# Properties

 Label 1248.2.a.p.1.1 Level $1248$ Weight $2$ Character 1248.1 Self dual yes Analytic conductor $9.965$ Analytic rank $0$ 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] = [1248,2,Mod(1,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, 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("1248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1248 = 2^{5} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1248.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: $$9.96533017226$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 1248.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.96239 q^{5} -3.35026 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.96239 q^{5} -3.35026 q^{7} +1.00000 q^{9} +1.61213 q^{11} +1.00000 q^{13} -2.96239 q^{15} +2.00000 q^{17} +3.35026 q^{19} -3.35026 q^{21} +6.70052 q^{23} +3.77575 q^{25} +1.00000 q^{27} +2.00000 q^{29} -6.57452 q^{31} +1.61213 q^{33} +9.92478 q^{35} +7.92478 q^{37} +1.00000 q^{39} +6.96239 q^{41} +0.775746 q^{43} -2.96239 q^{45} -2.38787 q^{47} +4.22425 q^{49} +2.00000 q^{51} +11.9248 q^{53} -4.77575 q^{55} +3.35026 q^{57} -0.312650 q^{59} +14.6253 q^{61} -3.35026 q^{63} -2.96239 q^{65} -8.12601 q^{67} +6.70052 q^{69} +4.31265 q^{71} +0.0752228 q^{73} +3.77575 q^{75} -5.40105 q^{77} -12.0000 q^{79} +1.00000 q^{81} +8.31265 q^{83} -5.92478 q^{85} +2.00000 q^{87} +8.88717 q^{89} -3.35026 q^{91} -6.57452 q^{93} -9.92478 q^{95} -7.92478 q^{97} +1.61213 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 2 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 2 * q^5 + 3 * q^9 $$3 q + 3 q^{3} + 2 q^{5} + 3 q^{9} + 4 q^{11} + 3 q^{13} + 2 q^{15} + 6 q^{17} + 13 q^{25} + 3 q^{27} + 6 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{35} + 2 q^{37} + 3 q^{39} + 10 q^{41} + 4 q^{43} + 2 q^{45} - 8 q^{47} + 11 q^{49} + 6 q^{51} + 14 q^{53} - 16 q^{55} + 20 q^{59} + 2 q^{61} + 2 q^{65} - 16 q^{67} - 8 q^{71} + 22 q^{73} + 13 q^{75} + 24 q^{77} - 36 q^{79} + 3 q^{81} + 4 q^{83} + 4 q^{85} + 6 q^{87} - 6 q^{89} - 8 q^{93} - 8 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 2 * q^5 + 3 * q^9 + 4 * q^11 + 3 * q^13 + 2 * q^15 + 6 * q^17 + 13 * q^25 + 3 * q^27 + 6 * q^29 - 8 * q^31 + 4 * q^33 + 8 * q^35 + 2 * q^37 + 3 * q^39 + 10 * q^41 + 4 * q^43 + 2 * q^45 - 8 * q^47 + 11 * q^49 + 6 * q^51 + 14 * q^53 - 16 * q^55 + 20 * q^59 + 2 * q^61 + 2 * q^65 - 16 * q^67 - 8 * q^71 + 22 * q^73 + 13 * q^75 + 24 * q^77 - 36 * q^79 + 3 * q^81 + 4 * q^83 + 4 * q^85 + 6 * q^87 - 6 * q^89 - 8 * q^93 - 8 * q^95 - 2 * q^97 + 4 * 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
$$5$$ −2.96239 −1.32482 −0.662410 0.749141i $$-0.730466\pi$$
−0.662410 + 0.749141i $$0.730466\pi$$
$$6$$ 0 0
$$7$$ −3.35026 −1.26628 −0.633140 0.774037i $$-0.718234\pi$$
−0.633140 + 0.774037i $$0.718234\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.61213 0.486075 0.243037 0.970017i $$-0.421856\pi$$
0.243037 + 0.970017i $$0.421856\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −2.96239 −0.764885
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 3.35026 0.768603 0.384301 0.923208i $$-0.374442\pi$$
0.384301 + 0.923208i $$0.374442\pi$$
$$20$$ 0 0
$$21$$ −3.35026 −0.731087
$$22$$ 0 0
$$23$$ 6.70052 1.39716 0.698578 0.715534i $$-0.253817\pi$$
0.698578 + 0.715534i $$0.253817\pi$$
$$24$$ 0 0
$$25$$ 3.77575 0.755149
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −6.57452 −1.18082 −0.590409 0.807104i $$-0.701034\pi$$
−0.590409 + 0.807104i $$0.701034\pi$$
$$32$$ 0 0
$$33$$ 1.61213 0.280635
$$34$$ 0 0
$$35$$ 9.92478 1.67759
$$36$$ 0 0
$$37$$ 7.92478 1.30283 0.651413 0.758724i $$-0.274177\pi$$
0.651413 + 0.758724i $$0.274177\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 6.96239 1.08734 0.543671 0.839298i $$-0.317034\pi$$
0.543671 + 0.839298i $$0.317034\pi$$
$$42$$ 0 0
$$43$$ 0.775746 0.118300 0.0591501 0.998249i $$-0.481161\pi$$
0.0591501 + 0.998249i $$0.481161\pi$$
$$44$$ 0 0
$$45$$ −2.96239 −0.441607
$$46$$ 0 0
$$47$$ −2.38787 −0.348307 −0.174154 0.984719i $$-0.555719\pi$$
−0.174154 + 0.984719i $$0.555719\pi$$
$$48$$ 0 0
$$49$$ 4.22425 0.603465
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 11.9248 1.63799 0.818997 0.573798i $$-0.194530\pi$$
