# Properties

 Label 1323.4.a.be.1.3 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ 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: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 40x^{4} + 453x^{2} - 1278$$ x^6 - 40*x^4 + 453*x^2 - 1278 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 1.3 Root $$-2.05936$$ 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)

 $$f(q)$$ $$=$$ $$q-2.05936 q^{2} -3.75905 q^{4} +14.5041 q^{5} +24.2161 q^{8} +O(q^{10})$$ $$q-2.05936 q^{2} -3.75905 q^{4} +14.5041 q^{5} +24.2161 q^{8} -29.8691 q^{10} -29.8094 q^{11} +13.3320 q^{13} -19.7971 q^{16} -64.2830 q^{17} +110.350 q^{19} -54.5216 q^{20} +61.3881 q^{22} -19.3456 q^{23} +85.3686 q^{25} -27.4552 q^{26} +111.113 q^{29} -192.938 q^{31} -152.959 q^{32} +132.382 q^{34} -71.5214 q^{37} -227.249 q^{38} +351.232 q^{40} -277.562 q^{41} -178.522 q^{43} +112.055 q^{44} +39.8395 q^{46} -531.874 q^{47} -175.804 q^{50} -50.1155 q^{52} +310.832 q^{53} -432.357 q^{55} -228.821 q^{58} +722.023 q^{59} -663.070 q^{61} +397.328 q^{62} +473.375 q^{64} +193.368 q^{65} -608.559 q^{67} +241.643 q^{68} +976.305 q^{71} -261.148 q^{73} +147.288 q^{74} -414.810 q^{76} +1236.15 q^{79} -287.139 q^{80} +571.600 q^{82} -1225.79 q^{83} -932.367 q^{85} +367.640 q^{86} -721.866 q^{88} +791.982 q^{89} +72.7211 q^{92} +1095.32 q^{94} +1600.52 q^{95} +935.253 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 32 q^{4}+O(q^{10})$$ 6 * q + 32 * q^4 $$6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58} - 886 q^{61} - 924 q^{64} - 2084 q^{67} + 2398 q^{73} - 3204 q^{76} + 984 q^{79} + 3892 q^{82} - 3600 q^{85} - 5796 q^{88} - 2772 q^{94} + 682 q^{97}+O(q^{100})$$ 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.05936 −0.728092 −0.364046 0.931381i $$-0.618605\pi$$
−0.364046 + 0.931381i $$0.618605\pi$$
$$3$$ 0 0
$$4$$ −3.75905 −0.469882
$$5$$ 14.5041 1.29729 0.648643 0.761093i $$-0.275337\pi$$
0.648643 + 0.761093i $$0.275337\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 24.2161 1.07021
$$9$$ 0 0
$$10$$ −29.8691 −0.944543
$$11$$ −29.8094 −0.817078 −0.408539 0.912741i $$-0.633962\pi$$
−0.408539 + 0.912741i $$0.633962\pi$$
$$12$$ 0 0
$$13$$ 13.3320 0.284432 0.142216 0.989836i $$-0.454577\pi$$
0.142216 + 0.989836i $$0.454577\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −19.7971 −0.309330
$$17$$ −64.2830 −0.917113 −0.458557 0.888665i $$-0.651633\pi$$
−0.458557 + 0.888665i $$0.651633\pi$$
$$18$$ 0 0
$$19$$ 110.350 1.33242 0.666209 0.745765i $$-0.267916\pi$$
0.666209 + 0.745765i $$0.267916\pi$$
$$20$$ −54.5216 −0.609570
$$21$$ 0 0
$$22$$ 61.3881 0.594908
$$23$$ −19.3456 −0.175384 −0.0876921 0.996148i $$-0.527949\pi$$
−0.0876921 + 0.996148i $$0.527949\pi$$
$$24$$ 0 0
$$25$$ 85.3686 0.682949
$$26$$ −27.4552 −0.207093
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 111.113 0.711487 0.355744 0.934584i $$-0.384228\pi$$
0.355744 + 0.934584i $$0.384228\pi$$
$$30$$ 0 0
$$31$$ −192.938 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$32$$ −152.959 −0.844989
$$33$$ 0 0
$$34$$ 132.382 0.667743
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −71.5214 −0.317785 −0.158892 0.987296i $$-0.550792\pi$$
−0.158892 + 0.987296i $$0.550792\pi$$
$$38$$ −227.249 −0.970123
$$39$$ 0 0
$$40$$ 351.232 1.38837
$$41$$ −277.562 −1.05727 −0.528634 0.848850i $$-0.677296\pi$$
−0.528634 + 0.848850i $$0.677296\pi$$
$$42$$ 0 0
$$43$$ −178.522 −0.633124 −0.316562 0.948572i $$-0.602528\pi$$
−0.316562 + 0.948572i $$0.602528\pi$$
$$44$$ 112.055 0.383930
$$45$$ 0 0
$$46$$ 39.8395 0.127696
$$47$$ −531.874 −1.65068 −0.825338 0.564638i $$-0.809016\pi$$
−0.825338 + 0.564638i $$0.809016\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −175.804 −0.497250
$$51$$ 0 0
$$52$$ −50.1155 −0.133649
$$53$$ 310.832 0.805586 0.402793 0.915291i $$-0.368039\pi$$
0.402793 + 0.915291i $$0.368039\pi$$
$$54$$ 0 0
$$55$$ −432.357 −1.05998
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −228.821 −0.518029
$$59$$ 722.023 1.59321 0.796605 0.604500i $$-0.206627\pi$$
0.796605 + 0.604500i $$0.206627\pi$$
$$60$$ 0 0
$$61$$ −663.070 −1.39176 −0.695880 0.718158i $$-0.744986\pi$$
−0.695880 + 0.718158i $$0.744986\pi$$
$$62$$ 397.328 0.813882
$$63$$ 0 0
$$64$$ 473.375 0.924560
$$65$$ 193.368 0.368990
$$66$$ 0 0
$$67$$ −608.559 −1.10966 −0.554830 0.831964i $$-0.687217\pi$$
−0.554830 + 0.831964i $$0.687217\pi$$
$$68$$ 241.643 0.430934
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 976.305 1.63192 0.815958 0.578111i $$-0.196210\pi$$
0.815958 + 0.578111i $$0.196210\pi$$
$$72$$ 0 0
