# Properties

 Label 7800.2.a.z.1.1 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.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: $$62.2833135766$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} -2.41421 q^{11} -1.00000 q^{13} -1.82843 q^{17} -2.00000 q^{19} +0.414214 q^{21} +4.82843 q^{23} -1.00000 q^{27} +2.65685 q^{29} +1.58579 q^{31} +2.41421 q^{33} +3.65685 q^{37} +1.00000 q^{39} -5.65685 q^{41} +6.48528 q^{43} +12.0711 q^{47} -6.82843 q^{49} +1.82843 q^{51} -0.171573 q^{53} +2.00000 q^{57} -3.58579 q^{59} +3.82843 q^{61} -0.414214 q^{63} +9.24264 q^{67} -4.82843 q^{69} +3.65685 q^{71} -13.3137 q^{73} +1.00000 q^{77} -3.17157 q^{79} +1.00000 q^{81} -11.7279 q^{83} -2.65685 q^{87} +0.414214 q^{91} -1.58579 q^{93} -18.9706 q^{97} -2.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} - 2 q^{21} + 4 q^{23} - 2 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 4 q^{37} + 2 q^{39} - 4 q^{43} + 10 q^{47} - 8 q^{49} - 2 q^{51} - 6 q^{53} + 4 q^{57} - 10 q^{59} + 2 q^{61} + 2 q^{63} + 10 q^{67} - 4 q^{69} - 4 q^{71} - 4 q^{73} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{87} - 2 q^{91} - 6 q^{93} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^17 - 4 * q^19 - 2 * q^21 + 4 * q^23 - 2 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 4 * q^37 + 2 * q^39 - 4 * q^43 + 10 * q^47 - 8 * q^49 - 2 * q^51 - 6 * q^53 + 4 * q^57 - 10 * q^59 + 2 * q^61 + 2 * q^63 + 10 * q^67 - 4 * q^69 - 4 * q^71 - 4 * q^73 + 2 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 + 6 * q^87 - 2 * q^91 - 6 * q^93 - 4 * q^97 - 2 * 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.414214 −0.156558 −0.0782790 0.996931i $$-0.524942\pi$$
−0.0782790 + 0.996931i $$0.524942\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.41421 −0.727913 −0.363956 0.931416i $$-0.618574\pi$$
−0.363956 + 0.931416i $$0.618574\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.82843 −0.443459 −0.221729 0.975108i $$-0.571170\pi$$
−0.221729 + 0.975108i $$0.571170\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0.414214 0.0903888
$$22$$ 0 0
$$23$$ 4.82843 1.00680 0.503398 0.864054i $$-0.332083\pi$$
0.503398 + 0.864054i $$0.332083\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.65685 0.493365 0.246683 0.969096i $$-0.420659\pi$$
0.246683 + 0.969096i $$0.420659\pi$$
$$30$$ 0 0
$$31$$ 1.58579 0.284816 0.142408 0.989808i $$-0.454516\pi$$
0.142408 + 0.989808i $$0.454516\pi$$
$$32$$ 0 0
$$33$$ 2.41421 0.420261
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.65685 0.601183 0.300592 0.953753i $$-0.402816\pi$$
0.300592 + 0.953753i $$0.402816\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −5.65685 −0.883452 −0.441726 0.897150i $$-0.645634\pi$$
−0.441726 + 0.897150i $$0.645634\pi$$
$$42$$ 0 0
$$43$$ 6.48528 0.988996 0.494498 0.869179i $$-0.335352\pi$$
0.494498 + 0.869179i $$0.335352\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0711 1.76075 0.880373 0.474282i $$-0.157292\pi$$
0.880373 + 0.474282i $$0.157292\pi$$
$$48$$ 0 0
$$49$$ −6.82843 −0.975490
$$50$$ 0 0
$$51$$ 1.82843 0.256031
$$52$$ 0 0
$$53$$ −0.171573 −0.0235673 −0.0117837 0.999931i $$-0.503751\pi$$
−0.0117837 + 0.999931i $$0.503751\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ −3.58579 −0.466830 −0.233415 0.972377i $$-0.574990\pi$$
−0.233415 + 0.972377i $$0.574990\pi$$
$$60$$ 0 0
$$61$$ 3.82843 0.490180 0.245090 0.969500i $$-0.421183\pi$$
0.245090 + 0.969500i $$0.421183\pi$$
$$62$$ 0 0
$$63$$ −0.414214 −0.0521860
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 9.24264 1.12917 0.564584 0.825376i $$-0.309037\pi$$
0.564584 + 0.825376i $$0.309037\pi$$
$$68$$ 0 0
$$69$$ −4.82843 −0.581274
$$70$$ 0 0
$$71$$ 3.65685 0.433989 0.216994 0.976173i $$-0.430375\pi$$
0.216994 + 0.976173i $$0.430375\pi$$
