# Properties

 Label 48.22.a.g.1.1 Level $48$ Weight $22$ Character 48.1 Self dual yes Analytic conductor $134.149$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$134.149125258$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 7$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-12.2377$$ of defining polynomial Character $$\chi$$ $$=$$ 48.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-59049.0 q^{3} -2.12776e7 q^{5} -6.32076e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$$ $$q-59049.0 q^{3} -2.12776e7 q^{5} -6.32076e8 q^{7} +3.48678e9 q^{9} -5.97585e10 q^{11} +7.38499e11 q^{13} +1.25642e12 q^{15} -8.35876e12 q^{17} -4.19061e13 q^{19} +3.73234e13 q^{21} -4.48926e13 q^{23} -2.41012e13 q^{25} -2.05891e14 q^{27} -2.76669e15 q^{29} -8.36452e15 q^{31} +3.52868e15 q^{33} +1.34490e16 q^{35} -1.77675e16 q^{37} -4.36077e16 q^{39} +1.45253e17 q^{41} -1.24744e17 q^{43} -7.41904e16 q^{45} -4.28566e17 q^{47} -1.59026e17 q^{49} +4.93577e17 q^{51} -4.77017e17 q^{53} +1.27152e18 q^{55} +2.47452e18 q^{57} -1.61959e18 q^{59} -3.76882e18 q^{61} -2.20391e18 q^{63} -1.57135e19 q^{65} +2.81797e18 q^{67} +2.65086e18 q^{69} -1.00228e19 q^{71} -1.72739e19 q^{73} +1.42315e18 q^{75} +3.77719e19 q^{77} +3.28276e19 q^{79} +1.21577e19 q^{81} -3.05240e17 q^{83} +1.77854e20 q^{85} +1.63370e20 q^{87} +2.34593e20 q^{89} -4.66787e20 q^{91} +4.93917e20 q^{93} +8.91662e20 q^{95} -5.92086e20 q^{97} -2.08365e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10})$$ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 $$2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100})$$ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 - 219869122968 * q^11 - 48468909956 * q^13 - 58864530924 * q^15 - 11333529041436 * q^17 - 11960585011624 * q^19 + 40147185517488 * q^21 + 146508390063504 * q^23 - 4786354247074 * q^25 - 411782264189298 * q^27 - 1798520043674052 * q^29 - 11169107526944992 * q^31 + 12983051842137432 * q^33 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 2862040663991844 * q^39 + 122972020616468052 * q^41 - 288455418162270040 * q^43 + 3475891686531276 * q^45 - 837243745741596960 * q^47 - 715285396941470670 * q^49 + 669233556367754364 * q^51 - 43007964012775764 * q^53 - 2294863118895313296 * q^55 + 706260584351385576 * q^57 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 - 2370651157622148912 * q^63 - 33242791066177513752 * q^65 + 16454068667621610296 * q^67 - 8651173924859847696 * q^69 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 + 282629431935472626 * q^75 + 45428450983816025664 * q^77 + 54055785594190591040 * q^79 + 24315330918113857602 * q^81 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 + 106200810058909096548 * q^87 + 226920767965448065524 * q^89 - 429154174685185000352 * q^91 + 659524630358574832608 * q^93 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 - 766636228226373222168 * 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
$$3$$ −59049.0 −0.577350
$$4$$ 0 0
$$5$$ −2.12776e7 −0.974400 −0.487200 0.873290i $$-0.661982\pi$$
−0.487200 + 0.873290i $$0.661982\pi$$
$$6$$ 0 0
$$7$$ −6.32076e8 −0.845745 −0.422873 0.906189i $$-0.638978\pi$$
−0.422873 + 0.906189i $$0.638978\pi$$
$$8$$ 0 0
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ −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 $$0.233462\pi$$
$$14$$ 0 0
$$15$$ 1.25642e12 0.562570
$$16$$ 0 0
$$17$$ −8.35876e12 −1.00561 −0.502804 0.864401i $$-0.667698\pi$$
−0.502804 + 0.864401i $$0.667698\pi$$
$$18$$ 0 0
$$19$$ −4.19061e13 −1.56807 −0.784034 0.620718i $$-0.786841\pi$$
−0.784034 + 0.620718i $$0.786841\pi$$
$$20$$ 0 0
$$21$$ 3.73234e13 0.488291
$$22$$ 0 0
$$23$$ −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
$$26$$ 0 0
$$27$$ −2.05891e14 −0.192450
$$28$$ 0 0
$$29$$ −2.76669e15 −1.22119 −0.610593 0.791945i $$-0.709069\pi$$
−0.610593 + 0.791945i $$0.709069\pi$$
$$30$$ 0 0
$$31$$ −8.36452e15 −1.83292 −0.916459 0.400128i $$-0.868966\pi$$
−0.916459 + 0.400128i $$0.868966\pi$$
