# Properties

 Label 4840.2.a.be.1.1 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $6$ 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.45753625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20$$ x^6 - x^5 - 13*x^4 + 11*x^3 + 41*x^2 - 30*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.87511$$ 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)

 $$f(q)$$ $$=$$ $$q-3.03399 q^{3} -1.00000 q^{5} -0.865927 q^{7} +6.20511 q^{9} +O(q^{10})$$ $$q-3.03399 q^{3} -1.00000 q^{5} -0.865927 q^{7} +6.20511 q^{9} +4.43509 q^{13} +3.03399 q^{15} +0.550031 q^{17} +0.258962 q^{19} +2.62722 q^{21} +7.92701 q^{23} +1.00000 q^{25} -9.72427 q^{27} +9.36210 q^{29} -5.31588 q^{31} +0.865927 q^{35} +6.38739 q^{37} -13.4560 q^{39} -8.74218 q^{41} -6.38622 q^{43} -6.20511 q^{45} +0.823208 q^{47} -6.25017 q^{49} -1.66879 q^{51} +2.40110 q^{53} -0.785690 q^{57} -13.5148 q^{59} +6.60761 q^{61} -5.37317 q^{63} -4.43509 q^{65} +7.17725 q^{67} -24.0505 q^{69} +2.79910 q^{71} +2.89074 q^{73} -3.03399 q^{75} +5.91452 q^{79} +10.8880 q^{81} +7.09821 q^{83} -0.550031 q^{85} -28.4046 q^{87} -16.6470 q^{89} -3.84047 q^{91} +16.1283 q^{93} -0.258962 q^{95} -7.83395 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 $$6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9} + 6 q^{13} - 2 q^{15} - 11 q^{17} + 11 q^{19} - 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} + 6 q^{29} + q^{31} + 6 q^{35} + 4 q^{37} - 27 q^{39} + 4 q^{41} - 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} - 31 q^{51} + 14 q^{53} + 5 q^{57} + 2 q^{59} + 4 q^{61} + 16 q^{63} - 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} + 9 q^{73} + 2 q^{75} - 36 q^{79} + 30 q^{81} + 45 q^{83} + 11 q^{85} - 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} - 11 q^{95} + 20 q^{97}+O(q^{100})$$ 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 + 6 * q^13 - 2 * q^15 - 11 * q^17 + 11 * q^19 - 2 * q^21 + 18 * q^23 + 6 * q^25 - q^27 + 6 * q^29 + q^31 + 6 * q^35 + 4 * q^37 - 27 * q^39 + 4 * q^41 - 3 * q^43 - 10 * q^45 + 14 * q^47 + 8 * q^49 - 31 * q^51 + 14 * q^53 + 5 * q^57 + 2 * q^59 + 4 * q^61 + 16 * q^63 - 6 * q^65 + 11 * q^67 + 8 * q^69 + 7 * q^71 + 9 * q^73 + 2 * q^75 - 36 * q^79 + 30 * q^81 + 45 * q^83 + 11 * q^85 - 25 * q^87 + q^89 - 8 * q^91 + 55 * q^93 - 11 * q^95 + 20 * 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
$$3$$ −3.03399 −1.75168 −0.875838 0.482605i $$-0.839691\pi$$
−0.875838 + 0.482605i $$0.839691\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.865927 −0.327290 −0.163645 0.986519i $$-0.552325\pi$$
−0.163645 + 0.986519i $$0.552325\pi$$
$$8$$ 0 0
$$9$$ 6.20511 2.06837
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 4.43509 1.23007 0.615037 0.788499i $$-0.289141\pi$$
0.615037 + 0.788499i $$0.289141\pi$$
$$14$$ 0 0
$$15$$ 3.03399 0.783373
$$16$$ 0 0
$$17$$ 0.550031 0.133402 0.0667010 0.997773i $$-0.478753\pi$$
0.0667010 + 0.997773i $$0.478753\pi$$
$$18$$ 0 0
$$19$$ 0.258962 0.0594100 0.0297050 0.999559i $$-0.490543\pi$$
0.0297050 + 0.999559i $$0.490543\pi$$
$$20$$ 0 0
$$21$$ 2.62722 0.573306
$$22$$ 0 0
$$23$$ 7.92701 1.65290 0.826448 0.563013i $$-0.190358\pi$$
0.826448 + 0.563013i $$0.190358\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −9.72427 −1.87144
$$28$$ 0 0
$$29$$ 9.36210 1.73850 0.869250 0.494374i $$-0.164602\pi$$
0.869250 + 0.494374i $$0.164602\pi$$
$$30$$ 0 0
$$31$$ −5.31588 −0.954760 −0.477380 0.878697i $$-0.658413\pi$$
−0.477380 + 0.878697i $$0.658413\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.865927 0.146368
$$36$$ 0 0
$$37$$ 6.38739 1.05008 0.525040 0.851077i $$-0.324050\pi$$
0.525040 + 0.851077i $$0.324050\pi$$
$$38$$ 0 0
$$39$$ −13.4560 −2.15469
$$40$$ 0 0
$$41$$ −8.74218 −1.36530 −0.682650 0.730746i $$-0.739173\pi$$
−0.682650 + 0.730746i $$0.739173\pi$$
$$42$$ 0 0
$$43$$ −6.38622 −0.973890 −0.486945 0.873433i $$-0.661889\pi$$
−0.486945 + 0.873433i $$0.661889\pi$$
$$44$$ 0 0
$$45$$ −6.20511 −0.925003
$$46$$ 0 0
$$47$$ 0.823208 0.120077 0.0600386 0.998196i $$-0.480878\pi$$
0.0600386 + 0.998196i $$0.480878\pi$$
$$48$$ 0 0
$$49$$ −6.25017 −0.892881
$$50$$ 0 0
$$51$$ −1.66879 −0.233677
$$52$$ 0 0
$$53$$ 2.40110 0.329816 0.164908 0.986309i $$-0.447267\pi$$
