Properties

 Label 2160.4.a.bn.1.1 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.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: $$127.444125612$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.47977.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 60x - 44$$ x^3 - x^2 - 60*x - 44 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-6.83575$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} -23.9261 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} -23.9261 q^{7} +57.9406 q^{11} -8.16237 q^{13} +50.0884 q^{17} -69.7782 q^{19} +4.92607 q^{23} +25.0000 q^{25} -79.4277 q^{29} -260.294 q^{31} -119.630 q^{35} -223.836 q^{37} +337.939 q^{41} -326.511 q^{43} +89.6851 q^{47} +229.457 q^{49} +543.672 q^{53} +289.703 q^{55} +92.0000 q^{59} +159.129 q^{61} -40.8119 q^{65} +910.561 q^{67} -293.232 q^{71} +142.022 q^{73} -1386.29 q^{77} -1106.50 q^{79} +813.367 q^{83} +250.442 q^{85} -956.666 q^{89} +195.294 q^{91} -348.891 q^{95} +106.363 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} - 9 q^{7}+O(q^{10})$$ 3 * q + 15 * q^5 - 9 * q^7 $$3 q + 15 q^{5} - 9 q^{7} - 18 q^{11} - 21 q^{13} + 84 q^{17} - 21 q^{19} - 48 q^{23} + 75 q^{25} - 36 q^{29} - 324 q^{31} - 45 q^{35} + 33 q^{37} + 114 q^{41} - 282 q^{43} - 282 q^{47} + 228 q^{49} + 222 q^{53} - 90 q^{55} + 276 q^{59} + 303 q^{61} - 105 q^{65} - 1035 q^{67} - 510 q^{71} + 447 q^{73} - 1578 q^{77} - 777 q^{79} - 78 q^{83} + 420 q^{85} - 324 q^{89} - 1995 q^{91} - 105 q^{95} + 1191 q^{97}+O(q^{100})$$ 3 * q + 15 * q^5 - 9 * q^7 - 18 * q^11 - 21 * q^13 + 84 * q^17 - 21 * q^19 - 48 * q^23 + 75 * q^25 - 36 * q^29 - 324 * q^31 - 45 * q^35 + 33 * q^37 + 114 * q^41 - 282 * q^43 - 282 * q^47 + 228 * q^49 + 222 * q^53 - 90 * q^55 + 276 * q^59 + 303 * q^61 - 105 * q^65 - 1035 * q^67 - 510 * q^71 + 447 * q^73 - 1578 * q^77 - 777 * q^79 - 78 * q^83 + 420 * q^85 - 324 * q^89 - 1995 * q^91 - 105 * q^95 + 1191 * 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
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −23.9261 −1.29189 −0.645943 0.763386i $$-0.723536\pi$$
−0.645943 + 0.763386i $$0.723536\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 57.9406 1.58816 0.794079 0.607814i $$-0.207954\pi$$
0.794079 + 0.607814i $$0.207954\pi$$
$$12$$ 0 0
$$13$$ −8.16237 −0.174141 −0.0870706 0.996202i $$-0.527751\pi$$
−0.0870706 + 0.996202i $$0.527751\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 50.0884 0.714602 0.357301 0.933989i $$-0.383697\pi$$
0.357301 + 0.933989i $$0.383697\pi$$
$$18$$ 0 0
$$19$$ −69.7782 −0.842538 −0.421269 0.906936i $$-0.638415\pi$$
−0.421269 + 0.906936i $$0.638415\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.92607 0.0446590 0.0223295 0.999751i $$-0.492892\pi$$
0.0223295 + 0.999751i $$0.492892\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −79.4277 −0.508598 −0.254299 0.967126i $$-0.581845\pi$$
−0.254299 + 0.967126i $$0.581845\pi$$
$$30$$ 0 0
$$31$$ −260.294 −1.50807 −0.754036 0.656833i $$-0.771896\pi$$
−0.754036 + 0.656833i $$0.771896\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −119.630 −0.577749
$$36$$ 0 0
$$37$$ −223.836 −0.994553 −0.497276 0.867592i $$-0.665666\pi$$
−0.497276 + 0.867592i $$0.665666\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 337.939 1.28725 0.643625 0.765341i $$-0.277430\pi$$
0.643625 + 0.765341i $$0.277430\pi$$
$$42$$ 0 0
$$43$$ −326.511 −1.15797 −0.578983 0.815340i $$-0.696550\pi$$
−0.578983 + 0.815340i $$0.696550\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 89.6851 0.278339 0.139169 0.990269i $$-0.455557\pi$$
0.139169 + 0.990269i $$0.455557\pi$$
$$48$$ 0 0
$$49$$ 229.457 0.668970
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 543.672 1.40904 0.704520 0.709684i $$-0.251162\pi$$
0.704520 + 0.709684i $$0.251162\pi$$
$$54$$ 0 0
$$55$$ 289.703 0.710246
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 92.0000 0.203006 0.101503 0.994835i $$-0.467635\pi$$
0.101503 + 0.994835i $$0.467635\pi$$
$$60$$ 0 0
$$61$$ 159.129 0.334006 0.167003 0.985956i $$-0.446591\pi$$
