# Properties

 Label 441.6.a.bb.1.4 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338$$ x^6 - 2*x^5 - 59*x^4 + 122*x^3 + 941*x^2 - 1856*x - 2338 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 7^{3}$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$4.27213$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +O(q^{10})$$ $$q+3.09163 q^{2} -22.4418 q^{4} +13.7926 q^{5} -168.314 q^{8} +42.6416 q^{10} +3.66659 q^{11} +780.229 q^{13} +197.776 q^{16} +50.0832 q^{17} -1063.38 q^{19} -309.532 q^{20} +11.3357 q^{22} -4102.45 q^{23} -2934.76 q^{25} +2412.18 q^{26} +1487.11 q^{29} -5519.97 q^{31} +5997.49 q^{32} +154.838 q^{34} +6143.27 q^{37} -3287.59 q^{38} -2321.49 q^{40} +10757.9 q^{41} +17696.7 q^{43} -82.2850 q^{44} -12683.3 q^{46} +29468.5 q^{47} -9073.19 q^{50} -17509.8 q^{52} +19255.9 q^{53} +50.5718 q^{55} +4597.58 q^{58} +6619.18 q^{59} -36750.3 q^{61} -17065.7 q^{62} +12213.2 q^{64} +10761.4 q^{65} +46909.2 q^{67} -1123.96 q^{68} +41693.7 q^{71} -29451.1 q^{73} +18992.7 q^{74} +23864.3 q^{76} +22124.4 q^{79} +2727.84 q^{80} +33259.5 q^{82} -3896.35 q^{83} +690.778 q^{85} +54711.6 q^{86} -617.138 q^{88} +20530.8 q^{89} +92066.6 q^{92} +91105.4 q^{94} -14666.8 q^{95} -17742.9 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10})$$ 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8 $$6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100})$$ 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8 - 864 * q^10 - 604 * q^11 - 1352 * q^13 + 4578 * q^16 + 3028 * q^17 - 1728 * q^19 + 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 + 14172 * q^26 + 5320 * q^29 - 3976 * q^31 + 37326 * q^32 + 16336 * q^34 + 22680 * q^37 + 52744 * q^38 - 100600 * q^40 + 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 + 51552 * q^47 + 40622 * q^50 - 119296 * q^52 - 80884 * q^53 - 11656 * q^55 - 70464 * q^58 + 8872 * q^59 - 50896 * q^61 + 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 + 37348 * q^68 + 110852 * q^71 - 64232 * q^73 + 27464 * q^74 + 194864 * q^76 + 111696 * q^79 - 308940 * q^80 + 189640 * q^82 + 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 - 35012 * q^89 + 449260 * q^92 + 121016 * q^94 + 119080 * q^95 - 70952 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.09163 0.546527 0.273264 0.961939i $$-0.411897\pi$$
0.273264 + 0.961939i $$0.411897\pi$$
$$3$$ 0 0
$$4$$ −22.4418 −0.701308
$$5$$ 13.7926 0.246730 0.123365 0.992361i $$-0.460631\pi$$
0.123365 + 0.992361i $$0.460631\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −168.314 −0.929811
$$9$$ 0 0
$$10$$ 42.6416 0.134845
$$11$$ 3.66659 0.00913651 0.00456826 0.999990i $$-0.498546\pi$$
0.00456826 + 0.999990i $$0.498546\pi$$
$$12$$ 0 0
$$13$$ 780.229 1.28045 0.640226 0.768186i $$-0.278841\pi$$
0.640226 + 0.768186i $$0.278841\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 197.776 0.193140
$$17$$ 50.0832 0.0420310 0.0210155 0.999779i $$-0.493310\pi$$
0.0210155 + 0.999779i $$0.493310\pi$$
$$18$$ 0 0
$$19$$ −1063.38 −0.675781 −0.337891 0.941185i $$-0.609713\pi$$
−0.337891 + 0.941185i $$0.609713\pi$$
$$20$$ −309.532 −0.173033
$$21$$ 0 0
$$22$$ 11.3357 0.00499336
$$23$$ −4102.45 −1.61705 −0.808526 0.588460i $$-0.799734\pi$$
−0.808526 + 0.588460i $$0.799734\pi$$
$$24$$ 0 0
$$25$$ −2934.76 −0.939124
$$26$$ 2412.18 0.699803
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1487.11 0.328358 0.164179 0.986431i $$-0.447503\pi$$
0.164179 + 0.986431i $$0.447503\pi$$
$$30$$ 0 0
$$31$$ −5519.97 −1.03165 −0.515825 0.856694i $$-0.672515\pi$$
−0.515825 + 0.856694i $$0.672515\pi$$
$$32$$ 5997.49 1.03537
$$33$$ 0 0
$$34$$ 154.838 0.0229711
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6143.27 0.737726 0.368863 0.929484i $$-0.379747\pi$$
0.368863 + 0.929484i $$0.379747\pi$$
$$38$$ −3287.59 −0.369333
$$39$$ 0 0
$$40$$ −2321.49 −0.229412
$$41$$ 10757.9 0.999468 0.499734 0.866179i $$-0.333431\pi$$
0.499734 + 0.866179i $$0.333431\pi$$
$$42$$ 0 0
$$43$$ 17696.7 1.45956 0.729779 0.683683i $$-0.239623\pi$$
0.729779 + 0.683683i $$0.239623\pi$$
$$44$$ −82.2850 −0.00640751
$$45$$ 0 0
$$46$$ −12683.3 −0.883763
$$47$$ 29468.5 1.94587 0.972933 0.231089i $$-0.0742289\pi$$
0.972933 + 0.231089i $$0.0742289\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −9073.19 −0.513257
$$51$$ 0 0
$$52$$ −17509.8 −0.897991
$$53$$ 19255.9 0.941616 0.470808 0.882236i $$-0.343962\pi$$
0.470808 + 0.882236i $$0.343962\pi$$
$$54$$ 0 0
$$55$$ 50.5718 0.00225425
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 4597.58 0.179457
$$59$$ 6619.18 0.247557 0.123778 0.992310i $$-0.460499\pi$$
0.123778 + 0.992310i $$0.460499\pi$$
$$60$$ 0 0
$$61$$ −36750.3 −1.26455 −0.632275 0.774744i $$-0.717879\pi$$
−0.632275 + 0.774744i $$0.717879\pi$$