$$73$$ −261.148 −0.418700 −0.209350 0.977841i $$-0.567135\pi$$
−0.209350 + 0.977841i $$0.567135\pi$$
$$74$$ 147.288 0.231377
$$75$$ 0 0
$$76$$ −414.810 −0.626078
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1236.15 1.76048 0.880239 0.474531i $$-0.157382\pi$$
0.880239 + 0.474531i $$0.157382\pi$$
$$80$$ −287.139 −0.401289
$$81$$ 0 0
$$82$$ 571.600 0.769789
$$83$$ −1225.79 −1.62106 −0.810529 0.585699i $$-0.800820\pi$$
−0.810529 + 0.585699i $$0.800820\pi$$
$$84$$ 0 0
$$85$$ −932.367 −1.18976
$$86$$ 367.640 0.460973
$$87$$ 0 0
$$88$$ −721.866 −0.874445
$$89$$ 791.982 0.943257 0.471629 0.881797i $$-0.343666\pi$$
0.471629 + 0.881797i $$0.343666\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 72.7211 0.0824098
$$93$$ 0 0
$$94$$ 1095.32 1.20185
$$95$$ 1600.52 1.72853
$$96$$ 0 0
$$97$$ 935.253 0.978975 0.489488 0.872010i $$-0.337184\pi$$
0.489488 + 0.872010i $$0.337184\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −320.905 −0.320905
$$101$$ 442.969 0.436407 0.218203 0.975903i $$-0.429980\pi$$
0.218203 + 0.975903i $$0.429980\pi$$
$$102$$ 0 0
$$103$$ 699.401 0.669068 0.334534 0.942384i $$-0.391421\pi$$
0.334534 + 0.942384i $$0.391421\pi$$
$$104$$ 322.848 0.304402
$$105$$ 0 0
$$106$$ −640.114 −0.586541
$$107$$ 1911.18 1.72674 0.863369 0.504573i $$-0.168350\pi$$
0.863369 + 0.504573i $$0.168350\pi$$
$$108$$ 0 0
$$109$$ −2071.42 −1.82024 −0.910118 0.414349i $$-0.864009\pi$$
−0.910118 + 0.414349i $$0.864009\pi$$
$$110$$ 890.378 0.771766
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2313.34 −1.92585 −0.962926 0.269767i $$-0.913053\pi$$
−0.962926 + 0.269767i $$0.913053\pi$$
$$114$$ 0 0
$$115$$ −280.590 −0.227523
$$116$$ −417.679 −0.334315
$$117$$ 0 0
$$118$$ −1486.90 −1.16000
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −442.403 −0.332384
$$122$$ 1365.50 1.01333
$$123$$ 0 0
$$124$$ 725.263 0.525246
$$125$$ −574.817 −0.411306
$$126$$ 0 0
$$127$$ −1317.50 −0.920542 −0.460271 0.887778i $$-0.652248\pi$$
−0.460271 + 0.887778i $$0.652248\pi$$
$$128$$ 248.828 0.171824
$$129$$ 0 0
$$130$$ −398.213 −0.268659
$$131$$ 458.841 0.306024 0.153012 0.988224i $$-0.451103\pi$$
0.153012 + 0.988224i $$0.451103\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 1253.24 0.807935
$$135$$ 0 0
$$136$$ −1556.68 −0.981503
$$137$$ 1332.76 0.831132 0.415566 0.909563i $$-0.363584\pi$$
0.415566 + 0.909563i $$0.363584\pi$$
$$138$$ 0 0
$$139$$ 406.835 0.248254 0.124127 0.992266i $$-0.460387\pi$$
0.124127 + 0.992266i $$0.460387\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2010.56 −1.18819
$$143$$ −397.417 −0.232403
$$144$$ 0 0
$$145$$ 1611.59 0.923002
$$146$$ 537.797 0.304852
$$147$$ 0 0
$$148$$ 268.853 0.149321
$$149$$ −435.523 −0.239459 −0.119730 0.992807i $$-0.538203\pi$$
−0.119730 + 0.992807i $$0.538203\pi$$
$$150$$ 0 0
$$151$$ −1965.34 −1.05918 −0.529592 0.848253i $$-0.677655\pi$$
−0.529592 + 0.848253i $$0.677655\pi$$
$$152$$ 2672.23 1.42597
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2798.39 −1.45014
$$156$$ 0 0
$$157$$ −831.483 −0.422672 −0.211336 0.977413i $$-0.567781\pi$$
−0.211336 + 0.977413i $$0.567781\pi$$
$$158$$ −2545.67 −1.28179
$$159$$ 0 0
$$160$$ −2218.54 −1.09619
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1176.12 −0.565159 −0.282579 0.959244i $$-0.591190\pi$$
−0.282579 + 0.959244i $$0.591190\pi$$
$$164$$ 1043.37 0.496791
$$165$$ 0 0
$$166$$ 2524.34 1.18028
$$167$$ −1674.29 −0.775809 −0.387905 0.921700i $$-0.626801\pi$$
−0.387905 + 0.921700i $$0.626801\pi$$
$$168$$ 0 0
$$169$$ −2019.26 −0.919098
$$170$$ 1920.08 0.866253
$$171$$ 0 0
$$172$$ 671.073 0.297493
$$173$$ −1432.92 −0.629728 −0.314864 0.949137i $$-0.601959\pi$$
−0.314864 + 0.949137i $$0.601959\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 590.139 0.252747
$$177$$ 0 0
$$178$$ −1630.97 −0.686778
$$179$$ −1194.53 −0.498790 −0.249395 0.968402i $$-0.580232\pi$$
−0.249395 + 0.968402i $$0.580232\pi$$
$$180$$ 0 0
$$181$$ −741.424 −0.304473 −0.152236 0.988344i $$-0.548648\pi$$
−0.152236 + 0.988344i $$0.548648\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −468.474 −0.187698
$$185$$ −1037.35 −0.412258
$$186$$ 0 0
$$187$$ 1916.24 0.749353
$$188$$ 1999.34 0.775623
$$189$$ 0 0
$$190$$ −3296.04 −1.25853
$$191$$ −3903.98 −1.47897 −0.739483 0.673175i $$-0.764930\pi$$
−0.739483 + 0.673175i $$0.764930\pi$$
$$192$$ 0 0
$$193$$ 1771.25 0.660609 0.330305 0.943874i $$-0.392849\pi$$
0.330305 + 0.943874i $$0.392849\pi$$
$$194$$ −1926.02 −0.712784
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 830.366 0.300310 0.150155 0.988662i $$-0.452023\pi$$
0.150155 + 0.988662i $$0.452023\pi$$
