# Properties

 Label 6552.2.a.bk.1.2 Level $6552$ Weight $2$ Character 6552.1 Self dual yes Analytic conductor $52.318$ 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] = [6552,2,Mod(1,6552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6552.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: $$52.3179834043$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2184) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 6552.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.56155 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+3.56155 q^{5} -1.00000 q^{7} -3.12311 q^{11} +1.00000 q^{13} +1.12311 q^{17} -1.56155 q^{19} -8.68466 q^{23} +7.68466 q^{25} -0.438447 q^{29} -5.56155 q^{31} -3.56155 q^{35} +1.12311 q^{37} +2.00000 q^{41} -9.56155 q^{43} -9.56155 q^{47} +1.00000 q^{49} -6.68466 q^{53} -11.1231 q^{55} +8.00000 q^{59} -2.00000 q^{61} +3.56155 q^{65} -7.12311 q^{67} +10.2462 q^{71} -8.43845 q^{73} +3.12311 q^{77} +8.68466 q^{79} +10.4384 q^{83} +4.00000 q^{85} -9.80776 q^{89} -1.00000 q^{91} -5.56155 q^{95} -6.68466 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 3 * q^5 - 2 * q^7 $$2 q + 3 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 6 q^{17} + q^{19} - 5 q^{23} + 3 q^{25} - 5 q^{29} - 7 q^{31} - 3 q^{35} - 6 q^{37} + 4 q^{41} - 15 q^{43} - 15 q^{47} + 2 q^{49} - q^{53} - 14 q^{55} + 16 q^{59} - 4 q^{61} + 3 q^{65} - 6 q^{67} + 4 q^{71} - 21 q^{73} - 2 q^{77} + 5 q^{79} + 25 q^{83} + 8 q^{85} + q^{89} - 2 q^{91} - 7 q^{95} - q^{97}+O(q^{100})$$ 2 * q + 3 * q^5 - 2 * q^7 + 2 * q^11 + 2 * q^13 - 6 * q^17 + q^19 - 5 * q^23 + 3 * q^25 - 5 * q^29 - 7 * q^31 - 3 * q^35 - 6 * q^37 + 4 * q^41 - 15 * q^43 - 15 * q^47 + 2 * q^49 - q^53 - 14 * q^55 + 16 * q^59 - 4 * q^61 + 3 * q^65 - 6 * q^67 + 4 * q^71 - 21 * q^73 - 2 * q^77 + 5 * q^79 + 25 * q^83 + 8 * q^85 + q^89 - 2 * q^91 - 7 * q^95 - 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
$$5$$ 3.56155 1.59277 0.796387 0.604787i $$-0.206742\pi$$
0.796387 + 0.604787i $$0.206742\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.12311 −0.941652 −0.470826 0.882226i $$-0.656044\pi$$
−0.470826 + 0.882226i $$0.656044\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.12311 0.272393 0.136197 0.990682i $$-0.456512\pi$$
0.136197 + 0.990682i $$0.456512\pi$$
$$18$$ 0 0
$$19$$ −1.56155 −0.358245 −0.179122 0.983827i $$-0.557326\pi$$
−0.179122 + 0.983827i $$0.557326\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.68466 −1.81088 −0.905438 0.424478i $$-0.860458\pi$$
−0.905438 + 0.424478i $$0.860458\pi$$
$$24$$ 0 0
$$25$$ 7.68466 1.53693
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −0.438447 −0.0814176 −0.0407088 0.999171i $$-0.512962\pi$$
−0.0407088 + 0.999171i $$0.512962\pi$$
$$30$$ 0 0
$$31$$ −5.56155 −0.998884 −0.499442 0.866347i $$-0.666462\pi$$
−0.499442 + 0.866347i $$0.666462\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.56155 −0.602012
$$36$$ 0 0
$$37$$ 1.12311 0.184637 0.0923187 0.995730i $$-0.470572\pi$$
0.0923187 + 0.995730i $$0.470572\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −9.56155 −1.45812 −0.729062 0.684448i $$-0.760043\pi$$
−0.729062 + 0.684448i $$0.760043\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.56155 −1.39470 −0.697348 0.716733i $$-0.745637\pi$$
−0.697348 + 0.716733i $$0.745637\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −6.68466 −0.918208 −0.459104 0.888382i $$-0.651830\pi$$
−0.459104 + 0.888382i $$0.651830\pi$$
$$54$$ 0 0
$$55$$ −11.1231 −1.49984
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.56155 0.441756
$$66$$ 0 0
$$67$$ −7.12311 −0.870226 −0.435113 0.900376i $$-0.643292\pi$$
−0.435113 + 0.900376i $$0.643292\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.2462 1.21600 0.608001 0.793936i $$-0.291972\pi$$
