# Properties

 Label 4140.2.a.m.1.2 Level $4140$ Weight $2$ Character 4140.1 Self dual yes Analytic conductor $33.058$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.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: $$33.0580664368$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 4140.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +2.56155 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +2.56155 q^{7} -2.00000 q^{11} -3.56155 q^{13} -2.56155 q^{17} +6.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} -6.12311 q^{29} +7.24621 q^{31} -2.56155 q^{35} -4.56155 q^{37} -4.12311 q^{41} -4.68466 q^{47} -0.438447 q^{49} +4.56155 q^{53} +2.00000 q^{55} +3.68466 q^{59} -7.12311 q^{61} +3.56155 q^{65} -8.56155 q^{67} -10.1231 q^{71} -4.43845 q^{73} -5.12311 q^{77} +4.87689 q^{79} +13.9309 q^{83} +2.56155 q^{85} -14.2462 q^{89} -9.12311 q^{91} -6.00000 q^{95} -13.1231 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + q^7 $$2 q - 2 q^{5} + q^{7} - 4 q^{11} - 3 q^{13} - q^{17} + 12 q^{19} - 2 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} - q^{35} - 5 q^{37} + 3 q^{47} - 5 q^{49} + 5 q^{53} + 4 q^{55} - 5 q^{59} - 6 q^{61} + 3 q^{65} - 13 q^{67} - 12 q^{71} - 13 q^{73} - 2 q^{77} + 18 q^{79} - q^{83} + q^{85} - 12 q^{89} - 10 q^{91} - 12 q^{95} - 18 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 - 4 * q^11 - 3 * q^13 - q^17 + 12 * q^19 - 2 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 - q^35 - 5 * q^37 + 3 * q^47 - 5 * q^49 + 5 * q^53 + 4 * q^55 - 5 * q^59 - 6 * q^61 + 3 * q^65 - 13 * q^67 - 12 * q^71 - 13 * q^73 - 2 * q^77 + 18 * q^79 - q^83 + q^85 - 12 * q^89 - 10 * q^91 - 12 * q^95 - 18 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.56155 0.968176 0.484088 0.875019i $$-0.339151\pi$$
0.484088 + 0.875019i $$0.339151\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −3.56155 −0.987797 −0.493899 0.869520i $$-0.664429\pi$$
−0.493899 + 0.869520i $$0.664429\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.56155 −0.621268 −0.310634 0.950530i $$-0.600541\pi$$
−0.310634 + 0.950530i $$0.600541\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.12311 −1.13703 −0.568516 0.822672i $$-0.692482\pi$$
−0.568516 + 0.822672i $$0.692482\pi$$
$$30$$ 0 0
$$31$$ 7.24621 1.30146 0.650729 0.759310i $$-0.274463\pi$$
0.650729 + 0.759310i $$0.274463\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.56155 −0.432981
$$36$$ 0 0
$$37$$ −4.56155 −0.749915 −0.374957 0.927042i $$-0.622343\pi$$
−0.374957 + 0.927042i $$0.622343\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.12311 −0.643921 −0.321960 0.946753i $$-0.604342\pi$$
−0.321960 + 0.946753i $$0.604342\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.68466 −0.683328 −0.341664 0.939822i $$-0.610990\pi$$
−0.341664 + 0.939822i $$0.610990\pi$$
$$48$$ 0 0
$$49$$ −0.438447 −0.0626353
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 4.56155 0.626577 0.313289 0.949658i $$-0.398569\pi$$
0.313289 + 0.949658i $$0.398569\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.68466 0.479702 0.239851 0.970810i $$-0.422901\pi$$
0.239851 + 0.970810i $$0.422901\pi$$
$$60$$ 0 0
$$61$$ −7.12311 −0.912020 −0.456010 0.889975i $$-0.650722\pi$$
−0.456010 + 0.889975i $$0.650722\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.56155 0.441756
$$66$$ 0 0
$$67$$ −8.56155 −1.04596 −0.522980 0.852345i $$-0.675180\pi$$
−0.522980 + 0.852345i $$0.675180\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.1231 −1.20139 −0.600696 0.799478i $$-0.705110\pi$$
−0.600696 + 0.799478i $$0.705110\pi$$
$$72$$ 0 0
$$73$$ −4.43845 −0.519481 −0.259740 0.965678i $$-0.583637\pi$$
−0.259740 + 0.965678i $$0.583637\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −5.12311 −0.583832
$$78$$ 0 0