$$198$$ 0 0
$$199$$ −1062.51 −0.378487 −0.189244 0.981930i $$-0.560604\pi$$
−0.189244 + 0.981930i $$0.560604\pi$$
$$200$$ 2067.29 0.730898
$$201$$ 0 0
$$202$$ −912.232 −0.317745
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −4025.79 −1.37158
$$206$$ −1440.32 −0.487143
$$207$$ 0 0
$$208$$ −263.934 −0.0879834
$$209$$ −3289.45 −1.08869
$$210$$ 0 0
$$211$$ −3848.67 −1.25570 −0.627851 0.778333i $$-0.716065\pi$$
−0.627851 + 0.778333i $$0.716065\pi$$
$$212$$ −1168.43 −0.378530
$$213$$ 0 0
$$214$$ −3935.81 −1.25722
$$215$$ −2589.30 −0.821342
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4265.79 1.32530
$$219$$ 0 0
$$220$$ 1625.25 0.498066
$$221$$ −857.018 −0.260856
$$222$$ 0 0
$$223$$ −5482.75 −1.64642 −0.823211 0.567736i $$-0.807819\pi$$
−0.823211 + 0.567736i $$0.807819\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 4764.00 1.40220
$$227$$ −394.837 −0.115446 −0.0577230 0.998333i $$-0.518384\pi$$
−0.0577230 + 0.998333i $$0.518384\pi$$
$$228$$ 0 0
$$229$$ 3823.57 1.10336 0.551678 0.834057i $$-0.313988\pi$$
0.551678 + 0.834057i $$0.313988\pi$$
$$230$$ 577.835 0.165658
$$231$$ 0 0
$$232$$ 2690.72 0.761441
$$233$$ −2537.54 −0.713475 −0.356738 0.934205i $$-0.616111\pi$$
−0.356738 + 0.934205i $$0.616111\pi$$
$$234$$ 0 0
$$235$$ −7714.35 −2.14140
$$236$$ −2714.12 −0.748620
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 5368.81 1.45305 0.726526 0.687139i $$-0.241134\pi$$
0.726526 + 0.687139i $$0.241134\pi$$
$$240$$ 0 0
$$241$$ −2108.02 −0.563442 −0.281721 0.959496i $$-0.590905\pi$$
−0.281721 + 0.959496i $$0.590905\pi$$
$$242$$ 911.064 0.242006
$$243$$ 0 0
$$244$$ 2492.51 0.653962
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1471.18 0.378982
$$248$$ −4672.20 −1.19631
$$249$$ 0 0
$$250$$ 1183.75 0.299469
$$251$$ 5717.94 1.43790 0.718951 0.695061i $$-0.244623\pi$$
0.718951 + 0.695061i $$0.244623\pi$$
$$252$$ 0 0
$$253$$ 576.680 0.143303
$$254$$ 2713.19 0.670240
$$255$$ 0 0
$$256$$ −4299.42 −1.04966
$$257$$ −4987.47 −1.21054 −0.605272 0.796019i $$-0.706936\pi$$
−0.605272 + 0.796019i $$0.706936\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −726.880 −0.173381
$$261$$ 0 0
$$262$$ −944.917 −0.222814
$$263$$ 615.273 0.144256 0.0721281 0.997395i $$-0.477021\pi$$
0.0721281 + 0.997395i $$0.477021\pi$$
$$264$$ 0 0
$$265$$ 4508.34 1.04508
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2287.60 0.521409
$$269$$ 793.962 0.179958 0.0899790 0.995944i $$-0.471320\pi$$
0.0899790 + 0.995944i $$0.471320\pi$$
$$270$$ 0 0
$$271$$ −3764.95 −0.843927 −0.421964 0.906613i $$-0.638659\pi$$
−0.421964 + 0.906613i $$0.638659\pi$$
$$272$$ 1272.62 0.283690
$$273$$ 0 0
$$274$$ −2744.62 −0.605141
$$275$$ −2544.78 −0.558022
$$276$$ 0 0
$$277$$ −3196.54 −0.693364 −0.346682 0.937983i $$-0.612692\pi$$
−0.346682 + 0.937983i $$0.612692\pi$$
$$278$$ −837.817 −0.180752
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6225.15 −1.32157 −0.660785 0.750575i $$-0.729777\pi$$
−0.660785 + 0.750575i $$0.729777\pi$$
$$282$$ 0 0
$$283$$ 3717.50 0.780857 0.390429 0.920633i $$-0.372327\pi$$
0.390429 + 0.920633i $$0.372327\pi$$
$$284$$ −3669.98 −0.766807
$$285$$ 0 0
$$286$$ 818.423 0.169211
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −780.693 −0.158904
$$290$$ −3318.84 −0.672031
$$291$$ 0 0
$$292$$ 981.669 0.196739
$$293$$ 6392.56 1.27460 0.637299 0.770616i $$-0.280052\pi$$
0.637299 + 0.770616i $$0.280052\pi$$
$$294$$ 0 0
$$295$$ 10472.3 2.06685
$$296$$ −1731.97 −0.340096
$$297$$ 0 0
$$298$$ 896.897 0.174348
$$299$$ −257.915 −0.0498849
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 4047.33 0.771183
$$303$$ 0 0
$$304$$ −2184.60 −0.412156
$$305$$ −9617.22 −1.80551
$$306$$ 0 0
$$307$$ −939.255 −0.174613 −0.0873063 0.996182i $$-0.527826\pi$$
−0.0873063 + 0.996182i $$0.527826\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 5762.88 1.05584
$$311$$ 2751.61 0.501703 0.250851 0.968026i $$-0.419289\pi$$
0.250851 + 0.968026i $$0.419289\pi$$
$$312$$ 0 0
$$313$$ 637.099 0.115051 0.0575255 0.998344i $$-0.481679\pi$$
0.0575255 + 0.998344i $$0.481679\pi$$
$$314$$ 1712.32 0.307744
$$315$$ 0 0
$$316$$ −4646.75 −0.827216
$$317$$ −2872.19 −0.508891 −0.254445 0.967087i $$-0.581893\pi$$
−0.254445 + 0.967087i $$0.581893\pi$$
$$318$$ 0 0
$$319$$ −3312.20 −0.581341
$$320$$ 6865.87 1.19942
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −7093.60 −1.22198
$$324$$ 0 0
$$325$$ 1138.13 0.194253
$$326$$ 2422.05 0.411488
$$327$$ 0 0
$$328$$ −6721.47 −1.13150
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6393.60 −1.06170 −0.530852 0.847464i $$-0.678128\pi$$
