Properties

 Label 441.6.a.n.1.2 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ 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] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$70.7292645375$$ 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 1.2 Root $$-2.54138$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.08276 q^{2} -6.16553 q^{4} -41.8276 q^{5} -193.986 q^{8} +O(q^{10})$$ $$q+5.08276 q^{2} -6.16553 q^{4} -41.8276 q^{5} -193.986 q^{8} -212.600 q^{10} -72.0965 q^{11} -632.317 q^{13} -788.689 q^{16} +1975.92 q^{17} -1864.93 q^{19} +257.889 q^{20} -366.449 q^{22} -413.711 q^{23} -1375.45 q^{25} -3213.92 q^{26} -731.934 q^{29} +6123.18 q^{31} +2198.84 q^{32} +10043.2 q^{34} +10350.4 q^{37} -9478.97 q^{38} +8113.99 q^{40} +3529.84 q^{41} -14515.2 q^{43} +444.513 q^{44} -2102.79 q^{46} +21423.3 q^{47} -6991.08 q^{50} +3898.57 q^{52} -12579.5 q^{53} +3015.62 q^{55} -3720.25 q^{58} +36133.9 q^{59} +4024.80 q^{61} +31122.6 q^{62} +36414.2 q^{64} +26448.3 q^{65} +15565.9 q^{67} -12182.6 q^{68} -12180.8 q^{71} +19589.1 q^{73} +52608.5 q^{74} +11498.2 q^{76} +36089.8 q^{79} +32989.0 q^{80} +17941.3 q^{82} +24572.6 q^{83} -82648.2 q^{85} -73777.5 q^{86} +13985.7 q^{88} +70243.3 q^{89} +2550.74 q^{92} +108890. q^{94} +78005.4 q^{95} +105758. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8 $$2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8} - 778 q^{10} - 424 q^{11} - 924 q^{13} - 2064 q^{16} + 2346 q^{17} - 360 q^{19} + 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} - 1148 q^{26} + 7052 q^{29} + 3548 q^{31} + 8096 q^{32} + 7422 q^{34} + 11090 q^{37} - 20138 q^{38} + 15936 q^{40} - 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} + 22956 q^{47} - 29992 q^{50} - 1400 q^{52} - 3042 q^{53} - 25076 q^{55} - 58852 q^{58} + 65808 q^{59} - 42486 q^{61} + 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} - 5460 q^{68} + 2208 q^{71} - 50506 q^{73} + 47370 q^{74} + 38836 q^{76} + 9004 q^{79} - 68816 q^{80} + 67732 q^{82} + 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} + 26666 q^{89} + 10284 q^{92} + 98034 q^{94} + 198140 q^{95} + 209132 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8 - 778 * q^10 - 424 * q^11 - 924 * q^13 - 2064 * q^16 + 2346 * q^17 - 360 * q^19 + 1708 * q^20 + 2126 * q^22 + 12 * q^23 + 1872 * q^25 - 1148 * q^26 + 7052 * q^29 + 3548 * q^31 + 8096 * q^32 + 7422 * q^34 + 11090 * q^37 - 20138 * q^38 + 15936 * q^40 - 3500 * q^41 - 12680 * q^43 - 5948 * q^44 - 5118 * q^46 + 22956 * q^47 - 29992 * q^50 - 1400 * q^52 - 3042 * q^53 - 25076 * q^55 - 58852 * q^58 + 65808 * q^59 - 42486 * q^61 + 49362 * q^62 + 35456 * q^64 + 3164 * q^65 + 42312 * q^67 - 5460 * q^68 + 2208 * q^71 - 50506 * q^73 + 47370 * q^74 + 38836 * q^76 + 9004 * q^79 - 68816 * q^80 + 67732 * q^82 + 104328 * q^83 - 53106 * q^85 - 86776 * q^86 - 20496 * q^88 + 26666 * q^89 + 10284 * q^92 + 98034 * q^94 + 198140 * q^95 + 209132 * q^97

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 $$-0.351689\pi$$
0.449257 + 0.893403i $$0.351689\pi$$
$$3$$ 0 0
$$4$$ −6.16553 −0.192673
$$5$$ −41.8276 −0.748235 −0.374118 0.927381i $$-0.622054\pi$$
−0.374118 + 0.927381i $$0.622054\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −193.986 −1.07163
$$9$$ 0 0
$$10$$ −212.600 −0.672300
$$11$$ −72.0965 −0.179652 −0.0898260 0.995957i $$-0.528631\pi$$
−0.0898260 + 0.995957i $$0.528631\pi$$
$$12$$ 0 0
$$13$$ −632.317 −1.03771 −0.518856 0.854862i $$-0.673642\pi$$
−0.518856 + 0.854862i $$0.673642\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −788.689 −0.770205
$$17$$ 1975.92 1.65824 0.829121 0.559069i $$-0.188841\pi$$
0.829121 + 0.559069i $$0.188841\pi$$
$$18$$ 0 0
$$19$$ −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 $$-0.525982\pi$$
−0.0815356 + 0.996670i $$0.525982\pi$$
$$24$$ 0 0
$$25$$ −1375.45 −0.440144
$$26$$ −3213.92 −0.932398
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −731.934 −0.161613 −0.0808066 0.996730i $$-0.525750\pi$$
−0.0808066 + 0.996730i $$0.525750\pi$$
$$30$$ 0 0
$$31$$ 6123.18 1.14439 0.572193 0.820119i $$-0.306093\pi$$
0.572193 + 0.820119i $$0.306093\pi$$
