Properties

 Label 9072.2.a.y.1.2 Level $9072$ Weight $2$ Character 9072.1 Self dual yes Analytic conductor $72.440$ Analytic rank $0$ 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] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9072 = 2^{4} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9072.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: $$72.4402847137$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1134) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.267949 q^{5} +1.00000 q^{7} +O(q^{10})$$ $$q-0.267949 q^{5} +1.00000 q^{7} +6.19615 q^{11} -6.46410 q^{13} -7.00000 q^{17} -0.732051 q^{19} -4.19615 q^{23} -4.92820 q^{25} -1.53590 q^{29} -8.19615 q^{31} -0.267949 q^{35} +10.6603 q^{37} -2.53590 q^{41} +1.46410 q^{43} +4.73205 q^{47} +1.00000 q^{49} +9.46410 q^{53} -1.66025 q^{55} +4.19615 q^{59} +3.92820 q^{61} +1.73205 q^{65} +6.73205 q^{67} +6.53590 q^{71} +8.26795 q^{73} +6.19615 q^{77} +9.12436 q^{79} +16.5885 q^{83} +1.87564 q^{85} -9.92820 q^{89} -6.46410 q^{91} +0.196152 q^{95} +10.9282 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - 4 * q^5 + 2 * q^7 $$2 q - 4 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} - 14 q^{17} + 2 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} - 6 q^{31} - 4 q^{35} + 4 q^{37} - 12 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} + 12 q^{53} + 14 q^{55} - 2 q^{59} - 6 q^{61} + 10 q^{67} + 20 q^{71} + 20 q^{73} + 2 q^{77} - 6 q^{79} + 2 q^{83} + 28 q^{85} - 6 q^{89} - 6 q^{91} - 10 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q - 4 * q^5 + 2 * q^7 + 2 * q^11 - 6 * q^13 - 14 * q^17 + 2 * q^19 + 2 * q^23 + 4 * q^25 - 10 * q^29 - 6 * q^31 - 4 * q^35 + 4 * q^37 - 12 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 + 12 * q^53 + 14 * q^55 - 2 * q^59 - 6 * q^61 + 10 * q^67 + 20 * q^71 + 20 * q^73 + 2 * q^77 - 6 * q^79 + 2 * q^83 + 28 * q^85 - 6 * q^89 - 6 * q^91 - 10 * q^95 + 8 * 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
$$5$$ −0.267949 −0.119831 −0.0599153 0.998203i $$-0.519083\pi$$
−0.0599153 + 0.998203i $$0.519083\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.19615 1.86821 0.934105 0.356998i $$-0.116200\pi$$
0.934105 + 0.356998i $$0.116200\pi$$
$$12$$ 0 0
$$13$$ −6.46410 −1.79282 −0.896410 0.443227i $$-0.853834\pi$$
−0.896410 + 0.443227i $$0.853834\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 0 0
$$19$$ −0.732051 −0.167944 −0.0839720 0.996468i $$-0.526761\pi$$
−0.0839720 + 0.996468i $$0.526761\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.19615 −0.874958 −0.437479 0.899229i $$-0.644129\pi$$
−0.437479 + 0.899229i $$0.644129\pi$$
$$24$$ 0 0
$$25$$ −4.92820 −0.985641
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.53590 −0.285209 −0.142605 0.989780i $$-0.545548\pi$$
−0.142605 + 0.989780i $$0.545548\pi$$
$$30$$ 0 0
$$31$$ −8.19615 −1.47207 −0.736036 0.676942i $$-0.763305\pi$$
−0.736036 + 0.676942i $$0.763305\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.267949 −0.0452917
$$36$$ 0 0
$$37$$ 10.6603 1.75253 0.876267 0.481825i $$-0.160026\pi$$
0.876267 + 0.481825i $$0.160026\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.53590 −0.396041 −0.198020 0.980198i $$-0.563451\pi$$
−0.198020 + 0.980198i $$0.563451\pi$$
$$42$$ 0 0
$$43$$ 1.46410 0.223273 0.111637 0.993749i $$-0.464391\pi$$
0.111637 + 0.993749i $$0.464391\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.73205 0.690241 0.345120 0.938558i $$-0.387838\pi$$
0.345120 + 0.938558i $$0.387838\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.46410 1.29999 0.649997 0.759937i $$-0.274770\pi$$
0.649997 + 0.759937i $$0.274770\pi$$
$$54$$ 0 0
$$55$$ −1.66025 −0.223869
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.19615 0.546293 0.273146 0.961973i $$-0.411936\pi$$
0.273146 + 0.961973i $$0.411936\pi$$
$$60$$ 0 0
$$61$$ 3.92820 0.502955 0.251477 0.967863i $$-0.419084\pi$$
