# Properties

 Label 147.6.a.i.1.1 Level $147$ Weight $6$ Character 147.1 Self dual yes Analytic conductor $23.576$ 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] = [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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{249})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 62$$ x^2 - x - 62 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$8.38987$$ 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)

 $$f(q)$$ $$=$$ $$q-9.38987 q^{2} -9.00000 q^{3} +56.1696 q^{4} -71.7291 q^{5} +84.5088 q^{6} -226.949 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-9.38987 q^{2} -9.00000 q^{3} +56.1696 q^{4} -71.7291 q^{5} +84.5088 q^{6} -226.949 q^{8} +81.0000 q^{9} +673.526 q^{10} -560.610 q^{11} -505.526 q^{12} +533.509 q^{13} +645.562 q^{15} +333.597 q^{16} +1005.70 q^{17} -760.579 q^{18} +1368.53 q^{19} -4028.99 q^{20} +5264.05 q^{22} +3228.08 q^{23} +2042.54 q^{24} +2020.06 q^{25} -5009.58 q^{26} -729.000 q^{27} -753.456 q^{29} -6061.74 q^{30} +8206.42 q^{31} +4129.95 q^{32} +5045.49 q^{33} -9443.39 q^{34} +4549.74 q^{36} -2808.66 q^{37} -12850.3 q^{38} -4801.58 q^{39} +16278.9 q^{40} +245.827 q^{41} -17504.5 q^{43} -31489.2 q^{44} -5810.05 q^{45} -30311.2 q^{46} -16345.5 q^{47} -3002.37 q^{48} -18968.1 q^{50} -9051.30 q^{51} +29967.0 q^{52} -29641.7 q^{53} +6845.21 q^{54} +40212.0 q^{55} -12316.7 q^{57} +7074.85 q^{58} -10356.1 q^{59} +36260.9 q^{60} +954.179 q^{61} -77057.2 q^{62} -49454.8 q^{64} -38268.1 q^{65} -47376.5 q^{66} -19815.2 q^{67} +56489.8 q^{68} -29052.7 q^{69} +62125.4 q^{71} -18382.9 q^{72} +27109.6 q^{73} +26373.0 q^{74} -18180.5 q^{75} +76869.6 q^{76} +45086.2 q^{78} +44687.4 q^{79} -23928.6 q^{80} +6561.00 q^{81} -2308.29 q^{82} +15606.6 q^{83} -72137.9 q^{85} +164365. q^{86} +6781.10 q^{87} +127230. q^{88} +13635.3 q^{89} +54555.6 q^{90} +181320. q^{92} -73857.8 q^{93} +153482. q^{94} -98163.1 q^{95} -37169.5 q^{96} -12919.5 q^{97} -45409.4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 18 q^{3} + 65 q^{4} - 33 q^{5} + 27 q^{6} - 375 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 18 * q^3 + 65 * q^4 - 33 * q^5 + 27 * q^6 - 375 * q^8 + 162 * q^9 $$2 q - 3 q^{2} - 18 q^{3} + 65 q^{4} - 33 q^{5} + 27 q^{6} - 375 q^{8} + 162 q^{9} + 921 q^{10} - 1137 q^{11} - 585 q^{12} + 925 q^{13} + 297 q^{15} - 895 q^{16} - 324 q^{17} - 243 q^{18} + 2311 q^{19} - 3687 q^{20} + 1581 q^{22} + 1596 q^{23} + 3375 q^{24} + 395 q^{25} - 2508 q^{26} - 1458 q^{27} - 2217 q^{29} - 8289 q^{30} + 4294 q^{31} + 1017 q^{32} + 10233 q^{33} - 17940 q^{34} + 5265 q^{36} - 19109 q^{37} - 6828 q^{38} - 8325 q^{39} + 10545 q^{40} - 12858 q^{41} - 2771 q^{43} - 36579 q^{44} - 2673 q^{45} - 40740 q^{46} - 23160 q^{47} + 8055 q^{48} - 29352 q^{50} + 2916 q^{51} + 33424 q^{52} - 31653 q^{53} + 2187 q^{54} + 17889 q^{55} - 20799 q^{57} - 2277 q^{58} + 41097 q^{59} + 33183 q^{60} + 42052 q^{61} - 102057 q^{62} - 30031 q^{64} - 23106 q^{65} - 14229 q^{66} + 30763 q^{67} + 44748 q^{68} - 14364 q^{69} + 102096 q^{71} - 30375 q^{72} - 28577 q^{73} - 77784 q^{74} - 3555 q^{75} + 85192 q^{76} + 22572 q^{78} - 18464 q^{79} - 71511 q^{80} + 13122 q^{81} - 86040 q^{82} + 61179 q^{83} - 123636 q^{85} + 258510 q^{86} + 19953 q^{87} + 212565 q^{88} + 29322 q^{89} + 74601 q^{90} + 166908 q^{92} - 38646 q^{93} + 109938 q^{94} - 61662 q^{95} - 9153 q^{96} - 9791 q^{97} - 92097 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 18 * q^3 + 65 * q^4 - 33 * q^5 + 27 * q^6 - 375 * q^8 + 162 * q^9 + 921 * q^10 - 1137 * q^11 - 585 * q^12 + 925 * q^13 + 297 * q^15 - 895 * q^16 - 324 * q^17 - 243 * q^18 + 2311 * q^19 - 3687 * q^20 + 1581 * q^22 + 1596 * q^23 + 3375 * q^24 + 395 * q^25 - 2508 * q^26 - 1458 * q^27 - 2217 * q^29 - 8289 * q^30 + 4294 * q^31 + 1017 * q^32 + 10233 * q^33 - 17940 * q^34 + 5265 * q^36 - 19109 * q^37 - 6828 * q^38 - 8325 * q^39 + 10545 * q^40 - 12858 * q^41 - 2771 * q^43 - 36579 * q^44 - 2673 * q^45 - 40740 * q^46 - 23160 * q^47 + 8055 * q^48 - 29352 * q^50 + 2916 * q^51 + 33424 * q^52 - 31653 * q^53 + 2187 * q^54 + 17889 * q^55 - 20799 * q^57 - 2277 * q^58 + 41097 * q^59 + 33183 * q^60 + 42052 * q^61 - 102057 * q^62 - 30031 * q^64 - 23106 * q^65 - 14229 * q^66 + 30763 * q^67 + 44748 * q^68 - 14364 * q^69 + 102096 * q^71 - 30375 * q^72 - 28577 * q^73 - 77784 * q^74 - 3555 * q^75 + 85192 * q^76 + 22572 * q^78 - 18464 * q^79 - 71511 * q^80 + 13122 * q^81 - 86040 * q^82 + 61179 * q^83 - 123636 * q^85 + 258510 * q^86 + 19953 * q^87 + 212565 * q^88 + 29322 * q^89 + 74601 * q^90 + 166908 * q^92 - 38646 * q^93 + 109938 * q^94 - 61662 * q^95 - 9153 * q^96 - 9791 * q^97 - 92097 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −9.38987 −1.65991 −0.829955 0.557831i $$-0.811634\pi$$
−0.829955 + 0.557831i $$0.811634\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 56.1696 1.75530
$$5$$ −71.7291 −1.28313 −0.641564 0.767069i $$-0.721714\pi$$
−0.641564 + 0.767069i $$0.721714\pi$$
$$6$$ 84.5088 0.958349
$$7$$ 0 0
$$8$$ −226.949 −1.25373
$$9$$ 81.0000 0.333333
$$10$$ 673.526 2.12988
$$11$$ −560.610 −1.39694 −0.698472 0.715637i $$-0.746137\pi$$
−0.698472 + 0.715637i $$0.746137\pi$$
$$12$$ −505.526 −1.01342
$$13$$ 533.509 0.875555 0.437777 0.899083i $$-0.355766\pi$$
0.437777 + 0.899083i $$0.355766\pi$$
$$14$$ 0 0
$$15$$ 645.562 0.740815
$$16$$ 333.597 0.325778
