# Properties

 Label 960.4.a.p.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.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: $$56.6418336055$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +20.0000 q^{11} +58.0000 q^{13} -15.0000 q^{15} -70.0000 q^{17} +92.0000 q^{19} -36.0000 q^{21} +112.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -66.0000 q^{29} -108.000 q^{31} -60.0000 q^{33} +60.0000 q^{35} +58.0000 q^{37} -174.000 q^{39} +66.0000 q^{41} +388.000 q^{43} +45.0000 q^{45} -408.000 q^{47} -199.000 q^{49} +210.000 q^{51} -474.000 q^{53} +100.000 q^{55} -276.000 q^{57} +540.000 q^{59} -14.0000 q^{61} +108.000 q^{63} +290.000 q^{65} +276.000 q^{67} -336.000 q^{69} -96.0000 q^{71} -790.000 q^{73} -75.0000 q^{75} +240.000 q^{77} +308.000 q^{79} +81.0000 q^{81} +1036.00 q^{83} -350.000 q^{85} +198.000 q^{87} +1210.00 q^{89} +696.000 q^{91} +324.000 q^{93} +460.000 q^{95} +1426.00 q^{97} +180.000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 12.0000 0.647939 0.323970 0.946068i $$-0.394982\pi$$
0.323970 + 0.946068i $$0.394982\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 20.0000 0.548202 0.274101 0.961701i $$-0.411620\pi$$
0.274101 + 0.961701i $$0.411620\pi$$
$$12$$ 0 0
$$13$$ 58.0000 1.23741 0.618704 0.785624i $$-0.287658\pi$$
0.618704 + 0.785624i $$0.287658\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ −70.0000 −0.998676 −0.499338 0.866407i $$-0.666423\pi$$
−0.499338 + 0.866407i $$0.666423\pi$$
$$18$$ 0 0
$$19$$ 92.0000 1.11086 0.555428 0.831565i $$-0.312555\pi$$
0.555428 + 0.831565i $$0.312555\pi$$
$$20$$ 0 0
$$21$$ −36.0000 −0.374088
$$22$$ 0 0
$$23$$ 112.000 1.01537 0.507687 0.861541i $$-0.330501\pi$$
0.507687 + 0.861541i $$0.330501\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −66.0000 −0.422617 −0.211308 0.977419i $$-0.567772\pi$$
−0.211308 + 0.977419i $$0.567772\pi$$
$$30$$ 0 0
$$31$$ −108.000 −0.625722 −0.312861 0.949799i $$-0.601287\pi$$
−0.312861 + 0.949799i $$0.601287\pi$$
$$32$$ 0 0
$$33$$ −60.0000 −0.316505
$$34$$ 0 0
$$35$$ 60.0000 0.289767
$$36$$ 0 0
$$37$$ 58.0000 0.257707 0.128853 0.991664i $$-0.458870\pi$$
0.128853 + 0.991664i $$0.458870\pi$$
$$38$$ 0 0
$$39$$ −174.000 −0.714418
$$40$$ 0 0
$$41$$ 66.0000 0.251402 0.125701 0.992068i $$-0.459882\pi$$
0.125701 + 0.992068i $$0.459882\pi$$
$$42$$ 0 0
$$43$$ 388.000 1.37603 0.688017 0.725695i $$-0.258482\pi$$
0.688017 + 0.725695i $$0.258482\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ −408.000 −1.26623 −0.633116 0.774057i $$-0.718224\pi$$
−0.633116 + 0.774057i $$0.718224\pi$$
$$48$$ 0 0
$$49$$ −199.000 −0.580175
$$50$$ 0 0
$$51$$ 210.000 0.576586
$$52$$ 0 0
$$53$$ −474.000 −1.22847 −0.614235 0.789123i $$-0.710535\pi$$
−0.614235 + 0.789123i $$0.710535\pi$$
$$54$$ 0 0
$$55$$ 100.000 0.245164
$$56$$ 0 0
$$57$$ −276.000 −0.641353
$$58$$ 0 0
$$59$$ 540.000 1.19156 0.595780 0.803148i $$-0.296843\pi$$
0.595780 + 0.803148i $$0.296843\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −0.0293855 −0.0146928 0.999892i $$-0.504677\pi$$
−0.0146928 + 0.999892i $$0.504677\pi$$
$$62$$ 0 0
$$63$$ 108.000 0.215980
$$64$$ 0 0
$$65$$ 290.000 0.553386
$$66$$ 0 0
$$67$$ 276.000 0.503265 0.251633 0.967823i $$-0.419033\pi$$
0.251633 + 0.967823i $$0.419033\pi$$
$$68$$ 0 0
$$69$$ −336.000 −0.586227
$$70$$ 0 0
$$71$$ −96.0000 −0.160466 −0.0802331 0.996776i $$-0.525566\pi$$
−0.0802331 + 0.996776i $$0.525566\pi$$
$$72$$ 0 0
$$73$$ −790.000 −1.26661 −0.633305 0.773902i $$-0.718302\pi$$
−0.633305 + 0.773902i $$0.718302\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ 240.000 0.355202
$$78$$ 0 0
$$79$$ 308.000 0.438642 0.219321 0.975653i $$-0.429616\pi$$
0.219321 + 0.975653i $$0.429616\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1036.00 1.37007 0.685035 0.728510i $$-0.259787\pi$$
0.685035 + 0.728510i $$0.259787\pi$$
$$84$$ 0 0
$$85$$ −350.000 −0.446622
$$86$$ 0 0
$$87$$ 198.000 0.243998
$$88$$ 0 0
$$89$$ 1210.00 1.44112 0.720560 0.693392i $$-0.243885\pi$$
0.720560 + 0.693392i $$0.243885\pi$$
$$90$$ 0 0
$$91$$ 696.000 0.801765
$$92$$ 0 0
$$93$$ 324.000 0.361261