0.251477 + 0.967863i $$0.419084\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.73205 0.214834
$$66$$ 0 0
$$67$$ 6.73205 0.822451 0.411225 0.911534i $$-0.365101\pi$$
0.411225 + 0.911534i $$0.365101\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.53590 0.775668 0.387834 0.921729i $$-0.373223\pi$$
0.387834 + 0.921729i $$0.373223\pi$$
$$72$$ 0 0
$$73$$ 8.26795 0.967690 0.483845 0.875154i $$-0.339240\pi$$
0.483845 + 0.875154i $$0.339240\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.19615 0.706117
$$78$$ 0 0
$$79$$ 9.12436 1.02657 0.513285 0.858218i $$-0.328428\pi$$
0.513285 + 0.858218i $$0.328428\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 16.5885 1.82082 0.910410 0.413707i $$-0.135766\pi$$
0.910410 + 0.413707i $$0.135766\pi$$
$$84$$ 0 0
$$85$$ 1.87564 0.203442
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.92820 −1.05239 −0.526194 0.850365i $$-0.676381\pi$$
−0.526194 + 0.850365i $$0.676381\pi$$
$$90$$ 0 0
$$91$$ −6.46410 −0.677622
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0.196152 0.0201248
$$96$$ 0 0
$$97$$ 10.9282 1.10959 0.554795 0.831987i $$-0.312797\pi$$
0.554795 + 0.831987i $$0.312797\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.92820 −0.888389 −0.444195 0.895930i $$-0.646510\pi$$
−0.444195 + 0.895930i $$0.646510\pi$$
$$102$$ 0 0
$$103$$ 8.39230 0.826918 0.413459 0.910523i $$-0.364320\pi$$
0.413459 + 0.910523i $$0.364320\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 3.19615 0.306136 0.153068 0.988216i $$-0.451085\pi$$
0.153068 + 0.988216i $$0.451085\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 5.73205 0.539226 0.269613 0.962969i $$-0.413104\pi$$
0.269613 + 0.962969i $$0.413104\pi$$
$$114$$ 0 0
$$115$$ 1.12436 0.104847
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −7.00000 −0.641689
$$120$$ 0 0
$$121$$ 27.3923 2.49021
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 2.66025 0.237940
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.5359 0.920526 0.460263 0.887783i $$-0.347755\pi$$
0.460263 + 0.887783i $$0.347755\pi$$
$$132$$ 0 0
$$133$$ −0.732051 −0.0634769
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 8.26795 0.706379 0.353189 0.935552i $$-0.385097\pi$$
0.353189 + 0.935552i $$0.385097\pi$$
$$138$$ 0 0
$$139$$ 3.26795 0.277184 0.138592 0.990350i $$-0.455742\pi$$
0.138592 + 0.990350i $$0.455742\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −40.0526 −3.34936
$$144$$ 0 0
$$145$$ 0.411543 0.0341768
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ 5.80385 0.472310 0.236155 0.971715i $$-0.424113\pi$$
0.236155 + 0.971715i $$0.424113\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 2.19615 0.176399
$$156$$ 0 0
$$157$$ −1.00000 −0.0798087 −0.0399043 0.999204i $$-0.512705\pi$$
−0.0399043 + 0.999204i $$0.512705\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.19615 −0.330703
$$162$$ 0 0
$$163$$ 13.4641 1.05459 0.527295 0.849682i $$-0.323206\pi$$
0.527295 + 0.849682i $$0.323206\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.80385 −0.139586 −0.0697930 0.997561i $$-0.522234\pi$$
−0.0697930 + 0.997561i $$0.522234\pi$$
$$168$$ 0 0
$$169$$ 28.7846 2.21420
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.26795 −0.476543 −0.238272 0.971199i $$-0.576581\pi$$
−0.238272 + 0.971199i $$0.576581\pi$$
$$174$$ 0 0
$$175$$ −4.92820 −0.372537
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2.19615 −0.164148 −0.0820741 0.996626i $$-0.526154\pi$$
−0.0820741 + 0.996626i $$0.526154\pi$$
$$180$$ 0 0
$$181$$ −16.3923 −1.21843 −0.609215 0.793005i $$-0.708515\pi$$
−0.609215 + 0.793005i $$0.708515\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −2.85641 −0.210007
$$186$$ 0 0
$$187$$ −43.3731 −3.17175
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −5.66025 −0.409562 −0.204781 0.978808i $$-0.565648\pi$$
−0.204781 + 0.978808i $$0.565648\pi$$
$$192$$ 0 0
$$193$$ 18.8564 1.35731 0.678657 0.734455i $$-0.262562\pi$$
