# Properties

 Label 147.6.a.m.1.4 Level $147$ Weight $6$ Character 147.1 Self dual yes Analytic conductor $23.576$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 97x^{2} + 7x + 294$$ x^4 - x^3 - 97*x^2 + 7*x + 294 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-9.22385$$ 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)

 $$f(q)$$ $$=$$ $$q+10.2239 q^{2} +9.00000 q^{3} +72.5272 q^{4} +23.7528 q^{5} +92.0147 q^{6} +414.344 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+10.2239 q^{2} +9.00000 q^{3} +72.5272 q^{4} +23.7528 q^{5} +92.0147 q^{6} +414.344 q^{8} +81.0000 q^{9} +242.845 q^{10} +465.526 q^{11} +652.744 q^{12} -1019.30 q^{13} +213.775 q^{15} +1915.32 q^{16} +561.757 q^{17} +828.132 q^{18} -1387.58 q^{19} +1722.72 q^{20} +4759.47 q^{22} +4113.62 q^{23} +3729.09 q^{24} -2560.80 q^{25} -10421.2 q^{26} +729.000 q^{27} -2381.37 q^{29} +2185.61 q^{30} +2950.66 q^{31} +6322.95 q^{32} +4189.73 q^{33} +5743.32 q^{34} +5874.70 q^{36} -9908.95 q^{37} -14186.4 q^{38} -9173.70 q^{39} +9841.83 q^{40} -4477.13 q^{41} +5181.48 q^{43} +33763.3 q^{44} +1923.98 q^{45} +42057.0 q^{46} -3121.59 q^{47} +17237.9 q^{48} -26181.3 q^{50} +5055.81 q^{51} -73926.9 q^{52} +1141.00 q^{53} +7453.19 q^{54} +11057.6 q^{55} -12488.2 q^{57} -24346.8 q^{58} +27497.1 q^{59} +15504.5 q^{60} -21103.5 q^{61} +30167.1 q^{62} +3354.67 q^{64} -24211.2 q^{65} +42835.2 q^{66} -55588.4 q^{67} +40742.6 q^{68} +37022.6 q^{69} -6076.90 q^{71} +33561.8 q^{72} -16779.6 q^{73} -101308. q^{74} -23047.2 q^{75} -100637. q^{76} -93790.5 q^{78} -4845.26 q^{79} +45494.2 q^{80} +6561.00 q^{81} -45773.5 q^{82} +60145.4 q^{83} +13343.3 q^{85} +52974.7 q^{86} -21432.4 q^{87} +192888. q^{88} -62497.4 q^{89} +19670.5 q^{90} +298349. q^{92} +26555.9 q^{93} -31914.7 q^{94} -32958.9 q^{95} +56906.5 q^{96} -63653.8 q^{97} +37707.6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9 $$4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22} + 6900 q^{23} + 1107 q^{24} + 2814 q^{25} - 15138 q^{26} + 2916 q^{27} + 540 q^{29} + 2547 q^{30} - 6410 q^{31} + 15519 q^{32} + 3618 q^{33} + 21144 q^{34} + 5589 q^{36} + 15250 q^{37} - 41250 q^{38} + 4158 q^{39} - 8547 q^{40} + 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 29457 q^{48} - 7302 q^{50} + 2484 q^{51} - 47476 q^{52} + 13692 q^{53} + 2187 q^{54} + 73124 q^{55} + 4590 q^{57} + 52309 q^{58} + 34830 q^{59} + 42471 q^{60} - 5364 q^{61} + 16029 q^{62} - 73487 q^{64} + 66864 q^{65} + 12375 q^{66} - 5994 q^{67} - 58272 q^{68} + 62100 q^{69} + 89268 q^{71} + 9963 q^{72} + 59638 q^{73} - 185442 q^{74} + 25326 q^{75} + 21308 q^{76} - 136242 q^{78} - 44062 q^{79} - 33381 q^{80} + 26244 q^{81} + 57596 q^{82} - 208446 q^{83} + 36324 q^{85} - 136968 q^{86} + 4860 q^{87} + 87597 q^{88} - 77520 q^{89} + 22923 q^{90} + 158256 q^{92} - 57690 q^{93} - 73722 q^{94} - 221376 q^{95} + 139671 q^{96} - 188630 q^{97} + 32562 q^{99}+O(q^{100})$$ 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9 + 283 * q^10 + 402 * q^11 + 621 * q^12 + 462 * q^13 + 3273 * q^16 + 276 * q^17 + 243 * q^18 + 510 * q^19 + 4719 * q^20 + 1375 * q^22 + 6900 * q^23 + 1107 * q^24 + 2814 * q^25 - 15138 * q^26 + 2916 * q^27 + 540 * q^29 + 2547 * q^30 - 6410 * q^31 + 15519 * q^32 + 3618 * q^33 + 21144 * q^34 + 5589 * q^36 + 15250 * q^37 - 41250 * q^38 + 4158 * q^39 - 8547 * q^40 + 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 29457 * q^48 - 7302 * q^50 + 2484 * q^51 - 47476 * q^52 + 13692 * q^53 + 2187 * q^54 + 73124 * q^55 + 4590 * q^57 + 52309 * q^58 + 34830 * q^59 + 42471 * q^60 - 5364 * q^61 + 16029 * q^62 - 73487 * q^64 + 66864 * q^65 + 12375 * q^66 - 5994 * q^67 - 58272 * q^68 + 62100 * q^69 + 89268 * q^71 + 9963 * q^72 + 59638 * q^73 - 185442 * q^74 + 25326 * q^75 + 21308 * q^76 - 136242 * q^78 - 44062 * q^79 - 33381 * q^80 + 26244 * q^81 + 57596 * q^82 - 208446 * q^83 + 36324 * q^85 - 136968 * q^86 + 4860 * q^87 + 87597 * q^88 - 77520 * q^89 + 22923 * q^90 + 158256 * q^92 - 57690 * q^93 - 73722 * q^94 - 221376 * q^95 + 139671 * q^96 - 188630 * q^97 + 32562 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 10.2239 1.80734 0.903669 0.428231i $$-0.140863\pi$$
0.903669 + 0.428231i $$0.140863\pi$$
$$3$$ 9.00000 0.577350
$$4$$ 72.5272 2.26647
$$5$$ 23.7528 0.424903 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$6$$ 92.0147 1.04347
$$7$$ 0 0
$$8$$ 414.344 2.28895
$$9$$ 81.0000 0.333333
$$10$$ 242.845 0.767944
$$11$$ 465.526 1.16001 0.580006 0.814612i $$-0.303050\pi$$
0.580006 + 0.814612i $$0.303050\pi$$
$$12$$ 652.744 1.30855
$$13$$ −1019.30 −1.67280 −0.836399 0.548121i $$-0.815343\pi$$
−0.836399 + 0.548121i $$0.815343\pi$$
$$14$$ 0 0
$$15$$ 213.775 0.245318
$$16$$ 1915.32 1.87043
$$17$$ 561.757 0.471440 0.235720 0.971821i $$-0.424255\pi$$
0.235720 + 0.971821i $$0.424255\pi$$
$$18$$ 828.132 0.602446
$$19$$ −1387.58 −0.881807 −0.440903 0.897555i $$-0.645342\pi$$
−0.440903 + 0.897555i $$0.645342\pi$$
$$20$$ 1722.72 0.963032
$$21$$ 0 0
$$22$$ 4759.47 2.09653
$$23$$ 4113.62 1.62145 0.810727 0.585425i $$-0.199072\pi$$
0.810727 + 0.585425i $$0.199072\pi$$
$$24$$ 3729.09 1.32152
$$25$$ −2560.80 −0.819457
$$26$$ −10421.2 −3.02331
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ −2381.37 −0.525814 −0.262907 0.964821i $$-0.584681\pi$$
