# Properties

 Label 147.6.a.k.1.2 Level $147$ Weight $6$ Character 147.1 Self dual yes Analytic conductor $23.576$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.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: $$23.5764215125$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{249})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 62$$ x^2 - x - 62 Coefficient ring: $$\Z[a_1, a_2]$$ 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 1.2 Root $$-7.38987$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.38987 q^{2} +9.00000 q^{3} +8.83040 q^{4} -38.7291 q^{5} +57.5088 q^{6} -148.051 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+6.38987 q^{2} +9.00000 q^{3} +8.83040 q^{4} -38.7291 q^{5} +57.5088 q^{6} -148.051 q^{8} +81.0000 q^{9} -247.474 q^{10} -576.390 q^{11} +79.4736 q^{12} -391.491 q^{13} -348.562 q^{15} -1228.60 q^{16} +1329.70 q^{17} +517.579 q^{18} -942.474 q^{19} -341.993 q^{20} -3683.05 q^{22} -1632.08 q^{23} -1332.46 q^{24} -1625.06 q^{25} -2501.58 q^{26} +729.000 q^{27} -1463.54 q^{29} -2227.26 q^{30} +3912.42 q^{31} -3112.95 q^{32} -5187.51 q^{33} +8496.61 q^{34} +715.262 q^{36} -16300.3 q^{37} -6022.28 q^{38} -3523.42 q^{39} +5733.86 q^{40} +13103.8 q^{41} +14733.5 q^{43} -5089.75 q^{44} -3137.05 q^{45} -10428.8 q^{46} +6814.52 q^{47} -11057.4 q^{48} -10383.9 q^{50} +11967.3 q^{51} -3457.02 q^{52} -2011.34 q^{53} +4658.21 q^{54} +22323.0 q^{55} -8482.26 q^{57} -9351.85 q^{58} -51453.1 q^{59} -3077.94 q^{60} -41097.8 q^{61} +24999.8 q^{62} +19423.8 q^{64} +15162.1 q^{65} -33147.5 q^{66} +50578.2 q^{67} +11741.8 q^{68} -14688.7 q^{69} +39970.6 q^{71} -11992.1 q^{72} +55686.6 q^{73} -104157. q^{74} -14625.5 q^{75} -8322.42 q^{76} -22514.2 q^{78} -63151.4 q^{79} +47582.4 q^{80} +6561.00 q^{81} +83731.7 q^{82} -45572.4 q^{83} -51498.1 q^{85} +94145.1 q^{86} -13171.9 q^{87} +85334.9 q^{88} -15686.7 q^{89} -20045.4 q^{90} -14411.9 q^{92} +35211.8 q^{93} +43543.9 q^{94} +36501.1 q^{95} -28016.5 q^{96} -3128.49 q^{97} -46687.6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 18 * q^3 + 65 * q^4 + 33 * q^5 - 27 * q^6 - 375 * q^8 + 162 * q^9 $$2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9} - 921 q^{10} - 1137 q^{11} + 585 q^{12} - 925 q^{13} + 297 q^{15} - 895 q^{16} + 324 q^{17} - 243 q^{18} - 2311 q^{19} + 3687 q^{20} + 1581 q^{22} + 1596 q^{23} - 3375 q^{24} + 395 q^{25} + 2508 q^{26} + 1458 q^{27} - 2217 q^{29} - 8289 q^{30} - 4294 q^{31} + 1017 q^{32} - 10233 q^{33} + 17940 q^{34} + 5265 q^{36} - 19109 q^{37} + 6828 q^{38} - 8325 q^{39} - 10545 q^{40} + 12858 q^{41} - 2771 q^{43} - 36579 q^{44} + 2673 q^{45} - 40740 q^{46} + 23160 q^{47} - 8055 q^{48} - 29352 q^{50} + 2916 q^{51} - 33424 q^{52} - 31653 q^{53} - 2187 q^{54} - 17889 q^{55} - 20799 q^{57} - 2277 q^{58} - 41097 q^{59} + 33183 q^{60} - 42052 q^{61} + 102057 q^{62} - 30031 q^{64} - 23106 q^{65} + 14229 q^{66} + 30763 q^{67} - 44748 q^{68} + 14364 q^{69} + 102096 q^{71} - 30375 q^{72} + 28577 q^{73} - 77784 q^{74} + 3555 q^{75} - 85192 q^{76} + 22572 q^{78} - 18464 q^{79} + 71511 q^{80} + 13122 q^{81} + 86040 q^{82} - 61179 q^{83} - 123636 q^{85} + 258510 q^{86} - 19953 q^{87} + 212565 q^{88} - 29322 q^{89} - 74601 q^{90} + 166908 q^{92} - 38646 q^{93} - 109938 q^{94} - 61662 q^{95} + 9153 q^{96} + 9791 q^{97} - 92097 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 18 * q^3 + 65 * q^4 + 33 * q^5 - 27 * q^6 - 375 * q^8 + 162 * q^9 - 921 * q^10 - 1137 * q^11 + 585 * q^12 - 925 * q^13 + 297 * q^15 - 895 * q^16 + 324 * q^17 - 243 * q^18 - 2311 * q^19 + 3687 * q^20 + 1581 * q^22 + 1596 * q^23 - 3375 * q^24 + 395 * q^25 + 2508 * q^26 + 1458 * q^27 - 2217 * q^29 - 8289 * q^30 - 4294 * q^31 + 1017 * q^32 - 10233 * q^33 + 17940 * q^34 + 5265 * q^36 - 19109 * q^37 + 6828 * q^38 - 8325 * q^39 - 10545 * q^40 + 12858 * q^41 - 2771 * q^43 - 36579 * q^44 + 2673 * q^45 - 40740 * q^46 + 23160 * q^47 - 8055 * q^48 - 29352 * q^50 + 2916 * q^51 - 33424 * q^52 - 31653 * q^53 - 2187 * q^54 - 17889 * q^55 - 20799 * q^57 - 2277 * q^58 - 41097 * q^59 + 33183 * q^60 - 42052 * q^61 + 102057 * q^62 - 30031 * q^64 - 23106 * q^65 + 14229 * q^66 + 30763 * q^67 - 44748 * q^68 + 14364 * q^69 + 102096 * q^71 - 30375 * q^72 + 28577 * q^73 - 77784 * q^74 + 3555 * q^75 - 85192 * q^76 + 22572 * q^78 - 18464 * q^79 + 71511 * q^80 + 13122 * q^81 + 86040 * q^82 - 61179 * q^83 - 123636 * q^85 + 258510 * q^86 - 19953 * q^87 + 212565 * q^88 - 29322 * q^89 - 74601 * q^90 + 166908 * q^92 - 38646 * q^93 - 109938 * q^94 - 61662 * q^95 + 9153 * q^96 + 9791 * q^97 - 92097 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 6.38987 1.12958 0.564790 0.825235i $$-0.308957\pi$$
0.564790 + 0.825235i $$0.308957\pi$$
$$3$$ 9.00000 0.577350
$$4$$ 8.83040 0.275950
$$5$$ −38.7291 −0.692807 −0.346403 0.938086i $$-0.612597\pi$$
−0.346403 + 0.938086i $$0.612597\pi$$
$$6$$ 57.5088 0.652163
$$7$$ 0 0
$$8$$ −148.051 −0.817872
$$9$$ 81.0000 0.333333
$$10$$ −247.474 −0.782580
$$11$$ −576.390 −1.43627 −0.718133 0.695906i $$-0.755003\pi$$
−0.718133 + 0.695906i $$0.755003\pi$$
$$12$$ 79.4736 0.159320
$$13$$ −391.491 −0.642486 −0.321243 0.946997i $$-0.604101\pi$$
−0.321243 + 0.946997i $$0.604101\pi$$
$$14$$ 0 0
$$15$$ −348.562 −0.399992
$$16$$ −1228.60 −1.19980
$$17$$ 1329.70 1.11592 0.557958 0.829869i $$-0.311585\pi$$
0.557958 + 0.829869i $$0.311585\pi$$
