# Properties

 Label 192.8.a.q.1.2 Level $192$ Weight $8$ Character 192.1 Self dual yes Analytic conductor $59.978$ 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] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 192.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: $$59.9779248930$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{46})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 46$$ x^2 - 46 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 96) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$6.78233$$ of defining polynomial Character $$\chi$$ $$=$$ 192.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-27.0000 q^{3} +227.552 q^{5} -724.656 q^{7} +729.000 q^{9} +O(q^{10})$$ $$q-27.0000 q^{3} +227.552 q^{5} -724.656 q^{7} +729.000 q^{9} +1125.31 q^{11} -2426.90 q^{13} -6143.90 q^{15} +28738.2 q^{17} -44674.2 q^{19} +19565.7 q^{21} +23822.7 q^{23} -26345.2 q^{25} -19683.0 q^{27} -11111.2 q^{29} +84678.5 q^{31} -30383.4 q^{33} -164897. q^{35} +199202. q^{37} +65526.2 q^{39} +272733. q^{41} -584037. q^{43} +165885. q^{45} +1.27056e6 q^{47} -298417. q^{49} -775932. q^{51} +371805. q^{53} +256067. q^{55} +1.20620e6 q^{57} -1.79416e6 q^{59} -2.22047e6 q^{61} -528274. q^{63} -552245. q^{65} -3.78620e6 q^{67} -643213. q^{69} -4.40108e6 q^{71} -4.62913e6 q^{73} +711319. q^{75} -815463. q^{77} -2.37397e6 q^{79} +531441. q^{81} -5.76865e6 q^{83} +6.53943e6 q^{85} +300002. q^{87} +1.61090e6 q^{89} +1.75866e6 q^{91} -2.28632e6 q^{93} -1.01657e7 q^{95} -81667.2 q^{97} +820352. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9}+O(q^{10})$$ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 $$2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9} - 1656 q^{11} - 6156 q^{13} + 5292 q^{15} + 17108 q^{17} + 504 q^{19} - 13608 q^{21} + 51552 q^{23} + 74926 q^{25} - 39366 q^{27} + 199804 q^{29} + 257256 q^{31} + 44712 q^{33} - 685296 q^{35} + 468724 q^{37} + 166212 q^{39} - 106940 q^{41} - 1617336 q^{43} - 142884 q^{45} + 646416 q^{47} + 387634 q^{49} - 461916 q^{51} - 1469492 q^{53} + 1434096 q^{55} - 13608 q^{57} - 4541544 q^{59} + 481412 q^{61} + 367416 q^{63} + 1027224 q^{65} - 4775256 q^{67} - 1391904 q^{69} - 1094400 q^{71} - 5731884 q^{73} - 2023002 q^{75} - 4232736 q^{77} - 10402776 q^{79} + 1062882 q^{81} - 2212200 q^{83} + 11465432 q^{85} - 5394708 q^{87} - 3604364 q^{89} - 2823120 q^{91} - 6945912 q^{93} - 29300976 q^{95} - 7156188 q^{97} - 1207224 q^{99}+O(q^{100})$$ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 - 1656 * q^11 - 6156 * q^13 + 5292 * q^15 + 17108 * q^17 + 504 * q^19 - 13608 * q^21 + 51552 * q^23 + 74926 * q^25 - 39366 * q^27 + 199804 * q^29 + 257256 * q^31 + 44712 * q^33 - 685296 * q^35 + 468724 * q^37 + 166212 * q^39 - 106940 * q^41 - 1617336 * q^43 - 142884 * q^45 + 646416 * q^47 + 387634 * q^49 - 461916 * q^51 - 1469492 * q^53 + 1434096 * q^55 - 13608 * q^57 - 4541544 * q^59 + 481412 * q^61 + 367416 * q^63 + 1027224 * q^65 - 4775256 * q^67 - 1391904 * q^69 - 1094400 * q^71 - 5731884 * q^73 - 2023002 * q^75 - 4232736 * q^77 - 10402776 * q^79 + 1062882 * q^81 - 2212200 * q^83 + 11465432 * q^85 - 5394708 * q^87 - 3604364 * q^89 - 2823120 * q^91 - 6945912 * q^93 - 29300976 * q^95 - 7156188 * q^97 - 1207224 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −27.0000 −0.577350
$$4$$ 0 0
$$5$$ 227.552 0.814114 0.407057 0.913403i $$-0.366555\pi$$
0.407057 + 0.913403i $$0.366555\pi$$
$$6$$ 0 0
$$7$$ −724.656 −0.798525 −0.399262 0.916837i $$-0.630734\pi$$
−0.399262 + 0.916837i $$0.630734\pi$$
$$8$$ 0 0
$$9$$ 729.000 0.333333
$$10$$ 0 0
$$11$$ 1125.31 0.254917 0.127458 0.991844i $$-0.459318\pi$$
0.127458 + 0.991844i $$0.459318\pi$$
$$12$$ 0 0
$$13$$ −2426.90 −0.306372 −0.153186 0.988197i $$-0.548953\pi$$
−0.153186 + 0.988197i $$0.548953\pi$$
$$14$$ 0 0
$$15$$ −6143.90 −0.470029
$$16$$ 0 0
$$17$$ 28738.2 1.41869 0.709347 0.704860i $$-0.248990\pi$$
0.709347 + 0.704860i $$0.248990\pi$$
$$18$$ 0 0
$$19$$ −44674.2 −1.49423 −0.747117 0.664692i $$-0.768563\pi$$
−0.747117 + 0.664692i $$0.768563\pi$$
$$20$$ 0 0
$$21$$ 19565.7 0.461029
$$22$$ 0 0
$$23$$ 23822.7 0.408266 0.204133 0.978943i $$-0.434562\pi$$
0.204133 + 0.978943i $$0.434562\pi$$
$$24$$ 0 0
$$25$$ −26345.2 −0.337218
$$26$$ 0 0
$$27$$ −19683.0 −0.192450
$$28$$ 0 0
$$29$$ −11111.2 −0.0845994 −0.0422997 0.999105i $$-0.513468\pi$$
−0.0422997 + 0.999105i $$0.513468\pi$$
$$30$$ 0 0
$$31$$ 84678.5 0.510514 0.255257 0.966873i $$-0.417840\pi$$
0.255257 + 0.966873i $$0.417840\pi$$
$$32$$ 0 0
$$33$$ −30383.4 −0.147176