−0.530852 + 0.847464i $$0.678128\pi$$
$$332$$ 4607.80 0.761705
$$333$$ 0 0
$$334$$ 3447.95 0.564861
$$335$$ −8826.59 −1.43955
$$336$$ 0 0
$$337$$ −12107.6 −1.95710 −0.978549 0.206016i $$-0.933950\pi$$
−0.978549 + 0.206016i $$0.933950\pi$$
$$338$$ 4158.37 0.669188
$$339$$ 0 0
$$340$$ 3504.81 0.559045
$$341$$ 5751.35 0.913352
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4323.10 −0.677575
$$345$$ 0 0
$$346$$ 2950.89 0.458500
$$347$$ 2710.14 0.419273 0.209636 0.977779i $$-0.432772\pi$$
0.209636 + 0.977779i $$0.432772\pi$$
$$348$$ 0 0
$$349$$ 1175.22 0.180252 0.0901261 0.995930i $$-0.471273\pi$$
0.0901261 + 0.995930i $$0.471273\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4559.62 0.690422
$$353$$ −5276.41 −0.795566 −0.397783 0.917480i $$-0.630220\pi$$
−0.397783 + 0.917480i $$0.630220\pi$$
$$354$$ 0 0
$$355$$ 14160.4 2.11706
$$356$$ −2977.10 −0.443219
$$357$$ 0 0
$$358$$ 2459.96 0.363165
$$359$$ −11286.5 −1.65928 −0.829639 0.558300i $$-0.811454\pi$$
−0.829639 + 0.558300i $$0.811454\pi$$
$$360$$ 0 0
$$361$$ 5318.03 0.775336
$$362$$ 1526.86 0.221684
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3787.72 −0.543173
$$366$$ 0 0
$$367$$ −7567.59 −1.07636 −0.538181 0.842829i $$-0.680888\pi$$
−0.538181 + 0.842829i $$0.680888\pi$$
$$368$$ 382.987 0.0542516
$$369$$ 0 0
$$370$$ 2136.28 0.300162
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6503.97 −0.902850 −0.451425 0.892309i $$-0.649084\pi$$
−0.451425 + 0.892309i $$0.649084\pi$$
$$374$$ −3946.21 −0.545598
$$375$$ 0 0
$$376$$ −12879.9 −1.76657
$$377$$ 1481.35 0.202370
$$378$$ 0 0
$$379$$ 10773.8 1.46019 0.730097 0.683344i $$-0.239475\pi$$
0.730097 + 0.683344i $$0.239475\pi$$
$$380$$ −6016.44 −0.812202
$$381$$ 0 0
$$382$$ 8039.70 1.07682
$$383$$ 6366.13 0.849331 0.424666 0.905350i $$-0.360392\pi$$
0.424666 + 0.905350i $$0.360392\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3647.64 −0.480985
$$387$$ 0 0
$$388$$ −3515.67 −0.460002
$$389$$ −5903.44 −0.769450 −0.384725 0.923031i $$-0.625704\pi$$
−0.384725 + 0.923031i $$0.625704\pi$$
$$390$$ 0 0
$$391$$ 1243.59 0.160847
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −1710.02 −0.218654
$$395$$ 17929.2 2.28384
$$396$$ 0 0
$$397$$ −3766.78 −0.476194 −0.238097 0.971241i $$-0.576524\pi$$
−0.238097 + 0.971241i $$0.576524\pi$$
$$398$$ 2188.08 0.275574
$$399$$ 0 0
$$400$$ −1690.05 −0.211256
$$401$$ 7252.00 0.903112 0.451556 0.892243i $$-0.350869\pi$$
0.451556 + 0.892243i $$0.350869\pi$$
$$402$$ 0 0
$$403$$ −2572.24 −0.317946
$$404$$ −1665.15 −0.205060
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2132.01 0.259655
$$408$$ 0 0
$$409$$ −12547.4 −1.51694 −0.758471 0.651707i $$-0.774053\pi$$
−0.758471 + 0.651707i $$0.774053\pi$$
$$410$$ 8290.54 0.998635
$$411$$ 0 0
$$412$$ −2629.08 −0.314383
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −17778.9 −2.10297
$$416$$ −2039.25 −0.240342
$$417$$ 0 0
$$418$$ 6774.15 0.792666
$$419$$ 12631.5 1.47276 0.736381 0.676567i $$-0.236533\pi$$
0.736381 + 0.676567i $$0.236533\pi$$
$$420$$ 0 0
$$421$$ −3495.94 −0.404707 −0.202354 0.979312i $$-0.564859\pi$$
−0.202354 + 0.979312i $$0.564859\pi$$
$$422$$ 7925.78 0.914267
$$423$$ 0 0
$$424$$ 7527.13 0.862146
$$425$$ −5487.75 −0.626341
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −7184.23 −0.811362
$$429$$ 0 0
$$430$$ 5332.28 0.598013
$$431$$ −2345.18 −0.262096 −0.131048 0.991376i $$-0.541834\pi$$
−0.131048 + 0.991376i $$0.541834\pi$$
$$432$$ 0 0
$$433$$ −3250.63 −0.360774 −0.180387 0.983596i $$-0.557735\pi$$
−0.180387 + 0.983596i $$0.557735\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7786.56 0.855295
$$437$$ −2134.78 −0.233685
$$438$$ 0 0
$$439$$ −16240.5 −1.76564 −0.882820 0.469711i $$-0.844358\pi$$
−0.882820 + 0.469711i $$0.844358\pi$$
$$440$$ −10470.0 −1.13440
$$441$$ 0 0
$$442$$ 1764.91 0.189928
$$443$$ 4849.04 0.520056 0.260028 0.965601i $$-0.416268\pi$$
0.260028 + 0.965601i $$0.416268\pi$$
$$444$$ 0 0
$$445$$ 11487.0 1.22367
$$446$$ 11290.9 1.19875
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 4139.93 0.435134 0.217567 0.976045i $$-0.430188\pi$$
0.217567 + 0.976045i $$0.430188\pi$$
$$450$$ 0 0
$$451$$ 8273.96 0.863870
$$452$$ 8695.98 0.904922
$$453$$ 0 0
$$454$$ 813.110 0.0840553
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9232.48 0.945027 0.472513 0.881323i $$-0.343347\pi$$
0.472513 + 0.881323i $$0.343347\pi$$
$$458$$ −7874.08 −0.803344
$$459$$ 0 0
$$460$$ 1054.75 0.106909
$$461$$ 1274.90 0.128803 0.0644013 0.997924i $$-0.479486\pi$$
0.0644013 + 0.997924i $$0.479486\pi$$
$$462$$ 0 0
