# Properties

 Label 560.8.a.i.1.1 Level $560$ Weight $8$ Character 560.1 Self dual yes Analytic conductor $174.936$ 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] = [560,8,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("560.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 560.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: $$174.935614271$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.31662$$ of defining polynomial Character $$\chi$$ $$=$$ 560.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-24.7995 q^{3} +125.000 q^{5} +343.000 q^{7} -1571.98 q^{9} +O(q^{10})$$ $$q-24.7995 q^{3} +125.000 q^{5} +343.000 q^{7} -1571.98 q^{9} +1432.37 q^{11} -6136.30 q^{13} -3099.94 q^{15} -15858.5 q^{17} +38567.5 q^{19} -8506.23 q^{21} +63987.4 q^{23} +15625.0 q^{25} +93220.9 q^{27} +94236.6 q^{29} -275990. q^{31} -35521.9 q^{33} +42875.0 q^{35} +156532. q^{37} +152177. q^{39} -303738. q^{41} -636818. q^{43} -196498. q^{45} -512021. q^{47} +117649. q^{49} +393282. q^{51} -201249. q^{53} +179046. q^{55} -956454. q^{57} +1.81196e6 q^{59} -982021. q^{61} -539191. q^{63} -767038. q^{65} +4.45336e6 q^{67} -1.58686e6 q^{69} -725436. q^{71} +2.17602e6 q^{73} -387492. q^{75} +491301. q^{77} +5.21525e6 q^{79} +1.12610e6 q^{81} -6.07921e6 q^{83} -1.98231e6 q^{85} -2.33702e6 q^{87} -1.06137e7 q^{89} -2.10475e6 q^{91} +6.84442e6 q^{93} +4.82093e6 q^{95} +6.64483e6 q^{97} -2.25166e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 30 q^{3} + 250 q^{5} + 686 q^{7} - 756 q^{9}+O(q^{10})$$ 2 * q + 30 * q^3 + 250 * q^5 + 686 * q^7 - 756 * q^9 $$2 q + 30 q^{3} + 250 q^{5} + 686 q^{7} - 756 q^{9} + 7906 q^{11} - 17818 q^{13} + 3750 q^{15} - 2398 q^{17} + 3612 q^{19} + 10290 q^{21} - 13844 q^{23} + 31250 q^{25} + 18090 q^{27} - 126898 q^{29} - 252768 q^{31} + 319230 q^{33} + 85750 q^{35} - 265860 q^{37} - 487974 q^{39} - 111920 q^{41} - 947572 q^{43} - 94500 q^{45} - 271274 q^{47} + 235298 q^{49} + 1130910 q^{51} - 1267792 q^{53} + 988250 q^{55} - 2871996 q^{57} + 1360120 q^{59} - 1813680 q^{61} - 259308 q^{63} - 2227250 q^{65} + 2189312 q^{67} - 5851980 q^{69} + 1494928 q^{71} + 7169788 q^{73} + 468750 q^{75} + 2711758 q^{77} + 7942974 q^{79} - 4775598 q^{81} + 304712 q^{83} - 299750 q^{85} - 14455086 q^{87} - 17943528 q^{89} - 6111574 q^{91} + 8116992 q^{93} + 451500 q^{95} + 4258074 q^{97} + 3030732 q^{99}+O(q^{100})$$ 2 * q + 30 * q^3 + 250 * q^5 + 686 * q^7 - 756 * q^9 + 7906 * q^11 - 17818 * q^13 + 3750 * q^15 - 2398 * q^17 + 3612 * q^19 + 10290 * q^21 - 13844 * q^23 + 31250 * q^25 + 18090 * q^27 - 126898 * q^29 - 252768 * q^31 + 319230 * q^33 + 85750 * q^35 - 265860 * q^37 - 487974 * q^39 - 111920 * q^41 - 947572 * q^43 - 94500 * q^45 - 271274 * q^47 + 235298 * q^49 + 1130910 * q^51 - 1267792 * q^53 + 988250 * q^55 - 2871996 * q^57 + 1360120 * q^59 - 1813680 * q^61 - 259308 * q^63 - 2227250 * q^65 + 2189312 * q^67 - 5851980 * q^69 + 1494928 * q^71 + 7169788 * q^73 + 468750 * q^75 + 2711758 * q^77 + 7942974 * q^79 - 4775598 * q^81 + 304712 * q^83 - 299750 * q^85 - 14455086 * q^87 - 17943528 * q^89 - 6111574 * q^91 + 8116992 * q^93 + 451500 * q^95 + 4258074 * q^97 + 3030732 * 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
$$3$$ −24.7995 −0.530296 −0.265148 0.964208i $$-0.585421\pi$$
−0.265148 + 0.964208i $$0.585421\pi$$
$$4$$ 0 0
$$5$$ 125.000 0.447214
$$6$$ 0 0
$$7$$ 343.000 0.377964
$$8$$ 0 0
$$9$$ −1571.98 −0.718786
$$10$$ 0 0
$$11$$ 1432.37 0.324474 0.162237 0.986752i $$-0.448129\pi$$
0.162237 + 0.986752i $$0.448129\pi$$
$$12$$ 0 0
$$13$$ −6136.30 −0.774649 −0.387325 0.921943i $$-0.626601\pi$$
−0.387325 + 0.921943i $$0.626601\pi$$
$$14$$ 0 0
$$15$$ −3099.94 −0.237156
$$16$$ 0 0
$$17$$ −15858.5 −0.782871 −0.391436 0.920205i $$-0.628021\pi$$
−0.391436 + 0.920205i $$0.628021\pi$$
$$18$$ 0 0
$$19$$ 38567.5 1.28998 0.644991 0.764190i $$-0.276861\pi$$
0.644991 + 0.764190i $$0.276861\pi$$
$$20$$ 0 0
$$21$$ −8506.23 −0.200433
$$22$$ 0 0
$$23$$ 63987.4 1.09660 0.548299 0.836282i $$-0.315276\pi$$
0.548299 + 0.836282i $$0.315276\pi$$
$$24$$ 0 0
$$25$$ 15625.0 0.200000
$$26$$ 0 0
$$27$$ 93220.9 0.911466
$$28$$ 0 0
$$29$$ 94236.6 0.717508 0.358754 0.933432i $$-0.383202\pi$$
0.358754 + 0.933432i $$0.383202\pi$$
$$30$$ 0 0
$$31$$ −275990. −1.66390 −0.831951 0.554849i $$-0.812776\pi$$
−0.831951 + 0.554849i $$0.812776\pi$$
$$32$$ 0 0
$$33$$ −35521.9 −0.172067
$$34$$ 0 0
$$35$$ 42875.0 0.169031
$$36$$ 0 0
$$37$$ 156532. 0.508038 0.254019 0.967199i $$-0.418247\pi$$
