# Properties

 Label 3724.2.a.i.1.1 Level $3724$ Weight $2$ Character 3724.1 Self dual yes Analytic conductor $29.736$ Analytic rank $0$ 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] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, 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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3724.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: $$29.7362897127$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.733.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 8$$ x^3 - x^2 - 7*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.69639$$ of defining polynomial Character $$\chi$$ $$=$$ 3724.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.69639 q^{3} -1.42586 q^{5} +4.27053 q^{9} +O(q^{10})$$ $$q-2.69639 q^{3} -1.42586 q^{5} +4.27053 q^{9} -3.27053 q^{11} -3.42586 q^{13} +3.84468 q^{15} -5.27053 q^{17} +1.00000 q^{19} +0.574141 q^{23} -2.96693 q^{25} -3.42586 q^{27} -0.122252 q^{29} -2.12225 q^{31} +8.81864 q^{33} -3.96693 q^{37} +9.23746 q^{39} +4.66332 q^{41} -11.9339 q^{43} -6.08918 q^{45} -3.11521 q^{47} +14.2114 q^{51} -6.08918 q^{53} +4.66332 q^{55} -2.69639 q^{57} +1.72242 q^{59} -12.2114 q^{61} +4.88479 q^{65} -5.27053 q^{67} -1.54811 q^{69} -6.81864 q^{71} +11.5481 q^{73} +8.00000 q^{75} +11.0821 q^{79} -3.57414 q^{81} -5.23746 q^{83} +7.51504 q^{85} +0.329638 q^{87} +1.45893 q^{89} +5.72242 q^{93} -1.42586 q^{95} -17.6563 q^{97} -13.9669 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 3 * q + q^3 - 2 * q^5 + 6 * q^9 $$3 q + q^{3} - 2 q^{5} + 6 q^{9} - 3 q^{11} - 8 q^{13} + 7 q^{15} - 9 q^{17} + 3 q^{19} + 4 q^{23} + 7 q^{25} - 8 q^{27} + 11 q^{29} + 5 q^{31} + 6 q^{33} + 4 q^{37} + 5 q^{39} - 11 q^{41} - 4 q^{43} + 9 q^{45} + 2 q^{47} + 4 q^{51} + 9 q^{53} - 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} + 26 q^{65} - 9 q^{67} + 9 q^{69} + 21 q^{73} + 24 q^{75} + 6 q^{79} - 13 q^{81} + 7 q^{83} - 7 q^{85} + 26 q^{87} + 18 q^{89} + 24 q^{93} - 2 q^{95} - 28 q^{97} - 26 q^{99}+O(q^{100})$$ 3 * q + q^3 - 2 * q^5 + 6 * q^9 - 3 * q^11 - 8 * q^13 + 7 * q^15 - 9 * q^17 + 3 * q^19 + 4 * q^23 + 7 * q^25 - 8 * q^27 + 11 * q^29 + 5 * q^31 + 6 * q^33 + 4 * q^37 + 5 * q^39 - 11 * q^41 - 4 * q^43 + 9 * q^45 + 2 * q^47 + 4 * q^51 + 9 * q^53 - 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 + 26 * q^65 - 9 * q^67 + 9 * q^69 + 21 * q^73 + 24 * q^75 + 6 * q^79 - 13 * q^81 + 7 * q^83 - 7 * q^85 + 26 * q^87 + 18 * q^89 + 24 * q^93 - 2 * q^95 - 28 * q^97 - 26 * q^99

## 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
$$3$$ −2.69639 −1.55676 −0.778382 0.627791i $$-0.783959\pi$$
−0.778382 + 0.627791i $$0.783959\pi$$
$$4$$ 0 0
$$5$$ −1.42586 −0.637663 −0.318832 0.947811i $$-0.603290\pi$$
−0.318832 + 0.947811i $$0.603290\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 4.27053 1.42351
$$10$$ 0 0
$$11$$ −3.27053 −0.986103 −0.493052 0.870000i $$-0.664119\pi$$
−0.493052 + 0.870000i $$0.664119\pi$$
$$12$$ 0 0
$$13$$ −3.42586 −0.950162 −0.475081 0.879942i $$-0.657581\pi$$
−0.475081 + 0.879942i $$0.657581\pi$$
$$14$$ 0 0
$$15$$ 3.84468 0.992691
$$16$$ 0 0
$$17$$ −5.27053 −1.27829 −0.639146 0.769085i $$-0.720712\pi$$
−0.639146 + 0.769085i $$0.720712\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.574141 0.119717 0.0598584 0.998207i $$-0.480935\pi$$
0.0598584 + 0.998207i $$0.480935\pi$$
$$24$$ 0 0
$$25$$ −2.96693 −0.593385
$$26$$ 0 0
$$27$$ −3.42586 −0.659307
$$28$$ 0 0
$$29$$ −0.122252 −0.0227015 −0.0113508 0.999936i $$-0.503613\pi$$
−0.0113508 + 0.999936i $$0.503613\pi$$
$$30$$ 0 0
$$31$$ −2.12225 −0.381168 −0.190584 0.981671i $$-0.561038\pi$$
−0.190584 + 0.981671i $$0.561038\pi$$
$$32$$ 0 0
$$33$$ 8.81864 1.53513
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.96693 −0.652159 −0.326079 0.945342i $$-0.605728\pi$$
−0.326079 + 0.945342i $$0.605728\pi$$
$$38$$ 0 0
$$39$$ 9.23746 1.47918
$$40$$ 0 0
$$41$$ 4.66332 0.728288 0.364144 0.931343i $$-0.381362\pi$$
0.364144 + 0.931343i $$0.381362\pi$$
$$42$$ 0 0
$$43$$ −11.9339 −1.81990 −0.909948 0.414723i $$-0.863879\pi$$
−0.909948 + 0.414723i $$0.863879\pi$$
$$44$$ 0 0
$$45$$ −6.08918 −0.907721
$$46$$ 0 0
$$47$$ −3.11521 −0.454400 −0.227200 0.973848i $$-0.572957\pi$$