$$463$$ 2061.80 0.206955 0.103477 0.994632i $$-0.467003\pi$$
0.103477 + 0.994632i $$0.467003\pi$$
$$464$$ −2199.71 −0.220084
$$465$$ 0 0
$$466$$ 5225.70 0.519476
$$467$$ −968.041 −0.0959221 −0.0479610 0.998849i $$-0.515272\pi$$
−0.0479610 + 0.998849i $$0.515272\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 15886.6 1.55914
$$471$$ 0 0
$$472$$ 17484.6 1.70507
$$473$$ 5321.62 0.517311
$$474$$ 0 0
$$475$$ 9420.39 0.909973
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −11056.3 −1.05796
$$479$$ 6200.05 0.591414 0.295707 0.955279i $$-0.404445\pi$$
0.295707 + 0.955279i $$0.404445\pi$$
$$480$$ 0 0
$$481$$ −953.519 −0.0903882
$$482$$ 4341.17 0.410238
$$483$$ 0 0
$$484$$ 1663.01 0.156181
$$485$$ 13565.0 1.27001
$$486$$ 0 0
$$487$$ −9829.66 −0.914629 −0.457315 0.889305i $$-0.651189\pi$$
−0.457315 + 0.889305i $$0.651189\pi$$
$$488$$ −16056.9 −1.48948
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 5053.57 0.464489 0.232245 0.972657i $$-0.425393\pi$$
0.232245 + 0.972657i $$0.425393\pi$$
$$492$$ 0 0
$$493$$ −7142.67 −0.652514
$$494$$ −3029.67 −0.275934
$$495$$ 0 0
$$496$$ 3819.61 0.345777
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 14061.6 1.26149 0.630747 0.775988i $$-0.282749\pi$$
0.630747 + 0.775988i $$0.282749\pi$$
$$500$$ 2160.77 0.193265
$$501$$ 0 0
$$502$$ −11775.3 −1.04693
$$503$$ 8001.67 0.709298 0.354649 0.934999i $$-0.384600\pi$$
0.354649 + 0.934999i $$0.384600\pi$$
$$504$$ 0 0
$$505$$ 6424.87 0.566144
$$506$$ −1187.59 −0.104337
$$507$$ 0 0
$$508$$ 4952.53 0.432546
$$509$$ −10913.8 −0.950386 −0.475193 0.879881i $$-0.657622\pi$$
−0.475193 + 0.879881i $$0.657622\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 6863.42 0.592428
$$513$$ 0 0
$$514$$ 10271.0 0.881388
$$515$$ 10144.2 0.867972
$$516$$ 0 0
$$517$$ 15854.8 1.34873
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4682.61 0.394896
$$521$$ −15803.4 −1.32890 −0.664452 0.747331i $$-0.731335\pi$$
−0.664452 + 0.747331i $$0.731335\pi$$
$$522$$ 0 0
$$523$$ 646.094 0.0540186 0.0270093 0.999635i $$-0.491402\pi$$
0.0270093 + 0.999635i $$0.491402\pi$$
$$524$$ −1724.81 −0.143795
$$525$$ 0 0
$$526$$ −1267.07 −0.105032
$$527$$ 12402.6 1.02517
$$528$$ 0 0
$$529$$ −11792.7 −0.969240
$$530$$ −9284.27 −0.760911
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3700.45 −0.300721
$$534$$ 0 0
$$535$$ 27720.0 2.24007
$$536$$ −14736.9 −1.18757
$$537$$ 0 0
$$538$$ −1635.05 −0.131026
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 10246.7 0.814303 0.407152 0.913361i $$-0.366522\pi$$
0.407152 + 0.913361i $$0.366522\pi$$
$$542$$ 7753.37 0.614457
$$543$$ 0 0
$$544$$ 9832.69 0.774950
$$545$$ −30044.0 −2.36137
$$546$$ 0 0
$$547$$ 4368.98 0.341506 0.170753 0.985314i $$-0.445380\pi$$
0.170753 + 0.985314i $$0.445380\pi$$
$$548$$ −5009.90 −0.390533
$$549$$ 0 0
$$550$$ 5240.61 0.406292
$$551$$ 12261.3 0.947998
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6582.82 0.504833
$$555$$ 0 0
$$556$$ −1529.31 −0.116650
$$557$$ 8396.43 0.638722 0.319361 0.947633i $$-0.396532\pi$$
0.319361 + 0.947633i $$0.396532\pi$$
$$558$$ 0 0
$$559$$ −2380.04 −0.180081
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 12819.8 0.962226
$$563$$ 12142.3 0.908944 0.454472 0.890761i $$-0.349828\pi$$
0.454472 + 0.890761i $$0.349828\pi$$
$$564$$ 0 0
$$565$$ −33553.0 −2.49838
$$566$$ −7655.66 −0.568536
$$567$$ 0 0
$$568$$ 23642.3 1.74649
$$569$$ −11196.1 −0.824896 −0.412448 0.910981i $$-0.635326\pi$$
−0.412448 + 0.910981i $$0.635326\pi$$
$$570$$ 0 0
$$571$$ 19659.6 1.44085 0.720427 0.693531i $$-0.243946\pi$$
0.720427 + 0.693531i $$0.243946\pi$$
$$572$$ 1493.91 0.109202
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1651.51 −0.119778
$$576$$ 0 0
$$577$$ −8516.45 −0.614462 −0.307231 0.951635i $$-0.599402\pi$$
−0.307231 + 0.951635i $$0.599402\pi$$
$$578$$ 1607.73 0.115696
$$579$$ 0 0
$$580$$ −6058.05 −0.433702
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9265.70 −0.658227
$$584$$ −6323.98 −0.448096
$$585$$ 0 0
$$586$$ −13164.6 −0.928025
$$587$$ −7115.53 −0.500323 −0.250161 0.968204i $$-0.580484\pi$$
−0.250161 + 0.968204i $$0.580484\pi$$
$$588$$ 0 0
$$589$$ −21290.6 −1.48941
$$590$$ −21566.2 −1.50486
$$591$$ 0 0
$$592$$ 1415.92 0.0983003
$$593$$ 6144.48 0.425504 0.212752 0.977106i $$-0.431757\pi$$
0.212752 + 0.977106i $$0.431757\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1637.15 0.112517
$$597$$ 0 0
$$598$$ 531.138 0.0363208
$$599$$ −9578.39 −0.653360 −0.326680 0.945135i $$-0.605930\pi$$
−0.326680 + 0.945135i $$0.605930\pi$$
$$600$$ 0 0
$$601$$ 19113.8 1.29728 0.648642 0.761094i $$-0.275337\pi$$
