# Properties

 Label 1620.2.a.g.1.1 Level $1620$ Weight $2$ Character 1620.1 Self dual yes Analytic conductor $12.936$ 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] = [1620,2,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.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: $$12.9357651274$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1620.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -2.73205 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -2.73205 q^{7} +1.73205 q^{11} +5.46410 q^{13} -4.73205 q^{17} -4.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} -7.73205 q^{29} +5.92820 q^{31} +2.73205 q^{35} -6.19615 q^{37} -11.1962 q^{41} +3.26795 q^{43} -1.26795 q^{47} +0.464102 q^{49} -7.26795 q^{53} -1.73205 q^{55} -7.73205 q^{59} -4.00000 q^{61} -5.46410 q^{65} +6.39230 q^{67} -11.1962 q^{71} -0.196152 q^{73} -4.73205 q^{77} -14.3923 q^{79} -15.1244 q^{83} +4.73205 q^{85} +5.19615 q^{89} -14.9282 q^{91} +4.46410 q^{95} +0.732051 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^7 $$2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65} - 8 q^{67} - 12 q^{71} + 10 q^{73} - 6 q^{77} - 8 q^{79} - 6 q^{83} + 6 q^{85} - 16 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 - 6 * q^17 - 2 * q^19 + 2 * q^25 - 12 * q^29 - 2 * q^31 + 2 * q^35 - 2 * q^37 - 12 * q^41 + 10 * q^43 - 6 * q^47 - 6 * q^49 - 18 * q^53 - 12 * q^59 - 8 * q^61 - 4 * q^65 - 8 * q^67 - 12 * q^71 + 10 * q^73 - 6 * q^77 - 8 * q^79 - 6 * q^83 + 6 * q^85 - 16 * q^91 + 2 * q^95 - 2 * 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
$$7$$ −2.73205 −1.03262 −0.516309 0.856402i $$-0.672694\pi$$
−0.516309 + 0.856402i $$0.672694\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.73205 0.522233 0.261116 0.965307i $$-0.415909\pi$$
0.261116 + 0.965307i $$0.415909\pi$$
$$12$$ 0 0
$$13$$ 5.46410 1.51547 0.757735 0.652563i $$-0.226306\pi$$
0.757735 + 0.652563i $$0.226306\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.73205 −1.14769 −0.573845 0.818964i $$-0.694549\pi$$
−0.573845 + 0.818964i $$0.694549\pi$$
$$18$$ 0 0
$$19$$ −4.46410 −1.02414 −0.512068 0.858945i $$-0.671120\pi$$
−0.512068 + 0.858945i $$0.671120\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.46410 0.722315 0.361158 0.932505i $$-0.382382\pi$$
0.361158 + 0.932505i $$0.382382\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.73205 −1.43581 −0.717903 0.696143i $$-0.754898\pi$$
−0.717903 + 0.696143i $$0.754898\pi$$
$$30$$ 0 0
$$31$$ 5.92820 1.06474 0.532368 0.846513i $$-0.321302\pi$$
0.532368 + 0.846513i $$0.321302\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.73205 0.461801
$$36$$ 0 0
$$37$$ −6.19615 −1.01864 −0.509321 0.860577i $$-0.670103\pi$$
−0.509321 + 0.860577i $$0.670103\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −11.1962 −1.74855 −0.874273 0.485435i $$-0.838661\pi$$
−0.874273 + 0.485435i $$0.838661\pi$$
$$42$$ 0 0
$$43$$ 3.26795 0.498358 0.249179 0.968458i $$-0.419839\pi$$
0.249179 + 0.968458i $$0.419839\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.26795 −0.184949 −0.0924747 0.995715i $$-0.529478\pi$$
−0.0924747 + 0.995715i $$0.529478\pi$$
$$48$$ 0 0
$$49$$ 0.464102 0.0663002
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −7.26795 −0.998330 −0.499165 0.866507i $$-0.666360\pi$$
−0.499165 + 0.866507i $$0.666360\pi$$
$$54$$ 0 0
$$55$$ −1.73205 −0.233550
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −7.73205 −1.00663 −0.503314 0.864104i $$-0.667886\pi$$
−0.503314 + 0.864104i $$0.667886\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −5.46410 −0.677738
$$66$$ 0 0
$$67$$ 6.39230 0.780944 0.390472 0.920615i $$-0.372312\pi$$
0.390472 + 0.920615i $$0.372312\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −11.1962 −1.32874 −0.664369 0.747404i $$-0.731300\pi$$
−0.664369 + 0.747404i $$0.731300\pi$$
$$72$$ 0 0
$$73$$ −0.196152 −0.0229579 −0.0114790 0.999934i $$-0.503654\pi$$