0.254019 + 0.967199i $$0.418247\pi$$
$$38$$ 0 0
$$39$$ 152177. 0.410793
$$40$$ 0 0
$$41$$ −303738. −0.688266 −0.344133 0.938921i $$-0.611827\pi$$
−0.344133 + 0.938921i $$0.611827\pi$$
$$42$$ 0 0
$$43$$ −636818. −1.22145 −0.610725 0.791843i $$-0.709122\pi$$
−0.610725 + 0.791843i $$0.709122\pi$$
$$44$$ 0 0
$$45$$ −196498. −0.321451
$$46$$ 0 0
$$47$$ −512021. −0.719358 −0.359679 0.933076i $$-0.617114\pi$$
−0.359679 + 0.933076i $$0.617114\pi$$
$$48$$ 0 0
$$49$$ 117649. 0.142857
$$50$$ 0 0
$$51$$ 393282. 0.415154
$$52$$ 0 0
$$53$$ −201249. −0.185681 −0.0928406 0.995681i $$-0.529595\pi$$
−0.0928406 + 0.995681i $$0.529595\pi$$
$$54$$ 0 0
$$55$$ 179046. 0.145109
$$56$$ 0 0
$$57$$ −956454. −0.684072
$$58$$ 0 0
$$59$$ 1.81196e6 1.14859 0.574296 0.818648i $$-0.305276\pi$$
0.574296 + 0.818648i $$0.305276\pi$$
$$60$$ 0 0
$$61$$ −982021. −0.553945 −0.276972 0.960878i $$-0.589331\pi$$
−0.276972 + 0.960878i $$0.589331\pi$$
$$62$$ 0 0
$$63$$ −539191. −0.271676
$$64$$ 0 0
$$65$$ −767038. −0.346434
$$66$$ 0 0
$$67$$ 4.45336e6 1.80895 0.904474 0.426528i $$-0.140264\pi$$
0.904474 + 0.426528i $$0.140264\pi$$
$$68$$ 0 0
$$69$$ −1.58686e6 −0.581522
$$70$$ 0 0
$$71$$ −725436. −0.240544 −0.120272 0.992741i $$-0.538377\pi$$
−0.120272 + 0.992741i $$0.538377\pi$$
$$72$$ 0 0
$$73$$ 2.17602e6 0.654685 0.327343 0.944906i $$-0.393847\pi$$
0.327343 + 0.944906i $$0.393847\pi$$
$$74$$ 0 0
$$75$$ −387492. −0.106059
$$76$$ 0 0
$$77$$ 491301. 0.122639
$$78$$ 0 0
$$79$$ 5.21525e6 1.19009 0.595045 0.803692i $$-0.297134\pi$$
0.595045 + 0.803692i $$0.297134\pi$$
$$80$$ 0 0
$$81$$ 1.12610e6 0.235439
$$82$$ 0 0
$$83$$ −6.07921e6 −1.16701 −0.583504 0.812110i $$-0.698319\pi$$
−0.583504 + 0.812110i $$0.698319\pi$$
$$84$$ 0 0
$$85$$ −1.98231e6 −0.350111
$$86$$ 0 0
$$87$$ −2.33702e6 −0.380492
$$88$$ 0 0
$$89$$ −1.06137e7 −1.59589 −0.797946 0.602729i $$-0.794080\pi$$
−0.797946 + 0.602729i $$0.794080\pi$$
$$90$$ 0 0
$$91$$ −2.10475e6 −0.292790
$$92$$ 0 0
$$93$$ 6.84442e6 0.882361
$$94$$ 0 0
$$95$$ 4.82093e6 0.576897
$$96$$ 0 0
$$97$$ 6.64483e6 0.739236 0.369618 0.929184i $$-0.379489\pi$$
0.369618 + 0.929184i $$0.379489\pi$$
$$98$$ 0 0
$$99$$ −2.25166e6 −0.233227
$$100$$ 0 0
$$101$$ 1.07531e7 1.03851 0.519254 0.854620i $$-0.326210\pi$$
0.519254 + 0.854620i $$0.326210\pi$$
$$102$$ 0 0
$$103$$ 1.05886e7 0.954788 0.477394 0.878689i $$-0.341581\pi$$
0.477394 + 0.878689i $$0.341581\pi$$
$$104$$ 0 0
$$105$$ −1.06328e6 −0.0896364
$$106$$ 0 0
$$107$$ 8.37234e6 0.660699 0.330349 0.943859i $$-0.392833\pi$$
0.330349 + 0.943859i $$0.392833\pi$$
$$108$$ 0 0
$$109$$ −1.95948e7 −1.44926 −0.724632 0.689136i $$-0.757990\pi$$
−0.724632 + 0.689136i $$0.757990\pi$$
$$110$$ 0 0
$$111$$ −3.88191e6 −0.269411
$$112$$ 0 0
$$113$$ 1.36310e7 0.888694 0.444347 0.895855i $$-0.353436\pi$$
0.444347 + 0.895855i $$0.353436\pi$$
$$114$$ 0 0
$$115$$ 7.99843e6 0.490413
$$116$$ 0 0
$$117$$ 9.64617e6 0.556807
$$118$$ 0 0
$$119$$ −5.43946e6 −0.295898
$$120$$ 0 0
$$121$$ −1.74355e7 −0.894717
$$122$$ 0 0
$$123$$ 7.53256e6 0.364985
$$124$$ 0 0
$$125$$ 1.95312e6 0.0894427
$$126$$ 0 0
$$127$$ −2.23763e7 −0.969336 −0.484668 0.874698i $$-0.661060\pi$$
−0.484668 + 0.874698i $$0.661060\pi$$
$$128$$ 0 0
$$129$$ 1.57928e7 0.647730
$$130$$ 0 0
$$131$$ 4.53330e6 0.176183 0.0880917 0.996112i $$-0.471923\pi$$
0.0880917 + 0.996112i $$0.471923\pi$$
$$132$$ 0 0
$$133$$ 1.32286e7 0.487567
$$134$$ 0 0
$$135$$ 1.16526e7 0.407620
$$136$$ 0 0
$$137$$ −5.07657e7 −1.68674 −0.843371 0.537332i $$-0.819432\pi$$
−0.843371 + 0.537332i $$0.819432\pi$$
$$138$$ 0 0
$$139$$ −1.05183e7 −0.332195 −0.166097 0.986109i $$-0.553117\pi$$
−0.166097 + 0.986109i $$0.553117\pi$$
$$140$$ 0 0
$$141$$ 1.26979e7 0.381473
$$142$$ 0 0
$$143$$ −8.78942e6 −0.251353
$$144$$ 0 0
$$145$$ 1.17796e7 0.320879
$$146$$ 0 0
$$147$$ −2.91764e6 −0.0757566
$$148$$ 0 0
$$149$$ 5.43497e7 1.34600 0.673000 0.739642i $$-0.265005\pi$$
0.673000 + 0.739642i $$0.265005\pi$$
$$150$$ 0 0
$$151$$ 2.23258e7 0.527700 0.263850 0.964564i $$-0.415008\pi$$
0.263850 + 0.964564i $$0.415008\pi$$
$$152$$ 0 0
$$153$$ 2.49293e7 0.562717
$$154$$ 0 0
$$155$$ −3.44988e7 −0.744120
$$156$$ 0 0
$$157$$ −4.37788e7 −0.902848 −0.451424 0.892310i $$-0.649084\pi$$
−0.451424 + 0.892310i $$0.649084\pi$$
$$158$$ 0 0
$$159$$ 4.99087e6 0.0984661
$$160$$ 0 0
$$161$$ 2.19477e7 0.414475
$$162$$ 0 0
$$163$$ −4.05451e7 −0.733300 −0.366650 0.930359i $$-0.619495\pi$$
−0.366650 + 0.930359i $$0.619495\pi$$
$$164$$ 0 0
$$165$$ −4.44024e6 −0.0769507
$$166$$ 0 0