−0.227200 + 0.973848i $$0.572957\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 14.2114 1.99000
$$52$$ 0 0
$$53$$ −6.08918 −0.836413 −0.418206 0.908352i $$-0.637341\pi$$
−0.418206 + 0.908352i $$0.637341\pi$$
$$54$$ 0 0
$$55$$ 4.66332 0.628802
$$56$$ 0 0
$$57$$ −2.69639 −0.357146
$$58$$ 0 0
$$59$$ 1.72242 0.224240 0.112120 0.993695i $$-0.464236\pi$$
0.112120 + 0.993695i $$0.464236\pi$$
$$60$$ 0 0
$$61$$ −12.2114 −1.56351 −0.781757 0.623584i $$-0.785676\pi$$
−0.781757 + 0.623584i $$0.785676\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.88479 0.605884
$$66$$ 0 0
$$67$$ −5.27053 −0.643898 −0.321949 0.946757i $$-0.604338\pi$$
−0.321949 + 0.946757i $$0.604338\pi$$
$$68$$ 0 0
$$69$$ −1.54811 −0.186371
$$70$$ 0 0
$$71$$ −6.81864 −0.809224 −0.404612 0.914488i $$-0.632593\pi$$
−0.404612 + 0.914488i $$0.632593\pi$$
$$72$$ 0 0
$$73$$ 11.5481 1.35160 0.675802 0.737083i $$-0.263797\pi$$
0.675802 + 0.737083i $$0.263797\pi$$
$$74$$ 0 0
$$75$$ 8.00000 0.923760
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.0821 1.24684 0.623419 0.781888i $$-0.285743\pi$$
0.623419 + 0.781888i $$0.285743\pi$$
$$80$$ 0 0
$$81$$ −3.57414 −0.397127
$$82$$ 0 0
$$83$$ −5.23746 −0.574886 −0.287443 0.957798i $$-0.592805\pi$$
−0.287443 + 0.957798i $$0.592805\pi$$
$$84$$ 0 0
$$85$$ 7.51504 0.815120
$$86$$ 0 0
$$87$$ 0.329638 0.0353409
$$88$$ 0 0
$$89$$ 1.45893 0.154646 0.0773232 0.997006i $$-0.475363\pi$$
0.0773232 + 0.997006i $$0.475363\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 5.72242 0.593388
$$94$$ 0 0
$$95$$ −1.42586 −0.146290
$$96$$ 0 0
$$97$$ −17.6563 −1.79272 −0.896362 0.443324i $$-0.853799\pi$$
−0.896362 + 0.443324i $$0.853799\pi$$
$$98$$ 0 0
$$99$$ −13.9669 −1.40373
$$100$$ 0 0
$$101$$ −17.0491 −1.69645 −0.848223 0.529640i $$-0.822327\pi$$
−0.848223 + 0.529640i $$0.822327\pi$$
$$102$$ 0 0
$$103$$ 11.3597 1.11931 0.559653 0.828727i $$-0.310934\pi$$
0.559653 + 0.828727i $$0.310934\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10.8186 1.04588 0.522939 0.852370i $$-0.324836\pi$$
0.522939 + 0.852370i $$0.324836\pi$$
$$108$$ 0 0
$$109$$ −12.2776 −1.17598 −0.587989 0.808869i $$-0.700080\pi$$
−0.587989 + 0.808869i $$0.700080\pi$$
$$110$$ 0 0
$$111$$ 10.6964 1.01526
$$112$$ 0 0
$$113$$ −0.696393 −0.0655111 −0.0327556 0.999463i $$-0.510428\pi$$
−0.0327556 + 0.999463i $$0.510428\pi$$
$$114$$ 0 0
$$115$$ −0.818644 −0.0763390
$$116$$ 0 0
$$117$$ −14.6302 −1.35257
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −0.303607 −0.0276007
$$122$$ 0 0
$$123$$ −12.5741 −1.13377
$$124$$ 0 0
$$125$$ 11.3597 1.01604
$$126$$ 0 0
$$127$$ −5.68935 −0.504848 −0.252424 0.967617i $$-0.581228\pi$$
−0.252424 + 0.967617i $$0.581228\pi$$
$$128$$ 0 0
$$129$$ 32.1784 2.83315
$$130$$ 0 0
$$131$$ 15.2044 1.32841 0.664207 0.747549i $$-0.268769\pi$$
0.664207 + 0.747549i $$0.268769\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 4.88479 0.420416
$$136$$ 0 0
$$137$$ 3.70344 0.316406 0.158203 0.987407i $$-0.449430\pi$$
0.158203 + 0.987407i $$0.449430\pi$$
$$138$$ 0 0
$$139$$ 11.6704 0.989867 0.494934 0.868931i $$-0.335192\pi$$
0.494934 + 0.868931i $$0.335192\pi$$
$$140$$ 0 0
$$141$$ 8.39983 0.707393
$$142$$ 0 0
$$143$$ 11.2044 0.936958
$$144$$ 0 0
$$145$$ 0.174313 0.0144759
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.8186 1.54168 0.770842 0.637027i $$-0.219836\pi$$
0.770842 + 0.637027i $$0.219836\pi$$
$$150$$ 0 0
$$151$$ −4.43290 −0.360744 −0.180372 0.983598i $$-0.557730\pi$$
−0.180372 + 0.983598i $$0.557730\pi$$
$$152$$ 0 0
$$153$$ −22.5080 −1.81966
$$154$$ 0 0
$$155$$ 3.02603 0.243057
$$156$$ 0 0
$$157$$ 9.82569 0.784175 0.392088 0.919928i $$-0.371753\pi$$
0.392088 + 0.919928i $$0.371753\pi$$
$$158$$ 0 0
$$159$$ 16.4188 1.30210
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.9409 −0.856957 −0.428479 0.903552i $$-0.640950\pi$$
−0.428479 + 0.903552i $$0.640950\pi$$
$$164$$ 0 0
$$165$$ −12.5741 −0.978896
$$166$$ 0 0
$$167$$ 12.7525 0.986818 0.493409 0.869797i $$-0.335751\pi$$
0.493409 + 0.869797i $$0.335751\pi$$
$$168$$ 0 0
$$169$$ −1.26349 −0.0971917
$$170$$ 0 0
$$171$$ 4.27053 0.326576
$$172$$ 0 0
$$173$$ −22.5411 −1.71377 −0.856883 0.515511i $$-0.827602\pi$$
−0.856883 + 0.515511i $$0.827602\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.64433 −0.349089
