# Properties

 Label 4840.2.a.t.1.2 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.65544$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.39593 q^{3} -1.00000 q^{5} +1.65544 q^{7} -1.05137 q^{9} +O(q^{10})$$ $$q+1.39593 q^{3} -1.00000 q^{5} +1.65544 q^{7} -1.05137 q^{9} +6.36226 q^{13} -1.39593 q^{15} -5.31088 q^{17} -4.36226 q^{19} +2.31088 q^{21} -5.57040 q^{23} +1.00000 q^{25} -5.65544 q^{27} -6.79186 q^{29} +0.259511 q^{31} -1.65544 q^{35} -0.791864 q^{37} +8.88128 q^{39} -2.74049 q^{41} -11.6554 q^{43} +1.05137 q^{45} +7.49868 q^{47} -4.25951 q^{49} -7.41363 q^{51} -1.84324 q^{53} -6.08942 q^{57} -7.15412 q^{59} +8.57040 q^{61} -1.74049 q^{63} -6.36226 q^{65} +15.6014 q^{67} -7.77589 q^{69} +1.05137 q^{71} -11.6865 q^{73} +1.39593 q^{75} -2.27284 q^{79} -4.74049 q^{81} -7.15412 q^{83} +5.31088 q^{85} -9.48098 q^{87} -8.31088 q^{89} +10.5324 q^{91} +0.362259 q^{93} +4.36226 q^{95} +14.3623 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 - q^7 + 6 * q^9 $$3 q + q^{3} - 3 q^{5} - q^{7} + 6 q^{9} - 2 q^{13} - q^{15} - 4 q^{17} + 8 q^{19} - 5 q^{21} - 2 q^{23} + 3 q^{25} - 11 q^{27} - 14 q^{29} - 2 q^{31} + q^{35} + 4 q^{37} - 11 q^{41} - 29 q^{43} - 6 q^{45} + q^{47} - 10 q^{49} + 8 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 11 q^{61} - 8 q^{63} + 2 q^{65} + 7 q^{67} + 28 q^{69} - 6 q^{71} - 4 q^{73} + q^{75} - 6 q^{79} - 17 q^{81} + 6 q^{83} + 4 q^{85} - 34 q^{87} - 13 q^{89} + 28 q^{91} - 20 q^{93} - 8 q^{95} + 22 q^{97}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 - q^7 + 6 * q^9 - 2 * q^13 - q^15 - 4 * q^17 + 8 * q^19 - 5 * q^21 - 2 * q^23 + 3 * q^25 - 11 * q^27 - 14 * q^29 - 2 * q^31 + q^35 + 4 * q^37 - 11 * q^41 - 29 * q^43 - 6 * q^45 + q^47 - 10 * q^49 + 8 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 11 * q^61 - 8 * q^63 + 2 * q^65 + 7 * q^67 + 28 * q^69 - 6 * q^71 - 4 * q^73 + q^75 - 6 * q^79 - 17 * q^81 + 6 * q^83 + 4 * q^85 - 34 * q^87 - 13 * q^89 + 28 * q^91 - 20 * q^93 - 8 * q^95 + 22 * q^97

## 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
$$3$$ 1.39593 0.805942 0.402971 0.915213i $$-0.367978\pi$$
0.402971 + 0.915213i $$0.367978\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.65544 0.625698 0.312849 0.949803i $$-0.398717\pi$$
0.312849 + 0.949803i $$0.398717\pi$$
$$8$$ 0 0
$$9$$ −1.05137 −0.350458
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 6.36226 1.76457 0.882287 0.470713i $$-0.156003\pi$$
0.882287 + 0.470713i $$0.156003\pi$$
$$14$$ 0 0
$$15$$ −1.39593 −0.360428
$$16$$ 0 0
$$17$$ −5.31088 −1.28808 −0.644039 0.764992i $$-0.722743\pi$$
−0.644039 + 0.764992i $$0.722743\pi$$
$$18$$ 0 0
$$19$$ −4.36226 −1.00077 −0.500385 0.865803i $$-0.666808\pi$$
−0.500385 + 0.865803i $$0.666808\pi$$
$$20$$ 0 0
$$21$$ 2.31088 0.504276
$$22$$ 0 0
$$23$$ −5.57040 −1.16151 −0.580754 0.814079i $$-0.697242\pi$$
−0.580754 + 0.814079i $$0.697242\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.65544 −1.08839
$$28$$ 0 0
$$29$$ −6.79186 −1.26122 −0.630609 0.776101i $$-0.717195\pi$$
−0.630609 + 0.776101i $$0.717195\pi$$
$$30$$ 0 0
$$31$$ 0.259511 0.0466095 0.0233047 0.999728i $$-0.492581\pi$$
0.0233047 + 0.999728i $$0.492581\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.65544 −0.279821
$$36$$ 0 0
$$37$$ −0.791864 −0.130182 −0.0650908 0.997879i $$-0.520734\pi$$
−0.0650908 + 0.997879i $$0.520734\pi$$
$$38$$ 0 0
$$39$$ 8.88128 1.42214
$$40$$ 0 0
$$41$$ −2.74049 −0.427993 −0.213996 0.976834i $$-0.568648\pi$$
−0.213996 + 0.976834i $$0.568648\pi$$
$$42$$ 0 0
$$43$$ −11.6554 −1.77744 −0.888719 0.458452i $$-0.848404\pi$$
−0.888719 + 0.458452i $$0.848404\pi$$
$$44$$ 0 0
$$45$$ 1.05137 0.156730
$$46$$ 0 0
$$47$$ 7.49868 1.09379 0.546897 0.837200i $$-0.315809\pi$$
0.546897 + 0.837200i $$0.315809\pi$$
$$48$$ 0 0
$$49$$ −4.25951 −0.608502
$$50$$ 0 0
$$51$$ −7.41363 −1.03812
$$52$$ 0 0
$$53$$ −1.84324 −0.253188 −0.126594 0.991955i $$-0.540405\pi$$
−0.126594 + 0.991955i $$0.540405\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.08942 −0.806563
$$58$$ 0 0
$$59$$ −7.15412 −0.931387 −0.465694 0.884946i $$-0.654195\pi$$
−0.465694 + 0.884946i $$0.654195\pi$$
$$60$$ 0 0
$$61$$ 8.57040 1.09733 0.548663 0.836043i $$-0.315137\pi$$
