# Properties

 Label 1521.4.a.v.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 25x^{2} + 24x + 78$$ x^4 - 2*x^3 - 25*x^2 + 24*x + 78 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.33039$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.33039 q^{2} +20.4131 q^{4} +16.4131 q^{5} +9.67968 q^{7} -66.1667 q^{8} +O(q^{10})$$ $$q-5.33039 q^{2} +20.4131 q^{4} +16.4131 q^{5} +9.67968 q^{7} -66.1667 q^{8} -87.4882 q^{10} -27.5882 q^{11} -51.5965 q^{14} +189.390 q^{16} -107.928 q^{17} -2.24723 q^{19} +335.042 q^{20} +147.056 q^{22} -41.8090 q^{23} +144.390 q^{25} +197.592 q^{28} -61.6213 q^{29} +191.932 q^{31} -480.187 q^{32} +575.300 q^{34} +158.874 q^{35} +98.4236 q^{37} +11.9786 q^{38} -1086.00 q^{40} +30.7452 q^{41} +238.325 q^{43} -563.160 q^{44} +222.858 q^{46} +511.482 q^{47} -249.304 q^{49} -769.653 q^{50} -492.825 q^{53} -452.807 q^{55} -640.472 q^{56} +328.466 q^{58} -484.179 q^{59} -444.021 q^{61} -1023.07 q^{62} +1044.47 q^{64} +190.114 q^{67} -2203.15 q^{68} -846.858 q^{70} -484.785 q^{71} -957.780 q^{73} -524.636 q^{74} -45.8729 q^{76} -267.045 q^{77} -375.216 q^{79} +3108.47 q^{80} -163.884 q^{82} +715.765 q^{83} -1771.43 q^{85} -1270.37 q^{86} +1825.42 q^{88} +1038.15 q^{89} -853.451 q^{92} -2726.40 q^{94} -36.8840 q^{95} +65.5636 q^{97} +1328.89 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 $$4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.33039 −1.88458 −0.942289 0.334800i $$-0.891331\pi$$
−0.942289 + 0.334800i $$0.891331\pi$$
$$3$$ 0 0
$$4$$ 20.4131 2.55164
$$5$$ 16.4131 1.46803 0.734016 0.679132i $$-0.237644\pi$$
0.734016 + 0.679132i $$0.237644\pi$$
$$6$$ 0 0
$$7$$ 9.67968 0.522654 0.261327 0.965250i $$-0.415840\pi$$
0.261327 + 0.965250i $$0.415840\pi$$
$$8$$ −66.1667 −2.92418
$$9$$ 0 0
$$10$$ −87.4882 −2.76662
$$11$$ −27.5882 −0.756195 −0.378098 0.925766i $$-0.623422\pi$$
−0.378098 + 0.925766i $$0.623422\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −51.5965 −0.984982
$$15$$ 0 0
$$16$$ 189.390 2.95921
$$17$$ −107.928 −1.53979 −0.769895 0.638171i $$-0.779691\pi$$
−0.769895 + 0.638171i $$0.779691\pi$$
$$18$$ 0 0
$$19$$ −2.24723 −0.0271342 −0.0135671 0.999908i $$-0.504319\pi$$
−0.0135671 + 0.999908i $$0.504319\pi$$
$$20$$ 335.042 3.74588
$$21$$ 0 0
$$22$$ 147.056 1.42511
$$23$$ −41.8090 −0.379034 −0.189517 0.981877i $$-0.560692\pi$$
−0.189517 + 0.981877i $$0.560692\pi$$
$$24$$ 0 0
$$25$$ 144.390 1.15512
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 197.592 1.33362
$$29$$ −61.6213 −0.394579 −0.197289 0.980345i $$-0.563214\pi$$
−0.197289 + 0.980345i $$0.563214\pi$$
$$30$$ 0 0
$$31$$ 191.932 1.11200 0.556000 0.831182i $$-0.312335\pi$$
0.556000 + 0.831182i $$0.312335\pi$$
$$32$$ −480.187 −2.65269
$$33$$ 0 0
$$34$$ 575.300 2.90185
$$35$$ 158.874 0.767272
$$36$$ 0 0
$$37$$ 98.4236 0.437317 0.218659 0.975801i $$-0.429832\pi$$
0.218659 + 0.975801i $$0.429832\pi$$
$$38$$ 11.9786 0.0511366
$$39$$ 0 0
$$40$$ −1086.00 −4.29279
$$41$$ 30.7452 0.117112 0.0585561 0.998284i $$-0.481350\pi$$
0.0585561 + 0.998284i $$0.481350\pi$$
$$42$$ 0 0
$$43$$ 238.325 0.845216 0.422608 0.906313i $$-0.361115\pi$$
0.422608 + 0.906313i $$0.361115\pi$$
$$44$$ −563.160 −1.92953
$$45$$ 0 0
$$46$$ 222.858 0.714319
$$47$$ 511.482 1.58739 0.793695 0.608316i $$-0.208155\pi$$
0.793695 + 0.608316i $$0.208155\pi$$
$$48$$ 0 0
$$49$$ −249.304 −0.726833
$$50$$ −769.653 −2.17691
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −492.825 −1.27726 −0.638630 0.769514i $$-0.720498\pi$$
−0.638630 + 0.769514i $$0.720498\pi$$
$$54$$ 0 0
$$55$$ −452.807 −1.11012
$$56$$ −640.472 −1.52833
$$57$$ 0 0
$$58$$ 328.466 0.743615
$$59$$ −484.179 −1.06838 −0.534192 0.845363i $$-0.679384\pi$$
−0.534192 + 0.845363i $$0.679384\pi$$
$$60$$ 0 0
$$61$$ −444.021 −0.931985 −0.465993 0.884789i $$-0.654303\pi$$
−0.465993 + 0.884789i $$0.654303\pi$$
$$62$$ −1023.07 −2.09565
$$63$$ 0 0
$$64$$ 1044.47 2.03998
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 190.114 0.346658 0.173329 0.984864i $$-0.444548\pi$$
0.173329 + 0.984864i $$0.444548\pi$$
$$68$$ −2203.15 −3.92898
$$69$$ 0 0
$$70$$ −846.858 −1.44598
$$71$$ −484.785 −0.810329 −0.405164 0.914244i $$-0.632786\pi$$
−0.405164 + 0.914244i $$0.632786\pi$$
$$72$$ 0 0
$$73$$ −957.780 −1.53561 −0.767806 0.640683i $$-0.778651\pi$$
−0.767806 + 0.640683i $$0.778651\pi$$
$$74$$ −524.636 −0.824159
$$75$$ 0 0
$$76$$ −45.8729 −0.0692367
$$77$$ −267.045 −0.395228
$$78$$ 0 0
$$79$$ −375.216 −0.534368 −0.267184 0.963646i $$-0.586093\pi$$
−0.267184 + 0.963646i $$0.586093\pi$$
