# Properties

 Label 1368.4.a.a.1.2 Level $1368$ Weight $4$ Character 1368.1 Self dual yes Analytic conductor $80.715$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,4,Mod(1,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, 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("1368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1368.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: $$80.7146128879$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 1368.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.27492 q^{5} -18.0997 q^{7} +O(q^{10})$$ $$q+6.27492 q^{5} -18.0997 q^{7} +33.9244 q^{11} -3.07558 q^{13} +14.2508 q^{17} -19.0000 q^{19} -114.072 q^{23} -85.6254 q^{25} +34.5257 q^{29} +107.698 q^{31} -113.574 q^{35} -181.698 q^{37} +444.743 q^{41} +120.323 q^{43} -306.371 q^{47} -15.4020 q^{49} -115.825 q^{53} +212.873 q^{55} -161.680 q^{59} +274.571 q^{61} -19.2990 q^{65} +81.6254 q^{67} -773.492 q^{71} +148.794 q^{73} -614.021 q^{77} -557.341 q^{79} -768.337 q^{83} +89.4228 q^{85} -457.884 q^{89} +55.6670 q^{91} -119.223 q^{95} -1500.82 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{5} - 6 q^{7}+O(q^{10})$$ 2 * q + 5 * q^5 - 6 * q^7 $$2 q + 5 q^{5} - 6 q^{7} + 15 q^{11} - 59 q^{13} + 104 q^{17} - 38 q^{19} + 21 q^{23} - 209 q^{25} + 137 q^{29} + 4 q^{31} - 129 q^{35} - 152 q^{37} + 210 q^{41} + 67 q^{43} - 273 q^{47} - 212 q^{49} - 209 q^{53} + 237 q^{55} - 799 q^{59} + 149 q^{61} + 52 q^{65} + 201 q^{67} - 792 q^{71} - 246 q^{73} - 843 q^{77} - 254 q^{79} - 374 q^{83} - 25 q^{85} + 564 q^{89} - 621 q^{91} - 95 q^{95} - 178 q^{97}+O(q^{100})$$ 2 * q + 5 * q^5 - 6 * q^7 + 15 * q^11 - 59 * q^13 + 104 * q^17 - 38 * q^19 + 21 * q^23 - 209 * q^25 + 137 * q^29 + 4 * q^31 - 129 * q^35 - 152 * q^37 + 210 * q^41 + 67 * q^43 - 273 * q^47 - 212 * q^49 - 209 * q^53 + 237 * q^55 - 799 * q^59 + 149 * q^61 + 52 * q^65 + 201 * q^67 - 792 * q^71 - 246 * q^73 - 843 * q^77 - 254 * q^79 - 374 * q^83 - 25 * q^85 + 564 * q^89 - 621 * q^91 - 95 * q^95 - 178 * 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
$$5$$ 6.27492 0.561246 0.280623 0.959818i $$-0.409459\pi$$
0.280623 + 0.959818i $$0.409459\pi$$
$$6$$ 0 0
$$7$$ −18.0997 −0.977290 −0.488645 0.872483i $$-0.662509\pi$$
−0.488645 + 0.872483i $$0.662509\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 33.9244 0.929873 0.464936 0.885344i $$-0.346077\pi$$
0.464936 + 0.885344i $$0.346077\pi$$
$$12$$ 0 0
$$13$$ −3.07558 −0.0656163 −0.0328082 0.999462i $$-0.510445\pi$$
−0.0328082 + 0.999462i $$0.510445\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 14.2508 0.203314 0.101657 0.994820i $$-0.467586\pi$$
0.101657 + 0.994820i $$0.467586\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −114.072 −1.03416 −0.517081 0.855937i $$-0.672981\pi$$
−0.517081 + 0.855937i $$0.672981\pi$$
$$24$$ 0 0
$$25$$ −85.6254 −0.685003
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 34.5257 0.221078 0.110539 0.993872i $$-0.464742\pi$$
0.110539 + 0.993872i $$0.464742\pi$$
$$30$$ 0 0
$$31$$ 107.698 0.623970 0.311985 0.950087i $$-0.399006\pi$$
0.311985 + 0.950087i $$0.399006\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −113.574 −0.548500
$$36$$ 0 0
$$37$$ −181.698 −0.807322 −0.403661 0.914909i $$-0.632262\pi$$
−0.403661 + 0.914909i $$0.632262\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 444.743 1.69408 0.847038 0.531532i $$-0.178384\pi$$
0.847038 + 0.531532i $$0.178384\pi$$
$$42$$ 0 0
$$43$$ 120.323 0.426723 0.213362 0.976973i $$-0.431559\pi$$
0.213362 + 0.976973i $$0.431559\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −306.371 −0.950826 −0.475413 0.879763i $$-0.657701\pi$$
−0.475413 + 0.879763i $$0.657701\pi$$
$$48$$ 0 0
$$49$$ −15.4020 −0.0449038
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −115.825 −0.300184 −0.150092 0.988672i $$-0.547957\pi$$
−0.150092 + 0.988672i $$0.547957\pi$$
$$54$$ 0 0
$$55$$ 212.873 0.521887
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −161.680 −0.356762 −0.178381 0.983961i $$-0.557086\pi$$
−0.178381 + 0.983961i $$0.557086\pi$$