$$18$$ 517.579 0.376527
$$19$$ −942.474 −0.598943 −0.299471 0.954105i $$-0.596810\pi$$
−0.299471 + 0.954105i $$0.596810\pi$$
$$20$$ −341.993 −0.191180
$$21$$ 0 0
$$22$$ −3683.05 −1.62238
$$23$$ −1632.08 −0.643312 −0.321656 0.946857i $$-0.604239\pi$$
−0.321656 + 0.946857i $$0.604239\pi$$
$$24$$ −1332.46 −0.472199
$$25$$ −1625.06 −0.520019
$$26$$ −2501.58 −0.725739
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ −1463.54 −0.323155 −0.161577 0.986860i $$-0.551658\pi$$
−0.161577 + 0.986860i $$0.551658\pi$$
$$30$$ −2227.26 −0.451823
$$31$$ 3912.42 0.731208 0.365604 0.930770i $$-0.380862\pi$$
0.365604 + 0.930770i $$0.380862\pi$$
$$32$$ −3112.95 −0.537399
$$33$$ −5187.51 −0.829228
$$34$$ 8496.61 1.26052
$$35$$ 0 0
$$36$$ 715.262 0.0919833
$$37$$ −16300.3 −1.95746 −0.978729 0.205160i $$-0.934229\pi$$
−0.978729 + 0.205160i $$0.934229\pi$$
$$38$$ −6022.28 −0.676553
$$39$$ −3523.42 −0.370939
$$40$$ 5733.86 0.566627
$$41$$ 13103.8 1.21741 0.608707 0.793395i $$-0.291688\pi$$
0.608707 + 0.793395i $$0.291688\pi$$
$$42$$ 0 0
$$43$$ 14733.5 1.21516 0.607582 0.794257i $$-0.292140\pi$$
0.607582 + 0.794257i $$0.292140\pi$$
$$44$$ −5089.75 −0.396337
$$45$$ −3137.05 −0.230936
$$46$$ −10428.8 −0.726672
$$47$$ 6814.52 0.449977 0.224989 0.974361i $$-0.427765\pi$$
0.224989 + 0.974361i $$0.427765\pi$$
$$48$$ −11057.4 −0.692706
$$49$$ 0 0
$$50$$ −10383.9 −0.587403
$$51$$ 11967.3 0.644274
$$52$$ −3457.02 −0.177294
$$53$$ −2011.34 −0.0983550 −0.0491775 0.998790i $$-0.515660\pi$$
−0.0491775 + 0.998790i $$0.515660\pi$$
$$54$$ 4658.21 0.217388
$$55$$ 22323.0 0.995054
$$56$$ 0 0
$$57$$ −8482.26 −0.345800
$$58$$ −9351.85 −0.365029
$$59$$ −51453.1 −1.92434 −0.962170 0.272451i $$-0.912166\pi$$
−0.962170 + 0.272451i $$0.912166\pi$$
$$60$$ −3077.94 −0.110378
$$61$$ −41097.8 −1.41415 −0.707073 0.707141i $$-0.749985\pi$$
−0.707073 + 0.707141i $$0.749985\pi$$
$$62$$ 24999.8 0.825958
$$63$$ 0 0
$$64$$ 19423.8 0.592766
$$65$$ 15162.1 0.445119
$$66$$ −33147.5 −0.936679
$$67$$ 50578.2 1.37650 0.688250 0.725473i $$-0.258379\pi$$
0.688250 + 0.725473i $$0.258379\pi$$
$$68$$ 11741.8 0.307937
$$69$$ −14688.7 −0.371416
$$70$$ 0 0
$$71$$ 39970.6 0.941012 0.470506 0.882397i $$-0.344071\pi$$
0.470506 + 0.882397i $$0.344071\pi$$
$$72$$ −11992.1 −0.272624
$$73$$ 55686.6 1.22305 0.611524 0.791226i $$-0.290557\pi$$
0.611524 + 0.791226i $$0.290557\pi$$
$$74$$ −104157. −2.21110
$$75$$ −14625.5 −0.300233
$$76$$ −8322.42 −0.165278
$$77$$ 0 0
$$78$$ −22514.2 −0.419006
$$79$$ −63151.4 −1.13845 −0.569226 0.822181i $$-0.692757\pi$$
−0.569226 + 0.822181i $$0.692757\pi$$
$$80$$ 47582.4 0.831231
$$81$$ 6561.00 0.111111
$$82$$ 83731.7 1.37517
$$83$$ −45572.4 −0.726116 −0.363058 0.931766i $$-0.618267\pi$$
−0.363058 + 0.931766i $$0.618267\pi$$
$$84$$ 0 0
$$85$$ −51498.1 −0.773114
$$86$$ 94145.1 1.37262
$$87$$ −13171.9 −0.186574
$$88$$ 85334.9 1.17468
$$89$$ −15686.7 −0.209921 −0.104961 0.994476i $$-0.533472\pi$$
−0.104961 + 0.994476i $$0.533472\pi$$
$$90$$ −20045.4 −0.260860
$$91$$ 0 0
$$92$$ −14411.9 −0.177522
$$93$$ 35211.8 0.422163
$$94$$ 43543.9 0.508285
$$95$$ 36501.1 0.414951
$$96$$ −28016.5 −0.310268
$$97$$ −3128.49 −0.0337603 −0.0168801 0.999858i $$-0.505373\pi$$
−0.0168801 + 0.999858i $$0.505373\pi$$
$$98$$ 0 0
$$99$$ −46687.6 −0.478755
$$100$$ −14349.9 −0.143499
$$101$$ 169011. 1.64858 0.824292 0.566164i $$-0.191573\pi$$
0.824292 + 0.566164i $$0.191573\pi$$
$$102$$ 76469.5 0.727759
$$103$$ −112820. −1.04784 −0.523918 0.851769i $$-0.675530\pi$$
−0.523918 + 0.851769i $$0.675530\pi$$
$$104$$ 57960.5 0.525471
$$105$$ 0 0
$$106$$ −12852.2 −0.111100
$$107$$ −22309.5 −0.188378 −0.0941890 0.995554i $$-0.530026\pi$$
−0.0941890 + 0.995554i $$0.530026\pi$$
$$108$$ 6437.36 0.0531066
$$109$$ −83819.5 −0.675739 −0.337869 0.941193i $$-0.609706\pi$$
−0.337869 + 0.941193i $$0.609706\pi$$
$$110$$ 142641. 1.12399
$$111$$ −146703. −1.13014
$$112$$ 0 0
$$113$$ −40928.4 −0.301529 −0.150764 0.988570i $$-0.548174\pi$$
−0.150764 + 0.988570i $$0.548174\pi$$
$$114$$ −54200.5 −0.390608
$$115$$ 63208.9 0.445691
$$116$$ −12923.7 −0.0891746
$$117$$ −31710.8 −0.214162
$$118$$ −328779. −2.17369
$$119$$ 0 0
$$120$$ 51604.8 0.327142
$$121$$ 171174. 1.06286
$$122$$ −262610. −1.59739
$$123$$ 117934. 0.702875
$$124$$ 34548.2 0.201777
$$125$$ 183965. 1.05308
$$126$$ 0 0
$$127$$ 83270.1 0.458120 0.229060 0.973412i $$-0.426435\pi$$
0.229060 + 0.973412i $$0.426435\pi$$
$$128$$ 223730. 1.20698
$$129$$ 132601. 0.701575
$$130$$ 96883.7 0.502797
$$131$$ 166875. 0.849596 0.424798 0.905288i $$-0.360345\pi$$
0.424798 + 0.905288i $$0.360345\pi$$
$$132$$ −45807.8 −0.228825
$$133$$ 0 0
$$134$$ 323188. 1.55487
$$135$$ −28233.5 −0.133331
$$136$$ −196863. −0.912676
$$137$$ 38223.9 0.173994 0.0869969 0.996209i $$-0.472273\pi$$
0.0869969 + 0.996209i $$0.472273\pi$$
$$138$$ −93858.9 −0.419544
$$139$$ −106263. −0.466492 −0.233246 0.972418i $$-0.574935\pi$$
−0.233246 + 0.972418i $$0.574935\pi$$
$$140$$ 0 0
$$141$$ 61330.7 0.259795
$$142$$ 255407. 1.06295
$$143$$ 225652. 0.922780
$$144$$ −99516.3 −0.399934
$$145$$ 56681.7 0.223884
$$146$$ 355830. 1.38153
$$147$$ 0 0
$$148$$ −143938. −0.540160
$$149$$ 192556. 0.710543 0.355271 0.934763i $$-0.384388\pi$$
0.355271 + 0.934763i $$0.384388\pi$$
$$150$$ −93455.2 −0.339137
$$151$$ 141699. 0.505735 0.252868 0.967501i $$-0.418626\pi$$
0.252868 + 0.967501i $$0.418626\pi$$
$$152$$ 139534. 0.489858