0.678657 + 0.734455i $$0.262562\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −15.7846 −1.12461 −0.562303 0.826931i $$-0.690085\pi$$
−0.562303 + 0.826931i $$0.690085\pi$$
$$198$$ 0 0
$$199$$ 19.1244 1.35569 0.677845 0.735205i $$-0.262914\pi$$
0.677845 + 0.735205i $$0.262914\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1.53590 −0.107799
$$204$$ 0 0
$$205$$ 0.679492 0.0474578
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.53590 −0.313755
$$210$$ 0 0
$$211$$ 17.2679 1.18877 0.594387 0.804179i $$-0.297395\pi$$
0.594387 + 0.804179i $$0.297395\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −0.392305 −0.0267550
$$216$$ 0 0
$$217$$ −8.19615 −0.556391
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 45.2487 3.04376
$$222$$ 0 0
$$223$$ 25.4641 1.70520 0.852601 0.522562i $$-0.175024\pi$$
0.852601 + 0.522562i $$0.175024\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −18.9282 −1.25631 −0.628154 0.778089i $$-0.716189\pi$$
−0.628154 + 0.778089i $$0.716189\pi$$
$$228$$ 0 0
$$229$$ −2.46410 −0.162832 −0.0814162 0.996680i $$-0.525944\pi$$
−0.0814162 + 0.996680i $$0.525944\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2.80385 0.183686 0.0918431 0.995773i $$-0.470724\pi$$
0.0918431 + 0.995773i $$0.470724\pi$$
$$234$$ 0 0
$$235$$ −1.26795 −0.0827119
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.0526 −0.650246 −0.325123 0.945672i $$-0.605406\pi$$
−0.325123 + 0.945672i $$0.605406\pi$$
$$240$$ 0 0
$$241$$ −14.2679 −0.919079 −0.459540 0.888157i $$-0.651986\pi$$
−0.459540 + 0.888157i $$0.651986\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.267949 −0.0171186
$$246$$ 0 0
$$247$$ 4.73205 0.301093
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −22.0526 −1.39195 −0.695973 0.718068i $$-0.745027\pi$$
−0.695973 + 0.718068i $$0.745027\pi$$
$$252$$ 0 0
$$253$$ −26.0000 −1.63461
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6.46410 −0.403220 −0.201610 0.979466i $$-0.564617\pi$$
−0.201610 + 0.979466i $$0.564617\pi$$
$$258$$ 0 0
$$259$$ 10.6603 0.662396
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −6.33975 −0.390925 −0.195463 0.980711i $$-0.562621\pi$$
−0.195463 + 0.980711i $$0.562621\pi$$
$$264$$ 0 0
$$265$$ −2.53590 −0.155779
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −5.58846 −0.340734 −0.170367 0.985381i $$-0.554495\pi$$
−0.170367 + 0.985381i $$0.554495\pi$$
$$270$$ 0 0
$$271$$ 19.5167 1.18555 0.592776 0.805367i $$-0.298032\pi$$
0.592776 + 0.805367i $$0.298032\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −30.5359 −1.84138
$$276$$ 0 0
$$277$$ −18.7846 −1.12866 −0.564329 0.825550i $$-0.690865\pi$$
−0.564329 + 0.825550i $$0.690865\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 13.1962 0.787216 0.393608 0.919278i $$-0.371227\pi$$
0.393608 + 0.919278i $$0.371227\pi$$
$$282$$ 0 0
$$283$$ −15.3205 −0.910710 −0.455355 0.890310i $$-0.650488\pi$$
−0.455355 + 0.890310i $$0.650488\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.53590 −0.149689
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −3.33975 −0.195110 −0.0975550 0.995230i $$-0.531102\pi$$
−0.0975550 + 0.995230i $$0.531102\pi$$
$$294$$ 0 0
$$295$$ −1.12436 −0.0654625
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 27.1244 1.56864
$$300$$ 0 0
$$301$$ 1.46410 0.0843894
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1.05256 −0.0602693
$$306$$ 0 0
$$307$$ 21.8564 1.24741 0.623706 0.781659i $$-0.285626\pi$$
0.623706 + 0.781659i $$0.285626\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 10.1962 0.578171 0.289085 0.957303i $$-0.406649\pi$$
0.289085 + 0.957303i $$0.406649\pi$$
$$312$$ 0 0
$$313$$ 25.5885 1.44635 0.723173 0.690667i $$-0.242683\pi$$
0.723173 + 0.690667i $$0.242683\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 31.3923 1.76317 0.881584 0.472028i $$-0.156478\pi$$
0.881584 + 0.472028i $$0.156478\pi$$
$$318$$ 0 0
$$319$$ −9.51666 −0.532831
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5.12436 0.285127
$$324$$ 0 0
