Properties

 Label 9.22.a.e.1.2 Level $9$ Weight $22$ Character 9.1 Self dual yes Analytic conductor $25.153$ Analytic rank $0$ 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] = [9,22,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 9.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: $$25.1529609858$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{649})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 162$$ x^2 - x - 162 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3^{2}\cdot 7$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-12.2377$$ of defining polynomial Character $$\chi$$ $$=$$ 9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1271.96 q^{2} -479282. q^{4} +2.12776e7 q^{5} +6.32076e8 q^{7} -3.27711e9 q^{8} +O(q^{10})$$ $$q+1271.96 q^{2} -479282. q^{4} +2.12776e7 q^{5} +6.32076e8 q^{7} -3.27711e9 q^{8} +2.70641e10 q^{10} -5.97585e10 q^{11} +7.38499e11 q^{13} +8.03972e11 q^{14} -3.16321e12 q^{16} +8.35876e12 q^{17} +4.19061e13 q^{19} -1.01980e13 q^{20} -7.60101e13 q^{22} -4.48926e13 q^{23} -2.41012e13 q^{25} +9.39338e14 q^{26} -3.02943e14 q^{28} +2.76669e15 q^{29} +8.36452e15 q^{31} +2.84914e15 q^{32} +1.06320e16 q^{34} +1.34490e16 q^{35} -1.77675e16 q^{37} +5.33027e16 q^{38} -6.97290e16 q^{40} -1.45253e17 q^{41} +1.24744e17 q^{43} +2.86412e16 q^{44} -5.71014e16 q^{46} -4.28566e17 q^{47} -1.59026e17 q^{49} -3.06556e16 q^{50} -3.53950e17 q^{52} +4.77017e17 q^{53} -1.27152e18 q^{55} -2.07138e18 q^{56} +3.51911e18 q^{58} -1.61959e18 q^{59} -3.76882e18 q^{61} +1.06393e19 q^{62} +1.02577e19 q^{64} +1.57135e19 q^{65} -2.81797e18 q^{67} -4.00621e18 q^{68} +1.71066e19 q^{70} -1.00228e19 q^{71} -1.72739e19 q^{73} -2.25995e19 q^{74} -2.00849e19 q^{76} -3.77719e19 q^{77} -3.28276e19 q^{79} -6.73054e19 q^{80} -1.84755e20 q^{82} -3.05240e17 q^{83} +1.77854e20 q^{85} +1.58669e20 q^{86} +1.95835e20 q^{88} -2.34593e20 q^{89} +4.66787e20 q^{91} +2.15162e19 q^{92} -5.45116e20 q^{94} +8.91662e20 q^{95} -5.92086e20 q^{97} -2.02274e20 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 666 q^{2} + 1179236 q^{4} - 996876 q^{5} + 679896112 q^{7} - 2427055848 q^{8}+O(q^{10})$$ 2 * q - 666 * q^2 + 1179236 * q^4 - 996876 * q^5 + 679896112 * q^7 - 2427055848 * q^8 $$2 q - 666 q^{2} + 1179236 q^{4} - 996876 q^{5} + 679896112 q^{7} - 2427055848 q^{8} + 70231066524 q^{10} - 219869122968 q^{11} - 48468909956 q^{13} + 711297706896 q^{14} - 8288736440560 q^{16} + 11333529041436 q^{17} + 11960585011624 q^{19} - 47140581172824 q^{20} + 234277148563128 q^{22} + 146508390063504 q^{23} - 4786354247074 q^{25} + 24\!\cdots\!16 q^{26}+ \cdots + 87\!\cdots\!14 q^{98}+O(q^{100})$$ 2 * q - 666 * q^2 + 1179236 * q^4 - 996876 * q^5 + 679896112 * q^7 - 2427055848 * q^8 + 70231066524 * q^10 - 219869122968 * q^11 - 48468909956 * q^13 + 711297706896 * q^14 - 8288736440560 * q^16 + 11333529041436 * q^17 + 11960585011624 * q^19 - 47140581172824 * q^20 + 234277148563128 * q^22 + 146508390063504 * q^23 - 4786354247074 * q^25 + 2464447363969716 * q^26 - 223631286954656 * q^28 + 1798520043674052 * q^29 + 11169107526944992 * q^31 + 10999491515934048 * q^32 + 4867007487413652 * q^34 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 111335877667820664 * q^38 - 88663459120547472 * q^40 - 122972020616468052 * q^41 + 288455418162270040 * q^43 - 236905239135592368 * q^44 - 428027981395161168 * q^46 - 837243745741596960 * q^47 - 715285396941470670 * q^49 - 68086886042031174 * q^50 - 1659150748242045896 * q^52 + 43007964012775764 * q^53 + 2294863118895313296 * q^55 - 2030730320050568640 * q^56 + 5395375030104291852 * q^58 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 + 5204128043677477536 * q^62 + 5211692013463482944 * q^64 + 33242791066177513752 * q^65 - 16454068667621610296 * q^67 + 927498335984326008 * q^68 + 19170849981674109600 * q^70 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 - 81714426393675287484 * q^74 - 69750112936092790064 * q^76 - 45428450983816025664 * q^77 - 54055785594190591040 * q^79 + 46862985896348738976 * q^80 - 227933719179803895996 * q^82 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 - 158596867265246585208 * q^86 + 59732519594715541728 * q^88 - 226920767965448065524 * q^89 + 429154174685185000352 * q^91 + 338958281810136649248 * q^92 + 246883115968835119776 * q^94 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 + 875730737371150330614 * q^98

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1271.96 0.878328 0.439164 0.898407i $$-0.355275\pi$$
0.439164 + 0.898407i $$0.355275\pi$$
$$3$$ 0 0
$$4$$ −479282. −0.228540
$$5$$ 2.12776e7 0.974400 0.487200 0.873290i $$-0.338018\pi$$
0.487200 + 0.873290i $$0.338018\pi$$
$$6$$ 0 0
$$7$$ 6.32076e8 0.845745 0.422873 0.906189i $$-0.361022\pi$$
0.422873 + 0.906189i $$0.361022\pi$$
$$8$$ −3.27711e9 −1.07906
$$9$$ 0 0
$$10$$ 2.70641e10 0.855843
$$11$$ −5.97585e10 −0.694666 −0.347333 0.937742i $$-0.612913\pi$$
−0.347333 + 0.937742i $$0.612913\pi$$
$$12$$ 0 0
$$13$$ 7.38499e11 1.48575 0.742874 0.669432i $$-0.233462\pi$$
0.742874 + 0.669432i $$0.233462\pi$$