$$79$$ 4.87689 0.548693 0.274347 0.961631i $$-0.411538\pi$$
0.274347 + 0.961631i $$0.411538\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 13.9309 1.52911 0.764556 0.644558i $$-0.222958\pi$$
0.764556 + 0.644558i $$0.222958\pi$$
$$84$$ 0 0
$$85$$ 2.56155 0.277839
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −14.2462 −1.51010 −0.755048 0.655670i $$-0.772386\pi$$
−0.755048 + 0.655670i $$0.772386\pi$$
$$90$$ 0 0
$$91$$ −9.12311 −0.956361
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −13.1231 −1.33245 −0.666225 0.745751i $$-0.732091\pi$$
−0.666225 + 0.745751i $$0.732091\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 17.6847 1.75969 0.879845 0.475261i $$-0.157647\pi$$
0.879845 + 0.475261i $$0.157647\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −3.43845 −0.332407 −0.166204 0.986091i $$-0.553151\pi$$
−0.166204 + 0.986091i $$0.553151\pi$$
$$108$$ 0 0
$$109$$ 15.3693 1.47211 0.736057 0.676920i $$-0.236686\pi$$
0.736057 + 0.676920i $$0.236686\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −12.8078 −1.20485 −0.602427 0.798174i $$-0.705799\pi$$
−0.602427 + 0.798174i $$0.705799\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.56155 −0.601497
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 7.80776 0.692827 0.346414 0.938082i $$-0.387399\pi$$
0.346414 + 0.938082i $$0.387399\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.192236 −0.0167957 −0.00839787 0.999965i $$-0.502673\pi$$
−0.00839787 + 0.999965i $$0.502673\pi$$
$$132$$ 0 0
$$133$$ 15.3693 1.33269
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.87689 −0.587533 −0.293766 0.955877i $$-0.594909\pi$$
−0.293766 + 0.955877i $$0.594909\pi$$
$$138$$ 0 0
$$139$$ −11.2462 −0.953891 −0.476946 0.878933i $$-0.658256\pi$$
−0.476946 + 0.878933i $$0.658256\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 7.12311 0.595664
$$144$$ 0 0
$$145$$ 6.12311 0.508496
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 9.36932 0.767564 0.383782 0.923424i $$-0.374621\pi$$
0.383782 + 0.923424i $$0.374621\pi$$
$$150$$ 0 0
$$151$$ −14.0540 −1.14370 −0.571848 0.820359i $$-0.693773\pi$$
−0.571848 + 0.820359i $$0.693773\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −7.24621 −0.582030
$$156$$ 0 0
$$157$$ −4.31534 −0.344402 −0.172201 0.985062i $$-0.555088\pi$$
−0.172201 + 0.985062i $$0.555088\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.56155 −0.201879
$$162$$ 0 0
$$163$$ 16.9309 1.32613 0.663064 0.748563i $$-0.269256\pi$$
0.663064 + 0.748563i $$0.269256\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −0.315342 −0.0242570
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −5.36932 −0.408222 −0.204111 0.978948i $$-0.565430\pi$$
−0.204111 + 0.978948i $$0.565430\pi$$
$$174$$ 0 0
$$175$$ 2.56155 0.193635
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 6.93087 0.518038 0.259019 0.965872i $$-0.416601\pi$$
0.259019 + 0.965872i $$0.416601\pi$$
$$180$$ 0 0
$$181$$ −5.36932 −0.399098 −0.199549 0.979888i $$-0.563948\pi$$
−0.199549 + 0.979888i $$0.563948\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.56155 0.335372
$$186$$ 0 0
$$187$$ 5.12311 0.374639
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.3693 −1.25680 −0.628400 0.777891i $$-0.716290\pi$$
−0.628400 + 0.777891i $$0.716290\pi$$
$$192$$ 0 0
$$193$$ 2.43845 0.175523 0.0877616 0.996142i $$-0.472029\pi$$
0.0877616 + 0.996142i $$0.472029\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.6847 1.33123 0.665613 0.746297i $$-0.268170\pi$$
0.665613 + 0.746297i $$0.268170\pi$$
$$198$$ 0 0
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −15.6847 −1.10085
$$204$$ 0 0
$$205$$ 4.12311 0.287970
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ 0.315342 0.0217090 0.0108545 0.999941i $$-0.496545\pi$$
