# Properties

 Label 49.6.a.e.1.1 Level $49$ Weight $6$ Character 49.1 Self dual yes Analytic conductor $7.859$ Analytic rank $0$ 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] = [49,6,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.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: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.54138$$ of defining polynomial Character $$\chi$$ $$=$$ 49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.08276 q^{2} -2.08276 q^{3} -6.16553 q^{4} -41.8276 q^{5} +10.5862 q^{6} +193.986 q^{8} -238.662 q^{9} +O(q^{10})$$ $$q-5.08276 q^{2} -2.08276 q^{3} -6.16553 q^{4} -41.8276 q^{5} +10.5862 q^{6} +193.986 q^{8} -238.662 q^{9} +212.600 q^{10} +72.0965 q^{11} +12.8413 q^{12} +632.317 q^{13} +87.1170 q^{15} -788.689 q^{16} +1975.92 q^{17} +1213.06 q^{18} +1864.93 q^{19} +257.889 q^{20} -366.449 q^{22} +413.711 q^{23} -404.027 q^{24} -1375.45 q^{25} -3213.92 q^{26} +1003.19 q^{27} +731.934 q^{29} -442.795 q^{30} -6123.18 q^{31} -2198.84 q^{32} -150.160 q^{33} -10043.2 q^{34} +1471.48 q^{36} +10350.4 q^{37} -9478.97 q^{38} -1316.97 q^{39} -8113.99 q^{40} +3529.84 q^{41} -14515.2 q^{43} -444.513 q^{44} +9982.67 q^{45} -2102.79 q^{46} +21423.3 q^{47} +1642.65 q^{48} +6991.08 q^{50} -4115.38 q^{51} -3898.57 q^{52} +12579.5 q^{53} -5098.97 q^{54} -3015.62 q^{55} -3884.20 q^{57} -3720.25 q^{58} +36133.9 q^{59} -537.122 q^{60} -4024.80 q^{61} +31122.6 q^{62} +36414.2 q^{64} -26448.3 q^{65} +763.227 q^{66} +15565.9 q^{67} -12182.6 q^{68} -861.661 q^{69} +12180.8 q^{71} -46297.2 q^{72} -19589.1 q^{73} -52608.5 q^{74} +2864.74 q^{75} -11498.2 q^{76} +6693.83 q^{78} +36089.8 q^{79} +32989.0 q^{80} +55905.5 q^{81} -17941.3 q^{82} +24572.6 q^{83} -82648.2 q^{85} +73777.5 q^{86} -1524.44 q^{87} +13985.7 q^{88} +70243.3 q^{89} -50739.5 q^{90} -2550.74 q^{92} +12753.1 q^{93} -108890. q^{94} -78005.4 q^{95} +4579.66 q^{96} -105758. q^{97} -17206.7 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 38 q^{5} + 82 q^{6} + 96 q^{8} - 380 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 8 * q^3 + 12 * q^4 + 38 * q^5 + 82 * q^6 + 96 * q^8 - 380 * q^9 $$2 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 38 q^{5} + 82 q^{6} + 96 q^{8} - 380 q^{9} + 778 q^{10} + 424 q^{11} + 196 q^{12} + 924 q^{13} + 892 q^{15} - 2064 q^{16} + 2346 q^{17} + 212 q^{18} + 360 q^{19} + 1708 q^{20} + 2126 q^{22} - 12 q^{23} - 1392 q^{24} + 1872 q^{25} - 1148 q^{26} - 2872 q^{27} - 7052 q^{29} + 5258 q^{30} - 3548 q^{31} - 8096 q^{32} + 3398 q^{33} - 7422 q^{34} - 1096 q^{36} + 11090 q^{37} - 20138 q^{38} + 1624 q^{39} - 15936 q^{40} - 3500 q^{41} - 12680 q^{43} + 5948 q^{44} - 1300 q^{45} - 5118 q^{46} + 22956 q^{47} - 11216 q^{48} + 29992 q^{50} - 384 q^{51} + 1400 q^{52} + 3042 q^{53} - 32546 q^{54} + 25076 q^{55} - 19058 q^{57} - 58852 q^{58} + 65808 q^{59} + 14084 q^{60} + 42486 q^{61} + 49362 q^{62} + 35456 q^{64} - 3164 q^{65} + 25894 q^{66} + 42312 q^{67} - 5460 q^{68} - 5154 q^{69} - 2208 q^{71} - 32448 q^{72} + 50506 q^{73} - 47370 q^{74} + 35608 q^{75} - 38836 q^{76} + 27524 q^{78} + 9004 q^{79} - 68816 q^{80} + 51178 q^{81} - 67732 q^{82} + 104328 q^{83} - 53106 q^{85} + 86776 q^{86} - 80008 q^{87} - 20496 q^{88} + 26666 q^{89} - 130652 q^{90} - 10284 q^{92} + 38718 q^{93} - 98034 q^{94} - 198140 q^{95} - 54880 q^{96} - 209132 q^{97} - 66944 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 8 * q^3 + 12 * q^4 + 38 * q^5 + 82 * q^6 + 96 * q^8 - 380 * q^9 + 778 * q^10 + 424 * q^11 + 196 * q^12 + 924 * q^13 + 892 * q^15 - 2064 * q^16 + 2346 * q^17 + 212 * q^18 + 360 * q^19 + 1708 * q^20 + 2126 * q^22 - 12 * q^23 - 1392 * q^24 + 1872 * q^25 - 1148 * q^26 - 2872 * q^27 - 7052 * q^29 + 5258 * q^30 - 3548 * q^31 - 8096 * q^32 + 3398 * q^33 - 7422 * q^34 - 1096 * q^36 + 11090 * q^37 - 20138 * q^38 + 1624 * q^39 - 15936 * q^40 - 3500 * q^41 - 12680 * q^43 + 5948 * q^44 - 1300 * q^45 - 5118 * q^46 + 22956 * q^47 - 11216 * q^48 + 29992 * q^50 - 384 * q^51 + 1400 * q^52 + 3042 * q^53 - 32546 * q^54 + 25076 * q^55 - 19058 * q^57 - 58852 * q^58 + 65808 * q^59 + 14084 * q^60 + 42486 * q^61 + 49362 * q^62 + 35456 * q^64 - 3164 * q^65 + 25894 * q^66 + 42312 * q^67 - 5460 * q^68 - 5154 * q^69 - 2208 * q^71 - 32448 * q^72 + 50506 * q^73 - 47370 * q^74 + 35608 * q^75 - 38836 * q^76 + 27524 * q^78 + 9004 * q^79 - 68816 * q^80 + 51178 * q^81 - 67732 * q^82 + 104328 * q^83 - 53106 * q^85 + 86776 * q^86 - 80008 * q^87 - 20496 * q^88 + 26666 * q^89 - 130652 * q^90 - 10284 * q^92 + 38718 * q^93 - 98034 * q^94 - 198140 * q^95 - 54880 * q^96 - 209132 * q^97 - 66944 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.08276 −0.898514 −0.449257 0.893403i $$-0.648311\pi$$
−0.449257 + 0.893403i $$0.648311\pi$$
$$3$$ −2.08276 −0.133609 −0.0668046 0.997766i $$-0.521280\pi$$
−0.0668046 + 0.997766i $$0.521280\pi$$
$$4$$ −6.16553 −0.192673
$$5$$ −41.8276 −0.748235 −0.374118 0.927381i $$-0.622054\pi$$
−0.374118 + 0.927381i $$0.622054\pi$$
$$6$$ 10.5862 0.120050
$$7$$ 0 0
$$8$$ 193.986 1.07163
$$9$$ −238.662 −0.982149
$$10$$ 212.600 0.672300
$$11$$ 72.0965 0.179652 0.0898260 0.995957i $$-0.471369\pi$$
0.0898260 + 0.995957i $$0.471369\pi$$
$$12$$ 12.8413 0.0257429
$$13$$ 632.317 1.03771 0.518856 0.854862i $$-0.326358\pi$$
0.518856 + 0.854862i $$0.326358\pi$$
$$14$$ 0 0
$$15$$ 87.1170 0.0999712
$$16$$ −788.689 −0.770205