$$34$$ 0 0
$$35$$ −164897. −0.650090
$$36$$ 0 0
$$37$$ 199202. 0.646530 0.323265 0.946309i $$-0.395220\pi$$
0.323265 + 0.946309i $$0.395220\pi$$
$$38$$ 0 0
$$39$$ 65526.2 0.176884
$$40$$ 0 0
$$41$$ 272733. 0.618008 0.309004 0.951061i $$-0.400004\pi$$
0.309004 + 0.951061i $$0.400004\pi$$
$$42$$ 0 0
$$43$$ −584037. −1.12021 −0.560107 0.828420i $$-0.689240\pi$$
−0.560107 + 0.828420i $$0.689240\pi$$
$$44$$ 0 0
$$45$$ 165885. 0.271371
$$46$$ 0 0
$$47$$ 1.27056e6 1.78506 0.892532 0.450983i $$-0.148927\pi$$
0.892532 + 0.450983i $$0.148927\pi$$
$$48$$ 0 0
$$49$$ −298417. −0.362358
$$50$$ 0 0
$$51$$ −775932. −0.819083
$$52$$ 0 0
$$53$$ 371805. 0.343044 0.171522 0.985180i $$-0.445132\pi$$
0.171522 + 0.985180i $$0.445132\pi$$
$$54$$ 0 0
$$55$$ 256067. 0.207531
$$56$$ 0 0
$$57$$ 1.20620e6 0.862697
$$58$$ 0 0
$$59$$ −1.79416e6 −1.13731 −0.568657 0.822575i $$-0.692537\pi$$
−0.568657 + 0.822575i $$0.692537\pi$$
$$60$$ 0 0
$$61$$ −2.22047e6 −1.25253 −0.626267 0.779608i $$-0.715418\pi$$
−0.626267 + 0.779608i $$0.715418\pi$$
$$62$$ 0 0
$$63$$ −528274. −0.266175
$$64$$ 0 0
$$65$$ −552245. −0.249422
$$66$$ 0 0
$$67$$ −3.78620e6 −1.53795 −0.768974 0.639280i $$-0.779232\pi$$
−0.768974 + 0.639280i $$0.779232\pi$$
$$68$$ 0 0
$$69$$ −643213. −0.235713
$$70$$ 0 0
$$71$$ −4.40108e6 −1.45934 −0.729668 0.683802i $$-0.760325\pi$$
−0.729668 + 0.683802i $$0.760325\pi$$
$$72$$ 0 0
$$73$$ −4.62913e6 −1.39274 −0.696369 0.717684i $$-0.745202\pi$$
−0.696369 + 0.717684i $$0.745202\pi$$
$$74$$ 0 0
$$75$$ 711319. 0.194693
$$76$$ 0 0
$$77$$ −815463. −0.203557
$$78$$ 0 0
$$79$$ −2.37397e6 −0.541727 −0.270863 0.962618i $$-0.587309\pi$$
−0.270863 + 0.962618i $$0.587309\pi$$
$$80$$ 0 0
$$81$$ 531441. 0.111111
$$82$$ 0 0
$$83$$ −5.76865e6 −1.10739 −0.553696 0.832719i $$-0.686783\pi$$
−0.553696 + 0.832719i $$0.686783\pi$$
$$84$$ 0 0
$$85$$ 6.53943e6 1.15498
$$86$$ 0 0
$$87$$ 300002. 0.0488435
$$88$$ 0 0
$$89$$ 1.61090e6 0.242217 0.121108 0.992639i $$-0.461355\pi$$
0.121108 + 0.992639i $$0.461355\pi$$
$$90$$ 0 0
$$91$$ 1.75866e6 0.244646
$$92$$ 0 0
$$93$$ −2.28632e6 −0.294745
$$94$$ 0 0
$$95$$ −1.01657e7 −1.21648
$$96$$ 0 0
$$97$$ −81667.2 −0.00908546 −0.00454273 0.999990i $$-0.501446\pi$$
−0.00454273 + 0.999990i $$0.501446\pi$$
$$98$$ 0 0
$$99$$ 820352. 0.0849722
$$100$$ 0 0
$$101$$ −9.09433e6 −0.878307 −0.439153 0.898412i $$-0.644722\pi$$
−0.439153 + 0.898412i $$0.644722\pi$$
$$102$$ 0 0
$$103$$ −1.97220e7 −1.77836 −0.889181 0.457556i $$-0.848725\pi$$
−0.889181 + 0.457556i $$0.848725\pi$$
$$104$$ 0 0
$$105$$ 4.45221e6 0.375330
$$106$$ 0 0
$$107$$ 8.09689e6 0.638962 0.319481 0.947593i $$-0.396491\pi$$
0.319481 + 0.947593i $$0.396491\pi$$
$$108$$ 0 0
$$109$$ 1.90660e7 1.41016 0.705078 0.709130i $$-0.250912\pi$$
0.705078 + 0.709130i $$0.250912\pi$$
$$110$$ 0 0
$$111$$ −5.37846e6 −0.373274
$$112$$ 0 0
$$113$$ 1.69299e7 1.10377 0.551886 0.833919i $$-0.313908\pi$$
0.551886 + 0.833919i $$0.313908\pi$$
$$114$$ 0 0
$$115$$ 5.42090e6 0.332375
$$116$$ 0 0
$$117$$ −1.76921e6 −0.102124
$$118$$ 0 0
$$119$$ −2.08253e7 −1.13286
$$120$$ 0 0
$$121$$ −1.82208e7 −0.935018
$$122$$ 0 0
$$123$$ −7.36379e6 −0.356807
$$124$$ 0 0
$$125$$ −2.37724e7 −1.08865
$$126$$ 0 0
$$127$$ −1.91925e7 −0.831418 −0.415709 0.909498i $$-0.636466\pi$$
−0.415709 + 0.909498i $$0.636466\pi$$
$$128$$ 0 0
$$129$$ 1.57690e7 0.646756
$$130$$ 0 0
$$131$$ −2.48634e7 −0.966298 −0.483149 0.875538i $$-0.660507\pi$$
−0.483149 + 0.875538i $$0.660507\pi$$
$$132$$ 0 0
$$133$$ 3.23734e7 1.19318
$$134$$ 0 0
$$135$$ −4.47890e6 −0.156676
$$136$$ 0 0
$$137$$ −2.43232e7 −0.808164 −0.404082 0.914723i $$-0.632409\pi$$
−0.404082 + 0.914723i $$0.632409\pi$$
$$138$$ 0 0
$$139$$ −3.83111e7 −1.20996 −0.604982 0.796239i $$-0.706820\pi$$
−0.604982 + 0.796239i $$0.706820\pi$$
$$140$$ 0 0
$$141$$ −3.43052e7 −1.03061
$$142$$ 0 0
$$143$$ −2.73101e6 −0.0780994
$$144$$ 0 0
$$145$$ −2.52837e6 −0.0688735
$$146$$ 0 0
$$147$$ 8.05727e6 0.209207
$$148$$ 0 0
$$149$$ 2.80329e6 0.0694250 0.0347125 0.999397i $$-0.488948\pi$$
0.0347125 + 0.999397i $$0.488948\pi$$
$$150$$ 0 0
$$151$$ 4.31756e7 1.02052 0.510258 0.860022i $$-0.329550\pi$$
0.510258 + 0.860022i $$0.329550\pi$$
$$152$$ 0 0
$$153$$ 2.09502e7 0.472898
$$154$$ 0 0
$$155$$ 1.92687e7 0.415616
$$156$$ 0 0
$$157$$ 7.86833e7 1.62268 0.811341 0.584573i $$-0.198738\pi$$
0.811341 + 0.584573i $$0.198738\pi$$
$$158$$ 0 0
$$159$$ −1.00387e7 −0.198056
$$160$$ 0 0
$$161$$ −1.72632e7 −0.326011
$$162$$ 0 0
$$163$$ −9.04599e6 −0.163606 −0.0818031 0.996649i $$-0.526068\pi$$
−0.0818031 + 0.996649i $$0.526068\pi$$
