# Properties

 Label 6840.2.a.bg.1.1 Level $6840$ Weight $2$ Character 6840.1 Self dual yes Analytic conductor $54.618$ 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] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.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: $$54.6176749826$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.254102$$ of defining polynomial Character $$\chi$$ $$=$$ 6840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -3.18953 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -3.18953 q^{7} -0.681331 q^{13} +1.18953 q^{17} +1.00000 q^{19} +2.17313 q^{23} +1.00000 q^{25} -2.81047 q^{29} +6.37907 q^{31} +3.18953 q^{35} +7.87086 q^{37} -0.983593 q^{41} +1.36266 q^{43} -11.7417 q^{47} +3.17313 q^{49} +1.69774 q^{53} -11.5358 q^{59} +7.36266 q^{61} +0.681331 q^{65} +7.02759 q^{67} +12.7581 q^{71} -5.53579 q^{73} +5.36266 q^{79} -2.37907 q^{83} -1.18953 q^{85} -3.01641 q^{89} +2.17313 q^{91} -1.00000 q^{95} +4.88727 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - q^7 $$3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - q^7 + 5 * q^13 - 5 * q^17 + 3 * q^19 + q^23 + 3 * q^25 - 17 * q^29 + 2 * q^31 + q^35 + 8 * q^37 - 6 * q^41 - 10 * q^43 - 4 * q^47 + 4 * q^49 - 5 * q^53 - 15 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 + 4 * q^71 + 3 * q^73 + 2 * q^79 + 10 * q^83 + 5 * q^85 - 6 * q^89 + q^91 - 3 * q^95 - 4 * 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.18953 −1.20553 −0.602765 0.797919i $$-0.705934\pi$$
−0.602765 + 0.797919i $$0.705934\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −0.681331 −0.188967 −0.0944836 0.995526i $$-0.530120\pi$$
−0.0944836 + 0.995526i $$0.530120\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.18953 0.288504 0.144252 0.989541i $$-0.453922\pi$$
0.144252 + 0.989541i $$0.453922\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.17313 0.453128 0.226564 0.973996i $$-0.427251\pi$$
0.226564 + 0.973996i $$0.427251\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.81047 −0.521890 −0.260945 0.965354i $$-0.584034\pi$$
−0.260945 + 0.965354i $$0.584034\pi$$
$$30$$ 0 0
$$31$$ 6.37907 1.14571 0.572857 0.819655i $$-0.305835\pi$$
0.572857 + 0.819655i $$0.305835\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.18953 0.539130
$$36$$ 0 0
$$37$$ 7.87086 1.29396 0.646981 0.762506i $$-0.276031\pi$$
0.646981 + 0.762506i $$0.276031\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −0.983593 −0.153611 −0.0768057 0.997046i $$-0.524472\pi$$
−0.0768057 + 0.997046i $$0.524472\pi$$
$$42$$ 0 0
$$43$$ 1.36266 0.207804 0.103902 0.994588i $$-0.466867\pi$$
0.103902 + 0.994588i $$0.466867\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −11.7417 −1.71271 −0.856354 0.516390i $$-0.827276\pi$$
−0.856354 + 0.516390i $$0.827276\pi$$
$$48$$ 0 0
$$49$$ 3.17313 0.453304
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.69774 0.233202 0.116601 0.993179i $$-0.462800\pi$$
0.116601 + 0.993179i $$0.462800\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −11.5358 −1.50183 −0.750916 0.660398i $$-0.770388\pi$$
−0.750916 + 0.660398i $$0.770388\pi$$
$$60$$ 0 0
$$61$$ 7.36266 0.942692 0.471346 0.881948i $$-0.343768\pi$$
0.471346 + 0.881948i $$0.343768\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.681331 0.0845087
$$66$$ 0 0
$$67$$ 7.02759 0.858556 0.429278 0.903172i $$-0.358768\pi$$
0.429278 + 0.903172i $$0.358768\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.7581 1.51411 0.757056 0.653350i $$-0.226637\pi$$
0.757056 + 0.653350i $$0.226637\pi$$
$$72$$ 0 0
$$73$$ −5.53579 −0.647915 −0.323958 0.946072i $$-0.605013\pi$$
−0.323958 + 0.946072i $$0.605013\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.36266 0.603347 0.301673 0.953411i $$-0.402455\pi$$
