$ −726234. −0.243334 −0.121667 0.992571i $$-0.538824\pi$$
−0.121667 + 0.992571i $$0.538824\pi$$
$$390$$ −1.86170e6 −0.619796
$$391$$ −529200. −0.175056
$$392$$ 403368. 0.132583
$$393$$ 1.39784e6 0.456538
$$394$$ −926532. −0.300691
$$395$$ −5.13053e6 −1.65451
$$396$$ 143856. 0.0460988
$$397$$ 4.57578e6 1.45710 0.728549 0.684993i $$-0.240195\pi$$
0.728549 + 0.684993i $$0.240195\pi$$
$$398$$ 2.20114e6 0.696529
$$399$$ −1.18364e6 −0.372211
$$400$$ −3.36142e6 −1.05045
$$401$$ −33870.0 −0.0105185 −0.00525926 0.999986i $$-0.501674\pi$$
−0.00525926 + 0.999986i $$0.501674\pi$$
$$402$$ −3.20522e6 −0.989221
$$403$$ −35360.0 −0.0108455
$$404$$ −10824.0 −0.00329940
$$405$$ 511758. 0.155034
$$406$$ 1.59995e6 0.481716
$$407$$ −2.41270e6 −0.721966
$$408$$ 190512. 0.0566594
$$409$$ −5.86178e6 −1.73269 −0.866346 0.499444i $$-0.833538\pi$$
−0.866346 + 0.499444i $$0.833538\pi$$
$$410$$ −3.72622e6 −1.09473
$$411$$ 2.37821e6 0.694459
$$412$$ 527072. 0.152977
$$413$$ 1.34711e6 0.388622
$$414$$ −2.04120e6 −0.585308
$$415$$ 3.13466e6 0.893451
$$416$$ −636480. −0.180323
$$417$$ −2.02151e6 −0.569292
$$418$$ −7.15018e6 −2.00159
$$419$$ 302748. 0.0842454 0.0421227 0.999112i $$-0.486588\pi$$
0.0421227 + 0.999112i $$0.486588\pi$$
$$420$$ −137592. −0.0380601
$$421$$ −5.36708e6 −1.47582 −0.737909 0.674900i $$-0.764187\pi$$
−0.737909 + 0.674900i $$0.764187\pi$$
$$422$$ −3.12146e6 −0.853252
$$423$$ −1.12752e6 −0.306389
$$424$$ −1.61179e6 −0.435406
$$425$$ −372834. −0.100125
$$426$$ 1.73016e6 0.461915
$$427$$ 2.42442e6 0.643485
$$428$$ −515664. −0.136068
$$429$$ 1.76623e6 0.463345
$$430$$ 5.39323e6 1.40662
$$431$$ 1.17706e6 0.305214 0.152607 0.988287i $$-0.451233\pi$$
0.152607 + 0.988287i $$0.451233\pi$$
$$432$$ 828144. 0.213499
$$433$$ −3.66249e6 −0.938766 −0.469383 0.882995i $$-0.655524\pi$$
−0.469383 + 0.882995i $$0.655524\pi$$
$$434$$ −23520.0 −0.00599395
$$435$$ 3.82028e6 0.967994
$$436$$ −403912. −0.101758
$$437$$ 1.12728e7 2.82376
$$438$$ −3.33968e6 −0.831802
$$439$$ −2.53674e6 −0.628225 −0.314113 0.949386i $$-0.601707\pi$$
−0.314113 + 0.949386i $$0.601707\pi$$
$$440$$ 5.81818e6 1.43270
$$441$$ 194481. 0.0476190
$$442$$ −334152. −0.0813558
$$443$$ 6.01504e6 1.45623 0.728113 0.685457i $$-0.240397\pi$$
0.728113 + 0.685457i $$0.240397\pi$$
$$444$$ 195624. 0.0470939
$$445$$ −621972. −0.148892
$$446$$ −1.82842e6 −0.435249
$$447$$ 738666. 0.174856
$$448$$ 1.35789e6 0.319646
$$449$$ 5.65965e6 1.32487 0.662436 0.749119i $$-0.269523\pi$$
0.662436 + 0.749119i $$0.269523\pi$$
$$450$$ −1.43807e6 −0.334773
$$451$$ 3.53513e6 0.818397
$$452$$ 880584. 0.202733
$$453$$ 2.58329e6 0.591463
$$454$$ −1.73153e6 −0.394267
$$455$$ −1.68932e6 −0.382547
$$456$$ −4.05821e6 −0.913949
$$457$$ −6.46159e6 −1.44727 −0.723634 0.690184i $$-0.757530\pi$$
−0.723634 + 0.690184i $$0.757530\pi$$
$$458$$ −4.63314e6 −1.03208
$$459$$ 91854.0 0.0203501
$$460$$ 1.31040e6 0.288742
$$461$$ −3.37353e6 −0.739320 −0.369660 0.929167i $$-0.620526\pi$$
−0.369660 + 0.929167i $$0.620526\pi$$
$$462$$ 1.17482e6 0.256075
$$463$$ −4.54974e6 −0.986358 −0.493179 0.869928i $$-0.664165\pi$$
