Properties

 Label 1323.4.a.ba.1.4 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $4$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 4$$ x^4 - 9*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$0.684742$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} +O(q^{10})$$ $$q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} -87.7802 q^{10} -30.3531 q^{11} +89.5603 q^{13} +132.967 q^{16} -13.4164 q^{17} +5.37355 q^{19} -316.496 q^{20} -156.527 q^{22} +167.449 q^{23} +164.747 q^{25} +461.852 q^{26} -135.085 q^{29} +18.9339 q^{31} +248.664 q^{32} -69.1868 q^{34} +402.560 q^{37} +27.7107 q^{38} -929.889 q^{40} +434.345 q^{41} +53.1868 q^{43} -564.367 q^{44} +863.516 q^{46} +155.218 q^{47} +849.581 q^{50} +1665.23 q^{52} -301.707 q^{53} +516.669 q^{55} -696.615 q^{58} -412.635 q^{59} +571.253 q^{61} +97.6396 q^{62} +218.593 q^{64} -1524.49 q^{65} +820.549 q^{67} -249.456 q^{68} -8.95825 q^{71} -21.9339 q^{73} +2075.95 q^{74} +99.9124 q^{76} +619.428 q^{79} -2263.36 q^{80} +2239.87 q^{82} +259.532 q^{83} +228.374 q^{85} +274.278 q^{86} -1658.15 q^{88} +484.329 q^{89} +3113.45 q^{92} +800.440 q^{94} -91.4683 q^{95} +252.747 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 26 q^{4}+O(q^{10})$$ 4 * q + 26 * q^4 $$4 q + 26 q^{4} - 206 q^{10} + 68 q^{13} + 290 q^{16} - 172 q^{19} - 94 q^{22} + 272 q^{25} - 408 q^{31} - 180 q^{34} + 1320 q^{37} - 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 3952 q^{52} - 352 q^{55} - 1432 q^{58} + 2672 q^{61} + 826 q^{64} + 1444 q^{67} + 396 q^{73} + 1222 q^{76} + 1220 q^{79} + 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 3492 q^{94} + 624 q^{97}+O(q^{100})$$ 4 * q + 26 * q^4 - 206 * q^10 + 68 * q^13 + 290 * q^16 - 172 * q^19 - 94 * q^22 + 272 * q^25 - 408 * q^31 - 180 * q^34 + 1320 * q^37 - 1446 * q^40 + 116 * q^43 + 1374 * q^46 + 3952 * q^52 - 352 * q^55 - 1432 * q^58 + 2672 * q^61 + 826 * q^64 + 1444 * q^67 + 396 * q^73 + 1222 * q^76 + 1220 * q^79 + 3590 * q^82 + 720 * q^85 - 6294 * q^88 + 3492 * q^94 + 624 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.15688 1.82323 0.911616 0.411043i $$-0.134836\pi$$
0.911616 + 0.411043i $$0.134836\pi$$
$$3$$ 0 0
$$4$$ 18.5934 2.32417
$$5$$ −17.0220 −1.52249 −0.761245 0.648464i $$-0.775412\pi$$
−0.761245 + 0.648464i $$0.775412\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 54.6288 2.41427
$$9$$ 0 0
$$10$$ −87.7802 −2.77585
$$11$$ −30.3531 −0.831982 −0.415991 0.909369i $$-0.636565\pi$$
−0.415991 + 0.909369i $$0.636565\pi$$
$$12$$ 0 0
$$13$$ 89.5603 1.91074 0.955368 0.295419i $$-0.0954592\pi$$
0.955368 + 0.295419i $$0.0954592\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 132.967 2.07761
$$17$$ −13.4164 −0.191409 −0.0957046 0.995410i $$-0.530510\pi$$
−0.0957046 + 0.995410i $$0.530510\pi$$
$$18$$ 0 0
$$19$$ 5.37355 0.0648830 0.0324415 0.999474i $$-0.489672\pi$$
0.0324415 + 0.999474i $$0.489672\pi$$
$$20$$ −316.496 −3.53853
$$21$$ 0 0
$$22$$ −156.527 −1.51690
$$23$$ 167.449 1.51807 0.759035 0.651050i $$-0.225671\pi$$
0.759035 + 0.651050i $$0.225671\pi$$
$$24$$ 0 0
$$25$$ 164.747 1.31798
$$26$$ 461.852 3.48371
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −135.085 −0.864986 −0.432493 0.901637i $$-0.642366\pi$$
−0.432493 + 0.901637i $$0.642366\pi$$
$$30$$ 0 0
$$31$$ 18.9339 0.109698 0.0548488 0.998495i $$-0.482532\pi$$
0.0548488 + 0.998495i $$0.482532\pi$$
$$32$$ 248.664 1.37369
$$33$$ 0 0
$$34$$ −69.1868 −0.348983
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 402.560 1.78866 0.894331 0.447406i $$-0.147652\pi$$
0.894331 + 0.447406i $$0.147652\pi$$
$$38$$ 27.7107 0.118297
$$39$$ 0 0
$$40$$ −929.889 −3.67571
$$41$$ 434.345 1.65447 0.827236 0.561855i $$-0.189912\pi$$
0.827236 + 0.561855i $$0.189912\pi$$
$$42$$ 0 0
$$43$$ 53.1868 0.188626 0.0943129 0.995543i $$-0.469935\pi$$
0.0943129 + 0.995543i $$0.469935\pi$$
$$44$$ −564.367 −1.93367
$$45$$ 0 0
$$46$$ 863.516 2.76779
$$47$$ 155.218 0.481720 0.240860 0.970560i $$-0.422570\pi$$
0.240860 + 0.970560i $$0.422570\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 849.581 2.40298
$$51$$ 0 0
$$52$$ 1665.23 4.44088
$$53$$ −301.707 −0.781938 −0.390969 0.920404i $$-0.627860\pi$$
−0.390969 + 0.920404i $$0.627860\pi$$
$$54$$ 0 0
$$55$$ 516.669 1.26669
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −696.615 −1.57707
$$59$$ −412.635 −0.910518 −0.455259 0.890359i $$-0.650453\pi$$
−0.455259 + 0.890359i $$0.650453\pi$$
$$60$$ 0 0
$$61$$ 571.253 1.19904 0.599520 0.800360i $$-0.295358\pi$$
0.599520 + 0.800360i $$0.295358\pi$$
$$62$$ 97.6396 0.200004
$$63$$ 0 0
