Properties

 Label 336.6.a.r.1.1 Level $336$ Weight $6$ Character 336.1 Self dual yes Analytic conductor $53.889$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.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: $$53.8889634572$$ 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 $$\chi$$ $$=$$ 336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.00000 q^{3} +78.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$$ $$q+9.00000 q^{3} +78.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} -444.000 q^{11} -442.000 q^{13} +702.000 q^{15} -126.000 q^{17} -2684.00 q^{19} -441.000 q^{21} -4200.00 q^{23} +2959.00 q^{25} +729.000 q^{27} -5442.00 q^{29} -80.0000 q^{31} -3996.00 q^{33} -3822.00 q^{35} -5434.00 q^{37} -3978.00 q^{39} +7962.00 q^{41} +11524.0 q^{43} +6318.00 q^{45} +13920.0 q^{47} +2401.00 q^{49} -1134.00 q^{51} -9594.00 q^{53} -34632.0 q^{55} -24156.0 q^{57} -27492.0 q^{59} +49478.0 q^{61} -3969.00 q^{63} -34476.0 q^{65} +59356.0 q^{67} -37800.0 q^{69} -32040.0 q^{71} -61846.0 q^{73} +26631.0 q^{75} +21756.0 q^{77} +65776.0 q^{79} +6561.00 q^{81} -40188.0 q^{83} -9828.00 q^{85} -48978.0 q^{87} -7974.00 q^{89} +21658.0 q^{91} -720.000 q^{93} -209352. q^{95} -143662. q^{97} -35964.0 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 9.00000 0.577350
$$4$$ 0 0
$$5$$ 78.0000 1.39531 0.697653 0.716436i $$-0.254228\pi$$
0.697653 + 0.716436i $$0.254228\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −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 $$-0.618141\pi$$
−0.362689 + 0.931910i $$0.618141\pi$$
$$14$$ 0 0
$$15$$ 702.000 0.805581
$$16$$ 0 0
$$17$$ −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 $$0.825123\pi$$
$$20$$ 0 0
$$21$$ −441.000 −0.218218
$$22$$ 0 0
$$23$$ −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
$$26$$ 0 0
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ −5442.00 −1.20161 −0.600805 0.799396i $$-0.705153\pi$$
−0.600805 + 0.799396i $$0.705153\pi$$
$$30$$ 0 0
$$31$$ −80.0000 −0.0149515 −0.00747577 0.999972i $$-0.502380\pi$$
−0.00747577 + 0.999972i $$0.502380\pi$$
$$32$$ 0 0
$$33$$ −3996.00 −0.638764
$$34$$ 0 0
$$35$$ −3822.00 −0.527376
$$36$$ 0 0
$$37$$ −5434.00 −0.652552 −0.326276 0.945274i $$-0.605794\pi$$
−0.326276 + 0.945274i $$0.605794\pi$$
$$38$$ 0 0
$$39$$ −3978.00 −0.418797
$$40$$ 0 0
$$41$$ 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 $$-0.342366\pi$$
0.475228 + 0.879863i $$0.342366\pi$$
$$44$$ 0 0
$$45$$ 6318.00 0.465102
$$46$$ 0 0
$$47$$ 13920.0 0.919167 0.459584 0.888134i $$-0.347999\pi$$
0.459584 + 0.888134i $$0.347999\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −1134.00 −0.0610503
$$52$$ 0 0
$$53$$ −9594.00 −0.469148 −0.234574 0.972098i $$-0.575370\pi$$
−0.234574 + 0.972098i $$0.575370\pi$$
$$54$$ 0 0
$$55$$ −34632.0 −1.54373
$$56$$ 0 0
$$57$$ −24156.0 −0.984777
$$58$$ 0 0
$$59$$ −27492.0 −1.02820 −0.514098 0.857731i $$-0.671873\pi$$
−0.514098 + 0.857731i $$0.671873\pi$$
$$60$$ 0 0
$$61$$ 49478.0 1.70250 0.851251 0.524759i $$-0.175845\pi$$
0.851251 + 0.524759i $$0.175845\pi$$
$$62$$ 0 0
$$63$$ −3969.00 −0.125988
$$64$$ 0 0
$$65$$ −34476.0 −1.01212
$$66$$ 0 0
$$67$$ 59356.0 1.61539 0.807695 0.589600i $$-0.200715\pi$$
0.807695 + 0.589600i $$0.200715\pi$$
$$68$$ 0 0
$$69$$ −37800.0 −0.955805
$$70$$ 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 $$-0.737657\pi$$
−0.679164 + 0.733987i $$0.737657\pi$$
$$74$$ 0 0
$$75$$ 26631.0 0.546681
$$76$$ 0 0
$$77$$ 21756.0 0.418169
$$78$$ 0 0
$$79$$ 65776.0 1.18577 0.592884 0.805288i $$-0.297989\pi$$
0.592884 + 0.805288i $$0.297989\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 0 0
$$83$$ −40188.0 −0.640326 −0.320163 0.947362i $$-0.603738\pi$$
−0.320163 + 0.947362i $$0.603738\pi$$
$$84$$ 0 0
$$85$$ −9828.00 −0.147543
$$86$$ 0 0