$$164$$ 0 0
$$165$$ −6.91380e6 −0.119818
$$166$$ 0 0
$$167$$ 1.02777e8 1.70761 0.853806 0.520592i $$-0.174289\pi$$
0.853806 + 0.520592i $$0.174289\pi$$
$$168$$ 0 0
$$169$$ −5.68587e7 −0.906136
$$170$$ 0 0
$$171$$ −3.25675e7 −0.498078
$$172$$ 0 0
$$173$$ −1.19796e8 −1.75906 −0.879532 0.475839i $$-0.842145\pi$$
−0.879532 + 0.475839i $$0.842145\pi$$
$$174$$ 0 0
$$175$$ 1.90912e7 0.269277
$$176$$ 0 0
$$177$$ 4.84424e7 0.656628
$$178$$ 0 0
$$179$$ −1.07336e7 −0.139882 −0.0699408 0.997551i $$-0.522281\pi$$
−0.0699408 + 0.997551i $$0.522281\pi$$
$$180$$ 0 0
$$181$$ −1.57289e7 −0.197162 −0.0985812 0.995129i $$-0.531430\pi$$
−0.0985812 + 0.995129i $$0.531430\pi$$
$$182$$ 0 0
$$183$$ 5.99526e7 0.723151
$$184$$ 0 0
$$185$$ 4.53289e7 0.526349
$$186$$ 0 0
$$187$$ 3.23394e7 0.361649
$$188$$ 0 0
$$189$$ 1.42634e7 0.153676
$$190$$ 0 0
$$191$$ 1.01986e8 1.05906 0.529532 0.848290i $$-0.322368\pi$$
0.529532 + 0.848290i $$0.322368\pi$$
$$192$$ 0 0
$$193$$ −8.34130e7 −0.835186 −0.417593 0.908634i $$-0.637126\pi$$
−0.417593 + 0.908634i $$0.637126\pi$$
$$194$$ 0 0
$$195$$ 1.49106e7 0.144004
$$196$$ 0 0
$$197$$ −5.81982e7 −0.542348 −0.271174 0.962530i $$-0.587412\pi$$
−0.271174 + 0.962530i $$0.587412\pi$$
$$198$$ 0 0
$$199$$ 1.61506e8 1.45279 0.726393 0.687280i $$-0.241195\pi$$
0.726393 + 0.687280i $$0.241195\pi$$
$$200$$ 0 0
$$201$$ 1.02227e8 0.887935
$$202$$ 0 0
$$203$$ 8.05178e6 0.0675547
$$204$$ 0 0
$$205$$ 6.20609e7 0.503129
$$206$$ 0 0
$$207$$ 1.73667e7 0.136089
$$208$$ 0 0
$$209$$ −5.02723e7 −0.380905
$$210$$ 0 0
$$211$$ 7.74548e7 0.567623 0.283811 0.958880i $$-0.408401\pi$$
0.283811 + 0.958880i $$0.408401\pi$$
$$212$$ 0 0
$$213$$ 1.18829e8 0.842548
$$214$$ 0 0
$$215$$ −1.32899e8 −0.911982
$$216$$ 0 0
$$217$$ −6.13627e7 −0.407658
$$218$$ 0 0
$$219$$ 1.24987e8 0.804098
$$220$$ 0 0
$$221$$ −6.97447e7 −0.434648
$$222$$ 0 0
$$223$$ −9.33465e7 −0.563678 −0.281839 0.959462i $$-0.590944\pi$$
−0.281839 + 0.959462i $$0.590944\pi$$
$$224$$ 0 0
$$225$$ −1.92056e7 −0.112406
$$226$$ 0 0
$$227$$ 4.28306e7 0.243032 0.121516 0.992589i $$-0.461224\pi$$
0.121516 + 0.992589i $$0.461224\pi$$
$$228$$ 0 0
$$229$$ 1.81117e8 0.996630 0.498315 0.866996i $$-0.333952\pi$$
0.498315 + 0.866996i $$0.333952\pi$$
$$230$$ 0 0
$$231$$ 2.20175e7 0.117524
$$232$$ 0 0
$$233$$ −2.73990e7 −0.141902 −0.0709512 0.997480i $$-0.522603\pi$$
−0.0709512 + 0.997480i $$0.522603\pi$$
$$234$$ 0 0
$$235$$ 2.89119e8 1.45325
$$236$$ 0 0
$$237$$ 6.40972e7 0.312766
$$238$$ 0 0
$$239$$ −7.76036e7 −0.367696 −0.183848 0.982955i $$-0.558855\pi$$
−0.183848 + 0.982955i $$0.558855\pi$$
$$240$$ 0 0
$$241$$ 4.21193e8 1.93830 0.969150 0.246471i $$-0.0792710\pi$$
0.969150 + 0.246471i $$0.0792710\pi$$
$$242$$ 0 0
$$243$$ −1.43489e7 −0.0641500
$$244$$ 0 0
$$245$$ −6.79054e7 −0.295001
$$246$$ 0 0
$$247$$ 1.08420e8 0.457792
$$248$$ 0 0
$$249$$ 1.55754e8 0.639353
$$250$$ 0 0
$$251$$ 4.75375e8 1.89748 0.948741 0.316053i $$-0.102358\pi$$
0.948741 + 0.316053i $$0.102358\pi$$
$$252$$ 0 0
$$253$$ 2.68079e7 0.104074
$$254$$ 0 0
$$255$$ −1.76565e8 −0.666827
$$256$$ 0 0
$$257$$ −5.12886e8 −1.88476 −0.942378 0.334551i $$-0.891416\pi$$
−0.942378 + 0.334551i $$0.891416\pi$$
$$258$$ 0 0
$$259$$ −1.44353e8 −0.516270
$$260$$ 0 0
$$261$$ −8.10005e6 −0.0281998
$$262$$ 0 0
$$263$$ 2.07182e8 0.702275 0.351138 0.936324i $$-0.385795\pi$$
0.351138 + 0.936324i $$0.385795\pi$$
$$264$$ 0 0
$$265$$ 8.46048e7 0.279277
$$266$$ 0 0
$$267$$ −4.34944e7 −0.139844
$$268$$ 0 0
$$269$$ −3.95082e7 −0.123753 −0.0618763 0.998084i $$-0.519708\pi$$
−0.0618763 + 0.998084i $$0.519708\pi$$
$$270$$ 0 0
$$271$$ −1.60355e8 −0.489430 −0.244715 0.969595i $$-0.578694\pi$$
−0.244715 + 0.969595i $$0.578694\pi$$
$$272$$ 0 0
$$273$$ −4.74839e7 −0.141246
$$274$$ 0 0
$$275$$ −2.96465e7 −0.0859625
$$276$$ 0 0
$$277$$ −6.47081e8 −1.82928 −0.914638 0.404274i $$-0.867524\pi$$
−0.914638 + 0.404274i $$0.867524\pi$$
$$278$$ 0 0
$$279$$ 6.17306e7 0.170171
$$280$$ 0 0
$$281$$ −3.78439e8 −1.01748 −0.508738 0.860922i $$-0.669888\pi$$
−0.508738 + 0.860922i $$0.669888\pi$$
$$282$$ 0 0
$$283$$ 4.46034e8 1.16981 0.584905 0.811102i $$-0.301132\pi$$
0.584905 + 0.811102i $$0.301132\pi$$
$$284$$ 0 0
$$285$$ 2.74474e8 0.702334
$$286$$ 0 0
$$287$$ −1.97637e8 −0.493495
$$288$$ 0 0
$$289$$ 4.15546e8 1.01269
$$290$$ 0 0
$$291$$ 2.20502e6 0.00524549
$$292$$ 0 0
$$293$$ −5.46621e8 −1.26955 −0.634775 0.772697i $$-0.718907\pi$$
−0.634775 + 0.772697i $$0.718907\pi$$
$$294$$ 0 0
$$295$$ −4.08265e8 −0.925903
$$296$$ 0 0
$$297$$ −2.21495e7 −0.0490587
$$298$$ 0 0