0.818997 + 0.573798i $$0.194530\pi$$
$$54$$ 0 0
$$55$$ −4.77575 −0.643961
$$56$$ 0 0
$$57$$ 3.35026 0.443753
$$58$$ 0 0
$$59$$ −0.312650 −0.0407036 −0.0203518 0.999793i $$-0.506479\pi$$
−0.0203518 + 0.999793i $$0.506479\pi$$
$$60$$ 0 0
$$61$$ 14.6253 1.87258 0.936289 0.351231i $$-0.114237\pi$$
0.936289 + 0.351231i $$0.114237\pi$$
$$62$$ 0 0
$$63$$ −3.35026 −0.422093
$$64$$ 0 0
$$65$$ −2.96239 −0.367439
$$66$$ 0 0
$$67$$ −8.12601 −0.992750 −0.496375 0.868108i $$-0.665336\pi$$
−0.496375 + 0.868108i $$0.665336\pi$$
$$68$$ 0 0
$$69$$ 6.70052 0.806648
$$70$$ 0 0
$$71$$ 4.31265 0.511817 0.255909 0.966701i $$-0.417625\pi$$
0.255909 + 0.966701i $$0.417625\pi$$
$$72$$ 0 0
$$73$$ 0.0752228 0.00880416 0.00440208 0.999990i $$-0.498599\pi$$
0.00440208 + 0.999990i $$0.498599\pi$$
$$74$$ 0 0
$$75$$ 3.77575 0.435986
$$76$$ 0 0
$$77$$ −5.40105 −0.615506
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 8.31265 0.912432 0.456216 0.889869i $$-0.349204\pi$$
0.456216 + 0.889869i $$0.349204\pi$$
$$84$$ 0 0
$$85$$ −5.92478 −0.642632
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 8.88717 0.942038 0.471019 0.882123i $$-0.343886\pi$$
0.471019 + 0.882123i $$0.343886\pi$$
$$90$$ 0 0
$$91$$ −3.35026 −0.351203
$$92$$ 0 0
$$93$$ −6.57452 −0.681745
$$94$$ 0 0
$$95$$ −9.92478 −1.01826
$$96$$ 0 0
$$97$$ −7.92478 −0.804639 −0.402320 0.915499i $$-0.631796\pi$$
−0.402320 + 0.915499i $$0.631796\pi$$
$$98$$ 0 0
$$99$$ 1.61213 0.162025
$$100$$ 0 0
$$101$$ 15.4010 1.53246 0.766231 0.642566i $$-0.222130\pi$$
0.766231 + 0.642566i $$0.222130\pi$$
$$102$$ 0 0
$$103$$ 4.62530 0.455744 0.227872 0.973691i $$-0.426823\pi$$
0.227872 + 0.973691i $$0.426823\pi$$
$$104$$ 0 0
$$105$$ 9.92478 0.968559
$$106$$ 0 0
$$107$$ −14.5501 −1.40661 −0.703305 0.710889i $$-0.748293\pi$$
−0.703305 + 0.710889i $$0.748293\pi$$
$$108$$ 0 0
$$109$$ −15.4010 −1.47515 −0.737576 0.675264i $$-0.764030\pi$$
−0.737576 + 0.675264i $$0.764030\pi$$
$$110$$ 0 0
$$111$$ 7.92478 0.752187
$$112$$ 0 0
$$113$$ −9.47627 −0.891452 −0.445726 0.895169i $$-0.647055\pi$$
−0.445726 + 0.895169i $$0.647055\pi$$
$$114$$ 0 0
$$115$$ −19.8496 −1.85098
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −6.70052 −0.614236
$$120$$ 0 0
$$121$$ −8.40105 −0.763732
$$122$$ 0 0
$$123$$ 6.96239 0.627777
$$124$$ 0 0
$$125$$ 3.62672 0.324383
$$126$$ 0 0
$$127$$ −5.29948 −0.470252 −0.235126 0.971965i $$-0.575550\pi$$
−0.235126 + 0.971965i $$0.575550\pi$$
$$128$$ 0 0
$$129$$ 0.775746 0.0683007
$$130$$ 0 0
$$131$$ −9.14903 −0.799355 −0.399677 0.916656i $$-0.630878\pi$$
−0.399677 + 0.916656i $$0.630878\pi$$
$$132$$ 0 0
$$133$$ −11.2243 −0.973266
$$134$$ 0 0
$$135$$ −2.96239 −0.254962
$$136$$ 0 0
$$137$$ 0.513881 0.0439038 0.0219519 0.999759i $$-0.493012\pi$$
0.0219519 + 0.999759i $$0.493012\pi$$
$$138$$ 0 0
$$139$$ 13.9248 1.18108 0.590542 0.807007i $$-0.298914\pi$$
0.590542 + 0.807007i $$0.298914\pi$$
$$140$$ 0 0
$$141$$ −2.38787 −0.201095
$$142$$ 0 0
$$143$$ 1.61213 0.134813
$$144$$ 0 0
$$145$$ −5.92478 −0.492026
$$146$$ 0 0
$$147$$ 4.22425 0.348411
$$148$$ 0 0
$$149$$ −12.8872 −1.05576 −0.527879 0.849320i $$-0.677013\pi$$
−0.527879 + 0.849320i $$0.677013\pi$$
$$150$$ 0 0
$$151$$ −13.2750 −1.08031 −0.540154 0.841566i $$-0.681634\pi$$
−0.540154 + 0.841566i $$0.681634\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 19.4763 1.56437
$$156$$ 0 0
$$157$$ 21.0738 1.68187 0.840936 0.541134i $$-0.182005\pi$$
0.840936 + 0.541134i $$0.182005\pi$$
$$158$$ 0 0
$$159$$ 11.9248 0.945696
$$160$$ 0 0
$$161$$ −22.4485 −1.76919
$$162$$ 0 0
$$163$$ 21.9003 1.71537 0.857683 0.514178i $$-0.171903\pi$$
0.857683 + 0.514178i $$0.171903\pi$$
$$164$$ 0 0
$$165$$ −4.77575 −0.371791
$$166$$ 0 0
$$167$$ 17.4617 1.35123 0.675613 0.737257i $$-0.263879\pi$$
0.675613 + 0.737257i $$0.263879\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 3.35026 0.256201
$$172$$ 0 0
$$173$$ 21.8496 1.66119 0.830595 0.556876i $$-0.188000\pi$$
0.830595 + 0.556876i $$0.188000\pi$$
$$174$$ 0 0
$$175$$ −12.6497 −0.956230
$$176$$ 0 0
$$177$$ −0.312650 −0.0235002
$$178$$ 0 0
$$179$$ 21.9248 1.63873 0.819367 0.573269i $$-0.194325\pi$$
0.819367 + 0.573269i $$0.194325\pi$$
$$180$$ 0 0
$$181$$ −18.6253 −1.38441 −0.692204 0.721702i $$-0.743360\pi$$
−0.692204 + 0.721702i $$0.743360\pi$$