$$87$$ −48978.0 −0.693750
$$88$$ 0 0
$$89$$ −7974.00 −0.106709 −0.0533545 0.998576i $$-0.516991\pi$$
−0.0533545 + 0.998576i $$0.516991\pi$$
$$90$$ 0 0
$$91$$ 21658.0 0.274167
$$92$$ 0 0
$$93$$ −720.000 −0.00863227
$$94$$ 0 0
$$95$$ −209352. −2.37995
$$96$$ 0 0
$$97$$ −143662. −1.55029 −0.775144 0.631784i $$-0.782323\pi$$
−0.775144 + 0.631784i $$0.782323\pi$$
$$98$$ 0 0
$$99$$ −35964.0 −0.368791
$$100$$ 0 0
$$101$$ −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 $$0.709598\pi$$
$$104$$ 0 0
$$105$$ −34398.0 −0.304481
$$106$$ 0 0
$$107$$ 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 $$0.633437\pi$$
$$110$$ 0 0
$$111$$ −48906.0 −0.376751
$$112$$ 0 0
$$113$$ 220146. 1.62186 0.810932 0.585140i $$-0.198960\pi$$
0.810932 + 0.585140i $$0.198960\pi$$
$$114$$ 0 0
$$115$$ −327600. −2.30993
$$116$$ 0 0
$$117$$ −35802.0 −0.241792
$$118$$ 0 0
$$119$$ 6174.00 0.0399668
$$120$$ 0 0
$$121$$ 36085.0 0.224059
$$122$$ 0 0
$$123$$ 71658.0 0.427073
$$124$$ 0 0
$$125$$ −12948.0 −0.0741187
$$126$$ 0 0
$$127$$ 74320.0 0.408880 0.204440 0.978879i $$-0.434463\pi$$
0.204440 + 0.978879i $$0.434463\pi$$
$$128$$ 0 0
$$129$$ 103716. 0.548746
$$130$$ 0 0
$$131$$ 155316. 0.790748 0.395374 0.918520i $$-0.370615\pi$$
0.395374 + 0.918520i $$0.370615\pi$$
$$132$$ 0 0
$$133$$ 131516. 0.644688
$$134$$ 0 0
$$135$$ 56862.0 0.268527
$$136$$ 0 0
$$137$$ −264246. −1.20284 −0.601419 0.798934i $$-0.705398\pi$$
−0.601419 + 0.798934i $$0.705398\pi$$
$$138$$ 0 0
$$139$$ −224612. −0.986043 −0.493022 0.870017i $$-0.664108\pi$$
−0.493022 + 0.870017i $$0.664108\pi$$
$$140$$ 0 0
$$141$$ 125280. 0.530682
$$142$$ 0 0
$$143$$ 196248. 0.802537
$$144$$ 0 0
$$145$$ −424476. −1.67661
$$146$$ 0 0
$$147$$ 21609.0 0.0824786
$$148$$ 0 0
$$149$$ −82074.0 −0.302859 −0.151429 0.988468i $$-0.548388\pi$$
−0.151429 + 0.988468i $$0.548388\pi$$
$$150$$ 0 0
$$151$$ 287032. 1.02444 0.512222 0.858853i $$-0.328823\pi$$
0.512222 + 0.858853i $$0.328823\pi$$
$$152$$ 0 0
$$153$$ −10206.0 −0.0352474
$$154$$ 0 0
$$155$$ −6240.00 −0.0208620
$$156$$ 0 0
$$157$$ 129878. 0.420520 0.210260 0.977646i $$-0.432569\pi$$
0.210260 + 0.977646i $$0.432569\pi$$
$$158$$ 0 0
$$159$$ −86346.0 −0.270863
$$160$$ 0 0
$$161$$ 205800. 0.625721
$$162$$ 0 0
$$163$$ −555284. −1.63699 −0.818495 0.574513i $$-0.805191\pi$$
−0.818495 + 0.574513i $$0.805191\pi$$
$$164$$ 0 0
$$165$$ −311688. −0.891272
$$166$$ 0 0
$$167$$ −43512.0 −0.120731 −0.0603654 0.998176i $$-0.519227\pi$$
−0.0603654 + 0.998176i $$0.519227\pi$$
$$168$$ 0 0
$$169$$ −175929. −0.473828
$$170$$ 0 0
$$171$$ −217404. −0.568561
$$172$$ 0 0
$$173$$ −18330.0 −0.0465637 −0.0232818 0.999729i $$-0.507412\pi$$
−0.0232818 + 0.999729i $$0.507412\pi$$
$$174$$ 0 0
$$175$$ −144991. −0.357887
$$176$$ 0 0
$$177$$ −247428. −0.593630
$$178$$ 0 0
$$179$$ 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 $$-0.642694\pi$$
−0.433423 + 0.901191i $$0.642694\pi$$
$$182$$ 0 0
$$183$$ 445302. 0.982940
$$184$$ 0 0
$$185$$ −423852. −0.910510
$$186$$ 0 0
$$187$$ 55944.0 0.116990
$$188$$ 0 0
$$189$$ −35721.0 −0.0727393
$$190$$ 0 0
$$191$$ 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 $$0.452583\pi$$
$$194$$ 0 0
$$195$$ −310284. −0.584350
$$196$$ 0 0
$$197$$ 154422. 0.283494 0.141747 0.989903i $$-0.454728\pi$$
0.141747 + 0.989903i $$0.454728\pi$$
$$198$$ 0 0
$$199$$ 366856. 0.656694 0.328347 0.944557i $$-0.393508\pi$$
0.328347 + 0.944557i $$0.393508\pi$$
$$200$$ 0 0
$$201$$ 534204. 0.932646
$$202$$ 0 0
$$203$$ 266658. 0.454166
$$204$$ 0 0
$$205$$ 621036. 1.03212
$$206$$ 0 0
$$207$$ −340200. −0.551834
$$208$$ 0 0
$$209$$ 1.19170e6 1.88712
$$210$$ 0 0
$$211$$ −520244. −0.804453 −0.402227 0.915540i $$-0.631764\pi$$
−0.402227 + 0.915540i $$0.631764\pi$$
$$212$$ 0 0
$$213$$ −288360. −0.435498
$$214$$ 0 0
$$215$$ 898872. 1.32618