0.301673 + 0.953411i $$0.402455\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −2.37907 −0.261137 −0.130568 0.991439i $$-0.541680\pi$$
−0.130568 + 0.991439i $$0.541680\pi$$
$$84$$ 0 0
$$85$$ −1.18953 −0.129023
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.01641 −0.319738 −0.159869 0.987138i $$-0.551107\pi$$
−0.159869 + 0.987138i $$0.551107\pi$$
$$90$$ 0 0
$$91$$ 2.17313 0.227806
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 4.88727 0.496227 0.248114 0.968731i $$-0.420189\pi$$
0.248114 + 0.968731i $$0.420189\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.1208 −1.20606 −0.603032 0.797717i $$-0.706041\pi$$
−0.603032 + 0.797717i $$0.706041\pi$$
$$102$$ 0 0
$$103$$ −17.8709 −1.76087 −0.880434 0.474168i $$-0.842749\pi$$
−0.880434 + 0.474168i $$0.842749\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.4231 −1.39433 −0.697165 0.716911i $$-0.745555\pi$$
−0.697165 + 0.716911i $$0.745555\pi$$
$$108$$ 0 0
$$109$$ 10.2059 0.977552 0.488776 0.872409i $$-0.337444\pi$$
0.488776 + 0.872409i $$0.337444\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.49180 0.140336 0.0701682 0.997535i $$-0.477646\pi$$
0.0701682 + 0.997535i $$0.477646\pi$$
$$114$$ 0 0
$$115$$ −2.17313 −0.202645
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −3.79406 −0.347801
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.24993 0.732063 0.366032 0.930602i $$-0.380716\pi$$
0.366032 + 0.930602i $$0.380716\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.75814 −0.765202 −0.382601 0.923914i $$-0.624972\pi$$
−0.382601 + 0.923914i $$0.624972\pi$$
$$132$$ 0 0
$$133$$ −3.18953 −0.276568
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 11.9149 1.01795 0.508977 0.860780i $$-0.330024\pi$$
0.508977 + 0.860780i $$0.330024\pi$$
$$138$$ 0 0
$$139$$ −23.4835 −1.99184 −0.995920 0.0902352i $$-0.971238\pi$$
−0.995920 + 0.0902352i $$0.971238\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 2.81047 0.233396
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.8461 1.54393 0.771967 0.635662i $$-0.219273\pi$$
0.771967 + 0.635662i $$0.219273\pi$$
$$150$$ 0 0
$$151$$ −16.0880 −1.30922 −0.654611 0.755966i $$-0.727167\pi$$
−0.654611 + 0.755966i $$0.727167\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.37907 −0.512379
$$156$$ 0 0
$$157$$ −13.7417 −1.09671 −0.548355 0.836246i $$-0.684746\pi$$
−0.548355 + 0.836246i $$0.684746\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.93126 −0.546260
$$162$$ 0 0
$$163$$ −14.6373 −1.14648 −0.573242 0.819386i $$-0.694315\pi$$
−0.573242 + 0.819386i $$0.694315\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −17.9588 −1.38970 −0.694849 0.719156i $$-0.744529\pi$$
−0.694849 + 0.719156i $$0.744529\pi$$
$$168$$ 0 0
$$169$$ −12.5358 −0.964291
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.85446 −0.521135 −0.260567 0.965456i $$-0.583910\pi$$
−0.260567 + 0.965456i $$0.583910\pi$$
$$174$$ 0 0
$$175$$ −3.18953 −0.241106
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −5.01641 −0.374944 −0.187472 0.982270i $$-0.560029\pi$$
−0.187472 + 0.982270i $$0.560029\pi$$
$$180$$ 0 0
$$181$$ 16.7253 1.24318 0.621592 0.783341i $$-0.286486\pi$$
0.621592 + 0.783341i $$0.286486\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −7.87086 −0.578677
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.2775 0.816013 0.408006 0.912979i $$-0.366224\pi$$
0.408006 + 0.912979i $$0.366224\pi$$
$$192$$ 0 0
$$193$$ 13.5798 0.977494 0.488747 0.872426i $$-0.337454\pi$$
0.488747 + 0.872426i $$0.337454\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 25.6566 1.81875 0.909374 0.415980i $$-0.136562\pi$$
0.909374 + 0.415980i $$0.136562\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 8.96408 0.629155
$$204$$ 0 0
$$205$$ 0.983593 0.0686971
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.2939 1.12172 0.560860 0.827911i $$-0.310471\pi$$