$$14$$ 8.03972e11 0.742842
$$15$$ 0 0
$$16$$ −3.16321e12 −0.719230
$$17$$ 8.35876e12 1.00561 0.502804 0.864401i $$-0.332302\pi$$
0.502804 + 0.864401i $$0.332302\pi$$
$$18$$ 0 0
$$19$$ 4.19061e13 1.56807 0.784034 0.620718i $$-0.213159\pi$$
0.784034 + 0.620718i $$0.213159\pi$$
$$20$$ −1.01980e13 −0.222689
$$21$$ 0 0
$$22$$ −7.60101e13 −0.610145
$$23$$ −4.48926e13 −0.225961 −0.112980 0.993597i $$-0.536040\pi$$
−0.112980 + 0.993597i $$0.536040\pi$$
$$24$$ 0 0
$$25$$ −2.41012e13 −0.0505438
$$26$$ 9.39338e14 1.30497
$$27$$ 0 0
$$28$$ −3.02943e14 −0.193286
$$29$$ 2.76669e15 1.22119 0.610593 0.791945i $$-0.290931\pi$$
0.610593 + 0.791945i $$0.290931\pi$$
$$30$$ 0 0
$$31$$ 8.36452e15 1.83292 0.916459 0.400128i $$-0.131034\pi$$
0.916459 + 0.400128i $$0.131034\pi$$
$$32$$ 2.84914e15 0.447341
$$33$$ 0 0
$$34$$ 1.06320e16 0.883253
$$35$$ 1.34490e16 0.824095
$$36$$ 0 0
$$37$$ −1.77675e16 −0.607447 −0.303724 0.952760i $$-0.598230\pi$$
−0.303724 + 0.952760i $$0.598230\pi$$
$$38$$ 5.33027e16 1.37728
$$39$$ 0 0
$$40$$ −6.97290e16 −1.05144
$$41$$ −1.45253e17 −1.69003 −0.845013 0.534745i $$-0.820408\pi$$
−0.845013 + 0.534745i $$0.820408\pi$$
$$42$$ 0 0
$$43$$ 1.24744e17 0.880239 0.440119 0.897939i $$-0.354936\pi$$
0.440119 + 0.897939i $$0.354936\pi$$
$$44$$ 2.86412e16 0.158759
$$45$$ 0 0
$$46$$ −5.71014e16 −0.198468
$$47$$ −4.28566e17 −1.18847 −0.594237 0.804290i $$-0.702546\pi$$
−0.594237 + 0.804290i $$0.702546\pi$$
$$48$$ 0 0
$$49$$ −1.59026e17 −0.284715
$$50$$ −3.06556e16 −0.0443941
$$51$$ 0 0
$$52$$ −3.53950e17 −0.339552
$$53$$ 4.77017e17 0.374660 0.187330 0.982297i $$-0.440017\pi$$
0.187330 + 0.982297i $$0.440017\pi$$
$$54$$ 0 0
$$55$$ −1.27152e18 −0.676883
$$56$$ −2.07138e18 −0.912611
$$57$$ 0 0
$$58$$ 3.51911e18 1.07260
$$59$$ −1.61959e18 −0.412533 −0.206267 0.978496i $$-0.566131\pi$$
−0.206267 + 0.978496i $$0.566131\pi$$
$$60$$ 0 0
$$61$$ −3.76882e18 −0.676461 −0.338230 0.941063i $$-0.609828\pi$$
−0.338230 + 0.941063i $$0.609828\pi$$
$$62$$ 1.06393e19 1.60990
$$63$$ 0 0
$$64$$ 1.02577e19 1.11214
$$65$$ 1.57135e19 1.44771
$$66$$ 0 0
$$67$$ −2.81797e18 −0.188865 −0.0944324 0.995531i $$-0.530104\pi$$
−0.0944324 + 0.995531i $$0.530104\pi$$
$$68$$ −4.00621e18 −0.229821
$$69$$ 0 0
$$70$$ 1.71066e19 0.723826
$$71$$ −1.00228e19 −0.365407 −0.182704 0.983168i $$-0.558485\pi$$
−0.182704 + 0.983168i $$0.558485\pi$$
$$72$$ 0 0
$$73$$ −1.72739e19 −0.470435 −0.235218 0.971943i $$-0.575580\pi$$
−0.235218 + 0.971943i $$0.575580\pi$$
$$74$$ −2.25995e19 −0.533538
$$75$$ 0 0
$$76$$ −2.00849e19 −0.358365
$$77$$ −3.77719e19 −0.587511
$$78$$ 0 0
$$79$$ −3.28276e19 −0.390080 −0.195040 0.980795i $$-0.562484\pi$$
−0.195040 + 0.980795i $$0.562484\pi$$
$$80$$ −6.73054e19 −0.700818
$$81$$ 0 0
$$82$$ −1.84755e20 −1.48440
$$83$$ −3.05240e17 −0.00215934 −0.00107967 0.999999i $$-0.500344\pi$$
−0.00107967 + 0.999999i $$0.500344\pi$$
$$84$$ 0 0
$$85$$ 1.77854e20 0.979864
$$86$$ 1.58669e20 0.773139
$$87$$ 0 0
$$88$$ 1.95835e20 0.749587
$$89$$ −2.34593e20 −0.797480 −0.398740 0.917064i $$-0.630552\pi$$
−0.398740 + 0.917064i $$0.630552\pi$$
$$90$$ 0 0
$$91$$ 4.66787e20 1.25656
$$92$$ 2.15162e19 0.0516409
$$93$$ 0 0
$$94$$ −5.45116e20 −1.04387
$$95$$ 8.91662e20 1.52793
$$96$$ 0 0
$$97$$ −5.92086e20 −0.815233 −0.407616 0.913153i $$-0.633640\pi$$
−0.407616 + 0.913153i $$0.633640\pi$$
$$98$$ −2.02274e20 −0.250073
$$99$$ 0 0
$$100$$ 1.15513e19 0.0115513
$$101$$ 1.66229e21 1.49739 0.748693 0.662917i $$-0.230682\pi$$
0.748693 + 0.662917i $$0.230682\pi$$
$$102$$ 0 0
$$103$$ 1.17919e21 0.864552 0.432276 0.901741i $$-0.357711\pi$$
0.432276 + 0.901741i $$0.357711\pi$$
$$104$$ −2.42014e21 −1.60321
$$105$$ 0 0
$$106$$ 6.06745e20 0.329075
$$107$$ 8.69160e20 0.427140 0.213570 0.976928i $$-0.431491\pi$$
0.213570 + 0.976928i $$0.431491\pi$$
$$108$$ 0 0
$$109$$ 4.12021e20 0.166702 0.0833510 0.996520i $$-0.473438\pi$$
0.0833510 + 0.996520i $$0.473438\pi$$
$$110$$ −1.61731e21 −0.594526
$$111$$ 0 0
$$112$$ −1.99939e21 −0.608286
$$113$$ −2.45943e21 −0.681569 −0.340785 0.940141i $$-0.610693\pi$$
−0.340785 + 0.940141i $$0.610693\pi$$
$$114$$ 0 0
$$115$$ −9.55207e20 −0.220176
$$116$$ −1.32602e21 −0.279089
$$117$$ 0 0
$$118$$ −2.06005e21 −0.362340
$$119$$ 5.28337e21 0.850487
$$120$$ 0 0
$$121$$ −3.82917e21 −0.517439
$$122$$ −4.79378e21 −0.594155
$$123$$ 0 0
$$124$$ −4.00896e21 −0.418894
$$125$$ −1.06588e22 −1.02365
$$126$$ 0 0
$$127$$ 5.53783e21 0.450195 0.225097 0.974336i $$-0.427730\pi$$
0.225097 + 0.974336i $$0.427730\pi$$
$$128$$ 7.07226e21 0.529485
$$129$$ 0 0
$$130$$ 1.99869e22 1.27157
$$131$$ 2.54630e22 1.49472 0.747362 0.664417i $$-0.231320\pi$$
0.747362 + 0.664417i $$0.231320\pi$$
$$132$$ 0 0
$$133$$ 2.64878e22 1.32619
$$134$$ −3.58433e21 −0.165885
$$135$$ 0 0
$$136$$ −2.73926e22 −1.08511
$$137$$ 4.82088e22 1.76832 0.884158 0.467188i $$-0.154733\pi$$
0.884158 + 0.467188i $$0.154733\pi$$
$$138$$ 0 0
$$139$$ −2.18194e22 −0.687363 −0.343682 0.939086i $$-0.611674\pi$$
−0.343682 + 0.939086i $$0.611674\pi$$
$$140$$ −6.44589e21 −0.188338
$$141$$ 0 0
$$142$$ −1.27486e22 −0.320948
$$143$$ −4.41316e22 −1.03210
$$144$$ 0 0
