# Properties

 Label 1323.4.a.ba.1.2 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 1.2 Root $$2.92081$$ 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)

 $$f(q)$$ $$=$$ $$q-1.55133 q^{2} -5.59339 q^{4} +9.81086 q^{5} +21.0878 q^{8} +O(q^{10})$$ $$q-1.55133 q^{2} -5.59339 q^{4} +9.81086 q^{5} +21.0878 q^{8} -15.2198 q^{10} -70.6023 q^{11} -55.5603 q^{13} +12.0331 q^{16} +13.4164 q^{17} -91.3735 q^{19} -54.8759 q^{20} +109.527 q^{22} +113.784 q^{23} -28.7471 q^{25} +86.1922 q^{26} +12.4959 q^{29} -222.934 q^{31} -187.369 q^{32} -20.8132 q^{34} +257.440 q^{37} +141.750 q^{38} +206.889 q^{40} +286.765 q^{41} +4.81323 q^{43} +394.906 q^{44} -176.516 q^{46} -609.517 q^{47} +44.5961 q^{50} +310.770 q^{52} +691.107 q^{53} -692.669 q^{55} -19.3852 q^{58} +217.936 q^{59} +764.747 q^{61} +345.843 q^{62} +194.407 q^{64} -545.094 q^{65} -98.5487 q^{67} -75.0432 q^{68} -921.274 q^{71} +219.934 q^{73} -399.373 q^{74} +511.088 q^{76} -9.42805 q^{79} +118.055 q^{80} -444.866 q^{82} -800.364 q^{83} +131.626 q^{85} -7.46689 q^{86} -1488.85 q^{88} -253.574 q^{89} -636.436 q^{92} +945.560 q^{94} -896.453 q^{95} +59.2529 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 26 q^{4}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.55133 −0.548477 −0.274238 0.961662i $$-0.588426\pi$$
−0.274238 + 0.961662i $$0.588426\pi$$
$$3$$ 0 0
$$4$$ −5.59339 −0.699173
$$5$$ 9.81086 0.877510 0.438755 0.898607i $$-0.355420\pi$$
0.438755 + 0.898607i $$0.355420\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 21.0878 0.931957
$$9$$ 0 0
$$10$$ −15.2198 −0.481294
$$11$$ −70.6023 −1.93522 −0.967609 0.252453i $$-0.918763\pi$$
−0.967609 + 0.252453i $$0.918763\pi$$
$$12$$ 0 0
$$13$$ −55.5603 −1.18536 −0.592679 0.805439i $$-0.701930\pi$$
−0.592679 + 0.805439i $$0.701930\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 12.0331 0.188017
$$17$$ 13.4164 0.191409 0.0957046 0.995410i $$-0.469490\pi$$
0.0957046 + 0.995410i $$0.469490\pi$$
$$18$$ 0 0
$$19$$ −91.3735 −1.10329 −0.551646 0.834079i $$-0.686000\pi$$
−0.551646 + 0.834079i $$0.686000\pi$$
$$20$$ −54.8759 −0.613531
$$21$$ 0 0
$$22$$ 109.527 1.06142
$$23$$ 113.784 1.03155 0.515773 0.856726i $$-0.327505\pi$$
0.515773 + 0.856726i $$0.327505\pi$$
$$24$$ 0 0
$$25$$ −28.7471 −0.229977
$$26$$ 86.1922 0.650141
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 12.4959 0.0800147 0.0400073 0.999199i $$-0.487262\pi$$
0.0400073 + 0.999199i $$0.487262\pi$$
$$30$$ 0 0
$$31$$ −222.934 −1.29162 −0.645808 0.763500i $$-0.723479\pi$$
−0.645808 + 0.763500i $$0.723479\pi$$
$$32$$ −187.369 −1.03508
$$33$$ 0 0
$$34$$ −20.8132 −0.104983
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 257.440 1.14386 0.571930 0.820302i $$-0.306195\pi$$
0.571930 + 0.820302i $$0.306195\pi$$
$$38$$ 141.750 0.605129
$$39$$ 0 0
$$40$$ 206.889 0.817801
$$41$$ 286.765 1.09232 0.546160 0.837681i $$-0.316089\pi$$
0.546160 + 0.837681i $$0.316089\pi$$
$$42$$ 0 0
$$43$$ 4.81323 0.0170700 0.00853500 0.999964i $$-0.497283\pi$$
0.00853500 + 0.999964i $$0.497283\pi$$
$$44$$ 394.906 1.35305
$$45$$ 0 0
$$46$$ −176.516 −0.565778
$$47$$ −609.517 −1.89164 −0.945822 0.324686i $$-0.894741\pi$$
−0.945822 + 0.324686i $$0.894741\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 44.5961 0.126137
$$51$$ 0 0
$$52$$ 310.770 0.828771
$$53$$ 691.107 1.79115 0.895574 0.444913i $$-0.146765\pi$$
0.895574 + 0.444913i $$0.146765\pi$$
$$54$$ 0 0
$$55$$ −692.669 −1.69817
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −19.3852 −0.0438862
$$59$$ 217.936 0.480895 0.240448 0.970662i $$-0.422706\pi$$
0.240448 + 0.970662i $$0.422706\pi$$
$$60$$ 0 0
$$61$$ 764.747 1.60518 0.802589 0.596533i $$-0.203455\pi$$
0.802589 + 0.596533i $$0.203455\pi$$
$$62$$ 345.843 0.708421
$$63$$ 0 0
$$64$$ 194.407 0.379700
$$65$$ −545.094 −1.04016
$$66$$ 0 0
$$67$$ −98.5487 −0.179696 −0.0898481 0.995955i $$-0.528638\pi$$
−0.0898481 + 0.995955i $$0.528638\pi$$
$$68$$ −75.0432 −0.133828
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −921.274 −1.53993 −0.769966 0.638085i $$-0.779727\pi$$
−0.769966 + 0.638085i $$0.779727\pi$$
$$72$$ 0 0
$$73$$ 219.934 0.352621 0.176310 0.984335i $$-0.443584\pi$$
0.176310 + 0.984335i $$0.443584\pi$$
$$74$$ −399.373 −0.627381
$$75$$ 0 0
$$76$$ 511.088 0.771392
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −9.42805 −0.0134271 −0.00671354 0.999977i $$-0.502137\pi$$
−0.00671354 + 0.999977i $$0.502137\pi$$
$$80$$ 118.055 0.164986
$$81$$ 0 0
$$82$$ −444.866 −0.599112