$$182$$ 0 0
$$183$$ 14.6253 1.08113
$$184$$ 0 0
$$185$$ −23.4763 −1.72601
$$186$$ 0 0
$$187$$ 3.22425 0.235781
$$188$$ 0 0
$$189$$ −3.35026 −0.243696
$$190$$ 0 0
$$191$$ −21.7743 −1.57554 −0.787768 0.615972i $$-0.788763\pi$$
−0.787768 + 0.615972i $$0.788763\pi$$
$$192$$ 0 0
$$193$$ 19.2506 1.38569 0.692844 0.721087i $$-0.256357\pi$$
0.692844 + 0.721087i $$0.256357\pi$$
$$194$$ 0 0
$$195$$ −2.96239 −0.212141
$$196$$ 0 0
$$197$$ −4.51388 −0.321601 −0.160800 0.986987i $$-0.551408\pi$$
−0.160800 + 0.986987i $$0.551408\pi$$
$$198$$ 0 0
$$199$$ 9.14903 0.648558 0.324279 0.945962i $$-0.394878\pi$$
0.324279 + 0.945962i $$0.394878\pi$$
$$200$$ 0 0
$$201$$ −8.12601 −0.573164
$$202$$ 0 0
$$203$$ −6.70052 −0.470285
$$204$$ 0 0
$$205$$ −20.6253 −1.44053
$$206$$ 0 0
$$207$$ 6.70052 0.465719
$$208$$ 0 0
$$209$$ 5.40105 0.373598
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 4.31265 0.295498
$$214$$ 0 0
$$215$$ −2.29806 −0.156727
$$216$$ 0 0
$$217$$ 22.0263 1.49525
$$218$$ 0 0
$$219$$ 0.0752228 0.00508308
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ −18.4241 −1.23377 −0.616883 0.787054i $$-0.711605\pi$$
−0.616883 + 0.787054i $$0.711605\pi$$
$$224$$ 0 0
$$225$$ 3.77575 0.251716
$$226$$ 0 0
$$227$$ −1.61213 −0.107001 −0.0535003 0.998568i $$-0.517038\pi$$
−0.0535003 + 0.998568i $$0.517038\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ −5.40105 −0.355363
$$232$$ 0 0
$$233$$ 0.0752228 0.00492801 0.00246400 0.999997i $$-0.499216\pi$$
0.00246400 + 0.999997i $$0.499216\pi$$
$$234$$ 0 0
$$235$$ 7.07381 0.461444
$$236$$ 0 0
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ −9.08840 −0.587880 −0.293940 0.955824i $$-0.594967\pi$$
−0.293940 + 0.955824i $$0.594967\pi$$
$$240$$ 0 0
$$241$$ 15.7743 1.01611 0.508057 0.861323i $$-0.330364\pi$$
0.508057 + 0.861323i $$0.330364\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −12.5139 −0.799483
$$246$$ 0 0
$$247$$ 3.35026 0.213172
$$248$$ 0 0
$$249$$ 8.31265 0.526793
$$250$$ 0 0
$$251$$ 15.2243 0.960946 0.480473 0.877009i $$-0.340465\pi$$
0.480473 + 0.877009i $$0.340465\pi$$
$$252$$ 0 0
$$253$$ 10.8021 0.679122
$$254$$ 0 0
$$255$$ −5.92478 −0.371024
$$256$$ 0 0
$$257$$ −4.07522 −0.254205 −0.127103 0.991890i $$-0.540568\pi$$
−0.127103 + 0.991890i $$0.540568\pi$$
$$258$$ 0 0
$$259$$ −26.5501 −1.64974
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 12.7757 0.787786 0.393893 0.919156i $$-0.371128\pi$$
0.393893 + 0.919156i $$0.371128\pi$$
$$264$$ 0 0
$$265$$ −35.3258 −2.17005
$$266$$ 0 0
$$267$$ 8.88717 0.543886
$$268$$ 0 0
$$269$$ 21.8496 1.33219 0.666095 0.745867i $$-0.267964\pi$$
0.666095 + 0.745867i $$0.267964\pi$$
$$270$$ 0 0
$$271$$ 0.499293 0.0303299 0.0151649 0.999885i $$-0.495173\pi$$
0.0151649 + 0.999885i $$0.495173\pi$$
$$272$$ 0 0
$$273$$ −3.35026 −0.202767
$$274$$ 0 0
$$275$$ 6.08698 0.367059
$$276$$ 0 0
$$277$$ −13.8496 −0.832139 −0.416070 0.909333i $$-0.636593\pi$$
−0.416070 + 0.909333i $$0.636593\pi$$
$$278$$ 0 0
$$279$$ −6.57452 −0.393606
$$280$$ 0 0
$$281$$ −14.4387 −0.861338 −0.430669 0.902510i $$-0.641722\pi$$
−0.430669 + 0.902510i $$0.641722\pi$$
$$282$$ 0 0
$$283$$ −6.55008 −0.389362 −0.194681 0.980867i $$-0.562367\pi$$
−0.194681 + 0.980867i $$0.562367\pi$$
$$284$$ 0 0
$$285$$ −9.92478 −0.587893
$$286$$ 0 0
$$287$$ −23.3258 −1.37688
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −7.92478 −0.464559
$$292$$ 0 0
$$293$$ −14.4387 −0.843515 −0.421758 0.906709i $$-0.638587\pi$$
−0.421758 + 0.906709i $$0.638587\pi$$
$$294$$ 0 0
$$295$$ 0.926192 0.0539250
$$296$$ 0 0
$$297$$ 1.61213 0.0935451
$$298$$ 0 0
$$299$$ 6.70052 0.387501
$$300$$ 0 0
$$301$$ −2.59895 −0.149801
$$302$$ 0 0
$$303$$ 15.4010 0.884767
$$304$$ 0 0
$$305$$ −43.3258 −2.48083
$$306$$ 0 0
$$307$$ −15.8740 −0.905977 −0.452988 0.891516i $$-0.649642\pi$$
−0.452988 + 0.891516i $$0.649642\pi$$
$$308$$ 0 0
$$309$$ 4.62530 0.263124
$$310$$ 0 0
$$311$$ −8.25202 −0.467929 −0.233964 0.972245i $$-0.575170\pi$$
−0.233964 + 0.972245i $$0.575170\pi$$
$$312$$ 0 0
$$313$$ −3.40105 −0.192239 −0.0961193 0.995370i $$-0.530643\pi$$
−0.0961193 + 0.995370i $$0.530643\pi$$
$$314$$ 0 0
$$315$$ 9.92478 0.559198
$$316$$ 0 0
$$317$$ 10.4387 0.586293 0.293147 0.956067i $$-0.405298\pi$$