$$216$$ 0 0
$$217$$ 3920.00 0.00565115
$$218$$ 0 0
$$219$$ −556614. −0.784231
$$220$$ 0 0
$$221$$ 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
$$225$$ 239679. 0.315627
$$226$$ 0 0
$$227$$ −288588. −0.371718 −0.185859 0.982576i $$-0.559507\pi$$
−0.185859 + 0.982576i $$0.559507\pi$$
$$228$$ 0 0
$$229$$ 772190. 0.973051 0.486525 0.873666i $$-0.338264\pi$$
0.486525 + 0.873666i $$0.338264\pi$$
$$230$$ 0 0
$$231$$ 195804. 0.241430
$$232$$ 0 0
$$233$$ 252234. 0.304378 0.152189 0.988351i $$-0.451368\pi$$
0.152189 + 0.988351i $$0.451368\pi$$
$$234$$ 0 0
$$235$$ 1.08576e6 1.28252
$$236$$ 0 0
$$237$$ 591984. 0.684603
$$238$$ 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 $$-0.525870\pi$$
−0.0811825 + 0.996699i $$0.525870\pi$$
$$242$$ 0 0
$$243$$ 59049.0 0.0641500
$$244$$ 0 0
$$245$$ 187278. 0.199329
$$246$$ 0 0
$$247$$ 1.18633e6 1.23726
$$248$$ 0 0
$$249$$ −361692. −0.369692
$$250$$ 0 0
$$251$$ −607860. −0.609003 −0.304501 0.952512i $$-0.598490\pi$$
−0.304501 + 0.952512i $$0.598490\pi$$
$$252$$ 0 0
$$253$$ 1.86480e6 1.83160
$$254$$ 0 0
$$255$$ −88452.0 −0.0851838
$$256$$ 0 0
$$257$$ 95586.0 0.0902737 0.0451369 0.998981i $$-0.485628\pi$$
0.0451369 + 0.998981i $$0.485628\pi$$
$$258$$ 0 0
$$259$$ 266266. 0.246642
$$260$$ 0 0
$$261$$ −440802. −0.400537
$$262$$ 0 0
$$263$$ 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
$$267$$ −71766.0 −0.0616085
$$268$$ 0 0
$$269$$ 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 $$0.470535\pi$$
$$272$$ 0 0
$$273$$ 194922. 0.158290
$$274$$ 0 0
$$275$$ −1.31380e6 −1.04760
$$276$$ 0 0
$$277$$ −342778. −0.268419 −0.134210 0.990953i $$-0.542850\pi$$
−0.134210 + 0.990953i $$0.542850\pi$$
$$278$$ 0 0
$$279$$ −6480.00 −0.00498384
$$280$$ 0 0
$$281$$ 480378. 0.362925 0.181463 0.983398i $$-0.441917\pi$$
0.181463 + 0.983398i $$0.441917\pi$$
$$282$$ 0 0
$$283$$ 29980.0 0.0222518 0.0111259 0.999938i $$-0.496458\pi$$
0.0111259 + 0.999938i $$0.496458\pi$$
$$284$$ 0 0
$$285$$ −1.88417e6 −1.37407
$$286$$ 0 0
$$287$$ −390138. −0.279585
$$288$$ 0 0
$$289$$ −1.40398e6 −0.988819
$$290$$ 0 0
$$291$$ −1.29296e6 −0.895060
$$292$$ 0 0
$$293$$ −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
$$296$$ 0 0
$$297$$ −323676. −0.212921
$$298$$ 0 0
$$299$$ 1.85640e6 1.20086
$$300$$ 0 0
$$301$$ −564676. −0.359239
$$302$$ 0 0
$$303$$ −24354.0 −0.0152393
$$304$$ 0 0
$$305$$ 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
$$309$$ −1.18591e6 −0.706572
$$310$$ 0 0
$$311$$ −1.83718e6 −1.07708 −0.538542 0.842598i $$-0.681025\pi$$
−0.538542 + 0.842598i $$0.681025\pi$$
$$312$$ 0 0
$$313$$ −365494. −0.210872 −0.105436 0.994426i $$-0.533624\pi$$
−0.105436 + 0.994426i $$0.533624\pi$$
$$314$$ 0 0
$$315$$ −309582. −0.175792
$$316$$ 0 0
$$317$$ −28338.0 −0.0158388 −0.00791938 0.999969i $$-0.502521\pi$$
−0.00791938 + 0.999969i $$0.502521\pi$$
$$318$$ 0 0
$$319$$ 2.41625e6 1.32943
$$320$$ 0 0
$$321$$ 1.16024e6 0.628473
$$322$$ 0 0
$$323$$ 338184. 0.180363
$$324$$ 0 0
$$325$$ −1.30788e6 −0.686845
$$326$$ 0 0
$$327$$ −908802. −0.470002
$$328$$ 0 0
$$329$$ −682080. −0.347413
$$330$$ 0 0
$$331$$ −1.93392e6 −0.970214 −0.485107 0.874455i $$-0.661219\pi$$
−0.485107 + 0.874455i $$0.661219\pi$$
$$332$$ 0 0
$$333$$ −440154. −0.217517
$$334$$ 0 0
$$335$$ 4.62977e6 2.25397
$$336$$ 0 0
$$337$$ −1.88817e6 −0.905664 −0.452832 0.891596i $$-0.649586\pi$$
−0.452832 + 0.891596i $$0.649586\pi$$
$$338$$ 0 0
$$339$$ 1.98131e6 0.936384
$$340$$ 0 0
$$341$$ 35520.0 0.0165420
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ −2.94840e6 −1.33364
$$346$$ 0 0
$$347$$ −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 $$-0.554876\pi$$
−0.171546 + 0.985176i $$0.554876\pi$$
$$350$$ 0 0
$$351$$ −322218. −0.139599
$$352$$ 0 0
$$353$$ 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