$$299$$ −5.78152e7 −0.125081
$$300$$ 0 0
$$301$$ 4.23226e8 0.894519
$$302$$ 0 0
$$303$$ 2.45547e8 0.507091
$$304$$ 0 0
$$305$$ −5.05271e8 −1.01971
$$306$$ 0 0
$$307$$ −6.30513e8 −1.24368 −0.621842 0.783143i $$-0.713615\pi$$
−0.621842 + 0.783143i $$0.713615\pi$$
$$308$$ 0 0
$$309$$ 5.32493e8 1.02674
$$310$$ 0 0
$$311$$ 3.23661e8 0.610139 0.305069 0.952330i $$-0.401320\pi$$
0.305069 + 0.952330i $$0.401320\pi$$
$$312$$ 0 0
$$313$$ 8.60648e8 1.58643 0.793214 0.608943i $$-0.208406\pi$$
0.793214 + 0.608943i $$0.208406\pi$$
$$314$$ 0 0
$$315$$ −1.20210e8 −0.216697
$$316$$ 0 0
$$317$$ −2.23939e8 −0.394840 −0.197420 0.980319i $$-0.563256\pi$$
−0.197420 + 0.980319i $$0.563256\pi$$
$$318$$ 0 0
$$319$$ −1.25035e7 −0.0215658
$$320$$ 0 0
$$321$$ −2.18616e8 −0.368905
$$322$$ 0 0
$$323$$ −1.28386e9 −2.11986
$$324$$ 0 0
$$325$$ 6.39370e7 0.103314
$$326$$ 0 0
$$327$$ −5.14782e8 −0.814154
$$328$$ 0 0
$$329$$ −9.20721e8 −1.42542
$$330$$ 0 0
$$331$$ −1.12017e9 −1.69780 −0.848902 0.528550i $$-0.822736\pi$$
−0.848902 + 0.528550i $$0.822736\pi$$
$$332$$ 0 0
$$333$$ 1.45219e8 0.215510
$$334$$ 0 0
$$335$$ −8.61556e8 −1.25207
$$336$$ 0 0
$$337$$ 3.53693e8 0.503410 0.251705 0.967804i $$-0.419009\pi$$
0.251705 + 0.967804i $$0.419009\pi$$
$$338$$ 0 0
$$339$$ −4.57107e8 −0.637263
$$340$$ 0 0
$$341$$ 9.52897e7 0.130138
$$342$$ 0 0
$$343$$ 8.13035e8 1.08788
$$344$$ 0 0
$$345$$ −1.46364e8 −0.191897
$$346$$ 0 0
$$347$$ −1.88230e8 −0.241845 −0.120922 0.992662i $$-0.538585\pi$$
−0.120922 + 0.992662i $$0.538585\pi$$
$$348$$ 0 0
$$349$$ 8.58810e7 0.108145 0.0540727 0.998537i $$-0.482780\pi$$
0.0540727 + 0.998537i $$0.482780\pi$$
$$350$$ 0 0
$$351$$ 4.77686e7 0.0589614
$$352$$ 0 0
$$353$$ 4.55346e7 0.0550973 0.0275487 0.999620i $$-0.491230\pi$$
0.0275487 + 0.999620i $$0.491230\pi$$
$$354$$ 0 0
$$355$$ −1.00147e9 −1.18807
$$356$$ 0 0
$$357$$ 5.62283e8 0.654058
$$358$$ 0 0
$$359$$ 1.01706e9 1.16015 0.580077 0.814562i $$-0.303022\pi$$
0.580077 + 0.814562i $$0.303022\pi$$
$$360$$ 0 0
$$361$$ 1.10191e9 1.23274
$$362$$ 0 0
$$363$$ 4.91963e8 0.539833
$$364$$ 0 0
$$365$$ −1.05337e9 −1.13385
$$366$$ 0 0
$$367$$ −9.15304e8 −0.966571 −0.483286 0.875463i $$-0.660557\pi$$
−0.483286 + 0.875463i $$0.660557\pi$$
$$368$$ 0 0
$$369$$ 1.98822e8 0.206003
$$370$$ 0 0
$$371$$ −2.69430e8 −0.273929
$$372$$ 0 0
$$373$$ −6.99424e8 −0.697846 −0.348923 0.937151i $$-0.613452\pi$$
−0.348923 + 0.937151i $$0.613452\pi$$
$$374$$ 0 0
$$375$$ 6.41854e8 0.628531
$$376$$ 0 0
$$377$$ 2.69657e7 0.0259189
$$378$$ 0 0
$$379$$ 1.35218e9 1.27584 0.637921 0.770102i $$-0.279795\pi$$
0.637921 + 0.770102i $$0.279795\pi$$
$$380$$ 0 0
$$381$$ 5.18198e8 0.480019
$$382$$ 0 0
$$383$$ −1.69797e9 −1.54431 −0.772155 0.635435i $$-0.780821\pi$$
−0.772155 + 0.635435i $$0.780821\pi$$
$$384$$ 0 0
$$385$$ −1.85560e8 −0.165719
$$386$$ 0 0
$$387$$ −4.25763e8 −0.373405
$$388$$ 0 0
$$389$$ 7.88879e8 0.679496 0.339748 0.940516i $$-0.389658\pi$$
0.339748 + 0.940516i $$0.389658\pi$$
$$390$$ 0 0
$$391$$ 6.84622e8 0.579204
$$392$$ 0 0
$$393$$ 6.71312e8 0.557892
$$394$$ 0 0
$$395$$ −5.40201e8 −0.441028
$$396$$ 0 0
$$397$$ −1.19931e9 −0.961978 −0.480989 0.876727i $$-0.659722\pi$$
−0.480989 + 0.876727i $$0.659722\pi$$
$$398$$ 0 0
$$399$$ −8.74081e8 −0.688885
$$400$$ 0 0
$$401$$ −1.04006e9 −0.805480 −0.402740 0.915314i $$-0.631942\pi$$
−0.402740 + 0.915314i $$0.631942\pi$$
$$402$$ 0 0
$$403$$ −2.05506e8 −0.156407
$$404$$ 0 0
$$405$$ 1.20930e8 0.0904571
$$406$$ 0 0
$$407$$ 2.24165e8 0.164811
$$408$$ 0 0
$$409$$ −4.04909e8 −0.292634 −0.146317 0.989238i $$-0.546742\pi$$
−0.146317 + 0.989238i $$0.546742\pi$$
$$410$$ 0 0
$$411$$ 6.56727e8 0.466594
$$412$$ 0 0
$$413$$ 1.30015e9 0.908173
$$414$$ 0 0
$$415$$ −1.31267e9 −0.901543
$$416$$ 0 0
$$417$$ 1.03440e9 0.698573
$$418$$ 0 0
$$419$$ 2.46116e8 0.163452 0.0817261 0.996655i $$-0.473957\pi$$
0.0817261 + 0.996655i $$0.473957\pi$$
$$420$$ 0 0
$$421$$ 1.07285e9 0.700731 0.350365 0.936613i $$-0.386057\pi$$
0.350365 + 0.936613i $$0.386057\pi$$
$$422$$ 0 0
$$423$$ 9.26241e8 0.595022
$$424$$ 0 0
$$425$$ −7.57113e8 −0.478409
$$426$$ 0 0
$$427$$ 1.60907e9 1.00018
$$428$$ 0 0
$$429$$ 7.37374e7 0.0450907
$$430$$ 0 0
$$431$$ −8.24317e8 −0.495934 −0.247967 0.968768i $$-0.579762\pi$$
−0.247967 + 0.968768i $$0.579762\pi$$
$$432$$ 0 0
$$433$$ −4.40778e8 −0.260923 −0.130462 0.991453i $$-0.541646\pi$$
−0.130462 + 0.991453i $$0.541646\pi$$
$$434$$ 0 0
$$435$$ 6.82660e7 0.0397642
$$436$$ 0 0
$$437$$ −1.06426e9 −0.610045
$$438$$ 0 0
$$439$$ −5.43490e8 −0.306596 −0.153298 0.988180i $$-0.548989\pi$$
