# Properties

 Label 1089.4.a.u.1.1 Level $1089$ Weight $4$ Character 1089.1 Self dual yes Analytic conductor $64.253$ Analytic rank $0$ 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] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1089.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: $$64.2530799963$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.42443$$ of defining polynomial Character $$\chi$$ $$=$$ 1089.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} -31.6977 q^{7} -15.8199 q^{8} +O(q^{10})$$ $$q-4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} -31.6977 q^{7} -15.8199 q^{8} +12.6046 q^{10} -5.15114 q^{13} +140.244 q^{14} -22.6107 q^{16} +121.942 q^{17} -34.8489 q^{19} -32.9772 q^{20} -116.244 q^{23} -116.884 q^{25} +22.7909 q^{26} -366.919 q^{28} -69.4534 q^{29} +140.605 q^{31} +226.598 q^{32} -539.524 q^{34} +90.3023 q^{35} -420.070 q^{37} +154.186 q^{38} +45.0685 q^{40} -322.058 q^{41} -321.035 q^{43} +514.315 q^{46} +231.408 q^{47} +661.745 q^{49} +517.145 q^{50} -59.6274 q^{52} -4.91916 q^{53} +501.453 q^{56} +307.292 q^{58} -406.443 q^{59} +556.431 q^{61} -622.095 q^{62} -821.683 q^{64} +14.6749 q^{65} +84.7452 q^{67} +1411.55 q^{68} -399.536 q^{70} -49.0808 q^{71} -785.884 q^{73} +1858.57 q^{74} -403.395 q^{76} +383.118 q^{79} +64.4147 q^{80} +1424.92 q^{82} -930.211 q^{83} -347.395 q^{85} +1420.40 q^{86} +732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} -1023.85 q^{94} +99.2794 q^{95} -1171.49 q^{97} -2927.84 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8}+O(q^{10})$$ 2 * q + q^2 + 33 * q^4 + 14 * q^5 - 24 * q^7 + 57 * q^8 $$2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8} + 104 q^{10} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} - 50 q^{19} + 328 q^{20} - 134 q^{23} + 42 q^{25} - 112 q^{26} - 202 q^{28} - 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 220 q^{35} - 328 q^{37} + 72 q^{38} + 1272 q^{40} - 782 q^{41} - 386 q^{43} + 418 q^{46} - 266 q^{47} + 378 q^{49} + 1379 q^{50} - 592 q^{52} + 522 q^{53} + 1062 q^{56} - 390 q^{58} + 172 q^{59} + 778 q^{61} + 568 q^{62} + 809 q^{64} - 404 q^{65} - 776 q^{67} + 1070 q^{68} + 304 q^{70} - 630 q^{71} - 1296 q^{73} + 2358 q^{74} - 728 q^{76} - 652 q^{79} + 3832 q^{80} - 1070 q^{82} - 324 q^{83} - 616 q^{85} + 1068 q^{86} + 756 q^{89} - 28 q^{91} - 1726 q^{92} - 3722 q^{94} - 156 q^{95} - 452 q^{97} - 4467 q^{98}+O(q^{100})$$ 2 * q + q^2 + 33 * q^4 + 14 * q^5 - 24 * q^7 + 57 * q^8 + 104 * q^10 - 30 * q^13 + 182 * q^14 + 201 * q^16 + 106 * q^17 - 50 * q^19 + 328 * q^20 - 134 * q^23 + 42 * q^25 - 112 * q^26 - 202 * q^28 - 198 * q^29 + 360 * q^31 + 857 * q^32 - 626 * q^34 + 220 * q^35 - 328 * q^37 + 72 * q^38 + 1272 * q^40 - 782 * q^41 - 386 * q^43 + 418 * q^46 - 266 * q^47 + 378 * q^49 + 1379 * q^50 - 592 * q^52 + 522 * q^53 + 1062 * q^56 - 390 * q^58 + 172 * q^59 + 778 * q^61 + 568 * q^62 + 809 * q^64 - 404 * q^65 - 776 * q^67 + 1070 * q^68 + 304 * q^70 - 630 * q^71 - 1296 * q^73 + 2358 * q^74 - 728 * q^76 - 652 * q^79 + 3832 * q^80 - 1070 * q^82 - 324 * q^83 - 616 * q^85 + 1068 * q^86 + 756 * q^89 - 28 * q^91 - 1726 * q^92 - 3722 * q^94 - 156 * q^95 - 452 * q^97 - 4467 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.42443 −1.56427 −0.782136 0.623108i $$-0.785870\pi$$
−0.782136 + 0.623108i $$0.785870\pi$$
$$3$$ 0 0
$$4$$ 11.5756 1.44695
$$5$$ −2.84886 −0.254810 −0.127405 0.991851i $$-0.540665\pi$$
−0.127405 + 0.991851i $$0.540665\pi$$
$$6$$ 0 0
$$7$$ −31.6977 −1.71152 −0.855758 0.517377i $$-0.826909\pi$$
−0.855758 + 0.517377i $$0.826909\pi$$
$$8$$ −15.8199 −0.699146
$$9$$ 0 0
$$10$$ 12.6046 0.398591
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −5.15114 −0.109898 −0.0549488 0.998489i $$-0.517500\pi$$
−0.0549488 + 0.998489i $$0.517500\pi$$
$$14$$ 140.244 2.67728
$$15$$ 0 0
$$16$$ −22.6107 −0.353293
$$17$$ 121.942 1.73972 0.869861 0.493297i $$-0.164208\pi$$
0.869861 + 0.493297i $$0.164208\pi$$
$$18$$ 0 0
$$19$$ −34.8489 −0.420783 −0.210391 0.977617i $$-0.567474\pi$$
−0.210391 + 0.977617i $$0.567474\pi$$
$$20$$ −32.9772 −0.368696
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −116.244 −1.05385 −0.526926 0.849911i $$-0.676656\pi$$
−0.526926 + 0.849911i $$0.676656\pi$$
$$24$$ 0 0
$$25$$ −116.884 −0.935072
$$26$$ 22.7909 0.171910
$$27$$ 0 0
$$28$$ −366.919 −2.47647
$$29$$ −69.4534 −0.444730 −0.222365 0.974963i $$-0.571378\pi$$
−0.222365 + 0.974963i $$0.571378\pi$$
$$30$$ 0 0
$$31$$ 140.605 0.814623 0.407312 0.913289i $$-0.366466\pi$$
0.407312 + 0.913289i $$0.366466\pi$$
$$32$$ 226.598 1.25179
$$33$$ 0 0
$$34$$ −539.524 −2.72140
$$35$$ 90.3023 0.436111
$$36$$ 0 0
$$37$$ −420.070 −1.86646 −0.933232 0.359276i $$-0.883024\pi$$
−0.933232 + 0.359276i $$0.883024\pi$$
$$38$$ 154.186 0.658219
$$39$$ 0 0