$$83$$ −800.364 −1.05845 −0.529225 0.848481i $$-0.677517\pi$$
−0.529225 + 0.848481i $$0.677517\pi$$
$$84$$ 0 0
$$85$$ 131.626 0.167963
$$86$$ −7.46689 −0.00936250
$$87$$ 0 0
$$88$$ −1488.85 −1.80354
$$89$$ −253.574 −0.302008 −0.151004 0.988533i $$-0.548251\pi$$
−0.151004 + 0.988533i $$0.548251\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −636.436 −0.721229
$$93$$ 0 0
$$94$$ 945.560 1.03752
$$95$$ −896.453 −0.968149
$$96$$ 0 0
$$97$$ 59.2529 0.0620229 0.0310114 0.999519i $$-0.490127\pi$$
0.0310114 + 0.999519i $$0.490127\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 160.794 0.160794
$$101$$ −683.043 −0.672924 −0.336462 0.941697i $$-0.609230\pi$$
−0.336462 + 0.941697i $$0.609230\pi$$
$$102$$ 0 0
$$103$$ 520.537 0.497962 0.248981 0.968508i $$-0.419904\pi$$
0.248981 + 0.968508i $$0.419904\pi$$
$$104$$ −1171.64 −1.10470
$$105$$ 0 0
$$106$$ −1072.13 −0.982403
$$107$$ −714.428 −0.645480 −0.322740 0.946488i $$-0.604604\pi$$
−0.322740 + 0.946488i $$0.604604\pi$$
$$108$$ 0 0
$$109$$ −182.802 −0.160635 −0.0803175 0.996769i $$-0.525593\pi$$
−0.0803175 + 0.996769i $$0.525593\pi$$
$$110$$ 1074.56 0.931408
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 154.433 0.128565 0.0642825 0.997932i $$-0.479524\pi$$
0.0642825 + 0.997932i $$0.479524\pi$$
$$114$$ 0 0
$$115$$ 1116.32 0.905191
$$116$$ −69.8942 −0.0559441
$$117$$ 0 0
$$118$$ −338.089 −0.263760
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 3653.69 2.74507
$$122$$ −1186.37 −0.880402
$$123$$ 0 0
$$124$$ 1246.96 0.903064
$$125$$ −1508.39 −1.07932
$$126$$ 0 0
$$127$$ 2072.00 1.44772 0.723858 0.689949i $$-0.242367\pi$$
0.723858 + 0.689949i $$0.242367\pi$$
$$128$$ 1197.37 0.826823
$$129$$ 0 0
$$130$$ 845.619 0.570505
$$131$$ 25.5712 0.0170547 0.00852736 0.999964i $$-0.497286\pi$$
0.00852736 + 0.999964i $$0.497286\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 152.881 0.0985591
$$135$$ 0 0
$$136$$ 282.922 0.178385
$$137$$ 581.202 0.362448 0.181224 0.983442i $$-0.441994\pi$$
0.181224 + 0.983442i $$0.441994\pi$$
$$138$$ 0 0
$$139$$ −1185.01 −0.723103 −0.361552 0.932352i $$-0.617753\pi$$
−0.361552 + 0.932352i $$0.617753\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1429.20 0.844616
$$143$$ 3922.69 2.29393
$$144$$ 0 0
$$145$$ 122.595 0.0702136
$$146$$ −341.189 −0.193404
$$147$$ 0 0
$$148$$ −1439.96 −0.799756
$$149$$ 471.892 0.259456 0.129728 0.991550i $$-0.458590\pi$$
0.129728 + 0.991550i $$0.458590\pi$$
$$150$$ 0 0
$$151$$ 2635.18 1.42019 0.710093 0.704108i $$-0.248653\pi$$
0.710093 + 0.704108i $$0.248653\pi$$
$$152$$ −1926.87 −1.02822
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2187.17 −1.13341
$$156$$ 0 0
$$157$$ 359.210 0.182599 0.0912996 0.995823i $$-0.470898\pi$$
0.0912996 + 0.995823i $$0.470898\pi$$
$$158$$ 14.6260 0.00736444
$$159$$ 0 0
$$160$$ −1838.25 −0.908292
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2659.91 1.27816 0.639081 0.769140i $$-0.279315\pi$$
0.639081 + 0.769140i $$0.279315\pi$$
$$164$$ −1603.99 −0.763722
$$165$$ 0 0
$$166$$ 1241.63 0.580536
$$167$$ −702.648 −0.325584 −0.162792 0.986660i $$-0.552050\pi$$
−0.162792 + 0.986660i $$0.552050\pi$$
$$168$$ 0 0
$$169$$ 889.949 0.405075
$$170$$ −204.196 −0.0921240
$$171$$ 0 0
$$172$$ −26.9222 −0.0119349
$$173$$ 738.814 0.324688 0.162344 0.986734i $$-0.448095\pi$$
0.162344 + 0.986734i $$0.448095\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −849.563 −0.363853
$$177$$ 0 0
$$178$$ 393.375 0.165645
$$179$$ 3011.14 1.25734 0.628668 0.777674i $$-0.283600\pi$$
0.628668 + 0.777674i $$0.283600\pi$$
$$180$$ 0 0
$$181$$ −204.288 −0.0838928 −0.0419464 0.999120i $$-0.513356\pi$$
−0.0419464 + 0.999120i $$0.513356\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2399.45 0.961356
$$185$$ 2525.70 1.00375
$$186$$ 0 0
$$187$$ −947.230 −0.370419
$$188$$ 3409.27 1.32259
$$189$$ 0 0
$$190$$ 1390.69 0.531007
$$191$$ 3802.34 1.44046 0.720229 0.693736i $$-0.244037\pi$$
0.720229 + 0.693736i $$0.244037\pi$$
$$192$$ 0 0
$$193$$ −3085.69 −1.15084 −0.575422 0.817857i $$-0.695162\pi$$
−0.575422 + 0.817857i $$0.695162\pi$$
$$194$$ −91.9206 −0.0340181
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2710.10 0.980134 0.490067 0.871685i $$-0.336972\pi$$
0.490067 + 0.871685i $$0.336972\pi$$
$$198$$ 0 0
$$199$$ 4807.21 1.71243 0.856217 0.516616i $$-0.172809\pi$$
0.856217 + 0.516616i $$0.172809\pi$$
$$200$$ −606.212 −0.214328
$$201$$ 0 0
$$202$$ 1059.62 0.369083
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2813.41 0.958522