$$153$$ 107706. 0.371972
$$154$$ 0 0
$$155$$ −151524. −0.506586
$$156$$ −31113.2 −0.102361
$$157$$ −565771. −1.83186 −0.915928 0.401342i $$-0.868544\pi$$
−0.915928 + 0.401342i $$0.868544\pi$$
$$158$$ −403529. −1.28597
$$159$$ −18102.1 −0.0567853
$$160$$ 120562. 0.372314
$$161$$ 0 0
$$162$$ 41923.9 0.125509
$$163$$ −430201. −1.26824 −0.634121 0.773233i $$-0.718638\pi$$
−0.634121 + 0.773233i $$0.718638\pi$$
$$164$$ 115712. 0.335946
$$165$$ 200907. 0.574495
$$166$$ −291201. −0.820206
$$167$$ 240265. 0.666653 0.333327 0.942811i $$-0.391829\pi$$
0.333327 + 0.942811i $$0.391829\pi$$
$$168$$ 0 0
$$169$$ −218028. −0.587212
$$170$$ −329066. −0.873294
$$171$$ −76340.4 −0.199648
$$172$$ 130103. 0.335324
$$173$$ 179300. 0.455476 0.227738 0.973722i $$-0.426867\pi$$
0.227738 + 0.973722i $$0.426867\pi$$
$$174$$ −84166.7 −0.210750
$$175$$ 0 0
$$176$$ 708151. 1.72323
$$177$$ −463078. −1.11102
$$178$$ −100236. −0.237123
$$179$$ −575559. −1.34263 −0.671317 0.741170i $$-0.734271\pi$$
−0.671317 + 0.741170i $$0.734271\pi$$
$$180$$ −27701.4 −0.0637267
$$181$$ −581006. −1.31821 −0.659105 0.752051i $$-0.729065\pi$$
−0.659105 + 0.752051i $$0.729065\pi$$
$$182$$ 0 0
$$183$$ −369880. −0.816458
$$184$$ 241630. 0.526147
$$185$$ 631297. 1.35614
$$186$$ 224998. 0.476867
$$187$$ −766426. −1.60275
$$188$$ 60174.9 0.124171
$$189$$ 0 0
$$190$$ 233237. 0.468721
$$191$$ −660560. −1.31017 −0.655087 0.755554i $$-0.727368\pi$$
−0.655087 + 0.755554i $$0.727368\pi$$
$$192$$ 174814. 0.342234
$$193$$ −557310. −1.07697 −0.538485 0.842635i $$-0.681003\pi$$
−0.538485 + 0.842635i $$0.681003\pi$$
$$194$$ −19990.7 −0.0381349
$$195$$ 136459. 0.256989
$$196$$ 0 0
$$197$$ −761400. −1.39781 −0.698904 0.715216i $$-0.746328\pi$$
−0.698904 + 0.715216i $$0.746328\pi$$
$$198$$ −298327. −0.540792
$$199$$ 135860. 0.243197 0.121598 0.992579i $$-0.461198\pi$$
0.121598 + 0.992579i $$0.461198\pi$$
$$200$$ 240591. 0.425309
$$201$$ 455204. 0.794723
$$202$$ 1.07996e6 1.86221
$$203$$ 0 0
$$204$$ 105676. 0.177787
$$205$$ −507499. −0.843433
$$206$$ −720906. −1.18361
$$207$$ −132198. −0.214437
$$208$$ 480985. 0.770856
$$209$$ 543232. 0.860240
$$210$$ 0 0
$$211$$ −991157. −1.53263 −0.766313 0.642467i $$-0.777911\pi$$
−0.766313 + 0.642467i $$0.777911\pi$$
$$212$$ −17761.0 −0.0271411
$$213$$ 359736. 0.543293
$$214$$ −142555. −0.212788
$$215$$ −570615. −0.841873
$$216$$ −107929. −0.157400
$$217$$ 0 0
$$218$$ −535596. −0.763301
$$219$$ 501180. 0.706128
$$220$$ 197121. 0.274585
$$221$$ −520566. −0.716960
$$222$$ −937413. −1.27658
$$223$$ −543344. −0.731666 −0.365833 0.930681i $$-0.619216\pi$$
−0.365833 + 0.930681i $$0.619216\pi$$
$$224$$ 0 0
$$225$$ −131630. −0.173340
$$226$$ −261527. −0.340601
$$227$$ −16.3341 −2.10393e−5 0 −1.05196e−5 1.00000i $$-0.500003\pi$$
−1.05196e−5 1.00000i $$0.500003\pi$$
$$228$$ −74901.8 −0.0954234
$$229$$ 77379.0 0.0975067 0.0487534 0.998811i $$-0.484475\pi$$
0.0487534 + 0.998811i $$0.484475\pi$$
$$230$$ 403896. 0.503443
$$231$$ 0 0
$$232$$ 216679. 0.264299
$$233$$ −103657. −0.125086 −0.0625432 0.998042i $$-0.519921\pi$$
−0.0625432 + 0.998042i $$0.519921\pi$$
$$234$$ −202628. −0.241913
$$235$$ −263920. −0.311747
$$236$$ −454351. −0.531021
$$237$$ −568362. −0.657286
$$238$$ 0 0
$$239$$ 689109. 0.780356 0.390178 0.920739i $$-0.372413\pi$$
0.390178 + 0.920739i $$0.372413\pi$$
$$240$$ 428242. 0.479911
$$241$$ −220296. −0.244323 −0.122161 0.992510i $$-0.538983\pi$$
−0.122161 + 0.992510i $$0.538983\pi$$
$$242$$ 1.09378e6 1.20058
$$243$$ 59049.0 0.0641500
$$244$$ −362910. −0.390234
$$245$$ 0 0
$$246$$ 753585. 0.793953
$$247$$ 368970. 0.384812
$$248$$ −579236. −0.598035
$$249$$ −410151. −0.419224
$$250$$ 1.17551e6 1.18954
$$251$$ −1.43641e6 −1.43912 −0.719558 0.694433i $$-0.755655\pi$$
−0.719558 + 0.694433i $$0.755655\pi$$
$$252$$ 0 0
$$253$$ 940714. 0.923966
$$254$$ 532085. 0.517483
$$255$$ −463482. −0.446358
$$256$$ 808042. 0.770609
$$257$$ 909197. 0.858668 0.429334 0.903146i $$-0.358748\pi$$
0.429334 + 0.903146i $$0.358748\pi$$
$$258$$ 847306. 0.792485
$$259$$ 0 0
$$260$$ 133887. 0.122830
$$261$$ −118547. −0.107718
$$262$$ 1.06631e6 0.959687
$$263$$ 749057. 0.667768 0.333884 0.942614i $$-0.391641\pi$$
0.333884 + 0.942614i $$0.391641\pi$$
$$264$$ 768014. 0.678203
$$265$$ 77897.4 0.0681410
$$266$$ 0 0
$$267$$ −141180. −0.121198
$$268$$ 446626. 0.379845
$$269$$ −669945. −0.564493 −0.282246 0.959342i $$-0.591080\pi$$
−0.282246 + 0.959342i $$0.591080\pi$$
$$270$$ −180408. −0.150608
$$271$$ −540659. −0.447199 −0.223599 0.974681i $$-0.571781\pi$$
−0.223599 + 0.974681i $$0.571781\pi$$
$$272$$ −1.63367e6 −1.33888
$$273$$ 0 0
$$274$$ 244246. 0.196540
$$275$$ 936668. 0.746885
$$276$$ −129707. −0.102492
$$277$$ −401910. −0.314723 −0.157362 0.987541i $$-0.550299\pi$$
−0.157362 + 0.987541i $$0.550299\pi$$
$$278$$ −679005. −0.526940
$$279$$ 316906. 0.243736
$$280$$ 0 0
$$281$$ −429139. −0.324214 −0.162107 0.986773i $$-0.551829\pi$$
−0.162107 + 0.986773i $$0.551829\pi$$
$$282$$ 391895. 0.293459
$$283$$ 340927. 0.253044 0.126522 0.991964i $$-0.459619\pi$$
0.126522 + 0.991964i $$0.459619\pi$$
$$284$$ 352957. 0.259672
$$285$$ 328510. 0.239572
$$286$$ 1.44188e6 1.04235
$$287$$ 0 0
$$288$$ −252149. −0.179133
$$289$$ 348246. 0.245268
$$290$$ 362188. 0.252895
$$291$$ −28156.5 −0.0194915
$$292$$ 491735. 0.337500
$$293$$ 388847. 0.264612 0.132306 0.991209i $$-0.457762\pi$$
