# Properties

 Label 575.2.a.a.1.1 Level $575$ Weight $2$ Character 575.1 Self dual yes Analytic conductor $4.591$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 575.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: $$4.59139811622$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 575.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +5.00000 q^{17} -2.00000 q^{18} +8.00000 q^{19} +2.00000 q^{21} +1.00000 q^{23} +4.00000 q^{26} +4.00000 q^{27} -2.00000 q^{28} -5.00000 q^{29} -5.00000 q^{31} +8.00000 q^{32} -10.0000 q^{34} +2.00000 q^{36} -7.00000 q^{37} -16.0000 q^{38} +4.00000 q^{39} -7.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -2.00000 q^{46} +2.00000 q^{47} +8.00000 q^{48} -6.00000 q^{49} -10.0000 q^{51} -4.00000 q^{52} +1.00000 q^{53} -8.00000 q^{54} -16.0000 q^{57} +10.0000 q^{58} +3.00000 q^{59} -6.00000 q^{61} +10.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -13.0000 q^{67} +10.0000 q^{68} -2.00000 q^{69} +13.0000 q^{71} -8.00000 q^{73} +14.0000 q^{74} +16.0000 q^{76} -8.00000 q^{78} -14.0000 q^{79} -11.0000 q^{81} +14.0000 q^{82} +3.00000 q^{83} +4.00000 q^{84} +8.00000 q^{86} +10.0000 q^{87} -14.0000 q^{89} +2.00000 q^{91} +2.00000 q^{92} +10.0000 q^{93} -4.00000 q^{94} -16.0000 q^{96} -14.0000 q^{97} +12.0000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 4.00000 1.63299
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −4.00000 −1.15470
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 4.00000 0.769800
$$28$$ −2.00000 −0.377964
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 0 0
$$34$$ −10.0000 −1.71499
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −16.0000 −2.59554
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −7.00000 −1.09322 −0.546608 0.837389i $$-0.684081\pi$$
−0.546608 + 0.837389i $$0.684081\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 8.00000 1.15470
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −10.0000 −1.40028
$$52$$ −4.00000 −0.554700
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ −8.00000 −1.08866
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −16.0000 −2.11925
$$58$$ 10.0000 1.31306
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 10.0000 1.27000
$$63$$ −1.00000 −0.125988
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.0000 −1.58820 −0.794101 0.607785i $$-0.792058\pi$$
−0.794101 + 0.607785i $$0.792058\pi$$
$$68$$ 10.0000 1.21268
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ 13.0000 1.54282 0.771408 0.636341i $$-0.219553\pi$$
0.771408 + 0.636341i $$0.219553\pi$$
$$72$$ 0 0
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ 14.0000 1.62747
$$75$$ 0 0
$$76$$ 16.0000 1.83533
$$77$$ 0 0
$$78$$ −8.00000 −0.905822
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 14.0000 1.54604
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 2.00000 0.208514
$$93$$ 10.0000 1.03695
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ −16.0000 −1.63299
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 12.0000 1.21218
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 20.0000 1.98030
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 8.00000 0.769800
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 14.0000 1.32882
$$112$$ 4.00000 0.377964
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ 32.0000 2.99707
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ −2.00000 −0.184900
$$118$$ −6.00000 −0.552345
$$119$$ −5.00000 −0.458349
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 12.0000 1.08643
$$123$$ 14.0000 1.26234
$$124$$ −10.0000 −0.898027
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 26.0000 2.24606
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 9.00000 0.763370 0.381685 0.924292i $$-0.375344\pi$$
0.381685 + 0.924292i $$0.375344\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ −26.0000 −2.18187