$$94$$ 0 0
$$95$$ 460.000 0.496790
$$96$$ 0 0
$$97$$ 1426.00 1.49266 0.746332 0.665574i $$-0.231813\pi$$
0.746332 + 0.665574i $$0.231813\pi$$
$$98$$ 0 0
$$99$$ 180.000 0.182734
$$100$$ 0 0
$$101$$ −74.0000 −0.0729037 −0.0364519 0.999335i $$-0.511606\pi$$
−0.0364519 + 0.999335i $$0.511606\pi$$
$$102$$ 0 0
$$103$$ −1436.00 −1.37372 −0.686861 0.726789i $$-0.741012\pi$$
−0.686861 + 0.726789i $$0.741012\pi$$
$$104$$ 0 0
$$105$$ −180.000 −0.167297
$$106$$ 0 0
$$107$$ −84.0000 −0.0758933 −0.0379467 0.999280i $$-0.512082\pi$$
−0.0379467 + 0.999280i $$0.512082\pi$$
$$108$$ 0 0
$$109$$ 250.000 0.219685 0.109842 0.993949i $$-0.464965\pi$$
0.109842 + 0.993949i $$0.464965\pi$$
$$110$$ 0 0
$$111$$ −174.000 −0.148787
$$112$$ 0 0
$$113$$ −654.000 −0.544453 −0.272226 0.962233i $$-0.587760\pi$$
−0.272226 + 0.962233i $$0.587760\pi$$
$$114$$ 0 0
$$115$$ 560.000 0.454089
$$116$$ 0 0
$$117$$ 522.000 0.412469
$$118$$ 0 0
$$119$$ −840.000 −0.647081
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ 0 0
$$123$$ −198.000 −0.145147
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 2572.00 1.79707 0.898536 0.438900i $$-0.144632\pi$$
0.898536 + 0.438900i $$0.144632\pi$$
$$128$$ 0 0
$$129$$ −1164.00 −0.794453
$$130$$ 0 0
$$131$$ 836.000 0.557570 0.278785 0.960354i $$-0.410068\pi$$
0.278785 + 0.960354i $$0.410068\pi$$
$$132$$ 0 0
$$133$$ 1104.00 0.719766
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ 1250.00 0.779523 0.389762 0.920916i $$-0.372557\pi$$
0.389762 + 0.920916i $$0.372557\pi$$
$$138$$ 0 0
$$139$$ 2428.00 1.48158 0.740792 0.671734i $$-0.234450\pi$$
0.740792 + 0.671734i $$0.234450\pi$$
$$140$$ 0 0
$$141$$ 1224.00 0.731060
$$142$$ 0 0
$$143$$ 1160.00 0.678350
$$144$$ 0 0
$$145$$ −330.000 −0.189000
$$146$$ 0 0
$$147$$ 597.000 0.334964
$$148$$ 0 0
$$149$$ −1746.00 −0.959986 −0.479993 0.877272i $$-0.659361\pi$$
−0.479993 + 0.877272i $$0.659361\pi$$
$$150$$ 0 0
$$151$$ 2092.00 1.12745 0.563724 0.825963i $$-0.309368\pi$$
0.563724 + 0.825963i $$0.309368\pi$$
$$152$$ 0 0
$$153$$ −630.000 −0.332892
$$154$$ 0 0
$$155$$ −540.000 −0.279831
$$156$$ 0 0
$$157$$ 2162.00 1.09902 0.549511 0.835487i $$-0.314814\pi$$
0.549511 + 0.835487i $$0.314814\pi$$
$$158$$ 0 0
$$159$$ 1422.00 0.709257
$$160$$ 0 0
$$161$$ 1344.00 0.657901
$$162$$ 0 0
$$163$$ −932.000 −0.447852 −0.223926 0.974606i $$-0.571887\pi$$
−0.223926 + 0.974606i $$0.571887\pi$$
$$164$$ 0 0
$$165$$ −300.000 −0.141545
$$166$$ 0 0
$$167$$ 3192.00 1.47907 0.739534 0.673119i $$-0.235046\pi$$
0.739534 + 0.673119i $$0.235046\pi$$
$$168$$ 0 0
$$169$$ 1167.00 0.531179
$$170$$ 0 0
$$171$$ 828.000 0.370285
$$172$$ 0 0
$$173$$ −2282.00 −1.00287 −0.501437 0.865194i $$-0.667195\pi$$
−0.501437 + 0.865194i $$0.667195\pi$$
$$174$$ 0 0
$$175$$ 300.000 0.129588
$$176$$ 0 0
$$177$$ −1620.00 −0.687947
$$178$$ 0 0
$$179$$ 2004.00 0.836793 0.418397 0.908264i $$-0.362592\pi$$
0.418397 + 0.908264i $$0.362592\pi$$
$$180$$ 0 0
$$181$$ 4226.00 1.73545 0.867724 0.497046i $$-0.165582\pi$$
0.867724 + 0.497046i $$0.165582\pi$$
$$182$$ 0 0
$$183$$ 42.0000 0.0169657
$$184$$ 0 0
$$185$$ 290.000 0.115250
$$186$$ 0 0
$$187$$ −1400.00 −0.547477
$$188$$ 0 0
$$189$$ −324.000 −0.124696
$$190$$ 0 0
$$191$$ 2656.00 1.00619 0.503093 0.864232i $$-0.332195\pi$$
0.503093 + 0.864232i $$0.332195\pi$$
$$192$$ 0 0
$$193$$ 2162.00 0.806343 0.403171 0.915124i $$-0.367908\pi$$
0.403171 + 0.915124i $$0.367908\pi$$
$$194$$ 0 0
$$195$$ −870.000 −0.319497
$$196$$ 0 0
$$197$$ −3514.00 −1.27087 −0.635437 0.772153i $$-0.719180\pi$$
−0.635437 + 0.772153i $$0.719180\pi$$
$$198$$ 0 0
$$199$$ −988.000 −0.351947 −0.175974 0.984395i $$-0.556307\pi$$
−0.175974 + 0.984395i $$0.556307\pi$$
$$200$$ 0 0
$$201$$ −828.000 −0.290560
$$202$$ 0 0
$$203$$ −792.000 −0.273830
$$204$$ 0 0
$$205$$ 330.000 0.112430
$$206$$ 0 0
$$207$$ 1008.00 0.338458
$$208$$ 0 0
$$209$$ 1840.00 0.608973
$$210$$ 0 0
$$211$$ −3548.00 −1.15760 −0.578802 0.815468i $$-0.696480\pi$$
−0.578802 + 0.815468i $$0.696480\pi$$
$$212$$ 0 0
$$213$$ 288.000 0.0926452
$$214$$ 0 0
$$215$$ 1940.00 0.615381
$$216$$ 0 0
$$217$$ −1296.00 −0.405430