0.164908 + 0.986309i $$0.447267\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.785690 −0.104067
$$58$$ 0 0
$$59$$ −13.5148 −1.75948 −0.879741 0.475453i $$-0.842284\pi$$
−0.879741 + 0.475453i $$0.842284\pi$$
$$60$$ 0 0
$$61$$ 6.60761 0.846018 0.423009 0.906126i $$-0.360974\pi$$
0.423009 + 0.906126i $$0.360974\pi$$
$$62$$ 0 0
$$63$$ −5.37317 −0.676956
$$64$$ 0 0
$$65$$ −4.43509 −0.550105
$$66$$ 0 0
$$67$$ 7.17725 0.876840 0.438420 0.898770i $$-0.355538\pi$$
0.438420 + 0.898770i $$0.355538\pi$$
$$68$$ 0 0
$$69$$ −24.0505 −2.89534
$$70$$ 0 0
$$71$$ 2.79910 0.332192 0.166096 0.986110i $$-0.446884\pi$$
0.166096 + 0.986110i $$0.446884\pi$$
$$72$$ 0 0
$$73$$ 2.89074 0.338335 0.169168 0.985587i $$-0.445892\pi$$
0.169168 + 0.985587i $$0.445892\pi$$
$$74$$ 0 0
$$75$$ −3.03399 −0.350335
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.91452 0.665436 0.332718 0.943026i $$-0.392034\pi$$
0.332718 + 0.943026i $$0.392034\pi$$
$$80$$ 0 0
$$81$$ 10.8880 1.20978
$$82$$ 0 0
$$83$$ 7.09821 0.779129 0.389565 0.920999i $$-0.372625\pi$$
0.389565 + 0.920999i $$0.372625\pi$$
$$84$$ 0 0
$$85$$ −0.550031 −0.0596592
$$86$$ 0 0
$$87$$ −28.4046 −3.04529
$$88$$ 0 0
$$89$$ −16.6470 −1.76458 −0.882291 0.470704i $$-0.844000\pi$$
−0.882291 + 0.470704i $$0.844000\pi$$
$$90$$ 0 0
$$91$$ −3.84047 −0.402590
$$92$$ 0 0
$$93$$ 16.1283 1.67243
$$94$$ 0 0
$$95$$ −0.258962 −0.0265690
$$96$$ 0 0
$$97$$ −7.83395 −0.795417 −0.397708 0.917512i $$-0.630194\pi$$
−0.397708 + 0.917512i $$0.630194\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.1989 1.81086 0.905431 0.424493i $$-0.139548\pi$$
0.905431 + 0.424493i $$0.139548\pi$$
$$102$$ 0 0
$$103$$ 11.8025 1.16294 0.581469 0.813569i $$-0.302478\pi$$
0.581469 + 0.813569i $$0.302478\pi$$
$$104$$ 0 0
$$105$$ −2.62722 −0.256390
$$106$$ 0 0
$$107$$ 0.520965 0.0503636 0.0251818 0.999683i $$-0.491984\pi$$
0.0251818 + 0.999683i $$0.491984\pi$$
$$108$$ 0 0
$$109$$ 6.22545 0.596290 0.298145 0.954521i $$-0.403632\pi$$
0.298145 + 0.954521i $$0.403632\pi$$
$$110$$ 0 0
$$111$$ −19.3793 −1.83940
$$112$$ 0 0
$$113$$ −9.87578 −0.929035 −0.464518 0.885564i $$-0.653772\pi$$
−0.464518 + 0.885564i $$0.653772\pi$$
$$114$$ 0 0
$$115$$ −7.92701 −0.739198
$$116$$ 0 0
$$117$$ 27.5202 2.54425
$$118$$ 0 0
$$119$$ −0.476287 −0.0436611
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 26.5237 2.39156
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −3.71934 −0.330038 −0.165019 0.986290i $$-0.552769\pi$$
−0.165019 + 0.986290i $$0.552769\pi$$
$$128$$ 0 0
$$129$$ 19.3758 1.70594
$$130$$ 0 0
$$131$$ −20.4259 −1.78462 −0.892308 0.451426i $$-0.850915\pi$$
−0.892308 + 0.451426i $$0.850915\pi$$
$$132$$ 0 0
$$133$$ −0.224243 −0.0194443
$$134$$ 0 0
$$135$$ 9.72427 0.836932
$$136$$ 0 0
$$137$$ 13.4216 1.14669 0.573344 0.819315i $$-0.305646\pi$$
0.573344 + 0.819315i $$0.305646\pi$$
$$138$$ 0 0
$$139$$ −14.0856 −1.19473 −0.597364 0.801970i $$-0.703785\pi$$
−0.597364 + 0.801970i $$0.703785\pi$$
$$140$$ 0 0
$$141$$ −2.49761 −0.210336
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −9.36210 −0.777480
$$146$$ 0 0
$$147$$ 18.9630 1.56404
$$148$$ 0 0
$$149$$ −1.79173 −0.146784 −0.0733920 0.997303i $$-0.523382\pi$$
−0.0733920 + 0.997303i $$0.523382\pi$$
$$150$$ 0 0
$$151$$ −19.0871 −1.55328 −0.776641 0.629943i $$-0.783078\pi$$
−0.776641 + 0.629943i $$0.783078\pi$$
$$152$$ 0 0
$$153$$ 3.41300 0.275925
$$154$$ 0 0
$$155$$ 5.31588 0.426982
$$156$$ 0 0
$$157$$ 9.78753 0.781130 0.390565 0.920575i $$-0.372280\pi$$
0.390565 + 0.920575i $$0.372280\pi$$
$$158$$ 0 0
$$159$$ −7.28492 −0.577732
$$160$$ 0 0
$$161$$ −6.86422 −0.540976
$$162$$ 0 0
$$163$$ 0.396843 0.0310831 0.0155416 0.999879i $$-0.495053\pi$$
0.0155416 + 0.999879i $$0.495053\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −13.9140 −1.07670 −0.538349 0.842722i $$-0.680952\pi$$
−0.538349 + 0.842722i $$0.680952\pi$$
$$168$$ 0 0
$$169$$ 6.67004 0.513080
$$170$$ 0 0
$$171$$ 1.60689 0.122882
$$172$$ 0 0
$$173$$ −5.33390 −0.405529 −0.202765 0.979228i $$-0.564993\pi$$
−0.202765 + 0.979228i $$0.564993\pi$$
$$174$$ 0 0
$$175$$ −0.865927 −0.0654580
$$176$$ 0 0
$$177$$ 41.0039 3.08204
$$178$$ 0 0
$$179$$ 7.63478 0.570650 0.285325 0.958431i $$-0.407898\pi$$
0.285325 + 0.958431i $$0.407898\pi$$
$$180$$ 0 0
