# Properties

 Label 1521.4.a.x.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5054412.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 29x^{2} + 48$$ x^4 - 29*x^2 + 48 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$-5.21898$$ 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)

 $$f(q)$$ $$=$$ $$q-5.21898 q^{2} +19.2377 q^{4} +5.83936 q^{5} -31.3139 q^{7} -58.6495 q^{8} +O(q^{10})$$ $$q-5.21898 q^{2} +19.2377 q^{4} +5.83936 q^{5} -31.3139 q^{7} -58.6495 q^{8} -30.4755 q^{10} +16.2773 q^{11} +163.426 q^{14} +152.189 q^{16} +54.0000 q^{17} +66.3500 q^{19} +112.336 q^{20} -84.9510 q^{22} +182.853 q^{23} -90.9019 q^{25} -602.408 q^{28} +164.853 q^{29} -58.9055 q^{31} -325.073 q^{32} -281.825 q^{34} -182.853 q^{35} -110.366 q^{37} -346.279 q^{38} -342.475 q^{40} -55.0357 q^{41} -113.147 q^{43} +313.139 q^{44} -954.305 q^{46} -514.089 q^{47} +637.559 q^{49} +474.415 q^{50} -242.559 q^{53} +95.0490 q^{55} +1836.54 q^{56} -860.364 q^{58} +265.036 q^{59} -468.098 q^{61} +307.426 q^{62} +479.042 q^{64} +852.919 q^{67} +1038.84 q^{68} +954.305 q^{70} -165.619 q^{71} +315.325 q^{73} +576.000 q^{74} +1276.42 q^{76} -509.706 q^{77} +479.608 q^{79} +888.684 q^{80} +287.230 q^{82} -574.235 q^{83} +315.325 q^{85} +590.512 q^{86} -954.657 q^{88} -66.7144 q^{89} +3517.68 q^{92} +2683.02 q^{94} +387.441 q^{95} -1438.25 q^{97} -3327.40 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 26 q^{4}+O(q^{10})$$ 4 * q + 26 * q^4 $$4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100})$$ 4 * q + 26 * q^4 - 20 * q^10 + 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 + 44 * q^25 + 48 * q^29 - 120 * q^35 - 468 * q^38 - 1268 * q^40 - 1064 * q^43 + 716 * q^49 + 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 + 1050 * q^64 + 1404 * q^68 + 2304 * q^74 - 816 * q^77 + 288 * q^79 + 28 * q^82 - 2392 * q^88 + 8568 * q^92 + 6656 * q^94 + 3384 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.21898 −1.84519 −0.922594 0.385773i $$-0.873935\pi$$
−0.922594 + 0.385773i $$0.873935\pi$$
$$3$$ 0 0
$$4$$ 19.2377 2.40472
$$5$$ 5.83936 0.522288 0.261144 0.965300i $$-0.415900\pi$$
0.261144 + 0.965300i $$0.415900\pi$$
$$6$$ 0 0
$$7$$ −31.3139 −1.69079 −0.845395 0.534141i $$-0.820635\pi$$
−0.845395 + 0.534141i $$0.820635\pi$$
$$8$$ −58.6495 −2.59197
$$9$$ 0 0
$$10$$ −30.4755 −0.963719
$$11$$ 16.2773 0.446163 0.223082 0.974800i $$-0.428388\pi$$
0.223082 + 0.974800i $$0.428388\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 163.426 3.11983
$$15$$ 0 0
$$16$$ 152.189 2.37795
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ 66.3500 0.801144 0.400572 0.916265i $$-0.368811\pi$$
0.400572 + 0.916265i $$0.368811\pi$$
$$20$$ 112.336 1.25595
$$21$$ 0 0
$$22$$ −84.9510 −0.823255
$$23$$ 182.853 1.65772 0.828858 0.559459i $$-0.188991\pi$$
0.828858 + 0.559459i $$0.188991\pi$$
$$24$$ 0 0
$$25$$ −90.9019 −0.727215
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −602.408 −4.06587
$$29$$ 164.853 1.05560 0.527800 0.849369i $$-0.323017\pi$$
0.527800 + 0.849369i $$0.323017\pi$$
$$30$$ 0 0
$$31$$ −58.9055 −0.341282 −0.170641 0.985333i $$-0.554584\pi$$
−0.170641 + 0.985333i $$0.554584\pi$$
$$32$$ −325.073 −1.79579
$$33$$ 0 0
$$34$$ −281.825 −1.42155
$$35$$ −182.853 −0.883079
$$36$$ 0 0
$$37$$ −110.366 −0.490382 −0.245191 0.969475i $$-0.578851\pi$$
−0.245191 + 0.969475i $$0.578851\pi$$
$$38$$ −346.279 −1.47826
$$39$$ 0 0
$$40$$ −342.475 −1.35375
$$41$$ −55.0357 −0.209637 −0.104819 0.994491i $$-0.533426\pi$$
−0.104819 + 0.994491i $$0.533426\pi$$
$$42$$ 0 0
$$43$$ −113.147 −0.401274 −0.200637 0.979666i $$-0.564301\pi$$
−0.200637 + 0.979666i $$0.564301\pi$$
$$44$$ 313.139 1.07290
$$45$$ 0 0
$$46$$ −954.305 −3.05880
$$47$$ −514.089 −1.59548 −0.797740 0.603001i $$-0.793971\pi$$
−0.797740 + 0.603001i $$0.793971\pi$$
$$48$$ 0 0
$$49$$ 637.559 1.85877
$$50$$ 474.415 1.34185
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −242.559 −0.628641 −0.314321 0.949317i $$-0.601777\pi$$
−0.314321 + 0.949317i $$0.601777\pi$$
$$54$$ 0 0
$$55$$ 95.0490 0.233026
$$56$$ 1836.54 4.38247
$$57$$ 0 0
$$58$$ −860.364 −1.94778
$$59$$ 265.036 0.584825 0.292413 0.956292i $$-0.405542\pi$$
0.292413 + 0.956292i $$0.405542\pi$$
$$60$$ 0 0
$$61$$ −468.098 −0.982522 −0.491261 0.871013i $$-0.663464\pi$$
−0.491261 + 0.871013i $$0.663464\pi$$
$$62$$ 307.426 0.629729
$$63$$ 0 0
$$64$$ 479.042 0.935628
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 852.919 1.55523 0.777617 0.628739i $$-0.216428\pi$$
0.777617 + 0.628739i $$0.216428\pi$$
$$68$$ 1038.84 1.85261
$$69$$ 0 0