0.167003 + 0.985956i $$0.446591\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −40.8119 −0.0778783
$$66$$ 0 0
$$67$$ 910.561 1.66034 0.830169 0.557511i $$-0.188244\pi$$
0.830169 + 0.557511i $$0.188244\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −293.232 −0.490144 −0.245072 0.969505i $$-0.578812\pi$$
−0.245072 + 0.969505i $$0.578812\pi$$
$$72$$ 0 0
$$73$$ 142.022 0.227705 0.113853 0.993498i $$-0.463681\pi$$
0.113853 + 0.993498i $$0.463681\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1386.29 −2.05172
$$78$$ 0 0
$$79$$ −1106.50 −1.57584 −0.787920 0.615777i $$-0.788842\pi$$
−0.787920 + 0.615777i $$0.788842\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 813.367 1.07565 0.537823 0.843058i $$-0.319247\pi$$
0.537823 + 0.843058i $$0.319247\pi$$
$$84$$ 0 0
$$85$$ 250.442 0.319580
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −956.666 −1.13940 −0.569699 0.821853i $$-0.692940\pi$$
−0.569699 + 0.821853i $$0.692940\pi$$
$$90$$ 0 0
$$91$$ 195.294 0.224971
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −348.891 −0.376794
$$96$$ 0 0
$$97$$ 106.363 0.111335 0.0556677 0.998449i $$-0.482271\pi$$
0.0556677 + 0.998449i $$0.482271\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 703.082 0.692666 0.346333 0.938112i $$-0.387427\pi$$
0.346333 + 0.938112i $$0.387427\pi$$
$$102$$ 0 0
$$103$$ −968.360 −0.926363 −0.463181 0.886263i $$-0.653292\pi$$
−0.463181 + 0.886263i $$0.653292\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 366.393 0.331033 0.165517 0.986207i $$-0.447071\pi$$
0.165517 + 0.986207i $$0.447071\pi$$
$$108$$ 0 0
$$109$$ 723.614 0.635869 0.317934 0.948113i $$-0.397011\pi$$
0.317934 + 0.948113i $$0.397011\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −978.279 −0.814413 −0.407207 0.913336i $$-0.633497\pi$$
−0.407207 + 0.913336i $$0.633497\pi$$
$$114$$ 0 0
$$115$$ 24.6303 0.0199721
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1198.42 −0.923184
$$120$$ 0 0
$$121$$ 2026.11 1.52225
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2821.70 −1.97154 −0.985770 0.168099i $$-0.946237\pi$$
−0.985770 + 0.168099i $$0.946237\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2131.82 −1.42182 −0.710908 0.703285i $$-0.751716\pi$$
−0.710908 + 0.703285i $$0.751716\pi$$
$$132$$ 0 0
$$133$$ 1669.52 1.08846
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 816.808 0.509377 0.254688 0.967023i $$-0.418027\pi$$
0.254688 + 0.967023i $$0.418027\pi$$
$$138$$ 0 0
$$139$$ −1876.80 −1.14524 −0.572620 0.819821i $$-0.694073\pi$$
−0.572620 + 0.819821i $$0.694073\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −472.933 −0.276564
$$144$$ 0 0
$$145$$ −397.139 −0.227452
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2542.71 −1.39803 −0.699016 0.715106i $$-0.746379\pi$$
−0.699016 + 0.715106i $$0.746379\pi$$
$$150$$ 0 0
$$151$$ −333.443 −0.179703 −0.0898516 0.995955i $$-0.528639\pi$$
−0.0898516 + 0.995955i $$0.528639\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1301.47 −0.674430
$$156$$ 0 0
$$157$$ 3650.02 1.85543 0.927716 0.373286i $$-0.121769\pi$$
0.927716 + 0.373286i $$0.121769\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −117.861 −0.0576943
$$162$$ 0 0
$$163$$ −745.580 −0.358272 −0.179136 0.983824i $$-0.557330\pi$$
−0.179136 + 0.983824i $$0.557330\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3321.73 −1.53918 −0.769591 0.638537i $$-0.779540\pi$$
−0.769591 + 0.638537i $$0.779540\pi$$
$$168$$ 0 0
$$169$$ −2130.38 −0.969675
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −1038.58 −0.456426 −0.228213 0.973611i $$-0.573288\pi$$
−0.228213 + 0.973611i $$0.573288\pi$$
$$174$$ 0 0
$$175$$ −598.152 −0.258377
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 250.181 0.104466 0.0522329 0.998635i $$-0.483366\pi$$
0.0522329 + 0.998635i $$0.483366\pi$$
$$180$$ 0 0
$$181$$ −1171.61 −0.481132 −0.240566 0.970633i $$-0.577333\pi$$
−0.240566 + 0.970633i $$0.577333\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1119.18 −0.444778
$$186$$ 0 0
$$187$$ 2902.15 1.13490