$$181$$ 1.30453 0.0969647 0.0484824 0.998824i $$-0.484562\pi$$
0.0484824 + 0.998824i $$0.484562\pi$$
$$182$$ 0 0
$$183$$ −20.0474 −1.48195
$$184$$ 0 0
$$185$$ −6.38739 −0.469610
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 8.42052 0.612502
$$190$$ 0 0
$$191$$ 25.0833 1.81497 0.907483 0.420088i $$-0.138001\pi$$
0.907483 + 0.420088i $$0.138001\pi$$
$$192$$ 0 0
$$193$$ 19.9561 1.43647 0.718235 0.695801i $$-0.244950\pi$$
0.718235 + 0.695801i $$0.244950\pi$$
$$194$$ 0 0
$$195$$ 13.4560 0.963607
$$196$$ 0 0
$$197$$ 1.31088 0.0933960 0.0466980 0.998909i $$-0.485130\pi$$
0.0466980 + 0.998909i $$0.485130\pi$$
$$198$$ 0 0
$$199$$ −4.99115 −0.353813 −0.176907 0.984228i $$-0.556609\pi$$
−0.176907 + 0.984228i $$0.556609\pi$$
$$200$$ 0 0
$$201$$ −21.7757 −1.53594
$$202$$ 0 0
$$203$$ −8.10690 −0.568993
$$204$$ 0 0
$$205$$ 8.74218 0.610581
$$206$$ 0 0
$$207$$ 49.1880 3.41880
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 6.41108 0.441357 0.220678 0.975347i $$-0.429173\pi$$
0.220678 + 0.975347i $$0.429173\pi$$
$$212$$ 0 0
$$213$$ −8.49245 −0.581893
$$214$$ 0 0
$$215$$ 6.38622 0.435537
$$216$$ 0 0
$$217$$ 4.60316 0.312483
$$218$$ 0 0
$$219$$ −8.77047 −0.592654
$$220$$ 0 0
$$221$$ 2.43944 0.164094
$$222$$ 0 0
$$223$$ −14.8484 −0.994321 −0.497160 0.867659i $$-0.665624\pi$$
−0.497160 + 0.867659i $$0.665624\pi$$
$$224$$ 0 0
$$225$$ 6.20511 0.413674
$$226$$ 0 0
$$227$$ 23.4761 1.55816 0.779081 0.626923i $$-0.215686\pi$$
0.779081 + 0.626923i $$0.215686\pi$$
$$228$$ 0 0
$$229$$ 19.6330 1.29739 0.648693 0.761050i $$-0.275316\pi$$
0.648693 + 0.761050i $$0.275316\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −20.9946 −1.37541 −0.687703 0.725993i $$-0.741381\pi$$
−0.687703 + 0.725993i $$0.741381\pi$$
$$234$$ 0 0
$$235$$ −0.823208 −0.0537002
$$236$$ 0 0
$$237$$ −17.9446 −1.16563
$$238$$ 0 0
$$239$$ 18.8533 1.21952 0.609761 0.792586i $$-0.291266\pi$$
0.609761 + 0.792586i $$0.291266\pi$$
$$240$$ 0 0
$$241$$ 13.8318 0.890987 0.445494 0.895285i $$-0.353028\pi$$
0.445494 + 0.895285i $$0.353028\pi$$
$$242$$ 0 0
$$243$$ −3.86143 −0.247711
$$244$$ 0 0
$$245$$ 6.25017 0.399309
$$246$$ 0 0
$$247$$ 1.14852 0.0730787
$$248$$ 0 0
$$249$$ −21.5359 −1.36478
$$250$$ 0 0
$$251$$ 15.2719 0.963956 0.481978 0.876183i $$-0.339919\pi$$
0.481978 + 0.876183i $$0.339919\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 1.66879 0.104504
$$256$$ 0 0
$$257$$ 28.7880 1.79575 0.897873 0.440254i $$-0.145112\pi$$
0.897873 + 0.440254i $$0.145112\pi$$
$$258$$ 0 0
$$259$$ −5.53101 −0.343681
$$260$$ 0 0
$$261$$ 58.0929 3.59586
$$262$$ 0 0
$$263$$ 9.23218 0.569281 0.284640 0.958634i $$-0.408126\pi$$
0.284640 + 0.958634i $$0.408126\pi$$
$$264$$ 0 0
$$265$$ −2.40110 −0.147498
$$266$$ 0 0
$$267$$ 50.5070 3.09098
$$268$$ 0 0
$$269$$ 26.8324 1.63600 0.817999 0.575220i $$-0.195084\pi$$
0.817999 + 0.575220i $$0.195084\pi$$
$$270$$ 0 0
$$271$$ −23.5612 −1.43124 −0.715620 0.698490i $$-0.753856\pi$$
−0.715620 + 0.698490i $$0.753856\pi$$
$$272$$ 0 0
$$273$$ 11.6519 0.705208
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −9.99241 −0.600386 −0.300193 0.953879i $$-0.597051\pi$$
−0.300193 + 0.953879i $$0.597051\pi$$
$$278$$ 0 0
$$279$$ −32.9856 −1.97480
$$280$$ 0 0
$$281$$ −4.09042 −0.244014 −0.122007 0.992529i $$-0.538933\pi$$
−0.122007 + 0.992529i $$0.538933\pi$$
$$282$$ 0 0
$$283$$ −4.77268 −0.283706 −0.141853 0.989888i $$-0.545306\pi$$
−0.141853 + 0.989888i $$0.545306\pi$$
$$284$$ 0 0
$$285$$ 0.785690 0.0465402
$$286$$ 0 0
$$287$$ 7.57010 0.446849
$$288$$ 0 0
$$289$$ −16.6975 −0.982204
$$290$$ 0 0
$$291$$ 23.7681 1.39331
$$292$$ 0 0
$$293$$ −0.144012 −0.00841327 −0.00420664 0.999991i $$-0.501339\pi$$
−0.00420664 + 0.999991i $$0.501339\pi$$
$$294$$ 0 0
$$295$$ 13.5148 0.786864
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 35.1570 2.03318
$$300$$ 0 0
$$301$$ 5.53001 0.318744
$$302$$ 0 0
$$303$$ −55.2154 −3.17204
$$304$$ 0 0
$$305$$ −6.60761 −0.378351
$$306$$ 0 0
$$307$$ 25.7576 1.47006 0.735031 0.678034i $$-0.237168\pi$$
0.735031 + 0.678034i $$0.237168\pi$$
$$308$$ 0 0
$$309$$ −35.8088 −2.03709
$$310$$ 0 0
$$311$$ 16.3915 0.929475 0.464738 0.885448i $$-0.346149\pi$$
0.464738 + 0.885448i $$0.346149\pi$$
$$312$$ 0 0
$$313$$ 26.5238 1.49921 0.749606 0.661885i $$-0.230243\pi$$