$$17$$ 1975.92 1.65824 0.829121 0.559069i $$-0.188841\pi$$
0.829121 + 0.559069i $$0.188841\pi$$
$$18$$ 1213.06 0.882474
$$19$$ 1864.93 1.18516 0.592581 0.805511i $$-0.298109\pi$$
0.592581 + 0.805511i $$0.298109\pi$$
$$20$$ 257.889 0.144164
$$21$$ 0 0
$$22$$ −366.449 −0.161420
$$23$$ 413.711 0.163071 0.0815356 0.996670i $$-0.474018\pi$$
0.0815356 + 0.996670i $$0.474018\pi$$
$$24$$ −404.027 −0.143180
$$25$$ −1375.45 −0.440144
$$26$$ −3213.92 −0.932398
$$27$$ 1003.19 0.264833
$$28$$ 0 0
$$29$$ 731.934 0.161613 0.0808066 0.996730i $$-0.474250\pi$$
0.0808066 + 0.996730i $$0.474250\pi$$
$$30$$ −442.795 −0.0898255
$$31$$ −6123.18 −1.14439 −0.572193 0.820119i $$-0.693907\pi$$
−0.572193 + 0.820119i $$0.693907\pi$$
$$32$$ −2198.84 −0.379593
$$33$$ −150.160 −0.0240032
$$34$$ −10043.2 −1.48995
$$35$$ 0 0
$$36$$ 1471.48 0.189233
$$37$$ 10350.4 1.24295 0.621473 0.783436i $$-0.286535\pi$$
0.621473 + 0.783436i $$0.286535\pi$$
$$38$$ −9478.97 −1.06488
$$39$$ −1316.97 −0.138648
$$40$$ −8113.99 −0.801834
$$41$$ 3529.84 0.327941 0.163970 0.986465i $$-0.447570\pi$$
0.163970 + 0.986465i $$0.447570\pi$$
$$42$$ 0 0
$$43$$ −14515.2 −1.19716 −0.598581 0.801062i $$-0.704269\pi$$
−0.598581 + 0.801062i $$0.704269\pi$$
$$44$$ −444.513 −0.0346140
$$45$$ 9982.67 0.734878
$$46$$ −2102.79 −0.146522
$$47$$ 21423.3 1.41463 0.707314 0.706900i $$-0.249907\pi$$
0.707314 + 0.706900i $$0.249907\pi$$
$$48$$ 1642.65 0.102906
$$49$$ 0 0
$$50$$ 6991.08 0.395475
$$51$$ −4115.38 −0.221557
$$52$$ −3898.57 −0.199939
$$53$$ 12579.5 0.615138 0.307569 0.951526i $$-0.400485\pi$$
0.307569 + 0.951526i $$0.400485\pi$$
$$54$$ −5098.97 −0.237957
$$55$$ −3015.62 −0.134422
$$56$$ 0 0
$$57$$ −3884.20 −0.158349
$$58$$ −3720.25 −0.145212
$$59$$ 36133.9 1.35140 0.675702 0.737175i $$-0.263841\pi$$
0.675702 + 0.737175i $$0.263841\pi$$
$$60$$ −537.122 −0.0192617
$$61$$ −4024.80 −0.138490 −0.0692451 0.997600i $$-0.522059\pi$$
−0.0692451 + 0.997600i $$0.522059\pi$$
$$62$$ 31122.6 1.02825
$$63$$ 0 0
$$64$$ 36414.2 1.11127
$$65$$ −26448.3 −0.776453
$$66$$ 763.227 0.0215672
$$67$$ 15565.9 0.423632 0.211816 0.977310i $$-0.432062\pi$$
0.211816 + 0.977310i $$0.432062\pi$$
$$68$$ −12182.6 −0.319498
$$69$$ −861.661 −0.0217878
$$70$$ 0 0
$$71$$ 12180.8 0.286766 0.143383 0.989667i $$-0.454202\pi$$
0.143383 + 0.989667i $$0.454202\pi$$
$$72$$ −46297.2 −1.05250
$$73$$ −19589.1 −0.430237 −0.215119 0.976588i $$-0.569014\pi$$
−0.215119 + 0.976588i $$0.569014\pi$$
$$74$$ −52608.5 −1.11680
$$75$$ 2864.74 0.0588073
$$76$$ −11498.2 −0.228348
$$77$$ 0 0
$$78$$ 6693.83 0.124577
$$79$$ 36089.8 0.650604 0.325302 0.945610i $$-0.394534\pi$$
0.325302 + 0.945610i $$0.394534\pi$$
$$80$$ 32989.0 0.576294
$$81$$ 55905.5 0.946764
$$82$$ −17941.3 −0.294659
$$83$$ 24572.6 0.391522 0.195761 0.980652i $$-0.437282\pi$$
0.195761 + 0.980652i $$0.437282\pi$$
$$84$$ 0 0
$$85$$ −82648.2 −1.24076
$$86$$ 73777.5 1.07567
$$87$$ −1524.44 −0.0215930
$$88$$ 13985.7 0.192521
$$89$$ 70243.3 0.940005 0.470002 0.882665i $$-0.344253\pi$$
0.470002 + 0.882665i $$0.344253\pi$$
$$90$$ −50739.5 −0.660298
$$91$$ 0 0
$$92$$ −2550.74 −0.0314193
$$93$$ 12753.1 0.152901
$$94$$ −108890. −1.27106
$$95$$ −78005.4 −0.886779
$$96$$ 4579.66 0.0507172
$$97$$ −105758. −1.14126 −0.570630 0.821207i $$-0.693301\pi$$
−0.570630 + 0.821207i $$0.693301\pi$$
$$98$$ 0 0
$$99$$ −17206.7 −0.176445
$$100$$ 8480.37 0.0848037
$$101$$ −36461.8 −0.355660 −0.177830 0.984061i $$-0.556908\pi$$
−0.177830 + 0.984061i $$0.556908\pi$$
$$102$$ 20917.5 0.199072
$$103$$ −64520.1 −0.599242 −0.299621 0.954058i $$-0.596860\pi$$
−0.299621 + 0.954058i $$0.596860\pi$$
$$104$$ 122661. 1.11205
$$105$$ 0 0
$$106$$ −63938.4 −0.552710
$$107$$ 66045.6 0.557679 0.278840 0.960338i $$-0.410050\pi$$
0.278840 + 0.960338i $$0.410050\pi$$
$$108$$ −6185.18 −0.0510262
$$109$$ −37938.0 −0.305850 −0.152925 0.988238i $$-0.548869\pi$$
−0.152925 + 0.988238i $$0.548869\pi$$
$$110$$ 15327.7 0.120780
$$111$$ −21557.4 −0.166069
$$112$$ 0 0
$$113$$ 123802. 0.912080 0.456040 0.889959i $$-0.349267\pi$$
0.456040 + 0.889959i $$0.349267\pi$$
$$114$$ 19742.4 0.142278
$$115$$ −17304.5 −0.122016
$$116$$ −4512.76 −0.0311384
$$117$$ −150910. −1.01919
$$118$$ −183660. −1.21426
$$119$$ 0 0
$$120$$ 16899.5 0.107132
$$121$$ −155853. −0.967725
$$122$$ 20457.1 0.124435
$$123$$ −7351.81 −0.0438159
$$124$$ 37752.6 0.220492
$$125$$ 188243. 1.07757
$$126$$ 0 0
$$127$$ 128724. 0.708189 0.354095 0.935210i $$-0.384789\pi$$
0.354095 + 0.935210i $$0.384789\pi$$
$$128$$ −114722. −0.618902
$$129$$ 30231.8 0.159952
$$130$$ 134431. 0.697653
$$131$$ 147902. 0.753003 0.376501 0.926416i $$-0.377127\pi$$
0.376501 + 0.926416i $$0.377127\pi$$
$$132$$ 925.814 0.00462476
$$133$$ 0 0
$$134$$ −79118.0 −0.380639
$$135$$ −41961.0 −0.198158
$$136$$ 383302. 1.77703
$$137$$ −91157.4 −0.414945 −0.207472 0.978241i $$-0.566524\pi$$
−0.207472 + 0.978241i $$0.566524\pi$$
$$138$$ 4379.62 0.0195767
$$139$$ 334657. 1.46914 0.734570 0.678533i $$-0.237384\pi$$
0.734570 + 0.678533i $$0.237384\pi$$
$$140$$ 0 0
$$141$$ −44619.7 −0.189007
$$142$$ −61911.9 −0.257664
$$143$$ 45587.8 0.186427
$$144$$ 188230. 0.756455
$$145$$ −30615.1 −0.120925
$$146$$ 99566.9 0.386574
$$147$$ 0 0
$$148$$ −63815.5 −0.239482
$$149$$ 138271. 0.510231 0.255115 0.966911i $$-0.417887\pi$$
0.255115 + 0.966911i $$0.417887\pi$$
$$150$$ −14560.8 −0.0528392
$$151$$ 111169. 0.396773 0.198386 0.980124i $$-0.436430\pi$$