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1091.59 −0.413533 −0.206766 0.978390i $$-0.566294\pi$$
−0.206766 + 0.978390i $$0.566294\pi$$
$$192$$ 0 0
$$193$$ −171.475 −0.0639537 −0.0319768 0.999489i $$-0.510180\pi$$
−0.0319768 + 0.999489i $$0.510180\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2368.70 −0.856665 −0.428332 0.903621i $$-0.640899\pi$$
−0.428332 + 0.903621i $$0.640899\pi$$
$$198$$ 0 0
$$199$$ −3373.82 −1.20183 −0.600914 0.799314i $$-0.705197\pi$$
−0.600914 + 0.799314i $$0.705197\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1900.39 0.657051
$$204$$ 0 0
$$205$$ 1689.70 0.575676
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4042.99 −1.33808
$$210$$ 0 0
$$211$$ −5148.57 −1.67982 −0.839910 0.542726i $$-0.817392\pi$$
−0.839910 + 0.542726i $$0.817392\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1632.56 −0.517858
$$216$$ 0 0
$$217$$ 6227.82 1.94826
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −408.841 −0.124442
$$222$$ 0 0
$$223$$ 3604.51 1.08240 0.541201 0.840893i $$-0.317970\pi$$
0.541201 + 0.840893i $$0.317970\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3324.89 0.972162 0.486081 0.873914i $$-0.338426\pi$$
0.486081 + 0.873914i $$0.338426\pi$$
$$228$$ 0 0
$$229$$ −2809.94 −0.810856 −0.405428 0.914127i $$-0.632877\pi$$
−0.405428 + 0.914127i $$0.632877\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 4200.85 1.18115 0.590573 0.806984i $$-0.298902\pi$$
0.590573 + 0.806984i $$0.298902\pi$$
$$234$$ 0 0
$$235$$ 448.426 0.124477
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5787.06 −1.56625 −0.783125 0.621865i $$-0.786375\pi$$
−0.783125 + 0.621865i $$0.786375\pi$$
$$240$$ 0 0
$$241$$ −2971.77 −0.794309 −0.397154 0.917752i $$-0.630002\pi$$
−0.397154 + 0.917752i $$0.630002\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1147.28 0.299172
$$246$$ 0 0
$$247$$ 569.556 0.146721
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4035.62 1.01484 0.507422 0.861698i $$-0.330599\pi$$
0.507422 + 0.861698i $$0.330599\pi$$
$$252$$ 0 0
$$253$$ 285.419 0.0709255
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2852.67 0.692392 0.346196 0.938162i $$-0.387473\pi$$
0.346196 + 0.938162i $$0.387473\pi$$
$$258$$ 0 0
$$259$$ 5355.52 1.28485
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2910.63 −0.682423 −0.341212 0.939987i $$-0.610837\pi$$
−0.341212 + 0.939987i $$0.610837\pi$$
$$264$$ 0 0
$$265$$ 2718.36 0.630142
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −476.299 −0.107957 −0.0539785 0.998542i $$-0.517190\pi$$
−0.0539785 + 0.998542i $$0.517190\pi$$
$$270$$ 0 0
$$271$$ 4186.19 0.938350 0.469175 0.883105i $$-0.344551\pi$$
0.469175 + 0.883105i $$0.344551\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1448.51 0.317632
$$276$$ 0 0
$$277$$ 4993.37 1.08311 0.541556 0.840664i $$-0.317835\pi$$
0.541556 + 0.840664i $$0.317835\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4480.61 −0.951212 −0.475606 0.879658i $$-0.657771\pi$$
−0.475606 + 0.879658i $$0.657771\pi$$
$$282$$ 0 0
$$283$$ −8117.91 −1.70516 −0.852579 0.522599i $$-0.824963\pi$$
−0.852579 + 0.522599i $$0.824963\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8085.56 −1.66298
$$288$$ 0 0
$$289$$ −2404.15 −0.489344
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −5999.46 −1.19622 −0.598109 0.801414i $$-0.704081\pi$$
−0.598109 + 0.801414i $$0.704081\pi$$
$$294$$ 0 0
$$295$$ 460.000 0.0907872
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −40.2084 −0.00777696
$$300$$ 0 0
$$301$$ 7812.14 1.49596
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 795.644 0.149372
$$306$$ 0 0
$$307$$ 1058.34 0.196751 0.0983753 0.995149i $$-0.468635\pi$$
0.0983753 + 0.995149i $$0.468635\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 3120.17 0.568902 0.284451 0.958691i $$-0.408189\pi$$
0.284451 + 0.958691i $$0.408189\pi$$
$$312$$ 0 0
$$313$$ −4425.91 −0.799257 −0.399628 0.916677i $$-0.630861\pi$$
−0.399628 + 0.916677i $$0.630861\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9032.06 1.60029 0.800144 0.599808i $$-0.204757\pi$$