$$80$$ 3108.47 4.34422
$$81$$ 0 0
$$82$$ −163.884 −0.220707
$$83$$ 715.765 0.946571 0.473286 0.880909i $$-0.343068\pi$$
0.473286 + 0.880909i $$0.343068\pi$$
$$84$$ 0 0
$$85$$ −1771.43 −2.26046
$$86$$ −1270.37 −1.59288
$$87$$ 0 0
$$88$$ 1825.42 2.21125
$$89$$ 1038.15 1.23645 0.618224 0.786002i $$-0.287852\pi$$
0.618224 + 0.786002i $$0.287852\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −853.451 −0.967157
$$93$$ 0 0
$$94$$ −2726.40 −2.99156
$$95$$ −36.8840 −0.0398339
$$96$$ 0 0
$$97$$ 65.5636 0.0686286 0.0343143 0.999411i $$-0.489075\pi$$
0.0343143 + 0.999411i $$0.489075\pi$$
$$98$$ 1328.89 1.36977
$$99$$ 0 0
$$100$$ 2947.44 2.94744
$$101$$ −531.798 −0.523920 −0.261960 0.965079i $$-0.584369\pi$$
−0.261960 + 0.965079i $$0.584369\pi$$
$$102$$ 0 0
$$103$$ −735.984 −0.704064 −0.352032 0.935988i $$-0.614509\pi$$
−0.352032 + 0.935988i $$0.614509\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 2626.95 2.40710
$$107$$ 783.265 0.707673 0.353837 0.935307i $$-0.384877\pi$$
0.353837 + 0.935307i $$0.384877\pi$$
$$108$$ 0 0
$$109$$ −532.339 −0.467788 −0.233894 0.972262i $$-0.575147\pi$$
−0.233894 + 0.972262i $$0.575147\pi$$
$$110$$ 2413.64 2.09210
$$111$$ 0 0
$$112$$ 1833.23 1.54664
$$113$$ 180.589 0.150340 0.0751699 0.997171i $$-0.476050\pi$$
0.0751699 + 0.997171i $$0.476050\pi$$
$$114$$ 0 0
$$115$$ −686.215 −0.556434
$$116$$ −1257.88 −1.00682
$$117$$ 0 0
$$118$$ 2580.86 2.01346
$$119$$ −1044.71 −0.804777
$$120$$ 0 0
$$121$$ −569.893 −0.428169
$$122$$ 2366.81 1.75640
$$123$$ 0 0
$$124$$ 3917.92 2.83742
$$125$$ 318.242 0.227716
$$126$$ 0 0
$$127$$ 1431.63 1.00029 0.500146 0.865941i $$-0.333280\pi$$
0.500146 + 0.865941i $$0.333280\pi$$
$$128$$ −1725.94 −1.19182
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2067.32 −1.37880 −0.689400 0.724381i $$-0.742126\pi$$
−0.689400 + 0.724381i $$0.742126\pi$$
$$132$$ 0 0
$$133$$ −21.7525 −0.0141818
$$134$$ −1013.38 −0.653304
$$135$$ 0 0
$$136$$ 7141.25 4.50262
$$137$$ 387.512 0.241660 0.120830 0.992673i $$-0.461444\pi$$
0.120830 + 0.992673i $$0.461444\pi$$
$$138$$ 0 0
$$139$$ 752.568 0.459223 0.229611 0.973282i $$-0.426254\pi$$
0.229611 + 0.973282i $$0.426254\pi$$
$$140$$ 3243.10 1.95780
$$141$$ 0 0
$$142$$ 2584.09 1.52713
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −1011.40 −0.579254
$$146$$ 5105.34 2.89398
$$147$$ 0 0
$$148$$ 2009.13 1.11587
$$149$$ 2636.72 1.44972 0.724862 0.688895i $$-0.241904\pi$$
0.724862 + 0.688895i $$0.241904\pi$$
$$150$$ 0 0
$$151$$ −3332.42 −1.79595 −0.897975 0.440046i $$-0.854962\pi$$
−0.897975 + 0.440046i $$0.854962\pi$$
$$152$$ 148.692 0.0793454
$$153$$ 0 0
$$154$$ 1423.45 0.744839
$$155$$ 3150.20 1.63245
$$156$$ 0 0
$$157$$ −1625.26 −0.826179 −0.413089 0.910690i $$-0.635550\pi$$
−0.413089 + 0.910690i $$0.635550\pi$$
$$158$$ 2000.05 1.00706
$$159$$ 0 0
$$160$$ −7881.36 −3.89423
$$161$$ −404.698 −0.198104
$$162$$ 0 0
$$163$$ −1835.37 −0.881944 −0.440972 0.897521i $$-0.645366\pi$$
−0.440972 + 0.897521i $$0.645366\pi$$
$$164$$ 627.605 0.298828
$$165$$ 0 0
$$166$$ −3815.31 −1.78389
$$167$$ −1945.00 −0.901248 −0.450624 0.892714i $$-0.648798\pi$$
−0.450624 + 0.892714i $$0.648798\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 9442.44 4.26001
$$171$$ 0 0
$$172$$ 4864.96 2.15668
$$173$$ −2531.63 −1.11258 −0.556289 0.830989i $$-0.687775\pi$$
−0.556289 + 0.830989i $$0.687775\pi$$
$$174$$ 0 0
$$175$$ 1397.65 0.603726
$$176$$ −5224.91 −2.23774
$$177$$ 0 0
$$178$$ −5533.76 −2.33018
$$179$$ −4263.01 −1.78007 −0.890035 0.455892i $$-0.849320\pi$$
−0.890035 + 0.455892i $$0.849320\pi$$
$$180$$ 0 0
$$181$$ 3944.61 1.61989 0.809946 0.586504i $$-0.199496\pi$$
0.809946 + 0.586504i $$0.199496\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2766.36 1.10836
$$185$$ 1615.43 0.641995
$$186$$ 0 0
$$187$$ 2977.54 1.16438
$$188$$ 10440.9 4.05044
$$189$$ 0 0
$$190$$ 196.606 0.0750701
$$191$$ 214.109 0.0811119 0.0405559 0.999177i $$-0.487087\pi$$
0.0405559 + 0.999177i $$0.487087\pi$$
$$192$$ 0 0
$$193$$ −1207.19 −0.450234 −0.225117 0.974332i $$-0.572276\pi$$
−0.225117 + 0.974332i $$0.572276\pi$$
$$194$$ −349.480 −0.129336
$$195$$ 0 0
$$196$$ −5089.06 −1.85461
$$197$$ 927.631 0.335487 0.167744 0.985831i $$-0.446352\pi$$
0.167744 + 0.985831i $$0.446352\pi$$
$$198$$ 0 0
$$199$$ 478.951 0.170613 0.0853064 0.996355i $$-0.472813\pi$$
0.0853064 + 0.996355i $$0.472813\pi$$
$$200$$ −9553.77 −3.37777
$$201$$ 0 0
$$202$$ 2834.69 0.987368
$$203$$ −596.474 −0.206228
$$204$$ 0 0
$$205$$ 504.624 0.171924
$$206$$ 3923.08 1.32686
