# Properties

 Label 49.6.a.d.1.1 Level $49$ Weight $6$ Character 49.1 Self dual yes Analytic conductor $7.859$ 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] = [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: $$1$$ 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 $$-0.478720\pi$$
0.0668046 + 0.997766i $$0.478720\pi$$
$$4$$ −6.16553 −0.192673
$$5$$ 41.8276 0.748235 0.374118 0.927381i $$-0.377946\pi$$
0.374118 + 0.927381i $$0.377946\pi$$
$$6$$ −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 $$-0.673642\pi$$
−0.518856 + 0.854862i $$0.673642\pi$$
$$14$$ 0 0
$$15$$ 87.1170 0.0999712
$$16$$ −788.689 −0.770205
$$17$$ −1975.92 −1.65824 −0.829121 0.559069i $$-0.811159\pi$$
−0.829121 + 0.559069i $$0.811159\pi$$
$$18$$ 1213.06 0.882474
$$19$$ −1864.93 −1.18516 −0.592581 0.805511i $$-0.701891\pi$$
−0.592581 + 0.805511i $$0.701891\pi$$
$$20$$ −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 $$-0.306093\pi$$
0.572193 + 0.820119i $$0.306093\pi$$
$$32$$ −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 $$-0.552430\pi$$
−0.163970 + 0.986465i $$0.552430\pi$$
$$42$$ 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 $$-0.750093\pi$$
−0.707314 + 0.706900i $$0.750093\pi$$
$$48$$ −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 $$-0.736159\pi$$
−0.675702 + 0.737175i $$0.736159\pi$$
$$60$$ −537.122 −0.0192617
$$61$$ 4024.80 0.138490 0.0692451 0.997600i $$-0.477941\pi$$
0.0692451 + 0.997600i $$0.477941\pi$$
$$62$$ −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 $$-0.430986\pi$$
0.215119 + 0.976588i $$0.430986\pi$$
$$74$$ −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 $$-0.562718\pi$$
−0.195761 + 0.980652i $$0.562718\pi$$
$$84$$ 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 $$-0.655747\pi$$
−0.470002 + 0.882665i $$0.655747\pi$$
$$90$$ 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 $$-0.306699\pi$$
0.570630 + 0.821207i $$0.306699\pi$$
$$98$$ 0 0
$$99$$ −17206.7 −0.176445
$$100$$ 8480.37 0.0848037
$$101$$ 36461.8 0.355660 0.177830 0.984061i $$-0.443092\pi$$
0.177830 + 0.984061i $$0.443092\pi$$
$$102$$ 20917.5 0.199072
$$103$$ 64520.1 0.599242 0.299621 0.954058i $$-0.403140\pi$$
0.299621 + 0.954058i $$0.403140\pi$$
$$104$$ −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 $$-0.622873\pi$$
−0.376501 + 0.926416i $$0.622873\pi$$
$$132$$ −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 $$-0.762616\pi$$
−0.734570 + 0.678533i $$0.762616\pi$$
$$140$$ 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 $$-0.519671\pi$$
−0.0617587 + 0.998091i $$0.519671\pi$$
$$158$$ −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 $$-0.553417\pi$$
−0.167028 + 0.985952i $$0.553417\pi$$
$$168$$ 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 $$-0.139491\pi$$
0.905507 + 0.424331i $$0.139491\pi$$
$$174$$ −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 $$-0.249797\pi$$
0.707557 + 0.706656i $$0.249797\pi$$
$$182$$ 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 $$-0.625346\pi$$
−0.383687 + 0.923463i $$0.625346\pi$$
$$200$$ −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 $$-0.578649\pi$$
−0.244575 + 0.969630i $$0.578649\pi$$
$$224$$ 0 0
$$225$$ 328268. 0.432287
$$226$$ −629258. −0.819517
$$227$$ −843041. −1.08589 −0.542943 0.839770i $$-0.682690\pi$$
−0.542943 + 0.839770i $$0.682690\pi$$
$$228$$ 23948.1 0.0305094
$$229$$ −568666. −0.716587 −0.358293 0.933609i $$-0.616641\pi$$
−0.358293 + 0.933609i $$0.616641\pi$$
$$230$$ −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 $$-0.569187\pi$$
−0.215651 + 0.976470i $$0.569187\pi$$
$$242$$ 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 $$-0.638129\pi$$
−0.420453 + 0.907314i $$0.638129\pi$$
$$252$$ 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 $$-0.455971\pi$$
0.137879 + 0.990449i $$0.455971\pi$$
$$258$$ 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 $$-0.534852\pi$$