$$62$$ −17065.7 −0.563825
$$63$$ 0 0
$$64$$ 12213.2 0.372717
$$65$$ 10761.4 0.315926
$$66$$ 0 0
$$67$$ 46909.2 1.27665 0.638323 0.769768i $$-0.279628\pi$$
0.638323 + 0.769768i $$0.279628\pi$$
$$68$$ −1123.96 −0.0294767
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 41693.7 0.981577 0.490789 0.871279i $$-0.336709\pi$$
0.490789 + 0.871279i $$0.336709\pi$$
$$72$$ 0 0
$$73$$ −29451.1 −0.646837 −0.323418 0.946256i $$-0.604832\pi$$
−0.323418 + 0.946256i $$0.604832\pi$$
$$74$$ 18992.7 0.403188
$$75$$ 0 0
$$76$$ 23864.3 0.473931
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 22124.4 0.398844 0.199422 0.979914i $$-0.436094\pi$$
0.199422 + 0.979914i $$0.436094\pi$$
$$80$$ 2727.84 0.0476535
$$81$$ 0 0
$$82$$ 33259.5 0.546237
$$83$$ −3896.35 −0.0620816 −0.0310408 0.999518i $$-0.509882\pi$$
−0.0310408 + 0.999518i $$0.509882\pi$$
$$84$$ 0 0
$$85$$ 690.778 0.0103703
$$86$$ 54711.6 0.797689
$$87$$ 0 0
$$88$$ −617.138 −0.00849523
$$89$$ 20530.8 0.274746 0.137373 0.990519i $$-0.456134\pi$$
0.137373 + 0.990519i $$0.456134\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 92066.6 1.13405
$$93$$ 0 0
$$94$$ 91105.4 1.06347
$$95$$ −14666.8 −0.166735
$$96$$ 0 0
$$97$$ −17742.9 −0.191468 −0.0957338 0.995407i $$-0.530520\pi$$
−0.0957338 + 0.995407i $$0.530520\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 65861.5 0.658615
$$101$$ −142085. −1.38594 −0.692971 0.720965i $$-0.743699\pi$$
−0.692971 + 0.720965i $$0.743699\pi$$
$$102$$ 0 0
$$103$$ 210538. 1.95540 0.977702 0.209996i $$-0.0673450\pi$$
0.977702 + 0.209996i $$0.0673450\pi$$
$$104$$ −131323. −1.19058
$$105$$ 0 0
$$106$$ 59532.0 0.514619
$$107$$ 202671. 1.71132 0.855660 0.517538i $$-0.173151\pi$$
0.855660 + 0.517538i $$0.173151\pi$$
$$108$$ 0 0
$$109$$ 139538. 1.12493 0.562466 0.826821i $$-0.309853\pi$$
0.562466 + 0.826821i $$0.309853\pi$$
$$110$$ 156.349 0.00123201
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 206316. 1.51998 0.759989 0.649936i $$-0.225204\pi$$
0.759989 + 0.649936i $$0.225204\pi$$
$$114$$ 0 0
$$115$$ −56583.5 −0.398975
$$116$$ −33373.4 −0.230280
$$117$$ 0 0
$$118$$ 20464.0 0.135296
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −161038. −0.999917
$$122$$ −113618. −0.691111
$$123$$ 0 0
$$124$$ 123878. 0.723504
$$125$$ −83580.0 −0.478440
$$126$$ 0 0
$$127$$ 89874.2 0.494454 0.247227 0.968958i $$-0.420481\pi$$
0.247227 + 0.968958i $$0.420481\pi$$
$$128$$ −154161. −0.831668
$$129$$ 0 0
$$130$$ 33270.2 0.172662
$$131$$ 320115. 1.62978 0.814889 0.579617i $$-0.196798\pi$$
0.814889 + 0.579617i $$0.196798\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 145026. 0.697722
$$135$$ 0 0
$$136$$ −8429.69 −0.0390809
$$137$$ −258971. −1.17883 −0.589414 0.807831i $$-0.700641\pi$$
−0.589414 + 0.807831i $$0.700641\pi$$
$$138$$ 0 0
$$139$$ −282366. −1.23958 −0.619791 0.784767i $$-0.712783\pi$$
−0.619791 + 0.784767i $$0.712783\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 128901. 0.536459
$$143$$ 2860.78 0.0116989
$$144$$ 0 0
$$145$$ 20511.1 0.0810156
$$146$$ −91051.9 −0.353514
$$147$$ 0 0
$$148$$ −137866. −0.517373
$$149$$ 319669. 1.17960 0.589800 0.807549i $$-0.299207\pi$$
0.589800 + 0.807549i $$0.299207\pi$$
$$150$$ 0 0
$$151$$ −84046.3 −0.299969 −0.149985 0.988688i $$-0.547922\pi$$
−0.149985 + 0.988688i $$0.547922\pi$$
$$152$$ 178982. 0.628349
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −76134.8 −0.254539
$$156$$ 0 0
$$157$$ −126195. −0.408596 −0.204298 0.978909i $$-0.565491\pi$$
−0.204298 + 0.978909i $$0.565491\pi$$
$$158$$ 68400.2 0.217979
$$159$$ 0 0
$$160$$ 82721.1 0.255456
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −235790. −0.695114 −0.347557 0.937659i $$-0.612989\pi$$
−0.347557 + 0.937659i $$0.612989\pi$$
$$164$$ −241428. −0.700935
$$165$$ 0 0
$$166$$ −12046.1 −0.0339293
$$167$$ 149843. 0.415761 0.207881 0.978154i $$-0.433343\pi$$
0.207881 + 0.978154i $$0.433343\pi$$
$$168$$ 0 0
$$169$$ 237464. 0.639559
$$170$$ 2135.63 0.00566765
$$171$$ 0 0
$$172$$ −397147. −1.02360
$$173$$ 704460. 1.78954 0.894769 0.446530i $$-0.147340\pi$$
0.894769 + 0.446530i $$0.147340\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 725.162 0.00176463
$$177$$ 0 0
$$178$$ 63473.6 0.150156
$$179$$ 557573. 1.30067 0.650337 0.759645i $$-0.274627\pi$$
0.650337 + 0.759645i $$0.274627\pi$$
$$180$$ 0 0
$$181$$ −443092. −1.00530 −0.502652 0.864489i $$-0.667642\pi$$
−0.502652 + 0.864489i $$0.667642\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 690500. 1.50355
$$185$$ 84731.7 0.182019
$$186$$ 0 0
$$187$$ 183.634 0.000384017 0
$$188$$ −661327. −1.36465
$$189$$ 0 0
$$190$$ −45344.4 −0.0911254
$$191$$ −157481. −0.312353 −0.156177 0.987729i $$-0.549917\pi$$
−0.156177 + 0.987729i $$0.549917\pi$$
$$192$$ 0 0