$$143$$ 0 0
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ 16.0000 1.32417
$$147$$ 12.0000 0.989743
$$148$$ −14.0000 −1.15079
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 5.00000 0.404226
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 8.00000 0.640513
$$157$$ 3.00000 0.239426 0.119713 0.992809i $$-0.461803\pi$$
0.119713 + 0.992809i $$0.461803\pi$$
$$158$$ 28.0000 2.22756
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 22.0000 1.72848
$$163$$ −24.0000 −1.87983 −0.939913 0.341415i $$-0.889094\pi$$
−0.939913 + 0.341415i $$0.889094\pi$$
$$164$$ −14.0000 −1.09322
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −20.0000 −1.51620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 28.0000 2.09869
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 12.0000 0.887066
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −20.0000 −1.46647
$$187$$ 0 0
$$188$$ 4.00000 0.291730
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 16.0000 1.15470
$$193$$ 12.0000 0.863779 0.431889 0.901927i $$-0.357847\pi$$
0.431889 + 0.901927i $$0.357847\pi$$
$$194$$ 28.0000 2.01028
$$195$$ 0 0
$$196$$ −12.0000 −0.857143
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ 26.0000 1.83390
$$202$$ −30.0000 −2.11079
$$203$$ 5.00000 0.350931
$$204$$ −20.0000 −1.40028
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 8.00000 0.554700
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −9.00000 −0.619586 −0.309793 0.950804i $$-0.600260\pi$$
−0.309793 + 0.950804i $$0.600260\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −26.0000 −1.78149
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ −36.0000 −2.43823
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ −10.0000 −0.672673
$$222$$ −28.0000 −1.87924
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ −8.00000 −0.534522
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ −32.0000 −2.11925
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 28.0000 1.81880
$$238$$ 10.0000 0.648204
$$239$$ −1.00000 −0.0646846 −0.0323423 0.999477i $$-0.510297\pi$$
−0.0323423 + 0.999477i $$0.510297\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 22.0000 1.41421
$$243$$ 10.0000 0.641500
$$244$$ −12.0000 −0.768221
$$245$$ 0 0
$$246$$ −28.0000 −1.78521
$$247$$ −16.0000 −1.01806
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ −40.0000 −2.50982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ −16.0000 −0.996116
$$259$$ 7.00000 0.434959
$$260$$ 0 0
$$261$$ −5.00000 −0.309492
$$262$$ 0 0
$$263$$ 13.0000 0.801614 0.400807 0.916162i $$-0.368730\pi$$
0.400807 + 0.916162i $$0.368730\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 28.0000 1.71357
$$268$$ −26.0000 −1.58820
$$269$$ −15.0000 −0.914566 −0.457283 0.889321i $$-0.651177\pi$$
−0.457283 + 0.889321i $$0.651177\pi$$
$$270$$ 0 0
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ −20.0000 −1.21268
$$273$$ −4.00000 −0.242091
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −18.0000 −1.07957
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 8.00000 0.476393
$$283$$ −11.0000 −0.653882 −0.326941 0.945045i $$-0.606018\pi$$
−0.326941 + 0.945045i $$0.606018\pi$$
$$284$$ 26.0000 1.54282
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 7.00000 0.413197
$$288$$ 8.00000 0.471405
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 28.0000 1.64139
$$292$$ −16.0000 −0.936329
$$293$$ −29.0000 −1.69420 −0.847099 0.531435i $$-0.821653\pi$$
−0.847099 + 0.531435i $$0.821653\pi$$
$$294$$ −24.0000 −1.39971
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ −16.0000 −0.920697
$$303$$ −30.0000 −1.72345
$$304$$ −32.0000 −1.83533
$$305$$ 0 0
$$306$$ −10.0000 −0.571662
$$307$$ −14.0000 −0.799022 −0.399511 0.916728i $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ −21.0000 −1.18699 −0.593495 0.804838i $$-0.702252\pi$$