0.608001 + 0.793936i $$0.291972\pi$$
$$72$$ 0 0
$$73$$ −8.43845 −0.987646 −0.493823 0.869563i $$-0.664401\pi$$
−0.493823 + 0.869563i $$0.664401\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.12311 0.355911
$$78$$ 0 0
$$79$$ 8.68466 0.977100 0.488550 0.872536i $$-0.337526\pi$$
0.488550 + 0.872536i $$0.337526\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 10.4384 1.14577 0.572884 0.819636i $$-0.305824\pi$$
0.572884 + 0.819636i $$0.305824\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.80776 −1.03962 −0.519810 0.854282i $$-0.673997\pi$$
−0.519810 + 0.854282i $$0.673997\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −5.56155 −0.570603
$$96$$ 0 0
$$97$$ −6.68466 −0.678724 −0.339362 0.940656i $$-0.610211\pi$$
−0.339362 + 0.940656i $$0.610211\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.24621 −0.422514 −0.211257 0.977431i $$-0.567756\pi$$
−0.211257 + 0.977431i $$0.567756\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.4924 1.59438 0.797191 0.603727i $$-0.206318\pi$$
0.797191 + 0.603727i $$0.206318\pi$$
$$108$$ 0 0
$$109$$ −16.2462 −1.55610 −0.778052 0.628199i $$-0.783792\pi$$
−0.778052 + 0.628199i $$0.783792\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.31534 −0.123737 −0.0618685 0.998084i $$-0.519706\pi$$
−0.0618685 + 0.998084i $$0.519706\pi$$
$$114$$ 0 0
$$115$$ −30.9309 −2.88432
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1.12311 −0.102955
$$120$$ 0 0
$$121$$ −1.24621 −0.113292
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 9.56155 0.855211
$$126$$ 0 0
$$127$$ −22.2462 −1.97403 −0.987016 0.160622i $$-0.948650\pi$$
−0.987016 + 0.160622i $$0.948650\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 1.56155 0.135404
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 13.1231 1.12118 0.560591 0.828093i $$-0.310574\pi$$
0.560591 + 0.828093i $$0.310574\pi$$
$$138$$ 0 0
$$139$$ −15.1231 −1.28273 −0.641363 0.767238i $$-0.721631\pi$$
−0.641363 + 0.767238i $$0.721631\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.12311 −0.261167
$$144$$ 0 0
$$145$$ −1.56155 −0.129680
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −8.24621 −0.675556 −0.337778 0.941226i $$-0.609675\pi$$
−0.337778 + 0.941226i $$0.609675\pi$$
$$150$$ 0 0
$$151$$ 12.4924 1.01662 0.508309 0.861174i $$-0.330271\pi$$
0.508309 + 0.861174i $$0.330271\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −19.8078 −1.59100
$$156$$ 0 0
$$157$$ 9.12311 0.728103 0.364052 0.931379i $$-0.381393\pi$$
0.364052 + 0.931379i $$0.381393\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.68466 0.684447
$$162$$ 0 0
$$163$$ 2.24621 0.175937 0.0879684 0.996123i $$-0.471963\pi$$
0.0879684 + 0.996123i $$0.471963\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.6847 −0.981568 −0.490784 0.871281i $$-0.663290\pi$$
−0.490784 + 0.871281i $$0.663290\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −7.36932 −0.560279 −0.280139 0.959959i $$-0.590381\pi$$
−0.280139 + 0.959959i $$0.590381\pi$$
$$174$$ 0 0
$$175$$ −7.68466 −0.580906
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 17.5616 1.31261 0.656306 0.754495i $$-0.272118\pi$$
0.656306 + 0.754495i $$0.272118\pi$$
$$180$$ 0 0
$$181$$ 12.2462 0.910254 0.455127 0.890427i $$-0.349594\pi$$
0.455127 + 0.890427i $$0.349594\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −3.50758 −0.256499
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.4924 0.903920 0.451960 0.892038i $$-0.350725\pi$$
0.451960 + 0.892038i $$0.350725\pi$$
$$192$$ 0 0
$$193$$ 8.24621 0.593575 0.296788 0.954944i $$-0.404085\pi$$
0.296788 + 0.954944i $$0.404085\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 0 0
$$199$$ 6.24621 0.442782 0.221391 0.975185i $$-0.428940\pi$$
0.221391 + 0.975185i $$0.428940\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0.438447 0.0307730