$$32$$ 2198.84 0.379593
$$33$$ 0 0
$$34$$ 10043.2 1.48995
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10350.4 1.24295 0.621473 0.783436i $$-0.286535\pi$$
0.621473 + 0.783436i $$0.286535\pi$$
$$38$$ −9478.97 −1.06488
$$39$$ 0 0
$$40$$ 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
$$45$$ 0 0
$$46$$ −2102.79 −0.146522
$$47$$ 21423.3 1.41463 0.707314 0.706900i $$-0.249907\pi$$
0.707314 + 0.706900i $$0.249907\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −6991.08 −0.395475
$$51$$ 0 0
$$52$$ 3898.57 0.199939
$$53$$ −12579.5 −0.615138 −0.307569 0.951526i $$-0.599515\pi$$
−0.307569 + 0.951526i $$0.599515\pi$$
$$54$$ 0 0
$$55$$ 3015.62 0.134422
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −3720.25 −0.145212
$$59$$ 36133.9 1.35140 0.675702 0.737175i $$-0.263841\pi$$
0.675702 + 0.737175i $$0.263841\pi$$
$$60$$ 0 0
$$61$$ 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
$$66$$ 0 0
$$67$$ 15565.9 0.423632 0.211816 0.977310i $$-0.432062\pi$$
0.211816 + 0.977310i $$0.432062\pi$$
$$68$$ −12182.6 −0.319498
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12180.8 −0.286766 −0.143383 0.989667i $$-0.545798\pi$$
−0.143383 + 0.989667i $$0.545798\pi$$
$$72$$ 0 0
$$73$$ 19589.1 0.430237 0.215119 0.976588i $$-0.430986\pi$$
0.215119 + 0.976588i $$0.430986\pi$$
$$74$$ 52608.5 1.11680
$$75$$ 0 0
$$76$$ 11498.2 0.228348
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 36089.8 0.650604 0.325302 0.945610i $$-0.394534\pi$$
0.325302 + 0.945610i $$0.394534\pi$$
$$80$$ 32989.0 0.576294
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 13985.7 0.192521
$$89$$ 70243.3 0.940005 0.470002 0.882665i $$-0.344253\pi$$
0.470002 + 0.882665i $$0.344253\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2550.74 0.0314193
$$93$$ 0 0
$$94$$ 108890. 1.27106
$$95$$ 78005.4 0.886779
$$96$$ 0 0
$$97$$ 105758. 1.14126 0.570630 0.821207i $$-0.306699\pi$$
0.570630 + 0.821207i $$0.306699\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 8480.37 0.0848037
$$101$$ −36461.8 −0.355660 −0.177830 0.984061i $$-0.556908\pi$$
−0.177830 + 0.984061i $$0.556908\pi$$
$$102$$ 0 0
$$103$$ 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 $$-0.589950\pi$$
−0.278840 + 0.960338i $$0.589950\pi$$
$$108$$ 0 0
$$109$$ −37938.0 −0.305850 −0.152925 0.988238i $$-0.548869\pi$$
−0.152925 + 0.988238i $$0.548869\pi$$
$$110$$ 15327.7 0.120780
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −123802. −0.912080 −0.456040 0.889959i $$-0.650733\pi$$
−0.456040 + 0.889959i $$0.650733\pi$$
$$114$$ 0 0
$$115$$ 17304.5 0.122016
$$116$$ 4512.76 0.0311384
$$117$$ 0 0
$$118$$ 183660. 1.21426
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −155853. −0.967725
$$122$$ 20457.1 0.124435
$$123$$ 0 0
$$124$$ −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
$$129$$ 0 0
$$130$$ 134431. 0.697653
$$131$$ 147902. 0.753003 0.376501 0.926416i $$-0.377127\pi$$
0.376501 + 0.926416i $$0.377127\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 79118.0 0.380639
$$135$$ 0 0
$$136$$ −383302. −1.77703
$$137$$ 91157.4 0.414945 0.207472 0.978241i $$-0.433476\pi$$
0.207472 + 0.978241i $$0.433476\pi$$
$$138$$ 0 0
$$139$$ −334657. −1.46914 −0.734570 0.678533i $$-0.762616\pi$$
−0.734570 + 0.678533i $$0.762616\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −61911.9 −0.257664
$$143$$ 45587.8 0.186427
$$144$$ 0 0
$$145$$ 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 $$-0.582113\pi$$
−0.255115 + 0.966911i $$0.582113\pi$$
$$150$$ 0 0
$$151$$ 111169. 0.396773 0.198386 0.980124i $$-0.436430\pi$$
0.198386 + 0.980124i $$0.436430\pi$$
$$152$$ 361770. 1.27006
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −256118. −0.856270
$$156$$ 0 0
$$157$$ −38148.5 −0.123517 −0.0617587 0.998091i $$-0.519671\pi$$
−0.0617587 + 0.998091i $$0.519671\pi$$
$$158$$ 183436. 0.584577
$$159$$ 0 0
$$160$$ −91972.3 −0.284025
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −212905. −0.627648 −0.313824 0.949481i $$-0.601610\pi$$
−0.313824 + 0.949481i $$0.601610\pi$$
$$164$$ −21763.3 −0.0631852
$$165$$ 0 0
$$166$$ 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
$$171$$ 0 0