0.198386 + 0.980124i $$0.436430\pi$$
$$152$$ 361770. 1.27006
$$153$$ −471578. −1.62864
$$154$$ 0 0
$$155$$ 256118. 0.856270
$$156$$ 8119.79 0.0267137
$$157$$ 38148.5 0.123517 0.0617587 0.998091i $$-0.480329\pi$$
0.0617587 + 0.998091i $$0.480329\pi$$
$$158$$ −183436. −0.584577
$$159$$ −26200.0 −0.0821881
$$160$$ 91972.3 0.284025
$$161$$ 0 0
$$162$$ −284154. −0.850681
$$163$$ −212905. −0.627648 −0.313824 0.949481i $$-0.601610\pi$$
−0.313824 + 0.949481i $$0.601610\pi$$
$$164$$ −21763.3 −0.0631852
$$165$$ 6280.83 0.0179600
$$166$$ −124897. −0.351788
$$167$$ 120396. 0.334057 0.167028 0.985952i $$-0.446583\pi$$
0.167028 + 0.985952i $$0.446583\pi$$
$$168$$ 0 0
$$169$$ 28532.2 0.0768456
$$170$$ 420081. 1.11484
$$171$$ −445087. −1.16400
$$172$$ 89494.0 0.230660
$$173$$ −712914. −1.81101 −0.905507 0.424331i $$-0.860509\pi$$
−0.905507 + 0.424331i $$0.860509\pi$$
$$174$$ 7748.39 0.0194016
$$175$$ 0 0
$$176$$ −56861.7 −0.138369
$$177$$ −75258.4 −0.180560
$$178$$ −357030. −0.844607
$$179$$ −749738. −1.74895 −0.874474 0.485072i $$-0.838793\pi$$
−0.874474 + 0.485072i $$0.838793\pi$$
$$180$$ −61548.4 −0.141591
$$181$$ −623718. −1.41511 −0.707557 0.706656i $$-0.750203\pi$$
−0.707557 + 0.706656i $$0.750203\pi$$
$$182$$ 0 0
$$183$$ 8382.69 0.0185036
$$184$$ 80254.2 0.174752
$$185$$ −432932. −0.930016
$$186$$ −64821.1 −0.137383
$$187$$ 142457. 0.297907
$$188$$ −132086. −0.272560
$$189$$ 0 0
$$190$$ 396483. 0.796784
$$191$$ 417726. 0.828530 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$192$$ −75842.2 −0.148477
$$193$$ 770700. 1.48933 0.744667 0.667436i $$-0.232608\pi$$
0.744667 + 0.667436i $$0.232608\pi$$
$$194$$ 537544. 1.02544
$$195$$ 55085.6 0.103741
$$196$$ 0 0
$$197$$ −479193. −0.879721 −0.439861 0.898066i $$-0.644972\pi$$
−0.439861 + 0.898066i $$0.644972\pi$$
$$198$$ 87457.5 0.158538
$$199$$ 428686. 0.767373 0.383687 0.923463i $$-0.374654\pi$$
0.383687 + 0.923463i $$0.374654\pi$$
$$200$$ −266818. −0.471673
$$201$$ −32420.2 −0.0566011
$$202$$ 185327. 0.319565
$$203$$ 0 0
$$204$$ 25373.5 0.0426879
$$205$$ −147645. −0.245377
$$206$$ 327940. 0.538427
$$207$$ −98737.0 −0.160160
$$208$$ −498702. −0.799250
$$209$$ 134455. 0.212917
$$210$$ 0 0
$$211$$ −588544. −0.910066 −0.455033 0.890475i $$-0.650373\pi$$
−0.455033 + 0.890475i $$0.650373\pi$$
$$212$$ −77559.0 −0.118520
$$213$$ −25369.6 −0.0383147
$$214$$ −335694. −0.501083
$$215$$ 607138. 0.895759
$$216$$ 194605. 0.283804
$$217$$ 0 0
$$218$$ 192830. 0.274811
$$219$$ 40799.5 0.0574837
$$220$$ 18592.9 0.0258994
$$221$$ 1.24941e6 1.72078
$$222$$ 109571. 0.149215
$$223$$ 363249. 0.489151 0.244575 0.969630i $$-0.421351\pi$$
0.244575 + 0.969630i $$0.421351\pi$$
$$224$$ 0 0
$$225$$ 328268. 0.432287
$$226$$ −629258. −0.819517
$$227$$ 843041. 1.08589 0.542943 0.839770i $$-0.317310\pi$$
0.542943 + 0.839770i $$0.317310\pi$$
$$228$$ 23948.1 0.0305094
$$229$$ 568666. 0.716587 0.358293 0.933609i $$-0.383359\pi$$
0.358293 + 0.933609i $$0.383359\pi$$
$$230$$ 87954.8 0.109633
$$231$$ 0 0
$$232$$ 141985. 0.173190
$$233$$ −1.05651e6 −1.27492 −0.637461 0.770482i $$-0.720015\pi$$
−0.637461 + 0.770482i $$0.720015\pi$$
$$234$$ 767041. 0.915754
$$235$$ −896086. −1.05847
$$236$$ −222785. −0.260379
$$237$$ −75166.5 −0.0869267
$$238$$ 0 0
$$239$$ 853715. 0.966759 0.483379 0.875411i $$-0.339409\pi$$
0.483379 + 0.875411i $$0.339409\pi$$
$$240$$ −68708.3 −0.0769983
$$241$$ 388888. 0.431302 0.215651 0.976470i $$-0.430813\pi$$
0.215651 + 0.976470i $$0.430813\pi$$
$$242$$ 792164. 0.869515
$$243$$ −360212. −0.391330
$$244$$ 24815.0 0.0266833
$$245$$ 0 0
$$246$$ 37367.5 0.0393692
$$247$$ 1.17922e6 1.22986
$$248$$ −1.18781e6 −1.22636
$$249$$ −51178.9 −0.0523109
$$250$$ −956795. −0.968209
$$251$$ 839328. 0.840906 0.420453 0.907314i $$-0.361871\pi$$
0.420453 + 0.907314i $$0.361871\pi$$
$$252$$ 0 0
$$253$$ 29827.1 0.0292961
$$254$$ −654272. −0.636318
$$255$$ 172137. 0.165776
$$256$$ −582151. −0.555182
$$257$$ −291986. −0.275759 −0.137879 0.990449i $$-0.544029\pi$$
−0.137879 + 0.990449i $$0.544029\pi$$
$$258$$ −153661. −0.143719
$$259$$ 0 0
$$260$$ 163068. 0.149601
$$261$$ −174685. −0.158728
$$262$$ −751752. −0.676584
$$263$$ −288495. −0.257187 −0.128594 0.991697i $$-0.541046\pi$$
−0.128594 + 0.991697i $$0.541046\pi$$
$$264$$ −29128.9 −0.0257226
$$265$$ −526169. −0.460268
$$266$$ 0 0
$$267$$ −146300. −0.125593
$$268$$ −95972.2 −0.0816222
$$269$$ 259370. 0.218544 0.109272 0.994012i $$-0.465148\pi$$
0.109272 + 0.994012i $$0.465148\pi$$
$$270$$ 213278. 0.178047
$$271$$ 2.19551e6 1.81599 0.907994 0.418984i $$-0.137614\pi$$
0.907994 + 0.418984i $$0.137614\pi$$
$$272$$ −1.55839e6 −1.27719
$$273$$ 0 0
$$274$$ 463331. 0.372834
$$275$$ −99165.1 −0.0790728
$$276$$ 5312.59 0.00419792
$$277$$ 126991. 0.0994426 0.0497213 0.998763i $$-0.484167\pi$$
0.0497213 + 0.998763i $$0.484167\pi$$
$$278$$ −1.70098e6 −1.32004
$$279$$ 1.46137e6 1.12396
$$280$$ 0 0
$$281$$ −2.22759e6 −1.68294 −0.841472 0.540301i $$-0.818310\pi$$
−0.841472 + 0.540301i $$0.818310\pi$$
$$282$$ 226791. 0.169826
$$283$$ −1.18895e6 −0.882463 −0.441231 0.897393i $$-0.645458\pi$$
−0.441231 + 0.897393i $$0.645458\pi$$
$$284$$ −75100.7 −0.0552520
$$285$$ 162467. 0.118482
$$286$$ −231712. −0.167507
$$287$$ 0 0
$$288$$ 524780. 0.372817
$$289$$ 2.48442e6 1.74977
$$290$$ 155609. 0.108653
$$291$$ 220269. 0.152483
$$292$$ 120777. 0.0828949
$$293$$ −1.83223e6 −1.24684 −0.623421 0.781886i $$-0.714258\pi$$
