# Properties

 Label 147.6.a.l.1.1 Level $147$ Weight $6$ Character 147.1 Self dual yes Analytic conductor $23.576$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 97x^{2} + 7x + 294$$ x^4 - x^3 - 97*x^2 + 7*x + 294 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$10.1812$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-9.18123 q^{2} -9.00000 q^{3} +52.2950 q^{4} -22.0716 q^{5} +82.6311 q^{6} -186.333 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-9.18123 q^{2} -9.00000 q^{3} +52.2950 q^{4} -22.0716 q^{5} +82.6311 q^{6} -186.333 q^{8} +81.0000 q^{9} +202.644 q^{10} +416.710 q^{11} -470.655 q^{12} -797.918 q^{13} +198.644 q^{15} +37.3245 q^{16} +1375.55 q^{17} -743.680 q^{18} -2313.03 q^{19} -1154.23 q^{20} -3825.91 q^{22} -955.402 q^{23} +1676.99 q^{24} -2637.84 q^{25} +7325.87 q^{26} -729.000 q^{27} -7035.29 q^{29} -1823.80 q^{30} -1261.19 q^{31} +5619.96 q^{32} -3750.39 q^{33} -12629.2 q^{34} +4235.89 q^{36} +9776.44 q^{37} +21236.4 q^{38} +7181.26 q^{39} +4112.66 q^{40} +5400.95 q^{41} +19686.6 q^{43} +21791.8 q^{44} -1787.80 q^{45} +8771.76 q^{46} -2056.56 q^{47} -335.921 q^{48} +24218.7 q^{50} -12380.0 q^{51} -41727.1 q^{52} +18022.7 q^{53} +6693.12 q^{54} -9197.45 q^{55} +20817.2 q^{57} +64592.6 q^{58} -7435.68 q^{59} +10388.1 q^{60} -3495.38 q^{61} +11579.3 q^{62} -52792.5 q^{64} +17611.3 q^{65} +34433.2 q^{66} +15856.4 q^{67} +71934.4 q^{68} +8598.62 q^{69} +58133.5 q^{71} -15093.0 q^{72} -39110.7 q^{73} -89759.8 q^{74} +23740.6 q^{75} -120960. q^{76} -65932.8 q^{78} +9760.69 q^{79} -823.812 q^{80} +6561.00 q^{81} -49587.4 q^{82} +70395.7 q^{83} -30360.6 q^{85} -180747. q^{86} +63317.6 q^{87} -77646.6 q^{88} -144306. q^{89} +16414.2 q^{90} -49962.7 q^{92} +11350.7 q^{93} +18881.8 q^{94} +51052.2 q^{95} -50579.7 q^{96} +79328.7 q^{97} +33753.5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 - 36 * q^3 + 69 * q^4 - 27 * q^6 + 123 * q^8 + 324 * q^9 $$4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9} - 283 q^{10} + 402 q^{11} - 621 q^{12} - 462 q^{13} + 3273 q^{16} - 276 q^{17} + 243 q^{18} - 510 q^{19} - 4719 q^{20} + 1375 q^{22} + 6900 q^{23} - 1107 q^{24} + 2814 q^{25} + 15138 q^{26} - 2916 q^{27} + 540 q^{29} + 2547 q^{30} + 6410 q^{31} + 15519 q^{32} - 3618 q^{33} - 21144 q^{34} + 5589 q^{36} + 15250 q^{37} + 41250 q^{38} + 4158 q^{39} + 8547 q^{40} - 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} + 15060 q^{47} - 29457 q^{48} - 7302 q^{50} + 2484 q^{51} + 47476 q^{52} + 13692 q^{53} - 2187 q^{54} - 73124 q^{55} + 4590 q^{57} + 52309 q^{58} - 34830 q^{59} + 42471 q^{60} + 5364 q^{61} - 16029 q^{62} - 73487 q^{64} + 66864 q^{65} - 12375 q^{66} - 5994 q^{67} + 58272 q^{68} - 62100 q^{69} + 89268 q^{71} + 9963 q^{72} - 59638 q^{73} - 185442 q^{74} - 25326 q^{75} - 21308 q^{76} - 136242 q^{78} - 44062 q^{79} + 33381 q^{80} + 26244 q^{81} - 57596 q^{82} + 208446 q^{83} + 36324 q^{85} - 136968 q^{86} - 4860 q^{87} + 87597 q^{88} + 77520 q^{89} - 22923 q^{90} + 158256 q^{92} - 57690 q^{93} + 73722 q^{94} - 221376 q^{95} - 139671 q^{96} + 188630 q^{97} + 32562 q^{99}+O(q^{100})$$ 4 * q + 3 * q^2 - 36 * q^3 + 69 * q^4 - 27 * q^6 + 123 * q^8 + 324 * q^9 - 283 * q^10 + 402 * q^11 - 621 * q^12 - 462 * q^13 + 3273 * q^16 - 276 * q^17 + 243 * q^18 - 510 * q^19 - 4719 * q^20 + 1375 * q^22 + 6900 * q^23 - 1107 * q^24 + 2814 * q^25 + 15138 * q^26 - 2916 * q^27 + 540 * q^29 + 2547 * q^30 + 6410 * q^31 + 15519 * q^32 - 3618 * q^33 - 21144 * q^34 + 5589 * q^36 + 15250 * q^37 + 41250 * q^38 + 4158 * q^39 + 8547 * q^40 - 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 + 15060 * q^47 - 29457 * q^48 - 7302 * q^50 + 2484 * q^51 + 47476 * q^52 + 13692 * q^53 - 2187 * q^54 - 73124 * q^55 + 4590 * q^57 + 52309 * q^58 - 34830 * q^59 + 42471 * q^60 + 5364 * q^61 - 16029 * q^62 - 73487 * q^64 + 66864 * q^65 - 12375 * q^66 - 5994 * q^67 + 58272 * q^68 - 62100 * q^69 + 89268 * q^71 + 9963 * q^72 - 59638 * q^73 - 185442 * q^74 - 25326 * q^75 - 21308 * q^76 - 136242 * q^78 - 44062 * q^79 + 33381 * q^80 + 26244 * q^81 - 57596 * q^82 + 208446 * q^83 + 36324 * q^85 - 136968 * q^86 - 4860 * q^87 + 87597 * q^88 + 77520 * q^89 - 22923 * q^90 + 158256 * q^92 - 57690 * q^93 + 73722 * q^94 - 221376 * q^95 - 139671 * q^96 + 188630 * q^97 + 32562 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −9.18123 −1.62303 −0.811514 0.584333i $$-0.801356\pi$$
−0.811514 + 0.584333i $$0.801356\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 52.2950 1.63422
$$5$$ −22.0716 −0.394829 −0.197414 0.980320i $$-0.563254\pi$$
−0.197414 + 0.980320i $$0.563254\pi$$
$$6$$ 82.6311 0.937055
$$7$$ 0 0
$$8$$ −186.333 −1.02935
$$9$$ 81.0000 0.333333
$$10$$ 202.644 0.640818
$$11$$ 416.710 1.03837 0.519184 0.854662i $$-0.326236\pi$$
0.519184 + 0.854662i $$0.326236\pi$$
$$12$$ −470.655 −0.943516
$$13$$ −797.918 −1.30948 −0.654742 0.755853i $$-0.727223\pi$$
−0.654742 + 0.755853i $$0.727223\pi$$
$$14$$ 0 0
$$15$$ 198.644 0.227955
$$16$$ 37.3245 0.0364497
$$17$$ 1375.55 1.15439 0.577197 0.816605i $$-0.304146\pi$$
0.577197 + 0.816605i $$0.304146\pi$$
$$18$$ −743.680 −0.541009
$$19$$ −2313.03 −1.46993 −0.734965 0.678105i $$-0.762801\pi$$
−0.734965 + 0.678105i $$0.762801\pi$$
$$20$$ −1154.23 −0.645236
$$21$$ 0 0
$$22$$ −3825.91 −1.68530
$$23$$ −955.402 −0.376588 −0.188294 0.982113i $$-0.560296\pi$$