$$70$$ 954.305 1.62945
$$71$$ −165.619 −0.276836 −0.138418 0.990374i $$-0.544202\pi$$
−0.138418 + 0.990374i $$0.544202\pi$$
$$72$$ 0 0
$$73$$ 315.325 0.505562 0.252781 0.967524i $$-0.418655\pi$$
0.252781 + 0.967524i $$0.418655\pi$$
$$74$$ 576.000 0.904846
$$75$$ 0 0
$$76$$ 1276.42 1.92653
$$77$$ −509.706 −0.754368
$$78$$ 0 0
$$79$$ 479.608 0.683039 0.341519 0.939875i $$-0.389058\pi$$
0.341519 + 0.939875i $$0.389058\pi$$
$$80$$ 888.684 1.24197
$$81$$ 0 0
$$82$$ 287.230 0.386820
$$83$$ −574.235 −0.759404 −0.379702 0.925109i $$-0.623973\pi$$
−0.379702 + 0.925109i $$0.623973\pi$$
$$84$$ 0 0
$$85$$ 315.325 0.402374
$$86$$ 590.512 0.740426
$$87$$ 0 0
$$88$$ −954.657 −1.15644
$$89$$ −66.7144 −0.0794575 −0.0397287 0.999211i $$-0.512649\pi$$
−0.0397287 + 0.999211i $$0.512649\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 3517.68 3.98634
$$93$$ 0 0
$$94$$ 2683.02 2.94396
$$95$$ 387.441 0.418428
$$96$$ 0 0
$$97$$ −1438.25 −1.50549 −0.752744 0.658313i $$-0.771270\pi$$
−0.752744 + 0.658313i $$0.771270\pi$$
$$98$$ −3327.40 −3.42978
$$99$$ 0 0
$$100$$ −1748.75 −1.74875
$$101$$ 896.264 0.882986 0.441493 0.897265i $$-0.354449\pi$$
0.441493 + 0.897265i $$0.354449\pi$$
$$102$$ 0 0
$$103$$ −22.2644 −0.0212988 −0.0106494 0.999943i $$-0.503390\pi$$
−0.0106494 + 0.999943i $$0.503390\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1265.91 1.15996
$$107$$ 351.441 0.317524 0.158762 0.987317i $$-0.449250\pi$$
0.158762 + 0.987317i $$0.449250\pi$$
$$108$$ 0 0
$$109$$ 967.008 0.849748 0.424874 0.905252i $$-0.360318\pi$$
0.424874 + 0.905252i $$0.360318\pi$$
$$110$$ −496.059 −0.429976
$$111$$ 0 0
$$112$$ −4765.62 −4.02061
$$113$$ −48.2943 −0.0402048 −0.0201024 0.999798i $$-0.506399\pi$$
−0.0201024 + 0.999798i $$0.506399\pi$$
$$114$$ 0 0
$$115$$ 1067.74 0.865805
$$116$$ 3171.40 2.53842
$$117$$ 0 0
$$118$$ −1383.22 −1.07911
$$119$$ −1690.95 −1.30260
$$120$$ 0 0
$$121$$ −1066.05 −0.800938
$$122$$ 2442.99 1.81294
$$123$$ 0 0
$$124$$ −1133.21 −0.820686
$$125$$ −1260.73 −0.902104
$$126$$ 0 0
$$127$$ −1763.02 −1.23183 −0.615916 0.787812i $$-0.711214\pi$$
−0.615916 + 0.787812i $$0.711214\pi$$
$$128$$ 100.479 0.0693844
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −955.970 −0.637584 −0.318792 0.947825i $$-0.603277\pi$$
−0.318792 + 0.947825i $$0.603277\pi$$
$$132$$ 0 0
$$133$$ −2077.68 −1.35457
$$134$$ −4451.37 −2.86970
$$135$$ 0 0
$$136$$ −3167.07 −1.99687
$$137$$ −15.9215 −0.00992894 −0.00496447 0.999988i $$-0.501580\pi$$
−0.00496447 + 0.999988i $$0.501580\pi$$
$$138$$ 0 0
$$139$$ 2074.26 1.26573 0.632866 0.774261i $$-0.281878\pi$$
0.632866 + 0.774261i $$0.281878\pi$$
$$140$$ −3517.68 −2.12356
$$141$$ 0 0
$$142$$ 864.362 0.510815
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 962.635 0.551327
$$146$$ −1645.68 −0.932857
$$147$$ 0 0
$$148$$ −2123.20 −1.17923
$$149$$ 2764.08 1.51975 0.759873 0.650071i $$-0.225261\pi$$
0.759873 + 0.650071i $$0.225261\pi$$
$$150$$ 0 0
$$151$$ −1618.46 −0.872239 −0.436120 0.899889i $$-0.643648\pi$$
−0.436120 + 0.899889i $$0.643648\pi$$
$$152$$ −3891.40 −2.07654
$$153$$ 0 0
$$154$$ 2660.14 1.39195
$$155$$ −343.970 −0.178247
$$156$$ 0 0
$$157$$ −1109.97 −0.564237 −0.282119 0.959380i $$-0.591037\pi$$
−0.282119 + 0.959380i $$0.591037\pi$$
$$158$$ −2503.06 −1.26034
$$159$$ 0 0
$$160$$ −1898.22 −0.937921
$$161$$ −5725.83 −2.80285
$$162$$ 0 0
$$163$$ 233.201 0.112060 0.0560299 0.998429i $$-0.482156\pi$$
0.0560299 + 0.998429i $$0.482156\pi$$
$$164$$ −1058.76 −0.504119
$$165$$ 0 0
$$166$$ 2996.92 1.40124
$$167$$ −215.405 −0.0998118 −0.0499059 0.998754i $$-0.515892\pi$$
−0.0499059 + 0.998754i $$0.515892\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −1645.68 −0.742456
$$171$$ 0 0
$$172$$ −2176.69 −0.964950
$$173$$ 1383.15 0.607854 0.303927 0.952695i $$-0.401702\pi$$
0.303927 + 0.952695i $$0.401702\pi$$
$$174$$ 0 0
$$175$$ 2846.49 1.22957
$$176$$ 2477.22 1.06095
$$177$$ 0 0
$$178$$ 348.181 0.146614
$$179$$ −3642.79 −1.52109 −0.760545 0.649285i $$-0.775068\pi$$
−0.760545 + 0.649285i $$0.775068\pi$$
$$180$$ 0 0
$$181$$ 2621.97 1.07674 0.538369 0.842709i $$-0.319041\pi$$
0.538369 + 0.842709i $$0.319041\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −10724.2 −4.29674
$$185$$ −644.469 −0.256121
$$186$$ 0 0
$$187$$ 878.975 0.343727
$$188$$ −9889.91 −3.83668
$$189$$ 0 0
$$190$$ −2022.05 −0.772078
$$191$$ 3419.32 1.29536 0.647679 0.761913i $$-0.275740\pi$$
0.647679 + 0.761913i $$0.275740\pi$$
$$192$$ 0 0
$$193$$ −1698.39 −0.633435 −0.316718 0.948520i $$-0.602581\pi$$
−0.316718 + 0.948520i $$0.602581\pi$$
$$194$$ 7506.20 2.77791