$$218$$ 0 0
$$219$$ 2370.00 0.731277
$$220$$ 0 0
$$221$$ −4060.00 −1.23577
$$222$$ 0 0
$$223$$ 732.000 0.219813 0.109907 0.993942i $$-0.464945\pi$$
0.109907 + 0.993942i $$0.464945\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −5492.00 −1.60580 −0.802901 0.596113i $$-0.796711\pi$$
−0.802901 + 0.596113i $$0.796711\pi$$
$$228$$ 0 0
$$229$$ −798.000 −0.230277 −0.115138 0.993349i $$-0.536731\pi$$
−0.115138 + 0.993349i $$0.536731\pi$$
$$230$$ 0 0
$$231$$ −720.000 −0.205076
$$232$$ 0 0
$$233$$ −2886.00 −0.811451 −0.405726 0.913995i $$-0.632981\pi$$
−0.405726 + 0.913995i $$0.632981\pi$$
$$234$$ 0 0
$$235$$ −2040.00 −0.566276
$$236$$ 0 0
$$237$$ −924.000 −0.253250
$$238$$ 0 0
$$239$$ 4096.00 1.10857 0.554285 0.832327i $$-0.312992\pi$$
0.554285 + 0.832327i $$0.312992\pi$$
$$240$$ 0 0
$$241$$ 2354.00 0.629189 0.314594 0.949226i $$-0.398132\pi$$
0.314594 + 0.949226i $$0.398132\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −995.000 −0.259462
$$246$$ 0 0
$$247$$ 5336.00 1.37458
$$248$$ 0 0
$$249$$ −3108.00 −0.791010
$$250$$ 0 0
$$251$$ 2916.00 0.733292 0.366646 0.930361i $$-0.380506\pi$$
0.366646 + 0.930361i $$0.380506\pi$$
$$252$$ 0 0
$$253$$ 2240.00 0.556631
$$254$$ 0 0
$$255$$ 1050.00 0.257857
$$256$$ 0 0
$$257$$ 882.000 0.214076 0.107038 0.994255i $$-0.465863\pi$$
0.107038 + 0.994255i $$0.465863\pi$$
$$258$$ 0 0
$$259$$ 696.000 0.166978
$$260$$ 0 0
$$261$$ −594.000 −0.140872
$$262$$ 0 0
$$263$$ −4456.00 −1.04475 −0.522374 0.852716i $$-0.674954\pi$$
−0.522374 + 0.852716i $$0.674954\pi$$
$$264$$ 0 0
$$265$$ −2370.00 −0.549388
$$266$$ 0 0
$$267$$ −3630.00 −0.832031
$$268$$ 0 0
$$269$$ 1486.00 0.336814 0.168407 0.985718i $$-0.446138\pi$$
0.168407 + 0.985718i $$0.446138\pi$$
$$270$$ 0 0
$$271$$ −4676.00 −1.04814 −0.524072 0.851674i $$-0.675588\pi$$
−0.524072 + 0.851674i $$0.675588\pi$$
$$272$$ 0 0
$$273$$ −2088.00 −0.462899
$$274$$ 0 0
$$275$$ 500.000 0.109640
$$276$$ 0 0
$$277$$ 2898.00 0.628606 0.314303 0.949323i $$-0.398229\pi$$
0.314303 + 0.949323i $$0.398229\pi$$
$$278$$ 0 0
$$279$$ −972.000 −0.208574
$$280$$ 0 0
$$281$$ 5194.00 1.10266 0.551331 0.834287i $$-0.314120\pi$$
0.551331 + 0.834287i $$0.314120\pi$$
$$282$$ 0 0
$$283$$ 5420.00 1.13846 0.569232 0.822177i $$-0.307240\pi$$
0.569232 + 0.822177i $$0.307240\pi$$
$$284$$ 0 0
$$285$$ −1380.00 −0.286822
$$286$$ 0 0
$$287$$ 792.000 0.162893
$$288$$ 0 0
$$289$$ −13.0000 −0.00264604
$$290$$ 0 0
$$291$$ −4278.00 −0.861790
$$292$$ 0 0
$$293$$ −9130.00 −1.82041 −0.910205 0.414157i $$-0.864076\pi$$
−0.910205 + 0.414157i $$0.864076\pi$$
$$294$$ 0 0
$$295$$ 2700.00 0.532882
$$296$$ 0 0
$$297$$ −540.000 −0.105502
$$298$$ 0 0
$$299$$ 6496.00 1.25643
$$300$$ 0 0
$$301$$ 4656.00 0.891586
$$302$$ 0 0
$$303$$ 222.000 0.0420910
$$304$$ 0 0
$$305$$ −70.0000 −0.0131416
$$306$$ 0 0
$$307$$ −6044.00 −1.12361 −0.561807 0.827269i $$-0.689894\pi$$
−0.561807 + 0.827269i $$0.689894\pi$$
$$308$$ 0 0
$$309$$ 4308.00 0.793118
$$310$$ 0 0
$$311$$ −6120.00 −1.11586 −0.557931 0.829887i $$-0.688405\pi$$
−0.557931 + 0.829887i $$0.688405\pi$$
$$312$$ 0 0
$$313$$ −614.000 −0.110880 −0.0554398 0.998462i $$-0.517656\pi$$
−0.0554398 + 0.998462i $$0.517656\pi$$
$$314$$ 0 0
$$315$$ 540.000 0.0965891
$$316$$ 0 0
$$317$$ −786.000 −0.139262 −0.0696312 0.997573i $$-0.522182\pi$$
−0.0696312 + 0.997573i $$0.522182\pi$$
$$318$$ 0 0
$$319$$ −1320.00 −0.231680
$$320$$ 0 0
$$321$$ 252.000 0.0438170
$$322$$ 0 0
$$323$$ −6440.00 −1.10938
$$324$$ 0 0
$$325$$ 1450.00 0.247482
$$326$$ 0 0
$$327$$ −750.000 −0.126835
$$328$$ 0 0
$$329$$ −4896.00 −0.820441
$$330$$ 0 0
$$331$$ −468.000 −0.0777148 −0.0388574 0.999245i $$-0.512372\pi$$
−0.0388574 + 0.999245i $$0.512372\pi$$
$$332$$ 0 0
$$333$$ 522.000 0.0859022
$$334$$ 0 0
$$335$$ 1380.00 0.225067
$$336$$ 0 0
$$337$$ 1538.00 0.248606 0.124303 0.992244i $$-0.460331\pi$$
0.124303 + 0.992244i $$0.460331\pi$$
$$338$$ 0 0
$$339$$ 1962.00 0.314340
$$340$$ 0 0
$$341$$ −2160.00 −0.343022
$$342$$ 0 0
$$343$$ −6504.00 −1.02386
$$344$$ 0 0
$$345$$ −1680.00 −0.262169
$$346$$ 0 0
$$347$$ −8396.00 −1.29891 −0.649454 0.760401i $$-0.725002\pi$$
