# Properties

 Label 7600.2.a.bq.1.1 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ 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] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ 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: $$\Z[a_1, a_2, a_3]$$ 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 $$\chi$$ $$=$$ 7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.68133 q^{3} -3.18953 q^{7} +4.18953 q^{9} +O(q^{10})$$ $$q-2.68133 q^{3} -3.18953 q^{7} +4.18953 q^{9} +0.681331 q^{13} +1.18953 q^{17} -1.00000 q^{19} +8.55220 q^{21} -2.17313 q^{23} -3.18953 q^{27} +2.81047 q^{29} -6.37907 q^{31} -7.87086 q^{37} -1.82687 q^{39} +0.983593 q^{41} +1.36266 q^{43} +11.7417 q^{47} +3.17313 q^{49} -3.18953 q^{51} +1.69774 q^{53} +2.68133 q^{57} -11.5358 q^{59} +7.36266 q^{61} -13.3627 q^{63} +7.02759 q^{67} +5.82687 q^{69} +12.7581 q^{71} +5.53579 q^{73} -5.36266 q^{79} -4.01641 q^{81} +2.37907 q^{83} -7.53579 q^{87} +3.01641 q^{89} -2.17313 q^{91} +17.1044 q^{93} -4.88727 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - q^{7} + 4 q^{9}+O(q^{10})$$ 3 * q - q^3 - q^7 + 4 * q^9 $$3 q - q^{3} - q^{7} + 4 q^{9} - 5 q^{13} - 5 q^{17} - 3 q^{19} + 3 q^{21} - q^{23} - q^{27} + 17 q^{29} - 2 q^{31} - 8 q^{37} - 11 q^{39} + 6 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{49} - q^{51} - 5 q^{53} + q^{57} - 15 q^{59} + 8 q^{61} - 26 q^{63} + 3 q^{67} + 23 q^{69} + 4 q^{71} - 3 q^{73} - 2 q^{79} - 9 q^{81} - 10 q^{83} - 3 q^{87} + 6 q^{89} - q^{91} + 6 q^{93} + 4 q^{97}+O(q^{100})$$ 3 * q - q^3 - q^7 + 4 * q^9 - 5 * q^13 - 5 * q^17 - 3 * q^19 + 3 * q^21 - q^23 - q^27 + 17 * q^29 - 2 * q^31 - 8 * q^37 - 11 * q^39 + 6 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^49 - q^51 - 5 * q^53 + q^57 - 15 * q^59 + 8 * q^61 - 26 * q^63 + 3 * q^67 + 23 * q^69 + 4 * q^71 - 3 * q^73 - 2 * q^79 - 9 * q^81 - 10 * q^83 - 3 * q^87 + 6 * q^89 - q^91 + 6 * q^93 + 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
$$3$$ −2.68133 −1.54807 −0.774033 0.633145i $$-0.781764\pi$$
−0.774033 + 0.633145i $$0.781764\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.18953 −1.20553 −0.602765 0.797919i $$-0.705934\pi$$
−0.602765 + 0.797919i $$0.705934\pi$$
$$8$$ 0 0
$$9$$ 4.18953 1.39651
$$10$$ 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 $$-0.469880\pi$$
0.0944836 + 0.995526i $$0.469880\pi$$
$$14$$ 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
$$21$$ 8.55220 1.86624
$$22$$ 0 0
$$23$$ −2.17313 −0.453128 −0.226564 0.973996i $$-0.572749\pi$$
−0.226564 + 0.973996i $$0.572749\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −3.18953 −0.613826
$$28$$ 0 0
$$29$$ 2.81047 0.521890 0.260945 0.965354i $$-0.415966\pi$$
0.260945 + 0.965354i $$0.415966\pi$$
$$30$$ 0 0
$$31$$ −6.37907 −1.14571 −0.572857 0.819655i $$-0.694165\pi$$
−0.572857 + 0.819655i $$0.694165\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.87086 −1.29396 −0.646981 0.762506i $$-0.723969\pi$$
−0.646981 + 0.762506i $$0.723969\pi$$
$$38$$ 0 0
$$39$$ −1.82687 −0.292534
$$40$$ 0 0
$$41$$ 0.983593 0.153611 0.0768057 0.997046i $$-0.475528\pi$$
0.0768057 + 0.997046i $$0.475528\pi$$