$$60$$ 0 0
$$61$$ 274.571 0.576314 0.288157 0.957583i $$-0.406957\pi$$
0.288157 + 0.957583i $$0.406957\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −19.2990 −0.0368269
$$66$$ 0 0
$$67$$ 81.6254 0.148838 0.0744189 0.997227i $$-0.476290\pi$$
0.0744189 + 0.997227i $$0.476290\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −773.492 −1.29291 −0.646455 0.762952i $$-0.723749\pi$$
−0.646455 + 0.762952i $$0.723749\pi$$
$$72$$ 0 0
$$73$$ 148.794 0.238562 0.119281 0.992861i $$-0.461941\pi$$
0.119281 + 0.992861i $$0.461941\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −614.021 −0.908755
$$78$$ 0 0
$$79$$ −557.341 −0.793743 −0.396872 0.917874i $$-0.629904\pi$$
−0.396872 + 0.917874i $$0.629904\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −768.337 −1.01610 −0.508048 0.861329i $$-0.669633\pi$$
−0.508048 + 0.861329i $$0.669633\pi$$
$$84$$ 0 0
$$85$$ 89.4228 0.114109
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −457.884 −0.545344 −0.272672 0.962107i $$-0.587907\pi$$
−0.272672 + 0.962107i $$0.587907\pi$$
$$90$$ 0 0
$$91$$ 55.6670 0.0641262
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −119.223 −0.128759
$$96$$ 0 0
$$97$$ −1500.82 −1.57098 −0.785490 0.618874i $$-0.787589\pi$$
−0.785490 + 0.618874i $$0.787589\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1731.06 1.70541 0.852707 0.522389i $$-0.174959\pi$$
0.852707 + 0.522389i $$0.174959\pi$$
$$102$$ 0 0
$$103$$ −1887.77 −1.80590 −0.902951 0.429743i $$-0.858604\pi$$
−0.902951 + 0.429743i $$0.858604\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 411.468 0.371758 0.185879 0.982573i $$-0.440487\pi$$
0.185879 + 0.982573i $$0.440487\pi$$
$$108$$ 0 0
$$109$$ 1186.86 1.04294 0.521469 0.853270i $$-0.325384\pi$$
0.521469 + 0.853270i $$0.325384\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −509.478 −0.424139 −0.212069 0.977255i $$-0.568020\pi$$
−0.212069 + 0.977255i $$0.568020\pi$$
$$114$$ 0 0
$$115$$ −715.794 −0.580419
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −257.935 −0.198697
$$120$$ 0 0
$$121$$ −180.134 −0.135337
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1321.66 −0.945701
$$126$$ 0 0
$$127$$ −1169.71 −0.817284 −0.408642 0.912695i $$-0.633998\pi$$
−0.408642 + 0.912695i $$0.633998\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1052.31 0.701838 0.350919 0.936406i $$-0.385869\pi$$
0.350919 + 0.936406i $$0.385869\pi$$
$$132$$ 0 0
$$133$$ 343.894 0.224206
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2116.92 1.32015 0.660075 0.751200i $$-0.270524\pi$$
0.660075 + 0.751200i $$0.270524\pi$$
$$138$$ 0 0
$$139$$ −1115.90 −0.680929 −0.340464 0.940257i $$-0.610584\pi$$
−0.340464 + 0.940257i $$0.610584\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −104.337 −0.0610148
$$144$$ 0 0
$$145$$ 216.646 0.124079
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −472.693 −0.259896 −0.129948 0.991521i $$-0.541481\pi$$
−0.129948 + 0.991521i $$0.541481\pi$$
$$150$$ 0 0
$$151$$ −1635.40 −0.881372 −0.440686 0.897661i $$-0.645265\pi$$
−0.440686 + 0.897661i $$0.645265\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 675.794 0.350201
$$156$$ 0 0
$$157$$ −785.904 −0.399503 −0.199751 0.979847i $$-0.564013\pi$$
−0.199751 + 0.979847i $$0.564013\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 2064.67 1.01068
$$162$$ 0 0
$$163$$ 1224.95 0.588623 0.294311 0.955710i $$-0.404910\pi$$
0.294311 + 0.955710i $$0.404910\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1071.26 −0.496386 −0.248193 0.968711i $$-0.579837\pi$$
−0.248193 + 0.968711i $$0.579837\pi$$
$$168$$ 0 0
$$169$$ −2187.54 −0.995694
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −825.821 −0.362925 −0.181462 0.983398i $$-0.558083\pi$$
−0.181462 + 0.983398i $$0.558083\pi$$
$$174$$ 0 0
$$175$$ 1549.79 0.669447
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 465.802 0.194501 0.0972506 0.995260i $$-0.468995\pi$$
0.0972506 + 0.995260i $$0.468995\pi$$
$$180$$ 0 0
$$181$$ −1214.74 −0.498847 −0.249423 0.968395i $$-0.580241\pi$$
−0.249423 + 0.968395i $$0.580241\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1140.14 −0.453106