$$193$$ −778040. −1.50352 −0.751759 0.659437i $$-0.770795\pi$$
−0.751759 + 0.659437i $$0.770795\pi$$
$$194$$ −54854.4 −0.104642
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 340283. 0.624704 0.312352 0.949966i $$-0.398883\pi$$
0.312352 + 0.949966i $$0.398883\pi$$
$$198$$ 0 0
$$199$$ −296509. −0.530768 −0.265384 0.964143i $$-0.585499\pi$$
−0.265384 + 0.964143i $$0.585499\pi$$
$$200$$ 493961. 0.873209
$$201$$ 0 0
$$202$$ −439274. −0.757456
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 148380. 0.246599
$$206$$ 650904. 1.06868
$$207$$ 0 0
$$208$$ 154310. 0.247307
$$209$$ −3898.99 −0.00617429
$$210$$ 0 0
$$211$$ 506728. 0.783554 0.391777 0.920060i $$-0.371860\pi$$
0.391777 + 0.920060i $$0.371860\pi$$
$$212$$ −432138. −0.660363
$$213$$ 0 0
$$214$$ 626582. 0.935283
$$215$$ 244084. 0.360116
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 431399. 0.614806
$$219$$ 0 0
$$220$$ −1134.93 −0.00158092
$$221$$ 39076.3 0.0538187
$$222$$ 0 0
$$223$$ 462362. 0.622615 0.311308 0.950309i $$-0.399233\pi$$
0.311308 + 0.950309i $$0.399233\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 637852. 0.830709
$$227$$ −98226.6 −0.126521 −0.0632607 0.997997i $$-0.520150\pi$$
−0.0632607 + 0.997997i $$0.520150\pi$$
$$228$$ 0 0
$$229$$ −501403. −0.631827 −0.315914 0.948788i $$-0.602311\pi$$
−0.315914 + 0.948788i $$0.602311\pi$$
$$230$$ −174935. −0.218051
$$231$$ 0 0
$$232$$ −250301. −0.305311
$$233$$ 613916. 0.740831 0.370415 0.928866i $$-0.379215\pi$$
0.370415 + 0.928866i $$0.379215\pi$$
$$234$$ 0 0
$$235$$ 406447. 0.480103
$$236$$ −148547. −0.173613
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1.10020e6 −1.24588 −0.622940 0.782269i $$-0.714062\pi$$
−0.622940 + 0.782269i $$0.714062\pi$$
$$240$$ 0 0
$$241$$ −89883.5 −0.0996868 −0.0498434 0.998757i $$-0.515872\pi$$
−0.0498434 + 0.998757i $$0.515872\pi$$
$$242$$ −497868. −0.546482
$$243$$ 0 0
$$244$$ 824744. 0.886839
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −829683. −0.865306
$$248$$ 929087. 0.959240
$$249$$ 0 0
$$250$$ −258398. −0.261480
$$251$$ 1.01050e6 1.01240 0.506202 0.862415i $$-0.331049\pi$$
0.506202 + 0.862415i $$0.331049\pi$$
$$252$$ 0 0
$$253$$ −15042.0 −0.0147742
$$254$$ 277858. 0.270233
$$255$$ 0 0
$$256$$ −867430. −0.827246
$$257$$ −1.21233e6 −1.14495 −0.572475 0.819922i $$-0.694017\pi$$
−0.572475 + 0.819922i $$0.694017\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −241505. −0.221561
$$261$$ 0 0
$$262$$ 989677. 0.890718
$$263$$ −384901. −0.343131 −0.171566 0.985173i $$-0.554883\pi$$
−0.171566 + 0.985173i $$0.554883\pi$$
$$264$$ 0 0
$$265$$ 265589. 0.232325
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −1.05273e6 −0.895322
$$269$$ −1.57455e6 −1.32671 −0.663355 0.748305i $$-0.730868\pi$$
−0.663355 + 0.748305i $$0.730868\pi$$
$$270$$ 0 0
$$271$$ −332270. −0.274832 −0.137416 0.990513i $$-0.543880\pi$$
−0.137416 + 0.990513i $$0.543880\pi$$
$$272$$ 9905.24 0.00811788
$$273$$ 0 0
$$274$$ −800643. −0.644262
$$275$$ −10760.6 −0.00858032
$$276$$ 0 0
$$277$$ 2.23543e6 1.75050 0.875251 0.483669i $$-0.160696\pi$$
0.875251 + 0.483669i $$0.160696\pi$$
$$278$$ −872970. −0.677465
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −69723.6 −0.0526761 −0.0263381 0.999653i $$-0.508385\pi$$
−0.0263381 + 0.999653i $$0.508385\pi$$
$$282$$ 0 0
$$283$$ 476452. 0.353633 0.176817 0.984244i $$-0.443420\pi$$
0.176817 + 0.984244i $$0.443420\pi$$
$$284$$ −935684. −0.688388
$$285$$ 0 0
$$286$$ 8844.46 0.00639376
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.41735e6 −0.998233
$$290$$ 63412.6 0.0442773
$$291$$ 0 0
$$292$$ 660938. 0.453632
$$293$$ 2.27589e6 1.54875 0.774377 0.632724i $$-0.218063\pi$$
0.774377 + 0.632724i $$0.218063\pi$$
$$294$$ 0 0
$$295$$ 91295.8 0.0610796
$$296$$ −1.03400e6 −0.685946
$$297$$ 0 0
$$298$$ 988297. 0.644684
$$299$$ −3.20085e6 −2.07056
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −259840. −0.163941
$$303$$ 0 0
$$304$$ −210312. −0.130521
$$305$$ −506882. −0.312002
$$306$$ 0 0
$$307$$ 1.61790e6 0.979731 0.489866 0.871798i $$-0.337046\pi$$
0.489866 + 0.871798i $$0.337046\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −235380. −0.139112
$$311$$ −1.02667e6 −0.601910 −0.300955 0.953638i $$-0.597305\pi$$
−0.300955 + 0.953638i $$0.597305\pi$$
$$312$$ 0 0
$$313$$ −317214. −0.183017 −0.0915085 0.995804i $$-0.529169\pi$$
−0.0915085 + 0.995804i $$0.529169\pi$$
$$314$$ −390149. −0.223309
$$315$$ 0 0
$$316$$ −496512. −0.279712
$$317$$ −1.57126e6 −0.878212 −0.439106 0.898435i $$-0.644705\pi$$
−0.439106 + 0.898435i $$0.644705\pi$$
$$318$$ 0 0
$$319$$ 5452.61 0.00300005
$$320$$ 168452. 0.0919602
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −53257.7 −0.0284038
$$324$$ 0 0
$$325$$ −2.28979e6 −1.20250
$$326$$ −728974. −0.379899