$$32$$ 0 0
$$33$$ 3.52868e15 0.401066
$$34$$ 0 0
$$35$$ 1.34490e16 0.824095
$$36$$ 0 0
$$37$$ −1.77675e16 −0.607447 −0.303724 0.952760i $$-0.598230\pi$$
−0.303724 + 0.952760i $$0.598230\pi$$
$$38$$ 0 0
$$39$$ −4.36077e16 −0.857797
$$40$$ 0 0
$$41$$ 1.45253e17 1.69003 0.845013 0.534745i $$-0.179592\pi$$
0.845013 + 0.534745i $$0.179592\pi$$
$$42$$ 0 0
$$43$$ −1.24744e17 −0.880239 −0.440119 0.897939i $$-0.645064\pi$$
−0.440119 + 0.897939i $$0.645064\pi$$
$$44$$ 0 0
$$45$$ −7.41904e16 −0.324800
$$46$$ 0 0
$$47$$ −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
$$50$$ 0 0
$$51$$ 4.93577e17 0.580588
$$52$$ 0 0
$$53$$ −4.77017e17 −0.374660 −0.187330 0.982297i $$-0.559983\pi$$
−0.187330 + 0.982297i $$0.559983\pi$$
$$54$$ 0 0
$$55$$ 1.27152e18 0.676883
$$56$$ 0 0
$$57$$ 2.47452e18 0.905324
$$58$$ 0 0
$$59$$ −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 $$0.609828\pi$$
$$62$$ 0 0
$$63$$ −2.20391e18 −0.281915
$$64$$ 0 0
$$65$$ −1.57135e19 −1.44771
$$66$$ 0 0
$$67$$ 2.81797e18 0.188865 0.0944324 0.995531i $$-0.469896\pi$$
0.0944324 + 0.995531i $$0.469896\pi$$
$$68$$ 0 0
$$69$$ 2.65086e18 0.130458
$$70$$ 0 0
$$71$$ −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 $$0.575580\pi$$
$$74$$ 0 0
$$75$$ 1.42315e18 0.0291815
$$76$$ 0 0
$$77$$ 3.77719e19 0.587511
$$78$$ 0 0
$$79$$ 3.28276e19 0.390080 0.195040 0.980795i $$-0.437516\pi$$
0.195040 + 0.980795i $$0.437516\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 0 0
$$83$$ −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
$$86$$ 0 0
$$87$$ 1.63370e20 0.705052
$$88$$ 0 0
$$89$$ 2.34593e20 0.797480 0.398740 0.917064i $$-0.369448\pi$$
0.398740 + 0.917064i $$0.369448\pi$$
$$90$$ 0 0
$$91$$ −4.66787e20 −1.25656
$$92$$ 0 0
$$93$$ 4.93917e20 1.05824
$$94$$ 0 0
$$95$$ 8.91662e20 1.52793
$$96$$ 0 0
$$97$$ −5.92086e20 −0.815233 −0.407616 0.913153i $$-0.633640\pi$$
−0.407616 + 0.913153i $$0.633640\pi$$
$$98$$ 0 0
$$99$$ −2.08365e20 −0.231555
$$100$$ 0 0
$$101$$ −1.66229e21 −1.49739 −0.748693 0.662917i $$-0.769318\pi$$
−0.748693 + 0.662917i $$0.769318\pi$$
$$102$$ 0 0
$$103$$ −1.17919e21 −0.864552 −0.432276 0.901741i $$-0.642289\pi$$
−0.432276 + 0.901741i $$0.642289\pi$$
$$104$$ 0 0
$$105$$ −7.94153e20 −0.475791
$$106$$ 0 0
$$107$$ 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 $$0.473438\pi$$
$$110$$ 0 0
$$111$$ 1.04915e21 0.350710
$$112$$ 0 0
$$113$$ 2.45943e21 0.681569 0.340785 0.940141i $$-0.389307\pi$$
0.340785 + 0.940141i $$0.389307\pi$$
$$114$$ 0 0
$$115$$ 9.55207e20 0.220176
$$116$$ 0 0
$$117$$ 2.57499e21 0.495249
$$118$$ 0 0
$$119$$ 5.28337e21 0.850487
$$120$$ 0 0
$$121$$ −3.82917e21 −0.517439
$$122$$ 0 0
$$123$$ −8.57702e21 −0.975737
$$124$$ 0 0
$$125$$ 1.06588e22 1.02365
$$126$$ 0 0
$$127$$ −5.53783e21 −0.450195 −0.225097 0.974336i $$-0.572270\pi$$
−0.225097 + 0.974336i $$0.572270\pi$$
$$128$$ 0 0
$$129$$ 7.36600e21 0.508206
$$130$$ 0 0
$$131$$ 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
$$134$$ 0 0
$$135$$ 4.38087e21 0.187523
$$136$$ 0 0
$$137$$ −4.82088e22 −1.76832 −0.884158 0.467188i $$-0.845267\pi$$
−0.884158 + 0.467188i $$0.845267\pi$$
$$138$$ 0 0
$$139$$ 2.18194e22 0.687363 0.343682 0.939086i $$-0.388326\pi$$
0.343682 + 0.939086i $$0.388326\pi$$
$$140$$ 0 0
$$141$$ 2.53064e22 0.686166
$$142$$ 0 0
$$143$$ −4.41316e22 −1.03210
$$144$$ 0 0
$$145$$ 5.88685e22 1.18992
$$146$$ 0 0
$$147$$ 9.39035e21 0.164380
$$148$$ 0 0
$$149$$ 4.84140e22 0.735386 0.367693 0.929947i $$-0.380148\pi$$
0.367693 + 0.929947i $$0.380148\pi$$
$$150$$ 0 0
$$151$$ 1.22948e23 1.62354 0.811770 0.583977i $$-0.198504\pi$$
0.811770 + 0.583977i $$0.198504\pi$$
$$152$$ 0 0
$$153$$ −2.91452e22 −0.335202
$$154$$ 0 0
$$155$$ 1.77977e23 1.78600
$$156$$ 0 0
$$157$$ 5.89828e22 0.517344 0.258672 0.965965i $$-0.416715\pi$$
0.258672 + 0.965965i $$0.416715\pi$$
$$158$$ 0 0
$$159$$ 2.81674e22 0.216310
$$160$$ 0 0
$$161$$ 2.83755e22 0.191105
$$162$$ 0 0
$$163$$ 3.18244e22 0.188274 0.0941369 0.995559i $$-0.469991\pi$$
0.0941369 + 0.995559i $$0.469991\pi$$