$$204$$ 0 0
$$205$$ 7.12311 0.497499
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.87689 0.337342
$$210$$ 0 0
$$211$$ −3.31534 −0.228238 −0.114119 0.993467i $$-0.536404\pi$$
−0.114119 + 0.993467i $$0.536404\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −34.0540 −2.32246
$$216$$ 0 0
$$217$$ 5.56155 0.377543
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.12311 0.0755483
$$222$$ 0 0
$$223$$ 11.8078 0.790706 0.395353 0.918529i $$-0.370622\pi$$
0.395353 + 0.918529i $$0.370622\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 22.2462 1.47653 0.738266 0.674509i $$-0.235645\pi$$
0.738266 + 0.674509i $$0.235645\pi$$
$$228$$ 0 0
$$229$$ −16.2462 −1.07358 −0.536790 0.843716i $$-0.680363\pi$$
−0.536790 + 0.843716i $$0.680363\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −26.6847 −1.74817 −0.874085 0.485773i $$-0.838538\pi$$
−0.874085 + 0.485773i $$0.838538\pi$$
$$234$$ 0 0
$$235$$ −34.0540 −2.22144
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 24.4924 1.58428 0.792142 0.610337i $$-0.208966\pi$$
0.792142 + 0.610337i $$0.208966\pi$$
$$240$$ 0 0
$$241$$ 5.80776 0.374111 0.187055 0.982349i $$-0.440106\pi$$
0.187055 + 0.982349i $$0.440106\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3.56155 0.227539
$$246$$ 0 0
$$247$$ −1.56155 −0.0993592
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 8.87689 0.560305 0.280152 0.959956i $$-0.409615\pi$$
0.280152 + 0.959956i $$0.409615\pi$$
$$252$$ 0 0
$$253$$ 27.1231 1.70522
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −1.12311 −0.0697864
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −13.5616 −0.836241 −0.418121 0.908392i $$-0.637311\pi$$
−0.418121 + 0.908392i $$0.637311\pi$$
$$264$$ 0 0
$$265$$ −23.8078 −1.46250
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 17.6155 1.07404 0.537019 0.843570i $$-0.319550\pi$$
0.537019 + 0.843570i $$0.319550\pi$$
$$270$$ 0 0
$$271$$ −12.4924 −0.758861 −0.379430 0.925220i $$-0.623880\pi$$
−0.379430 + 0.925220i $$0.623880\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −24.0000 −1.44725
$$276$$ 0 0
$$277$$ 16.0540 0.964590 0.482295 0.876009i $$-0.339803\pi$$
0.482295 + 0.876009i $$0.339803\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8.24621 0.491928 0.245964 0.969279i $$-0.420896\pi$$
0.245964 + 0.969279i $$0.420896\pi$$
$$282$$ 0 0
$$283$$ −8.49242 −0.504822 −0.252411 0.967620i $$-0.581224\pi$$
−0.252411 + 0.967620i $$0.581224\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ −15.7386 −0.925802
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 11.5616 0.675433 0.337717 0.941248i $$-0.390345\pi$$
0.337717 + 0.941248i $$0.390345\pi$$
$$294$$ 0 0
$$295$$ 28.4924 1.65889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −8.68466 −0.502247
$$300$$ 0 0
$$301$$ 9.56155 0.551119
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.12311 −0.407868
$$306$$ 0 0
$$307$$ −4.68466 −0.267368 −0.133684 0.991024i $$-0.542681\pi$$
−0.133684 + 0.991024i $$0.542681\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 29.8617 1.69330 0.846652 0.532147i $$-0.178615\pi$$
0.846652 + 0.532147i $$0.178615\pi$$
$$312$$ 0 0
$$313$$ −15.7538 −0.890457 −0.445228 0.895417i $$-0.646878\pi$$
−0.445228 + 0.895417i $$0.646878\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.3693 0.863227 0.431613 0.902059i $$-0.357944\pi$$
0.431613 + 0.902059i $$0.357944\pi$$
$$318$$ 0 0
$$319$$ 1.36932 0.0766670
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.75379 −0.0975834
$$324$$ 0 0
$$325$$ 7.68466 0.426268
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 9.56155 0.527145
$$330$$ 0 0
$$331$$ −15.1231 −0.831241 −0.415621 0.909538i $$-0.636436\pi$$