$$178$$ 0 0
$$179$$ 6.41882 0.479765 0.239882 0.970802i $$-0.422891\pi$$
0.239882 + 0.970802i $$0.422891\pi$$
$$180$$ 0 0
$$181$$ −9.81160 −0.729291 −0.364645 0.931146i $$-0.618810\pi$$
−0.364645 + 0.931146i $$0.618810\pi$$
$$182$$ 0 0
$$183$$ 32.9268 2.43402
$$184$$ 0 0
$$185$$ 5.65628 0.415858
$$186$$ 0 0
$$187$$ 17.2375 1.26053
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 15.2515 1.10356 0.551782 0.833989i $$-0.313948\pi$$
0.551782 + 0.833989i $$0.313948\pi$$
$$192$$ 0 0
$$193$$ −15.7455 −1.13338 −0.566691 0.823930i $$-0.691777\pi$$
−0.566691 + 0.823930i $$0.691777\pi$$
$$194$$ 0 0
$$195$$ −13.1713 −0.943217
$$196$$ 0 0
$$197$$ 24.8086 1.76754 0.883770 0.467922i $$-0.154997\pi$$
0.883770 + 0.467922i $$0.154997\pi$$
$$198$$ 0 0
$$199$$ 2.58823 0.183474 0.0917372 0.995783i $$-0.470758\pi$$
0.0917372 + 0.995783i $$0.470758\pi$$
$$200$$ 0 0
$$201$$ 14.2114 1.00240
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.64924 −0.464403
$$206$$ 0 0
$$207$$ 2.45189 0.170418
$$208$$ 0 0
$$209$$ −3.27053 −0.226228
$$210$$ 0 0
$$211$$ 21.7785 1.49930 0.749648 0.661837i $$-0.230223\pi$$
0.749648 + 0.661837i $$0.230223\pi$$
$$212$$ 0 0
$$213$$ 18.3857 1.25977
$$214$$ 0 0
$$215$$ 17.0160 1.16048
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −31.1382 −2.10413
$$220$$ 0 0
$$221$$ 18.0561 1.21459
$$222$$ 0 0
$$223$$ 9.90078 0.663005 0.331503 0.943454i $$-0.392444\pi$$
0.331503 + 0.943454i $$0.392444\pi$$
$$224$$ 0 0
$$225$$ −12.6704 −0.844691
$$226$$ 0 0
$$227$$ −2.71538 −0.180226 −0.0901131 0.995932i $$-0.528723\pi$$
−0.0901131 + 0.995932i $$0.528723\pi$$
$$228$$ 0 0
$$229$$ 8.29656 0.548252 0.274126 0.961694i $$-0.411611\pi$$
0.274126 + 0.961694i $$0.411611\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.7295 0.964959 0.482480 0.875907i $$-0.339736\pi$$
0.482480 + 0.875907i $$0.339736\pi$$
$$234$$ 0 0
$$235$$ 4.44185 0.289754
$$236$$ 0 0
$$237$$ −29.8818 −1.94103
$$238$$ 0 0
$$239$$ 28.9970 1.87566 0.937830 0.347095i $$-0.112832\pi$$
0.937830 + 0.347095i $$0.112832\pi$$
$$240$$ 0 0
$$241$$ 4.31065 0.277673 0.138837 0.990315i $$-0.455664\pi$$
0.138837 + 0.990315i $$0.455664\pi$$
$$242$$ 0 0
$$243$$ 19.9149 1.27754
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.42586 −0.217982
$$248$$ 0 0
$$249$$ 14.1223 0.894961
$$250$$ 0 0
$$251$$ 10.4329 0.658519 0.329259 0.944239i $$-0.393201\pi$$
0.329259 + 0.944239i $$0.393201\pi$$
$$252$$ 0 0
$$253$$ −1.87775 −0.118053
$$254$$ 0 0
$$255$$ −20.2635 −1.26895
$$256$$ 0 0
$$257$$ −23.7124 −1.47914 −0.739569 0.673081i $$-0.764971\pi$$
−0.739569 + 0.673081i $$0.764971\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −0.522079 −0.0323159
$$262$$ 0 0
$$263$$ 30.9409 1.90790 0.953949 0.299970i $$-0.0969766\pi$$
0.953949 + 0.299970i $$0.0969766\pi$$
$$264$$ 0 0
$$265$$ 8.68231 0.533350
$$266$$ 0 0
$$267$$ −3.93385 −0.240748
$$268$$ 0 0
$$269$$ 9.17131 0.559185 0.279592 0.960119i $$-0.409801\pi$$
0.279592 + 0.960119i $$0.409801\pi$$
$$270$$ 0 0
$$271$$ −20.0420 −1.21747 −0.608733 0.793375i $$-0.708322\pi$$
−0.608733 + 0.793375i $$0.708322\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 9.70344 0.585139
$$276$$ 0 0
$$277$$ −7.45893 −0.448164 −0.224082 0.974570i $$-0.571938\pi$$
−0.224082 + 0.974570i $$0.571938\pi$$
$$278$$ 0 0
$$279$$ −9.06315 −0.542596
$$280$$ 0 0
$$281$$ 12.2114 0.728473 0.364236 0.931307i $$-0.381330\pi$$
0.364236 + 0.931307i $$0.381330\pi$$
$$282$$ 0 0
$$283$$ 8.02303 0.476920 0.238460 0.971152i $$-0.423357\pi$$
0.238460 + 0.971152i $$0.423357\pi$$
$$284$$ 0 0
$$285$$ 3.84468 0.227739
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 10.7785 0.634031
$$290$$ 0 0
$$291$$ 47.6083 2.79085
$$292$$ 0 0
$$293$$ −14.0331 −0.819821 −0.409910 0.912126i $$-0.634440\pi$$
−0.409910 + 0.912126i $$0.634440\pi$$
$$294$$ 0 0
$$295$$ −2.45593 −0.142990
$$296$$ 0 0
$$297$$ 11.2044 0.650145
$$298$$ 0 0
$$299$$ −1.96693 −0.113750
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 45.9710 2.64096
$$304$$ 0 0
$$305$$ 17.4118 0.996995
$$306$$ 0 0
$$307$$ −16.0702 −0.917174 −0.458587 0.888649i $$-0.651644\pi$$
−0.458587 + 0.888649i $$0.651644\pi$$
$$308$$ 0 0
$$309$$ −30.6302 −1.74249
$$310$$ 0 0