$$17$$ 1005.70 0.844007 0.422004 0.906594i $$-0.361327\pi$$
0.422004 + 0.906594i $$0.361327\pi$$
$$18$$ −760.579 −0.553303
$$19$$ 1368.53 0.869699 0.434850 0.900503i $$-0.356801\pi$$
0.434850 + 0.900503i $$0.356801\pi$$
$$20$$ −4028.99 −2.25228
$$21$$ 0 0
$$22$$ 5264.05 2.31880
$$23$$ 3228.08 1.27240 0.636201 0.771523i $$-0.280505\pi$$
0.636201 + 0.771523i $$0.280505\pi$$
$$24$$ 2042.54 0.723841
$$25$$ 2020.06 0.646419
$$26$$ −5009.58 −1.45334
$$27$$ −729.000 −0.192450
$$28$$ 0 0
$$29$$ −753.456 −0.166365 −0.0831827 0.996534i $$-0.526508\pi$$
−0.0831827 + 0.996534i $$0.526508\pi$$
$$30$$ −6061.74 −1.22969
$$31$$ 8206.42 1.53373 0.766866 0.641807i $$-0.221815\pi$$
0.766866 + 0.641807i $$0.221815\pi$$
$$32$$ 4129.95 0.712968
$$33$$ 5045.49 0.806526
$$34$$ −9443.39 −1.40098
$$35$$ 0 0
$$36$$ 4549.74 0.585100
$$37$$ −2808.66 −0.337284 −0.168642 0.985677i $$-0.553938\pi$$
−0.168642 + 0.985677i $$0.553938\pi$$
$$38$$ −12850.3 −1.44362
$$39$$ −4801.58 −0.505502
$$40$$ 16278.9 1.60870
$$41$$ 245.827 0.0228387 0.0114193 0.999935i $$-0.496365\pi$$
0.0114193 + 0.999935i $$0.496365\pi$$
$$42$$ 0 0
$$43$$ −17504.5 −1.44371 −0.721853 0.692047i $$-0.756709\pi$$
−0.721853 + 0.692047i $$0.756709\pi$$
$$44$$ −31489.2 −2.45206
$$45$$ −5810.05 −0.427710
$$46$$ −30311.2 −2.11207
$$47$$ −16345.5 −1.07933 −0.539663 0.841881i $$-0.681449\pi$$
−0.539663 + 0.841881i $$0.681449\pi$$
$$48$$ −3002.37 −0.188088
$$49$$ 0 0
$$50$$ −18968.1 −1.07300
$$51$$ −9051.30 −0.487288
$$52$$ 29967.0 1.53686
$$53$$ −29641.7 −1.44948 −0.724741 0.689021i $$-0.758041\pi$$
−0.724741 + 0.689021i $$0.758041\pi$$
$$54$$ 6845.21 0.319450
$$55$$ 40212.0 1.79246
$$56$$ 0 0
$$57$$ −12316.7 −0.502121
$$58$$ 7074.85 0.276151
$$59$$ −10356.1 −0.387317 −0.193659 0.981069i $$-0.562035\pi$$
−0.193659 + 0.981069i $$0.562035\pi$$
$$60$$ 36260.9 1.30035
$$61$$ 954.179 0.0328326 0.0164163 0.999865i $$-0.494774\pi$$
0.0164163 + 0.999865i $$0.494774\pi$$
$$62$$ −77057.2 −2.54586
$$63$$ 0 0
$$64$$ −49454.8 −1.50924
$$65$$ −38268.1 −1.12345
$$66$$ −47376.5 −1.33876
$$67$$ −19815.2 −0.539276 −0.269638 0.962962i $$-0.586904\pi$$
−0.269638 + 0.962962i $$0.586904\pi$$
$$68$$ 56489.8 1.48149
$$69$$ −29052.7 −0.734622
$$70$$ 0 0
$$71$$ 62125.4 1.46259 0.731296 0.682060i $$-0.238916\pi$$
0.731296 + 0.682060i $$0.238916\pi$$
$$72$$ −18382.9 −0.417910
$$73$$ 27109.6 0.595410 0.297705 0.954658i $$-0.403779\pi$$
0.297705 + 0.954658i $$0.403779\pi$$
$$74$$ 26373.0 0.559861
$$75$$ −18180.5 −0.373210
$$76$$ 76869.6 1.52658
$$77$$ 0 0
$$78$$ 45086.2 0.839087
$$79$$ 44687.4 0.805595 0.402798 0.915289i $$-0.368038\pi$$
0.402798 + 0.915289i $$0.368038\pi$$
$$80$$ −23928.6 −0.418015
$$81$$ 6561.00 0.111111
$$82$$ −2308.29 −0.0379101
$$83$$ 15606.6 0.248665 0.124332 0.992241i $$-0.460321\pi$$
0.124332 + 0.992241i $$0.460321\pi$$
$$84$$ 0 0
$$85$$ −72137.9 −1.08297
$$86$$ 164365. 2.39642
$$87$$ 6781.10 0.0960511
$$88$$ 127230. 1.75139
$$89$$ 13635.3 0.182469 0.0912347 0.995829i $$-0.470919\pi$$
0.0912347 + 0.995829i $$0.470919\pi$$
$$90$$ 54555.6 0.709959
$$91$$ 0 0
$$92$$ 181320. 2.23345
$$93$$ −73857.8 −0.885500
$$94$$ 153482. 1.79159
$$95$$ −98163.1 −1.11594
$$96$$ −37169.5 −0.411632
$$97$$ −12919.5 −0.139417 −0.0697086 0.997567i $$-0.522207\pi$$
−0.0697086 + 0.997567i $$0.522207\pi$$
$$98$$ 0 0
$$99$$ −45409.4 −0.465648
$$100$$ 113466. 1.13466
$$101$$ 25142.9 0.245252 0.122626 0.992453i $$-0.460868\pi$$
0.122626 + 0.992453i $$0.460868\pi$$
$$102$$ 84990.5 0.808854
$$103$$ −160753. −1.49302 −0.746511 0.665373i $$-0.768273\pi$$
−0.746511 + 0.665373i $$0.768273\pi$$
$$104$$ −121079. −1.09771
$$105$$ 0 0
$$106$$ 278331. 2.40601
$$107$$ −94375.5 −0.796893 −0.398446 0.917192i $$-0.630451\pi$$
−0.398446 + 0.917192i $$0.630451\pi$$
$$108$$ −40947.6 −0.337808
$$109$$ −83393.5 −0.672304 −0.336152 0.941808i $$-0.609126\pi$$
−0.336152 + 0.941808i $$0.609126\pi$$
$$110$$ −377586. −2.97532
$$111$$ 25278.0 0.194731
$$112$$ 0 0
$$113$$ −179254. −1.32060 −0.660301 0.751001i $$-0.729571\pi$$
−0.660301 + 0.751001i $$0.729571\pi$$
$$114$$ 115653. 0.833476
$$115$$ −231547. −1.63266
$$116$$ −42321.3 −0.292021
$$117$$ 43214.2 0.291852
$$118$$ 97242.5 0.642911
$$119$$ 0 0
$$120$$ −146510. −0.928781
$$121$$ 153233. 0.951455
$$122$$ −8959.61 −0.0544991
$$123$$ −2212.45 −0.0131859
$$124$$ 460951. 2.69216
$$125$$ 79256.4 0.453690
$$126$$ 0 0
$$127$$ −143674. −0.790440 −0.395220 0.918586i $$-0.629332\pi$$
−0.395220 + 0.918586i $$0.629332\pi$$
$$128$$ 332215. 1.79223
$$129$$ 157540. 0.833524
$$130$$ 359332. 1.86482
$$131$$ 52289.9 0.266219 0.133110 0.991101i $$-0.457504\pi$$
0.133110 + 0.991101i $$0.457504\pi$$
$$132$$ 283403. 1.41570
$$133$$ 0 0
$$134$$ 186062. 0.895150
$$135$$ 52290.5 0.246938
$$136$$ −228243. −1.05816
$$137$$ 9410.10 0.0428344 0.0214172 0.999771i $$-0.493182\pi$$
0.0214172 + 0.999771i $$0.493182\pi$$
$$138$$ 272801. 1.21941
$$139$$ 183094. 0.803781 0.401890 0.915688i $$-0.368353\pi$$
0.401890 + 0.915688i $$0.368353\pi$$
$$140$$ 0 0
$$141$$ 147109. 0.623150
$$142$$ −583349. −2.42777
$$143$$ −299090. −1.22310
$$144$$ 27021.3 0.108593
$$145$$ 54044.7 0.213468
$$146$$ −254556. −0.988328
$$147$$ 0 0
$$148$$ −157762. −0.592034
$$149$$ −167002. −0.616247 −0.308123 0.951346i $$-0.599701\pi$$
−0.308123 + 0.951346i $$0.599701\pi$$
$$150$$ 170713. 0.619495
$$151$$ 376264. 1.34292 0.671461 0.741040i $$-0.265667\pi$$