−0.262907 + 0.964821i $$0.584681\pi$$
$$30$$ 2185.61 0.443373
$$31$$ 2950.66 0.551460 0.275730 0.961235i $$-0.411080\pi$$
0.275730 + 0.961235i $$0.411080\pi$$
$$32$$ 6322.95 1.09155
$$33$$ 4189.73 0.669733
$$34$$ 5743.32 0.852051
$$35$$ 0 0
$$36$$ 5874.70 0.755491
$$37$$ −9908.95 −1.18994 −0.594968 0.803750i $$-0.702835\pi$$
−0.594968 + 0.803750i $$0.702835\pi$$
$$38$$ −14186.4 −1.59372
$$39$$ −9173.70 −0.965791
$$40$$ 9841.83 0.972581
$$41$$ −4477.13 −0.415949 −0.207974 0.978134i $$-0.566687\pi$$
−0.207974 + 0.978134i $$0.566687\pi$$
$$42$$ 0 0
$$43$$ 5181.48 0.427349 0.213675 0.976905i $$-0.431457\pi$$
0.213675 + 0.976905i $$0.431457\pi$$
$$44$$ 33763.3 2.62914
$$45$$ 1923.98 0.141634
$$46$$ 42057.0 2.93052
$$47$$ −3121.59 −0.206125 −0.103063 0.994675i $$-0.532864\pi$$
−0.103063 + 0.994675i $$0.532864\pi$$
$$48$$ 17237.9 1.07989
$$49$$ 0 0
$$50$$ −26181.3 −1.48104
$$51$$ 5055.81 0.272186
$$52$$ −73926.9 −3.79135
$$53$$ 1141.00 0.0557950 0.0278975 0.999611i $$-0.491119\pi$$
0.0278975 + 0.999611i $$0.491119\pi$$
$$54$$ 7453.19 0.347823
$$55$$ 11057.6 0.492893
$$56$$ 0 0
$$57$$ −12488.2 −0.509111
$$58$$ −24346.8 −0.950325
$$59$$ 27497.1 1.02839 0.514194 0.857674i $$-0.328091\pi$$
0.514194 + 0.857674i $$0.328091\pi$$
$$60$$ 15504.5 0.556007
$$61$$ −21103.5 −0.726157 −0.363078 0.931759i $$-0.618274\pi$$
−0.363078 + 0.931759i $$0.618274\pi$$
$$62$$ 30167.1 0.996676
$$63$$ 0 0
$$64$$ 3354.67 0.102376
$$65$$ −24211.2 −0.710778
$$66$$ 42835.2 1.21043
$$67$$ −55588.4 −1.51286 −0.756428 0.654077i $$-0.773057\pi$$
−0.756428 + 0.654077i $$0.773057\pi$$
$$68$$ 40742.6 1.06851
$$69$$ 37022.6 0.936146
$$70$$ 0 0
$$71$$ −6076.90 −0.143066 −0.0715330 0.997438i $$-0.522789\pi$$
−0.0715330 + 0.997438i $$0.522789\pi$$
$$72$$ 33561.8 0.762982
$$73$$ −16779.6 −0.368532 −0.184266 0.982876i $$-0.558991\pi$$
−0.184266 + 0.982876i $$0.558991\pi$$
$$74$$ −101308. −2.15062
$$75$$ −23047.2 −0.473114
$$76$$ −100637. −1.99859
$$77$$ 0 0
$$78$$ −93790.5 −1.74551
$$79$$ −4845.26 −0.0873473 −0.0436737 0.999046i $$-0.513906\pi$$
−0.0436737 + 0.999046i $$0.513906\pi$$
$$80$$ 45494.2 0.794752
$$81$$ 6561.00 0.111111
$$82$$ −45773.5 −0.751760
$$83$$ 60145.4 0.958313 0.479156 0.877730i $$-0.340943\pi$$
0.479156 + 0.877730i $$0.340943\pi$$
$$84$$ 0 0
$$85$$ 13343.3 0.200316
$$86$$ 52974.7 0.772365
$$87$$ −21432.4 −0.303579
$$88$$ 192888. 2.65520
$$89$$ −62497.4 −0.836348 −0.418174 0.908367i $$-0.637330\pi$$
−0.418174 + 0.908367i $$0.637330\pi$$
$$90$$ 19670.5 0.255981
$$91$$ 0 0
$$92$$ 298349. 3.67498
$$93$$ 26555.9 0.318386
$$94$$ −31914.7 −0.372539
$$95$$ −32958.9 −0.374683
$$96$$ 56906.5 0.630208
$$97$$ −63653.8 −0.686903 −0.343451 0.939170i $$-0.611596\pi$$
−0.343451 + 0.939170i $$0.611596\pi$$
$$98$$ 0 0
$$99$$ 37707.6 0.386670
$$100$$ −185728. −1.85728
$$101$$ 184623. 1.80087 0.900434 0.434993i $$-0.143249\pi$$
0.900434 + 0.434993i $$0.143249\pi$$
$$102$$ 51689.9 0.491932
$$103$$ 52043.7 0.483365 0.241683 0.970355i $$-0.422301\pi$$
0.241683 + 0.970355i $$0.422301\pi$$
$$104$$ −422341. −3.82895
$$105$$ 0 0
$$106$$ 11665.4 0.100840
$$107$$ 48177.8 0.406806 0.203403 0.979095i $$-0.434800\pi$$
0.203403 + 0.979095i $$0.434800\pi$$
$$108$$ 52872.3 0.436183
$$109$$ −36435.6 −0.293737 −0.146869 0.989156i $$-0.546919\pi$$
−0.146869 + 0.989156i $$0.546919\pi$$
$$110$$ 113051. 0.890824
$$111$$ −89180.6 −0.687010
$$112$$ 0 0
$$113$$ −96711.1 −0.712492 −0.356246 0.934392i $$-0.615944\pi$$
−0.356246 + 0.934392i $$0.615944\pi$$
$$114$$ −127678. −0.920137
$$115$$ 97710.0 0.688961
$$116$$ −172714. −1.19174
$$117$$ −82563.3 −0.557599
$$118$$ 281126. 1.85864
$$119$$ 0 0
$$120$$ 88576.5 0.561520
$$121$$ 55663.5 0.345626
$$122$$ −215759. −1.31241
$$123$$ −40294.1 −0.240148
$$124$$ 214003. 1.24987
$$125$$ −135054. −0.773093
$$126$$ 0 0
$$127$$ 23322.9 0.128314 0.0641568 0.997940i $$-0.479564\pi$$
0.0641568 + 0.997940i $$0.479564\pi$$
$$128$$ −168037. −0.906524
$$129$$ 46633.4 0.246730
$$130$$ −247532. −1.28462
$$131$$ −338758. −1.72469 −0.862345 0.506321i $$-0.831005\pi$$
−0.862345 + 0.506321i $$0.831005\pi$$
$$132$$ 303870. 1.51793
$$133$$ 0 0
$$134$$ −568328. −2.73424
$$135$$ 17315.8 0.0817727
$$136$$ 232760. 1.07910
$$137$$ −62876.7 −0.286213 −0.143106 0.989707i $$-0.545709\pi$$
−0.143106 + 0.989707i $$0.545709\pi$$
$$138$$ 378513. 1.69193
$$139$$ 211927. 0.930356 0.465178 0.885217i $$-0.345990\pi$$
0.465178 + 0.885217i $$0.345990\pi$$
$$140$$ 0 0
$$141$$ −28094.3 −0.119007
$$142$$ −62129.4 −0.258569
$$143$$ −474511. −1.94047
$$144$$ 155141. 0.623477
$$145$$ −56564.3 −0.223420
$$146$$ −171553. −0.666062
$$147$$ 0 0
$$148$$ −718668. −2.69696
$$149$$ 140273. 0.517616 0.258808 0.965929i $$-0.416670\pi$$
0.258808 + 0.965929i $$0.416670\pi$$
$$150$$ −235632. −0.855077
$$151$$ −163991. −0.585300 −0.292650 0.956220i $$-0.594537\pi$$
−0.292650 + 0.956220i $$0.594537\pi$$
$$152$$ −574934. −2.01841
$$153$$ 45502.3 0.157147
$$154$$ 0 0
$$155$$ 70086.4 0.234317
$$156$$ −665342. −2.18894
$$157$$ 556543. 1.80198 0.900990 0.433840i $$-0.142842\pi$$
0.900990 + 0.433840i $$0.142842\pi$$
$$158$$ −49537.2 −0.157866
$$159$$ 10269.0 0.0322132
$$160$$ 150188. 0.463804
$$161$$ 0 0
$$162$$ 67078.7 0.200815
$$163$$ −19726.6 −0.0581546 −0.0290773 0.999577i $$-0.509257\pi$$
−0.0290773 + 0.999577i $$0.509257\pi$$