−0.109272 + 0.994012i $$0.534852\pi$$
$$270$$ 213278. 0.178047
$$271$$ −2.19551e6 −1.81599 −0.907994 0.418984i $$-0.862386\pi$$
−0.907994 + 0.418984i $$0.862386\pi$$
$$272$$ 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 $$-0.354542\pi$$
0.441231 + 0.897393i $$0.354542\pi$$
$$284$$ −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 $$-0.285742\pi$$
0.623421 + 0.781886i $$0.285742\pi$$
$$294$$ 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 $$-0.569720\pi$$
−0.217285 + 0.976108i $$0.569720\pi$$
$$308$$ 0 0
$$309$$ 134380. 0.0800642
$$310$$ −1.30179e6 −0.769370
$$311$$ −856892. −0.502372 −0.251186 0.967939i $$-0.580821\pi$$
−0.251186 + 0.967939i $$0.580821\pi$$
$$312$$ −255474. −0.148580
$$313$$ 1.61699e6 0.932924 0.466462 0.884541i $$-0.345528\pi$$
0.466462 + 0.884541i $$0.345528\pi$$
$$314$$ 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 $$-0.472556\pi$$
0.0861102 + 0.996286i $$0.472556\pi$$
$$350$$ 0 0
$$351$$ 634333. 0.274821
$$352$$ −158529. −0.0681948
$$353$$ 492407. 0.210323 0.105162 0.994455i $$-0.466464\pi$$
0.105162 + 0.994455i $$0.466464\pi$$
$$354$$ 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 $$-0.640030\pi$$
−0.425863 + 0.904788i $$0.640030\pi$$
$$368$$ −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 $$-0.308575\pi$$
0.565781 + 0.824555i $$0.308575\pi$$
$$384$$ −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 $$-0.559315\pi$$
−0.185268 + 0.982688i $$0.559315\pi$$
$$398$$ 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 $$-0.434288\pi$$
0.204978 + 0.978767i $$0.434288\pi$$
$$410$$ 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 $$-0.260688\pi$$
0.682971 + 0.730446i $$0.260688\pi$$
$$420$$ 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 $$-0.609442\pi$$
−0.337087 + 0.941473i $$0.609442\pi$$
$$434$$ 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 $$-0.397572\pi$$
0.316264 + 0.948671i $$0.397572\pi$$
$$440$$ 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 $$-0.733943\pi$$
−0.670553 + 0.741862i $$0.733943\pi$$
$$462$$ 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 $$-0.697089\pi$$
−0.580363 + 0.814358i $$0.697089\pi$$
$$468$$ −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 $$-0.730899\pi$$
−0.663428 + 0.748240i $$0.730899\pi$$
$$480$$ −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 $$-0.869021\pi$$
−0.916529 + 0.399968i $$0.869021\pi$$
$$504$$ 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 $$-0.585260\pi$$
−0.264661 + 0.964342i $$0.585260\pi$$
$$510$$ 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 $$-0.710224\pi$$
−0.613464 + 0.789723i $$0.710224\pi$$
$$522$$ 887882. 0.142619
$$523$$ 4.75669e6 0.760415 0.380208 0.924901i $$-0.375853\pi$$
0.380208 + 0.924901i $$0.375853\pi$$
$$524$$ 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 $$-0.462937\pi$$
0.116175 + 0.993229i $$0.462937\pi$$
$$564$$ 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 $$-0.363737\pi$$
0.415127 + 0.909763i $$0.363737\pi$$
$$578$$ −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 $$-0.485237\pi$$
0.0463628 + 0.998925i $$0.485237\pi$$
$$588$$ 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 $$-0.817081\pi$$
−0.839379 + 0.543546i $$0.817081\pi$$
$$594$$ 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 $$-0.394629\pi$$
0.325020 + 0.945707i $$0.394629\pi$$
$$602$$ 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 $$-0.425668\pi$$
0.231405 + 0.972858i $$0.425668\pi$$
$$608$$ 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 $$-0.234603\pi$$
0.740469 + 0.672090i $$0.234603\pi$$
$$620$$ −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 $$-0.292358\pi$$
0.607037 + 0.794674i $$0.292358\pi$$
$$644$$ 0 0
$$645$$ −1.26452e6 −0.119682
$$646$$ −1.87297e7 −1.76584
$$647$$ −1.61348e7 −1.51531 −0.757657 0.652653i $$-0.773656\pi$$
−0.757657 + 0.652653i $$0.773656\pi$$
$$648$$ 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 $$-0.339502\pi$$