$$206$$ −807.523 −0.273120
$$207$$ 0 0
$$208$$ −668.561 −0.222867
$$209$$ 6451.19 2.13511
$$210$$ 0 0
$$211$$ 5648.81 1.84303 0.921517 0.388338i $$-0.126951\pi$$
0.921517 + 0.388338i $$0.126951\pi$$
$$212$$ −3865.63 −1.25232
$$213$$ 0 0
$$214$$ 1108.31 0.354031
$$215$$ 47.2219 0.0149791
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 283.585 0.0881046
$$219$$ 0 0
$$220$$ 3874.37 1.18732
$$221$$ −745.420 −0.226889
$$222$$ 0 0
$$223$$ −4595.61 −1.38002 −0.690011 0.723799i $$-0.742394\pi$$
−0.690011 + 0.723799i $$0.742394\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −239.576 −0.0705150
$$227$$ −1856.53 −0.542829 −0.271414 0.962463i $$-0.587491\pi$$
−0.271414 + 0.962463i $$0.587491\pi$$
$$228$$ 0 0
$$229$$ 3830.37 1.10532 0.552659 0.833408i $$-0.313613\pi$$
0.552659 + 0.833408i $$0.313613\pi$$
$$230$$ −1731.77 −0.496476
$$231$$ 0 0
$$232$$ 263.510 0.0745702
$$233$$ −2244.80 −0.631166 −0.315583 0.948898i $$-0.602200\pi$$
−0.315583 + 0.948898i $$0.602200\pi$$
$$234$$ 0 0
$$235$$ −5979.89 −1.65994
$$236$$ −1219.00 −0.336229
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3184.89 0.861980 0.430990 0.902357i $$-0.358164\pi$$
0.430990 + 0.902357i $$0.358164\pi$$
$$240$$ 0 0
$$241$$ −1607.14 −0.429564 −0.214782 0.976662i $$-0.568904\pi$$
−0.214782 + 0.976662i $$0.568904\pi$$
$$242$$ −5668.06 −1.50561
$$243$$ 0 0
$$244$$ −4277.53 −1.12230
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5076.74 1.30780
$$248$$ −4701.18 −1.20373
$$249$$ 0 0
$$250$$ 2340.01 0.591980
$$251$$ −3398.32 −0.854582 −0.427291 0.904114i $$-0.640532\pi$$
−0.427291 + 0.904114i $$0.640532\pi$$
$$252$$ 0 0
$$253$$ −8033.39 −1.99627
$$254$$ −3214.34 −0.794039
$$255$$ 0 0
$$256$$ −3412.76 −0.833193
$$257$$ 3127.70 0.759147 0.379573 0.925162i $$-0.376071\pi$$
0.379573 + 0.925162i $$0.376071\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 3048.92 0.727255
$$261$$ 0 0
$$262$$ −39.6693 −0.00935412
$$263$$ −4470.12 −1.04806 −0.524030 0.851700i $$-0.675572\pi$$
−0.524030 + 0.851700i $$0.675572\pi$$
$$264$$ 0 0
$$265$$ 6780.35 1.57175
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 551.221 0.125639
$$269$$ −3021.24 −0.684789 −0.342394 0.939556i $$-0.611238\pi$$
−0.342394 + 0.939556i $$0.611238\pi$$
$$270$$ 0 0
$$271$$ 6635.95 1.48747 0.743736 0.668473i $$-0.233052\pi$$
0.743736 + 0.668473i $$0.233052\pi$$
$$272$$ 161.441 0.0359881
$$273$$ 0 0
$$274$$ −901.634 −0.198794
$$275$$ 2029.61 0.445055
$$276$$ 0 0
$$277$$ −2223.61 −0.482324 −0.241162 0.970485i $$-0.577529\pi$$
−0.241162 + 0.970485i $$0.577529\pi$$
$$278$$ 1838.34 0.396605
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6027.19 1.27954 0.639772 0.768565i $$-0.279029\pi$$
0.639772 + 0.768565i $$0.279029\pi$$
$$282$$ 0 0
$$283$$ 769.183 0.161566 0.0807830 0.996732i $$-0.474258\pi$$
0.0807830 + 0.996732i $$0.474258\pi$$
$$284$$ 5153.04 1.07668
$$285$$ 0 0
$$286$$ −6085.37 −1.25817
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4733.00 −0.963363
$$290$$ −190.185 −0.0385105
$$291$$ 0 0
$$292$$ −1230.18 −0.246543
$$293$$ 5232.52 1.04330 0.521650 0.853160i $$-0.325317\pi$$
0.521650 + 0.853160i $$0.325317\pi$$
$$294$$ 0 0
$$295$$ 2138.14 0.421990
$$296$$ 5428.83 1.06603
$$297$$ 0 0
$$298$$ −732.059 −0.142305
$$299$$ −6321.86 −1.22275
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4088.03 −0.778939
$$303$$ 0 0
$$304$$ −1099.50 −0.207437
$$305$$ 7502.82 1.40856
$$306$$ 0 0
$$307$$ 1705.68 0.317096 0.158548 0.987351i $$-0.449319\pi$$
0.158548 + 0.987351i $$0.449319\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 3393.02 0.621647
$$311$$ 7220.87 1.31659 0.658293 0.752762i $$-0.271279\pi$$
0.658293 + 0.752762i $$0.271279\pi$$
$$312$$ 0 0
$$313$$ 9173.62 1.65663 0.828313 0.560266i $$-0.189301\pi$$
0.828313 + 0.560266i $$0.189301\pi$$
$$314$$ −557.252 −0.100151
$$315$$ 0 0
$$316$$ 52.7347 0.00938785
$$317$$ −1082.48 −0.191793 −0.0958965 0.995391i $$-0.530572\pi$$
−0.0958965 + 0.995391i $$0.530572\pi$$
$$318$$ 0 0
$$319$$ −882.238 −0.154846
$$320$$ 1907.30 0.333191
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1225.90 −0.211180
$$324$$ 0 0
$$325$$ 1597.20 0.272605
$$326$$ −4126.39 −0.701042
$$327$$ 0 0
$$328$$ 6047.23 1.01800
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1753.27 −0.291143 −0.145572 0.989348i $$-0.546502\pi$$
−0.145572 + 0.989348i $$0.546502\pi$$
$$332$$ 4476.75 0.740041
$$333$$ 0 0
$$334$$ 1090.04 0.178575
$$335$$ −966.847 −0.157685
$$336$$ 0 0