0.0108545 + 0.999941i $$0.496545\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 18.5616 1.26004
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.12311 0.613686
$$222$$ 0 0
$$223$$ −17.6155 −1.17962 −0.589812 0.807541i $$-0.700798\pi$$
−0.589812 + 0.807541i $$0.700798\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −26.2462 −1.74202 −0.871011 0.491263i $$-0.836535\pi$$
−0.871011 + 0.491263i $$0.836535\pi$$
$$228$$ 0 0
$$229$$ −26.7386 −1.76694 −0.883469 0.468489i $$-0.844799\pi$$
−0.883469 + 0.468489i $$0.844799\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0.684658 0.0448535 0.0224267 0.999748i $$-0.492861\pi$$
0.0224267 + 0.999748i $$0.492861\pi$$
$$234$$ 0 0
$$235$$ 4.68466 0.305593
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2.75379 −0.178128 −0.0890639 0.996026i $$-0.528388\pi$$
−0.0890639 + 0.996026i $$0.528388\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0.438447 0.0280114
$$246$$ 0 0
$$247$$ −21.3693 −1.35970
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 7.36932 0.465147 0.232574 0.972579i $$-0.425285\pi$$
0.232574 + 0.972579i $$0.425285\pi$$
$$252$$ 0 0
$$253$$ 2.00000 0.125739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −31.8078 −1.98411 −0.992057 0.125790i $$-0.959853\pi$$
−0.992057 + 0.125790i $$0.959853\pi$$
$$258$$ 0 0
$$259$$ −11.6847 −0.726049
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −0.0691303 −0.00426276 −0.00213138 0.999998i $$-0.500678\pi$$
−0.00213138 + 0.999998i $$0.500678\pi$$
$$264$$ 0 0
$$265$$ −4.56155 −0.280214
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −23.2462 −1.41735 −0.708673 0.705537i $$-0.750706\pi$$
−0.708673 + 0.705537i $$0.750706\pi$$
$$270$$ 0 0
$$271$$ 21.9309 1.33221 0.666103 0.745860i $$-0.267961\pi$$
0.666103 + 0.745860i $$0.267961\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −6.68466 −0.401642 −0.200821 0.979628i $$-0.564361\pi$$
−0.200821 + 0.979628i $$0.564361\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.87689 0.410241 0.205121 0.978737i $$-0.434241\pi$$
0.205121 + 0.978737i $$0.434241\pi$$
$$282$$ 0 0
$$283$$ 14.8078 0.880230 0.440115 0.897941i $$-0.354938\pi$$
0.440115 + 0.897941i $$0.354938\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −10.5616 −0.623429
$$288$$ 0 0
$$289$$ −10.4384 −0.614026
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 16.8078 0.981920 0.490960 0.871182i $$-0.336646\pi$$
0.490960 + 0.871182i $$0.336646\pi$$
$$294$$ 0 0
$$295$$ −3.68466 −0.214529
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3.56155 0.205970
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 7.12311 0.407868
$$306$$ 0 0
$$307$$ −21.1231 −1.20556 −0.602780 0.797908i $$-0.705940\pi$$
−0.602780 + 0.797908i $$0.705940\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.68466 −0.492462 −0.246231 0.969211i $$-0.579192\pi$$
−0.246231 + 0.969211i $$0.579192\pi$$
$$312$$ 0 0
$$313$$ −0.807764 −0.0456575 −0.0228288 0.999739i $$-0.507267\pi$$
−0.0228288 + 0.999739i $$0.507267\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −15.1231 −0.849398 −0.424699 0.905335i $$-0.639620\pi$$
−0.424699 + 0.905335i $$0.639620\pi$$
$$318$$ 0 0
$$319$$ 12.2462 0.685656
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −15.3693 −0.855172
$$324$$ 0 0
$$325$$ −3.56155 −0.197559
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 28.6155 1.57285 0.786426 0.617685i $$-0.211929\pi$$
0.786426 + 0.617685i $$0.211929\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 8.56155 0.467768
$$336$$ 0 0
$$337$$ 13.6155 0.741685 0.370843 0.928696i $$-0.379069\pi$$
0.370843 + 0.928696i $$0.379069\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −14.4924 −0.784809
$$342$$ 0 0
$$343$$ −19.0540 −1.02882