−0.593495 + 0.804838i $$0.702252\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −28.0000 −1.57512
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ 4.00000 0.224309
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ 2.00000 0.111456
$$323$$ 40.0000 2.22566
$$324$$ −22.0000 −1.22222
$$325$$ 0 0
$$326$$ 48.0000 2.65847
$$327$$ −36.0000 −1.99080
$$328$$ 0 0
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ −5.00000 −0.274825 −0.137412 0.990514i $$-0.543879\pi$$
−0.137412 + 0.990514i $$0.543879\pi$$
$$332$$ 6.00000 0.329293
$$333$$ −7.00000 −0.383598
$$334$$ 32.0000 1.75096
$$335$$ 0 0
$$336$$ −8.00000 −0.436436
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 18.0000 0.979071
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −16.0000 −0.865181
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ −8.00000 −0.429463 −0.214731 0.976673i $$-0.568888\pi$$
−0.214731 + 0.976673i $$0.568888\pi$$
$$348$$ 20.0000 1.07211
$$349$$ 23.0000 1.23116 0.615581 0.788074i $$-0.288921\pi$$
0.615581 + 0.788074i $$0.288921\pi$$
$$350$$ 0 0
$$351$$ −8.00000 −0.427008
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −28.0000 −1.48400
$$357$$ 10.0000 0.529256
$$358$$ 8.00000 0.422813
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 28.0000 1.47165
$$363$$ 22.0000 1.15470
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ −24.0000 −1.25450
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −7.00000 −0.364405
$$370$$ 0 0
$$371$$ −1.00000 −0.0519174
$$372$$ 20.0000 1.03695
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.0000 0.515026
$$378$$ 8.00000 0.411476
$$379$$ 6.00000 0.308199 0.154100 0.988055i $$-0.450752\pi$$
0.154100 + 0.988055i $$0.450752\pi$$
$$380$$ 0 0
$$381$$ −40.0000 −2.04926
$$382$$ 16.0000 0.818631
$$383$$ −3.00000 −0.153293 −0.0766464 0.997058i $$-0.524421\pi$$
−0.0766464 + 0.997058i $$0.524421\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ −4.00000 −0.203331
$$388$$ −28.0000 −1.42148
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ 0 0
$$391$$ 5.00000 0.252861
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ −52.0000 −2.59352
$$403$$ 10.0000 0.498135
$$404$$ 30.0000 1.49256
$$405$$ 0 0
$$406$$ −10.0000 −0.496292
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ −3.00000 −0.147620
$$414$$ −2.00000 −0.0982946
$$415$$ 0 0
$$416$$ −16.0000 −0.784465
$$417$$ −18.0000 −0.881464
$$418$$ 0 0
$$419$$ −2.00000 −0.0977064 −0.0488532 0.998806i $$-0.515557\pi$$
−0.0488532 + 0.998806i $$0.515557\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 18.0000 0.876226
$$423$$ 2.00000 0.0972433
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 52.0000 2.51941
$$427$$ 6.00000 0.290360
$$428$$ −18.0000 −0.870063
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ −16.0000 −0.769800
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 0 0
$$436$$ 36.0000 1.72409
$$437$$ 8.00000 0.382692
$$438$$ −32.0000 −1.52902
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 20.0000 0.951303
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 28.0000 1.32882
$$445$$ 0 0
$$446$$ −28.0000 −1.32584
$$447$$ 0 0
$$448$$ 8.00000 0.377964
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 2.00000 0.0940721
$$453$$ −16.0000 −0.751746
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.00000 −0.0467780 −0.0233890 0.999726i $$-0.507446\pi$$
−0.0233890 + 0.999726i $$0.507446\pi$$
$$458$$ −8.00000 −0.373815
$$459$$ 20.0000 0.933520
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 20.0000 0.928477
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ 13.0000 0.600284
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −56.0000 −2.57217
$$475$$ 0 0
$$476$$ −10.0000 −0.458349
$$477$$ 1.00000 0.0457869
$$478$$ 2.00000 0.0914779