$$327$$ 0 0
$$328$$ −1.81071e6 −0.929317
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.53832e6 0.771751 0.385875 0.922551i $$-0.373899\pi$$
0.385875 + 0.922551i $$0.373899\pi$$
$$332$$ 87441.4 0.0435383
$$333$$ 0 0
$$334$$ 463257. 0.227225
$$335$$ 647000. 0.314987
$$336$$ 0 0
$$337$$ −2.86811e6 −1.37569 −0.687846 0.725857i $$-0.741444\pi$$
−0.687846 + 0.725857i $$0.741444\pi$$
$$338$$ 734149. 0.349537
$$339$$ 0 0
$$340$$ −15502.3 −0.00727277
$$341$$ −20239.5 −0.00942569
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −2.97860e6 −1.35711
$$345$$ 0 0
$$346$$ 2.17793e6 0.978031
$$347$$ 111966. 0.0499185 0.0249593 0.999688i $$-0.492054\pi$$
0.0249593 + 0.999688i $$0.492054\pi$$
$$348$$ 0 0
$$349$$ 3.75314e6 1.64942 0.824711 0.565555i $$-0.191338\pi$$
0.824711 + 0.565555i $$0.191338\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 21990.3 0.00945965
$$353$$ 3.22923e6 1.37931 0.689655 0.724138i $$-0.257762\pi$$
0.689655 + 0.724138i $$0.257762\pi$$
$$354$$ 0 0
$$355$$ 575065. 0.242184
$$356$$ −460749. −0.192681
$$357$$ 0 0
$$358$$ 1.72381e6 0.710855
$$359$$ −1.83659e6 −0.752100 −0.376050 0.926599i $$-0.622718\pi$$
−0.376050 + 0.926599i $$0.622718\pi$$
$$360$$ 0 0
$$361$$ −1.34531e6 −0.543319
$$362$$ −1.36988e6 −0.549427
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −406208. −0.159594
$$366$$ 0 0
$$367$$ 339025. 0.131391 0.0656957 0.997840i $$-0.479073\pi$$
0.0656957 + 0.997840i $$0.479073\pi$$
$$368$$ −811366. −0.312318
$$369$$ 0 0
$$370$$ 261959. 0.0994784
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −706622. −0.262975 −0.131488 0.991318i $$-0.541975\pi$$
−0.131488 + 0.991318i $$0.541975\pi$$
$$374$$ 567.729 0.000209876 0
$$375$$ 0 0
$$376$$ −4.95995e6 −1.80929
$$377$$ 1.16028e6 0.420447
$$378$$ 0 0
$$379$$ 648296. 0.231833 0.115917 0.993259i $$-0.463019\pi$$
0.115917 + 0.993259i $$0.463019\pi$$
$$380$$ 329151. 0.116933
$$381$$ 0 0
$$382$$ −486873. −0.170709
$$383$$ −3.07022e6 −1.06948 −0.534741 0.845016i $$-0.679591\pi$$
−0.534741 + 0.845016i $$0.679591\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.40541e6 −0.821714
$$387$$ 0 0
$$388$$ 398183. 0.134278
$$389$$ 4.59998e6 1.54128 0.770641 0.637270i $$-0.219936\pi$$
0.770641 + 0.637270i $$0.219936\pi$$
$$390$$ 0 0
$$391$$ −205464. −0.0679663
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 1.05203e6 0.341418
$$395$$ 305153. 0.0984066
$$396$$ 0 0
$$397$$ −2.64159e6 −0.841181 −0.420590 0.907251i $$-0.638177\pi$$
−0.420590 + 0.907251i $$0.638177\pi$$
$$398$$ −916694. −0.290079
$$399$$ 0 0
$$400$$ −580425. −0.181383
$$401$$ −4.89677e6 −1.52072 −0.760359 0.649503i $$-0.774977\pi$$
−0.760359 + 0.649503i $$0.774977\pi$$
$$402$$ 0 0
$$403$$ −4.30684e6 −1.32098
$$404$$ 3.18865e6 0.971972
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 22524.8 0.00674025
$$408$$ 0 0
$$409$$ −96336.3 −0.0284762 −0.0142381 0.999899i $$-0.504532\pi$$
−0.0142381 + 0.999899i $$0.504532\pi$$
$$410$$ 458735. 0.134773
$$411$$ 0 0
$$412$$ −4.72485e6 −1.37134
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −53740.9 −0.0153174
$$416$$ 4.67941e6 1.32574
$$417$$ 0 0
$$418$$ −12054.2 −0.00337442
$$419$$ 1.09657e6 0.305142 0.152571 0.988292i $$-0.451245\pi$$
0.152571 + 0.988292i $$0.451245\pi$$
$$420$$ 0 0
$$421$$ −1.93660e6 −0.532517 −0.266259 0.963902i $$-0.585788\pi$$
−0.266259 + 0.963902i $$0.585788\pi$$
$$422$$ 1.56661e6 0.428234
$$423$$ 0 0
$$424$$ −3.24103e6 −0.875526
$$425$$ −146982. −0.0394723
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4.54830e6 −1.20016
$$429$$ 0 0
$$430$$ 754616. 0.196813
$$431$$ 3.07330e6 0.796914 0.398457 0.917187i $$-0.369546\pi$$
0.398457 + 0.917187i $$0.369546\pi$$
$$432$$ 0 0
$$433$$ −3.80919e6 −0.976366 −0.488183 0.872741i $$-0.662340\pi$$
−0.488183 + 0.872741i $$0.662340\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −3.13149e6 −0.788923
$$437$$ 4.36248e6 1.09277
$$438$$ 0 0
$$439$$ −533019. −0.132002 −0.0660011 0.997820i $$-0.521024\pi$$
−0.0660011 + 0.997820i $$0.521024\pi$$
$$440$$ −8511.94 −0.00209603
$$441$$ 0 0
$$442$$ 120809. 0.0294134
$$443$$ 4.66745e6 1.12998 0.564990 0.825098i $$-0.308880\pi$$
0.564990 + 0.825098i $$0.308880\pi$$
$$444$$ 0 0
$$445$$ 283173. 0.0677879
$$446$$ 1.42945e6 0.340276
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.10134e6 1.42827 0.714133 0.700010i $$-0.246821\pi$$
0.714133 + 0.700010i $$0.246821\pi$$
$$450$$ 0 0
$$451$$ 39444.9 0.00913166
$$452$$ −4.63012e6 −1.06597
$$453$$ 0 0
$$454$$ −303680. −0.0691475
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.94498e6 1.33156 0.665779 0.746149i $$-0.268099\pi$$
0.665779 + 0.746149i $$0.268099\pi$$
$$458$$ −1.55015e6 −0.345311
$$459$$ 0 0
$$460$$ 1.26984e6 0.279804