$$195$$ 0 0
$$196$$ 12265.2 4.46982
$$197$$ 2293.72 0.829548 0.414774 0.909925i $$-0.363861\pi$$
0.414774 + 0.909925i $$0.363861\pi$$
$$198$$ 0 0
$$199$$ 900.981 0.320949 0.160474 0.987040i $$-0.448698\pi$$
0.160474 + 0.987040i $$0.448698\pi$$
$$200$$ 5331.35 1.88492
$$201$$ 0 0
$$202$$ −4677.58 −1.62928
$$203$$ −5162.18 −1.78480
$$204$$ 0 0
$$205$$ −321.373 −0.109491
$$206$$ 116.197 0.0393002
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1080.00 0.357441
$$210$$ 0 0
$$211$$ 431.019 0.140628 0.0703142 0.997525i $$-0.477600\pi$$
0.0703142 + 0.997525i $$0.477600\pi$$
$$212$$ −4666.28 −1.51170
$$213$$ 0 0
$$214$$ −1834.17 −0.585892
$$215$$ −660.706 −0.209580
$$216$$ 0 0
$$217$$ 1844.56 0.577036
$$218$$ −5046.79 −1.56794
$$219$$ 0 0
$$220$$ 1828.53 0.560361
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 4104.30 1.23249 0.616243 0.787556i $$-0.288654\pi$$
0.616243 + 0.787556i $$0.288654\pi$$
$$224$$ 10179.3 3.03631
$$225$$ 0 0
$$226$$ 252.047 0.0741854
$$227$$ 1809.11 0.528963 0.264482 0.964391i $$-0.414799\pi$$
0.264482 + 0.964391i $$0.414799\pi$$
$$228$$ 0 0
$$229$$ 5249.33 1.51478 0.757392 0.652961i $$-0.226473\pi$$
0.757392 + 0.652961i $$0.226473\pi$$
$$230$$ −5572.53 −1.59757
$$231$$ 0 0
$$232$$ −9668.54 −2.73608
$$233$$ 2808.88 0.789768 0.394884 0.918731i $$-0.370785\pi$$
0.394884 + 0.918731i $$0.370785\pi$$
$$234$$ 0 0
$$235$$ −3001.95 −0.833300
$$236$$ 5098.69 1.40634
$$237$$ 0 0
$$238$$ 8825.03 2.40354
$$239$$ 6712.01 1.81659 0.908293 0.418334i $$-0.137386\pi$$
0.908293 + 0.418334i $$0.137386\pi$$
$$240$$ 0 0
$$241$$ −2519.11 −0.673321 −0.336661 0.941626i $$-0.609297\pi$$
−0.336661 + 0.941626i $$0.609297\pi$$
$$242$$ 5563.69 1.47788
$$243$$ 0 0
$$244$$ −9005.15 −2.36269
$$245$$ 3722.93 0.970814
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 3454.78 0.884591
$$249$$ 0 0
$$250$$ 6579.71 1.66455
$$251$$ −828.000 −0.208219 −0.104109 0.994566i $$-0.533199\pi$$
−0.104109 + 0.994566i $$0.533199\pi$$
$$252$$ 0 0
$$253$$ 2976.35 0.739612
$$254$$ 9201.16 2.27296
$$255$$ 0 0
$$256$$ −4356.73 −1.06366
$$257$$ 5840.76 1.41765 0.708826 0.705383i $$-0.249225\pi$$
0.708826 + 0.705383i $$0.249225\pi$$
$$258$$ 0 0
$$259$$ 3456.00 0.829133
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4989.19 1.17646
$$263$$ 4064.06 0.952854 0.476427 0.879214i $$-0.341932\pi$$
0.476427 + 0.879214i $$0.341932\pi$$
$$264$$ 0 0
$$265$$ −1416.39 −0.328332
$$266$$ 10843.3 2.49943
$$267$$ 0 0
$$268$$ 16408.2 3.73990
$$269$$ −1845.44 −0.418285 −0.209142 0.977885i $$-0.567067\pi$$
−0.209142 + 0.977885i $$0.567067\pi$$
$$270$$ 0 0
$$271$$ 2106.78 0.472242 0.236121 0.971724i $$-0.424124\pi$$
0.236121 + 0.971724i $$0.424124\pi$$
$$272$$ 8218.19 1.83199
$$273$$ 0 0
$$274$$ 83.0939 0.0183208
$$275$$ −1479.64 −0.324457
$$276$$ 0 0
$$277$$ −4781.94 −1.03725 −0.518626 0.855001i $$-0.673556\pi$$
−0.518626 + 0.855001i $$0.673556\pi$$
$$278$$ −10825.5 −2.33551
$$279$$ 0 0
$$280$$ 10724.2 2.28891
$$281$$ 5865.81 1.24528 0.622642 0.782507i $$-0.286059\pi$$
0.622642 + 0.782507i $$0.286059\pi$$
$$282$$ 0 0
$$283$$ −6407.02 −1.34579 −0.672894 0.739739i $$-0.734949\pi$$
−0.672894 + 0.739739i $$0.734949\pi$$
$$284$$ −3186.14 −0.665713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1723.38 0.354453
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ −5023.97 −1.01730
$$291$$ 0 0
$$292$$ 6066.15 1.21573
$$293$$ −3010.24 −0.600204 −0.300102 0.953907i $$-0.597021\pi$$
−0.300102 + 0.953907i $$0.597021\pi$$
$$294$$ 0 0
$$295$$ 1547.64 0.305447
$$296$$ 6472.94 1.27105
$$297$$ 0 0
$$298$$ −14425.7 −2.80422
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 3543.07 0.678470
$$302$$ 8446.69 1.60944
$$303$$ 0 0
$$304$$ 10097.7 1.90508
$$305$$ −2733.39 −0.513159
$$306$$ 0 0
$$307$$ 3341.84 0.621266 0.310633 0.950530i $$-0.399459\pi$$
0.310633 + 0.950530i $$0.399459\pi$$
$$308$$ −9805.59 −1.81404
$$309$$ 0 0
$$310$$ 1795.17 0.328900
$$311$$ −8755.20 −1.59634 −0.798170 0.602432i $$-0.794199\pi$$
−0.798170 + 0.602432i $$0.794199\pi$$
$$312$$ 0 0
$$313$$ 1948.93 0.351949 0.175974 0.984395i $$-0.443692\pi$$
0.175974 + 0.984395i $$0.443692\pi$$
$$314$$ 5792.91 1.04112
$$315$$ 0 0
$$316$$ 9226.57 1.64252
$$317$$ −1940.43 −0.343802 −0.171901 0.985114i $$-0.554991\pi$$
−0.171901 + 0.985114i $$0.554991\pi$$
$$318$$ 0 0
$$319$$ 2683.36 0.470970
$$320$$ 2797.29 0.488667
$$321$$ 0 0
$$322$$ 29883.0 5.17178
$$323$$ 3582.90 0.617207
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −1217.07 −0.206771
$$327$$ 0 0
$$328$$ 3227.82 0.543373
$$329$$ 16098.1 2.69762
$$330$$ 0 0