−0.623421 + 0.781886i $$0.714258\pi$$
$$294$$ 0 0
$$295$$ −1.51140e6 −1.01117
$$296$$ 2.00783e6 1.33198
$$297$$ 72326.3 0.0475779
$$298$$ −702800. −0.458449
$$299$$ 261596. 0.169221
$$300$$ −17662.6 −0.0113306
$$301$$ 0 0
$$302$$ −565047. −0.356506
$$303$$ 75941.3 0.0475194
$$304$$ −1.47085e6 −0.912817
$$305$$ 168348. 0.103623
$$306$$ 2.39692e6 1.46336
$$307$$ 717638. 0.434569 0.217285 0.976108i $$-0.430280\pi$$
0.217285 + 0.976108i $$0.430280\pi$$
$$308$$ 0 0
$$309$$ 134380. 0.0800642
$$310$$ −1.30179e6 −0.769370
$$311$$ 856892. 0.502372 0.251186 0.967939i $$-0.419179\pi$$
0.251186 + 0.967939i $$0.419179\pi$$
$$312$$ −255474. −0.148580
$$313$$ −1.61699e6 −0.932924 −0.466462 0.884541i $$-0.654472\pi$$
−0.466462 + 0.884541i $$0.654472\pi$$
$$314$$ −193900. −0.110982
$$315$$ 0 0
$$316$$ −222512. −0.125354
$$317$$ −2.26559e6 −1.26629 −0.633145 0.774033i $$-0.718236\pi$$
−0.633145 + 0.774033i $$0.718236\pi$$
$$318$$ 133169. 0.0738472
$$319$$ 52769.8 0.0290341
$$320$$ −1.52312e6 −0.831495
$$321$$ −137557. −0.0745111
$$322$$ 0 0
$$323$$ 3.68495e6 1.96528
$$324$$ −344687. −0.182416
$$325$$ −869721. −0.456743
$$326$$ 1.08214e6 0.563950
$$327$$ 79015.9 0.0408644
$$328$$ 684740. 0.351432
$$329$$ 0 0
$$330$$ −31924.0 −0.0161373
$$331$$ 709650. 0.356020 0.178010 0.984029i $$-0.443034\pi$$
0.178010 + 0.984029i $$0.443034\pi$$
$$332$$ −151503. −0.0754355
$$333$$ −2.47024e6 −1.22076
$$334$$ −611943. −0.300155
$$335$$ −651086. −0.316976
$$336$$ 0 0
$$337$$ 603572. 0.289504 0.144752 0.989468i $$-0.453762\pi$$
0.144752 + 0.989468i $$0.453762\pi$$
$$338$$ −145023. −0.0690468
$$339$$ −257851. −0.121862
$$340$$ 509570. 0.239060
$$341$$ −441459. −0.205591
$$342$$ 2.26227e6 1.04587
$$343$$ 0 0
$$344$$ −2.81576e6 −1.28292
$$345$$ 36041.2 0.0163024
$$346$$ 3.62357e6 1.62722
$$347$$ 1.75731e6 0.783474 0.391737 0.920077i $$-0.371874\pi$$
0.391737 + 0.920077i $$0.371874\pi$$
$$348$$ 9399.00 0.00416038
$$349$$ −391875. −0.172220 −0.0861102 0.996286i $$-0.527444\pi$$
−0.0861102 + 0.996286i $$0.527444\pi$$
$$350$$ 0 0
$$351$$ 634333. 0.274821
$$352$$ −158529. −0.0681948
$$353$$ −492407. −0.210323 −0.105162 0.994455i $$-0.533536\pi$$
−0.105162 + 0.994455i $$0.533536\pi$$
$$354$$ 382521. 0.162236
$$355$$ −509492. −0.214569
$$356$$ −433087. −0.181113
$$357$$ 0 0
$$358$$ 3.81074e6 1.57145
$$359$$ −3.77032e6 −1.54398 −0.771991 0.635634i $$-0.780739\pi$$
−0.771991 + 0.635634i $$0.780739\pi$$
$$360$$ 1.93650e6 0.787520
$$361$$ 1.00185e6 0.404607
$$362$$ 3.17021e6 1.27150
$$363$$ 324605. 0.129297
$$364$$ 0 0
$$365$$ 819367. 0.321919
$$366$$ −42607.2 −0.0166257
$$367$$ 2.19768e6 0.851726 0.425863 0.904788i $$-0.359970\pi$$
0.425863 + 0.904788i $$0.359970\pi$$
$$368$$ −326289. −0.125598
$$369$$ −842439. −0.322086
$$370$$ 2.20049e6 0.835632
$$371$$ 0 0
$$372$$ −78629.7 −0.0294598
$$373$$ −1.65636e6 −0.616427 −0.308213 0.951317i $$-0.599731\pi$$
−0.308213 + 0.951317i $$0.599731\pi$$
$$374$$ −724076. −0.267673
$$375$$ −392066. −0.143973
$$376$$ 4.15583e6 1.51596
$$377$$ 462814. 0.167708
$$378$$ 0 0
$$379$$ −2.82050e6 −1.00862 −0.504310 0.863523i $$-0.668253\pi$$
−0.504310 + 0.863523i $$0.668253\pi$$
$$380$$ 480944. 0.170858
$$381$$ −268101. −0.0946206
$$382$$ −2.12320e6 −0.744446
$$383$$ −3.24845e6 −1.13156 −0.565781 0.824555i $$-0.691425\pi$$
−0.565781 + 0.824555i $$0.691425\pi$$
$$384$$ 238939. 0.0826911
$$385$$ 0 0
$$386$$ −3.91729e6 −1.33819
$$387$$ 3.46424e6 1.17579
$$388$$ 652055. 0.219890
$$389$$ 4.65348e6 1.55921 0.779604 0.626273i $$-0.215420\pi$$
0.779604 + 0.626273i $$0.215420\pi$$
$$390$$ −279987. −0.0932130
$$391$$ 817461. 0.270411
$$392$$ 0 0
$$393$$ −308045. −0.100608
$$394$$ 2.43563e6 0.790442
$$395$$ −1.50955e6 −0.486805
$$396$$ 106088. 0.0339961
$$397$$ 1.16361e6 0.370536 0.185268 0.982688i $$-0.440685\pi$$
0.185268 + 0.982688i $$0.440685\pi$$
$$398$$ −2.17891e6 −0.689495
$$399$$ 0 0
$$400$$ 1.08480e6 0.339001
$$401$$ 322380. 0.100117 0.0500584 0.998746i $$-0.484059\pi$$
0.0500584 + 0.998746i $$0.484059\pi$$
$$402$$ 164784. 0.0508569
$$403$$ −3.87179e6 −1.18754
$$404$$ 224806. 0.0685259
$$405$$ −2.33839e6 −0.708403
$$406$$ 0 0
$$407$$ 746226. 0.223298
$$408$$ −798328. −0.237427
$$409$$ −1.38690e6 −0.409956 −0.204978 0.978767i $$-0.565712\pi$$
−0.204978 + 0.978767i $$0.565712\pi$$
$$410$$ 750443. 0.220474
$$411$$ 189859. 0.0554405
$$412$$ 397800. 0.115457
$$413$$ 0 0
$$414$$ 501857. 0.143906
$$415$$ −1.02781e6 −0.292950
$$416$$ −1.39036e6 −0.393909
$$417$$ −697011. −0.196291
$$418$$ −683400. −0.191309
$$419$$ −4.90871e6 −1.36594 −0.682971 0.730446i $$-0.739312\pi$$
−0.682971 + 0.730446i $$0.739312\pi$$
$$420$$ 0 0
$$421$$ 2.43924e6 0.670733 0.335367 0.942088i $$-0.391140\pi$$
0.335367 + 0.942088i $$0.391140\pi$$
$$422$$ 2.99143e6 0.817707
$$423$$ −5.11293e6 −1.38937
$$424$$ 2.44024e6 0.659202
$$425$$ −2.71779e6 −0.729865
$$426$$ 128948. 0.0344263
$$427$$ 0 0
$$428$$ −407206. −0.107450
$$429$$ −94948.7 −0.0249084
$$430$$ −3.08594e6 −0.804852
$$431$$ −5.22752e6 −1.35551 −0.677755 0.735288i $$-0.737047\pi$$
−0.677755 + 0.735288i $$0.737047\pi$$
$$432$$ −791204. −0.203976
$$433$$ 2.63022e6 0.674174 0.337087 0.941473i $$-0.390558\pi$$
0.337087 + 0.941473i $$0.390558\pi$$
$$434$$ 0 0
$$435$$ 63763.9 0.0161567
$$436$$ 233908. 0.0589289
$$437$$ 771539. 0.193266
$$438$$ −207374. −0.0516499
$$439$$ −2.55412e6 −0.632527 −0.316264 0.948671i $$-0.602428\pi$$