0.749606 + 0.661885i $$0.230243\pi$$
$$314$$ 0 0
$$315$$ 5.37317 0.302744
$$316$$ 0 0
$$317$$ 16.7158 0.938852 0.469426 0.882972i $$-0.344461\pi$$
0.469426 + 0.882972i $$0.344461\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −1.58060 −0.0882206
$$322$$ 0 0
$$323$$ 0.142437 0.00792542
$$324$$ 0 0
$$325$$ 4.43509 0.246015
$$326$$ 0 0
$$327$$ −18.8880 −1.04451
$$328$$ 0 0
$$329$$ −0.712839 −0.0393001
$$330$$ 0 0
$$331$$ 4.39584 0.241617 0.120809 0.992676i $$-0.461451\pi$$
0.120809 + 0.992676i $$0.461451\pi$$
$$332$$ 0 0
$$333$$ 39.6344 2.17195
$$334$$ 0 0
$$335$$ −7.17725 −0.392135
$$336$$ 0 0
$$337$$ −12.6035 −0.686558 −0.343279 0.939233i $$-0.611538\pi$$
−0.343279 + 0.939233i $$0.611538\pi$$
$$338$$ 0 0
$$339$$ 29.9630 1.62737
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 11.4737 0.619521
$$344$$ 0 0
$$345$$ 24.0505 1.29484
$$346$$ 0 0
$$347$$ 23.9827 1.28746 0.643731 0.765252i $$-0.277386\pi$$
0.643731 + 0.765252i $$0.277386\pi$$
$$348$$ 0 0
$$349$$ −23.5901 −1.26275 −0.631376 0.775477i $$-0.717509\pi$$
−0.631376 + 0.775477i $$0.717509\pi$$
$$350$$ 0 0
$$351$$ −43.1281 −2.30201
$$352$$ 0 0
$$353$$ 6.46751 0.344231 0.172115 0.985077i $$-0.444940\pi$$
0.172115 + 0.985077i $$0.444940\pi$$
$$354$$ 0 0
$$355$$ −2.79910 −0.148561
$$356$$ 0 0
$$357$$ 1.44505 0.0764802
$$358$$ 0 0
$$359$$ −11.8241 −0.624053 −0.312026 0.950073i $$-0.601008\pi$$
−0.312026 + 0.950073i $$0.601008\pi$$
$$360$$ 0 0
$$361$$ −18.9329 −0.996470
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2.89074 −0.151308
$$366$$ 0 0
$$367$$ 8.38713 0.437805 0.218902 0.975747i $$-0.429752\pi$$
0.218902 + 0.975747i $$0.429752\pi$$
$$368$$ 0 0
$$369$$ −54.2462 −2.82394
$$370$$ 0 0
$$371$$ −2.07918 −0.107946
$$372$$ 0 0
$$373$$ −34.6263 −1.79288 −0.896441 0.443163i $$-0.853856\pi$$
−0.896441 + 0.443163i $$0.853856\pi$$
$$374$$ 0 0
$$375$$ 3.03399 0.156675
$$376$$ 0 0
$$377$$ 41.5218 2.13848
$$378$$ 0 0
$$379$$ 16.0726 0.825592 0.412796 0.910823i $$-0.364552\pi$$
0.412796 + 0.910823i $$0.364552\pi$$
$$380$$ 0 0
$$381$$ 11.2844 0.578120
$$382$$ 0 0
$$383$$ 8.41569 0.430022 0.215011 0.976612i $$-0.431021\pi$$
0.215011 + 0.976612i $$0.431021\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −39.6272 −2.01436
$$388$$ 0 0
$$389$$ 18.5917 0.942636 0.471318 0.881963i $$-0.343778\pi$$
0.471318 + 0.881963i $$0.343778\pi$$
$$390$$ 0 0
$$391$$ 4.36010 0.220500
$$392$$ 0 0
$$393$$ 61.9719 3.12607
$$394$$ 0 0
$$395$$ −5.91452 −0.297592
$$396$$ 0 0
$$397$$ −11.9897 −0.601743 −0.300872 0.953665i $$-0.597278\pi$$
−0.300872 + 0.953665i $$0.597278\pi$$
$$398$$ 0 0
$$399$$ 0.680350 0.0340601
$$400$$ 0 0
$$401$$ 18.7991 0.938781 0.469391 0.882991i $$-0.344474\pi$$
0.469391 + 0.882991i $$0.344474\pi$$
$$402$$ 0 0
$$403$$ −23.5764 −1.17442
$$404$$ 0 0
$$405$$ −10.8880 −0.541031
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 9.50429 0.469957 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$410$$ 0 0
$$411$$ −40.7211 −2.00863
$$412$$ 0 0
$$413$$ 11.7029 0.575860
$$414$$ 0 0
$$415$$ −7.09821 −0.348437
$$416$$ 0 0
$$417$$ 42.7357 2.09278
$$418$$ 0 0
$$419$$ 32.9152 1.60801 0.804007 0.594620i $$-0.202698\pi$$
0.804007 + 0.594620i $$0.202698\pi$$
$$420$$ 0 0
$$421$$ −1.03559 −0.0504717 −0.0252359 0.999682i $$-0.508034\pi$$
−0.0252359 + 0.999682i $$0.508034\pi$$
$$422$$ 0 0
$$423$$ 5.10810 0.248364
$$424$$ 0 0
$$425$$ 0.550031 0.0266804
$$426$$ 0 0
$$427$$ −5.72171 −0.276893
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −25.7613 −1.24088 −0.620439 0.784255i $$-0.713046\pi$$
−0.620439 + 0.784255i $$0.713046\pi$$
$$432$$ 0 0
$$433$$ −18.4406 −0.886201 −0.443100 0.896472i $$-0.646121\pi$$
−0.443100 + 0.896472i $$0.646121\pi$$
$$434$$ 0 0
$$435$$ 28.4046 1.36189
$$436$$ 0 0
$$437$$ 2.05280 0.0981986
$$438$$ 0 0
$$439$$ 4.44724 0.212255 0.106128 0.994353i $$-0.466155\pi$$
0.106128 + 0.994353i $$0.466155\pi$$
$$440$$ 0 0
$$441$$ −38.7830 −1.84681
$$442$$ 0 0
$$443$$ −25.0738 −1.19129 −0.595646 0.803247i $$-0.703104\pi$$
−0.595646 + 0.803247i $$0.703104\pi$$
$$444$$ 0 0
$$445$$ 16.6470 0.789145
$$446$$ 0 0
$$447$$ 5.43609 0.257118
$$448$$ 0 0
$$449$$ −18.6745 −0.881306 −0.440653 0.897678i $$-0.645253\pi$$
−0.440653 + 0.897678i $$0.645253\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 57.9100 2.72085