0.800144 + 0.599808i $$0.204757\pi$$
$$318$$ 0 0
$$319$$ −4602.09 −0.807735
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3495.08 −0.602079
$$324$$ 0 0
$$325$$ −204.059 −0.0348282
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2145.81 −0.359582
$$330$$ 0 0
$$331$$ −3153.85 −0.523720 −0.261860 0.965106i $$-0.584336\pi$$
−0.261860 + 0.965106i $$0.584336\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4552.80 0.742526
$$336$$ 0 0
$$337$$ −3438.53 −0.555811 −0.277906 0.960608i $$-0.589640\pi$$
−0.277906 + 0.960608i $$0.589640\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −15081.6 −2.39506
$$342$$ 0 0
$$343$$ 2716.64 0.427653
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −11108.0 −1.71848 −0.859238 0.511576i $$-0.829062\pi$$
−0.859238 + 0.511576i $$0.829062\pi$$
$$348$$ 0 0
$$349$$ 1687.51 0.258826 0.129413 0.991591i $$-0.458691\pi$$
0.129413 + 0.991591i $$0.458691\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −12048.9 −1.81671 −0.908356 0.418199i $$-0.862662\pi$$
−0.908356 + 0.418199i $$0.862662\pi$$
$$354$$ 0 0
$$355$$ −1466.16 −0.219199
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11017.0 1.61965 0.809827 0.586669i $$-0.199561\pi$$
0.809827 + 0.586669i $$0.199561\pi$$
$$360$$ 0 0
$$361$$ −1990.00 −0.290130
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 710.112 0.101833
$$366$$ 0 0
$$367$$ 10621.2 1.51069 0.755343 0.655329i $$-0.227470\pi$$
0.755343 + 0.655329i $$0.227470\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −13007.9 −1.82032
$$372$$ 0 0
$$373$$ 24.9392 0.00346194 0.00173097 0.999999i $$-0.499449\pi$$
0.00173097 + 0.999999i $$0.499449\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 648.319 0.0885679
$$378$$ 0 0
$$379$$ −4755.57 −0.644531 −0.322265 0.946649i $$-0.604444\pi$$
−0.322265 + 0.946649i $$0.604444\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1229.91 −0.164088 −0.0820440 0.996629i $$-0.526145\pi$$
−0.0820440 + 0.996629i $$0.526145\pi$$
$$384$$ 0 0
$$385$$ −6931.45 −0.917557
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −10616.3 −1.38372 −0.691862 0.722030i $$-0.743209\pi$$
−0.691862 + 0.722030i $$0.743209\pi$$
$$390$$ 0 0
$$391$$ 246.739 0.0319134
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −5532.52 −0.704738
$$396$$ 0 0
$$397$$ −13086.4 −1.65437 −0.827187 0.561927i $$-0.810060\pi$$
−0.827187 + 0.561927i $$0.810060\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 5253.36 0.654215 0.327107 0.944987i $$-0.393926\pi$$
0.327107 + 0.944987i $$0.393926\pi$$
$$402$$ 0 0
$$403$$ 2124.62 0.262618
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −12969.2 −1.57951
$$408$$ 0 0
$$409$$ 4778.69 0.577729 0.288865 0.957370i $$-0.406722\pi$$
0.288865 + 0.957370i $$0.406722\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2201.20 −0.262261
$$414$$ 0 0
$$415$$ 4066.84 0.481044
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7770.14 0.905958 0.452979 0.891521i $$-0.350361\pi$$
0.452979 + 0.891521i $$0.350361\pi$$
$$420$$ 0 0
$$421$$ −14440.5 −1.67170 −0.835852 0.548955i $$-0.815026\pi$$
−0.835852 + 0.548955i $$0.815026\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1252.21 0.142920
$$426$$ 0 0
$$427$$ −3807.32 −0.431497
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 13416.1 1.49937 0.749686 0.661794i $$-0.230205\pi$$
0.749686 + 0.661794i $$0.230205\pi$$
$$432$$ 0 0
$$433$$ −187.211 −0.0207778 −0.0103889 0.999946i $$-0.503307\pi$$
−0.0103889 + 0.999946i $$0.503307\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −343.732 −0.0376269
$$438$$ 0 0
$$439$$ −2171.47 −0.236079 −0.118039 0.993009i $$-0.537661\pi$$
−0.118039 + 0.993009i $$0.537661\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −11129.8 −1.19366 −0.596830 0.802368i $$-0.703573\pi$$
−0.596830 + 0.802368i $$0.703573\pi$$
$$444$$ 0 0
$$445$$ −4783.33 −0.509554
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 15602.0 1.63988 0.819938 0.572452i $$-0.194008\pi$$
0.819938 + 0.572452i $$0.194008\pi$$
$$450$$ 0 0
$$451$$ 19580.4 2.04436