$$186$$ 0 0
$$187$$ 483.451 0.189056
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −154.943 −0.0586980 −0.0293490 0.999569i $$-0.509343\pi$$
−0.0293490 + 0.999569i $$0.509343\pi$$
$$192$$ 0 0
$$193$$ −2776.64 −1.03558 −0.517790 0.855508i $$-0.673245\pi$$
−0.517790 + 0.855508i $$0.673245\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4865.09 −1.75951 −0.879754 0.475429i $$-0.842293\pi$$
−0.879754 + 0.475429i $$0.842293\pi$$
$$198$$ 0 0
$$199$$ −3014.43 −1.07381 −0.536903 0.843644i $$-0.680406\pi$$
−0.536903 + 0.843644i $$0.680406\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −624.905 −0.216058
$$204$$ 0 0
$$205$$ 2790.72 0.950793
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −644.564 −0.213327
$$210$$ 0 0
$$211$$ 3434.46 1.12056 0.560280 0.828303i $$-0.310693\pi$$
0.560280 + 0.828303i $$0.310693\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 755.017 0.239497
$$216$$ 0 0
$$217$$ −1949.29 −0.609800
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −43.8296 −0.0133407
$$222$$ 0 0
$$223$$ −2247.73 −0.674975 −0.337488 0.941330i $$-0.609577\pi$$
−0.337488 + 0.941330i $$0.609577\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −1883.81 −0.550806 −0.275403 0.961329i $$-0.588811\pi$$
−0.275403 + 0.961329i $$0.588811\pi$$
$$228$$ 0 0
$$229$$ 3594.49 1.03725 0.518626 0.855001i $$-0.326444\pi$$
0.518626 + 0.855001i $$0.326444\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5530.01 1.55486 0.777431 0.628968i $$-0.216522\pi$$
0.777431 + 0.628968i $$0.216522\pi$$
$$234$$ 0 0
$$235$$ −1922.45 −0.533647
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2587.82 −0.700387 −0.350193 0.936677i $$-0.613884\pi$$
−0.350193 + 0.936677i $$0.613884\pi$$
$$240$$ 0 0
$$241$$ −3576.45 −0.955932 −0.477966 0.878378i $$-0.658626\pi$$
−0.477966 + 0.878378i $$0.658626\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −96.6462 −0.0252020
$$246$$ 0 0
$$247$$ 58.4360 0.0150534
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5966.87 −1.50050 −0.750251 0.661154i $$-0.770067\pi$$
−0.750251 + 0.661154i $$0.770067\pi$$
$$252$$ 0 0
$$253$$ −3869.84 −0.961638
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3841.01 −0.932278 −0.466139 0.884712i $$-0.654355\pi$$
−0.466139 + 0.884712i $$0.654355\pi$$
$$258$$ 0 0
$$259$$ 3288.67 0.788988
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3956.41 −0.927614 −0.463807 0.885936i $$-0.653517\pi$$
−0.463807 + 0.885936i $$0.653517\pi$$
$$264$$ 0 0
$$265$$ −726.791 −0.168477
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −2158.66 −0.489279 −0.244639 0.969614i $$-0.578670\pi$$
−0.244639 + 0.969614i $$0.578670\pi$$
$$270$$ 0 0
$$271$$ −2321.77 −0.520435 −0.260217 0.965550i $$-0.583794\pi$$
−0.260217 + 0.965550i $$0.583794\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2904.79 −0.636966
$$276$$ 0 0
$$277$$ −6184.24 −1.34143 −0.670713 0.741717i $$-0.734012\pi$$
−0.670713 + 0.741717i $$0.734012\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 7981.21 1.69437 0.847187 0.531294i $$-0.178294\pi$$
0.847187 + 0.531294i $$0.178294\pi$$
$$282$$ 0 0
$$283$$ −4851.71 −1.01910 −0.509548 0.860442i $$-0.670187\pi$$
−0.509548 + 0.860442i $$0.670187\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8049.69 −1.65560
$$288$$ 0 0
$$289$$ −4709.91 −0.958664
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1269.75 0.253172 0.126586 0.991956i $$-0.459598\pi$$
0.126586 + 0.991956i $$0.459598\pi$$
$$294$$ 0 0
$$295$$ −1014.53 −0.200231
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 350.838 0.0678579
$$300$$ 0 0
$$301$$ −2177.81 −0.417032
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1722.91 0.323454
$$306$$ 0 0
$$307$$ −9699.35 −1.80316 −0.901581 0.432610i $$-0.857593\pi$$
−0.901581 + 0.432610i $$0.857593\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 323.284 0.0589446 0.0294723 0.999566i $$-0.490617\pi$$
0.0294723 + 0.999566i $$0.490617\pi$$
$$312$$ 0 0
$$313$$ −3251.29 −0.587137 −0.293569 0.955938i $$-0.594843\pi$$