$$331$$ 7402.13 1.22918 0.614589 0.788848i $$-0.289322\pi$$
0.614589 + 0.788848i $$0.289322\pi$$
$$332$$ −11047.0 −1.82615
$$333$$ 0 0
$$334$$ 1124.20 0.184171
$$335$$ 4980.50 0.812280
$$336$$ 0 0
$$337$$ −5494.05 −0.888071 −0.444035 0.896009i $$-0.646454\pi$$
−0.444035 + 0.896009i $$0.646454\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 6066.15 0.967597
$$341$$ −958.823 −0.152267
$$342$$ 0 0
$$343$$ −9223.77 −1.45200
$$344$$ 6636.03 1.04009
$$345$$ 0 0
$$346$$ −7218.62 −1.12160
$$347$$ −3410.76 −0.527664 −0.263832 0.964569i $$-0.584986\pi$$
−0.263832 + 0.964569i $$0.584986\pi$$
$$348$$ 0 0
$$349$$ 12629.8 1.93712 0.968562 0.248773i $$-0.0800272\pi$$
0.968562 + 0.248773i $$0.0800272\pi$$
$$350$$ −14855.8 −2.26878
$$351$$ 0 0
$$352$$ −5291.32 −0.801217
$$353$$ 2981.78 0.449586 0.224793 0.974407i $$-0.427829\pi$$
0.224793 + 0.974407i $$0.427829\pi$$
$$354$$ 0 0
$$355$$ −967.109 −0.144588
$$356$$ −1283.43 −0.191073
$$357$$ 0 0
$$358$$ 19011.7 2.80670
$$359$$ 8942.30 1.31464 0.657321 0.753611i $$-0.271690\pi$$
0.657321 + 0.753611i $$0.271690\pi$$
$$360$$ 0 0
$$361$$ −2456.68 −0.358168
$$362$$ −13684.0 −1.98678
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1841.30 0.264049
$$366$$ 0 0
$$367$$ −4735.26 −0.673511 −0.336756 0.941592i $$-0.609330\pi$$
−0.336756 + 0.941592i $$0.609330\pi$$
$$368$$ 27828.1 3.94196
$$369$$ 0 0
$$370$$ 3363.47 0.472590
$$371$$ 7595.45 1.06290
$$372$$ 0 0
$$373$$ −8304.01 −1.15272 −0.576361 0.817195i $$-0.695528\pi$$
−0.576361 + 0.817195i $$0.695528\pi$$
$$374$$ −4587.35 −0.634241
$$375$$ 0 0
$$376$$ 30151.1 4.13543
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4088.11 −0.554069 −0.277035 0.960860i $$-0.589352\pi$$
−0.277035 + 0.960860i $$0.589352\pi$$
$$380$$ 7453.50 1.00620
$$381$$ 0 0
$$382$$ −17845.4 −2.39018
$$383$$ 13951.5 1.86132 0.930662 0.365879i $$-0.119232\pi$$
0.930662 + 0.365879i $$0.119232\pi$$
$$384$$ 0 0
$$385$$ −2976.35 −0.393997
$$386$$ 8863.88 1.16881
$$387$$ 0 0
$$388$$ −27668.7 −3.62027
$$389$$ 2804.26 0.365506 0.182753 0.983159i $$-0.441499\pi$$
0.182753 + 0.983159i $$0.441499\pi$$
$$390$$ 0 0
$$391$$ 9874.06 1.27712
$$392$$ −37392.5 −4.81787
$$393$$ 0 0
$$394$$ −11970.9 −1.53067
$$395$$ 2800.60 0.356743
$$396$$ 0 0
$$397$$ −6556.18 −0.828830 −0.414415 0.910088i $$-0.636014\pi$$
−0.414415 + 0.910088i $$0.636014\pi$$
$$398$$ −4702.20 −0.592211
$$399$$ 0 0
$$400$$ −13834.2 −1.72928
$$401$$ −4730.95 −0.589157 −0.294579 0.955627i $$-0.595179\pi$$
−0.294579 + 0.955627i $$0.595179\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 17242.1 2.12333
$$405$$ 0 0
$$406$$ 26941.3 3.29329
$$407$$ −1796.47 −0.218790
$$408$$ 0 0
$$409$$ 12314.4 1.48878 0.744389 0.667746i $$-0.232741\pi$$
0.744389 + 0.667746i $$0.232741\pi$$
$$410$$ 1677.24 0.202032
$$411$$ 0 0
$$412$$ −428.316 −0.0512175
$$413$$ −8299.29 −0.988817
$$414$$ 0 0
$$415$$ −3353.16 −0.396627
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −5636.50 −0.659546
$$419$$ 5499.85 0.641254 0.320627 0.947206i $$-0.396106\pi$$
0.320627 + 0.947206i $$0.396106\pi$$
$$420$$ 0 0
$$421$$ −12629.7 −1.46208 −0.731039 0.682336i $$-0.760964\pi$$
−0.731039 + 0.682336i $$0.760964\pi$$
$$422$$ −2249.48 −0.259486
$$423$$ 0 0
$$424$$ 14225.9 1.62942
$$425$$ −4908.70 −0.560252
$$426$$ 0 0
$$427$$ 14658.0 1.66124
$$428$$ 6760.94 0.763557
$$429$$ 0 0
$$430$$ 3448.21 0.386715
$$431$$ 7191.15 0.803679 0.401840 0.915710i $$-0.368371\pi$$
0.401840 + 0.915710i $$0.368371\pi$$
$$432$$ 0 0
$$433$$ 6062.68 0.672873 0.336436 0.941706i $$-0.390778\pi$$
0.336436 + 0.941706i $$0.390778\pi$$
$$434$$ −9626.71 −1.06474
$$435$$ 0 0
$$436$$ 18603.0 2.04340
$$437$$ 12132.3 1.32807
$$438$$ 0 0
$$439$$ 11864.3 1.28986 0.644932 0.764240i $$-0.276886\pi$$
0.644932 + 0.764240i $$0.276886\pi$$
$$440$$ −5574.58 −0.603995
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −10560.5 −1.13261 −0.566303 0.824197i $$-0.691627\pi$$
−0.566303 + 0.824197i $$0.691627\pi$$
$$444$$ 0 0
$$445$$ −389.569 −0.0414997
$$446$$ −21420.3 −2.27417
$$447$$ 0 0
$$448$$ −15000.6 −1.58195
$$449$$ −12659.7 −1.33062 −0.665308 0.746569i $$-0.731700\pi$$
−0.665308 + 0.746569i $$0.731700\pi$$
$$450$$ 0 0
$$451$$ −895.834 −0.0935325
$$452$$ −929.072 −0.0966812
$$453$$ 0 0
$$454$$ −9441.69 −0.976037
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1544.24 0.158067 0.0790336 0.996872i $$-0.474817\pi$$
0.0790336 + 0.996872i $$0.474817\pi$$
$$458$$ −27396.1 −2.79506
$$459$$ 0 0
$$460$$ 20541.0 2.08202
$$461$$ −13196.8 −1.33327 −0.666635 0.745384i $$-0.732266\pi$$
−0.666635 + 0.745384i $$0.732266\pi$$