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 61.9970 0.0205188
$$210$$ 0 0
$$211$$ −1450.95 −0.473402 −0.236701 0.971583i $$-0.576066\pi$$
−0.236701 + 0.971583i $$0.576066\pi$$
$$212$$ −10060.1 −3.25910
$$213$$ 0 0
$$214$$ −4175.11 −1.33367
$$215$$ 3911.66 1.24080
$$216$$ 0 0
$$217$$ 1857.84 0.581191
$$218$$ 2837.58 0.881583
$$219$$ 0 0
$$220$$ −9243.19 −2.83262
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2059.79 −0.618536 −0.309268 0.950975i $$-0.600084\pi$$
−0.309268 + 0.950975i $$0.600084\pi$$
$$224$$ −4648.06 −1.38644
$$225$$ 0 0
$$226$$ −962.612 −0.283327
$$227$$ −4482.46 −1.31062 −0.655311 0.755359i $$-0.727463\pi$$
−0.655311 + 0.755359i $$0.727463\pi$$
$$228$$ 0 0
$$229$$ −1630.39 −0.470477 −0.235239 0.971938i $$-0.575587\pi$$
−0.235239 + 0.971938i $$0.575587\pi$$
$$230$$ 3657.80 1.04864
$$231$$ 0 0
$$232$$ 4077.27 1.15382
$$233$$ 1903.69 0.535258 0.267629 0.963522i $$-0.413760\pi$$
0.267629 + 0.963522i $$0.413760\pi$$
$$234$$ 0 0
$$235$$ 8395.00 2.33034
$$236$$ −9883.58 −2.72613
$$237$$ 0 0
$$238$$ 5568.72 1.51667
$$239$$ −3763.79 −1.01866 −0.509328 0.860572i $$-0.670106\pi$$
−0.509328 + 0.860572i $$0.670106\pi$$
$$240$$ 0 0
$$241$$ −3614.74 −0.966166 −0.483083 0.875575i $$-0.660483\pi$$
−0.483083 + 0.875575i $$0.660483\pi$$
$$242$$ 3037.75 0.806918
$$243$$ 0 0
$$244$$ −9063.85 −2.37809
$$245$$ −4091.84 −1.06701
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −12699.5 −3.25169
$$249$$ 0 0
$$250$$ −1696.36 −0.429148
$$251$$ 5729.77 1.44088 0.720438 0.693520i $$-0.243941\pi$$
0.720438 + 0.693520i $$0.243941\pi$$
$$252$$ 0 0
$$253$$ 1153.43 0.286624
$$254$$ −7631.18 −1.88513
$$255$$ 0 0
$$256$$ 844.191 0.206101
$$257$$ 5525.79 1.34120 0.670602 0.741818i $$-0.266036\pi$$
0.670602 + 0.741818i $$0.266036\pi$$
$$258$$ 0 0
$$259$$ 952.709 0.228565
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 11019.6 2.59846
$$263$$ −5223.21 −1.22463 −0.612313 0.790615i $$-0.709761\pi$$
−0.612313 + 0.790615i $$0.709761\pi$$
$$264$$ 0 0
$$265$$ −8088.78 −1.87506
$$266$$ 115.949 0.0267267
$$267$$ 0 0
$$268$$ 3880.81 0.884545
$$269$$ −7203.88 −1.63282 −0.816410 0.577473i $$-0.804039\pi$$
−0.816410 + 0.577473i $$0.804039\pi$$
$$270$$ 0 0
$$271$$ −8577.69 −1.92272 −0.961360 0.275293i $$-0.911225\pi$$
−0.961360 + 0.275293i $$0.911225\pi$$
$$272$$ −20440.5 −4.55656
$$273$$ 0 0
$$274$$ −2065.59 −0.455427
$$275$$ −3983.44 −0.873493
$$276$$ 0 0
$$277$$ 7169.19 1.55507 0.777536 0.628838i $$-0.216469\pi$$
0.777536 + 0.628838i $$0.216469\pi$$
$$278$$ −4011.48 −0.865442
$$279$$ 0 0
$$280$$ −10512.1 −2.24364
$$281$$ −849.157 −0.180272 −0.0901360 0.995929i $$-0.528730\pi$$
−0.0901360 + 0.995929i $$0.528730\pi$$
$$282$$ 0 0
$$283$$ 1115.37 0.234283 0.117141 0.993115i $$-0.462627\pi$$
0.117141 + 0.993115i $$0.462627\pi$$
$$284$$ −9895.95 −2.06766
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 297.604 0.0612091
$$288$$ 0 0
$$289$$ 6735.49 1.37095
$$290$$ 5391.14 1.09165
$$291$$ 0 0
$$292$$ −19551.2 −3.91832
$$293$$ 1863.53 0.371565 0.185782 0.982591i $$-0.440518\pi$$
0.185782 + 0.982591i $$0.440518\pi$$
$$294$$ 0 0
$$295$$ −7946.87 −1.56842
$$296$$ −6512.36 −1.27879
$$297$$ 0 0
$$298$$ −14054.8 −2.73212
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 2306.91 0.441755
$$302$$ 17763.1 3.38461
$$303$$ 0 0
$$304$$ −425.602 −0.0802959
$$305$$ −7287.76 −1.36818
$$306$$ 0 0
$$307$$ −6387.50 −1.18747 −0.593736 0.804660i $$-0.702348\pi$$
−0.593736 + 0.804660i $$0.702348\pi$$
$$308$$ −5451.21 −1.00848
$$309$$ 0 0
$$310$$ −16791.8 −3.07648
$$311$$ 3492.59 0.636806 0.318403 0.947955i $$-0.396853\pi$$
0.318403 + 0.947955i $$0.396853\pi$$
$$312$$ 0 0
$$313$$ −5912.01 −1.06762 −0.533812 0.845603i $$-0.679241\pi$$
−0.533812 + 0.845603i $$0.679241\pi$$
$$314$$ 8663.29 1.55700
$$315$$ 0 0
$$316$$ −7659.31 −1.36351
$$317$$ 1677.54 0.297224 0.148612 0.988896i $$-0.452519\pi$$
0.148612 + 0.988896i $$0.452519\pi$$
$$318$$ 0 0
$$319$$ 1700.02 0.298378
$$320$$ 17143.0 2.99476
$$321$$ 0 0
$$322$$ 2157.20 0.373342
$$323$$ 242.540 0.0417810
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9783.22 1.66209
$$327$$ 0 0
$$328$$ −2034.31 −0.342457
$$329$$ 4950.98 0.829655
$$330$$ 0 0
$$331$$ 2010.31 0.333827 0.166913 0.985972i $$-0.446620\pi$$
0.166913 + 0.985972i $$0.446620\pi$$
$$332$$ 14611.0 2.41531
$$333$$ 0 0
$$334$$ 10367.6 1.69847
$$335$$ 3120.35 0.508905
$$336$$ 0 0
$$337$$ 7139.24 1.15400 0.577002 0.816743i $$-0.304222\pi$$
0.577002 + 0.816743i $$0.304222\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −36160.5 −5.76787