0.132306 + 0.991209i $$0.457762\pi$$
$$294$$ 0 0
$$295$$ 1.99273e6 1.33319
$$296$$ 2.41328e6 1.60095
$$297$$ −420188. −0.276409
$$298$$ 1.23040e6 0.802615
$$299$$ 638945. 0.413319
$$300$$ −129149. −0.0828493
$$301$$ 0 0
$$302$$ 905435. 0.571268
$$303$$ 1.52110e6 0.951811
$$304$$ 1.15792e6 0.718612
$$305$$ 1.59168e6 0.979730
$$306$$ 688225. 0.420172
$$307$$ 2.35747e6 1.42758 0.713789 0.700361i $$-0.246978\pi$$
0.713789 + 0.700361i $$0.246978\pi$$
$$308$$ 0 0
$$309$$ −1.01538e6 −0.604969
$$310$$ −968220. −0.572229
$$311$$ 1.43663e6 0.842254 0.421127 0.907002i $$-0.361635\pi$$
0.421127 + 0.907002i $$0.361635\pi$$
$$312$$ 521645. 0.303381
$$313$$ 822800. 0.474715 0.237358 0.971422i $$-0.423719\pi$$
0.237358 + 0.971422i $$0.423719\pi$$
$$314$$ −3.61520e6 −2.06923
$$315$$ 0 0
$$316$$ −557652. −0.314156
$$317$$ −1.76693e6 −0.987580 −0.493790 0.869581i $$-0.664389\pi$$
−0.493790 + 0.869581i $$0.664389\pi$$
$$318$$ −115670. −0.0641435
$$319$$ 843572. 0.464136
$$320$$ −752264. −0.410672
$$321$$ −200785. −0.108760
$$322$$ 0 0
$$323$$ −1.25321e6 −0.668370
$$324$$ 57936.2 0.0306611
$$325$$ 636196. 0.334105
$$326$$ −2.74893e6 −1.43258
$$327$$ −754376. −0.390138
$$328$$ −1.94003e6 −0.995689
$$329$$ 0 0
$$330$$ 1.28377e6 0.648937
$$331$$ −3052.27 −0.00153128 −0.000765638 1.00000i $$-0.500244\pi$$
−0.000765638 1.00000i $$0.500244\pi$$
$$332$$ −402422. −0.200372
$$333$$ −1.32033e6 −0.652486
$$334$$ 1.53526e6 0.753038
$$335$$ −1.95885e6 −0.953649
$$336$$ 0 0
$$337$$ 2.02939e6 0.973398 0.486699 0.873570i $$-0.338201\pi$$
0.486699 + 0.873570i $$0.338201\pi$$
$$338$$ −1.39317e6 −0.663302
$$339$$ −368356. −0.174088
$$340$$ −454748. −0.213341
$$341$$ −2.25508e6 −1.05021
$$342$$ −487805. −0.225518
$$343$$ 0 0
$$344$$ −2.18130e6 −0.993848
$$345$$ 568880. 0.257320
$$346$$ 1.14570e6 0.514496
$$347$$ 3.78218e6 1.68624 0.843119 0.537727i $$-0.180717\pi$$
0.843119 + 0.537727i $$0.180717\pi$$
$$348$$ −116313. −0.0514850
$$349$$ −291147. −0.127953 −0.0639763 0.997951i $$-0.520378\pi$$
−0.0639763 + 0.997951i $$0.520378\pi$$
$$350$$ 0 0
$$351$$ −285397. −0.123646
$$352$$ 1.79427e6 0.771848
$$353$$ −385076. −0.164479 −0.0822394 0.996613i $$-0.526207\pi$$
−0.0822394 + 0.996613i $$0.526207\pi$$
$$354$$ −2.95901e6 −1.25498
$$355$$ −1.54803e6 −0.651939
$$356$$ −138520. −0.0579277
$$357$$ 0 0
$$358$$ −3.67775e6 −1.51661
$$359$$ −3.23014e6 −1.32277 −0.661385 0.750046i $$-0.730031\pi$$
−0.661385 + 0.750046i $$0.730031\pi$$
$$360$$ 464443. 0.188876
$$361$$ −1.58784e6 −0.641268
$$362$$ −3.71255e6 −1.48902
$$363$$ 1.54057e6 0.613641
$$364$$ 0 0
$$365$$ −2.15669e6 −0.847336
$$366$$ −2.36349e6 −0.922254
$$367$$ 479559. 0.185856 0.0929280 0.995673i $$-0.470377\pi$$
0.0929280 + 0.995673i $$0.470377\pi$$
$$368$$ 2.00517e6 0.771847
$$369$$ 1.06141e6 0.405805
$$370$$ 4.03390e6 1.53187
$$371$$ 0 0
$$372$$ 310934. 0.116496
$$373$$ 872666. 0.324770 0.162385 0.986727i $$-0.448081\pi$$
0.162385 + 0.986727i $$0.448081\pi$$
$$374$$ −4.89736e6 −1.81043
$$375$$ 1.65569e6 0.607996
$$376$$ −1.00889e6 −0.368024
$$377$$ 572965. 0.207622
$$378$$ 0 0
$$379$$ −2.43493e6 −0.870742 −0.435371 0.900251i $$-0.643383\pi$$
−0.435371 + 0.900251i $$0.643383\pi$$
$$380$$ 322320. 0.114506
$$381$$ 749431. 0.264496
$$382$$ −4.22089e6 −1.47995
$$383$$ 3.61169e6 1.25809 0.629047 0.777367i $$-0.283445\pi$$
0.629047 + 0.777367i $$0.283445\pi$$
$$384$$ 2.01357e6 0.696848
$$385$$ 0 0
$$386$$ −3.56114e6 −1.21652
$$387$$ 1.19341e6 0.405055
$$388$$ −27625.9 −0.00931615
$$389$$ −175232. −0.0587138 −0.0293569 0.999569i $$-0.509346\pi$$
−0.0293569 + 0.999569i $$0.509346\pi$$
$$390$$ 871954. 0.290290
$$391$$ −2.17018e6 −0.717882
$$392$$ 0 0
$$393$$ 1.50187e6 0.490515
$$394$$ −4.86525e6 −1.57893
$$395$$ 2.44579e6 0.788727
$$396$$ −412270. −0.132112
$$397$$ −1.88515e6 −0.600303 −0.300152 0.953892i $$-0.597037\pi$$
−0.300152 + 0.953892i $$0.597037\pi$$
$$398$$ 868126. 0.274710
$$399$$ 0 0
$$400$$ 1.99654e6 0.623920
$$401$$ 1.33983e6 0.416091 0.208046 0.978119i $$-0.433290\pi$$
0.208046 + 0.978119i $$0.433290\pi$$
$$402$$ 2.90869e6 0.897703
$$403$$ −1.53168e6 −0.469791
$$404$$ 1.49243e6 0.454927
$$405$$ −254101. −0.0769785
$$406$$ 0 0
$$407$$ 9.39535e6 2.81143
$$408$$ −1.77177e6 −0.526934
$$409$$ −6.58628e6 −1.94685 −0.973423 0.229013i $$-0.926450\pi$$
−0.973423 + 0.229013i $$0.926450\pi$$
$$410$$ −3.24285e6 −0.952725
$$411$$ 344015. 0.100455
$$412$$ −996247. −0.289150
$$413$$ 0 0
$$414$$ −844730. −0.242224
$$415$$ 1.76497e6 0.503058
$$416$$ 1.21869e6 0.345271
$$417$$ −956365. −0.269329
$$418$$ 3.47118e6 0.971710
$$419$$ 6.96869e6 1.93917 0.969585 0.244754i $$-0.0787071\pi$$
0.969585 + 0.244754i $$0.0787071\pi$$
$$420$$ 0 0
$$421$$ 3.84041e6 1.05602 0.528010 0.849238i $$-0.322938\pi$$
0.528010 + 0.849238i $$0.322938\pi$$
$$422$$ −6.33336e6 −1.73122
$$423$$ 551976. 0.149992
$$424$$ 297781. 0.0804418
$$425$$ −2.16084e6 −0.580297
$$426$$ 2.29866e6 0.613693
$$427$$ 0 0
$$428$$ −197002. −0.0519829
$$429$$ 2.03086e6 0.532767
$$430$$ −3.64615e6 −0.950963
$$431$$ 3.03636e6 0.787337 0.393668 0.919253i $$-0.371206\pi$$
0.393668 + 0.919253i $$0.371206\pi$$
$$432$$ −895647. −0.230902
$$433$$ 941529. 0.241332 0.120666 0.992693i $$-0.461497\pi$$
0.120666 + 0.992693i $$0.461497\pi$$
$$434$$ 0 0
$$435$$ 510135. 0.129259
$$436$$ −740160. −0.186470
$$437$$ 1.53819e6 0.385307
$$438$$ 3.20247e6 0.797627