$$64$$ 218.593 0.426940
$$65$$ −1524.49 −2.90908
$$66$$ 0 0
$$67$$ 820.549 1.49621 0.748104 0.663581i $$-0.230964\pi$$
0.748104 + 0.663581i $$0.230964\pi$$
$$68$$ −249.456 −0.444868
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.95825 −0.0149739 −0.00748696 0.999972i $$-0.502383\pi$$
−0.00748696 + 0.999972i $$0.502383\pi$$
$$72$$ 0 0
$$73$$ −21.9339 −0.0351666 −0.0175833 0.999845i $$-0.505597\pi$$
−0.0175833 + 0.999845i $$0.505597\pi$$
$$74$$ 2075.95 3.26115
$$75$$ 0 0
$$76$$ 99.9124 0.150799
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 619.428 0.882166 0.441083 0.897466i $$-0.354594\pi$$
0.441083 + 0.897466i $$0.354594\pi$$
$$80$$ −2263.36 −3.16314
$$81$$ 0 0
$$82$$ 2239.87 3.01649
$$83$$ 259.532 0.343221 0.171610 0.985165i $$-0.445103\pi$$
0.171610 + 0.985165i $$0.445103\pi$$
$$84$$ 0 0
$$85$$ 228.374 0.291419
$$86$$ 274.278 0.343908
$$87$$ 0 0
$$88$$ −1658.15 −2.00863
$$89$$ 484.329 0.576840 0.288420 0.957504i $$-0.406870\pi$$
0.288420 + 0.957504i $$0.406870\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 3113.45 3.52826
$$93$$ 0 0
$$94$$ 800.440 0.878288
$$95$$ −91.4683 −0.0987837
$$96$$ 0 0
$$97$$ 252.747 0.264563 0.132281 0.991212i $$-0.457770\pi$$
0.132281 + 0.991212i $$0.457770\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 3063.21 3.06321
$$101$$ −629.378 −0.620054 −0.310027 0.950728i $$-0.600338\pi$$
−0.310027 + 0.950728i $$0.600338\pi$$
$$102$$ 0 0
$$103$$ −1172.54 −1.12168 −0.560842 0.827923i $$-0.689523\pi$$
−0.560842 + 0.827923i $$0.689523\pi$$
$$104$$ 4892.57 4.61304
$$105$$ 0 0
$$106$$ −1555.87 −1.42565
$$107$$ 1002.87 0.906087 0.453044 0.891488i $$-0.350338\pi$$
0.453044 + 0.891488i $$0.350338\pi$$
$$108$$ 0 0
$$109$$ 542.802 0.476981 0.238491 0.971145i $$-0.423347\pi$$
0.238491 + 0.971145i $$0.423347\pi$$
$$110$$ 2664.40 2.30946
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1388.74 1.15612 0.578062 0.815993i $$-0.303809\pi$$
0.578062 + 0.815993i $$0.303809\pi$$
$$114$$ 0 0
$$115$$ −2850.32 −2.31125
$$116$$ −2511.68 −2.01038
$$117$$ 0 0
$$118$$ −2127.91 −1.66009
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −409.689 −0.307805
$$122$$ 2945.88 2.18613
$$123$$ 0 0
$$124$$ 352.045 0.254956
$$125$$ −676.573 −0.484117
$$126$$ 0 0
$$127$$ −2330.00 −1.62798 −0.813991 0.580877i $$-0.802710\pi$$
−0.813991 + 0.580877i $$0.802710\pi$$
$$128$$ −862.051 −0.595276
$$129$$ 0 0
$$130$$ −7861.62 −5.30392
$$131$$ 226.817 0.151276 0.0756379 0.997135i $$-0.475901\pi$$
0.0756379 + 0.997135i $$0.475901\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4231.47 2.72793
$$135$$ 0 0
$$136$$ −732.922 −0.462114
$$137$$ −1900.83 −1.18540 −0.592698 0.805425i $$-0.701937\pi$$
−0.592698 + 0.805425i $$0.701937\pi$$
$$138$$ 0 0
$$139$$ −1958.99 −1.19539 −0.597695 0.801724i $$-0.703917\pi$$
−0.597695 + 0.801724i $$0.703917\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −46.1966 −0.0273009
$$143$$ −2718.43 −1.58970
$$144$$ 0 0
$$145$$ 2299.40 1.31693
$$146$$ −113.110 −0.0641169
$$147$$ 0 0
$$148$$ 7484.96 4.15716
$$149$$ 1518.37 0.834831 0.417416 0.908716i $$-0.362936\pi$$
0.417416 + 0.908716i $$0.362936\pi$$
$$150$$ 0 0
$$151$$ −1815.18 −0.978262 −0.489131 0.872210i $$-0.662686\pi$$
−0.489131 + 0.872210i $$0.662686\pi$$
$$152$$ 293.550 0.156645
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −322.291 −0.167013
$$156$$ 0 0
$$157$$ 1858.79 0.944889 0.472445 0.881360i $$-0.343372\pi$$
0.472445 + 0.881360i $$0.343372\pi$$
$$158$$ 3194.31 1.60839
$$159$$ 0 0
$$160$$ −4232.75 −2.09142
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 870.089 0.418102 0.209051 0.977905i $$-0.432962\pi$$
0.209051 + 0.977905i $$0.432962\pi$$
$$164$$ 8075.95 3.84528
$$165$$ 0 0
$$166$$ 1338.37 0.625771
$$167$$ −1749.13 −0.810488 −0.405244 0.914209i $$-0.632813\pi$$
−0.405244 + 0.914209i $$0.632813\pi$$
$$168$$ 0 0
$$169$$ 5824.05 2.65091
$$170$$ 1177.69 0.531324
$$171$$ 0 0
$$172$$ 988.922 0.438399
$$173$$ −39.3374 −0.0172877 −0.00864383 0.999963i $$-0.502751\pi$$
−0.00864383 + 0.999963i $$0.502751\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4035.96 −1.72853
$$177$$ 0 0
$$178$$ 2497.62 1.05171
$$179$$ −2650.58 −1.10678 −0.553391 0.832922i $$-0.686666\pi$$
−0.553391 + 0.832922i $$0.686666\pi$$
$$180$$ 0 0
$$181$$ −2719.71 −1.11688 −0.558438 0.829546i $$-0.688599\pi$$
−0.558438 + 0.829546i $$0.688599\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 9147.55 3.66504
$$185$$ −6852.37 −2.72322
$$186$$ 0 0
$$187$$ 407.230 0.159249
$$188$$ 2886.03 1.11960
$$189$$ 0 0
$$190$$ −471.691 −0.180106
$$191$$ 4339.00 1.64376 0.821881 0.569659i $$-0.192925\pi$$