$$341$$ −5295.05 −0.840889
$$342$$ 0 0
$$343$$ −5733.31 −0.902536
$$344$$ −15769.2 −2.47156
$$345$$ 0 0
$$346$$ 13494.6 2.09674
$$347$$ −1.13990 −0.000176349 0 −8.81743e−5 1.00000i $$-0.500028\pi$$
−8.81743e−5 1.00000i $$0.500028\pi$$
$$348$$ 0 0
$$349$$ 12199.1 1.87107 0.935535 0.353235i $$-0.114918\pi$$
0.935535 + 0.353235i $$0.114918\pi$$
$$350$$ −7450.00 −1.13777
$$351$$ 0 0
$$352$$ 13247.5 2.00595
$$353$$ 10892.3 1.64232 0.821160 0.570698i $$-0.193327\pi$$
0.821160 + 0.570698i $$0.193327\pi$$
$$354$$ 0 0
$$355$$ −7956.81 −1.18959
$$356$$ 21191.9 3.15497
$$357$$ 0 0
$$358$$ 22723.5 3.35468
$$359$$ 3525.78 0.518339 0.259169 0.965832i $$-0.416551\pi$$
0.259169 + 0.965832i $$0.416551\pi$$
$$360$$ 0 0
$$361$$ −6853.95 −0.999264
$$362$$ −21026.3 −3.05281
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15720.1 −2.25433
$$366$$ 0 0
$$367$$ 2383.75 0.339049 0.169525 0.985526i $$-0.445777\pi$$
0.169525 + 0.985526i $$0.445777\pi$$
$$368$$ −7918.19 −1.12164
$$369$$ 0 0
$$370$$ −8610.90 −1.20989
$$371$$ −4770.39 −0.667564
$$372$$ 0 0
$$373$$ −13282.2 −1.84377 −0.921885 0.387463i $$-0.873352\pi$$
−0.921885 + 0.387463i $$0.873352\pi$$
$$374$$ −15871.5 −2.19437
$$375$$ 0 0
$$376$$ −33843.1 −4.64181
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4436.73 −0.601318 −0.300659 0.953732i $$-0.597207\pi$$
−0.300659 + 0.953732i $$0.597207\pi$$
$$380$$ −752.917 −0.101642
$$381$$ 0 0
$$382$$ −1141.28 −0.152862
$$383$$ 810.412 0.108120 0.0540602 0.998538i $$-0.482784\pi$$
0.0540602 + 0.998538i $$0.482784\pi$$
$$384$$ 0 0
$$385$$ −4383.03 −0.580207
$$386$$ 6434.78 0.848501
$$387$$ 0 0
$$388$$ 1338.35 0.175115
$$389$$ −3463.79 −0.451469 −0.225734 0.974189i $$-0.572478\pi$$
−0.225734 + 0.974189i $$0.572478\pi$$
$$390$$ 0 0
$$391$$ 4512.37 0.583633
$$392$$ 16495.6 2.12539
$$393$$ 0 0
$$394$$ −4944.64 −0.632252
$$395$$ −6158.45 −0.784469
$$396$$ 0 0
$$397$$ 425.405 0.0537796 0.0268898 0.999638i $$-0.491440\pi$$
0.0268898 + 0.999638i $$0.491440\pi$$
$$398$$ −2553.00 −0.321533
$$399$$ 0 0
$$400$$ 27345.9 3.41823
$$401$$ 1186.85 0.147801 0.0739007 0.997266i $$-0.476455\pi$$
0.0739007 + 0.997266i $$0.476455\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −10855.6 −1.33685
$$405$$ 0 0
$$406$$ 3179.44 0.388653
$$407$$ −2715.33 −0.330697
$$408$$ 0 0
$$409$$ 8007.42 0.968071 0.484036 0.875048i $$-0.339170\pi$$
0.484036 + 0.875048i $$0.339170\pi$$
$$410$$ −2689.84 −0.324005
$$411$$ 0 0
$$412$$ −15023.7 −1.79652
$$413$$ −4686.70 −0.558395
$$414$$ 0 0
$$415$$ 11747.9 1.38960
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −330.468 −0.0386692
$$419$$ −6832.46 −0.796629 −0.398314 0.917249i $$-0.630405\pi$$
−0.398314 + 0.917249i $$0.630405\pi$$
$$420$$ 0 0
$$421$$ 10739.6 1.24326 0.621632 0.783309i $$-0.286470\pi$$
0.621632 + 0.783309i $$0.286470\pi$$
$$422$$ 7734.16 0.892163
$$423$$ 0 0
$$424$$ 32608.6 3.73494
$$425$$ −15583.7 −1.77864
$$426$$ 0 0
$$427$$ −4297.99 −0.487106
$$428$$ 15988.9 1.80573
$$429$$ 0 0
$$430$$ −20850.7 −2.33839
$$431$$ −5214.45 −0.582763 −0.291382 0.956607i $$-0.594115\pi$$
−0.291382 + 0.956607i $$0.594115\pi$$
$$432$$ 0 0
$$433$$ 8642.24 0.959168 0.479584 0.877496i $$-0.340788\pi$$
0.479584 + 0.877496i $$0.340788\pi$$
$$434$$ −9903.02 −1.09530
$$435$$ 0 0
$$436$$ −10866.7 −1.19362
$$437$$ 93.9545 0.0102848
$$438$$ 0 0
$$439$$ −13026.2 −1.41619 −0.708097 0.706116i $$-0.750446\pi$$
−0.708097 + 0.706116i $$0.750446\pi$$
$$440$$ 29960.7 3.24619
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11533.0 1.23690 0.618450 0.785824i $$-0.287761\pi$$
0.618450 + 0.785824i $$0.287761\pi$$
$$444$$ 0 0
$$445$$ 17039.3 1.81515
$$446$$ 10979.5 1.16568
$$447$$ 0 0
$$448$$ 10110.2 1.06621
$$449$$ −9882.75 −1.03874 −0.519372 0.854548i $$-0.673834\pi$$
−0.519372 + 0.854548i $$0.673834\pi$$
$$450$$ 0 0
$$451$$ −848.204 −0.0885596
$$452$$ 3686.38 0.383613
$$453$$ 0 0
$$454$$ 23893.3 2.46997
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −15628.1 −1.59967 −0.799836 0.600218i $$-0.795080\pi$$
−0.799836 + 0.600218i $$0.795080\pi$$
$$458$$ 8690.63 0.886652
$$459$$ 0 0
$$460$$ −14007.8 −1.41982
$$461$$ 7747.46 0.782723 0.391361 0.920237i $$-0.372004\pi$$
0.391361 + 0.920237i $$0.372004\pi$$
$$462$$ 0 0
$$463$$ −333.422 −0.0334675 −0.0167337 0.999860i $$-0.505327\pi$$
−0.0167337 + 0.999860i $$0.505327\pi$$
$$464$$ −11670.4 −1.16764
$$465$$ 0 0
$$466$$ −10147.4 −1.00874
$$467$$ −8198.33 −0.812363 −0.406182 0.913792i $$-0.633140\pi$$
−0.406182 + 0.913792i $$0.633140\pi$$