$$40$$ 45.0685 0.178149
$$41$$ −322.058 −1.22676 −0.613378 0.789789i $$-0.710190\pi$$
−0.613378 + 0.789789i $$0.710190\pi$$
$$42$$ 0 0
$$43$$ −321.035 −1.13854 −0.569272 0.822149i $$-0.692775\pi$$
−0.569272 + 0.822149i $$0.692775\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 514.315 1.64851
$$47$$ 231.408 0.718176 0.359088 0.933304i $$-0.383088\pi$$
0.359088 + 0.933304i $$0.383088\pi$$
$$48$$ 0 0
$$49$$ 661.745 1.92929
$$50$$ 517.145 1.46271
$$51$$ 0 0
$$52$$ −59.6274 −0.159016
$$53$$ −4.91916 −0.0127490 −0.00637452 0.999980i $$-0.502029\pi$$
−0.00637452 + 0.999980i $$0.502029\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 501.453 1.19660
$$57$$ 0 0
$$58$$ 307.292 0.695679
$$59$$ −406.443 −0.896854 −0.448427 0.893820i $$-0.648016\pi$$
−0.448427 + 0.893820i $$0.648016\pi$$
$$60$$ 0 0
$$61$$ 556.431 1.16793 0.583964 0.811779i $$-0.301501\pi$$
0.583964 + 0.811779i $$0.301501\pi$$
$$62$$ −622.095 −1.27429
$$63$$ 0 0
$$64$$ −821.683 −1.60485
$$65$$ 14.6749 0.0280030
$$66$$ 0 0
$$67$$ 84.7452 0.154526 0.0772632 0.997011i $$-0.475382\pi$$
0.0772632 + 0.997011i $$0.475382\pi$$
$$68$$ 1411.55 2.51728
$$69$$ 0 0
$$70$$ −399.536 −0.682196
$$71$$ −49.0808 −0.0820398 −0.0410199 0.999158i $$-0.513061\pi$$
−0.0410199 + 0.999158i $$0.513061\pi$$
$$72$$ 0 0
$$73$$ −785.884 −1.26001 −0.630005 0.776591i $$-0.716947\pi$$
−0.630005 + 0.776591i $$0.716947\pi$$
$$74$$ 1858.57 2.91966
$$75$$ 0 0
$$76$$ −403.395 −0.608850
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 383.118 0.545622 0.272811 0.962068i $$-0.412047\pi$$
0.272811 + 0.962068i $$0.412047\pi$$
$$80$$ 64.4147 0.0900223
$$81$$ 0 0
$$82$$ 1424.92 1.91898
$$83$$ −930.211 −1.23017 −0.615084 0.788462i $$-0.710878\pi$$
−0.615084 + 0.788462i $$0.710878\pi$$
$$84$$ 0 0
$$85$$ −347.395 −0.443298
$$86$$ 1420.40 1.78099
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 732.559 0.872484 0.436242 0.899829i $$-0.356309\pi$$
0.436242 + 0.899829i $$0.356309\pi$$
$$90$$ 0 0
$$91$$ 163.279 0.188092
$$92$$ −1345.59 −1.52487
$$93$$ 0 0
$$94$$ −1023.85 −1.12342
$$95$$ 99.2794 0.107220
$$96$$ 0 0
$$97$$ −1171.49 −1.22626 −0.613128 0.789984i $$-0.710089\pi$$
−0.613128 + 0.789984i $$0.710089\pi$$
$$98$$ −2927.84 −3.01793
$$99$$ 0 0
$$100$$ −1353.00 −1.35300
$$101$$ −1221.27 −1.20318 −0.601589 0.798806i $$-0.705465\pi$$
−0.601589 + 0.798806i $$0.705465\pi$$
$$102$$ 0 0
$$103$$ 516.745 0.494334 0.247167 0.968973i $$-0.420500\pi$$
0.247167 + 0.968973i $$0.420500\pi$$
$$104$$ 81.4903 0.0768345
$$105$$ 0 0
$$106$$ 21.7645 0.0199430
$$107$$ −152.025 −0.137353 −0.0686765 0.997639i $$-0.521878\pi$$
−0.0686765 + 0.997639i $$0.521878\pi$$
$$108$$ 0 0
$$109$$ −2170.32 −1.90714 −0.953572 0.301164i $$-0.902625\pi$$
−0.953572 + 0.301164i $$0.902625\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 716.708 0.604666
$$113$$ 646.397 0.538123 0.269062 0.963123i $$-0.413286\pi$$
0.269062 + 0.963123i $$0.413286\pi$$
$$114$$ 0 0
$$115$$ 331.163 0.268532
$$116$$ −803.963 −0.643501
$$117$$ 0 0
$$118$$ 1798.28 1.40292
$$119$$ −3865.28 −2.97756
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −2461.89 −1.82696
$$123$$ 0 0
$$124$$ 1627.58 1.17872
$$125$$ 689.093 0.493075
$$126$$ 0 0
$$127$$ 993.304 0.694027 0.347014 0.937860i $$-0.387196\pi$$
0.347014 + 0.937860i $$0.387196\pi$$
$$128$$ 1822.69 1.25863
$$129$$ 0 0
$$130$$ −64.9279 −0.0438043
$$131$$ 385.814 0.257318 0.128659 0.991689i $$-0.458933\pi$$
0.128659 + 0.991689i $$0.458933\pi$$
$$132$$ 0 0
$$133$$ 1104.63 0.720177
$$134$$ −374.949 −0.241721
$$135$$ 0 0
$$136$$ −1929.11 −1.21632
$$137$$ −884.840 −0.551803 −0.275901 0.961186i $$-0.588976\pi$$
−0.275901 + 0.961186i $$0.588976\pi$$
$$138$$ 0 0
$$139$$ 1091.94 0.666312 0.333156 0.942872i $$-0.391886\pi$$
0.333156 + 0.942872i $$0.391886\pi$$
$$140$$ 1045.30 0.631029
$$141$$ 0 0
$$142$$ 217.155 0.128333
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 197.863 0.113322
$$146$$ 3477.09 1.97100
$$147$$ 0 0
$$148$$ −4862.55 −2.70067
$$149$$ 297.014 0.163304 0.0816522 0.996661i $$-0.473980\pi$$
0.0816522 + 0.996661i $$0.473980\pi$$
$$150$$ 0 0
$$151$$ 1887.86 1.01743 0.508716 0.860935i $$-0.330120\pi$$
0.508716 + 0.860935i $$0.330120\pi$$
$$152$$ 551.304 0.294189
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −400.562 −0.207574
$$156$$ 0 0
$$157$$ −56.5343 −0.0287384 −0.0143692 0.999897i $$-0.504574\pi$$
−0.0143692 + 0.999897i $$0.504574\pi$$
$$158$$ −1695.08 −0.853501
$$159$$ 0 0
$$160$$ −645.547 −0.318968
$$161$$ 3684.68 1.80369
$$162$$ 0 0
$$163$$ −49.2338 −0.0236582 −0.0118291 0.999930i $$-0.503765\pi$$
−0.0118291 + 0.999930i $$0.503765\pi$$
$$164$$ −3728.01 −1.77505
$$165$$ 0 0
$$166$$ 4115.65 1.92432
$$167$$ 2068.75 0.958589 0.479294 0.877654i $$-0.340893\pi$$
0.479294 + 0.877654i $$0.340893\pi$$
