# Properties

 Label 441.6.a.s.1.2 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ 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] = [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: $$1$$ 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 1.2 Root $$8.38987$$ 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)

 $$f(q)$$ $$=$$ $$q+9.38987 q^{2} +56.1696 q^{4} -71.7291 q^{5} +226.949 q^{8} +O(q^{10})$$ $$q+9.38987 q^{2} +56.1696 q^{4} -71.7291 q^{5} +226.949 q^{8} -673.526 q^{10} +560.610 q^{11} -533.509 q^{13} +333.597 q^{16} +1005.70 q^{17} -1368.53 q^{19} -4028.99 q^{20} +5264.05 q^{22} -3228.08 q^{23} +2020.06 q^{25} -5009.58 q^{26} +753.456 q^{29} -8206.42 q^{31} -4129.95 q^{32} +9443.39 q^{34} -2808.66 q^{37} -12850.3 q^{38} -16278.9 q^{40} +245.827 q^{41} -17504.5 q^{43} +31489.2 q^{44} -30311.2 q^{46} -16345.5 q^{47} +18968.1 q^{50} -29967.0 q^{52} +29641.7 q^{53} -40212.0 q^{55} +7074.85 q^{58} -10356.1 q^{59} -954.179 q^{61} -77057.2 q^{62} -49454.8 q^{64} +38268.1 q^{65} -19815.2 q^{67} +56489.8 q^{68} -62125.4 q^{71} -27109.6 q^{73} -26373.0 q^{74} -76869.6 q^{76} +44687.4 q^{79} -23928.6 q^{80} +2308.29 q^{82} +15606.6 q^{83} -72137.9 q^{85} -164365. q^{86} +127230. q^{88} +13635.3 q^{89} -181320. q^{92} -153482. q^{94} +98163.1 q^{95} +12919.5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 65 * q^4 - 33 * q^5 + 375 * q^8 $$2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29} - 4294 q^{31} - 1017 q^{32} + 17940 q^{34} - 19109 q^{37} - 6828 q^{38} - 10545 q^{40} - 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} - 23160 q^{47} + 29352 q^{50} - 33424 q^{52} + 31653 q^{53} - 17889 q^{55} - 2277 q^{58} + 41097 q^{59} - 42052 q^{61} - 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} + 44748 q^{68} - 102096 q^{71} + 28577 q^{73} + 77784 q^{74} - 85192 q^{76} - 18464 q^{79} - 71511 q^{80} + 86040 q^{82} + 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} + 29322 q^{89} - 166908 q^{92} - 109938 q^{94} + 61662 q^{95} + 9791 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 65 * q^4 - 33 * q^5 + 375 * q^8 - 921 * q^10 + 1137 * q^11 - 925 * q^13 - 895 * q^16 - 324 * q^17 - 2311 * q^19 - 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 - 2508 * q^26 + 2217 * q^29 - 4294 * q^31 - 1017 * q^32 + 17940 * q^34 - 19109 * q^37 - 6828 * q^38 - 10545 * q^40 - 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 - 23160 * q^47 + 29352 * q^50 - 33424 * q^52 + 31653 * q^53 - 17889 * q^55 - 2277 * q^58 + 41097 * q^59 - 42052 * q^61 - 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 + 44748 * q^68 - 102096 * q^71 + 28577 * q^73 + 77784 * q^74 - 85192 * q^76 - 18464 * q^79 - 71511 * q^80 + 86040 * q^82 + 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 + 29322 * q^89 - 166908 * q^92 - 109938 * q^94 + 61662 * q^95 + 9791 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 9.38987 1.65991 0.829955 0.557831i $$-0.188366\pi$$
0.829955 + 0.557831i $$0.188366\pi$$
$$3$$ 0 0
$$4$$ 56.1696 1.75530
$$5$$ −71.7291 −1.28313 −0.641564 0.767069i $$-0.721714\pi$$
−0.641564 + 0.767069i $$0.721714\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 226.949 1.25373
$$9$$ 0 0
$$10$$ −673.526 −2.12988
$$11$$ 560.610 1.39694 0.698472 0.715637i $$-0.253863\pi$$
0.698472 + 0.715637i $$0.253863\pi$$
$$12$$ 0 0
$$13$$ −533.509 −0.875555 −0.437777 0.899083i $$-0.644234\pi$$
−0.437777 + 0.899083i $$0.644234\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 333.597 0.325778
$$17$$ 1005.70 0.844007 0.422004 0.906594i $$-0.361327\pi$$
0.422004 + 0.906594i $$0.361327\pi$$
$$18$$ 0 0
$$19$$ −1368.53 −0.869699 −0.434850 0.900503i $$-0.643199\pi$$
−0.434850 + 0.900503i $$0.643199\pi$$
$$20$$ −4028.99 −2.25228
$$21$$ 0 0
$$22$$ 5264.05 2.31880
$$23$$ −3228.08 −1.27240 −0.636201 0.771523i $$-0.719495\pi$$
−0.636201 + 0.771523i $$0.719495\pi$$
$$24$$ 0 0
$$25$$ 2020.06 0.646419
$$26$$ −5009.58 −1.45334
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 753.456 0.166365 0.0831827 0.996534i $$-0.473492\pi$$
0.0831827 + 0.996534i $$0.473492\pi$$
$$30$$ 0 0
$$31$$ −8206.42 −1.53373 −0.766866 0.641807i $$-0.778185\pi$$
−0.766866 + 0.641807i $$0.778185\pi$$
$$32$$ −4129.95 −0.712968
$$33$$ 0 0
$$34$$ 9443.39 1.40098
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2808.66 −0.337284 −0.168642 0.985677i $$-0.553938\pi$$