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 14.0000 0.638345
$$482$$ −12.0000 −0.546585
$$483$$ 2.00000 0.0910032
$$484$$ −22.0000 −1.00000
$$485$$ 0 0
$$486$$ −20.0000 −0.907218
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 48.0000 2.17064
$$490$$ 0 0
$$491$$ −31.0000 −1.39901 −0.699505 0.714628i $$-0.746596\pi$$
−0.699505 + 0.714628i $$0.746596\pi$$
$$492$$ 28.0000 1.26234
$$493$$ −25.0000 −1.12594
$$494$$ 32.0000 1.43975
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ −13.0000 −0.583130
$$498$$ 12.0000 0.537733
$$499$$ −11.0000 −0.492428 −0.246214 0.969216i $$-0.579187\pi$$
−0.246214 + 0.969216i $$0.579187\pi$$
$$500$$ 0 0
$$501$$ 32.0000 1.42965
$$502$$ −36.0000 −1.60676
$$503$$ 9.00000 0.401290 0.200645 0.979664i $$-0.435696\pi$$
0.200645 + 0.979664i $$0.435696\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 18.0000 0.799408
$$508$$ 40.0000 1.77471
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ −32.0000 −1.41421
$$513$$ 32.0000 1.41283
$$514$$ 44.0000 1.94076
$$515$$ 0 0
$$516$$ 16.0000 0.704361
$$517$$ 0 0
$$518$$ −14.0000 −0.615125
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −20.0000 −0.876216 −0.438108 0.898922i $$-0.644351\pi$$
−0.438108 + 0.898922i $$0.644351\pi$$
$$522$$ 10.0000 0.437688
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −26.0000 −1.13365
$$527$$ −25.0000 −1.08902
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 3.00000 0.130189
$$532$$ −16.0000 −0.693688
$$533$$ 14.0000 0.606407
$$534$$ −56.0000 −2.42336
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 8.00000 0.345225
$$538$$ 30.0000 1.29339
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 30.0000 1.28861
$$543$$ 28.0000 1.20160
$$544$$ 40.0000 1.71499
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −40.0000 −1.70406
$$552$$ 0 0
$$553$$ 14.0000 0.595341
$$554$$ −52.0000 −2.20927
$$555$$ 0 0
$$556$$ 18.0000 0.763370
$$557$$ 33.0000 1.39825 0.699127 0.714997i $$-0.253572\pi$$
0.699127 + 0.714997i $$0.253572\pi$$
$$558$$ 10.0000 0.423334
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 24.0000 1.01238
$$563$$ 25.0000 1.05362 0.526812 0.849982i $$-0.323387\pi$$
0.526812 + 0.849982i $$0.323387\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ 11.0000 0.461957
$$568$$ 0 0
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 0 0
$$571$$ 18.0000 0.753277 0.376638 0.926360i $$-0.377080\pi$$
0.376638 + 0.926360i $$0.377080\pi$$
$$572$$ 0 0
$$573$$ 16.0000 0.668410
$$574$$ −14.0000 −0.584349
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ −16.0000 −0.665512
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ −56.0000 −2.32127
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 58.0000 2.39596
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 24.0000 0.989743
$$589$$ −40.0000 −1.64817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 28.0000 1.15079
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 4.00000 0.163572
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ 0 0
$$601$$ −25.0000 −1.01977 −0.509886 0.860242i $$-0.670312\pi$$
−0.509886 + 0.860242i $$0.670312\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −13.0000 −0.529401
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 60.0000 2.43733
$$607$$ 38.0000 1.54237 0.771186 0.636610i $$-0.219664\pi$$
0.771186 + 0.636610i $$0.219664\pi$$
$$608$$ 64.0000 2.59554
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ −4.00000 −0.161823
$$612$$ 10.0000 0.404226
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 47.0000 1.89215 0.946074 0.323949i $$-0.105011\pi$$
0.946074 + 0.323949i $$0.105011\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −8.00000 −0.320771
$$623$$ 14.0000 0.560898
$$624$$ −16.0000 −0.640513
$$625$$ 0 0
$$626$$ 42.0000 1.67866
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ −35.0000 −1.39554