$$42$$ 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 $$-0.172724\pi$$
0.856354 + 0.516390i $$0.172724\pi$$
$$48$$ 0 0
$$49$$ 3.17313 0.453304
$$50$$ 0 0
$$51$$ −3.18953 −0.446624
$$52$$ 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
$$57$$ 2.68133 0.355151
$$58$$ 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
$$63$$ −13.3627 −1.68354
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.02759 0.858556 0.429278 0.903172i $$-0.358768\pi$$
0.429278 + 0.903172i $$0.358768\pi$$
$$68$$ 0 0
$$69$$ 5.82687 0.701473
$$70$$ 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 $$-0.394987\pi$$
0.323958 + 0.946072i $$0.394987\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.36266 −0.603347 −0.301673 0.953411i $$-0.597545\pi$$
−0.301673 + 0.953411i $$0.597545\pi$$
$$80$$ 0 0
$$81$$ −4.01641 −0.446267
$$82$$ 0 0
$$83$$ 2.37907 0.261137 0.130568 0.991439i $$-0.458320\pi$$
0.130568 + 0.991439i $$0.458320\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −7.53579 −0.807921
$$88$$ 0 0
$$89$$ 3.01641 0.319738 0.159869 0.987138i $$-0.448893\pi$$
0.159869 + 0.987138i $$0.448893\pi$$
$$90$$ 0 0
$$91$$ −2.17313 −0.227806
$$92$$ 0 0
$$93$$ 17.1044 1.77364
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −4.88727 −0.496227 −0.248114 0.968731i $$-0.579811\pi$$
−0.248114 + 0.968731i $$0.579811\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.1208 1.20606 0.603032 0.797717i $$-0.293959\pi$$
0.603032 + 0.797717i $$0.293959\pi$$
$$102$$ 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 $$-0.254445\pi$$
0.697165 + 0.716911i $$0.254445\pi$$
$$108$$ 0 0
$$109$$ 10.2059 0.977552 0.488776 0.872409i $$-0.337444\pi$$
0.488776 + 0.872409i $$0.337444\pi$$
$$110$$ 0 0
$$111$$ 21.1044 2.00314
$$112$$ 0 0
$$113$$ 1.49180 0.140336 0.0701682 0.997535i $$-0.477646\pi$$
0.0701682 + 0.997535i $$0.477646\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.85446 0.263895
$$118$$ 0 0
$$119$$ −3.79406 −0.347801
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −2.63734 −0.237801
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.24993 0.732063 0.366032 0.930602i $$-0.380716\pi$$
0.366032 + 0.930602i $$0.380716\pi$$
$$128$$ 0 0
$$129$$ −3.65375 −0.321694
$$130$$ 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 $$-0.0287619\pi$$
0.995920 + 0.0902352i $$0.0287619\pi$$
$$140$$ 0 0
$$141$$ −31.4835 −2.65139
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −8.50820 −0.701745
$$148$$ 0 0
$$149$$ −18.8461 −1.54393 −0.771967 0.635662i $$-0.780727\pi$$
−0.771967 + 0.635662i $$0.780727\pi$$
$$150$$ 0 0
$$151$$ 16.0880 1.30922 0.654611 0.755966i $$-0.272833\pi$$
0.654611 + 0.755966i $$0.272833\pi$$
$$152$$ 0 0
$$153$$ 4.98359 0.402900
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.7417 1.09671 0.548355 0.836246i $$-0.315254\pi$$
0.548355 + 0.836246i $$0.315254\pi$$
$$158$$ 0 0
$$159$$ −4.55220 −0.361013
$$160$$ 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 $$-0.255471\pi$$
0.694849 + 0.719156i $$0.255471\pi$$
$$168$$ 0 0
$$169$$ −12.5358 −0.964291
$$170$$ 0 0
$$171$$ −4.18953 −0.320382
$$172$$ 0 0
$$173$$ −6.85446 −0.521135 −0.260567 0.965456i $$-0.583910\pi$$