$$164$$ 0 0
$$165$$ −7.50818e22 −0.390799
$$166$$ 0 0
$$167$$ 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
$$170$$ 0 0
$$171$$ −1.46118e23 −0.522689
$$172$$ 0 0
$$173$$ 3.21510e23 1.01791 0.508956 0.860792i $$-0.330031\pi$$
0.508956 + 0.860792i $$0.330031\pi$$
$$174$$ 0 0
$$175$$ 1.52338e22 0.0427472
$$176$$ 0 0
$$177$$ 9.56351e22 0.238176
$$178$$ 0 0
$$179$$ 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 $$0.501996\pi$$
$$182$$ 0 0
$$183$$ 2.22545e23 0.390555
$$184$$ 0 0
$$185$$ 3.78050e23 0.591897
$$186$$ 0 0
$$187$$ 4.99507e23 0.698561
$$188$$ 0 0
$$189$$ 1.30139e23 0.162764
$$190$$ 0 0
$$191$$ −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 $$0.528806\pi$$
$$194$$ 0 0
$$195$$ 9.27866e23 0.835837
$$196$$ 0 0
$$197$$ 1.65216e24 1.33708 0.668541 0.743675i $$-0.266919\pi$$
0.668541 + 0.743675i $$0.266919\pi$$
$$198$$ 0 0
$$199$$ 1.52537e24 1.11024 0.555120 0.831770i $$-0.312672\pi$$
0.555120 + 0.831770i $$0.312672\pi$$
$$200$$ 0 0
$$201$$ −1.66398e23 −0.109041
$$202$$ 0 0
$$203$$ 1.74876e24 1.03281
$$204$$ 0 0
$$205$$ −3.09063e24 −1.64676
$$206$$ 0 0
$$207$$ −1.56531e23 −0.0753202
$$208$$ 0 0
$$209$$ 2.50425e24 1.08928
$$210$$ 0 0
$$211$$ −1.35748e24 −0.534277 −0.267138 0.963658i $$-0.586078\pi$$
−0.267138 + 0.963658i $$0.586078\pi$$
$$212$$ 0 0
$$213$$ 5.91838e23 0.210968
$$214$$ 0 0
$$215$$ 2.65425e24 0.857705
$$216$$ 0 0
$$217$$ 5.28701e24 1.55018
$$218$$ 0 0
$$219$$ 1.02001e24 0.271606
$$220$$ 0 0
$$221$$ −6.17294e24 −1.49408
$$222$$ 0 0
$$223$$ −4.73192e24 −1.04192 −0.520962 0.853580i $$-0.674427\pi$$
−0.520962 + 0.853580i $$0.674427\pi$$
$$224$$ 0 0
$$225$$ −8.40356e22 −0.0168479
$$226$$ 0 0
$$227$$ −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 $$0.427208\pi$$
$$230$$ 0 0
$$231$$ −2.23039e24 −0.339200
$$232$$ 0 0
$$233$$ −6.60115e24 −0.917028 −0.458514 0.888687i $$-0.651618\pi$$
−0.458514 + 0.888687i $$0.651618\pi$$
$$234$$ 0 0
$$235$$ 9.11885e24 1.15805
$$236$$ 0 0
$$237$$ −1.93843e24 −0.225213
$$238$$ 0 0
$$239$$ −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 $$0.770571\pi$$
$$242$$ 0 0
$$243$$ −7.17898e23 −0.0641500
$$244$$ 0 0
$$245$$ 3.38370e24 0.277426
$$246$$ 0 0
$$247$$ −3.09477e25 −2.32975
$$248$$ 0 0
$$249$$ 1.80241e22 0.00124670
$$250$$ 0 0
$$251$$ 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
$$254$$ 0 0
$$255$$ −1.05021e25 −0.565725
$$256$$ 0 0
$$257$$ −3.06596e25 −1.52149 −0.760744 0.649052i $$-0.775166\pi$$
−0.760744 + 0.649052i $$0.775166\pi$$
$$258$$ 0 0
$$259$$ 1.12304e25 0.513746
$$260$$ 0 0
$$261$$ −9.64685e24 −0.407062
$$262$$ 0 0
$$263$$ 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
$$266$$ 0 0
$$267$$ −1.38525e25 −0.460425
$$268$$ 0 0
$$269$$ 2.29737e25 0.706043 0.353022 0.935615i $$-0.385154\pi$$
0.353022 + 0.935615i $$0.385154\pi$$
$$270$$ 0 0
$$271$$ 4.02085e25 1.14325 0.571624 0.820516i $$-0.306314\pi$$
0.571624 + 0.820516i $$0.306314\pi$$
$$272$$ 0 0
$$273$$ 2.75633e25 0.725477
$$274$$ 0 0
$$275$$ 1.44025e24 0.0351111
$$276$$ 0 0
$$277$$ −4.48722e25 −1.01377 −0.506886 0.862013i $$-0.669203\pi$$
−0.506886 + 0.862013i $$0.669203\pi$$
$$278$$ 0 0
$$279$$ −2.91653e25 −0.610973
$$280$$ 0 0
$$281$$ −6.06489e25 −1.17871 −0.589355 0.807874i $$-0.700618\pi$$
−0.589355 + 0.807874i $$0.700618\pi$$
$$282$$ 0 0
$$283$$ −1.04229e26 −1.88032 −0.940158 0.340739i $$-0.889323\pi$$
−0.940158 + 0.340739i $$0.889323\pi$$
$$284$$ 0 0
$$285$$ −5.26517e25 −0.882148
$$286$$ 0 0
$$287$$ −9.18107e25 −1.42933
$$288$$ 0 0
$$289$$ 7.76980e23 0.0112456
$$290$$ 0 0
$$291$$ 3.49621e25 0.470675
$$292$$ 0 0
$$293$$ 6.24634e25 0.782556 0.391278 0.920273i $$-0.372033\pi$$
0.391278 + 0.920273i $$0.372033\pi$$
$$294$$ 0 0
$$295$$ 3.44610e25 0.401972
$$296$$ 0 0
$$297$$ 1.23037e25 0.133689
$$298$$ 0 0
$$299$$ −3.31532e25 −0.335720
$$300$$ 0 0
$$301$$ 7.88476e25 0.744458
$$302$$ 0 0
$$303$$ 9.81568e25 0.864516
$$304$$ 0 0
$$305$$ 8.01915e25 0.659144
$$306$$ 0 0
$$307$$ −7.14411e25 −0.548271 −0.274135 0.961691i $$-0.588392\pi$$