−0.293569 + 0.955938i $$0.594843\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 7512.25 1.33101 0.665505 0.746393i $$-0.268216\pi$$
0.665505 + 0.746393i $$0.268216\pi$$
$$318$$ 0 0
$$319$$ 1171.27 0.205575
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −270.766 −0.0466434
$$324$$ 0 0
$$325$$ 263.348 0.0449474
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 5545.22 0.929233
$$330$$ 0 0
$$331$$ 2857.14 0.474449 0.237225 0.971455i $$-0.423762\pi$$
0.237225 + 0.971455i $$0.423762\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 512.193 0.0835346
$$336$$ 0 0
$$337$$ 10635.9 1.71921 0.859606 0.510958i $$-0.170709\pi$$
0.859606 + 0.510958i $$0.170709\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3653.58 0.580213
$$342$$ 0 0
$$343$$ 6486.96 1.02117
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8357.08 1.29289 0.646443 0.762962i $$-0.276256\pi$$
0.646443 + 0.762962i $$0.276256\pi$$
$$348$$ 0 0
$$349$$ 7741.28 1.18734 0.593669 0.804709i $$-0.297679\pi$$
0.593669 + 0.804709i $$0.297679\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2480.46 −0.373999 −0.186999 0.982360i $$-0.559876\pi$$
−0.186999 + 0.982360i $$0.559876\pi$$
$$354$$ 0 0
$$355$$ −4853.60 −0.725640
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4310.17 −0.633655 −0.316827 0.948483i $$-0.602618\pi$$
−0.316827 + 0.948483i $$0.602618\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 933.670 0.133892
$$366$$ 0 0
$$367$$ −8484.71 −1.20681 −0.603404 0.797436i $$-0.706189\pi$$
−0.603404 + 0.797436i $$0.706189\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2096.39 0.293367
$$372$$ 0 0
$$373$$ 8039.16 1.11596 0.557978 0.829855i $$-0.311577\pi$$
0.557978 + 0.829855i $$0.311577\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −106.187 −0.0145063
$$378$$ 0 0
$$379$$ −2976.22 −0.403372 −0.201686 0.979450i $$-0.564642\pi$$
−0.201686 + 0.979450i $$0.564642\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5118.15 0.682834 0.341417 0.939912i $$-0.389093\pi$$
0.341417 + 0.939912i $$0.389093\pi$$
$$384$$ 0 0
$$385$$ −3852.93 −0.510035
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 6758.74 0.880929 0.440465 0.897770i $$-0.354814\pi$$
0.440465 + 0.897770i $$0.354814\pi$$
$$390$$ 0 0
$$391$$ −1625.62 −0.210259
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −3497.27 −0.445485
$$396$$ 0 0
$$397$$ 14529.0 1.83675 0.918373 0.395716i $$-0.129504\pi$$
0.918373 + 0.395716i $$0.129504\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12416.4 −1.54625 −0.773124 0.634255i $$-0.781307\pi$$
−0.773124 + 0.634255i $$0.781307\pi$$
$$402$$ 0 0
$$403$$ −331.233 −0.0409426
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6163.99 −0.750707
$$408$$ 0 0
$$409$$ 11615.8 1.40432 0.702159 0.712020i $$-0.252220\pi$$
0.702159 + 0.712020i $$0.252220\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 2926.36 0.348660
$$414$$ 0 0
$$415$$ −4821.25 −0.570279
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −16739.9 −1.95179 −0.975893 0.218248i $$-0.929966\pi$$
−0.975893 + 0.218248i $$0.929966\pi$$
$$420$$ 0 0
$$421$$ 6261.41 0.724852 0.362426 0.932013i $$-0.381949\pi$$
0.362426 + 0.932013i $$0.381949\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −1220.23 −0.139271
$$426$$ 0 0
$$427$$ −4969.64 −0.563226
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8893.01 0.993878 0.496939 0.867786i $$-0.334457\pi$$
0.496939 + 0.867786i $$0.334457\pi$$
$$432$$ 0 0
$$433$$ 6326.61 0.702166 0.351083 0.936344i $$-0.385814\pi$$
0.351083 + 0.936344i $$0.385814\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2167.37 0.237253
$$438$$ 0 0
$$439$$ 15823.4 1.72030 0.860149 0.510043i $$-0.170370\pi$$
0.860149 + 0.510043i $$0.170370\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 14056.3 1.50753 0.753763 0.657147i $$-0.228237\pi$$
0.753763 + 0.657147i $$0.228237\pi$$
$$444$$ 0 0
$$445$$ −2873.18 −0.306072
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −15412.9 −1.62000 −0.809999 0.586431i $$-0.800532\pi$$
−0.809999 + 0.586431i $$0.800532\pi$$
$$450$$ 0 0
$$451$$ 15087.6 1.57527