$$167$$ −9.73453e7 −1.61736 −0.808682 0.588247i $$-0.799818\pi$$
−0.808682 + 0.588247i $$0.799818\pi$$
$$168$$ 0 0
$$169$$ −2.50943e7 −0.399919
$$170$$ 0 0
$$171$$ −6.06275e7 −0.927221
$$172$$ 0 0
$$173$$ −5.10607e7 −0.749765 −0.374882 0.927072i $$-0.622317\pi$$
−0.374882 + 0.927072i $$0.622317\pi$$
$$174$$ 0 0
$$175$$ 5.35938e6 0.0755929
$$176$$ 0 0
$$177$$ −4.49356e7 −0.609094
$$178$$ 0 0
$$179$$ −1.45811e8 −1.90023 −0.950113 0.311907i $$-0.899032\pi$$
−0.950113 + 0.311907i $$0.899032\pi$$
$$180$$ 0 0
$$181$$ −6.09656e7 −0.764205 −0.382102 0.924120i $$-0.624800\pi$$
−0.382102 + 0.924120i $$0.624800\pi$$
$$182$$ 0 0
$$183$$ 2.43536e7 0.293755
$$184$$ 0 0
$$185$$ 1.95665e7 0.227202
$$186$$ 0 0
$$187$$ −2.27151e7 −0.254021
$$188$$ 0 0
$$189$$ 3.19748e7 0.344502
$$190$$ 0 0
$$191$$ 1.52578e8 1.58444 0.792219 0.610237i $$-0.208926\pi$$
0.792219 + 0.610237i $$0.208926\pi$$
$$192$$ 0 0
$$193$$ −1.39277e8 −1.39453 −0.697267 0.716812i $$-0.745601\pi$$
−0.697267 + 0.716812i $$0.745601\pi$$
$$194$$ 0 0
$$195$$ 1.90221e7 0.183712
$$196$$ 0 0
$$197$$ 6.52480e7 0.608044 0.304022 0.952665i $$-0.401670\pi$$
0.304022 + 0.952665i $$0.401670\pi$$
$$198$$ 0 0
$$199$$ −1.93503e6 −0.0174061 −0.00870307 0.999962i $$-0.502770\pi$$
−0.00870307 + 0.999962i $$0.502770\pi$$
$$200$$ 0 0
$$201$$ −1.10441e8 −0.959278
$$202$$ 0 0
$$203$$ 3.23232e7 0.271192
$$204$$ 0 0
$$205$$ −3.79673e7 −0.307802
$$206$$ 0 0
$$207$$ −1.00587e8 −0.788219
$$208$$ 0 0
$$209$$ 5.52427e7 0.418565
$$210$$ 0 0
$$211$$ 5.17848e7 0.379502 0.189751 0.981832i $$-0.439232\pi$$
0.189751 + 0.981832i $$0.439232\pi$$
$$212$$ 0 0
$$213$$ 1.79904e7 0.127560
$$214$$ 0 0
$$215$$ −7.96023e7 −0.546249
$$216$$ 0 0
$$217$$ −9.46647e7 −0.628896
$$218$$ 0 0
$$219$$ −5.39642e7 −0.347177
$$220$$ 0 0
$$221$$ 9.73124e7 0.606450
$$222$$ 0 0
$$223$$ 1.25065e8 0.755209 0.377605 0.925967i $$-0.376748\pi$$
0.377605 + 0.925967i $$0.376748\pi$$
$$224$$ 0 0
$$225$$ −2.45623e7 −0.143757
$$226$$ 0 0
$$227$$ −1.92108e7 −0.109007 −0.0545036 0.998514i $$-0.517358\pi$$
−0.0545036 + 0.998514i $$0.517358\pi$$
$$228$$ 0 0
$$229$$ −1.05650e8 −0.581360 −0.290680 0.956820i $$-0.593882\pi$$
−0.290680 + 0.956820i $$0.593882\pi$$
$$230$$ 0 0
$$231$$ −1.21840e7 −0.0650353
$$232$$ 0 0
$$233$$ −2.31646e8 −1.19972 −0.599859 0.800106i $$-0.704776\pi$$
−0.599859 + 0.800106i $$0.704776\pi$$
$$234$$ 0 0
$$235$$ −6.40026e7 −0.321707
$$236$$ 0 0
$$237$$ −1.29336e8 −0.631101
$$238$$ 0 0
$$239$$ −1.09174e8 −0.517281 −0.258641 0.965974i $$-0.583275\pi$$
−0.258641 + 0.965974i $$0.583275\pi$$
$$240$$ 0 0
$$241$$ −8.25277e7 −0.379787 −0.189893 0.981805i $$-0.560814\pi$$
−0.189893 + 0.981805i $$0.560814\pi$$
$$242$$ 0 0
$$243$$ −2.31801e8 −1.03632
$$244$$ 0 0
$$245$$ 1.47061e7 0.0638877
$$246$$ 0 0
$$247$$ −2.36662e8 −0.999283
$$248$$ 0 0
$$249$$ 1.50761e8 0.618860
$$250$$ 0 0
$$251$$ 2.40987e7 0.0961912 0.0480956 0.998843i $$-0.484685\pi$$
0.0480956 + 0.998843i $$0.484685\pi$$
$$252$$ 0 0
$$253$$ 9.16534e7 0.355817
$$254$$ 0 0
$$255$$ 4.91603e7 0.185662
$$256$$ 0 0
$$257$$ 9.75049e7 0.358311 0.179156 0.983821i $$-0.442663\pi$$
0.179156 + 0.983821i $$0.442663\pi$$
$$258$$ 0 0
$$259$$ 5.36904e7 0.192020
$$260$$ 0 0
$$261$$ −1.48139e8 −0.515735
$$262$$ 0 0
$$263$$ −2.98637e8 −1.01228 −0.506138 0.862452i $$-0.668927\pi$$
−0.506138 + 0.862452i $$0.668927\pi$$
$$264$$ 0 0
$$265$$ −2.51561e7 −0.0830392
$$266$$ 0 0
$$267$$ 2.63216e8 0.846296
$$268$$ 0 0
$$269$$ −3.90722e8 −1.22387 −0.611934 0.790909i $$-0.709608\pi$$
−0.611934 + 0.790909i $$0.709608\pi$$
$$270$$ 0 0
$$271$$ −2.12098e8 −0.647357 −0.323678 0.946167i $$-0.604920\pi$$
−0.323678 + 0.946167i $$0.604920\pi$$
$$272$$ 0 0
$$273$$ 5.21968e7 0.155265
$$274$$ 0 0
$$275$$ 2.23807e7 0.0648947
$$276$$ 0 0
$$277$$ 1.86723e8 0.527861 0.263930 0.964542i $$-0.414981\pi$$
0.263930 + 0.964542i $$0.414981\pi$$
$$278$$ 0 0
$$279$$ 4.33853e8 1.19599
$$280$$ 0 0
$$281$$ −7.38791e8 −1.98632 −0.993161 0.116756i $$-0.962750\pi$$
−0.993161 + 0.116756i $$0.962750\pi$$
$$282$$ 0 0
$$283$$ 3.11903e8 0.818026 0.409013 0.912529i $$-0.365873\pi$$
0.409013 + 0.912529i $$0.365873\pi$$
$$284$$ 0 0
$$285$$ −1.19557e8 −0.305926
$$286$$ 0 0
$$287$$ −1.04182e8 −0.260140
$$288$$ 0 0
$$289$$ −1.58847e8 −0.387113
$$290$$ 0 0
$$291$$ −1.64789e8 −0.392014
$$292$$ 0 0
$$293$$ −5.05466e8 −1.17397 −0.586983 0.809599i $$-0.699684\pi$$
−0.586983 + 0.809599i $$0.699684\pi$$
$$294$$ 0 0
$$295$$ 2.26495e8 0.513666
$$296$$ 0 0
$$297$$ 1.33526e8 0.295747
$$298$$ 0 0
$$299$$ −3.92646e8 −0.849478
$$300$$ 0 0