$$462$$ 0 0
$$463$$ −16309.2 −1.63705 −0.818524 0.574472i $$-0.805208\pi$$
−0.818524 + 0.574472i $$0.805208\pi$$
$$464$$ 25088.7 2.51016
$$465$$ 0 0
$$466$$ −14659.5 −1.45727
$$467$$ 14260.8 1.41308 0.706541 0.707672i $$-0.250255\pi$$
0.706541 + 0.707672i $$0.250255\pi$$
$$468$$ 0 0
$$469$$ −26708.2 −2.62957
$$470$$ 15667.1 1.53760
$$471$$ 0 0
$$472$$ −15544.2 −1.51585
$$473$$ −1841.73 −0.179034
$$474$$ 0 0
$$475$$ −6031.34 −0.582604
$$476$$ −32530.0 −3.13238
$$477$$ 0 0
$$478$$ −35029.9 −3.35194
$$479$$ 18011.5 1.71809 0.859046 0.511899i $$-0.171058\pi$$
0.859046 + 0.511899i $$0.171058\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 13147.2 1.24240
$$483$$ 0 0
$$484$$ −20508.4 −1.92603
$$485$$ −8398.46 −0.786298
$$486$$ 0 0
$$487$$ 14043.3 1.30670 0.653351 0.757055i $$-0.273363\pi$$
0.653351 + 0.757055i $$0.273363\pi$$
$$488$$ 27453.7 2.54666
$$489$$ 0 0
$$490$$ −19429.9 −1.79133
$$491$$ 12966.1 1.19176 0.595878 0.803075i $$-0.296804\pi$$
0.595878 + 0.803075i $$0.296804\pi$$
$$492$$ 0 0
$$493$$ 8902.06 0.813242
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8964.75 −0.811551
$$497$$ 5186.17 0.468072
$$498$$ 0 0
$$499$$ 10215.5 0.916453 0.458227 0.888835i $$-0.348485\pi$$
0.458227 + 0.888835i $$0.348485\pi$$
$$500$$ −24253.6 −2.16930
$$501$$ 0 0
$$502$$ 4321.31 0.384203
$$503$$ 16632.0 1.47432 0.737161 0.675717i $$-0.236166\pi$$
0.737161 + 0.675717i $$0.236166\pi$$
$$504$$ 0 0
$$505$$ 5233.61 0.461173
$$506$$ −15533.5 −1.36472
$$507$$ 0 0
$$508$$ −33916.5 −2.96221
$$509$$ 15235.3 1.32671 0.663354 0.748306i $$-0.269132\pi$$
0.663354 + 0.748306i $$0.269132\pi$$
$$510$$ 0 0
$$511$$ −9874.06 −0.854799
$$512$$ 21933.9 1.89326
$$513$$ 0 0
$$514$$ −30482.8 −2.61584
$$515$$ −130.009 −0.0111241
$$516$$ 0 0
$$517$$ −8367.99 −0.711845
$$518$$ −18036.8 −1.52991
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2680.23 −0.225380 −0.112690 0.993630i $$-0.535947\pi$$
−0.112690 + 0.993630i $$0.535947\pi$$
$$522$$ 0 0
$$523$$ −2410.38 −0.201527 −0.100764 0.994910i $$-0.532129\pi$$
−0.100764 + 0.994910i $$0.532129\pi$$
$$524$$ −18390.7 −1.53321
$$525$$ 0 0
$$526$$ −21210.2 −1.75819
$$527$$ −3180.90 −0.262926
$$528$$ 0 0
$$529$$ 21268.2 1.74802
$$530$$ 7392.09 0.605834
$$531$$ 0 0
$$532$$ −39969.8 −3.25735
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 2052.19 0.165839
$$536$$ −50023.3 −4.03111
$$537$$ 0 0
$$538$$ 9631.32 0.771814
$$539$$ 10377.7 0.829315
$$540$$ 0 0
$$541$$ 9969.58 0.792284 0.396142 0.918189i $$-0.370349\pi$$
0.396142 + 0.918189i $$0.370349\pi$$
$$542$$ −10995.2 −0.871375
$$543$$ 0 0
$$544$$ −17554.0 −1.38349
$$545$$ 5646.70 0.443813
$$546$$ 0 0
$$547$$ 16848.8 1.31701 0.658505 0.752576i $$-0.271189\pi$$
0.658505 + 0.752576i $$0.271189\pi$$
$$548$$ −306.293 −0.0238763
$$549$$ 0 0
$$550$$ 7722.20 0.598683
$$551$$ 10938.0 0.845688
$$552$$ 0 0
$$553$$ −15018.4 −1.15488
$$554$$ 24956.8 1.91393
$$555$$ 0 0
$$556$$ 39904.2 3.04373
$$557$$ 3800.83 0.289132 0.144566 0.989495i $$-0.453821\pi$$
0.144566 + 0.989495i $$0.453821\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −27828.1 −2.09992
$$561$$ 0 0
$$562$$ −30613.5 −2.29778
$$563$$ 15750.2 1.17903 0.589513 0.807759i $$-0.299320\pi$$
0.589513 + 0.807759i $$0.299320\pi$$
$$564$$ 0 0
$$565$$ −282.007 −0.0209985
$$566$$ 33438.1 2.48323
$$567$$ 0 0
$$568$$ 9713.48 0.717550
$$569$$ −17753.2 −1.30800 −0.654002 0.756493i $$-0.726911\pi$$
−0.654002 + 0.756493i $$0.726911\pi$$
$$570$$ 0 0
$$571$$ 25293.1 1.85374 0.926868 0.375388i $$-0.122491\pi$$
0.926868 + 0.375388i $$0.122491\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −8994.29 −0.654032
$$575$$ −16621.7 −1.20552
$$576$$ 0 0
$$577$$ −18488.8 −1.33396 −0.666982 0.745073i $$-0.732414\pi$$
−0.666982 + 0.745073i $$0.732414\pi$$
$$578$$ 10422.3 0.750018
$$579$$ 0 0
$$580$$ 18518.9 1.32579
$$581$$ 17981.5 1.28399
$$582$$ 0 0
$$583$$ −3948.20 −0.280477
$$584$$ −18493.7 −1.31040
$$585$$ 0 0
$$586$$ 15710.4 1.10749
$$587$$ 17376.7 1.22183 0.610914 0.791697i $$-0.290802\pi$$
0.610914 + 0.791697i $$0.290802\pi$$
$$588$$ 0 0
$$589$$ −3908.38 −0.273416
$$590$$ −8077.09 −0.563608
$$591$$ 0 0
$$592$$ −16796.5 −1.16610
$$593$$ −7991.09 −0.553381 −0.276690 0.960959i $$-0.589238\pi$$
−0.276690 + 0.960959i $$0.589238\pi$$
$$594$$ 0 0
$$595$$ −9874.06 −0.680331
$$596$$ 53174.7 3.65456
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −10386.5 −0.708480 −0.354240 0.935154i $$-0.615260\pi$$
−0.354240 + 0.935154i $$0.615260\pi$$
$$600$$ 0 0
$$601$$ −9241.77 −0.627254 −0.313627 0.949546i $$-0.601544\pi$$