−0.493179 + 0.869928i $$0.664165\pi$$
$$464$$ 6.18211e6 1.33304
$$465$$ −56160.0 −0.0120447
$$466$$ −1.51340e6 −0.322842
$$467$$ 2.01136e6 0.426773 0.213386 0.976968i $$-0.431551\pi$$
0.213386 + 0.976968i $$0.431551\pi$$
$$468$$ −143208. −0.0302240
$$469$$ −2.90844e6 −0.610560
$$470$$ 6.51456e6 1.36032
$$471$$ −1.16890e6 −0.242787
$$472$$ 4.61866e6 0.954247
$$473$$ −5.11666e6 −1.05156
$$474$$ −3.55190e6 −0.726132
$$475$$ 7.94196e6 1.61508
$$476$$ −24696.0 −0.00499585
$$477$$ −777114. −0.156383
$$478$$ 8.70682e6 1.74297
$$479$$ −7.60402e6 −1.51427 −0.757137 0.653257i $$-0.773402\pi$$
−0.757137 + 0.653257i $$0.773402\pi$$
$$480$$ −1.01088e6 −0.200261
$$481$$ 2.40183e6 0.473347
$$482$$ 878388. 0.172214
$$483$$ −1.85220e6 −0.361260
$$484$$ 144340. 0.0280074
$$485$$ −1.12056e7 −2.16313
$$486$$ 354294. 0.0680414
$$487$$ 673112. 0.128607 0.0643035 0.997930i $$-0.479517\pi$$
0.0643035 + 0.997930i $$0.479517\pi$$
$$488$$ 8.31230e6 1.58005
$$489$$ −4.99756e6 −0.945117
$$490$$ −1.12367e6 −0.211421
$$491$$ −2.47170e6 −0.462692 −0.231346 0.972872i $$-0.574313\pi$$
−0.231346 + 0.972872i $$0.574313\pi$$
$$492$$ −286632. −0.0533841
$$493$$ 685692. 0.127061
$$494$$ 7.11797e6 1.31232
$$495$$ 2.80519e6 0.514576
$$496$$ −90880.0 −0.0165869
$$497$$ 1.56996e6 0.285100
$$498$$ 2.17015e6 0.392118
$$499$$ 6.08152e6 1.09335 0.546677 0.837343i $$-0.315892\pi$$
0.546677 + 0.837343i $$0.315892\pi$$
$$500$$ −51792.0 −0.00926483
$$501$$ −391608. −0.0697039
$$502$$ −3.64716e6 −0.645945
$$503$$ −846216. −0.149129 −0.0745644 0.997216i $$-0.523757\pi$$
−0.0745644 + 0.997216i $$0.523757\pi$$
$$504$$ 666792. 0.116927
$$505$$ −211068. −0.0368293
$$506$$ −1.11888e7 −1.94271
$$507$$ 1.58336e6 0.273565
$$508$$ −297280. −0.0511101
$$509$$ −7.66785e6 −1.31183 −0.655917 0.754833i $$-0.727718\pi$$
−0.655917 + 0.754833i $$0.727718\pi$$
$$510$$ −530712. −0.0903511
$$511$$ −3.03045e6 −0.513400
$$512$$ 4.47130e6 0.753804
$$513$$ −1.95664e6 −0.328259
$$514$$ −573516. −0.0957498
$$515$$ 1.02779e7 1.70760
$$516$$ 414864. 0.0685933
$$517$$ −6.18048e6 −1.01694
$$518$$ 1.59760e6 0.261603
$$519$$ 164970. 0.0268835
$$520$$ −5.79197e6 −0.939329
$$521$$ −9.68938e6 −1.56387 −0.781937 0.623357i $$-0.785768\pi$$
−0.781937 + 0.623357i $$0.785768\pi$$
$$522$$ 2.64481e6 0.424833
$$523$$ −7.51678e6 −1.20165 −0.600824 0.799381i $$-0.705161\pi$$
−0.600824 + 0.799381i $$0.705161\pi$$
$$524$$ −621264. −0.0988435
$$525$$ −1.30492e6 −0.206626
$$526$$ 1.32021e7 2.08055
$$527$$ −10080.0 −0.00158101
$$528$$ 4.53946e6 0.708629
$$529$$ 1.12037e7 1.74069
$$530$$ 4.48999e6 0.694314
$$531$$ 2.22685e6 0.342732
$$532$$ 526064. 0.0805860
$$533$$ −3.51920e6 −0.536570
$$534$$ −430596. −0.0653457
$$535$$ −1.00554e7 −1.51886
$$536$$ −9.97181e6 −1.49921
$$537$$ 1.37992e6 0.206499
$$538$$ −1.06215e7 −1.58208
$$539$$ 1.06604e6 0.158053
$$540$$ −227448. −0.0335659
$$541$$ 7.34325e6 1.07869 0.539343 0.842086i $$-0.318673\pi$$
0.539343 + 0.842086i $$0.318673\pi$$
$$542$$ 1.34102e6 0.196082
$$543$$ 3.43859e6 0.500474
$$544$$ −181440. −0.0262867
$$545$$ −7.87628e6 −1.13587
$$546$$ −1.16953e6 −0.167892