$$325$$ 31.8564 1.76708
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 4.73205 0.260886
$$330$$ 0 0
$$331$$ −12.3923 −0.681143 −0.340571 0.940219i $$-0.610620\pi$$
−0.340571 + 0.940219i $$0.610620\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −1.80385 −0.0985547
$$336$$ 0 0
$$337$$ −16.3923 −0.892946 −0.446473 0.894797i $$-0.647320\pi$$
−0.446473 + 0.894797i $$0.647320\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −50.7846 −2.75014
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 21.4641 1.15225 0.576127 0.817360i $$-0.304563\pi$$
0.576127 + 0.817360i $$0.304563\pi$$
$$348$$ 0 0
$$349$$ 1.46410 0.0783716 0.0391858 0.999232i $$-0.487524\pi$$
0.0391858 + 0.999232i $$0.487524\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −1.75129 −0.0929488
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −10.9282 −0.576769 −0.288384 0.957515i $$-0.593118\pi$$
−0.288384 + 0.957515i $$0.593118\pi$$
$$360$$ 0 0
$$361$$ −18.4641 −0.971795
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2.21539 −0.115959
$$366$$ 0 0
$$367$$ −11.1244 −0.580687 −0.290343 0.956923i $$-0.593770\pi$$
−0.290343 + 0.956923i $$0.593770\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 9.46410 0.491352
$$372$$ 0 0
$$373$$ −6.14359 −0.318103 −0.159052 0.987270i $$-0.550844\pi$$
−0.159052 + 0.987270i $$0.550844\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.92820 0.511328
$$378$$ 0 0
$$379$$ 27.5167 1.41344 0.706718 0.707495i $$-0.250175\pi$$
0.706718 + 0.707495i $$0.250175\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 19.7128 1.00728 0.503639 0.863914i $$-0.331994\pi$$
0.503639 + 0.863914i $$0.331994\pi$$
$$384$$ 0 0
$$385$$ −1.66025 −0.0846144
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 13.4641 0.682657 0.341329 0.939944i $$-0.389123\pi$$
0.341329 + 0.939944i $$0.389123\pi$$
$$390$$ 0 0
$$391$$ 29.3731 1.48546
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −2.44486 −0.123014
$$396$$ 0 0
$$397$$ −21.0000 −1.05396 −0.526980 0.849878i $$-0.676676\pi$$
−0.526980 + 0.849878i $$0.676676\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −10.5167 −0.525177 −0.262588 0.964908i $$-0.584576\pi$$
−0.262588 + 0.964908i $$0.584576\pi$$
$$402$$ 0 0
$$403$$ 52.9808 2.63916
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 66.0526 3.27410
$$408$$ 0 0
$$409$$ 17.3397 0.857395 0.428698 0.903448i $$-0.358973\pi$$
0.428698 + 0.903448i $$0.358973\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 4.19615 0.206479
$$414$$ 0 0
$$415$$ −4.44486 −0.218190
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −9.46410 −0.462352 −0.231176 0.972912i $$-0.574257\pi$$
−0.231176 + 0.972912i $$0.574257\pi$$
$$420$$ 0 0
$$421$$ 0.124356 0.00606072 0.00303036 0.999995i $$-0.499035\pi$$
0.00303036 + 0.999995i $$0.499035\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 34.4974 1.67337
$$426$$ 0 0
$$427$$ 3.92820 0.190099
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −14.5359 −0.700170 −0.350085 0.936718i $$-0.613847\pi$$
−0.350085 + 0.936718i $$0.613847\pi$$
$$432$$ 0 0
$$433$$ 15.7321 0.756034 0.378017 0.925799i $$-0.376606\pi$$
0.378017 + 0.925799i $$0.376606\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.07180 0.146944
$$438$$ 0 0
$$439$$ −23.3205 −1.11303 −0.556514 0.830839i $$-0.687861\pi$$
−0.556514 + 0.830839i $$0.687861\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −15.2679 −0.725402 −0.362701 0.931906i $$-0.618145\pi$$
−0.362701 + 0.931906i $$0.618145\pi$$
$$444$$ 0 0
$$445$$ 2.66025 0.126108
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −15.8564 −0.748310 −0.374155 0.927366i $$-0.622067\pi$$
−0.374155 + 0.927366i $$0.622067\pi$$
$$450$$ 0 0
$$451$$ −15.7128 −0.739887
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1.73205 0.0811998
$$456$$ 0 0
$$457$$ −6.85641 −0.320729 −0.160365 0.987058i $$-0.551267\pi$$
−0.160365 + 0.987058i $$0.551267\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.78461 0.315991 0.157995 0.987440i $$-0.449497\pi$$