$$72$$ 0 0
$$73$$ −13.3137 −1.55825 −0.779126 0.626868i $$-0.784337\pi$$
−0.779126 + 0.626868i $$0.784337\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ −3.17157 −0.356830 −0.178415 0.983955i $$-0.557097\pi$$
−0.178415 + 0.983955i $$0.557097\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −11.7279 −1.28731 −0.643653 0.765317i $$-0.722582\pi$$
−0.643653 + 0.765317i $$0.722582\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −2.65685 −0.284845
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0.414214 0.0434214
$$92$$ 0 0
$$93$$ −1.58579 −0.164438
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −18.9706 −1.92617 −0.963084 0.269200i $$-0.913241\pi$$
−0.963084 + 0.269200i $$0.913241\pi$$
$$98$$ 0 0
$$99$$ −2.41421 −0.242638
$$100$$ 0 0
$$101$$ −11.8284 −1.17697 −0.588486 0.808507i $$-0.700276\pi$$
−0.588486 + 0.808507i $$0.700276\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.51472 0.146433 0.0732167 0.997316i $$-0.476674\pi$$
0.0732167 + 0.997316i $$0.476674\pi$$
$$108$$ 0 0
$$109$$ 3.31371 0.317396 0.158698 0.987327i $$-0.449270\pi$$
0.158698 + 0.987327i $$0.449270\pi$$
$$110$$ 0 0
$$111$$ −3.65685 −0.347093
$$112$$ 0 0
$$113$$ 9.31371 0.876160 0.438080 0.898936i $$-0.355659\pi$$
0.438080 + 0.898936i $$0.355659\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 0.757359 0.0694270
$$120$$ 0 0
$$121$$ −5.17157 −0.470143
$$122$$ 0 0
$$123$$ 5.65685 0.510061
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.8284 1.49328 0.746641 0.665228i $$-0.231665\pi$$
0.746641 + 0.665228i $$0.231665\pi$$
$$128$$ 0 0
$$129$$ −6.48528 −0.570997
$$130$$ 0 0
$$131$$ −6.34315 −0.554203 −0.277102 0.960841i $$-0.589374\pi$$
−0.277102 + 0.960841i $$0.589374\pi$$
$$132$$ 0 0
$$133$$ 0.828427 0.0718337
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.9706 1.44989 0.724947 0.688805i $$-0.241864\pi$$
0.724947 + 0.688805i $$0.241864\pi$$
$$138$$ 0 0
$$139$$ −8.82843 −0.748817 −0.374409 0.927264i $$-0.622154\pi$$
−0.374409 + 0.927264i $$0.622154\pi$$
$$140$$ 0 0
$$141$$ −12.0711 −1.01657
$$142$$ 0 0
$$143$$ 2.41421 0.201887
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 6.82843 0.563199
$$148$$ 0 0
$$149$$ −5.31371 −0.435316 −0.217658 0.976025i $$-0.569842\pi$$
−0.217658 + 0.976025i $$0.569842\pi$$
$$150$$ 0 0
$$151$$ 1.24264 0.101125 0.0505623 0.998721i $$-0.483899\pi$$
0.0505623 + 0.998721i $$0.483899\pi$$
$$152$$ 0 0
$$153$$ −1.82843 −0.147820
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.31371 0.344271 0.172136 0.985073i $$-0.444933\pi$$
0.172136 + 0.985073i $$0.444933\pi$$
$$158$$ 0 0
$$159$$ 0.171573 0.0136066
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ −25.3137 −1.98272 −0.991361 0.131159i $$-0.958130\pi$$
−0.991361 + 0.131159i $$0.958130\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2.68629 0.207871 0.103936 0.994584i $$-0.466856\pi$$
0.103936 + 0.994584i $$0.466856\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ −13.4853 −1.02527 −0.512633 0.858608i $$-0.671330\pi$$
−0.512633 + 0.858608i $$0.671330\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.58579 0.269524
$$178$$ 0 0
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −14.6569 −1.08944 −0.544718 0.838619i $$-0.683363\pi$$
−0.544718 + 0.838619i $$0.683363\pi$$
$$182$$ 0 0
$$183$$ −3.82843 −0.283005
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.41421 0.322799
$$188$$ 0 0
$$189$$ 0.414214 0.0301296
$$190$$ 0 0
$$191$$ −2.48528 −0.179829 −0.0899143 0.995950i $$-0.528659\pi$$
−0.0899143 + 0.995950i $$0.528659\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.00000 −0.284988 −0.142494 0.989796i $$-0.545512\pi$$
−0.142494 + 0.989796i $$0.545512\pi$$
$$198$$ 0 0
$$199$$ 4.82843 0.342278 0.171139 0.985247i $$-0.445255\pi$$
0.171139 + 0.985247i $$0.445255\pi$$