$$547$$ 2.18296e6 0.311945 0.155973 0.987761i $$-0.450149\pi$$
0.155973 + 0.987761i $$0.450149\pi$$
$$548$$ −1.05698e6 −0.150355
$$549$$ 4.00772e6 0.567501
$$550$$ −7.88278e6 −1.11115
$$551$$ −1.46063e7 −2.04957
$$552$$ −6.35040e6 −0.887061
$$553$$ −3.22302e6 −0.448178
$$554$$ 2.05667e6 0.284702
$$555$$ 3.81467e6 0.525683
$$556$$ 898448. 0.123255
$$557$$ 1.25466e7 1.71351 0.856755 0.515724i $$-0.172477\pi$$
0.856755 + 0.515724i $$0.172477\pi$$
$$558$$ −38880.0 −0.00528617
$$559$$ 5.09361e6 0.689439
$$560$$ −4.34179e6 −0.585058
$$561$$ 503496. 0.0675443
$$562$$ −2.88227e6 −0.384940
$$563$$ 5.15972e6 0.686050 0.343025 0.939326i $$-0.388549\pi$$
0.343025 + 0.939326i $$0.388549\pi$$
$$564$$ 501120. 0.0663352
$$565$$ 1.71714e7 2.26300
$$566$$ 179880. 0.0236016
$$567$$ 321489. 0.0419961
$$568$$ 5.38272e6 0.700053
$$569$$ 1.17452e7 1.52083 0.760414 0.649439i $$-0.224996\pi$$
0.760414 + 0.649439i $$0.224996\pi$$
$$570$$ 1.13050e7 1.45742
$$571$$ −7.54728e6 −0.968725 −0.484362 0.874867i $$-0.660948\pi$$
−0.484362 + 0.874867i $$0.660948\pi$$
$$572$$ −784992. −0.100317
$$573$$ 2.46067e6 0.313089
$$574$$ −2.34083e6 −0.296544
$$575$$ 1.24278e7 1.56756
$$576$$ 2.24467e6 0.281901
$$577$$ 9.28483e6 1.16101 0.580503 0.814258i $$-0.302856\pi$$
0.580503 + 0.814258i $$0.302856\pi$$
$$578$$ 8.42389e6 1.04880
$$579$$ −1.38242e6 −0.171373
$$580$$ −1.69790e6 −0.209577
$$581$$ 1.96921e6 0.242020
$$582$$ −7.75775e6 −0.949354
$$583$$ −4.25974e6 −0.519053
$$584$$ −1.03901e7 −1.26063
$$585$$ −2.79256e6 −0.337374
$$586$$ 1.18840e6 0.142961
$$587$$ 1.47623e6 0.176831 0.0884155 0.996084i $$-0.471820\pi$$
0.0884155 + 0.996084i $$0.471820\pi$$
$$588$$ −86436.0 −0.0103098
$$589$$ 214720. 0.0255026
$$590$$ −1.28663e7 −1.52168
$$591$$ −1.38980e6 −0.163675
$$592$$ 6.17302e6 0.723925
$$593$$ −1.24007e7 −1.44813 −0.724067 0.689729i $$-0.757730\pi$$
−0.724067 + 0.689729i $$0.757730\pi$$
$$594$$ 1.94206e6 0.225837
$$595$$ −481572. −0.0557659
$$596$$ −328296. −0.0378573
$$597$$ 3.30170e6 0.379142
$$598$$ 1.11384e7 1.27371
$$599$$ −3.69127e6 −0.420348 −0.210174 0.977664i $$-0.567403\pi$$
−0.210174 + 0.977664i $$0.567403\pi$$
$$600$$ −4.47401e6 −0.507363
$$601$$ 9.12223e6 1.03018 0.515092 0.857135i $$-0.327758\pi$$
0.515092 + 0.857135i $$0.327758\pi$$
$$602$$ 3.38806e6 0.381030
$$603$$ −4.80784e6 −0.538464
$$604$$ −1.14813e6 −0.128055
$$605$$ 2.81463e6 0.312632
$$606$$ −146124. −0.0161637
$$607$$ −5.67914e6 −0.625620 −0.312810 0.949816i $$-0.601270\pi$$
−0.312810 + 0.949816i $$0.601270\pi$$
$$608$$ 3.86496e6 0.424020
$$609$$ 2.39992e6 0.262213
$$610$$ −2.31557e7 −2.51961
$$611$$ 6.15264e6 0.666743
$$612$$ −40824.0 −0.00440592
$$613$$ −1.40106e7 −1.50593 −0.752966 0.658060i $$-0.771377\pi$$
−0.752966 + 0.658060i $$0.771377\pi$$
$$614$$ 6.27382e6 0.671600
$$615$$ −5.58932e6 −0.595897
$$616$$ 3.65501e6 0.388094
$$617$$ −253686. −0.0268277 −0.0134139 0.999910i $$-0.504270\pi$$
−0.0134139 + 0.999910i $$0.504270\pi$$
$$618$$ 7.11547e6 0.749433
$$619$$ 4.30034e6 0.451103 0.225552 0.974231i $$-0.427582\pi$$
0.225552 + 0.974231i $$0.427582\pi$$
$$620$$ 24960.0 0.00260775