0.671461 + 0.741040i $$0.265667\pi$$
$$152$$ −310586. −1.09037
$$153$$ 81461.7 0.281336
$$154$$ 0 0
$$155$$ −588639. −1.96797
$$156$$ −269703. −0.887307
$$157$$ 39075.0 0.126517 0.0632587 0.997997i $$-0.479851\pi$$
0.0632587 + 0.997997i $$0.479851\pi$$
$$158$$ −419608. −1.33722
$$159$$ 266775. 0.836859
$$160$$ −296237. −0.914829
$$161$$ 0 0
$$162$$ −61606.9 −0.184434
$$163$$ −477919. −1.40892 −0.704458 0.709745i $$-0.748810\pi$$
−0.704458 + 0.709745i $$0.748810\pi$$
$$164$$ 13808.0 0.0400887
$$165$$ −361908. −1.03488
$$166$$ −146544. −0.412761
$$167$$ 39793.4 0.110413 0.0552064 0.998475i $$-0.482418\pi$$
0.0552064 + 0.998475i $$0.482418\pi$$
$$168$$ 0 0
$$169$$ −86661.4 −0.233404
$$170$$ 677366. 1.79763
$$171$$ 110851. 0.289900
$$172$$ −983221. −2.53414
$$173$$ 48338.2 0.122794 0.0613968 0.998113i $$-0.480445\pi$$
0.0613968 + 0.998113i $$0.480445\pi$$
$$174$$ −63673.7 −0.159436
$$175$$ 0 0
$$176$$ −187018. −0.455094
$$177$$ 93205.0 0.223618
$$178$$ −128034. −0.302883
$$179$$ −142911. −0.333374 −0.166687 0.986010i $$-0.553307\pi$$
−0.166687 + 0.986010i $$0.553307\pi$$
$$180$$ −326348. −0.750759
$$181$$ −77245.3 −0.175257 −0.0876285 0.996153i $$-0.527929\pi$$
−0.0876285 + 0.996153i $$0.527929\pi$$
$$182$$ 0 0
$$183$$ −8587.61 −0.0189559
$$184$$ −732610. −1.59525
$$185$$ 201463. 0.432778
$$186$$ 693515. 1.46985
$$187$$ −563806. −1.17903
$$188$$ −918119. −1.89454
$$189$$ 0 0
$$190$$ 921739. 1.85235
$$191$$ 272054. 0.539600 0.269800 0.962916i $$-0.413042\pi$$
0.269800 + 0.962916i $$0.413042\pi$$
$$192$$ 445093. 0.871360
$$193$$ −16033.8 −0.0309844 −0.0154922 0.999880i $$-0.504932\pi$$
−0.0154922 + 0.999880i $$0.504932\pi$$
$$194$$ 121312. 0.231420
$$195$$ 344413. 0.648624
$$196$$ 0 0
$$197$$ 1.03228e6 1.89510 0.947552 0.319603i $$-0.103549\pi$$
0.947552 + 0.319603i $$0.103549\pi$$
$$198$$ 426388. 0.772934
$$199$$ −881736. −1.57836 −0.789180 0.614162i $$-0.789494\pi$$
−0.789180 + 0.614162i $$0.789494\pi$$
$$200$$ −458451. −0.810435
$$201$$ 178337. 0.311351
$$202$$ −236089. −0.407096
$$203$$ 0 0
$$204$$ −508408. −0.855337
$$205$$ −17633.0 −0.0293049
$$206$$ 1.50945e6 2.47828
$$207$$ 261474. 0.424134
$$208$$ 177977. 0.285237
$$209$$ −767210. −1.21492
$$210$$ 0 0
$$211$$ −372813. −0.576480 −0.288240 0.957558i $$-0.593070\pi$$
−0.288240 + 0.957558i $$0.593070\pi$$
$$212$$ −1.66496e6 −2.54428
$$213$$ −559128. −0.844428
$$214$$ 886174. 1.32277
$$215$$ 1.25558e6 1.85246
$$216$$ 165446. 0.241280
$$217$$ 0 0
$$218$$ 783054. 1.11596
$$219$$ −243987. −0.343760
$$220$$ 2.25869e6 3.14630
$$221$$ 536550. 0.738975
$$222$$ −237357. −0.323236
$$223$$ −1.08205e6 −1.45708 −0.728541 0.685002i $$-0.759801\pi$$
−0.728541 + 0.685002i $$0.759801\pi$$
$$224$$ 0 0
$$225$$ 163625. 0.215473
$$226$$ 1.68317e6 2.19208
$$227$$ 553049. 0.712359 0.356179 0.934418i $$-0.384079\pi$$
0.356179 + 0.934418i $$0.384079\pi$$
$$228$$ −691826. −0.881373
$$229$$ 523024. 0.659072 0.329536 0.944143i $$-0.393108\pi$$
0.329536 + 0.944143i $$0.393108\pi$$
$$230$$ 2.17420e6 2.71006
$$231$$ 0 0
$$232$$ 170996. 0.208577
$$233$$ −364181. −0.439468 −0.219734 0.975560i $$-0.570519\pi$$
−0.219734 + 0.975560i $$0.570519\pi$$
$$234$$ −405776. −0.484447
$$235$$ 1.17245e6 1.38492
$$236$$ −581698. −0.679858
$$237$$ −402186. −0.465111
$$238$$ 0 0
$$239$$ 371841. 0.421078 0.210539 0.977585i $$-0.432478\pi$$
0.210539 + 0.977585i $$0.432478\pi$$
$$240$$ 215357. 0.241341
$$241$$ −1.71147e6 −1.89814 −0.949069 0.315067i $$-0.897973\pi$$
−0.949069 + 0.315067i $$0.897973\pi$$
$$242$$ −1.43883e6 −1.57933
$$243$$ −59049.0 −0.0641500
$$244$$ 53595.8 0.0576310
$$245$$ 0 0
$$246$$ 20774.6 0.0218874
$$247$$ 730121. 0.761469
$$248$$ −1.86244e6 −1.92289
$$249$$ −140460. −0.143567
$$250$$ −744207. −0.753084
$$251$$ 58134.1 0.0582434 0.0291217 0.999576i $$-0.490729\pi$$
0.0291217 + 0.999576i $$0.490729\pi$$
$$252$$ 0 0
$$253$$ −1.80969e6 −1.77748
$$254$$ 1.34908e6 1.31206
$$255$$ 649242. 0.625253
$$256$$ −1.53691e6 −1.46571
$$257$$ −311839. −0.294509 −0.147254 0.989099i $$-0.547044\pi$$
−0.147254 + 0.989099i $$0.547044\pi$$
$$258$$ −1.47928e6 −1.38357
$$259$$ 0 0
$$260$$ −2.14950e6 −1.97199
$$261$$ −61029.9 −0.0554551
$$262$$ −490995. −0.441900
$$263$$ 863965. 0.770206 0.385103 0.922874i $$-0.374166\pi$$
0.385103 + 0.922874i $$0.374166\pi$$
$$264$$ −1.14507e6 −1.01117
$$265$$ 2.12617e6 1.85987
$$266$$ 0 0
$$267$$ −122718. −0.105349
$$268$$ −1.11301e6 −0.946592
$$269$$ −1.12069e6 −0.944290 −0.472145 0.881521i $$-0.656520\pi$$
−0.472145 + 0.881521i $$0.656520\pi$$
$$270$$ −491001. −0.409895
$$271$$ 1.14012e6 0.943030 0.471515 0.881858i $$-0.343707\pi$$
0.471515 + 0.881858i $$0.343707\pi$$
$$272$$ 335498. 0.274959
$$273$$ 0 0
$$274$$ −88359.6 −0.0711013
$$275$$ −1.13247e6 −0.903012
$$276$$ −1.63188e6 −1.28948
$$277$$ −1.98801e6 −1.55675 −0.778375 0.627799i $$-0.783956\pi$$
−0.778375 + 0.627799i $$0.783956\pi$$
$$278$$ −1.71923e6 −1.33420
$$279$$ 664720. 0.511244
$$280$$ 0 0
$$281$$ 532321. 0.402168 0.201084 0.979574i $$-0.435554\pi$$
0.201084 + 0.979574i $$0.435554\pi$$
$$282$$ −1.38134e6 −1.03437
$$283$$ 2.62473e6 1.94813 0.974067 0.226259i $$-0.0726497\pi$$
0.974067 + 0.226259i $$0.0726497\pi$$
$$284$$ 3.48956e6 2.56729
$$285$$ 883468. 0.644286
$$286$$ 2.80842e6 2.03024
$$287$$ 0 0
$$288$$ 334526. 0.237656
$$289$$ −408424. −0.287651
$$290$$ −507473. −0.354338
$$291$$ 116275. 0.0804925