$$468$$ 0 0
$$469$$ 1840.24 0.181182
$$470$$ −44748.7 −4.39171
$$471$$ 0 0
$$472$$ 32036.5 3.12415
$$473$$ −6574.96 −0.639148
$$474$$ 0 0
$$475$$ −324.477 −0.0313432
$$476$$ −21325.8 −2.05350
$$477$$ 0 0
$$478$$ 20062.5 1.91974
$$479$$ 6435.88 0.613910 0.306955 0.951724i $$-0.400690\pi$$
0.306955 + 0.951724i $$0.400690\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 19268.0 1.82082
$$483$$ 0 0
$$484$$ −11633.3 −1.09253
$$485$$ 1076.10 0.100749
$$486$$ 0 0
$$487$$ 8095.37 0.753257 0.376629 0.926364i $$-0.377083\pi$$
0.376629 + 0.926364i $$0.377083\pi$$
$$488$$ 29379.4 2.72529
$$489$$ 0 0
$$490$$ 21811.1 2.01087
$$491$$ −5116.46 −0.470270 −0.235135 0.971963i $$-0.575553\pi$$
−0.235135 + 0.971963i $$0.575553\pi$$
$$492$$ 0 0
$$493$$ 6650.67 0.607568
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 36349.9 3.29064
$$497$$ −4692.56 −0.423521
$$498$$ 0 0
$$499$$ −18050.7 −1.61936 −0.809682 0.586870i $$-0.800360\pi$$
−0.809682 + 0.586870i $$0.800360\pi$$
$$500$$ 6496.31 0.581047
$$501$$ 0 0
$$502$$ −30541.9 −2.71544
$$503$$ −10531.1 −0.933512 −0.466756 0.884386i $$-0.654577\pi$$
−0.466756 + 0.884386i $$0.654577\pi$$
$$504$$ 0 0
$$505$$ −8728.45 −0.769131
$$506$$ −6148.25 −0.540165
$$507$$ 0 0
$$508$$ 29224.1 2.55238
$$509$$ 1963.31 0.170967 0.0854834 0.996340i $$-0.472757\pi$$
0.0854834 + 0.996340i $$0.472757\pi$$
$$510$$ 0 0
$$511$$ −9271.00 −0.802593
$$512$$ 9307.69 0.803409
$$513$$ 0 0
$$514$$ −29454.6 −2.52760
$$515$$ −12079.8 −1.03359
$$516$$ 0 0
$$517$$ −14110.9 −1.20038
$$518$$ −5078.31 −0.430750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7044.93 0.592407 0.296203 0.955125i $$-0.404279\pi$$
0.296203 + 0.955125i $$0.404279\pi$$
$$522$$ 0 0
$$523$$ −3213.29 −0.268657 −0.134328 0.990937i $$-0.542888\pi$$
−0.134328 + 0.990937i $$0.542888\pi$$
$$524$$ −42200.4 −3.51819
$$525$$ 0 0
$$526$$ 27841.7 2.30790
$$527$$ −20714.9 −1.71225
$$528$$ 0 0
$$529$$ −10419.0 −0.856333
$$530$$ 43116.4 3.53369
$$531$$ 0 0
$$532$$ −444.036 −0.0361868
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12855.8 1.03889
$$536$$ −12579.2 −1.01369
$$537$$ 0 0
$$538$$ 38399.5 3.07718
$$539$$ 6877.83 0.549627
$$540$$ 0 0
$$541$$ 11251.4 0.894150 0.447075 0.894497i $$-0.352466\pi$$
0.447075 + 0.894497i $$0.352466\pi$$
$$542$$ 45722.4 3.62352
$$543$$ 0 0
$$544$$ 51825.8 4.08458
$$545$$ −8737.33 −0.686727
$$546$$ 0 0
$$547$$ 1533.54 0.119871 0.0599353 0.998202i $$-0.480911\pi$$
0.0599353 + 0.998202i $$0.480911\pi$$
$$548$$ 7910.31 0.616628
$$549$$ 0 0
$$550$$ 21233.3 1.64617
$$551$$ 138.477 0.0107066
$$552$$ 0 0
$$553$$ −3631.97 −0.279289
$$554$$ −38214.6 −2.93066
$$555$$ 0 0
$$556$$ 15362.2 1.17177
$$557$$ −16845.7 −1.28146 −0.640731 0.767766i $$-0.721369\pi$$
−0.640731 + 0.767766i $$0.721369\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 30089.0 2.27052
$$561$$ 0 0
$$562$$ 4526.34 0.339737
$$563$$ 20820.1 1.55855 0.779273 0.626685i $$-0.215589\pi$$
0.779273 + 0.626685i $$0.215589\pi$$
$$564$$ 0 0
$$565$$ 2964.03 0.220704
$$566$$ −5945.37 −0.441524
$$567$$ 0 0
$$568$$ 32076.6 2.36955
$$569$$ −23636.6 −1.74147 −0.870735 0.491752i $$-0.836357\pi$$
−0.870735 + 0.491752i $$0.836357\pi$$
$$570$$ 0 0
$$571$$ −26955.1 −1.97554 −0.987771 0.155913i $$-0.950168\pi$$
−0.987771 + 0.155913i $$0.950168\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −1586.35 −0.115353
$$575$$ −6036.78 −0.437828
$$576$$ 0 0
$$577$$ 23499.8 1.69551 0.847755 0.530388i $$-0.177954\pi$$
0.847755 + 0.530388i $$0.177954\pi$$
$$578$$ −35902.8 −2.58367
$$579$$ 0 0
$$580$$ −20645.7 −1.47805
$$581$$ 6928.38 0.494729
$$582$$ 0 0
$$583$$ 13596.1 0.965857
$$584$$ 63373.1 4.49040
$$585$$ 0 0
$$586$$ −9933.33 −0.700243
$$587$$ −4637.50 −0.326082 −0.163041 0.986619i $$-0.552130\pi$$
−0.163041 + 0.986619i $$0.552130\pi$$
$$588$$ 0 0
$$589$$ −431.316 −0.0301733
$$590$$ 42359.9 2.95582
$$591$$ 0 0
$$592$$ 18640.4 1.29411
$$593$$ −12633.5 −0.874869 −0.437434 0.899250i $$-0.644113\pi$$
−0.437434 + 0.899250i $$0.644113\pi$$
$$594$$ 0 0
$$595$$ −17146.9 −1.18144
$$596$$ 53823.7 3.69917
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 18757.1 1.27946 0.639730 0.768600i $$-0.279046\pi$$
0.639730 + 0.768600i $$0.279046\pi$$
$$600$$ 0 0
$$601$$ −3632.98 −0.246576 −0.123288 0.992371i $$-0.539344\pi$$
−0.123288 + 0.992371i $$0.539344\pi$$
$$602$$ −12296.8 −0.832522
$$603$$ 0 0
$$604$$ −68025.0 −4.58261
$$605$$ −9353.71 −0.628566
$$606$$ 0 0
$$607$$ −12700.0 −0.849219 −0.424610 0.905377i $$-0.639589\pi$$
−0.424610 + 0.905377i $$0.639589\pi$$