0.821881 + 0.569659i $$0.192925\pi$$
$$192$$ 0 0
$$193$$ 977.689 0.364640 0.182320 0.983239i $$-0.441639\pi$$
0.182320 + 0.983239i $$0.441639\pi$$
$$194$$ 1303.39 0.482359
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1556.32 −0.562859 −0.281430 0.959582i $$-0.590809\pi$$
−0.281430 + 0.959582i $$0.590809\pi$$
$$198$$ 0 0
$$199$$ −1723.21 −0.613846 −0.306923 0.951734i $$-0.599299\pi$$
−0.306923 + 0.951734i $$0.599299\pi$$
$$200$$ 8999.94 3.18196
$$201$$ 0 0
$$202$$ −3245.62 −1.13050
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −7393.41 −2.51892
$$206$$ −6046.63 −2.04509
$$207$$ 0 0
$$208$$ 11908.6 3.96976
$$209$$ −163.104 −0.0539815
$$210$$ 0 0
$$211$$ 1295.19 0.422581 0.211290 0.977423i $$-0.432233\pi$$
0.211290 + 0.977423i $$0.432233\pi$$
$$212$$ −5609.76 −1.81736
$$213$$ 0 0
$$214$$ 5171.69 1.65201
$$215$$ −905.343 −0.287181
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2799.16 0.869647
$$219$$ 0 0
$$220$$ 9606.63 2.94400
$$221$$ −1201.58 −0.365732
$$222$$ 0 0
$$223$$ 483.611 0.145224 0.0726121 0.997360i $$-0.476866\pi$$
0.0726121 + 0.997360i $$0.476866\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 7161.58 2.10788
$$227$$ 4019.86 1.17536 0.587681 0.809092i $$-0.300041\pi$$
0.587681 + 0.809092i $$0.300041\pi$$
$$228$$ 0 0
$$229$$ −5070.37 −1.46314 −0.731570 0.681766i $$-0.761212\pi$$
−0.731570 + 0.681766i $$0.761212\pi$$
$$230$$ −14698.7 −4.21394
$$231$$ 0 0
$$232$$ −7379.51 −2.08831
$$233$$ −4418.26 −1.24227 −0.621137 0.783702i $$-0.713329\pi$$
−0.621137 + 0.783702i $$0.713329\pi$$
$$234$$ 0 0
$$235$$ −2642.11 −0.733415
$$236$$ −7672.29 −2.11620
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −3228.15 −0.873690 −0.436845 0.899537i $$-0.643904\pi$$
−0.436845 + 0.899537i $$0.643904\pi$$
$$240$$ 0 0
$$241$$ 1537.14 0.410854 0.205427 0.978672i $$-0.434142\pi$$
0.205427 + 0.978672i $$0.434142\pi$$
$$242$$ −2112.72 −0.561200
$$243$$ 0 0
$$244$$ 10621.5 2.78678
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 481.257 0.123974
$$248$$ 1034.33 0.264840
$$249$$ 0 0
$$250$$ −3489.01 −0.882657
$$251$$ −178.384 −0.0448587 −0.0224293 0.999748i $$-0.507140\pi$$
−0.0224293 + 0.999748i $$0.507140\pi$$
$$252$$ 0 0
$$253$$ −5082.61 −1.26301
$$254$$ −12015.5 −2.96819
$$255$$ 0 0
$$256$$ −6194.24 −1.51227
$$257$$ −3271.92 −0.794152 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −28345.5 −6.76120
$$261$$ 0 0
$$262$$ 1169.67 0.275811
$$263$$ −4738.45 −1.11097 −0.555486 0.831526i $$-0.687468\pi$$
−0.555486 + 0.831526i $$0.687468\pi$$
$$264$$ 0 0
$$265$$ 5135.65 1.19049
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 15256.8 3.47745
$$269$$ −418.456 −0.0948466 −0.0474233 0.998875i $$-0.515101\pi$$
−0.0474233 + 0.998875i $$0.515101\pi$$
$$270$$ 0 0
$$271$$ −5263.95 −1.17993 −0.589967 0.807428i $$-0.700859\pi$$
−0.589967 + 0.807428i $$0.700859\pi$$
$$272$$ −1783.94 −0.397673
$$273$$ 0 0
$$274$$ −9802.37 −2.16125
$$275$$ −5000.59 −1.09653
$$276$$ 0 0
$$277$$ 2855.61 0.619412 0.309706 0.950832i $$-0.399769\pi$$
0.309706 + 0.950832i $$0.399769\pi$$
$$278$$ −10102.3 −2.17947
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8589.72 1.82356 0.911779 0.410682i $$-0.134709\pi$$
0.911779 + 0.410682i $$0.134709\pi$$
$$282$$ 0 0
$$283$$ −3681.18 −0.773228 −0.386614 0.922242i $$-0.626355\pi$$
−0.386614 + 0.922242i $$0.626355\pi$$
$$284$$ −166.564 −0.0348020
$$285$$ 0 0
$$286$$ −14018.6 −2.89839
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4733.00 −0.963363
$$290$$ 11857.7 2.40107
$$291$$ 0 0
$$292$$ −407.825 −0.0817334
$$293$$ −1958.68 −0.390536 −0.195268 0.980750i $$-0.562558\pi$$
−0.195268 + 0.980750i $$0.562558\pi$$
$$294$$ 0 0
$$295$$ 7023.86 1.38625
$$296$$ 21991.4 4.31832
$$297$$ 0 0
$$298$$ 7830.06 1.52209
$$299$$ 14996.8 2.90063
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −9360.68 −1.78360
$$303$$ 0 0
$$304$$ 714.504 0.134801
$$305$$ −9723.84 −1.82553
$$306$$ 0 0
$$307$$ 5672.32 1.05452 0.527258 0.849706i $$-0.323220\pi$$
0.527258 + 0.849706i $$0.323220\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1662.02 −0.304504
$$311$$ −1271.71 −0.231872 −0.115936 0.993257i $$-0.536987\pi$$
−0.115936 + 0.993257i $$0.536987\pi$$
$$312$$ 0 0
$$313$$ 4868.38 0.879160 0.439580 0.898204i $$-0.355127\pi$$
0.439580 + 0.898204i $$0.355127\pi$$
$$314$$ 9585.55 1.72275
$$315$$ 0 0
$$316$$ 11517.3 2.05031
$$317$$ 3224.18 0.571256 0.285628 0.958341i $$-0.407798\pi$$
0.285628 + 0.958341i $$0.407798\pi$$
$$318$$ 0 0
$$319$$ 4100.24 0.719653
$$320$$ −3720.89 −0.650012