$$164$$ −324713. −0.942736
$$165$$ 99518.0 0.284572
$$166$$ 614918. 1.73200
$$167$$ 94776.2 0.262971 0.131486 0.991318i $$-0.458025\pi$$
0.131486 + 0.991318i $$0.458025\pi$$
$$168$$ 0 0
$$169$$ 667679. 1.79825
$$170$$ 136420. 0.362039
$$171$$ −112394. −0.293936
$$172$$ 375798. 0.968576
$$173$$ 338841. 0.860757 0.430379 0.902649i $$-0.358380\pi$$
0.430379 + 0.902649i $$0.358380\pi$$
$$174$$ −219121. −0.548670
$$175$$ 0 0
$$176$$ 891631. 2.16972
$$177$$ 247474. 0.593740
$$178$$ −638965. −1.51156
$$179$$ 776193. 1.81066 0.905330 0.424708i $$-0.139623\pi$$
0.905330 + 0.424708i $$0.139623\pi$$
$$180$$ 139541. 0.321011
$$181$$ −132697. −0.301067 −0.150534 0.988605i $$-0.548099\pi$$
−0.150534 + 0.988605i $$0.548099\pi$$
$$182$$ 0 0
$$183$$ −189932. −0.419247
$$184$$ 1.70445e6 3.71142
$$185$$ −235366. −0.505607
$$186$$ 271504. 0.575431
$$187$$ 261512. 0.546875
$$188$$ −226400. −0.467178
$$189$$ 0 0
$$190$$ −336967. −0.677178
$$191$$ 6637.41 0.0131648 0.00658242 0.999978i $$-0.497905\pi$$
0.00658242 + 0.999978i $$0.497905\pi$$
$$192$$ 30192.0 0.0591070
$$193$$ 452590. 0.874604 0.437302 0.899315i $$-0.355934\pi$$
0.437302 + 0.899315i $$0.355934\pi$$
$$194$$ −650787. −1.24147
$$195$$ −217901. −0.410368
$$196$$ 0 0
$$197$$ 816952. 1.49979 0.749896 0.661556i $$-0.230104\pi$$
0.749896 + 0.661556i $$0.230104\pi$$
$$198$$ 385517. 0.698845
$$199$$ 806417. 1.44353 0.721767 0.692136i $$-0.243330\pi$$
0.721767 + 0.692136i $$0.243330\pi$$
$$200$$ −1.06105e6 −1.87569
$$201$$ −500296. −0.873447
$$202$$ 1.88756e6 3.25478
$$203$$ 0 0
$$204$$ 366684. 0.616902
$$205$$ −106344. −0.176738
$$206$$ 532087. 0.873605
$$207$$ 333203. 0.540484
$$208$$ −1.95229e6 −3.12885
$$209$$ −645954. −1.02291
$$210$$ 0 0
$$211$$ 68773.9 0.106345 0.0531726 0.998585i $$-0.483067\pi$$
0.0531726 + 0.998585i $$0.483067\pi$$
$$212$$ 82753.3 0.126458
$$213$$ −54692.1 −0.0825992
$$214$$ 492563. 0.735237
$$215$$ 123075. 0.181582
$$216$$ 302057. 0.440508
$$217$$ 0 0
$$218$$ −372512. −0.530883
$$219$$ −151017. −0.212772
$$220$$ 801973. 1.11713
$$221$$ −572599. −0.788623
$$222$$ −911769. −1.24166
$$223$$ −620227. −0.835196 −0.417598 0.908632i $$-0.637128\pi$$
−0.417598 + 0.908632i $$0.637128\pi$$
$$224$$ 0 0
$$225$$ −207425. −0.273152
$$226$$ −988760. −1.28771
$$227$$ 1.00223e6 1.29093 0.645467 0.763788i $$-0.276663\pi$$
0.645467 + 0.763788i $$0.276663\pi$$
$$228$$ −905734. −1.15389
$$229$$ 885861. 1.11629 0.558145 0.829743i $$-0.311513\pi$$
0.558145 + 0.829743i $$0.311513\pi$$
$$230$$ 998973. 1.24519
$$231$$ 0 0
$$232$$ −986707. −1.20356
$$233$$ 596163. 0.719408 0.359704 0.933066i $$-0.382878\pi$$
0.359704 + 0.933066i $$0.382878\pi$$
$$234$$ −844115. −1.00777
$$235$$ −74146.6 −0.0875834
$$236$$ 1.99429e6 2.33081
$$237$$ −43607.4 −0.0504300
$$238$$ 0 0
$$239$$ 743111. 0.841509 0.420754 0.907175i $$-0.361765\pi$$
0.420754 + 0.907175i $$0.361765\pi$$
$$240$$ 409448. 0.458850
$$241$$ −1.17484e6 −1.30297 −0.651487 0.758660i $$-0.725854\pi$$
−0.651487 + 0.758660i $$0.725854\pi$$
$$242$$ 569095. 0.624664
$$243$$ 59049.0 0.0641500
$$244$$ −1.53058e6 −1.64582
$$245$$ 0 0
$$246$$ −411961. −0.434029
$$247$$ 1.41436e6 1.47508
$$248$$ 1.22259e6 1.26226
$$249$$ 541309. 0.553282
$$250$$ −1.38077e6 −1.39724
$$251$$ 352992. 0.353655 0.176828 0.984242i $$-0.443416\pi$$
0.176828 + 0.984242i $$0.443416\pi$$
$$252$$ 0 0
$$253$$ 1.91500e6 1.88090
$$254$$ 238450. 0.231906
$$255$$ 120090. 0.115653
$$256$$ −1.82533e6 −1.74077
$$257$$ 10012.7 0.00945621 0.00472811 0.999989i $$-0.498495\pi$$
0.00472811 + 0.999989i $$0.498495\pi$$
$$258$$ 476773. 0.445925
$$259$$ 0 0
$$260$$ −1.75597e6 −1.61096
$$261$$ −192891. −0.175271
$$262$$ −3.46341e6 −3.11710
$$263$$ 1.68180e6 1.49929 0.749644 0.661841i $$-0.230225\pi$$
0.749644 + 0.661841i $$0.230225\pi$$
$$264$$ 1.73599e6 1.53298
$$265$$ 27101.9 0.0237075
$$266$$ 0 0
$$267$$ −562477. −0.482866
$$268$$ −4.03167e6 −3.42885
$$269$$ −1.91717e6 −1.61540 −0.807702 0.589591i $$-0.799289\pi$$
−0.807702 + 0.589591i $$0.799289\pi$$
$$270$$ 177034. 0.147791
$$271$$ 2.29000e6 1.89414 0.947069 0.321029i $$-0.104029\pi$$
0.947069 + 0.321029i $$0.104029\pi$$
$$272$$ 1.07594e6 0.881795
$$273$$ 0 0
$$274$$ −642843. −0.517283
$$275$$ −1.19212e6 −0.950580
$$276$$ 2.68514e6 2.12175
$$277$$ −394802. −0.309157 −0.154579 0.987980i $$-0.549402\pi$$
−0.154579 + 0.987980i $$0.549402\pi$$
$$278$$ 2.16671e6 1.68147
$$279$$ 239003. 0.183820
$$280$$ 0 0
$$281$$ −1.77699e6 −1.34252 −0.671259 0.741223i $$-0.734246\pi$$
−0.671259 + 0.741223i $$0.734246\pi$$
$$282$$ −287232. −0.215085
$$283$$ 1.21524e6 0.901981 0.450991 0.892529i $$-0.351071\pi$$
0.450991 + 0.892529i $$0.351071\pi$$
$$284$$ −440741. −0.324255
$$285$$ −296630. −0.216323
$$286$$ −4.85133e6 −3.50708
$$287$$ 0 0
$$288$$ 512159. 0.363851
$$289$$ −1.10429e6 −0.777745
$$290$$ −578305. −0.403796
$$291$$ −572884. −0.396583
$$292$$ −1.21698e6 −0.835268
$$293$$ −1.48897e6 −1.01325 −0.506627 0.862165i $$-0.669108\pi$$
−0.506627 + 0.862165i $$0.669108\pi$$
$$294$$ 0 0
$$295$$ 653133. 0.436965
$$296$$ −4.10571e6 −2.72370
$$297$$ 339368. 0.223244
$$298$$ 1.43413e6 0.935508
$$299$$ −4.19301e6 −2.71236
$$300$$ −1.67155e6 −1.07230
$$301$$ 0 0
$$302$$ −1.67662e6 −1.05784
$$303$$ 1.66161e6 1.03973
$$304$$ −2.65766e6 −1.64936
$$305$$ −501268. −0.308546
$$306$$ 465209. 0.284017