0.560860 + 0.827911i $$0.310471\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1.36266 −0.0929327
$$216$$ 0 0
$$217$$ −20.3463 −1.38119
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −0.810466 −0.0545178
$$222$$ 0 0
$$223$$ −24.2499 −1.62390 −0.811948 0.583730i $$-0.801593\pi$$
−0.811948 + 0.583730i $$0.801593\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5.31867 −0.353012 −0.176506 0.984300i $$-0.556480\pi$$
−0.176506 + 0.984300i $$0.556480\pi$$
$$228$$ 0 0
$$229$$ 11.7089 0.773747 0.386873 0.922133i $$-0.373555\pi$$
0.386873 + 0.922133i $$0.373555\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −27.5163 −1.80265 −0.901325 0.433142i $$-0.857405\pi$$
−0.901325 + 0.433142i $$0.857405\pi$$
$$234$$ 0 0
$$235$$ 11.7417 0.765946
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −14.9313 −0.965823 −0.482912 0.875669i $$-0.660421\pi$$
−0.482912 + 0.875669i $$0.660421\pi$$
$$240$$ 0 0
$$241$$ −9.32985 −0.600988 −0.300494 0.953784i $$-0.597152\pi$$
−0.300494 + 0.953784i $$0.597152\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.17313 −0.202724
$$246$$ 0 0
$$247$$ −0.681331 −0.0433520
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.27468 −0.0804569 −0.0402285 0.999191i $$-0.512809\pi$$
−0.0402285 + 0.999191i $$0.512809\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −31.2887 −1.95174 −0.975868 0.218363i $$-0.929928\pi$$
−0.975868 + 0.218363i $$0.929928\pi$$
$$258$$ 0 0
$$259$$ −25.1044 −1.55991
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −25.5163 −1.57340 −0.786700 0.617335i $$-0.788212\pi$$
−0.786700 + 0.617335i $$0.788212\pi$$
$$264$$ 0 0
$$265$$ −1.69774 −0.104291
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −26.7581 −1.63147 −0.815736 0.578424i $$-0.803668\pi$$
−0.815736 + 0.578424i $$0.803668\pi$$
$$270$$ 0 0
$$271$$ 14.1731 0.860956 0.430478 0.902601i $$-0.358345\pi$$
0.430478 + 0.902601i $$0.358345\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6.08798 0.365791 0.182896 0.983132i $$-0.441453\pi$$
0.182896 + 0.983132i $$0.441453\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.7089 0.937115 0.468558 0.883433i $$-0.344774\pi$$
0.468558 + 0.883433i $$0.344774\pi$$
$$282$$ 0 0
$$283$$ 0.346255 0.0205827 0.0102913 0.999947i $$-0.496724\pi$$
0.0102913 + 0.999947i $$0.496724\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3.13720 0.185183
$$288$$ 0 0
$$289$$ −15.5850 −0.916765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6.71414 −0.392244 −0.196122 0.980579i $$-0.562835\pi$$
−0.196122 + 0.980579i $$0.562835\pi$$
$$294$$ 0 0
$$295$$ 11.5358 0.671640
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.48062 −0.0856264
$$300$$ 0 0
$$301$$ −4.34625 −0.250514
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.36266 −0.421585
$$306$$ 0 0
$$307$$ −17.6126 −1.00520 −0.502602 0.864518i $$-0.667624\pi$$
−0.502602 + 0.864518i $$0.667624\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6.58501 0.373402 0.186701 0.982417i $$-0.440221\pi$$
0.186701 + 0.982417i $$0.440221\pi$$
$$312$$ 0 0
$$313$$ −18.6178 −1.05234 −0.526171 0.850379i $$-0.676373\pi$$
−0.526171 + 0.850379i $$0.676373\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.7857 0.998946 0.499473 0.866330i $$-0.333527\pi$$
0.499473 + 0.866330i $$0.333527\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1.18953 0.0661874
$$324$$ 0 0
$$325$$ −0.681331 −0.0377934
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 37.4506 2.06472
$$330$$ 0 0
$$331$$ −3.53579 −0.194345 −0.0971723 0.995268i $$-0.530980\pi$$
−0.0971723 + 0.995268i $$0.530980\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −7.02759 −0.383958
$$336$$ 0 0
$$337$$ −7.87086 −0.428753 −0.214377 0.976751i $$-0.568772\pi$$
−0.214377 + 0.976751i $$0.568772\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 12.2059 0.659059