−0.0114790 + 0.999934i $$0.503654\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.73205 −0.539267
$$78$$ 0 0
$$79$$ −14.3923 −1.61926 −0.809630 0.586940i $$-0.800332\pi$$
−0.809630 + 0.586940i $$0.800332\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −15.1244 −1.66011 −0.830057 0.557679i $$-0.811692\pi$$
−0.830057 + 0.557679i $$0.811692\pi$$
$$84$$ 0 0
$$85$$ 4.73205 0.513263
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 5.19615 0.550791 0.275396 0.961331i $$-0.411191\pi$$
0.275396 + 0.961331i $$0.411191\pi$$
$$90$$ 0 0
$$91$$ −14.9282 −1.56490
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.46410 0.458007
$$96$$ 0 0
$$97$$ 0.732051 0.0743285 0.0371642 0.999309i $$-0.488168\pi$$
0.0371642 + 0.999309i $$0.488168\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.12436 0.609396 0.304698 0.952449i $$-0.401444\pi$$
0.304698 + 0.952449i $$0.401444\pi$$
$$102$$ 0 0
$$103$$ 18.3923 1.81225 0.906124 0.423013i $$-0.139027\pi$$
0.906124 + 0.423013i $$0.139027\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.46410 0.334887 0.167444 0.985882i $$-0.446449\pi$$
0.167444 + 0.985882i $$0.446449\pi$$
$$108$$ 0 0
$$109$$ −7.92820 −0.759384 −0.379692 0.925113i $$-0.623970\pi$$
−0.379692 + 0.925113i $$0.623970\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −0.339746 −0.0319606 −0.0159803 0.999872i $$-0.505087\pi$$
−0.0159803 + 0.999872i $$0.505087\pi$$
$$114$$ 0 0
$$115$$ −3.46410 −0.323029
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.9282 1.18513
$$120$$ 0 0
$$121$$ −8.00000 −0.727273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 4.19615 0.372348 0.186174 0.982517i $$-0.440391\pi$$
0.186174 + 0.982517i $$0.440391\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.2679 0.897115 0.448557 0.893754i $$-0.351938\pi$$
0.448557 + 0.893754i $$0.351938\pi$$
$$132$$ 0 0
$$133$$ 12.1962 1.05754
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.39230 0.375260 0.187630 0.982240i $$-0.439919\pi$$
0.187630 + 0.982240i $$0.439919\pi$$
$$138$$ 0 0
$$139$$ 15.3923 1.30556 0.652779 0.757548i $$-0.273603\pi$$
0.652779 + 0.757548i $$0.273603\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 9.46410 0.791428
$$144$$ 0 0
$$145$$ 7.73205 0.642112
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −17.3923 −1.41537 −0.707683 0.706530i $$-0.750259\pi$$
−0.707683 + 0.706530i $$0.750259\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −5.92820 −0.476165
$$156$$ 0 0
$$157$$ 3.26795 0.260811 0.130405 0.991461i $$-0.458372\pi$$
0.130405 + 0.991461i $$0.458372\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −9.46410 −0.745876
$$162$$ 0 0
$$163$$ 18.7321 1.46721 0.733604 0.679577i $$-0.237837\pi$$
0.733604 + 0.679577i $$0.237837\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −21.1244 −1.63465 −0.817326 0.576176i $$-0.804544\pi$$
−0.817326 + 0.576176i $$0.804544\pi$$
$$168$$ 0 0
$$169$$ 16.8564 1.29665
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −24.2487 −1.84360 −0.921798 0.387671i $$-0.873280\pi$$
−0.921798 + 0.387671i $$0.873280\pi$$
$$174$$ 0 0
$$175$$ −2.73205 −0.206524
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −12.1244 −0.906217 −0.453108 0.891455i $$-0.649685\pi$$
−0.453108 + 0.891455i $$0.649685\pi$$
$$180$$ 0 0
$$181$$ −16.4641 −1.22377 −0.611884 0.790948i $$-0.709588\pi$$
−0.611884 + 0.790948i $$0.709588\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 6.19615 0.455550
$$186$$ 0 0
$$187$$ −8.19615 −0.599362
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.0526 1.37859 0.689297 0.724479i $$-0.257919\pi$$
0.689297 + 0.724479i $$0.257919\pi$$
$$192$$ 0 0
$$193$$ −10.5885 −0.762174 −0.381087 0.924539i $$-0.624450\pi$$
−0.381087 + 0.924539i $$0.624450\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −13.8564 −0.987228 −0.493614 0.869681i $$-0.664324\pi$$
−0.493614 + 0.869681i $$0.664324\pi$$
$$198$$ 0 0
$$199$$ 15.8564 1.12403 0.562015 0.827127i $$-0.310026\pi$$
0.562015 + 0.827127i $$0.310026\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 21.1244 1.48264