0.293147 + 0.956067i $$0.405298\pi$$
$$318$$ 0 0
$$319$$ 3.22425 0.180524
$$320$$ 0 0
$$321$$ −14.5501 −0.812106
$$322$$ 0 0
$$323$$ 6.70052 0.372827
$$324$$ 0 0
$$325$$ 3.77575 0.209441
$$326$$ 0 0
$$327$$ −15.4010 −0.851680
$$328$$ 0 0
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −6.57452 −0.361368 −0.180684 0.983541i $$-0.557831\pi$$
−0.180684 + 0.983541i $$0.557831\pi$$
$$332$$ 0 0
$$333$$ 7.92478 0.434275
$$334$$ 0 0
$$335$$ 24.0724 1.31522
$$336$$ 0 0
$$337$$ 31.4010 1.71052 0.855262 0.518196i $$-0.173396\pi$$
0.855262 + 0.518196i $$0.173396\pi$$
$$338$$ 0 0
$$339$$ −9.47627 −0.514680
$$340$$ 0 0
$$341$$ −10.5990 −0.573965
$$342$$ 0 0
$$343$$ 9.29948 0.502125
$$344$$ 0 0
$$345$$ −19.8496 −1.06866
$$346$$ 0 0
$$347$$ 21.2506 1.14079 0.570396 0.821370i $$-0.306790\pi$$
0.570396 + 0.821370i $$0.306790\pi$$
$$348$$ 0 0
$$349$$ −19.9248 −1.06655 −0.533274 0.845942i $$-0.679039\pi$$
−0.533274 + 0.845942i $$0.679039\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −6.43866 −0.342695 −0.171348 0.985211i $$-0.554812\pi$$
−0.171348 + 0.985211i $$0.554812\pi$$
$$354$$ 0 0
$$355$$ −12.7757 −0.678066
$$356$$ 0 0
$$357$$ −6.70052 −0.354629
$$358$$ 0 0
$$359$$ 28.5647 1.50759 0.753793 0.657112i $$-0.228222\pi$$
0.753793 + 0.657112i $$0.228222\pi$$
$$360$$ 0 0
$$361$$ −7.77575 −0.409250
$$362$$ 0 0
$$363$$ −8.40105 −0.440941
$$364$$ 0 0
$$365$$ −0.222839 −0.0116639
$$366$$ 0 0
$$367$$ 19.3258 1.00880 0.504400 0.863470i $$-0.331714\pi$$
0.504400 + 0.863470i $$0.331714\pi$$
$$368$$ 0 0
$$369$$ 6.96239 0.362447
$$370$$ 0 0
$$371$$ −39.9511 −2.07416
$$372$$ 0 0
$$373$$ −25.6991 −1.33065 −0.665325 0.746554i $$-0.731707\pi$$
−0.665325 + 0.746554i $$0.731707\pi$$
$$374$$ 0 0
$$375$$ 3.62672 0.187283
$$376$$ 0 0
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 24.7513 1.27139 0.635695 0.771941i $$-0.280714\pi$$
0.635695 + 0.771941i $$0.280714\pi$$
$$380$$ 0 0
$$381$$ −5.29948 −0.271500
$$382$$ 0 0
$$383$$ −21.6121 −1.10433 −0.552164 0.833735i $$-0.686198\pi$$
−0.552164 + 0.833735i $$0.686198\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ 0.775746 0.0394334
$$388$$ 0 0
$$389$$ 25.3258 1.28407 0.642035 0.766675i $$-0.278090\pi$$
0.642035 + 0.766675i $$0.278090\pi$$
$$390$$ 0 0
$$391$$ 13.4010 0.677720
$$392$$ 0 0
$$393$$ −9.14903 −0.461508
$$394$$ 0 0
$$395$$ 35.5487 1.78865
$$396$$ 0 0
$$397$$ −3.92478 −0.196979 −0.0984895 0.995138i $$-0.531401\pi$$
−0.0984895 + 0.995138i $$0.531401\pi$$
$$398$$ 0 0
$$399$$ −11.2243 −0.561916
$$400$$ 0 0
$$401$$ −17.4109 −0.869459 −0.434729 0.900561i $$-0.643156\pi$$
−0.434729 + 0.900561i $$0.643156\pi$$
$$402$$ 0 0
$$403$$ −6.57452 −0.327500
$$404$$ 0 0
$$405$$ −2.96239 −0.147202
$$406$$ 0 0
$$407$$ 12.7757 0.633270
$$408$$ 0 0
$$409$$ −15.9248 −0.787430 −0.393715 0.919233i $$-0.628810\pi$$
−0.393715 + 0.919233i $$0.628810\pi$$
$$410$$ 0 0
$$411$$ 0.513881 0.0253479
$$412$$ 0 0
$$413$$ 1.04746 0.0515422
$$414$$ 0 0
$$415$$ −24.6253 −1.20881
$$416$$ 0 0
$$417$$ 13.9248 0.681899
$$418$$ 0 0
$$419$$ −36.9986 −1.80750 −0.903750 0.428062i $$-0.859197\pi$$
−0.903750 + 0.428062i $$0.859197\pi$$
$$420$$ 0 0
$$421$$ −39.4010 −1.92029 −0.960145 0.279503i $$-0.909830\pi$$
−0.960145 + 0.279503i $$0.909830\pi$$
$$422$$ 0 0
$$423$$ −2.38787 −0.116102
$$424$$ 0 0
$$425$$ 7.55149 0.366301
$$426$$ 0 0
$$427$$ −48.9986 −2.37121
$$428$$ 0 0
$$429$$ 1.61213 0.0778342
$$430$$ 0 0
$$431$$ 35.6385 1.71664 0.858322 0.513111i $$-0.171507\pi$$
0.858322 + 0.513111i $$0.171507\pi$$
$$432$$ 0 0
$$433$$ 33.0738 1.58943 0.794713 0.606986i $$-0.207621\pi$$
0.794713 + 0.606986i $$0.207621\pi$$
$$434$$ 0 0
$$435$$ −5.92478 −0.284071
$$436$$ 0 0
$$437$$ 22.4485 1.07386
$$438$$ 0 0
$$439$$ −32.4749 −1.54994 −0.774970 0.631998i $$-0.782235\pi$$
−0.774970 + 0.631998i $$0.782235\pi$$
$$440$$ 0 0
$$441$$ 4.22425 0.201155
$$442$$ 0 0
$$443$$ −9.14903 −0.434684 −0.217342 0.976096i $$-0.569739\pi$$
−0.217342 + 0.976096i $$0.569739\pi$$
$$444$$ 0 0
$$445$$ −26.3272 −1.24803
$$446$$ 0 0
$$447$$ −12.8872 −0.609542
$$448$$ 0 0
$$449$$ −24.3634 −1.14978 −0.574891 0.818230i $$-0.694955\pi$$
−0.574891 + 0.818230i $$0.694955\pi$$
$$450$$ 0 0
$$451$$ 11.2243 0.528529
$$452$$ 0 0
$$453$$ −13.2750 −0.623716
$$454$$ 0 0
$$455$$ 9.92478 0.465281