$$292$$ 1.52274e6 1.04512
$$293$$ 609962. 0.415082 0.207541 0.978226i $$-0.433454\pi$$
0.207541 + 0.978226i $$0.433454\pi$$
$$294$$ 0 0
$$295$$ 742834. 0.496978
$$296$$ 637424. 0.422863
$$297$$ 408685. 0.268842
$$298$$ 1.56812e6 1.02291
$$299$$ 1.72221e6 1.11406
$$300$$ −1.02119e6 −0.655096
$$301$$ 0 0
$$302$$ −3.53307e6 −2.22913
$$303$$ −226286. −0.141596
$$304$$ 456536. 0.283329
$$305$$ −68442.3 −0.0421284
$$306$$ −764915. −0.466992
$$307$$ −1.34843e6 −0.816551 −0.408275 0.912859i $$-0.633870\pi$$
−0.408275 + 0.912859i $$0.633870\pi$$
$$308$$ 0 0
$$309$$ 1.44678e6 0.861997
$$310$$ 5.52724e6 3.26666
$$311$$ −3.33061e6 −1.95264 −0.976320 0.216330i $$-0.930591\pi$$
−0.976320 + 0.216330i $$0.930591\pi$$
$$312$$ 1.08972e6 0.633762
$$313$$ −2.83670e6 −1.63664 −0.818320 0.574763i $$-0.805094\pi$$
−0.818320 + 0.574763i $$0.805094\pi$$
$$314$$ −366909. −0.210007
$$315$$ 0 0
$$316$$ 2.51007e6 1.41406
$$317$$ −1.08839e6 −0.608326 −0.304163 0.952620i $$-0.598377\pi$$
−0.304163 + 0.952620i $$0.598377\pi$$
$$318$$ −2.50498e6 −1.38911
$$319$$ 422395. 0.232403
$$320$$ 3.54734e6 1.93655
$$321$$ 849380. 0.460086
$$322$$ 0 0
$$323$$ 1.37633e6 0.734033
$$324$$ 368529. 0.195033
$$325$$ 1.07772e6 0.565975
$$326$$ 4.48760e6 2.33867
$$327$$ 750541. 0.388155
$$328$$ −55790.4 −0.0286335
$$329$$ 0 0
$$330$$ 3.39827e6 1.71780
$$331$$ 1.30555e6 0.654971 0.327485 0.944856i $$-0.393799\pi$$
0.327485 + 0.944856i $$0.393799\pi$$
$$332$$ 876619. 0.436481
$$333$$ −227502. −0.112428
$$334$$ −373654. −0.183275
$$335$$ 1.42133e6 0.691961
$$336$$ 0 0
$$337$$ −3.17016e6 −1.52057 −0.760285 0.649590i $$-0.774941\pi$$
−0.760285 + 0.649590i $$0.774941\pi$$
$$338$$ 813739. 0.387430
$$339$$ 1.61328e6 0.762450
$$340$$ −4.05196e6 −1.90094
$$341$$ −4.60060e6 −2.14254
$$342$$ −1.04087e6 −0.481207
$$343$$ 0 0
$$344$$ 3.97263e6 1.81002
$$345$$ 2.08392e6 0.942615
$$346$$ −453889. −0.203826
$$347$$ −1.71592e6 −0.765019 −0.382510 0.923951i $$-0.624940\pi$$
−0.382510 + 0.923951i $$0.624940\pi$$
$$348$$ 380892. 0.168598
$$349$$ −2.95822e6 −1.30007 −0.650034 0.759905i $$-0.725245\pi$$
−0.650034 + 0.759905i $$0.725245\pi$$
$$350$$ 0 0
$$351$$ −388928. −0.168501
$$352$$ −2.31529e6 −0.995976
$$353$$ 3.76980e6 1.61021 0.805103 0.593135i $$-0.202110\pi$$
0.805103 + 0.593135i $$0.202110\pi$$
$$354$$ −875182. −0.371185
$$355$$ −4.45620e6 −1.87669
$$356$$ 765890. 0.320289
$$357$$ 0 0
$$358$$ 1.34191e6 0.553371
$$359$$ 1.92987e6 0.790300 0.395150 0.918617i $$-0.370693\pi$$
0.395150 + 0.918617i $$0.370693\pi$$
$$360$$ 1.31859e6 0.536232
$$361$$ −603234. −0.243623
$$362$$ 725323. 0.290911
$$363$$ −1.37909e6 −0.549323
$$364$$ 0 0
$$365$$ −1.94455e6 −0.763988
$$366$$ 80636.5 0.0314651
$$367$$ 2.36742e6 0.917509 0.458754 0.888563i $$-0.348296\pi$$
0.458754 + 0.888563i $$0.348296\pi$$
$$368$$ 1.07688e6 0.414521
$$369$$ 19912.0 0.00761289
$$370$$ −1.89171e6 −0.718373
$$371$$ 0 0
$$372$$ −4.14856e6 −1.55432
$$373$$ 3.53829e6 1.31680 0.658402 0.752666i $$-0.271233\pi$$
0.658402 + 0.752666i $$0.271233\pi$$
$$374$$ 5.29406e6 1.95709
$$375$$ −713307. −0.261938
$$376$$ 3.70960e6 1.35318
$$377$$ −401975. −0.145662
$$378$$ 0 0
$$379$$ 1.79847e6 0.643139 0.321569 0.946886i $$-0.395790\pi$$
0.321569 + 0.946886i $$0.395790\pi$$
$$380$$ −5.51378e6 −1.95880
$$381$$ 1.29307e6 0.456361
$$382$$ −2.55455e6 −0.895686
$$383$$ 2.60815e6 0.908521 0.454261 0.890869i $$-0.349904\pi$$
0.454261 + 0.890869i $$0.349904\pi$$
$$384$$ −2.98994e6 −1.03475
$$385$$ 0 0
$$386$$ 150555. 0.0514313
$$387$$ −1.41786e6 −0.481235
$$388$$ −725683. −0.244719
$$389$$ −2.82995e6 −0.948211 −0.474106 0.880468i $$-0.657228\pi$$
−0.474106 + 0.880468i $$0.657228\pi$$
$$390$$ −3.23399e6 −1.07666
$$391$$ 3.24648e6 1.07392
$$392$$ 0 0
$$393$$ −470609. −0.153702
$$394$$ −9.69299e6 −3.14570
$$395$$ −3.20538e6 −1.03368
$$396$$ −2.55063e6 −0.817352
$$397$$ −2.43062e6 −0.773999 −0.387000 0.922080i $$-0.626489\pi$$
−0.387000 + 0.922080i $$0.626489\pi$$
$$398$$ 8.27939e6 2.61993
$$399$$ 0 0
$$400$$ 673885. 0.210589
$$401$$ −2.43184e6 −0.755222 −0.377611 0.925964i $$-0.623254\pi$$
−0.377611 + 0.925964i $$0.623254\pi$$
$$402$$ −1.67456e6 −0.516815
$$403$$ 4.37820e6 1.34287
$$404$$ 1.41227e6 0.430491
$$405$$ −470614. −0.142570
$$406$$ 0 0
$$407$$ 1.57457e6 0.471167
$$408$$ 2.05419e6 0.610927
$$409$$ −4.77466e6 −1.41135 −0.705674 0.708537i $$-0.749356\pi$$
−0.705674 + 0.708537i $$0.749356\pi$$
$$410$$ 165571. 0.0486436
$$411$$ −84690.9 −0.0247305
$$412$$ −9.02944e6 −2.62070
$$413$$ 0 0
$$414$$ −2.45521e6 −0.704024
$$415$$ −1.11945e6 −0.319069
$$416$$ 2.20336e6 0.624242
$$417$$ −1.64785e6 −0.464063
$$418$$ 7.20400e6 2.01666
$$419$$ −457181. −0.127219 −0.0636097 0.997975i $$-0.520261\pi$$
−0.0636097 + 0.997975i $$0.520261\pi$$
$$420$$ 0 0
$$421$$ −1.82396e6 −0.501545 −0.250773 0.968046i $$-0.580685\pi$$
−0.250773 + 0.968046i $$0.580685\pi$$
$$422$$ 3.50066e6 0.956905
$$423$$ −1.32398e6 −0.359776
$$424$$ 6.72715e6 1.81726
$$425$$ 2.03157e6 0.545582
$$426$$ 5.25014e6 1.40167
$$427$$ 0 0
$$428$$ −5.30104e6 −1.39879
$$429$$ 2.69181e6 0.706158
$$430$$ −1.17897e7 −3.07492
$$431$$ −3.38249e6 −0.877087 −0.438544 0.898710i $$-0.644506\pi$$
−0.438544 + 0.898710i $$0.644506\pi$$
$$432$$ −243192. −0.0626960
$$433$$ −285266. −0.0731190 −0.0365595 0.999331i $$-0.511640\pi$$
−0.0365595 + 0.999331i $$0.511640\pi$$
$$434$$ 0 0
$$435$$ −486402. −0.123246
$$436$$ −4.68418e6 −1.18010