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −72.0937 −0.0124192
$$324$$ 0 0
$$325$$ 14754.8 2.51831
$$326$$ 4486.94 0.762297
$$327$$ 0 0
$$328$$ 23727.8 3.99435
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5309.27 0.881643 0.440821 0.897595i $$-0.354687\pi$$
0.440821 + 0.897595i $$0.354687\pi$$
$$332$$ 4825.57 0.797704
$$333$$ 0 0
$$334$$ −9020.04 −1.47771
$$335$$ −13967.3 −2.27796
$$336$$ 0 0
$$337$$ −2304.33 −0.372478 −0.186239 0.982505i $$-0.559630\pi$$
−0.186239 + 0.982505i $$0.559630\pi$$
$$338$$ 30033.9 4.83322
$$339$$ 0 0
$$340$$ 4246.24 0.677308
$$341$$ −574.702 −0.0912664
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2905.53 0.455394
$$345$$ 0 0
$$346$$ −202.858 −0.0315194
$$347$$ −5196.20 −0.803881 −0.401940 0.915666i $$-0.631664\pi$$
−0.401940 + 0.915666i $$0.631664\pi$$
$$348$$ 0 0
$$349$$ 9030.60 1.38509 0.692546 0.721374i $$-0.256489\pi$$
0.692546 + 0.721374i $$0.256489\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −7547.72 −1.14288
$$353$$ 8885.51 1.33974 0.669869 0.742479i $$-0.266350\pi$$
0.669869 + 0.742479i $$0.266350\pi$$
$$354$$ 0 0
$$355$$ 152.487 0.0227976
$$356$$ 9005.31 1.34068
$$357$$ 0 0
$$358$$ −13668.7 −2.01792
$$359$$ −37.2983 −0.00548337 −0.00274169 0.999996i $$-0.500873\pi$$
−0.00274169 + 0.999996i $$0.500873\pi$$
$$360$$ 0 0
$$361$$ −6830.12 −0.995790
$$362$$ −14025.2 −2.03632
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 373.357 0.0535409
$$366$$ 0 0
$$367$$ −3335.79 −0.474460 −0.237230 0.971454i $$-0.576240\pi$$
−0.237230 + 0.971454i $$0.576240\pi$$
$$368$$ 22265.2 3.15395
$$369$$ 0 0
$$370$$ −35336.8 −4.96506
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3935.81 0.546349 0.273175 0.961964i $$-0.411926\pi$$
0.273175 + 0.961964i $$0.411926\pi$$
$$374$$ 2100.03 0.290348
$$375$$ 0 0
$$376$$ 8479.37 1.16301
$$377$$ −12098.2 −1.65276
$$378$$ 0 0
$$379$$ 2308.00 0.312808 0.156404 0.987693i $$-0.450010\pi$$
0.156404 + 0.987693i $$0.450010\pi$$
$$380$$ −1700.71 −0.229590
$$381$$ 0 0
$$382$$ 22375.7 2.99696
$$383$$ 2931.62 0.391119 0.195559 0.980692i $$-0.437348\pi$$
0.195559 + 0.980692i $$0.437348\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5041.82 0.664824
$$387$$ 0 0
$$388$$ 4699.42 0.614890
$$389$$ −7610.12 −0.991898 −0.495949 0.868352i $$-0.665180\pi$$
−0.495949 + 0.868352i $$0.665180\pi$$
$$390$$ 0 0
$$391$$ −2246.57 −0.290572
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −8025.76 −1.02622
$$395$$ −10543.9 −1.34309
$$396$$ 0 0
$$397$$ 4269.74 0.539778 0.269889 0.962891i $$-0.413013\pi$$
0.269889 + 0.962891i $$0.413013\pi$$
$$398$$ −8886.41 −1.11918
$$399$$ 0 0
$$400$$ 21905.9 2.73824
$$401$$ −11128.8 −1.38590 −0.692948 0.720988i $$-0.743688\pi$$
−0.692948 + 0.720988i $$0.743688\pi$$
$$402$$ 0 0
$$403$$ 1695.72 0.209603
$$404$$ −11702.3 −1.44111
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −12219.0 −1.48814
$$408$$ 0 0
$$409$$ −7220.39 −0.872922 −0.436461 0.899723i $$-0.643768\pi$$
−0.436461 + 0.899723i $$0.643768\pi$$
$$410$$ −38126.9 −4.59257
$$411$$ 0 0
$$412$$ −21801.4 −2.60699
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4417.74 −0.522550
$$416$$ 22270.4 2.62475
$$417$$ 0 0
$$418$$ −841.106 −0.0984207
$$419$$ 16631.2 1.93911 0.969556 0.244870i $$-0.0787454\pi$$
0.969556 + 0.244870i $$0.0787454\pi$$
$$420$$ 0 0
$$421$$ −654.498 −0.0757679 −0.0378839 0.999282i $$-0.512062\pi$$
−0.0378839 + 0.999282i $$0.512062\pi$$
$$422$$ 6679.14 0.770463
$$423$$ 0 0
$$424$$ −16481.9 −1.88781
$$425$$ −2210.31 −0.252273
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 18646.8 2.10590
$$429$$ 0 0
$$430$$ −4668.74 −0.523597
$$431$$ −3049.91 −0.340856 −0.170428 0.985370i $$-0.554515\pi$$
−0.170428 + 0.985370i $$0.554515\pi$$
$$432$$ 0 0
$$433$$ 1599.41 0.177512 0.0887562 0.996053i $$-0.471711\pi$$
0.0887562 + 0.996053i $$0.471711\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 10092.5 1.10859
$$437$$ 899.797 0.0984968
$$438$$ 0 0
$$439$$ 15893.9 1.72796 0.863979 0.503528i $$-0.167965\pi$$
0.863979 + 0.503528i $$0.167965\pi$$
$$440$$ 28225.0 3.05813
$$441$$ 0 0
$$442$$ −6196.39 −0.666815
$$443$$ −6160.22 −0.660679 −0.330339 0.943862i $$-0.607163\pi$$
−0.330339 + 0.943862i $$0.607163\pi$$
$$444$$ 0 0
$$445$$ −8244.23 −0.878233
$$446$$ 2493.92 0.264777
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 8742.47 0.918892 0.459446 0.888206i $$-0.348048\pi$$
0.459446 + 0.888206i $$0.348048\pi$$
$$450$$ 0 0
$$451$$ −13183.7 −1.37649
$$452$$ 25821.4 2.68703
$$453$$ 0 0
$$454$$ 20729.9 2.14296
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −580.112 −0.0593796 −0.0296898 0.999559i $$-0.509452\pi$$