−0.168642 + 0.985677i $$0.553938\pi$$
$$38$$ −12850.3 −1.44362
$$39$$ 0 0
$$40$$ −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
$$45$$ 0 0
$$46$$ −30311.2 −2.11207
$$47$$ −16345.5 −1.07933 −0.539663 0.841881i $$-0.681449\pi$$
−0.539663 + 0.841881i $$0.681449\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 18968.1 1.07300
$$51$$ 0 0
$$52$$ −29967.0 −1.53686
$$53$$ 29641.7 1.44948 0.724741 0.689021i $$-0.241959\pi$$
0.724741 + 0.689021i $$0.241959\pi$$
$$54$$ 0 0
$$55$$ −40212.0 −1.79246
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 7074.85 0.276151
$$59$$ −10356.1 −0.387317 −0.193659 0.981069i $$-0.562035\pi$$
−0.193659 + 0.981069i $$0.562035\pi$$
$$60$$ 0 0
$$61$$ −954.179 −0.0328326 −0.0164163 0.999865i $$-0.505226\pi$$
−0.0164163 + 0.999865i $$0.505226\pi$$
$$62$$ −77057.2 −2.54586
$$63$$ 0 0
$$64$$ −49454.8 −1.50924
$$65$$ 38268.1 1.12345
$$66$$ 0 0
$$67$$ −19815.2 −0.539276 −0.269638 0.962962i $$-0.586904\pi$$
−0.269638 + 0.962962i $$0.586904\pi$$
$$68$$ 56489.8 1.48149
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −62125.4 −1.46259 −0.731296 0.682060i $$-0.761084\pi$$
−0.731296 + 0.682060i $$0.761084\pi$$
$$72$$ 0 0
$$73$$ −27109.6 −0.595410 −0.297705 0.954658i $$-0.596221\pi$$
−0.297705 + 0.954658i $$0.596221\pi$$
$$74$$ −26373.0 −0.559861
$$75$$ 0 0
$$76$$ −76869.6 −1.52658
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 44687.4 0.805595 0.402798 0.915289i $$-0.368038\pi$$
0.402798 + 0.915289i $$0.368038\pi$$
$$80$$ −23928.6 −0.418015
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 127230. 1.75139
$$89$$ 13635.3 0.182469 0.0912347 0.995829i $$-0.470919\pi$$
0.0912347 + 0.995829i $$0.470919\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −181320. −2.23345
$$93$$ 0 0
$$94$$ −153482. −1.79159
$$95$$ 98163.1 1.11594
$$96$$ 0 0
$$97$$ 12919.5 0.139417 0.0697086 0.997567i $$-0.477793\pi$$
0.0697086 + 0.997567i $$0.477793\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 113466. 1.13466
$$101$$ 25142.9 0.245252 0.122626 0.992453i $$-0.460868\pi$$
0.122626 + 0.992453i $$0.460868\pi$$
$$102$$ 0 0
$$103$$ 160753. 1.49302 0.746511 0.665373i $$-0.231727\pi$$
0.746511 + 0.665373i $$0.231727\pi$$
$$104$$ −121079. −1.09771
$$105$$ 0 0
$$106$$ 278331. 2.40601
$$107$$ 94375.5 0.796893 0.398446 0.917192i $$-0.369549\pi$$
0.398446 + 0.917192i $$0.369549\pi$$
$$108$$ 0 0
$$109$$ −83393.5 −0.672304 −0.336152 0.941808i $$-0.609126\pi$$
−0.336152 + 0.941808i $$0.609126\pi$$
$$110$$ −377586. −2.97532
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 179254. 1.32060 0.660301 0.751001i $$-0.270429\pi$$
0.660301 + 0.751001i $$0.270429\pi$$
$$114$$ 0 0
$$115$$ 231547. 1.63266
$$116$$ 42321.3 0.292021
$$117$$ 0 0
$$118$$ −97242.5 −0.642911
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 153233. 0.951455
$$122$$ −8959.61 −0.0544991
$$123$$ 0 0
$$124$$ −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
$$129$$ 0 0
$$130$$ 359332. 1.86482
$$131$$ 52289.9 0.266219 0.133110 0.991101i $$-0.457504\pi$$
0.133110 + 0.991101i $$0.457504\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −186062. −0.895150
$$135$$ 0 0
$$136$$ 228243. 1.05816
$$137$$ −9410.10 −0.0428344 −0.0214172 0.999771i $$-0.506818\pi$$
−0.0214172 + 0.999771i $$0.506818\pi$$
$$138$$ 0 0
$$139$$ −183094. −0.803781 −0.401890 0.915688i $$-0.631647\pi$$
−0.401890 + 0.915688i $$0.631647\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −583349. −2.42777
$$143$$ −299090. −1.22310
$$144$$ 0 0
$$145$$ −54044.7 −0.213468
$$146$$ −254556. −0.988328
$$147$$ 0 0
$$148$$ −157762. −0.592034
$$149$$ 167002. 0.616247 0.308123 0.951346i $$-0.400299\pi$$
0.308123 + 0.951346i $$0.400299\pi$$
$$150$$ 0 0
$$151$$ 376264. 1.34292 0.671461 0.741040i $$-0.265667\pi$$
0.671461 + 0.741040i $$0.265667\pi$$
$$152$$ −310586. −1.09037
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 588639. 1.96797
$$156$$ 0 0
$$157$$ −39075.0 −0.126517 −0.0632587 0.997997i $$-0.520149\pi$$
−0.0632587 + 0.997997i $$0.520149\pi$$
$$158$$ 419608. 1.33722
$$159$$ 0 0
$$160$$ 296237. 0.914829
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −477919. −1.40892 −0.704458 0.709745i $$-0.748810\pi$$
−0.704458 + 0.709745i $$0.748810\pi$$
$$164$$ 13808.0 0.0400887
$$165$$ 0 0
$$166$$ 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