$$437$$ 4.41771e6 1.10661
$$438$$ 2.29100e6 0.570611
$$439$$ 4.35220e6 1.07782 0.538911 0.842363i $$-0.318836\pi$$
0.538911 + 0.842363i $$0.318836\pi$$
$$440$$ −9.12610e6 −2.24726
$$441$$ 0 0
$$442$$ −5.03813e6 −1.22663
$$443$$ −5.10560e6 −1.23605 −0.618027 0.786157i $$-0.712068\pi$$
−0.618027 + 0.786157i $$0.712068\pi$$
$$444$$ 1.41985e6 0.341811
$$445$$ −978049. −0.234132
$$446$$ 1.01603e7 2.41862
$$447$$ 1.50301e6 0.355790
$$448$$ 0 0
$$449$$ 3.04163e6 0.712016 0.356008 0.934483i $$-0.384138\pi$$
0.356008 + 0.934483i $$0.384138\pi$$
$$450$$ −1.53642e6 −0.357666
$$451$$ −137813. −0.0319043
$$452$$ −1.00686e7 −2.31805
$$453$$ −3.38638e6 −0.775336
$$454$$ −5.19305e6 −1.18245
$$455$$ 0 0
$$456$$ 2.79528e6 0.629524
$$457$$ −1.74432e6 −0.390694 −0.195347 0.980734i $$-0.562583\pi$$
−0.195347 + 0.980734i $$0.562583\pi$$
$$458$$ −4.91113e6 −1.09400
$$459$$ −733156. −0.162429
$$460$$ −1.30059e7 −2.86580
$$461$$ −6.85701e6 −1.50273 −0.751367 0.659884i $$-0.770605\pi$$
−0.751367 + 0.659884i $$0.770605\pi$$
$$462$$ 0 0
$$463$$ 5.13844e6 1.11398 0.556992 0.830518i $$-0.311955\pi$$
0.556992 + 0.830518i $$0.311955\pi$$
$$464$$ −251351. −0.0541982
$$465$$ 5.29775e6 1.13621
$$466$$ 3.41961e6 0.729477
$$467$$ −4.58171e6 −0.972154 −0.486077 0.873916i $$-0.661573\pi$$
−0.486077 + 0.873916i $$0.661573\pi$$
$$468$$ 2.42733e6 0.512287
$$469$$ 0 0
$$470$$ −1.10091e7 −2.29883
$$471$$ −351675. −0.0730448
$$472$$ 2.35031e6 0.485591
$$473$$ 9.81320e6 2.01678
$$474$$ 3.77647e6 0.772042
$$475$$ 2.76450e6 0.562190
$$476$$ 0 0
$$477$$ −2.40097e6 −0.483161
$$478$$ −3.49154e6 −0.698952
$$479$$ −288523. −0.0574568 −0.0287284 0.999587i $$-0.509146\pi$$
−0.0287284 + 0.999587i $$0.509146\pi$$
$$480$$ 2.66614e6 0.528177
$$481$$ −1.49845e6 −0.295310
$$482$$ 1.60705e7 3.15074
$$483$$ 0 0
$$484$$ 8.60702e6 1.67009
$$485$$ 926703. 0.178890
$$486$$ 554462. 0.106483
$$487$$ −7.81685e6 −1.49351 −0.746757 0.665097i $$-0.768390\pi$$
−0.746757 + 0.665097i $$0.768390\pi$$
$$488$$ −216550. −0.0411632
$$489$$ 4.30127e6 0.813438
$$490$$ 0 0
$$491$$ 3.14467e6 0.588669 0.294335 0.955702i $$-0.404902\pi$$
0.294335 + 0.955702i $$0.404902\pi$$
$$492$$ −124272. −0.0231452
$$493$$ −757751. −0.140414
$$494$$ −6.85574e6 −1.26397
$$495$$ 3.25718e6 0.597487
$$496$$ 2.73763e6 0.499656
$$497$$ 0 0
$$498$$ 1.31890e6 0.238308
$$499$$ 6.72563e6 1.20915 0.604577 0.796547i $$-0.293342\pi$$
0.604577 + 0.796547i $$0.293342\pi$$
$$500$$ 4.45180e6 0.796362
$$501$$ −358140. −0.0637469
$$502$$ −545871. −0.0966787
$$503$$ −9.45056e6 −1.66547 −0.832737 0.553669i $$-0.813227\pi$$
−0.832737 + 0.553669i $$0.813227\pi$$
$$504$$ 0 0
$$505$$ −1.80348e6 −0.314690
$$506$$ 1.69928e7 2.95045
$$507$$ 779952. 0.134756
$$508$$ −8.07011e6 −1.38746
$$509$$ −8.83702e6 −1.51186 −0.755930 0.654653i $$-0.772815\pi$$
−0.755930 + 0.654653i $$0.772815\pi$$
$$510$$ −6.09629e6 −1.03786
$$511$$ 0 0
$$512$$ 3.80044e6 0.640707
$$513$$ −997656. −0.167374
$$514$$ 2.92813e6 0.488858
$$515$$ 1.15307e7 1.91574
$$516$$ 8.84899e6 1.46308
$$517$$ 9.16344e6 1.50776
$$518$$ 0 0
$$519$$ −435044. −0.0708949
$$520$$ 8.68492e6 1.40850
$$521$$ 694985. 0.112171 0.0560855 0.998426i $$-0.482138\pi$$
0.0560855 + 0.998426i $$0.482138\pi$$
$$522$$ 573063. 0.0920505
$$523$$ 3.58210e6 0.572643 0.286321 0.958134i $$-0.407568\pi$$
0.286321 + 0.958134i $$0.407568\pi$$
$$524$$ 2.93710e6 0.467294
$$525$$ 0 0
$$526$$ −8.11252e6 −1.27847
$$527$$ 8.25320e6 1.29448
$$528$$ 1.68316e6 0.262749
$$529$$ 3.98415e6 0.619009
$$530$$ −1.99644e7 −3.08722
$$531$$ −838845. −0.129106
$$532$$ 0 0
$$533$$ 131151. 0.0199965
$$534$$ 1.15230e6 0.174869
$$535$$ 6.76947e6 1.02252
$$536$$ 4.49705e6 0.676107
$$537$$ 1.28620e6 0.192474
$$538$$ 1.05231e7 1.56744
$$539$$ 0 0
$$540$$ 2.93714e6 0.433451
$$541$$ −5.16846e6 −0.759220 −0.379610 0.925147i $$-0.623942\pi$$
−0.379610 + 0.925147i $$0.623942\pi$$
$$542$$ −1.07055e7 −1.56535
$$543$$ 695208. 0.101185
$$544$$ 4.15349e6 0.601750
$$545$$ 5.98174e6 0.862653
$$546$$ 0 0
$$547$$ 8.47489e6 1.21106 0.605530 0.795822i $$-0.292961\pi$$
0.605530 + 0.795822i $$0.292961\pi$$
$$548$$ 528562. 0.0751873
$$549$$ 77288.5 0.0109442
$$550$$ 1.06337e7 1.49892
$$551$$ −1.03112e6 −0.144688
$$552$$ 6.59349e6 0.921018
$$553$$ 0 0
$$554$$ 1.86671e7 2.58407
$$555$$ −1.81317e6 −0.249865
$$556$$ 1.02843e7 1.41088
$$557$$ −1.04213e7 −1.42325 −0.711627 0.702557i $$-0.752041\pi$$
−0.711627 + 0.702557i $$0.752041\pi$$
$$558$$ −6.24163e6 −0.848619
$$559$$ −9.33880e6 −1.26404
$$560$$ 0 0
$$561$$ 5.07425e6 0.680714
$$562$$ −4.99842e6 −0.667562
$$563$$ −7.24077e6 −0.962751 −0.481375 0.876515i $$-0.659863\pi$$
−0.481375 + 0.876515i $$0.659863\pi$$
$$564$$ 8.26307e6 1.09381
$$565$$ 1.28577e7 1.69450
$$566$$ −2.46459e7 −3.23373
$$567$$ 0 0
$$568$$ −1.40993e7 −1.83369
$$569$$ 916536. 0.118678 0.0593388 0.998238i $$-0.481101\pi$$
0.0593388 + 0.998238i $$0.481101\pi$$
$$570$$ −8.29565e6 −1.06946
$$571$$ 1.00708e7 1.29262 0.646312 0.763073i $$-0.276311\pi$$
0.646312 + 0.763073i $$0.276311\pi$$
$$572$$ −1.67998e7 −2.14691
$$573$$ −2.44849e6 −0.311538
$$574$$ 0 0
$$575$$ 6.52091e6 0.822505
$$576$$ −4.00584e6 −0.503080
$$577$$ 1.16997e7 1.46298 0.731488 0.681855i $$-0.238826\pi$$
0.731488 + 0.681855i $$0.238826\pi$$
$$578$$ 3.83505e6 0.477475
$$579$$ 144304. 0.0178888
$$580$$ 3.03567e6 0.374701
$$581$$ 0 0
$$582$$ −1.09181e6 −0.133610
$$583$$ 1.66174e7 2.02485