$$456$$ 0 0
$$457$$ 29.4763 1.37884 0.689421 0.724361i $$-0.257865\pi$$
0.689421 + 0.724361i $$0.257865\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −7.11283 −0.331278 −0.165639 0.986186i $$-0.552969\pi$$
−0.165639 + 0.986186i $$0.552969\pi$$
$$462$$ 0 0
$$463$$ 42.7974 1.98896 0.994481 0.104918i $$-0.0334580\pi$$
0.994481 + 0.104918i $$0.0334580\pi$$
$$464$$ 0 0
$$465$$ 19.4763 0.903190
$$466$$ 0 0
$$467$$ −12.6253 −0.584229 −0.292115 0.956383i $$-0.594359\pi$$
−0.292115 + 0.956383i $$0.594359\pi$$
$$468$$ 0 0
$$469$$ 27.2243 1.25710
$$470$$ 0 0
$$471$$ 21.0738 0.971030
$$472$$ 0 0
$$473$$ 1.25060 0.0575027
$$474$$ 0 0
$$475$$ 12.6497 0.580410
$$476$$ 0 0
$$477$$ 11.9248 0.545998
$$478$$ 0 0
$$479$$ 34.3390 1.56899 0.784494 0.620136i $$-0.212923\pi$$
0.784494 + 0.620136i $$0.212923\pi$$
$$480$$ 0 0
$$481$$ 7.92478 0.361339
$$482$$ 0 0
$$483$$ −22.4485 −1.02144
$$484$$ 0 0
$$485$$ 23.4763 1.06600
$$486$$ 0 0
$$487$$ −21.5271 −0.975484 −0.487742 0.872988i $$-0.662179\pi$$
−0.487742 + 0.872988i $$0.662179\pi$$
$$488$$ 0 0
$$489$$ 21.9003 0.990368
$$490$$ 0 0
$$491$$ −26.0263 −1.17455 −0.587276 0.809387i $$-0.699800\pi$$
−0.587276 + 0.809387i $$0.699800\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ −4.77575 −0.214654
$$496$$ 0 0
$$497$$ −14.4485 −0.648104
$$498$$ 0 0
$$499$$ 1.42548 0.0638135 0.0319067 0.999491i $$-0.489842\pi$$
0.0319067 + 0.999491i $$0.489842\pi$$
$$500$$ 0 0
$$501$$ 17.4617 0.780130
$$502$$ 0 0
$$503$$ 1.67276 0.0745847 0.0372924 0.999304i $$-0.488127\pi$$
0.0372924 + 0.999304i $$0.488127\pi$$
$$504$$ 0 0
$$505$$ −45.6239 −2.03024
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ 6.58910 0.292057 0.146028 0.989280i $$-0.453351\pi$$
0.146028 + 0.989280i $$0.453351\pi$$
$$510$$ 0 0
$$511$$ −0.252016 −0.0111485
$$512$$ 0 0
$$513$$ 3.35026 0.147918
$$514$$ 0 0
$$515$$ −13.7019 −0.603780
$$516$$ 0 0
$$517$$ −3.84955 −0.169303
$$518$$ 0 0
$$519$$ 21.8496 0.959089
$$520$$ 0 0
$$521$$ 26.3733 1.15543 0.577717 0.816237i $$-0.303944\pi$$
0.577717 + 0.816237i $$0.303944\pi$$
$$522$$ 0 0
$$523$$ 3.32582 0.145428 0.0727141 0.997353i $$-0.476834\pi$$
0.0727141 + 0.997353i $$0.476834\pi$$
$$524$$ 0 0
$$525$$ −12.6497 −0.552080
$$526$$ 0 0
$$527$$ −13.1490 −0.572781
$$528$$ 0 0
$$529$$ 21.8970 0.952044
$$530$$ 0 0
$$531$$ −0.312650 −0.0135679
$$532$$ 0 0
$$533$$ 6.96239 0.301575
$$534$$ 0 0
$$535$$ 43.1030 1.86350
$$536$$ 0 0
$$537$$ 21.9248 0.946124
$$538$$ 0 0
$$539$$ 6.81003 0.293329
$$540$$ 0 0
$$541$$ 33.8496 1.45531 0.727653 0.685945i $$-0.240611\pi$$
0.727653 + 0.685945i $$0.240611\pi$$
$$542$$ 0 0
$$543$$ −18.6253 −0.799288
$$544$$ 0 0
$$545$$ 45.6239 1.95431
$$546$$ 0 0
$$547$$ −25.7743 −1.10203 −0.551015 0.834495i $$-0.685759\pi$$
−0.551015 + 0.834495i $$0.685759\pi$$
$$548$$ 0 0
$$549$$ 14.6253 0.624193
$$550$$ 0 0
$$551$$ 6.70052 0.285452
$$552$$ 0 0
$$553$$ 40.2031 1.70961
$$554$$ 0 0
$$555$$ −23.4763 −0.996512
$$556$$ 0 0
$$557$$ −4.88717 −0.207076 −0.103538 0.994626i $$-0.533016\pi$$
−0.103538 + 0.994626i $$0.533016\pi$$
$$558$$ 0 0
$$559$$ 0.775746 0.0328106
$$560$$ 0 0
$$561$$ 3.22425 0.136128
$$562$$ 0 0
$$563$$ −32.2228 −1.35803 −0.679015 0.734124i $$-0.737593\pi$$
−0.679015 + 0.734124i $$0.737593\pi$$
$$564$$ 0 0
$$565$$ 28.0724 1.18101
$$566$$ 0 0
$$567$$ −3.35026 −0.140698
$$568$$ 0 0
$$569$$ −18.2228 −0.763941 −0.381971 0.924174i $$-0.624754\pi$$
−0.381971 + 0.924174i $$0.624754\pi$$
$$570$$ 0 0
$$571$$ 36.9986 1.54834 0.774171 0.632976i $$-0.218167\pi$$
0.774171 + 0.632976i $$0.218167\pi$$
$$572$$ 0 0
$$573$$ −21.7743 −0.909636
$$574$$ 0 0
$$575$$ 25.2995 1.05506
$$576$$ 0 0
$$577$$ −1.10299 −0.0459179 −0.0229589 0.999736i $$-0.507309\pi$$
−0.0229589 + 0.999736i $$0.507309\pi$$
$$578$$ 0 0
$$579$$ 19.2506 0.800028
$$580$$ 0 0
$$581$$ −27.8496 −1.15539
$$582$$ 0 0
$$583$$ 19.2243 0.796187
$$584$$ 0 0
$$585$$ −2.96239 −0.122480
$$586$$ 0 0
$$587$$ −37.0884 −1.53080 −0.765401 0.643554i $$-0.777459\pi$$
−0.765401 + 0.643554i $$0.777459\pi$$
$$588$$ 0 0
$$589$$ −22.0263 −0.907580
$$590$$ 0 0
$$591$$ −4.51388 −0.185676
$$592$$ 0 0
$$593$$ 39.8397 1.63602 0.818010 0.575204i $$-0.195077\pi$$
0.818010 + 0.575204i $$0.195077\pi$$
$$594$$ 0 0
$$595$$ 19.8496 0.813752
$$596$$ 0 0