$$311$$ 1.56710 0.0888620 0.0444310 0.999012i $$-0.485853\pi$$
0.0444310 + 0.999012i $$0.485853\pi$$
$$312$$ 0 0
$$313$$ −29.6232 −1.67440 −0.837201 0.546895i $$-0.815810\pi$$
−0.837201 + 0.546895i $$0.815810\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −22.4890 −1.26311 −0.631554 0.775332i $$-0.717583\pi$$
−0.631554 + 0.775332i $$0.717583\pi$$
$$318$$ 0 0
$$319$$ 0.399828 0.0223861
$$320$$ 0 0
$$321$$ −29.1713 −1.62818
$$322$$ 0 0
$$323$$ −5.27053 −0.293260
$$324$$ 0 0
$$325$$ 10.1643 0.563812
$$326$$ 0 0
$$327$$ 33.1052 1.83072
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.3857 1.56022 0.780111 0.625641i $$-0.215163\pi$$
0.780111 + 0.625641i $$0.215163\pi$$
$$332$$ 0 0
$$333$$ −16.9409 −0.928355
$$334$$ 0 0
$$335$$ 7.51504 0.410590
$$336$$ 0 0
$$337$$ 29.6272 1.61390 0.806950 0.590620i $$-0.201117\pi$$
0.806950 + 0.590620i $$0.201117\pi$$
$$338$$ 0 0
$$339$$ 1.87775 0.101985
$$340$$ 0 0
$$341$$ 6.94090 0.375871
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2.20739 0.118842
$$346$$ 0 0
$$347$$ −32.8607 −1.76405 −0.882026 0.471200i $$-0.843821\pi$$
−0.882026 + 0.471200i $$0.843821\pi$$
$$348$$ 0 0
$$349$$ 32.5641 1.74312 0.871558 0.490292i $$-0.163110\pi$$
0.871558 + 0.490292i $$0.163110\pi$$
$$350$$ 0 0
$$351$$ 11.7365 0.626448
$$352$$ 0 0
$$353$$ 30.6633 1.63204 0.816022 0.578021i $$-0.196175\pi$$
0.816022 + 0.578021i $$0.196175\pi$$
$$354$$ 0 0
$$355$$ 9.72242 0.516013
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.67740 −0.352420 −0.176210 0.984353i $$-0.556384\pi$$
−0.176210 + 0.984353i $$0.556384\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0.818644 0.0429677
$$364$$ 0 0
$$365$$ −16.4660 −0.861868
$$366$$ 0 0
$$367$$ −5.19544 −0.271200 −0.135600 0.990764i $$-0.543296\pi$$
−0.135600 + 0.990764i $$0.543296\pi$$
$$368$$ 0 0
$$369$$ 19.9149 1.03673
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.31769 −0.0682275 −0.0341137 0.999418i $$-0.510861\pi$$
−0.0341137 + 0.999418i $$0.510861\pi$$
$$374$$ 0 0
$$375$$ −30.6302 −1.58174
$$376$$ 0 0
$$377$$ 0.418817 0.0215701
$$378$$ 0 0
$$379$$ −15.6563 −0.804209 −0.402104 0.915594i $$-0.631721\pi$$
−0.402104 + 0.915594i $$0.631721\pi$$
$$380$$ 0 0
$$381$$ 15.3407 0.785929
$$382$$ 0 0
$$383$$ −29.0631 −1.48506 −0.742529 0.669814i $$-0.766374\pi$$
−0.742529 + 0.669814i $$0.766374\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −50.9639 −2.59064
$$388$$ 0 0
$$389$$ 10.7295 0.544006 0.272003 0.962296i $$-0.412314\pi$$
0.272003 + 0.962296i $$0.412314\pi$$
$$390$$ 0 0
$$391$$ −3.02603 −0.153033
$$392$$ 0 0
$$393$$ −40.9970 −2.06803
$$394$$ 0 0
$$395$$ −15.8016 −0.795063
$$396$$ 0 0
$$397$$ −10.7997 −0.542019 −0.271010 0.962577i $$-0.587358\pi$$
−0.271010 + 0.962577i $$0.587358\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.66332 −0.332750 −0.166375 0.986063i $$-0.553206\pi$$
−0.166375 + 0.986063i $$0.553206\pi$$
$$402$$ 0 0
$$403$$ 7.27053 0.362171
$$404$$ 0 0
$$405$$ 5.09622 0.253233
$$406$$ 0 0
$$407$$ 12.9740 0.643096
$$408$$ 0 0
$$409$$ 25.5481 1.26327 0.631636 0.775265i $$-0.282384\pi$$
0.631636 + 0.775265i $$0.282384\pi$$
$$410$$ 0 0
$$411$$ −9.98592 −0.492569
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 7.46788 0.366584
$$416$$ 0 0
$$417$$ −31.4679 −1.54099
$$418$$ 0 0
$$419$$ −33.2415 −1.62395 −0.811977 0.583690i $$-0.801608\pi$$
−0.811977 + 0.583690i $$0.801608\pi$$
$$420$$ 0 0
$$421$$ 2.95094 0.143820 0.0719099 0.997411i $$-0.477091\pi$$
0.0719099 + 0.997411i $$0.477091\pi$$
$$422$$ 0 0
$$423$$ −13.3036 −0.646844
$$424$$ 0 0
$$425$$ 15.6373 0.758520
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −30.2114 −1.45862
$$430$$ 0 0
$$431$$ 4.29656 0.206958 0.103479 0.994632i $$-0.467003\pi$$
0.103479 + 0.994632i $$0.467003\pi$$
$$432$$ 0 0
$$433$$ −5.96693 −0.286752 −0.143376 0.989668i $$-0.545796\pi$$
−0.143376 + 0.989668i $$0.545796\pi$$
$$434$$ 0 0
$$435$$ −0.470017 −0.0225356
$$436$$ 0 0
$$437$$ 0.574141 0.0274649
$$438$$ 0 0
$$439$$ −37.5711 −1.79317 −0.896586 0.442869i $$-0.853961\pi$$
−0.896586 + 0.442869i $$0.853961\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −15.2515 −0.724623 −0.362311 0.932057i $$-0.618012\pi$$
−0.362311 + 0.932057i $$0.618012\pi$$
$$444$$ 0 0
$$445$$ −2.08023 −0.0986124
$$446$$ 0 0