$$172$$ 89494.0 0.230660
$$173$$ −712914. −1.81101 −0.905507 0.424331i $$-0.860509\pi$$
−0.905507 + 0.424331i $$0.860509\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 56861.7 0.138369
$$177$$ 0 0
$$178$$ 357030. 0.844607
$$179$$ 749738. 1.74895 0.874474 0.485072i $$-0.161207\pi$$
0.874474 + 0.485072i $$0.161207\pi$$
$$180$$ 0 0
$$181$$ 623718. 1.41511 0.707557 0.706656i $$-0.249797\pi$$
0.707557 + 0.706656i $$0.249797\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 80254.2 0.174752
$$185$$ −432932. −0.930016
$$186$$ 0 0
$$187$$ −142457. −0.297907
$$188$$ −132086. −0.272560
$$189$$ 0 0
$$190$$ 396483. 0.796784
$$191$$ −417726. −0.828530 −0.414265 0.910156i $$-0.635961\pi$$
−0.414265 + 0.910156i $$0.635961\pi$$
$$192$$ 0 0
$$193$$ 770700. 1.48933 0.744667 0.667436i $$-0.232608\pi$$
0.744667 + 0.667436i $$0.232608\pi$$
$$194$$ 537544. 1.02544
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 479193. 0.879721 0.439861 0.898066i $$-0.355028\pi$$
0.439861 + 0.898066i $$0.355028\pi$$
$$198$$ 0 0
$$199$$ −428686. −0.767373 −0.383687 0.923463i $$-0.625346\pi$$
−0.383687 + 0.923463i $$0.625346\pi$$
$$200$$ 266818. 0.471673
$$201$$ 0 0
$$202$$ −185327. −0.319565
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −147645. −0.245377
$$206$$ 327940. 0.538427
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ −335694. −0.501083
$$215$$ 607138. 0.895759
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −192830. −0.274811
$$219$$ 0 0
$$220$$ −18592.9 −0.0258994
$$221$$ −1.24941e6 −1.72078
$$222$$ 0 0
$$223$$ −363249. −0.489151 −0.244575 0.969630i $$-0.578649\pi$$
−0.244575 + 0.969630i $$0.578649\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −629258. −0.819517
$$227$$ 843041. 1.08589 0.542943 0.839770i $$-0.317310\pi$$
0.542943 + 0.839770i $$0.317310\pi$$
$$228$$ 0 0
$$229$$ −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 $$-0.279985\pi$$
0.637461 + 0.770482i $$0.279985\pi$$
$$234$$ 0 0
$$235$$ −896086. −1.05847
$$236$$ −222785. −0.260379
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −853715. −0.966759 −0.483379 0.875411i $$-0.660591\pi$$
−0.483379 + 0.875411i $$0.660591\pi$$
$$240$$ 0 0
$$241$$ −388888. −0.431302 −0.215651 0.976470i $$-0.569187\pi$$
−0.215651 + 0.976470i $$0.569187\pi$$
$$242$$ −792164. −0.869515
$$243$$ 0 0
$$244$$ −24815.0 −0.0266833
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.17922e6 1.22986
$$248$$ −1.18781e6 −1.22636
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ −582151. −0.555182
$$257$$ −291986. −0.275759 −0.137879 0.990449i $$-0.544029\pi$$
−0.137879 + 0.990449i $$0.544029\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −163068. −0.149601
$$261$$ 0 0
$$262$$ 751752. 0.676584
$$263$$ 288495. 0.257187 0.128594 0.991697i $$-0.458954\pi$$
0.128594 + 0.991697i $$0.458954\pi$$
$$264$$ 0 0
$$265$$ 526169. 0.460268
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −95972.2 −0.0816222
$$269$$ 259370. 0.218544 0.109272 0.994012i $$-0.465148\pi$$
0.109272 + 0.994012i $$0.465148\pi$$
$$270$$ 0 0
$$271$$ −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
$$276$$ 0 0
$$277$$ 126991. 0.0994426 0.0497213 0.998763i $$-0.484167\pi$$
0.0497213 + 0.998763i $$0.484167\pi$$
$$278$$ −1.70098e6 −1.32004
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.22759e6 1.68294 0.841472 0.540301i $$-0.181690\pi$$
0.841472 + 0.540301i $$0.181690\pi$$
$$282$$ 0 0
$$283$$ 1.18895e6 0.882463 0.441231 0.897393i $$-0.354542\pi$$
0.441231 + 0.897393i $$0.354542\pi$$
$$284$$ 75100.7 0.0552520
$$285$$ 0 0
$$286$$ 231712. 0.167507
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.48442e6 1.74977
$$290$$ 155609. 0.108653
$$291$$ 0 0
$$292$$ −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
$$297$$ 0 0
$$298$$ −702800. −0.458449
$$299$$ 261596. 0.169221
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 565047. 0.356506
$$303$$ 0 0
$$304$$ 1.47085e6 0.912817
$$305$$ −168348. −0.103623
$$306$$ 0 0
$$307$$ −717638. −0.434569 −0.217285 0.976108i $$-0.569720\pi$$
−0.217285 + 0.976108i $$0.569720\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1.30179e6 −0.769370