−0.274135 + 0.961691i $$0.588392\pi$$
$$308$$ 0 0
$$309$$ 6.96297e25 0.499150
$$310$$ 0 0
$$311$$ 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 $$0.677575\pi$$
$$314$$ 0 0
$$315$$ 4.68939e25 0.274698
$$316$$ 0 0
$$317$$ −5.71691e25 −0.313357 −0.156678 0.987650i $$-0.550079\pi$$
−0.156678 + 0.987650i $$0.550079\pi$$
$$318$$ 0 0
$$319$$ 1.65333e26 0.848316
$$320$$ 0 0
$$321$$ −5.13230e25 −0.246609
$$322$$ 0 0
$$323$$ 3.50283e26 1.57686
$$324$$ 0 0
$$325$$ −1.77987e25 −0.0750954
$$326$$ 0 0
$$327$$ −2.43294e25 −0.0962454
$$328$$ 0 0
$$329$$ 2.70886e26 1.00515
$$330$$ 0 0
$$331$$ −6.67957e25 −0.232571 −0.116285 0.993216i $$-0.537099\pi$$
−0.116285 + 0.993216i $$0.537099\pi$$
$$332$$ 0 0
$$333$$ −6.19515e25 −0.202482
$$334$$ 0 0
$$335$$ −5.99597e25 −0.184030
$$336$$ 0 0
$$337$$ 3.55365e26 1.02461 0.512307 0.858802i $$-0.328791\pi$$
0.512307 + 0.858802i $$0.328791\pi$$
$$338$$ 0 0
$$339$$ −1.45227e26 −0.393504
$$340$$ 0 0
$$341$$ 4.99851e26 1.27327
$$342$$ 0 0
$$343$$ 4.53560e26 1.08654
$$344$$ 0 0
$$345$$ −5.64040e25 −0.127119
$$346$$ 0 0
$$347$$ 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 $$0.698263\pi$$
$$350$$ 0 0
$$351$$ −1.52050e26 −0.285932
$$352$$ 0 0
$$353$$ 4.96062e26 0.878823 0.439412 0.898286i $$-0.355187\pi$$
0.439412 + 0.898286i $$0.355187\pi$$
$$354$$ 0 0
$$355$$ 2.13262e26 0.356053
$$356$$ 0 0
$$357$$ −3.11978e26 −0.491029
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ 2.26109e26 0.298743
$$364$$ 0 0
$$365$$ 3.67547e26 0.458392
$$366$$ 0 0
$$367$$ −9.70198e25 −0.114253 −0.0571264 0.998367i $$-0.518194\pi$$
−0.0571264 + 0.998367i $$0.518194\pi$$
$$368$$ 0 0
$$369$$ 5.06465e26 0.563342
$$370$$ 0 0
$$371$$ 3.01511e26 0.316867
$$372$$ 0 0
$$373$$ 1.03252e27 1.02554 0.512771 0.858525i $$-0.328619\pi$$
0.512771 + 0.858525i $$0.328619\pi$$
$$374$$ 0 0
$$375$$ −6.29389e26 −0.591005
$$376$$ 0 0
$$377$$ −2.04320e27 −1.81437
$$378$$ 0 0
$$379$$ −2.20881e26 −0.185544 −0.0927721 0.995687i $$-0.529573\pi$$
−0.0927721 + 0.995687i $$0.529573\pi$$
$$380$$ 0 0
$$381$$ 3.27003e26 0.259920
$$382$$ 0 0
$$383$$ 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
$$386$$ 0 0
$$387$$ −4.34955e26 −0.293413
$$388$$ 0 0
$$389$$ −3.60474e26 −0.230358 −0.115179 0.993345i $$-0.536744\pi$$
−0.115179 + 0.993345i $$0.536744\pi$$
$$390$$ 0 0
$$391$$ 3.75247e26 0.227228
$$392$$ 0 0
$$393$$ −1.50357e27 −0.862980
$$394$$ 0 0
$$395$$ −6.98492e26 −0.380095
$$396$$ 0 0
$$397$$ −1.47191e27 −0.759595 −0.379798 0.925070i $$-0.624006\pi$$
−0.379798 + 0.925070i $$0.624006\pi$$
$$398$$ 0 0
$$399$$ −1.56408e27 −0.765674
$$400$$ 0 0
$$401$$ 1.00401e27 0.466360 0.233180 0.972434i $$-0.425087\pi$$
0.233180 + 0.972434i $$0.425087\pi$$
$$402$$ 0 0
$$403$$ −6.17719e27 −2.72325
$$404$$ 0 0
$$405$$ −2.58686e26 −0.108267
$$406$$ 0 0
$$407$$ 1.06176e27 0.421973
$$408$$ 0 0
$$409$$ −4.41838e27 −1.66789 −0.833947 0.551845i $$-0.813924\pi$$
−0.833947 + 0.551845i $$0.813924\pi$$
$$410$$ 0 0
$$411$$ 2.84668e27 1.02094
$$412$$ 0 0
$$413$$ 1.02370e27 0.348898
$$414$$ 0 0
$$415$$ 6.49477e24 0.00210406
$$416$$ 0 0
$$417$$ −1.28841e27 −0.396849
$$418$$ 0 0
$$419$$ 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 $$0.376255\pi$$
$$422$$ 0 0
$$423$$ −1.49432e27 −0.396158
$$424$$ 0 0
$$425$$ 2.01456e26 0.0508272
$$426$$ 0 0
$$427$$ 2.38218e27 0.572114
$$428$$ 0 0
$$429$$ 2.60593e27 0.595883
$$430$$ 0 0
$$431$$ −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 $$0.614712\pi$$
$$434$$ 0 0
$$435$$ −3.47613e27 −0.687003
$$436$$ 0 0
$$437$$ 1.88128e27 0.354322
$$438$$ 0 0
$$439$$ −2.29516e27 −0.412037 −0.206018 0.978548i $$-0.566051\pi$$
−0.206018 + 0.978548i $$0.566051\pi$$
$$440$$ 0 0
$$441$$ −5.54491e26 −0.0949050
$$442$$ 0 0
$$443$$ 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
$$446$$ 0 0
$$447$$ −2.85880e27 −0.424576
$$448$$ 0 0
$$449$$ 1.23324e28 1.74768 0.873839 0.486216i $$-0.161623\pi$$
0.873839 + 0.486216i $$0.161623\pi$$
$$450$$ 0 0
$$451$$ −8.68008e27 −1.17400
$$452$$ 0 0