−0.260567 + 0.965456i $$0.583910\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 30.9313 2.32494
$$178$$ 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
$$183$$ −19.7417 −1.45935
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 10.1731 0.739986
$$190$$ 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 $$-0.662546\pi$$
−0.488747 + 0.872426i $$0.662546\pi$$
$$194$$ 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 $$-0.863438\pi$$
−0.909374 + 0.415980i $$0.863438\pi$$
$$200$$ 0 0
$$201$$ −18.8433 −1.32910
$$202$$ 0 0
$$203$$ −8.96408 −0.629155
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −9.10439 −0.632799
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.2939 −1.12172 −0.560860 0.827911i $$-0.689529\pi$$
−0.560860 + 0.827911i $$0.689529\pi$$
$$212$$ 0 0
$$213$$ −34.2088 −2.34395
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 20.3463 1.38119
$$218$$ 0 0
$$219$$ −14.8433 −1.00302
$$220$$ 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 $$-0.443520\pi$$
0.176506 + 0.984300i $$0.443520\pi$$
$$228$$ 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
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 14.3791 0.934021
$$238$$ 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
$$243$$ 20.3379 1.30468
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.681331 −0.0433520
$$248$$ 0 0
$$249$$ −6.37907 −0.404257
$$250$$ 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
$$261$$ 11.7745 0.728826
$$262$$ 0 0
$$263$$ 25.5163 1.57340 0.786700 0.617335i $$-0.211788\pi$$
0.786700 + 0.617335i $$0.211788\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.08798 −0.494977
$$268$$ 0 0
$$269$$ 26.7581 1.63147 0.815736 0.578424i $$-0.196332\pi$$
0.815736 + 0.578424i $$0.196332\pi$$
$$270$$ 0 0
$$271$$ −14.1731 −0.860956 −0.430478 0.902601i $$-0.641655\pi$$
−0.430478 + 0.902601i $$0.641655\pi$$
$$272$$ 0 0
$$273$$ 5.82687 0.352658
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6.08798 −0.365791 −0.182896 0.983132i $$-0.558547\pi$$
−0.182896 + 0.983132i $$0.558547\pi$$
$$278$$ 0 0
$$279$$ −26.7253 −1.60000
$$280$$ 0 0
$$281$$ −15.7089 −0.937115 −0.468558 0.883433i $$-0.655226\pi$$
−0.468558 + 0.883433i $$0.655226\pi$$
$$282$$ 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
$$291$$ 13.1044 0.768193
$$292$$ 0 0
$$293$$ −6.71414 −0.392244 −0.196122 0.980579i $$-0.562835\pi$$
−0.196122 + 0.980579i $$0.562835\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.48062 −0.0856264
$$300$$ 0 0
$$301$$ −4.34625 −0.250514
$$302$$ 0 0
$$303$$ −32.4999 −1.86707
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −17.6126 −1.00520 −0.502602 0.864518i $$-0.667624\pi$$
−0.502602 + 0.864518i $$0.667624\pi$$
$$308$$ 0 0
$$309$$ 47.9177 2.72594
$$310$$ 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 $$-0.323627\pi$$
0.526171 + 0.850379i $$0.323627\pi$$
$$314$$ 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
$$321$$ −38.6730 −2.15852
$$322$$ 0 0
$$323$$ −1.18953 −0.0661874
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −27.3655 −1.51332
$$328$$ 0 0
$$329$$ −37.4506 −2.06472
$$330$$ 0 0