$$168$$ 0 0
$$169$$ −2170.47 −0.987923
$$170$$ 1537.03 0.693438
$$171$$ 0 0
$$172$$ −3716.17 −1.64741
$$173$$ −604.012 −0.265446 −0.132723 0.991153i $$-0.542372\pi$$
−0.132723 + 0.991153i $$0.542372\pi$$
$$174$$ 0 0
$$175$$ 3704.96 1.60039
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −3241.15 −1.36480
$$179$$ 2132.02 0.890251 0.445126 0.895468i $$-0.353159\pi$$
0.445126 + 0.895468i $$0.353159\pi$$
$$180$$ 0 0
$$181$$ −589.371 −0.242031 −0.121015 0.992651i $$-0.538615\pi$$
−0.121015 + 0.992651i $$0.538615\pi$$
$$182$$ −722.418 −0.294226
$$183$$ 0 0
$$184$$ 1838.97 0.736796
$$185$$ 1196.72 0.475593
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 2678.68 1.03916
$$189$$ 0 0
$$190$$ −439.255 −0.167720
$$191$$ 2160.90 0.818624 0.409312 0.912395i $$-0.365769\pi$$
0.409312 + 0.912395i $$0.365769\pi$$
$$192$$ 0 0
$$193$$ 1490.91 0.556052 0.278026 0.960574i $$-0.410320\pi$$
0.278026 + 0.960574i $$0.410320\pi$$
$$194$$ 5183.18 1.91820
$$195$$ 0 0
$$196$$ 7660.08 2.79157
$$197$$ −230.529 −0.0833732 −0.0416866 0.999131i $$-0.513273\pi$$
−0.0416866 + 0.999131i $$0.513273\pi$$
$$198$$ 0 0
$$199$$ 22.4007 0.00797963 0.00398982 0.999992i $$-0.498730\pi$$
0.00398982 + 0.999992i $$0.498730\pi$$
$$200$$ 1849.09 0.653752
$$201$$ 0 0
$$202$$ 5403.43 1.88210
$$203$$ 2201.51 0.761163
$$204$$ 0 0
$$205$$ 917.497 0.312589
$$206$$ −2286.30 −0.773273
$$207$$ 0 0
$$208$$ 116.471 0.0388260
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1051.64 0.343117 0.171558 0.985174i $$-0.445120\pi$$
0.171558 + 0.985174i $$0.445120\pi$$
$$212$$ −56.9421 −0.0184472
$$213$$ 0 0
$$214$$ 672.622 0.214857
$$215$$ 914.583 0.290112
$$216$$ 0 0
$$217$$ −4456.84 −1.39424
$$218$$ 9602.42 2.98329
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −628.141 −0.191191
$$222$$ 0 0
$$223$$ 3861.80 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$224$$ −7182.65 −2.14246
$$225$$ 0 0
$$226$$ −2859.94 −0.841771
$$227$$ −872.721 −0.255174 −0.127587 0.991827i $$-0.540723\pi$$
−0.127587 + 0.991827i $$0.540723\pi$$
$$228$$ 0 0
$$229$$ 1841.72 0.531459 0.265730 0.964048i $$-0.414387\pi$$
0.265730 + 0.964048i $$0.414387\pi$$
$$230$$ −1465.21 −0.420057
$$231$$ 0 0
$$232$$ 1098.74 0.310931
$$233$$ 3932.14 1.10559 0.552796 0.833317i $$-0.313561\pi$$
0.552796 + 0.833317i $$0.313561\pi$$
$$234$$ 0 0
$$235$$ −659.248 −0.182998
$$236$$ −4704.81 −1.29770
$$237$$ 0 0
$$238$$ 17101.7 4.65772
$$239$$ 4772.10 1.29155 0.645777 0.763526i $$-0.276534\pi$$
0.645777 + 0.763526i $$0.276534\pi$$
$$240$$ 0 0
$$241$$ −3988.84 −1.06616 −0.533078 0.846066i $$-0.678965\pi$$
−0.533078 + 0.846066i $$0.678965\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 6441.00 1.68993
$$245$$ −1885.22 −0.491601
$$246$$ 0 0
$$247$$ 179.511 0.0462431
$$248$$ −2224.34 −0.569540
$$249$$ 0 0
$$250$$ −3048.84 −0.771303
$$251$$ 5474.22 1.37661 0.688306 0.725421i $$-0.258355\pi$$
0.688306 + 0.725421i $$0.258355\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −4394.80 −1.08565
$$255$$ 0 0
$$256$$ −1490.90 −0.363989
$$257$$ 6434.01 1.56164 0.780822 0.624754i $$-0.214801\pi$$
0.780822 + 0.624754i $$0.214801\pi$$
$$258$$ 0 0
$$259$$ 13315.3 3.19448
$$260$$ 169.870 0.0405188
$$261$$ 0 0
$$262$$ −1707.01 −0.402516
$$263$$ 7589.00 1.77931 0.889654 0.456636i $$-0.150946\pi$$
0.889654 + 0.456636i $$0.150946\pi$$
$$264$$ 0 0
$$265$$ 14.0140 0.00324858
$$266$$ −4887.35 −1.12655
$$267$$ 0 0
$$268$$ 980.974 0.223591
$$269$$ −478.178 −0.108383 −0.0541914 0.998531i $$-0.517258\pi$$
−0.0541914 + 0.998531i $$0.517258\pi$$
$$270$$ 0 0
$$271$$ 122.323 0.0274192 0.0137096 0.999906i $$-0.495636\pi$$
0.0137096 + 0.999906i $$0.495636\pi$$
$$272$$ −2757.20 −0.614631
$$273$$ 0 0
$$274$$ 3914.91 0.863170
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8199.41 −1.77854 −0.889269 0.457385i $$-0.848786\pi$$
−0.889269 + 0.457385i $$0.848786\pi$$
$$278$$ −4831.22 −1.04229
$$279$$ 0 0
$$280$$ −1428.57 −0.304905
$$281$$ 6943.79 1.47413 0.737067 0.675820i $$-0.236210\pi$$
0.737067 + 0.675820i $$0.236210\pi$$
$$282$$ 0 0
$$283$$ −1035.14 −0.217429 −0.108715 0.994073i $$-0.534673\pi$$
−0.108715 + 0.994073i $$0.534673\pi$$
$$284$$ −568.139 −0.118707
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10208.5 2.09961
$$288$$ 0 0
$$289$$ 9956.85 2.02663
$$290$$ −875.430 −0.177266
$$291$$ 0 0
$$292$$ −9097.06 −1.82317
$$293$$ −6144.81 −1.22520 −0.612600 0.790393i $$-0.709876\pi$$
−0.612600 + 0.790393i $$0.709876\pi$$
$$294$$ 0 0
$$295$$ 1157.90 0.228527
$$296$$ 6645.45 1.30493
$$297$$ 0 0
$$298$$ −1314.12 −0.255452
$$299$$ 598.791 0.115816
$$300$$ 0 0
$$301$$ 10176.1 1.94864
$$302$$ −8352.72 −1.59154
$$303$$ 0 0
$$304$$ 787.958 0.148659
$$305$$ −1585.19 −0.297599
$$306$$ 0 0
$$307$$ 2186.09 0.406406 0.203203 0.979137i $$-0.434865\pi$$