$$356$$ 0 0
$$357$$ 55566.0 0.0230748
$$358$$ 0 0
$$359$$ −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
$$362$$ 0 0
$$363$$ 324765. 0.129361
$$364$$ 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 $$0.551899\pi$$
$$368$$ 0 0
$$369$$ 644922. 0.246571
$$370$$ 0 0
$$371$$ 470106. 0.177321
$$372$$ 0 0
$$373$$ −1.51993e6 −0.565655 −0.282827 0.959171i $$-0.591272\pi$$
−0.282827 + 0.959171i $$0.591272\pi$$
$$374$$ 0 0
$$375$$ −116532. −0.0427924
$$376$$ 0 0
$$377$$ 2.40536e6 0.871620
$$378$$ 0 0
$$379$$ −2.64465e6 −0.945737 −0.472869 0.881133i $$-0.656781\pi$$
−0.472869 + 0.881133i $$0.656781\pi$$
$$380$$ 0 0
$$381$$ 668880. 0.236067
$$382$$ 0 0
$$383$$ −2.01336e6 −0.701333 −0.350667 0.936500i $$-0.614045\pi$$
−0.350667 + 0.936500i $$0.614045\pi$$
$$384$$ 0 0
$$385$$ 1.69697e6 0.583474
$$386$$ 0 0
$$387$$ 933444. 0.316819
$$388$$ 0 0
$$389$$ −726234. −0.243334 −0.121667 0.992571i $$-0.538824\pi$$
−0.121667 + 0.992571i $$0.538824\pi$$
$$390$$ 0 0
$$391$$ 529200. 0.175056
$$392$$ 0 0
$$393$$ 1.39784e6 0.456538
$$394$$ 0 0
$$395$$ 5.13053e6 1.65451
$$396$$ 0 0
$$397$$ 4.57578e6 1.45710 0.728549 0.684993i $$-0.240195\pi$$
0.728549 + 0.684993i $$0.240195\pi$$
$$398$$ 0 0
$$399$$ 1.18364e6 0.372211
$$400$$ 0 0
$$401$$ −33870.0 −0.0105185 −0.00525926 0.999986i $$-0.501674\pi$$
−0.00525926 + 0.999986i $$0.501674\pi$$
$$402$$ 0 0
$$403$$ 35360.0 0.0108455
$$404$$ 0 0
$$405$$ 511758. 0.155034
$$406$$ 0 0
$$407$$ 2.41270e6 0.721966
$$408$$ 0 0
$$409$$ −5.86178e6 −1.73269 −0.866346 0.499444i $$-0.833538\pi$$
−0.866346 + 0.499444i $$0.833538\pi$$
$$410$$ 0 0
$$411$$ −2.37821e6 −0.694459
$$412$$ 0 0
$$413$$ 1.34711e6 0.388622
$$414$$ 0 0
$$415$$ −3.13466e6 −0.893451
$$416$$ 0 0
$$417$$ −2.02151e6 −0.569292
$$418$$ 0 0
$$419$$ −302748. −0.0842454 −0.0421227 0.999112i $$-0.513412\pi$$
−0.0421227 + 0.999112i $$0.513412\pi$$
$$420$$ 0 0
$$421$$ −5.36708e6 −1.47582 −0.737909 0.674900i $$-0.764187\pi$$
−0.737909 + 0.674900i $$0.764187\pi$$
$$422$$ 0 0
$$423$$ 1.12752e6 0.306389
$$424$$ 0 0
$$425$$ −372834. −0.100125
$$426$$ 0 0
$$427$$ −2.42442e6 −0.643485
$$428$$ 0 0
$$429$$ 1.76623e6 0.463345
$$430$$ 0 0
$$431$$ −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 $$-0.655524\pi$$
−0.469383 + 0.882995i $$0.655524\pi$$
$$434$$ 0 0
$$435$$ −3.82028e6 −0.967994
$$436$$ 0 0
$$437$$ 1.12728e7 2.82376
$$438$$ 0 0
$$439$$ 2.53674e6 0.628225 0.314113 0.949386i $$-0.398293\pi$$
0.314113 + 0.949386i $$0.398293\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ 0 0
$$443$$ −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
$$446$$ 0 0
$$447$$ −738666. −0.174856
$$448$$ 0 0
$$449$$ 5.65965e6 1.32487 0.662436 0.749119i $$-0.269523\pi$$
0.662436 + 0.749119i $$0.269523\pi$$
$$450$$ 0 0
$$451$$ −3.53513e6 −0.818397
$$452$$ 0 0
$$453$$ 2.58329e6 0.591463
$$454$$ 0 0
$$455$$ 1.68932e6 0.382547
$$456$$ 0 0
$$457$$ −6.46159e6 −1.44727 −0.723634 0.690184i $$-0.757530\pi$$
−0.723634 + 0.690184i $$0.757530\pi$$
$$458$$ 0 0
$$459$$ −91854.0 −0.0203501
$$460$$ 0 0
$$461$$ −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 $$-0.335835\pi$$
0.493179 + 0.869928i $$0.335835\pi$$
$$464$$ 0 0
$$465$$ −56160.0 −0.0120447
$$466$$ 0 0
$$467$$ −2.01136e6 −0.426773 −0.213386 0.976968i $$-0.568449\pi$$
−0.213386 + 0.976968i $$0.568449\pi$$
$$468$$ 0 0
$$469$$ −2.90844e6 −0.610560
$$470$$ 0 0
$$471$$ 1.16890e6 0.242787
$$472$$ 0 0
$$473$$ −5.11666e6 −1.05156
$$474$$ 0 0
$$475$$ −7.94196e6 −1.61508
$$476$$ 0 0
$$477$$ −777114. −0.156383
$$478$$ 0 0
$$479$$ 7.60402e6 1.51427 0.757137 0.653257i $$-0.226598\pi$$
0.757137 + 0.653257i $$0.226598\pi$$
$$480$$ 0 0
$$481$$ 2.40183e6 0.473347
$$482$$ 0 0
$$483$$ 1.85220e6 0.361260
$$484$$ 0 0
$$485$$ −1.12056e7 −2.16313
$$486$$ 0 0
$$487$$ −673112. −0.128607 −0.0643035 0.997930i $$-0.520483\pi$$