−0.153298 + 0.988180i $$0.548989\pi$$
$$440$$ 0 0
$$441$$ −2.17546e8 −0.120786
$$442$$ 0 0
$$443$$ 1.85228e9 1.01226 0.506130 0.862457i $$-0.331075\pi$$
0.506130 + 0.862457i $$0.331075\pi$$
$$444$$ 0 0
$$445$$ 3.66564e8 0.197192
$$446$$ 0 0
$$447$$ −7.56888e7 −0.0400825
$$448$$ 0 0
$$449$$ 1.41843e9 0.739515 0.369758 0.929128i $$-0.379441\pi$$
0.369758 + 0.929128i $$0.379441\pi$$
$$450$$ 0 0
$$451$$ 3.06909e8 0.157541
$$452$$ 0 0
$$453$$ −1.16574e9 −0.589195
$$454$$ 0 0
$$455$$ 4.00187e8 0.199170
$$456$$ 0 0
$$457$$ −1.89446e9 −0.928493 −0.464247 0.885706i $$-0.653675\pi$$
−0.464247 + 0.885706i $$0.653675\pi$$
$$458$$ 0 0
$$459$$ −5.65654e8 −0.273028
$$460$$ 0 0
$$461$$ −3.41152e9 −1.62179 −0.810895 0.585192i $$-0.801019\pi$$
−0.810895 + 0.585192i $$0.801019\pi$$
$$462$$ 0 0
$$463$$ −1.75923e9 −0.823736 −0.411868 0.911244i $$-0.635123\pi$$
−0.411868 + 0.911244i $$0.635123\pi$$
$$464$$ 0 0
$$465$$ −5.20256e8 −0.239956
$$466$$ 0 0
$$467$$ 2.76328e9 1.25550 0.627749 0.778416i $$-0.283976\pi$$
0.627749 + 0.778416i $$0.283976\pi$$
$$468$$ 0 0
$$469$$ 2.74369e9 1.22809
$$470$$ 0 0
$$471$$ −2.12445e9 −0.936856
$$472$$ 0 0
$$473$$ −6.57224e8 −0.285561
$$474$$ 0 0
$$475$$ 1.17695e9 0.503883
$$476$$ 0 0
$$477$$ 2.71046e8 0.114348
$$478$$ 0 0
$$479$$ 9.32149e8 0.387535 0.193767 0.981047i $$-0.437929\pi$$
0.193767 + 0.981047i $$0.437929\pi$$
$$480$$ 0 0
$$481$$ −4.83444e8 −0.198079
$$482$$ 0 0
$$483$$ 4.66108e8 0.188222
$$484$$ 0 0
$$485$$ −1.85835e7 −0.00739660
$$486$$ 0 0
$$487$$ 3.53986e9 1.38878 0.694392 0.719597i $$-0.255674\pi$$
0.694392 + 0.719597i $$0.255674\pi$$
$$488$$ 0 0
$$489$$ 2.44242e8 0.0944581
$$490$$ 0 0
$$491$$ 4.04042e9 1.54043 0.770213 0.637786i $$-0.220150\pi$$
0.770213 + 0.637786i $$0.220150\pi$$
$$492$$ 0 0
$$493$$ −3.19315e8 −0.120021
$$494$$ 0 0
$$495$$ 1.86673e8 0.0691771
$$496$$ 0 0
$$497$$ 3.18927e9 1.16532
$$498$$ 0 0
$$499$$ −1.83149e9 −0.659860 −0.329930 0.944005i $$-0.607025\pi$$
−0.329930 + 0.944005i $$0.607025\pi$$
$$500$$ 0 0
$$501$$ −2.77498e9 −0.985890
$$502$$ 0 0
$$503$$ 3.44153e9 1.20577 0.602884 0.797829i $$-0.294018\pi$$
0.602884 + 0.797829i $$0.294018\pi$$
$$504$$ 0 0
$$505$$ −2.06943e9 −0.715042
$$506$$ 0 0
$$507$$ 1.53518e9 0.523158
$$508$$ 0 0
$$509$$ 5.73694e8 0.192827 0.0964136 0.995341i $$-0.469263\pi$$
0.0964136 + 0.995341i $$0.469263\pi$$
$$510$$ 0 0
$$511$$ 3.35453e9 1.11214
$$512$$ 0 0
$$513$$ 8.79321e8 0.287566
$$514$$ 0 0
$$515$$ −4.48777e9 −1.44779
$$516$$ 0 0
$$517$$ 1.42978e9 0.455043
$$518$$ 0 0
$$519$$ 3.23450e9 1.01560
$$520$$ 0 0
$$521$$ 2.83121e9 0.877081 0.438540 0.898712i $$-0.355496\pi$$
0.438540 + 0.898712i $$0.355496\pi$$
$$522$$ 0 0
$$523$$ 3.02759e9 0.925426 0.462713 0.886508i $$-0.346876\pi$$
0.462713 + 0.886508i $$0.346876\pi$$
$$524$$ 0 0
$$525$$ −5.15461e8 −0.155467
$$526$$ 0 0
$$527$$ 2.43351e9 0.724262
$$528$$ 0 0
$$529$$ −2.83730e9 −0.833319
$$530$$ 0 0
$$531$$ −1.30795e9 −0.379104
$$532$$ 0 0
$$533$$ −6.61895e8 −0.189341
$$534$$ 0 0
$$535$$ 1.84246e9 0.520188
$$536$$ 0 0
$$537$$ 2.89808e8 0.0807606
$$538$$ 0 0
$$539$$ −3.35812e8 −0.0923711
$$540$$ 0 0
$$541$$ 4.46669e9 1.21282 0.606408 0.795153i $$-0.292610\pi$$
0.606408 + 0.795153i $$0.292610\pi$$
$$542$$ 0 0
$$543$$ 4.24681e8 0.113832
$$544$$ 0 0
$$545$$ 4.33851e9 1.14803
$$546$$ 0 0
$$547$$ 1.49405e9 0.390311 0.195155 0.980772i $$-0.437479\pi$$
0.195155 + 0.980772i $$0.437479\pi$$
$$548$$ 0 0
$$549$$ −1.61872e9 −0.417511
$$550$$ 0 0
$$551$$ 4.96382e8 0.126411
$$552$$ 0 0
$$553$$ 1.72031e9 0.432582
$$554$$ 0 0
$$555$$ −1.22388e9 −0.303888
$$556$$ 0 0
$$557$$ 5.02645e9 1.23245 0.616224 0.787571i $$-0.288662\pi$$
0.616224 + 0.787571i $$0.288662\pi$$
$$558$$ 0 0
$$559$$ 1.41740e9 0.343202
$$560$$ 0 0
$$561$$ −8.73165e8 −0.208798
$$562$$ 0 0
$$563$$ −4.95585e9 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$564$$ 0 0
$$565$$ 3.85243e9 0.898597
$$566$$ 0 0
$$567$$ −3.85112e8 −0.0887250
$$568$$ 0 0
$$569$$ −2.06899e9 −0.470831 −0.235415 0.971895i $$-0.575645\pi$$
−0.235415 + 0.971895i $$0.575645\pi$$
$$570$$ 0 0
$$571$$ −1.84156e9 −0.413961 −0.206980 0.978345i $$-0.566364\pi$$
−0.206980 + 0.978345i $$0.566364\pi$$
$$572$$ 0 0
$$573$$ −2.75361e9 −0.611451
$$574$$ 0 0
$$575$$ −6.27613e8 −0.137675
$$576$$ 0 0
$$577$$ −5.22939e9 −1.13328 −0.566638 0.823967i $$-0.691756\pi$$
−0.566638 + 0.823967i $$0.691756\pi$$
$$578$$ 0 0
$$579$$ 2.25215e9 0.482195
$$580$$ 0 0
$$581$$ 4.18029e9 0.884280
$$582$$ 0 0
$$583$$ 4.18396e8 0.0874475
$$584$$ 0 0
$$585$$ −4.02586e8 −0.0831407
$$586$$ 0 0