0.548663 + 0.836043i $$0.315137\pi$$
$$62$$ 0 0
$$63$$ −1.74049 −0.219281
$$64$$ 0 0
$$65$$ −6.36226 −0.789141
$$66$$ 0 0
$$67$$ 15.6014 1.90602 0.953009 0.302942i $$-0.0979689\pi$$
0.953009 + 0.302942i $$0.0979689\pi$$
$$68$$ 0 0
$$69$$ −7.77589 −0.936107
$$70$$ 0 0
$$71$$ 1.05137 0.124775 0.0623876 0.998052i $$-0.480129\pi$$
0.0623876 + 0.998052i $$0.480129\pi$$
$$72$$ 0 0
$$73$$ −11.6865 −1.36780 −0.683899 0.729576i $$-0.739717\pi$$
−0.683899 + 0.729576i $$0.739717\pi$$
$$74$$ 0 0
$$75$$ 1.39593 0.161188
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −2.27284 −0.255715 −0.127857 0.991793i $$-0.540810\pi$$
−0.127857 + 0.991793i $$0.540810\pi$$
$$80$$ 0 0
$$81$$ −4.74049 −0.526721
$$82$$ 0 0
$$83$$ −7.15412 −0.785267 −0.392633 0.919695i $$-0.628436\pi$$
−0.392633 + 0.919695i $$0.628436\pi$$
$$84$$ 0 0
$$85$$ 5.31088 0.576046
$$86$$ 0 0
$$87$$ −9.48098 −1.01647
$$88$$ 0 0
$$89$$ −8.31088 −0.880952 −0.440476 0.897764i $$-0.645190\pi$$
−0.440476 + 0.897764i $$0.645190\pi$$
$$90$$ 0 0
$$91$$ 10.5324 1.10409
$$92$$ 0 0
$$93$$ 0.362259 0.0375645
$$94$$ 0 0
$$95$$ 4.36226 0.447558
$$96$$ 0 0
$$97$$ 14.3623 1.45827 0.729133 0.684372i $$-0.239923\pi$$
0.729133 + 0.684372i $$0.239923\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.79186 0.377305 0.188652 0.982044i $$-0.439588\pi$$
0.188652 + 0.982044i $$0.439588\pi$$
$$102$$ 0 0
$$103$$ 9.77589 0.963247 0.481624 0.876378i $$-0.340047\pi$$
0.481624 + 0.876378i $$0.340047\pi$$
$$104$$ 0 0
$$105$$ −2.31088 −0.225519
$$106$$ 0 0
$$107$$ 14.9123 1.44163 0.720814 0.693129i $$-0.243768\pi$$
0.720814 + 0.693129i $$0.243768\pi$$
$$108$$ 0 0
$$109$$ −10.1541 −0.972589 −0.486294 0.873795i $$-0.661652\pi$$
−0.486294 + 0.873795i $$0.661652\pi$$
$$110$$ 0 0
$$111$$ −1.10539 −0.104919
$$112$$ 0 0
$$113$$ 12.6218 1.18736 0.593678 0.804703i $$-0.297675\pi$$
0.593678 + 0.804703i $$0.297675\pi$$
$$114$$ 0 0
$$115$$ 5.57040 0.519442
$$116$$ 0 0
$$117$$ −6.68912 −0.618409
$$118$$ 0 0
$$119$$ −8.79186 −0.805949
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −3.82554 −0.344937
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 1.29318 0.114751 0.0573757 0.998353i $$-0.481727\pi$$
0.0573757 + 0.998353i $$0.481727\pi$$
$$128$$ 0 0
$$129$$ −16.2702 −1.43251
$$130$$ 0 0
$$131$$ −5.74049 −0.501549 −0.250774 0.968046i $$-0.580685\pi$$
−0.250774 + 0.968046i $$0.580685\pi$$
$$132$$ 0 0
$$133$$ −7.22147 −0.626181
$$134$$ 0 0
$$135$$ 5.65544 0.486743
$$136$$ 0 0
$$137$$ −10.1161 −0.864275 −0.432138 0.901808i $$-0.642241\pi$$
−0.432138 + 0.901808i $$0.642241\pi$$
$$138$$ 0 0
$$139$$ −5.84324 −0.495617 −0.247808 0.968809i $$-0.579710\pi$$
−0.247808 + 0.968809i $$0.579710\pi$$
$$140$$ 0 0
$$141$$ 10.4676 0.881535
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.79186 0.564034
$$146$$ 0 0
$$147$$ −5.94599 −0.490417
$$148$$ 0 0
$$149$$ −22.0868 −1.80942 −0.904710 0.426029i $$-0.859912\pi$$
−0.904710 + 0.426029i $$0.859912\pi$$
$$150$$ 0 0
$$151$$ −7.46765 −0.607708 −0.303854 0.952719i $$-0.598274\pi$$
−0.303854 + 0.952719i $$0.598274\pi$$
$$152$$ 0 0
$$153$$ 5.58373 0.451418
$$154$$ 0 0
$$155$$ −0.259511 −0.0208444
$$156$$ 0 0
$$157$$ 11.1408 0.889132 0.444566 0.895746i $$-0.353358\pi$$
0.444566 + 0.895746i $$0.353358\pi$$
$$158$$ 0 0
$$159$$ −2.57303 −0.204055
$$160$$ 0 0
$$161$$ −9.22147 −0.726754
$$162$$ 0 0
$$163$$ −8.55005 −0.669692 −0.334846 0.942273i $$-0.608684\pi$$
−0.334846 + 0.942273i $$0.608684\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.0717 −0.779373 −0.389686 0.920948i $$-0.627417\pi$$
−0.389686 + 0.920948i $$0.627417\pi$$
$$168$$ 0 0
$$169$$ 27.4783 2.11372
$$170$$ 0 0
$$171$$ 4.58637 0.350728
$$172$$ 0 0
$$173$$ −12.7785 −0.971534 −0.485767 0.874088i $$-0.661460\pi$$
−0.485767 + 0.874088i $$0.661460\pi$$
$$174$$ 0 0
$$175$$ 1.65544 0.125140
$$176$$ 0 0
$$177$$ −9.98667 −0.750644
$$178$$ 0 0
$$179$$ 12.1027 0.904602 0.452301 0.891865i $$-0.350603\pi$$
0.452301 + 0.891865i $$0.350603\pi$$
$$180$$ 0 0
$$181$$ −19.1382 −1.42253 −0.711264 0.702925i $$-0.751877\pi$$
−0.711264 + 0.702925i $$0.751877\pi$$
$$182$$ 0 0
$$183$$ 11.9637 0.884381
$$184$$ 0 0
$$185$$ 0.791864 0.0582190