−0.188294 + 0.982113i $$0.560296\pi$$
$$24$$ 1676.99 0.594297
$$25$$ −2637.84 −0.844110
$$26$$ 7325.87 2.12533
$$27$$ −729.000 −0.192450
$$28$$ 0 0
$$29$$ −7035.29 −1.55341 −0.776707 0.629862i $$-0.783111\pi$$
−0.776707 + 0.629862i $$0.783111\pi$$
$$30$$ −1823.80 −0.369976
$$31$$ −1261.19 −0.235709 −0.117855 0.993031i $$-0.537602\pi$$
−0.117855 + 0.993031i $$0.537602\pi$$
$$32$$ 5619.96 0.970194
$$33$$ −3750.39 −0.599503
$$34$$ −12629.2 −1.87361
$$35$$ 0 0
$$36$$ 4235.89 0.544739
$$37$$ 9776.44 1.17402 0.587012 0.809579i $$-0.300304\pi$$
0.587012 + 0.809579i $$0.300304\pi$$
$$38$$ 21236.4 2.38574
$$39$$ 7181.26 0.756030
$$40$$ 4112.66 0.406418
$$41$$ 5400.95 0.501777 0.250888 0.968016i $$-0.419277\pi$$
0.250888 + 0.968016i $$0.419277\pi$$
$$42$$ 0 0
$$43$$ 19686.6 1.62367 0.811837 0.583885i $$-0.198468\pi$$
0.811837 + 0.583885i $$0.198468\pi$$
$$44$$ 21791.8 1.69692
$$45$$ −1787.80 −0.131610
$$46$$ 8771.76 0.611213
$$47$$ −2056.56 −0.135799 −0.0678997 0.997692i $$-0.521630\pi$$
−0.0678997 + 0.997692i $$0.521630\pi$$
$$48$$ −335.921 −0.0210443
$$49$$ 0 0
$$50$$ 24218.7 1.37001
$$51$$ −12380.0 −0.666490
$$52$$ −41727.1 −2.13998
$$53$$ 18022.7 0.881315 0.440658 0.897675i $$-0.354745\pi$$
0.440658 + 0.897675i $$0.354745\pi$$
$$54$$ 6693.12 0.312352
$$55$$ −9197.45 −0.409978
$$56$$ 0 0
$$57$$ 20817.2 0.848664
$$58$$ 64592.6 2.52123
$$59$$ −7435.68 −0.278093 −0.139047 0.990286i $$-0.544404\pi$$
−0.139047 + 0.990286i $$0.544404\pi$$
$$60$$ 10388.1 0.372527
$$61$$ −3495.38 −0.120274 −0.0601368 0.998190i $$-0.519154\pi$$
−0.0601368 + 0.998190i $$0.519154\pi$$
$$62$$ 11579.3 0.382563
$$63$$ 0 0
$$64$$ −52792.5 −1.61110
$$65$$ 17611.3 0.517022
$$66$$ 34433.2 0.973009
$$67$$ 15856.4 0.431537 0.215769 0.976445i $$-0.430774\pi$$
0.215769 + 0.976445i $$0.430774\pi$$
$$68$$ 71934.4 1.88653
$$69$$ 8598.62 0.217423
$$70$$ 0 0
$$71$$ 58133.5 1.36861 0.684306 0.729195i $$-0.260105\pi$$
0.684306 + 0.729195i $$0.260105\pi$$
$$72$$ −15093.0 −0.343118
$$73$$ −39110.7 −0.858990 −0.429495 0.903069i $$-0.641308\pi$$
−0.429495 + 0.903069i $$0.641308\pi$$
$$74$$ −89759.8 −1.90547
$$75$$ 23740.6 0.487347
$$76$$ −120960. −2.40218
$$77$$ 0 0
$$78$$ −65932.8 −1.22706
$$79$$ 9760.69 0.175960 0.0879798 0.996122i $$-0.471959\pi$$
0.0879798 + 0.996122i $$0.471959\pi$$
$$80$$ −823.812 −0.0143914
$$81$$ 6561.00 0.111111
$$82$$ −49587.4 −0.814397
$$83$$ 70395.7 1.12163 0.560816 0.827940i $$-0.310487\pi$$
0.560816 + 0.827940i $$0.310487\pi$$
$$84$$ 0 0
$$85$$ −30360.6 −0.455788
$$86$$ −180747. −2.63527
$$87$$ 63317.6 0.896864
$$88$$ −77646.6 −1.06885
$$89$$ −144306. −1.93112 −0.965562 0.260173i $$-0.916220\pi$$
−0.965562 + 0.260173i $$0.916220\pi$$
$$90$$ 16414.2 0.213606
$$91$$ 0 0
$$92$$ −49962.7 −0.615427
$$93$$ 11350.7 0.136087
$$94$$ 18881.8 0.220406
$$95$$ 51052.2 0.580371
$$96$$ −50579.7 −0.560142
$$97$$ 79328.7 0.856053 0.428027 0.903766i $$-0.359209\pi$$
0.428027 + 0.903766i $$0.359209\pi$$
$$98$$ 0 0
$$99$$ 33753.5 0.346123
$$100$$ −137946. −1.37946
$$101$$ 84833.7 0.827495 0.413747 0.910392i $$-0.364220\pi$$
0.413747 + 0.910392i $$0.364220\pi$$
$$102$$ 113663. 1.08173
$$103$$ −20332.3 −0.188839 −0.0944197 0.995532i $$-0.530100\pi$$
−0.0944197 + 0.995532i $$0.530100\pi$$
$$104$$ 148678. 1.34792
$$105$$ 0 0
$$106$$ −165471. −1.43040
$$107$$ 6962.19 0.0587877 0.0293938 0.999568i $$-0.490642\pi$$
0.0293938 + 0.999568i $$0.490642\pi$$
$$108$$ −38123.0 −0.314505
$$109$$ 112651. 0.908177 0.454088 0.890957i $$-0.349965\pi$$
0.454088 + 0.890957i $$0.349965\pi$$
$$110$$ 84443.9 0.665406
$$111$$ −87988.0 −0.677823
$$112$$ 0 0
$$113$$ 112005. 0.825167 0.412583 0.910920i $$-0.364627\pi$$
0.412583 + 0.910920i $$0.364627\pi$$
$$114$$ −191128. −1.37741
$$115$$ 21087.3 0.148688
$$116$$ −367910. −2.53862
$$117$$ −64631.4 −0.436494
$$118$$ 68268.7 0.451353
$$119$$ 0 0
$$120$$ −37014.0 −0.234646
$$121$$ 12595.8 0.0782102
$$122$$ 32091.9 0.195207
$$123$$ −48608.5 −0.289701
$$124$$ −65954.0 −0.385200
$$125$$ 127195. 0.728108
$$126$$ 0 0
$$127$$ 82224.5 0.452368 0.226184 0.974085i $$-0.427375\pi$$
0.226184 + 0.974085i $$0.427375\pi$$
$$128$$ 304862. 1.64467
$$129$$ −177179. −0.937428
$$130$$ −161694. −0.839140
$$131$$ 175812. 0.895097 0.447548 0.894260i $$-0.352297\pi$$
0.447548 + 0.894260i $$0.352297\pi$$
$$132$$ −196126. −0.979718
$$133$$ 0 0
$$134$$ −145581. −0.700397
$$135$$ 16090.2 0.0759849
$$136$$ −256310. −1.18828
$$137$$ 31330.3 0.142614 0.0713072 0.997454i $$-0.477283\pi$$
0.0713072 + 0.997454i $$0.477283\pi$$
$$138$$ −78945.9 −0.352884
$$139$$ −152234. −0.668305 −0.334152 0.942519i $$-0.608450\pi$$
−0.334152 + 0.942519i $$0.608450\pi$$
$$140$$ 0 0
$$141$$ 18509.1 0.0784038
$$142$$ −533737. −2.22129
$$143$$ −332500. −1.35973
$$144$$ 3023.29 0.0121499
$$145$$ 155280. 0.613332
$$146$$ 359084. 1.39416
$$147$$ 0 0
$$148$$ 511259. 1.91861
$$149$$ 362860. 1.33898 0.669489 0.742822i $$-0.266513\pi$$
0.669489 + 0.742822i $$0.266513\pi$$
$$150$$ −217968. −0.790978
$$151$$ 205126. 0.732112 0.366056 0.930593i $$-0.380708\pi$$
0.366056 + 0.930593i $$0.380708\pi$$
$$152$$ 430992. 1.51308
$$153$$ 111420. 0.384798
$$154$$ 0 0
$$155$$ 27836.5 0.0930648
$$156$$ 375544. 1.23552
$$157$$ 77272.0 0.250192 0.125096 0.992145i $$-0.460076\pi$$
0.125096 + 0.992145i $$0.460076\pi$$
$$158$$ −89615.1 −0.285587
$$159$$ −162205. −0.508828