$$597$$ 9.14903 0.374445
$$598$$ 0 0
$$599$$ −22.0263 −0.899972 −0.449986 0.893036i $$-0.648571\pi$$
−0.449986 + 0.893036i $$0.648571\pi$$
$$600$$ 0 0
$$601$$ −35.4010 −1.44404 −0.722019 0.691873i $$-0.756786\pi$$
−0.722019 + 0.691873i $$0.756786\pi$$
$$602$$ 0 0
$$603$$ −8.12601 −0.330917
$$604$$ 0 0
$$605$$ 24.8872 1.01181
$$606$$ 0 0
$$607$$ 10.0752 0.408941 0.204470 0.978873i $$-0.434453\pi$$
0.204470 + 0.978873i $$0.434453\pi$$
$$608$$ 0 0
$$609$$ −6.70052 −0.271519
$$610$$ 0 0
$$611$$ −2.38787 −0.0966030
$$612$$ 0 0
$$613$$ 14.3733 0.580532 0.290266 0.956946i $$-0.406256\pi$$
0.290266 + 0.956946i $$0.406256\pi$$
$$614$$ 0 0
$$615$$ −20.6253 −0.831692
$$616$$ 0 0
$$617$$ 38.6615 1.55645 0.778227 0.627984i $$-0.216119\pi$$
0.778227 + 0.627984i $$0.216119\pi$$
$$618$$ 0 0
$$619$$ 8.12601 0.326612 0.163306 0.986575i $$-0.447784\pi$$
0.163306 + 0.986575i $$0.447784\pi$$
$$620$$ 0 0
$$621$$ 6.70052 0.268883
$$622$$ 0 0
$$623$$ −29.7743 −1.19288
$$624$$ 0 0
$$625$$ −29.6225 −1.18490
$$626$$ 0 0
$$627$$ 5.40105 0.215697
$$628$$ 0 0
$$629$$ 15.8496 0.631963
$$630$$ 0 0
$$631$$ 22.5745 0.898677 0.449339 0.893362i $$-0.351660\pi$$
0.449339 + 0.893362i $$0.351660\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 0 0
$$635$$ 15.6991 0.623000
$$636$$ 0 0
$$637$$ 4.22425 0.167371
$$638$$ 0 0
$$639$$ 4.31265 0.170606
$$640$$ 0 0
$$641$$ −17.8496 −0.705015 −0.352508 0.935809i $$-0.614671\pi$$
−0.352508 + 0.935809i $$0.614671\pi$$
$$642$$ 0 0
$$643$$ 2.47295 0.0975234 0.0487617 0.998810i $$-0.484473\pi$$
0.0487617 + 0.998810i $$0.484473\pi$$
$$644$$ 0 0
$$645$$ −2.29806 −0.0904861
$$646$$ 0 0
$$647$$ 22.9525 0.902357 0.451179 0.892434i $$-0.351004\pi$$
0.451179 + 0.892434i $$0.351004\pi$$
$$648$$ 0 0
$$649$$ −0.504032 −0.0197850
$$650$$ 0 0
$$651$$ 22.0263 0.863281
$$652$$ 0 0
$$653$$ −19.4010 −0.759222 −0.379611 0.925146i $$-0.623942\pi$$
−0.379611 + 0.925146i $$0.623942\pi$$
$$654$$ 0 0
$$655$$ 27.1030 1.05900
$$656$$ 0 0
$$657$$ 0.0752228 0.00293472
$$658$$ 0 0
$$659$$ −15.8496 −0.617411 −0.308705 0.951158i $$-0.599896\pi$$
−0.308705 + 0.951158i $$0.599896\pi$$
$$660$$ 0 0
$$661$$ 30.8773 1.20099 0.600494 0.799629i $$-0.294971\pi$$
0.600494 + 0.799629i $$0.294971\pi$$
$$662$$ 0 0
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ 33.2506 1.28940
$$666$$ 0 0
$$667$$ 13.4010 0.518891
$$668$$ 0 0
$$669$$ −18.4241 −0.712316
$$670$$ 0 0
$$671$$ 23.5778 0.910212
$$672$$ 0 0
$$673$$ 16.3272 0.629369 0.314684 0.949196i $$-0.398101\pi$$
0.314684 + 0.949196i $$0.398101\pi$$
$$674$$ 0 0
$$675$$ 3.77575 0.145329
$$676$$ 0 0
$$677$$ 32.0752 1.23275 0.616375 0.787452i $$-0.288600\pi$$
0.616375 + 0.787452i $$0.288600\pi$$
$$678$$ 0 0
$$679$$ 26.5501 1.01890
$$680$$ 0 0
$$681$$ −1.61213 −0.0617768
$$682$$ 0 0
$$683$$ −15.1344 −0.579103 −0.289552 0.957162i $$-0.593506\pi$$
−0.289552 + 0.957162i $$0.593506\pi$$
$$684$$ 0 0
$$685$$ −1.52232 −0.0581647
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ 0 0
$$689$$ 11.9248 0.454298
$$690$$ 0 0
$$691$$ 19.7235 0.750319 0.375160 0.926960i $$-0.377588\pi$$
0.375160 + 0.926960i $$0.377588\pi$$
$$692$$ 0 0
$$693$$ −5.40105 −0.205169
$$694$$ 0 0
$$695$$ −41.2506 −1.56472
$$696$$ 0 0
$$697$$ 13.9248 0.527439
$$698$$ 0 0
$$699$$ 0.0752228 0.00284519
$$700$$ 0 0
$$701$$ −7.92478 −0.299315 −0.149657 0.988738i $$-0.547817\pi$$
−0.149657 + 0.988738i $$0.547817\pi$$
$$702$$ 0 0
$$703$$ 26.5501 1.00136
$$704$$ 0 0
$$705$$ 7.07381 0.266415
$$706$$ 0 0
$$707$$ −51.5975 −1.94053
$$708$$ 0 0
$$709$$ 15.5515 0.584049 0.292024 0.956411i $$-0.405671\pi$$
0.292024 + 0.956411i $$0.405671\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ 0 0
$$713$$ −44.0527 −1.64979
$$714$$ 0 0
$$715$$ −4.77575 −0.178603
$$716$$ 0 0
$$717$$ −9.08840 −0.339412
$$718$$ 0 0
$$719$$ −18.5501 −0.691801 −0.345901 0.938271i $$-0.612427\pi$$
−0.345901 + 0.938271i $$0.612427\pi$$
$$720$$ 0 0
$$721$$ −15.4960 −0.577100
$$722$$ 0 0
$$723$$ 15.7743 0.586654
$$724$$ 0 0
$$725$$ 7.55149 0.280455
$$726$$ 0 0
$$727$$ 11.9511 0.443243 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.55149 0.0573840
$$732$$ 0 0
$$733$$ 8.80209 0.325113 0.162556 0.986699i $$-0.448026\pi$$
0.162556 + 0.986699i $$0.448026\pi$$
$$734$$ 0 0
$$735$$ −12.5139 −0.461581