$$171$$ 0 0
$$172$$ −983221. −2.53414
$$173$$ 48338.2 0.122794 0.0613968 0.998113i $$-0.480445\pi$$
0.0613968 + 0.998113i $$0.480445\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 187018. 0.455094
$$177$$ 0 0
$$178$$ 128034. 0.302883
$$179$$ 142911. 0.333374 0.166687 0.986010i $$-0.446693\pi$$
0.166687 + 0.986010i $$0.446693\pi$$
$$180$$ 0 0
$$181$$ 77245.3 0.175257 0.0876285 0.996153i $$-0.472071\pi$$
0.0876285 + 0.996153i $$0.472071\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −732610. −1.59525
$$185$$ 201463. 0.432778
$$186$$ 0 0
$$187$$ 563806. 1.17903
$$188$$ −918119. −1.89454
$$189$$ 0 0
$$190$$ 921739. 1.85235
$$191$$ −272054. −0.539600 −0.269800 0.962916i $$-0.586958\pi$$
−0.269800 + 0.962916i $$0.586958\pi$$
$$192$$ 0 0
$$193$$ −16033.8 −0.0309844 −0.0154922 0.999880i $$-0.504932\pi$$
−0.0154922 + 0.999880i $$0.504932\pi$$
$$194$$ 121312. 0.231420
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.03228e6 −1.89510 −0.947552 0.319603i $$-0.896451\pi$$
−0.947552 + 0.319603i $$0.896451\pi$$
$$198$$ 0 0
$$199$$ 881736. 1.57836 0.789180 0.614162i $$-0.210506\pi$$
0.789180 + 0.614162i $$0.210506\pi$$
$$200$$ 458451. 0.810435
$$201$$ 0 0
$$202$$ 236089. 0.407096
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −17633.0 −0.0293049
$$206$$ 1.50945e6 2.47828
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ 886174. 1.32277
$$215$$ 1.25558e6 1.85246
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −783054. −1.11596
$$219$$ 0 0
$$220$$ −2.25869e6 −3.14630
$$221$$ −536550. −0.738975
$$222$$ 0 0
$$223$$ 1.08205e6 1.45708 0.728541 0.685002i $$-0.240199\pi$$
0.728541 + 0.685002i $$0.240199\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 1.68317e6 2.19208
$$227$$ 553049. 0.712359 0.356179 0.934418i $$-0.384079\pi$$
0.356179 + 0.934418i $$0.384079\pi$$
$$228$$ 0 0
$$229$$ −523024. −0.659072 −0.329536 0.944143i $$-0.606892\pi$$
−0.329536 + 0.944143i $$0.606892\pi$$
$$230$$ 2.17420e6 2.71006
$$231$$ 0 0
$$232$$ 170996. 0.208577
$$233$$ 364181. 0.439468 0.219734 0.975560i $$-0.429481\pi$$
0.219734 + 0.975560i $$0.429481\pi$$
$$234$$ 0 0
$$235$$ 1.17245e6 1.38492
$$236$$ −581698. −0.679858
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −371841. −0.421078 −0.210539 0.977585i $$-0.567522\pi$$
−0.210539 + 0.977585i $$0.567522\pi$$
$$240$$ 0 0
$$241$$ 1.71147e6 1.89814 0.949069 0.315067i $$-0.102027\pi$$
0.949069 + 0.315067i $$0.102027\pi$$
$$242$$ 1.43883e6 1.57933
$$243$$ 0 0
$$244$$ −53595.8 −0.0576310
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 730121. 0.761469
$$248$$ −1.86244e6 −1.92289
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ −1.53691e6 −1.46571
$$257$$ −311839. −0.294509 −0.147254 0.989099i $$-0.547044\pi$$
−0.147254 + 0.989099i $$0.547044\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 2.14950e6 1.97199
$$261$$ 0 0
$$262$$ 490995. 0.441900
$$263$$ −863965. −0.770206 −0.385103 0.922874i $$-0.625834\pi$$
−0.385103 + 0.922874i $$0.625834\pi$$
$$264$$ 0 0
$$265$$ −2.12617e6 −1.85987
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −1.11301e6 −0.946592
$$269$$ −1.12069e6 −0.944290 −0.472145 0.881521i $$-0.656520\pi$$
−0.472145 + 0.881521i $$0.656520\pi$$
$$270$$ 0 0
$$271$$ −1.14012e6 −0.943030 −0.471515 0.881858i $$-0.656293\pi$$
−0.471515 + 0.881858i $$0.656293\pi$$
$$272$$ 335498. 0.274959
$$273$$ 0 0
$$274$$ −88359.6 −0.0711013
$$275$$ 1.13247e6 0.903012
$$276$$ 0 0
$$277$$ −1.98801e6 −1.55675 −0.778375 0.627799i $$-0.783956\pi$$
−0.778375 + 0.627799i $$0.783956\pi$$
$$278$$ −1.71923e6 −1.33420
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −532321. −0.402168 −0.201084 0.979574i $$-0.564446\pi$$
−0.201084 + 0.979574i $$0.564446\pi$$
$$282$$ 0 0
$$283$$ −2.62473e6 −1.94813 −0.974067 0.226259i $$-0.927350\pi$$
−0.974067 + 0.226259i $$0.927350\pi$$
$$284$$ −3.48956e6 −2.56729
$$285$$ 0 0
$$286$$ −2.80842e6 −2.03024
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −408424. −0.287651
$$290$$ −507473. −0.354338
$$291$$ 0 0
$$292$$ −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
$$297$$ 0 0
$$298$$ 1.56812e6 1.02291
$$299$$ 1.72221e6 1.11406
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 3.53307e6 2.22913
$$303$$ 0 0
$$304$$ −456536. −0.283329
$$305$$ 68442.3 0.0421284
$$306$$ 0 0
$$307$$ 1.34843e6 0.816551 0.408275 0.912859i $$-0.366130\pi$$