$$311$$ 856892. 0.502372 0.251186 0.967939i $$-0.419179\pi$$
0.251186 + 0.967939i $$0.419179\pi$$
$$312$$ 0 0
$$313$$ 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 $$-0.281764\pi$$
0.633145 + 0.774033i $$0.281764\pi$$
$$318$$ 0 0
$$319$$ 52769.8 0.0290341
$$320$$ −1.52312e6 −0.831495
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.68495e6 −1.96528
$$324$$ 0 0
$$325$$ 869721. 0.456743
$$326$$ −1.08214e6 −0.563950
$$327$$ 0 0
$$328$$ −684740. −0.351432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 709650. 0.356020 0.178010 0.984029i $$-0.443034\pi$$
0.178010 + 0.984029i $$0.443034\pi$$
$$332$$ −151503. −0.0754355
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 509570. 0.239060
$$341$$ −441459. −0.205591
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.81576e6 1.28292
$$345$$ 0 0
$$346$$ −3.62357e6 −1.62722
$$347$$ −1.75731e6 −0.783474 −0.391737 0.920077i $$-0.628126\pi$$
−0.391737 + 0.920077i $$0.628126\pi$$
$$348$$ 0 0
$$349$$ 391875. 0.172220 0.0861102 0.996286i $$-0.472556\pi$$
0.0861102 + 0.996286i $$0.472556\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −158529. −0.0681948
$$353$$ −492407. −0.210323 −0.105162 0.994455i $$-0.533536\pi$$
−0.105162 + 0.994455i $$0.533536\pi$$
$$354$$ 0 0
$$355$$ 509492. 0.214569
$$356$$ −433087. −0.181113
$$357$$ 0 0
$$358$$ 3.81074e6 1.57145
$$359$$ 3.77032e6 1.54398 0.771991 0.635634i $$-0.219261\pi$$
0.771991 + 0.635634i $$0.219261\pi$$
$$360$$ 0 0
$$361$$ 1.00185e6 0.404607
$$362$$ 3.17021e6 1.27150
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −819367. −0.321919
$$366$$ 0 0
$$367$$ −2.19768e6 −0.851726 −0.425863 0.904788i $$-0.640030\pi$$
−0.425863 + 0.904788i $$0.640030\pi$$
$$368$$ 326289. 0.125598
$$369$$ 0 0
$$370$$ −2.20049e6 −0.835632
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.65636e6 −0.616427 −0.308213 0.951317i $$-0.599731\pi$$
−0.308213 + 0.951317i $$0.599731\pi$$
$$374$$ −724076. −0.267673
$$375$$ 0 0
$$376$$ −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
$$381$$ 0 0
$$382$$ −2.12320e6 −0.744446
$$383$$ −3.24845e6 −1.13156 −0.565781 0.824555i $$-0.691425\pi$$
−0.565781 + 0.824555i $$0.691425\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.91729e6 1.33819
$$387$$ 0 0
$$388$$ −652055. −0.219890
$$389$$ −4.65348e6 −1.55921 −0.779604 0.626273i $$-0.784580\pi$$
−0.779604 + 0.626273i $$0.784580\pi$$
$$390$$ 0 0
$$391$$ −817461. −0.270411
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2.43563e6 0.790442
$$395$$ −1.50955e6 −0.486805
$$396$$ 0 0
$$397$$ −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 $$-0.515941\pi$$
−0.0500584 + 0.998746i $$0.515941\pi$$
$$402$$ 0 0
$$403$$ −3.87179e6 −1.18754
$$404$$ 224806. 0.0685259
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −746226. −0.223298
$$408$$ 0 0
$$409$$ 1.38690e6 0.409956 0.204978 0.978767i $$-0.434288\pi$$
0.204978 + 0.978767i $$0.434288\pi$$
$$410$$ −750443. −0.220474
$$411$$ 0 0
$$412$$ −397800. −0.115457
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.02781e6 −0.292950
$$416$$ −1.39036e6 −0.393909
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ 2.44024e6 0.659202
$$425$$ −2.71779e6 −0.729865
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 407206. 0.107450
$$429$$ 0 0
$$430$$ 3.08594e6 0.804852
$$431$$ 5.22752e6 1.35551 0.677755 0.735288i $$-0.262953\pi$$
0.677755 + 0.735288i $$0.262953\pi$$
$$432$$ 0 0
$$433$$ −2.63022e6 −0.674174 −0.337087 0.941473i $$-0.609442\pi$$
−0.337087 + 0.941473i $$0.609442\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 233908. 0.0589289
$$437$$ 771539. 0.193266
$$438$$ 0 0
$$439$$ 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 $$-0.653840\pi$$
−0.464707 + 0.885465i $$0.653840\pi$$
$$444$$ 0 0
$$445$$ −2.93811e6 −0.703345
$$446$$ −1.84631e6 −0.439509
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1.49369e6 −0.349658 −0.174829 0.984599i $$-0.555937\pi$$
−0.174829 + 0.984599i $$0.555937\pi$$
$$450$$ 0 0
$$451$$ −254489. −0.0589152
$$452$$ 763307. 0.175733
$$453$$ 0 0
$$454$$ 4.28498e6 0.975684
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2.16221e6 −0.484293 −0.242146 0.970240i $$-0.577851\pi$$