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 976.468 0.100610
$$456$$ 0 0
$$457$$ −12065.3 −1.23499 −0.617495 0.786575i $$-0.711852\pi$$
−0.617495 + 0.786575i $$0.711852\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −9703.61 −0.980352 −0.490176 0.871623i $$-0.663067\pi$$
−0.490176 + 0.871623i $$0.663067\pi$$
$$462$$ 0 0
$$463$$ 9787.17 0.982394 0.491197 0.871048i $$-0.336559\pi$$
0.491197 + 0.871048i $$0.336559\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9051.97 −0.896949 −0.448474 0.893796i $$-0.648032\pi$$
−0.448474 + 0.893796i $$0.648032\pi$$
$$468$$ 0 0
$$469$$ −21786.1 −2.14497
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −18918.3 −1.83903
$$474$$ 0 0
$$475$$ −1744.46 −0.168508
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −8492.58 −0.810096 −0.405048 0.914295i $$-0.632745\pi$$
−0.405048 + 0.914295i $$0.632745\pi$$
$$480$$ 0 0
$$481$$ 1827.04 0.173193
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 531.816 0.0497907
$$486$$ 0 0
$$487$$ 7190.96 0.669104 0.334552 0.942377i $$-0.391415\pi$$
0.334552 + 0.942377i $$0.391415\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20304.9 −1.86629 −0.933146 0.359499i $$-0.882948\pi$$
−0.933146 + 0.359499i $$0.882948\pi$$
$$492$$ 0 0
$$493$$ −3978.41 −0.363445
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 7015.88 0.633210
$$498$$ 0 0
$$499$$ −5079.64 −0.455703 −0.227852 0.973696i $$-0.573170\pi$$
−0.227852 + 0.973696i $$0.573170\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1329.81 −0.117880 −0.0589398 0.998262i $$-0.518772\pi$$
−0.0589398 + 0.998262i $$0.518772\pi$$
$$504$$ 0 0
$$505$$ 3515.41 0.309770
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −6501.31 −0.566140 −0.283070 0.959099i $$-0.591353\pi$$
−0.283070 + 0.959099i $$0.591353\pi$$
$$510$$ 0 0
$$511$$ −3398.04 −0.294169
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −4841.80 −0.414282
$$516$$ 0 0
$$517$$ 5196.41 0.442046
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5846.09 0.491597 0.245798 0.969321i $$-0.420950\pi$$
0.245798 + 0.969321i $$0.420950\pi$$
$$522$$ 0 0
$$523$$ 2544.28 0.212722 0.106361 0.994328i $$-0.466080\pi$$
0.106361 + 0.994328i $$0.466080\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −13037.7 −1.07767
$$528$$ 0 0
$$529$$ −12142.7 −0.998006
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2758.39 −0.224163
$$534$$ 0 0
$$535$$ 1831.97 0.148043
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 13294.9 1.06243
$$540$$ 0 0
$$541$$ 19350.2 1.53776 0.768880 0.639393i $$-0.220814\pi$$
0.768880 + 0.639393i $$0.220814\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 3618.07 0.284369
$$546$$ 0 0
$$547$$ 4957.23 0.387488 0.193744 0.981052i $$-0.437937\pi$$
0.193744 + 0.981052i $$0.437937\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5542.32 0.428513
$$552$$ 0 0
$$553$$ 26474.3 2.03581
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1559.47 0.118630 0.0593148 0.998239i $$-0.481108\pi$$
0.0593148 + 0.998239i $$0.481108\pi$$
$$558$$ 0 0
$$559$$ 2665.11 0.201650
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 7533.60 0.563949 0.281974 0.959422i $$-0.409011\pi$$
0.281974 + 0.959422i $$0.409011\pi$$
$$564$$ 0 0
$$565$$ −4891.39 −0.364217
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −19257.9 −1.41886 −0.709432 0.704774i $$-0.751048\pi$$
−0.709432 + 0.704774i $$0.751048\pi$$
$$570$$ 0 0
$$571$$ 26102.2 1.91304 0.956518 0.291675i $$-0.0942125\pi$$
0.956518 + 0.291675i $$0.0942125\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 123.152 0.00893179
$$576$$ 0 0
$$577$$ −12553.3 −0.905721 −0.452860 0.891581i $$-0.649596\pi$$
−0.452860 + 0.891581i $$0.649596\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −19460.7 −1.38961
$$582$$ 0 0
$$583$$ 31500.7 2.23778
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 16778.4 1.17976 0.589881 0.807490i $$-0.299175\pi$$
0.589881 + 0.807490i $$0.299175\pi$$
$$588$$ 0 0
$$589$$ 18162.9 1.27061
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 21693.4 1.50226 0.751132 0.660152i $$-0.229508\pi$$