−0.0296898 + 0.999559i $$0.509452\pi$$
$$458$$ −26147.3 −2.66764
$$459$$ 0 0
$$460$$ −52997.0 −5.37174
$$461$$ −1957.05 −0.197720 −0.0988599 0.995101i $$-0.531520\pi$$
−0.0988599 + 0.995101i $$0.531520\pi$$
$$462$$ 0 0
$$463$$ 8111.80 0.814227 0.407114 0.913378i $$-0.366535\pi$$
0.407114 + 0.913378i $$0.366535\pi$$
$$464$$ −17961.8 −1.79710
$$465$$ 0 0
$$466$$ −22784.4 −2.26495
$$467$$ −1240.29 −0.122899 −0.0614493 0.998110i $$-0.519572\pi$$
−0.0614493 + 0.998110i $$0.519572\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −13625.1 −1.33718
$$471$$ 0 0
$$472$$ −22541.8 −2.19824
$$473$$ −1614.38 −0.156933
$$474$$ 0 0
$$475$$ 885.276 0.0855142
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −16647.2 −1.59294
$$479$$ −18605.0 −1.77470 −0.887352 0.461092i $$-0.847458\pi$$
−0.887352 + 0.461092i $$0.847458\pi$$
$$480$$ 0 0
$$481$$ 36053.4 3.41766
$$482$$ 7926.84 0.749083
$$483$$ 0 0
$$484$$ −7617.51 −0.715393
$$485$$ −4302.25 −0.402794
$$486$$ 0 0
$$487$$ 874.431 0.0813640 0.0406820 0.999172i $$-0.487047\pi$$
0.0406820 + 0.999172i $$0.487047\pi$$
$$488$$ 31206.9 2.89481
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 5849.95 0.537688 0.268844 0.963184i $$-0.413358\pi$$
0.268844 + 0.963184i $$0.413358\pi$$
$$492$$ 0 0
$$493$$ 1812.35 0.165566
$$494$$ 2481.78 0.226034
$$495$$ 0 0
$$496$$ 2517.58 0.227908
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −1095.76 −0.0983029 −0.0491514 0.998791i $$-0.515652\pi$$
−0.0491514 + 0.998791i $$0.515652\pi$$
$$500$$ −12579.8 −1.12517
$$501$$ 0 0
$$502$$ −919.907 −0.0817878
$$503$$ −4988.25 −0.442177 −0.221088 0.975254i $$-0.570961\pi$$
−0.221088 + 0.975254i $$0.570961\pi$$
$$504$$ 0 0
$$505$$ 10713.2 0.944025
$$506$$ −26210.4 −2.30275
$$507$$ 0 0
$$508$$ −43322.5 −3.78371
$$509$$ 16831.7 1.46572 0.732859 0.680381i $$-0.238186\pi$$
0.732859 + 0.680381i $$0.238186\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −25046.5 −2.16193
$$513$$ 0 0
$$514$$ −16872.9 −1.44792
$$515$$ 19958.9 1.70775
$$516$$ 0 0
$$517$$ −4711.35 −0.400783
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −83281.2 −7.02331
$$521$$ 12826.2 1.07855 0.539277 0.842129i $$-0.318698\pi$$
0.539277 + 0.842129i $$0.318698\pi$$
$$522$$ 0 0
$$523$$ 3377.82 0.282412 0.141206 0.989980i $$-0.454902\pi$$
0.141206 + 0.989980i $$0.454902\pi$$
$$524$$ 4217.30 0.351591
$$525$$ 0 0
$$526$$ −24435.6 −2.02556
$$527$$ −254.024 −0.0209971
$$528$$ 0 0
$$529$$ 15872.3 1.30453
$$530$$ 26483.9 2.17054
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 38900.1 3.16126
$$534$$ 0 0
$$535$$ −17070.8 −1.37951
$$536$$ 44825.6 3.61226
$$537$$ 0 0
$$538$$ −2157.93 −0.172927
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 4866.90 0.386774 0.193387 0.981123i $$-0.438053\pi$$
0.193387 + 0.981123i $$0.438053\pi$$
$$542$$ −27145.5 −2.15129
$$543$$ 0 0
$$544$$ −3336.18 −0.262936
$$545$$ −9239.55 −0.726199
$$546$$ 0 0
$$547$$ 6870.41 0.537034 0.268517 0.963275i $$-0.413467\pi$$
0.268517 + 0.963275i $$0.413467\pi$$
$$548$$ −35342.9 −2.75506
$$549$$ 0 0
$$550$$ −25787.4 −1.99923
$$551$$ −725.883 −0.0561228
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 14726.0 1.12933
$$555$$ 0 0
$$556$$ −36424.2 −2.77829
$$557$$ −19795.9 −1.50589 −0.752943 0.658086i $$-0.771366\pi$$
−0.752943 + 0.658086i $$0.771366\pi$$
$$558$$ 0 0
$$559$$ 4763.42 0.360414
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 44296.1 3.32477
$$563$$ −6981.58 −0.522626 −0.261313 0.965254i $$-0.584156\pi$$
−0.261313 + 0.965254i $$0.584156\pi$$
$$564$$ 0 0
$$565$$ −23639.1 −1.76019
$$566$$ −18983.4 −1.40977
$$567$$ 0 0
$$568$$ −489.378 −0.0361512
$$569$$ −3412.25 −0.251404 −0.125702 0.992068i $$-0.540118\pi$$
−0.125702 + 0.992068i $$0.540118\pi$$
$$570$$ 0 0
$$571$$ 20647.3 1.51325 0.756624 0.653850i $$-0.226847\pi$$
0.756624 + 0.653850i $$0.226847\pi$$
$$572$$ −50544.9 −3.69473
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 27586.8 2.00078
$$576$$ 0 0
$$577$$ −12938.7 −0.933530 −0.466765 0.884381i $$-0.654581\pi$$
−0.466765 + 0.884381i $$0.654581\pi$$
$$578$$ −24407.5 −1.75643
$$579$$ 0 0
$$580$$ 42753.7 3.06078
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 9157.75 0.650558
$$584$$ −1198.22 −0.0849019
$$585$$ 0 0
$$586$$ −10100.7 −0.712038
$$587$$ −14032.4 −0.986676 −0.493338 0.869838i $$-0.664223\pi$$
−0.493338 + 0.869838i $$0.664223\pi$$
$$588$$ 0 0
$$589$$ 101.742 0.00711750
$$590$$ 36221.2 2.52746
$$591$$ 0 0
$$592$$ 53527.2 3.71614
$$593$$ −23344.9 −1.61663 −0.808313 0.588752i $$-0.799619\pi$$