$$331$$ 3.53579 0.194345 0.0971723 0.995268i $$-0.469020\pi$$
0.0971723 + 0.995268i $$0.469020\pi$$
$$332$$ 0 0
$$333$$ −32.9753 −1.80703
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7.87086 0.428753 0.214377 0.976751i $$-0.431228\pi$$
0.214377 + 0.976751i $$0.431228\pi$$
$$338$$ 0 0
$$339$$ −4.00000 −0.217250
$$340$$ 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 $$-0.361971\pi$$
0.420169 + 0.907446i $$0.361971\pi$$
$$348$$ 0 0
$$349$$ −21.4178 −1.14647 −0.573235 0.819391i $$-0.694312\pi$$
−0.573235 + 0.819391i $$0.694312\pi$$
$$350$$ 0 0
$$351$$ −2.17313 −0.115993
$$352$$ 0 0
$$353$$ −3.15672 −0.168015 −0.0840076 0.996465i $$-0.526772\pi$$
−0.0840076 + 0.996465i $$0.526772\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 10.1731 0.538419
$$358$$ 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
$$363$$ 29.4946 1.54807
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 19.2252 1.00355 0.501773 0.864999i $$-0.332681\pi$$
0.501773 + 0.864999i $$0.332681\pi$$
$$368$$ 0 0
$$369$$ 4.12080 0.214520
$$370$$ 0 0
$$371$$ −5.41499 −0.281132
$$372$$ 0 0
$$373$$ −9.95601 −0.515503 −0.257751 0.966211i $$-0.582982\pi$$
−0.257751 + 0.966211i $$0.582982\pi$$
$$374$$ 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 $$-0.331860\pi$$
0.504003 + 0.863702i $$0.331860\pi$$
$$380$$ 0 0
$$381$$ −22.1208 −1.13328
$$382$$ 0 0
$$383$$ 11.9037 0.608250 0.304125 0.952632i $$-0.401636\pi$$
0.304125 + 0.952632i $$0.401636\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 5.70892 0.290201
$$388$$ 0 0
$$389$$ 10.3463 0.524576 0.262288 0.964990i $$-0.415523\pi$$
0.262288 + 0.964990i $$0.415523\pi$$
$$390$$ 0 0
$$391$$ −2.58501 −0.130730
$$392$$ 0 0
$$393$$ 23.4835 1.18458
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −26.1536 −1.31261 −0.656306 0.754495i $$-0.727882\pi$$
−0.656306 + 0.754495i $$0.727882\pi$$
$$398$$ 0 0
$$399$$ −8.55220 −0.428145
$$400$$ 0 0
$$401$$ −32.7253 −1.63422 −0.817112 0.576479i $$-0.804426\pi$$
−0.817112 + 0.576479i $$0.804426\pi$$
$$402$$ 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
$$411$$ −31.9477 −1.57586
$$412$$ 0 0
$$413$$ 36.7938 1.81050
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −62.9669 −3.08350
$$418$$ 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
$$423$$ 49.1924 2.39182
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 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 $$-0.519699\pi$$
−0.0618452 + 0.998086i $$0.519699\pi$$
$$434$$ 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 $$-0.721010\pi$$
−0.639865 + 0.768488i $$0.721010\pi$$
$$440$$ 0 0
$$441$$ 13.2939 0.633044
$$442$$ 0 0
$$443$$ 10.7909 0.512693 0.256347 0.966585i $$-0.417481\pi$$
0.256347 + 0.966585i $$0.417481\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 50.5327 2.39011
$$448$$ 0 0
$$449$$ −29.4835 −1.39141 −0.695705 0.718327i $$-0.744908\pi$$
−0.695705 + 0.718327i $$0.744908\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −43.1372 −2.02676
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11.8269 0.553238 0.276619 0.960980i $$-0.410786\pi$$
0.276619 + 0.960980i $$0.410786\pi$$
$$458$$ 0 0
$$459$$ −3.79406 −0.177092