$$461$$ 2.09415e6 0.458940 0.229470 0.973316i $$-0.426301\pi$$
0.229470 + 0.973316i $$0.426301\pi$$
$$462$$ 0 0
$$463$$ 2.41123e6 0.522741 0.261371 0.965239i $$-0.415826\pi$$
0.261371 + 0.965239i $$0.415826\pi$$
$$464$$ 294114. 0.0634191
$$465$$ 0 0
$$466$$ 1.89800e6 0.404884
$$467$$ 4.51833e6 0.958706 0.479353 0.877622i $$-0.340871\pi$$
0.479353 + 0.877622i $$0.340871\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1.25658e6 0.262389
$$471$$ 0 0
$$472$$ −1.11410e6 −0.230181
$$473$$ 64886.6 0.0133353
$$474$$ 0 0
$$475$$ 3.12078e6 0.634643
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −3.40140e6 −0.680908
$$479$$ −8.35928e6 −1.66468 −0.832339 0.554267i $$-0.812999\pi$$
−0.832339 + 0.554267i $$0.812999\pi$$
$$480$$ 0 0
$$481$$ 4.79315e6 0.944624
$$482$$ −277886. −0.0544815
$$483$$ 0 0
$$484$$ 3.61398e6 0.701249
$$485$$ −244721. −0.0472407
$$486$$ 0 0
$$487$$ 598304. 0.114314 0.0571570 0.998365i $$-0.481796\pi$$
0.0571570 + 0.998365i $$0.481796\pi$$
$$488$$ 6.18558e6 1.17579
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.76760e6 −0.518084 −0.259042 0.965866i $$-0.583407\pi$$
−0.259042 + 0.965866i $$0.583407\pi$$
$$492$$ 0 0
$$493$$ 74479.1 0.0138012
$$494$$ −2.56507e6 −0.472914
$$495$$ 0 0
$$496$$ −1.09172e6 −0.199253
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 3.03921e6 0.546398 0.273199 0.961958i $$-0.411918\pi$$
0.273199 + 0.961958i $$0.411918\pi$$
$$500$$ 1.87569e6 0.335533
$$501$$ 0 0
$$502$$ 3.12410e6 0.553307
$$503$$ −9.37896e6 −1.65286 −0.826428 0.563043i $$-0.809631\pi$$
−0.826428 + 0.563043i $$0.809631\pi$$
$$504$$ 0 0
$$505$$ −1.95973e6 −0.341953
$$506$$ −46504.3 −0.00807452
$$507$$ 0 0
$$508$$ −2.01694e6 −0.346764
$$509$$ −5.90139e6 −1.00962 −0.504812 0.863229i $$-0.668438\pi$$
−0.504812 + 0.863229i $$0.668438\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 2.25139e6 0.379555
$$513$$ 0 0
$$514$$ −3.74806e6 −0.625747
$$515$$ 2.90386e6 0.482456
$$516$$ 0 0
$$517$$ 108049. 0.0177784
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −1.81129e6 −0.293751
$$521$$ −9.37426e6 −1.51301 −0.756506 0.653986i $$-0.773095\pi$$
−0.756506 + 0.653986i $$0.773095\pi$$
$$522$$ 0 0
$$523$$ 6.46830e6 1.03404 0.517018 0.855975i $$-0.327042\pi$$
0.517018 + 0.855975i $$0.327042\pi$$
$$524$$ −7.18398e6 −1.14298
$$525$$ 0 0
$$526$$ −1.18997e6 −0.187531
$$527$$ −276458. −0.0433613
$$528$$ 0 0
$$529$$ 1.03938e7 1.61486
$$530$$ 821102. 0.126972
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 8.39364e6 1.27977
$$534$$ 0 0
$$535$$ 2.79536e6 0.422234
$$536$$ −7.89546e6 −1.18704
$$537$$ 0 0
$$538$$ −4.86792e6 −0.725083
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.46089e6 −0.361492 −0.180746 0.983530i $$-0.557851\pi$$
−0.180746 + 0.983530i $$0.557851\pi$$
$$542$$ −1.02725e6 −0.150203
$$543$$ 0 0
$$544$$ 300373. 0.0435175
$$545$$ 1.92459e6 0.277554
$$546$$ 0 0
$$547$$ 503726. 0.0719824 0.0359912 0.999352i $$-0.488541\pi$$
0.0359912 + 0.999352i $$0.488541\pi$$
$$548$$ 5.81180e6 0.826721
$$549$$ 0 0
$$550$$ −33267.7 −0.00468938
$$551$$ −1.58137e6 −0.221898
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6.91113e6 0.956697
$$555$$ 0 0
$$556$$ 6.33681e6 0.869328
$$557$$ −632648. −0.0864021 −0.0432011 0.999066i $$-0.513756\pi$$
−0.0432011 + 0.999066i $$0.513756\pi$$
$$558$$ 0 0
$$559$$ 1.38075e7 1.86890
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −215559. −0.0287889
$$563$$ −1.00780e7 −1.33999 −0.669995 0.742366i $$-0.733704\pi$$
−0.669995 + 0.742366i $$0.733704\pi$$
$$564$$ 0 0
$$565$$ 2.84564e6 0.375024
$$566$$ 1.47301e6 0.193270
$$567$$ 0 0
$$568$$ −7.01763e6 −0.912682
$$569$$ −2.01685e6 −0.261151 −0.130576 0.991438i $$-0.541683\pi$$
−0.130576 + 0.991438i $$0.541683\pi$$
$$570$$ 0 0
$$571$$ 4.09727e6 0.525902 0.262951 0.964809i $$-0.415304\pi$$
0.262951 + 0.964809i $$0.415304\pi$$
$$572$$ −64201.1 −0.00820451
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.20397e7 1.51861
$$576$$ 0 0
$$577$$ −8.14322e6 −1.01826 −0.509128 0.860691i $$-0.670032\pi$$
−0.509128 + 0.860691i $$0.670032\pi$$
$$578$$ −4.38191e6 −0.545562
$$579$$ 0 0
$$580$$ −460307. −0.0568169
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 70603.4 0.00860309
$$584$$ 4.95703e6 0.601436
$$585$$ 0 0
$$586$$ 7.03621e6 0.846437
$$587$$ 8.81465e6 1.05587 0.527934 0.849285i $$-0.322967\pi$$
0.527934 + 0.849285i $$0.322967\pi$$
$$588$$ 0 0
$$589$$ 5.86985e6 0.697170
$$590$$ 282253. 0.0333816
$$591$$ 0 0
$$592$$ 1.21499e6 0.142485
$$593$$ 1.15739e7 1.35158 0.675792 0.737093i $$-0.263802\pi$$
0.675792 + 0.737093i $$0.263802\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −7.17396e6 −0.827263
$$597$$ 0 0
$$598$$ −9.89584e6 −1.13162
$$599$$ 1.23551e7 1.40695 0.703474 0.710721i $$-0.251631\pi$$
0.703474 + 0.710721i $$0.251631\pi$$