$$200$$ 0 0
$$201$$ −9.24264 −0.651926
$$202$$ 0 0
$$203$$ −1.10051 −0.0772403
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.82843 0.335599
$$208$$ 0 0
$$209$$ 4.82843 0.333989
$$210$$ 0 0
$$211$$ −6.48528 −0.446465 −0.223233 0.974765i $$-0.571661\pi$$
−0.223233 + 0.974765i $$0.571661\pi$$
$$212$$ 0 0
$$213$$ −3.65685 −0.250564
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.656854 −0.0445902
$$218$$ 0 0
$$219$$ 13.3137 0.899657
$$220$$ 0 0
$$221$$ 1.82843 0.122993
$$222$$ 0 0
$$223$$ −12.3431 −0.826558 −0.413279 0.910604i $$-0.635617\pi$$
−0.413279 + 0.910604i $$0.635617\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 14.4142 0.956705 0.478352 0.878168i $$-0.341234\pi$$
0.478352 + 0.878168i $$0.341234\pi$$
$$228$$ 0 0
$$229$$ −15.6569 −1.03463 −0.517317 0.855794i $$-0.673069\pi$$
−0.517317 + 0.855794i $$0.673069\pi$$
$$230$$ 0 0
$$231$$ −1.00000 −0.0657952
$$232$$ 0 0
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3.17157 0.206016
$$238$$ 0 0
$$239$$ 3.24264 0.209749 0.104874 0.994485i $$-0.466556\pi$$
0.104874 + 0.994485i $$0.466556\pi$$
$$240$$ 0 0
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ 11.7279 0.743227
$$250$$ 0 0
$$251$$ −8.00000 −0.504956 −0.252478 0.967603i $$-0.581245\pi$$
−0.252478 + 0.967603i $$0.581245\pi$$
$$252$$ 0 0
$$253$$ −11.6569 −0.732860
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.00000 0.0623783 0.0311891 0.999514i $$-0.490071\pi$$
0.0311891 + 0.999514i $$0.490071\pi$$
$$258$$ 0 0
$$259$$ −1.51472 −0.0941200
$$260$$ 0 0
$$261$$ 2.65685 0.164455
$$262$$ 0 0
$$263$$ 19.3137 1.19093 0.595467 0.803380i $$-0.296967\pi$$
0.595467 + 0.803380i $$0.296967\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.3137 −0.628838 −0.314419 0.949284i $$-0.601810\pi$$
−0.314419 + 0.949284i $$0.601810\pi$$
$$270$$ 0 0
$$271$$ 14.8995 0.905080 0.452540 0.891744i $$-0.350518\pi$$
0.452540 + 0.891744i $$0.350518\pi$$
$$272$$ 0 0
$$273$$ −0.414214 −0.0250693
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.3137 0.799943 0.399972 0.916528i $$-0.369020\pi$$
0.399972 + 0.916528i $$0.369020\pi$$
$$278$$ 0 0
$$279$$ 1.58579 0.0949386
$$280$$ 0 0
$$281$$ −14.0000 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$282$$ 0 0
$$283$$ −23.4558 −1.39431 −0.697153 0.716923i $$-0.745550\pi$$
−0.697153 + 0.716923i $$0.745550\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.34315 0.138312
$$288$$ 0 0
$$289$$ −13.6569 −0.803344
$$290$$ 0 0
$$291$$ 18.9706 1.11207
$$292$$ 0 0
$$293$$ −17.3137 −1.01148 −0.505739 0.862687i $$-0.668780\pi$$
−0.505739 + 0.862687i $$0.668780\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.41421 0.140087
$$298$$ 0 0
$$299$$ −4.82843 −0.279235
$$300$$ 0 0
$$301$$ −2.68629 −0.154835
$$302$$ 0 0
$$303$$ 11.8284 0.679525
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1.51472 −0.0858918 −0.0429459 0.999077i $$-0.513674\pi$$
−0.0429459 + 0.999077i $$0.513674\pi$$
$$312$$ 0 0
$$313$$ −14.1716 −0.801025 −0.400512 0.916291i $$-0.631168\pi$$
−0.400512 + 0.916291i $$0.631168\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −30.9706 −1.73948 −0.869740 0.493510i $$-0.835714\pi$$
−0.869740 + 0.493510i $$0.835714\pi$$
$$318$$ 0 0
$$319$$ −6.41421 −0.359127
$$320$$ 0 0
$$321$$ −1.51472 −0.0845433
$$322$$ 0 0
$$323$$ 3.65685 0.203473
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −3.31371 −0.183248
$$328$$ 0 0
$$329$$ −5.00000 −0.275659
$$330$$ 0 0
$$331$$ −14.9706 −0.822857 −0.411428 0.911442i $$-0.634970\pi$$
−0.411428 + 0.911442i $$0.634970\pi$$
$$332$$ 0 0
$$333$$ 3.65685 0.200394
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 9.48528 0.516696 0.258348 0.966052i $$-0.416822\pi$$
0.258348 + 0.966052i $$0.416822\pi$$
$$338$$ 0 0
$$339$$ −9.31371 −0.505851
$$340$$ 0 0