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −9.36226 −0.681004
$$190$$ 0 0
$$191$$ −12.0894 −0.874759 −0.437380 0.899277i $$-0.644093\pi$$
−0.437380 + 0.899277i $$0.644093\pi$$
$$192$$ 0 0
$$193$$ 13.6191 0.980326 0.490163 0.871631i $$-0.336937\pi$$
0.490163 + 0.871631i $$0.336937\pi$$
$$194$$ 0 0
$$195$$ −8.88128 −0.636002
$$196$$ 0 0
$$197$$ −14.7919 −1.05388 −0.526938 0.849904i $$-0.676660\pi$$
−0.526938 + 0.849904i $$0.676660\pi$$
$$198$$ 0 0
$$199$$ −17.6865 −1.25376 −0.626881 0.779115i $$-0.715669\pi$$
−0.626881 + 0.779115i $$0.715669\pi$$
$$200$$ 0 0
$$201$$ 21.7785 1.53614
$$202$$ 0 0
$$203$$ −11.2435 −0.789142
$$204$$ 0 0
$$205$$ 2.74049 0.191404
$$206$$ 0 0
$$207$$ 5.85657 0.407060
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −27.0868 −1.86473 −0.932365 0.361518i $$-0.882259\pi$$
−0.932365 + 0.361518i $$0.882259\pi$$
$$212$$ 0 0
$$213$$ 1.46765 0.100562
$$214$$ 0 0
$$215$$ 11.6554 0.794895
$$216$$ 0 0
$$217$$ 0.429605 0.0291635
$$218$$ 0 0
$$219$$ −16.3135 −1.10237
$$220$$ 0 0
$$221$$ −33.7892 −2.27291
$$222$$ 0 0
$$223$$ −8.69348 −0.582159 −0.291079 0.956699i $$-0.594014\pi$$
−0.291079 + 0.956699i $$0.594014\pi$$
$$224$$ 0 0
$$225$$ −1.05137 −0.0700916
$$226$$ 0 0
$$227$$ 3.40926 0.226281 0.113140 0.993579i $$-0.463909\pi$$
0.113140 + 0.993579i $$0.463909\pi$$
$$228$$ 0 0
$$229$$ 13.3489 0.882122 0.441061 0.897477i $$-0.354602\pi$$
0.441061 + 0.897477i $$0.354602\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.6351 −0.958777 −0.479389 0.877603i $$-0.659142\pi$$
−0.479389 + 0.877603i $$0.659142\pi$$
$$234$$ 0 0
$$235$$ −7.49868 −0.489160
$$236$$ 0 0
$$237$$ −3.17273 −0.206091
$$238$$ 0 0
$$239$$ 8.36226 0.540910 0.270455 0.962733i $$-0.412826\pi$$
0.270455 + 0.962733i $$0.412826\pi$$
$$240$$ 0 0
$$241$$ 11.0487 0.711712 0.355856 0.934541i $$-0.384189\pi$$
0.355856 + 0.934541i $$0.384189\pi$$
$$242$$ 0 0
$$243$$ 10.3489 0.663884
$$244$$ 0 0
$$245$$ 4.25951 0.272130
$$246$$ 0 0
$$247$$ −27.7538 −1.76593
$$248$$ 0 0
$$249$$ −9.98667 −0.632879
$$250$$ 0 0
$$251$$ −13.5704 −0.856556 −0.428278 0.903647i $$-0.640880\pi$$
−0.428278 + 0.903647i $$0.640880\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 7.41363 0.464260
$$256$$ 0 0
$$257$$ −0.881280 −0.0549727 −0.0274864 0.999622i $$-0.508750\pi$$
−0.0274864 + 0.999622i $$0.508750\pi$$
$$258$$ 0 0
$$259$$ −1.31088 −0.0814544
$$260$$ 0 0
$$261$$ 7.14079 0.442004
$$262$$ 0 0
$$263$$ −14.3623 −0.885615 −0.442807 0.896617i $$-0.646017\pi$$
−0.442807 + 0.896617i $$0.646017\pi$$
$$264$$ 0 0
$$265$$ 1.84324 0.113229
$$266$$ 0 0
$$267$$ −11.6014 −0.709996
$$268$$ 0 0
$$269$$ 11.2188 0.684024 0.342012 0.939696i $$-0.388892\pi$$
0.342012 + 0.939696i $$0.388892\pi$$
$$270$$ 0 0
$$271$$ 14.3756 0.873255 0.436627 0.899642i $$-0.356173\pi$$
0.436627 + 0.899642i $$0.356173\pi$$
$$272$$ 0 0
$$273$$ 14.7024 0.889833
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 31.2029 1.87480 0.937399 0.348257i $$-0.113226\pi$$
0.937399 + 0.348257i $$0.113226\pi$$
$$278$$ 0 0
$$279$$ −0.272843 −0.0163347
$$280$$ 0 0
$$281$$ 17.1675 1.02412 0.512062 0.858948i $$-0.328882\pi$$
0.512062 + 0.858948i $$0.328882\pi$$
$$282$$ 0 0
$$283$$ 11.9150 0.708270 0.354135 0.935194i $$-0.384775\pi$$
0.354135 + 0.935194i $$0.384775\pi$$
$$284$$ 0 0
$$285$$ 6.08942 0.360706
$$286$$ 0 0
$$287$$ −4.53672 −0.267794
$$288$$ 0 0
$$289$$ 11.2055 0.659147
$$290$$ 0 0
$$291$$ 20.0487 1.17528
$$292$$ 0 0
$$293$$ −10.5324 −0.615307 −0.307653 0.951499i $$-0.599544\pi$$
−0.307653 + 0.951499i $$0.599544\pi$$
$$294$$ 0 0
$$295$$ 7.15412 0.416529
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −35.4403 −2.04957
$$300$$ 0 0
$$301$$ −19.2949 −1.11214
$$302$$ 0 0
$$303$$ 5.29318 0.304085
$$304$$ 0 0
$$305$$ −8.57040 −0.490739
$$306$$ 0 0
$$307$$ −11.1541 −0.636599 −0.318300 0.947990i $$-0.603112\pi$$
−0.318300 + 0.947990i $$0.603112\pi$$
$$308$$ 0 0
$$309$$ 13.6465 0.776321
$$310$$ 0 0
$$311$$ −10.9840 −0.622847 −0.311424 0.950271i $$-0.600806\pi$$
−0.311424 + 0.950271i $$0.600806\pi$$
$$312$$ 0 0
$$313$$ 12.3977 0.700757 0.350379 0.936608i $$-0.386053\pi$$
0.350379 + 0.936608i $$0.386053\pi$$
$$314$$ 0 0
$$315$$ 1.74049 0.0980655
$$316$$ 0 0