$$447$$ −50.7425 −2.40004
$$448$$ 0 0
$$449$$ −35.7926 −1.68916 −0.844579 0.535431i $$-0.820149\pi$$
−0.844579 + 0.535431i $$0.820149\pi$$
$$450$$ 0 0
$$451$$ −15.2515 −0.718167
$$452$$ 0 0
$$453$$ 11.9528 0.561594
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.6443 0.872145 0.436073 0.899912i $$-0.356369\pi$$
0.436073 + 0.899912i $$0.356369\pi$$
$$458$$ 0 0
$$459$$ 18.0561 0.842787
$$460$$ 0 0
$$461$$ 11.8306 0.551006 0.275503 0.961300i $$-0.411156\pi$$
0.275503 + 0.961300i $$0.411156\pi$$
$$462$$ 0 0
$$463$$ 0.752498 0.0349715 0.0174858 0.999847i $$-0.494434\pi$$
0.0174858 + 0.999847i $$0.494434\pi$$
$$464$$ 0 0
$$465$$ −8.15937 −0.378382
$$466$$ 0 0
$$467$$ 13.8588 0.641307 0.320653 0.947197i $$-0.396098\pi$$
0.320653 + 0.947197i $$0.396098\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −26.4939 −1.22077
$$472$$ 0 0
$$473$$ 39.0301 1.79460
$$474$$ 0 0
$$475$$ −2.96693 −0.136132
$$476$$ 0 0
$$477$$ −26.0040 −1.19064
$$478$$ 0 0
$$479$$ 7.58118 0.346393 0.173197 0.984887i $$-0.444590\pi$$
0.173197 + 0.984887i $$0.444590\pi$$
$$480$$ 0 0
$$481$$ 13.5901 0.619657
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 25.1754 1.14315
$$486$$ 0 0
$$487$$ −10.6864 −0.484245 −0.242122 0.970246i $$-0.577844\pi$$
−0.242122 + 0.970246i $$0.577844\pi$$
$$488$$ 0 0
$$489$$ 29.5010 1.33408
$$490$$ 0 0
$$491$$ −14.1643 −0.639225 −0.319612 0.947548i $$-0.603553\pi$$
−0.319612 + 0.947548i $$0.603553\pi$$
$$492$$ 0 0
$$493$$ 0.644331 0.0290192
$$494$$ 0 0
$$495$$ 19.9149 0.895107
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −23.9108 −1.07040 −0.535198 0.844727i $$-0.679763\pi$$
−0.535198 + 0.844727i $$0.679763\pi$$
$$500$$ 0 0
$$501$$ −34.3857 −1.53624
$$502$$ 0 0
$$503$$ 37.4388 1.66932 0.834658 0.550769i $$-0.185665\pi$$
0.834658 + 0.550769i $$0.185665\pi$$
$$504$$ 0 0
$$505$$ 24.3096 1.08176
$$506$$ 0 0
$$507$$ 3.40687 0.151304
$$508$$ 0 0
$$509$$ −31.3126 −1.38790 −0.693952 0.720021i $$-0.744132\pi$$
−0.693952 + 0.720021i $$0.744132\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −3.42586 −0.151255
$$514$$ 0 0
$$515$$ −16.1973 −0.713740
$$516$$ 0 0
$$517$$ 10.1884 0.448085
$$518$$ 0 0
$$519$$ 60.7796 2.66793
$$520$$ 0 0
$$521$$ −21.8346 −0.956593 −0.478296 0.878199i $$-0.658746\pi$$
−0.478296 + 0.878199i $$0.658746\pi$$
$$522$$ 0 0
$$523$$ −9.37380 −0.409888 −0.204944 0.978774i $$-0.565701\pi$$
−0.204944 + 0.978774i $$0.565701\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 11.1854 0.487244
$$528$$ 0 0
$$529$$ −22.6704 −0.985668
$$530$$ 0 0
$$531$$ 7.35567 0.319209
$$532$$ 0 0
$$533$$ −15.9759 −0.691992
$$534$$ 0 0
$$535$$ −15.4259 −0.666918
$$536$$ 0 0
$$537$$ −17.3077 −0.746880
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −30.4608 −1.30961 −0.654807 0.755796i $$-0.727250\pi$$
−0.654807 + 0.755796i $$0.727250\pi$$
$$542$$ 0 0
$$543$$ 26.4559 1.13533
$$544$$ 0 0
$$545$$ 17.5061 0.749878
$$546$$ 0 0
$$547$$ 34.2675 1.46517 0.732587 0.680673i $$-0.238313\pi$$
0.732587 + 0.680673i $$0.238313\pi$$
$$548$$ 0 0
$$549$$ −52.1493 −2.22568
$$550$$ 0 0
$$551$$ −0.122252 −0.00520809
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −15.2515 −0.647392
$$556$$ 0 0
$$557$$ −1.44889 −0.0613915 −0.0306957 0.999529i $$-0.509772\pi$$
−0.0306957 + 0.999529i $$0.509772\pi$$
$$558$$ 0 0
$$559$$ 40.8837 1.72920
$$560$$ 0 0
$$561$$ −46.4790 −1.96234
$$562$$ 0 0
$$563$$ 10.8707 0.458146 0.229073 0.973409i $$-0.426431\pi$$
0.229073 + 0.973409i $$0.426431\pi$$
$$564$$ 0 0
$$565$$ 0.992958 0.0417740
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 25.0772 1.05129 0.525646 0.850703i $$-0.323824\pi$$
0.525646 + 0.850703i $$0.323824\pi$$
$$570$$ 0 0
$$571$$ −11.6042 −0.485621 −0.242811 0.970074i $$-0.578069\pi$$
−0.242811 + 0.970074i $$0.578069\pi$$
$$572$$ 0 0
$$573$$ −41.1242 −1.71799
$$574$$ 0 0
$$575$$ −1.70344 −0.0710382
$$576$$ 0 0
$$577$$ −26.5782 −1.10646 −0.553232 0.833027i $$-0.686606\pi$$
−0.553232 + 0.833027i $$0.686606\pi$$
$$578$$ 0 0
$$579$$ 42.4559 1.76441
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 19.9149 0.824789
$$584$$ 0 0
$$585$$ 20.8607 0.862482
$$586$$ 0 0
$$587$$ 16.5552 0.683304 0.341652 0.939826i $$-0.389014\pi$$
0.341652 + 0.939826i $$0.389014\pi$$
$$588$$ 0 0