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.4924 −0.885360 −0.442680 0.896680i $$-0.645972\pi$$
−0.442680 + 0.896680i $$0.645972\pi$$
$$348$$ 0 0
$$349$$ −26.3693 −1.41152 −0.705759 0.708452i $$-0.749394\pi$$
−0.705759 + 0.708452i $$0.749394\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2.05398 0.109322 0.0546610 0.998505i $$-0.482592\pi$$
0.0546610 + 0.998505i $$0.482592\pi$$
$$354$$ 0 0
$$355$$ 10.1231 0.537279
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −15.6155 −0.824156 −0.412078 0.911149i $$-0.635197\pi$$
−0.412078 + 0.911149i $$0.635197\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.43845 0.232319
$$366$$ 0 0
$$367$$ −22.5616 −1.17770 −0.588852 0.808241i $$-0.700420\pi$$
−0.588852 + 0.808241i $$0.700420\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 11.6847 0.606637
$$372$$ 0 0
$$373$$ −20.2462 −1.04831 −0.524155 0.851623i $$-0.675619\pi$$
−0.524155 + 0.851623i $$0.675619\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 21.8078 1.12316
$$378$$ 0 0
$$379$$ −36.4924 −1.87449 −0.937245 0.348672i $$-0.886633\pi$$
−0.937245 + 0.348672i $$0.886633\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −3.19224 −0.163116 −0.0815578 0.996669i $$-0.525990\pi$$
−0.0815578 + 0.996669i $$0.525990\pi$$
$$384$$ 0 0
$$385$$ 5.12311 0.261098
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 10.8769 0.551480 0.275740 0.961232i $$-0.411077\pi$$
0.275740 + 0.961232i $$0.411077\pi$$
$$390$$ 0 0
$$391$$ 2.56155 0.129543
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.87689 −0.245383
$$396$$ 0 0
$$397$$ 34.5464 1.73383 0.866917 0.498453i $$-0.166098\pi$$
0.866917 + 0.498453i $$0.166098\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ −25.8078 −1.28558
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.12311 0.452216
$$408$$ 0 0
$$409$$ −35.9848 −1.77934 −0.889668 0.456608i $$-0.849064\pi$$
−0.889668 + 0.456608i $$0.849064\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 9.43845 0.464436
$$414$$ 0 0
$$415$$ −13.9309 −0.683839
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 18.8769 0.922197 0.461098 0.887349i $$-0.347456\pi$$
0.461098 + 0.887349i $$0.347456\pi$$
$$420$$ 0 0
$$421$$ 25.1231 1.22443 0.612213 0.790693i $$-0.290280\pi$$
0.612213 + 0.790693i $$0.290280\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.56155 −0.124254
$$426$$ 0 0
$$427$$ −18.2462 −0.882996
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 10.2462 0.493543 0.246771 0.969074i $$-0.420630\pi$$
0.246771 + 0.969074i $$0.420630\pi$$
$$432$$ 0 0
$$433$$ 39.9309 1.91896 0.959478 0.281785i $$-0.0909265\pi$$
0.959478 + 0.281785i $$0.0909265\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 19.8078 0.945373 0.472686 0.881231i $$-0.343284\pi$$
0.472686 + 0.881231i $$0.343284\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.8078 0.561004 0.280502 0.959853i $$-0.409499\pi$$
0.280502 + 0.959853i $$0.409499\pi$$
$$444$$ 0 0
$$445$$ 14.2462 0.675335
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 21.6847 1.02336 0.511681 0.859175i $$-0.329023\pi$$
0.511681 + 0.859175i $$0.329023\pi$$
$$450$$ 0 0
$$451$$ 8.24621 0.388299
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 9.12311 0.427698
$$456$$ 0 0
$$457$$ 9.43845 0.441512 0.220756 0.975329i $$-0.429148\pi$$
0.220756 + 0.975329i $$0.429148\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 40.9309 1.90634 0.953170 0.302434i $$-0.0977992\pi$$
0.953170 + 0.302434i $$0.0977992\pi$$
$$462$$ 0 0
$$463$$ −31.3693 −1.45786 −0.728928 0.684590i $$-0.759981\pi$$
−0.728928 + 0.684590i $$0.759981\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.3153 1.03263 0.516315 0.856398i $$-0.327303\pi$$
0.516315 + 0.856398i $$0.327303\pi$$
$$468$$ 0 0
$$469$$ −21.9309 −1.01267