$$337$$ 8918.33 1.44158 0.720790 0.693153i $$-0.243779\pi$$
0.720790 + 0.693153i $$0.243779\pi$$
$$338$$ −1380.60 −0.222174
$$339$$ 0 0
$$340$$ −736.238 −0.117436
$$341$$ 15739.7 2.49956
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 101.500 0.0159085
$$345$$ 0 0
$$346$$ −1146.14 −0.178084
$$347$$ 6811.49 1.05377 0.526887 0.849935i $$-0.323359\pi$$
0.526887 + 0.849935i $$0.323359\pi$$
$$348$$ 0 0
$$349$$ −9254.60 −1.41945 −0.709724 0.704480i $$-0.751180\pi$$
−0.709724 + 0.704480i $$0.751180\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 13228.7 2.00311
$$353$$ −8005.75 −1.20709 −0.603546 0.797329i $$-0.706246\pi$$
−0.603546 + 0.797329i $$0.706246\pi$$
$$354$$ 0 0
$$355$$ −9038.49 −1.35130
$$356$$ 1418.34 0.211156
$$357$$ 0 0
$$358$$ −4671.26 −0.689619
$$359$$ −6611.34 −0.971958 −0.485979 0.873970i $$-0.661537\pi$$
−0.485979 + 0.873970i $$0.661537\pi$$
$$360$$ 0 0
$$361$$ 1490.12 0.217251
$$362$$ 316.917 0.0460132
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2157.74 0.309428
$$366$$ 0 0
$$367$$ −1836.21 −0.261170 −0.130585 0.991437i $$-0.541686\pi$$
−0.130585 + 0.991437i $$0.541686\pi$$
$$368$$ 1369.17 0.193948
$$369$$ 0 0
$$370$$ −3918.19 −0.550533
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4819.81 −0.669062 −0.334531 0.942385i $$-0.608578\pi$$
−0.334531 + 0.942385i $$0.608578\pi$$
$$374$$ 1469.46 0.203166
$$375$$ 0 0
$$376$$ −12853.4 −1.76293
$$377$$ −694.275 −0.0948461
$$378$$ 0 0
$$379$$ 6710.00 0.909418 0.454709 0.890640i $$-0.349743\pi$$
0.454709 + 0.890640i $$0.349743\pi$$
$$380$$ 5014.21 0.676904
$$381$$ 0 0
$$382$$ −5898.67 −0.790058
$$383$$ −11584.9 −1.54559 −0.772797 0.634653i $$-0.781143\pi$$
−0.772797 + 0.634653i $$0.781143\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4786.91 0.631211
$$387$$ 0 0
$$388$$ −331.424 −0.0433648
$$389$$ 7523.59 0.980620 0.490310 0.871548i $$-0.336884\pi$$
0.490310 + 0.871548i $$0.336884\pi$$
$$390$$ 0 0
$$391$$ 1526.57 0.197447
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4204.24 −0.537581
$$395$$ −92.4973 −0.0117824
$$396$$ 0 0
$$397$$ 3302.26 0.417471 0.208735 0.977972i $$-0.433065\pi$$
0.208735 + 0.977972i $$0.433065\pi$$
$$398$$ −7457.56 −0.939230
$$399$$ 0 0
$$400$$ −345.916 −0.0432395
$$401$$ 6996.80 0.871331 0.435665 0.900109i $$-0.356513\pi$$
0.435665 + 0.900109i $$0.356513\pi$$
$$402$$ 0 0
$$403$$ 12386.3 1.53103
$$404$$ 3820.52 0.470491
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −18175.8 −2.21362
$$408$$ 0 0
$$409$$ 4534.39 0.548193 0.274097 0.961702i $$-0.411621\pi$$
0.274097 + 0.961702i $$0.411621\pi$$
$$410$$ −4364.52 −0.525727
$$411$$ 0 0
$$412$$ −2911.57 −0.348161
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −7852.26 −0.928801
$$416$$ 10410.3 1.22694
$$417$$ 0 0
$$418$$ −10007.9 −1.17106
$$419$$ 1497.50 0.174601 0.0873004 0.996182i $$-0.472176\pi$$
0.0873004 + 0.996182i $$0.472176\pi$$
$$420$$ 0 0
$$421$$ −4669.50 −0.540564 −0.270282 0.962781i $$-0.587117\pi$$
−0.270282 + 0.962781i $$0.587117\pi$$
$$422$$ −8763.15 −1.01086
$$423$$ 0 0
$$424$$ 14573.9 1.66927
$$425$$ −385.683 −0.0440197
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 3996.07 0.451302
$$429$$ 0 0
$$430$$ −73.2565 −0.00821568
$$431$$ −11985.2 −1.33946 −0.669732 0.742603i $$-0.733591\pi$$
−0.669732 + 0.742603i $$0.733591\pi$$
$$432$$ 0 0
$$433$$ 8226.59 0.913036 0.456518 0.889714i $$-0.349096\pi$$
0.456518 + 0.889714i $$0.349096\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 1022.48 0.112312
$$437$$ −10396.8 −1.13809
$$438$$ 0 0
$$439$$ 8154.12 0.886503 0.443251 0.896397i $$-0.353825\pi$$
0.443251 + 0.896397i $$0.353825\pi$$
$$440$$ −14606.9 −1.58262
$$441$$ 0 0
$$442$$ 1156.39 0.124443
$$443$$ −11406.0 −1.22329 −0.611644 0.791133i $$-0.709492\pi$$
−0.611644 + 0.791133i $$0.709492\pi$$
$$444$$ 0 0
$$445$$ −2487.77 −0.265015
$$446$$ 7129.29 0.756910
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2463.59 0.258940 0.129470 0.991583i $$-0.458672\pi$$
0.129470 + 0.991583i $$0.458672\pi$$
$$450$$ 0 0
$$451$$ −20246.3 −2.11388
$$452$$ −863.805 −0.0898893
$$453$$ 0 0
$$454$$ 2880.08 0.297729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −16349.9 −1.67356 −0.836778 0.547542i $$-0.815564\pi$$
−0.836778 + 0.547542i $$0.815564\pi$$
$$458$$ −5942.15 −0.606241
$$459$$ 0 0
$$460$$ −6243.98 −0.632885
$$461$$ 2872.86 0.290244 0.145122 0.989414i $$-0.453643\pi$$
0.145122 + 0.989414i $$0.453643\pi$$
$$462$$ 0 0
$$463$$ 2984.20 0.299541 0.149771 0.988721i $$-0.452146\pi$$
0.149771 + 0.988721i $$0.452146\pi$$