−0.242146 + 0.970240i $$0.577851\pi$$
$$458$$ −2.89040e6 −0.643863
$$459$$ 0 0
$$460$$ −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
$$465$$ 0 0
$$466$$ 5.36999e6 1.14554
$$467$$ 5.47044e6 1.16073 0.580363 0.814358i $$-0.302911\pi$$
0.580363 + 0.814358i $$0.302911\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −4.55459e6 −0.951054
$$471$$ 0 0
$$472$$ −7.00949e6 −1.44821
$$473$$ 1.04650e6 0.215073
$$474$$ 0 0
$$475$$ 2.56511e6 0.521641
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −4.33923e6 −0.868646
$$479$$ 6.66289e6 1.32686 0.663428 0.748240i $$-0.269101\pi$$
0.663428 + 0.748240i $$0.269101\pi$$
$$480$$ 0 0
$$481$$ −6.54473e6 −1.28982
$$482$$ −1.97663e6 −0.387531
$$483$$ 0 0
$$484$$ 960916. 0.186454
$$485$$ −4.42362e6 −0.853931
$$486$$ 0 0
$$487$$ 9.53693e6 1.82216 0.911079 0.412232i $$-0.135251\pi$$
0.911079 + 0.412232i $$0.135251\pi$$
$$488$$ −780755. −0.148411
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8.19294e6 1.53369 0.766843 0.641835i $$-0.221827\pi$$
0.766843 + 0.641835i $$0.221827\pi$$
$$492$$ 0 0
$$493$$ −1.44625e6 −0.267994
$$494$$ 5.99372e6 1.10504
$$495$$ 0 0
$$496$$ −4.82928e6 −0.881411
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.31437e6 −0.775650 −0.387825 0.921733i $$-0.626774\pi$$
−0.387825 + 0.921733i $$0.626774\pi$$
$$500$$ −1.16062e6 −0.207618
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ −793649. −0.136449
$$509$$ 3.09396e6 0.529322 0.264661 0.964342i $$-0.414740\pi$$
0.264661 + 0.964342i $$0.414740\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −6.63004e6 −1.11774
$$513$$ 0 0
$$514$$ −1.48410e6 −0.247773
$$515$$ −2.69872e6 −0.448374
$$516$$ 0 0
$$517$$ −1.54455e6 −0.254141
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −5.13061e6 −0.832072
$$521$$ 7.60175e6 1.22693 0.613464 0.789723i $$-0.289776\pi$$
0.613464 + 0.789723i $$0.289776\pi$$
$$522$$ 0 0
$$523$$ 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
$$528$$ 0 0
$$529$$ −6.26519e6 −0.973408
$$530$$ 2.67439e6 0.413557
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.23198e6 −0.340308
$$534$$ 0 0
$$535$$ 2.76253e6 0.417275
$$536$$ −3.01958e6 −0.453978
$$537$$ 0 0
$$538$$ 1.31832e6 0.196365
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.10052e7 1.61661 0.808305 0.588764i $$-0.200385\pi$$
0.808305 + 0.588764i $$0.200385\pi$$
$$542$$ −1.11593e7 −1.63169
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 504032. 0.0710480
$$551$$ 1.36500e6 0.191538
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 645463. 0.0893505
$$555$$ 0 0
$$556$$ 2.06334e6 0.283063
$$557$$ 6.45222e6 0.881194 0.440597 0.897705i $$-0.354767\pi$$
0.440597 + 0.897705i $$0.354767\pi$$
$$558$$ 0 0
$$559$$ 9.17823e6 1.24231
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.13223e7 1.51215
$$563$$ −1.74748e6 −0.232349 −0.116175 0.993229i $$-0.537063\pi$$
−0.116175 + 0.993229i $$0.537063\pi$$
$$564$$ 0 0
$$565$$ 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 $$-0.510570\pi$$
−0.0331992 + 0.999449i $$0.510570\pi$$
$$570$$ 0 0
$$571$$ 5.22364e6 0.670475 0.335238 0.942134i $$-0.391183\pi$$
0.335238 + 0.942134i $$0.391183\pi$$
$$572$$ −281073. −0.0359194
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 569038. 0.0717748
$$576$$ 0 0
$$577$$ 6.63973e6 0.830254 0.415127 0.909763i $$-0.363737\pi$$
0.415127 + 0.909763i $$0.363737\pi$$
$$578$$ 1.26277e7 1.57219
$$579$$ 0 0
$$580$$ −188758. −0.0232989
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 906935. 0.110511
$$584$$ −3.80002e6 −0.461056
$$585$$ 0 0
$$586$$ −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
$$591$$ 0 0
$$592$$ −8.16324e6 −0.957322
$$593$$ 1.43756e7 1.67876 0.839379 0.543546i $$-0.182919\pi$$
0.839379 + 0.543546i $$0.182919\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 852515. 0.0983075
$$597$$ 0 0
$$598$$ 1.32963e6 0.152047
$$599$$ −1.20835e7 −1.37602 −0.688010 0.725701i $$-0.741516\pi$$
−0.688010 + 0.725701i $$0.741516\pi$$
$$600$$ 0 0
$$601$$ 5.75607e6 0.650040 0.325020 0.945707i $$-0.394629\pi$$
0.325020 + 0.945707i $$0.394629\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −685416. −0.0764473