$$160$$ −124042. −0.383060
$$161$$ 0 0
$$162$$ −60238.0 −0.180336
$$163$$ 184931. 0.545182 0.272591 0.962130i $$-0.412119\pi$$
0.272591 + 0.962130i $$0.412119\pi$$
$$164$$ 282442. 0.820012
$$165$$ 82777.0 0.236701
$$166$$ −646319. −1.82044
$$167$$ −129262. −0.358657 −0.179329 0.983789i $$-0.557393\pi$$
−0.179329 + 0.983789i $$0.557393\pi$$
$$168$$ 0 0
$$169$$ 265380. 0.714746
$$170$$ 278748. 0.739757
$$171$$ −187355. −0.489977
$$172$$ 1.02951e6 2.65344
$$173$$ 507867. 1.29013 0.645067 0.764126i $$-0.276829\pi$$
0.645067 + 0.764126i $$0.276829\pi$$
$$174$$ −581334. −1.45563
$$175$$ 0 0
$$176$$ 15553.5 0.0378483
$$177$$ 66921.1 0.160557
$$178$$ 1.32491e6 3.13427
$$179$$ −132589. −0.309296 −0.154648 0.987970i $$-0.549424\pi$$
−0.154648 + 0.987970i $$0.549424\pi$$
$$180$$ −93492.9 −0.215079
$$181$$ 740060. 1.67908 0.839538 0.543301i $$-0.182826\pi$$
0.839538 + 0.543301i $$0.182826\pi$$
$$182$$ 0 0
$$183$$ 31458.5 0.0694400
$$184$$ 178023. 0.387642
$$185$$ −215782. −0.463538
$$186$$ −104214. −0.220873
$$187$$ 573205. 1.19869
$$188$$ −107548. −0.221926
$$189$$ 0 0
$$190$$ −468722. −0.941957
$$191$$ −582732. −1.15581 −0.577904 0.816105i $$-0.696129\pi$$
−0.577904 + 0.816105i $$0.696129\pi$$
$$192$$ 475133. 0.930169
$$193$$ −400904. −0.774725 −0.387362 0.921928i $$-0.626614\pi$$
−0.387362 + 0.921928i $$0.626614\pi$$
$$194$$ −728335. −1.38940
$$195$$ −158502. −0.298503
$$196$$ 0 0
$$197$$ 671589. 1.23293 0.616464 0.787383i $$-0.288564\pi$$
0.616464 + 0.787383i $$0.288564\pi$$
$$198$$ −309898. −0.561767
$$199$$ 455022. 0.814515 0.407258 0.913313i $$-0.366485\pi$$
0.407258 + 0.913313i $$0.366485\pi$$
$$200$$ 491517. 0.868887
$$201$$ −142708. −0.249148
$$202$$ −778878. −1.34305
$$203$$ 0 0
$$204$$ −647409. −1.08919
$$205$$ −119208. −0.198116
$$206$$ 186675. 0.306492
$$207$$ −77387.5 −0.125529
$$208$$ −29781.9 −0.0477303
$$209$$ −963860. −1.52633
$$210$$ 0 0
$$211$$ −1.19545e6 −1.84852 −0.924260 0.381764i $$-0.875317\pi$$
−0.924260 + 0.381764i $$0.875317\pi$$
$$212$$ 942499. 1.44026
$$213$$ −523201. −0.790168
$$214$$ −63921.4 −0.0954140
$$215$$ −434514. −0.641073
$$216$$ 135837. 0.198099
$$217$$ 0 0
$$218$$ −1.03428e6 −1.47400
$$219$$ 351996. 0.495938
$$220$$ −480980. −0.669993
$$221$$ −1.09758e6 −1.51166
$$222$$ 807838. 1.10012
$$223$$ −296529. −0.399305 −0.199653 0.979867i $$-0.563981\pi$$
−0.199653 + 0.979867i $$0.563981\pi$$
$$224$$ 0 0
$$225$$ −213665. −0.281370
$$226$$ −1.02834e6 −1.33927
$$227$$ −218146. −0.280985 −0.140492 0.990082i $$-0.544869\pi$$
−0.140492 + 0.990082i $$0.544869\pi$$
$$228$$ 1.08864e6 1.38690
$$229$$ 1.22920e6 1.54894 0.774471 0.632609i $$-0.218016\pi$$
0.774471 + 0.632609i $$0.218016\pi$$
$$230$$ −193607. −0.241324
$$231$$ 0 0
$$232$$ 1.31090e6 1.59901
$$233$$ −62944.2 −0.0759567 −0.0379784 0.999279i $$-0.512092\pi$$
−0.0379784 + 0.999279i $$0.512092\pi$$
$$234$$ 593395. 0.708442
$$235$$ 45391.7 0.0536175
$$236$$ −388849. −0.454465
$$237$$ −87846.2 −0.101590
$$238$$ 0 0
$$239$$ 219330. 0.248372 0.124186 0.992259i $$-0.460368\pi$$
0.124186 + 0.992259i $$0.460368\pi$$
$$240$$ 7414.31 0.00830888
$$241$$ −433864. −0.481183 −0.240592 0.970626i $$-0.577341\pi$$
−0.240592 + 0.970626i $$0.577341\pi$$
$$242$$ −115645. −0.126937
$$243$$ −59049.0 −0.0641500
$$244$$ −182791. −0.196553
$$245$$ 0 0
$$246$$ 446286. 0.470193
$$247$$ 1.84561e6 1.92485
$$248$$ 235001. 0.242628
$$249$$ −633561. −0.647575
$$250$$ −1.16781e6 −1.18174
$$251$$ 1.71109e6 1.71431 0.857155 0.515059i $$-0.172230\pi$$
0.857155 + 0.515059i $$0.172230\pi$$
$$252$$ 0 0
$$253$$ −398125. −0.391037
$$254$$ −754922. −0.734206
$$255$$ 273246. 0.263150
$$256$$ −1.10964e6 −1.05824
$$257$$ −984057. −0.929368 −0.464684 0.885477i $$-0.653832\pi$$
−0.464684 + 0.885477i $$0.653832\pi$$
$$258$$ 1.62672e6 1.52147
$$259$$ 0 0
$$260$$ 920984. 0.844926
$$261$$ −569859. −0.517804
$$262$$ −1.61417e6 −1.45277
$$263$$ 547095. 0.487724 0.243862 0.969810i $$-0.421586\pi$$
0.243862 + 0.969810i $$0.421586\pi$$
$$264$$ 698820. 0.617100
$$265$$ −397791. −0.347969
$$266$$ 0 0
$$267$$ 1.29876e6 1.11493
$$268$$ 829211. 0.705226
$$269$$ 641112. 0.540198 0.270099 0.962833i $$-0.412944\pi$$
0.270099 + 0.962833i $$0.412944\pi$$
$$270$$ −147728. −0.123325
$$271$$ 548012. 0.453280 0.226640 0.973979i $$-0.427226\pi$$
0.226640 + 0.973979i $$0.427226\pi$$
$$272$$ 51341.8 0.0420774
$$273$$ 0 0
$$274$$ −287651. −0.231467
$$275$$ −1.09921e6 −0.876498
$$276$$ 449664. 0.355317
$$277$$ 1.67414e6 1.31097 0.655485 0.755208i $$-0.272464\pi$$
0.655485 + 0.755208i $$0.272464\pi$$
$$278$$ 1.39769e6 1.08468
$$279$$ −102157. −0.0785698
$$280$$ 0 0
$$281$$ 1.81078e6 1.36804 0.684021 0.729462i $$-0.260230\pi$$
0.684021 + 0.729462i $$0.260230\pi$$
$$282$$ −169936. −0.127252
$$283$$ −2.51315e6 −1.86531 −0.932657 0.360764i $$-0.882516\pi$$
−0.932657 + 0.360764i $$0.882516\pi$$
$$284$$ 3.04009e6 2.23661
$$285$$ −459470. −0.335077
$$286$$ 3.05276e6 2.20687
$$287$$ 0 0
$$288$$ 455217. 0.323398
$$289$$ 472283. 0.332627
$$290$$ −1.42566e6 −0.995455
$$291$$ −713958. −0.494243
$$292$$ −2.04529e6 −1.40378
$$293$$ −107228. −0.0729691 −0.0364845 0.999334i $$-0.511616\pi$$
−0.0364845 + 0.999334i $$0.511616\pi$$
$$294$$ 0 0
$$295$$ 164117. 0.109799
$$296$$ −1.82167e6 −1.20848
$$297$$ −303781. −0.199834
$$298$$ −3.33150e6 −2.17320
$$299$$ 762332. 0.493136