$$608$$ 1079.09 0.0719786
$$609$$ 0 0
$$610$$ 38846.6 2.57845
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 21640.1 1.42584 0.712918 0.701248i $$-0.247373\pi$$
0.712918 + 0.701248i $$0.247373\pi$$
$$614$$ 34047.9 2.23788
$$615$$ 0 0
$$616$$ 17669.5 1.15572
$$617$$ 16541.7 1.07933 0.539663 0.841881i $$-0.318552\pi$$
0.539663 + 0.841881i $$0.318552\pi$$
$$618$$ 0 0
$$619$$ −21138.9 −1.37261 −0.686303 0.727316i $$-0.740767\pi$$
−0.686303 + 0.727316i $$0.740767\pi$$
$$620$$ 64305.2 4.16542
$$621$$ 0 0
$$622$$ −18616.9 −1.20011
$$623$$ 10049.0 0.646235
$$624$$ 0 0
$$625$$ −12825.4 −0.820823
$$626$$ 31513.3 2.01202
$$627$$ 0 0
$$628$$ −33176.6 −2.10811
$$629$$ −10622.7 −0.673377
$$630$$ 0 0
$$631$$ 5489.80 0.346348 0.173174 0.984891i $$-0.444598\pi$$
0.173174 + 0.984891i $$0.444598\pi$$
$$632$$ 24826.8 1.56259
$$633$$ 0 0
$$634$$ −8941.94 −0.560141
$$635$$ 23497.5 1.46846
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −9061.76 −0.562318
$$639$$ 0 0
$$640$$ −28328.1 −1.74963
$$641$$ −4297.04 −0.264778 −0.132389 0.991198i $$-0.542265\pi$$
−0.132389 + 0.991198i $$0.542265\pi$$
$$642$$ 0 0
$$643$$ 25696.9 1.57603 0.788016 0.615655i $$-0.211108\pi$$
0.788016 + 0.615655i $$0.211108\pi$$
$$644$$ −8261.14 −0.505488
$$645$$ 0 0
$$646$$ −1292.83 −0.0787396
$$647$$ −2174.98 −0.132160 −0.0660798 0.997814i $$-0.521049\pi$$
−0.0660798 + 0.997814i $$0.521049\pi$$
$$648$$ 0 0
$$649$$ 13357.6 0.807907
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −37465.5 −2.25040
$$653$$ 15454.5 0.926160 0.463080 0.886316i $$-0.346744\pi$$
0.463080 + 0.886316i $$0.346744\pi$$
$$654$$ 0 0
$$655$$ −33931.1 −2.02412
$$656$$ 5822.82 0.346560
$$657$$ 0 0
$$658$$ −26390.7 −1.56355
$$659$$ 3148.77 0.186129 0.0930643 0.995660i $$-0.470334\pi$$
0.0930643 + 0.995660i $$0.470334\pi$$
$$660$$ 0 0
$$661$$ −2099.70 −0.123553 −0.0617767 0.998090i $$-0.519677\pi$$
−0.0617767 + 0.998090i $$0.519677\pi$$
$$662$$ −10715.7 −0.629122
$$663$$ 0 0
$$664$$ −47359.8 −2.76795
$$665$$ −357.026 −0.0208193
$$666$$ 0 0
$$667$$ 2576.32 0.149559
$$668$$ −39703.4 −2.29966
$$669$$ 0 0
$$670$$ −16632.7 −0.959071
$$671$$ 12249.7 0.704763
$$672$$ 0 0
$$673$$ 30970.8 1.77390 0.886950 0.461865i $$-0.152819\pi$$
0.886950 + 0.461865i $$0.152819\pi$$
$$674$$ −38055.0 −2.17481
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −14640.6 −0.831141 −0.415570 0.909561i $$-0.636418\pi$$
−0.415570 + 0.909561i $$0.636418\pi$$
$$678$$ 0 0
$$679$$ 634.635 0.0358690
$$680$$ 117210. 6.60999
$$681$$ 0 0
$$682$$ 28224.7 1.58472
$$683$$ 6685.83 0.374563 0.187281 0.982306i $$-0.440032\pi$$
0.187281 + 0.982306i $$0.440032\pi$$
$$684$$ 0 0
$$685$$ 6360.27 0.354764
$$686$$ 30560.8 1.70090
$$687$$ 0 0
$$688$$ 45136.3 2.50117
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −30194.1 −1.66228 −0.831141 0.556062i $$-0.812312\pi$$
−0.831141 + 0.556062i $$0.812312\pi$$
$$692$$ −51678.4 −2.83890
$$693$$ 0 0
$$694$$ 6.07611 0.000332343 0
$$695$$ 12352.0 0.674154
$$696$$ 0 0
$$697$$ −3318.28 −0.180328
$$698$$ −65026.0 −3.52618
$$699$$ 0 0
$$700$$ 28530.3 1.54049
$$701$$ 30300.9 1.63260 0.816298 0.577631i $$-0.196023\pi$$
0.816298 + 0.577631i $$0.196023\pi$$
$$702$$ 0 0
$$703$$ −221.181 −0.0118663
$$704$$ −28815.1 −1.54263
$$705$$ 0 0
$$706$$ −58060.3 −3.09508
$$707$$ −5147.64 −0.273829
$$708$$ 0 0
$$709$$ 26123.2 1.38375 0.691875 0.722017i $$-0.256785\pi$$
0.691875 + 0.722017i $$0.256785\pi$$
$$710$$ 42412.9 2.24187
$$711$$ 0 0
$$712$$ −68691.1 −3.61560
$$713$$ −8024.48 −0.421486
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −87021.3 −4.54209
$$717$$ 0 0
$$718$$ −18793.8 −0.976850
$$719$$ −19325.7 −1.00240 −0.501200 0.865331i $$-0.667108\pi$$
−0.501200 + 0.865331i $$0.667108\pi$$
$$720$$ 0 0
$$721$$ −7124.09 −0.367982
$$722$$ 36534.2 1.88319
$$723$$ 0 0
$$724$$ 80521.7 4.13338
$$725$$ −8897.47 −0.455784
$$726$$ 0 0
$$727$$ 26065.8 1.32975 0.664875 0.746954i $$-0.268485\pi$$
0.664875 + 0.746954i $$0.268485\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 83794.4 4.24845
$$731$$ −25722.0 −1.30145
$$732$$ 0 0
$$733$$ 1055.45 0.0531843 0.0265921 0.999646i $$-0.491534\pi$$
0.0265921 + 0.999646i $$0.491534\pi$$
$$734$$ −12706.4 −0.638965
$$735$$ 0 0
$$736$$ 20076.2 1.00546
$$737$$ −5244.89 −0.262141
$$738$$ 0 0
$$739$$ 9410.40 0.468426 0.234213 0.972185i $$-0.424749\pi$$
0.234213 + 0.972185i $$0.424749\pi$$
$$740$$ 32976.0 1.63814
$$741$$ 0 0
$$742$$ 25428.1 1.25808
$$743$$ 7523.70 0.371491 0.185746 0.982598i $$-0.440530\pi$$
0.185746 + 0.982598i $$0.440530\pi$$