$$587$$ 3.29340e9 0.672065 0.336033 0.941850i $$-0.390915\pi$$
0.336033 + 0.941850i $$0.390915\pi$$
$$588$$ 0 0
$$589$$ −3.78294e9 −0.762827
$$590$$ 0 0
$$591$$ 1.57135e9 0.313125
$$592$$ 0 0
$$593$$ 2.35508e8 0.0463783 0.0231892 0.999731i $$-0.492618\pi$$
0.0231892 + 0.999731i $$0.492618\pi$$
$$594$$ 0 0
$$595$$ −4.73884e9 −0.922279
$$596$$ 0 0
$$597$$ −4.36065e9 −0.838766
$$598$$ 0 0
$$599$$ −1.02534e8 −0.0194928 −0.00974639 0.999953i $$-0.503102\pi$$
−0.00974639 + 0.999953i $$0.503102\pi$$
$$600$$ 0 0
$$601$$ 2.48675e9 0.467275 0.233637 0.972324i $$-0.424937\pi$$
0.233637 + 0.972324i $$0.424937\pi$$
$$602$$ 0 0
$$603$$ −2.76014e9 −0.512649
$$604$$ 0 0
$$605$$ −4.14619e9 −0.761211
$$606$$ 0 0
$$607$$ −2.87128e9 −0.521092 −0.260546 0.965461i $$-0.583903\pi$$
−0.260546 + 0.965461i $$0.583903\pi$$
$$608$$ 0 0
$$609$$ −2.17398e8 −0.0390027
$$610$$ 0 0
$$611$$ −3.08353e9 −0.546894
$$612$$ 0 0
$$613$$ −7.52444e9 −1.31936 −0.659679 0.751547i $$-0.729308\pi$$
−0.659679 + 0.751547i $$0.729308\pi$$
$$614$$ 0 0
$$615$$ −1.67564e9 −0.290482
$$616$$ 0 0
$$617$$ 6.34114e8 0.108685 0.0543425 0.998522i $$-0.482694\pi$$
0.0543425 + 0.998522i $$0.482694\pi$$
$$618$$ 0 0
$$619$$ −4.81321e9 −0.815675 −0.407838 0.913054i $$-0.633717\pi$$
−0.407838 + 0.913054i $$0.633717\pi$$
$$620$$ 0 0
$$621$$ −4.68902e8 −0.0785709
$$622$$ 0 0
$$623$$ −1.16735e9 −0.193416
$$624$$ 0 0
$$625$$ −3.35123e9 −0.549066
$$626$$ 0 0
$$627$$ 1.35735e9 0.219916
$$628$$ 0 0
$$629$$ 5.72472e9 0.917228
$$630$$ 0 0
$$631$$ −3.35568e9 −0.531714 −0.265857 0.964012i $$-0.585655\pi$$
−0.265857 + 0.964012i $$0.585655\pi$$
$$632$$ 0 0
$$633$$ −2.09128e9 −0.327717
$$634$$ 0 0
$$635$$ −4.36730e9 −0.676869
$$636$$ 0 0
$$637$$ 7.24228e8 0.111016
$$638$$ 0 0
$$639$$ −3.20839e9 −0.486445
$$640$$ 0 0
$$641$$ 1.01582e10 1.52339 0.761696 0.647935i $$-0.224367\pi$$
0.761696 + 0.647935i $$0.224367\pi$$
$$642$$ 0 0
$$643$$ 5.05703e9 0.750165 0.375083 0.926991i $$-0.377614\pi$$
0.375083 + 0.926991i $$0.377614\pi$$
$$644$$ 0 0
$$645$$ 3.58827e9 0.526533
$$646$$ 0 0
$$647$$ −1.05626e10 −1.53323 −0.766613 0.642109i $$-0.778060\pi$$
−0.766613 + 0.642109i $$0.778060\pi$$
$$648$$ 0 0
$$649$$ −2.01899e9 −0.289920
$$650$$ 0 0
$$651$$ 1.65679e9 0.235361
$$652$$ 0 0
$$653$$ −6.19593e8 −0.0870784 −0.0435392 0.999052i $$-0.513863\pi$$
−0.0435392 + 0.999052i $$0.513863\pi$$
$$654$$ 0 0
$$655$$ −5.65771e9 −0.786677
$$656$$ 0 0
$$657$$ −3.37464e9 −0.464246
$$658$$ 0 0
$$659$$ 6.32359e9 0.860725 0.430363 0.902656i $$-0.358386\pi$$
0.430363 + 0.902656i $$0.358386\pi$$
$$660$$ 0 0
$$661$$ −9.81233e9 −1.32150 −0.660750 0.750606i $$-0.729761\pi$$
−0.660750 + 0.750606i $$0.729761\pi$$
$$662$$ 0 0
$$663$$ 1.88311e9 0.250944
$$664$$ 0 0
$$665$$ 7.36662e9 0.971388
$$666$$ 0 0
$$667$$ −2.64698e8 −0.0345391
$$668$$ 0 0
$$669$$ 2.52036e9 0.325440
$$670$$ 0 0
$$671$$ −2.49871e9 −0.319292
$$672$$ 0 0
$$673$$ −1.37509e10 −1.73892 −0.869458 0.494006i $$-0.835532\pi$$
−0.869458 + 0.494006i $$0.835532\pi$$
$$674$$ 0 0
$$675$$ 5.18552e8 0.0648976
$$676$$ 0 0
$$677$$ 8.30590e9 1.02879 0.514395 0.857553i $$-0.328017\pi$$
0.514395 + 0.857553i $$0.328017\pi$$
$$678$$ 0 0
$$679$$ 5.91806e7 0.00725496
$$680$$ 0 0
$$681$$ −1.15643e9 −0.140315
$$682$$ 0 0
$$683$$ 2.10499e9 0.252800 0.126400 0.991979i $$-0.459658\pi$$
0.126400 + 0.991979i $$0.459658\pi$$
$$684$$ 0 0
$$685$$ −5.53480e9 −0.657938
$$686$$ 0 0
$$687$$ −4.89015e9 −0.575405
$$688$$ 0 0
$$689$$ −9.02331e8 −0.105099
$$690$$ 0 0
$$691$$ 4.48072e9 0.516624 0.258312 0.966061i $$-0.416834\pi$$
0.258312 + 0.966061i $$0.416834\pi$$
$$692$$ 0 0
$$693$$ −5.94472e8 −0.0678524
$$694$$ 0 0
$$695$$ −8.71775e9 −0.985049
$$696$$ 0 0
$$697$$ 7.83786e9 0.876764
$$698$$ 0 0
$$699$$ 7.39774e8 0.0819274
$$700$$ 0 0
$$701$$ 1.62398e10 1.78060 0.890300 0.455374i $$-0.150494\pi$$
0.890300 + 0.455374i $$0.150494\pi$$
$$702$$ 0 0
$$703$$ −8.89920e9 −0.966067
$$704$$ 0 0
$$705$$ −7.80622e9 −0.839032
$$706$$ 0 0
$$707$$ 6.59026e9 0.701350
$$708$$ 0 0
$$709$$ −8.62447e9 −0.908805 −0.454403 0.890796i $$-0.650147\pi$$
−0.454403 + 0.890796i $$0.650147\pi$$
$$710$$ 0 0
$$711$$ −1.73062e9 −0.180576
$$712$$ 0 0
$$713$$ 2.01727e9 0.208425
$$714$$ 0 0
$$715$$ −6.21447e8 −0.0635818
$$716$$ 0 0
$$717$$ 2.09530e9 0.212289
$$718$$ 0 0
$$719$$ 2.97175e9 0.298168 0.149084 0.988825i $$-0.452367\pi$$
0.149084 + 0.988825i $$0.452367\pi$$
$$720$$ 0 0
$$721$$ 1.42916e10 1.42007
$$722$$ 0 0
$$723$$ −1.13722e10 −1.11908
$$724$$ 0 0
$$725$$ 2.92726e8 0.0285284
$$726$$ 0 0
$$727$$ −1.52769e10 −1.47457 −0.737284 0.675583i $$-0.763892\pi$$