$$464$$ 150.364 0.0150441
$$465$$ 0 0
$$466$$ 3482.42 0.346180
$$467$$ −19419.5 −1.92426 −0.962129 0.272595i $$-0.912118\pi$$
−0.962129 + 0.272595i $$0.912118\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 9276.76 0.910436
$$471$$ 0 0
$$472$$ 4595.78 0.448174
$$473$$ −339.825 −0.0330342
$$474$$ 0 0
$$475$$ 2626.72 0.253731
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −4940.80 −0.472776
$$479$$ −18229.3 −1.73887 −0.869435 0.494047i $$-0.835517\pi$$
−0.869435 + 0.494047i $$0.835517\pi$$
$$480$$ 0 0
$$481$$ −14303.4 −1.35588
$$482$$ 2493.20 0.235606
$$483$$ 0 0
$$484$$ −20436.5 −1.91928
$$485$$ 581.322 0.0544257
$$486$$ 0 0
$$487$$ 17079.6 1.58922 0.794609 0.607122i $$-0.207676\pi$$
0.794609 + 0.607122i $$0.207676\pi$$
$$488$$ 16126.8 1.49596
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1087.13 0.0999214 0.0499607 0.998751i $$-0.484090\pi$$
0.0499607 + 0.998751i $$0.484090\pi$$
$$492$$ 0 0
$$493$$ 167.650 0.0153155
$$494$$ −7875.69 −0.717295
$$495$$ 0 0
$$496$$ −2682.58 −0.242845
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 18785.8 1.68530 0.842652 0.538459i $$-0.180993\pi$$
0.842652 + 0.538459i $$0.180993\pi$$
$$500$$ 8437.01 0.754629
$$501$$ 0 0
$$502$$ 5271.91 0.468719
$$503$$ 12278.7 1.08843 0.544214 0.838947i $$-0.316828\pi$$
0.544214 + 0.838947i $$0.316828\pi$$
$$504$$ 0 0
$$505$$ −6701.24 −0.590497
$$506$$ 12462.4 1.09490
$$507$$ 0 0
$$508$$ −11589.5 −1.01220
$$509$$ −1226.83 −0.106834 −0.0534168 0.998572i $$-0.517011\pi$$
−0.0534168 + 0.998572i $$0.517011\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −4284.63 −0.369836
$$513$$ 0 0
$$514$$ −4852.09 −0.416374
$$515$$ 5106.91 0.436966
$$516$$ 0 0
$$517$$ 43033.3 3.66074
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −11494.8 −0.969388
$$521$$ 12571.3 1.05712 0.528559 0.848897i $$-0.322733\pi$$
0.528559 + 0.848897i $$0.322733\pi$$
$$522$$ 0 0
$$523$$ 7828.18 0.654498 0.327249 0.944938i $$-0.393878\pi$$
0.327249 + 0.944938i $$0.393878\pi$$
$$524$$ −143.030 −0.0119242
$$525$$ 0 0
$$526$$ 6934.62 0.574836
$$527$$ −2990.97 −0.247227
$$528$$ 0 0
$$529$$ 779.727 0.0640854
$$530$$ −10518.5 −0.862068
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −15932.7 −1.29479
$$534$$ 0 0
$$535$$ −7009.15 −0.566415
$$536$$ −2078.17 −0.167469
$$537$$ 0 0
$$538$$ 4686.93 0.375591
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5726.90 −0.455118 −0.227559 0.973764i $$-0.573074\pi$$
−0.227559 + 0.973764i $$0.573074\pi$$
$$542$$ −10294.5 −0.815844
$$543$$ 0 0
$$544$$ −2513.82 −0.198124
$$545$$ −1793.44 −0.140959
$$546$$ 0 0
$$547$$ −7738.41 −0.604882 −0.302441 0.953168i $$-0.597801\pi$$
−0.302441 + 0.953168i $$0.597801\pi$$
$$548$$ −3250.89 −0.253414
$$549$$ 0 0
$$550$$ −3148.59 −0.244102
$$551$$ −1141.79 −0.0882795
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 3449.55 0.264544
$$555$$ 0 0
$$556$$ 6628.23 0.505575
$$557$$ 6714.94 0.510810 0.255405 0.966834i $$-0.417791\pi$$
0.255405 + 0.966834i $$0.417791\pi$$
$$558$$ 0 0
$$559$$ −267.424 −0.0202341
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −9350.13 −0.701800
$$563$$ 25016.6 1.87269 0.936343 0.351087i $$-0.114188\pi$$
0.936343 + 0.351087i $$0.114188\pi$$
$$564$$ 0 0
$$565$$ 1515.12 0.112817
$$566$$ −1193.25 −0.0886152
$$567$$ 0 0
$$568$$ −19427.6 −1.43515
$$569$$ 11748.3 0.865578 0.432789 0.901495i $$-0.357530\pi$$
0.432789 + 0.901495i $$0.357530\pi$$
$$570$$ 0 0
$$571$$ 2168.65 0.158941 0.0794705 0.996837i $$-0.474677\pi$$
0.0794705 + 0.996837i $$0.474677\pi$$
$$572$$ −21941.1 −1.60385
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3270.95 −0.237231
$$576$$ 0 0
$$577$$ −3941.26 −0.284362 −0.142181 0.989841i $$-0.545411\pi$$
−0.142181 + 0.989841i $$0.545411\pi$$
$$578$$ 7342.43 0.528382
$$579$$ 0 0
$$580$$ −685.722 −0.0490915
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −48793.8 −3.46626
$$584$$ 4637.92 0.328627
$$585$$ 0 0
$$586$$ −8117.34 −0.572226
$$587$$ 13873.7 0.975521 0.487760 0.872978i $$-0.337814\pi$$
0.487760 + 0.872978i $$0.337814\pi$$
$$588$$ 0 0
$$589$$ 20370.3 1.42503
$$590$$ −3316.95 −0.231452
$$591$$ 0 0
$$592$$ 3097.79 0.215065
$$593$$ −9754.07 −0.675467 −0.337733 0.941242i $$-0.609660\pi$$
−0.337733 + 0.941242i $$0.609660\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2639.48 −0.181405
$$597$$ 0 0
$$598$$ 9807.26 0.670650
$$599$$ −15994.0 −1.09098 −0.545491 0.838116i $$-0.683657\pi$$
−0.545491 + 0.838116i $$0.683657\pi$$
$$600$$ 0 0