$$145$$ 5.88685e22 1.18992
$$146$$ −2.19716e22 −0.413197
$$147$$ 0 0
$$148$$ 8.51565e21 0.138826
$$149$$ −4.84140e22 −0.735386 −0.367693 0.929947i $$-0.619852\pi$$
−0.367693 + 0.929947i $$0.619852\pi$$
$$150$$ 0 0
$$151$$ −1.22948e23 −1.62354 −0.811770 0.583977i $$-0.801496\pi$$
−0.811770 + 0.583977i $$0.801496\pi$$
$$152$$ −1.37331e23 −1.69204
$$153$$ 0 0
$$154$$ −4.80441e22 −0.516027
$$155$$ 1.77977e23 1.78600
$$156$$ 0 0
$$157$$ 5.89828e22 0.517344 0.258672 0.965965i $$-0.416715\pi$$
0.258672 + 0.965965i $$0.416715\pi$$
$$158$$ −4.17552e22 −0.342619
$$159$$ 0 0
$$160$$ 6.06228e22 0.435889
$$161$$ −2.83755e22 −0.191105
$$162$$ 0 0
$$163$$ −3.18244e22 −0.188274 −0.0941369 0.995559i $$-0.530009\pi$$
−0.0941369 + 0.995559i $$0.530009\pi$$
$$164$$ 6.96170e22 0.386238
$$165$$ 0 0
$$166$$ −3.88252e20 −0.00189661
$$167$$ 3.62822e22 0.166407 0.0832033 0.996533i $$-0.473485\pi$$
0.0832033 + 0.996533i $$0.473485\pi$$
$$168$$ 0 0
$$169$$ 2.98317e23 1.20744
$$170$$ 2.26223e23 0.860642
$$171$$ 0 0
$$172$$ −5.97875e22 −0.201169
$$173$$ −3.21510e23 −1.01791 −0.508956 0.860792i $$-0.669969\pi$$
−0.508956 + 0.860792i $$0.669969\pi$$
$$174$$ 0 0
$$175$$ −1.52338e22 −0.0427472
$$176$$ 1.89028e23 0.499625
$$177$$ 0 0
$$178$$ −2.98392e23 −0.700449
$$179$$ 1.80418e23 0.399321 0.199661 0.979865i $$-0.436016\pi$$
0.199661 + 0.979865i $$0.436016\pi$$
$$180$$ 0 0
$$181$$ −6.36646e21 −0.0125393 −0.00626965 0.999980i $$-0.501996\pi$$
−0.00626965 + 0.999980i $$0.501996\pi$$
$$182$$ 5.93733e23 1.10368
$$183$$ 0 0
$$184$$ 1.47118e23 0.243825
$$185$$ −3.78050e23 −0.591897
$$186$$ 0 0
$$187$$ −4.99507e23 −0.698561
$$188$$ 2.05404e23 0.271613
$$189$$ 0 0
$$190$$ 1.13415e24 1.34202
$$191$$ −8.17882e23 −0.915883 −0.457942 0.888982i $$-0.651413\pi$$
−0.457942 + 0.888982i $$0.651413\pi$$
$$192$$ 0 0
$$193$$ −1.80064e23 −0.180748 −0.0903742 0.995908i $$-0.528806\pi$$
−0.0903742 + 0.995908i $$0.528806\pi$$
$$194$$ −7.53107e23 −0.716042
$$195$$ 0 0
$$196$$ 7.62185e22 0.0650686
$$197$$ −1.65216e24 −1.33708 −0.668541 0.743675i $$-0.733081\pi$$
−0.668541 + 0.743675i $$0.733081\pi$$
$$198$$ 0 0
$$199$$ −1.52537e24 −1.11024 −0.555120 0.831770i $$-0.687328\pi$$
−0.555120 + 0.831770i $$0.687328\pi$$
$$200$$ 7.89822e22 0.0545399
$$201$$ 0 0
$$202$$ 2.11436e24 1.31520
$$203$$ 1.74876e24 1.03281
$$204$$ 0 0
$$205$$ −3.09063e24 −1.64676
$$206$$ 1.49987e24 0.759361
$$207$$ 0 0
$$208$$ −2.33603e24 −1.06859
$$209$$ −2.50425e24 −1.08928
$$210$$ 0 0
$$211$$ 1.35748e24 0.534277 0.267138 0.963658i $$-0.413922\pi$$
0.267138 + 0.963658i $$0.413922\pi$$
$$212$$ −2.28626e23 −0.0856247
$$213$$ 0 0
$$214$$ 1.10553e24 0.375169
$$215$$ 2.65425e24 0.857705
$$216$$ 0 0
$$217$$ 5.28701e24 1.55018
$$218$$ 5.24072e23 0.146419
$$219$$ 0 0
$$220$$ 6.09415e23 0.154695
$$221$$ 6.17294e24 1.49408
$$222$$ 0 0
$$223$$ 4.73192e24 1.04192 0.520962 0.853580i $$-0.325573\pi$$
0.520962 + 0.853580i $$0.325573\pi$$
$$224$$ 1.80087e24 0.378336
$$225$$ 0 0
$$226$$ −3.12828e24 −0.598642
$$227$$ −8.00226e24 −1.46198 −0.730989 0.682389i $$-0.760941\pi$$
−0.730989 + 0.682389i $$0.760941\pi$$
$$228$$ 0 0
$$229$$ 2.72111e24 0.453391 0.226696 0.973966i $$-0.427208\pi$$
0.226696 + 0.973966i $$0.427208\pi$$
$$230$$ −1.21498e24 −0.193387
$$231$$ 0 0
$$232$$ −9.06674e24 −1.31773
$$233$$ 6.60115e24 0.917028 0.458514 0.888687i $$-0.348382\pi$$
0.458514 + 0.888687i $$0.348382\pi$$
$$234$$ 0 0
$$235$$ −9.11885e24 −1.15805
$$236$$ 7.76240e23 0.0942801
$$237$$ 0 0
$$238$$ 6.72021e24 0.747007
$$239$$ −9.92224e24 −1.05544 −0.527718 0.849420i $$-0.676952\pi$$
−0.527718 + 0.849420i $$0.676952\pi$$
$$240$$ 0 0
$$241$$ −1.54177e25 −1.50259 −0.751295 0.659966i $$-0.770571\pi$$
−0.751295 + 0.659966i $$0.770571\pi$$
$$242$$ −4.87054e24 −0.454481
$$243$$ 0 0
$$244$$ 1.80633e24 0.154598
$$245$$ −3.38370e24 −0.277426
$$246$$ 0 0
$$247$$ 3.09477e25 2.32975
$$248$$ −2.74114e25 −1.97783
$$249$$ 0 0
$$250$$ −1.35575e25 −0.899101
$$251$$ 1.00009e25 0.636014 0.318007 0.948088i $$-0.396986\pi$$
0.318007 + 0.948088i $$0.396986\pi$$
$$252$$ 0 0
$$253$$ 2.68271e24 0.156967
$$254$$ 7.04387e24 0.395419
$$255$$ 0 0
$$256$$ −1.25164e25 −0.647080
$$257$$ 3.06596e25 1.52149 0.760744 0.649052i $$-0.224834\pi$$
0.760744 + 0.649052i $$0.224834\pi$$
$$258$$ 0 0
$$259$$ −1.12304e25 −0.513746
$$260$$ −7.53119e24 −0.330860
$$261$$ 0 0
$$262$$ 3.23878e25 1.31286
$$263$$ 2.33455e25 0.909216 0.454608 0.890692i $$-0.349779\pi$$
0.454608 + 0.890692i $$0.349779\pi$$
$$264$$ 0 0
$$265$$ 1.01498e25 0.365069
$$266$$ 3.36914e25 1.16483
$$267$$ 0 0
$$268$$ 1.35060e24 0.0431631
$$269$$ −2.29737e25 −0.706043 −0.353022 0.935615i $$-0.614846\pi$$
−0.353022 + 0.935615i $$0.614846\pi$$
$$270$$ 0 0
$$271$$ −4.02085e25 −1.14325 −0.571624 0.820516i $$-0.693686\pi$$
−0.571624 + 0.820516i $$0.693686\pi$$
$$272$$ −2.64405e25 −0.723263
$$273$$ 0 0
$$274$$ 6.13194e25 1.55316
$$275$$ 1.44025e24 0.0351111
$$276$$ 0 0
$$277$$ −4.48722e25 −1.01377 −0.506886 0.862013i $$-0.669203\pi$$
−0.506886 + 0.862013i $$0.669203\pi$$
$$278$$ −2.77532e25 −0.603731
$$279$$ 0 0
$$280$$ −4.40740e25 −0.889248