0.483123 + 0.875552i $$0.339502\pi$$
$$662$$ −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 $$-0.513417\pi$$
−0.0421373 + 0.999112i $$0.513417\pi$$
$$678$$ −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 $$-0.219113\pi$$
0.772286 + 0.635275i $$0.219113\pi$$
$$692$$ −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 $$-0.703097\pi$$
−0.595628 + 0.803261i $$0.703097\pi$$
$$720$$ 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 $$-0.514049\pi$$
−0.0441227 + 0.999026i $$0.514049\pi$$
$$728$$ 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 $$-0.521717\pi$$
−0.0681716 + 0.997674i $$0.521717\pi$$
$$734$$ 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 $$-0.252712\pi$$
0.701056 + 0.713106i $$0.252712\pi$$
$$762$$ −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 $$-0.694142\pi$$
−0.572799 + 0.819696i $$0.694142\pi$$
$$770$$ 0 0
$$771$$ 608138. 0.0368439
$$772$$ −4.75177e6 −0.286954
$$773$$ 9.30837e6 0.560306 0.280153 0.959955i $$-0.409615\pi$$
0.280153 + 0.959955i $$0.409615\pi$$
$$774$$ −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 $$-0.333680\pi$$
0.499056 + 0.866570i $$0.333680\pi$$
$$788$$ 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 $$-0.166948\pi$$
0.865584 + 0.500764i $$0.166948\pi$$
$$798$$ 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 $$-0.427838\pi$$
0.224767 + 0.974413i $$0.427838\pi$$
$$812$$ 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 $$-0.806342\pi$$
−0.820567 + 0.571550i $$0.806342\pi$$
$$830$$ 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 $$-0.598680\pi$$
−0.305069 + 0.952330i $$0.598680\pi$$
$$840$$ 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 $$-0.507485\pi$$
−0.0235135 + 0.999724i $$0.507485\pi$$
$$854$$ 0 0
$$855$$ 1.86169e7 0.870949
$$856$$ 1.28119e7 0.597628
$$857$$ −2.64465e7 −1.23003 −0.615016 0.788514i $$-0.710851\pi$$
−0.615016 + 0.788514i $$0.710851\pi$$
$$858$$ 482601. 0.0223805
$$859$$ −2.86716e7 −1.32577 −0.662887 0.748719i $$-0.730669\pi$$
−0.662887 + 0.748719i $$0.730669\pi$$
$$860$$ 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 $$-0.386022\pi$$
0.350468 + 0.936575i $$0.386022\pi$$
$$882$$ 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 $$-0.450277\pi$$
0.155575 + 0.987824i $$0.450277\pi$$
$$888$$ 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 $$-0.367412\pi$$
0.404596 + 0.914495i $$0.367412\pi$$
$$930$$ −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 $$-0.459636\pi$$
0.126466 + 0.991971i $$0.459636\pi$$
$$938$$ 0 0
$$939$$ 3.36781e6 0.124647
$$940$$ 5.52484e6 0.203939
$$941$$ 4.90883e7 1.80719 0.903595 0.428388i $$-0.140918\pi$$
0.903595 + 0.428388i $$0.140918\pi$$
$$942$$ 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 $$-0.474184\pi$$
0.0810143 + 0.996713i $$0.474184\pi$$
$$972$$ −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 $$-0.806228\pi$$
−0.820362 + 0.571845i $$0.806228\pi$$
$$984$$ −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 $$-0.426660\pi$$
0.228370 + 0.973574i $$0.426660\pi$$
$$998$$ 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.d.1.1 2
3.2 odd 2 441.6.a.n.1.2 2
4.3 odd 2 784.6.a.ba.1.1 2
7.2 even 3 7.6.c.a.4.2 yes 4
7.3 odd 6 49.6.c.f.30.2 4
7.4 even 3 7.6.c.a.2.2 4
7.5 odd 6 49.6.c.f.18.2 4
7.6 odd 2 49.6.a.e.1.1 2
21.2 odd 6 63.6.e.d.46.1 4
21.11 odd 6 63.6.e.d.37.1 4
21.20 even 2 441.6.a.m.1.2 2
28.11 odd 6 112.6.i.c.65.2 4
28.23 odd 6 112.6.i.c.81.2 4
28.27 even 2 784.6.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.2 4 7.4 even 3
7.6.c.a.4.2 yes 4 7.2 even 3
49.6.a.d.1.1 2 1.1 even 1 trivial
49.6.a.e.1.1 2 7.6 odd 2
49.6.c.f.18.2 4 7.5 odd 6
49.6.c.f.30.2 4 7.3 odd 6
63.6.e.d.37.1 4 21.11 odd 6
63.6.e.d.46.1 4 21.2 odd 6
112.6.i.c.65.2 4 28.11 odd 6
112.6.i.c.81.2 4 28.23 odd 6
441.6.a.m.1.2 2 21.20 even 2
441.6.a.n.1.2 2 3.2 odd 2
784.6.a.t.1.2 2 28.27 even 2
784.6.a.ba.1.1 2 4.3 odd 2