0.751132 + 0.660152i $$0.229508\pi$$
$$594$$ 0 0
$$595$$ −5992.10 −0.412861
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 22980.4 1.56753 0.783767 0.621055i $$-0.213296\pi$$
0.783767 + 0.621055i $$0.213296\pi$$
$$600$$ 0 0
$$601$$ −10048.9 −0.682038 −0.341019 0.940056i $$-0.610772\pi$$
−0.341019 + 0.940056i $$0.610772\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 10130.6 0.680770
$$606$$ 0 0
$$607$$ −24514.6 −1.63923 −0.819617 0.572911i $$-0.805814\pi$$
−0.819617 + 0.572911i $$0.805814\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −732.043 −0.0484702
$$612$$ 0 0
$$613$$ 1845.67 0.121609 0.0608043 0.998150i $$-0.480633\pi$$
0.0608043 + 0.998150i $$0.480633\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9365.42 0.611082 0.305541 0.952179i $$-0.401163\pi$$
0.305541 + 0.952179i $$0.401163\pi$$
$$618$$ 0 0
$$619$$ 19517.2 1.26731 0.633653 0.773618i $$-0.281555\pi$$
0.633653 + 0.773618i $$0.281555\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 22889.3 1.47197
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −11211.6 −0.710709
$$630$$ 0 0
$$631$$ −22314.1 −1.40778 −0.703892 0.710307i $$-0.748556\pi$$
−0.703892 + 0.710307i $$0.748556\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −14108.5 −0.881700
$$636$$ 0 0
$$637$$ −1872.91 −0.116495
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 20150.8 1.24167 0.620833 0.783943i $$-0.286795\pi$$
0.620833 + 0.783943i $$0.286795\pi$$
$$642$$ 0 0
$$643$$ −5345.25 −0.327832 −0.163916 0.986474i $$-0.552413\pi$$
−0.163916 + 0.986474i $$0.552413\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −13701.4 −0.832549 −0.416274 0.909239i $$-0.636664\pi$$
−0.416274 + 0.909239i $$0.636664\pi$$
$$648$$ 0 0
$$649$$ 5330.53 0.322406
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −713.119 −0.0427358 −0.0213679 0.999772i $$-0.506802\pi$$
−0.0213679 + 0.999772i $$0.506802\pi$$
$$654$$ 0 0
$$655$$ −10659.1 −0.635856
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 26491.2 1.56593 0.782967 0.622063i $$-0.213705\pi$$
0.782967 + 0.622063i $$0.213705\pi$$
$$660$$ 0 0
$$661$$ 645.746 0.0379979 0.0189989 0.999820i $$-0.493952\pi$$
0.0189989 + 0.999820i $$0.493952\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 8347.59 0.486775
$$666$$ 0 0
$$667$$ −391.266 −0.0227135
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 9220.01 0.530454
$$672$$ 0 0
$$673$$ −6602.56 −0.378172 −0.189086 0.981961i $$-0.560553\pi$$
−0.189086 + 0.981961i $$0.560553\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −4968.68 −0.282070 −0.141035 0.990005i $$-0.545043\pi$$
−0.141035 + 0.990005i $$0.545043\pi$$
$$678$$ 0 0
$$679$$ −2544.85 −0.143833
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24694.2 1.38345 0.691727 0.722159i $$-0.256850\pi$$
0.691727 + 0.722159i $$0.256850\pi$$
$$684$$ 0 0
$$685$$ 4084.04 0.227800
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −4437.66 −0.245372
$$690$$ 0 0
$$691$$ 19259.2 1.06028 0.530141 0.847910i $$-0.322139\pi$$
0.530141 + 0.847910i $$0.322139\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −9384.02 −0.512167
$$696$$ 0 0
$$697$$ 16926.8 0.919871
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 31090.7 1.67515 0.837575 0.546323i $$-0.183973\pi$$
0.837575 + 0.546323i $$0.183973\pi$$
$$702$$ 0 0
$$703$$ 15618.9 0.837948
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −16822.0 −0.894845
$$708$$ 0 0
$$709$$ −20832.0 −1.10347 −0.551736 0.834019i $$-0.686034\pi$$
−0.551736 + 0.834019i $$0.686034\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1282.23 −0.0673490
$$714$$ 0 0
$$715$$ −2364.66 −0.123683
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 21328.1 1.10626 0.553132 0.833093i $$-0.313432\pi$$
0.553132 + 0.833093i $$0.313432\pi$$
$$720$$ 0 0
$$721$$ 23169.1 1.19676
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1985.69 −0.101720
$$726$$ 0 0
$$727$$ −4121.56 −0.210262 −0.105131 0.994458i $$-0.533526\pi$$
−0.105131 + 0.994458i $$0.533526\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −16354.5 −0.827485