$$744$$ 0 0
$$745$$ 43276.8 2.12824
$$746$$ 70799.4 3.47473
$$747$$ 0 0
$$748$$ 60780.8 2.97108
$$749$$ 7581.76 0.369868
$$750$$ 0 0
$$751$$ −12984.1 −0.630886 −0.315443 0.948945i $$-0.602153\pi$$
−0.315443 + 0.948945i $$0.602153\pi$$
$$752$$ 96869.3 4.69742
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −54695.3 −2.63651
$$756$$ 0 0
$$757$$ −27934.6 −1.34122 −0.670609 0.741811i $$-0.733967\pi$$
−0.670609 + 0.741811i $$0.733967\pi$$
$$758$$ 23649.5 1.13323
$$759$$ 0 0
$$760$$ 2440.49 0.116482
$$761$$ 15519.3 0.739255 0.369627 0.929180i $$-0.379485\pi$$
0.369627 + 0.929180i $$0.379485\pi$$
$$762$$ 0 0
$$763$$ −5152.88 −0.244491
$$764$$ 4370.62 0.206968
$$765$$ 0 0
$$766$$ −4319.82 −0.203761
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 12885.2 0.604228 0.302114 0.953272i $$-0.402308\pi$$
0.302114 + 0.953272i $$0.402308\pi$$
$$770$$ 23363.3 1.09345
$$771$$ 0 0
$$772$$ −24642.4 −1.14883
$$773$$ −5892.04 −0.274155 −0.137078 0.990560i $$-0.543771\pi$$
−0.137078 + 0.990560i $$0.543771\pi$$
$$774$$ 0 0
$$775$$ 27713.0 1.28449
$$776$$ −4338.12 −0.200682
$$777$$ 0 0
$$778$$ 18463.4 0.850828
$$779$$ −69.0916 −0.00317775
$$780$$ 0 0
$$781$$ 13374.3 0.612766
$$782$$ −24052.7 −1.09990
$$783$$ 0 0
$$784$$ −47215.5 −2.15085
$$785$$ −26675.6 −1.21286
$$786$$ 0 0
$$787$$ −21020.4 −0.952091 −0.476045 0.879421i $$-0.657930\pi$$
−0.476045 + 0.879421i $$0.657930\pi$$
$$788$$ 18935.8 0.856042
$$789$$ 0 0
$$790$$ 32827.0 1.47839
$$791$$ 1748.05 0.0785757
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −2267.58 −0.101352
$$795$$ 0 0
$$796$$ 9776.87 0.435342
$$797$$ −31355.6 −1.39356 −0.696782 0.717283i $$-0.745386\pi$$
−0.696782 + 0.717283i $$0.745386\pi$$
$$798$$ 0 0
$$799$$ −55203.3 −2.44425
$$800$$ −69334.0 −3.06416
$$801$$ 0 0
$$802$$ −6326.37 −0.278543
$$803$$ 26423.4 1.16122
$$804$$ 0 0
$$805$$ −6642.34 −0.290822
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 35187.3 1.53204
$$809$$ 18132.5 0.788017 0.394009 0.919107i $$-0.371088\pi$$
0.394009 + 0.919107i $$0.371088\pi$$
$$810$$ 0 0
$$811$$ −24755.3 −1.07186 −0.535928 0.844263i $$-0.680038\pi$$
−0.535928 + 0.844263i $$0.680038\pi$$
$$812$$ −12175.9 −0.526219
$$813$$ 0 0
$$814$$ 14473.8 0.623225
$$815$$ −30124.0 −1.29472
$$816$$ 0 0
$$817$$ −535.572 −0.0229343
$$818$$ −42682.7 −1.82441
$$819$$ 0 0
$$820$$ 10300.9 0.438688
$$821$$ −4082.65 −0.173551 −0.0867755 0.996228i $$-0.527656\pi$$
−0.0867755 + 0.996228i $$0.527656\pi$$
$$822$$ 0 0
$$823$$ −34327.0 −1.45391 −0.726954 0.686687i $$-0.759065\pi$$
−0.726954 + 0.686687i $$0.759065\pi$$
$$824$$ 48697.6 2.05881
$$825$$ 0 0
$$826$$ 24981.9 1.05234
$$827$$ −3228.87 −0.135767 −0.0678833 0.997693i $$-0.521625\pi$$
−0.0678833 + 0.997693i $$0.521625\pi$$
$$828$$ 0 0
$$829$$ −10452.4 −0.437908 −0.218954 0.975735i $$-0.570265\pi$$
−0.218954 + 0.975735i $$0.570265\pi$$
$$830$$ −62621.0 −2.61880
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 26906.9 1.11917
$$834$$ 0 0
$$835$$ −31923.4 −1.32306
$$836$$ 1265.55 0.0523564
$$837$$ 0 0
$$838$$ 36419.7 1.50131
$$839$$ 28289.0 1.16406 0.582028 0.813169i $$-0.302259\pi$$
0.582028 + 0.813169i $$0.302259\pi$$
$$840$$ 0 0
$$841$$ −20591.8 −0.844308
$$842$$ −57246.1 −2.34303
$$843$$ 0 0
$$844$$ −29618.5 −1.20795
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −5516.39 −0.223784
$$848$$ −93335.9 −3.77968
$$849$$ 0 0
$$850$$ 83067.2 3.35198
$$851$$ −4114.99 −0.165758
$$852$$ 0 0
$$853$$ −26631.8 −1.06900 −0.534498 0.845170i $$-0.679499\pi$$
−0.534498 + 0.845170i $$0.679499\pi$$
$$854$$ 22910.0 0.917989
$$855$$ 0 0
$$856$$ −51826.0 −2.06936
$$857$$ −11796.7 −0.470209 −0.235104 0.971970i $$-0.575543\pi$$
−0.235104 + 0.971970i $$0.575543\pi$$
$$858$$ 0 0
$$859$$ −22672.8 −0.900567 −0.450283 0.892886i $$-0.648677\pi$$
−0.450283 + 0.892886i $$0.648677\pi$$
$$860$$ 79849.0 3.16608
$$861$$ 0 0
$$862$$ 27795.0 1.09826
$$863$$ −21421.1 −0.844940 −0.422470 0.906377i $$-0.638837\pi$$
−0.422470 + 0.906377i $$0.638837\pi$$
$$864$$ 0 0
$$865$$ −41551.9 −1.63330
$$866$$ −46066.6 −1.80763
$$867$$ 0 0
$$868$$ 37924.3 1.48299
$$869$$ 10351.5 0.404086
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 35223.1 1.36790
$$873$$ 0 0
$$874$$ −500.814 −0.0193825
$$875$$ 3080.48 0.119016
$$876$$ 0 0
$$877$$ −5155.20 −0.198493 −0.0992466 0.995063i $$-0.531643\pi$$
−0.0992466 + 0.995063i $$0.531643\pi$$
$$878$$ 69435.0 2.66893
$$879$$ 0 0
$$880$$ −85756.9 −3.28507
$$881$$ −23692.2 −0.906027 −0.453013 0.891504i $$-0.649651\pi$$
−0.453013 + 0.891504i $$0.649651\pi$$