0.203203 + 0.979137i $$0.434865\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 1772.26 0.324702
$$311$$ 7484.83 1.36471 0.682357 0.731019i $$-0.260955\pi$$
0.682357 + 0.731019i $$0.260955\pi$$
$$312$$ 0 0
$$313$$ −6833.33 −1.23400 −0.617001 0.786962i $$-0.711653\pi$$
−0.617001 + 0.786962i $$0.711653\pi$$
$$314$$ 250.132 0.0449546
$$315$$ 0 0
$$316$$ 4434.81 0.789485
$$317$$ −924.265 −0.163760 −0.0818800 0.996642i $$-0.526092\pi$$
−0.0818800 + 0.996642i $$0.526092\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 2340.86 0.408931
$$321$$ 0 0
$$322$$ −16302.6 −2.82145
$$323$$ −4249.54 −0.732046
$$324$$ 0 0
$$325$$ 602.086 0.102762
$$326$$ 217.831 0.0370078
$$327$$ 0 0
$$328$$ 5094.91 0.857681
$$329$$ −7335.10 −1.22917
$$330$$ 0 0
$$331$$ −9820.46 −1.63076 −0.815380 0.578927i $$-0.803472\pi$$
−0.815380 + 0.578927i $$0.803472\pi$$
$$332$$ −10767.7 −1.77999
$$333$$ 0 0
$$334$$ −9153.02 −1.49949
$$335$$ −241.427 −0.0393748
$$336$$ 0 0
$$337$$ −600.808 −0.0971161 −0.0485580 0.998820i $$-0.515463\pi$$
−0.0485580 + 0.998820i $$0.515463\pi$$
$$338$$ 9603.07 1.54538
$$339$$ 0 0
$$340$$ −4021.30 −0.641428
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −10103.5 −1.59049
$$344$$ 5078.73 0.796008
$$345$$ 0 0
$$346$$ 2672.41 0.415230
$$347$$ −3143.41 −0.486303 −0.243152 0.969988i $$-0.578181\pi$$
−0.243152 + 0.969988i $$0.578181\pi$$
$$348$$ 0 0
$$349$$ −720.663 −0.110533 −0.0552667 0.998472i $$-0.517601\pi$$
−0.0552667 + 0.998472i $$0.517601\pi$$
$$350$$ −16392.3 −2.50345
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1207.12 −0.182007 −0.0910034 0.995851i $$-0.529007\pi$$
−0.0910034 + 0.995851i $$0.529007\pi$$
$$354$$ 0 0
$$355$$ 139.824 0.0209045
$$356$$ 8479.79 1.26244
$$357$$ 0 0
$$358$$ −9432.99 −1.39260
$$359$$ 8748.31 1.28612 0.643062 0.765814i $$-0.277664\pi$$
0.643062 + 0.765814i $$0.277664\pi$$
$$360$$ 0 0
$$361$$ −5644.56 −0.822942
$$362$$ 2607.63 0.378602
$$363$$ 0 0
$$364$$ 1890.05 0.272158
$$365$$ 2238.87 0.321063
$$366$$ 0 0
$$367$$ −6730.45 −0.957293 −0.478647 0.878008i $$-0.658872\pi$$
−0.478647 + 0.878008i $$0.658872\pi$$
$$368$$ 2628.37 0.372318
$$369$$ 0 0
$$370$$ −5294.81 −0.743956
$$371$$ 155.926 0.0218202
$$372$$ 0 0
$$373$$ 227.394 0.0315657 0.0157828 0.999875i $$-0.494976\pi$$
0.0157828 + 0.999875i $$0.494976\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −3660.84 −0.502110
$$377$$ 357.764 0.0488748
$$378$$ 0 0
$$379$$ 11356.2 1.53913 0.769565 0.638568i $$-0.220473\pi$$
0.769565 + 0.638568i $$0.220473\pi$$
$$380$$ 1149.22 0.155141
$$381$$ 0 0
$$382$$ −9560.74 −1.28055
$$383$$ −10753.6 −1.43468 −0.717338 0.696725i $$-0.754640\pi$$
−0.717338 + 0.696725i $$0.754640\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6596.43 −0.869817
$$387$$ 0 0
$$388$$ −13560.7 −1.77433
$$389$$ 11727.1 1.52850 0.764252 0.644918i $$-0.223109\pi$$
0.764252 + 0.644918i $$0.223109\pi$$
$$390$$ 0 0
$$391$$ −14175.1 −1.83341
$$392$$ −10468.7 −1.34885
$$393$$ 0 0
$$394$$ 1019.96 0.130418
$$395$$ −1091.45 −0.139030
$$396$$ 0 0
$$397$$ −359.905 −0.0454990 −0.0227495 0.999741i $$-0.507242\pi$$
−0.0227495 + 0.999741i $$0.507242\pi$$
$$398$$ −99.1105 −0.0124823
$$399$$ 0 0
$$400$$ 2642.83 0.330354
$$401$$ 4066.71 0.506438 0.253219 0.967409i $$-0.418511\pi$$
0.253219 + 0.967409i $$0.418511\pi$$
$$402$$ 0 0
$$403$$ −724.274 −0.0895252
$$404$$ −14136.9 −1.74093
$$405$$ 0 0
$$406$$ −9740.45 −1.19067
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 13488.8 1.63076 0.815379 0.578927i $$-0.196528\pi$$
0.815379 + 0.578927i $$0.196528\pi$$
$$410$$ −4059.40 −0.488975
$$411$$ 0 0
$$412$$ 5981.62 0.715275
$$413$$ 12883.3 1.53498
$$414$$ 0 0
$$415$$ 2650.04 0.313459
$$416$$ −1167.24 −0.137569
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7040.12 0.820841 0.410420 0.911896i $$-0.365382\pi$$
0.410420 + 0.911896i $$0.365382\pi$$
$$420$$ 0 0
$$421$$ 9171.74 1.06177 0.530883 0.847445i $$-0.321860\pi$$
0.530883 + 0.847445i $$0.321860\pi$$
$$422$$ −4652.89 −0.536728
$$423$$ 0 0
$$424$$ 77.8204 0.00891343
$$425$$ −14253.1 −1.62677
$$426$$ 0 0
$$427$$ −17637.6 −1.99893
$$428$$ −1759.77 −0.198742
$$429$$ 0 0
$$430$$ −4046.51 −0.453814
$$431$$ 992.995 0.110976 0.0554882 0.998459i $$-0.482328\pi$$
0.0554882 + 0.998459i $$0.482328\pi$$
$$432$$ 0 0
$$433$$ 3790.21 0.420660 0.210330 0.977630i $$-0.432546\pi$$
0.210330 + 0.977630i $$0.432546\pi$$
$$434$$ 19719.0 2.18097
$$435$$ 0 0
$$436$$ −25122.7 −2.75954
$$437$$ 4050.98 0.443443
$$438$$ 0 0
$$439$$ 5136.97 0.558483 0.279242 0.960221i $$-0.409917\pi$$
0.279242 + 0.960221i $$0.409917\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 2779.16 0.299075
$$443$$ −10676.8 −1.14508 −0.572541 0.819876i $$-0.694042\pi$$
−0.572541 + 0.819876i $$0.694042\pi$$
$$444$$ 0 0
$$445$$ −2086.96 −0.222317
$$446$$ −17086.2 −1.81403