−0.808313 + 0.588752i $$0.799619\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 28231.7 1.94029
$$597$$ 0 0
$$598$$ 77336.7 5.28852
$$599$$ 21813.4 1.48793 0.743966 0.668217i $$-0.232942\pi$$
0.743966 + 0.668217i $$0.232942\pi$$
$$600$$ 0 0
$$601$$ −1010.28 −0.0685690 −0.0342845 0.999412i $$-0.510915\pi$$
−0.0342845 + 0.999412i $$0.510915\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −33750.4 −2.27365
$$605$$ 6973.71 0.468631
$$606$$ 0 0
$$607$$ 1763.19 0.117901 0.0589504 0.998261i $$-0.481225\pi$$
0.0589504 + 0.998261i $$0.481225\pi$$
$$608$$ 1336.21 0.0891288
$$609$$ 0 0
$$610$$ −50144.7 −3.32836
$$611$$ 13901.4 0.920440
$$612$$ 0 0
$$613$$ 264.802 0.0174474 0.00872368 0.999962i $$-0.497223\pi$$
0.00872368 + 0.999962i $$0.497223\pi$$
$$614$$ 29251.4 1.92263
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −17930.8 −1.16996 −0.584980 0.811048i $$-0.698898\pi$$
−0.584980 + 0.811048i $$0.698898\pi$$
$$618$$ 0 0
$$619$$ 19581.1 1.27146 0.635729 0.771912i $$-0.280700\pi$$
0.635729 + 0.771912i $$0.280700\pi$$
$$620$$ −5992.49 −0.388168
$$621$$ 0 0
$$622$$ −6558.07 −0.422757
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9076.78 −0.580914
$$626$$ 25105.6 1.60291
$$627$$ 0 0
$$628$$ 34561.2 2.19609
$$629$$ −5400.91 −0.342366
$$630$$ 0 0
$$631$$ 13595.9 0.857754 0.428877 0.903363i $$-0.358909\pi$$
0.428877 + 0.903363i $$0.358909\pi$$
$$632$$ 33838.6 2.12979
$$633$$ 0 0
$$634$$ 16626.7 1.04153
$$635$$ 39661.1 2.47859
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 21144.4 1.31209
$$639$$ 0 0
$$640$$ 14673.8 0.906301
$$641$$ −9626.58 −0.593178 −0.296589 0.955005i $$-0.595849\pi$$
−0.296589 + 0.955005i $$0.595849\pi$$
$$642$$ 0 0
$$643$$ −26571.5 −1.62967 −0.814834 0.579695i $$-0.803172\pi$$
−0.814834 + 0.579695i $$0.803172\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −371.778 −0.0226431
$$647$$ 29729.0 1.80644 0.903220 0.429179i $$-0.141197\pi$$
0.903220 + 0.429179i $$0.141197\pi$$
$$648$$ 0 0
$$649$$ 12524.8 0.757535
$$650$$ 76088.7 4.59145
$$651$$ 0 0
$$652$$ 16177.9 0.971742
$$653$$ −16796.1 −1.00656 −0.503278 0.864125i $$-0.667873\pi$$
−0.503278 + 0.864125i $$0.667873\pi$$
$$654$$ 0 0
$$655$$ −3860.88 −0.230316
$$656$$ 57753.6 3.43734
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −2078.96 −0.122891 −0.0614453 0.998110i $$-0.519571\pi$$
−0.0614453 + 0.998110i $$0.519571\pi$$
$$660$$ 0 0
$$661$$ 6600.40 0.388390 0.194195 0.980963i $$-0.437791\pi$$
0.194195 + 0.980963i $$0.437791\pi$$
$$662$$ 27379.2 1.60744
$$663$$ 0 0
$$664$$ 14177.9 0.828629
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −22619.8 −1.31311
$$668$$ −32522.2 −1.88372
$$669$$ 0 0
$$670$$ −72027.9 −4.15325
$$671$$ −17339.3 −0.997580
$$672$$ 0 0
$$673$$ −21644.8 −1.23974 −0.619871 0.784704i $$-0.712815\pi$$
−0.619871 + 0.784704i $$0.712815\pi$$
$$674$$ −11883.2 −0.679113
$$675$$ 0 0
$$676$$ 108289. 6.16118
$$677$$ −349.464 −0.0198390 −0.00991949 0.999951i $$-0.503158\pi$$
−0.00991949 + 0.999951i $$0.503158\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 12475.8 0.703565
$$681$$ 0 0
$$682$$ −2963.67 −0.166400
$$683$$ −14807.9 −0.829589 −0.414795 0.909915i $$-0.636147\pi$$
−0.414795 + 0.909915i $$0.636147\pi$$
$$684$$ 0 0
$$685$$ 32355.9 1.80475
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 7072.08 0.391890
$$689$$ −27021.0 −1.49408
$$690$$ 0 0
$$691$$ −21679.4 −1.19352 −0.596760 0.802420i $$-0.703546\pi$$
−0.596760 + 0.802420i $$0.703546\pi$$
$$692$$ −731.415 −0.0401795
$$693$$ 0 0
$$694$$ −26796.2 −1.46566
$$695$$ 33345.8 1.81997
$$696$$ 0 0
$$697$$ −5827.35 −0.316681
$$698$$ 46569.7 2.52534
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5065.92 0.272949 0.136474 0.990644i $$-0.456423\pi$$
0.136474 + 0.990644i $$0.456423\pi$$
$$702$$ 0 0
$$703$$ 2163.18 0.116054
$$704$$ −6634.99 −0.355207
$$705$$ 0 0
$$706$$ 45821.5 2.44265
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −16981.8 −0.899527 −0.449764 0.893148i $$-0.648492\pi$$
−0.449764 + 0.893148i $$0.648492\pi$$
$$710$$ 786.356 0.0415654
$$711$$ 0 0
$$712$$ 26458.3 1.39265
$$713$$ 3170.46 0.166528
$$714$$ 0 0
$$715$$ 46273.1 2.42030
$$716$$ −49283.3 −2.57235
$$717$$ 0 0
$$718$$ −192.343 −0.00999746
$$719$$ 31450.6 1.63130 0.815652 0.578543i $$-0.196378\pi$$
0.815652 + 0.578543i $$0.196378\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −35222.1 −1.81556
$$723$$ 0 0
$$724$$ −50568.7 −2.59581
$$725$$ −22254.8 −1.14003
$$726$$ 0 0
$$727$$ −14930.2 −0.761665 −0.380833 0.924644i $$-0.624363\pi$$
−0.380833 + 0.924644i $$0.624363\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 1925.36 0.0976174