$$281$$ 6.06489e25 1.17871 0.589355 0.807874i $$-0.299382\pi$$
0.589355 + 0.807874i $$0.299382\pi$$
$$282$$ 0 0
$$283$$ 1.04229e26 1.88032 0.940158 0.340739i $$-0.110677\pi$$
0.940158 + 0.340739i $$0.110677\pi$$
$$284$$ 4.80376e24 0.0835100
$$285$$ 0 0
$$286$$ −5.61334e25 −0.906521
$$287$$ −9.18107e25 −1.42933
$$288$$ 0 0
$$289$$ 7.76980e23 0.0112456
$$290$$ 7.48781e25 1.04514
$$291$$ 0 0
$$292$$ 8.27907e24 0.107513
$$293$$ −6.24634e25 −0.782556 −0.391278 0.920273i $$-0.627967\pi$$
−0.391278 + 0.920273i $$0.627967\pi$$
$$294$$ 0 0
$$295$$ −3.44610e25 −0.401972
$$296$$ 5.82261e25 0.655472
$$297$$ 0 0
$$298$$ −6.15805e25 −0.645911
$$299$$ −3.31532e25 −0.335720
$$300$$ 0 0
$$301$$ 7.88476e25 0.744458
$$302$$ −1.56384e26 −1.42600
$$303$$ 0 0
$$304$$ −1.32558e26 −1.12780
$$305$$ −8.01915e25 −0.659144
$$306$$ 0 0
$$307$$ 7.14411e25 0.548271 0.274135 0.961691i $$-0.411608\pi$$
0.274135 + 0.961691i $$0.411608\pi$$
$$308$$ 1.81034e25 0.134269
$$309$$ 0 0
$$310$$ 2.26379e26 1.56869
$$311$$ 6.01915e25 0.403228 0.201614 0.979465i $$-0.435381\pi$$
0.201614 + 0.979465i $$0.435381\pi$$
$$312$$ 0 0
$$313$$ −1.69049e26 −1.05876 −0.529379 0.848385i $$-0.677575\pi$$
−0.529379 + 0.848385i $$0.677575\pi$$
$$314$$ 7.50235e25 0.454398
$$315$$ 0 0
$$316$$ 1.57337e25 0.0891488
$$317$$ 5.71691e25 0.313357 0.156678 0.987650i $$-0.449921\pi$$
0.156678 + 0.987650i $$0.449921\pi$$
$$318$$ 0 0
$$319$$ −1.65333e26 −0.848316
$$320$$ 2.18259e26 1.08367
$$321$$ 0 0
$$322$$ −3.60924e25 −0.167853
$$323$$ 3.50283e26 1.57686
$$324$$ 0 0
$$325$$ −1.77987e25 −0.0750954
$$326$$ −4.04792e25 −0.165366
$$327$$ 0 0
$$328$$ 4.76009e26 1.82364
$$329$$ −2.70886e26 −1.00515
$$330$$ 0 0
$$331$$ 6.67957e25 0.232571 0.116285 0.993216i $$-0.462901\pi$$
0.116285 + 0.993216i $$0.462901\pi$$
$$332$$ 1.46296e23 0.000493495 0
$$333$$ 0 0
$$334$$ 4.61493e25 0.146160
$$335$$ −5.99597e25 −0.184030
$$336$$ 0 0
$$337$$ 3.55365e26 1.02461 0.512307 0.858802i $$-0.328791\pi$$
0.512307 + 0.858802i $$0.328791\pi$$
$$338$$ 3.79446e26 1.06053
$$339$$ 0 0
$$340$$ −8.52424e25 −0.223938
$$341$$ −4.99851e26 −1.27327
$$342$$ 0 0
$$343$$ −4.53560e26 −1.08654
$$344$$ −4.08799e26 −0.949832
$$345$$ 0 0
$$346$$ −4.08947e26 −0.894062
$$347$$ 3.94238e25 0.0836179 0.0418090 0.999126i $$-0.486688\pi$$
0.0418090 + 0.999126i $$0.486688\pi$$
$$348$$ 0 0
$$349$$ −5.84299e26 −1.16672 −0.583362 0.812212i $$-0.698263\pi$$
−0.583362 + 0.812212i $$0.698263\pi$$
$$350$$ −1.93767e25 −0.0375461
$$351$$ 0 0
$$352$$ −1.70260e26 −0.310753
$$353$$ −4.96062e26 −0.878823 −0.439412 0.898286i $$-0.644813\pi$$
−0.439412 + 0.898286i $$0.644813\pi$$
$$354$$ 0 0
$$355$$ −2.13262e26 −0.356053
$$356$$ 1.12436e26 0.182256
$$357$$ 0 0
$$358$$ 2.29483e26 0.350735
$$359$$ 3.35136e26 0.497426 0.248713 0.968577i $$-0.419992\pi$$
0.248713 + 0.968577i $$0.419992\pi$$
$$360$$ 0 0
$$361$$ 1.04192e27 1.45884
$$362$$ −8.09785e24 −0.0110136
$$363$$ 0 0
$$364$$ −2.23723e26 −0.287174
$$365$$ −3.67547e26 −0.458392
$$366$$ 0 0
$$367$$ 9.70198e25 0.114253 0.0571264 0.998367i $$-0.481806\pi$$
0.0571264 + 0.998367i $$0.481806\pi$$
$$368$$ 1.42005e26 0.162518
$$369$$ 0 0
$$370$$ −4.80862e26 −0.519880
$$371$$ 3.01511e26 0.316867
$$372$$ 0 0
$$373$$ 1.03252e27 1.02554 0.512771 0.858525i $$-0.328619\pi$$
0.512771 + 0.858525i $$0.328619\pi$$
$$374$$ −6.35350e26 −0.613566
$$375$$ 0 0
$$376$$ 1.40446e27 1.28244
$$377$$ 2.04320e27 1.81437
$$378$$ 0 0
$$379$$ 2.20881e26 0.185544 0.0927721 0.995687i $$-0.470427\pi$$
0.0927721 + 0.995687i $$0.470427\pi$$
$$380$$ −4.27358e26 −0.349191
$$381$$ 0 0
$$382$$ −1.04031e27 −0.804446
$$383$$ 1.59547e27 1.20033 0.600165 0.799876i $$-0.295101\pi$$
0.600165 + 0.799876i $$0.295101\pi$$
$$384$$ 0 0
$$385$$ −8.03695e26 −0.572471
$$386$$ −2.29033e26 −0.158756
$$387$$ 0 0
$$388$$ 2.83776e26 0.186313
$$389$$ 3.60474e26 0.230358 0.115179 0.993345i $$-0.463256\pi$$
0.115179 + 0.993345i $$0.463256\pi$$
$$390$$ 0 0
$$391$$ −3.75247e26 −0.227228
$$392$$ 5.21147e26 0.307225
$$393$$ 0 0
$$394$$ −2.10148e27 −1.17440
$$395$$ −6.98492e26 −0.380095
$$396$$ 0 0
$$397$$ −1.47191e27 −0.759595 −0.379798 0.925070i $$-0.624006\pi$$
−0.379798 + 0.925070i $$0.624006\pi$$
$$398$$ −1.94020e27 −0.975156
$$399$$ 0 0
$$400$$ 7.62370e25 0.0363527
$$401$$ −1.00401e27 −0.466360 −0.233180 0.972434i $$-0.574913\pi$$
−0.233180 + 0.972434i $$0.574913\pi$$
$$402$$ 0 0
$$403$$ 6.17719e27 2.72325
$$404$$ −7.96708e26 −0.342212
$$405$$ 0 0
$$406$$ 2.22434e27 0.907148
$$407$$ 1.06176e27 0.421973
$$408$$ 0 0
$$409$$ −4.41838e27 −1.66789 −0.833947 0.551845i $$-0.813924\pi$$
−0.833947 + 0.551845i $$0.813924\pi$$
$$410$$ −3.93114e27 −1.44640
$$411$$ 0 0
$$412$$ −5.65163e26 −0.197584
$$413$$ −1.02370e27 −0.348898
$$414$$ 0 0
$$415$$ −6.49477e24 −0.00210406
$$416$$ 2.10409e27 0.664635
$$417$$ 0 0
$$418$$ −3.18529e27 −0.956749
$$419$$ 2.36639e27 0.693169 0.346584 0.938019i $$-0.387341\pi$$
0.346584 + 0.938019i $$0.387341\pi$$
$$420$$ 0 0
$$421$$ 2.72067e27 0.758078 0.379039 0.925381i $$-0.376255\pi$$
0.379039 + 0.925381i $$0.376255\pi$$
$$422$$ 1.72665e27 0.469271
$$423$$ 0 0
$$424$$ −1.56324e27 −0.404281