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 43.2311 1.97528 0.987639 0.156748i $$-0.0501009\pi$$
0.987639 + 0.156748i $$0.0501009\pi$$
$$480$$ 0 0
$$481$$ 16.2462 0.740763
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 13.1231 0.595890
$$486$$ 0 0
$$487$$ 7.80776 0.353804 0.176902 0.984229i $$-0.443393\pi$$
0.176902 + 0.984229i $$0.443393\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −11.4924 −0.518646 −0.259323 0.965791i $$-0.583499\pi$$
−0.259323 + 0.965791i $$0.583499\pi$$
$$492$$ 0 0
$$493$$ 15.6847 0.706401
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −25.9309 −1.16316
$$498$$ 0 0
$$499$$ 22.6155 1.01241 0.506205 0.862413i $$-0.331048\pi$$
0.506205 + 0.862413i $$0.331048\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.93087 0.175269 0.0876344 0.996153i $$-0.472069\pi$$
0.0876344 + 0.996153i $$0.472069\pi$$
$$504$$ 0 0
$$505$$ −17.6847 −0.786957
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 27.1771 1.20460 0.602301 0.798269i $$-0.294251\pi$$
0.602301 + 0.798269i $$0.294251\pi$$
$$510$$ 0 0
$$511$$ −11.3693 −0.502949
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 9.36932 0.412062
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 13.6155 0.596507 0.298254 0.954487i $$-0.403596\pi$$
0.298254 + 0.954487i $$0.403596\pi$$
$$522$$ 0 0
$$523$$ −14.7386 −0.644475 −0.322238 0.946659i $$-0.604435\pi$$
−0.322238 + 0.946659i $$0.604435\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −18.5616 −0.808554
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 14.6847 0.636063
$$534$$ 0 0
$$535$$ 3.43845 0.148657
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.876894 0.0377705
$$540$$ 0 0
$$541$$ −2.68466 −0.115422 −0.0577112 0.998333i $$-0.518380\pi$$
−0.0577112 + 0.998333i $$0.518380\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −15.3693 −0.658349
$$546$$ 0 0
$$547$$ −21.1771 −0.905467 −0.452733 0.891646i $$-0.649551\pi$$
−0.452733 + 0.891646i $$0.649551\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −36.7386 −1.56512
$$552$$ 0 0
$$553$$ 12.4924 0.531232
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15.6847 0.664580 0.332290 0.943177i $$-0.392179\pi$$
0.332290 + 0.943177i $$0.392179\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −32.6695 −1.37686 −0.688428 0.725305i $$-0.741699\pi$$
−0.688428 + 0.725305i $$0.741699\pi$$
$$564$$ 0 0
$$565$$ 12.8078 0.538827
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −16.7386 −0.701720 −0.350860 0.936428i $$-0.614111\pi$$
−0.350860 + 0.936428i $$0.614111\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −29.5616 −1.23066 −0.615332 0.788268i $$-0.710978\pi$$
−0.615332 + 0.788268i $$0.710978\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 35.6847 1.48045
$$582$$ 0 0
$$583$$ −9.12311 −0.377840
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −7.06913 −0.291774 −0.145887 0.989301i $$-0.546604\pi$$
−0.145887 + 0.989301i $$0.546604\pi$$
$$588$$ 0 0
$$589$$ 43.4773 1.79145
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −15.6155 −0.641253 −0.320626 0.947206i $$-0.603893\pi$$
−0.320626 + 0.947206i $$0.603893\pi$$
$$594$$ 0 0
$$595$$ 6.56155 0.268997
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 32.9848 1.34772 0.673862 0.738857i $$-0.264634\pi$$
0.673862 + 0.738857i $$0.264634\pi$$
$$600$$ 0 0
$$601$$ 36.3693 1.48354 0.741768 0.670657i $$-0.233988\pi$$
0.741768 + 0.670657i $$0.233988\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 8.49242 0.344697 0.172348 0.985036i $$-0.444865\pi$$
0.172348 + 0.985036i $$0.444865\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 16.6847 0.674989
$$612$$ 0 0
$$613$$ −5.61553 −0.226809 −0.113405 0.993549i $$-0.536176\pi$$
−0.113405 + 0.993549i $$0.536176\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −0.561553 −0.0226073 −0.0113036 0.999936i $$-0.503598\pi$$