$$204$$ 0 0
$$205$$ 11.1962 0.781973
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −7.73205 −0.534837
$$210$$ 0 0
$$211$$ −19.9282 −1.37191 −0.685957 0.727642i $$-0.740616\pi$$
−0.685957 + 0.727642i $$0.740616\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −3.26795 −0.222872
$$216$$ 0 0
$$217$$ −16.1962 −1.09947
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −25.8564 −1.73929
$$222$$ 0 0
$$223$$ −5.85641 −0.392174 −0.196087 0.980587i $$-0.562823\pi$$
−0.196087 + 0.980587i $$0.562823\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 26.1962 1.73870 0.869350 0.494197i $$-0.164538\pi$$
0.869350 + 0.494197i $$0.164538\pi$$
$$228$$ 0 0
$$229$$ −17.8564 −1.17998 −0.589992 0.807409i $$-0.700869\pi$$
−0.589992 + 0.807409i $$0.700869\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.5885 1.61084 0.805422 0.592702i $$-0.201939\pi$$
0.805422 + 0.592702i $$0.201939\pi$$
$$234$$ 0 0
$$235$$ 1.26795 0.0827119
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.53590 0.552141 0.276071 0.961137i $$-0.410968\pi$$
0.276071 + 0.961137i $$0.410968\pi$$
$$240$$ 0 0
$$241$$ 10.3205 0.664802 0.332401 0.943138i $$-0.392141\pi$$
0.332401 + 0.943138i $$0.392141\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.464102 −0.0296504
$$246$$ 0 0
$$247$$ −24.3923 −1.55205
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.5359 1.29621 0.648107 0.761549i $$-0.275561\pi$$
0.648107 + 0.761549i $$0.275561\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 15.4641 0.964624 0.482312 0.875999i $$-0.339797\pi$$
0.482312 + 0.875999i $$0.339797\pi$$
$$258$$ 0 0
$$259$$ 16.9282 1.05187
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −24.2487 −1.49524 −0.747620 0.664127i $$-0.768803\pi$$
−0.747620 + 0.664127i $$0.768803\pi$$
$$264$$ 0 0
$$265$$ 7.26795 0.446467
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −16.2679 −0.991874 −0.495937 0.868358i $$-0.665175\pi$$
−0.495937 + 0.868358i $$0.665175\pi$$
$$270$$ 0 0
$$271$$ 16.7846 1.01959 0.509796 0.860295i $$-0.329721\pi$$
0.509796 + 0.860295i $$0.329721\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.73205 0.104447
$$276$$ 0 0
$$277$$ 5.12436 0.307893 0.153946 0.988079i $$-0.450802\pi$$
0.153946 + 0.988079i $$0.450802\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 29.3205 1.74911 0.874557 0.484922i $$-0.161152\pi$$
0.874557 + 0.484922i $$0.161152\pi$$
$$282$$ 0 0
$$283$$ 16.5359 0.982957 0.491479 0.870890i $$-0.336457\pi$$
0.491479 + 0.870890i $$0.336457\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 30.5885 1.80558
$$288$$ 0 0
$$289$$ 5.39230 0.317194
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 24.5885 1.43647 0.718237 0.695799i $$-0.244950\pi$$
0.718237 + 0.695799i $$0.244950\pi$$
$$294$$ 0 0
$$295$$ 7.73205 0.450177
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 18.9282 1.09465
$$300$$ 0 0
$$301$$ −8.92820 −0.514613
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −1.80385 −0.102951 −0.0514755 0.998674i $$-0.516392\pi$$
−0.0514755 + 0.998674i $$0.516392\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.5167 −0.936574 −0.468287 0.883576i $$-0.655129\pi$$
−0.468287 + 0.883576i $$0.655129\pi$$
$$312$$ 0 0
$$313$$ 14.9282 0.843792 0.421896 0.906644i $$-0.361365\pi$$
0.421896 + 0.906644i $$0.361365\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.12436 0.175481 0.0877406 0.996143i $$-0.472035\pi$$
0.0877406 + 0.996143i $$0.472035\pi$$
$$318$$ 0 0
$$319$$ −13.3923 −0.749825
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 21.1244 1.17539
$$324$$ 0 0
$$325$$ 5.46410 0.303094
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3.46410 0.190982
$$330$$ 0 0
$$331$$ −29.3923 −1.61555 −0.807774 0.589493i $$-0.799328\pi$$
−0.807774 + 0.589493i $$0.799328\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −6.39230 −0.349249
$$336$$ 0 0
$$337$$ −34.2487 −1.86565 −0.932823 0.360335i $$-0.882662\pi$$