$$425$$ −2.01456e26 −0.0508272
$$426$$ 0 0
$$427$$ −2.38218e27 −0.572114
$$428$$ −4.16573e26 −0.0976183
$$429$$ 0 0
$$430$$ 3.37609e27 0.753347
$$431$$ −8.87397e27 −1.93244 −0.966222 0.257713i $$-0.917031\pi$$
−0.966222 + 0.257713i $$0.917031\pi$$
$$432$$ 0 0
$$433$$ −3.39994e27 −0.705258 −0.352629 0.935763i $$-0.614712\pi$$
−0.352629 + 0.935763i $$0.614712\pi$$
$$434$$ 6.72484e27 1.36157
$$435$$ 0 0
$$436$$ −1.97474e26 −0.0380980
$$437$$ −1.88128e27 −0.354322
$$438$$ 0 0
$$439$$ 2.29516e27 0.412037 0.206018 0.978548i $$-0.433949\pi$$
0.206018 + 0.978548i $$0.433949\pi$$
$$440$$ 4.16690e27 0.730398
$$441$$ 0 0
$$442$$ 7.85170e27 1.31229
$$443$$ 5.65265e27 0.922599 0.461299 0.887245i $$-0.347383\pi$$
0.461299 + 0.887245i $$0.347383\pi$$
$$444$$ 0 0
$$445$$ −4.99157e27 −0.777065
$$446$$ 6.01879e27 0.915152
$$447$$ 0 0
$$448$$ 6.48364e27 0.940589
$$449$$ −1.23324e28 −1.74768 −0.873839 0.486216i $$-0.838377\pi$$
−0.873839 + 0.486216i $$0.838377\pi$$
$$450$$ 0 0
$$451$$ 8.68008e27 1.17400
$$452$$ 1.17876e27 0.155766
$$453$$ 0 0
$$454$$ −1.01785e28 −1.28410
$$455$$ 9.93211e27 1.22440
$$456$$ 0 0
$$457$$ −7.62646e27 −0.897848 −0.448924 0.893570i $$-0.648193\pi$$
−0.448924 + 0.893570i $$0.648193\pi$$
$$458$$ 3.46113e27 0.398226
$$459$$ 0 0
$$460$$ 4.57814e26 0.0503189
$$461$$ 9.04851e26 0.0972114 0.0486057 0.998818i $$-0.484522\pi$$
0.0486057 + 0.998818i $$0.484522\pi$$
$$462$$ 0 0
$$463$$ −1.19705e27 −0.122889 −0.0614443 0.998111i $$-0.519571\pi$$
−0.0614443 + 0.998111i $$0.519571\pi$$
$$464$$ −8.75161e27 −0.878313
$$465$$ 0 0
$$466$$ 8.39637e27 0.805452
$$467$$ 1.16951e28 1.09693 0.548464 0.836174i $$-0.315213\pi$$
0.548464 + 0.836174i $$0.315213\pi$$
$$468$$ 0 0
$$469$$ −1.78117e27 −0.159732
$$470$$ −1.15988e28 −1.01715
$$471$$ 0 0
$$472$$ 5.30757e27 0.445148
$$473$$ −7.45451e27 −0.611472
$$474$$ 0 0
$$475$$ −1.00999e27 −0.0792562
$$476$$ −2.53222e27 −0.194370
$$477$$ 0 0
$$478$$ −1.26206e28 −0.927019
$$479$$ −2.04856e28 −1.47206 −0.736029 0.676950i $$-0.763301\pi$$
−0.736029 + 0.676950i $$0.763301\pi$$
$$480$$ 0 0
$$481$$ −1.31213e28 −0.902513
$$482$$ −1.96106e28 −1.31977
$$483$$ 0 0
$$484$$ 1.83525e27 0.118255
$$485$$ −1.25982e28 −0.794363
$$486$$ 0 0
$$487$$ −3.75693e27 −0.226871 −0.113436 0.993545i $$-0.536186\pi$$
−0.113436 + 0.993545i $$0.536186\pi$$
$$488$$ 1.23508e28 0.729942
$$489$$ 0 0
$$490$$ −4.30391e27 −0.243671
$$491$$ 1.53173e28 0.848841 0.424420 0.905465i $$-0.360478\pi$$
0.424420 + 0.905465i $$0.360478\pi$$
$$492$$ 0 0
$$493$$ 2.31261e28 1.22803
$$494$$ 3.93640e28 2.04629
$$495$$ 0 0
$$496$$ −2.64587e28 −1.31829
$$497$$ −6.33518e27 −0.309042
$$498$$ 0 0
$$499$$ 3.84836e27 0.179978 0.0899890 0.995943i $$-0.471317\pi$$
0.0899890 + 0.995943i $$0.471317\pi$$
$$500$$ 5.10855e27 0.233945
$$501$$ 0 0
$$502$$ 1.27208e28 0.558629
$$503$$ 2.25177e28 0.968411 0.484205 0.874954i $$-0.339109\pi$$
0.484205 + 0.874954i $$0.339109\pi$$
$$504$$ 0 0
$$505$$ 3.53696e28 1.45905
$$506$$ 3.41229e27 0.137869
$$507$$ 0 0
$$508$$ −2.65418e27 −0.102887
$$509$$ −2.09122e27 −0.0794078 −0.0397039 0.999211i $$-0.512641\pi$$
−0.0397039 + 0.999211i $$0.512641\pi$$
$$510$$ 0 0
$$511$$ −1.09184e28 −0.397869
$$512$$ −3.07519e28 −1.09783
$$513$$ 0 0
$$514$$ 3.89976e28 1.33637
$$515$$ 2.50902e28 0.842420
$$516$$ 0 0
$$517$$ 2.56104e28 0.825594
$$518$$ −1.42846e28 −0.451237
$$519$$ 0 0
$$520$$ −5.14948e28 −1.56217
$$521$$ 3.76486e27 0.111932 0.0559659 0.998433i $$-0.482176\pi$$
0.0559659 + 0.998433i $$0.482176\pi$$
$$522$$ 0 0
$$523$$ −1.74944e28 −0.499611 −0.249806 0.968296i $$-0.580367\pi$$
−0.249806 + 0.968296i $$0.580367\pi$$
$$524$$ −1.22040e28 −0.341604
$$525$$ 0 0
$$526$$ 2.96944e28 0.798590
$$527$$ 6.99170e28 1.84320
$$528$$ 0 0
$$529$$ −3.74562e28 −0.948942
$$530$$ 1.29101e28 0.320650
$$531$$ 0 0
$$532$$ −1.26952e28 −0.303086
$$533$$ −1.07269e29 −2.51095
$$534$$ 0 0
$$535$$ 1.84936e28 0.416205
$$536$$ 9.23480e27 0.203797
$$537$$ 0 0
$$538$$ −2.92215e28 −0.620138
$$539$$ 9.50317e27 0.197782
$$540$$ 0 0
$$541$$ −6.99960e28 −1.40121 −0.700603 0.713552i $$-0.747085\pi$$
−0.700603 + 0.713552i $$0.747085\pi$$
$$542$$ −5.11434e28 −1.00415
$$543$$ 0 0
$$544$$ 2.38153e28 0.449849
$$545$$ 8.76680e27 0.162434
$$546$$ 0 0
$$547$$ 3.00896e28 0.536476 0.268238 0.963353i $$-0.413559\pi$$
0.268238 + 0.963353i $$0.413559\pi$$
$$548$$ −2.31056e28 −0.404130
$$549$$ 0 0
$$550$$ 1.83193e27 0.0308391
$$551$$ 1.15941e29 1.91490
$$552$$ 0 0
$$553$$ −2.07495e28 −0.329909
$$554$$ −5.70754e28 −0.890424
$$555$$ 0 0
$$556$$ 1.04576e28 0.157090
$$557$$ −1.17263e29 −1.72855 −0.864277 0.503017i $$-0.832223\pi$$
−0.864277 + 0.503017i $$0.832223\pi$$
$$558$$ 0 0
$$559$$ 9.21233e28 1.30781
$$560$$ −4.25421e28 −0.592714
$$561$$ 0 0
$$562$$ 7.71427e28 1.03529
$$563$$ −6.46986e28 −0.852230 −0.426115 0.904669i $$-0.640118\pi$$
−0.426115 + 0.904669i $$0.640118\pi$$
$$564$$ 0 0
$$565$$ −5.23307e28 −0.664121
$$566$$ 1.32575e29 1.65153
$$567$$ 0 0
$$568$$ 3.28459e28 0.394297
$$569$$ 1.35009e29 1.59105 0.795527 0.605918i $$-0.207194\pi$$
0.795527 + 0.605918i $$0.207194\pi$$
$$570$$ 0 0