$$300$$ 1.24151e6 0.796431
$$301$$ 0 0
$$302$$ −1.88330e6 −1.18824
$$303$$ −763504. −0.477754
$$304$$ −86332.6 −0.0535785
$$305$$ 77148.7 0.0474875
$$306$$ −1.02297e6 −0.624538
$$307$$ −1.49622e6 −0.906042 −0.453021 0.891500i $$-0.649654\pi$$
−0.453021 + 0.891500i $$0.649654\pi$$
$$308$$ 0 0
$$309$$ 182990. 0.109027
$$310$$ −255573. −0.151047
$$311$$ −861202. −0.504899 −0.252449 0.967610i $$-0.581236\pi$$
−0.252449 + 0.967610i $$0.581236\pi$$
$$312$$ −1.33810e6 −0.778222
$$313$$ 503937. 0.290747 0.145374 0.989377i $$-0.453562\pi$$
0.145374 + 0.989377i $$0.453562\pi$$
$$314$$ −709452. −0.406068
$$315$$ 0 0
$$316$$ 510435. 0.287556
$$317$$ −480009. −0.268288 −0.134144 0.990962i $$-0.542828\pi$$
−0.134144 + 0.990962i $$0.542828\pi$$
$$318$$ 1.48924e6 0.825841
$$319$$ −2.93167e6 −1.61302
$$320$$ 1.16522e6 0.636109
$$321$$ −62659.7 −0.0339411
$$322$$ 0 0
$$323$$ −3.18168e6 −1.69688
$$324$$ 343107. 0.181580
$$325$$ 2.10478e6 1.10535
$$326$$ −1.69790e6 −0.884845
$$327$$ −1.01386e6 −0.524336
$$328$$ −1.00637e6 −0.516505
$$329$$ 0 0
$$330$$ −759995. −0.384172
$$331$$ −2.19922e6 −1.10331 −0.551656 0.834072i $$-0.686004\pi$$
−0.551656 + 0.834072i $$0.686004\pi$$
$$332$$ 3.68134e6 1.83299
$$333$$ 791892. 0.391341
$$334$$ 1.18678e6 0.582110
$$335$$ −349977. −0.170383
$$336$$ 0 0
$$337$$ −1.35725e6 −0.651008 −0.325504 0.945541i $$-0.605534\pi$$
−0.325504 + 0.945541i $$0.605534\pi$$
$$338$$ −2.43652e6 −1.16005
$$339$$ −1.00805e6 −0.476410
$$340$$ −1.58771e6 −0.744857
$$341$$ −525550. −0.244753
$$342$$ 1.72015e6 0.795245
$$343$$ 0 0
$$344$$ −3.66825e6 −1.67133
$$345$$ −189785. −0.0858449
$$346$$ −4.66285e6 −2.09392
$$347$$ 3.79941e6 1.69392 0.846959 0.531659i $$-0.178431\pi$$
0.846959 + 0.531659i $$0.178431\pi$$
$$348$$ 3.31119e6 1.46567
$$349$$ 1.31753e6 0.579024 0.289512 0.957174i $$-0.406507\pi$$
0.289512 + 0.957174i $$0.406507\pi$$
$$350$$ 0 0
$$351$$ 581682. 0.252010
$$352$$ 2.34189e6 1.00742
$$353$$ 3.42505e6 1.46295 0.731477 0.681866i $$-0.238832\pi$$
0.731477 + 0.681866i $$0.238832\pi$$
$$354$$ −614418. −0.260589
$$355$$ −1.28310e6 −0.540367
$$356$$ −7.54649e6 −3.15588
$$357$$ 0 0
$$358$$ 1.21733e6 0.501997
$$359$$ 1.47084e6 0.602322 0.301161 0.953573i $$-0.402626\pi$$
0.301161 + 0.953573i $$0.402626\pi$$
$$360$$ 333126. 0.135473
$$361$$ 2.87399e6 1.16069
$$362$$ −6.79466e6 −2.72519
$$363$$ −113362. −0.0451547
$$364$$ 0 0
$$365$$ 863235. 0.339154
$$366$$ −288827. −0.112703
$$367$$ 4.98079e6 1.93034 0.965168 0.261630i $$-0.0842601\pi$$
0.965168 + 0.261630i $$0.0842601\pi$$
$$368$$ −35659.9 −0.0137265
$$369$$ 437477. 0.167259
$$370$$ 1.98114e6 0.752335
$$371$$ 0 0
$$372$$ 593586. 0.222396
$$373$$ 3.96047e6 1.47392 0.736962 0.675934i $$-0.236260\pi$$
0.736962 + 0.675934i $$0.236260\pi$$
$$374$$ −5.26273e6 −1.94550
$$375$$ −1.14476e6 −0.420373
$$376$$ 383205. 0.139785
$$377$$ 5.61359e6 2.03417
$$378$$ 0 0
$$379$$ −1.75155e6 −0.626359 −0.313179 0.949694i $$-0.601394\pi$$
−0.313179 + 0.949694i $$0.601394\pi$$
$$380$$ 2.66977e6 0.948452
$$381$$ −740020. −0.261175
$$382$$ 5.35020e6 1.87591
$$383$$ −3.13834e6 −1.09321 −0.546604 0.837391i $$-0.684080\pi$$
−0.546604 + 0.837391i $$0.684080\pi$$
$$384$$ −2.74375e6 −0.949549
$$385$$ 0 0
$$386$$ 3.68079e6 1.25740
$$387$$ 1.59461e6 0.541224
$$388$$ 4.14849e6 1.39898
$$389$$ −1.05252e6 −0.352661 −0.176331 0.984331i $$-0.556423\pi$$
−0.176331 + 0.984331i $$0.556423\pi$$
$$390$$ 1.45524e6 0.484478
$$391$$ −1.31420e6 −0.434731
$$392$$ 0 0
$$393$$ −1.58231e6 −0.516784
$$394$$ −6.16601e6 −2.00108
$$395$$ −215434. −0.0694739
$$396$$ 1.76514e6 0.565640
$$397$$ −454724. −0.144801 −0.0724005 0.997376i $$-0.523066\pi$$
−0.0724005 + 0.997376i $$0.523066\pi$$
$$398$$ −4.17766e6 −1.32198
$$399$$ 0 0
$$400$$ −98456.3 −0.0307676
$$401$$ −2.88431e6 −0.895739 −0.447870 0.894099i $$-0.647817\pi$$
−0.447870 + 0.894099i $$0.647817\pi$$
$$402$$ 1.31023e6 0.404374
$$403$$ 1.00633e6 0.308657
$$404$$ 4.43638e6 1.35231
$$405$$ −144812. −0.0438699
$$406$$ 0 0
$$407$$ 4.07394e6 1.21907
$$408$$ 2.30679e6 0.686053
$$409$$ 225914. 0.0667782 0.0333891 0.999442i $$-0.489370\pi$$
0.0333891 + 0.999442i $$0.489370\pi$$
$$410$$ 1.09447e6 0.321548
$$411$$ −281973. −0.0823385
$$412$$ −1.06328e6 −0.308605
$$413$$ 0 0
$$414$$ 710513. 0.203738
$$415$$ −1.55375e6 −0.442853
$$416$$ −4.48427e6 −1.27045
$$417$$ 1.37011e6 0.385846
$$418$$ 8.84942e6 2.47727
$$419$$ −4.31027e6 −1.19941 −0.599707 0.800220i $$-0.704716\pi$$
−0.599707 + 0.800220i $$0.704716\pi$$
$$420$$ 0 0
$$421$$ 1.25088e6 0.343962 0.171981 0.985100i $$-0.444983\pi$$
0.171981 + 0.985100i $$0.444983\pi$$
$$422$$ 1.09757e7 3.00020
$$423$$ −166582. −0.0452665
$$424$$ −3.35823e6 −0.907184
$$425$$ −3.62849e6 −0.974436
$$426$$ 4.80363e6 1.28246
$$427$$ 0 0
$$428$$ 364087. 0.0960719
$$429$$ 2.99250e6 0.785039
$$430$$ 3.98937e6 1.04048
$$431$$ −4.40793e6 −1.14299 −0.571494 0.820606i $$-0.693636\pi$$
−0.571494 + 0.820606i $$0.693636\pi$$
$$432$$ −27209.6 −0.00701475
$$433$$ 1.60951e6 0.412549 0.206274 0.978494i $$-0.433866\pi$$
0.206274 + 0.978494i $$0.433866\pi$$
$$434$$ 0 0
$$435$$ −1.39752e6 −0.354108
$$436$$ 5.89110e6 1.48416
$$437$$ 2.20987e6 0.553558
$$438$$ −3.23176e6 −0.804921
$$439$$ 4.42452e6 1.09573 0.547867 0.836566i $$-0.315440\pi$$
0.547867 + 0.836566i $$0.315440\pi$$
$$440$$ 1.71379e6 0.422012
$$441$$ 0 0
$$442$$ 1.00771e7 2.45347