−0.649454 + 0.760401i $$0.725002\pi$$
$$348$$ 0 0
$$349$$ 11090.0 1.70096 0.850479 0.526010i $$-0.176312\pi$$
0.850479 + 0.526010i $$0.176312\pi$$
$$350$$ 0 0
$$351$$ −1566.00 −0.238139
$$352$$ 0 0
$$353$$ 3298.00 0.497266 0.248633 0.968598i $$-0.420019\pi$$
0.248633 + 0.968598i $$0.420019\pi$$
$$354$$ 0 0
$$355$$ −480.000 −0.0717627
$$356$$ 0 0
$$357$$ 2520.00 0.373593
$$358$$ 0 0
$$359$$ −10720.0 −1.57599 −0.787994 0.615682i $$-0.788880\pi$$
−0.787994 + 0.615682i $$0.788880\pi$$
$$360$$ 0 0
$$361$$ 1605.00 0.233999
$$362$$ 0 0
$$363$$ 2793.00 0.403842
$$364$$ 0 0
$$365$$ −3950.00 −0.566445
$$366$$ 0 0
$$367$$ 4116.00 0.585432 0.292716 0.956199i $$-0.405441\pi$$
0.292716 + 0.956199i $$0.405441\pi$$
$$368$$ 0 0
$$369$$ 594.000 0.0838006
$$370$$ 0 0
$$371$$ −5688.00 −0.795974
$$372$$ 0 0
$$373$$ −3590.00 −0.498346 −0.249173 0.968459i $$-0.580159\pi$$
−0.249173 + 0.968459i $$0.580159\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ −3828.00 −0.522950
$$378$$ 0 0
$$379$$ 12452.0 1.68764 0.843821 0.536625i $$-0.180301\pi$$
0.843821 + 0.536625i $$0.180301\pi$$
$$380$$ 0 0
$$381$$ −7716.00 −1.03754
$$382$$ 0 0
$$383$$ −12416.0 −1.65647 −0.828235 0.560381i $$-0.810655\pi$$
−0.828235 + 0.560381i $$0.810655\pi$$
$$384$$ 0 0
$$385$$ 1200.00 0.158851
$$386$$ 0 0
$$387$$ 3492.00 0.458678
$$388$$ 0 0
$$389$$ −2370.00 −0.308904 −0.154452 0.988000i $$-0.549361\pi$$
−0.154452 + 0.988000i $$0.549361\pi$$
$$390$$ 0 0
$$391$$ −7840.00 −1.01403
$$392$$ 0 0
$$393$$ −2508.00 −0.321913
$$394$$ 0 0
$$395$$ 1540.00 0.196167
$$396$$ 0 0
$$397$$ −9486.00 −1.19922 −0.599608 0.800294i $$-0.704677\pi$$
−0.599608 + 0.800294i $$0.704677\pi$$
$$398$$ 0 0
$$399$$ −3312.00 −0.415557
$$400$$ 0 0
$$401$$ −10630.0 −1.32378 −0.661891 0.749600i $$-0.730246\pi$$
−0.661891 + 0.749600i $$0.730246\pi$$
$$402$$ 0 0
$$403$$ −6264.00 −0.774273
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 1160.00 0.141275
$$408$$ 0 0
$$409$$ 12890.0 1.55836 0.779180 0.626800i $$-0.215636\pi$$
0.779180 + 0.626800i $$0.215636\pi$$
$$410$$ 0 0
$$411$$ −3750.00 −0.450058
$$412$$ 0 0
$$413$$ 6480.00 0.772058
$$414$$ 0 0
$$415$$ 5180.00 0.612714
$$416$$ 0 0
$$417$$ −7284.00 −0.855393
$$418$$ 0 0
$$419$$ 11196.0 1.30539 0.652697 0.757619i $$-0.273637\pi$$
0.652697 + 0.757619i $$0.273637\pi$$
$$420$$ 0 0
$$421$$ 8594.00 0.994883 0.497442 0.867497i $$-0.334273\pi$$
0.497442 + 0.867497i $$0.334273\pi$$
$$422$$ 0 0
$$423$$ −3672.00 −0.422077
$$424$$ 0 0
$$425$$ −1750.00 −0.199735
$$426$$ 0 0
$$427$$ −168.000 −0.0190400
$$428$$ 0 0
$$429$$ −3480.00 −0.391646
$$430$$ 0 0
$$431$$ 3544.00 0.396075 0.198038 0.980194i $$-0.436543\pi$$
0.198038 + 0.980194i $$0.436543\pi$$
$$432$$ 0 0
$$433$$ 5810.00 0.644829 0.322414 0.946599i $$-0.395506\pi$$
0.322414 + 0.946599i $$0.395506\pi$$
$$434$$ 0 0
$$435$$ 990.000 0.109119
$$436$$ 0 0
$$437$$ 10304.0 1.12793
$$438$$ 0 0
$$439$$ −9628.00 −1.04674 −0.523371 0.852105i $$-0.675326\pi$$
−0.523371 + 0.852105i $$0.675326\pi$$
$$440$$ 0 0
$$441$$ −1791.00 −0.193392
$$442$$ 0 0
$$443$$ 6100.00 0.654221 0.327110 0.944986i $$-0.393925\pi$$
0.327110 + 0.944986i $$0.393925\pi$$
$$444$$ 0 0
$$445$$ 6050.00 0.644489
$$446$$ 0 0
$$447$$ 5238.00 0.554248
$$448$$ 0 0
$$449$$ −7750.00 −0.814577 −0.407289 0.913300i $$-0.633526\pi$$
−0.407289 + 0.913300i $$0.633526\pi$$
$$450$$ 0 0
$$451$$ 1320.00 0.137819
$$452$$ 0 0
$$453$$ −6276.00 −0.650932
$$454$$ 0 0
$$455$$ 3480.00 0.358560
$$456$$ 0 0
$$457$$ 18314.0 1.87460 0.937301 0.348522i $$-0.113316\pi$$
0.937301 + 0.348522i $$0.113316\pi$$
$$458$$ 0 0
$$459$$ 1890.00 0.192195
$$460$$ 0 0
$$461$$ −6122.00 −0.618503 −0.309252 0.950980i $$-0.600079\pi$$
−0.309252 + 0.950980i $$0.600079\pi$$
$$462$$ 0 0
$$463$$ −10420.0 −1.04591 −0.522957 0.852359i $$-0.675171\pi$$
−0.522957 + 0.852359i $$0.675171\pi$$
$$464$$ 0 0
$$465$$ 1620.00 0.161561
$$466$$ 0 0
$$467$$ −12612.0 −1.24971 −0.624854 0.780742i $$-0.714842\pi$$
−0.624854 + 0.780742i $$0.714842\pi$$
$$468$$ 0 0
$$469$$ 3312.00 0.326085
$$470$$ 0 0
$$471$$ −6486.00 −0.634520
$$472$$ 0 0
$$473$$ 7760.00 0.754345