$$600$$ 0 0
$$601$$ −8.33752e6 −0.941566 −0.470783 0.882249i $$-0.656029\pi$$
−0.470783 + 0.882249i $$0.656029\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1.88616e6 0.210371
$$605$$ −2.22113e6 −0.246709
$$606$$ 0 0
$$607$$ −8.67189e6 −0.955305 −0.477653 0.878549i $$-0.658512\pi$$
−0.477653 + 0.878549i $$0.658512\pi$$
$$608$$ −6.37764e6 −0.699682
$$609$$ 0 0
$$610$$ −1.56709e6 −0.170518
$$611$$ 2.29921e7 2.49159
$$612$$ 0 0
$$613$$ 5.24031e6 0.563256 0.281628 0.959524i $$-0.409126\pi$$
0.281628 + 0.959524i $$0.409126\pi$$
$$614$$ 5.00196e6 0.535450
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.95644e6 0.735655 0.367828 0.929894i $$-0.380102\pi$$
0.367828 + 0.929894i $$0.380102\pi$$
$$618$$ 0 0
$$619$$ 1.82763e7 1.91717 0.958586 0.284802i $$-0.0919278\pi$$
0.958586 + 0.284802i $$0.0919278\pi$$
$$620$$ 1.70861e6 0.178510
$$621$$ 0 0
$$622$$ −3.17409e6 −0.328960
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 8.01835e6 0.821079
$$626$$ −980707. −0.100024
$$627$$ 0 0
$$628$$ 2.83205e6 0.286551
$$629$$ 307674. 0.0310074
$$630$$ 0 0
$$631$$ 1.90708e7 1.90676 0.953379 0.301777i $$-0.0975798\pi$$
0.953379 + 0.301777i $$0.0975798\pi$$
$$632$$ −3.72384e6 −0.370850
$$633$$ 0 0
$$634$$ −4.85774e6 −0.479967
$$635$$ 1.23960e6 0.121997
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 16857.4 0.00163961
$$639$$ 0 0
$$640$$ −2.12628e6 −0.205197
$$641$$ −3.20624e6 −0.308213 −0.154106 0.988054i $$-0.549250\pi$$
−0.154106 + 0.988054i $$0.549250\pi$$
$$642$$ 0 0
$$643$$ 4.35153e6 0.415063 0.207532 0.978228i $$-0.433457\pi$$
0.207532 + 0.978228i $$0.433457\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −164653. −0.0155234
$$647$$ 5.59976e6 0.525907 0.262953 0.964809i $$-0.415303\pi$$
0.262953 + 0.964809i $$0.415303\pi$$
$$648$$ 0 0
$$649$$ 24269.8 0.00226180
$$650$$ −7.07917e6 −0.657202
$$651$$ 0 0
$$652$$ 5.29156e6 0.487489
$$653$$ 6.74495e6 0.619007 0.309504 0.950898i $$-0.399837\pi$$
0.309504 + 0.950898i $$0.399837\pi$$
$$654$$ 0 0
$$655$$ 4.41523e6 0.402115
$$656$$ 2.12766e6 0.193038
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −2.09622e6 −0.188029 −0.0940143 0.995571i $$-0.529970\pi$$
−0.0940143 + 0.995571i $$0.529970\pi$$
$$660$$ 0 0
$$661$$ −1.94185e7 −1.72867 −0.864335 0.502917i $$-0.832260\pi$$
−0.864335 + 0.502917i $$0.832260\pi$$
$$662$$ 4.75591e6 0.421783
$$663$$ 0 0
$$664$$ 655810. 0.0577242
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.10079e6 −0.530972
$$668$$ −3.36274e6 −0.291577
$$669$$ 0 0
$$670$$ 2.00028e6 0.172149
$$671$$ −134748. −0.0115536
$$672$$ 0 0
$$673$$ 1.27627e7 1.08619 0.543094 0.839672i $$-0.317253\pi$$
0.543094 + 0.839672i $$0.317253\pi$$
$$674$$ −8.86713e6 −0.751853
$$675$$ 0 0
$$676$$ −5.32913e6 −0.448528
$$677$$ 3.86502e6 0.324101 0.162050 0.986782i $$-0.448189\pi$$
0.162050 + 0.986782i $$0.448189\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −116267. −0.00964241
$$681$$ 0 0
$$682$$ −62572.8 −0.00515140
$$683$$ 2.06418e7 1.69316 0.846578 0.532265i $$-0.178659\pi$$
0.846578 + 0.532265i $$0.178659\pi$$
$$684$$ 0 0
$$685$$ −3.57189e6 −0.290852
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 3.49998e6 0.281900
$$689$$ 1.50240e7 1.20570
$$690$$ 0 0
$$691$$ −1.62329e7 −1.29330 −0.646651 0.762786i $$-0.723831\pi$$
−0.646651 + 0.762786i $$0.723831\pi$$
$$692$$ −1.58094e7 −1.25502
$$693$$ 0 0
$$694$$ 346156. 0.0272818
$$695$$ −3.89456e6 −0.305842
$$696$$ 0 0
$$697$$ 538791. 0.0420086
$$698$$ 1.16033e7 0.901454
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8.27590e6 0.636092 0.318046 0.948075i $$-0.396973\pi$$
0.318046 + 0.948075i $$0.396973\pi$$
$$702$$ 0 0
$$703$$ −6.53266e6 −0.498542
$$704$$ 44780.7 0.00340533
$$705$$ 0 0
$$706$$ 9.98356e6 0.753830
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.38973e7 1.78539 0.892696 0.450660i $$-0.148811\pi$$
0.892696 + 0.450660i $$0.148811\pi$$
$$710$$ 1.77789e6 0.132360
$$711$$ 0 0
$$712$$ −3.45562e6 −0.255462
$$713$$ 2.26454e7 1.66823
$$714$$ 0 0
$$715$$ 39457.6 0.00288646
$$716$$ −1.25130e7 −0.912173
$$717$$ 0 0
$$718$$ −5.67804e6 −0.411043
$$719$$ 784349. 0.0565832 0.0282916 0.999600i $$-0.490993\pi$$
0.0282916 + 0.999600i $$0.490993\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −4.15920e6 −0.296939
$$723$$ 0 0
$$724$$ 9.94381e6 0.705028
$$725$$ −4.36431e6 −0.308369
$$726$$ 0 0
$$727$$ −1.61766e7 −1.13514 −0.567571 0.823324i $$-0.692117\pi$$
−0.567571 + 0.823324i $$0.692117\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −1.25584e6 −0.0872224
$$731$$ 886307. 0.0613467
$$732$$ 0 0
$$733$$ −1.36978e7 −0.941654 −0.470827 0.882226i $$-0.656044\pi$$
−0.470827 + 0.882226i $$0.656044\pi$$
$$734$$ 1.04814e6 0.0718090
$$735$$ 0 0
$$736$$ −2.46044e7 −1.67424