$$317$$ 6.54569 0.367642 0.183821 0.982960i $$-0.441153\pi$$
0.183821 + 0.982960i $$0.441153\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 20.8166 1.16187
$$322$$ 0 0
$$323$$ 23.1675 1.28907
$$324$$ 0 0
$$325$$ 6.36226 0.352915
$$326$$ 0 0
$$327$$ −14.1745 −0.783850
$$328$$ 0 0
$$329$$ 12.4136 0.684386
$$330$$ 0 0
$$331$$ −29.8786 −1.64228 −0.821139 0.570728i $$-0.806661\pi$$
−0.821139 + 0.570728i $$0.806661\pi$$
$$332$$ 0 0
$$333$$ 0.832545 0.0456232
$$334$$ 0 0
$$335$$ −15.6014 −0.852397
$$336$$ 0 0
$$337$$ −3.19480 −0.174032 −0.0870160 0.996207i $$-0.527733\pi$$
−0.0870160 + 0.996207i $$0.527733\pi$$
$$338$$ 0 0
$$339$$ 17.6191 0.956940
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −18.6395 −1.00644
$$344$$ 0 0
$$345$$ 7.77589 0.418640
$$346$$ 0 0
$$347$$ 21.3773 1.14759 0.573797 0.818997i $$-0.305470\pi$$
0.573797 + 0.818997i $$0.305470\pi$$
$$348$$ 0 0
$$349$$ 0.894612 0.0478875 0.0239437 0.999713i $$-0.492378\pi$$
0.0239437 + 0.999713i $$0.492378\pi$$
$$350$$ 0 0
$$351$$ −35.9814 −1.92054
$$352$$ 0 0
$$353$$ −7.05137 −0.375307 −0.187653 0.982235i $$-0.560088\pi$$
−0.187653 + 0.982235i $$0.560088\pi$$
$$354$$ 0 0
$$355$$ −1.05137 −0.0558012
$$356$$ 0 0
$$357$$ −12.2728 −0.649548
$$358$$ 0 0
$$359$$ −10.5324 −0.555876 −0.277938 0.960599i $$-0.589651\pi$$
−0.277938 + 0.960599i $$0.589651\pi$$
$$360$$ 0 0
$$361$$ 0.0293036 0.00154230
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 11.6865 0.611698
$$366$$ 0 0
$$367$$ 33.5341 1.75046 0.875232 0.483703i $$-0.160708\pi$$
0.875232 + 0.483703i $$0.160708\pi$$
$$368$$ 0 0
$$369$$ 2.88128 0.149993
$$370$$ 0 0
$$371$$ −3.05137 −0.158419
$$372$$ 0 0
$$373$$ −10.1922 −0.527730 −0.263865 0.964560i $$-0.584997\pi$$
−0.263865 + 0.964560i $$0.584997\pi$$
$$374$$ 0 0
$$375$$ −1.39593 −0.0720856
$$376$$ 0 0
$$377$$ −43.2116 −2.22551
$$378$$ 0 0
$$379$$ −0.170094 −0.00873715 −0.00436858 0.999990i $$-0.501391\pi$$
−0.00436858 + 0.999990i $$0.501391\pi$$
$$380$$ 0 0
$$381$$ 1.80520 0.0924830
$$382$$ 0 0
$$383$$ −12.8052 −0.654315 −0.327157 0.944970i $$-0.606091\pi$$
−0.327157 + 0.944970i $$0.606091\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 12.2542 0.622918
$$388$$ 0 0
$$389$$ 30.1134 1.52681 0.763406 0.645919i $$-0.223526\pi$$
0.763406 + 0.645919i $$0.223526\pi$$
$$390$$ 0 0
$$391$$ 29.5837 1.49611
$$392$$ 0 0
$$393$$ −8.01333 −0.404219
$$394$$ 0 0
$$395$$ 2.27284 0.114359
$$396$$ 0 0
$$397$$ −10.1027 −0.507042 −0.253521 0.967330i $$-0.581589\pi$$
−0.253521 + 0.967330i $$0.581589\pi$$
$$398$$ 0 0
$$399$$ −10.0807 −0.504665
$$400$$ 0 0
$$401$$ 36.2435 1.80992 0.904958 0.425501i $$-0.139902\pi$$
0.904958 + 0.425501i $$0.139902\pi$$
$$402$$ 0 0
$$403$$ 1.65107 0.0822458
$$404$$ 0 0
$$405$$ 4.74049 0.235557
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −12.9460 −0.640138 −0.320069 0.947394i $$-0.603706\pi$$
−0.320069 + 0.947394i $$0.603706\pi$$
$$410$$ 0 0
$$411$$ −14.1214 −0.696555
$$412$$ 0 0
$$413$$ −11.8432 −0.582768
$$414$$ 0 0
$$415$$ 7.15412 0.351182
$$416$$ 0 0
$$417$$ −8.15676 −0.399438
$$418$$ 0 0
$$419$$ −22.2188 −1.08546 −0.542730 0.839907i $$-0.682609\pi$$
−0.542730 + 0.839907i $$0.682609\pi$$
$$420$$ 0 0
$$421$$ 27.3623 1.33355 0.666777 0.745257i $$-0.267673\pi$$
0.666777 + 0.745257i $$0.267673\pi$$
$$422$$ 0 0
$$423$$ −7.88392 −0.383329
$$424$$ 0 0
$$425$$ −5.31088 −0.257616
$$426$$ 0 0
$$427$$ 14.1878 0.686596
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 34.1382 1.64438 0.822188 0.569215i $$-0.192753\pi$$
0.822188 + 0.569215i $$0.192753\pi$$
$$432$$ 0 0
$$433$$ 3.63774 0.174819 0.0874093 0.996172i $$-0.472141\pi$$
0.0874093 + 0.996172i $$0.472141\pi$$
$$434$$ 0 0
$$435$$ 9.48098 0.454578
$$436$$ 0 0
$$437$$ 24.2995 1.16240
$$438$$ 0 0
$$439$$ 18.0621 0.862055 0.431028 0.902339i $$-0.358151\pi$$
0.431028 + 0.902339i $$0.358151\pi$$
$$440$$ 0 0
$$441$$ 4.47834 0.213254
$$442$$ 0 0
$$443$$ 40.3126 1.91531 0.957655 0.287918i $$-0.0929631\pi$$
0.957655 + 0.287918i $$0.0929631\pi$$
$$444$$ 0 0
$$445$$ 8.31088 0.393974
$$446$$ 0 0
$$447$$ −30.8316 −1.45829
$$448$$ 0 0
$$449$$ 10.8839 0.513644 0.256822 0.966459i $$-0.417325\pi$$