−0.0643035 + 0.997930i $$0.520483\pi$$
$$488$$ 0 0
$$489$$ −4.99756e6 −0.945117
$$490$$ 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
$$494$$ 0 0
$$495$$ −2.80519e6 −0.514576
$$496$$ 0 0
$$497$$ 1.56996e6 0.285100
$$498$$ 0 0
$$499$$ −6.08152e6 −1.09335 −0.546677 0.837343i $$-0.684108\pi$$
−0.546677 + 0.837343i $$0.684108\pi$$
$$500$$ 0 0
$$501$$ −391608. −0.0697039
$$502$$ 0 0
$$503$$ 846216. 0.149129 0.0745644 0.997216i $$-0.476243\pi$$
0.0745644 + 0.997216i $$0.476243\pi$$
$$504$$ 0 0
$$505$$ −211068. −0.0368293
$$506$$ 0 0
$$507$$ −1.58336e6 −0.273565
$$508$$ 0 0
$$509$$ −7.66785e6 −1.31183 −0.655917 0.754833i $$-0.727718\pi$$
−0.655917 + 0.754833i $$0.727718\pi$$
$$510$$ 0 0
$$511$$ 3.03045e6 0.513400
$$512$$ 0 0
$$513$$ −1.95664e6 −0.328259
$$514$$ 0 0
$$515$$ −1.02779e7 −1.70760
$$516$$ 0 0
$$517$$ −6.18048e6 −1.01694
$$518$$ 0 0
$$519$$ −164970. −0.0268835
$$520$$ 0 0
$$521$$ −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 $$0.294839\pi$$
$$524$$ 0 0
$$525$$ −1.30492e6 −0.206626
$$526$$ 0 0
$$527$$ 10080.0 0.00158101
$$528$$ 0 0
$$529$$ 1.12037e7 1.74069
$$530$$ 0 0
$$531$$ −2.22685e6 −0.342732
$$532$$ 0 0
$$533$$ −3.51920e6 −0.536570
$$534$$ 0 0
$$535$$ 1.00554e7 1.51886
$$536$$ 0 0
$$537$$ 1.37992e6 0.206499
$$538$$ 0 0
$$539$$ −1.06604e6 −0.158053
$$540$$ 0 0
$$541$$ 7.34325e6 1.07869 0.539343 0.842086i $$-0.318673\pi$$
0.539343 + 0.842086i $$0.318673\pi$$
$$542$$ 0 0
$$543$$ −3.43859e6 −0.500474
$$544$$ 0 0
$$545$$ −7.87628e6 −1.13587
$$546$$ 0 0
$$547$$ −2.18296e6 −0.311945 −0.155973 0.987761i $$-0.549851\pi$$
−0.155973 + 0.987761i $$0.549851\pi$$
$$548$$ 0 0
$$549$$ 4.00772e6 0.567501
$$550$$ 0 0
$$551$$ 1.46063e7 2.04957
$$552$$ 0 0
$$553$$ −3.22302e6 −0.448178
$$554$$ 0 0
$$555$$ −3.81467e6 −0.525683
$$556$$ 0 0
$$557$$ 1.25466e7 1.71351 0.856755 0.515724i $$-0.172477\pi$$
0.856755 + 0.515724i $$0.172477\pi$$
$$558$$ 0 0
$$559$$ −5.09361e6 −0.689439
$$560$$ 0 0
$$561$$ 503496. 0.0675443
$$562$$ 0 0
$$563$$ −5.15972e6 −0.686050 −0.343025 0.939326i $$-0.611451\pi$$
−0.343025 + 0.939326i $$0.611451\pi$$
$$564$$ 0 0
$$565$$ 1.71714e7 2.26300
$$566$$ 0 0
$$567$$ −321489. −0.0419961
$$568$$ 0 0
$$569$$ 1.17452e7 1.52083 0.760414 0.649439i $$-0.224996\pi$$
0.760414 + 0.649439i $$0.224996\pi$$
$$570$$ 0 0
$$571$$ 7.54728e6 0.968725 0.484362 0.874867i $$-0.339052\pi$$
0.484362 + 0.874867i $$0.339052\pi$$
$$572$$ 0 0
$$573$$ 2.46067e6 0.313089
$$574$$ 0 0
$$575$$ −1.24278e7 −1.56756
$$576$$ 0 0
$$577$$ 9.28483e6 1.16101 0.580503 0.814258i $$-0.302856\pi$$
0.580503 + 0.814258i $$0.302856\pi$$
$$578$$ 0 0
$$579$$ 1.38242e6 0.171373
$$580$$ 0 0
$$581$$ 1.96921e6 0.242020
$$582$$ 0 0
$$583$$ 4.25974e6 0.519053
$$584$$ 0 0
$$585$$ −2.79256e6 −0.337374
$$586$$ 0 0
$$587$$ −1.47623e6 −0.176831 −0.0884155 0.996084i $$-0.528180\pi$$
−0.0884155 + 0.996084i $$0.528180\pi$$
$$588$$ 0 0
$$589$$ 214720. 0.0255026
$$590$$ 0 0
$$591$$ 1.38980e6 0.163675
$$592$$ 0 0
$$593$$ −1.24007e7 −1.44813 −0.724067 0.689729i $$-0.757730\pi$$
−0.724067 + 0.689729i $$0.757730\pi$$
$$594$$ 0 0
$$595$$ 481572. 0.0557659
$$596$$ 0 0
$$597$$ 3.30170e6 0.379142
$$598$$ 0 0
$$599$$ 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 $$-0.327758\pi$$
0.515092 + 0.857135i $$0.327758\pi$$
$$602$$ 0 0
$$603$$ 4.80784e6 0.538464
$$604$$ 0 0
$$605$$ 2.81463e6 0.312632
$$606$$ 0 0
$$607$$ 5.67914e6 0.625620 0.312810 0.949816i $$-0.398730\pi$$
0.312810 + 0.949816i $$0.398730\pi$$
$$608$$ 0 0
$$609$$ 2.39992e6 0.262213
$$610$$ 0 0
$$611$$ −6.15264e6 −0.666743
$$612$$ 0 0
$$613$$ −1.40106e7 −1.50593 −0.752966 0.658060i $$-0.771377\pi$$
−0.752966 + 0.658060i $$0.771377\pi$$
$$614$$ 0 0
$$615$$ 5.58932e6 0.595897
$$616$$ 0 0
$$617$$ −253686. −0.0268277 −0.0134139 0.999910i $$-0.504270\pi$$