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −15.6537 −0.840337 −0.420169 0.907446i $$-0.638029\pi$$
−0.420169 + 0.907446i $$0.638029\pi$$
$$348$$ 0 0
$$349$$ −21.4178 −1.14647 −0.573235 0.819391i $$-0.694312\pi$$
−0.573235 + 0.819391i $$0.694312\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3.15672 −0.168015 −0.0840076 0.996465i $$-0.526772\pi$$
−0.0840076 + 0.996465i $$0.526772\pi$$
$$354$$ 0 0
$$355$$ −12.7581 −0.677132
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −17.8269 −0.940866 −0.470433 0.882436i $$-0.655902\pi$$
−0.470433 + 0.882436i $$0.655902\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5.53579 0.289756
$$366$$ 0 0
$$367$$ 19.2252 1.00355 0.501773 0.864999i $$-0.332681\pi$$
0.501773 + 0.864999i $$0.332681\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5.41499 −0.281132
$$372$$ 0 0
$$373$$ 9.95601 0.515503 0.257751 0.966211i $$-0.417018\pi$$
0.257751 + 0.966211i $$0.417018\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.91486 0.0986201
$$378$$ 0 0
$$379$$ −19.6238 −1.00801 −0.504003 0.863702i $$-0.668140\pi$$
−0.504003 + 0.863702i $$0.668140\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −11.9037 −0.608250 −0.304125 0.952632i $$-0.598364\pi$$
−0.304125 + 0.952632i $$0.598364\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −10.3463 −0.524576 −0.262288 0.964990i $$-0.584477\pi$$
−0.262288 + 0.964990i $$0.584477\pi$$
$$390$$ 0 0
$$391$$ 2.58501 0.130730
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −5.36266 −0.269825
$$396$$ 0 0
$$397$$ 26.1536 1.31261 0.656306 0.754495i $$-0.272118\pi$$
0.656306 + 0.754495i $$0.272118\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 32.7253 1.63422 0.817112 0.576479i $$-0.195574\pi$$
0.817112 + 0.576479i $$0.195574\pi$$
$$402$$ 0 0
$$403$$ −4.34625 −0.216502
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 4.03281 0.199410 0.0997049 0.995017i $$-0.468210\pi$$
0.0997049 + 0.995017i $$0.468210\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 36.7938 1.81050
$$414$$ 0 0
$$415$$ 2.37907 0.116784
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 15.8297 0.773332 0.386666 0.922220i $$-0.373627\pi$$
0.386666 + 0.922220i $$0.373627\pi$$
$$420$$ 0 0
$$421$$ 6.05233 0.294973 0.147486 0.989064i $$-0.452882\pi$$
0.147486 + 0.989064i $$0.452882\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.18953 0.0577009
$$426$$ 0 0
$$427$$ −23.4835 −1.13644
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −25.7089 −1.23835 −0.619177 0.785251i $$-0.712534\pi$$
−0.619177 + 0.785251i $$0.712534\pi$$
$$432$$ 0 0
$$433$$ 2.57383 0.123690 0.0618452 0.998086i $$-0.480301\pi$$
0.0618452 + 0.998086i $$0.480301\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.17313 0.103955
$$438$$ 0 0
$$439$$ 26.8133 1.27973 0.639865 0.768488i $$-0.278990\pi$$
0.639865 + 0.768488i $$0.278990\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −10.7909 −0.512693 −0.256347 0.966585i $$-0.582519\pi$$
−0.256347 + 0.966585i $$0.582519\pi$$
$$444$$ 0 0
$$445$$ 3.01641 0.142991
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 29.4835 1.39141 0.695705 0.718327i $$-0.255092\pi$$
0.695705 + 0.718327i $$0.255092\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −2.17313 −0.101878
$$456$$ 0 0
$$457$$ −11.8269 −0.553238 −0.276619 0.960980i $$-0.589214\pi$$
−0.276619 + 0.960980i $$0.589214\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 7.45065 0.347011 0.173506 0.984833i $$-0.444491\pi$$
0.173506 + 0.984833i $$0.444491\pi$$
$$462$$ 0 0
$$463$$ −37.4506 −1.74048 −0.870240 0.492629i $$-0.836036\pi$$
−0.870240 + 0.492629i $$0.836036\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.4835 1.27178 0.635891 0.771779i $$-0.280633\pi$$
0.635891 + 0.771779i $$0.280633\pi$$