−0.0113036 + 0.999936i $$0.503598\pi$$
$$618$$ 0 0
$$619$$ 12.4924 0.502113 0.251056 0.967972i $$-0.419222\pi$$
0.251056 + 0.967972i $$0.419222\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −36.4924 −1.46204
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 11.6847 0.465898
$$630$$ 0 0
$$631$$ 30.2462 1.20408 0.602041 0.798465i $$-0.294354\pi$$
0.602041 + 0.798465i $$0.294354\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −7.80776 −0.309842
$$636$$ 0 0
$$637$$ 1.56155 0.0618710
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6.87689 −0.271621 −0.135810 0.990735i $$-0.543364\pi$$
−0.135810 + 0.990735i $$0.543364\pi$$
$$642$$ 0 0
$$643$$ 20.4233 0.805416 0.402708 0.915328i $$-0.368069\pi$$
0.402708 + 0.915328i $$0.368069\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −5.31534 −0.208968 −0.104484 0.994527i $$-0.533319\pi$$
−0.104484 + 0.994527i $$0.533319\pi$$
$$648$$ 0 0
$$649$$ −7.36932 −0.289271
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −39.6695 −1.55239 −0.776194 0.630494i $$-0.782852\pi$$
−0.776194 + 0.630494i $$0.782852\pi$$
$$654$$ 0 0
$$655$$ 0.192236 0.00751128
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −15.3693 −0.598704 −0.299352 0.954143i $$-0.596770\pi$$
−0.299352 + 0.954143i $$0.596770\pi$$
$$660$$ 0 0
$$661$$ −6.49242 −0.252526 −0.126263 0.991997i $$-0.540298\pi$$
−0.126263 + 0.991997i $$0.540298\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −15.3693 −0.595997
$$666$$ 0 0
$$667$$ 6.12311 0.237088
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 14.2462 0.549969
$$672$$ 0 0
$$673$$ 34.5464 1.33167 0.665833 0.746101i $$-0.268076\pi$$
0.665833 + 0.746101i $$0.268076\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −26.8078 −1.03031 −0.515153 0.857098i $$-0.672265\pi$$
−0.515153 + 0.857098i $$0.672265\pi$$
$$678$$ 0 0
$$679$$ −33.6155 −1.29005
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −42.0540 −1.60915 −0.804575 0.593851i $$-0.797607\pi$$
−0.804575 + 0.593851i $$0.797607\pi$$
$$684$$ 0 0
$$685$$ 6.87689 0.262753
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −16.2462 −0.618931
$$690$$ 0 0
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 11.2462 0.426593
$$696$$ 0 0
$$697$$ 10.5616 0.400047
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.75379 0.0662397 0.0331198 0.999451i $$-0.489456\pi$$
0.0331198 + 0.999451i $$0.489456\pi$$
$$702$$ 0 0
$$703$$ −27.3693 −1.03225
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 45.3002 1.70369
$$708$$ 0 0
$$709$$ 29.7538 1.11743 0.558713 0.829361i $$-0.311295\pi$$
0.558713 + 0.829361i $$0.311295\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −7.24621 −0.271373
$$714$$ 0 0
$$715$$ −7.12311 −0.266389
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −19.0540 −0.710593 −0.355297 0.934754i $$-0.615620\pi$$
−0.355297 + 0.934754i $$0.615620\pi$$
$$720$$ 0 0
$$721$$ −40.9848 −1.52636
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −6.12311 −0.227406
$$726$$ 0 0
$$727$$ −21.1922 −0.785977 −0.392988 0.919543i $$-0.628559\pi$$
−0.392988 + 0.919543i $$0.628559\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0.315342 0.0116474 0.00582370 0.999983i $$-0.498146\pi$$
0.00582370 + 0.999983i $$0.498146\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 17.1231 0.630738
$$738$$ 0 0
$$739$$ 0.615528 0.0226426 0.0113213 0.999936i $$-0.496396\pi$$
0.0113213 + 0.999936i $$0.496396\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −3.50758 −0.128681 −0.0643403 0.997928i $$-0.520494\pi$$
−0.0643403 + 0.997928i $$0.520494\pi$$
$$744$$ 0 0
$$745$$ −9.36932 −0.343265
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −8.80776 −0.321829
$$750$$ 0 0