$$460$$ 0 0
$$461$$ −7.45065 −0.347011 −0.173506 0.984833i $$-0.555509\pi$$
−0.173506 + 0.984833i $$0.555509\pi$$
$$462$$ 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 $$-0.719367\pi$$
−0.635891 + 0.771779i $$0.719367\pi$$
$$468$$ 0 0
$$469$$ −22.4147 −1.03502
$$470$$ 0 0
$$471$$ −36.8461 −1.69778
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 7.11273 0.325669
$$478$$ 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
$$483$$ −18.5850 −0.845647
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.92604 0.359163 0.179581 0.983743i $$-0.442526\pi$$
0.179581 + 0.983743i $$0.442526\pi$$
$$488$$ 0 0
$$489$$ 39.2475 1.77484
$$490$$ 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 $$-0.349492\pi$$
0.455412 + 0.890281i $$0.349492\pi$$
$$500$$ 0 0
$$501$$ −48.1536 −2.15134
$$502$$ 0 0
$$503$$ 1.06874 0.0476526 0.0238263 0.999716i $$-0.492415\pi$$
0.0238263 + 0.999716i $$0.492415\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 33.6126 1.49279
$$508$$ 0 0
$$509$$ −19.0164 −0.842887 −0.421444 0.906855i $$-0.638476\pi$$
−0.421444 + 0.906855i $$0.638476\pi$$
$$510$$ 0 0
$$511$$ −17.6566 −0.781081
$$512$$ 0 0
$$513$$ 3.18953 0.140821
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 18.3791 0.806752
$$520$$ 0 0
$$521$$ 9.07158 0.397433 0.198717 0.980057i $$-0.436323\pi$$
0.198717 + 0.980057i $$0.436323\pi$$
$$522$$ 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
$$531$$ −48.3296 −2.09733
$$532$$ 0 0
$$533$$ 0.670152 0.0290275
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 13.4506 0.580438
$$538$$ 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
$$543$$ −44.8461 −1.92453
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −18.1291 −0.775146 −0.387573 0.921839i $$-0.626686\pi$$
−0.387573 + 0.921839i $$0.626686\pi$$
$$548$$ 0 0
$$549$$ 30.8461 1.31648
$$550$$ 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 $$-0.730040\pi$$
−0.661405 + 0.750029i $$0.730040\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 12.8105 0.537989
$$568$$ 0 0
$$569$$ −31.7969 −1.33300 −0.666498 0.745507i $$-0.732207\pi$$
−0.666498 + 0.745507i $$0.732207\pi$$
$$570$$ 0 0
$$571$$ −18.3134 −0.766394 −0.383197 0.923667i $$-0.625177\pi$$
−0.383197 + 0.923667i $$0.625177\pi$$
$$572$$ 0 0
$$573$$ −30.2388 −1.26324
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −26.7282 −1.11271 −0.556354 0.830945i $$-0.687800\pi$$
−0.556354 + 0.830945i $$0.687800\pi$$
$$578$$ 0 0
$$579$$ 36.4119 1.51323
$$580$$ 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 $$-0.332436\pi$$
0.502439 + 0.864613i $$0.332436\pi$$
$$588$$ 0 0
$$589$$ 6.37907 0.262845
$$590$$ 0 0
$$591$$ −5.36266 −0.220590
$$592$$ 0 0
$$593$$ −12.7253 −0.522566 −0.261283 0.965262i $$-0.584146\pi$$
−0.261283 + 0.965262i $$0.584146\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 68.7938 2.81554
$$598$$ 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
$$603$$ 29.4423 1.19898
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −18.2827 −0.742074 −0.371037 0.928618i $$-0.620998\pi$$
−0.371037 + 0.928618i $$0.620998\pi$$
$$608$$ 0 0
$$609$$ 24.0357 0.973974