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 349.306 0.0359906
$$456$$ 0 0
$$457$$ 13336.9 1.36515 0.682573 0.730817i $$-0.260861\pi$$
0.682573 + 0.730817i $$0.260861\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10226.9 1.03322 0.516612 0.856220i $$-0.327193\pi$$
0.516612 + 0.856220i $$0.327193\pi$$
$$462$$ 0 0
$$463$$ −17797.4 −1.78642 −0.893212 0.449636i $$-0.851554\pi$$
−0.893212 + 0.449636i $$0.851554\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3100.03 0.307179 0.153589 0.988135i $$-0.450917\pi$$
0.153589 + 0.988135i $$0.450917\pi$$
$$468$$ 0 0
$$469$$ −1477.39 −0.145458
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4081.89 0.396798
$$474$$ 0 0
$$475$$ 1626.88 0.157151
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −6280.67 −0.599105 −0.299552 0.954080i $$-0.596837\pi$$
−0.299552 + 0.954080i $$0.596837\pi$$
$$480$$ 0 0
$$481$$ 558.826 0.0529735
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −9417.52 −0.881706
$$486$$ 0 0
$$487$$ 7940.07 0.738807 0.369403 0.929269i $$-0.379562\pi$$
0.369403 + 0.929269i $$0.379562\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −17276.3 −1.58792 −0.793959 0.607971i $$-0.791984\pi$$
−0.793959 + 0.607971i $$0.791984\pi$$
$$492$$ 0 0
$$493$$ 492.020 0.0449482
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 13999.9 1.26355
$$498$$ 0 0
$$499$$ 14939.6 1.34026 0.670128 0.742246i $$-0.266239\pi$$
0.670128 + 0.742246i $$0.266239\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 5559.39 0.492805 0.246402 0.969168i $$-0.420752\pi$$
0.246402 + 0.969168i $$0.420752\pi$$
$$504$$ 0 0
$$505$$ 10862.3 0.957157
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −11029.4 −0.960452 −0.480226 0.877145i $$-0.659445\pi$$
−0.480226 + 0.877145i $$0.659445\pi$$
$$510$$ 0 0
$$511$$ −2693.12 −0.233144
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −11845.6 −1.01355
$$516$$ 0 0
$$517$$ −10393.5 −0.884147
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7802.02 0.656070 0.328035 0.944665i $$-0.393614\pi$$
0.328035 + 0.944665i $$0.393614\pi$$
$$522$$ 0 0
$$523$$ −23625.2 −1.97525 −0.987627 0.156820i $$-0.949876\pi$$
−0.987627 + 0.156820i $$0.949876\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1534.78 0.126862
$$528$$ 0 0
$$529$$ 845.482 0.0694898
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1367.84 −0.111159
$$534$$ 0 0
$$535$$ 2581.93 0.208647
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −522.503 −0.0417548
$$540$$ 0 0
$$541$$ −24753.3 −1.96715 −0.983573 0.180509i $$-0.942226\pi$$
−0.983573 + 0.180509i $$0.942226\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 7447.43 0.585344
$$546$$ 0 0
$$547$$ 5655.38 0.442060 0.221030 0.975267i $$-0.429058\pi$$
0.221030 + 0.975267i $$0.429058\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −655.989 −0.0507188
$$552$$ 0 0
$$553$$ 10087.7 0.775717
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14716.3 1.11948 0.559741 0.828668i $$-0.310901\pi$$
0.559741 + 0.828668i $$0.310901\pi$$
$$558$$ 0 0
$$559$$ −370.063 −0.0280000
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 4400.91 0.329443 0.164721 0.986340i $$-0.447328\pi$$
0.164721 + 0.986340i $$0.447328\pi$$
$$564$$ 0 0
$$565$$ −3196.94 −0.238046
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10176.3 −0.749758 −0.374879 0.927074i $$-0.622316\pi$$
−0.374879 + 0.927074i $$0.622316\pi$$
$$570$$ 0 0
$$571$$ 6775.74 0.496595 0.248298 0.968684i $$-0.420129\pi$$
0.248298 + 0.968684i $$0.420129\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 9767.49 0.708404
$$576$$ 0 0
$$577$$ −3714.70 −0.268016 −0.134008 0.990980i $$-0.542785\pi$$
−0.134008 + 0.990980i $$0.542785\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13906.6 0.993021
$$582$$ 0 0
$$583$$ −3929.29 −0.279133
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9362.24 0.658298 0.329149 0.944278i $$-0.393238\pi$$
0.329149 + 0.944278i $$0.393238\pi$$
$$588$$ 0 0
$$589$$ −2046.26 −0.143149
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 14437.5 0.999793 0.499897 0.866085i $$-0.333371\pi$$