$$453$$ −7.25995e27 −0.937351
$$454$$ 0 0
$$455$$ 9.93211e27 1.22440
$$456$$ 0 0
$$457$$ −7.62646e27 −0.897848 −0.448924 0.893570i $$-0.648193\pi$$
−0.448924 + 0.893570i $$0.648193\pi$$
$$458$$ 0 0
$$459$$ 1.72100e27 0.193529
$$460$$ 0 0
$$461$$ −9.04851e26 −0.0972114 −0.0486057 0.998818i $$-0.515478\pi$$
−0.0486057 + 0.998818i $$0.515478\pi$$
$$462$$ 0 0
$$463$$ 1.19705e27 0.122889 0.0614443 0.998111i $$-0.480429\pi$$
0.0614443 + 0.998111i $$0.480429\pi$$
$$464$$ 0 0
$$465$$ −1.05094e28 −1.03115
$$466$$ 0 0
$$467$$ 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
$$470$$ 0 0
$$471$$ −3.48288e27 −0.298689
$$472$$ 0 0
$$473$$ 7.45451e27 0.611472
$$474$$ 0 0
$$475$$ 1.00999e27 0.0792562
$$476$$ 0 0
$$477$$ −1.66326e27 −0.124887
$$478$$ 0 0
$$479$$ −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
$$482$$ 0 0
$$483$$ −1.67555e27 −0.110335
$$484$$ 0 0
$$485$$ 1.25982e28 0.794363
$$486$$ 0 0
$$487$$ 3.75693e27 0.226871 0.113436 0.993545i $$-0.463814\pi$$
0.113436 + 0.993545i $$0.463814\pi$$
$$488$$ 0 0
$$489$$ −1.87920e27 −0.108700
$$490$$ 0 0
$$491$$ 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
$$494$$ 0 0
$$495$$ 4.43350e27 0.225628
$$496$$ 0 0
$$497$$ 6.33518e27 0.309042
$$498$$ 0 0
$$499$$ −3.84836e27 −0.179978 −0.0899890 0.995943i $$-0.528683\pi$$
−0.0899890 + 0.995943i $$0.528683\pi$$
$$500$$ 0 0
$$501$$ −2.14243e27 −0.0960749
$$502$$ 0 0
$$503$$ 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
$$506$$ 0 0
$$507$$ −1.76153e28 −0.697119
$$508$$ 0 0
$$509$$ 2.09122e27 0.0794078 0.0397039 0.999211i $$-0.487359\pi$$
0.0397039 + 0.999211i $$0.487359\pi$$
$$510$$ 0 0
$$511$$ 1.09184e28 0.397869
$$512$$ 0 0
$$513$$ 8.62810e27 0.301775
$$514$$ 0 0
$$515$$ 2.50902e28 0.842420
$$516$$ 0 0
$$517$$ 2.56104e28 0.825594
$$518$$ 0 0
$$519$$ −1.89849e28 −0.587692
$$520$$ 0 0
$$521$$ −3.76486e27 −0.111932 −0.0559659 0.998433i $$-0.517824\pi$$
−0.0559659 + 0.998433i $$0.517824\pi$$
$$522$$ 0 0
$$523$$ 1.74944e28 0.499611 0.249806 0.968296i $$-0.419633\pi$$
0.249806 + 0.968296i $$0.419633\pi$$
$$524$$ 0 0
$$525$$ −8.99539e26 −0.0246801
$$526$$ 0 0
$$527$$ 6.99170e28 1.84320
$$528$$ 0 0
$$529$$ −3.74562e28 −0.948942
$$530$$ 0 0
$$531$$ −5.64716e27 −0.137511
$$532$$ 0 0
$$533$$ 1.07269e29 2.51095
$$534$$ 0 0
$$535$$ −1.84936e28 −0.416205
$$536$$ 0 0
$$537$$ −1.06535e28 −0.230548
$$538$$ 0 0
$$539$$ 9.50317e27 0.197782
$$540$$ 0 0
$$541$$ −6.99960e28 −1.40121 −0.700603 0.713552i $$-0.747085\pi$$
−0.700603 + 0.713552i $$0.747085\pi$$
$$542$$ 0 0
$$543$$ 3.75933e26 0.00723956
$$544$$ 0 0
$$545$$ −8.76680e27 −0.162434
$$546$$ 0 0
$$547$$ −3.00896e28 −0.536476 −0.268238 0.963353i $$-0.586441\pi$$
−0.268238 + 0.963353i $$0.586441\pi$$
$$548$$ 0 0
$$549$$ −1.31411e28 −0.225487
$$550$$ 0 0
$$551$$ 1.15941e29 1.91490
$$552$$ 0 0
$$553$$ −2.07495e28 −0.329909
$$554$$ 0 0
$$555$$ −2.23235e28 −0.341732
$$556$$ 0 0
$$557$$ 1.17263e29 1.72855 0.864277 0.503017i $$-0.167777\pi$$
0.864277 + 0.503017i $$0.167777\pi$$
$$558$$ 0 0
$$559$$ −9.21233e28 −1.30781
$$560$$ 0 0
$$561$$ −2.94954e28 −0.403315
$$562$$ 0 0
$$563$$ −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
$$566$$ 0 0
$$567$$ −7.68456e27 −0.0939717
$$568$$ 0 0
$$569$$ −1.35009e29 −1.59105 −0.795527 0.605918i $$-0.792806\pi$$
−0.795527 + 0.605918i $$0.792806\pi$$
$$570$$ 0 0
$$571$$ −2.55750e28 −0.290494 −0.145247 0.989395i $$-0.546398\pi$$
−0.145247 + 0.989395i $$0.546398\pi$$
$$572$$ 0 0
$$573$$ 4.82951e28 0.528786
$$574$$ 0 0
$$575$$ 1.08197e27 0.0114209
$$576$$ 0 0
$$577$$ −1.52580e29 −1.55293 −0.776465 0.630160i $$-0.782989\pi$$
−0.776465 + 0.630160i $$0.782989\pi$$
$$578$$ 0 0
$$579$$ 1.06326e28 0.104355
$$580$$ 0 0
$$581$$ 1.92935e26 0.00182625
$$582$$ 0 0
$$583$$ 2.85058e28 0.260264
$$584$$ 0 0
$$585$$ −5.47896e28 −0.482571
$$586$$ 0 0
$$587$$ 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
$$590$$ 0 0
$$591$$ −9.75587e28 −0.771965
$$592$$ 0 0
$$593$$ 8.82363e28 0.673865 0.336933 0.941529i $$-0.390611\pi$$