0.157995 + 0.987440i $$0.449497\pi$$
$$462$$ 0 0
$$463$$ 1.41154 0.0656000 0.0328000 0.999462i $$-0.489558\pi$$
0.0328000 + 0.999462i $$0.489558\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −16.5885 −0.767622 −0.383811 0.923412i $$-0.625389\pi$$
−0.383811 + 0.923412i $$0.625389\pi$$
$$468$$ 0 0
$$469$$ 6.73205 0.310857
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.07180 0.417122
$$474$$ 0 0
$$475$$ 3.60770 0.165532
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −21.5167 −0.983121 −0.491561 0.870843i $$-0.663573\pi$$
−0.491561 + 0.870843i $$0.663573\pi$$
$$480$$ 0 0
$$481$$ −68.9090 −3.14198
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2.92820 −0.132963
$$486$$ 0 0
$$487$$ −2.58846 −0.117294 −0.0586471 0.998279i $$-0.518679\pi$$
−0.0586471 + 0.998279i $$0.518679\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 26.5359 1.19755 0.598774 0.800918i $$-0.295655\pi$$
0.598774 + 0.800918i $$0.295655\pi$$
$$492$$ 0 0
$$493$$ 10.7513 0.484214
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.53590 0.293175
$$498$$ 0 0
$$499$$ −19.8038 −0.886542 −0.443271 0.896388i $$-0.646182\pi$$
−0.443271 + 0.896388i $$0.646182\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 40.0526 1.78586 0.892928 0.450200i $$-0.148647\pi$$
0.892928 + 0.450200i $$0.148647\pi$$
$$504$$ 0 0
$$505$$ 2.39230 0.106456
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −31.8564 −1.41201 −0.706005 0.708207i $$-0.749504\pi$$
−0.706005 + 0.708207i $$0.749504\pi$$
$$510$$ 0 0
$$511$$ 8.26795 0.365753
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −2.24871 −0.0990901
$$516$$ 0 0
$$517$$ 29.3205 1.28951
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −33.1769 −1.45073 −0.725363 0.688367i $$-0.758328\pi$$
−0.725363 + 0.688367i $$0.758328\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 57.3731 2.49921
$$528$$ 0 0
$$529$$ −5.39230 −0.234448
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 16.3923 0.710030
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.19615 0.266887
$$540$$ 0 0
$$541$$ 3.33975 0.143587 0.0717934 0.997420i $$-0.477128\pi$$
0.0717934 + 0.997420i $$0.477128\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −0.856406 −0.0366844
$$546$$ 0 0
$$547$$ 22.7321 0.971952 0.485976 0.873972i $$-0.338464\pi$$
0.485976 + 0.873972i $$0.338464\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.12436 0.0478992
$$552$$ 0 0
$$553$$ 9.12436 0.388007
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −23.9282 −1.01387 −0.506935 0.861984i $$-0.669222\pi$$
−0.506935 + 0.861984i $$0.669222\pi$$
$$558$$ 0 0
$$559$$ −9.46410 −0.400289
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 19.7128 0.830796 0.415398 0.909640i $$-0.363642\pi$$
0.415398 + 0.909640i $$0.363642\pi$$
$$564$$ 0 0
$$565$$ −1.53590 −0.0646157
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −23.1962 −0.972433 −0.486217 0.873838i $$-0.661623\pi$$
−0.486217 + 0.873838i $$0.661623\pi$$
$$570$$ 0 0
$$571$$ 22.7321 0.951307 0.475653 0.879633i $$-0.342212\pi$$
0.475653 + 0.879633i $$0.342212\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 20.6795 0.862394
$$576$$ 0 0
$$577$$ 24.6603 1.02662 0.513310 0.858203i $$-0.328419\pi$$
0.513310 + 0.858203i $$0.328419\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 16.5885 0.688205
$$582$$ 0 0
$$583$$ 58.6410 2.42866
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.7321 0.608057 0.304028 0.952663i $$-0.401668\pi$$
0.304028 + 0.952663i $$0.401668\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 22.1769 0.910697 0.455348 0.890313i $$-0.349515\pi$$
0.455348 + 0.890313i $$0.349515\pi$$
$$594$$ 0 0
$$595$$ 1.87564 0.0768939
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −15.1244 −0.617964 −0.308982 0.951068i $$-0.599988\pi$$
−0.308982 + 0.951068i $$0.599988\pi$$
$$600$$ 0 0
$$601$$ 19.1962 0.783027 0.391514 0.920172i $$-0.371952\pi$$