$$307$$ 2.03109e6 1.22994 0.614968 0.788552i $$-0.289169\pi$$
0.614968 + 0.788552i $$0.289169\pi$$
$$308$$ 0 0
$$309$$ 468394. 0.279071
$$310$$ 716553. 0.423491
$$311$$ 289785. 0.169893 0.0849463 0.996386i $$-0.472928\pi$$
0.0849463 + 0.996386i $$0.472928\pi$$
$$312$$ −3.80106e6 −2.21064
$$313$$ 218211. 0.125897 0.0629484 0.998017i $$-0.479950\pi$$
0.0629484 + 0.998017i $$0.479950\pi$$
$$314$$ 5.69002e6 3.25679
$$315$$ 0 0
$$316$$ −351413. −0.197970
$$317$$ 1.29063e6 0.721363 0.360681 0.932689i $$-0.382544\pi$$
0.360681 + 0.932689i $$0.382544\pi$$
$$318$$ 104988. 0.0582202
$$319$$ −1.10859e6 −0.609951
$$320$$ 79682.8 0.0435000
$$321$$ 433600. 0.234870
$$322$$ 0 0
$$323$$ −779481. −0.415719
$$324$$ 475851. 0.251830
$$325$$ 2.61023e6 1.37079
$$326$$ −201682. −0.105105
$$327$$ −327920. −0.169589
$$328$$ −1.85507e6 −0.952084
$$329$$ 0 0
$$330$$ 1.01746e6 0.514318
$$331$$ −3.48561e6 −1.74867 −0.874336 0.485321i $$-0.838703\pi$$
−0.874336 + 0.485321i $$0.838703\pi$$
$$332$$ 4.36218e6 2.17199
$$333$$ −802625. −0.396645
$$334$$ 968978. 0.475278
$$335$$ −1.32038e6 −0.642817
$$336$$ 0 0
$$337$$ 249198. 0.119528 0.0597641 0.998213i $$-0.480965\pi$$
0.0597641 + 0.998213i $$0.480965\pi$$
$$338$$ 6.82625e6 3.25005
$$339$$ −870400. −0.411358
$$340$$ 967752. 0.454012
$$341$$ 1.37361e6 0.639700
$$342$$ −1.14910e6 −0.531241
$$343$$ 0 0
$$344$$ 2.14692e6 0.978180
$$345$$ 879390. 0.397772
$$346$$ 3.46426e6 1.55568
$$347$$ −1.50601e6 −0.671437 −0.335719 0.941962i $$-0.608979\pi$$
−0.335719 + 0.941962i $$0.608979\pi$$
$$348$$ −1.55443e6 −0.688054
$$349$$ 1.54370e6 0.678423 0.339212 0.940710i $$-0.389840\pi$$
0.339212 + 0.940710i $$0.389840\pi$$
$$350$$ 0 0
$$351$$ −743070. −0.321930
$$352$$ 2.94350e6 1.26621
$$353$$ −1.67796e6 −0.716710 −0.358355 0.933585i $$-0.616662\pi$$
−0.358355 + 0.933585i $$0.616662\pi$$
$$354$$ 2.53014e6 1.07309
$$355$$ −144344. −0.0607892
$$356$$ −4.53276e6 −1.89556
$$357$$ 0 0
$$358$$ 7.93568e6 3.27248
$$359$$ −2.49973e6 −1.02366 −0.511832 0.859086i $$-0.671033\pi$$
−0.511832 + 0.859086i $$0.671033\pi$$
$$360$$ 797188. 0.324194
$$361$$ −550727. −0.222417
$$362$$ −1.35667e6 −0.544130
$$363$$ 500971. 0.199547
$$364$$ 0 0
$$365$$ −398564. −0.156591
$$366$$ −1.94183e6 −0.757721
$$367$$ −2.10741e6 −0.816740 −0.408370 0.912816i $$-0.633903\pi$$
−0.408370 + 0.912816i $$0.633903\pi$$
$$368$$ 7.87890e6 3.03281
$$369$$ −362647. −0.138650
$$370$$ −2.40634e6 −0.913804
$$371$$ 0 0
$$372$$ 1.92602e6 0.721613
$$373$$ 2.73225e6 1.01683 0.508414 0.861113i $$-0.330232\pi$$
0.508414 + 0.861113i $$0.330232\pi$$
$$374$$ 2.67366e6 0.988389
$$375$$ −1.21548e6 −0.446346
$$376$$ −1.29341e6 −0.471810
$$377$$ 2.42733e6 0.879581
$$378$$ 0 0
$$379$$ −2.04295e6 −0.730567 −0.365283 0.930896i $$-0.619028\pi$$
−0.365283 + 0.930896i $$0.619028\pi$$
$$380$$ −2.39041e6 −0.849208
$$381$$ 209906. 0.0740819
$$382$$ 67859.9 0.0237933
$$383$$ 1.50899e6 0.525643 0.262821 0.964845i $$-0.415347\pi$$
0.262821 + 0.964845i $$0.415347\pi$$
$$384$$ −1.51233e6 −0.523382
$$385$$ 0 0
$$386$$ 4.62721e6 1.58071
$$387$$ 419700. 0.142450
$$388$$ −4.61663e6 −1.55685
$$389$$ 4.50065e6 1.50800 0.754000 0.656874i $$-0.228122\pi$$
0.754000 + 0.656874i $$0.228122\pi$$
$$390$$ −2.22779e6 −0.741673
$$391$$ 2.31085e6 0.764417
$$392$$ 0 0
$$393$$ −3.04882e6 −0.995751
$$394$$ 8.35240e6 2.71063
$$395$$ −115089. −0.0371142
$$396$$ 2.73483e6 0.876378
$$397$$ −3.52136e6 −1.12133 −0.560665 0.828042i $$-0.689455\pi$$
−0.560665 + 0.828042i $$0.689455\pi$$
$$398$$ 8.24469e6 2.60895
$$399$$ 0 0
$$400$$ −4.90476e6 −1.53274
$$401$$ −907107. −0.281707 −0.140853 0.990030i $$-0.544985\pi$$
−0.140853 + 0.990030i $$0.544985\pi$$
$$402$$ −5.11495e6 −1.57862
$$403$$ −3.00760e6 −0.922482
$$404$$ 1.33902e7 4.08162
$$405$$ 155842. 0.0472115
$$406$$ 0 0
$$407$$ −4.61287e6 −1.38034
$$408$$ 2.09484e6 0.623019
$$409$$ 4.31853e6 1.27652 0.638260 0.769821i $$-0.279654\pi$$
0.638260 + 0.769821i $$0.279654\pi$$
$$410$$ −1.08725e6 −0.319425
$$411$$ −565891. −0.165245
$$412$$ 3.77458e6 1.09553
$$413$$ 0 0
$$414$$ 3.40662e6 0.976838
$$415$$ 1.42862e6 0.407190
$$416$$ −6.44498e6 −1.82595
$$417$$ 1.90734e6 0.537141
$$418$$ −6.60413e6 −1.84874
$$419$$ −1.52041e6 −0.423083 −0.211541 0.977369i $$-0.567848\pi$$
−0.211541 + 0.977369i $$0.567848\pi$$
$$420$$ 0 0
$$421$$ −4.42050e6 −1.21553 −0.607766 0.794116i $$-0.707934\pi$$
−0.607766 + 0.794116i $$0.707934\pi$$
$$422$$ 703134. 0.192202
$$423$$ −252849. −0.0687085
$$424$$ 472765. 0.127712
$$425$$ −1.43855e6 −0.386325
$$426$$ −559164. −0.149285
$$427$$ 0 0
$$428$$ 3.49420e6 0.922016
$$429$$ −4.27060e6 −1.12033
$$430$$ 1.25830e6 0.328180
$$431$$ −6.84785e6 −1.77567 −0.887833 0.460166i $$-0.847790\pi$$
−0.887833 + 0.460166i $$0.847790\pi$$
$$432$$ 1.39627e6 0.359964
$$433$$ −3.99328e6 −1.02355 −0.511777 0.859119i $$-0.671012\pi$$
−0.511777 + 0.859119i $$0.671012\pi$$
$$434$$ 0 0
$$435$$ −509079. −0.128992
$$436$$ −2.64257e6 −0.665748
$$437$$ −5.70797e6 −1.42981
$$438$$ −1.54397e6 −0.384551
$$439$$ 1.48557e6 0.367901 0.183950 0.982936i $$-0.441111\pi$$
0.183950 + 0.982936i $$0.441111\pi$$
$$440$$ 4.58163e6 1.12821
$$441$$ 0 0
$$442$$ −5.85416e6 −1.42531
$$443$$ −5.62193e6 −1.36106 −0.680528 0.732722i $$-0.738250\pi$$
−0.680528 + 0.732722i $$0.738250\pi$$
$$444$$ −6.46801e6 −1.55709