$$454$$ 0 0
$$455$$ 3.84047 0.180044
$$456$$ 0 0
$$457$$ 14.5569 0.680941 0.340471 0.940255i $$-0.389414\pi$$
0.340471 + 0.940255i $$0.389414\pi$$
$$458$$ 0 0
$$459$$ −5.34865 −0.249654
$$460$$ 0 0
$$461$$ −20.6525 −0.961884 −0.480942 0.876752i $$-0.659705\pi$$
−0.480942 + 0.876752i $$0.659705\pi$$
$$462$$ 0 0
$$463$$ −21.8605 −1.01594 −0.507971 0.861374i $$-0.669604\pi$$
−0.507971 + 0.861374i $$0.669604\pi$$
$$464$$ 0 0
$$465$$ −16.1283 −0.747933
$$466$$ 0 0
$$467$$ 20.2984 0.939296 0.469648 0.882854i $$-0.344381\pi$$
0.469648 + 0.882854i $$0.344381\pi$$
$$468$$ 0 0
$$469$$ −6.21498 −0.286981
$$470$$ 0 0
$$471$$ −29.6953 −1.36829
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0.258962 0.0118820
$$476$$ 0 0
$$477$$ 14.8991 0.682182
$$478$$ 0 0
$$479$$ 9.89152 0.451955 0.225978 0.974132i $$-0.427442\pi$$
0.225978 + 0.974132i $$0.427442\pi$$
$$480$$ 0 0
$$481$$ 28.3287 1.29168
$$482$$ 0 0
$$483$$ 20.8260 0.947615
$$484$$ 0 0
$$485$$ 7.83395 0.355721
$$486$$ 0 0
$$487$$ 5.93879 0.269112 0.134556 0.990906i $$-0.457039\pi$$
0.134556 + 0.990906i $$0.457039\pi$$
$$488$$ 0 0
$$489$$ −1.20402 −0.0544476
$$490$$ 0 0
$$491$$ −27.3348 −1.23360 −0.616801 0.787119i $$-0.711572\pi$$
−0.616801 + 0.787119i $$0.711572\pi$$
$$492$$ 0 0
$$493$$ 5.14945 0.231919
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −2.42382 −0.108723
$$498$$ 0 0
$$499$$ −12.1678 −0.544705 −0.272353 0.962198i $$-0.587802\pi$$
−0.272353 + 0.962198i $$0.587802\pi$$
$$500$$ 0 0
$$501$$ 42.2150 1.88603
$$502$$ 0 0
$$503$$ 9.49180 0.423218 0.211609 0.977354i $$-0.432130\pi$$
0.211609 + 0.977354i $$0.432130\pi$$
$$504$$ 0 0
$$505$$ −18.1989 −0.809842
$$506$$ 0 0
$$507$$ −20.2369 −0.898750
$$508$$ 0 0
$$509$$ −0.957696 −0.0424491 −0.0212246 0.999775i $$-0.506756\pi$$
−0.0212246 + 0.999775i $$0.506756\pi$$
$$510$$ 0 0
$$511$$ −2.50317 −0.110734
$$512$$ 0 0
$$513$$ −2.51822 −0.111182
$$514$$ 0 0
$$515$$ −11.8025 −0.520082
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 16.1830 0.710356
$$520$$ 0 0
$$521$$ −18.2415 −0.799176 −0.399588 0.916695i $$-0.630847\pi$$
−0.399588 + 0.916695i $$0.630847\pi$$
$$522$$ 0 0
$$523$$ −28.3274 −1.23867 −0.619335 0.785127i $$-0.712598\pi$$
−0.619335 + 0.785127i $$0.712598\pi$$
$$524$$ 0 0
$$525$$ 2.62722 0.114661
$$526$$ 0 0
$$527$$ −2.92390 −0.127367
$$528$$ 0 0
$$529$$ 39.8375 1.73207
$$530$$ 0 0
$$531$$ −83.8610 −3.63926
$$532$$ 0 0
$$533$$ −38.7724 −1.67942
$$534$$ 0 0
$$535$$ −0.520965 −0.0225233
$$536$$ 0 0
$$537$$ −23.1639 −0.999594
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 31.7569 1.36534 0.682668 0.730729i $$-0.260819\pi$$
0.682668 + 0.730729i $$0.260819\pi$$
$$542$$ 0 0
$$543$$ −3.95792 −0.169851
$$544$$ 0 0
$$545$$ −6.22545 −0.266669
$$546$$ 0 0
$$547$$ −3.58429 −0.153253 −0.0766265 0.997060i $$-0.524415\pi$$
−0.0766265 + 0.997060i $$0.524415\pi$$
$$548$$ 0 0
$$549$$ 41.0009 1.74988
$$550$$ 0 0
$$551$$ 2.42443 0.103284
$$552$$ 0 0
$$553$$ −5.12155 −0.217790
$$554$$ 0 0
$$555$$ 19.3793 0.822605
$$556$$ 0 0
$$557$$ 16.6556 0.705719 0.352860 0.935676i $$-0.385209\pi$$
0.352860 + 0.935676i $$0.385209\pi$$
$$558$$ 0 0
$$559$$ −28.3235 −1.19796
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −39.8978 −1.68149 −0.840745 0.541431i $$-0.817883\pi$$
−0.840745 + 0.541431i $$0.817883\pi$$
$$564$$ 0 0
$$565$$ 9.87578 0.415477
$$566$$ 0 0
$$567$$ −9.42826 −0.395950
$$568$$ 0 0
$$569$$ −45.4644 −1.90597 −0.952983 0.303024i $$-0.902004\pi$$
−0.952983 + 0.303024i $$0.902004\pi$$
$$570$$ 0 0
$$571$$ 15.8877 0.664880 0.332440 0.943124i $$-0.392128\pi$$
0.332440 + 0.943124i $$0.392128\pi$$
$$572$$ 0 0
$$573$$ −76.1027 −3.17923
$$574$$ 0 0
$$575$$ 7.92701 0.330579
$$576$$ 0 0
$$577$$ −13.1413 −0.547078 −0.273539 0.961861i $$-0.588194\pi$$
−0.273539 + 0.961861i $$0.588194\pi$$
$$578$$ 0 0
$$579$$ −60.5466 −2.51623
$$580$$ 0 0
$$581$$ −6.14653 −0.255001
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −27.5202 −1.13782
$$586$$ 0 0
$$587$$ 14.9640 0.617629 0.308814 0.951122i $$-0.400068\pi$$
0.308814 + 0.951122i $$0.400068\pi$$
$$588$$ 0 0
$$589$$ −1.37661 −0.0567223
$$590$$ 0 0
$$591$$ −3.97719 −0.163600
$$592$$ 0 0
$$593$$ 13.7967 0.566564 0.283282 0.959037i $$-0.408577\pi$$