$$447$$ 0 0
$$448$$ 26045.5 2.74672
$$449$$ −10529.9 −1.10676 −0.553379 0.832929i $$-0.686662\pi$$
−0.553379 + 0.832929i $$0.686662\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 7482.42 0.778636
$$453$$ 0 0
$$454$$ 3861.29 0.399162
$$455$$ −465.160 −0.0479275
$$456$$ 0 0
$$457$$ 14072.5 1.44045 0.720225 0.693741i $$-0.244039\pi$$
0.720225 + 0.693741i $$0.244039\pi$$
$$458$$ −8148.55 −0.831347
$$459$$ 0 0
$$460$$ 3833.41 0.388551
$$461$$ −30.8173 −0.00311346 −0.00155673 0.999999i $$-0.500496\pi$$
−0.00155673 + 0.999999i $$0.500496\pi$$
$$462$$ 0 0
$$463$$ 17591.3 1.76573 0.882867 0.469622i $$-0.155610\pi$$
0.882867 + 0.469622i $$0.155610\pi$$
$$464$$ 1570.39 0.157120
$$465$$ 0 0
$$466$$ −17397.5 −1.72945
$$467$$ −13273.1 −1.31522 −0.657609 0.753360i $$-0.728432\pi$$
−0.657609 + 0.753360i $$0.728432\pi$$
$$468$$ 0 0
$$469$$ −2686.23 −0.264474
$$470$$ 2916.79 0.286259
$$471$$ 0 0
$$472$$ 6429.87 0.627031
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4073.27 0.393462
$$476$$ −44742.9 −4.30837
$$477$$ 0 0
$$478$$ −21113.8 −2.02034
$$479$$ 2496.68 0.238155 0.119077 0.992885i $$-0.462006\pi$$
0.119077 + 0.992885i $$0.462006\pi$$
$$480$$ 0 0
$$481$$ 2163.84 0.205120
$$482$$ 17648.3 1.66776
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3337.41 0.312462
$$486$$ 0 0
$$487$$ −3464.42 −0.322357 −0.161178 0.986925i $$-0.551529\pi$$
−0.161178 + 0.986925i $$0.551529\pi$$
$$488$$ −8802.65 −0.816552
$$489$$ 0 0
$$490$$ 8341.01 0.768997
$$491$$ −16224.6 −1.49125 −0.745625 0.666366i $$-0.767849\pi$$
−0.745625 + 0.666366i $$0.767849\pi$$
$$492$$ 0 0
$$493$$ −8469.29 −0.773707
$$494$$ −794.236 −0.0723367
$$495$$ 0 0
$$496$$ −3179.17 −0.287800
$$497$$ 1555.75 0.140412
$$498$$ 0 0
$$499$$ 9993.81 0.896562 0.448281 0.893893i $$-0.352036\pi$$
0.448281 + 0.893893i $$0.352036\pi$$
$$500$$ 7976.65 0.713453
$$501$$ 0 0
$$502$$ −24220.3 −2.15340
$$503$$ −15334.8 −1.35933 −0.679667 0.733520i $$-0.737876\pi$$
−0.679667 + 0.733520i $$0.737876\pi$$
$$504$$ 0 0
$$505$$ 3479.23 0.306581
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 11498.1 1.00422
$$509$$ 7291.23 0.634927 0.317464 0.948270i $$-0.397169\pi$$
0.317464 + 0.948270i $$0.397169\pi$$
$$510$$ 0 0
$$511$$ 24910.7 2.15653
$$512$$ −7985.14 −0.689251
$$513$$ 0 0
$$514$$ −28466.8 −2.44283
$$515$$ −1472.13 −0.125961
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −58912.5 −4.99704
$$519$$ 0 0
$$520$$ −232.154 −0.0195782
$$521$$ −16794.3 −1.41223 −0.706114 0.708098i $$-0.749553\pi$$
−0.706114 + 0.708098i $$0.749553\pi$$
$$522$$ 0 0
$$523$$ 21009.4 1.75655 0.878275 0.478157i $$-0.158695\pi$$
0.878275 + 0.478157i $$0.158695\pi$$
$$524$$ 4466.01 0.372326
$$525$$ 0 0
$$526$$ −33577.0 −2.78332
$$527$$ 17145.6 1.41722
$$528$$ 0 0
$$529$$ 1345.73 0.110605
$$530$$ −62.0039 −0.00508166
$$531$$ 0 0
$$532$$ 12786.7 1.04206
$$533$$ 1658.97 0.134818
$$534$$ 0 0
$$535$$ 433.097 0.0349989
$$536$$ −1340.66 −0.108036
$$537$$ 0 0
$$538$$ 2115.66 0.169540
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 16802.8 1.33532 0.667662 0.744464i $$-0.267295\pi$$
0.667662 + 0.744464i $$0.267295\pi$$
$$542$$ −541.211 −0.0428911
$$543$$ 0 0
$$544$$ 27631.9 2.17777
$$545$$ 6182.93 0.485959
$$546$$ 0 0
$$547$$ −16784.5 −1.31198 −0.655990 0.754770i $$-0.727749\pi$$
−0.655990 + 0.754770i $$0.727749\pi$$
$$548$$ −10242.5 −0.798429
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2420.37 0.187135
$$552$$ 0 0
$$553$$ −12144.0 −0.933840
$$554$$ 36277.7 2.78212
$$555$$ 0 0
$$556$$ 12639.9 0.964117
$$557$$ 18127.0 1.37893 0.689467 0.724317i $$-0.257845\pi$$
0.689467 + 0.724317i $$0.257845\pi$$
$$558$$ 0 0
$$559$$ 1653.70 0.125123
$$560$$ −2041.80 −0.154075
$$561$$ 0 0
$$562$$ −30722.3 −2.30595
$$563$$ 2090.88 0.156518 0.0782592 0.996933i $$-0.475064\pi$$
0.0782592 + 0.996933i $$0.475064\pi$$
$$564$$ 0 0
$$565$$ −1841.49 −0.137119
$$566$$ 4579.89 0.340119
$$567$$ 0 0
$$568$$ 776.452 0.0573578
$$569$$ 6249.23 0.460424 0.230212 0.973140i $$-0.426058\pi$$
0.230212 + 0.973140i $$0.426058\pi$$
$$570$$ 0 0
$$571$$ −6048.79 −0.443317 −0.221659 0.975124i $$-0.571147\pi$$
−0.221659 + 0.975124i $$0.571147\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −45166.8 −3.28437
$$575$$ 13587.1 0.985428
$$576$$ 0 0
$$577$$ −15729.1 −1.13486 −0.567429 0.823423i $$-0.692062\pi$$
−0.567429 + 0.823423i $$0.692062\pi$$
$$578$$ −44053.4 −3.17021
$$579$$ 0 0
$$580$$ 2290.38 0.163970
$$581$$ 29485.6 2.10545
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 12432.6 0.880931
$$585$$ 0 0
$$586$$ 27187.3 1.91655
$$587$$ −15620.5 −1.09835 −0.549173 0.835709i $$-0.685057\pi$$
−0.549173 + 0.835709i $$0.685057\pi$$
$$588$$ 0 0
$$589$$ −4899.91 −0.342780
$$590$$ −5123.04 −0.357478
$$591$$ 0 0
$$592$$ 9498.09 0.659407