$$732$$ 0 0
$$733$$ −11102.2 −0.559439 −0.279720 0.960082i $$-0.590241\pi$$
−0.279720 + 0.960082i $$0.590241\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 52758.4 2.63688
$$738$$ 0 0
$$739$$ 13749.5 0.684414 0.342207 0.939625i $$-0.388826\pi$$
0.342207 + 0.939625i $$0.388826\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −13962.9 −0.689433 −0.344717 0.938707i $$-0.612025\pi$$
−0.344717 + 0.938707i $$0.612025\pi$$
$$744$$ 0 0
$$745$$ −12713.5 −0.625219
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −8766.35 −0.427657
$$750$$ 0 0
$$751$$ −32801.6 −1.59380 −0.796902 0.604108i $$-0.793530\pi$$
−0.796902 + 0.604108i $$0.793530\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1667.21 −0.0803657
$$756$$ 0 0
$$757$$ −14664.9 −0.704102 −0.352051 0.935981i $$-0.614516\pi$$
−0.352051 + 0.935981i $$0.614516\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3304.79 0.157422 0.0787112 0.996897i $$-0.474919\pi$$
0.0787112 + 0.996897i $$0.474919\pi$$
$$762$$ 0 0
$$763$$ −17313.2 −0.821470
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −750.938 −0.0353518
$$768$$ 0 0
$$769$$ −33085.5 −1.55149 −0.775743 0.631049i $$-0.782625\pi$$
−0.775743 + 0.631049i $$0.782625\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −27880.8 −1.29729 −0.648643 0.761093i $$-0.724663\pi$$
−0.648643 + 0.761093i $$0.724663\pi$$
$$774$$ 0 0
$$775$$ −6507.36 −0.301614
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −23580.8 −1.08456
$$780$$ 0 0
$$781$$ −16990.0 −0.778426
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 18250.1 0.829775
$$786$$ 0 0
$$787$$ −17813.9 −0.806856 −0.403428 0.915011i $$-0.632181\pi$$
−0.403428 + 0.915011i $$0.632181\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 23406.4 1.05213
$$792$$ 0 0
$$793$$ −1298.87 −0.0581641
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −8005.10 −0.355778 −0.177889 0.984051i $$-0.556927\pi$$
−0.177889 + 0.984051i $$0.556927\pi$$
$$798$$ 0 0
$$799$$ 4492.19 0.198901
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8228.86 0.361632
$$804$$ 0 0
$$805$$ −589.307 −0.0258017
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −10140.7 −0.440700 −0.220350 0.975421i $$-0.570720\pi$$
−0.220350 + 0.975421i $$0.570720\pi$$
$$810$$ 0 0
$$811$$ 14662.7 0.634865 0.317433 0.948281i $$-0.397179\pi$$
0.317433 + 0.948281i $$0.397179\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −3727.90 −0.160224
$$816$$ 0 0
$$817$$ 22783.4 0.975630
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 36216.1 1.53953 0.769763 0.638330i $$-0.220374\pi$$
0.769763 + 0.638330i $$0.220374\pi$$
$$822$$ 0 0
$$823$$ 9695.36 0.410643 0.205321 0.978695i $$-0.434176\pi$$
0.205321 + 0.978695i $$0.434176\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 21471.9 0.902844 0.451422 0.892311i $$-0.350917\pi$$
0.451422 + 0.892311i $$0.350917\pi$$
$$828$$ 0 0
$$829$$ 10447.9 0.437723 0.218861 0.975756i $$-0.429766\pi$$
0.218861 + 0.975756i $$0.429766\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 11493.1 0.478047
$$834$$ 0 0
$$835$$ −16608.7 −0.688343
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −4917.61 −0.202354 −0.101177 0.994868i $$-0.532261\pi$$
−0.101177 + 0.994868i $$0.532261\pi$$
$$840$$ 0 0
$$841$$ −18080.2 −0.741328
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −10651.9 −0.433652
$$846$$ 0 0
$$847$$ −48476.9 −1.96657
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1102.63 −0.0444157
$$852$$ 0 0
$$853$$ 45031.3 1.80755 0.903777 0.428004i $$-0.140783\pi$$
0.903777 + 0.428004i $$0.140783\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 14826.6 0.590976 0.295488 0.955346i $$-0.404518\pi$$
0.295488 + 0.955346i $$0.404518\pi$$
$$858$$ 0 0
$$859$$ −43287.9 −1.71940 −0.859700 0.510799i $$-0.829350\pi$$
−0.859700 + 0.510799i $$0.829350\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 40830.8 1.61054 0.805270 0.592908i $$-0.202020\pi$$
0.805270 + 0.592908i $$0.202020\pi$$
$$864$$ 0 0
$$865$$ −5192.89 −0.204120