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 6.84612 0.276512 0.138256 0.990397i $$-0.455850\pi$$
0.138256 + 0.990397i $$0.455850\pi$$
$$614$$ 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 $$-0.171311\pi$$
0.858638 + 0.512583i $$0.171311\pi$$
$$620$$ 0 0
$$621$$ 6.93126 0.278142
$$622$$ 0 0
$$623$$ −9.62093 −0.385454
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 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 $$-0.679035\pi$$
−0.533265 + 0.845948i $$0.679035\pi$$
$$632$$ 0 0
$$633$$ 43.6894 1.73650
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.16195 0.0856595
$$638$$ 0 0
$$639$$ 53.4506 2.11447
$$640$$ 0 0
$$641$$ 28.1208 1.11070 0.555352 0.831615i $$-0.312583\pi$$
0.555352 + 0.831615i $$0.312583\pi$$
$$642$$ 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 $$-0.493312\pi$$
0.0210082 + 0.999779i $$0.493312\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −54.5550 −2.13818
$$652$$ 0 0
$$653$$ 21.3298 0.834701 0.417351 0.908745i $$-0.362959\pi$$
0.417351 + 0.908745i $$0.362959\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 23.1924 0.904821
$$658$$ 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
$$663$$ −2.17313 −0.0843973
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.10750 −0.236483
$$668$$ 0 0
$$669$$ 65.0221 2.51390
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −47.1840 −1.81881 −0.909405 0.415911i $$-0.863463\pi$$
−0.909405 + 0.415911i $$0.863463\pi$$
$$674$$ 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
$$681$$ −14.2611 −0.546487
$$682$$ 0 0
$$683$$ −17.2887 −0.661534 −0.330767 0.943713i $$-0.607307\pi$$
−0.330767 + 0.943713i $$0.607307\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −31.3955 −1.19781
$$688$$ 0 0
$$689$$ 1.15672 0.0440675
$$690$$ 0 0
$$691$$ 35.8297 1.36303 0.681513 0.731806i $$-0.261322\pi$$
0.681513 + 0.731806i $$0.261322\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.17002 0.0443176
$$698$$ 0 0
$$699$$ 73.7802 2.79062
$$700$$ 0 0
$$701$$ −38.0880 −1.43856 −0.719282 0.694719i $$-0.755529\pi$$
−0.719282 + 0.694719i $$0.755529\pi$$
$$702$$ 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
$$711$$ −22.4671 −0.842580
$$712$$ 0 0
$$713$$ 13.8625 0.519156
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 40.0357 1.49516
$$718$$ 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
$$723$$ 25.0164 0.930370
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −39.4311 −1.46242 −0.731210 0.682153i $$-0.761044\pi$$
−0.731210 + 0.682153i $$0.761044\pi$$
$$728$$ 0 0
$$729$$ −42.4835 −1.57346
$$730$$ 0 0
$$731$$ 1.62093 0.0599523
$$732$$ 0 0
$$733$$ −33.1372 −1.22395 −0.611975 0.790877i $$-0.709625\pi$$
−0.611975 + 0.790877i $$0.709625\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 25.8625 0.951368 0.475684 0.879616i $$-0.342201\pi$$
0.475684 + 0.879616i $$0.342201\pi$$
$$740$$ 0 0
$$741$$ 1.82687 0.0671118
$$742$$ 0 0
$$743$$ −28.6842 −1.05232 −0.526160 0.850386i $$-0.676369\pi$$
−0.526160 + 0.850386i $$0.676369\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 9.96719 0.364680
$$748$$ 0 0
$$749$$ −46.0028 −1.68091
$$750$$ 0 0
$$751$$ 34.2968 1.25151 0.625753 0.780021i $$-0.284792\pi$$