0.256822 + 0.966459i $$0.417325\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −10.4243 −0.489778
$$454$$ 0 0
$$455$$ −10.5324 −0.493764
$$456$$ 0 0
$$457$$ −38.6438 −1.80768 −0.903841 0.427868i $$-0.859265\pi$$
−0.903841 + 0.427868i $$0.859265\pi$$
$$458$$ 0 0
$$459$$ 30.0354 1.40193
$$460$$ 0 0
$$461$$ 25.6891 1.19646 0.598231 0.801324i $$-0.295871\pi$$
0.598231 + 0.801324i $$0.295871\pi$$
$$462$$ 0 0
$$463$$ −2.41190 −0.112091 −0.0560453 0.998428i $$-0.517849\pi$$
−0.0560453 + 0.998428i $$0.517849\pi$$
$$464$$ 0 0
$$465$$ −0.362259 −0.0167994
$$466$$ 0 0
$$467$$ −39.8423 −1.84368 −0.921842 0.387567i $$-0.873316\pi$$
−0.921842 + 0.387567i $$0.873316\pi$$
$$468$$ 0 0
$$469$$ 25.8273 1.19259
$$470$$ 0 0
$$471$$ 15.5518 0.716588
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.36226 −0.200154
$$476$$ 0 0
$$477$$ 1.93793 0.0887319
$$478$$ 0 0
$$479$$ −26.4517 −1.20861 −0.604304 0.796754i $$-0.706549\pi$$
−0.604304 + 0.796754i $$0.706549\pi$$
$$480$$ 0 0
$$481$$ −5.03804 −0.229715
$$482$$ 0 0
$$483$$ −12.8725 −0.585721
$$484$$ 0 0
$$485$$ −14.3623 −0.652157
$$486$$ 0 0
$$487$$ −28.5324 −1.29292 −0.646462 0.762946i $$-0.723752\pi$$
−0.646462 + 0.762946i $$0.723752\pi$$
$$488$$ 0 0
$$489$$ −11.9353 −0.539733
$$490$$ 0 0
$$491$$ 28.0354 1.26522 0.632610 0.774471i $$-0.281984\pi$$
0.632610 + 0.774471i $$0.281984\pi$$
$$492$$ 0 0
$$493$$ 36.0708 1.62455
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.74049 0.0780716
$$498$$ 0 0
$$499$$ −31.0380 −1.38945 −0.694727 0.719274i $$-0.744475\pi$$
−0.694727 + 0.719274i $$0.744475\pi$$
$$500$$ 0 0
$$501$$ −14.0594 −0.628129
$$502$$ 0 0
$$503$$ −37.0505 −1.65200 −0.825999 0.563671i $$-0.809389\pi$$
−0.825999 + 0.563671i $$0.809389\pi$$
$$504$$ 0 0
$$505$$ −3.79186 −0.168736
$$506$$ 0 0
$$507$$ 38.3579 1.70353
$$508$$ 0 0
$$509$$ 36.7219 1.62767 0.813834 0.581097i $$-0.197376\pi$$
0.813834 + 0.581097i $$0.197376\pi$$
$$510$$ 0 0
$$511$$ −19.3463 −0.855829
$$512$$ 0 0
$$513$$ 24.6705 1.08923
$$514$$ 0 0
$$515$$ −9.77589 −0.430777
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −17.8380 −0.783000
$$520$$ 0 0
$$521$$ 23.6572 1.03644 0.518220 0.855247i $$-0.326595\pi$$
0.518220 + 0.855247i $$0.326595\pi$$
$$522$$ 0 0
$$523$$ 16.7785 0.733674 0.366837 0.930285i $$-0.380441\pi$$
0.366837 + 0.930285i $$0.380441\pi$$
$$524$$ 0 0
$$525$$ 2.31088 0.100855
$$526$$ 0 0
$$527$$ −1.37823 −0.0600367
$$528$$ 0 0
$$529$$ 8.02930 0.349100
$$530$$ 0 0
$$531$$ 7.52166 0.326412
$$532$$ 0 0
$$533$$ −17.4357 −0.755224
$$534$$ 0 0
$$535$$ −14.9123 −0.644716
$$536$$ 0 0
$$537$$ 16.8946 0.729056
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.75646 0.247490 0.123745 0.992314i $$-0.460510\pi$$
0.123745 + 0.992314i $$0.460510\pi$$
$$542$$ 0 0
$$543$$ −26.7156 −1.14647
$$544$$ 0 0
$$545$$ 10.1541 0.434955
$$546$$ 0 0
$$547$$ 10.9486 0.468129 0.234065 0.972221i $$-0.424797\pi$$
0.234065 + 0.972221i $$0.424797\pi$$
$$548$$ 0 0
$$549$$ −9.01069 −0.384567
$$550$$ 0 0
$$551$$ 29.6279 1.26219
$$552$$ 0 0
$$553$$ −3.76256 −0.160000
$$554$$ 0 0
$$555$$ 1.10539 0.0469211
$$556$$ 0 0
$$557$$ 1.82991 0.0775356 0.0387678 0.999248i $$-0.487657\pi$$
0.0387678 + 0.999248i $$0.487657\pi$$
$$558$$ 0 0
$$559$$ −74.1549 −3.13642
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.61476 0.0680541 0.0340270 0.999421i $$-0.489167\pi$$
0.0340270 + 0.999421i $$0.489167\pi$$
$$564$$ 0 0
$$565$$ −12.6218 −0.531002
$$566$$ 0 0
$$567$$ −7.84761 −0.329569
$$568$$ 0 0
$$569$$ −23.4296 −0.982220 −0.491110 0.871098i $$-0.663409\pi$$
−0.491110 + 0.871098i $$0.663409\pi$$
$$570$$ 0 0
$$571$$ 30.1382 1.26124 0.630621 0.776091i $$-0.282800\pi$$
0.630621 + 0.776091i $$0.282800\pi$$
$$572$$ 0 0
$$573$$ −16.8760 −0.705005
$$574$$ 0 0
$$575$$ −5.57040 −0.232302
$$576$$ 0 0
$$577$$ −37.5651 −1.56386 −0.781928 0.623369i $$-0.785764\pi$$
−0.781928 + 0.623369i $$0.785764\pi$$
$$578$$ 0 0
$$579$$ 19.0114 0.790086
$$580$$ 0 0
$$581$$ −11.8432 −0.491340
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 6.68912 0.276561
$$586$$ 0 0
$$587$$ 24.7962 1.02345 0.511725 0.859149i $$-0.329007\pi$$
0.511725 + 0.859149i $$0.329007\pi$$
$$588$$ 0 0
$$589$$ −1.13205 −0.0466454
$$590$$ 0 0
$$591$$ −20.6484 −0.849363
$$592$$ 0 0