$$468$$ 0 0
$$469$$ −22.4147 −1.03502
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −10.6597 −0.487054 −0.243527 0.969894i $$-0.578304\pi$$
−0.243527 + 0.969894i $$0.578304\pi$$
$$480$$ 0 0
$$481$$ −5.36266 −0.244516
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.88727 −0.221920
$$486$$ 0 0
$$487$$ 7.92604 0.359163 0.179581 0.983743i $$-0.442526\pi$$
0.179581 + 0.983743i $$0.442526\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.68656 0.0761133 0.0380567 0.999276i $$-0.487883\pi$$
0.0380567 + 0.999276i $$0.487883\pi$$
$$492$$ 0 0
$$493$$ −3.34314 −0.150568
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −40.6925 −1.82531
$$498$$ 0 0
$$499$$ −20.3463 −0.910823 −0.455412 0.890281i $$-0.650508\pi$$
−0.455412 + 0.890281i $$0.650508\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1.06874 −0.0476526 −0.0238263 0.999716i $$-0.507585\pi$$
−0.0238263 + 0.999716i $$0.507585\pi$$
$$504$$ 0 0
$$505$$ 12.1208 0.539368
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 19.0164 0.842887 0.421444 0.906855i $$-0.361524\pi$$
0.421444 + 0.906855i $$0.361524\pi$$
$$510$$ 0 0
$$511$$ 17.6566 0.781081
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 17.8709 0.787484
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −9.07158 −0.397433 −0.198717 0.980057i $$-0.563677\pi$$
−0.198717 + 0.980057i $$0.563677\pi$$
$$522$$ 0 0
$$523$$ −13.4946 −0.590079 −0.295040 0.955485i $$-0.595333\pi$$
−0.295040 + 0.955485i $$0.595333\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.58812 0.330544
$$528$$ 0 0
$$529$$ −18.2775 −0.794675
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0.670152 0.0290275
$$534$$ 0 0
$$535$$ 14.4231 0.623563
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 4.63734 0.199375 0.0996874 0.995019i $$-0.468216\pi$$
0.0996874 + 0.995019i $$0.468216\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −10.2059 −0.437174
$$546$$ 0 0
$$547$$ −18.1291 −0.775146 −0.387573 0.921839i $$-0.626686\pi$$
−0.387573 + 0.921839i $$0.626686\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.81047 −0.119730
$$552$$ 0 0
$$553$$ −17.1044 −0.727353
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −32.2968 −1.36846 −0.684229 0.729267i $$-0.739861\pi$$
−0.684229 + 0.729267i $$0.739861\pi$$
$$558$$ 0 0
$$559$$ −0.928423 −0.0392681
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 31.3871 1.32281 0.661405 0.750029i $$-0.269960\pi$$
0.661405 + 0.750029i $$0.269960\pi$$
$$564$$ 0 0
$$565$$ −1.49180 −0.0627604
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 31.7969 1.33300 0.666498 0.745507i $$-0.267793\pi$$
0.666498 + 0.745507i $$0.267793\pi$$
$$570$$ 0 0
$$571$$ 18.3134 0.766394 0.383197 0.923667i $$-0.374823\pi$$
0.383197 + 0.923667i $$0.374823\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.17313 0.0906257
$$576$$ 0 0
$$577$$ 26.7282 1.11271 0.556354 0.830945i $$-0.312200\pi$$
0.556354 + 0.830945i $$0.312200\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7.58812 0.314808
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −24.3463 −1.00488 −0.502439 0.864613i $$-0.667564\pi$$
−0.502439 + 0.864613i $$0.667564\pi$$
$$588$$ 0 0
$$589$$ 6.37907 0.262845
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −12.7253 −0.522566 −0.261283 0.965262i $$-0.584146\pi$$
−0.261283 + 0.965262i $$0.584146\pi$$
$$594$$ 0 0
$$595$$ 3.79406 0.155541
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −10.6373 −0.434630 −0.217315 0.976102i $$-0.569730\pi$$
−0.217315 + 0.976102i $$0.569730\pi$$
$$600$$ 0 0
$$601$$ 11.5329 0.470439 0.235219 0.971942i $$-0.424419\pi$$
0.235219 + 0.971942i $$0.424419\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −18.2827 −0.742074 −0.371037 0.928618i $$-0.620998\pi$$