0.648642 + 0.761094i $$0.275337\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 7387.80 0.497691
$$605$$ −6416.65 −0.431196
$$606$$ 0 0
$$607$$ −20442.6 −1.36695 −0.683477 0.729972i $$-0.739533\pi$$
−0.683477 + 0.729972i $$0.739533\pi$$
$$608$$ −16879.0 −1.12588
$$609$$ 0 0
$$610$$ 19805.3 1.31458
$$611$$ −7090.92 −0.469506
$$612$$ 0 0
$$613$$ 17422.7 1.14796 0.573978 0.818870i $$-0.305399\pi$$
0.573978 + 0.818870i $$0.305399\pi$$
$$614$$ 1934.26 0.127134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1805.17 0.117785 0.0588926 0.998264i $$-0.481243\pi$$
0.0588926 + 0.998264i $$0.481243\pi$$
$$618$$ 0 0
$$619$$ −8142.23 −0.528698 −0.264349 0.964427i $$-0.585157\pi$$
−0.264349 + 0.964427i $$0.585157\pi$$
$$620$$ 10519.3 0.681394
$$621$$ 0 0
$$622$$ −5666.55 −0.365286
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19008.3 −1.21653
$$626$$ −1312.01 −0.0837678
$$627$$ 0 0
$$628$$ 3125.59 0.198606
$$629$$ 4597.61 0.291445
$$630$$ 0 0
$$631$$ 3630.07 0.229019 0.114509 0.993422i $$-0.463470\pi$$
0.114509 + 0.993422i $$0.463470\pi$$
$$632$$ 29934.7 1.88408
$$633$$ 0 0
$$634$$ 5914.87 0.370519
$$635$$ −19109.1 −1.19421
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 6821.00 0.423270
$$639$$ 0 0
$$640$$ 3609.02 0.222905
$$641$$ −22223.6 −1.36939 −0.684696 0.728829i $$-0.740065\pi$$
−0.684696 + 0.728829i $$0.740065\pi$$
$$642$$ 0 0
$$643$$ 5013.00 0.307455 0.153727 0.988113i $$-0.450872\pi$$
0.153727 + 0.988113i $$0.450872\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 14608.3 0.889712
$$647$$ 7716.16 0.468862 0.234431 0.972133i $$-0.424677\pi$$
0.234431 + 0.972133i $$0.424677\pi$$
$$648$$ 0 0
$$649$$ −21523.0 −1.30178
$$650$$ −2343.82 −0.141434
$$651$$ 0 0
$$652$$ 4421.10 0.265558
$$653$$ 5644.40 0.338258 0.169129 0.985594i $$-0.445905\pi$$
0.169129 + 0.985594i $$0.445905\pi$$
$$654$$ 0 0
$$655$$ 6655.07 0.397000
$$656$$ 5494.94 0.327044
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −8812.27 −0.520906 −0.260453 0.965487i $$-0.583872\pi$$
−0.260453 + 0.965487i $$0.583872\pi$$
$$660$$ 0 0
$$661$$ 6191.02 0.364300 0.182150 0.983271i $$-0.441694\pi$$
0.182150 + 0.983271i $$0.441694\pi$$
$$662$$ 13166.7 0.773019
$$663$$ 0 0
$$664$$ −29683.8 −1.73487
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2149.54 −0.124784
$$668$$ 6293.73 0.364538
$$669$$ 0 0
$$670$$ 18177.1 1.04812
$$671$$ 19765.7 1.13718
$$672$$ 0 0
$$673$$ 29580.7 1.69428 0.847141 0.531368i $$-0.178322\pi$$
0.847141 + 0.531368i $$0.178322\pi$$
$$674$$ 24933.8 1.42495
$$675$$ 0 0
$$676$$ 7590.50 0.431867
$$677$$ 26290.0 1.49248 0.746239 0.665678i $$-0.231857\pi$$
0.746239 + 0.665678i $$0.231857\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −22578.3 −1.27329
$$681$$ 0 0
$$682$$ −11844.1 −0.665005
$$683$$ 810.684 0.0454172 0.0227086 0.999742i $$-0.492771\pi$$
0.0227086 + 0.999742i $$0.492771\pi$$
$$684$$ 0 0
$$685$$ 19330.4 1.07821
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 3534.22 0.195844
$$689$$ 4144.00 0.229135
$$690$$ 0 0
$$691$$ −1171.38 −0.0644883 −0.0322442 0.999480i $$-0.510265\pi$$
−0.0322442 + 0.999480i $$0.510265\pi$$
$$692$$ 5386.42 0.295897
$$693$$ 0 0
$$694$$ −5581.14 −0.305269
$$695$$ 5900.76 0.322056
$$696$$ 0 0
$$697$$ 17842.6 0.969634
$$698$$ −2420.19 −0.131240
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1057.70 0.0569885 0.0284943 0.999594i $$-0.490929\pi$$
0.0284943 + 0.999594i $$0.490929\pi$$
$$702$$ 0 0
$$703$$ −7892.35 −0.423422
$$704$$ −14111.0 −0.755437
$$705$$ 0 0
$$706$$ 10866.0 0.579245
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −29243.7 −1.54904 −0.774521 0.632548i $$-0.782009\pi$$
−0.774521 + 0.632548i $$0.782009\pi$$
$$710$$ −29161.3 −1.54142
$$711$$ 0 0
$$712$$ 19178.7 1.00948
$$713$$ 3732.50 0.196049
$$714$$ 0 0
$$715$$ −5764.17 −0.301493
$$716$$ 4490.30 0.234372
$$717$$ 0 0
$$718$$ 23243.0 1.20811
$$719$$ 24557.7 1.27378 0.636890 0.770955i $$-0.280221\pi$$
0.636890 + 0.770955i $$0.280221\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −10951.7 −0.564516
$$723$$ 0 0
$$724$$ 2787.05 0.143066
$$725$$ 9485.55 0.485909
$$726$$ 0 0
$$727$$ 23444.2 1.19601 0.598003 0.801494i $$-0.295961\pi$$
0.598003 + 0.801494i $$0.295961\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 7800.25 0.395480
$$731$$ 11475.9 0.580646
$$732$$ 0 0
$$733$$ 35971.3 1.81259 0.906296 0.422644i $$-0.138898\pi$$
0.906296 + 0.422644i $$0.138898\pi$$
$$734$$ 15584.4 0.783692
$$735$$ 0 0
$$736$$ 2959.09 0.148198
$$737$$ 18140.7 0.906679
$$738$$ 0 0
$$739$$ 12068.6 0.600746 0.300373 0.953822i $$-0.402889\pi$$