$$474$$ 0 0
$$475$$ 2300.00 0.222171
$$476$$ 0 0
$$477$$ −4266.00 −0.409490
$$478$$ 0 0
$$479$$ −3352.00 −0.319743 −0.159871 0.987138i $$-0.551108\pi$$
−0.159871 + 0.987138i $$0.551108\pi$$
$$480$$ 0 0
$$481$$ 3364.00 0.318888
$$482$$ 0 0
$$483$$ −4032.00 −0.379839
$$484$$ 0 0
$$485$$ 7130.00 0.667539
$$486$$ 0 0
$$487$$ −17108.0 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$488$$ 0 0
$$489$$ 2796.00 0.258567
$$490$$ 0 0
$$491$$ 11388.0 1.04671 0.523354 0.852116i $$-0.324681\pi$$
0.523354 + 0.852116i $$0.324681\pi$$
$$492$$ 0 0
$$493$$ 4620.00 0.422057
$$494$$ 0 0
$$495$$ 900.000 0.0817212
$$496$$ 0 0
$$497$$ −1152.00 −0.103972
$$498$$ 0 0
$$499$$ 8996.00 0.807047 0.403523 0.914969i $$-0.367785\pi$$
0.403523 + 0.914969i $$0.367785\pi$$
$$500$$ 0 0
$$501$$ −9576.00 −0.853940
$$502$$ 0 0
$$503$$ −19504.0 −1.72891 −0.864454 0.502713i $$-0.832335\pi$$
−0.864454 + 0.502713i $$0.832335\pi$$
$$504$$ 0 0
$$505$$ −370.000 −0.0326035
$$506$$ 0 0
$$507$$ −3501.00 −0.306676
$$508$$ 0 0
$$509$$ −8306.00 −0.723295 −0.361647 0.932315i $$-0.617786\pi$$
−0.361647 + 0.932315i $$0.617786\pi$$
$$510$$ 0 0
$$511$$ −9480.00 −0.820686
$$512$$ 0 0
$$513$$ −2484.00 −0.213784
$$514$$ 0 0
$$515$$ −7180.00 −0.614347
$$516$$ 0 0
$$517$$ −8160.00 −0.694152
$$518$$ 0 0
$$519$$ 6846.00 0.579010
$$520$$ 0 0
$$521$$ 14850.0 1.24873 0.624367 0.781131i $$-0.285357\pi$$
0.624367 + 0.781131i $$0.285357\pi$$
$$522$$ 0 0
$$523$$ −20044.0 −1.67584 −0.837919 0.545795i $$-0.816228\pi$$
−0.837919 + 0.545795i $$0.816228\pi$$
$$524$$ 0 0
$$525$$ −900.000 −0.0748176
$$526$$ 0 0
$$527$$ 7560.00 0.624893
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ 0 0
$$531$$ 4860.00 0.397187
$$532$$ 0 0
$$533$$ 3828.00 0.311086
$$534$$ 0 0
$$535$$ −420.000 −0.0339405
$$536$$ 0 0
$$537$$ −6012.00 −0.483123
$$538$$ 0 0
$$539$$ −3980.00 −0.318053
$$540$$ 0 0
$$541$$ 21930.0 1.74278 0.871390 0.490590i $$-0.163219\pi$$
0.871390 + 0.490590i $$0.163219\pi$$
$$542$$ 0 0
$$543$$ −12678.0 −1.00196
$$544$$ 0 0
$$545$$ 1250.00 0.0982461
$$546$$ 0 0
$$547$$ −19988.0 −1.56239 −0.781193 0.624290i $$-0.785389\pi$$
−0.781193 + 0.624290i $$0.785389\pi$$
$$548$$ 0 0
$$549$$ −126.000 −0.00979517
$$550$$ 0 0
$$551$$ −6072.00 −0.469466
$$552$$ 0 0
$$553$$ 3696.00 0.284213
$$554$$ 0 0
$$555$$ −870.000 −0.0665395
$$556$$ 0 0
$$557$$ 11718.0 0.891396 0.445698 0.895183i $$-0.352956\pi$$
0.445698 + 0.895183i $$0.352956\pi$$
$$558$$ 0 0
$$559$$ 22504.0 1.70272
$$560$$ 0 0
$$561$$ 4200.00 0.316086
$$562$$ 0 0
$$563$$ −12756.0 −0.954887 −0.477443 0.878662i $$-0.658436\pi$$
−0.477443 + 0.878662i $$0.658436\pi$$
$$564$$ 0 0
$$565$$ −3270.00 −0.243487
$$566$$ 0 0
$$567$$ 972.000 0.0719932
$$568$$ 0 0
$$569$$ −15790.0 −1.16336 −0.581679 0.813418i $$-0.697604\pi$$
−0.581679 + 0.813418i $$0.697604\pi$$
$$570$$ 0 0
$$571$$ −2500.00 −0.183225 −0.0916127 0.995795i $$-0.529202\pi$$
−0.0916127 + 0.995795i $$0.529202\pi$$
$$572$$ 0 0
$$573$$ −7968.00 −0.580921
$$574$$ 0 0
$$575$$ 2800.00 0.203075
$$576$$ 0 0
$$577$$ 13778.0 0.994083 0.497041 0.867727i $$-0.334420\pi$$
0.497041 + 0.867727i $$0.334420\pi$$
$$578$$ 0 0
$$579$$ −6486.00 −0.465542
$$580$$ 0 0
$$581$$ 12432.0 0.887722
$$582$$ 0 0
$$583$$ −9480.00 −0.673450
$$584$$ 0 0
$$585$$ 2610.00 0.184462
$$586$$ 0 0
$$587$$ 11724.0 0.824363 0.412182 0.911102i $$-0.364767\pi$$
0.412182 + 0.911102i $$0.364767\pi$$
$$588$$ 0 0
$$589$$ −9936.00 −0.695086
$$590$$ 0 0
$$591$$ 10542.0 0.733739
$$592$$ 0 0
$$593$$ −27054.0 −1.87348 −0.936741 0.350024i $$-0.886174\pi$$
−0.936741 + 0.350024i $$0.886174\pi$$
$$594$$ 0 0
$$595$$ −4200.00 −0.289384
$$596$$ 0 0
$$597$$ 2964.00 0.203197
$$598$$ 0 0
$$599$$ 15328.0 1.04555 0.522776 0.852470i $$-0.324897\pi$$
0.522776 + 0.852470i $$0.324897\pi$$
$$600$$ 0 0
$$601$$ −25286.0 −1.71620 −0.858101 0.513481i $$-0.828356\pi$$
−0.858101 + 0.513481i $$0.828356\pi$$
$$602$$ 0 0
$$603$$ 2484.00 0.167755
$$604$$ 0 0
$$605$$ −4655.00 −0.312814
$$606$$ 0 0
$$607$$ 22060.0 1.47510 0.737552 0.675291i $$-0.235982\pi$$
0.737552 + 0.675291i $$0.235982\pi$$
$$608$$ 0 0
$$609$$ 2376.00 0.158096