0.336933 + 0.941529i $$0.390611\pi$$
$$594$$ 0 0
$$595$$ −1.12417e29 −0.828715
$$596$$ 0 0
$$597$$ −9.00714e28 −0.640998
$$598$$ 0 0
$$599$$ −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 $$0.577934\pi$$
$$602$$ 0 0
$$603$$ 9.82566e27 0.0629550
$$604$$ 0 0
$$605$$ 8.14756e28 0.504192
$$606$$ 0 0
$$607$$ −8.07544e28 −0.482709 −0.241354 0.970437i $$-0.577592\pi$$
−0.241354 + 0.970437i $$0.577592\pi$$
$$608$$ 0 0
$$609$$ −1.03262e29 −0.596294
$$610$$ 0 0
$$611$$ −3.16496e29 −1.76577
$$612$$ 0 0
$$613$$ −2.21326e29 −1.19316 −0.596578 0.802555i $$-0.703473\pi$$
−0.596578 + 0.802555i $$0.703473\pi$$
$$614$$ 0 0
$$615$$ 1.82498e29 0.950759
$$616$$ 0 0
$$617$$ −2.16026e29 −1.08771 −0.543854 0.839180i $$-0.683035\pi$$
−0.543854 + 0.839180i $$0.683035\pi$$
$$618$$ 0 0
$$619$$ 2.09908e29 1.02159 0.510795 0.859702i $$-0.329351\pi$$
0.510795 + 0.859702i $$0.329351\pi$$
$$620$$ 0 0
$$621$$ 9.24299e27 0.0434861
$$622$$ 0 0
$$623$$ −1.48280e29 −0.674465
$$624$$ 0 0
$$625$$ −2.15300e29 −0.946901
$$626$$ 0 0
$$627$$ −1.47873e29 −0.628899
$$628$$ 0 0
$$629$$ 1.48514e29 0.610853
$$630$$ 0 0
$$631$$ −4.81094e28 −0.191391 −0.0956955 0.995411i $$-0.530508\pi$$
−0.0956955 + 0.995411i $$0.530508\pi$$
$$632$$ 0 0
$$633$$ 8.01576e28 0.308465
$$634$$ 0 0
$$635$$ 1.17832e29 0.438670
$$636$$ 0 0
$$637$$ −1.17441e29 −0.423014
$$638$$ 0 0
$$639$$ −3.49474e28 −0.121802
$$640$$ 0 0
$$641$$ −5.04867e28 −0.170282 −0.0851408 0.996369i $$-0.527134\pi$$
−0.0851408 + 0.996369i $$0.527134\pi$$
$$642$$ 0 0
$$643$$ 3.93930e29 1.28589 0.642945 0.765912i $$-0.277712\pi$$
0.642945 + 0.765912i $$0.277712\pi$$
$$644$$ 0 0
$$645$$ −1.56731e29 −0.495196
$$646$$ 0 0
$$647$$ 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
$$650$$ 0 0
$$651$$ −3.12193e29 −0.894998
$$652$$ 0 0
$$653$$ 1.16655e29 0.323830 0.161915 0.986805i $$-0.448233\pi$$
0.161915 + 0.986805i $$0.448233\pi$$
$$654$$ 0 0
$$655$$ −5.41792e29 −1.45646
$$656$$ 0 0
$$657$$ −6.02303e28 −0.156812
$$658$$ 0 0
$$659$$ 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 $$0.323329\pi$$
$$662$$ 0 0
$$663$$ 3.64506e29 0.862606
$$664$$ 0 0
$$665$$ −5.63598e29 −1.29224
$$666$$ 0 0
$$667$$ 1.24204e29 0.275940
$$668$$ 0 0
$$669$$ 2.79415e29 0.601555
$$670$$ 0 0
$$671$$ 2.25219e29 0.469915
$$672$$ 0 0
$$673$$ −1.19710e29 −0.242088 −0.121044 0.992647i $$-0.538624\pi$$
−0.121044 + 0.992647i $$0.538624\pi$$
$$674$$ 0 0
$$675$$ 4.96222e27 0.00972717
$$676$$ 0 0
$$677$$ 7.71503e29 1.46608 0.733038 0.680188i $$-0.238102\pi$$
0.733038 + 0.680188i $$0.238102\pi$$
$$678$$ 0 0
$$679$$ 3.74243e29 0.689479
$$680$$ 0 0
$$681$$ 4.72525e29 0.844074
$$682$$ 0 0
$$683$$ −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
$$686$$ 0 0
$$687$$ −1.60679e29 −0.261766
$$688$$ 0 0
$$689$$ −3.52277e29 −0.556650
$$690$$ 0 0
$$691$$ −1.07309e30 −1.64482 −0.822408 0.568898i $$-0.807370\pi$$
−0.822408 + 0.568898i $$0.807370\pi$$
$$692$$ 0 0
$$693$$ 1.31702e29 0.195837
$$694$$ 0 0
$$695$$ −4.64263e29 −0.669767
$$696$$ 0 0
$$697$$ −1.21413e30 −1.69950
$$698$$ 0 0
$$699$$ 3.89791e29 0.529446
$$700$$ 0 0
$$701$$ −7.18115e29 −0.946574 −0.473287 0.880908i $$-0.656933\pi$$
−0.473287 + 0.880908i $$0.656933\pi$$
$$702$$ 0 0
$$703$$ 7.44568e29 0.952518
$$704$$ 0 0
$$705$$ −5.38459e29 −0.668601
$$706$$ 0 0
$$707$$ 1.05070e30 1.26641
$$708$$ 0 0
$$709$$ 4.24360e29 0.496534 0.248267 0.968692i $$-0.420139\pi$$
0.248267 + 0.968692i $$0.420139\pi$$
$$710$$ 0 0
$$711$$ 1.14463e29 0.130027
$$712$$ 0 0
$$713$$ 3.75505e29 0.414167
$$714$$ 0 0
$$715$$ 9.39014e29 1.00568
$$716$$ 0 0
$$717$$ 5.85898e29 0.609356
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ 9.10400e29 0.867521
$$724$$ 0 0
$$725$$ 6.66805e28 0.0617234
$$726$$ 0 0
$$727$$ −1.42987e30 −1.28583 −0.642916 0.765937i $$-0.722275\pi$$
−0.642916 + 0.765937i $$0.722275\pi$$
$$728$$ 0 0
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ 1.04270e30 0.885175
$$732$$ 0 0
$$733$$ 1.99143e30 1.64276 0.821380 0.570381i $$-0.193205\pi$$