0.408275 + 0.912859i $$0.366130\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 5.52724e6 3.26666
$$311$$ −3.33061e6 −1.95264 −0.976320 0.216330i $$-0.930591\pi$$
−0.976320 + 0.216330i $$0.930591\pi$$
$$312$$ 0 0
$$313$$ 2.83670e6 1.63664 0.818320 0.574763i $$-0.194906\pi$$
0.818320 + 0.574763i $$0.194906\pi$$
$$314$$ −366909. −0.210007
$$315$$ 0 0
$$316$$ 2.51007e6 1.41406
$$317$$ 1.08839e6 0.608326 0.304163 0.952620i $$-0.401623\pi$$
0.304163 + 0.952620i $$0.401623\pi$$
$$318$$ 0 0
$$319$$ 422395. 0.232403
$$320$$ 3.54734e6 1.93655
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.37633e6 −0.734033
$$324$$ 0 0
$$325$$ −1.07772e6 −0.565975
$$326$$ −4.48760e6 −2.33867
$$327$$ 0 0
$$328$$ 55790.4 0.0286335
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.30555e6 0.654971 0.327485 0.944856i $$-0.393799\pi$$
0.327485 + 0.944856i $$0.393799\pi$$
$$332$$ 876619. 0.436481
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ −4.05196e6 −1.90094
$$341$$ −4.60060e6 −2.14254
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −3.97263e6 −1.81002
$$345$$ 0 0
$$346$$ 453889. 0.203826
$$347$$ 1.71592e6 0.765019 0.382510 0.923951i $$-0.375060\pi$$
0.382510 + 0.923951i $$0.375060\pi$$
$$348$$ 0 0
$$349$$ 2.95822e6 1.30007 0.650034 0.759905i $$-0.274755\pi$$
0.650034 + 0.759905i $$0.274755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.31529e6 −0.995976
$$353$$ 3.76980e6 1.61021 0.805103 0.593135i $$-0.202110\pi$$
0.805103 + 0.593135i $$0.202110\pi$$
$$354$$ 0 0
$$355$$ 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 $$-0.629307\pi$$
−0.395150 + 0.918617i $$0.629307\pi$$
$$360$$ 0 0
$$361$$ −603234. −0.243623
$$362$$ 725323. 0.290911
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.94455e6 0.763988
$$366$$ 0 0
$$367$$ −2.36742e6 −0.917509 −0.458754 0.888563i $$-0.651704\pi$$
−0.458754 + 0.888563i $$0.651704\pi$$
$$368$$ −1.07688e6 −0.414521
$$369$$ 0 0
$$370$$ 1.89171e6 0.718373
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.53829e6 1.31680 0.658402 0.752666i $$-0.271233\pi$$
0.658402 + 0.752666i $$0.271233\pi$$
$$374$$ 5.29406e6 1.95709
$$375$$ 0 0
$$376$$ −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
$$381$$ 0 0
$$382$$ −2.55455e6 −0.895686
$$383$$ 2.60815e6 0.908521 0.454261 0.890869i $$-0.349904\pi$$
0.454261 + 0.890869i $$0.349904\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −150555. −0.0514313
$$387$$ 0 0
$$388$$ 725683. 0.244719
$$389$$ 2.82995e6 0.948211 0.474106 0.880468i $$-0.342772\pi$$
0.474106 + 0.880468i $$0.342772\pi$$
$$390$$ 0 0
$$391$$ −3.24648e6 −1.07392
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −9.69299e6 −3.14570
$$395$$ −3.20538e6 −1.03368
$$396$$ 0 0
$$397$$ 2.43062e6 0.773999 0.387000 0.922080i $$-0.373511\pi$$
0.387000 + 0.922080i $$0.373511\pi$$
$$398$$ 8.27939e6 2.61993
$$399$$ 0 0
$$400$$ 673885. 0.210589
$$401$$ 2.43184e6 0.755222 0.377611 0.925964i $$-0.376746\pi$$
0.377611 + 0.925964i $$0.376746\pi$$
$$402$$ 0 0
$$403$$ 4.37820e6 1.34287
$$404$$ 1.41227e6 0.430491
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.57457e6 −0.471167
$$408$$ 0 0
$$409$$ 4.77466e6 1.41135 0.705674 0.708537i $$-0.250644\pi$$
0.705674 + 0.708537i $$0.250644\pi$$
$$410$$ −165571. −0.0486436
$$411$$ 0 0
$$412$$ 9.02944e6 2.62070
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.11945e6 −0.319069
$$416$$ 2.20336e6 0.624242
$$417$$ 0 0
$$418$$ −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
$$423$$ 0 0
$$424$$ 6.72715e6 1.81726
$$425$$ 2.03157e6 0.545582
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 5.30104e6 1.39879
$$429$$ 0 0
$$430$$ 1.17897e7 3.07492
$$431$$ 3.38249e6 0.877087 0.438544 0.898710i $$-0.355494\pi$$
0.438544 + 0.898710i $$0.355494\pi$$
$$432$$ 0 0
$$433$$ 285266. 0.0731190 0.0365595 0.999331i $$-0.488360\pi$$
0.0365595 + 0.999331i $$0.488360\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4.68418e6 −1.18010
$$437$$ 4.41771e6 1.10661
$$438$$ 0 0
$$439$$ −4.35220e6 −1.07782 −0.538911 0.842363i $$-0.681164\pi$$
−0.538911 + 0.842363i $$0.681164\pi$$
$$440$$ −9.12610e6 −2.24726
$$441$$ 0 0
$$442$$ −5.03813e6 −1.22663
$$443$$ 5.10560e6 1.23605 0.618027 0.786157i $$-0.287932\pi$$
0.618027 + 0.786157i $$0.287932\pi$$
$$444$$ 0 0
$$445$$ −978049. −0.234132
$$446$$ 1.01603e7 2.41862
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3.04163e6 −0.712016 −0.356008 0.934483i $$-0.615862\pi$$