$$443$$ −3.61438e6 −0.875034 −0.437517 0.899210i $$-0.644142\pi$$
−0.437517 + 0.899210i $$0.644142\pi$$
$$444$$ −4.60133e6 −1.10771
$$445$$ 3.18507e6 0.762464
$$446$$ 2.72250e6 0.648084
$$447$$ −3.26574e6 −0.773060
$$448$$ 0 0
$$449$$ −467024. −0.109326 −0.0546630 0.998505i $$-0.517408\pi$$
−0.0546630 + 0.998505i $$0.517408\pi$$
$$450$$ 1.96171e6 0.456671
$$451$$ 2.25063e6 0.521029
$$452$$ 5.85730e6 1.34850
$$453$$ −1.84613e6 −0.422685
$$454$$ 2.00285e6 0.456046
$$455$$ 0 0
$$456$$ −3.87893e6 −0.873575
$$457$$ −601252. −0.134668 −0.0673342 0.997730i $$-0.521449\pi$$
−0.0673342 + 0.997730i $$0.521449\pi$$
$$458$$ −1.12856e7 −2.51398
$$459$$ −1.00278e6 −0.222163
$$460$$ 1.10276e6 0.242988
$$461$$ −2.87193e6 −0.629392 −0.314696 0.949192i $$-0.601903\pi$$
−0.314696 + 0.949192i $$0.601903\pi$$
$$462$$ 0 0
$$463$$ −2.91502e6 −0.631959 −0.315979 0.948766i $$-0.602333\pi$$
−0.315979 + 0.948766i $$0.602333\pi$$
$$464$$ −262589. −0.0566215
$$465$$ −250529. −0.0537310
$$466$$ 577906. 0.123280
$$467$$ 7.19376e6 1.52638 0.763192 0.646172i $$-0.223631\pi$$
0.763192 + 0.646172i $$0.223631\pi$$
$$468$$ −3.37989e6 −0.713327
$$469$$ 0 0
$$470$$ −416751. −0.0870227
$$471$$ −695448. −0.144448
$$472$$ 1.38551e6 0.286256
$$473$$ 8.20358e6 1.68597
$$474$$ 806536. 0.164884
$$475$$ 6.10140e6 1.24078
$$476$$ 0 0
$$477$$ 1.45984e6 0.293772
$$478$$ −2.01372e6 −0.403115
$$479$$ 2.79649e6 0.556896 0.278448 0.960451i $$-0.410180\pi$$
0.278448 + 0.960451i $$0.410180\pi$$
$$480$$ 1.11637e6 0.221160
$$481$$ −7.80080e6 −1.53736
$$482$$ 3.98340e6 0.780974
$$483$$ 0 0
$$484$$ 658698. 0.127812
$$485$$ −1.75091e6 −0.337995
$$486$$ 542142. 0.104117
$$487$$ −2.93247e6 −0.560289 −0.280144 0.959958i $$-0.590382\pi$$
−0.280144 + 0.959958i $$0.590382\pi$$
$$488$$ 651305. 0.123804
$$489$$ −1.66438e6 −0.314761
$$490$$ 0 0
$$491$$ −3.06121e6 −0.573046 −0.286523 0.958073i $$-0.592499\pi$$
−0.286523 + 0.958073i $$0.592499\pi$$
$$492$$ −2.54198e6 −0.473434
$$493$$ −9.67740e6 −1.79325
$$494$$ −1.69449e7 −3.12408
$$495$$ −744993. −0.136659
$$496$$ −47073.4 −0.00859154
$$497$$ 0 0
$$498$$ 5.81687e6 1.05103
$$499$$ −6.55154e6 −1.17786 −0.588928 0.808186i $$-0.700450\pi$$
−0.588928 + 0.808186i $$0.700450\pi$$
$$500$$ 6.65167e6 1.18989
$$501$$ 1.16336e6 0.207071
$$502$$ −1.57099e7 −2.78237
$$503$$ 1.58524e6 0.279367 0.139684 0.990196i $$-0.455391\pi$$
0.139684 + 0.990196i $$0.455391\pi$$
$$504$$ 0 0
$$505$$ −1.87242e6 −0.326719
$$506$$ 3.65528e6 0.634664
$$507$$ −2.38842e6 −0.412659
$$508$$ 4.29993e6 0.739268
$$509$$ −6.77981e6 −1.15991 −0.579953 0.814650i $$-0.696929\pi$$
−0.579953 + 0.814650i $$0.696929\pi$$
$$510$$ −2.50873e6 −0.427099
$$511$$ 0 0
$$512$$ 432315. 0.0728829
$$513$$ 1.68620e6 0.282888
$$514$$ 9.03486e6 1.50839
$$515$$ 448766. 0.0745593
$$516$$ −9.26557e6 −1.53196
$$517$$ −856990. −0.141010
$$518$$ 0 0
$$519$$ −4.57081e6 −0.744859
$$520$$ −3.28157e6 −0.532198
$$521$$ 1.01384e7 1.63634 0.818170 0.574977i $$-0.194989\pi$$
0.818170 + 0.574977i $$0.194989\pi$$
$$522$$ 5.23200e6 0.840411
$$523$$ −99997.3 −0.0159858 −0.00799289 0.999968i $$-0.502544\pi$$
−0.00799289 + 0.999968i $$0.502544\pi$$
$$524$$ 9.19408e6 1.46278
$$525$$ 0 0
$$526$$ −5.02301e6 −0.791589
$$527$$ −1.73483e6 −0.272102
$$528$$ −139981. −0.0218517
$$529$$ −5.52355e6 −0.858181
$$530$$ 3.65221e6 0.564763
$$531$$ −602290. −0.0926978
$$532$$ 0 0
$$533$$ −4.30952e6 −0.657068
$$534$$ −1.19242e7 −1.80957
$$535$$ −153667. −0.0232111
$$536$$ −2.95457e6 −0.444204
$$537$$ 1.19330e6 0.178572
$$538$$ −5.88620e6 −0.876756
$$539$$ 0 0
$$540$$ 841436. 0.124176
$$541$$ −119743. −0.0175896 −0.00879481 0.999961i $$-0.502800\pi$$
−0.00879481 + 0.999961i $$0.502800\pi$$
$$542$$ −5.03143e6 −0.735687
$$543$$ −6.66054e6 −0.969415
$$544$$ 7.73054e6 1.11999
$$545$$ −2.48640e6 −0.358574
$$546$$ 0 0
$$547$$ 236568. 0.0338056 0.0169028 0.999857i $$-0.494619\pi$$
0.0169028 + 0.999857i $$0.494619\pi$$
$$548$$ 1.63842e6 0.233063
$$549$$ −283126. −0.0400912
$$550$$ 1.00921e7 1.42258
$$551$$ 1.62728e7 2.28341
$$552$$ −1.60220e6 −0.223805
$$553$$ 0 0
$$554$$ −1.53707e7 −2.12774
$$555$$ 1.94204e6 0.267624
$$556$$ −7.96107e6 −1.09216
$$557$$ −4.83666e6 −0.660553 −0.330277 0.943884i $$-0.607142\pi$$
−0.330277 + 0.943884i $$0.607142\pi$$
$$558$$ 937922. 0.127521
$$559$$ −1.57083e7 −2.12617
$$560$$ 0 0
$$561$$ −5.15885e6 −0.692063
$$562$$ −1.66252e7 −2.22037
$$563$$ 5.48295e6 0.729026 0.364513 0.931198i $$-0.381235\pi$$
0.364513 + 0.931198i $$0.381235\pi$$
$$564$$ 967932. 0.128129
$$565$$ −2.47213e6 −0.325800
$$566$$ 2.30738e7 3.02746
$$567$$ 0 0
$$568$$ −1.08322e7 −1.40878
$$569$$ −6.97660e6 −0.903364 −0.451682 0.892179i $$-0.649176\pi$$
−0.451682 + 0.892179i $$0.649176\pi$$
$$570$$ 4.21850e6 0.543839
$$571$$ −1.17715e7 −1.51092 −0.755458 0.655197i $$-0.772585\pi$$
−0.755458 + 0.655197i $$0.772585\pi$$
$$572$$ −1.73881e7 −2.22209
$$573$$ 5.24459e6 0.667306
$$574$$ 0 0
$$575$$ 2.52020e6 0.317882
$$576$$ −4.27620e6 −0.537034
$$577$$ −6.77292e6 −0.846909 −0.423454 0.905917i $$-0.639183\pi$$
−0.423454 + 0.905917i $$0.639183\pi$$
$$578$$ −4.33614e6 −0.539863
$$579$$ 3.60814e6 0.447288
$$580$$ 8.12037e6 1.00232
$$581$$ 0 0
$$582$$ 6.55501e6 0.802169
$$583$$ 7.51025e6 0.915130
$$584$$ 7.28760e6 0.884203
$$585$$ 1.42652e6 0.172341
$$586$$ 984484. 0.118431
$$587$$ 1.05020e7 1.25799 0.628996 0.777408i $$-0.283466\pi$$