−0.316264 + 0.948671i $$0.602428\pi$$
$$440$$ −584990. −0.144051
$$441$$ 0 0
$$442$$ −6.35046e6 −1.54614
$$443$$ 3.83900e6 0.929414 0.464707 0.885465i $$-0.346160\pi$$
0.464707 + 0.885465i $$0.346160\pi$$
$$444$$ 132913. 0.0319970
$$445$$ −2.93811e6 −0.703345
$$446$$ −1.84631e6 −0.439509
$$447$$ −287986. −0.0681715
$$448$$ 0 0
$$449$$ 1.49369e6 0.349658 0.174829 0.984599i $$-0.444063\pi$$
0.174829 + 0.984599i $$0.444063\pi$$
$$450$$ −1.66851e6 −0.388416
$$451$$ 254489. 0.0589152
$$452$$ −763307. −0.175733
$$453$$ −231539. −0.0530126
$$454$$ −4.28498e6 −0.975684
$$455$$ 0 0
$$456$$ −753481. −0.169692
$$457$$ −2.16221e6 −0.484293 −0.242146 0.970240i $$-0.577851\pi$$
−0.242146 + 0.970240i $$0.577851\pi$$
$$458$$ −2.89040e6 −0.643863
$$459$$ 1.98222e6 0.439158
$$460$$ 106692. 0.0235091
$$461$$ 6.11949e6 1.34111 0.670553 0.741862i $$-0.266057\pi$$
0.670553 + 0.741862i $$0.266057\pi$$
$$462$$ 0 0
$$463$$ 3.93615e6 0.853335 0.426667 0.904409i $$-0.359687\pi$$
0.426667 + 0.904409i $$0.359687\pi$$
$$464$$ −577268. −0.124475
$$465$$ −533433. −0.114406
$$466$$ 5.36999e6 1.14554
$$467$$ 5.47044e6 1.16073 0.580363 0.814358i $$-0.302911\pi$$
0.580363 + 0.814358i $$0.302911\pi$$
$$468$$ 930441. 0.196369
$$469$$ 0 0
$$470$$ 4.55459e6 0.951054
$$471$$ −79454.3 −0.0165031
$$472$$ 7.00949e6 1.44821
$$473$$ −1.04650e6 −0.215073
$$474$$ 382053. 0.0781049
$$475$$ −2.56511e6 −0.521641
$$476$$ 0 0
$$477$$ −3.00224e6 −0.604157
$$478$$ −4.33923e6 −0.868646
$$479$$ 6.66289e6 1.32686 0.663428 0.748240i $$-0.269101\pi$$
0.663428 + 0.748240i $$0.269101\pi$$
$$480$$ −191556. −0.0379484
$$481$$ 6.54473e6 1.28982
$$482$$ −1.97663e6 −0.387531
$$483$$ 0 0
$$484$$ 960916. 0.186454
$$485$$ 4.42362e6 0.853931
$$486$$ 1.83087e6 0.351615
$$487$$ 9.53693e6 1.82216 0.911079 0.412232i $$-0.135251\pi$$
0.911079 + 0.412232i $$0.135251\pi$$
$$488$$ −780755. −0.148411
$$489$$ 443430. 0.0838595
$$490$$ 0 0
$$491$$ −8.19294e6 −1.53369 −0.766843 0.641835i $$-0.778173\pi$$
−0.766843 + 0.641835i $$0.778173\pi$$
$$492$$ 45327.8 0.00844213
$$493$$ 1.44625e6 0.267994
$$494$$ −5.99372e6 −1.10504
$$495$$ 719715. 0.132022
$$496$$ 4.82928e6 0.881411
$$497$$ 0 0
$$498$$ 260130. 0.0470021
$$499$$ −4.31437e6 −0.775650 −0.387825 0.921733i $$-0.626774\pi$$
−0.387825 + 0.921733i $$0.626774\pi$$
$$500$$ −1.16062e6 −0.207618
$$501$$ −250756. −0.0446331
$$502$$ −4.26610e6 −0.755566
$$503$$ 1.04015e7 1.83306 0.916529 0.399968i $$-0.130979\pi$$
0.916529 + 0.399968i $$0.130979\pi$$
$$504$$ 0 0
$$505$$ 1.52511e6 0.266117
$$506$$ −151604. −0.0263229
$$507$$ −59425.9 −0.0102673
$$508$$ −793649. −0.136449
$$509$$ 3.09396e6 0.529322 0.264661 0.964342i $$-0.414740\pi$$
0.264661 + 0.964342i $$0.414740\pi$$
$$510$$ −874930. −0.148952
$$511$$ 0 0
$$512$$ 6.63004e6 1.11774
$$513$$ 1.87087e6 0.313870
$$514$$ 1.48410e6 0.247773
$$515$$ 2.69872e6 0.448374
$$516$$ −186395. −0.0308184
$$517$$ 1.54455e6 0.254141
$$518$$ 0 0
$$519$$ 1.48483e6 0.241968
$$520$$ −5.13061e6 −0.832072
$$521$$ 7.60175e6 1.22693 0.613464 0.789723i $$-0.289776\pi$$
0.613464 + 0.789723i $$0.289776\pi$$
$$522$$ 887882. 0.142619
$$523$$ −4.75669e6 −0.760415 −0.380208 0.924901i $$-0.624147\pi$$
−0.380208 + 0.924901i $$0.624147\pi$$
$$524$$ −911895. −0.145083
$$525$$ 0 0
$$526$$ 1.46635e6 0.231086
$$527$$ −1.20989e7 −1.89767
$$528$$ 118429. 0.0184874
$$529$$ −6.26519e6 −0.973408
$$530$$ 2.67439e6 0.413557
$$531$$ −8.62380e6 −1.32728
$$532$$ 0 0
$$533$$ 2.23198e6 0.340308
$$534$$ 743609. 0.112847
$$535$$ −2.76253e6 −0.417275
$$536$$ 3.01958e6 0.453978
$$537$$ 1.56153e6 0.233676
$$538$$ −1.31832e6 −0.196365
$$539$$ 0 0
$$540$$ 258711. 0.0381796
$$541$$ 1.10052e7 1.61661 0.808305 0.588764i $$-0.200385\pi$$
0.808305 + 0.588764i $$0.200385\pi$$
$$542$$ −1.11593e7 −1.63169
$$543$$ 1.29906e6 0.189072
$$544$$ −4.34474e6 −0.629458
$$545$$ 1.58686e6 0.228848
$$546$$ 0 0
$$547$$ −4.46311e6 −0.637778 −0.318889 0.947792i $$-0.603310\pi$$
−0.318889 + 0.947792i $$0.603310\pi$$
$$548$$ 562033. 0.0799485
$$549$$ 960566. 0.136018
$$550$$ 504032. 0.0710480
$$551$$ 1.36500e6 0.191538
$$552$$ −167150. −0.0233485
$$553$$ 0 0
$$554$$ −645463. −0.0893505
$$555$$ 901694. 0.124259
$$556$$ −2.06334e6 −0.283063
$$557$$ −6.45222e6 −0.881194 −0.440597 0.897705i $$-0.645233\pi$$
−0.440597 + 0.897705i $$0.645233\pi$$
$$558$$ −7.42780e6 −1.00989
$$559$$ −9.17823e6 −1.24231
$$560$$ 0 0
$$561$$ −296704. −0.0398031
$$562$$ 1.13223e7 1.51215
$$563$$ −1.74748e6 −0.232349 −0.116175 0.993229i $$-0.537063\pi$$
−0.116175 + 0.993229i $$0.537063\pi$$
$$564$$ 275104. 0.0364165
$$565$$ −5.17836e6 −0.682450
$$566$$ 6.04313e6 0.792905
$$567$$ 0 0
$$568$$ 2.36290e6 0.307308
$$569$$ 512789. 0.0663985 0.0331992 0.999449i $$-0.489430\pi$$
0.0331992 + 0.999449i $$0.489430\pi$$
$$570$$ −825780. −0.106458
$$571$$ 5.22364e6 0.670475 0.335238 0.942134i $$-0.391183\pi$$
0.335238 + 0.942134i $$0.391183\pi$$
$$572$$ −281073. −0.0359194
$$573$$ −870024. −0.110699
$$574$$ 0 0
$$575$$ −569038. −0.0717748
$$576$$ −8.69070e6 −1.09144
$$577$$ −6.63973e6 −0.830254 −0.415127 0.909763i $$-0.636263\pi$$
−0.415127 + 0.909763i $$0.636263\pi$$
$$578$$ −1.26277e7 −1.57219
$$579$$ −1.60519e6 −0.198989
$$580$$ 188758. 0.0232989
$$581$$ 0 0
$$582$$ −1.11958e6 −0.137008
$$583$$ 906935. 0.110511
$$584$$ −3.80002e6 −0.461056
$$585$$ 6.31221e6 0.762592
$$586$$ 9.31280e6 1.12030
$$587$$ −774096. −0.0927256 −0.0463628 0.998925i $$-0.514763\pi$$