$$731$$ −713.575 −0.0361047
$$732$$ 0 0
$$733$$ −26799.5 −1.35043 −0.675213 0.737623i $$-0.735948\pi$$
−0.675213 + 0.737623i $$0.735948\pi$$
$$734$$ −17202.3 −0.865050
$$735$$ 0 0
$$736$$ 41638.6 2.08535
$$737$$ −24906.2 −1.24482
$$738$$ 0 0
$$739$$ 22511.2 1.12055 0.560275 0.828307i $$-0.310695\pi$$
0.560275 + 0.828307i $$0.310695\pi$$
$$740$$ −127409. −6.32924
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −23783.0 −1.17431 −0.587155 0.809474i $$-0.699752\pi$$
−0.587155 + 0.809474i $$0.699752\pi$$
$$744$$ 0 0
$$745$$ −25845.7 −1.27102
$$746$$ 20296.5 0.996121
$$747$$ 0 0
$$748$$ 7571.78 0.370123
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −6729.39 −0.326976 −0.163488 0.986545i $$-0.552274\pi$$
−0.163488 + 0.986545i $$0.552274\pi$$
$$752$$ 20638.8 1.00083
$$753$$ 0 0
$$754$$ −62389.0 −3.01336
$$755$$ 30898.0 1.48939
$$756$$ 0 0
$$757$$ 34646.1 1.66345 0.831726 0.555187i $$-0.187353\pi$$
0.831726 + 0.555187i $$0.187353\pi$$
$$758$$ 11902.1 0.570321
$$759$$ 0 0
$$760$$ −4996.80 −0.238491
$$761$$ −24096.3 −1.14782 −0.573908 0.818919i $$-0.694573\pi$$
−0.573908 + 0.818919i $$0.694573\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 80676.6 3.82039
$$765$$ 0 0
$$766$$ 15118.0 0.713100
$$767$$ −36955.8 −1.73976
$$768$$ 0 0
$$769$$ 28099.8 1.31769 0.658846 0.752278i $$-0.271045\pi$$
0.658846 + 0.752278i $$0.271045\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18178.5 0.847487
$$773$$ −9455.98 −0.439984 −0.219992 0.975502i $$-0.570603\pi$$
−0.219992 + 0.975502i $$0.570603\pi$$
$$774$$ 0 0
$$775$$ 3119.30 0.144579
$$776$$ 13807.3 0.638727
$$777$$ 0 0
$$778$$ −39244.5 −1.80846
$$779$$ 2333.98 0.107347
$$780$$ 0 0
$$781$$ 271.911 0.0124580
$$782$$ −11585.3 −0.529781
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −31640.2 −1.43858
$$786$$ 0 0
$$787$$ 16846.0 0.763017 0.381509 0.924365i $$-0.375405\pi$$
0.381509 + 0.924365i $$0.375405\pi$$
$$788$$ −28937.3 −1.30818
$$789$$ 0 0
$$790$$ −54373.5 −2.44876
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 51161.6 2.29105
$$794$$ 22018.5 0.984140
$$795$$ 0 0
$$796$$ −32040.4 −1.42669
$$797$$ 35955.5 1.59801 0.799003 0.601327i $$-0.205361\pi$$
0.799003 + 0.601327i $$0.205361\pi$$
$$798$$ 0 0
$$799$$ −2082.47 −0.0922057
$$800$$ 40966.6 1.81049
$$801$$ 0 0
$$802$$ −57389.7 −2.52681
$$803$$ 665.761 0.0292580
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8744.64 0.382155
$$807$$ 0 0
$$808$$ −34382.1 −1.49698
$$809$$ 13680.6 0.594543 0.297272 0.954793i $$-0.403923\pi$$
0.297272 + 0.954793i $$0.403923\pi$$
$$810$$ 0 0
$$811$$ −10215.1 −0.442294 −0.221147 0.975241i $$-0.570980\pi$$
−0.221147 + 0.975241i $$0.570980\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −63011.7 −2.71322
$$815$$ −14810.6 −0.636557
$$816$$ 0 0
$$817$$ 285.802 0.0122386
$$818$$ −37234.6 −1.59154
$$819$$ 0 0
$$820$$ −137469. −5.85440
$$821$$ 14255.6 0.605996 0.302998 0.952991i $$-0.402012\pi$$
0.302998 + 0.952991i $$0.402012\pi$$
$$822$$ 0 0
$$823$$ −31349.9 −1.32781 −0.663905 0.747817i $$-0.731102\pi$$
−0.663905 + 0.747817i $$0.731102\pi$$
$$824$$ −64054.3 −2.70805
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6114.87 −0.257116 −0.128558 0.991702i $$-0.541035\pi$$
−0.128558 + 0.991702i $$0.541035\pi$$
$$828$$ 0 0
$$829$$ −32418.4 −1.35819 −0.679093 0.734052i $$-0.737627\pi$$
−0.679093 + 0.734052i $$0.737627\pi$$
$$830$$ −22781.7 −0.952730
$$831$$ 0 0
$$832$$ 19577.3 0.815770
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 29773.6 1.23396
$$836$$ −3032.65 −0.125462
$$837$$ 0 0
$$838$$ 85765.1 3.53545
$$839$$ 44683.5 1.83867 0.919336 0.393474i $$-0.128727\pi$$
0.919336 + 0.393474i $$0.128727\pi$$
$$840$$ 0 0
$$841$$ −6141.15 −0.251800
$$842$$ −3375.17 −0.138142
$$843$$ 0 0
$$844$$ 24082.0 0.982151
$$845$$ −99136.8 −4.03599
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −40117.1 −1.62456
$$849$$ 0 0
$$850$$ −11398.3 −0.459952
$$851$$ 67408.4 2.71531
$$852$$ 0 0
$$853$$ 9965.17 0.400001 0.200001 0.979796i $$-0.435906\pi$$
0.200001 + 0.979796i $$0.435906\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 54785.7 2.18754
$$857$$ −8639.89 −0.344379 −0.172190 0.985064i $$-0.555084\pi$$
−0.172190 + 0.985064i $$0.555084\pi$$
$$858$$ 0 0
$$859$$ −15551.5 −0.617707 −0.308853 0.951110i $$-0.599945\pi$$
−0.308853 + 0.951110i $$0.599945\pi$$
$$860$$ −16833.4 −0.667458
$$861$$ 0 0
$$862$$ −15728.0 −0.621460
$$863$$ −47746.4 −1.88332 −0.941661 0.336563i $$-0.890735\pi$$
−0.941661 + 0.336563i $$0.890735\pi$$
$$864$$ 0 0
$$865$$ 669.599 0.0263203
$$866$$ 8247.97 0.323646