−0.356008 + 0.934483i $$0.615862\pi$$
$$450$$ 0 0
$$451$$ 137813. 0.0319043
$$452$$ 1.00686e7 2.31805
$$453$$ 0 0
$$454$$ 5.19305e6 1.18245
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.74432e6 −0.390694 −0.195347 0.980734i $$-0.562583\pi$$
−0.195347 + 0.980734i $$0.562583\pi$$
$$458$$ −4.91113e6 −1.09400
$$459$$ 0 0
$$460$$ 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
$$465$$ 0 0
$$466$$ 3.41961e6 0.729477
$$467$$ −4.58171e6 −0.972154 −0.486077 0.873916i $$-0.661573\pi$$
−0.486077 + 0.873916i $$0.661573\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1.10091e7 2.29883
$$471$$ 0 0
$$472$$ −2.35031e6 −0.485591
$$473$$ −9.81320e6 −2.01678
$$474$$ 0 0
$$475$$ −2.76450e6 −0.562190
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −3.49154e6 −0.698952
$$479$$ −288523. −0.0574568 −0.0287284 0.999587i $$-0.509146\pi$$
−0.0287284 + 0.999587i $$0.509146\pi$$
$$480$$ 0 0
$$481$$ 1.49845e6 0.295310
$$482$$ 1.60705e7 3.15074
$$483$$ 0 0
$$484$$ 8.60702e6 1.67009
$$485$$ −926703. −0.178890
$$486$$ 0 0
$$487$$ −7.81685e6 −1.49351 −0.746757 0.665097i $$-0.768390\pi$$
−0.746757 + 0.665097i $$0.768390\pi$$
$$488$$ −216550. −0.0411632
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.14467e6 −0.588669 −0.294335 0.955702i $$-0.595098\pi$$
−0.294335 + 0.955702i $$0.595098\pi$$
$$492$$ 0 0
$$493$$ 757751. 0.140414
$$494$$ 6.85574e6 1.26397
$$495$$ 0 0
$$496$$ −2.73763e6 −0.499656
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 6.72563e6 1.20915 0.604577 0.796547i $$-0.293342\pi$$
0.604577 + 0.796547i $$0.293342\pi$$
$$500$$ 4.45180e6 0.796362
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ −8.07011e6 −1.38746
$$509$$ −8.83702e6 −1.51186 −0.755930 0.654653i $$-0.772815\pi$$
−0.755930 + 0.654653i $$0.772815\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −3.80044e6 −0.640707
$$513$$ 0 0
$$514$$ −2.92813e6 −0.488858
$$515$$ −1.15307e7 −1.91574
$$516$$ 0 0
$$517$$ −9.16344e6 −1.50776
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 8.68492e6 1.40850
$$521$$ 694985. 0.112171 0.0560855 0.998426i $$-0.482138\pi$$
0.0560855 + 0.998426i $$0.482138\pi$$
$$522$$ 0 0
$$523$$ −3.58210e6 −0.572643 −0.286321 0.958134i $$-0.592432\pi$$
−0.286321 + 0.958134i $$0.592432\pi$$
$$524$$ 2.93710e6 0.467294
$$525$$ 0 0
$$526$$ −8.11252e6 −1.27847
$$527$$ −8.25320e6 −1.29448
$$528$$ 0 0
$$529$$ 3.98415e6 0.619009
$$530$$ −1.99644e7 −3.08722
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −131151. −0.0199965
$$534$$ 0 0
$$535$$ −6.76947e6 −1.02252
$$536$$ −4.49705e6 −0.676107
$$537$$ 0 0
$$538$$ −1.05231e7 −1.56744
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5.16846e6 −0.759220 −0.379610 0.925147i $$-0.623942\pi$$
−0.379610 + 0.925147i $$0.623942\pi$$
$$542$$ −1.07055e7 −1.56535
$$543$$ 0 0
$$544$$ −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
$$549$$ 0 0
$$550$$ 1.06337e7 1.49892
$$551$$ −1.03112e6 −0.144688
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −1.86671e7 −2.58407
$$555$$ 0 0
$$556$$ −1.02843e7 −1.41088
$$557$$ 1.04213e7 1.42325 0.711627 0.702557i $$-0.247959\pi$$
0.711627 + 0.702557i $$0.247959\pi$$
$$558$$ 0 0
$$559$$ 9.33880e6 1.26404
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −4.99842e6 −0.667562
$$563$$ −7.24077e6 −0.962751 −0.481375 0.876515i $$-0.659863\pi$$
−0.481375 + 0.876515i $$0.659863\pi$$
$$564$$ 0 0
$$565$$ −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 $$-0.518899\pi$$
−0.0593388 + 0.998238i $$0.518899\pi$$
$$570$$ 0 0
$$571$$ 1.00708e7 1.29262 0.646312 0.763073i $$-0.276311\pi$$
0.646312 + 0.763073i $$0.276311\pi$$
$$572$$ −1.67998e7 −2.14691
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.52091e6 −0.822505
$$576$$ 0 0
$$577$$ −1.16997e7 −1.46298 −0.731488 0.681855i $$-0.761174\pi$$
−0.731488 + 0.681855i $$0.761174\pi$$
$$578$$ −3.83505e6 −0.477475
$$579$$ 0 0
$$580$$ −3.03567e6 −0.374701
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.66174e7 2.02485
$$584$$ −6.15251e6 −0.746484
$$585$$ 0 0
$$586$$ 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
$$591$$ 0 0
$$592$$ −936961. −0.109880
$$593$$ 1.57770e7 1.84242 0.921208 0.389069i $$-0.127203\pi$$
0.921208 + 0.389069i $$0.127203\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 9.38041e6 1.08170
$$597$$ 0 0
$$598$$ 1.61713e7 1.84924