$$571$$ 2.55750e28 0.290494 0.145247 0.989395i $$-0.453602\pi$$
0.145247 + 0.989395i $$0.453602\pi$$
$$572$$ 2.11515e28 0.235875
$$573$$ 0 0
$$574$$ −1.16779e29 −1.25542
$$575$$ 1.08197e27 0.0114209
$$576$$ 0 0
$$577$$ −1.52580e29 −1.55293 −0.776465 0.630160i $$-0.782989\pi$$
−0.776465 + 0.630160i $$0.782989\pi$$
$$578$$ 9.88284e26 0.00987733
$$579$$ 0 0
$$580$$ −2.82146e28 −0.271944
$$581$$ −1.92935e26 −0.00182625
$$582$$ 0 0
$$583$$ −2.85058e28 −0.260264
$$584$$ 5.66084e28 0.507629
$$585$$ 0 0
$$586$$ −7.94506e28 −0.687341
$$587$$ 1.08485e29 0.921868 0.460934 0.887434i $$-0.347514\pi$$
0.460934 + 0.887434i $$0.347514\pi$$
$$588$$ 0 0
$$589$$ 3.50525e29 2.87414
$$590$$ −4.38328e28 −0.353064
$$591$$ 0 0
$$592$$ 5.62023e28 0.436894
$$593$$ −8.82363e28 −0.673865 −0.336933 0.941529i $$-0.609389\pi$$
−0.336933 + 0.941529i $$0.609389\pi$$
$$594$$ 0 0
$$595$$ 1.12417e29 0.828715
$$596$$ 2.32040e28 0.168065
$$597$$ 0 0
$$598$$ −4.21694e28 −0.294873
$$599$$ −9.23342e28 −0.634426 −0.317213 0.948354i $$-0.602747\pi$$
−0.317213 + 0.948354i $$0.602747\pi$$
$$600$$ 0 0
$$601$$ −7.30699e28 −0.484793 −0.242397 0.970177i $$-0.577934\pi$$
−0.242397 + 0.970177i $$0.577934\pi$$
$$602$$ 1.00291e29 0.653878
$$603$$ 0 0
$$604$$ 5.89267e28 0.371043
$$605$$ −8.14756e28 −0.504192
$$606$$ 0 0
$$607$$ 8.07544e28 0.482709 0.241354 0.970437i $$-0.422408\pi$$
0.241354 + 0.970437i $$0.422408\pi$$
$$608$$ 1.19396e29 0.701461
$$609$$ 0 0
$$610$$ −1.02000e29 −0.578945
$$611$$ −3.16496e29 −1.76577
$$612$$ 0 0
$$613$$ −2.21326e29 −1.19316 −0.596578 0.802555i $$-0.703473\pi$$
−0.596578 + 0.802555i $$0.703473\pi$$
$$614$$ 9.08699e28 0.481562
$$615$$ 0 0
$$616$$ 1.23783e29 0.633960
$$617$$ 2.16026e29 1.08771 0.543854 0.839180i $$-0.316965\pi$$
0.543854 + 0.839180i $$0.316965\pi$$
$$618$$ 0 0
$$619$$ −2.09908e29 −1.02159 −0.510795 0.859702i $$-0.670649\pi$$
−0.510795 + 0.859702i $$0.670649\pi$$
$$620$$ −8.53011e28 −0.408171
$$621$$ 0 0
$$622$$ 7.65609e28 0.354167
$$623$$ −1.48280e29 −0.674465
$$624$$ 0 0
$$625$$ −2.15300e29 −0.946901
$$626$$ −2.15023e29 −0.929938
$$627$$ 0 0
$$628$$ −2.82694e28 −0.118234
$$629$$ −1.48514e29 −0.610853
$$630$$ 0 0
$$631$$ 4.81094e28 0.191391 0.0956955 0.995411i $$-0.469492\pi$$
0.0956955 + 0.995411i $$0.469492\pi$$
$$632$$ 1.07580e29 0.420921
$$633$$ 0 0
$$634$$ 7.27166e28 0.275230
$$635$$ 1.17832e29 0.438670
$$636$$ 0 0
$$637$$ −1.17441e29 −0.423014
$$638$$ −2.10296e29 −0.745100
$$639$$ 0 0
$$640$$ 1.50481e29 0.515930
$$641$$ 5.04867e28 0.170282 0.0851408 0.996369i $$-0.472866\pi$$
0.0851408 + 0.996369i $$0.472866\pi$$
$$642$$ 0 0
$$643$$ −3.93930e29 −1.28589 −0.642945 0.765912i $$-0.722288\pi$$
−0.642945 + 0.765912i $$0.722288\pi$$
$$644$$ 1.35999e28 0.0436751
$$645$$ 0 0
$$646$$ 4.45545e29 1.38500
$$647$$ 1.38475e29 0.423523 0.211762 0.977321i $$-0.432080\pi$$
0.211762 + 0.977321i $$0.432080\pi$$
$$648$$ 0 0
$$649$$ 9.67842e28 0.286573
$$650$$ −2.26392e28 −0.0659584
$$651$$ 0 0
$$652$$ 1.52528e28 0.0430280
$$653$$ −1.16655e29 −0.323830 −0.161915 0.986805i $$-0.551767\pi$$
−0.161915 + 0.986805i $$0.551767\pi$$
$$654$$ 0 0
$$655$$ 5.41792e29 1.45646
$$656$$ 4.59464e29 1.21552
$$657$$ 0 0
$$658$$ −3.44555e29 −0.882849
$$659$$ 3.36819e29 0.849377 0.424688 0.905340i $$-0.360384\pi$$
0.424688 + 0.905340i $$0.360384\pi$$
$$660$$ 0 0
$$661$$ 4.31449e29 1.05394 0.526968 0.849885i $$-0.323329\pi$$
0.526968 + 0.849885i $$0.323329\pi$$
$$662$$ 8.49612e28 0.204273
$$663$$ 0 0
$$664$$ 1.00030e27 0.00233006
$$665$$ 5.63598e29 1.29224
$$666$$ 0 0
$$667$$ −1.24204e29 −0.275940
$$668$$ −1.73894e28 −0.0380305
$$669$$ 0 0
$$670$$ −7.62660e28 −0.161639
$$671$$ 2.25219e29 0.469915
$$672$$ 0 0
$$673$$ −1.19710e29 −0.242088 −0.121044 0.992647i $$-0.538624\pi$$
−0.121044 + 0.992647i $$0.538624\pi$$
$$674$$ 4.52008e29 0.899948
$$675$$ 0 0
$$676$$ −1.42978e29 −0.275949
$$677$$ −7.71503e29 −1.46608 −0.733038 0.680188i $$-0.761898\pi$$
−0.733038 + 0.680188i $$0.761898\pi$$
$$678$$ 0 0
$$679$$ −3.74243e29 −0.689479
$$680$$ −5.82848e29 −1.05733
$$681$$ 0 0
$$682$$ −6.35788e29 −1.11835
$$683$$ −6.40488e29 −1.10941 −0.554707 0.832046i $$-0.687170\pi$$
−0.554707 + 0.832046i $$0.687170\pi$$
$$684$$ 0 0
$$685$$ 1.02577e30 1.72305
$$686$$ −5.76908e29 −0.954340
$$687$$ 0 0
$$688$$ −3.94591e29 −0.633094
$$689$$ 3.52277e29 0.556650
$$690$$ 0 0
$$691$$ 1.07309e30 1.64482 0.822408 0.568898i $$-0.192630\pi$$
0.822408 + 0.568898i $$0.192630\pi$$
$$692$$ 1.54094e29 0.232633
$$693$$ 0 0
$$694$$ 5.01454e28 0.0734440
$$695$$ −4.64263e29 −0.669767
$$696$$ 0 0
$$697$$ −1.21413e30 −1.69950
$$698$$ −7.43202e29 −1.02477
$$699$$ 0 0
$$700$$ 7.30127e27 0.00976943
$$701$$ 7.18115e29 0.946574 0.473287 0.880908i $$-0.343067\pi$$
0.473287 + 0.880908i $$0.343067\pi$$
$$702$$ 0 0
$$703$$ −7.44568e29 −0.952518
$$704$$ −6.12985e29 −0.772568
$$705$$ 0 0
$$706$$ −6.30969e29 −0.771895
$$707$$ 1.05070e30 1.26641
$$708$$ 0 0
$$709$$ 4.24360e29 0.496534 0.248267 0.968692i $$-0.420139\pi$$
0.248267 + 0.968692i $$0.420139\pi$$
$$710$$ −2.71259e29 −0.312732
$$711$$ 0 0
$$712$$ 7.68786e29 0.860529
$$713$$ −3.75505e29 −0.414167
$$714$$ 0 0