$$621$$ −3.06180e6 −0.318602
$$622$$ −1.10231e7 −1.14242
$$623$$ −390726. −0.0403322
$$624$$ −4.51901e6 −0.464603
$$625$$ −1.02568e7 −1.05030
$$626$$ 2.19296e6 0.223664
$$627$$ −1.07253e7 −1.08953
$$628$$ 519512. 0.0525650
$$629$$ 684684. 0.0690023
$$630$$ −1.85749e6 −0.186456
$$631$$ 1.04150e7 1.04132 0.520662 0.853763i $$-0.325685\pi$$
0.520662 + 0.853763i $$0.325685\pi$$
$$632$$ −1.10504e7 −1.10048
$$633$$ −4.68220e6 −0.464451
$$634$$ 170028. 0.0167995
$$635$$ −5.79696e6 −0.570514
$$636$$ 345384. 0.0338579
$$637$$ −1.06124e6 −0.103625
$$638$$ 1.44975e7 1.41007
$$639$$ 2.59524e6 0.251435
$$640$$ −1.65635e7 −1.59846
$$641$$ 4.52714e6 0.435190 0.217595 0.976039i $$-0.430179\pi$$
0.217595 + 0.976039i $$0.430179\pi$$
$$642$$ −6.96146e6 −0.666596
$$643$$ 1.49687e7 1.42776 0.713882 0.700266i $$-0.246935\pi$$
0.713882 + 0.700266i $$0.246935\pi$$
$$644$$ 823200. 0.0782151
$$645$$ 8.08985e6 0.765669
$$646$$ 2.02910e6 0.191304
$$647$$ −1.73020e7 −1.62493 −0.812465 0.583010i $$-0.801875\pi$$
−0.812465 + 0.583010i $$0.801875\pi$$
$$648$$ 1.10225e6 0.103120
$$649$$ 1.22064e7 1.13757
$$650$$ 7.84727e6 0.728509
$$651$$ −35280.0 −0.00326269
$$652$$ 2.22114e6 0.204624
$$653$$ 4.07470e6 0.373949 0.186975 0.982365i $$-0.440132\pi$$
0.186975 + 0.982365i $$0.440132\pi$$
$$654$$ −5.45281e6 −0.498513
$$655$$ −1.21146e7 −1.10334
$$656$$ −9.04483e6 −0.820618
$$657$$ −5.00953e6 −0.452776
$$658$$ 4.09248e6 0.368487
$$659$$ −3.79475e6 −0.340384 −0.170192 0.985411i $$-0.554439\pi$$
−0.170192 + 0.985411i $$0.554439\pi$$
$$660$$ −1.24675e6 −0.111409
$$661$$ 1.64261e7 1.46228 0.731142 0.682225i $$-0.238988\pi$$
0.731142 + 0.682225i $$0.238988\pi$$
$$662$$ −1.16035e7 −1.02907
$$663$$ −501228. −0.0442845
$$664$$ 6.75158e6 0.594272
$$665$$ 1.02582e7 0.899537
$$666$$ 2.64092e6 0.230712
$$667$$ −2.28564e7 −1.98927
$$668$$ 174048. 0.0150913
$$669$$ −2.74262e6 −0.236920
$$670$$ 2.77786e7 2.39069
$$671$$ 2.19682e7 1.88360
$$672$$ −635040. −0.0542473
$$673$$ 5.50675e6 0.468660 0.234330 0.972157i $$-0.424710\pi$$
0.234330 + 0.972157i $$0.424710\pi$$
$$674$$ 1.13290e7 0.960602
$$675$$ −2.15711e6 −0.182227
$$676$$ −703716. −0.0592285
$$677$$ 1.83957e7 1.54257 0.771286 0.636488i $$-0.219614\pi$$
0.771286 + 0.636488i $$0.219614\pi$$
$$678$$ 1.18879e7 0.993185
$$679$$ −7.03944e6 −0.585954
$$680$$ −1.65110e6 −0.136931
$$681$$ −2.59729e6 −0.214612
$$682$$ −213120. −0.0175454
$$683$$ 1.75835e6 0.144229 0.0721146 0.997396i $$-0.477025\pi$$
0.0721146 + 0.997396i $$0.477025\pi$$
$$684$$ 869616. 0.0710702
$$685$$ −2.06112e7 −1.67833
$$686$$ −705894. −0.0572703
$$687$$ −6.94971e6 −0.561791
$$688$$ 1.30913e7 1.05441
$$689$$ 4.24055e6 0.340309
$$690$$ 1.76904e7 1.41454
$$691$$ −5.36314e6 −0.427291 −0.213646 0.976911i $$-0.568534\pi$$
−0.213646 + 0.976911i $$0.568534\pi$$
$$692$$ −73320.0 −0.00582046
$$693$$ 1.76224e6 0.139390
$$694$$ −1.75162e7 −1.38052
$$695$$ 1.75197e7 1.37583
$$696$$ 8.22830e6 0.643854
$$697$$ −1.00321e6 −0.0782187
$$698$$ 4.68409e6 0.363904
$$699$$ −2.27011e6 −0.175733
$$700$$ 579964. 0.0447359
$$701$$ −2.12606e7 −1.63411 −0.817054 0.576561i $$-0.804394\pi$$