$$439$$ −1.34109e6 −0.332122 −0.166061 0.986116i $$-0.553105\pi$$
−0.166061 + 0.986116i $$0.553105\pi$$
$$440$$ −3.30494e6 −0.813827
$$441$$ 0 0
$$442$$ −3.32635e6 −0.809864
$$443$$ −772341. −0.186982 −0.0934910 0.995620i $$-0.529803\pi$$
−0.0934910 + 0.995620i $$0.529803\pi$$
$$444$$ −1.29545e6 −0.311862
$$445$$ 607531. 0.145435
$$446$$ −3.47190e6 −0.826475
$$447$$ 1.73300e6 0.410232
$$448$$ 0 0
$$449$$ 2.25684e6 0.528304 0.264152 0.964481i $$-0.414908\pi$$
0.264152 + 0.964481i $$0.414908\pi$$
$$450$$ −841097. −0.195801
$$451$$ −7.55291e6 −1.74853
$$452$$ −361414. −0.0832069
$$453$$ 1.27529e6 0.291986
$$454$$ −104.373 −2.37655e−5 0
$$455$$ 0 0
$$456$$ 1.25580e6 0.282820
$$457$$ 4.28470e6 0.959688 0.479844 0.877354i $$-0.340693\pi$$
0.479844 + 0.877354i $$0.340693\pi$$
$$458$$ 494442. 0.110142
$$459$$ 969352. 0.214758
$$460$$ 558160. 0.122988
$$461$$ 3.10462e6 0.680387 0.340193 0.940355i $$-0.389507\pi$$
0.340193 + 0.940355i $$0.389507\pi$$
$$462$$ 0 0
$$463$$ −3.53386e6 −0.766121 −0.383060 0.923723i $$-0.625130\pi$$
−0.383060 + 0.923723i $$0.625130\pi$$
$$464$$ 1.79811e6 0.387722
$$465$$ −1.36372e6 −0.292477
$$466$$ −662356. −0.141295
$$467$$ −2.72459e6 −0.578109 −0.289054 0.957313i $$-0.593341\pi$$
−0.289054 + 0.957313i $$0.593341\pi$$
$$468$$ −280019. −0.0590980
$$469$$ 0 0
$$470$$ −1.68641e6 −0.352143
$$471$$ −5.09194e6 −1.05762
$$472$$ 7.61767e6 1.57386
$$473$$ −8.49224e6 −1.74530
$$474$$ −3.63176e6 −0.742457
$$475$$ 1.53158e6 0.311461
$$476$$ 0 0
$$477$$ −162919. −0.0327850
$$478$$ 4.40331e6 0.881475
$$479$$ −978685. −0.194896 −0.0974482 0.995241i $$-0.531068\pi$$
−0.0974482 + 0.995241i $$0.531068\pi$$
$$480$$ 1.08505e6 0.214955
$$481$$ 6.38144e6 1.25764
$$482$$ −1.40766e6 −0.275982
$$483$$ 0 0
$$484$$ 1.51154e6 0.293296
$$485$$ 121164. 0.0233893
$$486$$ 377315. 0.0724626
$$487$$ 3.92744e6 0.750390 0.375195 0.926946i $$-0.377576\pi$$
0.375195 + 0.926946i $$0.377576\pi$$
$$488$$ 6.08456e6 1.15659
$$489$$ −3.87181e6 −0.732220
$$490$$ 0 0
$$491$$ 2.63241e6 0.492777 0.246388 0.969171i $$-0.420756\pi$$
0.246388 + 0.969171i $$0.420756\pi$$
$$492$$ 1.04141e6 0.193958
$$493$$ −1.94607e6 −0.360614
$$494$$ 2.35767e6 0.434676
$$495$$ 1.80817e6 0.331685
$$496$$ −4.80678e6 −0.877305
$$497$$ 0 0
$$498$$ −2.62081e6 −0.473546
$$499$$ 2.12544e6 0.382118 0.191059 0.981579i $$-0.438808\pi$$
0.191059 + 0.981579i $$0.438808\pi$$
$$500$$ 1.62449e6 0.290597
$$501$$ 2.16239e6 0.384892
$$502$$ −9.17850e6 −1.62560
$$503$$ −2.60929e6 −0.459835 −0.229917 0.973210i $$-0.573846\pi$$
−0.229917 + 0.973210i $$0.573846\pi$$
$$504$$ 0 0
$$505$$ −6.54564e6 −1.14215
$$506$$ 6.01104e6 1.04369
$$507$$ −1.96225e6 −0.339027
$$508$$ 735308. 0.126418
$$509$$ 1.00182e7 1.71394 0.856970 0.515366i $$-0.172344\pi$$
0.856970 + 0.515366i $$0.172344\pi$$
$$510$$ −2.96159e6 −0.504196
$$511$$ 0 0
$$512$$ −1.99607e6 −0.336512
$$513$$ −687063. −0.115267
$$514$$ 5.80965e6 0.969933
$$515$$ 4.36942e6 0.725948
$$516$$ 1.17092e6 0.193600
$$517$$ −3.92782e6 −0.646287
$$518$$ 0 0
$$519$$ 1.61370e6 0.262969
$$520$$ −2.24476e6 −0.364050
$$521$$ 4.17941e6 0.674560 0.337280 0.941404i $$-0.390493\pi$$
0.337280 + 0.941404i $$0.390493\pi$$
$$522$$ −757500. −0.121676
$$523$$ −3.60525e6 −0.576343 −0.288172 0.957579i $$-0.593047\pi$$
−0.288172 + 0.957579i $$0.593047\pi$$
$$524$$ 1.47357e6 0.234446
$$525$$ 0 0
$$526$$ 4.78637e6 0.754297
$$527$$ 5.20234e6 0.815967
$$528$$ 6.37336e6 0.994909
$$529$$ −3.77266e6 −0.586150
$$530$$ 497754. 0.0769707
$$531$$ −4.16770e6 −0.641446
$$532$$ 0 0
$$533$$ −5.13003e6 −0.782172
$$534$$ −902122. −0.136903
$$535$$ 864025. 0.130509
$$536$$ −7.48814e6 −1.12580
$$537$$ −5.18003e6 −0.775170
$$538$$ −4.28086e6 −0.637640
$$539$$ 0 0
$$540$$ −249313. −0.0367926
$$541$$ −6.68166e6 −0.981501 −0.490751 0.871300i $$-0.663277\pi$$
−0.490751 + 0.871300i $$0.663277\pi$$
$$542$$ −3.45474e6 −0.505146
$$543$$ −5.22906e6 −0.761069
$$544$$ −4.13929e6 −0.599692
$$545$$ 3.24625e6 0.468156
$$546$$ 0 0
$$547$$ 8.69076e6 1.24191 0.620954 0.783847i $$-0.286745\pi$$
0.620954 + 0.783847i $$0.286745\pi$$
$$548$$ 337532. 0.0480136
$$549$$ −3.32892e6 −0.471382
$$550$$ 5.98518e6 0.843666
$$551$$ 1.37935e6 0.193551
$$552$$ 2.17467e6 0.303771
$$553$$ 0 0
$$554$$ −2.56815e6 −0.355505
$$555$$ 5.68167e6 0.782967
$$556$$ −938342. −0.128728
$$557$$ 6.24742e6 0.853223 0.426612 0.904435i $$-0.359707\pi$$
0.426612 + 0.904435i $$0.359707\pi$$
$$558$$ 2.02499e6 0.275319
$$559$$ −5.76803e6 −0.780725
$$560$$ 0 0
$$561$$ −6.89783e6 −0.925349
$$562$$ −2.74214e6 −0.366226
$$563$$ 1.19045e7 1.58285 0.791423 0.611269i $$-0.209341\pi$$
0.791423 + 0.611269i $$0.209341\pi$$
$$564$$ 541574. 0.0716903
$$565$$ 1.58512e6 0.208901
$$566$$ 2.17848e6 0.285833
$$567$$ 0 0
$$568$$ −5.91768e6 −0.769627
$$569$$ 21414.4 0.00277284 0.00138642 0.999999i $$-0.499559\pi$$
0.00138642 + 0.999999i $$0.499559\pi$$
$$570$$ 2.09914e6 0.270616
$$571$$ 7.11647e6 0.913428 0.456714 0.889614i $$-0.349026\pi$$
0.456714 + 0.889614i $$0.349026\pi$$
$$572$$ 1.99259e6 0.254641
$$573$$ −5.94504e6 −0.756429
$$574$$ 0 0
$$575$$ 2.65223e6 0.334534
$$576$$ 1.57333e6 0.197589
$$577$$ 1.06652e7 1.33361 0.666805 0.745232i $$-0.267661\pi$$
0.666805 + 0.745232i $$0.267661\pi$$
$$578$$ 2.22524e6 0.277050
$$579$$ −5.01579e6 −0.621789
$$580$$ 500522. 0.0617808
$$581$$ 0 0
$$582$$ −179916. −0.0220172
$$583$$ 1.15932e6 0.141264
$$584$$ −8.24444e6 −1.00030