0.499897 + 0.866085i $$0.333371\pi$$
$$594$$ 0 0
$$595$$ −1618.52 −0.111518
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 19992.7 1.36374 0.681871 0.731473i $$-0.261167\pi$$
0.681871 + 0.731473i $$0.261167\pi$$
$$600$$ 0 0
$$601$$ 17371.4 1.17902 0.589511 0.807760i $$-0.299320\pi$$
0.589511 + 0.807760i $$0.299320\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1130.32 −0.0759574
$$606$$ 0 0
$$607$$ −14626.3 −0.978025 −0.489013 0.872277i $$-0.662643\pi$$
−0.489013 + 0.872277i $$0.662643\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 942.269 0.0623897
$$612$$ 0 0
$$613$$ −21052.2 −1.38710 −0.693549 0.720409i $$-0.743954\pi$$
−0.693549 + 0.720409i $$0.743954\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1605.72 −0.104771 −0.0523855 0.998627i $$-0.516682\pi$$
−0.0523855 + 0.998627i $$0.516682\pi$$
$$618$$ 0 0
$$619$$ 21457.6 1.39330 0.696652 0.717410i $$-0.254672\pi$$
0.696652 + 0.717410i $$0.254672\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 8287.54 0.532959
$$624$$ 0 0
$$625$$ 2409.89 0.154233
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −2589.34 −0.164140
$$630$$ 0 0
$$631$$ 9732.44 0.614013 0.307007 0.951707i $$-0.400673\pi$$
0.307007 + 0.951707i $$0.400673\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −7339.84 −0.458697
$$636$$ 0 0
$$637$$ 47.3700 0.00294642
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 25594.0 1.57707 0.788536 0.614989i $$-0.210839\pi$$
0.788536 + 0.614989i $$0.210839\pi$$
$$642$$ 0 0
$$643$$ −11938.5 −0.732207 −0.366104 0.930574i $$-0.619308\pi$$
−0.366104 + 0.930574i $$0.619308\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −15685.4 −0.953105 −0.476552 0.879146i $$-0.658114\pi$$
−0.476552 + 0.879146i $$0.658114\pi$$
$$648$$ 0 0
$$649$$ −5484.91 −0.331743
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11414.3 −0.684037 −0.342019 0.939693i $$-0.611111\pi$$
−0.342019 + 0.939693i $$0.611111\pi$$
$$654$$ 0 0
$$655$$ 6603.16 0.393903
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24500.2 −1.44824 −0.724122 0.689672i $$-0.757755\pi$$
−0.724122 + 0.689672i $$0.757755\pi$$
$$660$$ 0 0
$$661$$ 24821.0 1.46055 0.730276 0.683152i $$-0.239391\pi$$
0.730276 + 0.683152i $$0.239391\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2157.90 0.125835
$$666$$ 0 0
$$667$$ −3938.43 −0.228631
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 9314.65 0.535899
$$672$$ 0 0
$$673$$ −14862.9 −0.851297 −0.425649 0.904888i $$-0.639954\pi$$
−0.425649 + 0.904888i $$0.639954\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1218.79 0.0691902 0.0345951 0.999401i $$-0.488986\pi$$
0.0345951 + 0.999401i $$0.488986\pi$$
$$678$$ 0 0
$$679$$ 27164.3 1.53530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 10180.0 0.570319 0.285159 0.958480i $$-0.407953\pi$$
0.285159 + 0.958480i $$0.407953\pi$$
$$684$$ 0 0
$$685$$ 13283.5 0.740929
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 356.228 0.0196970
$$690$$ 0 0
$$691$$ 13712.9 0.754941 0.377470 0.926022i $$-0.376794\pi$$
0.377470 + 0.926022i $$0.376794\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −7002.16 −0.382168
$$696$$ 0 0
$$697$$ 6337.95 0.344429
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 29853.2 1.60848 0.804238 0.594307i $$-0.202574\pi$$
0.804238 + 0.594307i $$0.202574\pi$$
$$702$$ 0 0
$$703$$ 3452.26 0.185212
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −31331.6 −1.66669
$$708$$ 0 0
$$709$$ −11488.4 −0.608541 −0.304271 0.952586i $$-0.598413\pi$$
−0.304271 + 0.952586i $$0.598413\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −12285.3 −0.645286
$$714$$ 0 0
$$715$$ −654.708 −0.0342443
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 17536.5 0.909596 0.454798 0.890595i $$-0.349711\pi$$
0.454798 + 0.890595i $$0.349711\pi$$
$$720$$ 0 0
$$721$$ 34168.1 1.76489
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −2956.28 −0.151439
$$726$$ 0 0
$$727$$ 27607.0 1.40837 0.704185 0.710016i $$-0.251312\pi$$
0.704185 + 0.710016i $$0.251312\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1714.70 0.0867587