−0.817054 + 0.576561i $$0.804394\pi$$
$$702$$ −1.93331e6 −0.148067
$$703$$ −1.45849e7 −1.11305
$$704$$ 1.23041e7 0.935662
$$705$$ 9.77184e6 0.740463
$$706$$ −8.00622e6 −0.604527
$$707$$ −132594. −0.00997643
$$708$$ −989712. −0.0742037
$$709$$ 2.07729e6 0.155196 0.0775980 0.996985i $$-0.475275\pi$$
0.0775980 + 0.996985i $$0.475275\pi$$
$$710$$ −1.49947e7 −1.11633
$$711$$ −5.32786e6 −0.395256
$$712$$ −1.33963e6 −0.0990343
$$713$$ 336000. 0.0247523
$$714$$ −333396. −0.0244746
$$715$$ −1.53073e7 −1.11979
$$716$$ −613296. −0.0447082
$$717$$ 1.30602e7 0.948752
$$718$$ −6.10459e6 −0.441922
$$719$$ 4.23619e6 0.305600 0.152800 0.988257i $$-0.451171\pi$$
0.152800 + 0.988257i $$0.451171\pi$$
$$720$$ −7.17725e6 −0.515973
$$721$$ 6.45663e6 0.462560
$$722$$ −2.83665e7 −2.02518
$$723$$ 1.31758e6 0.0937415
$$724$$ −1.52826e6 −0.108356
$$725$$ −1.61029e7 −1.13778
$$726$$ 1.94859e6 0.137208
$$727$$ 2.14524e7 1.50536 0.752678 0.658389i $$-0.228762\pi$$
0.752678 + 0.658389i $$0.228762\pi$$
$$728$$ −3.63854e6 −0.254448
$$729$$ 531441. 0.0370370
$$730$$ 2.89439e7 2.01025
$$731$$ 1.45202e6 0.100503
$$732$$ −1.78121e6 −0.122867
$$733$$ −1.48892e7 −1.02355 −0.511777 0.859118i $$-0.671013\pi$$
−0.511777 + 0.859118i $$0.671013\pi$$
$$734$$ −5.02608e6 −0.344341
$$735$$ −1.68550e6 −0.115083
$$736$$ 6.04800e6 0.411545
$$737$$ −2.63541e7 −1.78722
$$738$$ −3.86953e6 −0.261528
$$739$$ 6.99324e6 0.471050 0.235525 0.971868i $$-0.424319\pi$$
0.235525 + 0.971868i $$0.424319\pi$$
$$740$$ −1.69541e6 −0.113814
$$741$$ 1.06770e7 0.714335
$$742$$ 2.82064e6 0.188078
$$743$$ 1.90428e6 0.126549 0.0632745 0.997996i $$-0.479846\pi$$
0.0632745 + 0.997996i $$0.479846\pi$$
$$744$$ −120960. −0.00801142
$$745$$ −6.40177e6 −0.422581
$$746$$ 9.11958e6 0.599968
$$747$$ 3.25523e6 0.213442
$$748$$ −223776. −0.0146238
$$749$$ −6.31688e6 −0.411432
$$750$$ −699192. −0.0453882
$$751$$ 1.95361e7 1.26398 0.631988 0.774978i $$-0.282239\pi$$
0.631988 + 0.774978i $$0.282239\pi$$
$$752$$ 1.58131e7 1.01970
$$753$$ −5.47074e6 −0.351608
$$754$$ −1.44322e7 −0.924493
$$755$$ −2.23885e7 −1.42941
$$756$$ −142884. −0.00909241
$$757$$ 1.25183e6 0.0793973 0.0396986 0.999212i $$-0.487360\pi$$
0.0396986 + 0.999212i $$0.487360\pi$$
$$758$$ −1.58679e7 −1.00311
$$759$$ −1.67832e7 −1.05748
$$760$$ 3.51711e7 2.20878
$$761$$ 2.04472e7 1.27989 0.639944 0.768422i $$-0.278958\pi$$
0.639944 + 0.768422i $$0.278958\pi$$
$$762$$ −4.01328e6 −0.250387
$$763$$ −4.94792e6 −0.307689
$$764$$ −1.09363e6 −0.0677857
$$765$$ −796068. −0.0491809
$$766$$ −1.20802e7 −0.743876
$$767$$ −1.21515e7 −0.745831
$$768$$ −3.48595e6 −0.213264
$$769$$ 2.21064e6 0.134804 0.0674020 0.997726i $$-0.478529\pi$$
0.0674020 + 0.997726i $$0.478529\pi$$
$$770$$ −1.01818e7 −0.618868
$$771$$ −860274. −0.0521196
$$772$$ 614408. 0.0371034
$$773$$ 1.29151e7 0.777405 0.388703 0.921363i $$-0.372923\pi$$
0.388703 + 0.921363i $$0.372923\pi$$
$$774$$ 5.60066e6 0.336037
$$775$$ 236720. 0.0141573
$$776$$ −2.41352e7 −1.43879
$$777$$ 2.39639e6 0.142399
$$778$$ 4.35740e6 0.258095
$$779$$ 2.13700e7 1.26171
$$780$$ 1.24114e6 0.0730437