$$585$$ 1.22813e6 0.148373
$$586$$ 2.48468e6 0.298901
$$587$$ 1.30101e7 1.55843 0.779213 0.626759i $$-0.215619\pi$$
0.779213 + 0.626759i $$0.215619\pi$$
$$588$$ 0 0
$$589$$ −3.68735e6 −0.437952
$$590$$ 1.27333e7 1.50595
$$591$$ −6.85260e6 −0.807025
$$592$$ 2.00265e7 2.34856
$$593$$ −4.26086e6 −0.497578 −0.248789 0.968558i $$-0.580033\pi$$
−0.248789 + 0.968558i $$0.580033\pi$$
$$594$$ −2.68495e6 −0.312226
$$595$$ 0 0
$$596$$ 1.70034e6 0.196074
$$597$$ 1.22274e6 0.140410
$$598$$ 4.08277e6 0.466877
$$599$$ −1.37958e7 −1.57101 −0.785507 0.618853i $$-0.787598\pi$$
−0.785507 + 0.618853i $$0.787598\pi$$
$$600$$ 2.16532e6 0.245552
$$601$$ −4.99695e6 −0.564311 −0.282155 0.959369i $$-0.591049\pi$$
−0.282155 + 0.959369i $$0.591049\pi$$
$$602$$ 0 0
$$603$$ 4.09683e6 0.458833
$$604$$ 1.25126e6 0.139558
$$605$$ −6.62942e6 −0.736355
$$606$$ 9.71962e6 1.07515
$$607$$ 3.04946e6 0.335932 0.167966 0.985793i $$-0.446280\pi$$
0.167966 + 0.985793i $$0.446280\pi$$
$$608$$ 2.93387e6 0.321871
$$609$$ 0 0
$$610$$ 1.01706e7 1.10668
$$611$$ −2.66782e6 −0.289104
$$612$$ 951085. 0.102646
$$613$$ −7.03625e6 −0.756293 −0.378147 0.925746i $$-0.623438\pi$$
−0.378147 + 0.925746i $$0.623438\pi$$
$$614$$ 1.50639e7 1.61256
$$615$$ −4.56749e6 −0.486956
$$616$$ 0 0
$$617$$ 1.00066e7 1.05822 0.529108 0.848554i $$-0.322527\pi$$
0.529108 + 0.848554i $$0.322527\pi$$
$$618$$ −6.48815e6 −0.683360
$$619$$ 6.55067e6 0.687161 0.343581 0.939123i $$-0.388360\pi$$
0.343581 + 0.939123i $$0.388360\pi$$
$$620$$ −1.33802e6 −0.139792
$$621$$ −1.18979e6 −0.123805
$$622$$ 9.17986e6 0.951393
$$623$$ 0 0
$$624$$ 4.32886e6 0.445054
$$625$$ −2.04650e6 −0.209561
$$626$$ 5.25758e6 0.536229
$$627$$ 4.88909e6 0.496660
$$628$$ −4.99598e6 −0.505501
$$629$$ −2.16746e7 −2.18436
$$630$$ 0 0
$$631$$ 2.22672e6 0.222635 0.111317 0.993785i $$-0.464493\pi$$
0.111317 + 0.993785i $$0.464493\pi$$
$$632$$ 9.34960e6 0.931109
$$633$$ −8.92042e6 −0.884863
$$634$$ −1.12905e7 −1.11555
$$635$$ −3.22497e6 −0.317389
$$636$$ −159849. −0.0156699
$$637$$ 0 0
$$638$$ 5.39031e6 0.524279
$$639$$ 3.23762e6 0.313671
$$640$$ −8.66484e6 −0.836201
$$641$$ −1.59340e7 −1.53172 −0.765859 0.643008i $$-0.777686\pi$$
−0.765859 + 0.643008i $$0.777686\pi$$
$$642$$ −1.28299e6 −0.122853
$$643$$ 1.49933e7 1.43011 0.715056 0.699067i $$-0.246401\pi$$
0.715056 + 0.699067i $$0.246401\pi$$
$$644$$ 0 0
$$645$$ −5.13553e6 −0.486056
$$646$$ −8.00783e6 −0.754977
$$647$$ −1.29805e7 −1.21908 −0.609540 0.792755i $$-0.708646\pi$$
−0.609540 + 0.792755i $$0.708646\pi$$
$$648$$ −971360. −0.0908747
$$649$$ 2.96571e7 2.76386
$$650$$ 4.06521e6 0.377398
$$651$$ 0 0
$$652$$ −3.79885e6 −0.349972
$$653$$ −1.65492e7 −1.51878 −0.759389 0.650637i $$-0.774502\pi$$
−0.759389 + 0.650637i $$0.774502\pi$$
$$654$$ −4.82036e6 −0.440692
$$655$$ −6.46291e6 −0.588606
$$656$$ −1.60993e7 −1.46066
$$657$$ 4.51062e6 0.407683
$$658$$ 0 0
$$659$$ 5.86879e6 0.526423 0.263212 0.964738i $$-0.415218\pi$$
0.263212 + 0.964738i $$0.415218\pi$$
$$660$$ 1.77409e6 0.158532
$$661$$ −7.27687e6 −0.647800 −0.323900 0.946091i $$-0.604994\pi$$
−0.323900 + 0.946091i $$0.604994\pi$$
$$662$$ −19503.6 −0.00172970
$$663$$ −4.68509e6 −0.413937
$$664$$ 6.74702e6 0.593870
$$665$$ 0 0
$$666$$ −8.43672e6 −0.737035
$$667$$ 2.38862e6 0.207889
$$668$$ 2.12164e6 0.183963
$$669$$ −4.89010e6 −0.422428
$$670$$ −1.25168e7 −1.07722
$$671$$ 2.36884e7 2.03109
$$672$$ 0 0
$$673$$ −1.82417e7 −1.55248 −0.776241 0.630437i $$-0.782876\pi$$
−0.776241 + 0.630437i $$0.782876\pi$$
$$674$$ 1.29675e7 1.09953
$$675$$ −1.18467e6 −0.100078
$$676$$ −1.92527e6 −0.162041
$$677$$ 7.76406e6 0.651054 0.325527 0.945533i $$-0.394458\pi$$
0.325527 + 0.945533i $$0.394458\pi$$
$$678$$ −2.35374e6 −0.196646
$$679$$ 0 0
$$680$$ 7.62432e6 0.632308
$$681$$ −147.007 −1.21470e−5 0
$$682$$ −1.44096e7 −1.18629
$$683$$ 8.56324e6 0.702403 0.351201 0.936300i $$-0.385773\pi$$
0.351201 + 0.936300i $$0.385773\pi$$
$$684$$ −674116. −0.0550927
$$685$$ −1.48038e6 −0.120544
$$686$$ 0 0
$$687$$ 696411. 0.0562955
$$688$$ −1.81015e7 −1.45796
$$689$$ 787423. 0.0631917
$$690$$ 3.63507e6 0.290663
$$691$$ −1.64509e7 −1.31068 −0.655338 0.755336i $$-0.727474\pi$$
−0.655338 + 0.755336i $$0.727474\pi$$
$$692$$ 1.58329e6 0.125689
$$693$$ 0 0
$$694$$ 2.41677e7 1.90474
$$695$$ 4.11546e6 0.323189
$$696$$ 1.95011e6 0.152593
$$697$$ 1.74242e7 1.35853
$$698$$ −1.86039e6 −0.144533
$$699$$ −932916. −0.0722187
$$700$$ 0 0
$$701$$ −1.66928e7 −1.28302 −0.641512 0.767113i $$-0.721693\pi$$
−0.641512 + 0.767113i $$0.721693\pi$$
$$702$$ −1.82365e6 −0.139669
$$703$$ 1.53626e7 1.17240
$$704$$ −1.11957e7 −0.851370
$$705$$ −2.37528e6 −0.179987
$$706$$ −2.46059e6 −0.185792
$$707$$ 0 0
$$708$$ −4.08916e6 −0.306585
$$709$$ 5.61779e6 0.419711 0.209855 0.977732i $$-0.432701\pi$$
0.209855 + 0.977732i $$0.432701\pi$$
$$710$$ −9.89167e6 −0.736417
$$711$$ −5.11526e6 −0.379484
$$712$$ 2.32242e6 0.171689
$$713$$ −6.38537e6 −0.470395
$$714$$ 0 0
$$715$$ −8.73927e6 −0.639308
$$716$$ −5.08242e6 −0.370500
$$717$$ 6.20198e6 0.450539
$$718$$ −2.06401e7 −1.49417
$$719$$ −1.03718e7 −0.748227 −0.374113 0.927383i $$-0.622053\pi$$
−0.374113 + 0.927383i $$0.622053\pi$$
$$720$$ 3.85418e6 0.277077
$$721$$ 0 0
$$722$$ −1.01461e7 −0.724363
$$723$$ −1.98266e6 −0.141060
$$724$$ −5.13052e6 −0.363760
$$725$$ 2.37835e6 0.168047
$$726$$ 9.84403e6 0.693156
$$727$$ −1.15369e7 −0.809565 −0.404783 0.914413i $$-0.632653\pi$$