$$584$$ −6.15251e6 −0.746484
$$585$$ −3.09972e6 −0.374483
$$586$$ −5.72747e6 −0.688999
$$587$$ −6.92367e6 −0.829357 −0.414678 0.909968i $$-0.636106\pi$$
−0.414678 + 0.909968i $$0.636106\pi$$
$$588$$ 0 0
$$589$$ 1.12307e7 1.33389
$$590$$ −6.97511e6 −0.824938
$$591$$ −9.29054e6 −1.09414
$$592$$ −936961. −0.109880
$$593$$ 1.57770e7 1.84242 0.921208 0.389069i $$-0.127203\pi$$
0.921208 + 0.389069i $$0.127203\pi$$
$$594$$ −3.83750e6 −0.446254
$$595$$ 0 0
$$596$$ −9.38041e6 −1.08170
$$597$$ 7.93563e6 0.911266
$$598$$ −1.61713e7 −1.84924
$$599$$ 1.42708e7 1.62511 0.812553 0.582887i $$-0.198077\pi$$
0.812553 + 0.582887i $$0.198077\pi$$
$$600$$ 4.12606e6 0.467905
$$601$$ 7.63222e6 0.861916 0.430958 0.902372i $$-0.358176\pi$$
0.430958 + 0.902372i $$0.358176\pi$$
$$602$$ 0 0
$$603$$ −1.60503e6 −0.179759
$$604$$ 2.11346e7 2.35723
$$605$$ −1.09912e7 −1.22084
$$606$$ 2.12480e6 0.235037
$$607$$ −3.56035e6 −0.392212 −0.196106 0.980583i $$-0.562830\pi$$
−0.196106 + 0.980583i $$0.562830\pi$$
$$608$$ 5.65194e6 0.620067
$$609$$ 0 0
$$610$$ 642664. 0.0699294
$$611$$ −8.72046e6 −0.945010
$$612$$ 4.57567e6 0.493829
$$613$$ −1.37284e7 −1.47560 −0.737800 0.675019i $$-0.764135\pi$$
−0.737800 + 0.675019i $$0.764135\pi$$
$$614$$ 1.26616e7 1.35540
$$615$$ 158697. 0.0169192
$$616$$ 0 0
$$617$$ −6.51173e6 −0.688626 −0.344313 0.938855i $$-0.611888\pi$$
−0.344313 + 0.938855i $$0.611888\pi$$
$$618$$ −1.35851e7 −1.43084
$$619$$ 8.85110e6 0.928476 0.464238 0.885711i $$-0.346328\pi$$
0.464238 + 0.885711i $$0.346328\pi$$
$$620$$ −3.30636e7 −3.45439
$$621$$ −2.35327e6 −0.244874
$$622$$ 3.12739e7 3.24121
$$623$$ 0 0
$$624$$ −1.60179e6 −0.164681
$$625$$ −1.19977e7 −1.22856
$$626$$ 2.66363e7 2.71667
$$627$$ 6.90489e6 0.701436
$$628$$ 2.19483e6 0.222076
$$629$$ −2.82467e6 −0.284670
$$630$$ 0 0
$$631$$ −6.89663e6 −0.689546 −0.344773 0.938686i $$-0.612044\pi$$
−0.344773 + 0.938686i $$0.612044\pi$$
$$632$$ −1.01418e7 −1.01000
$$633$$ 3.35531e6 0.332831
$$634$$ 1.02198e7 1.00977
$$635$$ 1.03056e7 1.01424
$$636$$ 1.49846e7 1.46894
$$637$$ 0 0
$$638$$ −3.96623e6 −0.385768
$$639$$ 5.03216e6 0.487531
$$640$$ −2.38295e7 −2.29967
$$641$$ −1.66695e7 −1.60242 −0.801210 0.598383i $$-0.795810\pi$$
−0.801210 + 0.598383i $$0.795810\pi$$
$$642$$ −7.97556e6 −0.763702
$$643$$ −1.28697e7 −1.22756 −0.613779 0.789478i $$-0.710351\pi$$
−0.613779 + 0.789478i $$0.710351\pi$$
$$644$$ 0 0
$$645$$ −1.13002e7 −1.06952
$$646$$ −1.29235e7 −1.21843
$$647$$ −1.14731e7 −1.07751 −0.538754 0.842463i $$-0.681105\pi$$
−0.538754 + 0.842463i $$0.681105\pi$$
$$648$$ −1.48901e6 −0.139303
$$649$$ 5.80574e6 0.541060
$$650$$ −1.01196e7 −0.939467
$$651$$ 0 0
$$652$$ −2.68445e7 −2.47307
$$653$$ −1.31801e7 −1.20958 −0.604791 0.796384i $$-0.706743\pi$$
−0.604791 + 0.796384i $$0.706743\pi$$
$$654$$ −7.04748e6 −0.644302
$$655$$ −3.75070e6 −0.341593
$$656$$ 82007.2 0.00744034
$$657$$ 2.19588e6 0.198470
$$658$$ 0 0
$$659$$ −1.13068e7 −1.01421 −0.507104 0.861885i $$-0.669284\pi$$
−0.507104 + 0.861885i $$0.669284\pi$$
$$660$$ −2.03282e7 −1.81652
$$661$$ 2.20319e6 0.196132 0.0980661 0.995180i $$-0.468734\pi$$
0.0980661 + 0.995180i $$0.468734\pi$$
$$662$$ −1.22589e7 −1.08719
$$663$$ −4.82895e6 −0.426647
$$664$$ −3.54192e6 −0.311758
$$665$$ 0 0
$$666$$ 2.13621e6 0.186620
$$667$$ −2.43222e6 −0.211684
$$668$$ 2.23518e6 0.193808
$$669$$ 9.73843e6 0.841247
$$670$$ −1.33461e7 −1.14859
$$671$$ −534922. −0.0458653
$$672$$ 0 0
$$673$$ −1.89787e7 −1.61521 −0.807606 0.589723i $$-0.799237\pi$$
−0.807606 + 0.589723i $$0.799237\pi$$
$$674$$ 2.97674e7 2.52401
$$675$$ −1.47262e6 −0.124403
$$676$$ −4.86773e6 −0.409694
$$677$$ −1.96475e7 −1.64754 −0.823771 0.566923i $$-0.808134\pi$$
−0.823771 + 0.566923i $$0.808134\pi$$
$$678$$ −1.51485e7 −1.26560
$$679$$ 0 0
$$680$$ 1.63717e7 1.35775
$$681$$ −4.97744e6 −0.411280
$$682$$ 4.31990e7 3.55642
$$683$$ 1.62705e7 1.33459 0.667295 0.744793i $$-0.267452\pi$$
0.667295 + 0.744793i $$0.267452\pi$$
$$684$$ 6.22644e6 0.508861
$$685$$ −674978. −0.0549621
$$686$$ 0 0
$$687$$ −4.70722e6 −0.380515
$$688$$ −5.83944e6 −0.470328
$$689$$ −1.58141e7 −1.26910
$$690$$ −1.95678e7 −1.56465
$$691$$ 1.94467e7 1.54935 0.774677 0.632357i $$-0.217912\pi$$
0.774677 + 0.632357i $$0.217912\pi$$
$$692$$ 2.71514e6 0.215539
$$693$$ 0 0
$$694$$ 1.61122e7 1.26986
$$695$$ −1.31332e7 −1.03135
$$696$$ −1.53897e6 −0.120422
$$697$$ 247229. 0.0192760
$$698$$ 2.77772e7 2.15800
$$699$$ 3.27763e6 0.253727
$$700$$ 0 0
$$701$$ 1.49625e7 1.15003 0.575014 0.818144i $$-0.304997\pi$$
0.575014 + 0.818144i $$0.304997\pi$$
$$702$$ 3.65198e6 0.279696
$$703$$ −3.84373e6 −0.293335
$$704$$ 2.77248e7 2.10832
$$705$$ −1.05520e7 −0.799581
$$706$$ −3.53979e7 −2.67280
$$707$$ 0 0
$$708$$ 5.23529e6 0.392516
$$709$$ 1.58639e6 0.118521 0.0592603 0.998243i $$-0.481126\pi$$
0.0592603 + 0.998243i $$0.481126\pi$$
$$710$$ 4.18431e7 3.11514
$$711$$ 3.61968e6 0.268532
$$712$$ −3.09453e6 −0.228767
$$713$$ 2.64910e7 1.95152
$$714$$ 0 0
$$715$$ 2.14535e7 1.56940
$$716$$ −8.02723e6 −0.585172
$$717$$ −3.34657e6 −0.243110
$$718$$ −1.81212e7 −1.31183
$$719$$ −1.81675e7 −1.31061 −0.655305 0.755364i $$-0.727460\pi$$
−0.655305 + 0.755364i $$0.727460\pi$$
$$720$$ −1.93822e6 −0.139338
$$721$$ 0 0
$$722$$ 5.66429e6 0.404392
$$723$$ 1.54033e7 1.09589
$$724$$ −4.33884e6 −0.307629
$$725$$ −1.52203e6 −0.107542
$$726$$ 1.29495e7 0.911826