$$781$$ 1.42258e7 0.834541
$$782$$ 3.17520e6 0.185675
$$783$$ 3.96722e6 0.231250
$$784$$ −2.72754e6 −0.158482
$$785$$ 1.01305e7 0.586754
$$786$$ −8.38706e6 −0.484232
$$787$$ −1.35499e7 −0.779830 −0.389915 0.920851i $$-0.627496\pi$$
−0.389915 + 0.920851i $$0.627496\pi$$
$$788$$ 617688. 0.0354367
$$789$$ 1.98031e7 1.13251
$$790$$ 3.07832e7 1.75487
$$791$$ 1.07872e7 0.613007
$$792$$ 6.04195e6 0.342266
$$793$$ −2.18693e7 −1.23496
$$794$$ −2.74547e7 −1.54549
$$795$$ 6.73499e6 0.377937
$$796$$ −1.46742e6 −0.0820867
$$797$$ −2.45956e7 −1.37155 −0.685776 0.727813i $$-0.740537\pi$$
−0.685776 + 0.727813i $$0.740537\pi$$
$$798$$ 7.10186e6 0.394789
$$799$$ 1.75392e6 0.0971948
$$800$$ 4.26096e6 0.235387
$$801$$ −645894. −0.0355697
$$802$$ 203220. 0.0111566
$$803$$ −2.74596e7 −1.50282
$$804$$ 2.13682e6 0.116581
$$805$$ 1.60524e7 0.873072
$$806$$ 212160. 0.0115034
$$807$$ −1.59322e7 −0.861177
$$808$$ −454608. −0.0244968
$$809$$ 1.55237e7 0.833920 0.416960 0.908925i $$-0.363095\pi$$
0.416960 + 0.908925i $$0.363095\pi$$
$$810$$ −3.07055e6 −0.164438
$$811$$ −2.66262e7 −1.42153 −0.710766 0.703429i $$-0.751651\pi$$
−0.710766 + 0.703429i $$0.751651\pi$$
$$812$$ −1.06663e6 −0.0567707
$$813$$ 2.01154e6 0.106734
$$814$$ 1.44762e7 0.765760
$$815$$ 4.33122e7 2.28410
$$816$$ −1.28822e6 −0.0677276
$$817$$ −3.09304e7 −1.62118
$$818$$ 3.51707e7 1.83780
$$819$$ −1.75430e6 −0.0913889
$$820$$ 2.48414e6 0.129016
$$821$$ −1.23891e7 −0.641477 −0.320739 0.947168i $$-0.603931\pi$$
−0.320739 + 0.947168i $$0.603931\pi$$
$$822$$ −1.42693e7 −0.736585
$$823$$ −3.65630e6 −0.188166 −0.0940831 0.995564i $$-0.529992\pi$$
−0.0940831 + 0.995564i $$0.529992\pi$$
$$824$$ 2.21370e7 1.13580
$$825$$ −1.18242e7 −0.604833
$$826$$ −8.08265e6 −0.412196
$$827$$ 2.80463e7 1.42597 0.712987 0.701178i $$-0.247342\pi$$
0.712987 + 0.701178i $$0.247342\pi$$
$$828$$ 1.36080e6 0.0689792
$$829$$ 2.11153e7 1.06712 0.533558 0.845763i $$-0.320855\pi$$
0.533558 + 0.845763i $$0.320855\pi$$
$$830$$ −1.88080e7 −0.947648
$$831$$ 3.08500e6 0.154972
$$832$$ −1.22487e7 −0.613454
$$833$$ −302526. −0.0151060
$$834$$ 1.21290e7 0.603826
$$835$$ 3.39394e6 0.168456
$$836$$ 4.76678e6 0.235890
$$837$$ −58320.0 −0.00287742
$$838$$ −1.81649e6 −0.0893557
$$839$$ 1.33947e7 0.656944 0.328472 0.944514i $$-0.393466\pi$$
0.328472 + 0.944514i $$0.393466\pi$$
$$840$$ −5.77886e6 −0.282582
$$841$$ 9.10422e6 0.443867
$$842$$ 3.22025e7 1.56534
$$843$$ −4.32340e6 −0.209535
$$844$$ 2.08098e6 0.100557
$$845$$ −1.37225e7 −0.661135
$$846$$ 6.76512e6 0.324975
$$847$$ 1.76816e6 0.0846865
$$848$$ 1.08988e7 0.520461
$$849$$ 269820. 0.0128471
$$850$$ 2.23700e6 0.106199
$$851$$ −2.28228e7 −1.08030
$$852$$ −1.15344e6 −0.0544372
$$853$$ 3.01513e7 1.41884 0.709420 0.704786i $$-0.248957\pi$$
0.709420 + 0.704786i $$0.248957\pi$$
$$854$$ −1.45465e7 −0.682519
$$855$$ 1.69575e7 0.793317
$$856$$ −2.16579e7 −1.01026
$$857$$ 2.39894e7 1.11575 0.557875 0.829925i $$-0.311617\pi$$
0.557875 + 0.829925i $$0.311617\pi$$
$$858$$ −1.05974e7 −0.491452
$$859$$ −8.87576e6 −0.410414 −0.205207 0.978719i $$-0.565787\pi$$