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ 18.0000 0.715436
$$634$$ −48.0000 −1.90632
$$635$$ 0 0
$$636$$ −4.00000 −0.158610
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ 13.0000 0.514272
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ −36.0000 −1.42081
$$643$$ −37.0000 −1.45914 −0.729569 0.683907i $$-0.760279\pi$$
−0.729569 + 0.683907i $$0.760279\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ 0 0
$$646$$ −80.0000 −3.14756
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ −48.0000 −1.87983
$$653$$ −2.00000 −0.0782660 −0.0391330 0.999234i $$-0.512460\pi$$
−0.0391330 + 0.999234i $$0.512460\pi$$
$$654$$ 72.0000 2.81542
$$655$$ 0 0
$$656$$ 28.0000 1.09322
$$657$$ −8.00000 −0.312110
$$658$$ 4.00000 0.155936
$$659$$ 22.0000 0.856998 0.428499 0.903542i $$-0.359042\pi$$
0.428499 + 0.903542i $$0.359042\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 20.0000 0.776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 14.0000 0.542489
$$667$$ −5.00000 −0.193601
$$668$$ −32.0000 −1.23812
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 16.0000 0.617213
$$673$$ −24.0000 −0.925132 −0.462566 0.886585i $$-0.653071\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ 52.0000 2.00297
$$675$$ 0 0
$$676$$ −18.0000 −0.692308
$$677$$ 3.00000 0.115299 0.0576497 0.998337i $$-0.481639\pi$$
0.0576497 + 0.998337i $$0.481639\pi$$
$$678$$ 4.00000 0.153619
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −46.0000 −1.76014 −0.880071 0.474843i $$-0.842505\pi$$
−0.880071 + 0.474843i $$0.842505\pi$$
$$684$$ 16.0000 0.611775
$$685$$ 0 0
$$686$$ −26.0000 −0.992685
$$687$$ −8.00000 −0.305219
$$688$$ 16.0000 0.609994
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 16.0000 0.607352
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −35.0000 −1.32572
$$698$$ −46.0000 −1.74113
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ −28.0000 −1.05755 −0.528773 0.848763i $$-0.677348\pi$$
−0.528773 + 0.848763i $$0.677348\pi$$
$$702$$ 16.0000 0.603881
$$703$$ −56.0000 −2.11208
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −15.0000 −0.564133
$$708$$ −12.0000 −0.450988
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ −14.0000 −0.525041
$$712$$ 0 0
$$713$$ −5.00000 −0.187251
$$714$$ −20.0000 −0.748481
$$715$$ 0 0
$$716$$ −8.00000 −0.298974
$$717$$ 2.00000 0.0746914
$$718$$ 12.0000 0.447836
$$719$$ −45.0000 −1.67822 −0.839108 0.543964i $$-0.816923\pi$$
−0.839108 + 0.543964i $$0.816923\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −90.0000 −3.34945
$$723$$ −12.0000 −0.446285
$$724$$ −28.0000 −1.04061
$$725$$ 0 0
$$726$$ −44.0000 −1.63299
$$727$$ −27.0000 −1.00137 −0.500687 0.865628i $$-0.666919\pi$$
−0.500687 + 0.865628i $$0.666919\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 24.0000 0.887066
$$733$$ −7.00000 −0.258551 −0.129275 0.991609i $$-0.541265\pi$$
−0.129275 + 0.991609i $$0.541265\pi$$
$$734$$ −26.0000 −0.959678
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ 14.0000 0.515347
$$739$$ 7.00000 0.257499 0.128750 0.991677i $$-0.458904\pi$$
0.128750 + 0.991677i $$0.458904\pi$$
$$740$$ 0 0
$$741$$ 32.0000 1.17555
$$742$$ 2.00000 0.0734223
$$743$$ 32.0000 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −12.0000 −0.439351
$$747$$ 3.00000 0.109764
$$748$$ 0 0
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −36.0000 −1.31191
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ −8.00000 −0.290957
$$757$$ 35.0000 1.27210 0.636048 0.771649i $$-0.280568\pi$$
0.636048 + 0.771649i $$0.280568\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 25.0000 0.906249 0.453125 0.891447i $$-0.350309\pi$$
0.453125 + 0.891447i $$0.350309\pi$$
$$762$$ 80.0000 2.89809
$$763$$ −18.0000 −0.651644
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 6.00000 0.216789