−0.415621 + 0.909538i $$0.636436\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −25.3693 −1.38607
$$336$$ 0 0
$$337$$ 28.0540 1.52820 0.764099 0.645099i $$-0.223184\pi$$
0.764099 + 0.645099i $$0.223184\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 17.3693 0.940601
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.24621 0.120583 0.0602915 0.998181i $$-0.480797\pi$$
0.0602915 + 0.998181i $$0.480797\pi$$
$$348$$ 0 0
$$349$$ −24.9309 −1.33452 −0.667259 0.744825i $$-0.732533\pi$$
−0.667259 + 0.744825i $$0.732533\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −12.2462 −0.651800 −0.325900 0.945404i $$-0.605667\pi$$
−0.325900 + 0.945404i $$0.605667\pi$$
$$354$$ 0 0
$$355$$ 36.4924 1.93682
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.4924 −1.29266 −0.646330 0.763058i $$-0.723697\pi$$
−0.646330 + 0.763058i $$0.723697\pi$$
$$360$$ 0 0
$$361$$ −16.5616 −0.871661
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −30.0540 −1.57310
$$366$$ 0 0
$$367$$ 7.61553 0.397527 0.198764 0.980047i $$-0.436307\pi$$
0.198764 + 0.980047i $$0.436307\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.68466 0.347050
$$372$$ 0 0
$$373$$ 34.4924 1.78595 0.892975 0.450106i $$-0.148614\pi$$
0.892975 + 0.450106i $$0.148614\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −0.438447 −0.0225812
$$378$$ 0 0
$$379$$ −13.3693 −0.686736 −0.343368 0.939201i $$-0.611568\pi$$
−0.343368 + 0.939201i $$0.611568\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 0 0
$$385$$ 11.1231 0.566886
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 26.9848 1.36819 0.684093 0.729395i $$-0.260198\pi$$
0.684093 + 0.729395i $$0.260198\pi$$
$$390$$ 0 0
$$391$$ −9.75379 −0.493270
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 30.9309 1.55630
$$396$$ 0 0
$$397$$ −37.4233 −1.87822 −0.939111 0.343615i $$-0.888348\pi$$
−0.939111 + 0.343615i $$0.888348\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.87689 −0.143665 −0.0718326 0.997417i $$-0.522885\pi$$
−0.0718326 + 0.997417i $$0.522885\pi$$
$$402$$ 0 0
$$403$$ −5.56155 −0.277041
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.50758 −0.173864
$$408$$ 0 0
$$409$$ 23.5616 1.16504 0.582522 0.812815i $$-0.302066\pi$$
0.582522 + 0.812815i $$0.302066\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 37.1771 1.82495
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −21.3693 −1.04396 −0.521980 0.852958i $$-0.674806\pi$$
−0.521980 + 0.852958i $$0.674806\pi$$
$$420$$ 0 0
$$421$$ −6.87689 −0.335159 −0.167580 0.985859i $$-0.553595\pi$$
−0.167580 + 0.985859i $$0.553595\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 8.63068 0.418650
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 21.3693 1.02932 0.514662 0.857393i $$-0.327917\pi$$
0.514662 + 0.857393i $$0.327917\pi$$
$$432$$ 0 0
$$433$$ −21.6155 −1.03878 −0.519388 0.854539i $$-0.673840\pi$$
−0.519388 + 0.854539i $$0.673840\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 13.5616 0.648737
$$438$$ 0 0
$$439$$ 0.384472 0.0183498 0.00917492 0.999958i $$-0.497079\pi$$
0.00917492 + 0.999958i $$0.497079\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 23.8078 1.13114 0.565571 0.824700i $$-0.308656\pi$$
0.565571 + 0.824700i $$0.308656\pi$$
$$444$$ 0 0
$$445$$ −34.9309 −1.65588
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −27.8617 −1.31488 −0.657438 0.753508i $$-0.728360\pi$$
−0.657438 + 0.753508i $$0.728360\pi$$
$$450$$ 0 0
$$451$$ −6.24621 −0.294123
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.56155 −0.166968
$$456$$ 0 0
$$457$$ −5.61553 −0.262683 −0.131342 0.991337i $$-0.541928\pi$$
−0.131342 + 0.991337i $$0.541928\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 36.2462 1.68815 0.844077 0.536222i $$-0.180149\pi$$
0.844077 + 0.536222i $$0.180149\pi$$
$$462$$ 0 0