$$601$$ −15183.7 −1.03054 −0.515272 0.857027i $$-0.672309\pi$$
−0.515272 + 0.857027i $$0.672309\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −14739.6 −0.992957
$$605$$ 35845.8 2.40883
$$606$$ 0 0
$$607$$ −18747.2 −1.25358 −0.626792 0.779187i $$-0.715632\pi$$
−0.626792 + 0.779187i $$0.715632\pi$$
$$608$$ 17120.6 1.14199
$$609$$ 0 0
$$610$$ −11639.3 −0.772562
$$611$$ 33865.0 2.24228
$$612$$ 0 0
$$613$$ −460.802 −0.0303615 −0.0151808 0.999885i $$-0.504832\pi$$
−0.0151808 + 0.999885i $$0.504832\pi$$
$$614$$ −2646.07 −0.173920
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −27174.7 −1.77311 −0.886557 0.462620i $$-0.846910\pi$$
−0.886557 + 0.462620i $$0.846910\pi$$
$$618$$ 0 0
$$619$$ −29663.1 −1.92611 −0.963055 0.269305i $$-0.913206\pi$$
−0.963055 + 0.269305i $$0.913206\pi$$
$$620$$ 12233.7 0.792447
$$621$$ 0 0
$$622$$ −11201.9 −0.722117
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11205.2 −0.717134
$$626$$ −14231.3 −0.908621
$$627$$ 0 0
$$628$$ −2009.20 −0.127669
$$629$$ 3453.92 0.218945
$$630$$ 0 0
$$631$$ 8710.14 0.549517 0.274758 0.961513i $$-0.411402\pi$$
0.274758 + 0.961513i $$0.411402\pi$$
$$632$$ −198.817 −0.0125135
$$633$$ 0 0
$$634$$ 1679.29 0.105194
$$635$$ 20328.1 1.27038
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1368.64 0.0849293
$$639$$ 0 0
$$640$$ 11747.2 0.725545
$$641$$ −14429.7 −0.889138 −0.444569 0.895745i $$-0.646643\pi$$
−0.444569 + 0.895745i $$0.646643\pi$$
$$642$$ 0 0
$$643$$ 25865.5 1.58637 0.793184 0.608982i $$-0.208422\pi$$
0.793184 + 0.608982i $$0.208422\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 1901.78 0.115827
$$647$$ −5059.78 −0.307451 −0.153725 0.988114i $$-0.549127\pi$$
−0.153725 + 0.988114i $$0.549127\pi$$
$$648$$ 0 0
$$649$$ −15386.8 −0.930637
$$650$$ −2477.77 −0.149517
$$651$$ 0 0
$$652$$ −14877.9 −0.893656
$$653$$ 15389.9 0.922287 0.461143 0.887326i $$-0.347439\pi$$
0.461143 + 0.887326i $$0.347439\pi$$
$$654$$ 0 0
$$655$$ 250.876 0.0149657
$$656$$ 3450.66 0.205375
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 30375.3 1.79553 0.897765 0.440474i $$-0.145190\pi$$
0.897765 + 0.440474i $$0.145190\pi$$
$$660$$ 0 0
$$661$$ −12410.4 −0.730270 −0.365135 0.930955i $$-0.618977\pi$$
−0.365135 + 0.930955i $$0.618977\pi$$
$$662$$ 2719.89 0.159685
$$663$$ 0 0
$$664$$ −16877.9 −0.986431
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1421.83 0.0825387
$$668$$ 3930.18 0.227640
$$669$$ 0 0
$$670$$ 1499.90 0.0864866
$$671$$ −53992.9 −3.10637
$$672$$ 0 0
$$673$$ 23632.8 1.35361 0.676804 0.736163i $$-0.263364\pi$$
0.676804 + 0.736163i $$0.263364\pi$$
$$674$$ −13835.2 −0.790673
$$675$$ 0 0
$$676$$ −4977.83 −0.283217
$$677$$ −7460.16 −0.423511 −0.211756 0.977323i $$-0.567918\pi$$
−0.211756 + 0.977323i $$0.567918\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 2775.71 0.156535
$$681$$ 0 0
$$682$$ −24417.3 −1.37095
$$683$$ 7798.73 0.436911 0.218455 0.975847i $$-0.429898\pi$$
0.218455 + 0.975847i $$0.429898\pi$$
$$684$$ 0 0
$$685$$ 5702.09 0.318052
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 57.9179 0.00320945
$$689$$ −38398.1 −2.12315
$$690$$ 0 0
$$691$$ −4748.63 −0.261427 −0.130714 0.991420i $$-0.541727\pi$$
−0.130714 + 0.991420i $$0.541727\pi$$
$$692$$ −4132.47 −0.227013
$$693$$ 0 0
$$694$$ −10566.8 −0.577971
$$695$$ −11626.0 −0.634530
$$696$$ 0 0
$$697$$ 3847.35 0.209080
$$698$$ 14356.9 0.778534
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13784.1 −0.742681 −0.371341 0.928497i $$-0.621102\pi$$
−0.371341 + 0.928497i $$0.621102\pi$$
$$702$$ 0 0
$$703$$ −23523.2 −1.26201
$$704$$ −13725.6 −0.734803
$$705$$ 0 0
$$706$$ 12419.5 0.662061
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 577.799 0.0306060 0.0153030 0.999883i $$-0.495129\pi$$
0.0153030 + 0.999883i $$0.495129\pi$$
$$710$$ 14021.6 0.741159
$$711$$ 0 0
$$712$$ −5347.30 −0.281459
$$713$$ −25366.2 −1.33236
$$714$$ 0 0
$$715$$ 38484.9 2.01294
$$716$$ −16842.5 −0.879096
$$717$$ 0 0
$$718$$ 10256.3 0.533097
$$719$$ 20274.7 1.05162 0.525812 0.850601i $$-0.323761\pi$$
0.525812 + 0.850601i $$0.323761\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −2311.67 −0.119157
$$723$$ 0 0
$$724$$ 1142.66 0.0586556
$$725$$ −359.220 −0.0184015
$$726$$ 0 0
$$727$$ −28861.8 −1.47239 −0.736193 0.676772i $$-0.763378\pi$$
−0.736193 + 0.676772i $$0.763378\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −3347.36 −0.169714
$$731$$ 64.5762 0.00326736
$$732$$ 0 0
$$733$$ 26169.5 1.31868 0.659340 0.751844i $$-0.270836\pi$$
0.659340 + 0.751844i $$0.270836\pi$$