0.283282 + 0.959037i $$0.408577\pi$$
$$594$$ 0 0
$$595$$ 0.476287 0.0195259
$$596$$ 0 0
$$597$$ 15.1431 0.619766
$$598$$ 0 0
$$599$$ 31.3893 1.28253 0.641267 0.767318i $$-0.278409\pi$$
0.641267 + 0.767318i $$0.278409\pi$$
$$600$$ 0 0
$$601$$ 2.55635 0.104276 0.0521379 0.998640i $$-0.483396\pi$$
0.0521379 + 0.998640i $$0.483396\pi$$
$$602$$ 0 0
$$603$$ 44.5356 1.81363
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 37.8289 1.53543 0.767713 0.640794i $$-0.221395\pi$$
0.767713 + 0.640794i $$0.221395\pi$$
$$608$$ 0 0
$$609$$ 24.5963 0.996692
$$610$$ 0 0
$$611$$ 3.65100 0.147704
$$612$$ 0 0
$$613$$ −11.6599 −0.470940 −0.235470 0.971882i $$-0.575663\pi$$
−0.235470 + 0.971882i $$0.575663\pi$$
$$614$$ 0 0
$$615$$ −26.5237 −1.06954
$$616$$ 0 0
$$617$$ 38.5910 1.55361 0.776807 0.629739i $$-0.216838\pi$$
0.776807 + 0.629739i $$0.216838\pi$$
$$618$$ 0 0
$$619$$ −8.16365 −0.328125 −0.164062 0.986450i $$-0.552460\pi$$
−0.164062 + 0.986450i $$0.552460\pi$$
$$620$$ 0 0
$$621$$ −77.0844 −3.09329
$$622$$ 0 0
$$623$$ 14.4151 0.577530
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3.51326 0.140083
$$630$$ 0 0
$$631$$ −13.0813 −0.520759 −0.260379 0.965506i $$-0.583848\pi$$
−0.260379 + 0.965506i $$0.583848\pi$$
$$632$$ 0 0
$$633$$ −19.4512 −0.773114
$$634$$ 0 0
$$635$$ 3.71934 0.147597
$$636$$ 0 0
$$637$$ −27.7201 −1.09831
$$638$$ 0 0
$$639$$ 17.3687 0.687096
$$640$$ 0 0
$$641$$ 35.6499 1.40809 0.704044 0.710156i $$-0.251376\pi$$
0.704044 + 0.710156i $$0.251376\pi$$
$$642$$ 0 0
$$643$$ −7.65614 −0.301929 −0.150964 0.988539i $$-0.548238\pi$$
−0.150964 + 0.988539i $$0.548238\pi$$
$$644$$ 0 0
$$645$$ −19.3758 −0.762919
$$646$$ 0 0
$$647$$ 9.18246 0.361000 0.180500 0.983575i $$-0.442228\pi$$
0.180500 + 0.983575i $$0.442228\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −13.9660 −0.547369
$$652$$ 0 0
$$653$$ −5.85282 −0.229039 −0.114519 0.993421i $$-0.536533\pi$$
−0.114519 + 0.993421i $$0.536533\pi$$
$$654$$ 0 0
$$655$$ 20.4259 0.798105
$$656$$ 0 0
$$657$$ 17.9373 0.699802
$$658$$ 0 0
$$659$$ 22.6530 0.882434 0.441217 0.897400i $$-0.354547\pi$$
0.441217 + 0.897400i $$0.354547\pi$$
$$660$$ 0 0
$$661$$ 1.68482 0.0655320 0.0327660 0.999463i $$-0.489568\pi$$
0.0327660 + 0.999463i $$0.489568\pi$$
$$662$$ 0 0
$$663$$ −7.40123 −0.287440
$$664$$ 0 0
$$665$$ 0.224243 0.00869575
$$666$$ 0 0
$$667$$ 74.2135 2.87356
$$668$$ 0 0
$$669$$ 45.0499 1.74173
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 9.48493 0.365617 0.182809 0.983149i $$-0.441481\pi$$
0.182809 + 0.983149i $$0.441481\pi$$
$$674$$ 0 0
$$675$$ −9.72427 −0.374288
$$676$$ 0 0
$$677$$ 4.11952 0.158326 0.0791631 0.996862i $$-0.474775\pi$$
0.0791631 + 0.996862i $$0.474775\pi$$
$$678$$ 0 0
$$679$$ 6.78363 0.260332
$$680$$ 0 0
$$681$$ −71.2262 −2.72940
$$682$$ 0 0
$$683$$ 29.0594 1.11193 0.555963 0.831207i $$-0.312349\pi$$
0.555963 + 0.831207i $$0.312349\pi$$
$$684$$ 0 0
$$685$$ −13.4216 −0.512814
$$686$$ 0 0
$$687$$ −59.5664 −2.27260
$$688$$ 0 0
$$689$$ 10.6491 0.405698
$$690$$ 0 0
$$691$$ 19.4633 0.740418 0.370209 0.928949i $$-0.379286\pi$$
0.370209 + 0.928949i $$0.379286\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 14.0856 0.534298
$$696$$ 0 0
$$697$$ −4.80847 −0.182134
$$698$$ 0 0
$$699$$ 63.6976 2.40926
$$700$$ 0 0
$$701$$ 15.1359 0.571675 0.285838 0.958278i $$-0.407728\pi$$
0.285838 + 0.958278i $$0.407728\pi$$
$$702$$ 0 0
$$703$$ 1.65409 0.0623853
$$704$$ 0 0
$$705$$ 2.49761 0.0940653
$$706$$ 0 0
$$707$$ −15.7590 −0.592677
$$708$$ 0 0
$$709$$ −27.0377 −1.01542 −0.507712 0.861527i $$-0.669508\pi$$
−0.507712 + 0.861527i $$0.669508\pi$$
$$710$$ 0 0
$$711$$ 36.7003 1.37637
$$712$$ 0 0
$$713$$ −42.1390 −1.57812
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −57.2009 −2.13621
$$718$$ 0 0
$$719$$ 3.74967 0.139839 0.0699195 0.997553i $$-0.477726\pi$$
0.0699195 + 0.997553i $$0.477726\pi$$
$$720$$ 0 0
$$721$$ −10.2201 −0.380618
$$722$$ 0 0
$$723$$ −41.9657 −1.56072
$$724$$ 0 0
$$725$$ 9.36210 0.347700
$$726$$ 0 0
$$727$$ 45.5685 1.69004 0.845022 0.534732i $$-0.179587\pi$$
0.845022 + 0.534732i $$0.179587\pi$$
$$728$$ 0 0
$$729$$ −20.9486 −0.775874
$$730$$ 0 0
$$731$$ −3.51262 −0.129919
$$732$$ 0 0
$$733$$ 37.1793 1.37325 0.686625 0.727012i $$-0.259091\pi$$
0.686625 + 0.727012i $$0.259091\pi$$