−0.404783 + 0.914413i $$0.632653\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ −1.37810e7 −0.957134
$$731$$ 1.95911e7 1.35602
$$732$$ −3.26619e6 −0.225301
$$733$$ −1.49470e7 −1.02753 −0.513764 0.857932i $$-0.671749\pi$$
−0.513764 + 0.857932i $$0.671749\pi$$
$$734$$ 3.06432e6 0.209939
$$735$$ 0 0
$$736$$ 5.08058e6 0.345715
$$737$$ −2.91528e7 −1.97702
$$738$$ 6.78227e6 0.458389
$$739$$ 9.01364e6 0.607140 0.303570 0.952809i $$-0.401821\pi$$
0.303570 + 0.952809i $$0.401821\pi$$
$$740$$ 5.57460e6 0.374227
$$741$$ 3.32073e6 0.222171
$$742$$ 0 0
$$743$$ −2.10239e7 −1.39714 −0.698571 0.715541i $$-0.746180\pi$$
−0.698571 + 0.715541i $$0.746180\pi$$
$$744$$ −5.21312e6 −0.345275
$$745$$ −7.45750e6 −0.492269
$$746$$ 5.57622e6 0.366854
$$747$$ −3.69136e6 −0.242039
$$748$$ −6.76785e6 −0.442279
$$749$$ 0 0
$$750$$ 1.05796e7 0.686779
$$751$$ 4.04219e6 0.261527 0.130764 0.991414i $$-0.458257\pi$$
0.130764 + 0.991414i $$0.458257\pi$$
$$752$$ −8.37230e6 −0.539884
$$753$$ −1.29277e7 −0.830874
$$754$$ 3.66117e6 0.234526
$$755$$ −5.48786e6 −0.350377
$$756$$ 0 0
$$757$$ −1.82059e7 −1.15471 −0.577353 0.816495i $$-0.695914\pi$$
−0.577353 + 0.816495i $$0.695914\pi$$
$$758$$ −1.55589e7 −0.983572
$$759$$ 8.46642e6 0.533452
$$760$$ −5.40402e6 −0.339377
$$761$$ 1.89200e7 1.18429 0.592146 0.805831i $$-0.298281\pi$$
0.592146 + 0.805831i $$0.298281\pi$$
$$762$$ 4.78876e6 0.298769
$$763$$ 0 0
$$764$$ −5.83301e6 −0.361542
$$765$$ −4.17134e6 −0.257705
$$766$$ 2.30782e7 1.42112
$$767$$ 2.01434e7 1.23636
$$768$$ 7.27238e6 0.444911
$$769$$ 1.12831e7 0.688037 0.344019 0.938963i $$-0.388212\pi$$
0.344019 + 0.938963i $$0.388212\pi$$
$$770$$ 0 0
$$771$$ 8.18277e6 0.495752
$$772$$ −4.92127e6 −0.297190
$$773$$ 3.68565e6 0.221853 0.110926 0.993829i $$-0.464618\pi$$
0.110926 + 0.993829i $$0.464618\pi$$
$$774$$ 7.62575e6 0.457541
$$775$$ −6.35791e6 −0.380242
$$776$$ 463176. 0.0276116
$$777$$ 0 0
$$778$$ −1.11971e6 −0.0663219
$$779$$ −1.23500e7 −0.729161
$$780$$ 1.20499e6 0.0709162
$$781$$ −2.30387e7 −1.35154
$$782$$ −1.38671e7 −0.810905
$$783$$ −1.06692e6 −0.0621912
$$784$$ 0 0
$$785$$ 2.19118e7 1.26912
$$786$$ 9.59677e6 0.554075
$$787$$ 1.40748e7 0.810039 0.405019 0.914308i $$-0.367265\pi$$
0.405019 + 0.914308i $$0.367265\pi$$
$$788$$ −6.72347e6 −0.385725
$$789$$ 6.74151e6 0.385536
$$790$$ 1.56283e7 0.890930
$$791$$ 0 0
$$792$$ 6.91213e6 0.391560
$$793$$ 1.60894e7 0.908569
$$794$$ −1.20459e7 −0.678090
$$795$$ 701077. 0.0393412
$$796$$ 1.19970e6 0.0671102
$$797$$ 1.75191e7 0.976937 0.488469 0.872582i $$-0.337556\pi$$
0.488469 + 0.872582i $$0.337556\pi$$
$$798$$ 0 0
$$799$$ 9.06127e6 0.502137
$$800$$ 5.05873e6 0.279458
$$801$$ −1.27062e6 −0.0699737
$$802$$ 8.56134e6 0.470008
$$803$$ −3.20972e7 −1.75662
$$804$$ 4.01963e6 0.219304
$$805$$ 0 0
$$806$$ −9.78721e6 −0.530666
$$807$$ −6.02950e6 −0.325910
$$808$$ −2.50222e7 −1.34833
$$809$$ 814521. 0.0437553 0.0218777 0.999761i $$-0.493036\pi$$
0.0218777 + 0.999761i $$0.493036\pi$$
$$810$$ −1.62367e6 −0.0869534
$$811$$ −1.26533e7 −0.675540 −0.337770 0.941229i $$-0.609673\pi$$
−0.337770 + 0.941229i $$0.609673\pi$$
$$812$$ 0 0
$$813$$ −4.86593e6 −0.258190
$$814$$ 6.00350e7 3.17573
$$815$$ 1.66613e7 0.878647
$$816$$ −1.47030e7 −0.773001
$$817$$ −1.38859e7 −0.727813
$$818$$ −4.20854e7 −2.19912
$$819$$ 0 0
$$820$$ −4.48142e6 −0.232745
$$821$$ 4.27393e6 0.221294 0.110647 0.993860i $$-0.464708\pi$$
0.110647 + 0.993860i $$0.464708\pi$$
$$822$$ 2.19821e6 0.113472
$$823$$ 1.70463e7 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$824$$ 1.67031e7 0.856996
$$825$$ 8.43001e6 0.431214
$$826$$ 0 0
$$827$$ 2.60828e6 0.132614 0.0663071 0.997799i $$-0.478878\pi$$
0.0663071 + 0.997799i $$0.478878\pi$$
$$828$$ −1.16736e6 −0.0591740
$$829$$ −2.13865e7 −1.08082 −0.540409 0.841403i $$-0.681730\pi$$
−0.540409 + 0.841403i $$0.681730\pi$$
$$830$$ 1.12780e7 0.568244
$$831$$ −3.61719e6 −0.181706
$$832$$ −7.60423e6 −0.380844
$$833$$ 0 0
$$834$$ −6.11104e6 −0.304229
$$835$$ −9.30525e6 −0.461862
$$836$$ 4.79696e6 0.237383
$$837$$ 2.85215e6 0.140721
$$838$$ 4.45290e7 2.19045
$$839$$ −771393. −0.0378330 −0.0189165 0.999821i $$-0.506022\pi$$
−0.0189165 + 0.999821i $$0.506022\pi$$
$$840$$ 0 0
$$841$$ −1.83692e7 −0.895571
$$842$$ 2.45397e7 1.19286
$$843$$ −3.86225e6 −0.187185
$$844$$ −8.75231e6 −0.422928
$$845$$ 8.44401e6 0.406824
$$846$$ 3.52705e6 0.169428
$$847$$ 0 0
$$848$$ 2.47113e6 0.118006
$$849$$ 3.06834e6 0.146095
$$850$$ −1.38075e7 −0.655492
$$851$$ 2.66034e7 1.25926
$$852$$ 3.17661e6 0.149922
$$853$$ −2.94032e7 −1.38364 −0.691818 0.722072i $$-0.743190\pi$$
−0.691818 + 0.722072i $$0.743190\pi$$
$$854$$ 0 0
$$855$$ 2.95659e6 0.138317
$$856$$ 3.30293e6 0.154069
$$857$$ 2.65010e7 1.23257 0.616283 0.787525i $$-0.288638\pi$$
0.616283 + 0.787525i $$0.288638\pi$$
$$858$$ 1.29770e7 0.601803
$$859$$ 3.67519e7 1.69940 0.849702 0.527264i $$-0.176782\pi$$
0.849702 + 0.527264i $$0.176782\pi$$
$$860$$ −5.03876e6 −0.232315
$$861$$ 0 0
$$862$$ 1.94020e7 0.889360
$$863$$ 1.18189e7 0.540196 0.270098 0.962833i $$-0.412944\pi$$
0.270098 + 0.962833i $$0.412944\pi$$
$$864$$ −2.26934e6 −0.103423
$$865$$ −6.94413e6 −0.315557
$$866$$ 6.01625e6 0.272603
$$867$$ 3.13421e6 0.141606
$$868$$ 0 0
$$869$$ 3.63998e7 1.63512
$$870$$ 3.25970e6 0.146009
$$871$$ −1.98009e7 −0.884382
$$872$$ 1.24095e7 0.552668
$$873$$ −253408. −0.0112534
$$874$$ 9.82884e6 0.435235