$$605$$ 6.51897e6 0.724086
$$606$$ 0 0
$$607$$ 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
$$612$$ 0 0
$$613$$ −2.64543e6 −0.284344 −0.142172 0.989842i $$-0.545409\pi$$
−0.142172 + 0.989842i $$0.545409\pi$$
$$614$$ −3.64758e6 −0.390467
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.43533e6 −0.680546 −0.340273 0.940327i $$-0.610520\pi$$
−0.340273 + 0.940327i $$0.610520\pi$$
$$618$$ 0 0
$$619$$ 1.41177e7 1.48094 0.740469 0.672090i $$-0.234603\pi$$
0.740469 + 0.672090i $$0.234603\pi$$
$$620$$ 1.57910e6 0.164980
$$621$$ 0 0
$$622$$ 4.35538e6 0.451388
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −3.57548e6 −0.366129
$$626$$ 8.21877e6 0.838246
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 1.15155e7 1.13778
$$635$$ −5.38421e6 −0.529892
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 268217. 0.0260876
$$639$$ 0 0
$$640$$ −4.79855e6 −0.463085
$$641$$ 1.04174e7 1.00141 0.500707 0.865617i $$-0.333073\pi$$
0.500707 + 0.865617i $$0.333073\pi$$
$$642$$ 0 0
$$643$$ 1.27284e7 1.21407 0.607037 0.794674i $$-0.292358\pi$$
0.607037 + 0.794674i $$0.292358\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −1.87297e7 −1.76584
$$647$$ 1.61348e7 1.51531 0.757657 0.652653i $$-0.226344\pi$$
0.757657 + 0.652653i $$0.226344\pi$$
$$648$$ 0 0
$$649$$ −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 $$-0.257763\pi$$
0.689654 + 0.724139i $$0.257763\pi$$
$$654$$ 0 0
$$655$$ −6.18640e6 −0.563423
$$656$$ −2.78395e6 −0.252581
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1.67927e7 −1.50628 −0.753140 0.657860i $$-0.771462\pi$$
−0.753140 + 0.657860i $$0.771462\pi$$
$$660$$ 0 0
$$661$$ 1.08540e7 0.966246 0.483123 0.875552i $$-0.339502\pi$$
0.483123 + 0.875552i $$0.339502\pi$$
$$662$$ 3.60698e6 0.319889
$$663$$ 0 0
$$664$$ −4.76675e6 −0.419567
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 302809. 0.0263544
$$668$$ −742303. −0.0643636
$$669$$ 0 0
$$670$$ −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
$$675$$ 0 0
$$676$$ −175916. −0.0148060
$$677$$ 1.00501e6 0.0842746 0.0421373 0.999112i $$-0.486583\pi$$
0.0421373 + 0.999112i $$0.486583\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1.60326e7 1.32963
$$681$$ 0 0
$$682$$ −2.24383e6 −0.184727
$$683$$ −1.87019e6 −0.153403 −0.0767014 0.997054i $$-0.524439\pi$$
−0.0767014 + 0.997054i $$0.524439\pi$$
$$684$$ 0 0
$$685$$ −3.81290e6 −0.310476
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 1.14480e7 0.922060
$$689$$ 7.95421e6 0.638336
$$690$$ 0 0
$$691$$ 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
$$696$$ 0 0
$$697$$ 6.97469e6 0.543805
$$698$$ 1.99181e6 0.154742
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.17488e7 −0.903024 −0.451512 0.892265i $$-0.649115\pi$$
−0.451512 + 0.892265i $$0.649115\pi$$
$$702$$ 0 0
$$703$$ −1.93027e7 −1.47309
$$704$$ −2.62534e6 −0.199643
$$705$$ 0 0
$$706$$ −2.50279e6 −0.188978
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.67948e7 1.25475 0.627377 0.778716i $$-0.284129\pi$$
0.627377 + 0.778716i $$0.284129\pi$$
$$710$$ 2.58963e6 0.192793
$$711$$ 0 0
$$712$$ −1.36262e7 −1.00734
$$713$$ −2.53322e6 −0.186616
$$714$$ 0 0
$$715$$ −1.90683e6 −0.139491
$$716$$ −4.62253e6 −0.336974
$$717$$ 0 0
$$718$$ 1.91636e7 1.38729
$$719$$ 1.65130e7 1.19126 0.595628 0.803261i $$-0.296903\pi$$
0.595628 + 0.803261i $$0.296903\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 5.09215e6 0.363545
$$723$$ 0 0
$$724$$ −3.84555e6 −0.272654
$$725$$ 1.00674e6 0.0711331
$$726$$ 0 0
$$727$$ −1.25756e6 −0.0882453 −0.0441227 0.999026i $$-0.514049\pi$$
−0.0441227 + 0.999026i $$0.514049\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −4.16465e6 −0.289248
$$731$$ −2.86810e7 −1.98518
$$732$$ 0 0
$$733$$ −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
$$738$$ 0 0
$$739$$ −2.38807e7 −1.60856 −0.804278 0.594253i $$-0.797448\pi$$
−0.804278 + 0.594253i $$0.797448\pi$$
$$740$$ 2.66925e6 0.179189
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.90819e7 1.26809 0.634043 0.773298i $$-0.281394\pi$$
0.634043 + 0.773298i $$0.281394\pi$$
$$744$$ 0 0
$$745$$ 5.78356e6 0.381773
$$746$$ −8.41886e6 −0.553868