$$593$$ 46.2542 1.89943 0.949717 0.313110i $$-0.101371\pi$$
0.949717 + 0.313110i $$0.101371\pi$$
$$594$$ 0 0
$$595$$ 8.79186 0.360431
$$596$$ 0 0
$$597$$ −24.6891 −1.01046
$$598$$ 0 0
$$599$$ −3.01069 −0.123014 −0.0615068 0.998107i $$-0.519591\pi$$
−0.0615068 + 0.998107i $$0.519591\pi$$
$$600$$ 0 0
$$601$$ 5.52166 0.225233 0.112617 0.993639i $$-0.464077\pi$$
0.112617 + 0.993639i $$0.464077\pi$$
$$602$$ 0 0
$$603$$ −16.4029 −0.667979
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −14.3356 −0.581864 −0.290932 0.956744i $$-0.593965\pi$$
−0.290932 + 0.956744i $$0.593965\pi$$
$$608$$ 0 0
$$609$$ −15.6952 −0.636002
$$610$$ 0 0
$$611$$ 47.7085 1.93008
$$612$$ 0 0
$$613$$ 41.3056 1.66832 0.834159 0.551524i $$-0.185954\pi$$
0.834159 + 0.551524i $$0.185954\pi$$
$$614$$ 0 0
$$615$$ 3.82554 0.154261
$$616$$ 0 0
$$617$$ 40.1869 1.61786 0.808932 0.587903i $$-0.200046\pi$$
0.808932 + 0.587903i $$0.200046\pi$$
$$618$$ 0 0
$$619$$ −47.3463 −1.90301 −0.951504 0.307636i $$-0.900462\pi$$
−0.951504 + 0.307636i $$0.900462\pi$$
$$620$$ 0 0
$$621$$ 31.5030 1.26417
$$622$$ 0 0
$$623$$ −13.7582 −0.551210
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4.20550 0.167684
$$630$$ 0 0
$$631$$ −33.9681 −1.35225 −0.676124 0.736788i $$-0.736341\pi$$
−0.676124 + 0.736788i $$0.736341\pi$$
$$632$$ 0 0
$$633$$ −37.8113 −1.50286
$$634$$ 0 0
$$635$$ −1.29318 −0.0513184
$$636$$ 0 0
$$637$$ −27.1001 −1.07375
$$638$$ 0 0
$$639$$ −1.10539 −0.0437285
$$640$$ 0 0
$$641$$ 24.3082 0.960118 0.480059 0.877236i $$-0.340615\pi$$
0.480059 + 0.877236i $$0.340615\pi$$
$$642$$ 0 0
$$643$$ 38.6209 1.52306 0.761529 0.648131i $$-0.224449\pi$$
0.761529 + 0.648131i $$0.224449\pi$$
$$644$$ 0 0
$$645$$ 16.2702 0.640639
$$646$$ 0 0
$$647$$ −34.9663 −1.37467 −0.687334 0.726341i $$-0.741219\pi$$
−0.687334 + 0.726341i $$0.741219\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0.599699 0.0235041
$$652$$ 0 0
$$653$$ −9.44030 −0.369427 −0.184714 0.982792i $$-0.559136\pi$$
−0.184714 + 0.982792i $$0.559136\pi$$
$$654$$ 0 0
$$655$$ 5.74049 0.224299
$$656$$ 0 0
$$657$$ 12.2869 0.479356
$$658$$ 0 0
$$659$$ −14.7652 −0.575171 −0.287585 0.957755i $$-0.592852\pi$$
−0.287585 + 0.957755i $$0.592852\pi$$
$$660$$ 0 0
$$661$$ −8.32422 −0.323775 −0.161887 0.986809i $$-0.551758\pi$$
−0.161887 + 0.986809i $$0.551758\pi$$
$$662$$ 0 0
$$663$$ −47.1675 −1.83183
$$664$$ 0 0
$$665$$ 7.22147 0.280037
$$666$$ 0 0
$$667$$ 37.8334 1.46491
$$668$$ 0 0
$$669$$ −12.1355 −0.469186
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −0.388923 −0.0149919 −0.00749595 0.999972i $$-0.502386\pi$$
−0.00749595 + 0.999972i $$0.502386\pi$$
$$674$$ 0 0
$$675$$ −5.65544 −0.217678
$$676$$ 0 0
$$677$$ −41.5837 −1.59819 −0.799096 0.601203i $$-0.794688\pi$$
−0.799096 + 0.601203i $$0.794688\pi$$
$$678$$ 0 0
$$679$$ 23.7759 0.912435
$$680$$ 0 0
$$681$$ 4.75910 0.182369
$$682$$ 0 0
$$683$$ −13.3153 −0.509494 −0.254747 0.967008i $$-0.581992\pi$$
−0.254747 + 0.967008i $$0.581992\pi$$
$$684$$ 0 0
$$685$$ 10.1161 0.386516
$$686$$ 0 0
$$687$$ 18.6342 0.710939
$$688$$ 0 0
$$689$$ −11.7272 −0.446769
$$690$$ 0 0
$$691$$ 3.54373 0.134810 0.0674049 0.997726i $$-0.478528\pi$$
0.0674049 + 0.997726i $$0.478528\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 5.84324 0.221647
$$696$$ 0 0
$$697$$ 14.5544 0.551288
$$698$$ 0 0
$$699$$ −20.4296 −0.772719
$$700$$ 0 0
$$701$$ −24.3489 −0.919646 −0.459823 0.888011i $$-0.652087\pi$$
−0.459823 + 0.888011i $$0.652087\pi$$
$$702$$ 0 0
$$703$$ 3.45431 0.130282
$$704$$ 0 0
$$705$$ −10.4676 −0.394234
$$706$$ 0 0
$$707$$ 6.27721 0.236079
$$708$$ 0 0
$$709$$ 21.6484 0.813024 0.406512 0.913645i $$-0.366745\pi$$
0.406512 + 0.913645i $$0.366745\pi$$
$$710$$ 0 0
$$711$$ 2.38961 0.0896173
$$712$$ 0 0
$$713$$ −1.44558 −0.0541373
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 11.6731 0.435942
$$718$$ 0 0
$$719$$ 35.2302 1.31387 0.656933 0.753949i $$-0.271854\pi$$
0.656933 + 0.753949i $$0.271854\pi$$
$$720$$ 0 0
$$721$$ 16.1834 0.602702
$$722$$ 0 0
$$723$$ 15.4233 0.573598
$$724$$ 0 0
$$725$$ −6.79186 −0.252243
$$726$$ 0 0
$$727$$ 22.3446 0.828714 0.414357 0.910114i $$-0.364007\pi$$
0.414357 + 0.910114i $$0.364007\pi$$
$$728$$ 0 0