−0.737284 + 0.675583i $$0.763892\pi$$
$$728$$ 0 0
$$729$$ 3.87420e8 0.0370370
$$730$$ 0 0
$$731$$ −1.67842e10 −1.58924
$$732$$ 0 0
$$733$$ 2.55786e9 0.239891 0.119945 0.992780i $$-0.461728\pi$$
0.119945 + 0.992780i $$0.461728\pi$$
$$734$$ 0 0
$$735$$ 1.83345e9 0.170319
$$736$$ 0 0
$$737$$ −4.26065e9 −0.392048
$$738$$ 0 0
$$739$$ −5.76099e9 −0.525099 −0.262550 0.964919i $$-0.584563\pi$$
−0.262550 + 0.964919i $$0.584563\pi$$
$$740$$ 0 0
$$741$$ −2.92733e9 −0.264306
$$742$$ 0 0
$$743$$ 8.79171e9 0.786344 0.393172 0.919465i $$-0.371378\pi$$
0.393172 + 0.919465i $$0.371378\pi$$
$$744$$ 0 0
$$745$$ 6.37894e8 0.0565199
$$746$$ 0 0
$$747$$ −4.20535e9 −0.369130
$$748$$ 0 0
$$749$$ −5.86746e9 −0.510227
$$750$$ 0 0
$$751$$ 6.14505e9 0.529402 0.264701 0.964330i $$-0.414727\pi$$
0.264701 + 0.964330i $$0.414727\pi$$
$$752$$ 0 0
$$753$$ −1.28351e10 −1.09551
$$754$$ 0 0
$$755$$ 9.82470e9 0.830816
$$756$$ 0 0
$$757$$ −7.09591e9 −0.594528 −0.297264 0.954795i $$-0.596074\pi$$
−0.297264 + 0.954795i $$0.596074\pi$$
$$758$$ 0 0
$$759$$ −7.23814e8 −0.0600871
$$760$$ 0 0
$$761$$ −1.44986e10 −1.19256 −0.596280 0.802776i $$-0.703355\pi$$
−0.596280 + 0.802776i $$0.703355\pi$$
$$762$$ 0 0
$$763$$ −1.38163e10 −1.12604
$$764$$ 0 0
$$765$$ 4.76725e9 0.384993
$$766$$ 0 0
$$767$$ 4.35425e9 0.348441
$$768$$ 0 0
$$769$$ 1.48341e10 1.17630 0.588151 0.808751i $$-0.299856\pi$$
0.588151 + 0.808751i $$0.299856\pi$$
$$770$$ 0 0
$$771$$ 1.38479e10 1.08816
$$772$$ 0 0
$$773$$ 1.64984e10 1.28473 0.642366 0.766398i $$-0.277953\pi$$
0.642366 + 0.766398i $$0.277953\pi$$
$$774$$ 0 0
$$775$$ −2.23087e9 −0.172154
$$776$$ 0 0
$$777$$ 3.89753e9 0.298069
$$778$$ 0 0
$$779$$ −1.21841e10 −0.923449
$$780$$ 0 0
$$781$$ −4.95259e9 −0.372009
$$782$$ 0 0
$$783$$ 2.18701e8 0.0162812
$$784$$ 0 0
$$785$$ 1.79045e10 1.32105
$$786$$ 0 0
$$787$$ 4.05010e9 0.296179 0.148090 0.988974i $$-0.452688\pi$$
0.148090 + 0.988974i $$0.452688\pi$$
$$788$$ 0 0
$$789$$ −5.59392e9 −0.405459
$$790$$ 0 0
$$791$$ −1.22683e10 −0.881390
$$792$$ 0 0
$$793$$ 5.38884e9 0.383742
$$794$$ 0 0
$$795$$ −2.28433e9 −0.161240
$$796$$ 0 0
$$797$$ −1.61834e9 −0.113231 −0.0566156 0.998396i $$-0.518031\pi$$
−0.0566156 + 0.998396i $$0.518031\pi$$
$$798$$ 0 0
$$799$$ 3.65137e10 2.53246
$$800$$ 0 0
$$801$$ 1.17435e9 0.0807390
$$802$$ 0 0
$$803$$ −5.20921e9 −0.355032
$$804$$ 0 0
$$805$$ −3.92828e9 −0.265410
$$806$$ 0 0
$$807$$ 1.06672e9 0.0714486
$$808$$ 0 0
$$809$$ −1.32510e10 −0.879889 −0.439945 0.898025i $$-0.645002\pi$$
−0.439945 + 0.898025i $$0.645002\pi$$
$$810$$ 0 0
$$811$$ −5.29612e9 −0.348646 −0.174323 0.984689i $$-0.555774\pi$$
−0.174323 + 0.984689i $$0.555774\pi$$
$$812$$ 0 0
$$813$$ 4.32959e9 0.282573
$$814$$ 0 0
$$815$$ −2.05843e9 −0.133194
$$816$$ 0 0
$$817$$ 2.60914e10 1.67386
$$818$$ 0 0
$$819$$ 1.28207e9 0.0815486
$$820$$ 0 0
$$821$$ 1.34687e10 0.849425 0.424712 0.905328i $$-0.360375\pi$$
0.424712 + 0.905328i $$0.360375\pi$$
$$822$$ 0 0
$$823$$ 5.91545e9 0.369903 0.184952 0.982748i $$-0.440787\pi$$
0.184952 + 0.982748i $$0.440787\pi$$
$$824$$ 0 0
$$825$$ 8.00455e8 0.0496305
$$826$$ 0 0
$$827$$ −2.16343e9 −0.133007 −0.0665033 0.997786i $$-0.521184\pi$$
−0.0665033 + 0.997786i $$0.521184\pi$$
$$828$$ 0 0
$$829$$ −1.03884e9 −0.0633301 −0.0316650 0.999499i $$-0.510081\pi$$
−0.0316650 + 0.999499i $$0.510081\pi$$
$$830$$ 0 0
$$831$$ 1.74712e10 1.05613
$$832$$ 0 0
$$833$$ −8.57598e9 −0.514075
$$834$$ 0 0
$$835$$ 2.33871e10 1.39019
$$836$$ 0 0
$$837$$ −1.66673e9 −0.0982484
$$838$$ 0 0
$$839$$ 2.96059e10 1.73066 0.865329 0.501205i $$-0.167110\pi$$
0.865329 + 0.501205i $$0.167110\pi$$
$$840$$ 0 0
$$841$$ −1.71264e10 −0.992843
$$842$$ 0 0
$$843$$ 1.02179e10 0.587440
$$844$$ 0 0
$$845$$ −1.29383e10 −0.737698
$$846$$ 0 0
$$847$$ 1.32038e10 0.746635
$$848$$ 0 0
$$849$$ −1.20429e10 −0.675391
$$850$$ 0 0
$$851$$ 4.74554e9 0.263956
$$852$$ 0 0
$$853$$ −1.48670e10 −0.820163 −0.410082 0.912049i $$-0.634500\pi$$
−0.410082 + 0.912049i $$0.634500\pi$$
$$854$$ 0 0
$$855$$ −7.41078e9 −0.405492
$$856$$ 0 0
$$857$$ 7.25984e8 0.0393998 0.0196999 0.999806i $$-0.493729\pi$$
0.0196999 + 0.999806i $$0.493729\pi$$
$$858$$ 0 0
$$859$$ −9.14667e9 −0.492365 −0.246182 0.969224i $$-0.579176\pi$$
−0.246182 + 0.969224i $$0.579176\pi$$
$$860$$ 0 0
$$861$$ 5.33621e9 0.284919
$$862$$ 0 0
$$863$$ 3.17870e10 1.68349 0.841746 0.539873i $$-0.181528\pi$$
0.841746 + 0.539873i $$0.181528\pi$$
$$864$$ 0 0
$$865$$ −2.72599e10 −1.43208
$$866$$ 0 0
$$867$$ −1.12197e10 −0.584677
$$868$$ 0 0
$$869$$ −2.67145e9 −0.138095
$$870$$ 0 0