−0.0134139 + 0.999910i $$0.504270\pi$$
$$618$$ 0 0
$$619$$ −4.30034e6 −0.451103 −0.225552 0.974231i $$-0.572418\pi$$
−0.225552 + 0.974231i $$0.572418\pi$$
$$620$$ 0 0
$$621$$ −3.06180e6 −0.318602
$$622$$ 0 0
$$623$$ 390726. 0.0403322
$$624$$ 0 0
$$625$$ −1.02568e7 −1.05030
$$626$$ 0 0
$$627$$ 1.07253e7 1.08953
$$628$$ 0 0
$$629$$ 684684. 0.0690023
$$630$$ 0 0
$$631$$ −1.04150e7 −1.04132 −0.520662 0.853763i $$-0.674315\pi$$
−0.520662 + 0.853763i $$0.674315\pi$$
$$632$$ 0 0
$$633$$ −4.68220e6 −0.464451
$$634$$ 0 0
$$635$$ 5.79696e6 0.570514
$$636$$ 0 0
$$637$$ −1.06124e6 −0.103625
$$638$$ 0 0
$$639$$ −2.59524e6 −0.251435
$$640$$ 0 0
$$641$$ 4.52714e6 0.435190 0.217595 0.976039i $$-0.430179\pi$$
0.217595 + 0.976039i $$0.430179\pi$$
$$642$$ 0 0
$$643$$ −1.49687e7 −1.42776 −0.713882 0.700266i $$-0.753065\pi$$
−0.713882 + 0.700266i $$0.753065\pi$$
$$644$$ 0 0
$$645$$ 8.08985e6 0.765669
$$646$$ 0 0
$$647$$ 1.73020e7 1.62493 0.812465 0.583010i $$-0.198125\pi$$
0.812465 + 0.583010i $$0.198125\pi$$
$$648$$ 0 0
$$649$$ 1.22064e7 1.13757
$$650$$ 0 0
$$651$$ 35280.0 0.00326269
$$652$$ 0 0
$$653$$ 4.07470e6 0.373949 0.186975 0.982365i $$-0.440132\pi$$
0.186975 + 0.982365i $$0.440132\pi$$
$$654$$ 0 0
$$655$$ 1.21146e7 1.10334
$$656$$ 0 0
$$657$$ −5.00953e6 −0.452776
$$658$$ 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 $$-0.238988\pi$$
0.731142 + 0.682225i $$0.238988\pi$$
$$662$$ 0 0
$$663$$ 501228. 0.0442845
$$664$$ 0 0
$$665$$ 1.02582e7 0.899537
$$666$$ 0 0
$$667$$ 2.28564e7 1.98927
$$668$$ 0 0
$$669$$ −2.74262e6 −0.236920
$$670$$ 0 0
$$671$$ −2.19682e7 −1.88360
$$672$$ 0 0
$$673$$ 5.50675e6 0.468660 0.234330 0.972157i $$-0.424710\pi$$
0.234330 + 0.972157i $$0.424710\pi$$
$$674$$ 0 0
$$675$$ 2.15711e6 0.182227
$$676$$ 0 0
$$677$$ 1.83957e7 1.54257 0.771286 0.636488i $$-0.219614\pi$$
0.771286 + 0.636488i $$0.219614\pi$$
$$678$$ 0 0
$$679$$ 7.03944e6 0.585954
$$680$$ 0 0
$$681$$ −2.59729e6 −0.214612
$$682$$ 0 0
$$683$$ −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
$$687$$ 6.94971e6 0.561791
$$688$$ 0 0
$$689$$ 4.24055e6 0.340309
$$690$$ 0 0
$$691$$ 5.36314e6 0.427291 0.213646 0.976911i $$-0.431466\pi$$
0.213646 + 0.976911i $$0.431466\pi$$
$$692$$ 0 0
$$693$$ 1.76224e6 0.139390
$$694$$ 0 0
$$695$$ −1.75197e7 −1.37583
$$696$$ 0 0
$$697$$ −1.00321e6 −0.0782187
$$698$$ 0 0
$$699$$ 2.27011e6 0.175733
$$700$$ 0 0
$$701$$ −2.12606e7 −1.63411 −0.817054 0.576561i $$-0.804394\pi$$
−0.817054 + 0.576561i $$0.804394\pi$$
$$702$$ 0 0
$$703$$ 1.45849e7 1.11305
$$704$$ 0 0
$$705$$ 9.77184e6 0.740463
$$706$$ 0 0
$$707$$ 132594. 0.00997643
$$708$$ 0 0
$$709$$ 2.07729e6 0.155196 0.0775980 0.996985i $$-0.475275\pi$$
0.0775980 + 0.996985i $$0.475275\pi$$
$$710$$ 0 0
$$711$$ 5.32786e6 0.395256
$$712$$ 0 0
$$713$$ 336000. 0.0247523
$$714$$ 0 0
$$715$$ 1.53073e7 1.11979
$$716$$ 0 0
$$717$$ 1.30602e7 0.948752
$$718$$ 0 0
$$719$$ −4.23619e6 −0.305600 −0.152800 0.988257i $$-0.548829\pi$$
−0.152800 + 0.988257i $$0.548829\pi$$
$$720$$ 0 0
$$721$$ 6.45663e6 0.462560
$$722$$ 0 0
$$723$$ −1.31758e6 −0.0937415
$$724$$ 0 0
$$725$$ −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
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −1.45202e6 −0.100503
$$732$$ 0 0
$$733$$ −1.48892e7 −1.02355 −0.511777 0.859118i $$-0.671013\pi$$
−0.511777 + 0.859118i $$0.671013\pi$$
$$734$$ 0 0
$$735$$ 1.68550e6 0.115083
$$736$$ 0 0
$$737$$ −2.63541e7 −1.78722
$$738$$ 0 0
$$739$$ −6.99324e6 −0.471050 −0.235525 0.971868i $$-0.575681\pi$$
−0.235525 + 0.971868i $$0.575681\pi$$
$$740$$ 0 0
$$741$$ 1.06770e7 0.714335
$$742$$ 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
$$746$$ 0 0
$$747$$ −3.25523e6 −0.213442
$$748$$ 0 0
$$749$$ −6.31688e6 −0.411432
$$750$$ 0 0
$$751$$ −1.95361e7 −1.26398 −0.631988 0.774978i $$-0.717761\pi$$