$$882$$ 0 0
$$883$$ −14591.5 −0.556108 −0.278054 0.960565i $$-0.589689\pi$$
−0.278054 + 0.960565i $$0.589689\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −61475.2 −2.33104
$$887$$ 9722.26 0.368029 0.184014 0.982924i $$-0.441091\pi$$
0.184014 + 0.982924i $$0.441091\pi$$
$$888$$ 0 0
$$889$$ 13857.8 0.522806
$$890$$ −90826.1 −3.42078
$$891$$ 0 0
$$892$$ −42046.6 −1.57828
$$893$$ −1149.42 −0.0430726
$$894$$ 0 0
$$895$$ −69969.2 −2.61320
$$896$$ −16706.6 −0.622911
$$897$$ 0 0
$$898$$ 52678.9 1.95759
$$899$$ −11827.1 −0.438771
$$900$$ 0 0
$$901$$ 53189.7 1.96671
$$902$$ 4521.26 0.166898
$$903$$ 0 0
$$904$$ −11949.0 −0.439621
$$905$$ 64743.2 2.37805
$$906$$ 0 0
$$907$$ 11799.0 0.431951 0.215975 0.976399i $$-0.430707\pi$$
0.215975 + 0.976399i $$0.430707\pi$$
$$908$$ −91500.9 −3.34423
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 43012.4 1.56429 0.782143 0.623099i $$-0.214127\pi$$
0.782143 + 0.623099i $$0.214127\pi$$
$$912$$ 0 0
$$913$$ −19746.6 −0.715793
$$914$$ 83303.8 3.01471
$$915$$ 0 0
$$916$$ −33281.3 −1.20049
$$917$$ −20011.0 −0.720635
$$918$$ 0 0
$$919$$ −4951.41 −0.177728 −0.0888639 0.996044i $$-0.528324\pi$$
−0.0888639 + 0.996044i $$0.528324\pi$$
$$920$$ 45404.5 1.62711
$$921$$ 0 0
$$922$$ −41297.0 −1.47510
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 14211.3 0.505152
$$926$$ 1777.27 0.0630721
$$927$$ 0 0
$$928$$ 29589.8 1.04669
$$929$$ 8934.86 0.315547 0.157774 0.987475i $$-0.449568\pi$$
0.157774 + 0.987475i $$0.449568\pi$$
$$930$$ 0 0
$$931$$ 560.243 0.0197221
$$932$$ 38860.2 1.36578
$$933$$ 0 0
$$934$$ 43700.3 1.53096
$$935$$ 48870.6 1.70935
$$936$$ 0 0
$$937$$ −13182.8 −0.459620 −0.229810 0.973235i $$-0.573811\pi$$
−0.229810 + 0.973235i $$0.573811\pi$$
$$938$$ −9809.20 −0.341452
$$939$$ 0 0
$$940$$ 171368. 5.94618
$$941$$ 21693.7 0.751536 0.375768 0.926714i $$-0.377379\pi$$
0.375768 + 0.926714i $$0.377379\pi$$
$$942$$ 0 0
$$943$$ −1285.43 −0.0443895
$$944$$ −91698.4 −3.16158
$$945$$ 0 0
$$946$$ 35047.1 1.20452
$$947$$ 49790.0 1.70851 0.854254 0.519856i $$-0.174014\pi$$
0.854254 + 0.519856i $$0.174014\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 1729.59 0.0590687
$$951$$ 0 0
$$952$$ 69125.0 2.35331
$$953$$ −4217.93 −0.143371 −0.0716853 0.997427i $$-0.522838\pi$$
−0.0716853 + 0.997427i $$0.522838\pi$$
$$954$$ 0 0
$$955$$ 3514.19 0.119075
$$956$$ −76830.5 −2.59924
$$957$$ 0 0
$$958$$ −34305.8 −1.15696
$$959$$ 3750.99 0.126304
$$960$$ 0 0
$$961$$ 7046.87 0.236544
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −73788.1 −2.46530
$$965$$ −19813.7 −0.660958
$$966$$ 0 0
$$967$$ −40927.9 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$968$$ 37707.9 1.25204
$$969$$ 0 0
$$970$$ −5736.04 −0.189869
$$971$$ 17114.8 0.565645 0.282822 0.959172i $$-0.408729\pi$$
0.282822 + 0.959172i $$0.408729\pi$$
$$972$$ 0 0
$$973$$ 7284.62 0.240015
$$974$$ −43151.5 −1.41957
$$975$$ 0 0
$$976$$ −84093.0 −2.75794
$$977$$ 118.470 0.00387940 0.00193970 0.999998i $$-0.499383\pi$$
0.00193970 + 0.999998i $$0.499383\pi$$
$$978$$ 0 0
$$979$$ −28640.7 −0.934996
$$980$$ −83527.2 −2.72263
$$981$$ 0 0
$$982$$ 27272.8 0.886261
$$983$$ −26002.8 −0.843705 −0.421852 0.906665i $$-0.638620\pi$$
−0.421852 + 0.906665i $$0.638620\pi$$
$$984$$ 0 0
$$985$$ 15225.3 0.492506
$$986$$ −35450.7 −1.14501
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9964.14 −0.320365
$$990$$ 0 0
$$991$$ −16062.0 −0.514860 −0.257430 0.966297i $$-0.582876\pi$$
−0.257430 + 0.966297i $$0.582876\pi$$
$$992$$ −92163.3 −2.94979
$$993$$ 0 0
$$994$$ 25013.2 0.798159
$$995$$ 7861.07 0.250465
$$996$$ 0 0
$$997$$ 1361.54 0.0432503 0.0216251 0.999766i $$-0.493116\pi$$
0.0216251 + 0.999766i $$0.493116\pi$$
$$998$$ 96217.5 3.05182
$$999$$ 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 1521.4.a.v.1.1 4
3.2 odd 2 507.4.a.m.1.4 4
13.3 even 3 117.4.g.e.100.4 8
13.9 even 3 117.4.g.e.55.4 8
13.12 even 2 1521.4.a.bb.1.4 4
39.5 even 4 507.4.b.h.337.1 8
39.8 even 4 507.4.b.h.337.8 8
39.29 odd 6 39.4.e.c.22.1 yes 8
39.35 odd 6 39.4.e.c.16.1 8
39.38 odd 2 507.4.a.i.1.1 4
156.35 even 6 624.4.q.i.289.1 8
156.107 even 6 624.4.q.i.529.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 39.35 odd 6
39.4.e.c.22.1 yes 8 39.29 odd 6
117.4.g.e.55.4 8 13.9 even 3
117.4.g.e.100.4 8 13.3 even 3
507.4.a.i.1.1 4 39.38 odd 2
507.4.a.m.1.4 4 3.2 odd 2
507.4.b.h.337.1 8 39.5 even 4
507.4.b.h.337.8 8 39.8 even 4
624.4.q.i.289.1 8 156.35 even 6
624.4.q.i.529.1 8 156.107 even 6
1521.4.a.v.1.1 4 1.1 even 1 trivial
1521.4.a.bb.1.4 4 13.12 even 2