$$727$$ 1.26903e7 0.890506 0.445253 0.895405i $$-0.353114\pi$$
0.445253 + 0.895405i $$0.353114\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ 1.82591e7 1.26815
$$731$$ −1.76043e7 −1.21850
$$732$$ −482362. −0.0332733
$$733$$ −3.11401e6 −0.214072 −0.107036 0.994255i $$-0.534136\pi$$
−0.107036 + 0.994255i $$0.534136\pi$$
$$734$$ −2.22298e7 −1.52298
$$735$$ 0 0
$$736$$ 1.33318e7 0.907182
$$737$$ 1.11086e7 0.753339
$$738$$ −186971. −0.0126367
$$739$$ 9.09899e6 0.612890 0.306445 0.951888i $$-0.400861\pi$$
0.306445 + 0.951888i $$0.400861\pi$$
$$740$$ 1.13161e7 0.759656
$$741$$ −6.57109e6 −0.439634
$$742$$ 0 0
$$743$$ −8.79218e6 −0.584285 −0.292142 0.956375i $$-0.594368\pi$$
−0.292142 + 0.956375i $$0.594368\pi$$
$$744$$ 1.67620e7 1.11018
$$745$$ 1.19789e7 0.790724
$$746$$ −3.32241e7 −2.18578
$$747$$ 1.26414e6 0.0828883
$$748$$ −3.16687e7 −2.06955
$$749$$ 0 0
$$750$$ 6.69786e6 0.434793
$$751$$ −2.80918e7 −1.81752 −0.908759 0.417320i $$-0.862969\pi$$
−0.908759 + 0.417320i $$0.862969\pi$$
$$752$$ −5.45280e6 −0.351621
$$753$$ −523207. −0.0336268
$$754$$ 3.77450e6 0.241786
$$755$$ −2.69891e7 −1.72314
$$756$$ 0 0
$$757$$ −4.18815e6 −0.265634 −0.132817 0.991141i $$-0.542402\pi$$
−0.132817 + 0.991141i $$0.542402\pi$$
$$758$$ −1.68874e7 −1.06755
$$759$$ 1.62872e7 1.02623
$$760$$ 2.22781e7 1.39908
$$761$$ −393182. −0.0246111 −0.0123056 0.999924i $$-0.503917\pi$$
−0.0123056 + 0.999924i $$0.503917\pi$$
$$762$$ −1.21417e7 −0.757518
$$763$$ 0 0
$$764$$ 1.52812e7 0.947159
$$765$$ −5.84317e6 −0.360990
$$766$$ −2.44901e7 −1.50806
$$767$$ −5.52508e6 −0.339117
$$768$$ 1.38321e7 0.846226
$$769$$ 1.57517e7 0.960530 0.480265 0.877124i $$-0.340541\pi$$
0.480265 + 0.877124i $$0.340541\pi$$
$$770$$ 0 0
$$771$$ 2.80655e6 0.170035
$$772$$ −900611. −0.0543869
$$773$$ 2.88472e6 0.173642 0.0868210 0.996224i $$-0.472329\pi$$
0.0868210 + 0.996224i $$0.472329\pi$$
$$774$$ 1.33136e7 0.798807
$$775$$ 1.65775e7 0.991433
$$776$$ 2.93207e6 0.174791
$$777$$ 0 0
$$778$$ 2.65729e7 1.57394
$$779$$ 336421. 0.0198628
$$780$$ 1.93455e7 1.13853
$$781$$ −3.48281e7 −2.04316
$$782$$ −3.04840e7 −1.78261
$$783$$ 549269. 0.0320170
$$784$$ 0 0
$$785$$ −2.80281e6 −0.162338
$$786$$ 4.41895e6 0.255131
$$787$$ −2.97866e7 −1.71429 −0.857145 0.515076i $$-0.827764\pi$$
−0.857145 + 0.515076i $$0.827764\pi$$
$$788$$ 5.79829e7 3.32647
$$789$$ −7.77569e6 −0.444679
$$790$$ 3.00981e7 1.71582
$$791$$ 0 0
$$792$$ 1.03056e7 0.583797
$$793$$ 509063. 0.0287467
$$794$$ 2.28232e7 1.28477
$$795$$ −1.91355e7 −1.07380
$$796$$ −4.95268e7 −2.77049
$$797$$ 8.11321e6 0.452425 0.226213 0.974078i $$-0.427366\pi$$
0.226213 + 0.974078i $$0.427366\pi$$
$$798$$ 0 0
$$799$$ −1.64387e7 −0.910960
$$800$$ 8.34274e6 0.460876
$$801$$ 1.10446e6 0.0608232
$$802$$ 2.28347e7 1.25360
$$803$$ −1.51979e7 −0.831756
$$804$$ 1.00171e7 0.546515
$$805$$ 0 0
$$806$$ −4.11107e7 −2.22904
$$807$$ 1.00862e7 0.545186
$$808$$ −5.70617e6 −0.307480
$$809$$ 111597. 0.00599489 0.00299744 0.999996i $$-0.499046\pi$$
0.00299744 + 0.999996i $$0.499046\pi$$
$$810$$ 4.41901e6 0.236653
$$811$$ −209250. −0.0111716 −0.00558578 0.999984i $$-0.501778\pi$$
−0.00558578 + 0.999984i $$0.501778\pi$$
$$812$$ 0 0
$$813$$ −1.02610e7 −0.544459
$$814$$ −1.47850e7 −0.782094
$$815$$ 3.42807e7 1.80782
$$816$$ −3.01949e6 −0.158748
$$817$$ −2.39554e7 −1.25559
$$818$$ 4.48334e7 2.34271
$$819$$ 0 0
$$820$$ −990437. −0.0514390
$$821$$ 6.84614e6 0.354477 0.177238 0.984168i $$-0.443284\pi$$
0.177238 + 0.984168i $$0.443284\pi$$
$$822$$ 795237. 0.0410503
$$823$$ −5.61210e6 −0.288819 −0.144409 0.989518i $$-0.546128\pi$$
−0.144409 + 0.989518i $$0.546128\pi$$
$$824$$ 3.64828e7 1.87185
$$825$$ 1.01922e7 0.521354
$$826$$ 0 0
$$827$$ −2.15868e7 −1.09755 −0.548776 0.835969i $$-0.684906\pi$$
−0.548776 + 0.835969i $$0.684906\pi$$
$$828$$ 1.46869e7 0.744483
$$829$$ 1.92427e7 0.972478 0.486239 0.873826i $$-0.338368\pi$$
0.486239 + 0.873826i $$0.338368\pi$$
$$830$$ 1.05115e7 0.529625
$$831$$ 1.78921e7 0.898790
$$832$$ −2.63846e7 −1.32142
$$833$$ 0 0
$$834$$ 1.54731e7 0.770303
$$835$$ −2.85434e6 −0.141674
$$836$$ −4.30939e7 −2.13255
$$837$$ −5.98248e6 −0.295167
$$838$$ 4.29287e6 0.211173
$$839$$ 2.47678e7 1.21474 0.607369 0.794420i $$-0.292225\pi$$
0.607369 + 0.794420i $$0.292225\pi$$
$$840$$ 0 0
$$841$$ −1.99435e7 −0.972323
$$842$$ 1.71267e7 0.832520
$$843$$ −4.79089e6 −0.232192
$$844$$ −2.09407e7 −1.01190
$$845$$ 6.21614e6 0.299488
$$846$$ 1.24320e7 0.597195
$$847$$ 0 0
$$848$$ −9.88836e6 −0.472210
$$849$$ −2.36226e7 −1.12476
$$850$$ −1.90762e7 −0.905618
$$851$$ −9.06659e6 −0.429161
$$852$$ −3.14060e7 −1.48222
$$853$$ −2.91267e7 −1.37062 −0.685312 0.728250i $$-0.740334\pi$$
−0.685312 + 0.728250i $$0.740334\pi$$
$$854$$ 0 0
$$855$$ −7.95121e6 −0.371979
$$856$$ 2.14185e7 0.999088
$$857$$ −3.22586e6 −0.150035 −0.0750175 0.997182i $$-0.523901\pi$$
−0.0750175 + 0.997182i $$0.523901\pi$$
$$858$$ −2.52758e7 −1.17216
$$859$$ 2.64671e7 1.22384 0.611918 0.790921i $$-0.290398\pi$$
0.611918 + 0.790921i $$0.290398\pi$$
$$860$$ 7.05255e7 3.25162
$$861$$ 0 0
$$862$$ 3.17611e7 1.45589
$$863$$ 1.58510e7 0.724487 0.362243 0.932084i $$-0.382011\pi$$
0.362243 + 0.932084i $$0.382011\pi$$
$$864$$ −3.01073e6 −0.137211
$$865$$ −3.46726e6 −0.157560
$$866$$ 2.67861e6 0.121371
$$867$$ 3.67582e6 0.166076
$$868$$ 0 0
$$869$$ −2.50522e7 −1.12537
$$870$$ 4.56725e6 0.204577
$$871$$ −1.05716e7 −0.472166
$$872$$ 1.89261e7 0.842888