−0.0463628 + 0.998925i $$0.514763\pi$$
$$588$$ 0 0
$$589$$ −1.14193e7 −1.35628
$$590$$ 7.68207e6 0.908549
$$591$$ 998046. 0.117539
$$592$$ −8.16324e6 −0.957322
$$593$$ 1.43756e7 1.67876 0.839379 0.543546i $$-0.182919\pi$$
0.839379 + 0.543546i $$0.182919\pi$$
$$594$$ −367617. −0.0427494
$$595$$ 0 0
$$596$$ −852515. −0.0983075
$$597$$ −892851. −0.102528
$$598$$ −1.32963e6 −0.152047
$$599$$ 1.20835e7 1.37602 0.688010 0.725701i $$-0.258484\pi$$
0.688010 + 0.725701i $$0.258484\pi$$
$$600$$ 555719. 0.0630199
$$601$$ −5.75607e6 −0.650040 −0.325020 0.945707i $$-0.605371\pi$$
−0.325020 + 0.945707i $$0.605371\pi$$
$$602$$ 0 0
$$603$$ −3.71500e6 −0.416069
$$604$$ −685416. −0.0764473
$$605$$ 6.51897e6 0.724086
$$606$$ −385991. −0.0426969
$$607$$ −4.20121e6 −0.462810 −0.231405 0.972858i $$-0.574332\pi$$
−0.231405 + 0.972858i $$0.574332\pi$$
$$608$$ −4.10067e6 −0.449879
$$609$$ 0 0
$$610$$ −855671. −0.0931070
$$611$$ 1.35463e7 1.46798
$$612$$ 2.90753e6 0.313795
$$613$$ −2.64543e6 −0.284344 −0.142172 0.989842i $$-0.545409\pi$$
−0.142172 + 0.989842i $$0.545409\pi$$
$$614$$ −3.64758e6 −0.390467
$$615$$ 307509. 0.0327846
$$616$$ 0 0
$$617$$ 6.43533e6 0.680546 0.340273 0.940327i $$-0.389480\pi$$
0.340273 + 0.940327i $$0.389480\pi$$
$$618$$ −683022. −0.0719388
$$619$$ −1.41177e7 −1.48094 −0.740469 0.672090i $$-0.765397\pi$$
−0.740469 + 0.672090i $$0.765397\pi$$
$$620$$ −1.57910e6 −0.164980
$$621$$ 415029. 0.0431867
$$622$$ −4.35538e6 −0.451388
$$623$$ 0 0
$$624$$ 1.03868e6 0.106787
$$625$$ −3.57548e6 −0.366129
$$626$$ 8.21877e6 0.838246
$$627$$ −280037. −0.0284476
$$628$$ −235206. −0.0237984
$$629$$ 2.04516e7 2.06111
$$630$$ 0 0
$$631$$ −4.70856e6 −0.470777 −0.235388 0.971901i $$-0.575636\pi$$
−0.235388 + 0.971901i $$0.575636\pi$$
$$632$$ 7.00092e6 0.697208
$$633$$ 1.22580e6 0.121593
$$634$$ 1.15155e7 1.13778
$$635$$ −5.38421e6 −0.529892
$$636$$ 161537. 0.0158354
$$637$$ 0 0
$$638$$ −268217. −0.0260876
$$639$$ −2.90708e6 −0.281647
$$640$$ 4.79855e6 0.463085
$$641$$ −1.04174e7 −1.00141 −0.500707 0.865617i $$-0.666927\pi$$
−0.500707 + 0.865617i $$0.666927\pi$$
$$642$$ 699171. 0.0669493
$$643$$ −1.27284e7 −1.21407 −0.607037 0.794674i $$-0.707642\pi$$
−0.607037 + 0.794674i $$0.707642\pi$$
$$644$$ 0 0
$$645$$ −1.26452e6 −0.119682
$$646$$ −1.87297e7 −1.76584
$$647$$ 1.61348e7 1.51531 0.757657 0.652653i $$-0.226344\pi$$
0.757657 + 0.652653i $$0.226344\pi$$
$$648$$ 1.08449e7 1.01458
$$649$$ 2.60513e6 0.242783
$$650$$ 4.42058e6 0.410390
$$651$$ 0 0
$$652$$ 1.31267e6 0.120931
$$653$$ −1.50295e7 −1.37931 −0.689654 0.724139i $$-0.742237\pi$$
−0.689654 + 0.724139i $$0.742237\pi$$
$$654$$ −401619. −0.0367172
$$655$$ −6.18640e6 −0.563423
$$656$$ −2.78395e6 −0.252581
$$657$$ 4.67518e6 0.422557
$$658$$ 0 0
$$659$$ 1.67927e7 1.50628 0.753140 0.657860i $$-0.228538\pi$$
0.753140 + 0.657860i $$0.228538\pi$$
$$660$$ −38724.6 −0.00346041
$$661$$ −1.08540e7 −0.966246 −0.483123 0.875552i $$-0.660498\pi$$
−0.483123 + 0.875552i $$0.660498\pi$$
$$662$$ −3.60698e6 −0.319889
$$663$$ −2.60223e6 −0.229912
$$664$$ 4.76675e6 0.419567
$$665$$ 0 0
$$666$$ 1.25557e7 1.09687
$$667$$ 302809. 0.0263544
$$668$$ −742303. −0.0643636
$$669$$ −756562. −0.0653551
$$670$$ 3.30932e6 0.284807
$$671$$ −290174. −0.0248801
$$672$$ 0 0
$$673$$ 1.23697e7 1.05274 0.526371 0.850255i $$-0.323552\pi$$
0.526371 + 0.850255i $$0.323552\pi$$
$$674$$ −3.06781e6 −0.260123
$$675$$ −1.37983e6 −0.116565
$$676$$ −175916. −0.0148060
$$677$$ 1.00501e6 0.0842746 0.0421373 0.999112i $$-0.486583\pi$$
0.0421373 + 0.999112i $$0.486583\pi$$
$$678$$ 1.31060e6 0.109495
$$679$$ 0 0
$$680$$ −1.60326e7 −1.32963
$$681$$ −1.75586e6 −0.145084
$$682$$ 2.24383e6 0.184727
$$683$$ 1.87019e6 0.153403 0.0767014 0.997054i $$-0.475561\pi$$
0.0767014 + 0.997054i $$0.475561\pi$$
$$684$$ 2.74419e6 0.224272
$$685$$ 3.81290e6 0.310476
$$686$$ 0 0
$$687$$ −1.18440e6 −0.0957426
$$688$$ 1.14480e7 0.922060
$$689$$ 7.95421e6 0.638336
$$690$$ −183189. −0.0146479
$$691$$ −1.93867e7 −1.54457 −0.772286 0.635275i $$-0.780887\pi$$
−0.772286 + 0.635275i $$0.780887\pi$$
$$692$$ 4.39549e6 0.348933
$$693$$ 0 0
$$694$$ −8.93199e6 −0.703963
$$695$$ −1.39979e7 −1.09926
$$696$$ −295721. −0.0231398
$$697$$ 6.97469e6 0.543805
$$698$$ 1.99181e6 0.154742
$$699$$ 2.20046e6 0.170342
$$700$$ 0 0
$$701$$ 1.17488e7 0.903024 0.451512 0.892265i $$-0.350885\pi$$
0.451512 + 0.892265i $$0.350885\pi$$
$$702$$ −3.22416e6 −0.246930
$$703$$ 1.93027e7 1.47309
$$704$$ 2.62534e6 0.199643
$$705$$ 1.86634e6 0.141422
$$706$$ 2.50279e6 0.188978
$$707$$ 0 0
$$708$$ 464008. 0.0347890
$$709$$ 1.67948e7 1.25475 0.627377 0.778716i $$-0.284129\pi$$
0.627377 + 0.778716i $$0.284129\pi$$
$$710$$ 2.58963e6 0.192793
$$711$$ −8.61326e6 −0.638990
$$712$$ 1.36262e7 1.00734
$$713$$ −2.53322e6 −0.186616
$$714$$ 0 0
$$715$$ −1.90683e6 −0.139491
$$716$$ 4.62253e6 0.336974
$$717$$ −1.77809e6 −0.129168
$$718$$ 1.91636e7 1.38729
$$719$$ 1.65130e7 1.19126 0.595628 0.803261i $$-0.296903\pi$$
0.595628 + 0.803261i $$0.296903\pi$$
$$720$$ −7.87323e6 −0.566007
$$721$$ 0 0
$$722$$ −5.09215e6 −0.363545
$$723$$ −809961. −0.0576260
$$724$$ 3.84555e6 0.272654
$$725$$ −1.00674e6 −0.0711331
$$726$$ −1.64989e6 −0.116175
$$727$$ 1.25756e6 0.0882453 0.0441227 0.999026i $$-0.485951\pi$$
0.0441227 + 0.999026i $$0.485951\pi$$
$$728$$ 0 0
$$729$$ −1.28348e7 −0.894479
$$730$$ −4.16465e6 −0.289248