$$593$$ −493.541 −0.0341776 −0.0170888 0.999854i $$-0.505440\pi$$
−0.0170888 + 0.999854i $$0.505440\pi$$
$$594$$ 0 0
$$595$$ 11011.6 0.758711
$$596$$ 3438.11 0.236293
$$597$$ 0 0
$$598$$ −2649.31 −0.181168
$$599$$ 12455.1 0.849585 0.424793 0.905291i $$-0.360347\pi$$
0.424793 + 0.905291i $$0.360347\pi$$
$$600$$ 0 0
$$601$$ −12454.8 −0.845329 −0.422664 0.906286i $$-0.638905\pi$$
−0.422664 + 0.906286i $$0.638905\pi$$
$$602$$ −45023.3 −3.04820
$$603$$ 0 0
$$604$$ 21853.1 1.47217
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4243.19 0.283733 0.141867 0.989886i $$-0.454690\pi$$
0.141867 + 0.989886i $$0.454690\pi$$
$$608$$ −7896.70 −0.526732
$$609$$ 0 0
$$610$$ 7013.57 0.465526
$$611$$ −1192.01 −0.0789259
$$612$$ 0 0
$$613$$ −5733.14 −0.377748 −0.188874 0.982001i $$-0.560484\pi$$
−0.188874 + 0.982001i $$0.560484\pi$$
$$614$$ −9672.18 −0.635729
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −15642.1 −1.02063 −0.510314 0.859988i $$-0.670471\pi$$
−0.510314 + 0.859988i $$0.670471\pi$$
$$618$$ 0 0
$$619$$ −7467.40 −0.484879 −0.242440 0.970167i $$-0.577948\pi$$
−0.242440 + 0.970167i $$0.577948\pi$$
$$620$$ −4636.74 −0.300348
$$621$$ 0 0
$$622$$ −33116.1 −2.13478
$$623$$ −23220.4 −1.49327
$$624$$ 0 0
$$625$$ 12647.4 0.809432
$$626$$ 30233.6 1.93031
$$627$$ 0 0
$$628$$ −654.416 −0.0415829
$$629$$ −51224.2 −3.24713
$$630$$ 0 0
$$631$$ −1486.38 −0.0937745 −0.0468872 0.998900i $$-0.514930\pi$$
−0.0468872 + 0.998900i $$0.514930\pi$$
$$632$$ −6060.87 −0.381469
$$633$$ 0 0
$$634$$ 4089.35 0.256165
$$635$$ −2829.78 −0.176845
$$636$$ 0 0
$$637$$ −3408.74 −0.212024
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −5192.58 −0.320711
$$641$$ −12386.0 −0.763211 −0.381606 0.924325i $$-0.624629\pi$$
−0.381606 + 0.924325i $$0.624629\pi$$
$$642$$ 0 0
$$643$$ −14458.1 −0.886737 −0.443369 0.896339i $$-0.646217\pi$$
−0.443369 + 0.896339i $$0.646217\pi$$
$$644$$ 42652.3 2.60984
$$645$$ 0 0
$$646$$ 18801.8 1.14512
$$647$$ −15792.8 −0.959625 −0.479813 0.877371i $$-0.659295\pi$$
−0.479813 + 0.877371i $$0.659295\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −2663.89 −0.160748
$$651$$ 0 0
$$652$$ −569.909 −0.0342321
$$653$$ 3179.93 0.190567 0.0952837 0.995450i $$-0.469624\pi$$
0.0952837 + 0.995450i $$0.469624\pi$$
$$654$$ 0 0
$$655$$ −1099.13 −0.0655672
$$656$$ 7281.96 0.433404
$$657$$ 0 0
$$658$$ 32453.6 1.92276
$$659$$ 11593.5 0.685308 0.342654 0.939462i $$-0.388674\pi$$
0.342654 + 0.939462i $$0.388674\pi$$
$$660$$ 0 0
$$661$$ 3233.88 0.190293 0.0951464 0.995463i $$-0.469668\pi$$
0.0951464 + 0.995463i $$0.469668\pi$$
$$662$$ 43449.9 2.55095
$$663$$ 0 0
$$664$$ 14715.8 0.860066
$$665$$ −3146.93 −0.183508
$$666$$ 0 0
$$667$$ 8073.56 0.468680
$$668$$ 23946.9 1.38703
$$669$$ 0 0
$$670$$ 1068.18 0.0615929
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 5495.72 0.314776 0.157388 0.987537i $$-0.449693\pi$$
0.157388 + 0.987537i $$0.449693\pi$$
$$674$$ 2658.23 0.151916
$$675$$ 0 0
$$676$$ −25124.4 −1.42947
$$677$$ 33836.7 1.92090 0.960451 0.278448i $$-0.0898200\pi$$
0.960451 + 0.278448i $$0.0898200\pi$$
$$678$$ 0 0
$$679$$ 37133.6 2.09876
$$680$$ 5495.75 0.309930
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 21080.3 1.18099 0.590493 0.807043i $$-0.298933\pi$$
0.590493 + 0.807043i $$0.298933\pi$$
$$684$$ 0 0
$$685$$ 2520.78 0.140605
$$686$$ 44702.2 2.48796
$$687$$ 0 0
$$688$$ 7258.84 0.402239
$$689$$ 25.3393 0.00140109
$$690$$ 0 0
$$691$$ 11811.3 0.650253 0.325127 0.945671i $$-0.394593\pi$$
0.325127 + 0.945671i $$0.394593\pi$$
$$692$$ −6991.79 −0.384087
$$693$$ 0 0
$$694$$ 13907.8 0.760711
$$695$$ −3110.79 −0.169783
$$696$$ 0 0
$$697$$ −39272.4 −2.13422
$$698$$ 3188.52 0.172904
$$699$$ 0 0
$$700$$ 42887.0 2.31568
$$701$$ 4244.99 0.228718 0.114359 0.993440i $$-0.463519\pi$$
0.114359 + 0.993440i $$0.463519\pi$$
$$702$$ 0 0
$$703$$ 14639.0 0.785376
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 5340.81 0.284708
$$707$$ 38711.5 2.05926
$$708$$ 0 0
$$709$$ −898.822 −0.0476107 −0.0238053 0.999717i $$-0.507578\pi$$
−0.0238053 + 0.999717i $$0.507578\pi$$
$$710$$ −618.643 −0.0327004
$$711$$ 0 0
$$712$$ −11589.0 −0.609993
$$713$$ −16344.5 −0.858493
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24679.4 1.28815
$$717$$ 0 0
$$718$$ −38706.3 −2.01185
$$719$$ −10741.8 −0.557165 −0.278582 0.960412i $$-0.589865\pi$$
−0.278582 + 0.960412i $$0.589865\pi$$
$$720$$ 0 0
$$721$$ −16379.6 −0.846061
$$722$$ 24973.9 1.28730
$$723$$ 0 0
$$724$$ −6822.30 −0.350206
$$725$$ 8117.99 0.415855
$$726$$ 0 0
$$727$$ 16794.2 0.856758 0.428379 0.903599i $$-0.359085\pi$$
0.428379 + 0.903599i $$0.359085\pi$$
$$728$$ −2583.06 −0.131503
$$729$$ 0 0
$$730$$ −9905.73 −0.502229
$$731$$ −39147.7 −1.98075
$$732$$ 0 0
$$733$$ −8659.40 −0.436347 −0.218173 0.975910i $$-0.570010\pi$$
−0.218173 + 0.975910i $$0.570010\pi$$