$$341$$ −3.82843 −0.207321
$$342$$ 0 0
$$343$$ 5.72792 0.309279
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 30.4853 1.63654 0.818268 0.574837i $$-0.194935\pi$$
0.818268 + 0.574837i $$0.194935\pi$$
$$348$$ 0 0
$$349$$ 7.65685 0.409862 0.204931 0.978776i $$-0.434303\pi$$
0.204931 + 0.978776i $$0.434303\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ 23.3137 1.24086 0.620432 0.784260i $$-0.286957\pi$$
0.620432 + 0.784260i $$0.286957\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −0.757359 −0.0400837
$$358$$ 0 0
$$359$$ −20.8995 −1.10303 −0.551517 0.834164i $$-0.685951\pi$$
−0.551517 + 0.834164i $$0.685951\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 5.17157 0.271437
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.65685 0.0864871 0.0432435 0.999065i $$-0.486231\pi$$
0.0432435 + 0.999065i $$0.486231\pi$$
$$368$$ 0 0
$$369$$ −5.65685 −0.294484
$$370$$ 0 0
$$371$$ 0.0710678 0.00368966
$$372$$ 0 0
$$373$$ −21.3431 −1.10511 −0.552553 0.833478i $$-0.686346\pi$$
−0.552553 + 0.833478i $$0.686346\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.65685 −0.136835
$$378$$ 0 0
$$379$$ −14.0711 −0.722782 −0.361391 0.932414i $$-0.617698\pi$$
−0.361391 + 0.932414i $$0.617698\pi$$
$$380$$ 0 0
$$381$$ −16.8284 −0.862146
$$382$$ 0 0
$$383$$ −7.65685 −0.391247 −0.195623 0.980679i $$-0.562673\pi$$
−0.195623 + 0.980679i $$0.562673\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 6.48528 0.329665
$$388$$ 0 0
$$389$$ 28.6274 1.45147 0.725734 0.687976i $$-0.241500\pi$$
0.725734 + 0.687976i $$0.241500\pi$$
$$390$$ 0 0
$$391$$ −8.82843 −0.446473
$$392$$ 0 0
$$393$$ 6.34315 0.319969
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −22.2843 −1.11842 −0.559208 0.829028i $$-0.688895\pi$$
−0.559208 + 0.829028i $$0.688895\pi$$
$$398$$ 0 0
$$399$$ −0.828427 −0.0414732
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 0 0
$$403$$ −1.58579 −0.0789936
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.82843 −0.437609
$$408$$ 0 0
$$409$$ −3.31371 −0.163852 −0.0819262 0.996638i $$-0.526107\pi$$
−0.0819262 + 0.996638i $$0.526107\pi$$
$$410$$ 0 0
$$411$$ −16.9706 −0.837096
$$412$$ 0 0
$$413$$ 1.48528 0.0730859
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 8.82843 0.432330
$$418$$ 0 0
$$419$$ 13.6569 0.667181 0.333590 0.942718i $$-0.391740\pi$$
0.333590 + 0.942718i $$0.391740\pi$$
$$420$$ 0 0
$$421$$ 11.3137 0.551396 0.275698 0.961244i $$-0.411091\pi$$
0.275698 + 0.961244i $$0.411091\pi$$
$$422$$ 0 0
$$423$$ 12.0711 0.586915
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.58579 −0.0767416
$$428$$ 0 0
$$429$$ −2.41421 −0.116559
$$430$$ 0 0
$$431$$ −26.9706 −1.29913 −0.649563 0.760308i $$-0.725048\pi$$
−0.649563 + 0.760308i $$0.725048\pi$$
$$432$$ 0 0
$$433$$ −2.68629 −0.129095 −0.0645475 0.997915i $$-0.520560\pi$$
−0.0645475 + 0.997915i $$0.520560\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −9.65685 −0.461950
$$438$$ 0 0
$$439$$ −1.51472 −0.0722936 −0.0361468 0.999346i $$-0.511508\pi$$
−0.0361468 + 0.999346i $$0.511508\pi$$
$$440$$ 0 0
$$441$$ −6.82843 −0.325163
$$442$$ 0 0
$$443$$ −27.4558 −1.30447 −0.652233 0.758018i $$-0.726168\pi$$
−0.652233 + 0.758018i $$0.726168\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 5.31371 0.251330
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 13.6569 0.643076
$$452$$ 0 0
$$453$$ −1.24264 −0.0583844
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.3431 −0.858056 −0.429028 0.903291i $$-0.641144\pi$$
−0.429028 + 0.903291i $$0.641144\pi$$
$$458$$ 0 0
$$459$$ 1.82843 0.0853437
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ 1.72792 0.0803033 0.0401517 0.999194i $$-0.487216\pi$$
0.0401517 + 0.999194i $$0.487216\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 0 0
$$469$$ −3.82843 −0.176780