$$767$$ −6.00000 −0.216647
$$768$$ −32.0000 −1.15470
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 44.0000 1.58462
$$772$$ 24.0000 0.863779
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −14.0000 −0.502247
$$778$$ −44.0000 −1.57748
$$779$$ −56.0000 −2.00641
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −10.0000 −0.357599
$$783$$ −20.0000 −0.714742
$$784$$ 24.0000 0.857143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −17.0000 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$788$$ 0 0
$$789$$ −26.0000 −0.925625
$$790$$ 0 0
$$791$$ −1.00000 −0.0355559
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 32.0000 1.13564
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ 15.0000 0.531327 0.265664 0.964066i $$-0.414409\pi$$
0.265664 + 0.964066i $$0.414409\pi$$
$$798$$ −32.0000 −1.13279
$$799$$ 10.0000 0.353775
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ −4.00000 −0.141245
$$803$$ 0 0
$$804$$ 52.0000 1.83390
$$805$$ 0 0
$$806$$ −20.0000 −0.704470
$$807$$ 30.0000 1.05605
$$808$$ 0 0
$$809$$ −25.0000 −0.878953 −0.439477 0.898254i $$-0.644836\pi$$
−0.439477 + 0.898254i $$0.644836\pi$$
$$810$$ 0 0
$$811$$ 31.0000 1.08856 0.544279 0.838905i $$-0.316803\pi$$
0.544279 + 0.838905i $$0.316803\pi$$
$$812$$ 10.0000 0.350931
$$813$$ 30.0000 1.05215
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 40.0000 1.40028
$$817$$ −32.0000 −1.11954
$$818$$ 38.0000 1.32864
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ −24.0000 −0.837096
$$823$$ 34.0000 1.18517 0.592583 0.805510i $$-0.298108\pi$$
0.592583 + 0.805510i $$0.298108\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ 9.00000 0.312961 0.156480 0.987681i $$-0.449985\pi$$
0.156480 + 0.987681i $$0.449985\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ 37.0000 1.28506 0.642532 0.766259i $$-0.277884\pi$$
0.642532 + 0.766259i $$0.277884\pi$$
$$830$$ 0 0
$$831$$ −52.0000 −1.80386
$$832$$ 16.0000 0.554700
$$833$$ −30.0000 −1.03944
$$834$$ 36.0000 1.24658
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −20.0000 −0.691301
$$838$$ 4.00000 0.138178
$$839$$ 14.0000 0.483334 0.241667 0.970359i $$-0.422306\pi$$
0.241667 + 0.970359i $$0.422306\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −20.0000 −0.689246
$$843$$ 24.0000 0.826604
$$844$$ −18.0000 −0.619586
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ 11.0000 0.377964
$$848$$ −4.00000 −0.137361
$$849$$ 22.0000 0.755038
$$850$$ 0 0
$$851$$ −7.00000 −0.239957
$$852$$ −52.0000 −1.78149
$$853$$ −8.00000 −0.273915 −0.136957 0.990577i $$-0.543732\pi$$
−0.136957 + 0.990577i $$0.543732\pi$$
$$854$$ −12.0000 −0.410632
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −24.0000 −0.819824 −0.409912 0.912125i $$-0.634441\pi$$
−0.409912 + 0.912125i $$0.634441\pi$$
$$858$$ 0 0
$$859$$ 43.0000 1.46714 0.733571 0.679613i $$-0.237852\pi$$
0.733571 + 0.679613i $$0.237852\pi$$
$$860$$ 0 0
$$861$$ −14.0000 −0.477119
$$862$$ 24.0000 0.817443
$$863$$ 18.0000 0.612727 0.306364 0.951915i $$-0.400888\pi$$
0.306364 + 0.951915i $$0.400888\pi$$
$$864$$ 32.0000 1.08866
$$865$$ 0 0
$$866$$ 38.0000 1.29129
$$867$$ −16.0000 −0.543388
$$868$$ 10.0000 0.339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 26.0000 0.880976
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 32.0000 1.08118
$$877$$ −8.00000 −0.270141 −0.135070 0.990836i $$-0.543126\pi$$
−0.135070 + 0.990836i $$0.543126\pi$$
$$878$$ 56.0000 1.88991
$$879$$ 58.0000 1.95629
$$880$$ 0 0
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ 12.0000 0.404061
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −20.0000 −0.672673
$$885$$ 0 0
$$886$$ −48.0000 −1.61259
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ 0 0
$$889$$ −20.0000 −0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 28.0000 0.937509