$$731$$ −2.86810e7 −1.98518
$$732$$ −51683.7 −0.00356513
$$733$$ 1.98332e6 0.136343 0.0681716 0.997674i $$-0.478283\pi$$
0.0681716 + 0.997674i $$0.478283\pi$$
$$734$$ −1.11703e7 −0.765288
$$735$$ 0 0
$$736$$ −909684. −0.0619007
$$737$$ 1.12225e6 0.0761063
$$738$$ 4.28191e6 0.289399
$$739$$ −2.38807e7 −1.60856 −0.804278 0.594253i $$-0.797448\pi$$
−0.804278 + 0.594253i $$0.797448\pi$$
$$740$$ 2.66925e6 0.179189
$$741$$ −2.45604e6 −0.164320
$$742$$ 0 0
$$743$$ −1.90819e7 −1.26809 −0.634043 0.773298i $$-0.718606\pi$$
−0.634043 + 0.773298i $$0.718606\pi$$
$$744$$ 2.47393e6 0.163853
$$745$$ −5.78356e6 −0.381773
$$746$$ 8.41886e6 0.553868
$$747$$ −5.86455e6 −0.384532
$$748$$ −878323. −0.0573985
$$749$$ 0 0
$$750$$ 1.99278e6 0.129362
$$751$$ −3.75805e6 −0.243144 −0.121572 0.992583i $$-0.538794\pi$$
−0.121572 + 0.992583i $$0.538794\pi$$
$$752$$ −1.68963e7 −1.08955
$$753$$ −1.74812e6 −0.112353
$$754$$ −2.35238e6 −0.150688
$$755$$ −4.64994e6 −0.296880
$$756$$ 0 0
$$757$$ 1.69904e7 1.07761 0.538807 0.842429i $$-0.318875\pi$$
0.538807 + 0.842429i $$0.318875\pi$$
$$758$$ 1.43359e7 0.906259
$$759$$ −62122.7 −0.00391423
$$760$$ −1.51320e7 −0.950302
$$761$$ −2.23998e7 −1.40211 −0.701056 0.713106i $$-0.747288\pi$$
−0.701056 + 0.713106i $$0.747288\pi$$
$$762$$ 1.36269e6 0.0850180
$$763$$ 0 0
$$764$$ −2.57550e6 −0.159635
$$765$$ 1.97250e7 1.21861
$$766$$ 1.65111e7 1.01672
$$767$$ 2.28481e7 1.40237
$$768$$ 1.21248e6 0.0741775
$$769$$ 1.87866e7 1.14560 0.572799 0.819696i $$-0.305858\pi$$
0.572799 + 0.819696i $$0.305858\pi$$
$$770$$ 0 0
$$771$$ 608138. 0.0368439
$$772$$ −4.75177e6 −0.286954
$$773$$ −9.30837e6 −0.560306 −0.280153 0.959955i $$-0.590385\pi$$
−0.280153 + 0.959955i $$0.590385\pi$$
$$774$$ −1.76079e7 −1.05646
$$775$$ 8.42212e6 0.503694
$$776$$ −2.05156e7 −1.22301
$$777$$ 0 0
$$778$$ −2.36525e7 −1.40097
$$779$$ 6.58288e6 0.388662
$$780$$ −339632. −0.0199881
$$781$$ 878189. 0.0515182
$$782$$ −4.15496e6 −0.242969
$$783$$ 734267. 0.0428006
$$784$$ 0 0
$$785$$ −1.59566e6 −0.0924201
$$786$$ 1.56572e6 0.0903979
$$787$$ −1.73427e7 −0.998111 −0.499056 0.866570i $$-0.666320\pi$$
−0.499056 + 0.866570i $$0.666320\pi$$
$$788$$ 2.95448e6 0.169498
$$789$$ 600867. 0.0343626
$$790$$ 7.67268e6 0.437401
$$791$$ 0 0
$$792$$ −3.33786e6 −0.189084
$$793$$ −2.54495e6 −0.143713
$$794$$ −5.91434e6 −0.332932
$$795$$ 1.09589e6 0.0614960
$$796$$ −2.64307e6 −0.147852
$$797$$ −3.10445e7 −1.73117 −0.865584 0.500764i $$-0.833052\pi$$
−0.865584 + 0.500764i $$0.833052\pi$$
$$798$$ 0 0
$$799$$ 4.23309e7 2.34580
$$800$$ 3.02439e6 0.167076
$$801$$ −1.67644e7 −0.923224
$$802$$ −1.63858e6 −0.0899563
$$803$$ −1.41231e6 −0.0772930
$$804$$ 199887. 0.0109055
$$805$$ 0 0
$$806$$ 1.96794e7 1.06702
$$807$$ −540206. −0.0291995
$$808$$ −7.07309e6 −0.381137
$$809$$ −2.47038e7 −1.32707 −0.663533 0.748147i $$-0.730944\pi$$
−0.663533 + 0.748147i $$0.730944\pi$$
$$810$$ 1.18855e7 0.636510
$$811$$ −8.42005e6 −0.449534 −0.224767 0.974413i $$-0.572162\pi$$
−0.224767 + 0.974413i $$0.572162\pi$$
$$812$$ 0 0
$$813$$ −4.57273e6 −0.242633
$$814$$ −3.79289e6 −0.200636
$$815$$ 8.90529e6 0.469628
$$816$$ 3.24576e6 0.170644
$$817$$ −2.70698e7 −1.41883
$$818$$ 7.04930e6 0.368352
$$819$$ 0 0
$$820$$ 910307. 0.0472774
$$821$$ −2.58827e7 −1.34014 −0.670071 0.742297i $$-0.733736\pi$$
−0.670071 + 0.742297i $$0.733736\pi$$
$$822$$ −965009. −0.0498141
$$823$$ 1.72004e7 0.885195 0.442597 0.896720i $$-0.354057\pi$$
0.442597 + 0.896720i $$0.354057\pi$$
$$824$$ −1.25160e7 −0.642167
$$825$$ 206537. 0.0105649
$$826$$ 0 0
$$827$$ −2.40337e7 −1.22196 −0.610979 0.791647i $$-0.709224\pi$$
−0.610979 + 0.791647i $$0.709224\pi$$
$$828$$ 608766. 0.0308585
$$829$$ 3.24736e7 1.64113 0.820567 0.571550i $$-0.193658\pi$$
0.820567 + 0.571550i $$0.193658\pi$$
$$830$$ 5.22413e6 0.263220
$$831$$ −264491. −0.0132865
$$832$$ 2.30254e7 1.15318
$$833$$ 0 0
$$834$$ 3.54274e6 0.176370
$$835$$ −5.03587e6 −0.249953
$$836$$ −828983. −0.0410232
$$837$$ −6.14269e6 −0.303072
$$838$$ 2.49498e7 1.22732
$$839$$ 1.24404e7 0.610139 0.305069 0.952330i $$-0.401320\pi$$
0.305069 + 0.952330i $$0.401320\pi$$
$$840$$ 0 0
$$841$$ −1.99754e7 −0.973881
$$842$$ −1.23981e7 −0.602663
$$843$$ 4.63954e6 0.224857
$$844$$ 3.62868e6 0.175345
$$845$$ −1.19344e6 −0.0574986
$$846$$ 2.59878e7 1.24837
$$847$$ 0 0
$$848$$ −9.92129e6 −0.473782
$$849$$ 2.47629e6 0.117905
$$850$$ 1.38139e7 0.655794
$$851$$ 4.28206e6 0.202689
$$852$$ 156417. 0.00738219
$$853$$ 999355. 0.0470270 0.0235135 0.999724i $$-0.492515\pi$$
0.0235135 + 0.999724i $$0.492515\pi$$
$$854$$ 0 0
$$855$$ 1.86169e7 0.870949
$$856$$ 1.28119e7 0.597628
$$857$$ 2.64465e7 1.23003 0.615016 0.788514i $$-0.289149\pi$$
0.615016 + 0.788514i $$0.289149\pi$$
$$858$$ 482601. 0.0223805
$$859$$ 2.86716e7 1.32577 0.662887 0.748719i $$-0.269331\pi$$
0.662887 + 0.748719i $$0.269331\pi$$
$$860$$ −3.74332e6 −0.172588
$$861$$ 0 0
$$862$$ 2.65703e7 1.21794
$$863$$ 4.08173e6 0.186560 0.0932798 0.995640i $$-0.470265\pi$$
0.0932798 + 0.995640i $$0.470265\pi$$
$$864$$ −2.20585e6 −0.100529
$$865$$ 2.98195e7 1.35506
$$866$$ −1.33688e7 −0.605755
$$867$$ −5.17446e6 −0.233785
$$868$$ 0 0
$$869$$ 2.60195e6 0.116882
$$870$$ −324097. −0.0145170
$$871$$ 9.84261e6 0.439608
$$872$$ −7.35946e6 −0.327759
$$873$$ 2.52405e7 1.12089
$$874$$ −3.92155e6 −0.173652
$$875$$ 0 0
$$876$$ −251550. −0.0110755
$$877$$ −2.82405e7 −1.23986 −0.619931 0.784657i $$-0.712839\pi$$