−0.932823 + 0.360335i $$0.882662\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.2679 0.556041
$$342$$ 0 0
$$343$$ 17.8564 0.964155
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 21.8038 1.17049 0.585246 0.810856i $$-0.300998\pi$$
0.585246 + 0.810856i $$0.300998\pi$$
$$348$$ 0 0
$$349$$ −25.0000 −1.33822 −0.669110 0.743164i $$-0.733324\pi$$
−0.669110 + 0.743164i $$0.733324\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −28.9808 −1.54249 −0.771245 0.636538i $$-0.780366\pi$$
−0.771245 + 0.636538i $$0.780366\pi$$
$$354$$ 0 0
$$355$$ 11.1962 0.594230
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 9.33975 0.492933 0.246466 0.969151i $$-0.420730\pi$$
0.246466 + 0.969151i $$0.420730\pi$$
$$360$$ 0 0
$$361$$ 0.928203 0.0488528
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0.196152 0.0102671
$$366$$ 0 0
$$367$$ −26.3923 −1.37767 −0.688834 0.724920i $$-0.741877\pi$$
−0.688834 + 0.724920i $$0.741877\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 19.8564 1.03089
$$372$$ 0 0
$$373$$ −14.0526 −0.727614 −0.363807 0.931474i $$-0.618523\pi$$
−0.363807 + 0.931474i $$0.618523\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −42.2487 −2.17592
$$378$$ 0 0
$$379$$ 4.53590 0.232993 0.116497 0.993191i $$-0.462834\pi$$
0.116497 + 0.993191i $$0.462834\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 18.2487 0.932466 0.466233 0.884662i $$-0.345611\pi$$
0.466233 + 0.884662i $$0.345611\pi$$
$$384$$ 0 0
$$385$$ 4.73205 0.241168
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 27.4641 1.39249 0.696243 0.717807i $$-0.254854\pi$$
0.696243 + 0.717807i $$0.254854\pi$$
$$390$$ 0 0
$$391$$ −16.3923 −0.828994
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 14.3923 0.724155
$$396$$ 0 0
$$397$$ 37.3205 1.87306 0.936531 0.350584i $$-0.114017\pi$$
0.936531 + 0.350584i $$0.114017\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.7846 −0.738308 −0.369154 0.929368i $$-0.620353\pi$$
−0.369154 + 0.929368i $$0.620353\pi$$
$$402$$ 0 0
$$403$$ 32.3923 1.61358
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.7321 −0.531968
$$408$$ 0 0
$$409$$ −17.8564 −0.882942 −0.441471 0.897275i $$-0.645543\pi$$
−0.441471 + 0.897275i $$0.645543\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 21.1244 1.03946
$$414$$ 0 0
$$415$$ 15.1244 0.742425
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 21.4641 1.04859 0.524295 0.851537i $$-0.324329\pi$$
0.524295 + 0.851537i $$0.324329\pi$$
$$420$$ 0 0
$$421$$ 13.7846 0.671821 0.335910 0.941894i $$-0.390956\pi$$
0.335910 + 0.941894i $$0.390956\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −4.73205 −0.229538
$$426$$ 0 0
$$427$$ 10.9282 0.528853
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.5885 −1.61790 −0.808950 0.587878i $$-0.799963\pi$$
−0.808950 + 0.587878i $$0.799963\pi$$
$$432$$ 0 0
$$433$$ 11.4641 0.550930 0.275465 0.961311i $$-0.411168\pi$$
0.275465 + 0.961311i $$0.411168\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −15.4641 −0.739748
$$438$$ 0 0
$$439$$ 6.60770 0.315368 0.157684 0.987490i $$-0.449597\pi$$
0.157684 + 0.987490i $$0.449597\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 37.2679 1.77065 0.885327 0.464969i $$-0.153935\pi$$
0.885327 + 0.464969i $$0.153935\pi$$
$$444$$ 0 0
$$445$$ −5.19615 −0.246321
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 24.1244 1.13850 0.569249 0.822165i $$-0.307234\pi$$
0.569249 + 0.822165i $$0.307234\pi$$
$$450$$ 0 0
$$451$$ −19.3923 −0.913148
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 14.9282 0.699845
$$456$$ 0 0
$$457$$ 22.1962 1.03829 0.519146 0.854686i $$-0.326250\pi$$
0.519146 + 0.854686i $$0.326250\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.58846 0.446579 0.223289 0.974752i $$-0.428320\pi$$
0.223289 + 0.974752i $$0.428320\pi$$
$$462$$ 0 0
$$463$$ −2.39230 −0.111180 −0.0555899 0.998454i $$-0.517704\pi$$
−0.0555899 + 0.998454i $$0.517704\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 26.1962 1.21221 0.606107 0.795383i $$-0.292730\pi$$