−0.205207 + 0.978719i $$0.565787\pi$$
$$860$$ −3.59549e6 −0.165772
$$861$$ −3.51124e6 −0.161418
$$862$$ −7.06234e6 −0.323728
$$863$$ −8.71286e6 −0.398230 −0.199115 0.979976i $$-0.563807\pi$$
−0.199115 + 0.979976i $$0.563807\pi$$
$$864$$ −1.04976e6 −0.0478416
$$865$$ −1.42974e6 −0.0649706
$$866$$ 2.19750e7 0.995711
$$867$$ 1.26358e7 0.570895
$$868$$ 15680.0 0.000706394 0
$$869$$ −2.92045e7 −1.31190
$$870$$ −2.29217e7 −1.02671
$$871$$ 2.62354e7 1.17177
$$872$$ −1.69643e7 −0.755518
$$873$$ −1.16366e7 −0.516763
$$874$$ −6.76368e7 −2.99505
$$875$$ −634452. −0.0280142
$$876$$ 2.22646e6 0.0980288
$$877$$ −2.95788e7 −1.29862 −0.649310 0.760524i $$-0.724942\pi$$
−0.649310 + 0.760524i $$0.724942\pi$$
$$878$$ 1.52205e7 0.666333
$$879$$ 1.78259e6 0.0778180
$$880$$ −3.93420e7 −1.71257
$$881$$ 2.45670e7 1.06638 0.533190 0.845995i $$-0.320993\pi$$
0.533190 + 0.845995i $$0.320993\pi$$
$$882$$ −1.16689e6 −0.0505076
$$883$$ 1.45682e7 0.628788 0.314394 0.949293i $$-0.398199\pi$$
0.314394 + 0.949293i $$0.398199\pi$$
$$884$$ 222768. 0.00958787
$$885$$ −1.92994e7 −0.828295
$$886$$ −3.60902e7 −1.54456
$$887$$ 1.61714e7 0.690141 0.345070 0.938577i $$-0.387855\pi$$
0.345070 + 0.938577i $$0.387855\pi$$
$$888$$ 8.21621e6 0.349654
$$889$$ −3.64168e6 −0.154542
$$890$$ 3.73183e6 0.157924
$$891$$ 2.91308e6 0.122930
$$892$$ 1.21894e6 0.0512946
$$893$$ −3.73613e7 −1.56781
$$894$$ −4.43200e6 −0.185462
$$895$$ −1.19593e7 −0.499054
$$896$$ −1.04052e7 −0.432995
$$897$$ 1.67076e7 0.693319
$$898$$ −3.39579e7 −1.40524
$$899$$ −435360. −0.0179659
$$900$$ 958716. 0.0394533
$$901$$ 1.20884e6 0.0496087
$$902$$ −2.12108e7 −0.868041
$$903$$ 5.08208e6 0.207407
$$904$$ 3.69845e7 1.50522
$$905$$ −2.98011e7 −1.20952
$$906$$ −1.54997e7 −0.627341
$$907$$ 3.14446e7 1.26919 0.634596 0.772844i $$-0.281167\pi$$
0.634596 + 0.772844i $$0.281167\pi$$
$$908$$ 1.15435e6 0.0464648
$$909$$ −219186. −0.00879839
$$910$$ 1.01359e7 0.405752
$$911$$ 1.51427e7 0.604514 0.302257 0.953227i $$-0.402260\pi$$
0.302257 + 0.953227i $$0.402260\pi$$
$$912$$ 2.74412e7 1.09249
$$913$$ 1.78435e7 0.708439
$$914$$ 3.87695e7 1.53506
$$915$$ −3.47336e7 −1.37150
$$916$$ 3.08876e6 0.121631
$$917$$ −7.61048e6 −0.298875
$$918$$ −551124. −0.0215845
$$919$$ 4.14876e7 1.62043 0.810214 0.586134i $$-0.199351\pi$$
0.810214 + 0.586134i $$0.199351\pi$$
$$920$$ 5.50368e7 2.14380
$$921$$ 9.41072e6 0.365573
$$922$$ 2.02412e7 0.784167
$$923$$ −1.41617e7 −0.547155
$$924$$ −783216. −0.0301788
$$925$$ −1.60792e7 −0.617889
$$926$$ 2.72985e7 1.04619
$$927$$ 1.06732e7 0.407939
$$928$$ −7.83648e6 −0.298711
$$929$$ −1.78495e7 −0.678556 −0.339278 0.940686i $$-0.610183\pi$$
−0.339278 + 0.940686i $$0.610183\pi$$
$$930$$ 336960. 0.0127753
$$931$$ 6.44428e6 0.243669
$$932$$ 1.00894e6 0.0380473
$$933$$ −1.65346e7 −0.621855
$$934$$ −1.20681e7 −0.452661
$$935$$ −4.36363e6 −0.163237
$$936$$ −6.01474e6 −0.224402
$$937$$ 2.96399e7 1.10288 0.551439 0.834215i $$-0.314079\pi$$
0.551439 + 0.834215i $$0.314079\pi$$
$$938$$ 1.74507e7 0.647597
$$939$$ 3.28945e6 0.121747
$$940$$ −4.34304e6 −0.160315
$$941$$ −3.22282e7 −1.18648 −0.593242 0.805024i $$-0.702152\pi$$