0.625753 + 0.780021i $$0.284792\pi$$
$$752$$ 0 0
$$753$$ 3.41783 0.124553
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −20.6597 −0.750889 −0.375445 0.926845i $$-0.622510\pi$$
−0.375445 + 0.926845i $$0.622510\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.1236 1.38198 0.690990 0.722864i $$-0.257175\pi$$
0.690990 + 0.722864i $$0.257175\pi$$
$$762$$ 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
$$771$$ 83.8953 3.02142
$$772$$ 0 0
$$773$$ −26.4559 −0.951552 −0.475776 0.879567i $$-0.657833\pi$$
−0.475776 + 0.879567i $$0.657833\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −67.3132 −2.41485
$$778$$ 0 0
$$779$$ −0.983593 −0.0352409
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −8.96408 −0.320350
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −27.3738 −0.975772 −0.487886 0.872907i $$-0.662232\pi$$
−0.487886 + 0.872907i $$0.662232\pi$$
$$788$$ 0 0
$$789$$ −68.4176 −2.43573
$$790$$ 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
$$801$$ 12.6373 0.446518
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −71.7474 −2.52563
$$808$$ 0 0
$$809$$ 52.2444 1.83682 0.918408 0.395634i $$-0.129475\pi$$
0.918408 + 0.395634i $$0.129475\pi$$
$$810$$ 0 0
$$811$$ −32.8984 −1.15522 −0.577610 0.816313i $$-0.696015\pi$$
−0.577610 + 0.816313i $$0.696015\pi$$
$$812$$ 0 0
$$813$$ 38.0028 1.33282
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.36266 −0.0476735
$$818$$ 0 0
$$819$$ −9.10439 −0.318133
$$820$$ 0 0
$$821$$ −4.44470 −0.155121 −0.0775605 0.996988i $$-0.524713\pi$$
−0.0775605 + 0.996988i $$0.524713\pi$$
$$822$$ 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 $$-0.200186\pi$$
0.808674 + 0.588257i $$0.200186\pi$$
$$828$$ 0 0
$$829$$ 28.2388 0.980772 0.490386 0.871505i $$-0.336856\pi$$
0.490386 + 0.871505i $$0.336856\pi$$
$$830$$ 0 0
$$831$$ 16.3239 0.566270
$$832$$ 0 0
$$833$$ 3.77454 0.130780
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.3463 0.703269
$$838$$ 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
$$843$$ 42.1208 1.45072
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 35.0849 1.20553
$$848$$ 0 0
$$849$$ −0.928423 −0.0318634
$$850$$ 0 0
$$851$$ 17.1044 0.586331
$$852$$ 0 0
$$853$$ −5.30749 −0.181725 −0.0908625 0.995863i $$-0.528962\pi$$
−0.0908625 + 0.995863i $$0.528962\pi$$
$$854$$ 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 $$-0.611849\pi$$
−0.344198 + 0.938897i $$0.611849\pi$$
$$860$$ 0 0
$$861$$ 8.41188 0.286676
$$862$$ 0 0
$$863$$ −30.3051 −1.03160 −0.515799 0.856710i $$-0.672505\pi$$
−0.515799 + 0.856710i $$0.672505\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 41.7886 1.41921
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.78811 0.162239
$$872$$ 0 0
$$873$$ −20.4754 −0.692987
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −9.43947 −0.318748 −0.159374 0.987218i $$-0.550948\pi$$
−0.159374 + 0.987218i $$0.550948\pi$$
$$878$$ 0 0
$$879$$ 18.0028 0.607221
$$880$$ 0 0
$$881$$ −24.4671 −0.824316 −0.412158 0.911112i $$-0.635225\pi$$
−0.412158 + 0.911112i $$0.635225\pi$$
$$882$$ 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 $$-0.223959\pi$$