0.391514 + 0.920172i $$0.371952\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −7.33975 −0.298403
$$606$$ 0 0
$$607$$ 24.5885 0.998015 0.499007 0.866598i $$-0.333698\pi$$
0.499007 + 0.866598i $$0.333698\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −30.5885 −1.23748
$$612$$ 0 0
$$613$$ 26.7846 1.08182 0.540910 0.841080i $$-0.318080\pi$$
0.540910 + 0.841080i $$0.318080\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11.9808 −0.482327 −0.241164 0.970484i $$-0.577529\pi$$
−0.241164 + 0.970484i $$0.577529\pi$$
$$618$$ 0 0
$$619$$ 23.7128 0.953098 0.476549 0.879148i $$-0.341887\pi$$
0.476549 + 0.879148i $$0.341887\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −9.92820 −0.397765
$$624$$ 0 0
$$625$$ 23.9282 0.957128
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −74.6218 −2.97537
$$630$$ 0 0
$$631$$ 3.66025 0.145712 0.0728562 0.997342i $$-0.476789\pi$$
0.0728562 + 0.997342i $$0.476789\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 3.21539 0.127599
$$636$$ 0 0
$$637$$ −6.46410 −0.256117
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −39.4449 −1.55798 −0.778989 0.627037i $$-0.784267\pi$$
−0.778989 + 0.627037i $$0.784267\pi$$
$$642$$ 0 0
$$643$$ 9.41154 0.371155 0.185578 0.982630i $$-0.440584\pi$$
0.185578 + 0.982630i $$0.440584\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −4.39230 −0.172679 −0.0863397 0.996266i $$-0.527517\pi$$
−0.0863397 + 0.996266i $$0.527517\pi$$
$$648$$ 0 0
$$649$$ 26.0000 1.02059
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 30.2487 1.18372 0.591862 0.806039i $$-0.298393\pi$$
0.591862 + 0.806039i $$0.298393\pi$$
$$654$$ 0 0
$$655$$ −2.82309 −0.110307
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.3923 −1.41764 −0.708821 0.705388i $$-0.750773\pi$$
−0.708821 + 0.705388i $$0.750773\pi$$
$$660$$ 0 0
$$661$$ −12.8564 −0.500056 −0.250028 0.968239i $$-0.580440\pi$$
−0.250028 + 0.968239i $$0.580440\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0.196152 0.00760646
$$666$$ 0 0
$$667$$ 6.44486 0.249546
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24.3397 0.939625
$$672$$ 0 0
$$673$$ −18.3205 −0.706204 −0.353102 0.935585i $$-0.614873\pi$$
−0.353102 + 0.935585i $$0.614873\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ 0 0
$$679$$ 10.9282 0.419386
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.85641 −0.0710334 −0.0355167 0.999369i $$-0.511308\pi$$
−0.0355167 + 0.999369i $$0.511308\pi$$
$$684$$ 0 0
$$685$$ −2.21539 −0.0846457
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −61.1769 −2.33065
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −0.875644 −0.0332151
$$696$$ 0 0
$$697$$ 17.7513 0.672378
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −27.3923 −1.03459 −0.517297 0.855806i $$-0.673062\pi$$
−0.517297 + 0.855806i $$0.673062\pi$$
$$702$$ 0 0
$$703$$ −7.80385 −0.294328
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −8.92820 −0.335780
$$708$$ 0 0
$$709$$ −3.87564 −0.145553 −0.0727764 0.997348i $$-0.523186\pi$$
−0.0727764 + 0.997348i $$0.523186\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 34.3923 1.28800
$$714$$ 0 0
$$715$$ 10.7321 0.401356
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −9.46410 −0.352951 −0.176476 0.984305i $$-0.556470\pi$$
−0.176476 + 0.984305i $$0.556470\pi$$
$$720$$ 0 0
$$721$$ 8.39230 0.312546
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 7.56922 0.281114
$$726$$ 0 0
$$727$$ −51.3205 −1.90337 −0.951686 0.307072i $$-0.900651\pi$$
−0.951686 + 0.307072i $$0.900651\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −10.2487 −0.379062
$$732$$ 0 0
$$733$$ 47.3205 1.74782 0.873911 0.486085i $$-0.161576\pi$$
0.873911 + 0.486085i $$0.161576\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 41.7128 1.53651
$$738$$ 0 0
$$739$$ 13.2679 0.488069 0.244035 0.969767i $$-0.421529\pi$$
0.244035 + 0.969767i $$0.421529\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −40.3923 −1.48185 −0.740925 0.671588i $$-0.765613\pi$$