0.821380 + 0.570381i $$0.193205\pi$$
$$734$$ 0 0
$$735$$ −1.99804e29 −0.160172
$$736$$ 0 0
$$737$$ −1.68398e29 −0.131198
$$738$$ 0 0
$$739$$ 2.11124e30 1.59872 0.799359 0.600854i $$-0.205173\pi$$
0.799359 + 0.600854i $$0.205173\pi$$
$$740$$ 0 0
$$741$$ 1.82743e30 1.34508
$$742$$ 0 0
$$743$$ −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
$$746$$ 0 0
$$747$$ −1.06431e27 −0.000719781 0
$$748$$ 0 0
$$749$$ −5.49375e29 −0.361251
$$750$$ 0 0
$$751$$ 2.28371e30 1.46023 0.730116 0.683323i $$-0.239466\pi$$
0.730116 + 0.683323i $$0.239466\pi$$
$$752$$ 0 0
$$753$$ −5.90546e29 −0.367203
$$754$$ 0 0
$$755$$ −2.61603e30 −1.58198
$$756$$ 0 0
$$757$$ 9.73411e29 0.572519 0.286259 0.958152i $$-0.407588\pi$$
0.286259 + 0.958152i $$0.407588\pi$$
$$758$$ 0 0
$$759$$ −1.58412e29 −0.0906251
$$760$$ 0 0
$$761$$ −3.06985e30 −1.70836 −0.854179 0.519979i $$-0.825940\pi$$
−0.854179 + 0.519979i $$0.825940\pi$$
$$762$$ 0 0
$$763$$ −2.60428e29 −0.140987
$$764$$ 0 0
$$765$$ 6.20140e29 0.326621
$$766$$ 0 0
$$767$$ −1.19607e30 −0.612920
$$768$$ 0 0
$$769$$ −7.51816e29 −0.374874 −0.187437 0.982277i $$-0.560018\pi$$
−0.187437 + 0.982277i $$0.560018\pi$$
$$770$$ 0 0
$$771$$ 1.81042e30 0.878432
$$772$$ 0 0
$$773$$ 1.23771e30 0.584434 0.292217 0.956352i $$-0.405607\pi$$
0.292217 + 0.956352i $$0.405607\pi$$
$$774$$ 0 0
$$775$$ 2.01595e29 0.0926427
$$776$$ 0 0
$$777$$ −6.63144e29 −0.296611
$$778$$ 0 0
$$779$$ −6.08698e30 −2.65008
$$780$$ 0 0
$$781$$ 5.98949e29 0.253836
$$782$$ 0 0
$$783$$ 5.69637e29 0.235017
$$784$$ 0 0
$$785$$ −1.25501e30 −0.504100
$$786$$ 0 0
$$787$$ −2.14352e30 −0.838288 −0.419144 0.907920i $$-0.637670\pi$$
−0.419144 + 0.907920i $$0.637670\pi$$
$$788$$ 0 0
$$789$$ −1.37853e30 −0.524936
$$790$$ 0 0
$$791$$ −1.55454e30 −0.576434
$$792$$ 0 0
$$793$$ −2.78327e30 −1.00505
$$794$$ 0 0
$$795$$ −5.99334e29 −0.210773
$$796$$ 0 0
$$797$$ 3.01280e30 1.03195 0.515974 0.856605i $$-0.327430\pi$$
0.515974 + 0.856605i $$0.327430\pi$$
$$798$$ 0 0
$$799$$ 3.58228e30 1.19514
$$800$$ 0 0
$$801$$ 8.17975e29 0.265827
$$802$$ 0 0
$$803$$ 1.03226e30 0.326796
$$804$$ 0 0
$$805$$ −6.03763e29 −0.186213
$$806$$ 0 0
$$807$$ −1.35657e30 −0.407634
$$808$$ 0 0
$$809$$ −5.76883e30 −1.68900 −0.844498 0.535559i $$-0.820101\pi$$
−0.844498 + 0.535559i $$0.820101\pi$$
$$810$$ 0 0
$$811$$ −3.98728e30 −1.13752 −0.568759 0.822504i $$-0.692576\pi$$
−0.568759 + 0.822504i $$0.692576\pi$$
$$812$$ 0 0
$$813$$ −2.37427e30 −0.660055
$$814$$ 0 0
$$815$$ −6.77146e29 −0.183454
$$816$$ 0 0
$$817$$ 5.22754e30 1.38027
$$818$$ 0 0
$$819$$ −1.62759e30 −0.418855
$$820$$ 0 0
$$821$$ −4.09615e30 −1.02748 −0.513739 0.857946i $$-0.671740\pi$$
−0.513739 + 0.857946i $$0.671740\pi$$
$$822$$ 0 0
$$823$$ −1.43384e30 −0.350592 −0.175296 0.984516i $$-0.556088\pi$$
−0.175296 + 0.984516i $$0.556088\pi$$
$$824$$ 0 0
$$825$$ −8.50453e28 −0.0202714
$$826$$ 0 0
$$827$$ 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 $$0.667892\pi$$
$$830$$ 0 0
$$831$$ 2.64966e30 0.585301
$$832$$ 0 0
$$833$$ 1.32926e30 0.286311
$$834$$ 0 0
$$835$$ −7.71998e29 −0.162147
$$836$$ 0 0
$$837$$ 1.72218e30 0.352745
$$838$$ 0 0
$$839$$ 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
$$842$$ 0 0
$$843$$ 3.58126e30 0.680528
$$844$$ 0 0
$$845$$ −6.34746e30 −1.17653
$$846$$ 0 0
$$847$$ 2.42033e30 0.437621
$$848$$ 0 0
$$849$$ 6.15462e30 1.08560
$$850$$ 0 0
$$851$$ 7.97630e29 0.137259
$$852$$ 0 0
$$853$$ −2.27471e30 −0.381911 −0.190955 0.981599i $$-0.561159\pi$$
−0.190955 + 0.981599i $$0.561159\pi$$
$$854$$ 0 0
$$855$$ 3.10903e30 0.509309
$$856$$ 0 0
$$857$$ 3.05003e30 0.487534 0.243767 0.969834i $$-0.421617\pi$$
0.243767 + 0.969834i $$0.421617\pi$$
$$858$$ 0 0
$$859$$ −1.34073e30 −0.209128 −0.104564 0.994518i $$-0.533345\pi$$
−0.104564 + 0.994518i $$0.533345\pi$$
$$860$$ 0 0
$$861$$ 5.42133e30 0.825225
$$862$$ 0 0
$$863$$ −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
$$866$$ 0 0
$$867$$ −4.58799e28 −0.00649265
$$868$$ 0 0
$$869$$ −1.96173e30 −0.270976
$$870$$ 0 0
$$871$$ 2.08107e30 0.280606