0.606107 + 0.795383i $$0.292730\pi$$
$$468$$ 0 0
$$469$$ −17.4641 −0.806417
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.66025 0.260259
$$474$$ 0 0
$$475$$ −4.46410 −0.204827
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −3.33975 −0.152597 −0.0762984 0.997085i $$-0.524310\pi$$
−0.0762984 + 0.997085i $$0.524310\pi$$
$$480$$ 0 0
$$481$$ −33.8564 −1.54372
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −0.732051 −0.0332407
$$486$$ 0 0
$$487$$ −30.5359 −1.38371 −0.691857 0.722035i $$-0.743207\pi$$
−0.691857 + 0.722035i $$0.743207\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.26795 −0.192610 −0.0963049 0.995352i $$-0.530702\pi$$
−0.0963049 + 0.995352i $$0.530702\pi$$
$$492$$ 0 0
$$493$$ 36.5885 1.64786
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 30.5885 1.37208
$$498$$ 0 0
$$499$$ 27.3923 1.22625 0.613124 0.789987i $$-0.289913\pi$$
0.613124 + 0.789987i $$0.289913\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −35.3205 −1.57486 −0.787432 0.616402i $$-0.788590\pi$$
−0.787432 + 0.616402i $$0.788590\pi$$
$$504$$ 0 0
$$505$$ −6.12436 −0.272530
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −26.7846 −1.18721 −0.593603 0.804758i $$-0.702295\pi$$
−0.593603 + 0.804758i $$0.702295\pi$$
$$510$$ 0 0
$$511$$ 0.535898 0.0237067
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −18.3923 −0.810462
$$516$$ 0 0
$$517$$ −2.19615 −0.0965867
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.3923 −0.455295 −0.227648 0.973744i $$-0.573103\pi$$
−0.227648 + 0.973744i $$0.573103\pi$$
$$522$$ 0 0
$$523$$ 3.60770 0.157753 0.0788767 0.996884i $$-0.474867\pi$$
0.0788767 + 0.996884i $$0.474867\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −28.0526 −1.22199
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −61.1769 −2.64987
$$534$$ 0 0
$$535$$ −3.46410 −0.149766
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.803848 0.0346242
$$540$$ 0 0
$$541$$ 2.46410 0.105940 0.0529700 0.998596i $$-0.483131\pi$$
0.0529700 + 0.998596i $$0.483131\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 7.92820 0.339607
$$546$$ 0 0
$$547$$ 4.78461 0.204575 0.102288 0.994755i $$-0.467384\pi$$
0.102288 + 0.994755i $$0.467384\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 34.5167 1.47046
$$552$$ 0 0
$$553$$ 39.3205 1.67208
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −4.39230 −0.186108 −0.0930540 0.995661i $$-0.529663\pi$$
−0.0930540 + 0.995661i $$0.529663\pi$$
$$558$$ 0 0
$$559$$ 17.8564 0.755246
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 22.7321 0.958042 0.479021 0.877804i $$-0.340992\pi$$
0.479021 + 0.877804i $$0.340992\pi$$
$$564$$ 0 0
$$565$$ 0.339746 0.0142932
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −43.0526 −1.80486 −0.902429 0.430839i $$-0.858218\pi$$
−0.902429 + 0.430839i $$0.858218\pi$$
$$570$$ 0 0
$$571$$ 0.856406 0.0358395 0.0179197 0.999839i $$-0.494296\pi$$
0.0179197 + 0.999839i $$0.494296\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.46410 0.144463
$$576$$ 0 0
$$577$$ 11.8038 0.491401 0.245700 0.969346i $$-0.420982\pi$$
0.245700 + 0.969346i $$0.420982\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 41.3205 1.71426
$$582$$ 0 0
$$583$$ −12.5885 −0.521361
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 3.80385 0.157002 0.0785008 0.996914i $$-0.474987\pi$$
0.0785008 + 0.996914i $$0.474987\pi$$
$$588$$ 0 0
$$589$$ −26.4641 −1.09043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −23.0718 −0.947445 −0.473723 0.880674i $$-0.657090\pi$$
−0.473723 + 0.880674i $$0.657090\pi$$
$$594$$ 0 0
$$595$$ −12.9282 −0.530005
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −14.4115 −0.588840 −0.294420 0.955676i $$-0.595126\pi$$
−0.294420 + 0.955676i $$0.595126\pi$$
$$600$$ 0 0
$$601$$ −24.3205 −0.992054 −0.496027 0.868307i $$-0.665208\pi$$
−0.496027 + 0.868307i $$0.665208\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 8.00000 0.325246