$$736$$ 0 0
$$737$$ −13.1002 −0.482550
$$738$$ 0 0
$$739$$ 23.8251 0.876421 0.438211 0.898872i $$-0.355612\pi$$
0.438211 + 0.898872i $$0.355612\pi$$
$$740$$ 0 0
$$741$$ 3.35026 0.123075
$$742$$ 0 0
$$743$$ −7.53690 −0.276502 −0.138251 0.990397i $$-0.544148\pi$$
−0.138251 + 0.990397i $$0.544148\pi$$
$$744$$ 0 0
$$745$$ 38.1768 1.39869
$$746$$ 0 0
$$747$$ 8.31265 0.304144
$$748$$ 0 0
$$749$$ 48.7466 1.78116
$$750$$ 0 0
$$751$$ −44.6253 −1.62840 −0.814200 0.580584i $$-0.802824\pi$$
−0.814200 + 0.580584i $$0.802824\pi$$
$$752$$ 0 0
$$753$$ 15.2243 0.554803
$$754$$ 0 0
$$755$$ 39.3258 1.43121
$$756$$ 0 0
$$757$$ −11.6728 −0.424254 −0.212127 0.977242i $$-0.568039\pi$$
−0.212127 + 0.977242i $$0.568039\pi$$
$$758$$ 0 0
$$759$$ 10.8021 0.392091
$$760$$ 0 0
$$761$$ −19.8397 −0.719189 −0.359594 0.933109i $$-0.617085\pi$$
−0.359594 + 0.933109i $$0.617085\pi$$
$$762$$ 0 0
$$763$$ 51.5975 1.86796
$$764$$ 0 0
$$765$$ −5.92478 −0.214211
$$766$$ 0 0
$$767$$ −0.312650 −0.0112891
$$768$$ 0 0
$$769$$ −35.1002 −1.26574 −0.632872 0.774256i $$-0.718124\pi$$
−0.632872 + 0.774256i $$0.718124\pi$$
$$770$$ 0 0
$$771$$ −4.07522 −0.146766
$$772$$ 0 0
$$773$$ −34.2882 −1.23326 −0.616631 0.787253i $$-0.711503\pi$$
−0.616631 + 0.787253i $$0.711503\pi$$
$$774$$ 0 0
$$775$$ −24.8237 −0.891694
$$776$$ 0 0
$$777$$ −26.5501 −0.952479
$$778$$ 0 0
$$779$$ 23.3258 0.835734
$$780$$ 0 0
$$781$$ 6.95254 0.248781
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ −62.4288 −2.22818
$$786$$ 0 0
$$787$$ 31.8251 1.13444 0.567221 0.823565i $$-0.308018\pi$$
0.567221 + 0.823565i $$0.308018\pi$$
$$788$$ 0 0
$$789$$ 12.7757 0.454829
$$790$$ 0 0
$$791$$ 31.7480 1.12883
$$792$$ 0 0
$$793$$ 14.6253 0.519360
$$794$$ 0 0
$$795$$ −35.3258 −1.25288
$$796$$ 0 0
$$797$$ 16.0752 0.569414 0.284707 0.958615i $$-0.408104\pi$$
0.284707 + 0.958615i $$0.408104\pi$$
$$798$$ 0 0
$$799$$ −4.77575 −0.168954
$$800$$ 0 0
$$801$$ 8.88717 0.314013
$$802$$ 0 0
$$803$$ 0.121269 0.00427948
$$804$$ 0 0
$$805$$ 66.5012 2.34386
$$806$$ 0 0
$$807$$ 21.8496 0.769141
$$808$$ 0 0
$$809$$ 1.32582 0.0466135 0.0233067 0.999728i $$-0.492581\pi$$
0.0233067 + 0.999728i $$0.492581\pi$$
$$810$$ 0 0
$$811$$ −12.3488 −0.433627 −0.216813 0.976213i $$-0.569566\pi$$
−0.216813 + 0.976213i $$0.569566\pi$$
$$812$$ 0 0
$$813$$ 0.499293 0.0175110
$$814$$ 0 0
$$815$$ −64.8773 −2.27255
$$816$$ 0 0
$$817$$ 2.59895 0.0909259
$$818$$ 0 0
$$819$$ −3.35026 −0.117068
$$820$$ 0 0
$$821$$ −54.1378 −1.88942 −0.944711 0.327905i $$-0.893657\pi$$
−0.944711 + 0.327905i $$0.893657\pi$$
$$822$$ 0 0
$$823$$ 15.7283 0.548254 0.274127 0.961694i $$-0.411611\pi$$
0.274127 + 0.961694i $$0.411611\pi$$
$$824$$ 0 0
$$825$$ 6.08698 0.211922
$$826$$ 0 0
$$827$$ −27.4880 −0.955852 −0.477926 0.878400i $$-0.658611\pi$$
−0.477926 + 0.878400i $$0.658611\pi$$
$$828$$ 0 0
$$829$$ 15.6728 0.544337 0.272169 0.962250i $$-0.412259\pi$$
0.272169 + 0.962250i $$0.412259\pi$$
$$830$$ 0 0
$$831$$ −13.8496 −0.480436
$$832$$ 0 0
$$833$$ 8.44851 0.292723
$$834$$ 0 0
$$835$$ −51.7283 −1.79013
$$836$$ 0 0
$$837$$ −6.57452 −0.227248
$$838$$ 0 0
$$839$$ −29.8641 −1.03102 −0.515512 0.856882i $$-0.672398\pi$$
−0.515512 + 0.856882i $$0.672398\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −14.4387 −0.497294
$$844$$ 0 0
$$845$$ −2.96239 −0.101909
$$846$$ 0 0
$$847$$ 28.1457 0.967098
$$848$$ 0 0
$$849$$ −6.55008 −0.224798
$$850$$ 0 0
$$851$$ 53.1002 1.82025
$$852$$ 0 0
$$853$$ 34.2228 1.17177 0.585884 0.810395i $$-0.300747\pi$$
0.585884 + 0.810395i $$0.300747\pi$$
$$854$$ 0 0
$$855$$ −9.92478 −0.339420
$$856$$ 0 0
$$857$$ 17.6267 0.602117 0.301059 0.953606i $$-0.402660\pi$$
0.301059 + 0.953606i $$0.402660\pi$$
$$858$$ 0 0
$$859$$ −6.80209 −0.232084 −0.116042 0.993244i $$-0.537021\pi$$
−0.116042 + 0.993244i $$0.537021\pi$$
$$860$$ 0 0
$$861$$ −23.3258 −0.794942
$$862$$ 0 0
$$863$$ −32.4142 −1.10339 −0.551696 0.834045i $$-0.686019\pi$$
−0.551696 + 0.834045i $$0.686019\pi$$
$$864$$ 0 0
$$865$$ −64.7269 −2.20078
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ −19.3455 −0.656252
$$870$$ 0 0
$$871$$ −8.12601 −0.275339
$$872$$ 0 0
$$873$$ −7.92478 −0.268213
$$874$$ 0 0
$$875$$ −12.1504 −0.410760
$$876$$ 0 0
$$877$$ −26.8773 −0.907582 −0.453791 0.891108i $$-0.649929\pi$$