$$751$$ −13.1231 −0.478869 −0.239434 0.970913i $$-0.576962\pi$$
−0.239434 + 0.970913i $$0.576962\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 14.0540 0.511477
$$756$$ 0 0
$$757$$ 12.5616 0.456557 0.228279 0.973596i $$-0.426690\pi$$
0.228279 + 0.973596i $$0.426690\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 43.9848 1.59445 0.797225 0.603683i $$-0.206301\pi$$
0.797225 + 0.603683i $$0.206301\pi$$
$$762$$ 0 0
$$763$$ 39.3693 1.42526
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −13.1231 −0.473848
$$768$$ 0 0
$$769$$ −10.6307 −0.383352 −0.191676 0.981458i $$-0.561392\pi$$
−0.191676 + 0.981458i $$0.561392\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 41.1231 1.47910 0.739548 0.673104i $$-0.235039\pi$$
0.739548 + 0.673104i $$0.235039\pi$$
$$774$$ 0 0
$$775$$ 7.24621 0.260292
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −24.7386 −0.886354
$$780$$ 0 0
$$781$$ 20.2462 0.724466
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 4.31534 0.154021
$$786$$ 0 0
$$787$$ −39.6847 −1.41461 −0.707303 0.706911i $$-0.750088\pi$$
−0.707303 + 0.706911i $$0.750088\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −32.8078 −1.16651
$$792$$ 0 0
$$793$$ 25.3693 0.900891
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 52.4233 1.85693 0.928464 0.371422i $$-0.121130\pi$$
0.928464 + 0.371422i $$0.121130\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8.87689 0.313259
$$804$$ 0 0
$$805$$ 2.56155 0.0902829
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 54.1771 1.90476 0.952382 0.304906i $$-0.0986251\pi$$
0.952382 + 0.304906i $$0.0986251\pi$$
$$810$$ 0 0
$$811$$ 47.2462 1.65904 0.829519 0.558478i $$-0.188614\pi$$
0.829519 + 0.558478i $$0.188614\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −16.9309 −0.593062
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 34.4924 1.20379 0.601897 0.798574i $$-0.294412\pi$$
0.601897 + 0.798574i $$0.294412\pi$$
$$822$$ 0 0
$$823$$ −38.0540 −1.32648 −0.663239 0.748408i $$-0.730819\pi$$
−0.663239 + 0.748408i $$0.730819\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −19.6847 −0.684503 −0.342251 0.939608i $$-0.611189\pi$$
−0.342251 + 0.939608i $$0.611189\pi$$
$$828$$ 0 0
$$829$$ −29.5464 −1.02619 −0.513094 0.858332i $$-0.671501\pi$$
−0.513094 + 0.858332i $$0.671501\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.12311 0.0389133
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 42.8769 1.48027 0.740137 0.672456i $$-0.234760\pi$$
0.740137 + 0.672456i $$0.234760\pi$$
$$840$$ 0 0
$$841$$ 8.49242 0.292842
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0.315342 0.0108481
$$846$$ 0 0
$$847$$ −17.9309 −0.616112
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4.56155 0.156368
$$852$$ 0 0
$$853$$ −20.2462 −0.693217 −0.346609 0.938010i $$-0.612667\pi$$
−0.346609 + 0.938010i $$0.612667\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 33.3153 1.13803 0.569015 0.822327i $$-0.307325\pi$$
0.569015 + 0.822327i $$0.307325\pi$$
$$858$$ 0 0
$$859$$ −14.5076 −0.494992 −0.247496 0.968889i $$-0.579608\pi$$
−0.247496 + 0.968889i $$0.579608\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 7.56155 0.257398 0.128699 0.991684i $$-0.458920\pi$$
0.128699 + 0.991684i $$0.458920\pi$$
$$864$$ 0 0
$$865$$ 5.36932 0.182562
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −9.75379 −0.330875
$$870$$ 0 0
$$871$$ 30.4924 1.03320
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.56155 −0.0865963
$$876$$ 0 0
$$877$$ −18.9848 −0.641073 −0.320536 0.947236i $$-0.603863\pi$$
−0.320536 + 0.947236i $$0.603863\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −41.4773 −1.39740 −0.698702 0.715413i $$-0.746239\pi$$
−0.698702 + 0.715413i $$0.746239\pi$$
$$882$$ 0 0
$$883$$ 10.7386 0.361384 0.180692 0.983540i $$-0.442166\pi$$