$$301$$ −2.18429e8 −0.461665
$$302$$ 0 0
$$303$$ −2.66672e8 −0.550717
$$304$$ 0 0
$$305$$ −1.22753e8 −0.247732
$$306$$ 0 0
$$307$$ −4.67463e8 −0.922067 −0.461034 0.887383i $$-0.652521\pi$$
−0.461034 + 0.887383i $$0.652521\pi$$
$$308$$ 0 0
$$309$$ −2.62591e8 −0.506321
$$310$$ 0 0
$$311$$ −1.16022e7 −0.0218714 −0.0109357 0.999940i $$-0.503481\pi$$
−0.0109357 + 0.999940i $$0.503481\pi$$
$$312$$ 0 0
$$313$$ 8.23197e8 1.51740 0.758698 0.651443i $$-0.225836\pi$$
0.758698 + 0.651443i $$0.225836\pi$$
$$314$$ 0 0
$$315$$ −6.73989e7 −0.121497
$$316$$ 0 0
$$317$$ 3.89154e8 0.686142 0.343071 0.939309i $$-0.388533\pi$$
0.343071 + 0.939309i $$0.388533\pi$$
$$318$$ 0 0
$$319$$ 1.34981e8 0.232812
$$320$$ 0 0
$$321$$ −2.07630e8 −0.350366
$$322$$ 0 0
$$323$$ −6.11621e8 −1.00989
$$324$$ 0 0
$$325$$ −9.58797e7 −0.154930
$$326$$ 0 0
$$327$$ 4.85940e8 0.768539
$$328$$ 0 0
$$329$$ −1.75623e8 −0.271892
$$330$$ 0 0
$$331$$ −1.48582e8 −0.225199 −0.112600 0.993640i $$-0.535918\pi$$
−0.112600 + 0.993640i $$0.535918\pi$$
$$332$$ 0 0
$$333$$ −2.46066e8 −0.365171
$$334$$ 0 0
$$335$$ 5.56670e8 0.808986
$$336$$ 0 0
$$337$$ 1.23379e8 0.175605 0.0878023 0.996138i $$-0.472016\pi$$
0.0878023 + 0.996138i $$0.472016\pi$$
$$338$$ 0 0
$$339$$ −3.38041e8 −0.471271
$$340$$ 0 0
$$341$$ −3.95319e8 −0.539892
$$342$$ 0 0
$$343$$ 4.03536e7 0.0539949
$$344$$ 0 0
$$345$$ −1.98357e8 −0.260064
$$346$$ 0 0
$$347$$ −1.31658e9 −1.69159 −0.845793 0.533511i $$-0.820872\pi$$
−0.845793 + 0.533511i $$0.820872\pi$$
$$348$$ 0 0
$$349$$ 2.64521e8 0.333097 0.166549 0.986033i $$-0.446738\pi$$
0.166549 + 0.986033i $$0.446738\pi$$
$$350$$ 0 0
$$351$$ −5.72032e8 −0.706066
$$352$$ 0 0
$$353$$ 1.30271e9 1.57629 0.788144 0.615490i $$-0.211042\pi$$
0.788144 + 0.615490i $$0.211042\pi$$
$$354$$ 0 0
$$355$$ −9.06795e7 −0.107575
$$356$$ 0 0
$$357$$ 1.34896e8 0.156913
$$358$$ 0 0
$$359$$ 1.03262e9 1.17790 0.588952 0.808168i $$-0.299541\pi$$
0.588952 + 0.808168i $$0.299541\pi$$
$$360$$ 0 0
$$361$$ 5.93578e8 0.664053
$$362$$ 0 0
$$363$$ 4.32392e8 0.474465
$$364$$ 0 0
$$365$$ 2.72002e8 0.292784
$$366$$ 0 0
$$367$$ −1.13124e9 −1.19460 −0.597302 0.802017i $$-0.703760\pi$$
−0.597302 + 0.802017i $$0.703760\pi$$
$$368$$ 0 0
$$369$$ 4.77472e8 0.494716
$$370$$ 0 0
$$371$$ −6.90284e7 −0.0701809
$$372$$ 0 0
$$373$$ 5.38130e8 0.536916 0.268458 0.963291i $$-0.413486\pi$$
0.268458 + 0.963291i $$0.413486\pi$$
$$374$$ 0 0
$$375$$ −4.84365e7 −0.0474311
$$376$$ 0 0
$$377$$ −5.78264e8 −0.555817
$$378$$ 0 0
$$379$$ −7.83114e8 −0.738904 −0.369452 0.929250i $$-0.620455\pi$$
−0.369452 + 0.929250i $$0.620455\pi$$
$$380$$ 0 0
$$381$$ 5.54920e8 0.514035
$$382$$ 0 0
$$383$$ −8.22468e8 −0.748038 −0.374019 0.927421i $$-0.622020\pi$$
−0.374019 + 0.927421i $$0.622020\pi$$
$$384$$ 0 0
$$385$$ 6.14127e7 0.0548460
$$386$$ 0 0
$$387$$ 1.00107e9 0.877961
$$388$$ 0 0
$$389$$ −1.07007e9 −0.921696 −0.460848 0.887479i $$-0.652455\pi$$
−0.460848 + 0.887479i $$0.652455\pi$$
$$390$$ 0 0
$$391$$ −1.01474e9 −0.858495
$$392$$ 0 0
$$393$$ −1.12424e8 −0.0934293
$$394$$ 0 0
$$395$$ 6.51906e8 0.532225
$$396$$ 0 0
$$397$$ −9.64552e8 −0.773676 −0.386838 0.922148i $$-0.626433\pi$$
−0.386838 + 0.922148i $$0.626433\pi$$
$$398$$ 0 0
$$399$$ −3.28064e8 −0.258555
$$400$$ 0 0
$$401$$ −1.94810e9 −1.50871 −0.754357 0.656465i $$-0.772051\pi$$
−0.754357 + 0.656465i $$0.772051\pi$$
$$402$$ 0 0
$$403$$ 1.69356e9 1.28894
$$404$$ 0 0
$$405$$ 1.40762e8 0.105292
$$406$$ 0 0
$$407$$ 2.24211e8 0.164845
$$408$$ 0 0
$$409$$ 8.63865e8 0.624330 0.312165 0.950028i $$-0.398946\pi$$
0.312165 + 0.950028i $$0.398946\pi$$
$$410$$ 0 0
$$411$$ 1.25896e9 0.894473
$$412$$ 0 0
$$413$$ 6.21501e8 0.434127
$$414$$ 0 0
$$415$$ −7.59901e8 −0.521902
$$416$$ 0 0
$$417$$ 2.60848e8 0.176162
$$418$$ 0 0
$$419$$ −2.21337e9 −1.46996 −0.734978 0.678091i $$-0.762808\pi$$
−0.734978 + 0.678091i $$0.762808\pi$$
$$420$$ 0 0
$$421$$ −2.89866e9 −1.89326 −0.946631 0.322321i $$-0.895537\pi$$
−0.946631 + 0.322321i $$0.895537\pi$$
$$422$$ 0 0
$$423$$ 8.04889e8 0.517065
$$424$$ 0 0
$$425$$ −2.47789e8 −0.156574
$$426$$ 0 0
$$427$$ −3.36833e8 −0.209371
$$428$$ 0 0
$$429$$ 2.17973e8 0.133292
$$430$$ 0 0
$$431$$ 2.42056e9 1.45628 0.728142 0.685426i $$-0.240384\pi$$
0.728142 + 0.685426i $$0.240384\pi$$
$$432$$ 0 0
$$433$$ −2.26686e9 −1.34189 −0.670946 0.741506i $$-0.734112\pi$$
−0.670946 + 0.741506i $$0.734112\pi$$
$$434$$ 0 0
$$435$$ −2.92128e8 −0.170161
$$436$$ 0 0
$$437$$ 2.46783e9 1.41459
$$438$$ 0 0
$$439$$ −1.98911e9 −1.12210 −0.561052 0.827780i $$-0.689603\pi$$