$$734$$ 0 0
$$735$$ −18.9630 −0.699460
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 3.86380 0.142132 0.0710661 0.997472i $$-0.477360\pi$$
0.0710661 + 0.997472i $$0.477360\pi$$
$$740$$ 0 0
$$741$$ −3.48461 −0.128010
$$742$$ 0 0
$$743$$ 7.73510 0.283773 0.141887 0.989883i $$-0.454683\pi$$
0.141887 + 0.989883i $$0.454683\pi$$
$$744$$ 0 0
$$745$$ 1.79173 0.0656438
$$746$$ 0 0
$$747$$ 44.0451 1.61153
$$748$$ 0 0
$$749$$ −0.451118 −0.0164835
$$750$$ 0 0
$$751$$ −19.7386 −0.720271 −0.360135 0.932900i $$-0.617269\pi$$
−0.360135 + 0.932900i $$0.617269\pi$$
$$752$$ 0 0
$$753$$ −46.3349 −1.68854
$$754$$ 0 0
$$755$$ 19.0871 0.694649
$$756$$ 0 0
$$757$$ 10.5828 0.384637 0.192319 0.981333i $$-0.438399\pi$$
0.192319 + 0.981333i $$0.438399\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −50.6698 −1.83678 −0.918390 0.395677i $$-0.870510\pi$$
−0.918390 + 0.395677i $$0.870510\pi$$
$$762$$ 0 0
$$763$$ −5.39079 −0.195160
$$764$$ 0 0
$$765$$ −3.41300 −0.123397
$$766$$ 0 0
$$767$$ −59.9395 −2.16429
$$768$$ 0 0
$$769$$ 16.2588 0.586306 0.293153 0.956066i $$-0.405295\pi$$
0.293153 + 0.956066i $$0.405295\pi$$
$$770$$ 0 0
$$771$$ −87.3426 −3.14557
$$772$$ 0 0
$$773$$ 16.4897 0.593095 0.296547 0.955018i $$-0.404165\pi$$
0.296547 + 0.955018i $$0.404165\pi$$
$$774$$ 0 0
$$775$$ −5.31588 −0.190952
$$776$$ 0 0
$$777$$ 16.7811 0.602017
$$778$$ 0 0
$$779$$ −2.26390 −0.0811125
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −91.0397 −3.25349
$$784$$ 0 0
$$785$$ −9.78753 −0.349332
$$786$$ 0 0
$$787$$ −25.7184 −0.916762 −0.458381 0.888756i $$-0.651570\pi$$
−0.458381 + 0.888756i $$0.651570\pi$$
$$788$$ 0 0
$$789$$ −28.0104 −0.997196
$$790$$ 0 0
$$791$$ 8.55171 0.304064
$$792$$ 0 0
$$793$$ 29.3054 1.04066
$$794$$ 0 0
$$795$$ 7.28492 0.258369
$$796$$ 0 0
$$797$$ 5.59752 0.198274 0.0991372 0.995074i $$-0.468392\pi$$
0.0991372 + 0.995074i $$0.468392\pi$$
$$798$$ 0 0
$$799$$ 0.452790 0.0160186
$$800$$ 0 0
$$801$$ −103.297 −3.64981
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 6.86422 0.241932
$$806$$ 0 0
$$807$$ −81.4092 −2.86574
$$808$$ 0 0
$$809$$ −19.6788 −0.691871 −0.345936 0.938258i $$-0.612438\pi$$
−0.345936 + 0.938258i $$0.612438\pi$$
$$810$$ 0 0
$$811$$ 7.41499 0.260375 0.130188 0.991489i $$-0.458442\pi$$
0.130188 + 0.991489i $$0.458442\pi$$
$$812$$ 0 0
$$813$$ 71.4844 2.50707
$$814$$ 0 0
$$815$$ −0.396843 −0.0139008
$$816$$ 0 0
$$817$$ −1.65379 −0.0578588
$$818$$ 0 0
$$819$$ −23.8305 −0.832706
$$820$$ 0 0
$$821$$ 5.30773 0.185241 0.0926205 0.995701i $$-0.470476\pi$$
0.0926205 + 0.995701i $$0.470476\pi$$
$$822$$ 0 0
$$823$$ −16.1952 −0.564528 −0.282264 0.959337i $$-0.591085\pi$$
−0.282264 + 0.959337i $$0.591085\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.97372 0.346820 0.173410 0.984850i $$-0.444521\pi$$
0.173410 + 0.984850i $$0.444521\pi$$
$$828$$ 0 0
$$829$$ −26.4859 −0.919893 −0.459947 0.887947i $$-0.652131\pi$$
−0.459947 + 0.887947i $$0.652131\pi$$
$$830$$ 0 0
$$831$$ 30.3169 1.05168
$$832$$ 0 0
$$833$$ −3.43779 −0.119112
$$834$$ 0 0
$$835$$ 13.9140 0.481514
$$836$$ 0 0
$$837$$ 51.6931 1.78677
$$838$$ 0 0
$$839$$ 24.0591 0.830613 0.415307 0.909681i $$-0.363674\pi$$
0.415307 + 0.909681i $$0.363674\pi$$
$$840$$ 0 0
$$841$$ 58.6490 2.02238
$$842$$ 0 0
$$843$$ 12.4103 0.427433
$$844$$ 0 0
$$845$$ −6.67004 −0.229456
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 14.4803 0.496962
$$850$$ 0 0
$$851$$ 50.6329 1.73567
$$852$$ 0 0
$$853$$ −22.5471 −0.771997 −0.385999 0.922499i $$-0.626143\pi$$
−0.385999 + 0.922499i $$0.626143\pi$$
$$854$$ 0 0
$$855$$ −1.60689 −0.0549545
$$856$$ 0 0
$$857$$ −10.0453 −0.343143 −0.171571 0.985172i $$-0.554884\pi$$
−0.171571 + 0.985172i $$0.554884\pi$$
$$858$$ 0 0
$$859$$ 15.7921 0.538819 0.269409 0.963026i $$-0.413172\pi$$
0.269409 + 0.963026i $$0.413172\pi$$
$$860$$ 0 0
$$861$$ −22.9676 −0.782734
$$862$$ 0 0
$$863$$ −22.6412 −0.770717 −0.385358 0.922767i $$-0.625922\pi$$
−0.385358 + 0.922767i $$0.625922\pi$$
$$864$$ 0 0
$$865$$ 5.33390 0.181358
$$866$$ 0 0
$$867$$ 50.6600 1.72050
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 31.8318 1.07858
$$872$$ 0 0
$$873$$ −48.6105 −1.64522
$$874$$ 0 0
$$875$$ 0.865927 0.0292737
$$876$$ 0 0
$$877$$ −23.7591 −0.802288 −0.401144 0.916015i $$-0.631387\pi$$