$$872$$ 0 0
$$873$$ −2.06448e30 −0.271744
$$874$$ 0 0
$$875$$ −6.73714e30 −0.865747
$$876$$ 0 0
$$877$$ 9.06320e30 1.13707 0.568533 0.822660i $$-0.307511\pi$$
0.568533 + 0.822660i $$0.307511\pi$$
$$878$$ 0 0
$$879$$ −3.68840e30 −0.451809
$$880$$ 0 0
$$881$$ −6.50767e30 −0.778356 −0.389178 0.921163i $$-0.627241\pi$$
−0.389178 + 0.921163i $$0.627241\pi$$
$$882$$ 0 0
$$883$$ −5.90742e30 −0.689939 −0.344969 0.938614i $$-0.612111\pi$$
−0.344969 + 0.938614i $$0.612111\pi$$
$$884$$ 0 0
$$885$$ −2.03489e30 −0.232079
$$886$$ 0 0
$$887$$ 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
$$890$$ 0 0
$$891$$ −7.26524e29 −0.0771852
$$892$$ 0 0
$$893$$ 1.79595e31 1.86361
$$894$$ 0 0
$$895$$ −3.83886e30 −0.389099
$$896$$ 0 0
$$897$$ 1.95766e30 0.193828
$$898$$ 0 0
$$899$$ 2.31420e31 2.23833
$$900$$ 0 0
$$901$$ 3.98728e30 0.376761
$$902$$ 0 0
$$903$$ −4.65587e30 −0.429813
$$904$$ 0 0
$$905$$ 1.35463e29 0.0122183
$$906$$ 0 0
$$907$$ 1.16251e31 1.02452 0.512262 0.858829i $$-0.328808\pi$$
0.512262 + 0.858829i $$0.328808\pi$$
$$908$$ 0 0
$$909$$ −5.79606e30 −0.499128
$$910$$ 0 0
$$911$$ 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
$$914$$ 0 0
$$915$$ −4.73523e30 −0.380557
$$916$$ 0 0
$$917$$ −1.60945e31 −1.26416
$$918$$ 0 0
$$919$$ 8.55562e30 0.656808 0.328404 0.944537i $$-0.393489\pi$$
0.328404 + 0.944537i $$0.393489\pi$$
$$920$$ 0 0
$$921$$ 4.21853e30 0.316544
$$922$$ 0 0
$$923$$ −7.40185e30 −0.542903
$$924$$ 0 0
$$925$$ 4.28218e29 0.0307027
$$926$$ 0 0
$$927$$ −4.11157e30 −0.288184
$$928$$ 0 0
$$929$$ 9.93712e30 0.680919 0.340460 0.940259i $$-0.389417\pi$$
0.340460 + 0.940259i $$0.389417\pi$$
$$930$$ 0 0
$$931$$ 6.66418e30 0.446452
$$932$$ 0 0
$$933$$ −3.55425e30 −0.232804
$$934$$ 0 0
$$935$$ −1.06283e31 −0.680679
$$936$$ 0 0
$$937$$ −2.14272e30 −0.134183 −0.0670917 0.997747i $$-0.521372\pi$$
−0.0670917 + 0.997747i $$0.521372\pi$$
$$938$$ 0 0
$$939$$ 9.98217e30 0.611275
$$940$$ 0 0
$$941$$ 1.69148e31 1.01292 0.506459 0.862264i $$-0.330954\pi$$
0.506459 + 0.862264i $$0.330954\pi$$
$$942$$ 0 0
$$943$$ −6.52077e30 −0.381879
$$944$$ 0 0
$$945$$ −2.76904e30 −0.158597
$$946$$ 0 0
$$947$$ 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
$$950$$ 0 0
$$951$$ 3.37578e30 0.180917
$$952$$ 0 0
$$953$$ 2.39001e31 1.25292 0.626462 0.779452i $$-0.284502\pi$$
0.626462 + 0.779452i $$0.284502\pi$$
$$954$$ 0 0
$$955$$ 1.74026e31 0.892437
$$956$$ 0 0
$$957$$ −9.76276e30 −0.489776
$$958$$ 0 0
$$959$$ 3.04716e31 1.49555
$$960$$ 0 0
$$961$$ 4.91397e31 2.35959
$$962$$ 0 0
$$963$$ 3.03057e30 0.142380
$$964$$ 0 0
$$965$$ 3.83132e30 0.176121
$$966$$ 0 0
$$967$$ −1.63072e31 −0.733504 −0.366752 0.930319i $$-0.619530\pi$$
−0.366752 + 0.930319i $$0.619530\pi$$
$$968$$ 0 0
$$969$$ −2.06839e31 −0.910401
$$970$$ 0 0
$$971$$ −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
$$974$$ 0 0
$$975$$ 1.05100e30 0.0433563
$$976$$ 0 0
$$977$$ 1.98044e31 0.799592 0.399796 0.916604i $$-0.369081\pi$$
0.399796 + 0.916604i $$0.369081\pi$$
$$978$$ 0 0
$$979$$ −1.40189e31 −0.553982
$$980$$ 0 0
$$981$$ 1.43663e30 0.0555673
$$982$$ 0 0
$$983$$ 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
$$986$$ 0 0
$$987$$ −1.59955e31 −0.580322
$$988$$ 0 0
$$989$$ 5.60008e30 0.198899
$$990$$ 0 0
$$991$$ −4.70795e30 −0.163704 −0.0818518 0.996645i $$-0.526083\pi$$
−0.0818518 + 0.996645i $$0.526083\pi$$
$$992$$ 0 0
$$993$$ 3.94422e30 0.134275
$$994$$ 0 0
$$995$$ −3.24561e31 −1.08182
$$996$$ 0 0
$$997$$ −1.58934e31 −0.518700 −0.259350 0.965783i $$-0.583508\pi$$
−0.259350 + 0.965783i $$0.583508\pi$$
$$998$$ 0 0
$$999$$ 3.65817e30 0.116903
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 48.22.a.g.1.1 2
4.3 odd 2 3.22.a.c.1.1 2
12.11 even 2 9.22.a.e.1.2 2
20.3 even 4 75.22.b.d.49.3 4
20.7 even 4 75.22.b.d.49.2 4
20.19 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 4.3 odd 2
9.22.a.e.1.2 2 12.11 even 2
48.22.a.g.1.1 2 1.1 even 1 trivial
75.22.a.d.1.2 2 20.19 odd 2
75.22.b.d.49.2 4 20.7 even 4
75.22.b.d.49.3 4 20.3 even 4