$$610$$ 0 0
$$611$$ −23664.0 −1.56685
$$612$$ 0 0
$$613$$ 8810.00 0.580477 0.290239 0.956954i $$-0.406265\pi$$
0.290239 + 0.956954i $$0.406265\pi$$
$$614$$ 0 0
$$615$$ −990.000 −0.0649116
$$616$$ 0 0
$$617$$ −11766.0 −0.767717 −0.383858 0.923392i $$-0.625405\pi$$
−0.383858 + 0.923392i $$0.625405\pi$$
$$618$$ 0 0
$$619$$ 28316.0 1.83864 0.919318 0.393515i $$-0.128741\pi$$
0.919318 + 0.393515i $$0.128741\pi$$
$$620$$ 0 0
$$621$$ −3024.00 −0.195409
$$622$$ 0 0
$$623$$ 14520.0 0.933758
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −5520.00 −0.351591
$$628$$ 0 0
$$629$$ −4060.00 −0.257365
$$630$$ 0 0
$$631$$ 1388.00 0.0875680 0.0437840 0.999041i $$-0.486059\pi$$
0.0437840 + 0.999041i $$0.486059\pi$$
$$632$$ 0 0
$$633$$ 10644.0 0.668343
$$634$$ 0 0
$$635$$ 12860.0 0.803675
$$636$$ 0 0
$$637$$ −11542.0 −0.717913
$$638$$ 0 0
$$639$$ −864.000 −0.0534888
$$640$$ 0 0
$$641$$ −15974.0 −0.984298 −0.492149 0.870511i $$-0.663788\pi$$
−0.492149 + 0.870511i $$0.663788\pi$$
$$642$$ 0 0
$$643$$ −13044.0 −0.800008 −0.400004 0.916513i $$-0.630991\pi$$
−0.400004 + 0.916513i $$0.630991\pi$$
$$644$$ 0 0
$$645$$ −5820.00 −0.355290
$$646$$ 0 0
$$647$$ −18016.0 −1.09472 −0.547359 0.836898i $$-0.684367\pi$$
−0.547359 + 0.836898i $$0.684367\pi$$
$$648$$ 0 0
$$649$$ 10800.0 0.653216
$$650$$ 0 0
$$651$$ 3888.00 0.234075
$$652$$ 0 0
$$653$$ 17830.0 1.06852 0.534259 0.845321i $$-0.320591\pi$$
0.534259 + 0.845321i $$0.320591\pi$$
$$654$$ 0 0
$$655$$ 4180.00 0.249353
$$656$$ 0 0
$$657$$ −7110.00 −0.422203
$$658$$ 0 0
$$659$$ 20740.0 1.22597 0.612986 0.790094i $$-0.289968\pi$$
0.612986 + 0.790094i $$0.289968\pi$$
$$660$$ 0 0
$$661$$ −12070.0 −0.710240 −0.355120 0.934821i $$-0.615560\pi$$
−0.355120 + 0.934821i $$0.615560\pi$$
$$662$$ 0 0
$$663$$ 12180.0 0.713472
$$664$$ 0 0
$$665$$ 5520.00 0.321889
$$666$$ 0 0
$$667$$ −7392.00 −0.429115
$$668$$ 0 0
$$669$$ −2196.00 −0.126909
$$670$$ 0 0
$$671$$ −280.000 −0.0161092
$$672$$ 0 0
$$673$$ 20514.0 1.17497 0.587486 0.809234i $$-0.300118\pi$$
0.587486 + 0.809234i $$0.300118\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ 13326.0 0.756514 0.378257 0.925701i $$-0.376524\pi$$
0.378257 + 0.925701i $$0.376524\pi$$
$$678$$ 0 0
$$679$$ 17112.0 0.967155
$$680$$ 0 0
$$681$$ 16476.0 0.927110
$$682$$ 0 0
$$683$$ −2988.00 −0.167398 −0.0836989 0.996491i $$-0.526673\pi$$
−0.0836989 + 0.996491i $$0.526673\pi$$
$$684$$ 0 0
$$685$$ 6250.00 0.348613
$$686$$ 0 0
$$687$$ 2394.00 0.132950
$$688$$ 0 0
$$689$$ −27492.0 −1.52012
$$690$$ 0 0
$$691$$ −16628.0 −0.915425 −0.457713 0.889100i $$-0.651331\pi$$
−0.457713 + 0.889100i $$0.651331\pi$$
$$692$$ 0 0
$$693$$ 2160.00 0.118401
$$694$$ 0 0
$$695$$ 12140.0 0.662585
$$696$$ 0 0
$$697$$ −4620.00 −0.251069
$$698$$ 0 0
$$699$$ 8658.00 0.468492
$$700$$ 0 0
$$701$$ −28082.0 −1.51304 −0.756521 0.653969i $$-0.773103\pi$$
−0.756521 + 0.653969i $$0.773103\pi$$
$$702$$ 0 0
$$703$$ 5336.00 0.286275
$$704$$ 0 0
$$705$$ 6120.00 0.326940
$$706$$ 0 0
$$707$$ −888.000 −0.0472372
$$708$$ 0 0
$$709$$ −20158.0 −1.06777 −0.533885 0.845557i $$-0.679269\pi$$
−0.533885 + 0.845557i $$0.679269\pi$$
$$710$$ 0 0
$$711$$ 2772.00 0.146214
$$712$$ 0 0
$$713$$ −12096.0 −0.635342
$$714$$ 0 0
$$715$$ 5800.00 0.303367
$$716$$ 0 0
$$717$$ −12288.0 −0.640033
$$718$$ 0 0
$$719$$ 30536.0 1.58387 0.791934 0.610607i $$-0.209075\pi$$
0.791934 + 0.610607i $$0.209075\pi$$
$$720$$ 0 0
$$721$$ −17232.0 −0.890088
$$722$$ 0 0
$$723$$ −7062.00 −0.363262
$$724$$ 0 0
$$725$$ −1650.00 −0.0845234
$$726$$ 0 0
$$727$$ −7204.00 −0.367512 −0.183756 0.982972i $$-0.558826\pi$$
−0.183756 + 0.982972i $$0.558826\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −27160.0 −1.37421
$$732$$ 0 0
$$733$$ −5398.00 −0.272005 −0.136003 0.990708i $$-0.543426\pi$$
−0.136003 + 0.990708i $$0.543426\pi$$
$$734$$ 0 0
$$735$$ 2985.00 0.149801
$$736$$ 0 0
$$737$$ 5520.00 0.275891
$$738$$ 0 0
$$739$$ −5660.00 −0.281741 −0.140870 0.990028i $$-0.544990\pi$$
−0.140870 + 0.990028i $$0.544990\pi$$
$$740$$ 0 0
$$741$$ −16008.0 −0.793615
$$742$$ 0 0
$$743$$ 3624.00 0.178939 0.0894695 0.995990i $$-0.471483\pi$$