−0.371037 + 0.928618i $$0.620998\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ −6.84612 −0.276512 −0.138256 0.990397i $$-0.544150\pi$$
−0.138256 + 0.990397i $$0.544150\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −15.0820 −0.607180 −0.303590 0.952803i $$-0.598185\pi$$
−0.303590 + 0.952803i $$0.598185\pi$$
$$618$$ 0 0
$$619$$ −42.7253 −1.71728 −0.858638 0.512583i $$-0.828689\pi$$
−0.858638 + 0.512583i $$0.828689\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 9.62093 0.385454
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 9.36266 0.373314
$$630$$ 0 0
$$631$$ 26.7909 1.06653 0.533265 0.845948i $$-0.320965\pi$$
0.533265 + 0.845948i $$0.320965\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −8.24993 −0.327389
$$636$$ 0 0
$$637$$ −2.16195 −0.0856595
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −28.1208 −1.11070 −0.555352 0.831615i $$-0.687417\pi$$
−0.555352 + 0.831615i $$0.687417\pi$$
$$642$$ 0 0
$$643$$ 15.1372 0.596953 0.298477 0.954417i $$-0.403522\pi$$
0.298477 + 0.954417i $$0.403522\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −1.06874 −0.0420164 −0.0210082 0.999779i $$-0.506688\pi$$
−0.0210082 + 0.999779i $$0.506688\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 21.3298 0.834701 0.417351 0.908745i $$-0.362959\pi$$
0.417351 + 0.908745i $$0.362959\pi$$
$$654$$ 0 0
$$655$$ 8.75814 0.342209
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −45.4639 −1.77102 −0.885512 0.464617i $$-0.846192\pi$$
−0.885512 + 0.464617i $$0.846192\pi$$
$$660$$ 0 0
$$661$$ 28.6074 1.11270 0.556349 0.830949i $$-0.312202\pi$$
0.556349 + 0.830949i $$0.312202\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 3.18953 0.123685
$$666$$ 0 0
$$667$$ −6.10750 −0.236483
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 47.1840 1.81881 0.909405 0.415911i $$-0.136537\pi$$
0.909405 + 0.415911i $$0.136537\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −20.8573 −0.801611 −0.400806 0.916163i $$-0.631270\pi$$
−0.400806 + 0.916163i $$0.631270\pi$$
$$678$$ 0 0
$$679$$ −15.5881 −0.598217
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 17.2887 0.661534 0.330767 0.943713i $$-0.392693\pi$$
0.330767 + 0.943713i $$0.392693\pi$$
$$684$$ 0 0
$$685$$ −11.9149 −0.455243
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1.15672 −0.0440675
$$690$$ 0 0
$$691$$ −35.8297 −1.36303 −0.681513 0.731806i $$-0.738678\pi$$
−0.681513 + 0.731806i $$0.738678\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 23.4835 0.890778
$$696$$ 0 0
$$697$$ −1.17002 −0.0443176
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 38.0880 1.43856 0.719282 0.694719i $$-0.244471\pi$$
0.719282 + 0.694719i $$0.244471\pi$$
$$702$$ 0 0
$$703$$ 7.87086 0.296855
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 38.6597 1.45395
$$708$$ 0 0
$$709$$ −15.2747 −0.573653 −0.286826 0.957983i $$-0.592600\pi$$
−0.286826 + 0.957983i $$0.592600\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 13.8625 0.519156
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 27.8597 1.03899 0.519495 0.854473i $$-0.326120\pi$$
0.519495 + 0.854473i $$0.326120\pi$$
$$720$$ 0 0
$$721$$ 56.9997 2.12278
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −2.81047 −0.104378
$$726$$ 0 0
$$727$$ −39.4311 −1.46242 −0.731210 0.682153i $$-0.761044\pi$$
−0.731210 + 0.682153i $$0.761044\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.62093 0.0599523
$$732$$ 0 0
$$733$$ 33.1372 1.22395 0.611975 0.790877i $$-0.290375\pi$$
0.611975 + 0.790877i $$0.290375\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −25.8625 −0.951368 −0.475684 0.879616i $$-0.657799\pi$$
−0.475684 + 0.879616i $$0.657799\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 28.6842 1.05232 0.526160 0.850386i $$-0.323631\pi$$
0.526160 + 0.850386i $$0.323631\pi$$
$$744$$ 0 0
$$745$$ −18.8461 −0.690468