−0.453791 + 0.891108i $$0.649929\pi$$
$$878$$ 0 0
$$879$$ −14.4387 −0.487004
$$880$$ 0 0
$$881$$ −33.4763 −1.12784 −0.563922 0.825828i $$-0.690708\pi$$
−0.563922 + 0.825828i $$0.690708\pi$$
$$882$$ 0 0
$$883$$ −26.3996 −0.888418 −0.444209 0.895923i $$-0.646515\pi$$
−0.444209 + 0.895923i $$0.646515\pi$$
$$884$$ 0 0
$$885$$ 0.926192 0.0311336
$$886$$ 0 0
$$887$$ −13.7743 −0.462497 −0.231248 0.972895i $$-0.574281\pi$$
−0.231248 + 0.972895i $$0.574281\pi$$
$$888$$ 0 0
$$889$$ 17.7546 0.595471
$$890$$ 0 0
$$891$$ 1.61213 0.0540083
$$892$$ 0 0
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ −64.9497 −2.17103
$$896$$ 0 0
$$897$$ 6.70052 0.223724
$$898$$ 0 0
$$899$$ −13.1490 −0.438545
$$900$$ 0 0
$$901$$ 23.8496 0.794544
$$902$$ 0 0
$$903$$ −2.59895 −0.0864877
$$904$$ 0 0
$$905$$ 55.1754 1.83409
$$906$$ 0 0
$$907$$ −3.74798 −0.124450 −0.0622249 0.998062i $$-0.519820\pi$$
−0.0622249 + 0.998062i $$0.519820\pi$$
$$908$$ 0 0
$$909$$ 15.4010 0.510820
$$910$$ 0 0
$$911$$ −1.42075 −0.0470714 −0.0235357 0.999723i $$-0.507492\pi$$
−0.0235357 + 0.999723i $$0.507492\pi$$
$$912$$ 0 0
$$913$$ 13.4010 0.443510
$$914$$ 0 0
$$915$$ −43.3258 −1.43231
$$916$$ 0 0
$$917$$ 30.6516 1.01221
$$918$$ 0 0
$$919$$ 55.5487 1.83238 0.916191 0.400743i $$-0.131248\pi$$
0.916191 + 0.400743i $$0.131248\pi$$
$$920$$ 0 0
$$921$$ −15.8740 −0.523066
$$922$$ 0 0
$$923$$ 4.31265 0.141953
$$924$$ 0 0
$$925$$ 29.9219 0.983828
$$926$$ 0 0
$$927$$ 4.62530 0.151915
$$928$$ 0 0
$$929$$ 49.4636 1.62285 0.811424 0.584458i $$-0.198693\pi$$
0.811424 + 0.584458i $$0.198693\pi$$
$$930$$ 0 0
$$931$$ 14.1524 0.463825
$$932$$ 0 0
$$933$$ −8.25202 −0.270159
$$934$$ 0 0
$$935$$ −9.55149 −0.312367
$$936$$ 0 0
$$937$$ −5.07381 −0.165754 −0.0828770 0.996560i $$-0.526411\pi$$
−0.0828770 + 0.996560i $$0.526411\pi$$
$$938$$ 0 0
$$939$$ −3.40105 −0.110989
$$940$$ 0 0
$$941$$ 18.8119 0.613252 0.306626 0.951830i $$-0.400800\pi$$
0.306626 + 0.951830i $$0.400800\pi$$
$$942$$ 0 0
$$943$$ 46.6516 1.51919
$$944$$ 0 0
$$945$$ 9.92478 0.322853
$$946$$ 0 0
$$947$$ −15.2652 −0.496052 −0.248026 0.968753i $$-0.579782\pi$$
−0.248026 + 0.968753i $$0.579782\pi$$
$$948$$ 0 0
$$949$$ 0.0752228 0.00244183
$$950$$ 0 0
$$951$$ 10.4387 0.338497
$$952$$ 0 0
$$953$$ 5.47627 0.177394 0.0886969 0.996059i $$-0.471730\pi$$
0.0886969 + 0.996059i $$0.471730\pi$$
$$954$$ 0 0
$$955$$ 64.5040 2.08730
$$956$$ 0 0
$$957$$ 3.22425 0.104225
$$958$$ 0 0
$$959$$ −1.72164 −0.0555945
$$960$$ 0 0
$$961$$ 12.2243 0.394331
$$962$$ 0 0
$$963$$ −14.5501 −0.468870
$$964$$ 0 0
$$965$$ −57.0278 −1.83579
$$966$$ 0 0
$$967$$ 46.0216 1.47996 0.739978 0.672632i $$-0.234836\pi$$
0.739978 + 0.672632i $$0.234836\pi$$
$$968$$ 0 0
$$969$$ 6.70052 0.215252
$$970$$ 0 0
$$971$$ −18.4485 −0.592041 −0.296020 0.955182i $$-0.595660\pi$$
−0.296020 + 0.955182i $$0.595660\pi$$
$$972$$ 0 0
$$973$$ −46.6516 −1.49558
$$974$$ 0 0
$$975$$ 3.77575 0.120921
$$976$$ 0 0
$$977$$ −0.160295 −0.00512828 −0.00256414 0.999997i $$-0.500816\pi$$
−0.00256414 + 0.999997i $$0.500816\pi$$
$$978$$ 0 0
$$979$$ 14.3272 0.457901
$$980$$ 0 0
$$981$$ −15.4010 −0.491718
$$982$$ 0 0
$$983$$ −28.8167 −0.919109 −0.459555 0.888149i $$-0.651991\pi$$
−0.459555 + 0.888149i $$0.651991\pi$$
$$984$$ 0 0
$$985$$ 13.3719 0.426063
$$986$$ 0 0
$$987$$ 8.00000 0.254643
$$988$$ 0 0
$$989$$ 5.19791 0.165284
$$990$$ 0 0
$$991$$ −40.2228 −1.27772 −0.638860 0.769323i $$-0.720594\pi$$
−0.638860 + 0.769323i $$0.720594\pi$$
$$992$$ 0 0
$$993$$ −6.57452 −0.208636
$$994$$ 0 0
$$995$$ −27.1030 −0.859222
$$996$$ 0 0
$$997$$ −12.8021 −0.405446 −0.202723 0.979236i $$-0.564979\pi$$
−0.202723 + 0.979236i $$0.564979\pi$$
$$998$$ 0 0
$$999$$ 7.92478 0.250729
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 1248.2.a.p.1.1 yes 3
3.2 odd 2 3744.2.a.z.1.3 3
4.3 odd 2 1248.2.a.o.1.1 3
8.3 odd 2 2496.2.a.bl.1.3 3
8.5 even 2 2496.2.a.bk.1.3 3
12.11 even 2 3744.2.a.ba.1.3 3
24.5 odd 2 7488.2.a.cy.1.1 3
24.11 even 2 7488.2.a.cx.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.1 3 4.3 odd 2
1248.2.a.p.1.1 yes 3 1.1 even 1 trivial
2496.2.a.bk.1.3 3 8.5 even 2
2496.2.a.bl.1.3 3 8.3 odd 2
3744.2.a.z.1.3 3 3.2 odd 2
3744.2.a.ba.1.3 3 12.11 even 2
7488.2.a.cx.1.1 3 24.11 even 2
7488.2.a.cy.1.1 3 24.5 odd 2