−0.561052 + 0.827780i $$0.689603\pi$$
$$440$$ 0 0
$$441$$ −1.84942e8 −0.102684
$$442$$ 0 0
$$443$$ 8.78038e8 0.479844 0.239922 0.970792i $$-0.422878\pi$$
0.239922 + 0.970792i $$0.422878\pi$$
$$444$$ 0 0
$$445$$ −1.32672e9 −0.713705
$$446$$ 0 0
$$447$$ −1.34785e9 −0.713779
$$448$$ 0 0
$$449$$ 1.53113e8 0.0798270 0.0399135 0.999203i $$-0.487292\pi$$
0.0399135 + 0.999203i $$0.487292\pi$$
$$450$$ 0 0
$$451$$ −4.35064e8 −0.223324
$$452$$ 0 0
$$453$$ −5.53668e8 −0.279837
$$454$$ 0 0
$$455$$ −2.63094e8 −0.130940
$$456$$ 0 0
$$457$$ −2.39624e9 −1.17442 −0.587210 0.809435i $$-0.699774\pi$$
−0.587210 + 0.809435i $$0.699774\pi$$
$$458$$ 0 0
$$459$$ −1.47834e9 −0.713560
$$460$$ 0 0
$$461$$ 1.61913e9 0.769713 0.384856 0.922977i $$-0.374251\pi$$
0.384856 + 0.922977i $$0.374251\pi$$
$$462$$ 0 0
$$463$$ −1.16133e9 −0.543778 −0.271889 0.962329i $$-0.587648\pi$$
−0.271889 + 0.962329i $$0.587648\pi$$
$$464$$ 0 0
$$465$$ 8.55553e8 0.394604
$$466$$ 0 0
$$467$$ −2.83969e9 −1.29021 −0.645107 0.764092i $$-0.723187\pi$$
−0.645107 + 0.764092i $$0.723187\pi$$
$$468$$ 0 0
$$469$$ 1.52750e9 0.683718
$$470$$ 0 0
$$471$$ 1.08569e9 0.478777
$$472$$ 0 0
$$473$$ −9.12156e8 −0.396328
$$474$$ 0 0
$$475$$ 6.02617e8 0.257996
$$476$$ 0 0
$$477$$ 3.16360e8 0.133465
$$478$$ 0 0
$$479$$ −2.38771e9 −0.992676 −0.496338 0.868129i $$-0.665322\pi$$
−0.496338 + 0.868129i $$0.665322\pi$$
$$480$$ 0 0
$$481$$ −9.60526e8 −0.393551
$$482$$ 0 0
$$483$$ −5.44292e8 −0.219794
$$484$$ 0 0
$$485$$ 8.30604e8 0.330596
$$486$$ 0 0
$$487$$ 2.20508e9 0.865113 0.432556 0.901607i $$-0.357612\pi$$
0.432556 + 0.901607i $$0.357612\pi$$
$$488$$ 0 0
$$489$$ 1.00550e9 0.388866
$$490$$ 0 0
$$491$$ −4.28064e8 −0.163201 −0.0816006 0.996665i $$-0.526003\pi$$
−0.0816006 + 0.996665i $$0.526003\pi$$
$$492$$ 0 0
$$493$$ −1.49445e9 −0.561716
$$494$$ 0 0
$$495$$ −2.81457e8 −0.104302
$$496$$ 0 0
$$497$$ −2.48824e8 −0.0909171
$$498$$ 0 0
$$499$$ 2.95178e9 1.06349 0.531743 0.846906i $$-0.321537\pi$$
0.531743 + 0.846906i $$0.321537\pi$$
$$500$$ 0 0
$$501$$ 2.41412e9 0.857681
$$502$$ 0 0
$$503$$ 5.22380e9 1.83020 0.915099 0.403229i $$-0.132112\pi$$
0.915099 + 0.403229i $$0.132112\pi$$
$$504$$ 0 0
$$505$$ 1.34414e9 0.464435
$$506$$ 0 0
$$507$$ 6.22326e8 0.212075
$$508$$ 0 0
$$509$$ −2.80532e9 −0.942911 −0.471455 0.881890i $$-0.656271\pi$$
−0.471455 + 0.881890i $$0.656271\pi$$
$$510$$ 0 0
$$511$$ 7.46374e8 0.247448
$$512$$ 0 0
$$513$$ 3.59530e9 1.17577
$$514$$ 0 0
$$515$$ 1.32357e9 0.426994
$$516$$ 0 0
$$517$$ −7.33401e8 −0.233413
$$518$$ 0 0
$$519$$ 1.26628e9 0.397597
$$520$$ 0 0
$$521$$ 1.40563e8 0.0435450 0.0217725 0.999763i $$-0.493069\pi$$
0.0217725 + 0.999763i $$0.493069\pi$$
$$522$$ 0 0
$$523$$ 1.81127e9 0.553638 0.276819 0.960922i $$-0.410720\pi$$
0.276819 + 0.960922i $$0.410720\pi$$
$$524$$ 0 0
$$525$$ −1.32910e8 −0.0400866
$$526$$ 0 0
$$527$$ 4.37679e9 1.30262
$$528$$ 0 0
$$529$$ 6.89567e8 0.202526
$$530$$ 0 0
$$531$$ −2.84837e9 −0.825592
$$532$$ 0 0
$$533$$ 1.86383e9 0.533164
$$534$$ 0 0
$$535$$ 1.04654e9 0.295474
$$536$$ 0 0
$$537$$ 3.61604e9 1.00768
$$538$$ 0 0
$$539$$ 1.68516e8 0.0463534
$$540$$ 0 0
$$541$$ −7.11633e9 −1.93226 −0.966130 0.258058i $$-0.916918\pi$$
−0.966130 + 0.258058i $$0.916918\pi$$
$$542$$ 0 0
$$543$$ 1.51192e9 0.405255
$$544$$ 0 0
$$545$$ −2.44935e9 −0.648130
$$546$$ 0 0
$$547$$ 6.02390e9 1.57370 0.786850 0.617144i $$-0.211710\pi$$
0.786850 + 0.617144i $$0.211710\pi$$
$$548$$ 0 0
$$549$$ 1.54372e9 0.398168
$$550$$ 0 0
$$551$$ 3.63447e9 0.925572
$$552$$ 0 0
$$553$$ 1.78883e9 0.449812
$$554$$ 0 0
$$555$$ −4.85239e8 −0.120484
$$556$$ 0 0
$$557$$ 3.55726e9 0.872214 0.436107 0.899895i $$-0.356357\pi$$
0.436107 + 0.899895i $$0.356357\pi$$
$$558$$ 0 0
$$559$$ 3.90771e9 0.946195
$$560$$ 0 0
$$561$$ 5.63324e8 0.134706
$$562$$ 0 0
$$563$$ −2.51240e9 −0.593347 −0.296673 0.954979i $$-0.595877\pi$$
−0.296673 + 0.954979i $$0.595877\pi$$
$$564$$ 0 0
$$565$$ 1.70387e9 0.397436
$$566$$ 0 0
$$567$$ 3.86252e8 0.0889877
$$568$$ 0 0
$$569$$ 3.02191e9 0.687683 0.343841 0.939028i $$-0.388272\pi$$
0.343841 + 0.939028i $$0.388272\pi$$
$$570$$ 0 0
$$571$$ −4.13151e9 −0.928716 −0.464358 0.885648i $$-0.653715\pi$$
−0.464358 + 0.885648i $$0.653715\pi$$
$$572$$ 0 0
$$573$$ −3.78386e9 −0.840221
$$574$$ 0 0
$$575$$ 9.99804e8 0.219320
$$576$$ 0 0
$$577$$ −3.66048e9 −0.793274 −0.396637 0.917976i $$-0.629823\pi$$
−0.396637 + 0.917976i $$0.629823\pi$$
$$578$$ 0 0
$$579$$ 3.45400e9 0.739516
$$580$$ 0 0
$$581$$ −2.08517e9 −0.441088
$$582$$ 0 0
$$583$$ −2.88262e8 −0.0602487