0.180692 + 0.983540i $$0.442166\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −42.7926 −1.43684 −0.718418 0.695612i $$-0.755133\pi$$
−0.718418 + 0.695612i $$0.755133\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −28.1080 −0.940597
$$894$$ 0 0
$$895$$ −6.93087 −0.231673
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −44.3693 −1.47980
$$900$$ 0 0
$$901$$ −11.6847 −0.389272
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 5.36932 0.178482
$$906$$ 0 0
$$907$$ 21.6847 0.720027 0.360014 0.932947i $$-0.382772\pi$$
0.360014 + 0.932947i $$0.382772\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 3.12311 0.103473 0.0517366 0.998661i $$-0.483524\pi$$
0.0517366 + 0.998661i $$0.483524\pi$$
$$912$$ 0 0
$$913$$ −27.8617 −0.922089
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −0.492423 −0.0162612
$$918$$ 0 0
$$919$$ 46.7386 1.54177 0.770883 0.636977i $$-0.219815\pi$$
0.770883 + 0.636977i $$0.219815\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 36.0540 1.18673
$$924$$ 0 0
$$925$$ −4.56155 −0.149983
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 56.7235 1.86104 0.930518 0.366245i $$-0.119357\pi$$
0.930518 + 0.366245i $$0.119357\pi$$
$$930$$ 0 0
$$931$$ −2.63068 −0.0862172
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −5.12311 −0.167543
$$936$$ 0 0
$$937$$ −16.2462 −0.530741 −0.265370 0.964147i $$-0.585494\pi$$
−0.265370 + 0.964147i $$0.585494\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 37.7538 1.23074 0.615369 0.788239i $$-0.289007\pi$$
0.615369 + 0.788239i $$0.289007\pi$$
$$942$$ 0 0
$$943$$ 4.12311 0.134267
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −56.6847 −1.84200 −0.921002 0.389558i $$-0.872628\pi$$
−0.921002 + 0.389558i $$0.872628\pi$$
$$948$$ 0 0
$$949$$ 15.8078 0.513142
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 38.4924 1.24689 0.623446 0.781866i $$-0.285732\pi$$
0.623446 + 0.781866i $$0.285732\pi$$
$$954$$ 0 0
$$955$$ 17.3693 0.562058
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −17.6155 −0.568835
$$960$$ 0 0
$$961$$ 21.5076 0.693793
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −2.43845 −0.0784964
$$966$$ 0 0
$$967$$ −56.6847 −1.82286 −0.911428 0.411460i $$-0.865019\pi$$
−0.911428 + 0.411460i $$0.865019\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.6307 0.726253 0.363127 0.931740i $$-0.381709\pi$$
0.363127 + 0.931740i $$0.381709\pi$$
$$972$$ 0 0
$$973$$ −28.8078 −0.923535
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −59.7926 −1.91294 −0.956468 0.291839i $$-0.905733\pi$$
−0.956468 + 0.291839i $$0.905733\pi$$
$$978$$ 0 0
$$979$$ 28.4924 0.910622
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 37.0540 1.18184 0.590919 0.806731i $$-0.298765\pi$$
0.590919 + 0.806731i $$0.298765\pi$$
$$984$$ 0 0
$$985$$ −18.6847 −0.595343
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 53.3002 1.69314 0.846568 0.532280i $$-0.178665\pi$$
0.846568 + 0.532280i $$0.178665\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 10.0000 0.317021
$$996$$ 0 0
$$997$$ 55.6155 1.76136 0.880681 0.473710i $$-0.157086\pi$$
0.880681 + 0.473710i $$0.157086\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 4140.2.a.m.1.2 2
3.2 odd 2 460.2.a.e.1.1 2
12.11 even 2 1840.2.a.m.1.2 2
15.2 even 4 2300.2.c.h.1749.3 4
15.8 even 4 2300.2.c.h.1749.2 4
15.14 odd 2 2300.2.a.i.1.2 2
24.5 odd 2 7360.2.a.bi.1.2 2
24.11 even 2 7360.2.a.bo.1.1 2
60.59 even 2 9200.2.a.bv.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.1 2 3.2 odd 2
1840.2.a.m.1.2 2 12.11 even 2
2300.2.a.i.1.2 2 15.14 odd 2
2300.2.c.h.1749.2 4 15.8 even 4
2300.2.c.h.1749.3 4 15.2 even 4
4140.2.a.m.1.2 2 1.1 even 1 trivial
7360.2.a.bi.1.2 2 24.5 odd 2
7360.2.a.bo.1.1 2 24.11 even 2
9200.2.a.bv.1.1 2 60.59 even 2