−0.740925 + 0.671588i $$0.765613\pi$$
$$744$$ 0 0
$$745$$ 2.41154 0.0883521
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 22.1436 0.808031 0.404016 0.914752i $$-0.367614\pi$$
0.404016 + 0.914752i $$0.367614\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.55514 −0.0565972
$$756$$ 0 0
$$757$$ 20.7846 0.755429 0.377715 0.925922i $$-0.376710\pi$$
0.377715 + 0.925922i $$0.376710\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −37.0000 −1.34125 −0.670624 0.741797i $$-0.733974\pi$$
−0.670624 + 0.741797i $$0.733974\pi$$
$$762$$ 0 0
$$763$$ 3.19615 0.115708
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −27.1244 −0.979404
$$768$$ 0 0
$$769$$ −4.41154 −0.159084 −0.0795421 0.996832i $$-0.525346\pi$$
−0.0795421 + 0.996832i $$0.525346\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 4.12436 0.148343 0.0741714 0.997246i $$-0.476369\pi$$
0.0741714 + 0.997246i $$0.476369\pi$$
$$774$$ 0 0
$$775$$ 40.3923 1.45093
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1.85641 0.0665127
$$780$$ 0 0
$$781$$ 40.4974 1.44911
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0.267949 0.00956352
$$786$$ 0 0
$$787$$ 28.3923 1.01208 0.506038 0.862511i $$-0.331109\pi$$
0.506038 + 0.862511i $$0.331109\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.73205 0.203808
$$792$$ 0 0
$$793$$ −25.3923 −0.901707
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 29.4449 1.04299 0.521495 0.853254i $$-0.325374\pi$$
0.521495 + 0.853254i $$0.325374\pi$$
$$798$$ 0 0
$$799$$ −33.1244 −1.17186
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 51.2295 1.80785
$$804$$ 0 0
$$805$$ 1.12436 0.0396283
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −32.1244 −1.12943 −0.564716 0.825285i $$-0.691014\pi$$
−0.564716 + 0.825285i $$0.691014\pi$$
$$810$$ 0 0
$$811$$ −18.1962 −0.638953 −0.319477 0.947594i $$-0.603507\pi$$
−0.319477 + 0.947594i $$0.603507\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −3.60770 −0.126372
$$816$$ 0 0
$$817$$ −1.07180 −0.0374974
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25.9282 −0.904901 −0.452450 0.891790i $$-0.649450\pi$$
−0.452450 + 0.891790i $$0.649450\pi$$
$$822$$ 0 0
$$823$$ −0.784610 −0.0273498 −0.0136749 0.999906i $$-0.504353\pi$$
−0.0136749 + 0.999906i $$0.504353\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −23.3205 −0.810934 −0.405467 0.914110i $$-0.632891\pi$$
−0.405467 + 0.914110i $$0.632891\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −7.00000 −0.242536
$$834$$ 0 0
$$835$$ 0.483340 0.0167267
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1.46410 0.0505464 0.0252732 0.999681i $$-0.491954\pi$$
0.0252732 + 0.999681i $$0.491954\pi$$
$$840$$ 0 0
$$841$$ −26.6410 −0.918656
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −7.71281 −0.265329
$$846$$ 0 0
$$847$$ 27.3923 0.941211
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −44.7321 −1.53339
$$852$$ 0 0
$$853$$ −49.7128 −1.70213 −0.851067 0.525057i $$-0.824044\pi$$
−0.851067 + 0.525057i $$0.824044\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.8564 0.575804 0.287902 0.957660i $$-0.407042\pi$$
0.287902 + 0.957660i $$0.407042\pi$$
$$858$$ 0 0
$$859$$ −24.3923 −0.832255 −0.416127 0.909306i $$-0.636613\pi$$
−0.416127 + 0.909306i $$0.636613\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −7.12436 −0.242516 −0.121258 0.992621i $$-0.538693\pi$$
−0.121258 + 0.992621i $$0.538693\pi$$
$$864$$ 0 0
$$865$$ 1.67949 0.0571044
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 56.5359 1.91785
$$870$$ 0 0
$$871$$ −43.5167 −1.47451
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2.66025 0.0899330
$$876$$ 0 0
$$877$$ −36.5167 −1.23308 −0.616540 0.787324i $$-0.711466\pi$$
−0.616540 + 0.787324i $$0.711466\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.2487 −1.01910 −0.509552 0.860440i $$-0.670189\pi$$
−0.509552 + 0.860440i $$0.670189\pi$$