−0.401144 + 0.916015i $$0.631387\pi$$
$$878$$ 0 0
$$879$$ 0.436931 0.0147373
$$880$$ 0 0
$$881$$ −21.6772 −0.730323 −0.365161 0.930944i $$-0.618986\pi$$
−0.365161 + 0.930944i $$0.618986\pi$$
$$882$$ 0 0
$$883$$ −40.6198 −1.36697 −0.683483 0.729967i $$-0.739535\pi$$
−0.683483 + 0.729967i $$0.739535\pi$$
$$884$$ 0 0
$$885$$ −41.0039 −1.37833
$$886$$ 0 0
$$887$$ −34.1509 −1.14668 −0.573338 0.819319i $$-0.694352\pi$$
−0.573338 + 0.819319i $$0.694352\pi$$
$$888$$ 0 0
$$889$$ 3.22068 0.108018
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0.213180 0.00713379
$$894$$ 0 0
$$895$$ −7.63478 −0.255202
$$896$$ 0 0
$$897$$ −106.666 −3.56148
$$898$$ 0 0
$$899$$ −49.7678 −1.65985
$$900$$ 0 0
$$901$$ 1.32068 0.0439982
$$902$$ 0 0
$$903$$ −16.7780 −0.558337
$$904$$ 0 0
$$905$$ −1.30453 −0.0433639
$$906$$ 0 0
$$907$$ 11.5541 0.383647 0.191824 0.981429i $$-0.438560\pi$$
0.191824 + 0.981429i $$0.438560\pi$$
$$908$$ 0 0
$$909$$ 112.926 3.74553
$$910$$ 0 0
$$911$$ −14.4057 −0.477284 −0.238642 0.971108i $$-0.576702\pi$$
−0.238642 + 0.971108i $$0.576702\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 20.0474 0.662748
$$916$$ 0 0
$$917$$ 17.6873 0.584087
$$918$$ 0 0
$$919$$ 49.1155 1.62017 0.810085 0.586312i $$-0.199421\pi$$
0.810085 + 0.586312i $$0.199421\pi$$
$$920$$ 0 0
$$921$$ −78.1482 −2.57507
$$922$$ 0 0
$$923$$ 12.4143 0.408621
$$924$$ 0 0
$$925$$ 6.38739 0.210016
$$926$$ 0 0
$$927$$ 73.2360 2.40539
$$928$$ 0 0
$$929$$ 23.9888 0.787048 0.393524 0.919314i $$-0.371256\pi$$
0.393524 + 0.919314i $$0.371256\pi$$
$$930$$ 0 0
$$931$$ −1.61856 −0.0530461
$$932$$ 0 0
$$933$$ −49.7316 −1.62814
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −13.1022 −0.428029 −0.214014 0.976831i $$-0.568654\pi$$
−0.214014 + 0.976831i $$0.568654\pi$$
$$938$$ 0 0
$$939$$ −80.4729 −2.62613
$$940$$ 0 0
$$941$$ 36.6826 1.19582 0.597909 0.801564i $$-0.295998\pi$$
0.597909 + 0.801564i $$0.295998\pi$$
$$942$$ 0 0
$$943$$ −69.2994 −2.25670
$$944$$ 0 0
$$945$$ −8.42052 −0.273919
$$946$$ 0 0
$$947$$ 20.6112 0.669774 0.334887 0.942258i $$-0.391302\pi$$
0.334887 + 0.942258i $$0.391302\pi$$
$$948$$ 0 0
$$949$$ 12.8207 0.416177
$$950$$ 0 0
$$951$$ −50.7156 −1.64457
$$952$$ 0 0
$$953$$ −17.7949 −0.576433 −0.288216 0.957565i $$-0.593062\pi$$
−0.288216 + 0.957565i $$0.593062\pi$$
$$954$$ 0 0
$$955$$ −25.0833 −0.811678
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −11.6222 −0.375299
$$960$$ 0 0
$$961$$ −2.74144 −0.0884337
$$962$$ 0 0
$$963$$ 3.23264 0.104170
$$964$$ 0 0
$$965$$ −19.9561 −0.642409
$$966$$ 0 0
$$967$$ 0.828077 0.0266292 0.0133146 0.999911i $$-0.495762\pi$$
0.0133146 + 0.999911i $$0.495762\pi$$
$$968$$ 0 0
$$969$$ −0.432154 −0.0138828
$$970$$ 0 0
$$971$$ −0.124535 −0.00399653 −0.00199827 0.999998i $$-0.500636\pi$$
−0.00199827 + 0.999998i $$0.500636\pi$$
$$972$$ 0 0
$$973$$ 12.1971 0.391022
$$974$$ 0 0
$$975$$ −13.4560 −0.430938
$$976$$ 0 0
$$977$$ −21.4664 −0.686770 −0.343385 0.939195i $$-0.611573\pi$$
−0.343385 + 0.939195i $$0.611573\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 38.6296 1.23335
$$982$$ 0 0
$$983$$ 31.7566 1.01288 0.506440 0.862275i $$-0.330961\pi$$
0.506440 + 0.862275i $$0.330961\pi$$
$$984$$ 0 0
$$985$$ −1.31088 −0.0417680
$$986$$ 0 0
$$987$$ 2.16275 0.0688410
$$988$$ 0 0
$$989$$ −50.6237 −1.60974
$$990$$ 0 0
$$991$$ 46.7101 1.48379 0.741897 0.670514i $$-0.233926\pi$$
0.741897 + 0.670514i $$0.233926\pi$$
$$992$$ 0 0
$$993$$ −13.3370 −0.423236
$$994$$ 0 0
$$995$$ 4.99115 0.158230
$$996$$ 0 0
$$997$$ −51.6548 −1.63592 −0.817962 0.575273i $$-0.804896\pi$$
−0.817962 + 0.575273i $$0.804896\pi$$
$$998$$ 0 0
$$999$$ −62.1127 −1.96516
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.be.1.1 6
4.3 odd 2 9680.2.a.cy.1.6 6
11.7 odd 10 440.2.y.b.401.3 yes 12
11.8 odd 10 440.2.y.b.361.3 12
11.10 odd 2 4840.2.a.bf.1.1 6
44.7 even 10 880.2.bo.j.401.1 12
44.19 even 10 880.2.bo.j.801.1 12
44.43 even 2 9680.2.a.cx.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.3 12 11.8 odd 10
440.2.y.b.401.3 yes 12 11.7 odd 10
880.2.bo.j.401.1 12 44.7 even 10
880.2.bo.j.801.1 12 44.19 even 10
4840.2.a.be.1.1 6 1.1 even 1 trivial
4840.2.a.bf.1.1 6 11.10 odd 2
9680.2.a.cx.1.6 6 44.43 even 2
9680.2.a.cy.1.6 6 4.3 odd 2