$$606$$ 0 0
$$607$$ 2.58846 0.105062 0.0525311 0.998619i $$-0.483271\pi$$
0.0525311 + 0.998619i $$0.483271\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.92820 −0.280285
$$612$$ 0 0
$$613$$ 24.3923 0.985196 0.492598 0.870257i $$-0.336047\pi$$
0.492598 + 0.870257i $$0.336047\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −14.1962 −0.568757
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 29.3205 1.16909
$$630$$ 0 0
$$631$$ −0.0717968 −0.00285818 −0.00142909 0.999999i $$-0.500455\pi$$
−0.00142909 + 0.999999i $$0.500455\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −4.19615 −0.166519
$$636$$ 0 0
$$637$$ 2.53590 0.100476
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.8038 −0.505722 −0.252861 0.967503i $$-0.581371\pi$$
−0.252861 + 0.967503i $$0.581371\pi$$
$$642$$ 0 0
$$643$$ 14.5885 0.575313 0.287656 0.957734i $$-0.407124\pi$$
0.287656 + 0.957734i $$0.407124\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.248711 0.00977785 0.00488893 0.999988i $$-0.498444\pi$$
0.00488893 + 0.999988i $$0.498444\pi$$
$$648$$ 0 0
$$649$$ −13.3923 −0.525694
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −2.53590 −0.0992374 −0.0496187 0.998768i $$-0.515801\pi$$
−0.0496187 + 0.998768i $$0.515801\pi$$
$$654$$ 0 0
$$655$$ −10.2679 −0.401202
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −2.53590 −0.0987846 −0.0493923 0.998779i $$-0.515728\pi$$
−0.0493923 + 0.998779i $$0.515728\pi$$
$$660$$ 0 0
$$661$$ 15.3923 0.598691 0.299346 0.954145i $$-0.403232\pi$$
0.299346 + 0.954145i $$0.403232\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −12.1962 −0.472947
$$666$$ 0 0
$$667$$ −26.7846 −1.03710
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.92820 −0.267460
$$672$$ 0 0
$$673$$ −38.3923 −1.47991 −0.739957 0.672654i $$-0.765154\pi$$
−0.739957 + 0.672654i $$0.765154\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 40.6410 1.56196 0.780981 0.624555i $$-0.214720\pi$$
0.780981 + 0.624555i $$0.214720\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 21.4641 0.821301 0.410651 0.911793i $$-0.365302\pi$$
0.410651 + 0.911793i $$0.365302\pi$$
$$684$$ 0 0
$$685$$ −4.39230 −0.167821
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −39.7128 −1.51294
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −15.3923 −0.583863
$$696$$ 0 0
$$697$$ 52.9808 2.00679
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −17.8756 −0.675154 −0.337577 0.941298i $$-0.609607\pi$$
−0.337577 + 0.941298i $$0.609607\pi$$
$$702$$ 0 0
$$703$$ 27.6603 1.04323
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −16.7321 −0.629274
$$708$$ 0 0
$$709$$ 10.5359 0.395684 0.197842 0.980234i $$-0.436607\pi$$
0.197842 + 0.980234i $$0.436607\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 20.5359 0.769075
$$714$$ 0 0
$$715$$ −9.46410 −0.353937
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −17.1962 −0.641308 −0.320654 0.947196i $$-0.603903\pi$$
−0.320654 + 0.947196i $$0.603903\pi$$
$$720$$ 0 0
$$721$$ −50.2487 −1.87136
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.73205 −0.287161
$$726$$ 0 0
$$727$$ −29.1769 −1.08211 −0.541056 0.840987i $$-0.681975\pi$$
−0.541056 + 0.840987i $$0.681975\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −15.4641 −0.571960
$$732$$ 0 0
$$733$$ 43.5692 1.60927 0.804633 0.593773i $$-0.202362\pi$$
0.804633 + 0.593773i $$0.202362\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 11.0718 0.407835
$$738$$ 0 0
$$739$$ −26.1769 −0.962933 −0.481467 0.876464i $$-0.659896\pi$$
−0.481467 + 0.876464i $$0.659896\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5.41154 −0.198530 −0.0992651 0.995061i $$-0.531649\pi$$
−0.0992651 + 0.995061i $$0.531649\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −9.46410 −0.345811
$$750$$ 0 0
$$751$$ 40.7846 1.48825 0.744126 0.668040i $$-0.232866\pi$$
0.744126 + 0.668040i $$0.232866\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 17.3923 0.632971