0.300373 + 0.953822i $$0.402889\pi$$
$$740$$ 3899.46 0.193712
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 5475.09 0.270339 0.135169 0.990823i $$-0.456842\pi$$
0.135169 + 0.990823i $$0.456842\pi$$
$$744$$ 0 0
$$745$$ −6316.86 −0.310647
$$746$$ 13394.0 0.657358
$$747$$ 0 0
$$748$$ −7203.23 −0.352107
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 25274.6 1.22807 0.614036 0.789278i $$-0.289545\pi$$
0.614036 + 0.789278i $$0.289545\pi$$
$$752$$ 10529.6 0.510604
$$753$$ 0 0
$$754$$ −3050.63 −0.147344
$$755$$ −28505.4 −1.37406
$$756$$ 0 0
$$757$$ −23583.7 −1.13232 −0.566159 0.824296i $$-0.691571\pi$$
−0.566159 + 0.824296i $$0.691571\pi$$
$$758$$ −22187.1 −1.06316
$$759$$ 0 0
$$760$$ 38758.3 1.84988
$$761$$ −16202.7 −0.771810 −0.385905 0.922539i $$-0.626111\pi$$
−0.385905 + 0.922539i $$0.626111\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 14675.3 0.694939
$$765$$ 0 0
$$766$$ −13110.1 −0.618392
$$767$$ 9625.98 0.453160
$$768$$ 0 0
$$769$$ 18603.1 0.872362 0.436181 0.899859i $$-0.356331\pi$$
0.436181 + 0.899859i $$0.356331\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6658.23 −0.310408
$$773$$ 27281.3 1.26939 0.634697 0.772761i $$-0.281125\pi$$
0.634697 + 0.772761i $$0.281125\pi$$
$$774$$ 0 0
$$775$$ −16470.8 −0.763419
$$776$$ 22648.2 1.04771
$$777$$ 0 0
$$778$$ 12157.3 0.560231
$$779$$ −30628.9 −1.40872
$$780$$ 0 0
$$781$$ −29103.0 −1.33340
$$782$$ −2561.00 −0.117112
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −12059.9 −0.548326
$$786$$ 0 0
$$787$$ −571.719 −0.0258953 −0.0129476 0.999916i $$-0.504121\pi$$
−0.0129476 + 0.999916i $$0.504121\pi$$
$$788$$ −3121.39 −0.141110
$$789$$ 0 0
$$790$$ −36922.7 −1.66285
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −8840.01 −0.395861
$$794$$ 7757.14 0.346714
$$795$$ 0 0
$$796$$ 3994.01 0.177844
$$797$$ −33080.5 −1.47023 −0.735113 0.677944i $$-0.762871\pi$$
−0.735113 + 0.677944i $$0.762871\pi$$
$$798$$ 0 0
$$799$$ 34190.5 1.51386
$$800$$ −13057.9 −0.577084
$$801$$ 0 0
$$802$$ −14934.5 −0.657549
$$803$$ 7784.66 0.342110
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5297.15 0.231494
$$807$$ 0 0
$$808$$ 10727.0 0.467047
$$809$$ 32838.2 1.42711 0.713554 0.700601i $$-0.247085\pi$$
0.713554 + 0.700601i $$0.247085\pi$$
$$810$$ 0 0
$$811$$ 20453.4 0.885592 0.442796 0.896622i $$-0.353987\pi$$
0.442796 + 0.896622i $$0.353987\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −4390.56 −0.189053
$$815$$ −17058.6 −0.733172
$$816$$ 0 0
$$817$$ −19699.8 −0.843585
$$818$$ 25839.6 1.10447
$$819$$ 0 0
$$820$$ 15133.2 0.644479
$$821$$ 32462.4 1.37996 0.689980 0.723828i $$-0.257619\pi$$
0.689980 + 0.723828i $$0.257619\pi$$
$$822$$ 0 0
$$823$$ −47088.3 −1.99440 −0.997202 0.0747505i $$-0.976184\pi$$
−0.997202 + 0.0747505i $$0.976184\pi$$
$$824$$ 16936.7 0.716043
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 17641.7 0.741792 0.370896 0.928674i $$-0.379051\pi$$
0.370896 + 0.928674i $$0.379051\pi$$
$$828$$ 0 0
$$829$$ 1583.53 0.0663427 0.0331714 0.999450i $$-0.489439\pi$$
0.0331714 + 0.999450i $$0.489439\pi$$
$$830$$ 36613.2 1.53116
$$831$$ 0 0
$$832$$ 6311.01 0.262975
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −24284.0 −1.00645
$$836$$ 12365.2 0.511555
$$837$$ 0 0
$$838$$ −26012.7 −1.07231
$$839$$ 39612.3 1.63000 0.815000 0.579461i $$-0.196737\pi$$
0.815000 + 0.579461i $$0.196737\pi$$
$$840$$ 0 0
$$841$$ −12042.9 −0.493786
$$842$$ 7199.39 0.294664
$$843$$ 0 0
$$844$$ 14467.3 0.590031
$$845$$ −29287.5 −1.19233
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6153.58 −0.249192
$$849$$ 0 0
$$850$$ 11301.2 0.456034
$$851$$ 1383.62 0.0557344
$$852$$ 0 0
$$853$$ 9443.21 0.379049 0.189525 0.981876i $$-0.439305\pi$$
0.189525 + 0.981876i $$0.439305\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 46281.3 1.84797
$$857$$ −20138.7 −0.802713 −0.401356 0.915922i $$-0.631461\pi$$
−0.401356 + 0.915922i $$0.631461\pi$$
$$858$$ 0 0
$$859$$ 18457.4 0.733129 0.366564 0.930393i $$-0.380534\pi$$
0.366564 + 0.930393i $$0.380534\pi$$
$$860$$ 9733.30 0.385933
$$861$$ 0 0
$$862$$ 4829.57 0.190830
$$863$$ −48840.2 −1.92646 −0.963232 0.268670i $$-0.913416\pi$$
−0.963232 + 0.268670i $$0.913416\pi$$
$$864$$ 0 0
$$865$$ −20783.2 −0.816936
$$866$$ 6694.20 0.262677
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −36848.8 −1.43845
$$870$$ 0 0
$$871$$ −8113.27 −0.315623
$$872$$ −50161.6 −1.94803
$$873$$ 0 0
$$874$$ 4396.27 0.170144
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −5584.39 −0.215019 −0.107509 0.994204i $$-0.534288\pi$$
−0.107509 + 0.994204i $$0.534288\pi$$