0.0894695 + 0.995990i $$0.471483\pi$$
$$744$$ 0 0
$$745$$ −8730.00 −0.429319
$$746$$ 0 0
$$747$$ 9324.00 0.456690
$$748$$ 0 0
$$749$$ −1008.00 −0.0491743
$$750$$ 0 0
$$751$$ 12532.0 0.608920 0.304460 0.952525i $$-0.401524\pi$$
0.304460 + 0.952525i $$0.401524\pi$$
$$752$$ 0 0
$$753$$ −8748.00 −0.423366
$$754$$ 0 0
$$755$$ 10460.0 0.504210
$$756$$ 0 0
$$757$$ 9026.00 0.433363 0.216681 0.976242i $$-0.430477\pi$$
0.216681 + 0.976242i $$0.430477\pi$$
$$758$$ 0 0
$$759$$ −6720.00 −0.321371
$$760$$ 0 0
$$761$$ 4674.00 0.222644 0.111322 0.993784i $$-0.464491\pi$$
0.111322 + 0.993784i $$0.464491\pi$$
$$762$$ 0 0
$$763$$ 3000.00 0.142342
$$764$$ 0 0
$$765$$ −3150.00 −0.148874
$$766$$ 0 0
$$767$$ 31320.0 1.47445
$$768$$ 0 0
$$769$$ 38386.0 1.80004 0.900022 0.435843i $$-0.143550\pi$$
0.900022 + 0.435843i $$0.143550\pi$$
$$770$$ 0 0
$$771$$ −2646.00 −0.123597
$$772$$ 0 0
$$773$$ 16774.0 0.780490 0.390245 0.920711i $$-0.372390\pi$$
0.390245 + 0.920711i $$0.372390\pi$$
$$774$$ 0 0
$$775$$ −2700.00 −0.125144
$$776$$ 0 0
$$777$$ −2088.00 −0.0964049
$$778$$ 0 0
$$779$$ 6072.00 0.279271
$$780$$ 0 0
$$781$$ −1920.00 −0.0879680
$$782$$ 0 0
$$783$$ 1782.00 0.0813327
$$784$$ 0 0
$$785$$ 10810.0 0.491497
$$786$$ 0 0
$$787$$ 27116.0 1.22818 0.614092 0.789234i $$-0.289522\pi$$
0.614092 + 0.789234i $$0.289522\pi$$
$$788$$ 0 0
$$789$$ 13368.0 0.603186
$$790$$ 0 0
$$791$$ −7848.00 −0.352772
$$792$$ 0 0
$$793$$ −812.000 −0.0363619
$$794$$ 0 0
$$795$$ 7110.00 0.317190
$$796$$ 0 0
$$797$$ 17494.0 0.777502 0.388751 0.921343i $$-0.372907\pi$$
0.388751 + 0.921343i $$0.372907\pi$$
$$798$$ 0 0
$$799$$ 28560.0 1.26456
$$800$$ 0 0
$$801$$ 10890.0 0.480374
$$802$$ 0 0
$$803$$ −15800.0 −0.694359
$$804$$ 0 0
$$805$$ 6720.00 0.294222
$$806$$ 0 0
$$807$$ −4458.00 −0.194460
$$808$$ 0 0
$$809$$ 9298.00 0.404079 0.202040 0.979377i $$-0.435243\pi$$
0.202040 + 0.979377i $$0.435243\pi$$
$$810$$ 0 0
$$811$$ 21252.0 0.920171 0.460085 0.887875i $$-0.347819\pi$$
0.460085 + 0.887875i $$0.347819\pi$$
$$812$$ 0 0
$$813$$ 14028.0 0.605146
$$814$$ 0 0
$$815$$ −4660.00 −0.200285
$$816$$ 0 0
$$817$$ 35696.0 1.52857
$$818$$ 0 0
$$819$$ 6264.00 0.267255
$$820$$ 0 0
$$821$$ −1578.00 −0.0670799 −0.0335399 0.999437i $$-0.510678\pi$$
−0.0335399 + 0.999437i $$0.510678\pi$$
$$822$$ 0 0
$$823$$ −9652.00 −0.408806 −0.204403 0.978887i $$-0.565525\pi$$
−0.204403 + 0.978887i $$0.565525\pi$$
$$824$$ 0 0
$$825$$ −1500.00 −0.0633010
$$826$$ 0 0
$$827$$ 15612.0 0.656448 0.328224 0.944600i $$-0.393550\pi$$
0.328224 + 0.944600i $$0.393550\pi$$
$$828$$ 0 0
$$829$$ 13194.0 0.552770 0.276385 0.961047i $$-0.410863\pi$$
0.276385 + 0.961047i $$0.410863\pi$$
$$830$$ 0 0
$$831$$ −8694.00 −0.362926
$$832$$ 0 0
$$833$$ 13930.0 0.579407
$$834$$ 0 0
$$835$$ 15960.0 0.661459
$$836$$ 0 0
$$837$$ 2916.00 0.120420
$$838$$ 0 0
$$839$$ 22512.0 0.926342 0.463171 0.886269i $$-0.346712\pi$$
0.463171 + 0.886269i $$0.346712\pi$$
$$840$$ 0 0
$$841$$ −20033.0 −0.821395
$$842$$ 0 0
$$843$$ −15582.0 −0.636622
$$844$$ 0 0
$$845$$ 5835.00 0.237550
$$846$$ 0 0
$$847$$ −11172.0 −0.453217
$$848$$ 0 0
$$849$$ −16260.0 −0.657293
$$850$$ 0 0
$$851$$ 6496.00 0.261669
$$852$$ 0 0
$$853$$ 3114.00 0.124996 0.0624978 0.998045i $$-0.480093\pi$$
0.0624978 + 0.998045i $$0.480093\pi$$
$$854$$ 0 0
$$855$$ 4140.00 0.165597
$$856$$ 0 0
$$857$$ −36254.0 −1.44506 −0.722528 0.691342i $$-0.757020\pi$$
−0.722528 + 0.691342i $$0.757020\pi$$
$$858$$ 0 0
$$859$$ −27644.0 −1.09802 −0.549011 0.835815i $$-0.684996\pi$$
−0.549011 + 0.835815i $$0.684996\pi$$
$$860$$ 0 0
$$861$$ −2376.00 −0.0940463
$$862$$ 0 0
$$863$$ 8608.00 0.339536 0.169768 0.985484i $$-0.445698\pi$$
0.169768 + 0.985484i $$0.445698\pi$$
$$864$$ 0 0
$$865$$ −11410.0 −0.448499
$$866$$ 0 0
$$867$$ 39.0000 0.00152769
$$868$$ 0 0
$$869$$ 6160.00 0.240465
$$870$$ 0 0
$$871$$ 16008.0 0.622744
$$872$$ 0 0
$$873$$ 12834.0 0.497555
$$874$$ 0 0
$$875$$ 1500.00 0.0579534
$$876$$ 0 0
$$877$$ −37294.0 −1.43595 −0.717975 0.696068i $$-0.754931\pi$$
−0.717975 + 0.696068i $$0.754931\pi$$
$$878$$ 0 0
$$879$$ 27390.0 1.05101
$$880$$ 0 0