$$599$$ −1.42708e7 −1.62511 −0.812553 0.582887i $$-0.801923\pi$$
−0.812553 + 0.582887i $$0.801923\pi$$
$$600$$ 0 0
$$601$$ −7.63222e6 −0.861916 −0.430958 0.902372i $$-0.641824\pi$$
−0.430958 + 0.902372i $$0.641824\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 2.11346e7 2.35723
$$605$$ −1.09912e7 −1.22084
$$606$$ 0 0
$$607$$ 3.56035e6 0.392212 0.196106 0.980583i $$-0.437170\pi$$
0.196106 + 0.980583i $$0.437170\pi$$
$$608$$ 5.65194e6 0.620067
$$609$$ 0 0
$$610$$ 642664. 0.0699294
$$611$$ 8.72046e6 0.945010
$$612$$ 0 0
$$613$$ −1.37284e7 −1.47560 −0.737800 0.675019i $$-0.764135\pi$$
−0.737800 + 0.675019i $$0.764135\pi$$
$$614$$ 1.26616e7 1.35540
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.51173e6 0.688626 0.344313 0.938855i $$-0.388112\pi$$
0.344313 + 0.938855i $$0.388112\pi$$
$$618$$ 0 0
$$619$$ −8.85110e6 −0.928476 −0.464238 0.885711i $$-0.653672\pi$$
−0.464238 + 0.885711i $$0.653672\pi$$
$$620$$ 3.30636e7 3.45439
$$621$$ 0 0
$$622$$ −3.12739e7 −3.24121
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.19977e7 −1.22856
$$626$$ 2.66363e7 2.71667
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ 1.02198e7 1.00977
$$635$$ 1.03056e7 1.01424
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 3.96623e6 0.385768
$$639$$ 0 0
$$640$$ 2.38295e7 2.29967
$$641$$ 1.66695e7 1.60242 0.801210 0.598383i $$-0.204190\pi$$
0.801210 + 0.598383i $$0.204190\pi$$
$$642$$ 0 0
$$643$$ 1.28697e7 1.22756 0.613779 0.789478i $$-0.289649\pi$$
0.613779 + 0.789478i $$0.289649\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −1.29235e7 −1.21843
$$647$$ −1.14731e7 −1.07751 −0.538754 0.842463i $$-0.681105\pi$$
−0.538754 + 0.842463i $$0.681105\pi$$
$$648$$ 0 0
$$649$$ −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 $$-0.293257\pi$$
0.604791 + 0.796384i $$0.293257\pi$$
$$654$$ 0 0
$$655$$ −3.75070e6 −0.341593
$$656$$ 82007.2 0.00744034
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.13068e7 1.01421 0.507104 0.861885i $$-0.330716\pi$$
0.507104 + 0.861885i $$0.330716\pi$$
$$660$$ 0 0
$$661$$ −2.20319e6 −0.196132 −0.0980661 0.995180i $$-0.531266\pi$$
−0.0980661 + 0.995180i $$0.531266\pi$$
$$662$$ 1.22589e7 1.08719
$$663$$ 0 0
$$664$$ 3.54192e6 0.311758
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.43222e6 −0.211684
$$668$$ 2.23518e6 0.193808
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ −4.86773e6 −0.409694
$$677$$ −1.96475e7 −1.64754 −0.823771 0.566923i $$-0.808134\pi$$
−0.823771 + 0.566923i $$0.808134\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −1.63717e7 −1.35775
$$681$$ 0 0
$$682$$ −4.31990e7 −3.55642
$$683$$ −1.62705e7 −1.33459 −0.667295 0.744793i $$-0.732548\pi$$
−0.667295 + 0.744793i $$0.732548\pi$$
$$684$$ 0 0
$$685$$ 674978. 0.0549621
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −5.83944e6 −0.470328
$$689$$ −1.58141e7 −1.26910
$$690$$ 0 0
$$691$$ −1.94467e7 −1.54935 −0.774677 0.632357i $$-0.782088\pi$$
−0.774677 + 0.632357i $$0.782088\pi$$
$$692$$ 2.71514e6 0.215539
$$693$$ 0 0
$$694$$ 1.61122e7 1.26986
$$695$$ 1.31332e7 1.03135
$$696$$ 0 0
$$697$$ 247229. 0.0192760
$$698$$ 2.77772e7 2.15800
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.49625e7 −1.15003 −0.575014 0.818144i $$-0.695003\pi$$
−0.575014 + 0.818144i $$0.695003\pi$$
$$702$$ 0 0
$$703$$ 3.84373e6 0.293335
$$704$$ −2.77248e7 −2.10832
$$705$$ 0 0
$$706$$ 3.53979e7 2.67280
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.58639e6 0.118521 0.0592603 0.998243i $$-0.481126\pi$$
0.0592603 + 0.998243i $$0.481126\pi$$
$$710$$ 4.18431e7 3.11514
$$711$$ 0 0
$$712$$ 3.09453e6 0.228767
$$713$$ 2.64910e7 1.95152
$$714$$ 0 0
$$715$$ 2.14535e7 1.56940
$$716$$ 8.02723e6 0.585172
$$717$$ 0 0
$$718$$ −1.81212e7 −1.31183
$$719$$ −1.81675e7 −1.31061 −0.655305 0.755364i $$-0.727460\pi$$
−0.655305 + 0.755364i $$0.727460\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −5.66429e6 −0.404392
$$723$$ 0 0
$$724$$ 4.33884e6 0.307629
$$725$$ 1.52203e6 0.107542
$$726$$ 0 0
$$727$$ −1.26903e7 −0.890506 −0.445253 0.895405i $$-0.646886\pi$$
−0.445253 + 0.895405i $$0.646886\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 1.82591e7 1.26815
$$731$$ −1.76043e7 −1.21850
$$732$$ 0 0
$$733$$ 3.11401e6 0.214072 0.107036 0.994255i $$-0.465864\pi$$
0.107036 + 0.994255i $$0.465864\pi$$
$$734$$ −2.22298e7 −1.52298
$$735$$ 0 0