$$463$$ 14.6307 0.679946 0.339973 0.940435i $$-0.389582\pi$$
0.339973 + 0.940435i $$0.389582\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −37.3693 −1.72925 −0.864623 0.502421i $$-0.832443\pi$$
−0.864623 + 0.502421i $$0.832443\pi$$
$$468$$ 0 0
$$469$$ 7.12311 0.328914
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 29.8617 1.37304
$$474$$ 0 0
$$475$$ −12.0000 −0.550598
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −28.6847 −1.31064 −0.655318 0.755353i $$-0.727465\pi$$
−0.655318 + 0.755353i $$0.727465\pi$$
$$480$$ 0 0
$$481$$ 1.12311 0.0512092
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −23.8078 −1.08105
$$486$$ 0 0
$$487$$ −33.3693 −1.51211 −0.756054 0.654509i $$-0.772875\pi$$
−0.756054 + 0.654509i $$0.772875\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −8.49242 −0.383258 −0.191629 0.981467i $$-0.561377\pi$$
−0.191629 + 0.981467i $$0.561377\pi$$
$$492$$ 0 0
$$493$$ −0.492423 −0.0221776
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10.2462 −0.459605
$$498$$ 0 0
$$499$$ 16.4924 0.738302 0.369151 0.929369i $$-0.379648\pi$$
0.369151 + 0.929369i $$0.379648\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 1.36932 0.0610548 0.0305274 0.999534i $$-0.490281\pi$$
0.0305274 + 0.999534i $$0.490281\pi$$
$$504$$ 0 0
$$505$$ −15.1231 −0.672969
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −4.43845 −0.196731 −0.0983654 0.995150i $$-0.531361\pi$$
−0.0983654 + 0.995150i $$0.531361\pi$$
$$510$$ 0 0
$$511$$ 8.43845 0.373295
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −28.4924 −1.25553
$$516$$ 0 0
$$517$$ 29.8617 1.31332
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14.8769 −0.651769 −0.325884 0.945410i $$-0.605662\pi$$
−0.325884 + 0.945410i $$0.605662\pi$$
$$522$$ 0 0
$$523$$ 7.12311 0.311472 0.155736 0.987799i $$-0.450225\pi$$
0.155736 + 0.987799i $$0.450225\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.24621 −0.272089
$$528$$ 0 0
$$529$$ 52.4233 2.27927
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2.00000 0.0866296
$$534$$ 0 0
$$535$$ 58.7386 2.53949
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −3.12311 −0.134522
$$540$$ 0 0
$$541$$ −43.3693 −1.86459 −0.932296 0.361696i $$-0.882198\pi$$
−0.932296 + 0.361696i $$0.882198\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −57.8617 −2.47852
$$546$$ 0 0
$$547$$ 38.0540 1.62707 0.813535 0.581516i $$-0.197540\pi$$
0.813535 + 0.581516i $$0.197540\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.684658 0.0291674
$$552$$ 0 0
$$553$$ −8.68466 −0.369309
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 7.36932 0.312248 0.156124 0.987737i $$-0.450100\pi$$
0.156124 + 0.987737i $$0.450100\pi$$
$$558$$ 0 0
$$559$$ −9.56155 −0.404411
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −24.4924 −1.03223 −0.516116 0.856519i $$-0.672623\pi$$
−0.516116 + 0.856519i $$0.672623\pi$$
$$564$$ 0 0
$$565$$ −4.68466 −0.197085
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7.17708 −0.300879 −0.150439 0.988619i $$-0.548069\pi$$
−0.150439 + 0.988619i $$0.548069\pi$$
$$570$$ 0 0
$$571$$ −30.0540 −1.25772 −0.628860 0.777519i $$-0.716478\pi$$
−0.628860 + 0.777519i $$0.716478\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −66.7386 −2.78319
$$576$$ 0 0
$$577$$ −12.2462 −0.509816 −0.254908 0.966965i $$-0.582045\pi$$
−0.254908 + 0.966965i $$0.582045\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −10.4384 −0.433060
$$582$$ 0 0
$$583$$ 20.8769 0.864633
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.9309 0.616263 0.308131 0.951344i $$-0.400296\pi$$
0.308131 + 0.951344i $$0.400296\pi$$
$$588$$ 0 0
$$589$$ 8.68466 0.357845
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 3.06913 0.126034 0.0630170 0.998012i $$-0.479928\pi$$
0.0630170 + 0.998012i $$0.479928\pi$$