$$756$$ 0 0
$$757$$ −20.3923 −0.741171 −0.370585 0.928798i $$-0.620843\pi$$
−0.370585 + 0.928798i $$0.620843\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.1244 −1.09201 −0.546004 0.837783i $$-0.683851\pi$$
−0.546004 + 0.837783i $$0.683851\pi$$
$$762$$ 0 0
$$763$$ 21.6603 0.784154
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −42.2487 −1.52551
$$768$$ 0 0
$$769$$ 35.2487 1.27110 0.635551 0.772059i $$-0.280773\pi$$
0.635551 + 0.772059i $$0.280773\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 17.6603 0.635195 0.317598 0.948226i $$-0.397124\pi$$
0.317598 + 0.948226i $$0.397124\pi$$
$$774$$ 0 0
$$775$$ 5.92820 0.212947
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 49.9808 1.79075
$$780$$ 0 0
$$781$$ −19.3923 −0.693911
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −3.26795 −0.116638
$$786$$ 0 0
$$787$$ −36.1962 −1.29025 −0.645127 0.764076i $$-0.723195\pi$$
−0.645127 + 0.764076i $$0.723195\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.928203 0.0330031
$$792$$ 0 0
$$793$$ −21.8564 −0.776144
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 23.3205 0.826055 0.413027 0.910719i $$-0.364471\pi$$
0.413027 + 0.910719i $$0.364471\pi$$
$$798$$ 0 0
$$799$$ 6.00000 0.212265
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −0.339746 −0.0119894
$$804$$ 0 0
$$805$$ 9.46410 0.333566
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −8.41154 −0.295734 −0.147867 0.989007i $$-0.547241\pi$$
−0.147867 + 0.989007i $$0.547241\pi$$
$$810$$ 0 0
$$811$$ −25.2487 −0.886602 −0.443301 0.896373i $$-0.646193\pi$$
−0.443301 + 0.896373i $$0.646193\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −18.7321 −0.656155
$$816$$ 0 0
$$817$$ −14.5885 −0.510386
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.33975 0.116558 0.0582790 0.998300i $$-0.481439\pi$$
0.0582790 + 0.998300i $$0.481439\pi$$
$$822$$ 0 0
$$823$$ −16.9282 −0.590080 −0.295040 0.955485i $$-0.595333\pi$$
−0.295040 + 0.955485i $$0.595333\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −45.4641 −1.58094 −0.790471 0.612500i $$-0.790164\pi$$
−0.790471 + 0.612500i $$0.790164\pi$$
$$828$$ 0 0
$$829$$ −51.7846 −1.79855 −0.899277 0.437380i $$-0.855907\pi$$
−0.899277 + 0.437380i $$0.855907\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −2.19615 −0.0760922
$$834$$ 0 0
$$835$$ 21.1244 0.731038
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −8.41154 −0.290399 −0.145199 0.989402i $$-0.546382\pi$$
−0.145199 + 0.989402i $$0.546382\pi$$
$$840$$ 0 0
$$841$$ 30.7846 1.06154
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −16.8564 −0.579878
$$846$$ 0 0
$$847$$ 21.8564 0.750995
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −21.4641 −0.735780
$$852$$ 0 0
$$853$$ −0.196152 −0.00671613 −0.00335807 0.999994i $$-0.501069\pi$$
−0.00335807 + 0.999994i $$0.501069\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −29.0718 −0.993074 −0.496537 0.868016i $$-0.665395\pi$$
−0.496537 + 0.868016i $$0.665395\pi$$
$$858$$ 0 0
$$859$$ 7.78461 0.265607 0.132804 0.991142i $$-0.457602\pi$$
0.132804 + 0.991142i $$0.457602\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −49.5167 −1.68557 −0.842783 0.538253i $$-0.819085\pi$$
−0.842783 + 0.538253i $$0.819085\pi$$
$$864$$ 0 0
$$865$$ 24.2487 0.824481
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −24.9282 −0.845631
$$870$$ 0 0
$$871$$ 34.9282 1.18350
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2.73205 0.0923602
$$876$$ 0 0
$$877$$ 14.2487 0.481145 0.240572 0.970631i $$-0.422665\pi$$
0.240572 + 0.970631i $$0.422665\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.80385 0.229227 0.114614 0.993410i $$-0.463437\pi$$
0.114614 + 0.993410i $$0.463437\pi$$
$$882$$ 0 0
$$883$$ −46.8372 −1.57620 −0.788098 0.615550i $$-0.788934\pi$$
−0.788098 + 0.615550i $$0.788934\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 13.2679 0.445494 0.222747 0.974876i $$-0.428498\pi$$
0.222747 + 0.974876i $$0.428498\pi$$