$$747$$ 0 0
$$748$$ 878323. 0.0573985
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −3.75805e6 −0.243144 −0.121572 0.992583i $$-0.538794\pi$$
−0.121572 + 0.992583i $$0.538794\pi$$
$$752$$ −1.68963e7 −1.08955
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ −1.51320e7 −0.950302
$$761$$ −2.23998e7 −1.40211 −0.701056 0.713106i $$-0.747288\pi$$
−0.701056 + 0.713106i $$0.747288\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 2.57550e6 0.159635
$$765$$ 0 0
$$766$$ −1.65111e7 −1.01672
$$767$$ −2.28481e7 −1.40237
$$768$$ 0 0
$$769$$ −1.87866e7 −1.14560 −0.572799 0.819696i $$-0.694142\pi$$
−0.572799 + 0.819696i $$0.694142\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.75177e6 −0.286954
$$773$$ −9.30837e6 −0.560306 −0.280153 0.959955i $$-0.590385\pi$$
−0.280153 + 0.959955i $$0.590385\pi$$
$$774$$ 0 0
$$775$$ −8.42212e6 −0.503694
$$776$$ −2.05156e7 −1.22301
$$777$$ 0 0
$$778$$ −2.36525e7 −1.40097
$$779$$ −6.58288e6 −0.388662
$$780$$ 0 0
$$781$$ 878189. 0.0515182
$$782$$ −4.15496e6 −0.242969
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.59566e6 0.0924201
$$786$$ 0 0
$$787$$ 1.73427e7 0.998111 0.499056 0.866570i $$-0.333680\pi$$
0.499056 + 0.866570i $$0.333680\pi$$
$$788$$ −2.95448e6 −0.169498
$$789$$ 0 0
$$790$$ −7.67268e6 −0.437401
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.54495e6 −0.143713
$$794$$ −5.91434e6 −0.332932
$$795$$ 0 0
$$796$$ 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
$$801$$ 0 0
$$802$$ −1.63858e6 −0.0899563
$$803$$ −1.41231e6 −0.0772930
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.96794e7 −1.06702
$$807$$ 0 0
$$808$$ 7.07309e6 0.381137
$$809$$ 2.47038e7 1.32707 0.663533 0.748147i $$-0.269056\pi$$
0.663533 + 0.748147i $$0.269056\pi$$
$$810$$ 0 0
$$811$$ 8.42005e6 0.449534 0.224767 0.974413i $$-0.427838\pi$$
0.224767 + 0.974413i $$0.427838\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −3.79289e6 −0.200636
$$815$$ 8.90529e6 0.469628
$$816$$ 0 0
$$817$$ 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 $$-0.266264\pi$$
0.670071 + 0.742297i $$0.266264\pi$$
$$822$$ 0 0
$$823$$ 1.72004e7 0.885195 0.442597 0.896720i $$-0.354057\pi$$
0.442597 + 0.896720i $$0.354057\pi$$
$$824$$ −1.25160e7 −0.642167
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.40337e7 1.22196 0.610979 0.791647i $$-0.290776\pi$$
0.610979 + 0.791647i $$0.290776\pi$$
$$828$$ 0 0
$$829$$ −3.24736e7 −1.64113 −0.820567 0.571550i $$-0.806342\pi$$
−0.820567 + 0.571550i $$0.806342\pi$$
$$830$$ −5.22413e6 −0.263220
$$831$$ 0 0
$$832$$ −2.30254e7 −1.15318
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −5.03587e6 −0.249953
$$836$$ −828983. −0.0410232
$$837$$ 0 0
$$838$$ −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
$$843$$ 0 0
$$844$$ 3.62868e6 0.175345
$$845$$ −1.19344e6 −0.0574986
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 9.92129e6 0.473782
$$849$$ 0 0
$$850$$ −1.38139e7 −0.655794
$$851$$ −4.28206e6 −0.202689
$$852$$ 0 0
$$853$$ −999355. −0.0470270 −0.0235135 0.999724i $$-0.507485\pi$$
−0.0235135 + 0.999724i $$0.507485\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 1.28119e7 0.597628
$$857$$ 2.64465e7 1.23003 0.615016 0.788514i $$-0.289149\pi$$
0.615016 + 0.788514i $$0.289149\pi$$
$$858$$ 0 0
$$859$$ −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 $$-0.529735\pi$$
−0.0932798 + 0.995640i $$0.529735\pi$$
$$864$$ 0 0
$$865$$ 2.98195e7 1.35506
$$866$$ −1.33688e7 −0.605755
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2.60195e6 −0.116882
$$870$$ 0 0
$$871$$ −9.84261e6 −0.439608
$$872$$ 7.35946e6 0.327759
$$873$$ 0 0
$$874$$ 3.92155e6 0.173652
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.82405e7 −1.23986 −0.619931 0.784657i $$-0.712839\pi$$
−0.619931 + 0.784657i $$0.712839\pi$$
$$878$$ 1.29820e7 0.568335
$$879$$ 0 0
$$880$$ −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
$$885$$ 0 0
$$886$$ −1.95127e7 −0.835091
$$887$$ −7.29088e6 −0.311151 −0.155575 0.987824i $$-0.549723\pi$$
−0.155575 + 0.987824i $$0.549723\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −1.49337e7 −0.631965
$$891$$ 0 0
$$892$$ 2.23962e6 0.0942460