$$445$$ −1.48449e6 −0.355367
$$446$$ −6.34111e6 −1.50948
$$447$$ 1.26246e6 0.298846
$$448$$ 0 0
$$449$$ 1.94883e6 0.456202 0.228101 0.973637i $$-0.426748\pi$$
0.228101 + 0.973637i $$0.426748\pi$$
$$450$$ −2.12068e6 −0.493679
$$451$$ −2.08422e6 −0.482505
$$452$$ −7.01418e6 −1.61484
$$453$$ −1.47592e6 −0.337923
$$454$$ 1.02467e7 2.33315
$$455$$ 0 0
$$456$$ −5.17441e6 −1.16533
$$457$$ −2.67312e6 −0.598726 −0.299363 0.954139i $$-0.596774\pi$$
−0.299363 + 0.954139i $$0.596774\pi$$
$$458$$ 9.05691e6 2.01751
$$459$$ 409521. 0.0907286
$$460$$ 7.08663e6 1.56151
$$461$$ −4.65262e6 −1.01964 −0.509819 0.860282i $$-0.670287\pi$$
−0.509819 + 0.860282i $$0.670287\pi$$
$$462$$ 0 0
$$463$$ −5.18586e6 −1.12426 −0.562132 0.827047i $$-0.690019\pi$$
−0.562132 + 0.827047i $$0.690019\pi$$
$$464$$ −4.56109e6 −0.983499
$$465$$ 630777. 0.135283
$$466$$ 6.09509e6 1.30021
$$467$$ −5.05980e6 −1.07360 −0.536799 0.843710i $$-0.680367\pi$$
−0.536799 + 0.843710i $$0.680367\pi$$
$$468$$ −5.98808e6 −1.26378
$$469$$ 0 0
$$470$$ −758064. −0.158293
$$471$$ 5.00889e6 1.04037
$$472$$ 1.13932e7 2.35392
$$473$$ 2.41212e6 0.495730
$$474$$ −445835. −0.0911441
$$475$$ 3.55331e6 0.722603
$$476$$ 0 0
$$477$$ 92420.8 0.0185983
$$478$$ 7.59745e6 1.52089
$$479$$ −5.55085e6 −1.10540 −0.552702 0.833379i $$-0.686403\pi$$
−0.552702 + 0.833379i $$0.686403\pi$$
$$480$$ 1.35169e6 0.267778
$$481$$ 1.01002e7 1.99052
$$482$$ −1.20114e7 −2.35491
$$483$$ 0 0
$$484$$ 4.03711e6 0.783353
$$485$$ −1.51196e6 −0.291867
$$486$$ 603708. 0.115941
$$487$$ −6.87810e6 −1.31415 −0.657077 0.753824i $$-0.728207\pi$$
−0.657077 + 0.753824i $$0.728207\pi$$
$$488$$ −8.74411e6 −1.66213
$$489$$ −177540. −0.0335756
$$490$$ 0 0
$$491$$ 7.44459e6 1.39360 0.696798 0.717267i $$-0.254607\pi$$
0.696798 + 0.717267i $$0.254607\pi$$
$$492$$ −2.92242e6 −0.544289
$$493$$ −1.33775e6 −0.247890
$$494$$ 1.44602e7 2.66598
$$495$$ 895662. 0.164298
$$496$$ 5.65145e6 1.03147
$$497$$ 0 0
$$498$$ 5.53426e6 0.999968
$$499$$ 2.15329e6 0.387126 0.193563 0.981088i $$-0.437996\pi$$
0.193563 + 0.981088i $$0.437996\pi$$
$$500$$ −9.79507e6 −1.75220
$$501$$ 852986. 0.151826
$$502$$ 3.60894e6 0.639175
$$503$$ 8.61012e6 1.51736 0.758681 0.651463i $$-0.225844\pi$$
0.758681 + 0.651463i $$0.225844\pi$$
$$504$$ 0 0
$$505$$ 4.38531e6 0.765195
$$506$$ 1.95786e7 3.39943
$$507$$ 6.00911e6 1.03822
$$508$$ 1.69154e6 0.290820
$$509$$ 3.34201e6 0.571759 0.285879 0.958266i $$-0.407714\pi$$
0.285879 + 0.958266i $$0.407714\pi$$
$$510$$ 1.22778e6 0.209024
$$511$$ 0 0
$$512$$ −1.32848e7 −2.23964
$$513$$ −1.01154e6 −0.169704
$$514$$ 102368. 0.0170906
$$515$$ 1.23619e6 0.205383
$$516$$ 3.38219e6 0.559208
$$517$$ −1.45318e6 −0.239108
$$518$$ 0 0
$$519$$ 3.04957e6 0.496958
$$520$$ −1.00318e7 −1.62693
$$521$$ −2.68305e6 −0.433046 −0.216523 0.976278i $$-0.569472\pi$$
−0.216523 + 0.976278i $$0.569472\pi$$
$$522$$ −1.97209e6 −0.316775
$$523$$ −5.42355e6 −0.867021 −0.433511 0.901148i $$-0.642725\pi$$
−0.433511 + 0.901148i $$0.642725\pi$$
$$524$$ −2.45692e7 −3.90897
$$525$$ 0 0
$$526$$ 1.71945e7 2.70972
$$527$$ 1.65755e6 0.259980
$$528$$ 8.02468e6 1.25269
$$529$$ 1.04855e7 1.62911
$$530$$ 277086. 0.0428474
$$531$$ 2.22726e6 0.342796
$$532$$ 0 0
$$533$$ 4.56353e6 0.695798
$$534$$ −5.75068e6 −0.872702
$$535$$ 1.14436e6 0.172853
$$536$$ −2.30327e7 −3.46285
$$537$$ 6.98574e6 1.04539
$$538$$ −1.96009e7 −2.91958
$$539$$ 0 0
$$540$$ 1.25587e6 0.185336
$$541$$ 2.18456e6 0.320900 0.160450 0.987044i $$-0.448705\pi$$
0.160450 + 0.987044i $$0.448705\pi$$
$$542$$ 2.34126e7 3.42335
$$543$$ −1.19427e6 −0.173821
$$544$$ 3.55196e6 0.514601
$$545$$ −865447. −0.124810
$$546$$ 0 0
$$547$$ −691437. −0.0988062 −0.0494031 0.998779i $$-0.515732\pi$$
−0.0494031 + 0.998779i $$0.515732\pi$$
$$548$$ −4.56027e6 −0.648693
$$549$$ −1.70939e6 −0.242052
$$550$$ −1.21881e7 −1.71802
$$551$$ 3.30434e6 0.463667
$$552$$ 1.53401e7 2.14279
$$553$$ 0 0
$$554$$ −4.03639e6 −0.558752
$$555$$ −2.11829e6 −0.291913
$$556$$ 1.53705e7 2.10863
$$557$$ 5.47656e6 0.747946 0.373973 0.927440i $$-0.377995\pi$$
0.373973 + 0.927440i $$0.377995\pi$$
$$558$$ 2.44353e6 0.332225
$$559$$ −5.28149e6 −0.714869
$$560$$ 0 0
$$561$$ 2.35361e6 0.315739
$$562$$ −1.81677e7 −2.42638
$$563$$ 398976. 0.0530488 0.0265244 0.999648i $$-0.491556\pi$$
0.0265244 + 0.999648i $$0.491556\pi$$
$$564$$ −2.03760e6 −0.269725
$$565$$ −2.29716e6 −0.302740
$$566$$ 1.24245e7 1.63019
$$567$$ 0 0
$$568$$ −2.51793e6 −0.327471
$$569$$ −4.23113e6 −0.547868 −0.273934 0.961749i $$-0.588325\pi$$
−0.273934 + 0.961749i $$0.588325\pi$$
$$570$$ −3.03270e6 −0.390969
$$571$$ 8.75102e6 1.12323 0.561615 0.827399i $$-0.310180\pi$$
0.561615 + 0.827399i $$0.310180\pi$$
$$572$$ −3.44149e7 −4.39801
$$573$$ 59736.7 0.00760072
$$574$$ 0 0
$$575$$ −1.05342e7 −1.32871
$$576$$ 271728. 0.0341254
$$577$$ −1.76712e6 −0.220966 −0.110483 0.993878i $$-0.535240\pi$$
−0.110483 + 0.993878i $$0.535240\pi$$
$$578$$ −1.12901e7 −1.40565
$$579$$ 4.07331e6 0.504953
$$580$$ −4.10245e6 −0.506376
$$581$$ 0 0
$$582$$ −5.85709e6 −0.716761
$$583$$ 531164. 0.0647228
$$584$$ −6.95254e6 −0.843551
$$585$$ −1.96111e6 −0.236926
$$586$$ −1.52231e7 −1.83129
$$587$$ −8.65009e6 −1.03616 −0.518078 0.855333i $$-0.673352\pi$$
−0.518078 + 0.855333i $$0.673352\pi$$
$$588$$ 0 0
$$589$$ −4.09426e6 −0.486281
$$590$$ 6.67754e6 0.789744