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.00000 0.133556
$$898$$ −30.0000 −1.00111
$$899$$ 25.0000 0.833797
$$900$$ 0 0
$$901$$ 5.00000 0.166574
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 32.0000 1.06313
$$907$$ −9.00000 −0.298840 −0.149420 0.988774i $$-0.547741\pi$$
−0.149420 + 0.988774i $$0.547741\pi$$
$$908$$ 0 0
$$909$$ 15.0000 0.497519
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 64.0000 2.11925
$$913$$ 0 0
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 8.00000 0.264327
$$917$$ 0 0
$$918$$ −40.0000 −1.32020
$$919$$ 4.00000 0.131948 0.0659739 0.997821i $$-0.478985\pi$$
0.0659739 + 0.997821i $$0.478985\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 36.0000 1.18560
$$923$$ −26.0000 −0.855800
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 0 0
$$928$$ −40.0000 −1.31306
$$929$$ 35.0000 1.14831 0.574156 0.818746i $$-0.305330\pi$$
0.574156 + 0.818746i $$0.305330\pi$$
$$930$$ 0 0
$$931$$ −48.0000 −1.57314
$$932$$ −12.0000 −0.393073
$$933$$ −8.00000 −0.261908
$$934$$ −54.0000 −1.76693
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −54.0000 −1.76410 −0.882052 0.471153i $$-0.843838\pi$$
−0.882052 + 0.471153i $$0.843838\pi$$
$$938$$ −26.0000 −0.848930
$$939$$ 42.0000 1.37062
$$940$$ 0 0
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 12.0000 0.390981
$$943$$ −7.00000 −0.227951
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −14.0000 −0.454939 −0.227469 0.973785i $$-0.573045\pi$$
−0.227469 + 0.973785i $$0.573045\pi$$
$$948$$ 56.0000 1.81880
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ −48.0000 −1.55651
$$952$$ 0 0
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −2.00000 −0.0646846
$$957$$ 0 0
$$958$$ 56.0000 1.80928
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −28.0000 −0.902756
$$963$$ −9.00000 −0.290021
$$964$$ 12.0000 0.386494
$$965$$ 0 0
$$966$$ −4.00000 −0.128698
$$967$$ 18.0000 0.578841 0.289420 0.957202i $$-0.406537\pi$$
0.289420 + 0.957202i $$0.406537\pi$$
$$968$$ 0 0
$$969$$ −80.0000 −2.56997
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 20.0000 0.641500
$$973$$ −9.00000 −0.288527
$$974$$ −64.0000 −2.05069
$$975$$ 0 0
$$976$$ 24.0000 0.768221
$$977$$ 39.0000 1.24772 0.623860 0.781536i $$-0.285563\pi$$
0.623860 + 0.781536i $$0.285563\pi$$
$$978$$ −96.0000 −3.06974
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ 62.0000 1.97850
$$983$$ −21.0000 −0.669796 −0.334898 0.942254i $$-0.608702\pi$$
−0.334898 + 0.942254i $$0.608702\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 50.0000 1.59232
$$987$$ 4.00000 0.127321
$$988$$ −32.0000 −1.01806
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 19.0000 0.603555 0.301777 0.953378i $$-0.402420\pi$$
0.301777 + 0.953378i $$0.402420\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 10.0000 0.317340
$$994$$ 26.0000 0.824670
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ −20.0000 −0.633406 −0.316703 0.948525i $$-0.602576\pi$$
−0.316703 + 0.948525i $$0.602576\pi$$
$$998$$ 22.0000 0.696398
$$999$$ −28.0000 −0.885881
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 575.2.a.a.1.1 1
3.2 odd 2 5175.2.a.z.1.1 1
4.3 odd 2 9200.2.a.bg.1.1 1
5.2 odd 4 115.2.b.a.24.1 2
5.3 odd 4 115.2.b.a.24.2 yes 2
5.4 even 2 575.2.a.e.1.1 1
15.2 even 4 1035.2.b.a.829.2 2
15.8 even 4 1035.2.b.a.829.1 2
15.14 odd 2 5175.2.a.a.1.1 1
20.3 even 4 1840.2.e.b.369.2 2
20.7 even 4 1840.2.e.b.369.1 2
20.19 odd 2 9200.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.b.a.24.1 2 5.2 odd 4
115.2.b.a.24.2 yes 2 5.3 odd 4
575.2.a.a.1.1 1 1.1 even 1 trivial
575.2.a.e.1.1 1 5.4 even 2
1035.2.b.a.829.1 2 15.8 even 4
1035.2.b.a.829.2 2 15.2 even 4
1840.2.e.b.369.1 2 20.7 even 4
1840.2.e.b.369.2 2 20.3 even 4
5175.2.a.a.1.1 1 15.14 odd 2
5175.2.a.z.1.1 1 3.2 odd 2
9200.2.a.g.1.1 1 20.19 odd 2
9200.2.a.bg.1.1 1 4.3 odd 2