$$893$$ −3.99529e7 −1.67656
$$894$$ 0 0
$$895$$ −3.13598e7 −1.30862
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −7.59205e6 −0.314173
$$899$$ −4.48176e6 −0.184948
$$900$$ 0 0
$$901$$ −2.48561e7 −1.02005
$$902$$ −1.29351e6 −0.0529361
$$903$$ 0 0
$$904$$ 2.40160e7 0.977415
$$905$$ −2.60886e7 −1.05884
$$906$$ 0 0
$$907$$ −3.73335e7 −1.50689 −0.753443 0.657514i $$-0.771608\pi$$
−0.753443 + 0.657514i $$0.771608\pi$$
$$908$$ −5.19779e6 −0.209221
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2475.17 9.88120e−5 0 4.94060e−5 1.00000i $$-0.499984\pi$$
4.94060e−5 1.00000i $$0.499984\pi$$
$$912$$ 0 0
$$913$$ −1.77160e6 −0.0703377
$$914$$ −1.09900e7 −0.435144
$$915$$ 0 0
$$916$$ 3.50613e6 0.138067
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.48238e6 −0.175073 −0.0875366 0.996161i $$-0.527899\pi$$
−0.0875366 + 0.996161i $$0.527899\pi$$
$$920$$ −3.35684e6 −0.130756
$$921$$ 0 0
$$922$$ 3.11039e7 1.20500
$$923$$ 7.70210e6 0.297581
$$924$$ 0 0
$$925$$ −1.42364e7 −0.547075
$$926$$ 2.00065e7 0.766733
$$927$$ 0 0
$$928$$ −1.60941e6 −0.0613473
$$929$$ −2.12859e7 −0.809193 −0.404596 0.914495i $$-0.632588\pi$$
−0.404596 + 0.914495i $$0.632588\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6.51394e6 −0.245643
$$933$$ 0 0
$$934$$ 2.78049e7 1.04293
$$935$$ 5.95865e6 0.222904
$$936$$ 0 0
$$937$$ 6.79757e6 0.252932 0.126466 0.991971i $$-0.459636\pi$$
0.126466 + 0.991971i $$0.459636\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 5.52484e6 0.203939
$$941$$ −4.90883e7 −1.80719 −0.903595 0.428388i $$-0.859082\pi$$
−0.903595 + 0.428388i $$0.859082\pi$$
$$942$$ 0 0
$$943$$ −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 $$-0.646708\pi$$
−0.444753 + 0.895653i $$0.646708\pi$$
$$948$$ 0 0
$$949$$ −1.23865e7 −0.446462
$$950$$ 1.30378e7 0.468702
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 513120. 0.0183015 0.00915075 0.999958i $$-0.497087\pi$$
0.00915075 + 0.999958i $$0.497087\pi$$
$$954$$ 0 0
$$955$$ 1.74725e7 0.619935
$$956$$ 5.26360e6 0.186268
$$957$$ 0 0
$$958$$ 3.38659e7 1.19220
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 8.86412e6 0.309619
$$962$$ −3.32653e7 −1.15892
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ −2.24842e7 −0.767269
$$971$$ −4.76036e6 −0.162029 −0.0810143 0.996713i $$-0.525816\pi$$
−0.0810143 + 0.996713i $$0.525816\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 4.84739e7 1.63723
$$975$$ 0 0
$$976$$ −3.17431e6 −0.106666
$$977$$ −2.87338e7 −0.963067 −0.481534 0.876428i $$-0.659920\pi$$
−0.481534 + 0.876428i $$0.659920\pi$$
$$978$$ 0 0
$$979$$ −5.06430e6 −0.168874
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 4.16428e7 1.37804
$$983$$ 4.97072e7 1.64072 0.820362 0.571845i $$-0.193772\pi$$
0.820362 + 0.571845i $$0.193772\pi$$
$$984$$ 0 0
$$985$$ −2.00435e7 −0.658239
$$986$$ −7.35092e6 −0.240796
$$987$$ 0 0
$$988$$ −7.27054e6 −0.236960
$$989$$ 6.00511e6 0.195223
$$990$$ 0 0
$$991$$ −2.91066e6 −0.0941471 −0.0470736 0.998891i $$-0.514990\pi$$
−0.0470736 + 0.998891i $$0.514990\pi$$
$$992$$ 1.34639e7 0.434401
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1.79309e7 0.574176
$$996$$ 0 0
$$997$$ 1.43353e7 0.456740 0.228370 0.973574i $$-0.426660\pi$$
0.228370 + 0.973574i $$0.426660\pi$$
$$998$$ −2.19289e7 −0.696933
$$999$$ 0 0
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 441.6.a.n.1.2 2
3.2 odd 2 49.6.a.d.1.1 2
7.2 even 3 63.6.e.d.46.1 4
7.4 even 3 63.6.e.d.37.1 4
7.6 odd 2 441.6.a.m.1.2 2
12.11 even 2 784.6.a.ba.1.1 2
21.2 odd 6 7.6.c.a.4.2 yes 4
21.5 even 6 49.6.c.f.18.2 4
21.11 odd 6 7.6.c.a.2.2 4
21.17 even 6 49.6.c.f.30.2 4
21.20 even 2 49.6.a.e.1.1 2
84.11 even 6 112.6.i.c.65.2 4
84.23 even 6 112.6.i.c.81.2 4
84.83 odd 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 21.11 odd 6
7.6.c.a.4.2 yes 4 21.2 odd 6
49.6.a.d.1.1 2 3.2 odd 2
49.6.a.e.1.1 2 21.20 even 2
49.6.c.f.18.2 4 21.5 even 6
49.6.c.f.30.2 4 21.17 even 6
63.6.e.d.37.1 4 7.4 even 3
63.6.e.d.46.1 4 7.2 even 3
112.6.i.c.65.2 4 84.11 even 6
112.6.i.c.81.2 4 84.23 even 6
441.6.a.m.1.2 2 7.6 odd 2
441.6.a.n.1.2 2 1.1 even 1 trivial
784.6.a.t.1.2 2 84.83 odd 2
784.6.a.ba.1.1 2 12.11 even 2