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 46.0028 1.68091
$$750$$ 0 0
$$751$$ −34.2968 −1.25151 −0.625753 0.780021i $$-0.715208\pi$$
−0.625753 + 0.780021i $$0.715208\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 16.0880 0.585502
$$756$$ 0 0
$$757$$ 20.6597 0.750889 0.375445 0.926845i $$-0.377490\pi$$
0.375445 + 0.926845i $$0.377490\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −38.1236 −1.38198 −0.690990 0.722864i $$-0.742825\pi$$
−0.690990 + 0.722864i $$0.742825\pi$$
$$762$$ 0 0
$$763$$ −32.5522 −1.17847
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7.85969 0.283797
$$768$$ 0 0
$$769$$ 20.3267 0.733001 0.366500 0.930418i $$-0.380556\pi$$
0.366500 + 0.930418i $$0.380556\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −26.4559 −0.951552 −0.475776 0.879567i $$-0.657833\pi$$
−0.475776 + 0.879567i $$0.657833\pi$$
$$774$$ 0 0
$$775$$ 6.37907 0.229143
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −0.983593 −0.0352409
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 13.7417 0.490463
$$786$$ 0 0
$$787$$ −27.3738 −0.975772 −0.487886 0.872907i $$-0.662232\pi$$
−0.487886 + 0.872907i $$0.662232\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.75814 −0.169180
$$792$$ 0 0
$$793$$ −5.01641 −0.178138
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −36.4454 −1.29096 −0.645481 0.763776i $$-0.723343\pi$$
−0.645481 + 0.763776i $$0.723343\pi$$
$$798$$ 0 0
$$799$$ −13.9672 −0.494124
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 6.93126 0.244295
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −52.2444 −1.83682 −0.918408 0.395634i $$-0.870525\pi$$
−0.918408 + 0.395634i $$0.870525\pi$$
$$810$$ 0 0
$$811$$ 32.8984 1.15522 0.577610 0.816313i $$-0.303985\pi$$
0.577610 + 0.816313i $$0.303985\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 14.6373 0.512724
$$816$$ 0 0
$$817$$ 1.36266 0.0476735
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 4.44470 0.155121 0.0775605 0.996988i $$-0.475287\pi$$
0.0775605 + 0.996988i $$0.475287\pi$$
$$822$$ 0 0
$$823$$ −27.7774 −0.968259 −0.484129 0.874996i $$-0.660864\pi$$
−0.484129 + 0.874996i $$0.660864\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −46.5110 −1.61735 −0.808674 0.588257i $$-0.799814\pi$$
−0.808674 + 0.588257i $$0.799814\pi$$
$$828$$ 0 0
$$829$$ 28.2388 0.980772 0.490386 0.871505i $$-0.336856\pi$$
0.490386 + 0.871505i $$0.336856\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 3.77454 0.130780
$$834$$ 0 0
$$835$$ 17.9588 0.621492
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −9.01641 −0.311281 −0.155640 0.987814i $$-0.549744\pi$$
−0.155640 + 0.987814i $$0.549744\pi$$
$$840$$ 0 0
$$841$$ −21.1013 −0.727630
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 12.5358 0.431244
$$846$$ 0 0
$$847$$ 35.0849 1.20553
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 17.1044 0.586331
$$852$$ 0 0
$$853$$ 5.30749 0.181725 0.0908625 0.995863i $$-0.471038\pi$$
0.0908625 + 0.995863i $$0.471038\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 38.9258 1.32968 0.664839 0.746986i $$-0.268500\pi$$
0.664839 + 0.746986i $$0.268500\pi$$
$$858$$ 0 0
$$859$$ 20.1760 0.688395 0.344198 0.938897i $$-0.388151\pi$$
0.344198 + 0.938897i $$0.388151\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 30.3051 1.03160 0.515799 0.856710i $$-0.327495\pi$$
0.515799 + 0.856710i $$0.327495\pi$$
$$864$$ 0 0
$$865$$ 6.85446 0.233059
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −4.78811 −0.162239
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.18953 0.107826
$$876$$ 0 0
$$877$$ 9.43947 0.318748 0.159374 0.987218i $$-0.449052\pi$$
0.159374 + 0.987218i $$0.449052\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 24.4671 0.824316 0.412158 0.911112i $$-0.364775\pi$$