$$729$$ 28.6679 1.06177
$$730$$ 0 0
$$731$$ 61.9007 2.28948
$$732$$ 0 0
$$733$$ −51.5111 −1.90261 −0.951303 0.308257i $$-0.900254\pi$$
−0.951303 + 0.308257i $$0.900254\pi$$
$$734$$ 0 0
$$735$$ 5.94599 0.219321
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 15.7219 0.578339 0.289169 0.957278i $$-0.406621\pi$$
0.289169 + 0.957278i $$0.406621\pi$$
$$740$$ 0 0
$$741$$ −38.7424 −1.42324
$$742$$ 0 0
$$743$$ −22.6041 −0.829263 −0.414631 0.909989i $$-0.636089\pi$$
−0.414631 + 0.909989i $$0.636089\pi$$
$$744$$ 0 0
$$745$$ 22.0868 0.809197
$$746$$ 0 0
$$747$$ 7.52166 0.275203
$$748$$ 0 0
$$749$$ 24.6865 0.902024
$$750$$ 0 0
$$751$$ 27.4490 1.00163 0.500815 0.865554i $$-0.333034\pi$$
0.500815 + 0.865554i $$0.333034\pi$$
$$752$$ 0 0
$$753$$ −18.9433 −0.690334
$$754$$ 0 0
$$755$$ 7.46765 0.271775
$$756$$ 0 0
$$757$$ 13.5704 0.493224 0.246612 0.969114i $$-0.420683\pi$$
0.246612 + 0.969114i $$0.420683\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.7892 −1.00736 −0.503679 0.863891i $$-0.668021\pi$$
−0.503679 + 0.863891i $$0.668021\pi$$
$$762$$ 0 0
$$763$$ −16.8096 −0.608547
$$764$$ 0 0
$$765$$ −5.58373 −0.201880
$$766$$ 0 0
$$767$$ −45.5164 −1.64350
$$768$$ 0 0
$$769$$ −6.13815 −0.221347 −0.110674 0.993857i $$-0.535301\pi$$
−0.110674 + 0.993857i $$0.535301\pi$$
$$770$$ 0 0
$$771$$ −1.23021 −0.0443048
$$772$$ 0 0
$$773$$ −21.9867 −0.790805 −0.395403 0.918508i $$-0.629395\pi$$
−0.395403 + 0.918508i $$0.629395\pi$$
$$774$$ 0 0
$$775$$ 0.259511 0.00932189
$$776$$ 0 0
$$777$$ −1.82991 −0.0656475
$$778$$ 0 0
$$779$$ 11.9547 0.428322
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 38.4110 1.37270
$$784$$ 0 0
$$785$$ −11.1408 −0.397632
$$786$$ 0 0
$$787$$ −19.3959 −0.691390 −0.345695 0.938347i $$-0.612357\pi$$
−0.345695 + 0.938347i $$0.612357\pi$$
$$788$$ 0 0
$$789$$ −20.0487 −0.713754
$$790$$ 0 0
$$791$$ 20.8946 0.742927
$$792$$ 0 0
$$793$$ 54.5271 1.93631
$$794$$ 0 0
$$795$$ 2.57303 0.0912561
$$796$$ 0 0
$$797$$ 41.2516 1.46121 0.730603 0.682802i $$-0.239239\pi$$
0.730603 + 0.682802i $$0.239239\pi$$
$$798$$ 0 0
$$799$$ −39.8246 −1.40889
$$800$$ 0 0
$$801$$ 8.73785 0.308737
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 9.22147 0.325014
$$806$$ 0 0
$$807$$ 15.6607 0.551283
$$808$$ 0 0
$$809$$ 31.0682 1.09230 0.546149 0.837688i $$-0.316093\pi$$
0.546149 + 0.837688i $$0.316093\pi$$
$$810$$ 0 0
$$811$$ −3.29227 −0.115607 −0.0578037 0.998328i $$-0.518410\pi$$
−0.0578037 + 0.998328i $$0.518410\pi$$
$$812$$ 0 0
$$813$$ 20.0673 0.703793
$$814$$ 0 0
$$815$$ 8.55005 0.299495
$$816$$ 0 0
$$817$$ 50.8441 1.77881
$$818$$ 0 0
$$819$$ −11.0734 −0.386937
$$820$$ 0 0
$$821$$ −31.2276 −1.08985 −0.544925 0.838485i $$-0.683442\pi$$
−0.544925 + 0.838485i $$0.683442\pi$$
$$822$$ 0 0
$$823$$ −8.04437 −0.280409 −0.140204 0.990123i $$-0.544776\pi$$
−0.140204 + 0.990123i $$0.544776\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −55.3286 −1.92396 −0.961982 0.273114i $$-0.911946\pi$$
−0.961982 + 0.273114i $$0.911946\pi$$
$$828$$ 0 0
$$829$$ 52.1895 1.81262 0.906309 0.422617i $$-0.138888\pi$$
0.906309 + 0.422617i $$0.138888\pi$$
$$830$$ 0 0
$$831$$ 43.5571 1.51098
$$832$$ 0 0
$$833$$ 22.6218 0.783798
$$834$$ 0 0
$$835$$ 10.0717 0.348546
$$836$$ 0 0
$$837$$ −1.46765 −0.0507293
$$838$$ 0 0
$$839$$ −51.1355 −1.76539 −0.882697 0.469943i $$-0.844275\pi$$
−0.882697 + 0.469943i $$0.844275\pi$$
$$840$$ 0 0
$$841$$ 17.1294 0.590669
$$842$$ 0 0
$$843$$ 23.9646 0.825385
$$844$$ 0 0
$$845$$ −27.4783 −0.945284
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 16.6325 0.570825
$$850$$ 0 0
$$851$$ 4.41099 0.151207
$$852$$ 0 0
$$853$$ −13.7892 −0.472134 −0.236067 0.971737i $$-0.575858\pi$$
−0.236067 + 0.971737i $$0.575858\pi$$
$$854$$ 0 0
$$855$$ −4.58637 −0.156850
$$856$$ 0 0
$$857$$ 20.3756 0.696017 0.348008 0.937491i $$-0.386858\pi$$
0.348008 + 0.937491i $$0.386858\pi$$
$$858$$ 0 0
$$859$$ 39.2302 1.33852 0.669259 0.743029i $$-0.266612\pi$$
0.669259 + 0.743029i $$0.266612\pi$$
$$860$$ 0 0
$$861$$ −6.33296 −0.215827
$$862$$ 0 0
$$863$$ 36.6395 1.24722 0.623611 0.781735i $$-0.285665\pi$$
0.623611 + 0.781735i $$0.285665\pi$$
$$864$$ 0 0
$$865$$ 12.7785 0.434483
$$866$$ 0 0
$$867$$ 15.6421 0.531234