$$734$$ 2848.56 0.143246
$$735$$ 0 0
$$736$$ −21319.6 −1.06773
$$737$$ 6957.77 0.347751
$$738$$ 0 0
$$739$$ −8351.16 −0.415700 −0.207850 0.978161i $$-0.566647\pi$$
−0.207850 + 0.978161i $$0.566647\pi$$
$$740$$ −14127.2 −0.701794
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20732.7 1.02370 0.511849 0.859075i $$-0.328961\pi$$
0.511849 + 0.859075i $$0.328961\pi$$
$$744$$ 0 0
$$745$$ 4629.67 0.227675
$$746$$ 7477.09 0.366965
$$747$$ 0 0
$$748$$ 5298.22 0.258987
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 13055.4 0.634351 0.317176 0.948367i $$-0.397265\pi$$
0.317176 + 0.948367i $$0.397265\pi$$
$$752$$ −7334.36 −0.355661
$$753$$ 0 0
$$754$$ 1077.05 0.0520208
$$755$$ 25853.4 1.24623
$$756$$ 0 0
$$757$$ 14425.9 0.692628 0.346314 0.938119i $$-0.387433\pi$$
0.346314 + 0.938119i $$0.387433\pi$$
$$758$$ −10409.4 −0.498794
$$759$$ 0 0
$$760$$ −18904.2 −0.902273
$$761$$ 5232.01 0.249225 0.124613 0.992205i $$-0.460231\pi$$
0.124613 + 0.992205i $$0.460231\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −21268.0 −1.00713
$$765$$ 0 0
$$766$$ 17972.0 0.847723
$$767$$ −12108.6 −0.570033
$$768$$ 0 0
$$769$$ 18570.2 0.870818 0.435409 0.900233i $$-0.356604\pi$$
0.435409 + 0.900233i $$0.356604\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 17259.5 0.804639
$$773$$ −23462.7 −1.09171 −0.545857 0.837878i $$-0.683796\pi$$
−0.545857 + 0.837878i $$0.683796\pi$$
$$774$$ 0 0
$$775$$ 6408.70 0.297042
$$776$$ 1249.51 0.0578027
$$777$$ 0 0
$$778$$ −11671.5 −0.537847
$$779$$ −26202.7 −1.20515
$$780$$ 0 0
$$781$$ 65044.1 2.98010
$$782$$ −2368.21 −0.108295
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3524.16 0.160233
$$786$$ 0 0
$$787$$ −25626.0 −1.16070 −0.580348 0.814369i $$-0.697083\pi$$
−0.580348 + 0.814369i $$0.697083\pi$$
$$788$$ −15158.6 −0.685284
$$789$$ 0 0
$$790$$ 143.493 0.00646236
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −42489.6 −1.90271
$$794$$ −5122.89 −0.228973
$$795$$ 0 0
$$796$$ −26888.6 −1.19729
$$797$$ −12209.4 −0.542633 −0.271316 0.962490i $$-0.587459\pi$$
−0.271316 + 0.962490i $$0.587459\pi$$
$$798$$ 0 0
$$799$$ −8177.53 −0.362078
$$800$$ 5386.33 0.238044
$$801$$ 0 0
$$802$$ −10854.3 −0.477905
$$803$$ −15527.8 −0.682398
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −19215.2 −0.839733
$$807$$ 0 0
$$808$$ −14403.9 −0.627136
$$809$$ 7844.50 0.340912 0.170456 0.985365i $$-0.445476\pi$$
0.170456 + 0.985365i $$0.445476\pi$$
$$810$$ 0 0
$$811$$ −7602.91 −0.329192 −0.164596 0.986361i $$-0.552632\pi$$
−0.164596 + 0.986361i $$0.552632\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 28196.7 1.21412
$$815$$ 26096.0 1.12160
$$816$$ 0 0
$$817$$ −439.802 −0.0188332
$$818$$ −7034.31 −0.300671
$$819$$ 0 0
$$820$$ −15736.5 −0.670173
$$821$$ −29146.5 −1.23900 −0.619501 0.784996i $$-0.712665\pi$$
−0.619501 + 0.784996i $$0.712665\pi$$
$$822$$ 0 0
$$823$$ −22062.1 −0.934432 −0.467216 0.884143i $$-0.654743\pi$$
−0.467216 + 0.884143i $$0.654743\pi$$
$$824$$ 10977.0 0.464079
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −4571.98 −0.192241 −0.0961206 0.995370i $$-0.530643\pi$$
−0.0961206 + 0.995370i $$0.530643\pi$$
$$828$$ 0 0
$$829$$ −10311.6 −0.432012 −0.216006 0.976392i $$-0.569303\pi$$
−0.216006 + 0.976392i $$0.569303\pi$$
$$830$$ 12181.4 0.509426
$$831$$ 0 0
$$832$$ −10801.3 −0.450081
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −6893.57 −0.285703
$$836$$ −36084.0 −1.49281
$$837$$ 0 0
$$838$$ −2323.11 −0.0957645
$$839$$ −3414.34 −0.140496 −0.0702480 0.997530i $$-0.522379\pi$$
−0.0702480 + 0.997530i $$0.522379\pi$$
$$840$$ 0 0
$$841$$ −24232.9 −0.993598
$$842$$ 7243.92 0.296487
$$843$$ 0 0
$$844$$ −31596.0 −1.28860
$$845$$ 8731.16 0.355457
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 8316.14 0.336766
$$849$$ 0 0
$$850$$ 598.320 0.0241438
$$851$$ 29292.4 1.17994
$$852$$ 0 0
$$853$$ −12093.2 −0.485419 −0.242709 0.970099i $$-0.578036\pi$$
−0.242709 + 0.970099i $$0.578036\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −15065.7 −0.601560
$$857$$ 35003.7 1.39522 0.697610 0.716478i $$-0.254247\pi$$
0.697610 + 0.716478i $$0.254247\pi$$
$$858$$ 0 0
$$859$$ 1669.49 0.0663123 0.0331562 0.999450i $$-0.489444\pi$$
0.0331562 + 0.999450i $$0.489444\pi$$
$$860$$ −264.130 −0.0104730
$$861$$ 0 0
$$862$$ 18593.0 0.734664
$$863$$ −1620.80 −0.0639312 −0.0319656 0.999489i $$-0.510177\pi$$
−0.0319656 + 0.999489i $$0.510177\pi$$
$$864$$ 0 0
$$865$$ 7248.40 0.284917
$$866$$ −12762.1 −0.500779