$$594$$ 0 0
$$595$$ −4.00000 −0.163984
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 35.8078 1.46307 0.731533 0.681806i $$-0.238805\pi$$
0.731533 + 0.681806i $$0.238805\pi$$
$$600$$ 0 0
$$601$$ −18.4924 −0.754322 −0.377161 0.926148i $$-0.623100\pi$$
−0.377161 + 0.926148i $$0.623100\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −4.43845 −0.180449
$$606$$ 0 0
$$607$$ 11.1231 0.451473 0.225736 0.974188i $$-0.427521\pi$$
0.225736 + 0.974188i $$0.427521\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −9.56155 −0.386819
$$612$$ 0 0
$$613$$ −14.8769 −0.600872 −0.300436 0.953802i $$-0.597132\pi$$
−0.300436 + 0.953802i $$0.597132\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 8.49242 0.341339 0.170670 0.985328i $$-0.445407\pi$$
0.170670 + 0.985328i $$0.445407\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 9.80776 0.392940
$$624$$ 0 0
$$625$$ −4.36932 −0.174773
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.26137 0.0502940
$$630$$ 0 0
$$631$$ −42.3542 −1.68609 −0.843046 0.537841i $$-0.819240\pi$$
−0.843046 + 0.537841i $$0.819240\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −79.2311 −3.14419
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.94602 0.155859 0.0779293 0.996959i $$-0.475169\pi$$
0.0779293 + 0.996959i $$0.475169\pi$$
$$642$$ 0 0
$$643$$ 22.7386 0.896724 0.448362 0.893852i $$-0.352008\pi$$
0.448362 + 0.893852i $$0.352008\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.6307 0.889704 0.444852 0.895604i $$-0.353256\pi$$
0.444852 + 0.895604i $$0.353256\pi$$
$$648$$ 0 0
$$649$$ −24.9848 −0.980741
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −36.2462 −1.41842 −0.709212 0.704995i $$-0.750949\pi$$
−0.709212 + 0.704995i $$0.750949\pi$$
$$654$$ 0 0
$$655$$ −14.2462 −0.556646
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 9.94602 0.387442 0.193721 0.981057i $$-0.437944\pi$$
0.193721 + 0.981057i $$0.437944\pi$$
$$660$$ 0 0
$$661$$ −14.1922 −0.552014 −0.276007 0.961156i $$-0.589011\pi$$
−0.276007 + 0.961156i $$0.589011\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 5.56155 0.215668
$$666$$ 0 0
$$667$$ 3.80776 0.147437
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.24621 0.241132
$$672$$ 0 0
$$673$$ 3.06913 0.118306 0.0591531 0.998249i $$-0.481160\pi$$
0.0591531 + 0.998249i $$0.481160\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 22.4924 0.864454 0.432227 0.901765i $$-0.357728\pi$$
0.432227 + 0.901765i $$0.357728\pi$$
$$678$$ 0 0
$$679$$ 6.68466 0.256534
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 32.9848 1.26213 0.631065 0.775730i $$-0.282618\pi$$
0.631065 + 0.775730i $$0.282618\pi$$
$$684$$ 0 0
$$685$$ 46.7386 1.78579
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −6.68466 −0.254665
$$690$$ 0 0
$$691$$ 17.5616 0.668073 0.334036 0.942560i $$-0.391589\pi$$
0.334036 + 0.942560i $$0.391589\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −53.8617 −2.04309
$$696$$ 0 0
$$697$$ 2.24621 0.0850813
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −38.3002 −1.44658 −0.723289 0.690545i $$-0.757371\pi$$
−0.723289 + 0.690545i $$0.757371\pi$$
$$702$$ 0 0
$$703$$ −1.75379 −0.0661454
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 4.24621 0.159695
$$708$$ 0 0
$$709$$ 1.50758 0.0566183 0.0283091 0.999599i $$-0.490988\pi$$
0.0283091 + 0.999599i $$0.490988\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 48.3002 1.80886
$$714$$ 0 0
$$715$$ −11.1231 −0.415981
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −22.6307 −0.843982 −0.421991 0.906600i $$-0.638669\pi$$
−0.421991 + 0.906600i $$0.638669\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −3.36932 −0.125133
$$726$$ 0 0
$$727$$ −52.1080 −1.93258 −0.966288 0.257462i $$-0.917114\pi$$