−0.593242 + 0.805024i $$0.702152\pi$$
$$942$$ 7.01341e6 0.257515
$$943$$ 3.34404e7 1.22459
$$944$$ −3.12309e7 −1.14066
$$945$$ −2.78624e6 −0.101494
$$946$$ 3.06999e7 1.11535
$$947$$ 4.84885e7 1.75697 0.878484 0.477772i $$-0.158556\pi$$
0.878484 + 0.477772i $$0.158556\pi$$
$$948$$ 2.36794e6 0.0855754
$$949$$ 2.73359e7 0.985300
$$950$$ −4.76517e7 −1.71305
$$951$$ 255042. 0.00914451
$$952$$ −1.03723e6 −0.0370923
$$953$$ −2.03264e7 −0.724983 −0.362491 0.931987i $$-0.618074\pi$$
−0.362491 + 0.931987i $$0.618074\pi$$
$$954$$ 4.66268e6 0.165869
$$955$$ −2.13258e7 −0.756654
$$956$$ −5.80454e6 −0.205411
$$957$$ 2.17462e7 0.767546
$$958$$ 4.56241e7 1.60613
$$959$$ −1.29481e7 −0.454630
$$960$$ −1.94538e7 −0.681282
$$961$$ −2.86228e7 −0.999776
$$962$$ −1.44110e7 −0.502060
$$963$$ −1.04422e7 −0.362849
$$964$$ −585592. −0.0202956
$$965$$ 1.19810e7 0.414165
$$966$$ 1.11132e7 0.383174
$$967$$ −3.66292e6 −0.125968 −0.0629841 0.998015i $$-0.520062\pi$$
−0.0629841 + 0.998015i $$0.520062\pi$$
$$968$$ 6.06228e6 0.207945
$$969$$ 3.04366e6 0.104132
$$970$$ 6.72338e7 2.29434
$$971$$ 1.48741e6 0.0506271 0.0253136 0.999680i $$-0.491942\pi$$
0.0253136 + 0.999680i $$0.491942\pi$$
$$972$$ −236196. −0.00801875
$$973$$ 1.10060e7 0.372689
$$974$$ −4.03867e6 −0.136408
$$975$$ 1.17709e7 0.396550
$$976$$ −5.62070e7 −1.88871
$$977$$ 4.07930e7 1.36725 0.683627 0.729831i $$-0.260401\pi$$
0.683627 + 0.729831i $$0.260401\pi$$
$$978$$ 2.99853e7 1.00245
$$979$$ −3.54046e6 −0.118060
$$980$$ 749112. 0.0249162
$$981$$ −8.17922e6 −0.271356
$$982$$ 1.48302e7 0.490759
$$983$$ −9.26326e6 −0.305759 −0.152880 0.988245i $$-0.548855\pi$$
−0.152880 + 0.988245i $$0.548855\pi$$
$$984$$ −1.20385e7 −0.396357
$$985$$ 1.20449e7 0.395561
$$986$$ −4.11415e6 −0.134768
$$987$$ 6.13872e6 0.200579
$$988$$ −4.74531e6 −0.154658
$$989$$ −4.84008e7 −1.57348
$$990$$ −1.68312e7 −0.545790
$$991$$ −5.22051e7 −1.68861 −0.844303 0.535866i $$-0.819985\pi$$
−0.844303 + 0.535866i $$0.819985\pi$$
$$992$$ 115200. 0.00371684
$$993$$ −1.74052e7 −0.560153
$$994$$ −9.41976e6 −0.302394
$$995$$ −2.86148e7 −0.916289
$$996$$ −1.44677e6 −0.0462116
$$997$$ −1.86609e7 −0.594560 −0.297280 0.954790i $$-0.596079\pi$$
−0.297280 + 0.954790i $$0.596079\pi$$
$$998$$ −3.64891e7 −1.15968
$$999$$ 3.96139e6 0.125584
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 21.6.a.a.1.1 1
3.2 odd 2 63.6.a.d.1.1 1
4.3 odd 2 336.6.a.r.1.1 1
5.2 odd 4 525.6.d.b.274.1 2
5.3 odd 4 525.6.d.b.274.2 2
5.4 even 2 525.6.a.d.1.1 1
7.2 even 3 147.6.e.j.67.1 2
7.3 odd 6 147.6.e.i.79.1 2
7.4 even 3 147.6.e.j.79.1 2
7.5 odd 6 147.6.e.i.67.1 2
7.6 odd 2 147.6.a.b.1.1 1
12.11 even 2 1008.6.a.c.1.1 1
21.20 even 2 441.6.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.a.1.1 1 1.1 even 1 trivial
63.6.a.d.1.1 1 3.2 odd 2
147.6.a.b.1.1 1 7.6 odd 2
147.6.e.i.67.1 2 7.5 odd 6
147.6.e.i.79.1 2 7.3 odd 6
147.6.e.j.67.1 2 7.2 even 3
147.6.e.j.79.1 2 7.4 even 3
336.6.a.r.1.1 1 4.3 odd 2
441.6.a.j.1.1 1 21.20 even 2
525.6.a.d.1.1 1 5.4 even 2
525.6.d.b.274.1 2 5.2 odd 4
525.6.d.b.274.2 2 5.3 odd 4
1008.6.a.c.1.1 1 12.11 even 2