$$591$$ 7.35257e6 0.865905
$$592$$ −1.89788e7 −2.22569
$$593$$ 1.54514e7 1.80439 0.902194 0.431329i $$-0.141955\pi$$
0.902194 + 0.431329i $$0.141955\pi$$
$$594$$ 3.46965e6 0.403478
$$595$$ 0 0
$$596$$ 1.01736e7 1.17316
$$597$$ 7.25775e6 0.833425
$$598$$ −4.28687e7 −4.90216
$$599$$ −3.84801e6 −0.438196 −0.219098 0.975703i $$-0.570312\pi$$
−0.219098 + 0.975703i $$0.570312\pi$$
$$600$$ −9.54948e6 −1.08293
$$601$$ 1.27578e7 1.44075 0.720376 0.693583i $$-0.243969\pi$$
0.720376 + 0.693583i $$0.243969\pi$$
$$602$$ 0 0
$$603$$ −4.50266e6 −0.504285
$$604$$ −1.18938e7 −1.32657
$$605$$ 1.32216e6 0.146858
$$606$$ 1.69880e7 1.87915
$$607$$ −1.36207e7 −1.50047 −0.750233 0.661173i $$-0.770059\pi$$
−0.750233 + 0.661173i $$0.770059\pi$$
$$608$$ −8.77359e6 −0.962539
$$609$$ 0 0
$$610$$ −5.12489e6 −0.557648
$$611$$ 3.18184e6 0.344806
$$612$$ 3.30015e6 0.356168
$$613$$ 1.53474e7 1.64962 0.824812 0.565407i $$-0.191281\pi$$
0.824812 + 0.565407i $$0.191281\pi$$
$$614$$ 2.07655e7 2.22291
$$615$$ −957099. −0.102040
$$616$$ 0 0
$$617$$ 1.18485e7 1.25300 0.626500 0.779421i $$-0.284487\pi$$
0.626500 + 0.779421i $$0.284487\pi$$
$$618$$ 4.78879e6 0.504376
$$619$$ −874460. −0.0917304 −0.0458652 0.998948i $$-0.514604\pi$$
−0.0458652 + 0.998948i $$0.514604\pi$$
$$620$$ 5.08317e6 0.531074
$$621$$ 2.99883e6 0.312049
$$622$$ 2.96272e6 0.307054
$$623$$ 0 0
$$624$$ −1.75706e7 −1.80644
$$625$$ 4.79460e6 0.490967
$$626$$ 2.23095e6 0.227538
$$627$$ −5.81358e6 −0.590575
$$628$$ 4.03645e7 4.08414
$$629$$ −5.56642e6 −0.560983
$$630$$ 0 0
$$631$$ −4.43859e6 −0.443784 −0.221892 0.975071i $$-0.571223\pi$$
−0.221892 + 0.975071i $$0.571223\pi$$
$$632$$ −2.00760e6 −0.199933
$$633$$ 618965. 0.0613984
$$634$$ 1.31952e7 1.30375
$$635$$ 553984. 0.0545209
$$636$$ 744780. 0.0730104
$$637$$ 0 0
$$638$$ −1.13341e7 −1.10239
$$639$$ −492229. −0.0476887
$$640$$ −3.99135e6 −0.385185
$$641$$ 812551. 0.0781098 0.0390549 0.999237i $$-0.487565\pi$$
0.0390549 + 0.999237i $$0.487565\pi$$
$$642$$ 4.43306e6 0.424489
$$643$$ −1.17941e7 −1.12496 −0.562481 0.826810i $$-0.690153\pi$$
−0.562481 + 0.826810i $$0.690153\pi$$
$$644$$ 0 0
$$645$$ 1.10767e6 0.104837
$$646$$ −7.96930e6 −0.751344
$$647$$ −2.43380e6 −0.228573 −0.114286 0.993448i $$-0.536458\pi$$
−0.114286 + 0.993448i $$0.536458\pi$$
$$648$$ 2.71851e6 0.254327
$$649$$ 1.28006e7 1.19294
$$650$$ 2.66866e7 2.47748
$$651$$ 0 0
$$652$$ −1.43072e6 −0.131806
$$653$$ −9.46664e6 −0.868786 −0.434393 0.900723i $$-0.643037\pi$$
−0.434393 + 0.900723i $$0.643037\pi$$
$$654$$ −3.35261e6 −0.306506
$$655$$ −8.04646e6 −0.732827
$$656$$ −8.57513e6 −0.778003
$$657$$ −1.35915e6 −0.122844
$$658$$ 0 0
$$659$$ 1.40662e7 1.26172 0.630860 0.775896i $$-0.282702\pi$$
0.630860 + 0.775896i $$0.282702\pi$$
$$660$$ 7.21776e6 0.644974
$$661$$ −1.71606e7 −1.52766 −0.763832 0.645415i $$-0.776685\pi$$
−0.763832 + 0.645415i $$0.776685\pi$$
$$662$$ −3.56363e7 −3.16044
$$663$$ −5.15339e6 −0.455312
$$664$$ 2.49209e7 2.19353
$$665$$ 0 0
$$666$$ −8.20592e6 −0.716872
$$667$$ −9.79606e6 −0.852583
$$668$$ 6.87385e6 0.596017
$$669$$ −5.58204e6 −0.482201
$$670$$ −1.34994e7 −1.16179
$$671$$ −9.82424e6 −0.842350
$$672$$ 0 0
$$673$$ 5.87113e6 0.499671 0.249836 0.968288i $$-0.419623\pi$$
0.249836 + 0.968288i $$0.419623\pi$$
$$674$$ 2.54777e6 0.216028
$$675$$ −1.86683e6 −0.157705
$$676$$ 4.84249e7 4.07570
$$677$$ 1.39274e7 1.16788 0.583938 0.811798i $$-0.301511\pi$$
0.583938 + 0.811798i $$0.301511\pi$$
$$678$$ −8.89884e6 −0.743462
$$679$$ 0 0
$$680$$ 5.52872e6 0.458513
$$681$$ 9.02009e6 0.745321
$$682$$ 1.40436e7 1.15615
$$683$$ 1.83101e7 1.50189 0.750945 0.660365i $$-0.229598\pi$$
0.750945 + 0.660365i $$0.229598\pi$$
$$684$$ −8.15160e6 −0.666197
$$685$$ −1.49350e6 −0.121613
$$686$$ 0 0
$$687$$ 7.97275e6 0.644490
$$688$$ 9.92420e6 0.799327
$$689$$ −1.16302e6 −0.0933337
$$690$$ 8.99076e6 0.718908
$$691$$ −1.76717e7 −1.40793 −0.703967 0.710233i $$-0.748590\pi$$
−0.703967 + 0.710233i $$0.748590\pi$$
$$692$$ 2.45752e7 1.95088
$$693$$ 0 0
$$694$$ −1.53973e7 −1.21351
$$695$$ 5.03386e6 0.395311
$$696$$ −8.88036e6 −0.694877
$$697$$ −2.51506e6 −0.196095
$$698$$ 1.57826e7 1.22614
$$699$$ 5.36547e6 0.415351
$$700$$ 0 0
$$701$$ 8.22993e6 0.632559 0.316279 0.948666i $$-0.397566\pi$$
0.316279 + 0.948666i $$0.397566\pi$$
$$702$$ −7.59703e6 −0.581837
$$703$$ 1.37494e7 1.04929
$$704$$ 1.56169e6 0.118758
$$705$$ −667320. −0.0505663
$$706$$ −1.71552e7 −1.29534
$$707$$ 0 0
$$708$$ 1.79486e7 1.34570
$$709$$ 2.46367e7 1.84063 0.920314 0.391180i $$-0.127933\pi$$
0.920314 + 0.391180i $$0.127933\pi$$
$$710$$ −1.47575e6 −0.109867
$$711$$ −392466. −0.0291158
$$712$$ −2.58954e7 −1.91436
$$713$$ 1.21379e7 0.894167
$$714$$ 0 0
$$715$$ −1.12710e7 −0.824510
$$716$$ 5.62951e7 4.10382
$$717$$ 6.68800e6 0.485845
$$718$$ −2.55569e7 −1.85011
$$719$$ −1.88254e7 −1.35807 −0.679034 0.734107i $$-0.737601\pi$$
−0.679034 + 0.734107i $$0.737601\pi$$
$$720$$ 3.68503e6 0.264917
$$721$$ 0 0
$$722$$ −5.63055e6 −0.401983
$$723$$ −1.05735e7 −0.752272
$$724$$ −9.62411e6 −0.682361
$$725$$ 6.09823e6 0.430882
$$726$$ 5.12186e6 0.360650
$$727$$ 6.77607e6 0.475491 0.237745 0.971328i $$-0.423592\pi$$
0.237745 + 0.971328i $$0.423592\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ −4.07486e6 −0.283012
$$731$$ 2.91073e6 0.201469