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 99.2603 3.36331
$$872$$ 0 0
$$873$$ −15.1001 −0.511061
$$874$$ 0 0
$$875$$ −1.65544 −0.0559642
$$876$$ 0 0
$$877$$ 46.1108 1.55705 0.778526 0.627613i $$-0.215968\pi$$
0.778526 + 0.627613i $$0.215968\pi$$
$$878$$ 0 0
$$879$$ −14.7024 −0.495901
$$880$$ 0 0
$$881$$ 1.32158 0.0445251 0.0222625 0.999752i $$-0.492913\pi$$
0.0222625 + 0.999752i $$0.492913\pi$$
$$882$$ 0 0
$$883$$ −17.2302 −0.579843 −0.289921 0.957050i $$-0.593629\pi$$
−0.289921 + 0.957050i $$0.593629\pi$$
$$884$$ 0 0
$$885$$ 9.98667 0.335698
$$886$$ 0 0
$$887$$ −17.7175 −0.594896 −0.297448 0.954738i $$-0.596135\pi$$
−0.297448 + 0.954738i $$0.596135\pi$$
$$888$$ 0 0
$$889$$ 2.14079 0.0717998
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −32.7112 −1.09464
$$894$$ 0 0
$$895$$ −12.1027 −0.404550
$$896$$ 0 0
$$897$$ −49.4722 −1.65183
$$898$$ 0 0
$$899$$ −1.76256 −0.0587847
$$900$$ 0 0
$$901$$ 9.78922 0.326126
$$902$$ 0 0
$$903$$ −26.9344 −0.896320
$$904$$ 0 0
$$905$$ 19.1382 0.636174
$$906$$ 0 0
$$907$$ −20.2011 −0.670767 −0.335384 0.942082i $$-0.608866\pi$$
−0.335384 + 0.942082i $$0.608866\pi$$
$$908$$ 0 0
$$909$$ −3.98667 −0.132229
$$910$$ 0 0
$$911$$ −45.1001 −1.49423 −0.747117 0.664693i $$-0.768562\pi$$
−0.747117 + 0.664693i $$0.768562\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −11.9637 −0.395507
$$916$$ 0 0
$$917$$ −9.50305 −0.313818
$$918$$ 0 0
$$919$$ 3.77589 0.124555 0.0622776 0.998059i $$-0.480164\pi$$
0.0622776 + 0.998059i $$0.480164\pi$$
$$920$$ 0 0
$$921$$ −15.5704 −0.513062
$$922$$ 0 0
$$923$$ 6.68912 0.220175
$$924$$ 0 0
$$925$$ −0.791864 −0.0260363
$$926$$ 0 0
$$927$$ −10.2781 −0.337578
$$928$$ 0 0
$$929$$ 26.7245 0.876803 0.438401 0.898779i $$-0.355545\pi$$
0.438401 + 0.898779i $$0.355545\pi$$
$$930$$ 0 0
$$931$$ 18.5811 0.608971
$$932$$ 0 0
$$933$$ −15.3330 −0.501978
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −6.68384 −0.218351 −0.109176 0.994022i $$-0.534821\pi$$
−0.109176 + 0.994022i $$0.534821\pi$$
$$938$$ 0 0
$$939$$ 17.3063 0.564769
$$940$$ 0 0
$$941$$ 13.1027 0.427137 0.213569 0.976928i $$-0.431491\pi$$
0.213569 + 0.976928i $$0.431491\pi$$
$$942$$ 0 0
$$943$$ 15.2656 0.497117
$$944$$ 0 0
$$945$$ 9.36226 0.304554
$$946$$ 0 0
$$947$$ 47.6058 1.54698 0.773490 0.633808i $$-0.218509\pi$$
0.773490 + 0.633808i $$0.218509\pi$$
$$948$$ 0 0
$$949$$ −74.3524 −2.41358
$$950$$ 0 0
$$951$$ 9.13733 0.296298
$$952$$ 0 0
$$953$$ −15.2348 −0.493504 −0.246752 0.969079i $$-0.579363\pi$$
−0.246752 + 0.969079i $$0.579363\pi$$
$$954$$ 0 0
$$955$$ 12.0894 0.391204
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −16.7466 −0.540776
$$960$$ 0 0
$$961$$ −30.9327 −0.997828
$$962$$ 0 0
$$963$$ −15.6784 −0.505230
$$964$$ 0 0
$$965$$ −13.6191 −0.438415
$$966$$ 0 0
$$967$$ 36.4243 1.17133 0.585664 0.810554i $$-0.300834\pi$$
0.585664 + 0.810554i $$0.300834\pi$$
$$968$$ 0 0
$$969$$ 32.3402 1.03892
$$970$$ 0 0
$$971$$ 35.1001 1.12642 0.563208 0.826315i $$-0.309567\pi$$
0.563208 + 0.826315i $$0.309567\pi$$
$$972$$ 0 0
$$973$$ −9.67314 −0.310107
$$974$$ 0 0
$$975$$ 8.88128 0.284429
$$976$$ 0 0
$$977$$ −42.8946 −1.37232 −0.686160 0.727451i $$-0.740705\pi$$
−0.686160 + 0.727451i $$0.740705\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.6758 0.340852
$$982$$ 0 0
$$983$$ −28.1559 −0.898032 −0.449016 0.893524i $$-0.648225\pi$$
−0.449016 + 0.893524i $$0.648225\pi$$
$$984$$ 0 0
$$985$$ 14.7919 0.471308
$$986$$ 0 0
$$987$$ 17.3286 0.551575
$$988$$ 0 0
$$989$$ 64.9254 2.06451
$$990$$ 0 0
$$991$$ −43.7085 −1.38845 −0.694224 0.719759i $$-0.744252\pi$$
−0.694224 + 0.719759i $$0.744252\pi$$
$$992$$ 0 0
$$993$$ −41.7085 −1.32358
$$994$$ 0 0
$$995$$ 17.6865 0.560699
$$996$$ 0 0
$$997$$ 13.0247 0.412497 0.206248 0.978500i $$-0.433875\pi$$
0.206248 + 0.978500i $$0.433875\pi$$
$$998$$ 0 0
$$999$$ 4.47834 0.141688
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 4840.2.a.t.1.2 3
4.3 odd 2 9680.2.a.cc.1.2 3
11.10 odd 2 4840.2.a.u.1.2 yes 3
44.43 even 2 9680.2.a.ca.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.t.1.2 3 1.1 even 1 trivial
4840.2.a.u.1.2 yes 3 11.10 odd 2
9680.2.a.ca.1.2 3 44.43 even 2
9680.2.a.cc.1.2 3 4.3 odd 2