$$737$$ 171997. 0.0116641
$$738$$ 0 0
$$739$$ −2.53712e7 −1.70895 −0.854477 0.519490i $$-0.826122\pi$$
−0.854477 + 0.519490i $$0.826122\pi$$
$$740$$ −1.90154e6 −0.127651
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −2.45336e7 −1.63038 −0.815192 0.579191i $$-0.803368\pi$$
−0.815192 + 0.579191i $$0.803368\pi$$
$$744$$ 0 0
$$745$$ 4.40907e6 0.291042
$$746$$ −2.18461e6 −0.143723
$$747$$ 0 0
$$748$$ −4121.10 −0.000269314 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1.84907e7 −1.19634 −0.598169 0.801370i $$-0.704105\pi$$
−0.598169 + 0.801370i $$0.704105\pi$$
$$752$$ 5.82815e6 0.375825
$$753$$ 0 0
$$754$$ 3.58716e6 0.229786
$$755$$ −1.15922e6 −0.0740113
$$756$$ 0 0
$$757$$ −2.94413e7 −1.86732 −0.933658 0.358166i $$-0.883402\pi$$
−0.933658 + 0.358166i $$0.883402\pi$$
$$758$$ 2.00429e6 0.126703
$$759$$ 0 0
$$760$$ 2.46863e6 0.155032
$$761$$ −1.86236e7 −1.16574 −0.582870 0.812565i $$-0.698070\pi$$
−0.582870 + 0.812565i $$0.698070\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 3.53417e6 0.219056
$$765$$ 0 0
$$766$$ −9.49199e6 −0.584501
$$767$$ 5.16448e6 0.316984
$$768$$ 0 0
$$769$$ 1.13326e7 0.691056 0.345528 0.938408i $$-0.387700\pi$$
0.345528 + 0.938408i $$0.387700\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.74607e7 1.05443
$$773$$ −8.22020e6 −0.494804 −0.247402 0.968913i $$-0.579577\pi$$
−0.247402 + 0.968913i $$0.579577\pi$$
$$774$$ 0 0
$$775$$ 1.61998e7 0.968848
$$776$$ 2.98637e6 0.178029
$$777$$ 0 0
$$778$$ 1.42214e7 0.842352
$$779$$ −1.14398e7 −0.675422
$$780$$ 0 0
$$781$$ 152874. 0.00896819
$$782$$ −635217. −0.0371454
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −1.74056e6 −0.100813
$$786$$ 0 0
$$787$$ 5.43081e6 0.312556 0.156278 0.987713i $$-0.450050\pi$$
0.156278 + 0.987713i $$0.450050\pi$$
$$788$$ −7.63657e6 −0.438110
$$789$$ 0 0
$$790$$ 943418. 0.0537819
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.86736e7 −1.61920
$$794$$ −8.16681e6 −0.459728
$$795$$ 0 0
$$796$$ 6.65421e6 0.372232
$$797$$ 1.78388e7 0.994765 0.497382 0.867531i $$-0.334295\pi$$
0.497382 + 0.867531i $$0.334295\pi$$
$$798$$ 0 0
$$799$$ 1.47587e6 0.0817866
$$800$$ −1.76012e7 −0.972339
$$801$$ 0 0
$$802$$ −1.51390e7 −0.831115
$$803$$ −107985. −0.00590983
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.33151e7 −0.721951
$$807$$ 0 0
$$808$$ 2.39149e7 1.28867
$$809$$ −1.18431e7 −0.636199 −0.318099 0.948057i $$-0.603045\pi$$
−0.318099 + 0.948057i $$0.603045\pi$$
$$810$$ 0 0
$$811$$ −618053. −0.0329969 −0.0164985 0.999864i $$-0.505252\pi$$
−0.0164985 + 0.999864i $$0.505252\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 69638.4 0.00368373
$$815$$ −3.25216e6 −0.171505
$$816$$ 0 0
$$817$$ −1.88184e7 −0.986342
$$818$$ −297836. −0.0155630
$$819$$ 0 0
$$820$$ −3.32992e6 −0.172941
$$821$$ −4.31256e6 −0.223294 −0.111647 0.993748i $$-0.535613\pi$$
−0.111647 + 0.993748i $$0.535613\pi$$
$$822$$ 0 0
$$823$$ 2.86620e7 1.47505 0.737524 0.675321i $$-0.235995\pi$$
0.737524 + 0.675321i $$0.235995\pi$$
$$824$$ −3.54364e7 −1.81816
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.86894e7 −0.950235 −0.475117 0.879922i $$-0.657594\pi$$
−0.475117 + 0.879922i $$0.657594\pi$$
$$828$$ 0 0
$$829$$ −1.32486e7 −0.669550 −0.334775 0.942298i $$-0.608660\pi$$
−0.334775 + 0.942298i $$0.608660\pi$$
$$830$$ −166147. −0.00837137
$$831$$ 0 0
$$832$$ 9.52907e6 0.477246
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.06672e6 0.102581
$$836$$ 87500.6 0.00433007
$$837$$ 0 0
$$838$$ 3.39019e6 0.166769
$$839$$ −2.52930e7 −1.24049 −0.620247 0.784406i $$-0.712968\pi$$
−0.620247 + 0.784406i $$0.712968\pi$$
$$840$$ 0 0
$$841$$ −1.82997e7 −0.892181
$$842$$ −5.98723e6 −0.291035
$$843$$ 0 0
$$844$$ −1.13719e7 −0.549512
$$845$$ 3.27525e6 0.157798
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 3.80835e6 0.181864
$$849$$ 0 0
$$850$$ −454414. −0.0215727
$$851$$ −2.52025e7 −1.19294
$$852$$ 0 0
$$853$$ −4.05806e7 −1.90962 −0.954808 0.297223i $$-0.903939\pi$$
−0.954808 + 0.297223i $$0.903939\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −3.41123e7 −1.59120
$$857$$ 1.94469e7 0.904480 0.452240 0.891896i $$-0.350625\pi$$
0.452240 + 0.891896i $$0.350625\pi$$
$$858$$ 0 0
$$859$$ 2.03242e7 0.939788 0.469894 0.882723i $$-0.344292\pi$$
0.469894 + 0.882723i $$0.344292\pi$$
$$860$$ −5.47769e6 −0.252552
$$861$$ 0 0
$$862$$ 9.50149e6 0.435536
$$863$$ 2.61202e7 1.19385 0.596926 0.802297i $$-0.296389\pi$$
0.596926 + 0.802297i $$0.296389\pi$$
$$864$$ 0 0
$$865$$ 9.71634e6 0.441532
$$866$$ −1.17766e7 −0.533611
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 81120.9 0.00364404
$$870$$ 0 0
$$871$$ 3.65999e7 1.63469
$$872$$ −2.34862e7 −1.04597
$$873$$ 0 0
$$874$$ 1.34872e7 0.597231