$$734$$ 29778.4 1.49747
$$735$$ 0 0
$$736$$ −26340.8 −1.31920
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −16705.7 −0.831567 −0.415783 0.909464i $$-0.636493\pi$$
−0.415783 + 0.909464i $$0.636493\pi$$
$$740$$ 13852.7 0.688157
$$741$$ 0 0
$$742$$ −689.884 −0.0341327
$$743$$ 1292.12 0.0637996 0.0318998 0.999491i $$-0.489844\pi$$
0.0318998 + 0.999491i $$0.489844\pi$$
$$744$$ 0 0
$$745$$ −846.151 −0.0416115
$$746$$ −1006.09 −0.0493773
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 4818.83 0.235082
$$750$$ 0 0
$$751$$ −14980.4 −0.727886 −0.363943 0.931421i $$-0.618570\pi$$
−0.363943 + 0.931421i $$0.618570\pi$$
$$752$$ −5232.30 −0.253726
$$753$$ 0 0
$$754$$ −1582.90 −0.0764535
$$755$$ −5378.25 −0.259251
$$756$$ 0 0
$$757$$ 3003.41 0.144202 0.0721010 0.997397i $$-0.477030\pi$$
0.0721010 + 0.997397i $$0.477030\pi$$
$$758$$ −50244.8 −2.40762
$$759$$ 0 0
$$760$$ −1570.59 −0.0749621
$$761$$ −20375.0 −0.970555 −0.485277 0.874360i $$-0.661281\pi$$
−0.485277 + 0.874360i $$0.661281\pi$$
$$762$$ 0 0
$$763$$ 68794.1 3.26411
$$764$$ 25013.6 1.18450
$$765$$ 0 0
$$766$$ 47578.4 2.24422
$$767$$ 2093.65 0.0985621
$$768$$ 0 0
$$769$$ 12372.4 0.580184 0.290092 0.956999i $$-0.406314\pi$$
0.290092 + 0.956999i $$0.406314\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 17258.1 0.804578
$$773$$ −21023.6 −0.978225 −0.489113 0.872221i $$-0.662679\pi$$
−0.489113 + 0.872221i $$0.662679\pi$$
$$774$$ 0 0
$$775$$ −16434.4 −0.761732
$$776$$ 18532.8 0.857331
$$777$$ 0 0
$$778$$ −51885.7 −2.39099
$$779$$ 11223.4 0.516198
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 62716.6 2.86795
$$783$$ 0 0
$$784$$ −14962.5 −0.681602
$$785$$ 161.058 0.00732281
$$786$$ 0 0
$$787$$ −30286.2 −1.37177 −0.685886 0.727709i $$-0.740585\pi$$
−0.685886 + 0.727709i $$0.740585\pi$$
$$788$$ −2668.51 −0.120637
$$789$$ 0 0
$$790$$ 4829.03 0.217480
$$791$$ −20489.3 −0.921007
$$792$$ 0 0
$$793$$ −2866.25 −0.128353
$$794$$ 1592.37 0.0711729
$$795$$ 0 0
$$796$$ 259.301 0.0115461
$$797$$ −32337.8 −1.43722 −0.718610 0.695413i $$-0.755221\pi$$
−0.718610 + 0.695413i $$0.755221\pi$$
$$798$$ 0 0
$$799$$ 28218.3 1.24943
$$800$$ −26485.7 −1.17052
$$801$$ 0 0
$$802$$ −17992.9 −0.792207
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −10497.1 −0.459596
$$806$$ 3204.50 0.140042
$$807$$ 0 0
$$808$$ 19320.3 0.841197
$$809$$ −891.707 −0.0387525 −0.0193762 0.999812i $$-0.506168\pi$$
−0.0193762 + 0.999812i $$0.506168\pi$$
$$810$$ 0 0
$$811$$ 10114.9 0.437957 0.218978 0.975730i $$-0.429728\pi$$
0.218978 + 0.975730i $$0.429728\pi$$
$$812$$ 25483.8 1.10136
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 140.260 0.00602833
$$816$$ 0 0
$$817$$ 11187.7 0.479080
$$818$$ −59680.4 −2.55095
$$819$$ 0 0
$$820$$ 10620.6 0.452300
$$821$$ 10833.5 0.460525 0.230262 0.973129i $$-0.426042\pi$$
0.230262 + 0.973129i $$0.426042\pi$$
$$822$$ 0 0
$$823$$ 31958.5 1.35359 0.676794 0.736173i $$-0.263369\pi$$
0.676794 + 0.736173i $$0.263369\pi$$
$$824$$ −8174.84 −0.345612
$$825$$ 0 0
$$826$$ −57001.3 −2.40112
$$827$$ 34847.3 1.46525 0.732624 0.680634i $$-0.238296\pi$$
0.732624 + 0.680634i $$0.238296\pi$$
$$828$$ 0 0
$$829$$ 6537.91 0.273910 0.136955 0.990577i $$-0.456268\pi$$
0.136955 + 0.990577i $$0.456268\pi$$
$$830$$ −11724.9 −0.490334
$$831$$ 0 0
$$832$$ 4232.60 0.176369
$$833$$ 80694.5 3.35642
$$834$$ 0 0
$$835$$ −5893.56 −0.244258
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −31148.5 −1.28402
$$839$$ −2710.34 −0.111527 −0.0557635 0.998444i $$-0.517759\pi$$
−0.0557635 + 0.998444i $$0.517759\pi$$
$$840$$ 0 0
$$841$$ −19565.2 −0.802215
$$842$$ −40579.7 −1.66089
$$843$$ 0 0
$$844$$ 12173.3 0.496471
$$845$$ 6183.35 0.251732
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 111.226 0.00450414
$$849$$ 0 0
$$850$$ 63061.7 2.54470
$$851$$ 48830.8 1.96698
$$852$$ 0 0
$$853$$ −9759.32 −0.391738 −0.195869 0.980630i $$-0.562753\pi$$
−0.195869 + 0.980630i $$0.562753\pi$$
$$854$$ 78036.2 3.12687
$$855$$ 0 0
$$856$$ 2405.01 0.0960298
$$857$$ −13649.8 −0.544072 −0.272036 0.962287i $$-0.587697\pi$$
−0.272036 + 0.962287i $$0.587697\pi$$
$$858$$ 0 0
$$859$$ 7796.42 0.309674 0.154837 0.987940i $$-0.450515\pi$$
0.154837 + 0.987940i $$0.450515\pi$$
$$860$$ 10586.8 0.419776
$$861$$ 0 0
$$862$$ −4393.43 −0.173597
$$863$$ −7183.57 −0.283350 −0.141675 0.989913i $$-0.545249\pi$$
−0.141675 + 0.989913i $$0.545249\pi$$
$$864$$ 0 0
$$865$$ 1720.75 0.0676383
$$866$$ −16769.5 −0.658026
$$867$$ 0 0
$$868$$ −51590.5 −2.01739
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −436.534 −0.0169821
$$872$$ 34334.1 1.33337
$$873$$ 0 0
$$874$$ −17923.3 −0.693666
$$875$$ −21842.7 −0.843905
$$876$$ 0 0
$$877$$ −17063.1 −0.656991 −0.328495 0.944506i $$-0.606542\pi$$
−0.328495 + 0.944506i $$0.606542\pi$$