$$584$$ 0 0
$$585$$ 1.20577e9 0.249012
$$586$$ 0 0
$$587$$ −8.93156e9 −1.82261 −0.911305 0.411731i $$-0.864924\pi$$
−0.911305 + 0.411731i $$0.864924\pi$$
$$588$$ 0 0
$$589$$ −1.06442e10 −2.14640
$$590$$ 0 0
$$591$$ −1.61812e9 −0.322444
$$592$$ 0 0
$$593$$ 8.00218e9 1.57586 0.787929 0.615766i $$-0.211153\pi$$
0.787929 + 0.615766i $$0.211153\pi$$
$$594$$ 0 0
$$595$$ −6.79932e8 −0.132329
$$596$$ 0 0
$$597$$ 4.79879e7 0.00923041
$$598$$ 0 0
$$599$$ −6.37081e9 −1.21116 −0.605579 0.795785i $$-0.707059\pi$$
−0.605579 + 0.795785i $$0.707059\pi$$
$$600$$ 0 0
$$601$$ 7.97677e9 1.49888 0.749439 0.662073i $$-0.230323\pi$$
0.749439 + 0.662073i $$0.230323\pi$$
$$602$$ 0 0
$$603$$ −7.00062e9 −1.30025
$$604$$ 0 0
$$605$$ −2.17944e9 −0.400130
$$606$$ 0 0
$$607$$ −5.42119e9 −0.983863 −0.491931 0.870634i $$-0.663709\pi$$
−0.491931 + 0.870634i $$0.663709\pi$$
$$608$$ 0 0
$$609$$ −8.01598e8 −0.143812
$$610$$ 0 0
$$611$$ 3.14191e9 0.557250
$$612$$ 0 0
$$613$$ 8.21824e9 1.44101 0.720505 0.693450i $$-0.243910\pi$$
0.720505 + 0.693450i $$0.243910\pi$$
$$614$$ 0 0
$$615$$ 9.41570e8 0.163226
$$616$$ 0 0
$$617$$ 8.15621e9 1.39795 0.698973 0.715148i $$-0.253641\pi$$
0.698973 + 0.715148i $$0.253641\pi$$
$$618$$ 0 0
$$619$$ 6.46052e9 1.09484 0.547420 0.836858i $$-0.315610\pi$$
0.547420 + 0.836858i $$0.315610\pi$$
$$620$$ 0 0
$$621$$ 5.96497e9 0.999511
$$622$$ 0 0
$$623$$ −3.64051e9 −0.603191
$$624$$ 0 0
$$625$$ 2.44141e8 0.0400000
$$626$$ 0 0
$$627$$ −1.36999e9 −0.221963
$$628$$ 0 0
$$629$$ −2.48236e9 −0.397729
$$630$$ 0 0
$$631$$ 8.82660e9 1.39859 0.699295 0.714833i $$-0.253497\pi$$
0.699295 + 0.714833i $$0.253497\pi$$
$$632$$ 0 0
$$633$$ −1.28424e9 −0.201248
$$634$$ 0 0
$$635$$ −2.79703e9 −0.433500
$$636$$ 0 0
$$637$$ −7.21930e8 −0.110664
$$638$$ 0 0
$$639$$ 1.14037e9 0.172900
$$640$$ 0 0
$$641$$ 8.54151e9 1.28095 0.640474 0.767980i $$-0.278738\pi$$
0.640474 + 0.767980i $$0.278738\pi$$
$$642$$ 0 0
$$643$$ 1.20342e10 1.78517 0.892585 0.450878i $$-0.148889\pi$$
0.892585 + 0.450878i $$0.148889\pi$$
$$644$$ 0 0
$$645$$ 1.97410e9 0.289674
$$646$$ 0 0
$$647$$ −1.89174e8 −0.0274598 −0.0137299 0.999906i $$-0.504370\pi$$
−0.0137299 + 0.999906i $$0.504370\pi$$
$$648$$ 0 0
$$649$$ 2.59539e9 0.372688
$$650$$ 0 0
$$651$$ 2.34764e9 0.333501
$$652$$ 0 0
$$653$$ 8.70977e9 1.22408 0.612041 0.790826i $$-0.290349\pi$$
0.612041 + 0.790826i $$0.290349\pi$$
$$654$$ 0 0
$$655$$ 5.66662e8 0.0787916
$$656$$ 0 0
$$657$$ −3.42067e9 −0.470579
$$658$$ 0 0
$$659$$ 7.48288e8 0.101852 0.0509260 0.998702i $$-0.483783\pi$$
0.0509260 + 0.998702i $$0.483783\pi$$
$$660$$ 0 0
$$661$$ 8.45586e9 1.13881 0.569407 0.822056i $$-0.307173\pi$$
0.569407 + 0.822056i $$0.307173\pi$$
$$662$$ 0 0
$$663$$ −2.41330e9 −0.321598
$$664$$ 0 0
$$665$$ 1.65358e9 0.218047
$$666$$ 0 0
$$667$$ 6.02996e9 0.786817
$$668$$ 0 0
$$669$$ −3.10154e9 −0.400485
$$670$$ 0 0
$$671$$ −1.40661e9 −0.179740
$$672$$ 0 0
$$673$$ 4.78543e9 0.605157 0.302578 0.953124i $$-0.402153\pi$$
0.302578 + 0.953124i $$0.402153\pi$$
$$674$$ 0 0
$$675$$ 1.45658e9 0.182293
$$676$$ 0 0
$$677$$ −1.29662e10 −1.60603 −0.803015 0.595958i $$-0.796772\pi$$
−0.803015 + 0.595958i $$0.796772\pi$$
$$678$$ 0 0
$$679$$ 2.27918e9 0.279405
$$680$$ 0 0
$$681$$ 4.76419e8 0.0578061
$$682$$ 0 0
$$683$$ 9.15988e9 1.10006 0.550031 0.835144i $$-0.314616\pi$$
0.550031 + 0.835144i $$0.314616\pi$$
$$684$$ 0 0
$$685$$ −6.34572e9 −0.754334
$$686$$ 0 0
$$687$$ 2.62007e9 0.308293
$$688$$ 0 0
$$689$$ 1.23492e9 0.143838
$$690$$ 0 0
$$691$$ −1.05298e10 −1.21407 −0.607037 0.794673i $$-0.707642\pi$$
−0.607037 + 0.794673i $$0.707642\pi$$
$$692$$ 0 0
$$693$$ −7.72318e8 −0.0881515
$$694$$ 0 0
$$695$$ −1.31478e9 −0.148562
$$696$$ 0 0
$$697$$ 4.81683e9 0.538824
$$698$$ 0 0
$$699$$ 5.74470e9 0.636205
$$700$$ 0 0
$$701$$ 1.27411e9 0.139699 0.0698497 0.997558i $$-0.477748\pi$$
0.0698497 + 0.997558i $$0.477748\pi$$
$$702$$ 0 0
$$703$$ 6.03703e9 0.655360
$$704$$ 0 0
$$705$$ 1.58723e9 0.170600
$$706$$ 0 0
$$707$$ 3.68832e9 0.392519
$$708$$ 0 0
$$709$$ −7.17795e9 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$710$$ 0 0
$$711$$ −8.19829e9 −0.855421
$$712$$ 0 0
$$713$$ −1.76599e10 −1.82463
$$714$$ 0 0
$$715$$ −1.09868e9 −0.112409
$$716$$ 0 0
$$717$$ 2.70746e9 0.274312
$$718$$ 0 0
$$719$$ −1.18502e10 −1.18898 −0.594488 0.804104i $$-0.702645\pi$$
−0.594488 + 0.804104i $$0.702645\pi$$
$$720$$ 0 0
$$721$$ 3.63188e9 0.360876
$$722$$ 0 0
$$723$$ 2.04665e9 0.201400
$$724$$ 0 0
$$725$$ 1.47245e9 0.143502
$$726$$ 0 0