$$470$$ 0 0
$$471$$ −4.31371 −0.198765
$$472$$ 0 0
$$473$$ −15.6569 −0.719903
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −0.171573 −0.00785578
$$478$$ 0 0
$$479$$ −27.0416 −1.23556 −0.617782 0.786350i $$-0.711969\pi$$
−0.617782 + 0.786350i $$0.711969\pi$$
$$480$$ 0 0
$$481$$ −3.65685 −0.166738
$$482$$ 0 0
$$483$$ 2.00000 0.0910032
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.24264 0.328195 0.164098 0.986444i $$-0.447529\pi$$
0.164098 + 0.986444i $$0.447529\pi$$
$$488$$ 0 0
$$489$$ 25.3137 1.14473
$$490$$ 0 0
$$491$$ −5.51472 −0.248876 −0.124438 0.992227i $$-0.539713\pi$$
−0.124438 + 0.992227i $$0.539713\pi$$
$$492$$ 0 0
$$493$$ −4.85786 −0.218787
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.51472 −0.0679444
$$498$$ 0 0
$$499$$ −21.5858 −0.966313 −0.483156 0.875534i $$-0.660510\pi$$
−0.483156 + 0.875534i $$0.660510\pi$$
$$500$$ 0 0
$$501$$ −2.68629 −0.120015
$$502$$ 0 0
$$503$$ −36.2843 −1.61784 −0.808918 0.587922i $$-0.799946\pi$$
−0.808918 + 0.587922i $$0.799946\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ −8.00000 −0.354594 −0.177297 0.984157i $$-0.556735\pi$$
−0.177297 + 0.984157i $$0.556735\pi$$
$$510$$ 0 0
$$511$$ 5.51472 0.243957
$$512$$ 0 0
$$513$$ 2.00000 0.0883022
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −29.1421 −1.28167
$$518$$ 0 0
$$519$$ 13.4853 0.591938
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ −6.48528 −0.283582 −0.141791 0.989897i $$-0.545286\pi$$
−0.141791 + 0.989897i $$0.545286\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.89949 −0.126304
$$528$$ 0 0
$$529$$ 0.313708 0.0136395
$$530$$ 0 0
$$531$$ −3.58579 −0.155610
$$532$$ 0 0
$$533$$ 5.65685 0.245026
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4.00000 0.172613
$$538$$ 0 0
$$539$$ 16.4853 0.710071
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 14.6569 0.628986
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −22.6274 −0.967478 −0.483739 0.875212i $$-0.660722\pi$$
−0.483739 + 0.875212i $$0.660722\pi$$
$$548$$ 0 0
$$549$$ 3.82843 0.163393
$$550$$ 0 0
$$551$$ −5.31371 −0.226372
$$552$$ 0 0
$$553$$ 1.31371 0.0558646
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ 0 0
$$559$$ −6.48528 −0.274298
$$560$$ 0 0
$$561$$ −4.41421 −0.186368
$$562$$ 0 0
$$563$$ −6.48528 −0.273322 −0.136661 0.990618i $$-0.543637\pi$$
−0.136661 + 0.990618i $$0.543637\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −0.414214 −0.0173953
$$568$$ 0 0
$$569$$ 44.3137 1.85773 0.928864 0.370422i $$-0.120787\pi$$
0.928864 + 0.370422i $$0.120787\pi$$
$$570$$ 0 0
$$571$$ −1.51472 −0.0633890 −0.0316945 0.999498i $$-0.510090\pi$$
−0.0316945 + 0.999498i $$0.510090\pi$$
$$572$$ 0 0
$$573$$ 2.48528 0.103824
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 14.2843 0.594662 0.297331 0.954774i $$-0.403904\pi$$
0.297331 + 0.954774i $$0.403904\pi$$
$$578$$ 0 0
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 4.85786 0.201538
$$582$$ 0 0
$$583$$ 0.414214 0.0171550
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 13.7279 0.566612 0.283306 0.959030i $$-0.408569\pi$$
0.283306 + 0.959030i $$0.408569\pi$$
$$588$$ 0 0
$$589$$ −3.17157 −0.130682
$$590$$ 0 0
$$591$$ 4.00000 0.164538
$$592$$ 0 0
$$593$$ −28.6274 −1.17559 −0.587794 0.809011i $$-0.700003\pi$$
−0.587794 + 0.809011i $$0.700003\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −4.82843 −0.197614
$$598$$ 0 0
$$599$$ 5.65685 0.231133 0.115566 0.993300i $$-0.463132\pi$$
0.115566 + 0.993300i $$0.463132\pi$$
$$600$$ 0 0
$$601$$ 1.14214 0.0465887 0.0232943 0.999729i $$-0.492585\pi$$
0.0232943 + 0.999729i $$0.492585\pi$$
$$602$$ 0 0
$$603$$ 9.24264 0.376389
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −12.1421 −0.492834 −0.246417 0.969164i $$-0.579253\pi$$