−0.631988 + 0.774978i $$0.717761\pi$$
$$752$$ 0 0
$$753$$ −5.47074e6 −0.351608
$$754$$ 0 0
$$755$$ 2.23885e7 1.42941
$$756$$ 0 0
$$757$$ 1.25183e6 0.0793973 0.0396986 0.999212i $$-0.487360\pi$$
0.0396986 + 0.999212i $$0.487360\pi$$
$$758$$ 0 0
$$759$$ 1.67832e7 1.05748
$$760$$ 0 0
$$761$$ 2.04472e7 1.27989 0.639944 0.768422i $$-0.278958\pi$$
0.639944 + 0.768422i $$0.278958\pi$$
$$762$$ 0 0
$$763$$ 4.94792e6 0.307689
$$764$$ 0 0
$$765$$ −796068. −0.0491809
$$766$$ 0 0
$$767$$ 1.21515e7 0.745831
$$768$$ 0 0
$$769$$ 2.21064e6 0.134804 0.0674020 0.997726i $$-0.478529\pi$$
0.0674020 + 0.997726i $$0.478529\pi$$
$$770$$ 0 0
$$771$$ 860274. 0.0521196
$$772$$ 0 0
$$773$$ 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
$$776$$ 0 0
$$777$$ 2.39639e6 0.142399
$$778$$ 0 0
$$779$$ −2.13700e7 −1.26171
$$780$$ 0 0
$$781$$ 1.42258e7 0.834541
$$782$$ 0 0
$$783$$ −3.96722e6 −0.231250
$$784$$ 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 $$0.372504\pi$$
$$788$$ 0 0
$$789$$ 1.98031e7 1.13251
$$790$$ 0 0
$$791$$ −1.07872e7 −0.613007
$$792$$ 0 0
$$793$$ −2.18693e7 −1.23496
$$794$$ 0 0
$$795$$ −6.73499e6 −0.377937
$$796$$ 0 0
$$797$$ −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
$$800$$ 0 0
$$801$$ −645894. −0.0355697
$$802$$ 0 0
$$803$$ 2.74596e7 1.50282
$$804$$ 0 0
$$805$$ 1.60524e7 0.873072
$$806$$ 0 0
$$807$$ 1.59322e7 0.861177
$$808$$ 0 0
$$809$$ 1.55237e7 0.833920 0.416960 0.908925i $$-0.363095\pi$$
0.416960 + 0.908925i $$0.363095\pi$$
$$810$$ 0 0
$$811$$ 2.66262e7 1.42153 0.710766 0.703429i $$-0.248349\pi$$
0.710766 + 0.703429i $$0.248349\pi$$
$$812$$ 0 0
$$813$$ 2.01154e6 0.106734
$$814$$ 0 0
$$815$$ −4.33122e7 −2.28410
$$816$$ 0 0
$$817$$ −3.09304e7 −1.62118
$$818$$ 0 0
$$819$$ 1.75430e6 0.0913889
$$820$$ 0 0
$$821$$ −1.23891e7 −0.641477 −0.320739 0.947168i $$-0.603931\pi$$
−0.320739 + 0.947168i $$0.603931\pi$$
$$822$$ 0 0
$$823$$ 3.65630e6 0.188166 0.0940831 0.995564i $$-0.470008\pi$$
0.0940831 + 0.995564i $$0.470008\pi$$
$$824$$ 0 0
$$825$$ −1.18242e7 −0.604833
$$826$$ 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 $$-0.320855\pi$$
0.533558 + 0.845763i $$0.320855\pi$$
$$830$$ 0 0
$$831$$ −3.08500e6 −0.154972
$$832$$ 0 0
$$833$$ −302526. −0.0151060
$$834$$ 0 0
$$835$$ −3.39394e6 −0.168456
$$836$$ 0 0
$$837$$ −58320.0 −0.00287742
$$838$$ 0 0
$$839$$ −1.33947e7 −0.656944 −0.328472 0.944514i $$-0.606534\pi$$
−0.328472 + 0.944514i $$0.606534\pi$$
$$840$$ 0 0
$$841$$ 9.10422e6 0.443867
$$842$$ 0 0
$$843$$ 4.32340e6 0.209535
$$844$$ 0 0
$$845$$ −1.37225e7 −0.661135
$$846$$ 0 0
$$847$$ −1.76816e6 −0.0846865
$$848$$ 0 0
$$849$$ 269820. 0.0128471
$$850$$ 0 0
$$851$$ 2.28228e7 1.08030
$$852$$ 0 0
$$853$$ 3.01513e7 1.41884 0.709420 0.704786i $$-0.248957\pi$$
0.709420 + 0.704786i $$0.248957\pi$$
$$854$$ 0 0
$$855$$ −1.69575e7 −0.793317
$$856$$ 0 0
$$857$$ 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 $$0.434213\pi$$
$$860$$ 0 0
$$861$$ −3.51124e6 −0.161418
$$862$$ 0 0
$$863$$ 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
$$866$$ 0 0
$$867$$ −1.26358e7 −0.570895
$$868$$ 0 0
$$869$$ −2.92045e7 −1.31190
$$870$$ 0 0
$$871$$ −2.62354e7 −1.17177
$$872$$ 0 0
$$873$$ −1.16366e7 −0.516763
$$874$$ 0 0
$$875$$ 634452. 0.0280142
$$876$$ 0 0
$$877$$ −2.95788e7 −1.29862 −0.649310 0.760524i $$-0.724942\pi$$
−0.649310 + 0.760524i $$0.724942\pi$$
$$878$$ 0 0
$$879$$ −1.78259e6 −0.0778180
$$880$$ 0 0
$$881$$ 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 $$-0.601801\pi$$
−0.314394 + 0.949293i $$0.601801\pi$$
$$884$$ 0 0
$$885$$ −1.92994e7 −0.828295
$$886$$ 0 0
$$887$$ −1.61714e7 −0.690141 −0.345070 0.938577i $$-0.612145\pi$$
−0.345070 + 0.938577i $$0.612145\pi$$
$$888$$ 0 0
$$889$$ −3.64168e6 −0.154542
$$890$$ 0 0
$$891$$ −2.91308e6 −0.122930