0.628996 + 0.777408i $$0.283466\pi$$
$$588$$ 0 0
$$589$$ 2.91717e6 0.346476
$$590$$ −1.50680e6 −0.178207
$$591$$ −6.04430e6 −0.711832
$$592$$ 364901. 0.0427928
$$593$$ 7.59074e6 0.886436 0.443218 0.896414i $$-0.353837\pi$$
0.443218 + 0.896414i $$0.353837\pi$$
$$594$$ 2.78909e6 0.324336
$$595$$ 0 0
$$596$$ 1.89758e7 2.18818
$$597$$ −4.09519e6 −0.470261
$$598$$ −6.99915e6 −0.800373
$$599$$ −1.30198e7 −1.48265 −0.741325 0.671146i $$-0.765802\pi$$
−0.741325 + 0.671146i $$0.765802\pi$$
$$600$$ −4.42365e6 −0.501652
$$601$$ −1.41821e7 −1.60160 −0.800801 0.598931i $$-0.795592\pi$$
−0.800801 + 0.598931i $$0.795592\pi$$
$$602$$ 0 0
$$603$$ 1.28437e6 0.143846
$$604$$ 1.07270e7 1.19643
$$605$$ −278010. −0.0308796
$$606$$ 7.00990e6 0.775408
$$607$$ −1.20772e7 −1.33044 −0.665221 0.746646i $$-0.731663\pi$$
−0.665221 + 0.746646i $$0.731663\pi$$
$$608$$ −1.29991e7 −1.42612
$$609$$ 0 0
$$610$$ −708320. −0.0770735
$$611$$ 1.64097e6 0.177827
$$612$$ 5.82668e6 0.628844
$$613$$ −2.72530e6 −0.292929 −0.146465 0.989216i $$-0.546789\pi$$
−0.146465 + 0.989216i $$0.546789\pi$$
$$614$$ 1.37371e7 1.47053
$$615$$ 1.07287e6 0.114382
$$616$$ 0 0
$$617$$ 8.47094e6 0.895816 0.447908 0.894080i $$-0.352169\pi$$
0.447908 + 0.894080i $$0.352169\pi$$
$$618$$ −1.68008e6 −0.176953
$$619$$ −1.73835e7 −1.82352 −0.911759 0.410726i $$-0.865275\pi$$
−0.911759 + 0.410726i $$0.865275\pi$$
$$620$$ 1.45571e6 0.152088
$$621$$ 696488. 0.0724744
$$622$$ 7.90690e6 0.819464
$$623$$ 0 0
$$624$$ 268037. 0.0275571
$$625$$ 5.43586e6 0.556632
$$626$$ −4.62676e6 −0.471891
$$627$$ 8.67474e6 0.881227
$$628$$ 4.04093e6 0.408868
$$629$$ 1.34480e7 1.35529
$$630$$ 0 0
$$631$$ −6.45149e6 −0.645040 −0.322520 0.946563i $$-0.604530\pi$$
−0.322520 + 0.946563i $$0.604530\pi$$
$$632$$ −1.81874e6 −0.181124
$$633$$ 1.07590e7 1.06724
$$634$$ 4.40707e6 0.435438
$$635$$ −1.81483e6 −0.178608
$$636$$ −8.48249e6 −0.831535
$$637$$ 0 0
$$638$$ 2.69164e7 2.61797
$$639$$ 4.70881e6 0.456204
$$640$$ −6.72879e6 −0.649362
$$641$$ 1.77716e7 1.70837 0.854185 0.519969i $$-0.174057\pi$$
0.854185 + 0.519969i $$0.174057\pi$$
$$642$$ 575293. 0.0550873
$$643$$ 9.34806e6 0.891649 0.445825 0.895120i $$-0.352910\pi$$
0.445825 + 0.895120i $$0.352910\pi$$
$$644$$ 0 0
$$645$$ 3.91063e6 0.370124
$$646$$ 2.92118e7 2.75408
$$647$$ −5.34386e6 −0.501874 −0.250937 0.968003i $$-0.580739\pi$$
−0.250937 + 0.968003i $$0.580739\pi$$
$$648$$ −1.22253e6 −0.114373
$$649$$ −3.09852e6 −0.288764
$$650$$ −1.93245e7 −1.79401
$$651$$ 0 0
$$652$$ 9.67098e6 0.890946
$$653$$ −1.19701e6 −0.109854 −0.0549270 0.998490i $$-0.517493\pi$$
−0.0549270 + 0.998490i $$0.517493\pi$$
$$654$$ 9.30851e6 0.851012
$$655$$ −3.88045e6 −0.353410
$$656$$ 201588. 0.0182896
$$657$$ −3.16796e6 −0.286330
$$658$$ 0 0
$$659$$ −1.17541e7 −1.05433 −0.527163 0.849764i $$-0.676744\pi$$
−0.527163 + 0.849764i $$0.676744\pi$$
$$660$$ 4.32882e6 0.386821
$$661$$ 1.80115e7 1.60341 0.801706 0.597719i $$-0.203926\pi$$
0.801706 + 0.597719i $$0.203926\pi$$
$$662$$ 2.01915e7 1.79071
$$663$$ 9.87819e6 0.872758
$$664$$ −1.31170e7 −1.15456
$$665$$ 0 0
$$666$$ −7.27054e6 −0.635157
$$667$$ 6.72153e6 0.584997
$$668$$ −6.75975e6 −0.586124
$$669$$ 2.66876e6 0.230539
$$670$$ 3.21322e6 0.276537
$$671$$ −1.45656e6 −0.124888
$$672$$ 0 0
$$673$$ 1.40977e7 1.19981 0.599904 0.800072i $$-0.295206\pi$$
0.599904 + 0.800072i $$0.295206\pi$$
$$674$$ 1.24613e7 1.05660
$$675$$ 1.92299e6 0.162449
$$676$$ 1.38780e7 1.16805
$$677$$ −7.65587e6 −0.641982 −0.320991 0.947082i $$-0.604016\pi$$
−0.320991 + 0.947082i $$0.604016\pi$$
$$678$$ 9.25510e6 0.773227
$$679$$ 0 0
$$680$$ 5.65718e6 0.469167
$$681$$ 1.96331e6 0.162227
$$682$$ 4.82520e6 0.397241
$$683$$ −1.08676e7 −0.891415 −0.445708 0.895179i $$-0.647048\pi$$
−0.445708 + 0.895179i $$0.647048\pi$$
$$684$$ −9.79773e6 −0.800728
$$685$$ −691510. −0.0563083
$$686$$ 0 0
$$687$$ −1.10628e7 −0.894283
$$688$$ 734791. 0.0591825
$$689$$ −1.43807e7 −1.15407
$$690$$ 1.74246e6 0.139329
$$691$$ 1.46118e7 1.16415 0.582073 0.813136i $$-0.302242\pi$$
0.582073 + 0.813136i $$0.302242\pi$$
$$692$$ 2.65589e7 2.10836
$$693$$ 0 0
$$694$$ −3.48832e7 −2.74927
$$695$$ 3.36005e6 0.263866
$$696$$ −1.17981e7 −0.923189
$$697$$ 7.42928e6 0.579248
$$698$$ −1.20965e7 −0.939772
$$699$$ 566498. 0.0438536
$$700$$ 0 0
$$701$$ 1.90104e7 1.46115 0.730577 0.682830i $$-0.239251\pi$$
0.730577 + 0.682830i $$0.239251\pi$$
$$702$$ −5.34056e6 −0.409019
$$703$$ −2.26132e7 −1.72573
$$704$$ −2.19992e7 −1.67292
$$705$$ −408525. −0.0309561
$$706$$ −3.14462e7 −2.37441
$$707$$ 0 0
$$708$$ 3.49964e6 0.262386
$$709$$ −1.15929e7 −0.866116 −0.433058 0.901366i $$-0.642565\pi$$
−0.433058 + 0.901366i $$0.642565\pi$$
$$710$$ 1.17804e7 0.877031
$$711$$ 790616. 0.0586532
$$712$$ 2.68890e7 1.98781
$$713$$ 1.20494e6 0.0887653
$$714$$ 0 0
$$715$$ 7.33881e6 0.536859
$$716$$ −6.93374e6 −0.505458
$$717$$ −1.97397e6 −0.143398
$$718$$ −1.35041e7 −0.977585
$$719$$ 1.02861e7 0.742040 0.371020 0.928625i $$-0.379008\pi$$
0.371020 + 0.928625i $$0.379008\pi$$
$$720$$ −66728.8 −0.00479713
$$721$$ 0 0
$$722$$ −2.63868e7 −1.88384
$$723$$ 3.90477e6 0.277811
$$724$$ 3.87014e7 2.74398
$$725$$ 1.85580e7 1.31125
$$726$$ 1.04081e6 0.0732873
$$727$$ −1.00970e7 −0.708526 −0.354263 0.935146i $$-0.615268\pi$$
−0.354263 + 0.935146i $$0.615268\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ −7.92556e6 −0.550456