−0.246417 + 0.969164i $$0.579253\pi$$
$$608$$ 0 0
$$609$$ 1.10051 0.0445947
$$610$$ 0 0
$$611$$ −12.0711 −0.488343
$$612$$ 0 0
$$613$$ −35.6569 −1.44017 −0.720083 0.693888i $$-0.755896\pi$$
−0.720083 + 0.693888i $$0.755896\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −5.02944 −0.202478 −0.101239 0.994862i $$-0.532281\pi$$
−0.101239 + 0.994862i $$0.532281\pi$$
$$618$$ 0 0
$$619$$ 0.627417 0.0252180 0.0126090 0.999921i $$-0.495986\pi$$
0.0126090 + 0.999921i $$0.495986\pi$$
$$620$$ 0 0
$$621$$ −4.82843 −0.193758
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −4.82843 −0.192829
$$628$$ 0 0
$$629$$ −6.68629 −0.266600
$$630$$ 0 0
$$631$$ 14.0000 0.557331 0.278666 0.960388i $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ 0 0
$$633$$ 6.48528 0.257767
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.82843 0.270552
$$638$$ 0 0
$$639$$ 3.65685 0.144663
$$640$$ 0 0
$$641$$ −35.4853 −1.40158 −0.700792 0.713365i $$-0.747170\pi$$
−0.700792 + 0.713365i $$0.747170\pi$$
$$642$$ 0 0
$$643$$ 4.62742 0.182488 0.0912438 0.995829i $$-0.470916\pi$$
0.0912438 + 0.995829i $$0.470916\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 45.9411 1.80613 0.903066 0.429502i $$-0.141311\pi$$
0.903066 + 0.429502i $$0.141311\pi$$
$$648$$ 0 0
$$649$$ 8.65685 0.339811
$$650$$ 0 0
$$651$$ 0.656854 0.0257441
$$652$$ 0 0
$$653$$ 34.6569 1.35623 0.678114 0.734957i $$-0.262798\pi$$
0.678114 + 0.734957i $$0.262798\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −13.3137 −0.519417
$$658$$ 0 0
$$659$$ −32.1421 −1.25208 −0.626040 0.779791i $$-0.715325\pi$$
−0.626040 + 0.779791i $$0.715325\pi$$
$$660$$ 0 0
$$661$$ −21.9411 −0.853411 −0.426705 0.904391i $$-0.640326\pi$$
−0.426705 + 0.904391i $$0.640326\pi$$
$$662$$ 0 0
$$663$$ −1.82843 −0.0710102
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.8284 0.496719
$$668$$ 0 0
$$669$$ 12.3431 0.477214
$$670$$ 0 0
$$671$$ −9.24264 −0.356808
$$672$$ 0 0
$$673$$ −21.1421 −0.814969 −0.407485 0.913212i $$-0.633594\pi$$
−0.407485 + 0.913212i $$0.633594\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ 7.85786 0.301557
$$680$$ 0 0
$$681$$ −14.4142 −0.552354
$$682$$ 0 0
$$683$$ 33.1838 1.26974 0.634871 0.772618i $$-0.281053\pi$$
0.634871 + 0.772618i $$0.281053\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 15.6569 0.597346
$$688$$ 0 0
$$689$$ 0.171573 0.00653641
$$690$$ 0 0
$$691$$ 0.757359 0.0288113 0.0144057 0.999896i $$-0.495414\pi$$
0.0144057 + 0.999896i $$0.495414\pi$$
$$692$$ 0 0
$$693$$ 1.00000 0.0379869
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 10.3431 0.391775
$$698$$ 0 0
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ −9.68629 −0.365846 −0.182923 0.983127i $$-0.558556\pi$$
−0.182923 + 0.983127i $$0.558556\pi$$
$$702$$ 0 0
$$703$$ −7.31371 −0.275842
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 4.89949 0.184264
$$708$$ 0 0
$$709$$ −10.6274 −0.399121 −0.199561 0.979886i $$-0.563951\pi$$
−0.199561 + 0.979886i $$0.563951\pi$$
$$710$$ 0 0
$$711$$ −3.17157 −0.118943
$$712$$ 0 0
$$713$$ 7.65685 0.286751
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −3.24264 −0.121099
$$718$$ 0 0
$$719$$ 2.62742 0.0979861 0.0489931 0.998799i $$-0.484399\pi$$
0.0489931 + 0.998799i $$0.484399\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 4.00000 0.148762
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 23.3137 0.864658 0.432329 0.901716i $$-0.357692\pi$$
0.432329 + 0.901716i $$0.357692\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −11.8579 −0.438579
$$732$$ 0 0
$$733$$ −51.3137 −1.89532 −0.947658 0.319289i $$-0.896556\pi$$
−0.947658 + 0.319289i $$0.896556\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −22.3137 −0.821936
$$738$$ 0 0
$$739$$ 3.44365 0.126677 0.0633384 0.997992i $$-0.479825\pi$$