$$732$$ −1.37752e7 −0.950212
$$733$$ −1.72227e7 −1.18397 −0.591986 0.805948i $$-0.701656\pi$$
−0.591986 + 0.805948i $$0.701656\pi$$
$$734$$ −2.15459e7 −1.47613
$$735$$ 0 0
$$736$$ 2.60102e7 1.76990
$$737$$ −2.58779e7 −1.75493
$$738$$ −3.70765e6 −0.250587
$$739$$ −1.76534e7 −1.18910 −0.594548 0.804060i $$-0.702669\pi$$
−0.594548 + 0.804060i $$0.702669\pi$$
$$740$$ −1.70704e7 −1.14595
$$741$$ 1.27292e7 0.851641
$$742$$ 0 0
$$743$$ −1.36977e7 −0.910281 −0.455141 0.890420i $$-0.650411\pi$$
−0.455141 + 0.890420i $$0.650411\pi$$
$$744$$ 1.10033e7 0.728768
$$745$$ 3.33188e6 0.219937
$$746$$ 2.79341e7 1.83775
$$747$$ 4.87178e6 0.319438
$$748$$ 1.89668e7 1.23948
$$749$$ 0 0
$$750$$ −1.24269e7 −0.806698
$$751$$ 1.16089e7 0.751090 0.375545 0.926804i $$-0.377456\pi$$
0.375545 + 0.926804i $$0.377456\pi$$
$$752$$ −5.97885e6 −0.385543
$$753$$ 3.17693e6 0.204183
$$754$$ 2.48167e7 1.58970
$$755$$ −3.89526e6 −0.248696
$$756$$ 0 0
$$757$$ 6.25226e6 0.396550 0.198275 0.980146i $$-0.436466\pi$$
0.198275 + 0.980146i $$0.436466\pi$$
$$758$$ −2.08868e7 −1.32038
$$759$$ 1.72350e7 1.08594
$$760$$ −1.36563e7 −0.857629
$$761$$ −3.02125e7 −1.89115 −0.945574 0.325406i $$-0.894499\pi$$
−0.945574 + 0.325406i $$0.894499\pi$$
$$762$$ 2.14605e6 0.133891
$$763$$ 0 0
$$764$$ 481393. 0.0298378
$$765$$ 1.08081e6 0.0667721
$$766$$ 1.54277e7 0.950014
$$767$$ −2.80278e7 −1.72028
$$768$$ −1.64280e7 −1.00504
$$769$$ −2.58756e7 −1.57788 −0.788940 0.614470i $$-0.789370\pi$$
−0.788940 + 0.614470i $$0.789370\pi$$
$$770$$ 0 0
$$771$$ 90114.1 0.00545955
$$772$$ 3.28251e7 1.98227
$$773$$ 2.55409e7 1.53740 0.768701 0.639608i $$-0.220903\pi$$
0.768701 + 0.639608i $$0.220903\pi$$
$$774$$ 4.29095e6 0.257455
$$775$$ −7.55605e6 −0.451898
$$776$$ −2.63746e7 −1.57228
$$777$$ 0 0
$$778$$ 4.60140e7 2.72547
$$779$$ 6.21236e6 0.366786
$$780$$ −1.58038e7 −0.930087
$$781$$ −2.82896e6 −0.165958
$$782$$ 2.36258e7 1.38156
$$783$$ −1.73602e6 −0.101193
$$784$$ 0 0
$$785$$ 1.32195e7 0.765667
$$786$$ −3.11707e7 −1.79966
$$787$$ −8.81179e6 −0.507139 −0.253570 0.967317i $$-0.581605\pi$$
−0.253570 + 0.967317i $$0.581605\pi$$
$$788$$ 5.92512e7 3.39924
$$789$$ 1.51362e7 0.865614
$$790$$ −1.17665e6 −0.0670779
$$791$$ 0 0
$$792$$ 1.56239e7 0.885068
$$793$$ 2.15108e7 1.21471
$$794$$ −3.60018e7 −2.02662
$$795$$ 243917. 0.0136875
$$796$$ 5.84871e7 3.27173
$$797$$ −2.07624e7 −1.15780 −0.578899 0.815399i $$-0.696517\pi$$
−0.578899 + 0.815399i $$0.696517\pi$$
$$798$$ 0 0
$$799$$ −1.75358e6 −0.0971757
$$800$$ −1.61918e7 −0.894481
$$801$$ −5.06229e6 −0.278783
$$802$$ −9.27413e6 −0.509140
$$803$$ −7.81136e6 −0.427501
$$804$$ −3.62850e7 −1.97965
$$805$$ 0 0
$$806$$ −3.07493e7 −1.66724
$$807$$ −1.72546e7 −0.932653
$$808$$ 7.64973e7 4.12209
$$809$$ 1.37021e7 0.736067 0.368034 0.929813i $$-0.380031\pi$$
0.368034 + 0.929813i $$0.380031\pi$$
$$810$$ 1.59331e6 0.0853271
$$811$$ 3.30799e7 1.76609 0.883043 0.469292i $$-0.155491\pi$$
0.883043 + 0.469292i $$0.155491\pi$$
$$812$$ 0 0
$$813$$ 2.06100e7 1.09358
$$814$$ −4.71613e7 −2.49474
$$815$$ −468563. −0.0247101
$$816$$ 9.68350e6 0.509104
$$817$$ −7.18971e6 −0.376840
$$818$$ 4.41520e7 2.30711
$$819$$ 0 0
$$820$$ −7.71285e6 −0.400572
$$821$$ 1.48323e7 0.767982 0.383991 0.923337i $$-0.374549\pi$$
0.383991 + 0.923337i $$0.374549\pi$$
$$822$$ −5.78558e6 −0.298653
$$823$$ 1.08722e7 0.559525 0.279762 0.960069i $$-0.409744\pi$$
0.279762 + 0.960069i $$0.409744\pi$$
$$824$$ 2.15640e7 1.10640
$$825$$ −1.07291e7 −0.548817
$$826$$ 0 0
$$827$$ −1.34267e7 −0.682662 −0.341331 0.939943i $$-0.610878\pi$$
−0.341331 + 0.939943i $$0.610878\pi$$
$$828$$ 2.41663e7 1.22499
$$829$$ −1.46754e7 −0.741658 −0.370829 0.928701i $$-0.620926\pi$$
−0.370829 + 0.928701i $$0.620926\pi$$
$$830$$ 1.46060e7 0.735931
$$831$$ −3.55322e6 −0.178492
$$832$$ −3.41941e6 −0.171255
$$833$$ 0 0
$$834$$ 1.95004e7 0.970796
$$835$$ 2.25120e6 0.111737
$$836$$ −4.68492e7 −2.31839
$$837$$ 2.15103e6 0.106129
$$838$$ −1.55444e7 −0.764654
$$839$$ 2.76635e7 1.35676 0.678379 0.734712i $$-0.262683\pi$$
0.678379 + 0.734712i $$0.262683\pi$$
$$840$$ 0 0
$$841$$ −1.48402e7 −0.723519
$$842$$ −4.51946e7 −2.19688
$$843$$ −1.59929e7 −0.775103
$$844$$ 4.98798e6 0.241028
$$845$$ 1.58593e7 0.764084
$$846$$ −2.58509e6 −0.124180
$$847$$ 0 0
$$848$$ 2.18537e6 0.104361
$$849$$ 1.09372e7 0.520759
$$850$$ −1.47075e7 −0.698219
$$851$$ −4.07616e7 −1.92942
$$852$$ −3.96667e6 −0.187209
$$853$$ 3.43515e6 0.161649 0.0808244 0.996728i $$-0.474245\pi$$
0.0808244 + 0.996728i $$0.474245\pi$$
$$854$$ 0 0
$$855$$ −2.66967e6 −0.124894
$$856$$ 1.99622e7 0.931158
$$857$$ −6.40626e6 −0.297956 −0.148978 0.988840i $$-0.547598\pi$$
−0.148978 + 0.988840i $$0.547598\pi$$
$$858$$ −4.36619e7 −2.02481
$$859$$ 1.81739e7 0.840360 0.420180 0.907441i $$-0.361967\pi$$
0.420180 + 0.907441i $$0.361967\pi$$
$$860$$ 8.92627e6 0.411551
$$861$$ 0 0
$$862$$ −7.00114e7 −3.20923
$$863$$ 818774. 0.0374229 0.0187114 0.999825i $$-0.494044\pi$$
0.0187114 + 0.999825i $$0.494044\pi$$
$$864$$ 4.60943e6 0.210069
$$865$$ 8.04843e6 0.365739
$$866$$ −4.08267e7 −1.84991
$$867$$ −9.93858e6 −0.449031
$$868$$ 0 0
$$869$$ −2.25560e6 −0.101324
$$870$$ −5.20475e6 −0.233132
$$871$$ 5.66613e7 2.53070
$$872$$ −1.50968e7 −0.672350
$$873$$ −5.15596e6 −0.228968
$$874$$ −5.83574e7 −2.58415
$$875$$ 0 0
$$876$$ −1.09528e7 −0.482242