$$715$$ −9.39014e29 −1.00568
$$716$$ −8.64710e28 −0.0912607
$$717$$ 0 0
$$718$$ 4.26277e29 0.436904
$$719$$ 5.73272e29 0.579038 0.289519 0.957172i $$-0.406505\pi$$
0.289519 + 0.957172i $$0.406505\pi$$
$$720$$ 0 0
$$721$$ 7.45334e29 0.731191
$$722$$ 1.32527e30 1.28134
$$723$$ 0 0
$$724$$ 3.05133e27 0.00286572
$$725$$ −6.66805e28 −0.0617234
$$726$$ 0 0
$$727$$ 1.42987e30 1.28583 0.642916 0.765937i $$-0.277725\pi$$
0.642916 + 0.765937i $$0.277725\pi$$
$$728$$ −1.52971e30 −1.35591
$$729$$ 0 0
$$730$$ −4.67503e29 −0.402619
$$731$$ 1.04270e30 0.885175
$$732$$ 0 0
$$733$$ 1.99143e30 1.64276 0.821380 0.570381i $$-0.193205\pi$$
0.821380 + 0.570381i $$0.193205\pi$$
$$734$$ 1.23405e29 0.100351
$$735$$ 0 0
$$736$$ −1.27905e29 −0.101081
$$737$$ 1.68398e29 0.131198
$$738$$ 0 0
$$739$$ −2.11124e30 −1.59872 −0.799359 0.600854i $$-0.794827\pi$$
−0.799359 + 0.600854i $$0.794827\pi$$
$$740$$ 1.81193e29 0.135272
$$741$$ 0 0
$$742$$ 3.83509e29 0.278313
$$743$$ −1.11538e30 −0.798067 −0.399033 0.916936i $$-0.630654\pi$$
−0.399033 + 0.916936i $$0.630654\pi$$
$$744$$ 0 0
$$745$$ −1.03013e30 −0.716561
$$746$$ 1.31331e30 0.900763
$$747$$ 0 0
$$748$$ 2.39405e29 0.159649
$$749$$ 5.49375e29 0.361251
$$750$$ 0 0
$$751$$ −2.28371e30 −1.46023 −0.730116 0.683323i $$-0.760534\pi$$
−0.730116 + 0.683323i $$0.760534\pi$$
$$752$$ 1.35564e30 0.854787
$$753$$ 0 0
$$754$$ 2.59886e30 1.59361
$$755$$ −2.61603e30 −1.58198
$$756$$ 0 0
$$757$$ 9.73411e29 0.572519 0.286259 0.958152i $$-0.407588\pi$$
0.286259 + 0.958152i $$0.407588\pi$$
$$758$$ 2.80951e29 0.162969
$$759$$ 0 0
$$760$$ −2.92207e30 −1.64873
$$761$$ 3.06985e30 1.70836 0.854179 0.519979i $$-0.174060\pi$$
0.854179 + 0.519979i $$0.174060\pi$$
$$762$$ 0 0
$$763$$ 2.60428e29 0.140987
$$764$$ 3.91996e29 0.209316
$$765$$ 0 0
$$766$$ 2.02936e30 1.05428
$$767$$ −1.19607e30 −0.612920
$$768$$ 0 0
$$769$$ −7.51816e29 −0.374874 −0.187437 0.982277i $$-0.560018\pi$$
−0.187437 + 0.982277i $$0.560018\pi$$
$$770$$ −1.02226e30 −0.502817
$$771$$ 0 0
$$772$$ 8.63013e28 0.0413081
$$773$$ −1.23771e30 −0.584434 −0.292217 0.956352i $$-0.594393\pi$$
−0.292217 + 0.956352i $$0.594393\pi$$
$$774$$ 0 0
$$775$$ −2.01595e29 −0.0926427
$$776$$ 1.94033e30 0.879686
$$777$$ 0 0
$$778$$ 4.58507e29 0.202330
$$779$$ −6.08698e30 −2.65008
$$780$$ 0 0
$$781$$ 5.98949e29 0.253836
$$782$$ −4.77297e29 −0.199580
$$783$$ 0 0
$$784$$ 5.03033e29 0.204776
$$785$$ 1.25501e30 0.504100
$$786$$ 0 0
$$787$$ 2.14352e30 0.838288 0.419144 0.907920i $$-0.362330\pi$$
0.419144 + 0.907920i $$0.362330\pi$$
$$788$$ 7.91853e29 0.305576
$$789$$ 0 0
$$790$$ −8.88450e29 −0.333848
$$791$$ −1.55454e30 −0.576434
$$792$$ 0 0
$$793$$ −2.78327e30 −1.00505
$$794$$ −1.87221e30 −0.667174
$$795$$ 0 0
$$796$$ 7.31081e29 0.253734
$$797$$ −3.01280e30 −1.03195 −0.515974 0.856605i $$-0.672570\pi$$
−0.515974 + 0.856605i $$0.672570\pi$$
$$798$$ 0 0
$$799$$ −3.58228e30 −1.19514
$$800$$ −6.86676e28 −0.0226103
$$801$$ 0 0
$$802$$ −1.27705e30 −0.409617
$$803$$ 1.03226e30 0.326796
$$804$$ 0 0
$$805$$ −6.03763e29 −0.186213
$$806$$ 7.85711e30 2.39191
$$807$$ 0 0
$$808$$ −5.44752e30 −1.61577
$$809$$ 5.76883e30 1.68900 0.844498 0.535559i $$-0.179899\pi$$
0.844498 + 0.535559i $$0.179899\pi$$
$$810$$ 0 0
$$811$$ 3.98728e30 1.13752 0.568759 0.822504i $$-0.307424\pi$$
0.568759 + 0.822504i $$0.307424\pi$$
$$812$$ −8.38148e29 −0.236038
$$813$$ 0 0
$$814$$ 1.35051e30 0.370631
$$815$$ −6.77146e29 −0.183454
$$816$$ 0 0
$$817$$ 5.22754e30 1.38027
$$818$$ −5.61998e30 −1.46496
$$819$$ 0 0
$$820$$ 1.48128e30 0.376350
$$821$$ 4.09615e30 1.02748 0.513739 0.857946i $$-0.328260\pi$$
0.513739 + 0.857946i $$0.328260\pi$$
$$822$$ 0 0
$$823$$ 1.43384e30 0.350592 0.175296 0.984516i $$-0.443912\pi$$
0.175296 + 0.984516i $$0.443912\pi$$
$$824$$ −3.86432e30 −0.932905
$$825$$ 0 0
$$826$$ −1.30210e30 −0.306447
$$827$$ 3.13142e30 0.727668 0.363834 0.931464i $$-0.381468\pi$$
0.363834 + 0.931464i $$0.381468\pi$$
$$828$$ 0 0
$$829$$ −4.44329e30 −1.00666 −0.503329 0.864095i $$-0.667892\pi$$
−0.503329 + 0.864095i $$0.667892\pi$$
$$830$$ −8.26106e27 −0.00184806
$$831$$ 0 0
$$832$$ 7.57531e30 1.65236
$$833$$ −1.32926e30 −0.286311
$$834$$ 0 0
$$835$$ 7.71998e29 0.162147
$$836$$ 1.20024e30 0.248944
$$837$$ 0 0
$$838$$ 3.00994e30 0.608829
$$839$$ 9.30057e29 0.185784 0.0928922 0.995676i $$-0.470389\pi$$
0.0928922 + 0.995676i $$0.470389\pi$$
$$840$$ 0 0
$$841$$ 2.52173e30 0.491293
$$842$$ 3.46057e30 0.665842
$$843$$ 0 0
$$844$$ −6.50614e29 −0.122103
$$845$$ 6.34746e30 1.17653
$$846$$ 0 0
$$847$$ −2.42033e30 −0.437621
$$848$$ −1.50891e30 −0.269467
$$849$$ 0 0
$$850$$ −2.56243e29 −0.0446430
$$851$$ 7.97630e29 0.137259
$$852$$ 0 0
$$853$$ −2.27471e30 −0.381911 −0.190955 0.981599i $$-0.561159\pi$$
−0.190955 + 0.981599i $$0.561159\pi$$
$$854$$ −3.03003e30 −0.502503
$$855$$ 0 0
$$856$$ −2.84833e30 −0.460910
$$857$$ −3.05003e30 −0.487534 −0.243767 0.969834i $$-0.578383\pi$$
−0.243767 + 0.969834i $$0.578383\pi$$
$$858$$ 0 0
$$859$$ 1.34073e30 0.209128 0.104564 0.994518i $$-0.466655\pi$$
0.104564 + 0.994518i $$0.466655\pi$$
$$860$$ −1.27213e30 −0.196020
$$861$$ 0 0