$$589$$ −2.12225 −0.0874459
$$590$$ 0 0
$$591$$ −66.8937 −2.75164
$$592$$ 0 0
$$593$$ −26.6864 −1.09588 −0.547939 0.836519i $$-0.684587\pi$$
−0.547939 + 0.836519i $$0.684587\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6.97887 −0.285626
$$598$$ 0 0
$$599$$ −19.6603 −0.803299 −0.401649 0.915793i $$-0.631563\pi$$
−0.401649 + 0.915793i $$0.631563\pi$$
$$600$$ 0 0
$$601$$ 23.1854 0.945752 0.472876 0.881129i $$-0.343216\pi$$
0.472876 + 0.881129i $$0.343216\pi$$
$$602$$ 0 0
$$603$$ −22.5080 −0.916596
$$604$$ 0 0
$$605$$ 0.432901 0.0175999
$$606$$ 0 0
$$607$$ −0.441848 −0.0179341 −0.00896704 0.999960i $$-0.502854\pi$$
−0.00896704 + 0.999960i $$0.502854\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10.6723 0.431754
$$612$$ 0 0
$$613$$ −28.3478 −1.14496 −0.572478 0.819920i $$-0.694018\pi$$
−0.572478 + 0.819920i $$0.694018\pi$$
$$614$$ 0 0
$$615$$ 17.9289 0.722965
$$616$$ 0 0
$$617$$ 31.7455 1.27802 0.639012 0.769197i $$-0.279343\pi$$
0.639012 + 0.769197i $$0.279343\pi$$
$$618$$ 0 0
$$619$$ −22.2535 −0.894442 −0.447221 0.894424i $$-0.647586\pi$$
−0.447221 + 0.894424i $$0.647586\pi$$
$$620$$ 0 0
$$621$$ −1.96693 −0.0789301
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.36271 −0.0545085
$$626$$ 0 0
$$627$$ 8.81864 0.352183
$$628$$ 0 0
$$629$$ 20.9078 0.833649
$$630$$ 0 0
$$631$$ −19.4719 −0.775165 −0.387583 0.921835i $$-0.626690\pi$$
−0.387583 + 0.921835i $$0.626690\pi$$
$$632$$ 0 0
$$633$$ −58.7235 −2.33405
$$634$$ 0 0
$$635$$ 8.11221 0.321923
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −29.1193 −1.15194
$$640$$ 0 0
$$641$$ 40.2345 1.58917 0.794583 0.607156i $$-0.207690\pi$$
0.794583 + 0.607156i $$0.207690\pi$$
$$642$$ 0 0
$$643$$ 17.1011 0.674403 0.337201 0.941433i $$-0.390520\pi$$
0.337201 + 0.941433i $$0.390520\pi$$
$$644$$ 0 0
$$645$$ −45.8818 −1.80659
$$646$$ 0 0
$$647$$ 5.37380 0.211266 0.105633 0.994405i $$-0.466313\pi$$
0.105633 + 0.994405i $$0.466313\pi$$
$$648$$ 0 0
$$649$$ −5.63325 −0.221124
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11.6183 −0.454659 −0.227330 0.973818i $$-0.572999\pi$$
−0.227330 + 0.973818i $$0.572999\pi$$
$$654$$ 0 0
$$655$$ −21.6793 −0.847081
$$656$$ 0 0
$$657$$ 49.3166 1.92402
$$658$$ 0 0
$$659$$ −41.8718 −1.63109 −0.815546 0.578692i $$-0.803563\pi$$
−0.815546 + 0.578692i $$0.803563\pi$$
$$660$$ 0 0
$$661$$ −6.71942 −0.261355 −0.130678 0.991425i $$-0.541715\pi$$
−0.130678 + 0.991425i $$0.541715\pi$$
$$662$$ 0 0
$$663$$ −48.6864 −1.89082
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.0701896 −0.00271775
$$668$$ 0 0
$$669$$ −26.6964 −1.03214
$$670$$ 0 0
$$671$$ 39.9379 1.54179
$$672$$ 0 0
$$673$$ 43.5280 1.67788 0.838941 0.544222i $$-0.183175\pi$$
0.838941 + 0.544222i $$0.183175\pi$$
$$674$$ 0 0
$$675$$ 10.1643 0.391223
$$676$$ 0 0
$$677$$ −18.1033 −0.695765 −0.347882 0.937538i $$-0.613099\pi$$
−0.347882 + 0.937538i $$0.613099\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 7.32173 0.280569
$$682$$ 0 0
$$683$$ 27.3456 1.04635 0.523176 0.852225i $$-0.324747\pi$$
0.523176 + 0.852225i $$0.324747\pi$$
$$684$$ 0 0
$$685$$ −5.28058 −0.201760
$$686$$ 0 0
$$687$$ −22.3708 −0.853499
$$688$$ 0 0
$$689$$ 20.8607 0.794728
$$690$$ 0 0
$$691$$ 30.8316 1.17289 0.586445 0.809989i $$-0.300527\pi$$
0.586445 + 0.809989i $$0.300527\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −16.6403 −0.631202
$$696$$ 0 0
$$697$$ −24.5782 −0.930965
$$698$$ 0 0
$$699$$ −39.7164 −1.50221
$$700$$ 0 0
$$701$$ −8.23042 −0.310859 −0.155429 0.987847i $$-0.549676\pi$$
−0.155429 + 0.987847i $$0.549676\pi$$
$$702$$ 0 0
$$703$$ −3.96693 −0.149615
$$704$$ 0 0
$$705$$ −11.9770 −0.451079
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −15.6183 −0.586558 −0.293279 0.956027i $$-0.594746\pi$$
−0.293279 + 0.956027i $$0.594746\pi$$
$$710$$ 0 0
$$711$$ 47.3266 1.77489
$$712$$ 0 0
$$713$$ −1.21847 −0.0456321
$$714$$ 0 0
$$715$$ −15.9759 −0.597464
$$716$$ 0 0
$$717$$ −78.1873 −2.91996
$$718$$ 0 0
$$719$$ −16.3567 −0.610002 −0.305001 0.952352i $$-0.598657\pi$$
−0.305001 + 0.952352i $$0.598657\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −11.6232 −0.432272
$$724$$ 0 0
$$725$$ 0.362711 0.0134708
$$726$$ 0 0
$$727$$ 17.6183 0.653427 0.326713 0.945123i $$-0.394059\pi$$