$$877$$ −3.33962e7 −1.46622 −0.733108 0.680113i $$-0.761931\pi$$
−0.733108 + 0.680113i $$0.761931\pi$$
$$878$$ 1.51882e7 0.664921
$$879$$ −1.34008e7 −0.585002
$$880$$ 2.11788e7 0.921921
$$881$$ −1.24325e7 −0.539658 −0.269829 0.962908i $$-0.586967\pi$$
−0.269829 + 0.962908i $$0.586967\pi$$
$$882$$ 0 0
$$883$$ −2.33298e7 −1.00695 −0.503476 0.864009i $$-0.667946\pi$$
−0.503476 + 0.864009i $$0.667946\pi$$
$$884$$ −4.15290e7 −1.78739
$$885$$ 5.87820e6 0.252282
$$886$$ −5.74778e7 −2.45989
$$887$$ −4.75528e6 −0.202940 −0.101470 0.994839i $$-0.532355\pi$$
−0.101470 + 0.994839i $$0.532355\pi$$
$$888$$ −3.69514e7 −1.57253
$$889$$ 0 0
$$890$$ −1.51772e7 −0.642269
$$891$$ 3.05432e6 0.128890
$$892$$ −4.49833e7 −1.89295
$$893$$ 4.33145e6 0.181763
$$894$$ 1.29072e7 0.540116
$$895$$ 1.84368e7 0.769356
$$896$$ 0 0
$$897$$ −3.77371e7 −1.56598
$$898$$ 1.99245e7 0.824512
$$899$$ −7.02661e6 −0.289966
$$900$$ −1.50440e7 −0.619093
$$901$$ 640963. 0.0263040
$$902$$ −2.13087e7 −0.872050
$$903$$ 0 0
$$904$$ −4.00716e7 −1.63086
$$905$$ −3.15192e6 −0.127924
$$906$$ −1.50896e7 −0.610742
$$907$$ −1.90902e7 −0.770537 −0.385268 0.922805i $$-0.625891\pi$$
−0.385268 + 0.922805i $$0.625891\pi$$
$$908$$ 7.26891e7 2.92587
$$909$$ 1.49544e7 0.600289
$$910$$ 0 0
$$911$$ −1.22591e6 −0.0489399 −0.0244700 0.999701i $$-0.507790\pi$$
−0.0244700 + 0.999701i $$0.507790\pi$$
$$912$$ −2.39189e7 −0.952257
$$913$$ 2.79993e7 1.11165
$$914$$ −2.73296e7 −1.08210
$$915$$ −4.51141e6 −0.178139
$$916$$ 6.42490e7 2.53004
$$917$$ 0 0
$$918$$ 4.18688e6 0.163977
$$919$$ −1.69227e7 −0.660970 −0.330485 0.943811i $$-0.607212\pi$$
−0.330485 + 0.943811i $$0.607212\pi$$
$$920$$ 4.04855e7 1.57699
$$921$$ 1.82798e7 0.710103
$$922$$ −4.75677e7 −1.84283
$$923$$ 6.19419e6 0.239321
$$924$$ 0 0
$$925$$ 2.53749e7 0.975101
$$926$$ −5.30195e7 −2.03193
$$927$$ 4.21554e6 0.161122
$$928$$ −1.50573e7 −0.573954
$$929$$ 6.77240e6 0.257456 0.128728 0.991680i $$-0.458911\pi$$
0.128728 + 0.991680i $$0.458911\pi$$
$$930$$ 6.44897e6 0.244502
$$931$$ 0 0
$$932$$ 4.32380e7 1.63052
$$933$$ 2.60806e6 0.0980875
$$934$$ −5.17307e7 −1.94035
$$935$$ 6.21166e6 0.232369
$$936$$ −3.42096e7 −1.27632
$$937$$ −1.41035e7 −0.524779 −0.262389 0.964962i $$-0.584511\pi$$
−0.262389 + 0.964962i $$0.584511\pi$$
$$938$$ 0 0
$$939$$ 1.96389e6 0.0726866
$$940$$ −5.37765e6 −0.198505
$$941$$ −5.62524e6 −0.207094 −0.103547 0.994625i $$-0.533019\pi$$
−0.103547 + 0.994625i $$0.533019\pi$$
$$942$$ 5.12102e7 1.88031
$$943$$ −1.84172e7 −0.674441
$$944$$ 5.26657e7 1.92353
$$945$$ 0 0
$$946$$ 2.46611e7 0.895952
$$947$$ 2.64126e7 0.957052 0.478526 0.878073i $$-0.341171\pi$$
0.478526 + 0.878073i $$0.341171\pi$$
$$948$$ −3.16272e6 −0.114298
$$949$$ 1.71035e7 0.616480
$$950$$ 3.63286e7 1.30599
$$951$$ 1.16157e7 0.416479
$$952$$ 0 0
$$953$$ 4.21824e7 1.50452 0.752262 0.658864i $$-0.228963\pi$$
0.752262 + 0.658864i $$0.228963\pi$$
$$954$$ 944896. 0.0336135
$$955$$ 157657. 0.00559378
$$956$$ 5.38957e7 1.90726
$$957$$ −9.97732e6 −0.352155
$$958$$ −5.67511e7 −1.99784
$$959$$ 0 0
$$960$$ 717145. 0.0251148
$$961$$ −1.99228e7 −0.695892
$$962$$ 1.03263e8 3.59755
$$963$$ 3.90240e6 0.135602
$$964$$ −8.52077e7 −2.95315
$$965$$ 1.07503e7 0.371622
$$966$$ 0 0
$$967$$ 4.01501e7 1.38077 0.690384 0.723443i $$-0.257441\pi$$
0.690384 + 0.723443i $$0.257441\pi$$
$$968$$ 2.30638e7 0.791120
$$969$$ −7.01533e6 −0.240015
$$970$$ −1.54580e7 −0.527503
$$971$$ 1.64551e7 0.560084 0.280042 0.959988i $$-0.409652\pi$$
0.280042 + 0.959988i $$0.409652\pi$$
$$972$$ 4.28266e6 0.145394
$$973$$ 0 0
$$974$$ −7.03207e7 −2.37512
$$975$$ 2.34920e7 0.791424
$$976$$ −4.04200e7 −1.35823
$$977$$ 1.71176e7 0.573729 0.286865 0.957971i $$-0.407387\pi$$
0.286865 + 0.957971i $$0.407387\pi$$
$$978$$ −1.81514e6 −0.0606825
$$979$$ −2.90942e7 −0.970174
$$980$$ 0 0
$$981$$ −2.95128e6 −0.0979125
$$982$$ 7.61124e7 2.51870
$$983$$ −2.33803e7 −0.771730 −0.385865 0.922555i $$-0.626097\pi$$
−0.385865 + 0.922555i $$0.626097\pi$$
$$984$$ −1.66956e7 −0.549686
$$985$$ 1.94049e7 0.637266
$$986$$ −1.36770e7 −0.448021
$$987$$ 0 0
$$988$$ 1.02579e8 3.34324
$$989$$ 2.13146e7 0.692927
$$990$$ 9.15712e6 0.296941
$$991$$ −4.70955e6 −0.152333 −0.0761667 0.997095i $$-0.524268\pi$$
−0.0761667 + 0.997095i $$0.524268\pi$$
$$992$$ 1.86568e7 0.601948
$$993$$ −3.13705e7 −1.00960
$$994$$ 0 0
$$995$$ 1.91547e7 0.613362
$$996$$ 3.92596e7 1.25400
$$997$$ −4.55428e7 −1.45105 −0.725524 0.688197i $$-0.758403\pi$$
−0.725524 + 0.688197i $$0.758403\pi$$
$$998$$ 2.20150e7 0.699667
$$999$$ −7.22363e6 −0.229003
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.m.1.4 4
3.2 odd 2 441.6.a.w.1.1 4
7.2 even 3 21.6.e.c.4.1 8
7.3 odd 6 147.6.e.o.79.1 8
7.4 even 3 21.6.e.c.16.1 yes 8
7.5 odd 6 147.6.e.o.67.1 8
7.6 odd 2 147.6.a.l.1.4 4
21.2 odd 6 63.6.e.e.46.4 8
21.11 odd 6 63.6.e.e.37.4 8
21.20 even 2 441.6.a.v.1.1 4
28.11 odd 6 336.6.q.j.289.2 8
28.23 odd 6 336.6.q.j.193.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.1 8 7.2 even 3
21.6.e.c.16.1 yes 8 7.4 even 3
63.6.e.e.37.4 8 21.11 odd 6
63.6.e.e.46.4 8 21.2 odd 6
147.6.a.l.1.4 4 7.6 odd 2
147.6.a.m.1.4 4 1.1 even 1 trivial
147.6.e.o.67.1 8 7.5 odd 6
147.6.e.o.79.1 8 7.3 odd 6
336.6.q.j.193.2 8 28.23 odd 6
336.6.q.j.289.2 8 28.11 odd 6
441.6.a.v.1.1 4 21.20 even 2
441.6.a.w.1.1 4 3.2 odd 2