$$862$$ −1.12873e31 −1.69732
$$863$$ −1.14916e31 −1.70713 −0.853566 0.520985i $$-0.825565\pi$$
−0.853566 + 0.520985i $$0.825565\pi$$
$$864$$ 0 0
$$865$$ −6.84096e30 −0.991855
$$866$$ −4.32457e30 −0.619448
$$867$$ 0 0
$$868$$ −2.53397e30 −0.354278
$$869$$ 1.96173e30 0.270976
$$870$$ 0 0
$$871$$ −2.08107e30 −0.280606
$$872$$ −1.35024e30 −0.179882
$$873$$ 0 0
$$874$$ −2.39290e30 −0.311211
$$875$$ −6.73714e30 −0.865747
$$876$$ 0 0
$$877$$ 9.06320e30 1.13707 0.568533 0.822660i $$-0.307511\pi$$
0.568533 + 0.822660i $$0.307511\pi$$
$$878$$ 2.91934e30 0.361904
$$879$$ 0 0
$$880$$ 4.02207e30 0.486835
$$881$$ 6.50767e30 0.778356 0.389178 0.921163i $$-0.372759\pi$$
0.389178 + 0.921163i $$0.372759\pi$$
$$882$$ 0 0
$$883$$ 5.90742e30 0.689939 0.344969 0.938614i $$-0.387889\pi$$
0.344969 + 0.938614i $$0.387889\pi$$
$$884$$ −2.95858e30 −0.341456
$$885$$ 0 0
$$886$$ 7.18992e30 0.810344
$$887$$ 3.42312e30 0.381263 0.190632 0.981662i $$-0.438946\pi$$
0.190632 + 0.981662i $$0.438946\pi$$
$$888$$ 0 0
$$889$$ 3.50032e30 0.380750
$$890$$ −6.34905e30 −0.682518
$$891$$ 0 0
$$892$$ −2.26792e30 −0.238121
$$893$$ −1.79595e31 −1.86361
$$894$$ 0 0
$$895$$ 3.83886e30 0.389099
$$896$$ 4.47020e30 0.447810
$$897$$ 0 0
$$898$$ −1.56863e31 −1.53503
$$899$$ 2.31420e31 2.23833
$$900$$ 0 0
$$901$$ 3.98728e30 0.376761
$$902$$ 1.10407e31 1.03116
$$903$$ 0 0
$$904$$ 8.05980e30 0.735455
$$905$$ −1.35463e29 −0.0122183
$$906$$ 0 0
$$907$$ −1.16251e31 −1.02452 −0.512262 0.858829i $$-0.671192\pi$$
−0.512262 + 0.858829i $$0.671192\pi$$
$$908$$ 3.83534e30 0.334120
$$909$$ 0 0
$$910$$ 1.26332e31 1.07542
$$911$$ 1.34510e31 1.13191 0.565957 0.824435i $$-0.308507\pi$$
0.565957 + 0.824435i $$0.308507\pi$$
$$912$$ 0 0
$$913$$ 1.82407e28 0.00150002
$$914$$ −9.70051e30 −0.788605
$$915$$ 0 0
$$916$$ −1.30418e30 −0.103618
$$917$$ 1.60945e31 1.26416
$$918$$ 0 0
$$919$$ −8.55562e30 −0.656808 −0.328404 0.944537i $$-0.606511\pi$$
−0.328404 + 0.944537i $$0.606511\pi$$
$$920$$ 3.13032e30 0.237583
$$921$$ 0 0
$$922$$ 1.15093e30 0.0853835
$$923$$ −7.40185e30 −0.542903
$$924$$ 0 0
$$925$$ 4.28218e29 0.0307027
$$926$$ −1.52259e30 −0.107937
$$927$$ 0 0
$$928$$ 7.88268e30 0.546286
$$929$$ −9.93712e30 −0.680919 −0.340460 0.940259i $$-0.610583\pi$$
−0.340460 + 0.940259i $$0.610583\pi$$
$$930$$ 0 0
$$931$$ −6.66418e30 −0.446452
$$932$$ −3.16381e30 −0.209577
$$933$$ 0 0
$$934$$ 1.48757e31 0.963462
$$935$$ −1.06283e31 −0.680679
$$936$$ 0 0
$$937$$ −2.14272e30 −0.134183 −0.0670917 0.997747i $$-0.521372\pi$$
−0.0670917 + 0.997747i $$0.521372\pi$$
$$938$$ −2.26557e30 −0.140297
$$939$$ 0 0
$$940$$ 4.37050e30 0.264660
$$941$$ −1.69148e31 −1.01292 −0.506459 0.862264i $$-0.669046\pi$$
−0.506459 + 0.862264i $$0.669046\pi$$
$$942$$ 0 0
$$943$$ 6.52077e30 0.381879
$$944$$ 5.12310e30 0.296706
$$945$$ 0 0
$$946$$ −9.48180e30 −0.537074
$$947$$ 1.96530e31 1.10092 0.550459 0.834862i $$-0.314453\pi$$
0.550459 + 0.834862i $$0.314453\pi$$
$$948$$ 0 0
$$949$$ −1.27568e31 −0.698948
$$950$$ −1.28466e30 −0.0696129
$$951$$ 0 0
$$952$$ −1.73142e31 −0.917728
$$953$$ −2.39001e31 −1.25292 −0.626462 0.779452i $$-0.715498\pi$$
−0.626462 + 0.779452i $$0.715498\pi$$
$$954$$ 0 0
$$955$$ −1.74026e31 −0.892437
$$956$$ 4.75555e30 0.241209
$$957$$ 0 0
$$958$$ −2.60567e31 −1.29295
$$959$$ 3.04716e31 1.49555
$$960$$ 0 0
$$961$$ 4.91397e31 2.35959
$$962$$ −1.66897e31 −0.792703
$$963$$ 0 0
$$964$$ 7.38943e30 0.343401
$$965$$ −3.83132e30 −0.176121
$$966$$ 0 0
$$967$$ 1.63072e31 0.733504 0.366752 0.930319i $$-0.380470\pi$$
0.366752 + 0.930319i $$0.380470\pi$$
$$968$$ 1.25486e31 0.558348
$$969$$ 0 0
$$970$$ −1.60243e31 −0.697712
$$971$$ −3.09343e31 −1.33241 −0.666207 0.745767i $$-0.732083\pi$$
−0.666207 + 0.745767i $$0.732083\pi$$
$$972$$ 0 0
$$973$$ −1.37915e31 −0.581334
$$974$$ −4.77865e30 −0.199267
$$975$$ 0 0
$$976$$ 1.19216e31 0.486531
$$977$$ −1.98044e31 −0.799592 −0.399796 0.916604i $$-0.630919\pi$$
−0.399796 + 0.916604i $$0.630919\pi$$
$$978$$ 0 0
$$979$$ 1.40189e31 0.553982
$$980$$ 1.62175e30 0.0634029
$$981$$ 0 0
$$982$$ 1.94829e31 0.745561
$$983$$ 4.34284e30 0.164423 0.0822114 0.996615i $$-0.473802\pi$$
0.0822114 + 0.996615i $$0.473802\pi$$
$$984$$ 0 0
$$985$$ −3.51541e31 −1.30285
$$986$$ 2.94154e31 1.07862
$$987$$ 0 0
$$988$$ −1.48327e31 −0.532441
$$989$$ −5.60008e30 −0.198899
$$990$$ 0 0
$$991$$ 4.70795e30 0.163704 0.0818518 0.996645i $$-0.473917\pi$$
0.0818518 + 0.996645i $$0.473917\pi$$
$$992$$ 2.38317e31 0.819939
$$993$$ 0 0
$$994$$ −8.05807e30 −0.271440
$$995$$ −3.24561e31 −1.08182
$$996$$ 0 0
$$997$$ −1.58934e31 −0.518700 −0.259350 0.965783i $$-0.583508\pi$$
−0.259350 + 0.965783i $$0.583508\pi$$
$$998$$ 4.89494e30 0.158080
$$999$$ 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 9.22.a.e.1.2 2
3.2 odd 2 3.22.a.c.1.1 2
12.11 even 2 48.22.a.g.1.1 2
15.2 even 4 75.22.b.d.49.2 4
15.8 even 4 75.22.b.d.49.3 4
15.14 odd 2 75.22.a.d.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.c.1.1 2 3.2 odd 2
9.22.a.e.1.2 2 1.1 even 1 trivial
48.22.a.g.1.1 2 12.11 even 2
75.22.a.d.1.2 2 15.14 odd 2
75.22.b.d.49.2 4 15.2 even 4
75.22.b.d.49.3 4 15.8 even 4