0.326713 + 0.945123i $$0.394059\pi$$
$$728$$ 0 0
$$729$$ −42.9759 −1.59170
$$730$$ 0 0
$$731$$ 62.8978 2.32636
$$732$$ 0 0
$$733$$ 42.8837 1.58395 0.791973 0.610556i $$-0.209054\pi$$
0.791973 + 0.610556i $$0.209054\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 17.2375 0.634950
$$738$$ 0 0
$$739$$ 19.4259 0.714592 0.357296 0.933991i $$-0.383699\pi$$
0.357296 + 0.933991i $$0.383699\pi$$
$$740$$ 0 0
$$741$$ 9.23746 0.339347
$$742$$ 0 0
$$743$$ 51.7305 1.89781 0.948904 0.315564i $$-0.102194\pi$$
0.948904 + 0.315564i $$0.102194\pi$$
$$744$$ 0 0
$$745$$ −26.8327 −0.983075
$$746$$ 0 0
$$747$$ −22.3668 −0.818357
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 23.9428 0.873685 0.436843 0.899538i $$-0.356097\pi$$
0.436843 + 0.899538i $$0.356097\pi$$
$$752$$ 0 0
$$753$$ −28.1312 −1.02516
$$754$$ 0 0
$$755$$ 6.32069 0.230033
$$756$$ 0 0
$$757$$ −2.48901 −0.0904645 −0.0452322 0.998976i $$-0.514403\pi$$
−0.0452322 + 0.998976i $$0.514403\pi$$
$$758$$ 0 0
$$759$$ 5.06315 0.183781
$$760$$ 0 0
$$761$$ 4.14528 0.150266 0.0751332 0.997174i $$-0.476062\pi$$
0.0751332 + 0.997174i $$0.476062\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 32.0932 1.16033
$$766$$ 0 0
$$767$$ −5.90078 −0.213065
$$768$$ 0 0
$$769$$ 11.0442 0.398263 0.199131 0.979973i $$-0.436188\pi$$
0.199131 + 0.979973i $$0.436188\pi$$
$$770$$ 0 0
$$771$$ 63.9379 2.30267
$$772$$ 0 0
$$773$$ 42.3426 1.52296 0.761479 0.648189i $$-0.224473\pi$$
0.761479 + 0.648189i $$0.224473\pi$$
$$774$$ 0 0
$$775$$ 6.29656 0.226179
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.66332 0.167081
$$780$$ 0 0
$$781$$ 22.3006 0.797979
$$782$$ 0 0
$$783$$ 0.418817 0.0149673
$$784$$ 0 0
$$785$$ −14.0100 −0.500040
$$786$$ 0 0
$$787$$ −49.5711 −1.76702 −0.883510 0.468412i $$-0.844826\pi$$
−0.883510 + 0.468412i $$0.844826\pi$$
$$788$$ 0 0
$$789$$ −83.4288 −2.97014
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 41.8346 1.48559
$$794$$ 0 0
$$795$$ −23.4109 −0.830300
$$796$$ 0 0
$$797$$ −38.8276 −1.37534 −0.687672 0.726022i $$-0.741367\pi$$
−0.687672 + 0.726022i $$0.741367\pi$$
$$798$$ 0 0
$$799$$ 16.4188 0.580856
$$800$$ 0 0
$$801$$ 6.23042 0.220141
$$802$$ 0 0
$$803$$ −37.7685 −1.33282
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −24.7295 −0.870518
$$808$$ 0 0
$$809$$ 13.4920 0.474354 0.237177 0.971466i $$-0.423778\pi$$
0.237177 + 0.971466i $$0.423778\pi$$
$$810$$ 0 0
$$811$$ −24.5552 −0.862248 −0.431124 0.902293i $$-0.641883\pi$$
−0.431124 + 0.902293i $$0.641883\pi$$
$$812$$ 0 0
$$813$$ 54.0412 1.89531
$$814$$ 0 0
$$815$$ 15.6002 0.546450
$$816$$ 0 0
$$817$$ −11.9339 −0.417513
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −37.9339 −1.32390 −0.661950 0.749548i $$-0.730271\pi$$
−0.661950 + 0.749548i $$0.730271\pi$$
$$822$$ 0 0
$$823$$ −10.0141 −0.349069 −0.174535 0.984651i $$-0.555842\pi$$
−0.174535 + 0.984651i $$0.555842\pi$$
$$824$$ 0 0
$$825$$ −26.1643 −0.910923
$$826$$ 0 0
$$827$$ −35.7685 −1.24379 −0.621896 0.783100i $$-0.713637\pi$$
−0.621896 + 0.783100i $$0.713637\pi$$
$$828$$ 0 0
$$829$$ −25.6753 −0.891739 −0.445869 0.895098i $$-0.647105\pi$$
−0.445869 + 0.895098i $$0.647105\pi$$
$$830$$ 0 0
$$831$$ 20.1122 0.697685
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −18.1833 −0.629258
$$836$$ 0 0
$$837$$ 7.27053 0.251306
$$838$$ 0 0
$$839$$ 42.7856 1.47712 0.738561 0.674187i $$-0.235506\pi$$
0.738561 + 0.674187i $$0.235506\pi$$
$$840$$ 0 0
$$841$$ −28.9851 −0.999485
$$842$$ 0 0
$$843$$ −32.9268 −1.13406
$$844$$ 0 0
$$845$$ 1.80156 0.0619756
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −21.6332 −0.742451
$$850$$ 0 0
$$851$$ −2.27758 −0.0780743
$$852$$ 0 0
$$853$$ −17.8447 −0.610990 −0.305495 0.952194i $$-0.598822\pi$$
−0.305495 + 0.952194i $$0.598822\pi$$
$$854$$ 0 0
$$855$$ −6.08918 −0.208246
$$856$$ 0 0
$$857$$ −37.8116 −1.29162 −0.645810 0.763498i $$-0.723480\pi$$
−0.645810 + 0.763498i $$0.723480\pi$$
$$858$$ 0 0
$$859$$ −22.8558 −0.779828 −0.389914 0.920851i $$-0.627495\pi$$
−0.389914 + 0.920851i $$0.627495\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 13.3177 0.453339 0.226670 0.973972i $$-0.427216\pi$$
0.226670 + 0.973972i $$0.427216\pi$$
$$864$$ 0 0
$$865$$ 32.1404 1.09281
$$866$$ 0 0
$$867$$ −29.0631 −0.987036
$$868$$ 0 0