0.412158 + 0.911112i $$0.364775\pi$$
$$882$$ 0 0
$$883$$ 6.12080 0.205981 0.102991 0.994682i $$-0.467159\pi$$
0.102991 + 0.994682i $$0.467159\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −45.4200 −1.52505 −0.762526 0.646957i $$-0.776041\pi$$
−0.762526 + 0.646957i $$0.776041\pi$$
$$888$$ 0 0
$$889$$ −26.3134 −0.882524
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −11.7417 −0.392922
$$894$$ 0 0
$$895$$ 5.01641 0.167680
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −17.9282 −0.597937
$$900$$ 0 0
$$901$$ 2.01952 0.0672798
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16.7253 −0.555969
$$906$$ 0 0
$$907$$ −10.2694 −0.340991 −0.170496 0.985358i $$-0.554537\pi$$
−0.170496 + 0.985358i $$0.554537\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 10.9180 0.361728 0.180864 0.983508i $$-0.442111\pi$$
0.180864 + 0.983508i $$0.442111\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 27.9344 0.922474
$$918$$ 0 0
$$919$$ −38.9313 −1.28422 −0.642112 0.766611i $$-0.721942\pi$$
−0.642112 + 0.766611i $$0.721942\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −8.69251 −0.286117
$$924$$ 0 0
$$925$$ 7.87086 0.258792
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −21.9700 −0.720813 −0.360407 0.932795i $$-0.617362\pi$$
−0.360407 + 0.932795i $$0.617362\pi$$
$$930$$ 0 0
$$931$$ 3.17313 0.103995
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 10.6402 0.347600 0.173800 0.984781i $$-0.444395\pi$$
0.173800 + 0.984781i $$0.444395\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 20.8656 0.680200 0.340100 0.940389i $$-0.389539\pi$$
0.340100 + 0.940389i $$0.389539\pi$$
$$942$$ 0 0
$$943$$ −2.13747 −0.0696057
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.6925 −0.412451 −0.206226 0.978504i $$-0.566118\pi$$
−0.206226 + 0.978504i $$0.566118\pi$$
$$948$$ 0 0
$$949$$ 3.77170 0.122435
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −16.7337 −0.542056 −0.271028 0.962571i $$-0.587364\pi$$
−0.271028 + 0.962571i $$0.587364\pi$$
$$954$$ 0 0
$$955$$ −11.2775 −0.364932
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −38.0028 −1.22718
$$960$$ 0 0
$$961$$ 9.69251 0.312662
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −13.5798 −0.437149
$$966$$ 0 0
$$967$$ 4.62688 0.148790 0.0743952 0.997229i $$-0.476297\pi$$
0.0743952 + 0.997229i $$0.476297\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 49.4506 1.58695 0.793473 0.608605i $$-0.208271\pi$$
0.793473 + 0.608605i $$0.208271\pi$$
$$972$$ 0 0
$$973$$ 74.9013 2.40123
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −22.5962 −0.722916 −0.361458 0.932388i $$-0.617721\pi$$
−0.361458 + 0.932388i $$0.617721\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −32.9424 −1.05070 −0.525350 0.850886i $$-0.676066\pi$$
−0.525350 + 0.850886i $$0.676066\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 2.96124 0.0941618
$$990$$ 0 0
$$991$$ −49.7802 −1.58132 −0.790660 0.612255i $$-0.790263\pi$$
−0.790660 + 0.612255i $$0.790263\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −25.6566 −0.813368
$$996$$ 0 0
$$997$$ −36.4014 −1.15284 −0.576422 0.817152i $$-0.695552\pi$$
−0.576422 + 0.817152i $$0.695552\pi$$
$$998$$ 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 6840.2.a.bg.1.1 3
3.2 odd 2 760.2.a.j.1.1 3
12.11 even 2 1520.2.a.s.1.3 3
15.2 even 4 3800.2.d.l.3649.6 6
15.8 even 4 3800.2.d.l.3649.1 6
15.14 odd 2 3800.2.a.x.1.3 3
24.5 odd 2 6080.2.a.bv.1.3 3
24.11 even 2 6080.2.a.bq.1.1 3
60.59 even 2 7600.2.a.bq.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.1 3 3.2 odd 2
1520.2.a.s.1.3 3 12.11 even 2
3800.2.a.x.1.3 3 15.14 odd 2
3800.2.d.l.3649.1 6 15.8 even 4
3800.2.d.l.3649.6 6 15.2 even 4
6080.2.a.bq.1.1 3 24.11 even 2
6080.2.a.bv.1.3 3 24.5 odd 2
6840.2.a.bg.1.1 3 1.1 even 1 trivial
7600.2.a.bq.1.1 3 60.59 even 2