$$871$$ 9.18871e9 0.471185
$$872$$ 0 0
$$873$$ −5.95354e7 −0.00302849
$$874$$ 0 0
$$875$$ 1.72268e10 0.869313
$$876$$ 0 0
$$877$$ 3.37560e10 1.68987 0.844933 0.534873i $$-0.179640\pi$$
0.844933 + 0.534873i $$0.179640\pi$$
$$878$$ 0 0
$$879$$ 1.47588e10 0.732975
$$880$$ 0 0
$$881$$ −1.35501e10 −0.667617 −0.333809 0.942641i $$-0.608334\pi$$
−0.333809 + 0.942641i $$0.608334\pi$$
$$882$$ 0 0
$$883$$ −3.32363e9 −0.162461 −0.0812307 0.996695i $$-0.525885\pi$$
−0.0812307 + 0.996695i $$0.525885\pi$$
$$884$$ 0 0
$$885$$ 1.10232e10 0.534570
$$886$$ 0 0
$$887$$ −2.18728e10 −1.05238 −0.526190 0.850367i $$-0.676380\pi$$
−0.526190 + 0.850367i $$0.676380\pi$$
$$888$$ 0 0
$$889$$ 1.39080e10 0.663908
$$890$$ 0 0
$$891$$ 5.98036e8 0.0283241
$$892$$ 0 0
$$893$$ −5.67614e10 −2.66731
$$894$$ 0 0
$$895$$ −2.44245e9 −0.113880
$$896$$ 0 0
$$897$$ 1.56101e9 0.0722158
$$898$$ 0 0
$$899$$ −9.40878e8 −0.0431891
$$900$$ 0 0
$$901$$ 1.06850e10 0.486674
$$902$$ 0 0
$$903$$ −1.14271e10 −0.516451
$$904$$ 0 0
$$905$$ −3.57915e9 −0.160513
$$906$$ 0 0
$$907$$ 2.92742e10 1.30275 0.651373 0.758757i $$-0.274193\pi$$
0.651373 + 0.758757i $$0.274193\pi$$
$$908$$ 0 0
$$909$$ −6.62977e9 −0.292769
$$910$$ 0 0
$$911$$ −7.06839e9 −0.309747 −0.154873 0.987934i $$-0.549497\pi$$
−0.154873 + 0.987934i $$0.549497\pi$$
$$912$$ 0 0
$$913$$ −6.49153e9 −0.282292
$$914$$ 0 0
$$915$$ 1.36423e10 0.588728
$$916$$ 0 0
$$917$$ 1.80174e10 0.771613
$$918$$ 0 0
$$919$$ −3.15970e10 −1.34289 −0.671446 0.741054i $$-0.734326\pi$$
−0.671446 + 0.741054i $$0.734326\pi$$
$$920$$ 0 0
$$921$$ 1.70239e10 0.718041
$$922$$ 0 0
$$923$$ 1.06810e10 0.447100
$$924$$ 0 0
$$925$$ −5.24802e9 −0.218022
$$926$$ 0 0
$$927$$ −1.43773e10 −0.592787
$$928$$ 0 0
$$929$$ 4.01061e10 1.64118 0.820589 0.571519i $$-0.193646\pi$$
0.820589 + 0.571519i $$0.193646\pi$$
$$930$$ 0 0
$$931$$ 1.33315e10 0.541448
$$932$$ 0 0
$$933$$ −8.73884e9 −0.352264
$$934$$ 0 0
$$935$$ 7.35890e9 0.294423
$$936$$ 0 0
$$937$$ 1.56498e10 0.621470 0.310735 0.950497i $$-0.399425\pi$$
0.310735 + 0.950497i $$0.399425\pi$$
$$938$$ 0 0
$$939$$ −2.32375e10 −0.915925
$$940$$ 0 0
$$941$$ −7.90172e9 −0.309142 −0.154571 0.987982i $$-0.549400\pi$$
−0.154571 + 0.987982i $$0.549400\pi$$
$$942$$ 0 0
$$943$$ 6.49723e9 0.252312
$$944$$ 0 0
$$945$$ 3.24566e9 0.125110
$$946$$ 0 0
$$947$$ 5.04304e10 1.92960 0.964800 0.262985i $$-0.0847069\pi$$
0.964800 + 0.262985i $$0.0847069\pi$$
$$948$$ 0 0
$$949$$ 1.12344e10 0.426696
$$950$$ 0 0
$$951$$ 6.04634e9 0.227961
$$952$$ 0 0
$$953$$ −5.01845e10 −1.87821 −0.939107 0.343626i $$-0.888345\pi$$
−0.939107 + 0.343626i $$0.888345\pi$$
$$954$$ 0 0
$$955$$ 2.32070e10 0.862199
$$956$$ 0 0
$$957$$ 3.37595e8 0.0124510
$$958$$ 0 0
$$959$$ 1.76260e10 0.645339
$$960$$ 0 0
$$961$$ −2.03422e10 −0.739376
$$962$$ 0 0
$$963$$ 5.90264e9 0.212987
$$964$$ 0 0
$$965$$ −1.89808e10 −0.679937
$$966$$ 0 0
$$967$$ −2.67558e10 −0.951535 −0.475767 0.879571i $$-0.657830\pi$$
−0.475767 + 0.879571i $$0.657830\pi$$
$$968$$ 0 0
$$969$$ 3.46641e10 1.22390
$$970$$ 0 0
$$971$$ −3.60085e10 −1.26223 −0.631115 0.775690i $$-0.717402\pi$$
−0.631115 + 0.775690i $$0.717402\pi$$
$$972$$ 0 0
$$973$$ 2.77623e10 0.966186
$$974$$ 0 0
$$975$$ −1.72630e9 −0.0596485
$$976$$ 0 0
$$977$$ 2.09028e10 0.717089 0.358544 0.933513i $$-0.383273\pi$$
0.358544 + 0.933513i $$0.383273\pi$$
$$978$$ 0 0
$$979$$ 1.81277e9 0.0617451
$$980$$ 0 0
$$981$$ 1.38991e10 0.470052
$$982$$ 0 0
$$983$$ −4.52039e10 −1.51788 −0.758942 0.651158i $$-0.774284\pi$$
−0.758942 + 0.651158i $$0.774284\pi$$
$$984$$ 0 0
$$985$$ −1.32431e10 −0.441533
$$986$$ 0 0
$$987$$ 2.48595e10 0.822966
$$988$$ 0 0
$$989$$ −1.39133e10 −0.457345
$$990$$ 0 0
$$991$$ 1.67128e10 0.545495 0.272747 0.962086i $$-0.412068\pi$$
0.272747 + 0.962086i $$0.412068\pi$$
$$992$$ 0 0
$$993$$ 3.02447e10 0.980227
$$994$$ 0 0
$$995$$ 3.67509e10 1.18273
$$996$$ 0 0
$$997$$ 8.89725e9 0.284330 0.142165 0.989843i $$-0.454594\pi$$
0.142165 + 0.989843i $$0.454594\pi$$
$$998$$ 0 0
$$999$$ −3.92090e9 −0.124425
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 192.8.a.q.1.2 2
3.2 odd 2 576.8.a.bo.1.1 2
4.3 odd 2 192.8.a.t.1.2 2
8.3 odd 2 96.8.a.e.1.1 2
8.5 even 2 96.8.a.h.1.1 yes 2
12.11 even 2 576.8.a.bn.1.1 2
24.5 odd 2 288.8.a.h.1.2 2
24.11 even 2 288.8.a.g.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
96.8.a.e.1.1 2 8.3 odd 2
96.8.a.h.1.1 yes 2 8.5 even 2
192.8.a.q.1.2 2 1.1 even 1 trivial
192.8.a.t.1.2 2 4.3 odd 2
288.8.a.g.1.2 2 24.11 even 2
288.8.a.h.1.2 2 24.5 odd 2
576.8.a.bn.1.1 2 12.11 even 2
576.8.a.bo.1.1 2 3.2 odd 2