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.62956e7 −0.715436 −0.357718 0.933830i $$-0.616445\pi$$
−0.357718 + 0.933830i $$0.616445\pi$$
$$878$$ −1.64789e6 −0.0721428
$$879$$ 0 0
$$880$$ 10001.9 0.000435387 0
$$881$$ 2.93722e7 1.27496 0.637479 0.770467i $$-0.279977\pi$$
0.637479 + 0.770467i $$0.279977\pi$$
$$882$$ 0 0
$$883$$ 1.21821e7 0.525800 0.262900 0.964823i $$-0.415321\pi$$
0.262900 + 0.964823i $$0.415321\pi$$
$$884$$ −876945. −0.0377435
$$885$$ 0 0
$$886$$ 1.44300e7 0.617565
$$887$$ −2.38021e7 −1.01579 −0.507897 0.861418i $$-0.669577\pi$$
−0.507897 + 0.861418i $$0.669577\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 875466. 0.0370480
$$891$$ 0 0
$$892$$ −1.03763e7 −0.436645
$$893$$ −3.13363e7 −1.31498
$$894$$ 0 0
$$895$$ 7.69038e6 0.320915
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 1.88631e7 0.780587
$$899$$ −8.20879e6 −0.338750
$$900$$ 0 0
$$901$$ 964396. 0.0395771
$$902$$ 121949. 0.00499070
$$903$$ 0 0
$$904$$ −3.47259e7 −1.41329
$$905$$ −6.11140e6 −0.248039
$$906$$ 0 0
$$907$$ 2.10870e7 0.851133 0.425566 0.904927i $$-0.360075\pi$$
0.425566 + 0.904927i $$0.360075\pi$$
$$908$$ 2.20439e6 0.0887305
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1.71528e7 −0.684761 −0.342381 0.939561i $$-0.611233\pi$$
−0.342381 + 0.939561i $$0.611233\pi$$
$$912$$ 0 0
$$913$$ −14286.3 −0.000567210 0
$$914$$ 1.83797e7 0.727733
$$915$$ 0 0
$$916$$ 1.12524e7 0.443105
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 7.35541e6 0.287289 0.143644 0.989629i $$-0.454118\pi$$
0.143644 + 0.989629i $$0.454118\pi$$
$$920$$ 9.52379e6 0.370971
$$921$$ 0 0
$$922$$ 6.47433e6 0.250823
$$923$$ 3.25306e7 1.25686
$$924$$ 0 0
$$925$$ −1.80290e7 −0.692817
$$926$$ 7.45463e6 0.285692
$$927$$ 0 0
$$928$$ 8.91891e6 0.339971
$$929$$ −3.34199e7 −1.27047 −0.635236 0.772318i $$-0.719097\pi$$
−0.635236 + 0.772318i $$0.719097\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −1.37774e7 −0.519550
$$933$$ 0 0
$$934$$ 1.39690e7 0.523959
$$935$$ 2532.80 9.47483e−5 0
$$936$$ 0 0
$$937$$ 2.87696e7 1.07049 0.535247 0.844696i $$-0.320219\pi$$
0.535247 + 0.844696i $$0.320219\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −9.12142e6 −0.336700
$$941$$ 2.16379e7 0.796601 0.398300 0.917255i $$-0.369600\pi$$
0.398300 + 0.917255i $$0.369600\pi$$
$$942$$ 0 0
$$943$$ −4.41339e7 −1.61619
$$944$$ 1.30911e6 0.0478132
$$945$$ 0 0
$$946$$ 200605. 0.00728809
$$947$$ 6.31254e6 0.228733 0.114367 0.993439i $$-0.463516\pi$$
0.114367 + 0.993439i $$0.463516\pi$$
$$948$$ 0 0
$$949$$ −2.29786e7 −0.828244
$$950$$ 9.64829e6 0.346850
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 5.59599e6 0.199593 0.0997963 0.995008i $$-0.468181\pi$$
0.0997963 + 0.995008i $$0.468181\pi$$
$$954$$ 0 0
$$955$$ −2.17208e6 −0.0770668
$$956$$ 2.46905e7 0.873746
$$957$$ 0 0
$$958$$ −2.58438e7 −0.909792
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.84091e6 0.0643021
$$962$$ 1.48186e7 0.516263
$$963$$ 0 0
$$964$$ 2.01715e6 0.0699111
$$965$$ −1.07312e7 −0.370963
$$966$$ 0 0
$$967$$ 1.33277e7 0.458340 0.229170 0.973386i $$-0.426399\pi$$
0.229170 + 0.973386i $$0.426399\pi$$
$$968$$ 2.71048e7 0.929734
$$969$$ 0 0
$$970$$ −756585. −0.0258183
$$971$$ −3.63257e7 −1.23642 −0.618210 0.786013i $$-0.712142\pi$$
−0.618210 + 0.786013i $$0.712142\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 1.84973e6 0.0624758
$$975$$ 0 0
$$976$$ −7.26831e6 −0.244236
$$977$$ 9.03739e6 0.302905 0.151453 0.988465i $$-0.451605\pi$$
0.151453 + 0.988465i $$0.451605\pi$$
$$978$$ 0 0
$$979$$ 75278.0 0.00251022
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −8.55639e6 −0.283147
$$983$$ 1.98539e7 0.655332 0.327666 0.944794i $$-0.393738\pi$$
0.327666 + 0.944794i $$0.393738\pi$$
$$984$$ 0 0
$$985$$ 4.69339e6 0.154133
$$986$$ 230261. 0.00754273
$$987$$ 0 0
$$988$$ 1.86196e7 0.606846
$$989$$ −7.25999e7 −2.36018
$$990$$ 0 0
$$991$$ −4.55557e7 −1.47353 −0.736764 0.676150i $$-0.763647\pi$$
−0.736764 + 0.676150i $$0.763647\pi$$
$$992$$ −3.31060e7 −1.06814
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.08963e6 −0.130956
$$996$$ 0 0
$$997$$ 4.51665e7 1.43906 0.719529 0.694462i $$-0.244358\pi$$
0.719529 + 0.694462i $$0.244358\pi$$
$$998$$ 9.39610e6 0.298622
$$999$$ 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.bb.1.4 6
3.2 odd 2 147.6.a.n.1.3 6
7.6 odd 2 441.6.a.ba.1.4 6
21.2 odd 6 147.6.e.q.67.4 12
21.5 even 6 147.6.e.p.67.4 12
21.11 odd 6 147.6.e.q.79.4 12
21.17 even 6 147.6.e.p.79.4 12
21.20 even 2 147.6.a.o.1.3 yes 6

By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.3 6 3.2 odd 2
147.6.a.o.1.3 yes 6 21.20 even 2
147.6.e.p.67.4 12 21.5 even 6
147.6.e.p.79.4 12 21.17 even 6
147.6.e.q.67.4 12 21.2 odd 6
147.6.e.q.79.4 12 21.11 odd 6
441.6.a.ba.1.4 6 7.6 odd 2
441.6.a.bb.1.4 6 1.1 even 1 trivial