$$882$$ 0 0
$$883$$ 9.66025 0.325093 0.162547 0.986701i $$-0.448029\pi$$
0.162547 + 0.986701i $$0.448029\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −56.4449 −1.89523 −0.947617 0.319410i $$-0.896515\pi$$
−0.947617 + 0.319410i $$0.896515\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −3.46410 −0.115922
$$894$$ 0 0
$$895$$ 0.588457 0.0196700
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 12.5885 0.419849
$$900$$ 0 0
$$901$$ −66.2487 −2.20706
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4.39230 0.146005
$$906$$ 0 0
$$907$$ 36.0000 1.19536 0.597680 0.801735i $$-0.296089\pi$$
0.597680 + 0.801735i $$0.296089\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 42.2487 1.39976 0.699881 0.714259i $$-0.253236\pi$$
0.699881 + 0.714259i $$0.253236\pi$$
$$912$$ 0 0
$$913$$ 102.785 3.40167
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 10.5359 0.347926
$$918$$ 0 0
$$919$$ −26.9808 −0.890013 −0.445007 0.895527i $$-0.646799\pi$$
−0.445007 + 0.895527i $$0.646799\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −42.2487 −1.39063
$$924$$ 0 0
$$925$$ −52.5359 −1.72737
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 51.4974 1.68958 0.844788 0.535101i $$-0.179727\pi$$
0.844788 + 0.535101i $$0.179727\pi$$
$$930$$ 0 0
$$931$$ −0.732051 −0.0239920
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 11.6218 0.380073
$$936$$ 0 0
$$937$$ −25.8372 −0.844064 −0.422032 0.906581i $$-0.638683\pi$$
−0.422032 + 0.906581i $$0.638683\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 34.1244 1.11242 0.556211 0.831041i $$-0.312255\pi$$
0.556211 + 0.831041i $$0.312255\pi$$
$$942$$ 0 0
$$943$$ 10.6410 0.346519
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −46.2487 −1.50288 −0.751441 0.659801i $$-0.770641\pi$$
−0.751441 + 0.659801i $$0.770641\pi$$
$$948$$ 0 0
$$949$$ −53.4449 −1.73489
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 41.5885 1.34718 0.673591 0.739104i $$-0.264751\pi$$
0.673591 + 0.739104i $$0.264751\pi$$
$$954$$ 0 0
$$955$$ 1.51666 0.0490780
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 8.26795 0.266986
$$960$$ 0 0
$$961$$ 36.1769 1.16700
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −5.05256 −0.162648
$$966$$ 0 0
$$967$$ 3.66025 0.117706 0.0588529 0.998267i $$-0.481256\pi$$
0.0588529 + 0.998267i $$0.481256\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 8.87564 0.284833 0.142416 0.989807i $$-0.454513\pi$$
0.142416 + 0.989807i $$0.454513\pi$$
$$972$$ 0 0
$$973$$ 3.26795 0.104766
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −57.7128 −1.84640 −0.923198 0.384324i $$-0.874435\pi$$
−0.923198 + 0.384324i $$0.874435\pi$$
$$978$$ 0 0
$$979$$ −61.5167 −1.96608
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 58.6410 1.87036 0.935179 0.354176i $$-0.115238\pi$$
0.935179 + 0.354176i $$0.115238\pi$$
$$984$$ 0 0
$$985$$ 4.22947 0.134762
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.14359 −0.195355
$$990$$ 0 0
$$991$$ 27.6603 0.878657 0.439328 0.898327i $$-0.355216\pi$$
0.439328 + 0.898327i $$0.355216\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −5.12436 −0.162453
$$996$$ 0 0
$$997$$ 41.2487 1.30636 0.653180 0.757203i $$-0.273435\pi$$
0.653180 + 0.757203i $$0.273435\pi$$
$$998$$ 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 9072.2.a.y.1.2 2
3.2 odd 2 9072.2.a.bp.1.1 2
4.3 odd 2 1134.2.a.m.1.2 yes 2
12.11 even 2 1134.2.a.l.1.1 2
28.27 even 2 7938.2.a.bt.1.1 2
36.7 odd 6 1134.2.f.r.757.1 4
36.11 even 6 1134.2.f.s.757.2 4
36.23 even 6 1134.2.f.s.379.2 4
36.31 odd 6 1134.2.f.r.379.1 4
84.83 odd 2 7938.2.a.bg.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 2 12.11 even 2
1134.2.a.m.1.2 yes 2 4.3 odd 2
1134.2.f.r.379.1 4 36.31 odd 6
1134.2.f.r.757.1 4 36.7 odd 6
1134.2.f.s.379.2 4 36.23 even 6
1134.2.f.s.757.2 4 36.11 even 6
7938.2.a.bg.1.2 2 84.83 odd 2
7938.2.a.bt.1.1 2 28.27 even 2
9072.2.a.y.1.2 2 1.1 even 1 trivial
9072.2.a.bp.1.1 2 3.2 odd 2