−0.619931 + 0.784657i $$0.712839\pi$$
$$878$$ 1.29820e7 0.568335
$$879$$ 3.81610e6 0.166590
$$880$$ 2.37839e6 0.103532
$$881$$ −1.61480e7 −0.700936 −0.350468 0.936575i $$-0.613978\pi$$
−0.350468 + 0.936575i $$0.613978\pi$$
$$882$$ 0 0
$$883$$ −3.86021e7 −1.66613 −0.833065 0.553174i $$-0.813416\pi$$
−0.833065 + 0.553174i $$0.813416\pi$$
$$884$$ −7.70328e6 −0.331547
$$885$$ 3.14788e6 0.135102
$$886$$ −1.95127e7 −0.835091
$$887$$ −7.29088e6 −0.311151 −0.155575 0.987824i $$-0.549723\pi$$
−0.155575 + 0.987824i $$0.549723\pi$$
$$888$$ −4.18184e6 −0.177965
$$889$$ 0 0
$$890$$ 1.49337e7 0.631965
$$891$$ 4.03059e6 0.170088
$$892$$ −2.23962e6 −0.0942460
$$893$$ 3.99529e7 1.67656
$$894$$ 1.46377e6 0.0612531
$$895$$ 3.13598e7 1.30862
$$896$$ 0 0
$$897$$ −544843. −0.0226095
$$898$$ −7.59205e6 −0.314173
$$899$$ −4.48176e6 −0.184948
$$900$$ −2.02394e6 −0.0832898
$$901$$ 2.48561e7 1.02005
$$902$$ −1.29351e6 −0.0529361
$$903$$ 0 0
$$904$$ 2.40160e7 0.977415
$$905$$ 2.60886e7 1.05884
$$906$$ 1.17686e6 0.0476325
$$907$$ −3.73335e7 −1.50689 −0.753443 0.657514i $$-0.771608\pi$$
−0.753443 + 0.657514i $$0.771608\pi$$
$$908$$ −5.19779e6 −0.209221
$$909$$ 8.70205e6 0.349311
$$910$$ 0 0
$$911$$ −2475.17 −9.88120e−5 0 −4.94060e−5 1.00000i $$-0.500016\pi$$
−4.94060e−5 1.00000i $$0.500016\pi$$
$$912$$ 3.06342e6 0.121961
$$913$$ 1.77160e6 0.0703377
$$914$$ 1.09900e7 0.435144
$$915$$ −350628. −0.0138450
$$916$$ −3.50613e6 −0.138067
$$917$$ 0 0
$$918$$ −1.00752e7 −0.394590
$$919$$ −4.48238e6 −0.175073 −0.0875366 0.996161i $$-0.527899\pi$$
−0.0875366 + 0.996161i $$0.527899\pi$$
$$920$$ −3.35684e6 −0.130756
$$921$$ −1.49467e6 −0.0580625
$$922$$ −3.11039e7 −1.20500
$$923$$ 7.70210e6 0.297581
$$924$$ 0 0
$$925$$ −1.42364e7 −0.547075
$$926$$ −2.00065e7 −0.766733
$$927$$ 1.53985e7 0.588544
$$928$$ −1.60941e6 −0.0613473
$$929$$ −2.12859e7 −0.809193 −0.404596 0.914495i $$-0.632588\pi$$
−0.404596 + 0.914495i $$0.632588\pi$$
$$930$$ 2.71131e6 0.102795
$$931$$ 0 0
$$932$$ 6.51394e6 0.245643
$$933$$ −1.78470e6 −0.0671215
$$934$$ −2.78049e7 −1.04293
$$935$$ −5.95865e6 −0.222904
$$936$$ −2.92745e7 −1.09219
$$937$$ −6.79757e6 −0.252932 −0.126466 0.991971i $$-0.540364\pi$$
−0.126466 + 0.991971i $$0.540364\pi$$
$$938$$ 0 0
$$939$$ 3.36781e6 0.124647
$$940$$ 5.52484e6 0.203939
$$941$$ −4.90883e7 −1.80719 −0.903595 0.428388i $$-0.859082\pi$$
−0.903595 + 0.428388i $$0.859082\pi$$
$$942$$ 403847. 0.0148283
$$943$$ 1.46033e6 0.0534776
$$944$$ −2.84985e7 −1.04086
$$945$$ 0 0
$$946$$ 5.31910e6 0.193246
$$947$$ 2.45484e7 0.889505 0.444753 0.895653i $$-0.353292\pi$$
0.444753 + 0.895653i $$0.353292\pi$$
$$948$$ 463441. 0.0167484
$$949$$ −1.23865e7 −0.446462
$$950$$ 1.30378e7 0.468702
$$951$$ 4.71869e6 0.169188
$$952$$ 0 0
$$953$$ −513120. −0.0183015 −0.00915075 0.999958i $$-0.502913\pi$$
−0.00915075 + 0.999958i $$0.502913\pi$$
$$954$$ 1.52597e7 0.542843
$$955$$ −1.74725e7 −0.619935
$$956$$ −5.26360e6 −0.186268
$$957$$ −109907. −0.00387923
$$958$$ −3.38659e7 −1.19220
$$959$$ 0 0
$$960$$ 3.17230e6 0.111095
$$961$$ 8.86412e6 0.309619
$$962$$ −3.32653e7 −1.15892
$$963$$ −1.57626e7 −0.547724
$$964$$ −2.39770e6 −0.0831002
$$965$$ −3.22366e7 −1.11437
$$966$$ 0 0
$$967$$ 3.34818e7 1.15144 0.575722 0.817645i $$-0.304721\pi$$
0.575722 + 0.817645i $$0.304721\pi$$
$$968$$ −3.02334e7 −1.03705
$$969$$ −7.67488e6 −0.262580
$$970$$ −2.24842e7 −0.767269
$$971$$ −4.76036e6 −0.162029 −0.0810143 0.996713i $$-0.525816\pi$$
−0.0810143 + 0.996713i $$0.525816\pi$$
$$972$$ 2.22090e6 0.0753986
$$973$$ 0 0
$$974$$ −4.84739e7 −1.63723
$$975$$ 1.81142e6 0.0610250
$$976$$ 3.17431e6 0.106666
$$977$$ 2.87338e7 0.963067 0.481534 0.876428i $$-0.340080\pi$$
0.481534 + 0.876428i $$0.340080\pi$$
$$978$$ −2.25385e6 −0.0753490
$$979$$ 5.06430e6 0.168874
$$980$$ 0 0
$$981$$ 9.05437e6 0.300390
$$982$$ 4.16428e7 1.37804
$$983$$ 4.97072e7 1.64072 0.820362 0.571845i $$-0.193772\pi$$
0.820362 + 0.571845i $$0.193772\pi$$
$$984$$ −1.42615e6 −0.0469546
$$985$$ 2.00435e7 0.658239
$$986$$ −7.35092e6 −0.240796
$$987$$ 0 0
$$988$$ −7.27054e6 −0.236960
$$989$$ −6.00511e6 −0.195223
$$990$$ −3.65814e6 −0.118624
$$991$$ −2.91066e6 −0.0941471 −0.0470736 0.998891i $$-0.514990\pi$$
−0.0470736 + 0.998891i $$0.514990\pi$$
$$992$$ 1.34639e7 0.434401
$$993$$ −1.47803e6 −0.0475676
$$994$$ 0 0
$$995$$ −1.79309e7 −0.574176
$$996$$ 315545. 0.0100789
$$997$$ −1.43353e7 −0.456740 −0.228370 0.973574i $$-0.573340\pi$$
−0.228370 + 0.973574i $$0.573340\pi$$
$$998$$ 2.19289e7 0.696933
$$999$$ 1.03834e7 0.329174
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 49.6.a.e.1.1 2
3.2 odd 2 441.6.a.m.1.2 2
4.3 odd 2 784.6.a.t.1.2 2
7.2 even 3 49.6.c.f.18.2 4
7.3 odd 6 7.6.c.a.2.2 4
7.4 even 3 49.6.c.f.30.2 4
7.5 odd 6 7.6.c.a.4.2 yes 4
7.6 odd 2 49.6.a.d.1.1 2
21.5 even 6 63.6.e.d.46.1 4
21.17 even 6 63.6.e.d.37.1 4
21.20 even 2 441.6.a.n.1.2 2
28.3 even 6 112.6.i.c.65.2 4
28.19 even 6 112.6.i.c.81.2 4
28.27 even 2 784.6.a.ba.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.2 4 7.3 odd 6
7.6.c.a.4.2 yes 4 7.5 odd 6
49.6.a.d.1.1 2 7.6 odd 2
49.6.a.e.1.1 2 1.1 even 1 trivial
49.6.c.f.18.2 4 7.2 even 3
49.6.c.f.30.2 4 7.4 even 3
63.6.e.d.37.1 4 21.17 even 6
63.6.e.d.46.1 4 21.5 even 6
112.6.i.c.65.2 4 28.3 even 6
112.6.i.c.81.2 4 28.19 even 6
441.6.a.m.1.2 2 3.2 odd 2
441.6.a.n.1.2 2 21.20 even 2
784.6.a.t.1.2 2 4.3 odd 2
784.6.a.ba.1.1 2 28.27 even 2