$$732$$ 0 0
$$733$$ −3519.63 −0.177354 −0.0886770 0.996060i $$-0.528264\pi$$
−0.0886770 + 0.996060i $$0.528264\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2769.09 0.138400
$$738$$ 0 0
$$739$$ 4557.41 0.226857 0.113428 0.993546i $$-0.463817\pi$$
0.113428 + 0.993546i $$0.463817\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5330.53 −0.263201 −0.131600 0.991303i $$-0.542012\pi$$
−0.131600 + 0.991303i $$0.542012\pi$$
$$744$$ 0 0
$$745$$ −2966.11 −0.145866
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −7447.43 −0.363315
$$750$$ 0 0
$$751$$ 22386.8 1.08776 0.543879 0.839163i $$-0.316955\pi$$
0.543879 + 0.839163i $$0.316955\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −10262.0 −0.494666
$$756$$ 0 0
$$757$$ 25315.6 1.21547 0.607736 0.794139i $$-0.292078\pi$$
0.607736 + 0.794139i $$0.292078\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18715.0 0.891485 0.445742 0.895161i $$-0.352940\pi$$
0.445742 + 0.895161i $$0.352940\pi$$
$$762$$ 0 0
$$763$$ −21481.7 −1.01925
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 497.260 0.0234094
$$768$$ 0 0
$$769$$ 20985.3 0.984070 0.492035 0.870575i $$-0.336253\pi$$
0.492035 + 0.870575i $$0.336253\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −6302.27 −0.293243 −0.146622 0.989193i $$-0.546840\pi$$
−0.146622 + 0.989193i $$0.546840\pi$$
$$774$$ 0 0
$$775$$ −9221.66 −0.427422
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8450.11 −0.388648
$$780$$ 0 0
$$781$$ −26240.3 −1.20224
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4931.48 −0.224219
$$786$$ 0 0
$$787$$ 4124.73 0.186825 0.0934123 0.995628i $$-0.470223\pi$$
0.0934123 + 0.995628i $$0.470223\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9221.39 0.414507
$$792$$ 0 0
$$793$$ −844.464 −0.0378156
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −8090.48 −0.359573 −0.179786 0.983706i $$-0.557541\pi$$
−0.179786 + 0.983706i $$0.557541\pi$$
$$798$$ 0 0
$$799$$ −4366.04 −0.193316
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 5047.75 0.221832
$$804$$ 0 0
$$805$$ 12955.6 0.567237
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 16735.9 0.727323 0.363662 0.931531i $$-0.381526\pi$$
0.363662 + 0.931531i $$0.381526\pi$$
$$810$$ 0 0
$$811$$ 8476.59 0.367020 0.183510 0.983018i $$-0.441254\pi$$
0.183510 + 0.983018i $$0.441254\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 7686.46 0.330362
$$816$$ 0 0
$$817$$ −2286.14 −0.0978970
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −902.399 −0.0383605 −0.0191802 0.999816i $$-0.506106\pi$$
−0.0191802 + 0.999816i $$0.506106\pi$$
$$822$$ 0 0
$$823$$ −23266.2 −0.985431 −0.492716 0.870190i $$-0.663996\pi$$
−0.492716 + 0.870190i $$0.663996\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 17103.5 0.719160 0.359580 0.933114i $$-0.382920\pi$$
0.359580 + 0.933114i $$0.382920\pi$$
$$828$$ 0 0
$$829$$ −6125.30 −0.256623 −0.128312 0.991734i $$-0.540956\pi$$
−0.128312 + 0.991734i $$0.540956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −219.491 −0.00912955
$$834$$ 0 0
$$835$$ −6722.05 −0.278594
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −19918.5 −0.819620 −0.409810 0.912171i $$-0.634405\pi$$
−0.409810 + 0.912171i $$0.634405\pi$$
$$840$$ 0 0
$$841$$ −23197.0 −0.951124
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −13726.6 −0.558829
$$846$$ 0 0
$$847$$ 3260.36 0.132264
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20726.7 0.834901
$$852$$ 0 0
$$853$$ 18785.3 0.754042 0.377021 0.926205i $$-0.376948\pi$$
0.377021 + 0.926205i $$0.376948\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 33483.4 1.33462 0.667311 0.744779i $$-0.267445\pi$$
0.667311 + 0.744779i $$0.267445\pi$$
$$858$$ 0 0
$$859$$ 7322.74 0.290860 0.145430 0.989369i $$-0.453543\pi$$
0.145430 + 0.989369i $$0.453543\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −905.563 −0.0357193 −0.0178596 0.999841i $$-0.505685\pi$$
−0.0178596 + 0.999841i $$0.505685\pi$$
$$864$$ 0 0
$$865$$ −5181.96 −0.203690
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −18907.5 −0.738080