0.762526 + 0.646957i $$0.223959\pi$$
$$888$$ 0 0
$$889$$ −26.3134 −0.882524
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −11.7417 −0.392922
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3.97003 0.132555
$$898$$ 0 0
$$899$$ −17.9282 −0.597937
$$900$$ 0 0
$$901$$ 2.01952 0.0672798
$$902$$ 0 0
$$903$$ 11.6537 0.387812
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −10.2694 −0.340991 −0.170496 0.985358i $$-0.554537\pi$$
−0.170496 + 0.985358i $$0.554537\pi$$
$$908$$ 0 0
$$909$$ 50.7805 1.68428
$$910$$ 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 $$-0.278058\pi$$
0.642112 + 0.766611i $$0.278058\pi$$
$$920$$ 0 0
$$921$$ 47.2252 1.55612
$$922$$ 0 0
$$923$$ 8.69251 0.286117
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −74.8706 −2.45907
$$928$$ 0 0
$$929$$ 21.9700 0.720813 0.360407 0.932795i $$-0.382638\pi$$
0.360407 + 0.932795i $$0.382638\pi$$
$$930$$ 0 0
$$931$$ −3.17313 −0.103995
$$932$$ 0 0
$$933$$ −17.6566 −0.578051
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.6402 −0.347600 −0.173800 0.984781i $$-0.555605\pi$$
−0.173800 + 0.984781i $$0.555605\pi$$
$$938$$ 0 0
$$939$$ −49.9205 −1.62910
$$940$$ 0 0
$$941$$ −20.8656 −0.680200 −0.340100 0.940389i $$-0.610461\pi$$
−0.340100 + 0.940389i $$0.610461\pi$$
$$942$$ 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 $$-0.433882\pi$$
0.206226 + 0.978504i $$0.433882\pi$$
$$948$$ 0 0
$$949$$ 3.77170 0.122435
$$950$$ 0 0
$$951$$ −47.6894 −1.54643
$$952$$ 0 0
$$953$$ −16.7337 −0.542056 −0.271028 0.962571i $$-0.587364\pi$$
−0.271028 + 0.962571i $$0.587364\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −38.0028 −1.22718
$$960$$ 0 0
$$961$$ 9.69251 0.312662
$$962$$ 0 0
$$963$$ 60.4259 1.94720
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.62688 0.148790 0.0743952 0.997229i $$-0.476297\pi$$
0.0743952 + 0.997229i $$0.476297\pi$$
$$968$$ 0 0
$$969$$ 3.18953 0.102463
$$970$$ 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
$$981$$ 42.7581 1.36516
$$982$$ 0 0
$$983$$ 32.9424 1.05070 0.525350 0.850886i $$-0.323934\pi$$
0.525350 + 0.850886i $$0.323934\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 100.418 3.19633
$$988$$ 0 0
$$989$$ −2.96124 −0.0941618
$$990$$ 0 0
$$991$$ 49.7802 1.58132 0.790660 0.612255i $$-0.209737\pi$$
0.790660 + 0.612255i $$0.209737\pi$$
$$992$$ 0 0
$$993$$ −9.48062 −0.300858
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 36.4014 1.15284 0.576422 0.817152i $$-0.304448\pi$$
0.576422 + 0.817152i $$0.304448\pi$$
$$998$$ 0 0
$$999$$ 25.1044 0.794268
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 7600.2.a.bq.1.1 3
4.3 odd 2 3800.2.a.x.1.3 3
5.4 even 2 1520.2.a.s.1.3 3
20.3 even 4 3800.2.d.l.3649.6 6
20.7 even 4 3800.2.d.l.3649.1 6
20.19 odd 2 760.2.a.j.1.1 3
40.19 odd 2 6080.2.a.bv.1.3 3
40.29 even 2 6080.2.a.bq.1.1 3
60.59 even 2 6840.2.a.bg.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.1 3 20.19 odd 2
1520.2.a.s.1.3 3 5.4 even 2
3800.2.a.x.1.3 3 4.3 odd 2
3800.2.d.l.3649.1 6 20.7 even 4
3800.2.d.l.3649.6 6 20.3 even 4
6080.2.a.bq.1.1 3 40.29 even 2
6080.2.a.bv.1.3 3 40.19 odd 2
6840.2.a.bg.1.1 3 60.59 even 2
7600.2.a.bq.1.1 3 1.1 even 1 trivial