$$727$$ 4.67874e9 0.451605 0.225802 0.974173i $$-0.427500\pi$$
0.225802 + 0.974173i $$0.427500\pi$$
$$728$$ 0 0
$$729$$ 3.28577e9 0.314116
$$730$$ 0 0
$$731$$ 1.00990e10 0.956238
$$732$$ 0 0
$$733$$ 1.28552e9 0.120563 0.0602817 0.998181i $$-0.480800\pi$$
0.0602817 + 0.998181i $$0.480800\pi$$
$$734$$ 0 0
$$735$$ −3.64705e8 −0.0338794
$$736$$ 0 0
$$737$$ 6.37884e9 0.586956
$$738$$ 0 0
$$739$$ 5.26720e9 0.480091 0.240046 0.970762i $$-0.422838\pi$$
0.240046 + 0.970762i $$0.422838\pi$$
$$740$$ 0 0
$$741$$ 5.86909e9 0.529916
$$742$$ 0 0
$$743$$ −4.15012e9 −0.371193 −0.185596 0.982626i $$-0.559422\pi$$
−0.185596 + 0.982626i $$0.559422\pi$$
$$744$$ 0 0
$$745$$ 6.79371e9 0.601950
$$746$$ 0 0
$$747$$ 9.55643e9 0.838829
$$748$$ 0 0
$$749$$ 2.87171e9 0.249721
$$750$$ 0 0
$$751$$ 6.37970e9 0.549618 0.274809 0.961499i $$-0.411385\pi$$
0.274809 + 0.961499i $$0.411385\pi$$
$$752$$ 0 0
$$753$$ −5.97635e8 −0.0510098
$$754$$ 0 0
$$755$$ 2.79072e9 0.235995
$$756$$ 0 0
$$757$$ −1.19658e10 −1.00255 −0.501274 0.865289i $$-0.667135\pi$$
−0.501274 + 0.865289i $$0.667135\pi$$
$$758$$ 0 0
$$759$$ −2.27296e9 −0.188688
$$760$$ 0 0
$$761$$ −2.00959e10 −1.65296 −0.826479 0.562967i $$-0.809660\pi$$
−0.826479 + 0.562967i $$0.809660\pi$$
$$762$$ 0 0
$$763$$ −6.72100e9 −0.547770
$$764$$ 0 0
$$765$$ 3.11616e9 0.251655
$$766$$ 0 0
$$767$$ −1.11187e10 −0.889756
$$768$$ 0 0
$$769$$ 2.46683e10 1.95613 0.978064 0.208304i $$-0.0667944\pi$$
0.978064 + 0.208304i $$0.0667944\pi$$
$$770$$ 0 0
$$771$$ −2.41807e9 −0.190011
$$772$$ 0 0
$$773$$ 8.88824e9 0.692130 0.346065 0.938211i $$-0.387518\pi$$
0.346065 + 0.938211i $$0.387518\pi$$
$$774$$ 0 0
$$775$$ −4.31235e9 −0.332781
$$776$$ 0 0
$$777$$ −1.33149e9 −0.101828
$$778$$ 0 0
$$779$$ −1.17144e10 −0.887850
$$780$$ 0 0
$$781$$ −1.03909e9 −0.0780502
$$782$$ 0 0
$$783$$ 8.78483e9 0.653984
$$784$$ 0 0
$$785$$ −5.47235e9 −0.403766
$$786$$ 0 0
$$787$$ 4.65006e9 0.340053 0.170027 0.985439i $$-0.445615\pi$$
0.170027 + 0.985439i $$0.445615\pi$$
$$788$$ 0 0
$$789$$ 7.40606e9 0.536806
$$790$$ 0 0
$$791$$ 4.67542e9 0.335895
$$792$$ 0 0
$$793$$ 6.02598e9 0.429113
$$794$$ 0 0
$$795$$ 6.23859e8 0.0440354
$$796$$ 0 0
$$797$$ 1.42890e10 0.999762 0.499881 0.866094i $$-0.333377\pi$$
0.499881 + 0.866094i $$0.333377\pi$$
$$798$$ 0 0
$$799$$ 8.11987e9 0.563165
$$800$$ 0 0
$$801$$ 1.66846e10 1.14711
$$802$$ 0 0
$$803$$ 3.11685e9 0.212428
$$804$$ 0 0
$$805$$ 2.74346e9 0.185359
$$806$$ 0 0
$$807$$ 9.68971e9 0.649013
$$808$$ 0 0
$$809$$ −4.92320e9 −0.326909 −0.163455 0.986551i $$-0.552264\pi$$
−0.163455 + 0.986551i $$0.552264\pi$$
$$810$$ 0 0
$$811$$ −2.35801e10 −1.55229 −0.776145 0.630555i $$-0.782827\pi$$
−0.776145 + 0.630555i $$0.782827\pi$$
$$812$$ 0 0
$$813$$ 5.25992e9 0.343291
$$814$$ 0 0
$$815$$ −5.06813e9 −0.327942
$$816$$ 0 0
$$817$$ −2.45605e10 −1.57565
$$818$$ 0 0
$$819$$ 3.30864e9 0.210453
$$820$$ 0 0
$$821$$ −2.86630e10 −1.80768 −0.903838 0.427875i $$-0.859262\pi$$
−0.903838 + 0.427875i $$0.859262\pi$$
$$822$$ 0 0
$$823$$ 2.76897e10 1.73148 0.865742 0.500490i $$-0.166847\pi$$
0.865742 + 0.500490i $$0.166847\pi$$
$$824$$ 0 0
$$825$$ −5.55030e8 −0.0344134
$$826$$ 0 0
$$827$$ 1.27176e10 0.781873 0.390936 0.920418i $$-0.372151\pi$$
0.390936 + 0.920418i $$0.372151\pi$$
$$828$$ 0 0
$$829$$ −1.50770e10 −0.919127 −0.459563 0.888145i $$-0.651994\pi$$
−0.459563 + 0.888145i $$0.651994\pi$$
$$830$$ 0 0
$$831$$ −4.63064e9 −0.279923
$$832$$ 0 0
$$833$$ −1.86573e9 −0.111839
$$834$$ 0 0
$$835$$ −1.21682e10 −0.723307
$$836$$ 0 0
$$837$$ −2.57281e10 −1.51659
$$838$$ 0 0
$$839$$ −4.59511e9 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$840$$ 0 0
$$841$$ −8.36934e9 −0.485182
$$842$$ 0 0
$$843$$ 1.83216e10 1.05334
$$844$$ 0 0
$$845$$ −3.13679e9 −0.178849
$$846$$ 0 0
$$847$$ −5.98038e9 −0.338171
$$848$$ 0 0
$$849$$ −7.73504e9 −0.433796
$$850$$ 0 0
$$851$$ 1.00161e10 0.557114
$$852$$ 0 0
$$853$$ −1.13971e9 −0.0628740 −0.0314370 0.999506i $$-0.510008\pi$$
−0.0314370 + 0.999506i $$0.510008\pi$$
$$854$$ 0 0
$$855$$ −7.57844e9 −0.414666
$$856$$ 0 0
$$857$$ 7.79419e9 0.422998 0.211499 0.977378i $$-0.432166\pi$$
0.211499 + 0.977378i $$0.432166\pi$$
$$858$$ 0 0
$$859$$ −1.27280e10 −0.685147 −0.342573 0.939491i $$-0.611299\pi$$
−0.342573 + 0.939491i $$0.611299\pi$$
$$860$$ 0 0
$$861$$ 2.58367e9 0.137951
$$862$$ 0 0
$$863$$ −2.53204e9 −0.134101 −0.0670507 0.997750i $$-0.521359\pi$$
−0.0670507 + 0.997750i $$0.521359\pi$$
$$864$$ 0 0
$$865$$ −6.38258e9 −0.335305
$$866$$ 0 0
$$867$$ 3.93933e9 0.205284
$$868$$ 0 0
$$869$$ 7.47014e9 0.386153