$$731$$ 2.70799e7 1.87436
$$732$$ 1.64512e6 0.113480
$$733$$ −1.92402e6 −0.132266 −0.0661332 0.997811i $$-0.521066\pi$$
−0.0661332 + 0.997811i $$0.521066\pi$$
$$734$$ −4.57298e7 −3.13299
$$735$$ 0 0
$$736$$ −5.36932e6 −0.365363
$$737$$ 6.60752e6 0.448095
$$738$$ −4.01658e6 −0.271466
$$739$$ 4.20273e6 0.283087 0.141543 0.989932i $$-0.454793\pi$$
0.141543 + 0.989932i $$0.454793\pi$$
$$740$$ −1.12843e7 −0.757522
$$741$$ −1.66104e7 −1.11131
$$742$$ 0 0
$$743$$ −1.99659e7 −1.32684 −0.663418 0.748249i $$-0.730895\pi$$
−0.663418 + 0.748249i $$0.730895\pi$$
$$744$$ −2.11501e6 −0.140081
$$745$$ −8.00891e6 −0.528667
$$746$$ −3.63620e7 −2.39222
$$747$$ 5.70205e6 0.373878
$$748$$ 2.99757e7 1.95892
$$749$$ 0 0
$$750$$ 1.05103e7 0.682277
$$751$$ −3.63975e6 −0.235490 −0.117745 0.993044i $$-0.537567\pi$$
−0.117745 + 0.993044i $$0.537567\pi$$
$$752$$ −76760.3 −0.00494985
$$753$$ −1.53998e7 −0.989757
$$754$$ −5.15396e7 −3.30151
$$755$$ −4.52745e6 −0.289059
$$756$$ 0 0
$$757$$ 1.73429e7 1.09997 0.549986 0.835174i $$-0.314633\pi$$
0.549986 + 0.835174i $$0.314633\pi$$
$$758$$ 1.60813e7 1.01660
$$759$$ 3.58313e6 0.225765
$$760$$ −9.51270e6 −0.597406
$$761$$ 1.03568e7 0.648284 0.324142 0.946009i $$-0.394924\pi$$
0.324142 + 0.946009i $$0.394924\pi$$
$$762$$ 6.79430e6 0.423894
$$763$$ 0 0
$$764$$ −3.04740e7 −1.88884
$$765$$ −2.45921e6 −0.151929
$$766$$ 2.88138e7 1.77431
$$767$$ 5.93306e6 0.364159
$$768$$ 9.98679e6 0.610974
$$769$$ 1.81548e7 1.10707 0.553534 0.832826i $$-0.313279\pi$$
0.553534 + 0.832826i $$0.313279\pi$$
$$770$$ 0 0
$$771$$ 8.85652e6 0.536571
$$772$$ −2.09653e7 −1.26607
$$773$$ 1.63566e7 0.984565 0.492282 0.870435i $$-0.336163\pi$$
0.492282 + 0.870435i $$0.336163\pi$$
$$774$$ −1.46405e7 −0.878422
$$775$$ 3.32683e6 0.198965
$$776$$ −1.47815e7 −0.881181
$$777$$ 0 0
$$778$$ 9.66346e6 0.572379
$$779$$ −1.24925e7 −0.737576
$$780$$ −8.28886e6 −0.487818
$$781$$ 2.42248e7 1.42112
$$782$$ 1.20660e7 0.705581
$$783$$ 5.12873e6 0.298955
$$784$$ 0 0
$$785$$ −1.70552e6 −0.0987829
$$786$$ 1.45275e7 0.838755
$$787$$ 4.44728e6 0.255952 0.127976 0.991777i $$-0.459152\pi$$
0.127976 + 0.991777i $$0.459152\pi$$
$$788$$ 3.51207e7 2.01487
$$789$$ −4.92386e6 −0.281587
$$790$$ 1.97795e6 0.112758
$$791$$ 0 0
$$792$$ −6.28938e6 −0.356283
$$793$$ 2.78903e6 0.157496
$$794$$ 4.17493e6 0.235016
$$795$$ 3.58012e6 0.200900
$$796$$ 2.37953e7 1.33110
$$797$$ 9.15303e6 0.510410 0.255205 0.966887i $$-0.417857\pi$$
0.255205 + 0.966887i $$0.417857\pi$$
$$798$$ 0 0
$$799$$ −2.82891e6 −0.156766
$$800$$ −1.48246e7 −0.818950
$$801$$ −1.16888e7 −0.643708
$$802$$ 2.64815e7 1.45381
$$803$$ −1.62978e7 −0.891948
$$804$$ −7.46290e6 −0.407162
$$805$$ 0 0
$$806$$ −9.23932e6 −0.500959
$$807$$ −5.77001e6 −0.311884
$$808$$ −1.58073e7 −0.851784
$$809$$ 2.51923e7 1.35331 0.676654 0.736301i $$-0.263429\pi$$
0.676654 + 0.736301i $$0.263429\pi$$
$$810$$ 1.32955e6 0.0712020
$$811$$ 8.90585e6 0.475470 0.237735 0.971330i $$-0.423595\pi$$
0.237735 + 0.971330i $$0.423595\pi$$
$$812$$ 0 0
$$813$$ −4.93211e6 −0.261702
$$814$$ −3.74038e7 −1.97858
$$815$$ −4.08173e6 −0.215254
$$816$$ −462076. −0.0242934
$$817$$ −4.55355e7 −2.38669
$$818$$ −2.07417e6 −0.108383
$$819$$ 0 0
$$820$$ −6.23396e6 −0.323765
$$821$$ 3.35490e7 1.73709 0.868543 0.495613i $$-0.165057\pi$$
0.868543 + 0.495613i $$0.165057\pi$$
$$822$$ 2.58886e6 0.133638
$$823$$ −1.46001e7 −0.751376 −0.375688 0.926746i $$-0.622593\pi$$
−0.375688 + 0.926746i $$0.622593\pi$$
$$824$$ 3.78857e6 0.194382
$$825$$ 9.89293e6 0.506046
$$826$$ 0 0
$$827$$ 1.97530e7 1.00431 0.502156 0.864777i $$-0.332540\pi$$
0.502156 + 0.864777i $$0.332540\pi$$
$$828$$ −4.04698e6 −0.205142
$$829$$ 4.06255e6 0.205311 0.102655 0.994717i $$-0.467266\pi$$
0.102655 + 0.994717i $$0.467266\pi$$
$$830$$ 1.42653e7 0.718762
$$831$$ −1.50673e7 −0.756889
$$832$$ 4.21241e7 2.10971
$$833$$ 0 0
$$834$$ −1.25793e7 −0.626239
$$835$$ 2.85302e6 0.141608
$$836$$ −5.04050e7 −2.49435
$$837$$ 919409. 0.0453623
$$838$$ 3.95735e7 1.94668
$$839$$ 1.60509e7 0.787217 0.393609 0.919278i $$-0.371227\pi$$
0.393609 + 0.919278i $$0.371227\pi$$
$$840$$ 0 0
$$841$$ 2.89842e7 1.41309
$$842$$ −1.14846e7 −0.558260
$$843$$ −1.62970e7 −0.789839
$$844$$ −6.25159e7 −3.02088
$$845$$ −5.85737e6 −0.282202
$$846$$ 1.52943e6 0.0734687
$$847$$ 0 0
$$848$$ 672690. 0.0321237
$$849$$ 2.26183e7 1.07694
$$850$$ 3.33140e7 1.58154
$$851$$ −9.34043e6 −0.442123
$$852$$ −2.73608e7 −1.29131
$$853$$ 1.23887e7 0.582979 0.291489 0.956574i $$-0.405849\pi$$
0.291489 + 0.956574i $$0.405849\pi$$
$$854$$ 0 0
$$855$$ 4.13523e6 0.193457
$$856$$ −1.29728e6 −0.0605133
$$857$$ −2.63519e7 −1.22563 −0.612816 0.790226i $$-0.709963\pi$$
−0.612816 + 0.790226i $$0.709963\pi$$
$$858$$ −2.74748e7 −1.27414
$$859$$ 1.21249e7 0.560654 0.280327 0.959905i $$-0.409557\pi$$
0.280327 + 0.959905i $$0.409557\pi$$
$$860$$ −2.27229e7 −1.04765
$$861$$ 0 0
$$862$$ 4.04702e7 1.85510
$$863$$ 2.55952e6 0.116985 0.0584927 0.998288i $$-0.481371\pi$$
0.0584927 + 0.998288i $$0.481371\pi$$
$$864$$ −4.09695e6 −0.186714
$$865$$ −1.12094e7 −0.509382
$$866$$ −1.47773e7 −0.669578
$$867$$ −4.25055e6 −0.192042
$$868$$ 0 0
$$869$$ 4.06737e6 0.182711
$$870$$ 1.28310e7 0.574726
$$871$$ −1.26521e7 −0.565091
$$872$$ −2.09906e7 −0.934834
$$873$$ 6.42562e6 0.285351
$$874$$ −2.02893e7 −0.898439
$$875$$ 0 0