$$736$$ 1.33318e7 0.907182
$$737$$ −1.11086e7 −0.753339
$$738$$ 0 0
$$739$$ 9.09899e6 0.612890 0.306445 0.951888i $$-0.400861\pi$$
0.306445 + 0.951888i $$0.400861\pi$$
$$740$$ 1.13161e7 0.759656
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8.79218e6 0.584285 0.292142 0.956375i $$-0.405632\pi$$
0.292142 + 0.956375i $$0.405632\pi$$
$$744$$ 0 0
$$745$$ −1.19789e7 −0.790724
$$746$$ 3.32241e7 2.18578
$$747$$ 0 0
$$748$$ 3.16687e7 2.06955
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.80918e7 −1.81752 −0.908759 0.417320i $$-0.862969\pi$$
−0.908759 + 0.417320i $$0.862969\pi$$
$$752$$ −5.45280e6 −0.351621
$$753$$ 0 0
$$754$$ −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
$$759$$ 0 0
$$760$$ 2.22781e7 1.39908
$$761$$ −393182. −0.0246111 −0.0123056 0.999924i $$-0.503917\pi$$
−0.0123056 + 0.999924i $$0.503917\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −1.52812e7 −0.947159
$$765$$ 0 0
$$766$$ 2.44901e7 1.50806
$$767$$ 5.52508e6 0.339117
$$768$$ 0 0
$$769$$ −1.57517e7 −0.960530 −0.480265 0.877124i $$-0.659459\pi$$
−0.480265 + 0.877124i $$0.659459\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −900611. −0.0543869
$$773$$ 2.88472e6 0.173642 0.0868210 0.996224i $$-0.472329\pi$$
0.0868210 + 0.996224i $$0.472329\pi$$
$$774$$ 0 0
$$775$$ −1.65775e7 −0.991433
$$776$$ 2.93207e6 0.174791
$$777$$ 0 0
$$778$$ 2.65729e7 1.57394
$$779$$ −336421. −0.0198628
$$780$$ 0 0
$$781$$ −3.48281e7 −2.04316
$$782$$ −3.04840e7 −1.78261
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2.80281e6 0.162338
$$786$$ 0 0
$$787$$ 2.97866e7 1.71429 0.857145 0.515076i $$-0.172236\pi$$
0.857145 + 0.515076i $$0.172236\pi$$
$$788$$ −5.79829e7 −3.32647
$$789$$ 0 0
$$790$$ −3.00981e7 −1.71582
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 509063. 0.0287467
$$794$$ 2.28232e7 1.28477
$$795$$ 0 0
$$796$$ 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
$$801$$ 0 0
$$802$$ 2.28347e7 1.25360
$$803$$ −1.51979e7 −0.831756
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.11107e7 2.22904
$$807$$ 0 0
$$808$$ 5.70617e6 0.307480
$$809$$ −111597. −0.00599489 −0.00299744 0.999996i $$-0.500954\pi$$
−0.00299744 + 0.999996i $$0.500954\pi$$
$$810$$ 0 0
$$811$$ 209250. 0.0111716 0.00558578 0.999984i $$-0.498222\pi$$
0.00558578 + 0.999984i $$0.498222\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −1.47850e7 −0.782094
$$815$$ 3.42807e7 1.80782
$$816$$ 0 0
$$817$$ 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 $$-0.556716\pi$$
−0.177238 + 0.984168i $$0.556716\pi$$
$$822$$ 0 0
$$823$$ −5.61210e6 −0.288819 −0.144409 0.989518i $$-0.546128\pi$$
−0.144409 + 0.989518i $$0.546128\pi$$
$$824$$ 3.64828e7 1.87185
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.15868e7 1.09755 0.548776 0.835969i $$-0.315094\pi$$
0.548776 + 0.835969i $$0.315094\pi$$
$$828$$ 0 0
$$829$$ −1.92427e7 −0.972478 −0.486239 0.873826i $$-0.661632\pi$$
−0.486239 + 0.873826i $$0.661632\pi$$
$$830$$ −1.05115e7 −0.529625
$$831$$ 0 0
$$832$$ 2.63846e7 1.32142
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −2.85434e6 −0.141674
$$836$$ −4.30939e7 −2.13255
$$837$$ 0 0
$$838$$ −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
$$843$$ 0 0
$$844$$ −2.09407e7 −1.01190
$$845$$ 6.21614e6 0.299488
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 9.88836e6 0.472210
$$849$$ 0 0
$$850$$ 1.90762e7 0.905618
$$851$$ 9.06659e6 0.429161
$$852$$ 0 0
$$853$$ 2.91267e7 1.37062 0.685312 0.728250i $$-0.259666\pi$$
0.685312 + 0.728250i $$0.259666\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 2.14185e7 0.999088
$$857$$ −3.22586e6 −0.150035 −0.0750175 0.997182i $$-0.523901\pi$$
−0.0750175 + 0.997182i $$0.523901\pi$$
$$858$$ 0 0
$$859$$ −2.64671e7 −1.22384 −0.611918 0.790921i $$-0.709602\pi$$
−0.611918 + 0.790921i $$0.709602\pi$$
$$860$$ 7.05255e7 3.25162
$$861$$ 0 0
$$862$$ 3.17611e7 1.45589
$$863$$ −1.58510e7 −0.724487 −0.362243 0.932084i $$-0.617989\pi$$
−0.362243 + 0.932084i $$0.617989\pi$$
$$864$$ 0 0
$$865$$ −3.46726e6 −0.157560
$$866$$ 2.67861e6 0.121371
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2.50522e7 1.12537
$$870$$ 0 0
$$871$$ 1.05716e7 0.472166
$$872$$ −1.89261e7 −0.842888
$$873$$ 0 0
$$874$$ 4.14817e7 1.83687
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.49132e7 −0.654743 −0.327372 0.944896i $$-0.606163\pi$$
−0.327372 + 0.944896i $$0.606163\pi$$