0.0633384 + 0.997992i $$0.479825\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ −15.5858 −0.571787 −0.285894 0.958261i $$-0.592290\pi$$
−0.285894 + 0.958261i $$0.592290\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −11.7279 −0.429102
$$748$$ 0 0
$$749$$ −0.627417 −0.0229253
$$750$$ 0 0
$$751$$ 33.5147 1.22297 0.611485 0.791256i $$-0.290573\pi$$
0.611485 + 0.791256i $$0.290573\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8.79899 0.319805 0.159902 0.987133i $$-0.448882\pi$$
0.159902 + 0.987133i $$0.448882\pi$$
$$758$$ 0 0
$$759$$ 11.6569 0.423117
$$760$$ 0 0
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 0 0
$$763$$ −1.37258 −0.0496908
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.58579 0.129475
$$768$$ 0 0
$$769$$ −6.34315 −0.228740 −0.114370 0.993438i $$-0.536485\pi$$
−0.114370 + 0.993438i $$0.536485\pi$$
$$770$$ 0 0
$$771$$ −1.00000 −0.0360141
$$772$$ 0 0
$$773$$ 16.6274 0.598047 0.299023 0.954246i $$-0.403339\pi$$
0.299023 + 0.954246i $$0.403339\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1.51472 0.0543402
$$778$$ 0 0
$$779$$ 11.3137 0.405356
$$780$$ 0 0
$$781$$ −8.82843 −0.315906
$$782$$ 0 0
$$783$$ −2.65685 −0.0949482
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 33.3848 1.19004 0.595019 0.803711i $$-0.297144\pi$$
0.595019 + 0.803711i $$0.297144\pi$$
$$788$$ 0 0
$$789$$ −19.3137 −0.687586
$$790$$ 0 0
$$791$$ −3.85786 −0.137170
$$792$$ 0 0
$$793$$ −3.82843 −0.135951
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 16.6569 0.590016 0.295008 0.955495i $$-0.404678\pi$$
0.295008 + 0.955495i $$0.404678\pi$$
$$798$$ 0 0
$$799$$ −22.0711 −0.780818
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 32.1421 1.13427
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10.3137 0.363060
$$808$$ 0 0
$$809$$ 18.6863 0.656975 0.328488 0.944508i $$-0.393461\pi$$
0.328488 + 0.944508i $$0.393461\pi$$
$$810$$ 0 0
$$811$$ 53.7279 1.88664 0.943321 0.331881i $$-0.107683\pi$$
0.943321 + 0.331881i $$0.107683\pi$$
$$812$$ 0 0
$$813$$ −14.8995 −0.522548
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −12.9706 −0.453783
$$818$$ 0 0
$$819$$ 0.414214 0.0144738
$$820$$ 0 0
$$821$$ 10.6863 0.372954 0.186477 0.982459i $$-0.440293\pi$$
0.186477 + 0.982459i $$0.440293\pi$$
$$822$$ 0 0
$$823$$ −17.9411 −0.625388 −0.312694 0.949854i $$-0.601231\pi$$
−0.312694 + 0.949854i $$0.601231\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −19.3848 −0.674075 −0.337037 0.941491i $$-0.609425\pi$$
−0.337037 + 0.941491i $$0.609425\pi$$
$$828$$ 0 0
$$829$$ 1.82843 0.0635039 0.0317519 0.999496i $$-0.489891\pi$$
0.0317519 + 0.999496i $$0.489891\pi$$
$$830$$ 0 0
$$831$$ −13.3137 −0.461847
$$832$$ 0 0
$$833$$ 12.4853 0.432589
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.58579 −0.0548128
$$838$$ 0 0
$$839$$ −3.65685 −0.126249 −0.0631243 0.998006i $$-0.520106\pi$$
−0.0631243 + 0.998006i $$0.520106\pi$$
$$840$$ 0 0
$$841$$ −21.9411 −0.756591
$$842$$ 0 0
$$843$$ 14.0000 0.482186
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.14214 0.0736047
$$848$$ 0 0
$$849$$ 23.4558 0.805002
$$850$$ 0 0
$$851$$ 17.6569 0.605269
$$852$$ 0 0
$$853$$ −29.6569 −1.01543 −0.507716 0.861525i $$-0.669510\pi$$
−0.507716 + 0.861525i $$0.669510\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −32.6274 −1.11453 −0.557266 0.830334i $$-0.688150\pi$$
−0.557266 + 0.830334i $$0.688150\pi$$
$$858$$ 0 0
$$859$$ −44.8284 −1.52953 −0.764763 0.644312i $$-0.777144\pi$$
−0.764763 + 0.644312i $$0.777144\pi$$
$$860$$ 0 0
$$861$$ −2.34315 −0.0798542
$$862$$ 0 0
$$863$$ 32.4142 1.10339 0.551696 0.834045i $$-0.313981\pi$$
0.551696 + 0.834045i $$0.313981\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.6569 0.463811
$$868$$ 0 0
$$869$$ 7.65685 0.259741
$$870$$ 0 0
$$871$$ −9.24264 −0.313175