$$869$$ −36.2445 −1.22951
$$870$$ 0 0
$$871$$ 18.0561 0.611808
$$872$$ 0 0
$$873$$ −75.4017 −2.55196
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −33.5571 −1.13314 −0.566571 0.824013i $$-0.691730\pi$$
−0.566571 + 0.824013i $$0.691730\pi$$
$$878$$ 0 0
$$879$$ 37.8387 1.27627
$$880$$ 0 0
$$881$$ 52.5641 1.77093 0.885465 0.464707i $$-0.153840\pi$$
0.885465 + 0.464707i $$0.153840\pi$$
$$882$$ 0 0
$$883$$ −5.68445 −0.191297 −0.0956484 0.995415i $$-0.530492\pi$$
−0.0956484 + 0.995415i $$0.530492\pi$$
$$884$$ 0 0
$$885$$ 6.62216 0.222601
$$886$$ 0 0
$$887$$ −26.3898 −0.886082 −0.443041 0.896501i $$-0.646100\pi$$
−0.443041 + 0.896501i $$0.646100\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 11.6894 0.391608
$$892$$ 0 0
$$893$$ −3.11521 −0.104247
$$894$$ 0 0
$$895$$ −9.15233 −0.305929
$$896$$ 0 0
$$897$$ 5.30361 0.177082
$$898$$ 0 0
$$899$$ 0.259449 0.00865309
$$900$$ 0 0
$$901$$ 32.0932 1.06918
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 13.9900 0.465042
$$906$$ 0 0
$$907$$ 0.211430 0.00702041 0.00351021 0.999994i $$-0.498883\pi$$
0.00351021 + 0.999994i $$0.498883\pi$$
$$908$$ 0 0
$$909$$ −72.8086 −2.41491
$$910$$ 0 0
$$911$$ −21.3787 −0.708308 −0.354154 0.935187i $$-0.615231\pi$$
−0.354154 + 0.935187i $$0.615231\pi$$
$$912$$ 0 0
$$913$$ 17.1293 0.566897
$$914$$ 0 0
$$915$$ −46.9490 −1.55209
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.8186 −0.554796 −0.277398 0.960755i $$-0.589472\pi$$
−0.277398 + 0.960755i $$0.589472\pi$$
$$920$$ 0 0
$$921$$ 43.3315 1.42782
$$922$$ 0 0
$$923$$ 23.3597 0.768894
$$924$$ 0 0
$$925$$ 11.7696 0.386981
$$926$$ 0 0
$$927$$ 48.5120 1.59334
$$928$$ 0 0
$$929$$ 35.3497 1.15979 0.579893 0.814693i $$-0.303095\pi$$
0.579893 + 0.814693i $$0.303095\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −4.22551 −0.138337
$$934$$ 0 0
$$935$$ −24.5782 −0.803793
$$936$$ 0 0
$$937$$ −4.78153 −0.156206 −0.0781029 0.996945i $$-0.524886\pi$$
−0.0781029 + 0.996945i $$0.524886\pi$$
$$938$$ 0 0
$$939$$ 79.8758 2.60665
$$940$$ 0 0
$$941$$ 13.6373 0.444563 0.222281 0.974983i $$-0.428650\pi$$
0.222281 + 0.974983i $$0.428650\pi$$
$$942$$ 0 0
$$943$$ 2.67740 0.0871883
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −59.3638 −1.92906 −0.964531 0.263968i $$-0.914969\pi$$
−0.964531 + 0.263968i $$0.914969\pi$$
$$948$$ 0 0
$$949$$ −39.5622 −1.28424
$$950$$ 0 0
$$951$$ 60.6392 1.96636
$$952$$ 0 0
$$953$$ −52.4790 −1.69996 −0.849980 0.526815i $$-0.823386\pi$$
−0.849980 + 0.526815i $$0.823386\pi$$
$$954$$ 0 0
$$955$$ −21.7465 −0.703702
$$956$$ 0 0
$$957$$ −1.07809 −0.0348498
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −26.4960 −0.854711
$$962$$ 0 0
$$963$$ 46.2014 1.48882
$$964$$ 0 0
$$965$$ 22.4508 0.722717
$$966$$ 0 0
$$967$$ 21.9900 0.707149 0.353575 0.935406i $$-0.384966\pi$$
0.353575 + 0.935406i $$0.384966\pi$$
$$968$$ 0 0
$$969$$ 14.2114 0.456537
$$970$$ 0 0
$$971$$ 40.9309 1.31353 0.656767 0.754094i $$-0.271924\pi$$
0.656767 + 0.754094i $$0.271924\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −27.4069 −0.877722
$$976$$ 0 0
$$977$$ 8.56924 0.274154 0.137077 0.990560i $$-0.456229\pi$$
0.137077 + 0.990560i $$0.456229\pi$$
$$978$$ 0 0
$$979$$ −4.77149 −0.152497
$$980$$ 0 0
$$981$$ −52.4318 −1.67402
$$982$$ 0 0
$$983$$ −35.2654 −1.12479 −0.562396 0.826868i $$-0.690120\pi$$
−0.562396 + 0.826868i $$0.690120\pi$$
$$984$$ 0 0
$$985$$ −35.3736 −1.12710
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.85172 −0.217872
$$990$$ 0 0
$$991$$ 1.78367 0.0566600 0.0283300 0.999599i $$-0.490981\pi$$
0.0283300 + 0.999599i $$0.490981\pi$$
$$992$$ 0 0
$$993$$ −76.5391 −2.42890
$$994$$ 0 0
$$995$$ −3.69044 −0.116995
$$996$$ 0 0
$$997$$ 37.4299 1.18542 0.592708 0.805417i $$-0.298059\pi$$
0.592708 + 0.805417i $$0.298059\pi$$
$$998$$ 0 0
$$999$$ 13.5901 0.429973
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 3724.2.a.i.1.1 3
7.6 odd 2 532.2.a.e.1.3 3
21.20 even 2 4788.2.a.o.1.2 3
28.27 even 2 2128.2.a.r.1.1 3
56.13 odd 2 8512.2.a.bn.1.1 3
56.27 even 2 8512.2.a.bl.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.e.1.3 3 7.6 odd 2
2128.2.a.r.1.1 3 28.27 even 2
3724.2.a.i.1.1 3 1.1 even 1 trivial
4788.2.a.o.1.2 3 21.20 even 2
8512.2.a.bl.1.3 3 56.27 even 2
8512.2.a.bn.1.1 3 56.13 odd 2