−0.966288 + 0.257462i $$0.917114\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −10.7386 −0.397183
$$732$$ 0 0
$$733$$ −7.94602 −0.293493 −0.146747 0.989174i $$-0.546880\pi$$
−0.146747 + 0.989174i $$0.546880\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 22.2462 0.819450
$$738$$ 0 0
$$739$$ −26.6307 −0.979626 −0.489813 0.871828i $$-0.662935\pi$$
−0.489813 + 0.871828i $$0.662935\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −34.2462 −1.25637 −0.628186 0.778063i $$-0.716202\pi$$
−0.628186 + 0.778063i $$0.716202\pi$$
$$744$$ 0 0
$$745$$ −29.3693 −1.07601
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −16.4924 −0.602620
$$750$$ 0 0
$$751$$ 6.93087 0.252911 0.126456 0.991972i $$-0.459640\pi$$
0.126456 + 0.991972i $$0.459640\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 44.4924 1.61925
$$756$$ 0 0
$$757$$ 19.5616 0.710977 0.355488 0.934681i $$-0.384315\pi$$
0.355488 + 0.934681i $$0.384315\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 34.3002 1.24338 0.621690 0.783263i $$-0.286446\pi$$
0.621690 + 0.783263i $$0.286446\pi$$
$$762$$ 0 0
$$763$$ 16.2462 0.588152
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ 18.6847 0.673786 0.336893 0.941543i $$-0.390624\pi$$
0.336893 + 0.941543i $$0.390624\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −46.4924 −1.67222 −0.836108 0.548565i $$-0.815174\pi$$
−0.836108 + 0.548565i $$0.815174\pi$$
$$774$$ 0 0
$$775$$ −42.7386 −1.53522
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3.12311 −0.111897
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 32.4924 1.15970
$$786$$ 0 0
$$787$$ 47.4233 1.69046 0.845229 0.534404i $$-0.179464\pi$$
0.845229 + 0.534404i $$0.179464\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.31534 0.0467682
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ −10.7386 −0.379906
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 26.3542 0.930018
$$804$$ 0 0
$$805$$ 30.9309 1.09017
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −53.0388 −1.86475 −0.932373 0.361498i $$-0.882265\pi$$
−0.932373 + 0.361498i $$0.882265\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 14.9309 0.522365
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 29.2311 1.02017 0.510085 0.860124i $$-0.329614\pi$$
0.510085 + 0.860124i $$0.329614\pi$$
$$822$$ 0 0
$$823$$ −24.9848 −0.870917 −0.435458 0.900209i $$-0.643414\pi$$
−0.435458 + 0.900209i $$0.643414\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −52.4924 −1.82534 −0.912670 0.408697i $$-0.865983\pi$$
−0.912670 + 0.408697i $$0.865983\pi$$
$$828$$ 0 0
$$829$$ 20.2462 0.703180 0.351590 0.936154i $$-0.385641\pi$$
0.351590 + 0.936154i $$0.385641\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.12311 0.0389133
$$834$$ 0 0
$$835$$ −45.1771 −1.56342
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 22.7386 0.785025 0.392512 0.919747i $$-0.371606\pi$$
0.392512 + 0.919747i $$0.371606\pi$$
$$840$$ 0 0
$$841$$ −28.8078 −0.993371
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 3.56155 0.122521
$$846$$ 0 0
$$847$$ 1.24621 0.0428203
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −9.75379 −0.334356
$$852$$ 0 0
$$853$$ 31.6695 1.08434 0.542172 0.840268i $$-0.317602\pi$$
0.542172 + 0.840268i $$0.317602\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15.7538 0.538139 0.269070 0.963121i $$-0.413284\pi$$
0.269070 + 0.963121i $$0.413284\pi$$
$$858$$ 0 0
$$859$$ −39.1231 −1.33486 −0.667432 0.744671i $$-0.732606\pi$$
−0.667432 + 0.744671i $$0.732606\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 44.9848 1.53130 0.765651 0.643256i $$-0.222417\pi$$
0.765651 + 0.643256i $$0.222417\pi$$
$$864$$ 0 0
$$865$$ −26.2462 −0.892398
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −27.1231 −0.920088
$$870$$ 0 0
$$871$$ −7.12311 −0.241357