$$878$$ −22728.2 −0.873620
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 32174.9 1.23042 0.615210 0.788363i $$-0.289071\pi$$
0.615210 + 0.788363i $$0.289071\pi$$
$$882$$ 0 0
$$883$$ 2843.68 0.108378 0.0541889 0.998531i $$-0.482743\pi$$
0.0541889 + 0.998531i $$0.482743\pi$$
$$884$$ −7271.09 −0.276644
$$885$$ 0 0
$$886$$ 47238.9 1.79122
$$887$$ −31417.8 −1.18930 −0.594649 0.803985i $$-0.702709\pi$$
−0.594649 + 0.803985i $$0.702709\pi$$
$$888$$ 0 0
$$889$$ −31485.5 −1.18784
$$890$$ 9233.59 0.347765
$$891$$ 0 0
$$892$$ 44702.5 1.67797
$$893$$ −8064.30 −0.302196
$$894$$ 0 0
$$895$$ −6073.83 −0.226845
$$896$$ −57775.1 −2.15416
$$897$$ 0 0
$$898$$ 46588.6 1.73127
$$899$$ −9765.47 −0.362288
$$900$$ 0 0
$$901$$ −599.852 −0.0221798
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −10225.9 −0.376227
$$905$$ 1679.03 0.0616718
$$906$$ 0 0
$$907$$ 12253.1 0.448573 0.224287 0.974523i $$-0.427995\pi$$
0.224287 + 0.974523i $$0.427995\pi$$
$$908$$ −10102.2 −0.369223
$$909$$ 0 0
$$910$$ 2058.07 0.0749717
$$911$$ 48422.4 1.76104 0.880518 0.474012i $$-0.157195\pi$$
0.880518 + 0.474012i $$0.157195\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −62262.9 −2.25326
$$915$$ 0 0
$$916$$ 21318.9 0.768993
$$917$$ −12229.4 −0.440404
$$918$$ 0 0
$$919$$ −5546.18 −0.199077 −0.0995385 0.995034i $$-0.531737\pi$$
−0.0995385 + 0.995034i $$0.531737\pi$$
$$920$$ −5238.96 −0.187743
$$921$$ 0 0
$$922$$ 136.349 0.00487030
$$923$$ 252.822 0.00901598
$$924$$ 0 0
$$925$$ 49099.5 1.74528
$$926$$ −77831.3 −2.76209
$$927$$ 0 0
$$928$$ −15738.0 −0.556709
$$929$$ 35684.5 1.26025 0.630125 0.776494i $$-0.283004\pi$$
0.630125 + 0.776494i $$0.283004\pi$$
$$930$$ 0 0
$$931$$ −23061.1 −0.811811
$$932$$ 45516.7 1.59973
$$933$$ 0 0
$$934$$ 58726.0 2.05736
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 48903.6 1.70503 0.852514 0.522705i $$-0.175077\pi$$
0.852514 + 0.522705i $$0.175077\pi$$
$$938$$ 11885.0 0.413710
$$939$$ 0 0
$$940$$ −7631.17 −0.264789
$$941$$ −23741.9 −0.822490 −0.411245 0.911525i $$-0.634906\pi$$
−0.411245 + 0.911525i $$0.634906\pi$$
$$942$$ 0 0
$$943$$ 37437.4 1.29282
$$944$$ 9189.97 0.316852
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −37612.4 −1.29064 −0.645321 0.763911i $$-0.723276\pi$$
−0.645321 + 0.763911i $$0.723276\pi$$
$$948$$ 0 0
$$949$$ 4048.20 0.138472
$$950$$ −18021.9 −0.615482
$$951$$ 0 0
$$952$$ 61148.2 2.08175
$$953$$ −48294.3 −1.64156 −0.820779 0.571246i $$-0.806460\pi$$
−0.820779 + 0.571246i $$0.806460\pi$$
$$954$$ 0 0
$$955$$ −6156.09 −0.208593
$$956$$ 55239.8 1.86881
$$957$$ 0 0
$$958$$ −11046.4 −0.372539
$$959$$ 28047.4 0.944419
$$960$$ 0 0
$$961$$ −10021.4 −0.336389
$$962$$ −9573.76 −0.320863
$$963$$ 0 0
$$964$$ −46173.1 −1.54267
$$965$$ −4247.39 −0.141687
$$966$$ 0 0
$$967$$ −1840.92 −0.0612204 −0.0306102 0.999531i $$-0.509745\pi$$
−0.0306102 + 0.999531i $$0.509745\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −14766.1 −0.488775
$$971$$ −31461.8 −1.03981 −0.519906 0.854223i $$-0.674033\pi$$
−0.519906 + 0.854223i $$0.674033\pi$$
$$972$$ 0 0
$$973$$ −34612.1 −1.14040
$$974$$ 15328.1 0.504254
$$975$$ 0 0
$$976$$ −12581.3 −0.412620
$$977$$ 7040.11 0.230535 0.115268 0.993334i $$-0.463227\pi$$
0.115268 + 0.993334i $$0.463227\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −21822.5 −0.711320
$$981$$ 0 0
$$982$$ 71784.4 2.33272
$$983$$ 24610.9 0.798541 0.399270 0.916833i $$-0.369263\pi$$
0.399270 + 0.916833i $$0.369263\pi$$
$$984$$ 0 0
$$985$$ 656.744 0.0212443
$$986$$ 37471.8 1.21029
$$987$$ 0 0
$$988$$ 2077.95 0.0669112
$$989$$ 37318.5 1.19986
$$990$$ 0 0
$$991$$ −40003.3 −1.28229 −0.641144 0.767421i $$-0.721540\pi$$
−0.641144 + 0.767421i $$0.721540\pi$$
$$992$$ 31860.8 1.01974
$$993$$ 0 0
$$994$$ −6883.31 −0.219643
$$995$$ −63.8165 −0.00203329
$$996$$ 0 0
$$997$$ 7342.61 0.233242 0.116621 0.993176i $$-0.462794\pi$$
0.116621 + 0.993176i $$0.462794\pi$$
$$998$$ −44216.9 −1.40247
$$999$$ 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 1089.4.a.u.1.1 2
3.2 odd 2 363.4.a.i.1.2 2
11.10 odd 2 99.4.a.f.1.2 2
33.32 even 2 33.4.a.c.1.1 2
44.43 even 2 1584.4.a.bj.1.1 2
55.54 odd 2 2475.4.a.p.1.1 2
132.131 odd 2 528.4.a.p.1.2 2
165.32 odd 4 825.4.c.h.199.2 4
165.98 odd 4 825.4.c.h.199.3 4
165.164 even 2 825.4.a.l.1.2 2
231.230 odd 2 1617.4.a.k.1.1 2
264.131 odd 2 2112.4.a.bg.1.1 2
264.197 even 2 2112.4.a.bn.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 33.32 even 2
99.4.a.f.1.2 2 11.10 odd 2
363.4.a.i.1.2 2 3.2 odd 2
528.4.a.p.1.2 2 132.131 odd 2
825.4.a.l.1.2 2 165.164 even 2
825.4.c.h.199.2 4 165.32 odd 4
825.4.c.h.199.3 4 165.98 odd 4
1089.4.a.u.1.1 2 1.1 even 1 trivial
1584.4.a.bj.1.1 2 44.43 even 2
1617.4.a.k.1.1 2 231.230 odd 2
2112.4.a.bg.1.1 2 264.131 odd 2
2112.4.a.bn.1.1 2 264.197 even 2
2475.4.a.p.1.1 2 55.54 odd 2