$$881$$ −5742.00 −0.219583 −0.109792 0.993955i $$-0.535018\pi$$
−0.109792 + 0.993955i $$0.535018\pi$$
$$882$$ 0 0
$$883$$ 46028.0 1.75421 0.877104 0.480301i $$-0.159472\pi$$
0.877104 + 0.480301i $$0.159472\pi$$
$$884$$ 0 0
$$885$$ −8100.00 −0.307659
$$886$$ 0 0
$$887$$ −10136.0 −0.383691 −0.191845 0.981425i $$-0.561447\pi$$
−0.191845 + 0.981425i $$0.561447\pi$$
$$888$$ 0 0
$$889$$ 30864.0 1.16439
$$890$$ 0 0
$$891$$ 1620.00 0.0609114
$$892$$ 0 0
$$893$$ −37536.0 −1.40660
$$894$$ 0 0
$$895$$ 10020.0 0.374225
$$896$$ 0 0
$$897$$ −19488.0 −0.725402
$$898$$ 0 0
$$899$$ 7128.00 0.264441
$$900$$ 0 0
$$901$$ 33180.0 1.22684
$$902$$ 0 0
$$903$$ −13968.0 −0.514757
$$904$$ 0 0
$$905$$ 21130.0 0.776116
$$906$$ 0 0
$$907$$ 30900.0 1.13122 0.565611 0.824672i $$-0.308641\pi$$
0.565611 + 0.824672i $$0.308641\pi$$
$$908$$ 0 0
$$909$$ −666.000 −0.0243012
$$910$$ 0 0
$$911$$ −45152.0 −1.64210 −0.821050 0.570857i $$-0.806611\pi$$
−0.821050 + 0.570857i $$0.806611\pi$$
$$912$$ 0 0
$$913$$ 20720.0 0.751075
$$914$$ 0 0
$$915$$ 210.000 0.00758731
$$916$$ 0 0
$$917$$ 10032.0 0.361271
$$918$$ 0 0
$$919$$ 30044.0 1.07841 0.539206 0.842174i $$-0.318725\pi$$
0.539206 + 0.842174i $$0.318725\pi$$
$$920$$ 0 0
$$921$$ 18132.0 0.648718
$$922$$ 0 0
$$923$$ −5568.00 −0.198562
$$924$$ 0 0
$$925$$ 1450.00 0.0515413
$$926$$ 0 0
$$927$$ −12924.0 −0.457907
$$928$$ 0 0
$$929$$ −11382.0 −0.401971 −0.200986 0.979594i $$-0.564414\pi$$
−0.200986 + 0.979594i $$0.564414\pi$$
$$930$$ 0 0
$$931$$ −18308.0 −0.644490
$$932$$ 0 0
$$933$$ 18360.0 0.644244
$$934$$ 0 0
$$935$$ −7000.00 −0.244839
$$936$$ 0 0
$$937$$ −34758.0 −1.21184 −0.605920 0.795525i $$-0.707195\pi$$
−0.605920 + 0.795525i $$0.707195\pi$$
$$938$$ 0 0
$$939$$ 1842.00 0.0640164
$$940$$ 0 0
$$941$$ 13270.0 0.459713 0.229856 0.973225i $$-0.426174\pi$$
0.229856 + 0.973225i $$0.426174\pi$$
$$942$$ 0 0
$$943$$ 7392.00 0.255267
$$944$$ 0 0
$$945$$ −1620.00 −0.0557657
$$946$$ 0 0
$$947$$ 35620.0 1.22228 0.611138 0.791524i $$-0.290712\pi$$
0.611138 + 0.791524i $$0.290712\pi$$
$$948$$ 0 0
$$949$$ −45820.0 −1.56731
$$950$$ 0 0
$$951$$ 2358.00 0.0804031
$$952$$ 0 0
$$953$$ −16926.0 −0.575327 −0.287664 0.957731i $$-0.592879\pi$$
−0.287664 + 0.957731i $$0.592879\pi$$
$$954$$ 0 0
$$955$$ 13280.0 0.449980
$$956$$ 0 0
$$957$$ 3960.00 0.133760
$$958$$ 0 0
$$959$$ 15000.0 0.505084
$$960$$ 0 0
$$961$$ −18127.0 −0.608472
$$962$$ 0 0
$$963$$ −756.000 −0.0252978
$$964$$ 0 0
$$965$$ 10810.0 0.360607
$$966$$ 0 0
$$967$$ −27676.0 −0.920372 −0.460186 0.887822i $$-0.652217\pi$$
−0.460186 + 0.887822i $$0.652217\pi$$
$$968$$ 0 0
$$969$$ 19320.0 0.640503
$$970$$ 0 0
$$971$$ 46916.0 1.55057 0.775286 0.631610i $$-0.217606\pi$$
0.775286 + 0.631610i $$0.217606\pi$$
$$972$$ 0 0
$$973$$ 29136.0 0.959977
$$974$$ 0 0
$$975$$ −4350.00 −0.142884
$$976$$ 0 0
$$977$$ 27594.0 0.903593 0.451796 0.892121i $$-0.350783\pi$$
0.451796 + 0.892121i $$0.350783\pi$$
$$978$$ 0 0
$$979$$ 24200.0 0.790026
$$980$$ 0 0
$$981$$ 2250.00 0.0732283
$$982$$ 0 0
$$983$$ −33016.0 −1.07126 −0.535629 0.844453i $$-0.679925\pi$$
−0.535629 + 0.844453i $$0.679925\pi$$
$$984$$ 0 0
$$985$$ −17570.0 −0.568352
$$986$$ 0 0
$$987$$ 14688.0 0.473682
$$988$$ 0 0
$$989$$ 43456.0 1.39719
$$990$$ 0 0
$$991$$ 2276.00 0.0729561 0.0364781 0.999334i $$-0.488386\pi$$
0.0364781 + 0.999334i $$0.488386\pi$$
$$992$$ 0 0
$$993$$ 1404.00 0.0448687
$$994$$ 0 0
$$995$$ −4940.00 −0.157396
$$996$$ 0 0
$$997$$ −57654.0 −1.83141 −0.915707 0.401846i $$-0.868369\pi$$
−0.915707 + 0.401846i $$0.868369\pi$$
$$998$$ 0 0
$$999$$ −1566.00 −0.0495956
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 960.4.a.p.1.1 1
4.3 odd 2 960.4.a.be.1.1 1
8.3 odd 2 480.4.a.a.1.1 1
8.5 even 2 480.4.a.h.1.1 yes 1
24.5 odd 2 1440.4.a.q.1.1 1
24.11 even 2 1440.4.a.l.1.1 1
40.19 odd 2 2400.4.a.u.1.1 1
40.29 even 2 2400.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.a.1.1 1 8.3 odd 2
480.4.a.h.1.1 yes 1 8.5 even 2
960.4.a.p.1.1 1 1.1 even 1 trivial
960.4.a.be.1.1 1 4.3 odd 2
1440.4.a.l.1.1 1 24.11 even 2
1440.4.a.q.1.1 1 24.5 odd 2
2400.4.a.b.1.1 1 40.29 even 2
2400.4.a.u.1.1 1 40.19 odd 2