$$873$$ −1.04648e6 −0.0464724
$$874$$ −4.14817e7 −1.83687
$$875$$ 0 0
$$876$$ −1.37046e7 −0.603403
$$877$$ −1.49132e7 −0.654743 −0.327372 0.944896i $$-0.606163\pi$$
−0.327372 + 0.944896i $$0.606163\pi$$
$$878$$ −4.08665e7 −1.78909
$$879$$ −5.48966e6 −0.239648
$$880$$ 1.34146e7 0.583944
$$881$$ −3.70679e7 −1.60901 −0.804504 0.593947i $$-0.797569\pi$$
−0.804504 + 0.593947i $$0.797569\pi$$
$$882$$ 0 0
$$883$$ 1.50440e7 0.649325 0.324663 0.945830i $$-0.394749\pi$$
0.324663 + 0.945830i $$0.394749\pi$$
$$884$$ 3.01378e7 1.29712
$$885$$ −6.68551e6 −0.286930
$$886$$ 4.79409e7 2.05174
$$887$$ 1.78101e7 0.760075 0.380038 0.924971i $$-0.375911\pi$$
0.380038 + 0.924971i $$0.375911\pi$$
$$888$$ −5.73682e6 −0.244140
$$889$$ 0 0
$$890$$ 9.18375e6 0.388638
$$891$$ −3.67816e6 −0.155216
$$892$$ −6.07782e7 −2.55762
$$893$$ −2.23692e7 −0.938690
$$894$$ −1.41131e7 −0.590580
$$895$$ 1.02508e7 0.427762
$$896$$ 0 0
$$897$$ −1.54999e7 −0.643202
$$898$$ −2.85605e7 −1.18188
$$899$$ −6.18317e6 −0.255160
$$900$$ 9.19074e6 0.378220
$$901$$ −2.98106e7 −1.22337
$$902$$ 1.29405e6 0.0529583
$$903$$ 0 0
$$904$$ 4.06815e7 1.65568
$$905$$ 5.54073e6 0.224877
$$906$$ 3.17977e7 1.28699
$$907$$ −1.16936e6 −0.0471987 −0.0235993 0.999721i $$-0.507513\pi$$
−0.0235993 + 0.999721i $$0.507513\pi$$
$$908$$ 3.10645e7 1.25040
$$909$$ 2.03658e6 0.0817506
$$910$$ 0 0
$$911$$ 2.74389e7 1.09539 0.547697 0.836677i $$-0.315505\pi$$
0.547697 + 0.836677i $$0.315505\pi$$
$$912$$ −4.10882e6 −0.163580
$$913$$ −8.74924e6 −0.347371
$$914$$ 1.63790e7 0.648516
$$915$$ 615981. 0.0243229
$$916$$ 2.93781e7 1.15687
$$917$$ 0 0
$$918$$ 6.88423e6 0.269618
$$919$$ 1.10218e7 0.430492 0.215246 0.976560i $$-0.430945\pi$$
0.215246 + 0.976560i $$0.430945\pi$$
$$920$$ 5.25495e7 2.04691
$$921$$ 1.21359e7 0.471436
$$922$$ 6.43864e7 2.49440
$$923$$ 3.31444e7 1.28058
$$924$$ 0 0
$$925$$ −5.67367e6 −0.218027
$$926$$ −4.82493e7 −1.84911
$$927$$ −1.30210e7 −0.497674
$$928$$ −3.11173e6 −0.118613
$$929$$ 4.25051e6 0.161585 0.0807927 0.996731i $$-0.474255\pi$$
0.0807927 + 0.996731i $$0.474255\pi$$
$$930$$ −4.97452e7 −1.88601
$$931$$ 0 0
$$932$$ −2.04559e7 −0.771398
$$933$$ 2.99755e7 1.12736
$$934$$ 4.30216e7 1.61369
$$935$$ 4.04413e7 1.51285
$$936$$ −9.80744e6 −0.365903
$$937$$ 3.75738e7 1.39809 0.699046 0.715077i $$-0.253608\pi$$
0.699046 + 0.715077i $$0.253608\pi$$
$$938$$ 0 0
$$939$$ 2.55303e7 0.944915
$$940$$ 6.58558e7 2.43094
$$941$$ 6.87043e6 0.252936 0.126468 0.991971i $$-0.459636\pi$$
0.126468 + 0.991971i $$0.459636\pi$$
$$942$$ 3.30218e6 0.121248
$$943$$ 793550. 0.0290600
$$944$$ −3.45476e6 −0.126179
$$945$$ 0 0
$$946$$ −9.21446e7 −3.34767
$$947$$ −2.30923e7 −0.836744 −0.418372 0.908276i $$-0.637399\pi$$
−0.418372 + 0.908276i $$0.637399\pi$$
$$948$$ −2.25906e7 −0.816409
$$949$$ 1.44632e7 0.521314
$$950$$ −2.59583e7 −0.933185
$$951$$ 9.79551e6 0.351217
$$952$$ 0 0
$$953$$ −2.95399e7 −1.05360 −0.526802 0.849988i $$-0.676609\pi$$
−0.526802 + 0.849988i $$0.676609\pi$$
$$954$$ 2.25448e7 0.802003
$$955$$ −1.95142e7 −0.692376
$$956$$ 2.08862e7 0.739119
$$957$$ −3.80156e6 −0.134178
$$958$$ 2.70919e6 0.0953730
$$959$$ 0 0
$$960$$ −3.19261e7 −1.11807
$$961$$ 3.87161e7 1.35233
$$962$$ 1.40702e7 0.490188
$$963$$ −7.64442e6 −0.265631
$$964$$ −9.61329e7 −3.33180
$$965$$ 1.15009e6 0.0397569
$$966$$ 0 0
$$967$$ 1.49151e7 0.512931 0.256466 0.966553i $$-0.417442\pi$$
0.256466 + 0.966553i $$0.417442\pi$$
$$968$$ −3.47761e7 −1.19287
$$969$$ −1.23869e7 −0.423794
$$970$$ −8.70162e6 −0.296941
$$971$$ 1.54830e7 0.526995 0.263498 0.964660i $$-0.415124\pi$$
0.263498 + 0.964660i $$0.415124\pi$$
$$972$$ −3.31676e6 −0.112603
$$973$$ 0 0
$$974$$ 7.33992e7 2.47910
$$975$$ −9.69947e6 −0.326766
$$976$$ 318311. 0.0106961
$$977$$ −2.83428e7 −0.949961 −0.474980 0.879996i $$-0.657545\pi$$
−0.474980 + 0.879996i $$0.657545\pi$$
$$978$$ −4.03884e7 −1.35023
$$979$$ −7.64410e6 −0.254900
$$980$$ 0 0
$$981$$ −6.75487e6 −0.224101
$$982$$ −2.95280e7 −0.977138
$$983$$ 2.56993e7 0.848278 0.424139 0.905597i $$-0.360577\pi$$
0.424139 + 0.905597i $$0.360577\pi$$
$$984$$ 502113. 0.0165316
$$985$$ −7.40446e7 −2.43166
$$986$$ 7.11518e6 0.233074
$$987$$ 0 0
$$988$$ 4.10106e7 1.33661
$$989$$ −5.65059e7 −1.83697
$$990$$ −3.05844e7 −0.991774
$$991$$ 4.46405e7 1.44392 0.721962 0.691933i $$-0.243240\pi$$
0.721962 + 0.691933i $$0.243240\pi$$
$$992$$ 3.38921e7 1.09350
$$993$$ −1.17499e7 −0.378148
$$994$$ 0 0
$$995$$ 6.32461e7 2.02524
$$996$$ −7.88957e6 −0.252003
$$997$$ 2.30554e7 0.734572 0.367286 0.930108i $$-0.380287\pi$$
0.367286 + 0.930108i $$0.380287\pi$$
$$998$$ −6.31528e7 −2.00709
$$999$$ 2.04752e6 0.0649103
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.i.1.1 2
3.2 odd 2 441.6.a.t.1.2 2
7.2 even 3 21.6.e.b.4.2 4
7.3 odd 6 147.6.e.l.79.2 4
7.4 even 3 21.6.e.b.16.2 yes 4
7.5 odd 6 147.6.e.l.67.2 4
7.6 odd 2 147.6.a.k.1.1 2
21.2 odd 6 63.6.e.c.46.1 4
21.11 odd 6 63.6.e.c.37.1 4
21.20 even 2 441.6.a.s.1.2 2
28.11 odd 6 336.6.q.e.289.2 4
28.23 odd 6 336.6.q.e.193.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 4 7.2 even 3
21.6.e.b.16.2 yes 4 7.4 even 3
63.6.e.c.37.1 4 21.11 odd 6
63.6.e.c.46.1 4 21.2 odd 6
147.6.a.i.1.1 2 1.1 even 1 trivial
147.6.a.k.1.1 2 7.6 odd 2
147.6.e.l.67.2 4 7.5 odd 6
147.6.e.l.79.2 4 7.3 odd 6
336.6.q.e.193.2 4 28.23 odd 6
336.6.q.e.289.2 4 28.11 odd 6
441.6.a.s.1.2 2 21.20 even 2
441.6.a.t.1.2 2 3.2 odd 2