$$875$$ 0 0
$$876$$ 4.42562e6 0.194856
$$877$$ 7.93509e6 0.348380 0.174190 0.984712i $$-0.444269\pi$$
0.174190 + 0.984712i $$0.444269\pi$$
$$878$$ −8.56939e6 −0.375158
$$879$$ 3.49963e6 0.152774
$$880$$ −2.74260e7 −1.19387
$$881$$ −4.20152e7 −1.82375 −0.911877 0.410464i $$-0.865367\pi$$
−0.911877 + 0.410464i $$0.865367\pi$$
$$882$$ 0 0
$$883$$ 2.12461e7 0.917016 0.458508 0.888690i $$-0.348384\pi$$
0.458508 + 0.888690i $$0.348384\pi$$
$$884$$ −4.59681e6 −0.197845
$$885$$ 1.79346e7 0.769720
$$886$$ −4.93516e6 −0.211211
$$887$$ −1.88490e7 −0.804415 −0.402208 0.915548i $$-0.631757\pi$$
−0.402208 + 0.915548i $$0.631757\pi$$
$$888$$ 2.17195e7 0.924309
$$889$$ 0 0
$$890$$ 3.88204e6 0.164280
$$891$$ −3.78169e6 −0.159585
$$892$$ −4.79795e6 −0.201903
$$893$$ −6.42251e6 −0.269511
$$894$$ 1.10736e7 0.463390
$$895$$ 2.22909e7 0.930186
$$896$$ 0 0
$$897$$ 5.75050e6 0.238630
$$898$$ 1.44209e7 0.596762
$$899$$ −5.72600e6 −0.236294
$$900$$ −1.16234e6 −0.0478331
$$901$$ −2.67448e6 −0.109756
$$902$$ −4.82621e7 −1.97510
$$903$$ 0 0
$$904$$ 6.05948e6 0.246612
$$905$$ 2.25018e7 0.913264
$$906$$ 8.14892e6 0.329822
$$907$$ −6.19446e6 −0.250026 −0.125013 0.992155i $$-0.539897\pi$$
−0.125013 + 0.992155i $$0.539897\pi$$
$$908$$ −144.237 −5.80578e−6 0
$$909$$ 1.36899e7 0.549528
$$910$$ 0 0
$$911$$ −2.50171e7 −0.998712 −0.499356 0.866397i $$-0.666430\pi$$
−0.499356 + 0.866397i $$0.666430\pi$$
$$912$$ 1.04213e7 0.414891
$$913$$ 2.62674e7 1.04290
$$914$$ 2.73787e7 1.08404
$$915$$ 1.43251e7 0.565647
$$916$$ 683288. 0.0269070
$$917$$ 0 0
$$918$$ 6.19403e6 0.242586
$$919$$ 1.54992e7 0.605370 0.302685 0.953091i $$-0.402117\pi$$
0.302685 + 0.953091i $$0.402117\pi$$
$$920$$ −9.35812e6 −0.364518
$$921$$ 2.12172e7 0.824212
$$922$$ 1.98381e7 0.768551
$$923$$ −1.56481e7 −0.604587
$$924$$ 0 0
$$925$$ 2.64890e7 1.01791
$$926$$ −2.25809e7 −0.865394
$$927$$ −9.13843e6 −0.349279
$$928$$ 4.55594e6 0.173663
$$929$$ −3.75285e7 −1.42667 −0.713333 0.700826i $$-0.752815\pi$$
−0.713333 + 0.700826i $$0.752815\pi$$
$$930$$ −8.71398e6 −0.330377
$$931$$ 0 0
$$932$$ −915335. −0.0345176
$$933$$ 1.29297e7 0.486276
$$934$$ −1.74098e7 −0.653020
$$935$$ 2.96830e7 1.11040
$$936$$ 4.69480e6 0.175157
$$937$$ 1.08298e7 0.402969 0.201485 0.979492i $$-0.435423\pi$$
0.201485 + 0.979492i $$0.435423\pi$$
$$938$$ 0 0
$$939$$ 7.40520e6 0.274077
$$940$$ −2.33052e6 −0.0860267
$$941$$ −3.24591e7 −1.19498 −0.597492 0.801875i $$-0.703836\pi$$
−0.597492 + 0.801875i $$0.703836\pi$$
$$942$$ −3.25368e7 −1.19467
$$943$$ −2.13865e7 −0.783177
$$944$$ 6.32151e7 2.30883
$$945$$ 0 0
$$946$$ −5.42643e7 −1.97145
$$947$$ 7.53877e6 0.273165 0.136583 0.990629i $$-0.456388\pi$$
0.136583 + 0.990629i $$0.456388\pi$$
$$948$$ −5.01886e6 −0.181378
$$949$$ −2.18008e7 −0.785792
$$950$$ 9.78656e6 0.351821
$$951$$ −1.59024e7 −0.570180
$$952$$ 0 0
$$953$$ −3.01356e7 −1.07485 −0.537424 0.843312i $$-0.680602\pi$$
−0.537424 + 0.843312i $$0.680602\pi$$
$$954$$ −1.04103e6 −0.0370333
$$955$$ 2.55829e7 0.907697
$$956$$ 6.08510e6 0.215339
$$957$$ 7.59215e6 0.267969
$$958$$ −6.25366e6 −0.220151
$$959$$ 0 0
$$960$$ −6.77038e6 −0.237102
$$961$$ −1.33221e7 −0.465335
$$962$$ 4.07765e7 1.42060
$$963$$ −1.80707e6 −0.0627926
$$964$$ −1.94530e6 −0.0674208
$$965$$ 2.15841e7 0.746132
$$966$$ 0 0
$$967$$ 2.88021e6 0.0990509 0.0495255 0.998773i $$-0.484229\pi$$
0.0495255 + 0.998773i $$0.484229\pi$$
$$968$$ −2.53425e7 −0.869282
$$969$$ −1.12789e7 −0.385883
$$970$$ 774220. 0.0264201
$$971$$ 2.74490e7 0.934283 0.467142 0.884183i $$-0.345284\pi$$
0.467142 + 0.884183i $$0.345284\pi$$
$$972$$ 521426. 0.0177022
$$973$$ 0 0
$$974$$ 2.50958e7 0.847626
$$975$$ 5.72577e6 0.192896
$$976$$ 5.04927e7 1.69669
$$977$$ −5.88524e7 −1.97255 −0.986274 0.165118i $$-0.947200\pi$$
−0.986274 + 0.165118i $$0.947200\pi$$
$$978$$ −2.47403e7 −0.827101
$$979$$ 9.04164e6 0.301502
$$980$$ 0 0
$$981$$ −6.78938e6 −0.225246
$$982$$ 1.68208e7 0.556631
$$983$$ −1.86077e7 −0.614198 −0.307099 0.951678i $$-0.599358\pi$$
−0.307099 + 0.951678i $$0.599358\pi$$
$$984$$ −1.74603e7 −0.574862
$$985$$ 2.94883e7 0.968410
$$986$$ −1.24352e7 −0.407342
$$987$$ 0 0
$$988$$ 3.25815e6 0.106189
$$989$$ −2.40462e7 −0.781729
$$990$$ 1.15539e7 0.374664
$$991$$ 2.47154e7 0.799435 0.399718 0.916638i $$-0.369108\pi$$
0.399718 + 0.916638i $$0.369108\pi$$
$$992$$ −1.21792e7 −0.392951
$$993$$ −27470.5 −0.000884083 0
$$994$$ 0 0
$$995$$ −5.26172e6 −0.168488
$$996$$ −3.62180e6 −0.115685
$$997$$ 1.79581e7 0.572167 0.286084 0.958205i $$-0.407646\pi$$
0.286084 + 0.958205i $$0.407646\pi$$
$$998$$ 1.35813e7 0.431633
$$999$$ −1.18829e7 −0.376713
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 147.6.a.k.1.2 2
3.2 odd 2 441.6.a.s.1.1 2
7.2 even 3 147.6.e.l.67.1 4
7.3 odd 6 21.6.e.b.16.1 yes 4
7.4 even 3 147.6.e.l.79.1 4
7.5 odd 6 21.6.e.b.4.1 4
7.6 odd 2 147.6.a.i.1.2 2
21.5 even 6 63.6.e.c.46.2 4
21.17 even 6 63.6.e.c.37.2 4
21.20 even 2 441.6.a.t.1.1 2
28.3 even 6 336.6.q.e.289.1 4
28.19 even 6 336.6.q.e.193.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 4 7.5 odd 6
21.6.e.b.16.1 yes 4 7.3 odd 6
63.6.e.c.37.2 4 21.17 even 6
63.6.e.c.46.2 4 21.5 even 6
147.6.a.i.1.2 2 7.6 odd 2
147.6.a.k.1.2 2 1.1 even 1 trivial
147.6.e.l.67.1 4 7.2 even 3
147.6.e.l.79.1 4 7.4 even 3
336.6.q.e.193.1 4 28.19 even 6
336.6.q.e.289.1 4 28.3 even 6
441.6.a.s.1.1 2 3.2 odd 2
441.6.a.t.1.1 2 21.20 even 2