$$870$$ 0 0
$$871$$ −251.045 −0.00976619
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 23921.6 0.924224
$$876$$ 0 0
$$877$$ −27155.7 −1.04559 −0.522795 0.852459i $$-0.675111\pi$$
−0.522795 + 0.852459i $$0.675111\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −26094.1 −0.997881 −0.498941 0.866636i $$-0.666277\pi$$
−0.498941 + 0.866636i $$0.666277\pi$$
$$882$$ 0 0
$$883$$ 29674.6 1.13095 0.565476 0.824765i $$-0.308693\pi$$
0.565476 + 0.824765i $$0.308693\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1734.72 0.0656665 0.0328333 0.999461i $$-0.489547\pi$$
0.0328333 + 0.999461i $$0.489547\pi$$
$$888$$ 0 0
$$889$$ 21171.4 0.798724
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 5821.05 0.218135
$$894$$ 0 0
$$895$$ 2922.87 0.109163
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 3718.34 0.137946
$$900$$ 0 0
$$901$$ −1650.60 −0.0610315
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −7622.42 −0.279975
$$906$$ 0 0
$$907$$ 6755.01 0.247295 0.123647 0.992326i $$-0.460541\pi$$
0.123647 + 0.992326i $$0.460541\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 35150.7 1.27837 0.639184 0.769054i $$-0.279272\pi$$
0.639184 + 0.769054i $$0.279272\pi$$
$$912$$ 0 0
$$913$$ −26065.4 −0.944840
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −19046.5 −0.685899
$$918$$ 0 0
$$919$$ −28684.2 −1.02960 −0.514802 0.857309i $$-0.672134\pi$$
−0.514802 + 0.857309i $$0.672134\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 2378.94 0.0848360
$$924$$ 0 0
$$925$$ 15557.9 0.553018
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 3762.58 0.132881 0.0664405 0.997790i $$-0.478836\pi$$
0.0664405 + 0.997790i $$0.478836\pi$$
$$930$$ 0 0
$$931$$ 292.638 0.0103016
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 3033.62 0.106107
$$936$$ 0 0
$$937$$ 25465.7 0.887863 0.443932 0.896061i $$-0.353583\pi$$
0.443932 + 0.896061i $$0.353583\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −40865.6 −1.41571 −0.707854 0.706359i $$-0.750337\pi$$
−0.707854 + 0.706359i $$0.750337\pi$$
$$942$$ 0 0
$$943$$ −50732.8 −1.75195
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8047.62 0.276148 0.138074 0.990422i $$-0.455909\pi$$
0.138074 + 0.990422i $$0.455909\pi$$
$$948$$ 0 0
$$949$$ −457.628 −0.0156536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 21562.2 0.732915 0.366458 0.930435i $$-0.380570\pi$$
0.366458 + 0.930435i $$0.380570\pi$$
$$954$$ 0 0
$$955$$ −972.257 −0.0329440
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −38315.5 −1.29017
$$960$$ 0 0
$$961$$ −18192.2 −0.610661
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −17423.2 −0.581215
$$966$$ 0 0
$$967$$ −14667.4 −0.487767 −0.243883 0.969805i $$-0.578421\pi$$
−0.243883 + 0.969805i $$0.578421\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 60154.7 1.98811 0.994056 0.108867i $$-0.0347221\pi$$
0.994056 + 0.108867i $$0.0347221\pi$$
$$972$$ 0 0
$$973$$ 20197.4 0.665465
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 29474.6 0.965176 0.482588 0.875848i $$-0.339697\pi$$
0.482588 + 0.875848i $$0.339697\pi$$
$$978$$ 0 0
$$979$$ −15533.4 −0.507100
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −6813.42 −0.221073 −0.110536 0.993872i $$-0.535257\pi$$
−0.110536 + 0.993872i $$0.535257\pi$$
$$984$$ 0 0
$$985$$ −30528.0 −0.987517
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −13725.5 −0.441301
$$990$$ 0 0
$$991$$ −11660.9 −0.373786 −0.186893 0.982380i $$-0.559842\pi$$
−0.186893 + 0.982380i $$0.559842\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −18915.3 −0.602669
$$996$$ 0 0
$$997$$ −37537.3 −1.19240 −0.596198 0.802838i $$-0.703323\pi$$
−0.596198 + 0.802838i $$0.703323\pi$$
$$998$$ 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 1368.4.a.a.1.2 2
3.2 odd 2 152.4.a.a.1.2 2
12.11 even 2 304.4.a.e.1.1 2
24.5 odd 2 1216.4.a.m.1.1 2
24.11 even 2 1216.4.a.k.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.a.1.2 2 3.2 odd 2
304.4.a.e.1.1 2 12.11 even 2
1216.4.a.k.1.2 2 24.11 even 2
1216.4.a.m.1.1 2 24.5 odd 2
1368.4.a.a.1.2 2 1.1 even 1 trivial