$$878$$ 33445.0 1.28555
$$879$$ 0 0
$$880$$ 8559.43 0.327884
$$881$$ −24753.3 −0.946606 −0.473303 0.880900i $$-0.656938\pi$$
−0.473303 + 0.880900i $$0.656938\pi$$
$$882$$ 0 0
$$883$$ −12368.7 −0.471394 −0.235697 0.971827i $$-0.575737\pi$$
−0.235697 + 0.971827i $$0.575737\pi$$
$$884$$ 3221.58 0.122572
$$885$$ 0 0
$$886$$ −9985.90 −0.378649
$$887$$ 738.713 0.0279634 0.0139817 0.999902i $$-0.495549\pi$$
0.0139817 + 0.999902i $$0.495549\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −23655.8 −0.890947
$$891$$ 0 0
$$892$$ 20609.9 0.773623
$$893$$ −58692.1 −2.19939
$$894$$ 0 0
$$895$$ −17325.6 −0.647073
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −8525.58 −0.316818
$$899$$ −21437.9 −0.795320
$$900$$ 0 0
$$901$$ −19981.2 −0.738814
$$902$$ −17039.0 −0.628977
$$903$$ 0 0
$$904$$ −56020.1 −2.06106
$$905$$ −10753.7 −0.394988
$$906$$ 0 0
$$907$$ −776.937 −0.0284430 −0.0142215 0.999899i $$-0.504527\pi$$
−0.0142215 + 0.999899i $$0.504527\pi$$
$$908$$ 1484.21 0.0542459
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 31326.5 1.13929 0.569645 0.821891i $$-0.307081\pi$$
0.569645 + 0.821891i $$0.307081\pi$$
$$912$$ 0 0
$$913$$ 36540.0 1.32453
$$914$$ −19013.0 −0.688067
$$915$$ 0 0
$$916$$ −14373.0 −0.518446
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 6149.72 0.220741 0.110370 0.993891i $$-0.464796\pi$$
0.110370 + 0.993891i $$0.464796\pi$$
$$920$$ −6794.80 −0.243498
$$921$$ 0 0
$$922$$ −2625.47 −0.0937802
$$923$$ 13016.1 0.464170
$$924$$ 0 0
$$925$$ −6105.68 −0.217031
$$926$$ −4245.98 −0.150682
$$927$$ 0 0
$$928$$ −16995.7 −0.601199
$$929$$ −31050.1 −1.09658 −0.548288 0.836289i $$-0.684720\pi$$
−0.548288 + 0.836289i $$0.684720\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 9538.74 0.335249
$$933$$ 0 0
$$934$$ 1993.54 0.0698401
$$935$$ 27793.2 0.972124
$$936$$ 0 0
$$937$$ −22802.9 −0.795023 −0.397512 0.917597i $$-0.630126\pi$$
−0.397512 + 0.917597i $$0.630126\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 28998.6 1.00620
$$941$$ −46442.5 −1.60891 −0.804455 0.594014i $$-0.797542\pi$$
−0.804455 + 0.594014i $$0.797542\pi$$
$$942$$ 0 0
$$943$$ 5369.61 0.185428
$$944$$ −14294.0 −0.492828
$$945$$ 0 0
$$946$$ −10959.1 −0.376651
$$947$$ −29207.7 −1.00224 −0.501121 0.865377i $$-0.667079\pi$$
−0.501121 + 0.865377i $$0.667079\pi$$
$$948$$ 0 0
$$949$$ −3481.61 −0.119092
$$950$$ −19399.9 −0.662544
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −6829.15 −0.232128 −0.116064 0.993242i $$-0.537028\pi$$
−0.116064 + 0.993242i $$0.537028\pi$$
$$954$$ 0 0
$$955$$ −56623.7 −1.91864
$$956$$ −20181.6 −0.682762
$$957$$ 0 0
$$958$$ −12768.1 −0.430604
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 7433.99 0.249538
$$962$$ 1963.64 0.0658110
$$963$$ 0 0
$$964$$ 7924.16 0.264751
$$965$$ 25690.4 0.856999
$$966$$ 0 0
$$967$$ −2583.84 −0.0859263 −0.0429632 0.999077i $$-0.513680\pi$$
−0.0429632 + 0.999077i $$0.513680\pi$$
$$968$$ −10713.3 −0.355720
$$969$$ 0 0
$$970$$ −27935.2 −0.924684
$$971$$ −32515.6 −1.07464 −0.537321 0.843378i $$-0.680563\pi$$
−0.537321 + 0.843378i $$0.680563\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 20242.8 0.665935
$$975$$ 0 0
$$976$$ 13126.9 0.430513
$$977$$ 222.043 0.00727102 0.00363551 0.999993i $$-0.498843\pi$$
0.00363551 + 0.999993i $$0.498843\pi$$
$$978$$ 0 0
$$979$$ −23608.5 −0.770715
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −10407.1 −0.338191
$$983$$ 10916.7 0.354210 0.177105 0.984192i $$-0.443327\pi$$
0.177105 + 0.984192i $$0.443327\pi$$
$$984$$ 0 0
$$985$$ 12043.7 0.389588
$$986$$ 14709.3 0.475091
$$987$$ 0 0
$$988$$ −5530.22 −0.178077
$$989$$ 3453.61 0.111040
$$990$$ 0 0
$$991$$ 8015.59 0.256936 0.128468 0.991714i $$-0.458994\pi$$
0.128468 + 0.991714i $$0.458994\pi$$
$$992$$ 29511.6 0.944552
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −15410.7 −0.491006
$$996$$ 0 0
$$997$$ 21833.7 0.693559 0.346780 0.937947i $$-0.387275\pi$$
0.346780 + 0.937947i $$0.387275\pi$$
$$998$$ −28957.9 −0.918484
$$999$$ 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.be.1.3 6
3.2 odd 2 inner 1323.4.a.be.1.4 6
7.3 odd 6 189.4.e.e.163.4 yes 12
7.5 odd 6 189.4.e.e.109.4 yes 12
7.6 odd 2 1323.4.a.bd.1.3 6
21.5 even 6 189.4.e.e.109.3 12
21.17 even 6 189.4.e.e.163.3 yes 12
21.20 even 2 1323.4.a.bd.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.e.109.3 12 21.5 even 6
189.4.e.e.109.4 yes 12 7.5 odd 6
189.4.e.e.163.3 yes 12 21.17 even 6
189.4.e.e.163.4 yes 12 7.3 odd 6
1323.4.a.bd.1.3 6 7.6 odd 2
1323.4.a.bd.1.4 6 21.20 even 2
1323.4.a.be.1.3 6 1.1 even 1 trivial
1323.4.a.be.1.4 6 3.2 odd 2 inner