$$878$$ −4.08665e7 −1.78909
$$879$$ 0 0
$$880$$ −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
$$885$$ 0 0
$$886$$ 4.79409e7 2.05174
$$887$$ 1.78101e7 0.760075 0.380038 0.924971i $$-0.375911\pi$$
0.380038 + 0.924971i $$0.375911\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −9.18375e6 −0.388638
$$891$$ 0 0
$$892$$ 6.07782e7 2.55762
$$893$$ 2.23692e7 0.938690
$$894$$ 0 0
$$895$$ −1.02508e7 −0.427762
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −2.85605e7 −1.18188
$$899$$ −6.18317e6 −0.255160
$$900$$ 0 0
$$901$$ 2.98106e7 1.22337
$$902$$ 1.29405e6 0.0529583
$$903$$ 0 0
$$904$$ 4.06815e7 1.65568
$$905$$ −5.54073e6 −0.224877
$$906$$ 0 0
$$907$$ −1.16936e6 −0.0471987 −0.0235993 0.999721i $$-0.507513\pi$$
−0.0235993 + 0.999721i $$0.507513\pi$$
$$908$$ 3.10645e7 1.25040
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2.74389e7 −1.09539 −0.547697 0.836677i $$-0.684495\pi$$
−0.547697 + 0.836677i $$0.684495\pi$$
$$912$$ 0 0
$$913$$ 8.74924e6 0.347371
$$914$$ −1.63790e7 −0.648516
$$915$$ 0 0
$$916$$ −2.93781e7 −1.15687
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.10218e7 0.430492 0.215246 0.976560i $$-0.430945\pi$$
0.215246 + 0.976560i $$0.430945\pi$$
$$920$$ 5.25495e7 2.04691
$$921$$ 0 0
$$922$$ −6.43864e7 −2.49440
$$923$$ 3.31444e7 1.28058
$$924$$ 0 0
$$925$$ −5.67367e6 −0.218027
$$926$$ 4.82493e7 1.84911
$$927$$ 0 0
$$928$$ −3.11173e6 −0.118613
$$929$$ 4.25051e6 0.161585 0.0807927 0.996731i $$-0.474255\pi$$
0.0807927 + 0.996731i $$0.474255\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 2.04559e7 0.771398
$$933$$ 0 0
$$934$$ −4.30216e7 −1.61369
$$935$$ −4.04413e7 −1.51285
$$936$$ 0 0
$$937$$ −3.75738e7 −1.39809 −0.699046 0.715077i $$-0.746392\pi$$
−0.699046 + 0.715077i $$0.746392\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 6.58558e7 2.43094
$$941$$ 6.87043e6 0.252936 0.126468 0.991971i $$-0.459636\pi$$
0.126468 + 0.991971i $$0.459636\pi$$
$$942$$ 0 0
$$943$$ −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 $$-0.362601\pi$$
0.418372 + 0.908276i $$0.362601\pi$$
$$948$$ 0 0
$$949$$ 1.44632e7 0.521314
$$950$$ −2.59583e7 −0.933185
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.95399e7 1.05360 0.526802 0.849988i $$-0.323391\pi$$
0.526802 + 0.849988i $$0.323391\pi$$
$$954$$ 0 0
$$955$$ 1.95142e7 0.692376
$$956$$ −2.08862e7 −0.739119
$$957$$ 0 0
$$958$$ −2.70919e6 −0.0953730
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 3.87161e7 1.35233
$$962$$ 1.40702e7 0.490188
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ −8.70162e6 −0.296941
$$971$$ 1.54830e7 0.526995 0.263498 0.964660i $$-0.415124\pi$$
0.263498 + 0.964660i $$0.415124\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −7.33992e7 −2.47910
$$975$$ 0 0
$$976$$ −318311. −0.0106961
$$977$$ 2.83428e7 0.949961 0.474980 0.879996i $$-0.342455\pi$$
0.474980 + 0.879996i $$0.342455\pi$$
$$978$$ 0 0
$$979$$ 7.64410e6 0.254900
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −2.95280e7 −0.977138
$$983$$ 2.56993e7 0.848278 0.424139 0.905597i $$-0.360577\pi$$
0.424139 + 0.905597i $$0.360577\pi$$
$$984$$ 0 0
$$985$$ 7.40446e7 2.43166
$$986$$ 7.11518e6 0.233074
$$987$$ 0 0
$$988$$ 4.10106e7 1.33661
$$989$$ 5.65059e7 1.83697
$$990$$ 0 0
$$991$$ 4.46405e7 1.44392 0.721962 0.691933i $$-0.243240\pi$$
0.721962 + 0.691933i $$0.243240\pi$$
$$992$$ 3.38921e7 1.09350
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −6.32461e7 −2.02524
$$996$$ 0 0
$$997$$ −2.30554e7 −0.734572 −0.367286 0.930108i $$-0.619713\pi$$
−0.367286 + 0.930108i $$0.619713\pi$$
$$998$$ 6.31528e7 2.00709
$$999$$ 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.s.1.2 2
3.2 odd 2 147.6.a.k.1.1 2
7.3 odd 6 63.6.e.c.37.1 4
7.5 odd 6 63.6.e.c.46.1 4
7.6 odd 2 441.6.a.t.1.2 2
21.2 odd 6 147.6.e.l.67.2 4
21.5 even 6 21.6.e.b.4.2 4
21.11 odd 6 147.6.e.l.79.2 4
21.17 even 6 21.6.e.b.16.2 yes 4
21.20 even 2 147.6.a.i.1.1 2
84.47 odd 6 336.6.q.e.193.2 4
84.59 odd 6 336.6.q.e.289.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 4 21.5 even 6
21.6.e.b.16.2 yes 4 21.17 even 6
63.6.e.c.37.1 4 7.3 odd 6
63.6.e.c.46.1 4 7.5 odd 6
147.6.a.i.1.1 2 21.20 even 2
147.6.a.k.1.1 2 3.2 odd 2
147.6.e.l.67.2 4 21.2 odd 6
147.6.e.l.79.2 4 21.11 odd 6
336.6.q.e.193.2 4 84.47 odd 6
336.6.q.e.289.2 4 84.59 odd 6
441.6.a.s.1.2 2 1.1 even 1 trivial
441.6.a.t.1.2 2 7.6 odd 2