−0.313627 + 0.949546i $$0.601544\pi$$
$$602$$ −18491.2 −1.25190
$$603$$ 0 0
$$604$$ −31135.4 −2.09749
$$605$$ −6225.04 −0.418320
$$606$$ 0 0
$$607$$ 18921.5 1.26524 0.632619 0.774463i $$-0.281980\pi$$
0.632619 + 0.774463i $$0.281980\pi$$
$$608$$ −21568.6 −1.43869
$$609$$ 0 0
$$610$$ 14265.5 0.946875
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 17138.1 1.12921 0.564603 0.825363i $$-0.309029\pi$$
0.564603 + 0.825363i $$0.309029\pi$$
$$614$$ −17441.0 −1.14635
$$615$$ 0 0
$$616$$ 29894.0 1.95530
$$617$$ −18825.8 −1.22836 −0.614180 0.789166i $$-0.710513\pi$$
−0.614180 + 0.789166i $$0.710513\pi$$
$$618$$ 0 0
$$619$$ 1392.83 0.0904404 0.0452202 0.998977i $$-0.485601\pi$$
0.0452202 + 0.998977i $$0.485601\pi$$
$$620$$ −6617.21 −0.428635
$$621$$ 0 0
$$622$$ 45693.2 2.94555
$$623$$ 2089.09 0.134346
$$624$$ 0 0
$$625$$ 4000.90 0.256057
$$626$$ −10171.4 −0.649412
$$627$$ 0 0
$$628$$ −21353.3 −1.35683
$$629$$ −5959.79 −0.377794
$$630$$ 0 0
$$631$$ 25488.4 1.60804 0.804022 0.594599i $$-0.202689\pi$$
0.804022 + 0.594599i $$0.202689\pi$$
$$632$$ −28128.8 −1.77041
$$633$$ 0 0
$$634$$ 10127.1 0.634379
$$635$$ −10294.9 −0.643371
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −14004.4 −0.869028
$$639$$ 0 0
$$640$$ 586.735 0.0362387
$$641$$ 6066.41 0.373805 0.186902 0.982379i $$-0.440155\pi$$
0.186902 + 0.982379i $$0.440155\pi$$
$$642$$ 0 0
$$643$$ −1598.78 −0.0980554 −0.0490277 0.998797i $$-0.515612\pi$$
−0.0490277 + 0.998797i $$0.515612\pi$$
$$644$$ −110152. −6.74006
$$645$$ 0 0
$$646$$ −18699.1 −1.13886
$$647$$ −23067.2 −1.40164 −0.700822 0.713336i $$-0.747183\pi$$
−0.700822 + 0.713336i $$0.747183\pi$$
$$648$$ 0 0
$$649$$ 4314.07 0.260928
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4486.26 0.269472
$$653$$ 23743.9 1.42293 0.711463 0.702723i $$-0.248033\pi$$
0.711463 + 0.702723i $$0.248033\pi$$
$$654$$ 0 0
$$655$$ −5582.25 −0.333002
$$656$$ −8375.81 −0.498507
$$657$$ 0 0
$$658$$ −84015.7 −4.97762
$$659$$ 7497.65 0.443197 0.221598 0.975138i $$-0.428873\pi$$
0.221598 + 0.975138i $$0.428873\pi$$
$$660$$ 0 0
$$661$$ 1255.26 0.0738638 0.0369319 0.999318i $$-0.488242\pi$$
0.0369319 + 0.999318i $$0.488242\pi$$
$$662$$ −38631.5 −2.26806
$$663$$ 0 0
$$664$$ 33678.6 1.96835
$$665$$ −12132.3 −0.707474
$$666$$ 0 0
$$667$$ 30143.8 1.74989
$$668$$ −4143.91 −0.240019
$$669$$ 0 0
$$670$$ −25993.1 −1.49881
$$671$$ −7619.38 −0.438365
$$672$$ 0 0
$$673$$ 1505.97 0.0862569 0.0431284 0.999070i $$-0.486268\pi$$
0.0431284 + 0.999070i $$0.486268\pi$$
$$674$$ 28673.3 1.63866
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 16201.4 0.919751 0.459876 0.887983i $$-0.347894\pi$$
0.459876 + 0.887983i $$0.347894\pi$$
$$678$$ 0 0
$$679$$ 45037.2 2.54546
$$680$$ −18493.7 −1.04294
$$681$$ 0 0
$$682$$ 5004.08 0.280962
$$683$$ 29090.4 1.62974 0.814870 0.579644i $$-0.196808\pi$$
0.814870 + 0.579644i $$0.196808\pi$$
$$684$$ 0 0
$$685$$ −92.9712 −0.00518576
$$686$$ 48138.7 2.67922
$$687$$ 0 0
$$688$$ −17219.7 −0.954208
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −940.952 −0.0518025 −0.0259012 0.999665i $$-0.508246\pi$$
−0.0259012 + 0.999665i $$0.508246\pi$$
$$692$$ 26608.6 1.46172
$$693$$ 0 0
$$694$$ 17800.7 0.973639
$$695$$ 12112.4 0.661077
$$696$$ 0 0
$$697$$ −2971.93 −0.161506
$$698$$ −65914.5 −3.57436
$$699$$ 0 0
$$700$$ 54760.1 2.95676
$$701$$ 30713.9 1.65485 0.827424 0.561578i $$-0.189805\pi$$
0.827424 + 0.561578i $$0.189805\pi$$
$$702$$ 0 0
$$703$$ −7322.81 −0.392866
$$704$$ 7797.51 0.417443
$$705$$ 0 0
$$706$$ −15561.8 −0.829571
$$707$$ −28065.5 −1.49294
$$708$$ 0 0
$$709$$ −25640.5 −1.35818 −0.679089 0.734056i $$-0.737625\pi$$
−0.679089 + 0.734056i $$0.737625\pi$$
$$710$$ 5047.32 0.266792
$$711$$ 0 0
$$712$$ 3912.77 0.205951
$$713$$ −10771.0 −0.565748
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −70079.1 −3.65779
$$717$$ 0 0
$$718$$ −46669.7 −2.42576
$$719$$ −20842.5 −1.08108 −0.540538 0.841319i $$-0.681779\pi$$
−0.540538 + 0.841319i $$0.681779\pi$$
$$720$$ 0 0
$$721$$ 697.183 0.0360117
$$722$$ 12821.3 0.660888
$$723$$ 0 0
$$724$$ 50440.8 2.58925
$$725$$ −14985.4 −0.767649
$$726$$ 0 0
$$727$$ −263.608 −0.0134480 −0.00672398 0.999977i $$-0.502140\pi$$
−0.00672398 + 0.999977i $$0.502140\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −9609.69 −0.487220
$$731$$ −6109.94 −0.309144
$$732$$ 0 0
$$733$$ −21835.5 −1.10029 −0.550146 0.835069i $$-0.685428\pi$$
−0.550146 + 0.835069i $$0.685428\pi$$
$$734$$ 24713.2 1.24275
$$735$$ 0 0
$$736$$ −59440.6 −2.97692
$$737$$ 13883.2 0.693888
$$738$$ 0 0
$$739$$ 19536.4 0.972476 0.486238 0.873826i $$-0.338369\pi$$