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −18801.6 −0.733946
$$870$$ 0 0
$$871$$ 73488.6 2.85886
$$872$$ 29652.6 1.15156
$$873$$ 0 0
$$874$$ 4640.14 0.179583
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −37958.8 −1.46155 −0.730773 0.682620i $$-0.760840\pi$$
−0.730773 + 0.682620i $$0.760840\pi$$
$$878$$ 81962.8 3.15047
$$879$$ 0 0
$$880$$ 68699.9 2.63168
$$881$$ −32323.6 −1.23611 −0.618053 0.786136i $$-0.712078\pi$$
−0.618053 + 0.786136i $$0.712078\pi$$
$$882$$ 0 0
$$883$$ 27760.0 1.05798 0.528991 0.848628i $$-0.322571\pi$$
0.528991 + 0.848628i $$0.322571\pi$$
$$884$$ −22341.4 −0.850026
$$885$$ 0 0
$$886$$ −31767.5 −1.20457
$$887$$ 32759.6 1.24009 0.620044 0.784567i $$-0.287115\pi$$
0.620044 + 0.784567i $$0.287115\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −42514.5 −1.60122
$$891$$ 0 0
$$892$$ 8991.97 0.337526
$$893$$ 834.071 0.0312554
$$894$$ 0 0
$$895$$ 45118.1 1.68506
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 45083.8 1.67535
$$899$$ −2557.67 −0.0948868
$$900$$ 0 0
$$901$$ 4047.83 0.149670
$$902$$ −67986.9 −2.50966
$$903$$ 0 0
$$904$$ 75865.3 2.79120
$$905$$ 46294.8 1.70043
$$906$$ 0 0
$$907$$ −42682.6 −1.56257 −0.781286 0.624174i $$-0.785436\pi$$
−0.781286 + 0.624174i $$0.785436\pi$$
$$908$$ 74742.8 2.73175
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −50130.0 −1.82314 −0.911570 0.411144i $$-0.865129\pi$$
−0.911570 + 0.411144i $$0.865129\pi$$
$$912$$ 0 0
$$913$$ −7877.60 −0.285554
$$914$$ −2991.57 −0.108263
$$915$$ 0 0
$$916$$ −94275.3 −3.40059
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −33784.7 −1.21268 −0.606341 0.795205i $$-0.707363\pi$$
−0.606341 + 0.795205i $$0.707363\pi$$
$$920$$ −155709. −5.57998
$$921$$ 0 0
$$922$$ −10092.3 −0.360489
$$923$$ −802.303 −0.0286112
$$924$$ 0 0
$$925$$ 66320.6 2.35742
$$926$$ 41831.5 1.48452
$$927$$ 0 0
$$928$$ −33590.7 −1.18822
$$929$$ 50854.8 1.79601 0.898004 0.439987i $$-0.145017\pi$$
0.898004 + 0.439987i $$0.145017\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −82150.4 −2.88726
$$933$$ 0 0
$$934$$ −6396.01 −0.224073
$$935$$ −6931.85 −0.242455
$$936$$ 0 0
$$937$$ 3684.00 0.128443 0.0642215 0.997936i $$-0.479544\pi$$
0.0642215 + 0.997936i $$0.479544\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −49125.8 −1.70458
$$941$$ 28660.5 0.992886 0.496443 0.868069i $$-0.334639\pi$$
0.496443 + 0.868069i $$0.334639\pi$$
$$942$$ 0 0
$$943$$ 72730.8 2.51160
$$944$$ −54866.9 −1.89170
$$945$$ 0 0
$$946$$ −8325.18 −0.286126
$$947$$ 31873.0 1.09370 0.546849 0.837231i $$-0.315827\pi$$
0.546849 + 0.837231i $$0.315827\pi$$
$$948$$ 0 0
$$949$$ −1964.40 −0.0671942
$$950$$ 4565.26 0.155912
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −3310.08 −0.112512 −0.0562561 0.998416i $$-0.517916\pi$$
−0.0562561 + 0.998416i $$0.517916\pi$$
$$954$$ 0 0
$$955$$ −73858.2 −2.50261
$$956$$ −60022.3 −2.03061
$$957$$ 0 0
$$958$$ −95943.6 −3.23570
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29432.5 −0.987966
$$962$$ 185923. 6.23119
$$963$$ 0 0
$$964$$ 28580.6 0.954897
$$965$$ −16642.2 −0.555161
$$966$$ 0 0
$$967$$ −168.413 −0.00560061 −0.00280031 0.999996i $$-0.500891\pi$$
−0.00280031 + 0.999996i $$0.500891\pi$$
$$968$$ −22380.8 −0.743127
$$969$$ 0 0
$$970$$ −22186.2 −0.734387
$$971$$ −37669.9 −1.24499 −0.622495 0.782624i $$-0.713881\pi$$
−0.622495 + 0.782624i $$0.713881\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 4509.33 0.148345
$$975$$ 0 0
$$976$$ 75957.7 2.49114
$$977$$ 26688.7 0.873948 0.436974 0.899474i $$-0.356050\pi$$
0.436974 + 0.899474i $$0.356050\pi$$
$$978$$ 0 0
$$979$$ −14700.9 −0.479921
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 30167.5 0.980329
$$983$$ 1102.29 0.0357655 0.0178828 0.999840i $$-0.494307\pi$$
0.0178828 + 0.999840i $$0.494307\pi$$
$$984$$ 0 0
$$985$$ 26491.6 0.856948
$$986$$ 9346.07 0.301866
$$987$$ 0 0
$$988$$ 8948.19 0.288137
$$989$$ 8906.09 0.286347
$$990$$ 0 0
$$991$$ 16523.6 0.529656 0.264828 0.964296i $$-0.414685\pi$$
0.264828 + 0.964296i $$0.414685\pi$$
$$992$$ 4708.17 0.150690
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 29332.5 0.934575
$$996$$ 0 0
$$997$$ −17994.5 −0.571605 −0.285802 0.958289i $$-0.592260\pi$$
−0.285802 + 0.958289i $$0.592260\pi$$
$$998$$ −5650.72 −0.179229
$$999$$ 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 1323.4.a.ba.1.4 4
3.2 odd 2 inner 1323.4.a.ba.1.1 4
7.6 odd 2 189.4.a.l.1.4 yes 4
21.20 even 2 189.4.a.l.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.l.1.1 4 21.20 even 2
189.4.a.l.1.4 yes 4 7.6 odd 2
1323.4.a.ba.1.1 4 3.2 odd 2 inner
1323.4.a.ba.1.4 4 1.1 even 1 trivial