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 665.642 0.0259843
$$870$$ 0 0
$$871$$ 5475.40 0.213004
$$872$$ −3854.88 −0.149705
$$873$$ 0 0
$$874$$ 16128.9 0.624218
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14449.2 −0.556346 −0.278173 0.960531i $$-0.589729\pi$$
−0.278173 + 0.960531i $$0.589729\pi$$
$$878$$ −12649.7 −0.486226
$$879$$ 0 0
$$880$$ −8334.94 −0.319285
$$881$$ 40111.6 1.53393 0.766966 0.641688i $$-0.221766\pi$$
0.766966 + 0.641688i $$0.221766\pi$$
$$882$$ 0 0
$$883$$ 10152.0 0.386911 0.193456 0.981109i $$-0.438030\pi$$
0.193456 + 0.981109i $$0.438030\pi$$
$$884$$ 4169.42 0.158634
$$885$$ 0 0
$$886$$ 17694.5 0.670945
$$887$$ −18907.0 −0.715711 −0.357856 0.933777i $$-0.616492\pi$$
−0.357856 + 0.933777i $$0.616492\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 3859.35 0.145355
$$891$$ 0 0
$$892$$ 25705.0 0.964875
$$893$$ 55693.8 2.08703
$$894$$ 0 0
$$895$$ 29541.9 1.10332
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −3821.83 −0.142022
$$899$$ −2785.75 −0.103348
$$900$$ 0 0
$$901$$ 9272.17 0.342842
$$902$$ 31408.6 1.15941
$$903$$ 0 0
$$904$$ 3256.65 0.119817
$$905$$ −2004.24 −0.0736167
$$906$$ 0 0
$$907$$ −6789.41 −0.248554 −0.124277 0.992248i $$-0.539661\pi$$
−0.124277 + 0.992248i $$0.539661\pi$$
$$908$$ 10384.3 0.379531
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 45716.8 1.66264 0.831320 0.555794i $$-0.187586\pi$$
0.831320 + 0.555794i $$0.187586\pi$$
$$912$$ 0 0
$$913$$ 56507.6 2.04833
$$914$$ 25364.0 0.917907
$$915$$ 0 0
$$916$$ −21424.7 −0.772809
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 19232.7 0.690346 0.345173 0.938539i $$-0.387820\pi$$
0.345173 + 0.938539i $$0.387820\pi$$
$$920$$ 23540.6 0.843599
$$921$$ 0 0
$$922$$ −4456.74 −0.159192
$$923$$ 51186.3 1.82537
$$924$$ 0 0
$$925$$ −7400.64 −0.263061
$$926$$ −4629.47 −0.164291
$$927$$ 0 0
$$928$$ −2341.34 −0.0828215
$$929$$ 45139.4 1.59416 0.797081 0.603873i $$-0.206377\pi$$
0.797081 + 0.603873i $$0.206377\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 12556.0 0.441295
$$933$$ 0 0
$$934$$ 30126.0 1.05541
$$935$$ −9293.13 −0.325046
$$936$$ 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
$$940$$ 33447.8 1.16058
$$941$$ −3082.72 −0.106795 −0.0533973 0.998573i $$-0.517005\pi$$
−0.0533973 + 0.998573i $$0.517005\pi$$
$$942$$ 0 0
$$943$$ 32629.2 1.12678
$$944$$ 2622.43 0.0904163
$$945$$ 0 0
$$946$$ 527.180 0.0181185
$$947$$ 6475.69 0.222209 0.111104 0.993809i $$-0.464561\pi$$
0.111104 + 0.993809i $$0.464561\pi$$
$$948$$ 0 0
$$949$$ −12219.6 −0.417982
$$950$$ −4074.91 −0.139166
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −7643.58 −0.259811 −0.129906 0.991526i $$-0.541467\pi$$
−0.129906 + 0.991526i $$0.541467\pi$$
$$954$$ 0 0
$$955$$ 37304.2 1.26402
$$956$$ −17814.3 −0.602674
$$957$$ 0 0
$$958$$ 28279.6 0.953730
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 19908.5 0.668273
$$962$$ 22189.3 0.743671
$$963$$ 0 0
$$964$$ 8989.36 0.300340
$$965$$ −30273.3 −1.00988
$$966$$ 0 0
$$967$$ 5636.41 0.187440 0.0937202 0.995599i $$-0.470124\pi$$
0.0937202 + 0.995599i $$0.470124\pi$$
$$968$$ 77048.2 2.55829
$$969$$ 0 0
$$970$$ −901.820 −0.0298512
$$971$$ −12541.0 −0.414479 −0.207240 0.978290i $$-0.566448\pi$$
−0.207240 + 0.978290i $$0.566448\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −26496.0 −0.871649
$$975$$ 0 0
$$976$$ 9202.25 0.301800
$$977$$ −45250.1 −1.48176 −0.740879 0.671638i $$-0.765591\pi$$
−0.740879 + 0.671638i $$0.765591\pi$$
$$978$$ 0 0
$$979$$ 17902.9 0.584452
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1686.49 −0.0548045
$$983$$ −48847.0 −1.58492 −0.792460 0.609923i $$-0.791200\pi$$
−0.792460 + 0.609923i $$0.791200\pi$$
$$984$$ 0 0
$$985$$ 26588.4 0.860077
$$986$$ −260.079 −0.00840022
$$987$$ 0 0
$$988$$ −28396.2 −0.914376
$$989$$ 547.667 0.0176085
$$990$$ 0 0
$$991$$ −31801.6 −1.01939 −0.509693 0.860357i $$-0.670241\pi$$
−0.509693 + 0.860357i $$0.670241\pi$$
$$992$$ 41771.0 1.33693
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 47162.9 1.50268
$$996$$ 0 0
$$997$$ 18382.5 0.583930 0.291965 0.956429i $$-0.405691\pi$$
0.291965 + 0.956429i $$0.405691\pi$$
$$998$$ −29142.8 −0.924349
$$999$$ 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.1.2 4
3.2 odd 2 inner 1323.4.a.ba.1.3 4
7.6 odd 2 189.4.a.l.1.2 4
21.20 even 2 189.4.a.l.1.3 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.l.1.2 4 7.6 odd 2
189.4.a.l.1.3 yes 4 21.20 even 2
1323.4.a.ba.1.2 4 1.1 even 1 trivial
1323.4.a.ba.1.3 4 3.2 odd 2 inner