$$870$$ 0 0
$$871$$ −2.73272e10 −1.40130
$$872$$ 0 0
$$873$$ −1.04456e10 −0.531352
$$874$$ 0 0
$$875$$ 6.69922e8 0.0338062
$$876$$ 0 0
$$877$$ 5.00988e9 0.250800 0.125400 0.992106i $$-0.459979\pi$$
0.125400 + 0.992106i $$0.459979\pi$$
$$878$$ 0 0
$$879$$ 1.25353e10 0.622550
$$880$$ 0 0
$$881$$ 9.46900e9 0.466539 0.233270 0.972412i $$-0.425058\pi$$
0.233270 + 0.972412i $$0.425058\pi$$
$$882$$ 0 0
$$883$$ −1.11146e10 −0.543289 −0.271644 0.962398i $$-0.587567\pi$$
−0.271644 + 0.962398i $$0.587567\pi$$
$$884$$ 0 0
$$885$$ −5.61696e9 −0.272395
$$886$$ 0 0
$$887$$ 7.27986e9 0.350260 0.175130 0.984545i $$-0.443965\pi$$
0.175130 + 0.984545i $$0.443965\pi$$
$$888$$ 0 0
$$889$$ −7.67506e9 −0.366375
$$890$$ 0 0
$$891$$ 1.61298e9 0.0763938
$$892$$ 0 0
$$893$$ −1.97473e10 −0.927959
$$894$$ 0 0
$$895$$ −1.82264e10 −0.849807
$$896$$ 0 0
$$897$$ 9.73743e9 0.450475
$$898$$ 0 0
$$899$$ −2.60084e10 −1.19386
$$900$$ 0 0
$$901$$ 3.19150e9 0.145365
$$902$$ 0 0
$$903$$ 5.41692e9 0.244819
$$904$$ 0 0
$$905$$ −7.62070e9 −0.341763
$$906$$ 0 0
$$907$$ 1.39503e10 0.620809 0.310405 0.950605i $$-0.399535\pi$$
0.310405 + 0.950605i $$0.399535\pi$$
$$908$$ 0 0
$$909$$ −1.69038e10 −0.746465
$$910$$ 0 0
$$911$$ −2.98148e8 −0.0130653 −0.00653263 0.999979i $$-0.502079\pi$$
−0.00653263 + 0.999979i $$0.502079\pi$$
$$912$$ 0 0
$$913$$ −8.70765e9 −0.378663
$$914$$ 0 0
$$915$$ 3.04420e9 0.131371
$$916$$ 0 0
$$917$$ 1.55492e9 0.0665910
$$918$$ 0 0
$$919$$ −2.67202e10 −1.13563 −0.567814 0.823157i $$-0.692211\pi$$
−0.567814 + 0.823157i $$0.692211\pi$$
$$920$$ 0 0
$$921$$ 1.15928e10 0.488969
$$922$$ 0 0
$$923$$ 4.45149e9 0.186337
$$924$$ 0 0
$$925$$ 2.44581e9 0.101608
$$926$$ 0 0
$$927$$ −1.66451e10 −0.686288
$$928$$ 0 0
$$929$$ −3.66336e10 −1.49908 −0.749540 0.661959i $$-0.769725\pi$$
−0.749540 + 0.661959i $$0.769725\pi$$
$$930$$ 0 0
$$931$$ 4.53742e9 0.184283
$$932$$ 0 0
$$933$$ 2.87728e8 0.0115983
$$934$$ 0 0
$$935$$ −2.83939e9 −0.113602
$$936$$ 0 0
$$937$$ 1.28088e10 0.508649 0.254325 0.967119i $$-0.418147\pi$$
0.254325 + 0.967119i $$0.418147\pi$$
$$938$$ 0 0
$$939$$ −2.04149e10 −0.804669
$$940$$ 0 0
$$941$$ −1.20663e10 −0.472073 −0.236037 0.971744i $$-0.575849\pi$$
−0.236037 + 0.971744i $$0.575849\pi$$
$$942$$ 0 0
$$943$$ −1.94354e10 −0.754751
$$944$$ 0 0
$$945$$ 3.99685e9 0.154066
$$946$$ 0 0
$$947$$ −8.36023e9 −0.319885 −0.159942 0.987126i $$-0.551131\pi$$
−0.159942 + 0.987126i $$0.551131\pi$$
$$948$$ 0 0
$$949$$ −1.33527e10 −0.507151
$$950$$ 0 0
$$951$$ −9.65082e9 −0.363859
$$952$$ 0 0
$$953$$ 4.49530e10 1.68242 0.841209 0.540710i $$-0.181844\pi$$
0.841209 + 0.540710i $$0.181844\pi$$
$$954$$ 0 0
$$955$$ 1.90722e10 0.708582
$$956$$ 0 0
$$957$$ −3.34747e9 −0.123459
$$958$$ 0 0
$$959$$ −1.74126e10 −0.637528
$$960$$ 0 0
$$961$$ 4.86580e10 1.76857
$$962$$ 0 0
$$963$$ −1.31612e10 −0.474901
$$964$$ 0 0
$$965$$ −1.74096e10 −0.623654
$$966$$ 0 0
$$967$$ −1.34247e8 −0.00477432 −0.00238716 0.999997i $$-0.500760\pi$$
−0.00238716 + 0.999997i $$0.500760\pi$$
$$968$$ 0 0
$$969$$ 1.51679e10 0.535541
$$970$$ 0 0
$$971$$ 3.00377e10 1.05293 0.526465 0.850197i $$-0.323517\pi$$
0.526465 + 0.850197i $$0.323517\pi$$
$$972$$ 0 0
$$973$$ −3.60777e9 −0.125558
$$974$$ 0 0
$$975$$ 2.37777e9 0.0821587
$$976$$ 0 0
$$977$$ 4.52860e10 1.55358 0.776789 0.629761i $$-0.216847\pi$$
0.776789 + 0.629761i $$0.216847\pi$$
$$978$$ 0 0
$$979$$ −1.52028e10 −0.517825
$$980$$ 0 0
$$981$$ 3.08027e10 1.04171
$$982$$ 0 0
$$983$$ −4.61443e10 −1.54946 −0.774731 0.632290i $$-0.782115\pi$$
−0.774731 + 0.632290i $$0.782115\pi$$
$$984$$ 0 0
$$985$$ 8.15600e9 0.271926
$$986$$ 0 0
$$987$$ 4.35537e9 0.144183
$$988$$ 0 0
$$989$$ −4.07484e10 −1.33944
$$990$$ 0 0
$$991$$ 1.05400e10 0.344018 0.172009 0.985095i $$-0.444974\pi$$
0.172009 + 0.985095i $$0.444974\pi$$
$$992$$ 0 0
$$993$$ 3.68475e9 0.119422
$$994$$ 0 0
$$995$$ −2.41879e8 −0.00778427
$$996$$ 0 0
$$997$$ −5.00734e10 −1.60020 −0.800099 0.599868i $$-0.795220\pi$$
−0.800099 + 0.599868i $$0.795220\pi$$
$$998$$ 0 0
$$999$$ 1.45920e10 0.463059
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 560.8.a.i.1.1 2
4.3 odd 2 35.8.a.a.1.1 2
12.11 even 2 315.8.a.c.1.2 2
20.3 even 4 175.8.b.c.99.2 4
20.7 even 4 175.8.b.c.99.3 4
20.19 odd 2 175.8.a.b.1.2 2
28.27 even 2 245.8.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.1 2 4.3 odd 2
175.8.a.b.1.2 2 20.19 odd 2
175.8.b.c.99.2 4 20.3 even 4
175.8.b.c.99.3 4 20.7 even 4
245.8.a.b.1.1 2 28.27 even 2
315.8.a.c.1.2 2 12.11 even 2
560.8.a.i.1.1 2 1.1 even 1 trivial