$$872$$ 0 0
$$873$$ −18.9706 −0.642056
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −25.6569 −0.866370 −0.433185 0.901305i $$-0.642610\pi$$
−0.433185 + 0.901305i $$0.642610\pi$$
$$878$$ 0 0
$$879$$ 17.3137 0.583977
$$880$$ 0 0
$$881$$ 24.5980 0.828727 0.414363 0.910111i $$-0.364004\pi$$
0.414363 + 0.910111i $$0.364004\pi$$
$$882$$ 0 0
$$883$$ 19.3137 0.649958 0.324979 0.945721i $$-0.394643\pi$$
0.324979 + 0.945721i $$0.394643\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 8.82843 0.296430 0.148215 0.988955i $$-0.452647\pi$$
0.148215 + 0.988955i $$0.452647\pi$$
$$888$$ 0 0
$$889$$ −6.97056 −0.233785
$$890$$ 0 0
$$891$$ −2.41421 −0.0808792
$$892$$ 0 0
$$893$$ −24.1421 −0.807886
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.82843 0.161216
$$898$$ 0 0
$$899$$ 4.21320 0.140518
$$900$$ 0 0
$$901$$ 0.313708 0.0104511
$$902$$ 0 0
$$903$$ 2.68629 0.0893942
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −33.9411 −1.12700 −0.563498 0.826117i $$-0.690545\pi$$
−0.563498 + 0.826117i $$0.690545\pi$$
$$908$$ 0 0
$$909$$ −11.8284 −0.392324
$$910$$ 0 0
$$911$$ −15.1716 −0.502657 −0.251328 0.967902i $$-0.580867\pi$$
−0.251328 + 0.967902i $$0.580867\pi$$
$$912$$ 0 0
$$913$$ 28.3137 0.937047
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 2.62742 0.0867650
$$918$$ 0 0
$$919$$ −21.7990 −0.719082 −0.359541 0.933129i $$-0.617067\pi$$
−0.359541 + 0.933129i $$0.617067\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ 0 0
$$923$$ −3.65685 −0.120367
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 13.6569 0.447585
$$932$$ 0 0
$$933$$ 1.51472 0.0495897
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −32.5980 −1.06493 −0.532465 0.846452i $$-0.678734\pi$$
−0.532465 + 0.846452i $$0.678734\pi$$
$$938$$ 0 0
$$939$$ 14.1716 0.462472
$$940$$ 0 0
$$941$$ 22.6863 0.739552 0.369776 0.929121i $$-0.379434\pi$$
0.369776 + 0.929121i $$0.379434\pi$$
$$942$$ 0 0
$$943$$ −27.3137 −0.889457
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.5858 0.571461 0.285731 0.958310i $$-0.407764\pi$$
0.285731 + 0.958310i $$0.407764\pi$$
$$948$$ 0 0
$$949$$ 13.3137 0.432181
$$950$$ 0 0
$$951$$ 30.9706 1.00429
$$952$$ 0 0
$$953$$ 23.3431 0.756159 0.378079 0.925773i $$-0.376585\pi$$
0.378079 + 0.925773i $$0.376585\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.41421 0.207342
$$958$$ 0 0
$$959$$ −7.02944 −0.226992
$$960$$ 0 0
$$961$$ −28.4853 −0.918880
$$962$$ 0 0
$$963$$ 1.51472 0.0488111
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −58.0122 −1.86555 −0.932773 0.360464i $$-0.882618\pi$$
−0.932773 + 0.360464i $$0.882618\pi$$
$$968$$ 0 0
$$969$$ −3.65685 −0.117475
$$970$$ 0 0
$$971$$ −23.1716 −0.743611 −0.371806 0.928311i $$-0.621261\pi$$
−0.371806 + 0.928311i $$0.621261\pi$$
$$972$$ 0 0
$$973$$ 3.65685 0.117233
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −0.970563 −0.0310511 −0.0155255 0.999879i $$-0.504942\pi$$
−0.0155255 + 0.999879i $$0.504942\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 3.31371 0.105799
$$982$$ 0 0
$$983$$ 17.2426 0.549955 0.274977 0.961451i $$-0.411330\pi$$
0.274977 + 0.961451i $$0.411330\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 5.00000 0.159152
$$988$$ 0 0
$$989$$ 31.3137 0.995718
$$990$$ 0 0
$$991$$ 44.1421 1.40222 0.701111 0.713053i $$-0.252688\pi$$
0.701111 + 0.713053i $$0.252688\pi$$
$$992$$ 0 0
$$993$$ 14.9706 0.475076
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.2010 −0.608102 −0.304051 0.952656i $$-0.598339\pi$$
−0.304051 + 0.952656i $$0.598339\pi$$
$$998$$ 0 0
$$999$$ −3.65685 −0.115698
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 7800.2.a.z.1.1 2
5.4 even 2 7800.2.a.ba.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.z.1.1 2 1.1 even 1 trivial
7800.2.a.ba.1.2 yes 2 5.4 even 2