$$888$$ 0 0
$$889$$ −11.4641 −0.384494
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 5.66025 0.189413
$$894$$ 0 0
$$895$$ 12.1244 0.405273
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −45.8372 −1.52876
$$900$$ 0 0
$$901$$ 34.3923 1.14577
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 16.4641 0.547285
$$906$$ 0 0
$$907$$ −1.21539 −0.0403564 −0.0201782 0.999796i $$-0.506423\pi$$
−0.0201782 + 0.999796i $$0.506423\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −6.80385 −0.225422 −0.112711 0.993628i $$-0.535953\pi$$
−0.112711 + 0.993628i $$0.535953\pi$$
$$912$$ 0 0
$$913$$ −26.1962 −0.866966
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −28.0526 −0.926377
$$918$$ 0 0
$$919$$ 51.3923 1.69528 0.847638 0.530575i $$-0.178024\pi$$
0.847638 + 0.530575i $$0.178024\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −61.1769 −2.01366
$$924$$ 0 0
$$925$$ −6.19615 −0.203728
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1.48334 0.0486668 0.0243334 0.999704i $$-0.492254\pi$$
0.0243334 + 0.999704i $$0.492254\pi$$
$$930$$ 0 0
$$931$$ −2.07180 −0.0679004
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 8.19615 0.268043
$$936$$ 0 0
$$937$$ 19.0718 0.623048 0.311524 0.950238i $$-0.399160\pi$$
0.311524 + 0.950238i $$0.399160\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −8.53590 −0.278262 −0.139131 0.990274i $$-0.544431\pi$$
−0.139131 + 0.990274i $$0.544431\pi$$
$$942$$ 0 0
$$943$$ −38.7846 −1.26300
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −52.6410 −1.71060 −0.855302 0.518130i $$-0.826628\pi$$
−0.855302 + 0.518130i $$0.826628\pi$$
$$948$$ 0 0
$$949$$ −1.07180 −0.0347920
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.53590 −0.0821458 −0.0410729 0.999156i $$-0.513078\pi$$
−0.0410729 + 0.999156i $$0.513078\pi$$
$$954$$ 0 0
$$955$$ −19.0526 −0.616526
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 4.14359 0.133664
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 10.5885 0.340854
$$966$$ 0 0
$$967$$ 7.41154 0.238339 0.119170 0.992874i $$-0.461977\pi$$
0.119170 + 0.992874i $$0.461977\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.2679 0.714612 0.357306 0.933987i $$-0.383695\pi$$
0.357306 + 0.933987i $$0.383695\pi$$
$$972$$ 0 0
$$973$$ −42.0526 −1.34814
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 4.39230 0.140522 0.0702611 0.997529i $$-0.477617\pi$$
0.0702611 + 0.997529i $$0.477617\pi$$
$$978$$ 0 0
$$979$$ 9.00000 0.287641
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −52.7321 −1.68189 −0.840946 0.541120i $$-0.818001\pi$$
−0.840946 + 0.541120i $$0.818001\pi$$
$$984$$ 0 0
$$985$$ 13.8564 0.441502
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11.3205 0.359971
$$990$$ 0 0
$$991$$ 55.7846 1.77206 0.886028 0.463631i $$-0.153454\pi$$
0.886028 + 0.463631i $$0.153454\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −15.8564 −0.502682
$$996$$ 0 0
$$997$$ 52.1962 1.65307 0.826534 0.562886i $$-0.190309\pi$$
0.826534 + 0.562886i $$0.190309\pi$$
$$998$$ 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 1620.2.a.g.1.1 2
3.2 odd 2 1620.2.a.h.1.1 yes 2
4.3 odd 2 6480.2.a.bh.1.2 2
5.2 odd 4 8100.2.d.m.649.1 4
5.3 odd 4 8100.2.d.m.649.4 4
5.4 even 2 8100.2.a.t.1.2 2
9.2 odd 6 1620.2.i.m.1081.2 4
9.4 even 3 1620.2.i.n.541.2 4
9.5 odd 6 1620.2.i.m.541.2 4
9.7 even 3 1620.2.i.n.1081.2 4
12.11 even 2 6480.2.a.bp.1.2 2
15.2 even 4 8100.2.d.l.649.1 4
15.8 even 4 8100.2.d.l.649.4 4
15.14 odd 2 8100.2.a.s.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.1 2 1.1 even 1 trivial
1620.2.a.h.1.1 yes 2 3.2 odd 2
1620.2.i.m.541.2 4 9.5 odd 6
1620.2.i.m.1081.2 4 9.2 odd 6
1620.2.i.n.541.2 4 9.4 even 3
1620.2.i.n.1081.2 4 9.7 even 3
6480.2.a.bh.1.2 2 4.3 odd 2
6480.2.a.bp.1.2 2 12.11 even 2
8100.2.a.s.1.2 2 15.14 odd 2
8100.2.a.t.1.2 2 5.4 even 2
8100.2.d.l.649.1 4 15.2 even 4
8100.2.d.l.649.4 4 15.8 even 4
8100.2.d.m.649.1 4 5.2 odd 4
8100.2.d.m.649.4 4 5.3 odd 4