$$876$$ 1.84076e7 0.810471
$$877$$ 2.65820e7 1.16705 0.583523 0.812096i $$-0.301674\pi$$
0.583523 + 0.812096i $$0.301674\pi$$
$$878$$ −4.06225e7 −1.77841
$$879$$ 965051. 0.0421287
$$880$$ −343290. −0.0149436
$$881$$ 8.32262e6 0.361260 0.180630 0.983551i $$-0.442186\pi$$
0.180630 + 0.983551i $$0.442186\pi$$
$$882$$ 0 0
$$883$$ 1.81133e7 0.781798 0.390899 0.920434i $$-0.372164\pi$$
0.390899 + 0.920434i $$0.372164\pi$$
$$884$$ −5.73977e7 −2.47038
$$885$$ −1.47706e6 −0.0633927
$$886$$ 3.31845e7 1.42020
$$887$$ 2.04255e7 0.871694 0.435847 0.900021i $$-0.356449\pi$$
0.435847 + 0.900021i $$0.356449\pi$$
$$888$$ 1.63950e7 0.697718
$$889$$ 0 0
$$890$$ −2.92429e7 −1.23750
$$891$$ 2.73403e6 0.115374
$$892$$ −1.55070e7 −0.652552
$$893$$ 4.75689e6 0.199616
$$894$$ 2.99835e7 1.25470
$$895$$ 2.92645e6 0.122119
$$896$$ 0 0
$$897$$ −6.86099e6 −0.284712
$$898$$ 4.28785e6 0.177439
$$899$$ 8.87285e6 0.366154
$$900$$ −1.11736e7 −0.459820
$$901$$ 2.47912e7 1.01739
$$902$$ −2.06635e7 −0.845645
$$903$$ 0 0
$$904$$ −2.08702e7 −0.849388
$$905$$ −1.63343e7 −0.662948
$$906$$ 1.69497e7 0.686029
$$907$$ −2.01197e7 −0.812089 −0.406044 0.913853i $$-0.633092\pi$$
−0.406044 + 0.913853i $$0.633092\pi$$
$$908$$ −1.14079e7 −0.459190
$$909$$ 6.87153e6 0.275832
$$910$$ 0 0
$$911$$ 3.17075e7 1.26580 0.632902 0.774232i $$-0.281863\pi$$
0.632902 + 0.774232i $$0.281863\pi$$
$$912$$ 776993. 0.0309336
$$913$$ 2.93345e7 1.16467
$$914$$ 5.52023e6 0.218571
$$915$$ −694339. −0.0274169
$$916$$ 6.42812e7 2.53131
$$917$$ 0 0
$$918$$ 9.20672e6 0.360577
$$919$$ −8.70727e6 −0.340090 −0.170045 0.985436i $$-0.554391\pi$$
−0.170045 + 0.985436i $$0.554391\pi$$
$$920$$ −3.92924e6 −0.153052
$$921$$ 1.34659e7 0.523104
$$922$$ 2.63678e7 1.02152
$$923$$ −4.63857e7 −1.79217
$$924$$ 0 0
$$925$$ −2.57887e7 −0.991005
$$926$$ 2.67634e7 1.02569
$$927$$ −1.64691e6 −0.0629465
$$928$$ −3.95381e7 −1.50711
$$929$$ −3.24042e7 −1.23186 −0.615931 0.787800i $$-0.711220\pi$$
−0.615931 + 0.787800i $$0.711220\pi$$
$$930$$ 2.30016e6 0.0872069
$$931$$ 0 0
$$932$$ −3.29167e6 −0.124130
$$933$$ 7.75082e6 0.291503
$$934$$ −6.60475e7 −2.47736
$$935$$ −1.26516e7 −0.473277
$$936$$ 1.20429e7 0.449307
$$937$$ 2.19155e7 0.815458 0.407729 0.913103i $$-0.366321\pi$$
0.407729 + 0.913103i $$0.366321\pi$$
$$938$$ 0 0
$$939$$ −4.53544e6 −0.167863
$$940$$ 2.37376e6 0.0876227
$$941$$ −1.46019e7 −0.537571 −0.268785 0.963200i $$-0.586622\pi$$
−0.268785 + 0.963200i $$0.586622\pi$$
$$942$$ 6.38506e6 0.234443
$$943$$ −5.16008e6 −0.188963
$$944$$ −277533. −0.0101364
$$945$$ 0 0
$$946$$ −7.53189e7 −2.73638
$$947$$ −2.08378e7 −0.755053 −0.377527 0.925999i $$-0.623225\pi$$
−0.377527 + 0.925999i $$0.623225\pi$$
$$948$$ −4.59391e6 −0.166021
$$949$$ 3.12071e7 1.12483
$$950$$ −5.60184e7 −2.01382
$$951$$ 4.32008e6 0.154896
$$952$$ 0 0
$$953$$ −942012. −0.0335988 −0.0167994 0.999859i $$-0.505348\pi$$
−0.0167994 + 0.999859i $$0.505348\pi$$
$$954$$ −1.34031e7 −0.476800
$$955$$ 1.28618e7 0.456346
$$956$$ 1.14698e7 0.405894
$$957$$ 2.63851e7 0.931275
$$958$$ −2.56752e7 −0.903857
$$959$$ 0 0
$$960$$ −1.04869e7 −0.367258
$$961$$ −2.70385e7 −0.944441
$$962$$ 7.16209e7 2.49518
$$963$$ 563937. 0.0195959
$$964$$ −2.26889e7 −0.786359
$$965$$ 8.84860e6 0.305884
$$966$$ 0 0
$$967$$ −2.56570e7 −0.882346 −0.441173 0.897422i $$-0.645438\pi$$
−0.441173 + 0.897422i $$0.645438\pi$$
$$968$$ −2.34701e6 −0.0805059
$$969$$ 2.86352e7 0.979694
$$970$$ 1.60755e7 0.548575
$$971$$ 2.14274e7 0.729324 0.364662 0.931140i $$-0.381184\pi$$
0.364662 + 0.931140i $$0.381184\pi$$
$$972$$ −3.08797e6 −0.104835
$$973$$ 0 0
$$974$$ 2.69237e7 0.909364
$$975$$ −1.89431e7 −0.638173
$$976$$ −130464. −0.00438394
$$977$$ −2.04841e7 −0.686562 −0.343281 0.939233i $$-0.611538\pi$$
−0.343281 + 0.939233i $$0.611538\pi$$
$$978$$ 1.52811e7 0.510866
$$979$$ −6.01338e7 −2.00522
$$980$$ 0 0
$$981$$ 9.12476e6 0.302726
$$982$$ 2.81057e7 0.930069
$$983$$ −2.77650e7 −0.916459 −0.458230 0.888834i $$-0.651516\pi$$
−0.458230 + 0.888834i $$0.651516\pi$$
$$984$$ 9.05736e6 0.298204
$$985$$ −1.48230e7 −0.486796
$$986$$ 8.88504e7 2.91050
$$987$$ 0 0
$$988$$ 9.65159e7 3.14562
$$989$$ −1.88086e7 −0.611456
$$990$$ 6.83995e6 0.221802
$$991$$ 3.46832e7 1.12185 0.560926 0.827866i $$-0.310445\pi$$
0.560926 + 0.827866i $$0.310445\pi$$
$$992$$ −7.08785e6 −0.228684
$$993$$ 1.97930e7 0.636998
$$994$$ 0 0
$$995$$ −1.00431e7 −0.321594
$$996$$ −3.31320e7 −1.05828
$$997$$ 1.48572e7 0.473369 0.236685 0.971587i $$-0.423939\pi$$
0.236685 + 0.971587i $$0.423939\pi$$
$$998$$ 6.01512e7 1.91169
$$999$$ −7.12703e6 −0.225941
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 147.6.a.l.1.1 4
3.2 odd 2 441.6.a.v.1.4 4
7.2 even 3 147.6.e.o.67.4 8
7.3 odd 6 21.6.e.c.16.4 yes 8
7.4 even 3 147.6.e.o.79.4 8
7.5 odd 6 21.6.e.c.4.4 8
7.6 odd 2 147.6.a.m.1.1 4
21.5 even 6 63.6.e.e.46.1 8
21.17 even 6 63.6.e.e.37.1 8
21.20 even 2 441.6.a.w.1.4 4
28.3 even 6 336.6.q.j.289.3 8
28.19 even 6 336.6.q.j.193.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.4 8 7.5 odd 6
21.6.e.c.16.4 yes 8 7.3 odd 6
63.6.e.e.37.1 8 21.17 even 6
63.6.e.e.46.1 8 21.5 even 6
147.6.a.l.1.1 4 1.1 even 1 trivial
147.6.a.m.1.1 4 7.6 odd 2
147.6.e.o.67.4 8 7.2 even 3
147.6.e.o.79.4 8 7.4 even 3
336.6.q.j.193.3 8 28.19 even 6
336.6.q.j.289.3 8 28.3 even 6
441.6.a.v.1.4 4 3.2 odd 2
441.6.a.w.1.4 4 21.20 even 2