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −9.56155 −0.323239
$$876$$ 0 0
$$877$$ 50.1080 1.69203 0.846013 0.533163i $$-0.178997\pi$$
0.846013 + 0.533163i $$0.178997\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ 7.50758 0.252650 0.126325 0.991989i $$-0.459682\pi$$
0.126325 + 0.991989i $$0.459682\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −33.7538 −1.13334 −0.566671 0.823944i $$-0.691769\pi$$
−0.566671 + 0.823944i $$0.691769\pi$$
$$888$$ 0 0
$$889$$ 22.2462 0.746114
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 14.9309 0.499643
$$894$$ 0 0
$$895$$ 62.5464 2.09070
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 2.43845 0.0813268
$$900$$ 0 0
$$901$$ −7.50758 −0.250114
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 43.6155 1.44983
$$906$$ 0 0
$$907$$ −17.5616 −0.583122 −0.291561 0.956552i $$-0.594175\pi$$
−0.291561 + 0.956552i $$0.594175\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 5.94602 0.197001 0.0985003 0.995137i $$-0.468595\pi$$
0.0985003 + 0.995137i $$0.468595\pi$$
$$912$$ 0 0
$$913$$ −32.6004 −1.07891
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4.00000 0.132092
$$918$$ 0 0
$$919$$ −4.49242 −0.148191 −0.0740957 0.997251i $$-0.523607\pi$$
−0.0740957 + 0.997251i $$0.523607\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.2462 0.337258
$$924$$ 0 0
$$925$$ 8.63068 0.283775
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 39.5616 1.29797 0.648986 0.760800i $$-0.275193\pi$$
0.648986 + 0.760800i $$0.275193\pi$$
$$930$$ 0 0
$$931$$ −1.56155 −0.0511778
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −12.4924 −0.408546
$$936$$ 0 0
$$937$$ −13.6155 −0.444800 −0.222400 0.974956i $$-0.571389\pi$$
−0.222400 + 0.974956i $$0.571389\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −16.5464 −0.539397 −0.269699 0.962945i $$-0.586924\pi$$
−0.269699 + 0.962945i $$0.586924\pi$$
$$942$$ 0 0
$$943$$ −17.3693 −0.565623
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 37.8617 1.23034 0.615171 0.788394i $$-0.289087\pi$$
0.615171 + 0.788394i $$0.289087\pi$$
$$948$$ 0 0
$$949$$ −8.43845 −0.273924
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 53.9157 1.74650 0.873251 0.487271i $$-0.162008\pi$$
0.873251 + 0.487271i $$0.162008\pi$$
$$954$$ 0 0
$$955$$ 44.4924 1.43974
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −13.1231 −0.423767
$$960$$ 0 0
$$961$$ −0.0691303 −0.00223001
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 29.3693 0.945432
$$966$$ 0 0
$$967$$ 36.1080 1.16115 0.580577 0.814206i $$-0.302827\pi$$
0.580577 + 0.814206i $$0.302827\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 54.3542 1.74431 0.872154 0.489231i $$-0.162723\pi$$
0.872154 + 0.489231i $$0.162723\pi$$
$$972$$ 0 0
$$973$$ 15.1231 0.484825
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −17.5076 −0.560117 −0.280059 0.959983i $$-0.590354\pi$$
−0.280059 + 0.959983i $$0.590354\pi$$
$$978$$ 0 0
$$979$$ 30.6307 0.978961
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 42.5464 1.35702 0.678510 0.734591i $$-0.262626\pi$$
0.678510 + 0.734591i $$0.262626\pi$$
$$984$$ 0 0
$$985$$ −35.6155 −1.13481
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 83.0388 2.64048
$$990$$ 0 0
$$991$$ 10.7386 0.341124 0.170562 0.985347i $$-0.445442\pi$$
0.170562 + 0.985347i $$0.445442\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 22.2462 0.705252
$$996$$ 0 0
$$997$$ −58.6004 −1.85589 −0.927946 0.372714i $$-0.878427\pi$$
−0.927946 + 0.372714i $$0.878427\pi$$
$$998$$ 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 6552.2.a.bk.1.2 2
3.2 odd 2 2184.2.a.n.1.1 2
12.11 even 2 4368.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.n.1.1 2 3.2 odd 2
4368.2.a.bf.1.1 2 12.11 even 2
6552.2.a.bk.1.2 2 1.1 even 1 trivial