0.486238 + 0.873826i $$0.338369\pi$$
$$740$$ −12398.1 −0.615897
$$741$$ 0 0
$$742$$ −39640.5 −1.96125
$$743$$ −30353.1 −1.49872 −0.749360 0.662163i $$-0.769639\pi$$
−0.749360 + 0.662163i $$0.769639\pi$$
$$744$$ 0 0
$$745$$ 16140.5 0.793746
$$746$$ 43338.4 2.12699
$$747$$ 0 0
$$748$$ 16909.5 0.826567
$$749$$ −11005.0 −0.536867
$$750$$ 0 0
$$751$$ −3904.20 −0.189702 −0.0948510 0.995491i $$-0.530237\pi$$
−0.0948510 + 0.995491i $$0.530237\pi$$
$$752$$ −78238.5 −3.79397
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −9450.74 −0.455560
$$756$$ 0 0
$$757$$ −2900.52 −0.139262 −0.0696308 0.997573i $$-0.522182\pi$$
−0.0696308 + 0.997573i $$0.522182\pi$$
$$758$$ 21335.8 1.02236
$$759$$ 0 0
$$760$$ −22723.3 −1.08455
$$761$$ −33518.7 −1.59665 −0.798325 0.602227i $$-0.794280\pi$$
−0.798325 + 0.602227i $$0.794280\pi$$
$$762$$ 0 0
$$763$$ −30280.8 −1.43675
$$764$$ 65780.0 3.11497
$$765$$ 0 0
$$766$$ −72812.5 −3.43449
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 552.769 0.0259211 0.0129606 0.999916i $$-0.495874\pi$$
0.0129606 + 0.999916i $$0.495874\pi$$
$$770$$ 15533.5 0.726999
$$771$$ 0 0
$$772$$ −32673.3 −1.52323
$$773$$ 36498.8 1.69828 0.849141 0.528166i $$-0.177120\pi$$
0.849141 + 0.528166i $$0.177120\pi$$
$$774$$ 0 0
$$775$$ 5354.62 0.248185
$$776$$ 84352.8 3.90218
$$777$$ 0 0
$$778$$ −14635.4 −0.674427
$$779$$ −3651.62 −0.167950
$$780$$ 0 0
$$781$$ −2695.83 −0.123514
$$782$$ −51532.5 −2.35652
$$783$$ 0 0
$$784$$ 97029.2 4.42006
$$785$$ −6481.51 −0.294694
$$786$$ 0 0
$$787$$ −4284.79 −0.194074 −0.0970371 0.995281i $$-0.530937\pi$$
−0.0970371 + 0.995281i $$0.530937\pi$$
$$788$$ 44126.0 1.99483
$$789$$ 0 0
$$790$$ −14616.3 −0.658258
$$791$$ 1512.28 0.0679779
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 34216.6 1.52935
$$795$$ 0 0
$$796$$ 17332.8 0.771792
$$797$$ 29538.5 1.31281 0.656405 0.754409i $$-0.272076\pi$$
0.656405 + 0.754409i $$0.272076\pi$$
$$798$$ 0 0
$$799$$ −27760.8 −1.22917
$$800$$ 29549.8 1.30593
$$801$$ 0 0
$$802$$ 24690.7 1.08711
$$803$$ 5132.65 0.225563
$$804$$ 0 0
$$805$$ −33435.2 −1.46389
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −52565.5 −2.28867
$$809$$ 895.586 0.0389211 0.0194605 0.999811i $$-0.493805\pi$$
0.0194605 + 0.999811i $$0.493805\pi$$
$$810$$ 0 0
$$811$$ 20139.7 0.872011 0.436006 0.899944i $$-0.356393\pi$$
0.436006 + 0.899944i $$0.356393\pi$$
$$812$$ −99308.7 −4.29194
$$813$$ 0 0
$$814$$ 9375.73 0.403709
$$815$$ 1361.75 0.0585274
$$816$$ 0 0
$$817$$ −7507.31 −0.321478
$$818$$ −64268.8 −2.74707
$$819$$ 0 0
$$820$$ −6182.49 −0.263295
$$821$$ 17263.2 0.733848 0.366924 0.930251i $$-0.380411\pi$$
0.366924 + 0.930251i $$0.380411\pi$$
$$822$$ 0 0
$$823$$ 12114.5 0.513104 0.256552 0.966530i $$-0.417413\pi$$
0.256552 + 0.966530i $$0.417413\pi$$
$$824$$ 1305.79 0.0552057
$$825$$ 0 0
$$826$$ 43313.8 1.82455
$$827$$ −31450.9 −1.32243 −0.661217 0.750194i $$-0.729960\pi$$
−0.661217 + 0.750194i $$0.729960\pi$$
$$828$$ 0 0
$$829$$ 13760.4 0.576499 0.288250 0.957555i $$-0.406927\pi$$
0.288250 + 0.957555i $$0.406927\pi$$
$$830$$ 17500.1 0.731852
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 34428.2 1.43201
$$834$$ 0 0
$$835$$ −1257.83 −0.0521305
$$836$$ 20776.8 0.859545
$$837$$ 0 0
$$838$$ −28703.6 −1.18323
$$839$$ 9846.21 0.405160 0.202580 0.979266i $$-0.435067\pi$$
0.202580 + 0.979266i $$0.435067\pi$$
$$840$$ 0 0
$$841$$ 2787.47 0.114292
$$842$$ 65914.2 2.69781
$$843$$ 0 0
$$844$$ 8291.83 0.338171
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 33382.1 1.35422
$$848$$ −36914.7 −1.49488
$$849$$ 0 0
$$850$$ 25618.4 1.03377
$$851$$ −20180.8 −0.812914
$$852$$ 0 0
$$853$$ −27574.5 −1.10684 −0.553420 0.832903i $$-0.686677\pi$$
−0.553420 + 0.832903i $$0.686677\pi$$
$$854$$ −76499.6 −3.06530
$$855$$ 0 0
$$856$$ −20611.9 −0.823013
$$857$$ 8046.95 0.320745 0.160373 0.987057i $$-0.448730\pi$$
0.160373 + 0.987057i $$0.448730\pi$$
$$858$$ 0 0
$$859$$ 2898.13 0.115114 0.0575570 0.998342i $$-0.481669\pi$$
0.0575570 + 0.998342i $$0.481669\pi$$
$$860$$ −12710.5 −0.503982
$$861$$ 0 0
$$862$$ −37530.5 −1.48294
$$863$$ 4961.16 0.195689 0.0978447 0.995202i $$-0.468805\pi$$
0.0978447 + 0.995202i $$0.468805\pi$$
$$864$$ 0 0
$$865$$ 8076.69 0.317475
$$866$$ −31641.0 −1.24158
$$867$$ 0 0
$$868$$ 35485.1 1.38761
$$869$$ 7806.72 0.304747
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −56714.5 −2.20252
$$873$$ 0 0
$$874$$ −63318.2 −2.45054
$$875$$ 39478.3 1.52527
$$876$$ 0 0
$$877$$ −1386.66 −0.0533913 −0.0266957 0.999644i $$-0.508498\pi$$
−0.0266957 + 0.999644i $$0.508498\pi$$