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −64111.5 −2.50268
$$870$$ 0 0
$$871$$ −7432.34 −0.289133
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2990.76 −0.115550
$$876$$ 0 0
$$877$$ 2322.20 0.0894127 0.0447064 0.999000i $$-0.485765\pi$$
0.0447064 + 0.999000i $$0.485765\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −10407.8 −0.398010 −0.199005 0.979998i $$-0.563771\pi$$
−0.199005 + 0.979998i $$0.563771\pi$$
$$882$$ 0 0
$$883$$ 14346.9 0.546784 0.273392 0.961903i $$-0.411854\pi$$
0.273392 + 0.961903i $$0.411854\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −22740.9 −0.860838 −0.430419 0.902629i $$-0.641634\pi$$
−0.430419 + 0.902629i $$0.641634\pi$$
$$888$$ 0 0
$$889$$ 67512.2 2.54701
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −6258.07 −0.234511
$$894$$ 0 0
$$895$$ 1250.90 0.0467186
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 20674.6 0.767003
$$900$$ 0 0
$$901$$ 27231.7 1.00690
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −5858.04 −0.215169
$$906$$ 0 0
$$907$$ 14222.6 0.520678 0.260339 0.965517i $$-0.416166\pi$$
0.260339 + 0.965517i $$0.416166\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6825.69 0.248238 0.124119 0.992267i $$-0.460390\pi$$
0.124119 + 0.992267i $$0.460390\pi$$
$$912$$ 0 0
$$913$$ 47127.0 1.70830
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 51006.1 1.83683
$$918$$ 0 0
$$919$$ −18851.0 −0.676647 −0.338323 0.941030i $$-0.609860\pi$$
−0.338323 + 0.941030i $$0.609860\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 2393.47 0.0853542
$$924$$ 0 0
$$925$$ −5595.91 −0.198911
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 7109.70 0.251089 0.125544 0.992088i $$-0.459932\pi$$
0.125544 + 0.992088i $$0.459932\pi$$
$$930$$ 0 0
$$931$$ −16011.1 −0.563633
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 14510.8 0.507543
$$936$$ 0 0
$$937$$ −14107.6 −0.491864 −0.245932 0.969287i $$-0.579094\pi$$
−0.245932 + 0.969287i $$0.579094\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −49409.8 −1.71170 −0.855852 0.517221i $$-0.826966\pi$$
−0.855852 + 0.517221i $$0.826966\pi$$
$$942$$ 0 0
$$943$$ 1664.71 0.0574872
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −24070.5 −0.825963 −0.412982 0.910739i $$-0.635513\pi$$
−0.412982 + 0.910739i $$0.635513\pi$$
$$948$$ 0 0
$$949$$ −1159.24 −0.0396528
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 55644.1 1.89138 0.945692 0.325064i $$-0.105386\pi$$
0.945692 + 0.325064i $$0.105386\pi$$
$$954$$ 0 0
$$955$$ −5457.95 −0.184937
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −19543.0 −0.658057
$$960$$ 0 0
$$961$$ 37962.1 1.27428
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −857.377 −0.0286010
$$966$$ 0 0
$$967$$ 34638.9 1.15193 0.575963 0.817476i $$-0.304627\pi$$
0.575963 + 0.817476i $$0.304627\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 28705.9 0.948731 0.474365 0.880328i $$-0.342678\pi$$
0.474365 + 0.880328i $$0.342678\pi$$
$$972$$ 0 0
$$973$$ 44904.5 1.47952
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 45305.4 1.48357 0.741785 0.670638i $$-0.233979\pi$$
0.741785 + 0.670638i $$0.233979\pi$$
$$978$$ 0 0
$$979$$ −55429.8 −1.80954
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 9220.63 0.299178 0.149589 0.988748i $$-0.452205\pi$$
0.149589 + 0.988748i $$0.452205\pi$$
$$984$$ 0 0
$$985$$ −11843.5 −0.383112
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1608.42 −0.0517136
$$990$$ 0 0
$$991$$ 23444.0 0.751488 0.375744 0.926723i $$-0.377387\pi$$
0.375744 + 0.926723i $$0.377387\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −16869.1 −0.537474
$$996$$ 0 0
$$997$$ 9848.13 0.312832 0.156416 0.987691i $$-0.450006\pi$$
0.156416 + 0.987691i $$0.450006\pi$$
$$998$$ 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 2160.4.a.bn.1.1 3
3.2 odd 2 2160.4.a.bf.1.1 3
4.3 odd 2 1080.4.a.n.1.3 yes 3
12.11 even 2 1080.4.a.h.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.h.1.3 3 12.11 even 2
1080.4.a.n.1.3 yes 3 4.3 odd 2
2160.4.a.bf.1.1 3 3.2 odd 2
2160.4.a.bn.1.1 3 1.1 even 1 trivial