$$892$$ 0 0
$$893$$ −3.73613e7 −1.56781
$$894$$ 0 0
$$895$$ 1.19593e7 0.499054
$$896$$ 0 0
$$897$$ 1.67076e7 0.693319
$$898$$ 0 0
$$899$$ 435360. 0.0179659
$$900$$ 0 0
$$901$$ 1.20884e6 0.0496087
$$902$$ 0 0
$$903$$ −5.08208e6 −0.207407
$$904$$ 0 0
$$905$$ −2.98011e7 −1.20952
$$906$$ 0 0
$$907$$ −3.14446e7 −1.26919 −0.634596 0.772844i $$-0.718833\pi$$
−0.634596 + 0.772844i $$0.718833\pi$$
$$908$$ 0 0
$$909$$ −219186. −0.00879839
$$910$$ 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
$$914$$ 0 0
$$915$$ 3.47336e7 1.37150
$$916$$ 0 0
$$917$$ −7.61048e6 −0.298875
$$918$$ 0 0
$$919$$ −4.14876e7 −1.62043 −0.810214 0.586134i $$-0.800649\pi$$
−0.810214 + 0.586134i $$0.800649\pi$$
$$920$$ 0 0
$$921$$ 9.41072e6 0.365573
$$922$$ 0 0
$$923$$ 1.41617e7 0.547155
$$924$$ 0 0
$$925$$ −1.60792e7 −0.617889
$$926$$ 0 0
$$927$$ −1.06732e7 −0.407939
$$928$$ 0 0
$$929$$ −1.78495e7 −0.678556 −0.339278 0.940686i $$-0.610183\pi$$
−0.339278 + 0.940686i $$0.610183\pi$$
$$930$$ 0 0
$$931$$ −6.44428e6 −0.243669
$$932$$ 0 0
$$933$$ −1.65346e7 −0.621855
$$934$$ 0 0
$$935$$ 4.36363e6 0.163237
$$936$$ 0 0
$$937$$ 2.96399e7 1.10288 0.551439 0.834215i $$-0.314079\pi$$
0.551439 + 0.834215i $$0.314079\pi$$
$$938$$ 0 0
$$939$$ −3.28945e6 −0.121747
$$940$$ 0 0
$$941$$ −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
$$944$$ 0 0
$$945$$ −2.78624e6 −0.101494
$$946$$ 0 0
$$947$$ −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
$$950$$ 0 0
$$951$$ −255042. −0.00914451
$$952$$ 0 0
$$953$$ −2.03264e7 −0.724983 −0.362491 0.931987i $$-0.618074\pi$$
−0.362491 + 0.931987i $$0.618074\pi$$
$$954$$ 0 0
$$955$$ 2.13258e7 0.756654
$$956$$ 0 0
$$957$$ 2.17462e7 0.767546
$$958$$ 0 0
$$959$$ 1.29481e7 0.454630
$$960$$ 0 0
$$961$$ −2.86228e7 −0.999776
$$962$$ 0 0
$$963$$ 1.04422e7 0.362849
$$964$$ 0 0
$$965$$ 1.19810e7 0.414165
$$966$$ 0 0
$$967$$ 3.66292e6 0.125968 0.0629841 0.998015i $$-0.479938\pi$$
0.0629841 + 0.998015i $$0.479938\pi$$
$$968$$ 0 0
$$969$$ 3.04366e6 0.104132
$$970$$ 0 0
$$971$$ −1.48741e6 −0.0506271 −0.0253136 0.999680i $$-0.508058\pi$$
−0.0253136 + 0.999680i $$0.508058\pi$$
$$972$$ 0 0
$$973$$ 1.10060e7 0.372689
$$974$$ 0 0
$$975$$ −1.17709e7 −0.396550
$$976$$ 0 0
$$977$$ 4.07930e7 1.36725 0.683627 0.729831i $$-0.260401\pi$$
0.683627 + 0.729831i $$0.260401\pi$$
$$978$$ 0 0
$$979$$ 3.54046e6 0.118060
$$980$$ 0 0
$$981$$ −8.17922e6 −0.271356
$$982$$ 0 0
$$983$$ 9.26326e6 0.305759 0.152880 0.988245i $$-0.451145\pi$$
0.152880 + 0.988245i $$0.451145\pi$$
$$984$$ 0 0
$$985$$ 1.20449e7 0.395561
$$986$$ 0 0
$$987$$ −6.13872e6 −0.200579
$$988$$ 0 0
$$989$$ −4.84008e7 −1.57348
$$990$$ 0 0
$$991$$ 5.22051e7 1.68861 0.844303 0.535866i $$-0.180015\pi$$
0.844303 + 0.535866i $$0.180015\pi$$
$$992$$ 0 0
$$993$$ −1.74052e7 −0.560153
$$994$$ 0 0
$$995$$ 2.86148e7 0.916289
$$996$$ 0 0
$$997$$ −1.86609e7 −0.594560 −0.297280 0.954790i $$-0.596079\pi$$
−0.297280 + 0.954790i $$0.596079\pi$$
$$998$$ 0 0
$$999$$ −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 336.6.a.r.1.1 1
3.2 odd 2 1008.6.a.c.1.1 1
4.3 odd 2 21.6.a.a.1.1 1
12.11 even 2 63.6.a.d.1.1 1
20.3 even 4 525.6.d.b.274.2 2
20.7 even 4 525.6.d.b.274.1 2
20.19 odd 2 525.6.a.d.1.1 1
28.3 even 6 147.6.e.i.79.1 2
28.11 odd 6 147.6.e.j.79.1 2
28.19 even 6 147.6.e.i.67.1 2
28.23 odd 6 147.6.e.j.67.1 2
28.27 even 2 147.6.a.b.1.1 1
84.83 odd 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 4.3 odd 2
63.6.a.d.1.1 1 12.11 even 2
147.6.a.b.1.1 1 28.27 even 2
147.6.e.i.67.1 2 28.19 even 6
147.6.e.i.79.1 2 28.3 even 6
147.6.e.j.67.1 2 28.23 odd 6
147.6.e.j.79.1 2 28.11 odd 6
336.6.a.r.1.1 1 1.1 even 1 trivial
441.6.a.j.1.1 1 84.83 odd 2
525.6.a.d.1.1 1 20.19 odd 2
525.6.d.b.274.1 2 20.7 even 4
525.6.d.b.274.2 2 20.3 even 4
1008.6.a.c.1.1 1 3.2 odd 2