$$878$$ −61919.3 −2.38004
$$879$$ 0 0
$$880$$ 14465.4 0.554123
$$881$$ −9030.36 −0.345335 −0.172668 0.984980i $$-0.555239\pi$$
−0.172668 + 0.984980i $$0.555239\pi$$
$$882$$ 0 0
$$883$$ 15512.7 0.591216 0.295608 0.955309i $$-0.404478\pi$$
0.295608 + 0.955309i $$0.404478\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 55115.0 2.08987
$$887$$ −7431.21 −0.281303 −0.140651 0.990059i $$-0.544920\pi$$
−0.140651 + 0.990059i $$0.544920\pi$$
$$888$$ 0 0
$$889$$ 55207.0 2.08277
$$890$$ 2033.15 0.0765747
$$891$$ 0 0
$$892$$ 78957.5 2.96378
$$893$$ −34109.8 −1.27821
$$894$$ 0 0
$$895$$ −21271.6 −0.794447
$$896$$ −3146.40 −0.117315
$$897$$ 0 0
$$898$$ 66070.5 2.45524
$$899$$ −9710.74 −0.360257
$$900$$ 0 0
$$901$$ −13098.2 −0.484310
$$902$$ 4675.34 0.172585
$$903$$ 0 0
$$904$$ 2832.44 0.104210
$$905$$ 15310.6 0.562367
$$906$$ 0 0
$$907$$ 10550.2 0.386234 0.193117 0.981176i $$-0.438140\pi$$
0.193117 + 0.981176i $$0.438140\pi$$
$$908$$ 34803.1 1.27201
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −35703.3 −1.29847 −0.649234 0.760589i $$-0.724910\pi$$
−0.649234 + 0.760589i $$0.724910\pi$$
$$912$$ 0 0
$$913$$ −9347.01 −0.338818
$$914$$ −8059.38 −0.291664
$$915$$ 0 0
$$916$$ 100985. 3.64263
$$917$$ 29935.1 1.07802
$$918$$ 0 0
$$919$$ −42896.6 −1.53975 −0.769873 0.638197i $$-0.779681\pi$$
−0.769873 + 0.638197i $$0.779681\pi$$
$$920$$ −62622.6 −2.24414
$$921$$ 0 0
$$922$$ 68874.0 2.46013
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 10032.5 0.356613
$$926$$ 85117.5 3.02066
$$927$$ 0 0
$$928$$ −53589.3 −1.89564
$$929$$ 10366.7 0.366114 0.183057 0.983102i $$-0.441401\pi$$
0.183057 + 0.983102i $$0.441401\pi$$
$$930$$ 0 0
$$931$$ 42302.0 1.48914
$$932$$ 54036.6 1.89917
$$933$$ 0 0
$$934$$ −74426.6 −2.60740
$$935$$ 5132.65 0.179525
$$936$$ 0 0
$$937$$ 20289.8 0.707405 0.353702 0.935358i $$-0.384923\pi$$
0.353702 + 0.935358i $$0.384923\pi$$
$$938$$ 139390. 4.85206
$$939$$ 0 0
$$940$$ −57750.7 −2.00385
$$941$$ −37089.1 −1.28488 −0.642438 0.766337i $$-0.722077\pi$$
−0.642438 + 0.766337i $$0.722077\pi$$
$$942$$ 0 0
$$943$$ −10063.4 −0.347519
$$944$$ 40335.4 1.39068
$$945$$ 0 0
$$946$$ 9611.96 0.330351
$$947$$ −23458.2 −0.804952 −0.402476 0.915430i $$-0.631850\pi$$
−0.402476 + 0.915430i $$0.631850\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 31477.5 1.07501
$$951$$ 0 0
$$952$$ 99173.4 3.37629
$$953$$ −34695.5 −1.17933 −0.589663 0.807649i $$-0.700740\pi$$
−0.589663 + 0.807649i $$0.700740\pi$$
$$954$$ 0 0
$$955$$ 19966.6 0.676550
$$956$$ 129124. 4.36838
$$957$$ 0 0
$$958$$ −94001.6 −3.17020
$$959$$ 498.563 0.0167878
$$960$$ 0 0
$$961$$ −26321.1 −0.883527
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −48462.1 −1.61915
$$965$$ −9917.53 −0.330836
$$966$$ 0 0
$$967$$ 6289.66 0.209164 0.104582 0.994516i $$-0.466649\pi$$
0.104582 + 0.994516i $$0.466649\pi$$
$$968$$ 62523.3 2.07601
$$969$$ 0 0
$$970$$ 43831.4 1.45087
$$971$$ 20185.9 0.667145 0.333573 0.942724i $$-0.391746\pi$$
0.333573 + 0.942724i $$0.391746\pi$$
$$972$$ 0 0
$$973$$ −64953.2 −2.14009
$$974$$ −73291.8 −2.41111
$$975$$ 0 0
$$976$$ −71239.2 −2.33639
$$977$$ −44244.0 −1.44881 −0.724406 0.689373i $$-0.757886\pi$$
−0.724406 + 0.689373i $$0.757886\pi$$
$$978$$ 0 0
$$979$$ −1085.93 −0.0354510
$$980$$ 71620.8 2.33453
$$981$$ 0 0
$$982$$ −67669.9 −2.19901
$$983$$ −8835.11 −0.286670 −0.143335 0.989674i $$-0.545783\pi$$
−0.143335 + 0.989674i $$0.545783\pi$$
$$984$$ 0 0
$$985$$ 13393.9 0.433263
$$986$$ −46459.6 −1.50058
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −20689.3 −0.665198
$$990$$ 0 0
$$991$$ 34915.1 1.11919 0.559594 0.828767i $$-0.310957\pi$$
0.559594 + 0.828767i $$0.310957\pi$$
$$992$$ 19148.6 0.612872
$$993$$ 0 0
$$994$$ −27066.5 −0.863680
$$995$$ 5261.15 0.167628
$$996$$ 0 0
$$997$$ 37962.8 1.20591 0.602956 0.797774i $$-0.293989\pi$$
0.602956 + 0.797774i $$0.293989\pi$$
$$998$$ −53314.6 −1.69103
$$999$$ 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.x.1.1 4
3.2 odd 2 507.4.a.j.1.4 4
13.5 odd 4 117.4.b.d.64.4 4
13.8 odd 4 117.4.b.d.64.1 4
13.12 even 2 inner 1521.4.a.x.1.4 4
39.5 even 4 39.4.b.a.25.1 4
39.8 even 4 39.4.b.a.25.4 yes 4
39.38 odd 2 507.4.a.j.1.1 4
156.47 odd 4 624.4.c.e.337.2 4
156.83 odd 4 624.4.c.e.337.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.a.25.1 4 39.5 even 4
39.4.b.a.25.4 yes 4 39.8 even 4
117.4.b.d.64.1 4 13.8 odd 4
117.4.b.d.64.4 4 13.5 odd 4
507.4.a.j.1.1 4 39.38 odd 2
507.4.a.j.1.4 4 3.2 odd 2
624.4.c.e.337.2 4 156.47 odd 4
624.4.c.e.337.3 4 156.83 odd 4
1521.4.a.x.1.1 4 1.1 even 1 trivial
1521.4.a.x.1.4 4 13.12 even 2 inner